Exact Utilities under Alternative Monetary Rules in a Simple Macro Model with Optimizing Agents

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Language:: English

Print Publication Date:: 24 Jan 2000

Keywords:: BOOK; price contract; Taylor rule; price level; price equation; unemployment-inflation tradeoff; Unemployment; Real interest rates; Unemployment rate; Global

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Abstract

DALE W. HENDERSON

Federal Reserve Board

dale.henderson@frb.gov

JINILL KIM

University of Virginia

jk9n@virginia.edu

Abstract

We construct an optimizingagent model of a closed economy which is simple enough that we can use it to make exact utility calculations. There is a stabilization problem because there are one-period nominal contracts for wages, or prices, or both and shocks that are unknown at the time when contracts are signed. We evaluate alternative monetary policy rules using the utility function of the representative agent. Fully optimal policy can attain the Pareto-optimal equilibrium. Fully optimal policy is contrasted with both ‘naive’ and ‘sophisticated’ simple rules that involve, respectively, complete stabilization and optimal stabilization of one variable or a combination of two variables. With wage contracts, outcomes depend crucially on whether there are also price contracts. For example, if labor supply is relatively inelastic, for productivity shocks, nominal income stabilization yields higher welfare when there are no price contracts. However, with price contracts, outcomes are independent of whether there are wage contracts, except, of course, for nominal wage outcomes.

1. Introduction

Interest in improving the analytical foundations of monetary stabilization policy is at a cyclical peak. This paper is a contribution to that endeavor. We construct an optimizing-agent model of a closed economy which is simple enough that we can make exact utility calculations. In this model, there is a stabilization problem because there are one-period nominal contracts for wages, or prices, or both and shocks that are unknown at the time when contracts are signed. We evaluate alternative monetary policy rules using as a criterion the utility function of the representative agent.

One well known advantage of using exact utility calculations is that it makes it possible to analyze shocks with large as well as small variances. An unexpected advantage is that it actually simplifies the algebraic derivations in our model. However, when shocks have small variances, it yields no advantage for welfare analysis in our model; welfare rankings are the same with exact and approximate utility calculations.¹

We focus on two cases, (1) wage contracts and flexible prices and (2) wage and price contracts. If wages are fixed by contracts, for some shocks the attractiveness of some simple rules depends crucially on whether prices are also fixed by contracts. We can limit our focus to two cases because, as we show, the outcomes in the third case, price contracts and flexible wages, are the same as the outcomes in the case of wage and price contracts for all variables except, of course, for the nominal wage.²

We calculate the fully optimal rule under complete information for each of our two cases of interest. This rule can attain the Pareto-optimal equilibrium because we assume oneperiod nominal contracts, so the policymaker does not face a tradeoff.³ Then we contrast the performance of the fully optimal policy with both ‘naive’ and ‘sophisticated’ versions of some simple rules. Naive simple rules involve complete stabilization of one variable or a combination of two variables. Sophisticated simple rules involve optimal stabilization of one variable or a combination of two variables. We consider sophisticated versions of simple rules in an attempt to put these rules in the best possible light.

Our paper is closely related to two sets of recent studies. The studies in one set contain evaluations of alternative monetary policies using approximate solutions of models with optimizing-agents.⁴ Of course, the authors of these studies have used approximate solutions because their models are complex enough that obtaining exact solutions would be relatively difficult and costly if it were even feasible. It seems useful to supplement their analysis with analysis of models that are simple enough that obtaining exact solutions is relatively easy.

The studies in the other set are based on two-country models in which exact utility calculations are possible.⁵ Our emphasis differs from the emphasis in these studies. We focus on the welfare effects of alternative monetary stabilization rules in a stochastic model. In contrast, the other studies focus either on the welfare effects of a one-time increase in the money supply in a perfect foresight model, on the implications of alternative money supply processes for asset returns in a stochastic model, or on a welfare comparison of fixed and flexible exchange rates in a stochastic model. Another notable difference between our paper and the other studies is that for us the interest rate, not the money supply, is the instrument of monetary policy.

The rest of this paper is organized into five more sections. Section 2 is a description of our model. We devote Section 3 to the benchmark version with flexible wages and prices. In sections 4 and 5, we analyze alternative monetary policy rules in versions with wage contracts and flexible prices and with both wage and price contracts, respectively. Section 6 contains our conclusions. The demonstration that the version with price contracts and flexible wages yields the same outcomes as the version with both wage and price contracts (except for nominal wages) is in the Appendix.

2. The Model

In this section we describe our model. We discuss the behavior of firms, households, and the government in successive subsections.

2.1. Firms

A continuum of ‘identical’ monopolistically competitive firms is distributed on the unit interval, f ∈ [0,1]. With no price contracts, firms set their prices for period t based on period t information. With one-period price contracts, firms set prices for period t + 1 based on period t information and agree to supply whatever their customers demand at those prices. In either case, the problem of firm f in period t is to find the

\begin{matrix} \max_{{P_{f, t + j}}} & \begin{matrix} {\begin{matrix} ε \end{matrix}}_{t} {\tilde{Δ}}_{t, t + j} (s_{P} P_{f, t + j} Y_{f, t + j} - W_{t + j} L_{f, t + j}) & (1) \end{matrix} \end{matrix}

where capital letters without serifs represent choice variables of individual firms or households and capital letters with serifs represent indexes that include all firms or households. The subscript j takes on the value 0 if there are no price contracts and the value 1 if there are price contracts. In period t + j, firm f sets the price P_f_{, t+j}, produces output Y_f_{, t+j}, and employs the amount L_f_{, t+j} of a labor index L_t+j for which it pays the wage indexW_{, t+j} per unit:

\begin{matrix} \begin{matrix} L_{t + j} = \int_{0}^{1} L_{f, t + j} d f = {(\int_{0}^{1} L_{h, t + j}^{\frac{1}{θ_{w}}} d h)}^{θ_{w}} & W_{t + j} {(\int_{0}^{1} W_{h, t + j}^{\frac{1}{θ_{w}}} d h)}^{1 - θ_{w}} \end{matrix} & (2) \end{matrix}

where L_{h, t+j} is the amount of labor supplied by household h in period t + j, W_{h, t+j} is the wage charged by household h in period t + j, and θ_w > 1. Firm f chooses quantities of L_{h, t+j} to minimize the cost of producing a unit of L_{f, t+j} given the W_{h, t+j}, and W_t+j is the minimum cost. All firms receive an ad valorem output subsidy, s_P. Each element of the infinite dimensional vector ${\tilde{δ}}_{t, t + j}$ is a stochastic discount factor, the price of a claim to one dollar delivered in a particular state in period t + j divided by the probability of that state. We use ε_t to indicate an expectation taken over the states in period t + j based on period t information. The production function of firm f is⁶

\begin{matrix} Y_{f, t + j} = \frac{L_{f, t + j}^{(1 - α)} X_{t + j}}{1 - α} & (3) \end{matrix}

where X_t+j is a productivity shock that hits all firms, and x_t+j = lnX_t+j∼ N (0, 2σ_x²). An expression for L_{f, t+j} is obtained by inverting this production function. Relative demand for output of firm f is a decreasing function of its relative price:

\begin{matrix} \frac{Y_{f, t + j}}{Y_{t + j}} = {(\frac{P_{f, t + j}}{P_{t + j}})}^{- \frac{θ_{P}}{θ_{P} - 1}} & (4) \end{matrix}

where θ_P > 1. In equation (4), Y_t+j is an index made up of the output of all firms and P_t+j is a price index which is the price of a unit of the output index:

\begin{matrix} Y_{t + j} = \int_{0}^{1} Y_{h, t + j} d h = {(\int_{0}^{1} Y_{f, t + j}^{\frac{1}{θ_{P}}})}^{θ_{P}} & \begin{matrix} P_{t + j} = {(\int_{0}^{1} P_{f, t + j}^{\frac{1}{1 - θ_{P}}} d f)}^{1 - θ_{P}} & (5) \end{matrix} \end{matrix}

where Y_{h, t+j} is the amount of the output index purchased by household h in period t + j. Household h chooses quantities of Y_{f, t+j} to minimize the cost of producing a unit of Y_{h, t+j} given the P_{f, t+j}, and P_t+j is the minimum cost.

To maximize profits, a firm must set its price so that expected discounted marginal revenue equals expected discounted marginal cost:

\begin{matrix} S_{p} (\frac{θ_{P}}{θ_{P - 1}} - 1) ε_{t} ({\tilde{δ}}_{t, t + j} Y_{f, t + j}) = (\frac{θ_{P}}{θ_{P} - 1}) ε_{t} (\frac{{\tilde{δ}}_{t, t + j} W_{t + j} L_{f, t + j}^{α} Y_{f, t + j}}{P_{f, t + j} X_{t + j}}) & (\begin{matrix} 6 \end{matrix}) \end{matrix}

Since firms are identical,

\begin{matrix} L_{f, + j} = L_{+ j} & \begin{matrix} Y_{f, + j} = Y_{+ j} & \begin{matrix} P_{f, + j} = P_{+ j} & (\begin{matrix} 7 \end{matrix}) \end{matrix} \end{matrix} \end{matrix}

where we omit t subscripts in the rest of this subsection for simplicity. Therefore, the equalities in (7) imply that the ‘aggregate production function’ and ‘aggregate price equation’ are, respectively,

\begin{matrix} {\begin{matrix} Y \end{matrix}}_{+ j} = \frac{L_{+ j}^{(1 - α)} X_{+ j}}{1 - α} & (\begin{matrix} 8 \end{matrix}) \end{matrix}

\begin{matrix} {\begin{matrix} S \end{matrix}}_{P} ε ({\tilde{Δ}}_{+ j} Y_{+ j}) = θ_{P} ε (\frac{{\tilde{Δ}}_{+ j} W_{+ j} L_{+ j}^{α} Y_{+ j}}{P_{+ j} X_{+ j}}) & (9) \end{matrix}

When j = 0 so that period t prices are set on the basis of period t information, the aggregate price equation (9) can be rewritten as

\begin{matrix} (\frac{S_{P}}{θ_{P}}) \frac{X}{L^{α}} = \frac{W}{P} & (10) \end{matrix}

which states that P must be chosen so that the marginal value product of labor (the gross subsidy rate over the markup parameter times the marginal product of labor) equals the real wage. We assume that the government sets s_P = θ_P to offset the effect of the distortion associated with monopolistic competition in the goods market. Under this assumption, the ratio $\frac{S_{P}}{θ_{P}}$ equals one, so it does not appear in what follows, and the implied version of equation (10) states that the marginal product of labor must equal the real wage.

2.2. Households

A continuum of ‘identical’ households is distributed on the unit interval, h ∈ [0,1]. With no wage contracts, households set their wages for period t based on period t information, but with wage contracts they set their wages for period t + 1 based on period t information. The problem of household h in period t is to find the

\begin{matrix} \underset{C_{h, s}, M_{h, s}, B_{h, s}, B_{h, s}^{g}, W_{h, s + j}}{\max {}} ε_{t} \sum_{s = t}^{\infty} β^{s - t} (\frac{C_{h, s}^{1} - 1}{1 - ρ} - \frac{χ_{0} L_{h, s}^{1 + χ}}{Z_{s} (1 + χ)}) U_{s} & (11) \end{matrix}

subject to

C_{h, s} = \underset{\begin{matrix} - \frac{\begin{matrix} M_{h, s} - M_{h, s - 1} + Δ_{s, s + 1} B_{h, s} - B_{h, s - 1} + B_{h, s}^{g} - I_{s - 1} B_{h, s - 1}^{g} \end{matrix}}{P_{s}} & (\begin{matrix} 12 \end{matrix}) \end{matrix}}{\frac{s_{W} W_{h, s} L_{h, s}}{P_{s}} + \frac{Γ_{s}}{P_{s}} - T_{h, s}}

\begin{matrix} \begin{matrix} {\begin{matrix} C \end{matrix}}_{h, s} = \min (C_{h, s}, \frac{M_{h, s}}{P_{s} V_{s}}), & \begin{matrix} M_{h, s} = P_{s} C_{h, s} V_{s} & C_{h, s} = C_{h, s} \end{matrix} \end{matrix} & (13) \end{matrix}

\begin{matrix} \frac{L_{h, s}}{L_{s}} = {(\frac{W_{h, s}}{W_{s}})}^{- \frac{θ_{w}}{θ_{w} - 1}} & (\begin{matrix} 14 \end{matrix}) \end{matrix}

In period s, household h chooses its expenditure on the output index (C_{h, s} = Y_{h, s}) and its holdings of money, M_{h, s}, which imply a consumption realization, C_{h, s} Household h also chooses its wage rate in period _{s + j} W_{h, s+j}, and agrees to supply however many units of its labor, L_{h, s+j}, firms want at this wage where the subscript j takes on the value 0 if there are no wage contracts and the value 1 if there are wage contracts. In addition, in period s, household h chooses its holdings of claims to a unit of currency in the various states in period s + 1. Each element in the infinite-dimensional vector Δ_{s, s+1} represents the price of an asset that will pay one unit of currency in a particular state of nature in the subsequent period, while the corresponding element of the vector B_{h, s} represents the quantity of such claims purchased by the household. The scalar variable B_{h, s-1} represents the value of the households’s claims given the current state of nature. Household h also chooses its holding of government bonds B^g_{h, s}, which pay I_s units of currency in every state of nature in period s + 1. Household h receives an aliquot share, Γ_s, of aggregate profits and pays lump sum taxes, T_{h, s}.⁷ All households receive an ad valorem labor subsidy, s_W.There are goods demand, U_s, money demand, V_s, and labor supply shocks, Z_s, that hit all consumers. We assume that the shocks U_s, X_s, and Z_s have lognormal distributions.⁸ We impose the restrictions that O < β < l, ρ ≥ l, and X ≥ 0.⁹ ε_t indicates an expectation over the various states in period s based on period t information.

According to equation (11), period utility depends positively on the consumption realization and negatively on labor supply. The period budget constraint, equation (12), states that consumption expenditure must equal disposable income minus asset accumulation. According to the first equality in equation (13), the consumption realization is equal to the minimum of consumption expenditure and adjusted real balances (real balances divided by a money demand shock). It is optimal for household h to keep consumption expenditure and adjusted real balances equal to one another (the second equality in equation (13)) so that the consumption realization is always equal to consumption expenditure (the third equality in equation (13)). Each household is a monopolistically competitive supplier of its unique labor input. Relative demand for labor of household h is a decreasing function of its relative wage as shown in equation (14).

Substituting equation (13) into equation (11), substituting equation (14) into equation (12), constructing a Lagrangian expression with the multiplier η_{h, s} associated with the period budget constraint for period s, and differentiating yields the first order conditions for household h for consumption, contingent claims, and government bonds for period t and for the nominal wage in period t + j, j = 0 or 1:

\begin{matrix} \frac{U_{t}}{C_{h, t}^{ρ}} = η_{h, t} & (\begin{matrix} 15 \end{matrix}) \end{matrix}

\begin{matrix} \frac{{\tilde{δ}}_{t, t + 1} η_{h, t}}{P_{t}} = \frac{{β η}_{h, t + 1}}{P_{t + 1}} & (16) \end{matrix}

\begin{matrix} \frac{η_{h, t}}{P_{t}} = β I_{t} ε_{t} (\frac{η_{h, t + 1}}{P_{t + 1}}) & (17) \end{matrix}

\begin{matrix} X_{0} (\frac{θ_{W}}{θ_{W} - 1}) ε_{t} (\frac{{(L_{h, t + j})}^{X} L_{h, t + j} U_{t + j}}{W_{h, t + j} Z_{t}}) = s_{w} (\frac{θ_{W}}{θ_{W} - 1} - 1) ε_{t} (\frac{η_{h, t + j} L_{h, t + j}}{P_{t + j}}) & (18) \end{matrix}

\begin{matrix} M_{h, t} = P_{t} C_{h, t} V_{t} & (\begin{matrix} 19 \end{matrix}) \end{matrix}

where the condition that consumption must equal adjusted real balances is repeated for convenience. The gross nominal interest rate, I_t, one plus the nominal interest rate, i_t, must be equal to one over the cost of acquiring claims to one unit of currency in every state of nature in period t + 1:

\begin{matrix} I_{t} = 1 + i_{t} = \frac{1}{\int δ_{t, t + 1}} & (20) \end{matrix}

where the integral is over the states of nature in period t + 1. Hereafter, we refer to the gross nominal interest rate as the interest rate and omit all t subscripts.

These first order conditions have implications for relationships among aggregate variables. Since households are identical,

\begin{matrix} C_{h} = C, L_{h} = L, W_{h} = W, T_{h} = T, M_{h} = M, B_{h} = B, η_{h} = η & (\begin{matrix} 21 \end{matrix}) \end{matrix}

Eliminating η and η + 1 using the condition that in each period in each state

\begin{matrix} \frac{U_{+ j}}{C_{+ j}^{ρ}} = η_{+ j} & (22) \end{matrix}

yields the ‘aggregate first-order conditions for the state contingent contracts,’ the ‘aggregate consumption Euler equation,’ the ‘aggregate wage equation,’ and the money market equilibrium condition:

\begin{matrix} {\tilde{δ}}_{t, t + 1} (\frac{U}{P C^{ρ}}) = β (\frac{U_{+ 1}}{P_{+ 1} C_{+ 1}^{ρ}}) & (23) \end{matrix}

\begin{matrix} \frac{U}{P C^{ρ}} = β I ε (\frac{U_{+ 1}}{P_{+ 1} C_{+ 1}^{ρ}}) & (24) \end{matrix}

\begin{matrix} θ_{W} X_{0} ε (\frac{L_{+ j}^{1 + X} U_{+ j}}{W_{+ j} Z_{+ j}}) = s_{W} ε (\frac{L_{+ j} U_{+ j}}{P_{+ j} C_{+ j}^{ρ}}) & (25) \end{matrix}

\begin{matrix} M = P C V & (26) \end{matrix}

When j = 0 so that consumers act on the basis of current information, conditions (24) and (25) can be rewritten as

\begin{matrix} \frac{U}{P C^{ρ}} = β I ε (\frac{U_{+ 1}}{P_{+ 1} C_{+ 1}^{ρ}}) & (27) \end{matrix}

\begin{matrix} (\frac{s_{W}}{θ_{W}}) \frac{W}{P} = \frac{X_{0} L^{X} C^{ρ}}{Z} & (28) \end{matrix}

Equation (27) states that C must be chosen so that the utility forgone by not spending the marginal dollar on consumption today equals the discounted expected utility of investing that dollar in a riskless security and spending it on consumption tomorrow. Equation (28) states that W must be chosen so that the marginal return from work must equal the marginal rate of substitution of consumption for labor. We assume that the government sets s_w = θ_w to offset the effect of the distortion associated with monopolistic competition in the labor market. Under this assumption the ratio $\frac{s_{W}}{θ_{W}}$ equals one, so it does not appear in what follows, and the implied version of equation (28) states that the real wage must equal the marginal rate of substitution.

2.3. Government

The government budget constraint is

\begin{matrix} \frac{\begin{matrix} M - M_{- 1} + B^{g} - I_{- 1} B_{- 1}^{g} \end{matrix}}{P} = G + (s_{P} - 1) Y + (s_{W} - 1) \frac{W}{P} L - T & (29) \end{matrix}

where G is real government spending. We impose simple assumptions about the paths of government spending, interest payments, subsidy payments, and taxes under which we can study alternative monetary policy reaction functions.¹⁰ In particular, we assume that the government budget is balanced period by period and that real government spending is always zero, so the government budget constraint becomes¹¹

\begin{matrix} \frac{\begin{matrix} i_{- 1} B_{- 1}^{g} \end{matrix}}{P} + (s_{P} - 1) Y + (s_{W} - 1) \frac{W}{P} L - T = 0 & (30) \end{matrix}

We assume that the government follows a monetary policy rule in the class

\begin{matrix} I = β^{- 1} P^{λ_{P}} Y^{λ_{Y}} Y^{* λ_{Y} *} {\bar{Y}}^{λ_{\bar{Y}}} M^{λ_{M}} U^{λ_{U}} V^{λ_{V}} X^{λ_{X}} Z^{λ_{Z}} & (31) \end{matrix}

where $\bar{Y}$ is a target level of output. For rules in this class, either the price level or the money supply is the ‘nominal anchor;’ the sum of λ_P and λ_M must be non-zero in order for the price level to be determined with flexible wages and prices or one-period contracts for prices, wages, or both. We derive the optimal λ_j, the ones that maximize expected welfare. We also consider some alternative values of the λ_j.

3. Flexible Wages and Prices

We consider four versions of our model. To establish a benchmark, we begin by considering the version with flexible wages and prices.

3.1. Solution

In each version of the model six equations are used to determine the equilibrium values of the variables. With flexible wages and prices the forms of these six equations are

\begin{matrix} Y = \frac{L^{\tilde{α}} X}{\tilde{α}}, & (p r o d u c t i o n) \end{matrix}

\begin{matrix} P = \frac{L^{α} W}{X}, & (p r i c e) \end{matrix}

\begin{matrix} \frac{X_{0} L^{\bar{X}}}{W Z} = \frac{L}{Y^{ρ} P}, & (\begin{matrix} w a g e \end{matrix}) \end{matrix}

\begin{matrix} β I ε (\frac{U_{+ 1}}{Y_{+ 1}^{ρ} P_{+ 1}}) = \frac{U}{Y^{ρ} P}, & (d e m a n d) \end{matrix}

\begin{matrix} I = β^{- 1} P^{λ_{P}} Y^{λ_{Y}} Y^{* λ_{Y} *} {\bar{Y}}^{λ_{\bar{Y}}} M^{λ_{M}} U^{λ_{U}} V^{λ_{V}} X^{λ_{X}} Z^{λ_{Z}} & (r u l e) \end{matrix}

\begin{matrix} M = P Y V & (\begin{matrix} m o n e y \end{matrix}) \end{matrix}

where we have imposed the equilibrium conditions that C = Y and C₊₁ = Y₊₁ and where $\tilde{α} = 1 - α$ and $\tilde{X} = 1 - X$ . With flexible wages and prices, both wages and prices are set after the shocks are known and the only expected magnitudes are in the demand equation.

The solutions for selected variables are shown in Table 1. Substituting the solutions for these variables into the equations of the model yields the solutions for the other variables.¹²

Substituting the production and price equations into the wage equation and solving yields the solution for L in equation (T1.1) where $\tilde{ρ} = ρ - 1$ . To solve for the price level we use the method of undetermined coefficients. Suppose that P takes the form given in equation (T1.2). We find Ω, ω_U, ω_V, ω_X, and ω_Z by beginning with the demand equation and eliminating Y and Y₊₁ Using the solution for Y^* implied by the solution for L_* in equation (T1.1), eliminating P using the conjectured solution in equation (T1.2), and eliminating I using the rule equation to obtain

\underset{\underset{\begin{matrix} + (\frac{λ_{Z} D + \tilde{α} (λ_{Y *} + λ_{Y} + λ_{M} + ρ)}{D}) Z & (32) \end{matrix}}{\underset{+ (λ_{v} + λ_{M}) v + (\frac{λ_{X} D + \tilde{X} (λ_{Y *} + λ_{Y} + ρ)}{D}) x}{= (λ_{Y *} + λ_{Y} + λ_{M}) \ln ({\tilde{α}}^{- 1} H^{\tilde{α}}) + λ_{\bar{Y}} \bar{Y} + \ln ε (Q_{1}) + (λ_{U} - 1) u}}}{- (λ_{P} + λ_{M}) \ln Ω - (1 + λ_{P} + λ_{M}) (Ω_{U} u + Ω_{V} v + Ω_{X} x + Ω_{Z} z)}

where lower case letters represent logarithms, D is defined in equation (T1.1), and

\begin{matrix} Q_{1} = U_{+ 1}^{1 - Ω_{U}} V_{+ 1}^{- Ω_{V}} X_{+ 1}^{- Ω_{X} - \frac{ρ \tilde{X}}{D}} Z_{+ 1}^{- Ω_{Z} - \frac{ρ \tilde{α}}{D}} & (\begin{matrix} 33 \end{matrix}) \end{matrix}

If equation (32) is to hold for all U, V, X, and Z, it must be that the ω_j and Ω take on the values given in equations (T1.4) through (T1.6). Substituting the solution for L^* and the implied solution for Y^* into the period utility function and considerable rearranging yield the solution for utility. So that we can simplify expressions by using logarithms, we express utility in terms of loss, L, by defining

\begin{matrix} L = - (\frac{C_{h, s}^{1 - ρ} - 1}{1 - ρ} - \frac{X_{0} L_{h, s}^{1 + X}}{Z_{s} (1 + X)} + \frac{1}{1 - ρ}) U_{s} > 0 & (34) \end{matrix}

The solution for loss is given in equation (T1.8). Taking expectations of equation (T1.8) yields the solution for expected loss in equation (T1.9).

3.2. Discussion

We are now prepared to discuss the effects of the shocks on the variables and utility. It is clear from Table 1 that our model passes the sunrise test. With flexible wages and prices, employment, L, and output, Y, the real variables that enter utility are independent of the money demand shock, V, and of the parameters of the monetary rule. Expected utility is independent of σ²_u and depends on σ²_v only because U enters the utility function directly.

L and Y depend only on the productivity shock, X, and the labor supply shock, Z. The effects of a labor supply shock are easier to analyze than those of a productivity shock. The downward sloping marginal product of labor schedule, MPL, and the upward sloping marginal rate of substitution (of consumption for labor) schedule, MRS, implied by the price Under complete output stabilizationand wage equations, respectively are shown in the top panel of Figure 1 in logarithm space.

Table 1.

Flexible wages and prices

Figure 1.

Flexible wages and prices.

An increase in Z shifts the MRS schedule down from MRS₀ to MRS₁. The equilibrium real wage must fall and equilibrium l must rise from l₀ to l₁. The upward sloping production function schedule PF is plotted in the bottom panel of Figure 1 in logarithm space. The increase in Z does not affect the production function, so y rises from y₀ to y₁ as l rises from l₀ to l₁. An increase in Z raises utility because it results in both an increase in the utility from consumption and a net reduction in the disutility of labor since we assume that $\tilde{ρ} > 0.$ .

Under our assumptions, an increase in X increases y and lowers l. An increase in X shifts both the MPL and MRS schedules up from MPL₀ to MPL₀ and from MRS₀ to MRS₂, respectively. Under our assumption that $\tilde{ρ} > 0,$ , it shifts the MRS schedule up by more. Therefore, the equilibrium real wage must rise and equilibrium l must fall. An increase in X also shifts the production function to the left from PF₀ to PF₂ and by more than it shifts the MRS to the left because it takes more of a fall in l to keep output constant than to keep households content with the same real wage. Thus, even though equilibrium l falls, equilibrium y rises. An increase in X raises utility because it results in both an increase in the utility from consumption and a decrease in the disutility of labor.

L and Y do not depend on the goods demand shock, U, or the money demand shock, V. With flexible wages and prices, the model is recursive. The real variables, labor, output, and the real wage, are determined by the subsystem made up of the production, price, and wage equations. Given values of these variable, the nominal variables, the price level, the nominal interest rate, and the money supply, are determined by the subsystem made up of the demand, rule, and money equations. Neither U nor V enters the subsystem that determines the real variables. An increase in U affects the utility of consumption and the disutility of labor in exactly the same way, so households have no incentive to change their decisions. Both U and V enter the subsystem that determines the nominal variables through the policy rule.

Increases in σ²_u σ²_x, σ²z, the variances of the logarithms of U, X, and Z, respectively, increase expected loss.

4. Wage Contracts and Flexible Prices

In this section, we consider the version with wage contracts and flexible prices.

4.1. Solution

In this version, the price and wage equations are

\begin{matrix} P = \frac{L^{α} W}{X}, & (p r i c e) \end{matrix}

\begin{matrix} \frac{1}{W} ε (\frac{X_{0} L^{\tilde{X}} U}{Z}) = ε (\frac{L U}{Y^{ρ} P}), & (w a g e) \end{matrix}

The price equation is the same as in the case of perfectly flexible wages and prices, but the wage equation is different. With wage contracts, wages must be set one period in advance without knowledge of the current shocks, so the wage equation contains expectations.

As before, we solve the model using the method of undetermined coefficients. The solutions for selected variables are displayed in Table 2. The solutions for the other variables can be obtained using these solutions and the equations of the model. Suppose that solution for L takes the form given in equation (T2.1). We find Ξ by substituting the output and price equations into the wage equation and collecting terms to obtain

Table 2.

Wage contracts and flexible prices

\begin{matrix} X_{0} ε (\frac{L^{\tilde{X}} U}{Z}) = {\tilde{α}}^{ρ} ε (\frac{U}{L^{\tilde{α} \tilde{ρ}} X^{\tilde{ρ}}}) . & (\begin{matrix} 35 \end{matrix}) \end{matrix}

Substituting in the conjectured form of the solution for L in equation (T2.1) yields

\begin{matrix} X_{0} ξ^{\tilde{χ}} ε Q_{3} = {\tilde{α}}^{ρ} ξ^{- \tilde{α} \tilde{ρ}} ε Q_{2}, & (36) \end{matrix}

\begin{matrix} Q_{2} = U^{1 - ξ_{U} \tilde{α} \tilde{ρ}} - V^{- ξ_{V} \tilde{α} \tilde{ρ}} X^{- (ξ_{X} \tilde{α} + 1) \tilde{ρ}} Z^{- ξ_{Z} \tilde{α} \tilde{ρ}}, & Q_{3} = U^{ξ_{U} \tilde{X} + 1} V^{ξ_{V} \tilde{X}} X^{ξ_{X} \tilde{X}} Z^{ξ_{Z} \tilde{X} - 1}, \end{matrix}

Therefore, if equation (36) is to hold, Ξ must take on the value in equation (T2.3).

We can find the ξ_j and W by substituting the rule equation into the demand equation and collecting terms to obtain

\begin{matrix} U Y^{- ρ} P^{- 1} = P^{λ_{P}} Y^{λ_{Y}} Y^{{* λ}_{Y *}} {\bar{Y}}^{λ_{\bar{Y}}} M^{λ_{M}} U^{λ_{U}} V^{λ_{V}} X^{λ_{X}} Z^{λ_{Z}} ε (U_{+ 1} Y_{+ 1}^{- ρ} P_{+ 1}^{- 1}), & (37) \end{matrix}

In a stationary rational expectations equilibrium with a levels reaction function W₊₁ = W. Imposing this restriction and eliminating Y, P, M, and Y^* using the output, price, and money equations and the solution for Y^* implied by the solution for L^* in equation (T1.1), respectively, and collecting some terms yields

\begin{matrix} (λ_{M} + \tilde{α} (ρ + λ_{Y}) + α (1 + λ_{P})) (\ln ξ + ξ_{U} u + ξ_{V} v + ξ_{X} x + ξ_{Z} z) = - (λ_{Y} + λ_{M}) \ln {\tilde{α}}^{- 1} - λ_{Y *} \ln ({\tilde{α}}^{- 1} H^{\tilde{α}}) - λ_{\bar{Y}} \bar{y} + \ln ε (Q_{4}) - (λ_{P} + λ_{M}) w + (1 - λ_{U}) u - (λ_{V} + λ_{M}) v - (\frac{(λ_{X} + \tilde{ρ} - λ_{P} + λ_{Y}) D + \tilde{χ} λ_{Y *}}{D}) x - (\frac{λ_{Z} D + \tilde{α} λ_{Y *}}{D}) z & (\begin{matrix} 38 \end{matrix}) \end{matrix}

Q_{4} = U^{1 - ξ_{U} (\tilde{α} ρ + α)} V^{- ξ_{V} (\tilde{α} ρ + α)} X^{- (\tilde{ρ} + ξ_{X} (\tilde{α} ρ + α))} Z^{- ξ_{Z} (\tilde{α} ρ + α)}

If equation (38) is to hold for all U, V, X, and Z, then the ξ_j and W must take on the values given in equations (T2.2) and (T2.8), respectively.

4.2. Expected Loss

With wage contracts, the solutions for all the variables depend on the parameters of the monetary rule. In this subsection we derive the optimal rule with wage contracts and describe the effects of the shocks under that rule. Note that there is a one to one mapping from the parameters of the policy rule to the coefficients of the shocks in the solution for L. It is more convenient to determine the optimal shock coefficients for L and then infer the optimal policy rule parameters.

The (logarithm of the) policymaker’s expected loss is given by

\begin{matrix} \ln ε L = \ln K + (ξ_{U}^{2} \tilde{α} \tilde{ρ} \tilde{χ} + 1) σ_{u}^{2} + ξ_{V}^{2} \tilde{α} \tilde{ρ} \tilde{χ} σ_{v}^{2} + ({(ξ_{X} + \frac{\tilde{ρ}}{D})}^{2} \tilde{α} \tilde{ρ} \tilde{χ} + {(\frac{\tilde{ρ} \tilde{χ}}{D})}^{2}) σ_{x}^{2} + ({(ξ_{Z} + \frac{1}{D})}^{2} \tilde{α} \tilde{ρ} \tilde{χ} + {(\frac{\tilde{α} \tilde{ρ}}{D})}^{2}) σ_{z}^{2} & (39) \end{matrix}

The derivation of this exact expression is actually simpler than the derivation of the standard approximation.

It is more convenient to work with the deviation of the policymaker’s expected loss from Pareto optimal expected loss, Δ ln εL = ln εL—ln εL^*, where

\begin{matrix} \frac{Δ \ln ε L}{\tilde{α} \tilde{ρ} \tilde{χ}} = ξ_{U}^{2} σ_{u}^{2} + ξ_{V}^{2} σ_{v}^{2} + {(ξ_{X} + \frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(ξ_{Z} + \frac{1}{D})}^{2} σ_{z}^{2} & (40) \end{matrix}

obtained by subtracting the expression for Pareto optimal expected loss in equation (T1.9) from equation (39).

4.3. Optimal Policy

It is clear from inspection that the values of the shock coefficients in the solution for labor which minimize (40) are

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ξ_{U} = 0, & ξ_{V} = 0, \end{matrix} & ξ_{X} = - \frac{\tilde{ρ}}{D}, \end{matrix} & ξ_{Z} = \frac{1}{D} \end{matrix} & (41) \end{matrix}

and that if the shock coefficients take on these values expected loss with wage contracts is equal to the Pareto optimal level of expected loss.

In characterizing the optimal policy rule, we assume that the policymaker adjusts the nominal interest rate only in response to the price level and the shocks:

\begin{matrix} \begin{matrix} \begin{matrix} λ_{U}, λ_{V}, λ_{X}, λ_{Z} \frac{>}{<} 0, & λ_{P} > 0, \end{matrix} & λ_{M} = λ_{Y} = λ_{Y *} = 0 \end{matrix} & (42) \end{matrix}

and that λ_P is an arbitrary positive number. The optimal rule coefficients implied by the optimal labor coefficients are obtained by equating the expressions for the shock coefficients in equation (T2.2) to the optimal values of these coefficients given in equation (41) and solving for the policy rule parameters. The results are

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} λ_{U} = 1, & {\begin{matrix} λ \end{matrix}}_{V} = 0, \end{matrix} & λ_{X} = - \frac{\tilde{ρ} χ}{\tilde{α} \tilde{ρ} + \tilde{χ}} + (\frac{\tilde{ρ} + \tilde{χ}}{\tilde{α} \tilde{ρ} + \tilde{χ}}) λ_{P}, \end{matrix} & λ_{Z} = - \frac{\tilde{α} ρ + α}{\tilde{α} \tilde{ρ} + \tilde{χ}} - (\frac{α}{\tilde{α} \tilde{ρ} + \tilde{χ}}) λ_{P} \end{matrix} & (43) \end{matrix}

The model exhibits determinacy for any positive value of λ_P, so the value λ_P can be chosen arbitrarily. Once a value of λ_P is chosen, the values of the other policy rule parameters are determined. The policymaker should move the interest rate to exactly match any movements in U, but should not respond at all to movements in V. It should lower the interest rate if Z rises no matter what the positive value of λ_P because the marginal disutility of labor varies inversely with Z, so output and employment should be increased. Whether it should raise or lower the interest rate if X rises depends on the value of λ_P.

An alternative way of finding the optimal rule is less direct but more elegant. If wages and prices are perfectly flexible and the policymaker follows the optimal rule for which the coefficients are given in equation (43), then for all shocks the economy is at the Pareto optimum, and the wage is unaffected. The wage result can be confirmed by substituting the expressions for the λ_i in equation (43) into the solution for W^* in equation (T1.3). The wage result implies that when the policymaker follows the optimal rule, the outcomes for all the variables including wages are the same no matter whether wages are preset in contracts. That is, the requirement that wages must remain constant is not a constraint that prevents attainment of the Pareto optimum. It follows that an alternative way of finding the optimal rule in the version with wage contracts and flexible prices without ever calculating the solution for that version is to find the rule that keeps wages constant in the version with flexible wages and prices.¹³

4.4. Output Gap Stabilization

If the nominal interest rate responds only to the output gap, that is, only to deviations of output from its Pareto-optimal level, so that

\begin{matrix} \begin{matrix} \begin{matrix} λ_{Y} = - λ_{Y *} > 0, & λ_{P} > 0, \end{matrix} & λ_{M} = λ_{U} = λ_{V} = λ_{X} = λ_{Z} = 0 \end{matrix} & (44) \end{matrix}

the values of the shock coefficients in the solution for labor are

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ξ_{U} = \frac{1}{Γ_{Y}}, & ξ_{V} = 0, \end{matrix} & ξ_{X} = - \frac{\tilde{ρ} - λ_{P} + λ_{Y}}{Γ_{Y}} \end{matrix} + \frac{\tilde{χ} λ_{Y}}{Γ_{Y} D}, & ξ_{Z} = \frac{\tilde{α} λ_{Y}}{Γ_{Y} D}, \end{matrix} & (45) \end{matrix}

Γ_{y} = \bar{α} (ρ + λ_{y}) + α (1 + λ_{p})

where the subscript on Γ indicates the special case under consideration. In this case, for example, Γ_y is equal to Γ with λ_M = 0. Recall that there must always be a nominal anchor, so λ_P > 0 in Γ_Y. Clearly if λ_y =—λ_Y* → ∞, the values of the shock coefficients in the solution for labor are the Pareto-optimal equilibrium values given in equation (41). That is, complete stabilization of the output gap yields the same result as the optimal policy discussed in the preceding subsection. This result makes sense because loss can be written as a function of output and shocks and because we assume that the policymaker knows the shocks and, therefore, can calculate the Pareto-optimal value of output.

4.5. Nominal Income Stabilization and Related Hybrid Rules

If the nominal interest rate responds only to deviations of nominal income from a constant target value $\bar{Y}$ , so that

\begin{matrix} \begin{matrix} \begin{matrix} λ_{P} = λ_{Y} > 0, & λ_{Y} = - λ_{\bar{Y}}, \end{matrix} & λ_{Y *} = λ_{M} = λ_{U} = λ_{V} = λ_{X} = λ_{Z} = 0 \end{matrix} & (46) \end{matrix}

then the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{G}^{P Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{1}{\tilde{α} ρ + α + λ_{Y}})}^{2} σ_{u}^{2} + {(\frac{- \tilde{ρ}}{\tilde{α} ρ + α + λ_{Y}} + \frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (\begin{matrix} 47 \end{matrix}) \end{matrix}

where the superscript after the vertical bar indicates which variable is being stabilized and the subscript after the vertical bar can take on three values: G for general, C for complete stabilization, and O for optimal stabilization.

Under complete nominal income stabilization $(\begin{matrix} λ_{P} = λ_{Y} > 0, & λ_{Y} = - λ_{\bar{Y}} \to \infty \end{matrix})$ , the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{C}^{P Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (48) \end{matrix}

Note that the more inelastic is labor supply (the larger X and, therefore, the larger is D) the closer is complete nominal income stabilization to the fully optimal policy.14

The policy that is optimal within the class of nominal income stabilization policies is found by minimizing the expected loss deviation in equation (47) with respect to λ_y The first order condition for λ_y and the optimal λ_y and ξ are

\begin{matrix} 0 = D σ_{u}^{2} + {\tilde{ρ}}^{2} χ σ_{x}^{2} - λ_{Y} {\tilde{ρ}}^{2} σ_{x}^{2} & (\begin{matrix} 49 \end{matrix}) \end{matrix}

\begin{matrix} λ_{Y} = \frac{D σ_{u}^{2} + {\tilde{ρ}}^{2} χ σ_{x}^{2}}{{\tilde{ρ}}^{2} σ_{x}^{2}} & (50) \end{matrix}

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ξ_{U} = \frac{{\tilde{ρ}}^{2} σ_{x}^{2}}{D ({\tilde{ρ}}^{2} σ_{x}^{2} + σ_{u}^{2})}, & ξ_{V} = 0, \end{matrix} & ξ_{X} = - \frac{{\tilde{ρ}}^{3} σ_{x}^{2}}{D ({\tilde{ρ}}^{2} σ_{x}^{2} + σ_{u}^{2})}, \end{matrix} & ξ_{Z} = 0. \end{matrix} & (51) \end{matrix}

Therefore, the expected loss from optimal stabilization of output is a positive fraction of the loss associated with the productivity shock under complete stabilization of output plus the irreducible loss associated with the labor supply shock:

\begin{matrix} \frac{Δ \ln ε U |_{O}^{P Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = (\frac{σ_{u}^{2}}{{\tilde{ρ}}^{2} σ_{x}^{2} + σ_{u}^{2}}) {(\frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (52) \end{matrix}

The fraction rises from zero to one as the ratio $\frac{σ_{u}^{2}}{σ_{x}^{2}}$ increases from zero to infinity.

Welfare is higher than with optimal nominal income stabilization if the policymaker completely stabilizes a combination of the price level and output in which the weights on the two variables are not equal.¹⁵ In particular, if

\begin{matrix} \begin{matrix} \begin{matrix} \frac{{\begin{matrix} λ \end{matrix}}_{P}}{λ_{Y}} = \frac{\tilde{χ}}{\tilde{ρ} + \tilde{χ}} > 0, & λ_{Y} = - λ_{\bar{Y}} \to \infty, \end{matrix} & λ_{Y *} = λ_{M} = λ_{U} = λ_{V} = λ_{X} = λ_{Z} = 0 \end{matrix} & (53) \end{matrix}

then the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{O}^{P, Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{1}{D})}^{2} σ_{z}^{2} & (54) \end{matrix}

The optimal hybrid policy can achieve the Pareto optimal outcomes for three of the four shocks. With only wage contracts, there are four disturbance coefficients in the solution for labor, ξ_U, ξ_V, ξ_X and ξ_Z. When a combination of the price level and output are stabilized, ξ_V and ξ_Z are equal to zero no matter what the values of the rule coefficients, λ_P and λ_Y Zero is the optimal value for ξ_V, but not for ξ_Z, so there is some irreducible loss. The two remaining disturbance coefficients, ξ_U and ξ_X, are independent functions of the rule coefficients, λ_P and λ_Y, so they can be set at their optimal values by the appropriate choices of values for these coefficients. A hybrid rule can do nothing to offset labor supply shocks. The realization of the labor supply shock does not enter the solution for output and the price level because only the expectation of the labor supply equation is in the set of equations that determines the equilibrium values of these variables.

There is an alternative way of finding the optimal hybrid rule which is analogous to the alternative way of finding the fully optimal rule discussed in the subsection on optimal policy. The optimal hybrid rule in the version with wage contracts and flexible prices is the rule that would make the nominal wage invariant to demand, money, and productivity shocks (U, V, and X) in the version with flexible wages and prices. The solution for the nominal wage with flexible wages and prices is given in equation (T1.3) and with a hybrid rule the nominal wage is invariant to U, V, and X if and only if the λ_i are set at the values given in equation (53).

4.6. Price Level Stabilization

If the nominal interest rate responds only to deviations of the price from a constant target value, so that

\begin{matrix} \begin{matrix} λ_{P} > 0, & {\begin{matrix} λ \end{matrix}}_{Y} = {\begin{matrix} λ \end{matrix}}_{Y *} = {\begin{matrix} λ \end{matrix}}_{M} = {\begin{matrix} λ \end{matrix}}_{U} = {\begin{matrix} λ \end{matrix}}_{V} = {\begin{matrix} λ \end{matrix}}_{X} = {\begin{matrix} λ \end{matrix}}_{Z} = 0 \end{matrix} & (55) \end{matrix}

then the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{G}^{P}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{1}{Γ_{P}})}^{2} σ_{u}^{2} + {(\frac{λ_{P} - \tilde{ρ}}{Γ_{P}} + \frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (56) \end{matrix}

Γ_{P} = \tilde{α} ρ + α + α λ_{P}

Under complete price level stabilization, the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{C}^{P}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{1}{α} + \frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} = {(\frac{ρ + χ}{α D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (57) \end{matrix}

For productivity shocks, under price level stabilization, employment and, therefore, output are more volatile than under the optimal policy. For labor supply shocks, employment and, therefore, output are less volatile than under the optimal policy.

The policy that is optimal within the class of price stabilization policies is found by minimizing the expected loss deviation in equation (56) with respect to λ_P. The first order condition for λ_P and the optimal λ_P and ξ are

\begin{matrix} 0 = α σ_{u}^{2} + ((λ_{P} - \tilde{ρ}) + Γ_{P} (\frac{\tilde{ρ}}{D})) ((λ_{P} - \tilde{ρ}) α - Γ_{P}) σ_{x}^{2} & (58) \end{matrix}

\begin{matrix} λ_{P} = \frac{α D σ_{u}^{2} + \tilde{ρ} ρ χ σ_{x}^{2}}{ρ (ρ + χ) σ_{x}^{2}} & (59) \end{matrix}

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ξ_{U} = \frac{ρ (ρ + χ) σ_{x}^{2}}{(ρ^{2} σ_{x}^{2} + α^{2} σ_{u}^{2})}, & ξ_{V} = 0, \end{matrix} & ξ_{X} = \frac{α D σ_{u}^{2} - \tilde{ρ} ρ^{2} σ_{x}^{2}}{D (ρ^{2} σ_{x}^{2} + α^{2} σ_{u}^{2})}, \end{matrix} & ξ_{Z} = 0. \end{matrix} & (60) \end{matrix}

Therefore, the expected loss from optimal stabilization of the price level is a positive fraction of the loss associated with the productivity shock under complete stabilization of the price level plus the irreducible loss associated with the labor supply shock:

\begin{matrix} \frac{Δ \ln ε U |_{O}^{P}}{\tilde{α} \tilde{ρ} \tilde{χ}} = (\frac{α^{2} σ_{u}^{2}}{ρ^{2} σ_{x}^{2} + α^{2} σ_{u}^{2}}) {(\frac{ρ + χ}{α D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (\begin{matrix} 61 \end{matrix}) \end{matrix}

The fraction rises from zero to one as the ratio $\frac{σ_{u}^{2}}{σ_{x}^{2}}$ increases from zero to infinity.

4.7. Output Stabilization

If the nominal interest rate responds only to deviations of the output from a constant target value, so that

\begin{matrix} \begin{matrix} \begin{matrix} λ_{Y} = - λ_{\bar{Y}} > 0, & λ_{P} > 0, \end{matrix} & λ_{Y *} = λ_{M} = λ_{U} = λ_{V} = λ_{X} = λ_{Z} = 0 \end{matrix} & (62) \end{matrix}

then the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{G}^{Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{1}{Γ_{Y}})}^{2} σ_{u}^{2} + {(\frac{λ_{P} - λ_{Y} - \tilde{ρ}}{Γ_{Y}} + \frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (63) \end{matrix}

Γ_{Y} = \tilde{α} (ρ + λ_{Y}) + α (1 + λ_{P})

Under complete output stabilization $(λ_{Y} = - λ_{\bar{Y}} \to \infty, λ_{P} > 0)$ , the expected loss deviation is

\begin{matrix} \frac{\ln ε L |_{C}^{Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{\tilde{χ}}{\tilde{α} D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (64) \end{matrix}

The policy that is optimal within the class of real output stabilization policies is found by minimizing the expected loss deviation in equation (63) with respect to λ_Y. The first order condition for λ_Y and the optimal λ_Y and ξ are

\begin{matrix} 0 = \tilde{α} D σ_{u}^{2} + (1 + λ_{P}) [(ρ + χ) (1 + λ_{P}) - \tilde{χ} (ρ + λ_{Y})] σ_{x}^{2} & (65) \end{matrix}

\begin{matrix} λ_{Y} = - ρ + \frac{(ρ + χ) {\tilde{λ}}_{P}}{\tilde{χ}} + \frac{\tilde{α} D}{\tilde{χ} {\tilde{λ}}_{P}} \frac{σ_{u}^{2}}{σ_{x}^{2}} & (66) \end{matrix}

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ξ_{U} = \frac{\tilde{χ} {\tilde{λ}}_{ρ} σ_{x}^{2}}{D ({\tilde{λ}}_{P}^{2} σ_{x}^{2} + {\tilde{α}}^{2} σ_{u}^{2})}, & ξ_{V} = 0, \end{matrix} & ξ_{X} = - \frac{\tilde{ρ} {\tilde{λ}}_{P}^{2} σ_{x}^{2} + \tilde{α} D σ_{u}^{2}}{D ({\tilde{λ}}_{P}^{2} σ_{x}^{2} + {\tilde{α}}^{2} σ_{u}^{2})}, \end{matrix} & ξ_{Z} = 0 \end{matrix} & (67) \end{matrix}

where ${\tilde{λ}}_{P} = 1 + λ_{P}$ . Therefore, the expected loss from optimal stabilization of output is a positive fraction of the loss associated with the productivity shock under complete stabilization of output plus the irreducible loss associated with the labor supply shock:

\begin{matrix} \frac{Δ \ln ε U |_{O}^{Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = (\frac{{\tilde{α}}^{2} σ_{u}^{2}}{{\tilde{λ}}_{P}^{2} σ_{x}^{2} + {\tilde{α}}^{2} σ_{u}^{2}}) {(\frac{\tilde{χ}}{\tilde{α} D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (68) \end{matrix}

The fraction increases from zero to one as the ratio $\frac{σ_{u}^{2}}{σ_{x}^{2}}$ increases from zero to infinity

4.8. Money Supply Stabilization

If the nominal interest rate responds only to deviations of the money supply from a constant target value, so that

\begin{matrix} λ_{M} = - λ_{\bar{Y}} > 0, & λ_{P} = λ_{Y} = λ_{Y *} = λ_{U} = λ_{V} = λ_{X} = λ_{Z} = 0 & (69) \end{matrix}

then the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{G}^{M}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{1}{Γ_{M}})}^{2} σ_{u}^{2} + {(\frac{λ_{M}}{Γ_{M}})}^{2} σ_{v}^{2} + {(\frac{- \tilde{ρ}}{Γ_{M}} + \frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (70) \end{matrix}

Γ_{M} = λ_{M} + \tilde{α} ρ + α

Under complete money supply stabilization $(λ_{M} = - λ_{\bar{Y}} \to \infty),$ , the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{C}^{M}}{\tilde{α} \tilde{ρ} \tilde{χ}} = σ_{v}^{2} + {(\frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (71) \end{matrix}

The policy that is optimal within the class of money supply stabilization policies is found by minimizing the expected loss deviation in equation (70) with respect to λ_M The first order condition for λ_M and the optimal λ_M and ξ are

\begin{matrix} 0 = - D σ_{u}^{2} + λ_{M} D (\tilde{α} ρ + α) σ_{v}^{2} + \tilde{ρ} (- \tilde{ρ} D + \tilde{ρ} Γ_{M}) σ_{x}^{2} & (72) \end{matrix}

\begin{matrix} λ_{M} = \frac{D σ_{u}^{2} + {\tilde{ρ}}^{2} χ σ_{x}^{2}}{{\tilde{ρ}}^{2} σ_{x}^{2} + D (\tilde{α} ρ + α) σ_{v}^{2}} & (73) \end{matrix}

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ξ_{U} = \frac{J}{R}, & ξ_{V} = - \frac{D σ_{u}^{2} + {\tilde{ρ}}^{2} χ σ_{x}^{2}}{R}, \end{matrix} & ξ_{X} = - \frac{\tilde{ρ} J}{R}, \end{matrix} & ξ_{Z} = 0. \end{matrix} & (74) \end{matrix}

\begin{matrix} \begin{matrix} J = {\tilde{ρ}}^{2} σ_{x}^{2} + D A σ_{v}^{2}, & R = D (σ_{u}^{2} + {\tilde{ρ}}^{2} σ_{x}^{2} + A^{2} σ_{u}^{2}), \end{matrix} & A = \tilde{α} ρ + α \end{matrix}

The expected loss from optimal stabilization of the money supply is

\begin{matrix} \frac{Δ \ln ε U |_{O}^{M}}{\tilde{α} \tilde{ρ} \tilde{χ}} = \frac{(σ_{u}^{2} + {\tilde{ρ}}^{2} σ_{x}^{2}) σ_{u}^{2} {\tilde{ρ}}^{2} σ_{x}^{2}}{R^{2}} + \frac{({\tilde{ρ}}^{2} σ_{x}^{2} + A^{2} σ_{v}^{2}) {\tilde{ρ}}^{2} χ^{2} σ_{x}^{2} σ_{u}^{2}}{R^{2}} + \frac{(σ_{u}^{2} + A^{2} σ_{v}^{2}) D^{2} σ_{u}^{2} σ_{v}^{2}}{R^{2}} + \frac{{2 \tilde{ρ}}^{2} (A^{2} + A_{χ} + χ^{2}) σ_{u}^{2} σ_{v}^{2} σ_{x}^{2}}{R^{2}} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (75) \end{matrix}

Comparison of equation (75) with equation (52) confirms that if σ²_u,σ²_x > 0, but σ²v = 0, then the expected loss from optimal money supply stabilization is the same as the expected loss from optimal nominal income stabilization. However, if σ²_x,σ²_v > 0, but σ²u = 0 or σ²_u,σ²_v > 0, but σ²_x = 0, exPected loss from optimal money supply stabilization is larger than expected loss from optimal nominal income stabilization.

Although we have used our model to make clear the disadvantages of money supply stabilization, we cannot use it to evaluate claims about the advantages of this policy. In our model, all data become available simultaneously. However, in real-world economies money supply data become available more quickly than most, and it is sometimes claimed that money supply stabilization has an advantage because of this fact. In our model, the policymaker can achieve a desired value for any single variable. However, it is sometimes claimed that in real-world economies it is easier to achieve a desired value for the money supply than for some other variables.

5. Wage and Price Contracts

In this section we consider the version with both wage and price contracts.

5.1. Solution

In this version, both the wage and price equations are different from the case of perfectly flexible wages and prices:

\begin{matrix} ε (\frac{U}{Y^{\tilde{ρ}}}) = \frac{W}{P} ε (\frac{L^{α} U}{Y^{\tilde{ρ}} X}), & (p r i c e) \end{matrix}

\begin{matrix} ε (\frac{χ_{0} L^{\tilde{χ}} U}{Z}) = \frac{W}{P} ε (\frac{L U}{Y^{ρ}}), & (w a g e) \end{matrix}

Both wages and prices must be set one period in advance without knowledge of the current shocks so both the wage equation and the price equation contain expectations.

We solve the model using the method of undetermined coefficients. The solutions are displayed in Table 3. Suppose that the solution for L has the form given in equation (T3.1). We find Ψ by substituting the production equation into the price and wage equations, collecting terms, and dividing the price equation by the wage equation to eliminate $\frac{W}{P}$ to obtain

Table 3.

Wage and price contracts

\begin{matrix} \frac{ε (L^{- \tilde{α} \tilde{ρ}} U X^{- \tilde{p}})}{χ_{0} ε (L^{\tilde{χ}} U Z^{- 1})} = \frac{ε (L^{α - \tilde{α} \tilde{ρ}} U X^{- ρ})}{{\tilde{α}}^{ρ} ε (L^{α - \tilde{α} \tilde{ρ}} U X^{- ρ})} & (76) \end{matrix}

Substituting in the conjectured form of the solution for L in equation (T3.1) and rearranging yields

\begin{matrix} \frac{{\tilde{α}}^{ρ} ψ^{- \tilde{α} \tilde{ρ}} ε (Q_{5})}{χ_{0} ψ^{\tilde{χ}} ε (Q_{6})} = 1 & (77) \end{matrix}

\begin{matrix} Q_{5} = U^{1 - ψ_{U} \tilde{α} \tilde{ρ}} V^{- ψ_{V} \tilde{α} \tilde{ρ}} X^{- (ψ_{X} \tilde{α} \tilde{ρ} + \tilde{ρ})} Z^{- ψ_{Z} \tilde{α} \tilde{ρ}}, & Q_{6} = U^{ψ_{U} \tilde{χ} + 1} V^{ψ_{V} \tilde{χ}} X^{ψ_{X} \tilde{χ}} Z^{ψ_{Z X} \tilde{χ} - 1} \end{matrix}

If equation (77) is to hold Ψ must take on the value in equation (T3.3).

We find the Ψ_j, P and W by substituting the rule equation into the demand equation to obtain

\begin{matrix} Y^{- ρ} P^{- 1} U = P^{λ_{P}} Y^{λ_{Y}} Y^{{* λ}_{Y *}} {\bar{Y}}^{λ_{\bar{Y}}} M^{λ_{M}} U^{λ_{U}} V^{λ_{V}} X^{λ_{X}} Z^{λ_{Z}} ε_{t} (Y_{+ 1}^{- ρ} P_{+ 1}^{- 1} U_{+ 1}) & (78) \end{matrix}

In a stationary rational expectations equilibrium with a levels reaction function P₊₁ = P. Imposing this restriction and eliminating Y, M, and Y^* using the production and money equations and the solution for Y^* implied by the solution for L^* in equation (T l.l), respectively, and collecting some terms yield

\begin{matrix} \tilde{α} (ρ + λ_{M} + λ_{Y}) (\ln ψ + ψ_{U} u + ψ_{V} v + ψ_{X} x + ψ_{Z} z) = - (λ_{Y} + λ_{M}) \ln {\tilde{α}}^{- 1} - λ_{Y *} \ln ({\tilde{α}}^{- 1} H^{\tilde{α}}) - λ_{\bar{Y}} \bar{y} - \ln ε_{t} (Q_{7}) - (λ_{P} + λ_{M}) p + (1 - λ_{U}) u - (λ_{V} + λ_{M}) v - (\frac{(λ_{X} + ρ + λ_{Y} + λ_{M}) D + \tilde{χ} λ_{Y *}}{D}) x - (\frac{λ_{Z} D + \tilde{α} λ_{Y *}}{D}) z & (79) \end{matrix}

Q_{7} = U^{1 - ψ_{U} \tilde{α} ρ} V^{- ψ_{V} \tilde{α} ρ} X^{- ψ_{X} \tilde{α} ρ} Z^{- ψ_{Z} \tilde{α} ρ}

If equation (79) is to hold for all U, V, X, and Z, it must be that the ψ_j and P, respectively, must take on the values given in equations (T3.2) and (T3.8). Given the solution for P, the price equation can be used to obtain the solution for W in equation (T3.10).¹⁶

5.2. Optimal Policy and Output Gap Stabilization

In this subsection we discuss the optimal policy with wage and price contracts. As in the case of wage contracts and flexible prices, we state the policymaker’s optimization problem in terms of the labor coefficients and then infer the optimal rule coefficients. It is clear from Tables 2 and 3 that the solutions for L and, therefore, the solutions for Y have exactly the same form with wage and price contracts as they do with wage contracts alone with ψ_j, j = U, V, X, Z replacing ξ^j, j = U, V, X, Z whereever they appear. It follows that the expressions for expected loss and, therefore, the optimal values of the shock coefficients in the solution for L are the same with wage and price contracts as they are with wage contracts alone. That is,

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ψ_{U} = 0, & ψ_{V} = 0, \end{matrix} & ψ_{X} = - \frac{\tilde{ρ}}{D}, \end{matrix} & ψ_{Z} = \frac{1}{D} \end{matrix} & (80) \end{matrix}

In characterizing the optimal policy rule, as before we assume that the policymaker responds only to the price level and the shocks:

\begin{matrix} \begin{matrix} \begin{matrix} λ_{U}, λ_{V,} λ_{X}, λ_{Z} \frac{>}{<} 0, & λ_{P} > 0, \end{matrix} & λ_{M} = λ_{Y} = λ_{Y *} = 0 \end{matrix} & (81) \end{matrix}

and that λ_P is an arbitrary positive number. The optimal rule coefficients implied by the optimal labor coefficients are

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} λ_{U} = 1, & λ_{V} = 0, \end{matrix} & λ_{X} = - \frac{ρ \tilde{χ}}{D}, \end{matrix} & λ_{Z} = - \frac{\tilde{α} ρ}{D} \end{matrix} & (82) \end{matrix}

In contrast to the results for wage contracts alone, with wage and price contracts the optimal λ_j, j = U, V, X, Z are independent of XP. The only role played by λ_P is to guarantee determinacy, in particular, to insure that agents can calculate the expected future price level. The contract price for the current period is set before the shocks are drawn so there can be no movements in the current price level induced by the shocks and therefore nothing for the policymaker to respond to.

With wage and price contracts, just as with wage contracts alone, complete stabilization of the output gap yields the optimal outcome and for the same reason.

5.3. Simple Policy Rules

Given one-period wage and price contracts and the list of variables we have included in the policy rule, there are really only two simple rules to consider: output stabilization and money supply stabilization. Since prices are set before uncertainty is resolved, the price level is always completely stabilized. As a consequence, stabilizing nominal income is the same thing as stabilizing output. Given the simple form of our money demand function, output stabilization and money supply stabilization have very similar implications. Stabilizing the money supply is the same thing as stabilizing output except that there is some increase in loss because shifts in money demand are not fully accommodated.

If the nominal interest rate responds only to deviations of output from the constant target value $\bar{Y}$ , so that

\begin{matrix} λ_{Y} = - λ_{\bar{Y}} > 0, & \begin{matrix} λ_{P} > 0, & \begin{matrix} λ_{Y *} = λ_{M} = λ_{U} = λ_{V} = λ_{X} = λ_{Z} = 0 & (83) \end{matrix} \end{matrix} \end{matrix}

then the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{G}^{P Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{1}{\tilde{α} (ρ + λ_{Y})})}^{2} σ_{u}^{2} + {(- \frac{\tilde{ρ} + λ_{Y}}{\tilde{α} (ρ + λ_{Y})} + \frac{\tilde{ρ}}{D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (84) \end{matrix}

Under complete output stabilization $(λ_{Y} = - λ_{\bar{Y}} \to \infty, λ_{P} > 0)$ , the solutions for the ψ_j are

\begin{matrix} ψ_{U} = 0, & \begin{matrix} ψ_{V} = 0, & \begin{matrix} \begin{matrix} ψ_{X} = - \frac{1}{\tilde{α}}, \end{matrix} & \begin{matrix} ψ_{Z} = 0, & (85) \end{matrix} \end{matrix} \end{matrix} \end{matrix}

and the expected loss deviation is

\begin{matrix} \frac{Δ \ln ε L |_{C}^{P Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = {(\frac{\tilde{χ}}{\tilde{α} D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (86) \end{matrix}

As is clear from a comparison of equations (86) and (48), if $\frac{\tilde{χ}}{\tilde{α}} > \tilde{ρ}$ , that is, if the ratio of the elasticity of the disutility of labor to the labor elasticity of production exceeds the elasticity of the utility of consumption, complete nominal income stabilization increases loss more when there are price contracts.

The policy that is optimal within the class of output stabilization policies is found by minimizing the expected loss deviation in equation (84) with respect to λ_Y. The first order condition and the optimal λ_y and ξ’s are

\begin{matrix} 0 = - \tilde{α} D σ_{u}^{2} + \tilde{ρ} (α + χ) σ_{x}^{2} + \tilde{α} \tilde{χ} σ_{x}^{2} λ_{Y} & (87) \end{matrix}

\begin{matrix} λ_{Y} = \frac{D σ_{u}^{2} - \tilde{ρ} (α + χ) σ_{x}^{2}}{\tilde{χ} σ_{x}^{2}} & (88) \end{matrix}

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} ξ_{U} = \frac{\tilde{χ} σ_{x}^{2}}{\tilde{α} D (σ_{u}^{2} + σ_{x}^{2})}, & ξ_{V} = 0, \end{matrix} & ξ_{X} = - \frac{D σ_{u}^{2} + \tilde{ρ} \tilde{α} σ_{x}^{2}}{\tilde{α} D (σ_{u}^{2} + σ_{x}^{2})}, \end{matrix} & ξ_{Z} = 0. \end{matrix} & (89) \end{matrix}

\begin{matrix} \frac{Δ \ln ε U |_{O}^{P Y}}{\tilde{α} \tilde{ρ} \tilde{χ}} = (\frac{σ_{u}^{2}}{σ_{u}^{2} + σ_{x}^{2}}) {(\frac{\tilde{χ}}{\tilde{α} D})}^{2} σ_{x}^{2} + {(\frac{1}{D})}^{2} σ_{z}^{2} & (90) \end{matrix}

The fraction rises from zero to one as the ratio $\frac{σ_{u}^{2}}{σ_{x}^{2}}$ increases from zero to infinity.

6. Conclusions

In this paper we construct an optimizing-agent model with one-period nominal contracts which is simple enough that we can make exact utility calculations. We evaluate alternative monetary policy rules using as a criterion the utility function of the representative agent. We focus on the two cases of (1) wage contracts and flexible prices and (2) wage and price contracts because, as we show, the outcomes in the third case, price contracts and flexible wages, are the same as the outcomes in the case of wage and price contracts for all variables except the nominal wage.

The fully optimal rule under complete information can attain the Pareto-optimal equilibrium because we assume one-period nominal contracts. We contrast the performance of the fully optimal policy with both ‘naive (complete)’ stabilization and ‘sophisticated (constrained optimal)’ stabilization of one variable or a combination of two variables. The simple rules we consider can never achieve the Pareto-optimal outcome because they imply no response to labor supply shocks. However, if there are no labor supply shocks, in a few special cases, naive and optimal simple rules are as good as fully optimal rules. Of course, in general, they are not.

A number of our conclusions regarding simple rules depend critically on the relative importance of productivity disturbances. For example, with only wage contracts, the more important are productivity disturbances, the worse are all forms of nominal income targeting and the greater the difference between the naive and sophisticated versions. Another critical parameter is the elasticity of the disutility of labor (which, of course, is inversely related to the elasticity of labor supply). For example, if the elasticity of the disutility of labor is high with wage contracts alone naive nominal income targeting performs very well but with both wage and price contracts it performs very badly.

Just how much further it is worthwhile to push the analysis of one-period nominal contract models is an open question. In this paper, we reaffirm that such models are tractable, but we show that some of their results are quite special, for example the result that if there are price contracts the existence of wage contracts is of no consequence. In Henderson and Kim (1999) we determine the effects of targeting money growth, inflation, and combinations of inflation and output on employment, output, and inflation. At a minimum, we plan to use the model of this paper to analyze the welfare implications of simple and optimal forms of these and related types of targeting.

Appendix A

In this appendix we summarize the properties of log normal distributions that are used in this paper.

Suppose that the variable Q has a log normal distribution; that is, suppose that q = ln Q ∼ N (μ_Q, 2σ_Q²). Now In Q^k = kq so Q^k = e^kq. It follows that the E (Q^k) = E (e^kq) = M (q, k) where M (q, k) is the moment generating function for q and is given by

\begin{matrix} M (q, k) = \int_{- \infty}^{\infty} e^{k q} [\frac{1}{2 \sqrt{π σ_{Q}}} e^{- \frac{{(q - μ_{Q})}^{2}}{4 σ_{Q}^{2}}}] d q = e^{k μ_{Q} + k^{2} σ_{Q}^{2}} & (A .1) \end{matrix}

Table 4.

Price contracts and flexible wages.

that is

\begin{matrix} E (Q^{k}) = e^{k μ_{Q} + k^{2} σ_{Q}^{2}} & (A .2) \end{matrix}

Note that if μ_Q = 0, then E (Q) = e^σ_Q² ≠ 1 and E (Q²) = e^4σ_Q². However, if E (Q) = 1 = e^μ_Q+σ_Q², then 0 = μ_Q+ σ_Q² so μ_Q = -σ_Q² and E (Q²) = e^{2μ_Q+4σ_Q²} = e^2σ_Q². We have assumed that μ_Q = 0 in order to simplify our calculations. However, we can understand why others might prefer the alternative assumption.

Now suppose that the variables U, V, and X are independently and log normally distributed; that is, suppose that u = ln U ∼ N (μ_u, 2σ_u²), v = ln V ∼ N (μ_v, 2σ_v²), and x = ln X ∼ N (μ_x, 2σ_x²). It follows that

\begin{matrix} E (U^{k_{U}} V^{k_{V}} X^{k_{X}}) = e^{k_{U} μ_{u} + k_{U}^{2} σ_{u}^{2} + k_{V} μ_{v} + k_{V}^{2} σ_{v}^{2} + k_{X} μ_{x} + k_{X}^{2} σ_{x}^{2}} . & (A .3) \end{matrix}

Appendix B

In this Appendix we show that the solutions with price contracts and flexible wages are the same as those with wage and price contracts for all variables except the nominal wages, as can be confirmed by comparing Table 4 with Table 3. With price contracts and flexible wages the wage and price equations are

\begin{matrix} ε (\frac{U}{Y^{\bar{ρ}}}) = \frac{1}{P} ε (\frac{W L^{α} U}{Y^{\bar{ρ}} X}) & (p r i c e) \end{matrix}

\begin{matrix} \frac{W}{P} = \frac{χ_{0} L^{χ} Y^{ρ}}{Z} & (w a g e) \end{matrix}

Suppose the solution for L takes the form given in equation (T4.1). To find Φ we substitute the production and wage equations into the price equation, and collect terms:

\begin{matrix} χ_{0} ε (L^{\bar{χ}} U Z^{- 1}) = {\tilde{α}}^{ρ} ε (L^{- \begin{matrix} \bar{α} & \bar{ρ} \end{matrix}} U X^{- \bar{ρ}}) & (B .1) \end{matrix}

Substituting in the conjectured form for L in equation (T4.1) in Table 4 yields

\begin{matrix} χ_{0} Φ^{\bar{χ}} ε Q_{6} = {\tilde{α}}^{ρ} Φ^{- \begin{matrix} \bar{α} & \bar{ρ} \end{matrix}} ε Q_{5} & (B .2) \end{matrix}

\begin{matrix} Q_{9} = U^{1 - Φ_{U} \bar{α} \bar{ρ}} V^{- Φ_{V} \bar{α} \bar{ρ}} X^{- (Φ_{χ} \bar{α} \bar{ρ} + \bar{ρ})} Z^{- Φ_{Z} \bar{α} \bar{ρ}}, & Q_{10} = U^{Φ_{U} \bar{χ} + 1} V^{Φ_{V} \bar{χ}} X^{Φ_{χ} \bar{χ}} Z^{Φ_{Z} \bar{χ} - 1} \end{matrix}

If equation (B.2) is to hold Φ must take on the value given by equation (T4.3). Note that Q₉, Q₁₀, and Φ are identical to Q₂, Q₃, and Ξ respectively except that ξ_j is replaced by φ_j for j = U, V, X, and Z.

To find the φ_j and P we substitute the rule equation into the demand equation:

\begin{matrix} Y^{- ρ} P^{- 1} U = P^{λ_{P}} Y^{λ_{Y}} Y^{{* λ}_{Y *}} {\bar{Y}}^{λ_{\bar{Y}}} M^{λ_{M}} U^{λ_{U}} V^{λ_{V}} X^{λ_{X}} Z^{λ_{Z}} (Y_{+ 1}^{- ρ} P_{+ 1}^{- 1} U_{+ 1}) & (B .3) \end{matrix}

Imposing the restriction that P₊₁ = P and eliminating Y, W, M, and Y^* using the production, wage, and money equations, and the solution for Y^* implied by the solution for L^* in equation (T l.l), respectively, and collecting terms yield

\begin{matrix} \tilde{α} (ρ + λ_{M} + λ_{Y}) (\ln Φ + ϕ_{U} u + ϕ_{V} v + ϕ_{X} x + ϕ_{Z} z) = - (λ_{Y} + λ_{M}) \ln ({\tilde{α}}^{- 1}) - λ_{Y *} \ln ({\tilde{α}}^{- 1} H^{\bar{α}}) - λ_{\bar{Y}} \bar{y} - \ln ε (Q_{11}) - (λ_{P} + λ_{M}) P + (1 - λ_{U}) u - (λ_{V} + λ_{M}) v - (\frac{(λ_{X} + ρ + λ_{Y} + λ_{M}) D + \tilde{χ} λ_{Y *}}{D}) x - (\frac{λ_{Z} D + \tilde{α} λ_{Y *}}{D}) z & (B .4) \end{matrix}

where ln ε(Q₁₁) is given by equation (T4.10). If equation (B.4) is to hold for all U, V, X, and Z, the φ_j and P must take on the values given in equations (T4.2) and (T4.8), respectively. The solution for W is found by substituting the solutions for L and P given by equations (T4.1) and (T4.8), respectively, and the solution for Y implied by the solution for L in equation (T4.1) into the wage equation (T4.10).

Acknowledgments

We would like to thank Jo Anna Gray, our discussant, for helpful comments and Charles Engel for suggesting that we change our specification of the objective function of firms to the current one. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System.

Simple Monetary Policy Rules Under Model Uncertainty

PETER ISARD

International Monetary Fund, Washington, DC 20431

pisard@imf.org

DOUGLAS LAXTON

International Monetary Fund, Washington, DC 20431

dlaxton@imf.org

ANN-CHARLOTTE ELIASSON

Stockholm School of Economics, Stockholm, Sweden

ann-charlotte.eliasson@seb.se

Abstract

Using stochastic simulations and stability analysis, the paper compares how different monetary policy rules perform in a moderately nonlinear model with a time-varying NAIRU. Rules that perform well in linear models but implicitly embody backward-looking measures of real interest rates (such as conventional Taylor rules) or substantial interest rate smoothing perform very poorly in models with moderate nonlinearities, particularly when policymakers tend to make serially-correlated errors in estimating the NAIRU. This challenges the practice of evaluating policy rules within linear models, in which the consequences of responding myopically to significant overheating are extremely unrealistic.

I. Introduction and Overview

This paper employs stochastic simulations and stability analysis to compare the performances of several types of simple monetary policy rules in a small model of the U.S. economy. The model, which is estimated with quarterly data for the post-1968 period, exhibits a moderate degree of nonlinearity, assumes that inflation expectations have a model-consistent component, and treats the non-accelerating-inflation rate of unemployment (NAIRU) as a time-varying and unobservable parameter. The simulation framework assumes that policymakers update their estimates of the NAIRU period by period, using their information about the macroeconomic model, and in a manner that implicitly recognizes the tendency to make serially-correlated errors in estimating the NAIRU.

The simulations and stability analysis demonstrate that several classes of rules that have been shown to perform well in linear models of the U.S. inflation process perform very poorly in our moderately-nonlinear model. These include conventional Taylor rules, as advocated by Taylor (1993, 1999a) and others; a class of forward-looking rules with a high degree of interest rate smoothing, as proposed by Clarida, Gali, and Gertler (1998); and a first-difference rule for the interest rate, as proposed by Levin, Wieland, and Williams (1999). One of the main conclusions is that rules that implicitly embody backward-looking measures of real interest rates (such as conventional Taylor rules) or too much interest rate smoothing can be too myopic to meet the stability conditions for macro models with moderate nonlinearities, particularly in a world in which policymakers tend to make seriallycorrelated errors in estimating the NAIRU. This finding in turn suggests, as a second main conclusion, that the propensity of economists to analyze the properties of monetary policy rules within the confines of linear models is difficult to defend as a research strategy. Linear models, in which bad policy rules affect the variances but not the means of inflation and unemployment, are fundamentally inappropriate for policy analysis because they fail to capture the fact that policymakers who allow economies to overheat significantly can fall behind “shifts in the curve” and fail to provide an anchor for inflation expectations, with first-order welfare consequences.

The paper also reports simulation results for inflation-forecast-based rules without explicit interest rate smoothing ¹ and explores how their optimal calibrations vary with the degree of NAIRU uncertainty, the shape of the Phillips curve, and the nature of inflation expectations. These results support the view that optimally-calibrated simple rules can deliver attractive macroeconomic performances in small empirical macro models. Indeed, a third main conclusion is that optimally-calibrated linear rules in which the interest rate is a function of the inflation forecast—when applied to model-consistent inflation forecasts—can stabilize our nonlinear model with approximately the same welfare outcome as a strategy that explicitly optimizes the policy loss (welfare) function. However, echoing the theme of Flood and Isard (1989), we stress that policymakers face considerable difficulties in attempting to identify the true macroeconomic model and the optimal calibration for any proposed policy rule.

In recognizing the contributions of Bob Flood, it is useful to reflect on the extent to which research on monetary policy strategies has shifted focus over the past decade (Box 1). Ten years ago, the academic discussion centered on the time inconsistency problem and the issue of monetary policy credibility.² The rules-versus-discretion debate continued to rage, and new thinking had emerged on the roles of institutional mechanisms other than rules (e.g., independent central banks and conservative central bankers), along with reputation, as vehicles for mitigating credibility problems.³ In this setting, Bob Flood became intrigued by the observation that central banks found it appealing to adopt simple policy rules, such as target growth rates for monetary aggregates, but to periodically modify the rules. This led to Flood and Isard (1989), which was interpreted as a contribution to both normative and positive economics.

Box 1.

Research on monetary policy: 1989 vs. 1999

The 1989 setting

concern with time inconsistency
focus on rules and other institutional arrangements for mitigating credibility problems
interest in robustness of rules—i.e., in simple rules that perform well across the spectrum of plausible macro models

Key points in Flood and Isard (1989)

Fully-state-contingent rules are not relevant possibilities in practice.
Since partially-state-contingent rules and discretion cannot be unambiguously ranked, it seems attractive to consider mixed strategies that combine a simple rule with discretion (an “escape clause”) and to establish institutional arrangements that provide incentives for policymakers not to overuse or underuse discretion.
In evaluating a simple policy rule, it is not valid to base counterfactual historical simulations on the assumption that rational market participants would have expected the authorities to completely adhere to the rule when policymakers, had they actually been confronted with the counterfactual history, would have sometimes had incentives to deviate from the rule.

The 1999 setting

consensus that simple rules cannot and should not be mechanically followed by policymakers
notion that research can nevertheless be useful for identifying the types and calibrations of rules that are relatively attractive as guidelines for policy
extensive reliance on stochastic simulation analysis with some attention to the robustness issue, little attention to modeling the process that the authorities use to update their information on key model parameters, and little explicit allowance for the fact that rational market participants might not find an announced rule fully credible

Flood and Isard (1989) started from the premise that an optimal fully-state-contingent rule for monetary policy is not a relevant possibility in a world in which knowledge about the macroeconomic structure and the nature of disturbances is incomplete. Since simple rules (including partially-state-contingent rules) and discretion cannot be unambiguously ranked, a mixed strategy of combining a simple rule with discretion can be preferable both to rigid adherence to the rule and to complete discretion.⁴ The paper showed formally that a mixed strategy under which the authorities adhered to a simple rule in “normal circumstances,” but overrode the rule when there were relatively large payoffs from doing so, could increase social welfare (relative to either the case of complete discretion or the case of rigid adherence to the rule) by providing a mechanism for both enhancing credibility during normal times and allowing for flexibility when it was most needed. It was also suggested that, by establishing well-designed institutional mechanisms, society could motivate the monetary authority to avoid both the overuse and the underuse of its override option. By 1990-91 such mixed strategies were referred to as “rules with escape clauses.”⁵

Along with the conceptual analysis that had emerged ten years ago, a second strand of literature was oriented toward simulating and comparing the performances of different types of simple monetary policy rules in empirically-estimated models of macroeconomic behavior. A primary objective of this literature, spearheaded by McCallum (1988), was to find a simple rule that performed reasonably well across the spectrum of plausible models. Although the search for robustness seemed appropriate in the context of model uncertainty, Bob Flood recognized a serious flaw in the methodology that was typically used to evaluate how well the rules performed. In particular, as Flood and Isard (1989) pointed out, it is not generally valid to base counterfactual historical simulations on the assumption that rational market participants would have expected the authorities to adhere rigidly to a given monetary rule when policymakers, had they actually been confronted with the counterfactual history, would have sometimes had incentives to deviate from the rule. While some economies have experienced prolonged periods of stable non-inflationary growth guided by transparent and predictable monetary policy behavior, no economy is insulated from occasional strong unanticipated shocks (such as the oil price shocks of the 1970s, or the current global financial crisis) that create situations in which the pursuit of short-run economic objectives would require a departure from any simple rule that the monetary authorities might have been following, and would therefore call into question the credibility of the rule.

Compared with the situation a decade ago, the academic literature today has become more extensively dominated by simulation studies, with credibility issues no longer at center stage. The shift in emphasis has obviously been facilitated by advances in computational technology, but it also reflects changes in the practice of monetary policy along with the widespread success that the industrial countries have had in subduing inflation during the 1990s. Monetary authorities in a number of industrial countries today are pursuing strategies of inflation targeting, broadly defined to encompass objectives for both the inflation rate and output/employment. To help guide the formulation of monetary policy strategies in both the inflation targeting cases and other countries, economists at central banks and elsewhere have been generating a large volume of research that simulates and compares the performances of selected forms of simple policy rules in different macroeconometric models.⁶

For the most part, contributors to the current stream of research on monetary policy rules implicitly accept the “escape clause” notion that monetary authorities should have a certain degree of flexibility to deviate from simple rules. In particular, few economists today seriously suggest that central banks should adhere mechanically to simple policy rules. In a world in which the structure of macroeconomic relationships and the distribution of shocks is imperfectly known ex ante, central banks need to be prepared to adjust their reaction patterns, and to exercise discretion intelligently,⁷ when macroeconomic behavior deviates substantially from the model on which previous reaction patterns were conditioned.

That being said, however, there remains considerable interest in analyzing how different types and calibrations of well-defined policy reaction functions would perform in hypothetical macroeconomic models, reflecting sentiment that such analysis can provide useful insights for monetary policy. Most central bankers and academic economists also believe that it is important for monetary policy to be transparent, and many have argued that the adoption of policy rules as guidelines can be helpful for communication, accountability, and credibility. Svensson (1999b) argues that this is particularly true for “targeting rules” that correspond to the first-order conditions of policy optimization problems.⁸

The recent literature on monetary rules has taken several directions (Box 2). Some researchers have sought to derive optimal rules (first-order conditions) for relatively simple macro models.⁹ Others have compared the performances of different simple rules in macro models with optimizing agents.¹⁰ Still others have looked for simple rules that exhibit robustness in performing relatively well across a spectrum of plausible macro models.¹¹

Box 2.

Alternative lines of research on monetary policy rules.

Alternative research objectives

identify properties of optimal rules (first-order conditions) for particular macro models
analyze performances of simple rules in macro models with optimizing agents
look for simple rules that exhibit robustness in performing well across a spectrum of plausible macro models
analyze how the optimal calibration of simple rules varies with key characteristics of macro models

Characteristics of macro models used in recent research

most of the models embody Phillips curves
some assume backward-looking inflation expectations; others embody a forward- looking model-consistent component of inflation expectations
most of the models are linear, such that policy-rule evaluation with quadratic loss functions focuses (almost) exclusively on the variances of inflation, output/unemployment, and in some cases the policy instrument (i.e., the nominal interest rate)
some studies allow explicitly for uncertainty about key model parameters—in particular, the NAIRU—but most of these studies simply treat the implications of this uncertainty as white noise rather than extending the model to allow the authorities to update their estimates of parameters period-by-period in a model-consistent manner that mimics the policymaking process and recognizes that policymakers in reality tend to make serially-correlated errors
almost all studies either implicitly assume that the candidate policy rules are fully credible or treat the degree of credibility as exogenous

A fourth approach, as reflected in the first set of simulation experiments reported in this paper, looks for insights from the somewhat different tack of exploring how the optimal calibration of simple policy rules varies with key characteristics of the macro model. This approach provides perspectives that may be useful in suggesting how policymakers should adapt the overall aggressiveness of their policy reactions, and the relative strengths of their reactions to inflation and unemployment, to the specification and parameters of the macroeconomic “model” they confront,¹² including such characteristics as the degree of NAIRU uncertainty, the degree of nonlinearity in the model, and the nature of inflation expectations.

The main conclusion from our stochastic simulations, however, relates to the robustness properties of rules that have been shown to perform well in linear models of the U.S. inflation process. In particular, we find that rules that perform well in linear models but implicitly embody backward-looking measures of real interest rates (such as conventional Taylor rules) or high degrees of interest rate smoothing, can fail to provide a nominal anchor for inflation expectations in models with moderate nonlinearities.

The analysis is developed by focusing on a small macro model in which certain key characteristics can be varied. The model resembles most of the others that have been used to analyze monetary policy rules insofar as it embodies Phillips curves as a “fixed” characteristic;¹³ beyond that, the treatment of inflation expectations, the shape of the Phillips curve, and the degree of uncertainty about the NAIRU are variable characteristics.

The relevance of models that rely on the Phillips curve paradigm has been a topic of active debate in recent years.¹⁴ Casual inference from the failure of inflation to accelerate in the United States through mid-year 1999, despite unemployment rates in the vicinity of 4¼ percent, suggests that the U.S. NAIRU may have declined over time to well below the 6 percent neighborhood in which it was thought to reside several years ago. Recent empirical work supports the view that the NAIRU for the United States has declined over the past decade, but it also suggests that a 95 percent confidence interval around the current value of the NAIRU may be as wide as 3 percentage points.¹⁵

Such time variation and imprecision in estimates of the NAIRU have led some economists to conclude that it is time to abandon the Phillips curve paradigm.¹⁶ We regard this position as premature in the absence of a stronger consensus on an alternative analytic framework. It may also be noted that most industrial-country central banks continue to rely on the Phillips curve framework and to condition the nature and strength of their policy reactions on such analytic frameworks. That being said, however, a central premise of this paper is that monetary policy analysis based on the Phillips curve paradigm can be strengthened considerably by taking account of the nature of ex ante uncertainty about the NAIRU and by updating estimates of the NAIRU regularly and in a model-consistent manner—that is, by explicitly modeling the process through which the monetary authorities rationally update their estimates of the NAIRU period by period, based on new observations of unemployment and inflation along with their information about the structure of the model. This approach recognizes that errors in estimating the NAIRU tend to be serially correlated rather than white noise.¹⁷

In the tradition of most other recent simulation studies of policy rules, the model variants we use in this paper assume that adherence to the policy rule is fully credible; in other papers we have simulated the performances of simple policy rules under a crude but empirically-based representation of imperfect and endogenous credibility.¹⁸ We demonstrate, however, that analysis based on the full credibility assumption is internally consistent in the following limited sense: When the macro model is well defined and known to the policymaker, when inflation expectations are either backward looking or model consistent, and when institutional arrangements motivate the policymaker to optimize over a long horizon, then the realized means and variances of inflation and unemployment are essentially independent of the loss-function parameter to which the credibility problem has traditionally been ascribed, so the announced calibration of a simple rule is time consistent. While this might be taken to justify the assumption that adherence to the announced calibration of the rule is fully credible, it does not generally imply—in a stochastic world with nonlinear behavior—that the prospect of achieving the inflation target embodied in the monetary policy rule is fully credible.

These considerations suggest that if policymakers are motivated by appropriate institutional arrangements cum reputation, credibility problems can be primarily attributed to the shortcomings of the analytic frameworks on which policies are based—that is, to the limitations of the authorities’ understanding of macroeconomic behavior. By the same token, they emphasize that institutional arrangements (commitment mechanisms) alone cannot make announced policy objectives fully credible when the authorities have imperfect information about the nature of macroeconomic behavior. Thus, although a number of simulation studies have now shown that simple policy rules, when optimally calibrated, are capable of generating an impressive degree of macroeconomic stability in well-defined macro models,¹⁹ economists should not be quick to take comfort in these results. Such findings need to be weighed against the realization that policymakers confront difficulties in trying to arrive at optimal calibrations of policy rules when the “true” macro model is not well defined, and even more so, against analysis suggesting that several rules that have been advocated on the basis of good performances in linear models perform very poorly in models with moderate nonlinearities.

The remainder of the paper is structured as follows. Section II presents the equations and estimated parameters of the base-case model, along with details on the other model variants. The model describes a closed economy and is estimated with quarterly data for the United States. It includes: short-run Phillips curves that link observed inflation rates (for the CPI and the CPI excluding food and energy) to both the expected rate of inflation and the gap between the NAIRU and the observed unemployment rate; an equation describing the behavior of survey data on inflation expectations; a description of the dynamics of the unemployment rate as a function of the real interest rate; and a model-consistent process for generating and updating estimates of the NAIRU. We consider several model variants (linear and nonlinear short-run Phillips curves paired with forward- and backward-looking inflation expectations), each of which is consistent with the long-run natural rate hypothesis. The incorporation of uncertainty is limited to simple additive uncertainty about the NAIRU and ex ante uncertainty about various exogenous shocks in an environment where all model parameters and frequency distributions of shocks are known and dynamic learning occurs only through the process of updating estimates of the NAIRU.

Section III describes the simple policy rules, the loss function, the monetary authority’s behavior, and the stochastic simulation framework. We distinguish between two classes of inflation forecast based (IFB) rules: IFB1 rules, in which a forward-looking measure of the real interest rate—in particular, a measure that embodies a model-consistent inflation forecast—is adjusted in response to both the deviation of inflation from target and a measure of the unemployment gap; and IFB2 rules, in which the same measure of the real interest rate is adjusted in response to deviations of an inflation forecast from target as well as the unemployment gap. Most of our simulations involve IFB1 rules. The Monte Carlo experiments employ a conventional quadratic loss function in searching for the optimal calibrations of the simple policy rules, but we focus in addition on a longer list of performance indicators, including the standard deviations of the unemployment, inflation, and the nominal interest rate and, for the nonlinear model variants, also the means of the unemployment and inflation rates.

Section IV reports the simulation results, which are presented in two subsections, each addressing a different set of issues. Subsection IV.A describes and compares the optimal calibrations of IFB1 rules under different well-defined model variants. In particular, it explores how the optimal calibrations of these rules depend on policy preferences, the degree of NAIRU uncertainty, the extent to which inflation expectations are backward looking, and the shape of the Phillips curve.

Subsection IV.B then addresses several specific rules that have been proposed in the literature and compares their performances with the performances of optimally calibrated IFB 1 rules. The additional rules on which we focus are: (i) Taylor’s (1993, 1999a) conventional Taylor rule; (ii) an inflation-forecast-based rule with interest rate smoothing, as estimated for the United States by Clarida, Gali, and Gertler (1998); (iii) the IFB2 rule analyzed by Isard and Laxton (1998); and (iv) a first-difference rule for the interest rate, as proposed by Levin, Wieland, and Williams (1999). The stochastic simulation results, supplemented by stability analysis (Appendix II), demonstrate that in a world in which inflation expectations have a forward-looking model-consistent component, monetary policy guided by a myopic rule that incorporates a backward-looking measure of the real interest rate, such as a conventional Taylor rule, can be destabilizing in our moderately nonlinear model. Similarly, rules with high degrees of interest rate smoothing, such as certain calibrations of forward-looking Clarida, Gali, and Gertler (CGG) rules and the first-difference rule proposed by Levin, Wieland, and Williams (LWW), can lead to instability in our model.

Section V summarizes the key messages of the paper.

II. A Model of the Unemployment-Inflation Process

Our model is a somewhat extended version of the framework developed in Laxton, Rose, and Tambakis (1999). It includes four estimated equations: two Phillips curves (one focusing on the CPI, the other on the CPI excluding food and energy), an equation describing the dynamics of inflation expectations, and an equation describing the dynamics of the unemployment rate. The inclusion of two Phillips curves allows us to exploit a larger data set when drawing inferences about the NAIRU. The model estimates are based on quarterly data for the United States over the period since 1968:Q1. The model is closed with a monetary policy reaction function and a model-consistent procedure for updating estimates of the NAIRU (both described in Section III). In the “base-case” version of the model, the Phillips curve specifications are convex and inflation expectations include a forwardlooking model-consistent component. Other model variants include linear Phillips curves and entirely-backward-looking inflation expectations.

A. The Short-Run Phillips Curves

The convex versions of our Phillips curves are broadly similar to the specification used in Debelle and Laxton (1997):

\begin{matrix} π_{t} = λ {\bar{π}}_{t}^{e} + (1 - λ) π_{t - 1} + γ (u_{t}^{*} - u_{t}) / (u_{t} - Φ_{t}) + ε_{t}^{π} & (1) \end{matrix}

\begin{matrix} π_{t}^{x} = λ^{x} {\bar{π}}_{t}^{e} + (1 - λ^{x}) π_{t - 1}^{x} + γ (u_{t}^{*} - u_{t}) / (u_{t} - Φ_{t}) + ε_{t}^{π^{x}} & (2) \end{matrix}

where

\begin{matrix} {\bar{π}}_{t}^{e} = (\sum_{i = 1}^{N} E_{t - i} {π 4}_{t + 4 - i}) / N & (\begin{matrix} 3 \end{matrix}) \end{matrix}

Here π_t denotes the rate of consumer price inflation during quarter t, measured at an annual rate; π4_t+4 denotes the rate of inflation over the year through quarter t + 4; E_tπ4_t+4 is the public’s expectation in quarter t of the rate of inflation over the year through quarter t + 4 π^x_t denotes the annualized rate of change during quarter t of the consumer price index excluding food and energy; u is the unemployment rate; and λ, λ^x, and γ are parameters to be estimated, (u^* and Φ will be defined below.)

The model of how expectations influence inflation dynamics is meant to reflect a bargaining framework that is capable of generating significant persistence in the inflation process.²⁰ The implicit underlying assumption is that a standard contract has an TV -quarter horizon, with one-Nth of the contracts respecified every quarter.²¹ Thus, equation (3) defines ${\bar{π}}_{t}^{e}$ as an average of one-year ahead inflation expectations that economic agents held during the N quarters in which currently-prevailing contracts were written. Inflation dynamics are also assumed to depend on the lagged inflation rate, which can be viewed as a summary indicator of the strength of incentives to incur the costs of revising price or wage contracts before their specified expiration dates.

Note that the coefficients on the first two right-hand-side terms in equations (1) and (2) are constrained to sum to unity, consistent with the long-run natural rate hypothesis. We refer to the sums of these terms as the core rates of inflation, π^c and π^cx.

Figure 1.

The convex Phillips curve.

\begin{matrix} π_{t}^{c} = λ {\bar{π}}_{t}^{e} + (1 - λ) π_{t - 1} & (4) \end{matrix}

\begin{matrix} π_{t}^{c x} = λ^{x} {\bar{π}}_{t}^{e} + (1 - λ^{x}) π_{t - 1} & (5) \end{matrix}

Figure 1 plots the difference between observed inflation and core inflation (vertical axis) against the unemployment rate (horizontal axis). For purposes of the discussion here, we interpret core inflation as synonymous with expected inflation, so the figure can be viewed as an expectations-augmented Phillips curve. Consistent with the specification in equation (1), the short-run Phillips curve is convex with horizontal asymptote at π—π^c =—γ and vertical asymptote at u = Φ. Following Laxton, Meredith, and Rose (1995), Φ can be interpreted as a “wall parameter,” reflecting short-run constraints on how far rising aggregate demand can lower unemployment before capacity constraints become absolutely binding and inflationary pressure becomes unbounded. The magnitude of u^* corresponds to the unemployment rate at which actual inflation and expected inflation coincide, such that there would be no systematic pressure for inflation to rise or fall in the absence of stochastic shocks.

This corresponds to the non-accelerating-inflation rate of unemployment in a deterministic world. We refer to u^* as the DNAIRU (deterministic NAIRU).²²

An important point is that the DNAIRU is not a feasible stable equilibrium in a stochastic world with a convex Phillips curve. The average rate of unemployment consistent with nonaccelerating-inflation in a stochastic world, denoted by $\bar{u}$ and referred to as the NAIRU, must be greater than the DNAIRU when the Phillips curve is convex. This can be illustrated in Figure 1 by assuming that actual inflation turned out to be uniformly distributed between plus and minus one percentage point of core (or expected) inflation, which would imply an average rate of unemployment of $\bar{u} = 0.5 (u_{1} + u_{2})$ . It can easily be seen that with a wider distribution of the actual inflation rate around core inflation, the average rate of unemployment would be even greater. The fact that the difference between the NAIRU and DNAIRU—and hence the average rate of unemployment—depends, in a nonlinear world, on the degree to which the authorities succeed in mitigating the variance of inflation has important implications for monetary policy.

Following Debelle and Laxton (1997) and others, the Phillips curve equations are estimated jointly with an equation that describes a time-varying DNAIRU.²³ We assume here that the latter follows a bounded random walk and arbitrarily set a floor at 4 percent and a ceiling at 8 percent, such that

\begin{matrix} u_{t}^{*} = {\begin{matrix} \begin{matrix} \begin{matrix} 8 & i f u_{t - 1}^{*} + ε_{t}^{u^{*}} \geq 8 \end{matrix} \\ \begin{matrix} 4 & i f u_{t - 1}^{*} + ε_{t}^{u^{*}} \leq 4 \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} u_{t - 1}^{*} + ε_{t}^{u^{*}} \end{matrix} & o t h e r w i s e \end{matrix} \end{matrix}} & (6) \end{matrix}

where ε_t^{u^*} is drawn from a normal distribution with mean zero. We also extend the formulation of the estimation problem beyond the approach used in previous studies by adding the assumptions that the business cycle component of unemployment is a stationary (and presumably highly autocorrelated) process, ε_t^u, and that the difference between the NAIRU and the DNAIRU is a constant:²⁴

\begin{matrix} u_{t} = {\bar{u}}_{t} + ε_{t}^{u} & (7) \end{matrix}

\begin{matrix} α = {\bar{u}}_{t} - u_{t}^{*} & (8) \end{matrix}

If we rewrite equations (1) and (2) for heuristic purposes as

\begin{matrix} π_{t} - π_{t}^{c} = Δ_{t} [1 / (u_{t} - Φ_{t})] - γ [u_{t} / (u_{t} - Φ_{t})] + ε_{t}^{π} & (9) \end{matrix}

\begin{matrix} π_{t}^{x} - π_{t}^{c x} = Δ_{t} [1 / (u_{t} - Φ_{t})] - γ [u_{t} / (u_{t} - Φ_{t})] + ε_{t}^{π^{x}} & (10) \end{matrix}

where δ_t = γ u^*_t is a time-vary ing parameter, equations (6)-(10) provide a nonlinear estimation problem that can be solved using the Kalman filter technique.²⁵

Table 1 reports the estimation results. The estimated parameters of the Phillips curves have the correct signs and are statistically significant. The gap between the NAIRU and the DNAIRU is estimated to be two-tenths of a percentage point. To remain consistent with these parameter estimates, the Phillips curves that are used in analyzing the linear variants of the model are calibrated as:²⁶

\begin{matrix} π_{t} = π_{t}^{c} + 0.80 (u_{t}^{*} - u_{t}) + ε_{t}^{π} & (1 a) \end{matrix}

\begin{matrix} π_{t}^{x} = π_{t}^{c x} + 0.80 (u_{t}^{*} - u_{t}) + ε_{t}^{π^{x}} & (2 a) \end{matrix}

Table 1.

Phillips curves and the time-varying NAIRU.

^{^1/}t-values in parentheses.

B. The Dynamics of Inflation Expectations

Within the sample period over which the Phillips curves are estimated, inflation expectations are based on the mean responses from the Michigan survey of expectations about one-year–ahead changes in the consumer price index. For purposes of the simulation analysis, however, we require a model of how expectations evolve. An important issue is the extent to which expectations are forward looking and model consistent.

The approach here, following Laxton, Rose, and Tambakis (1999), is based on an investigation of three alternative equations for explaining the historical survey data. The different specifications are described in Table 2. The first two include a forward-looking model–consistent component, π4^mc, which was constructed from a proxy for the model—namely, as the fitted values of an auxiliary equation that predicts observed inflation over the year ahead using four lagged values each of the unemployment rate, a long-term interest rate, the survey measure of inflation expectations, and the inflation rate.

As can be seen from Table 2, the constrained model has almost the same fit as the basic unconstrained model and slightly outperforms the overfitted model with inflation lags. The latter result indicates that, conditional on the presence of the forward-looking proxy, the estimation prefers the lagged dependent variable to lagged data on observed inflation. This suggests that expectations are not inherently backward looking; the lags of inflation are useful in explaining the expectations data only to the extent that they help predict the future, as reflected in their contribution to π4^mc. The estimates also point to substantial inertia in inflation expectations, as reflected in a relatively high coefficient on the lagged dependent variable.

Table 2.

Dynamics of inflation expectations.

Note: Figures in parentheses are t-statistics.

Table 2.

Dynamics of inflation expectations.

Estimated equation:
	E_tπ4_t+4 = α + Δ₁π4^mc_t + Δ₂E_t-1π4_t+3 + ρ₁π_t-1 + ρ₂π_t-2 + ρ₃π_t-3 + ρ₄π_t-4
E_tπ4_t+4 Expected inflation, CPI, one-year-ahead measure from the Michigan survey
π4^mc_t Fitted values from auxiliary regression to forecast one-year-ahead CPI inflation
π_t Inflation, CPI, quarterly measure at annual rate
Estimation period 1968:Q1 to 1997:Q2
Coefficient	Basic Model	Constrained	Overfilled, With
	Unconstrained	(Δ₂ = 1 - Δ₁)	Inflation Lags
α			0.470 (2.3)
Δ₁	0.258 (3.8)	0.261 (3.8)	0.346 (4.5)
Δ₂	0.726(10.5)
ρ₁			-0.092 (1.7)
ρ₂			-0.072 (1.2)
ρ₃			0.076 (1.2)
ρ₄			0.011 (0.2)
${\bar{R}}^{2}$	0.743	0.745	0.732
DW statistic	2.33	2.35	2.27
Residual variance	1.08	1.08	1.03

Note: Figures in parentheses are t-statistics.

Table 2.

Dynamics of inflation expectations.

Estimated equation:
	E_tπ4_t+4 = α + Δ₁π4^mc_t + Δ₂E_t-1π4_t+3 + ρ₁π_t-1 + ρ₂π_t-2 + ρ₃π_t-3 + ρ₄π_t-4
E_tπ4_t+4 Expected inflation, CPI, one-year-ahead measure from the Michigan survey
π4^mc_t Fitted values from auxiliary regression to forecast one-year-ahead CPI inflation
π_t Inflation, CPI, quarterly measure at annual rate
Estimation period 1968:Q1 to 1997:Q2
Coefficient	Basic Model	Constrained	Overfilled, With
	Unconstrained	(Δ₂ = 1 - Δ₁)	Inflation Lags
α			0.470 (2.3)
Δ₁	0.258 (3.8)	0.261 (3.8)	0.346 (4.5)
Δ₂	0.726(10.5)
ρ₁			-0.092 (1.7)
ρ₂			-0.072 (1.2)
ρ₃			0.076 (1.2)
ρ₄			0.011 (0.2)
${\bar{R}}^{2}$	0.743	0.745	0.732
DW statistic	2.33	2.35	2.27
Residual variance	1.08	1.08	1.03

Note: Figures in parentheses are t-statistics.

In light of the estimates reported in Table 2, most of our simulation analysis reflects the following base-case assumption about inflation expectations:

\begin{matrix} E_{t} {π 4}_{t + 4} = 0.261 {π 4}_{t}^{m c} + 0.739 E_{t - 1} {π 4}_{t + 3} + ε_{t}^{π^{e}} & (11) \end{matrix}

As alternatives, we also consider a fairly traditional forward-and-backward-looking components model with equal weights of 0.5 on the forward-and-backward-looking components,²⁷ along with two extreme cases in which inflation expectations are entirely backward looking and entirely forward looking. These correspond, respectively, to the following specifications:

\begin{matrix} E_{t} {π 4}_{t + 4} = 0.5 {π 4}_{t + 4}^{m c} + 0.5 {π 4}_{t - 1} + ε_{t}^{π^{e}} & (11 a) \end{matrix}

\begin{matrix} E_{t} {π 4}_{t + 4} = {π 4}_{t - 1} & (11 b) \end{matrix}

\begin{matrix} E_{t} {π 4}_{t + 4} = {π 4}_{t + 4}^{m c} & (11 c) \end{matrix}

Specification (11), which we consider more realistic than the other three, presents a case in which shocks to inflation expectations can be more persistent than under the traditional forward-and-backward-looking components model, thereby presenting a more difficult challenge for monetary policy.

C. The Dynamics of the Unemployment Rate

We draw again on Laxton, Rose, and Tambakis (1999) in modeling the behavior of the unemployment rate, which reflects the influence of monetary policy as transmitted through aggregate demand. The estimated equation has the form

\begin{matrix} u_{t} = c_{t} + \sum_{i = 1}^{2} η_{i} u_{t - i} + \sum_{i = 1}^{3} Φ_{i} r_{t - i} + ε_{t}^{u} & (12) \end{matrix}

where

\begin{matrix} r_{t} = {r s}_{t} - E_{t} {π 4}_{t + 4} & (13) \end{matrix}

is our measure of the real interest rate.²⁸ The time-varying “constant,” c^t, is assumed to follow a random walk to capture the combined effects of any changes in the trend levels of unemployment and the real interest rate.

Table 3 reports the fit of equation (12), which is also estimated using a Kalman filter. The lag structure is written in its final form, following testing down from specifications with longer lags.²⁹ The results reflect two stylized facts concerning the monetary authority’s ability to control the economy. First, there are important lags between changes in interest rates and their effects on aggregate demand. Second, there is persistence in movements in the unemployment rate, implying that shocks to aggregate demand propagate into future periods. The coefficients on the unemployment lags imply some augmenting propagation, but with relatively speedy reversion to the mean.³⁰

Table 3.

Dynamics of the unemployment rate.

III. The Policy Rules and Stochastic Simulation Framework

A. The Simple Policy Rules

We focus on several classes of simple policy rules. Part of the motivation for focusing on simple forms of policy reaction functions is pragmatic; particularly in the nonlinear variants of our model, the task of deriving the optimal rule associated with conventional specifications of policy loss functions would be horrendous. In addition, simple classes of rules are transparent and relatively appealing to policymakers.

Most prominent among the simple policy rules that have received attention in the recent literature are conventional Taylor rules. Under Taylor rules the monetary authorities adjust the short-term nominal interest rate in response to both the deviation of the current inflation rate from target and either the deviation of current output from potential output or the deviation of unemployment from the NAIRU.³¹ The conventional specification of Taylor rules, when expressed in terms of the unemployment gap, is:

\begin{matrix} r_{t} = r^{*} + {π 4}_{t} + w_{π} ({π 4}_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) & (14) \end{matrix}

where: rs_t is the nominal interest rate setting at time t; π4_t and u_t represent the rates of inflation and unemployment; π^TAR denotes a target rate of inflation; ${\bar{u}}_{t}$ is the authorities’ estimate of the NAIRU based on observed data through period t—1; r^* is a constant corresponding to the equilibrium real interest rate; and ω_π and ω_u are parameters.

Note that, in the second term on the right-hand side of (14), the inflation rate over the four quarters through period t appears as a backward-looking measure of the expected rate of inflation. As discussed below, in the context of our moderately nonlinear model of the U.S. economy, the precise form of the rule suggested by Taylor (1993,1998,1999a) is a very myopic rule that in some situations is not sufficient to ensure stability in the inflation process.³²

The second class of rules that we examine—which can be regarded as a class of inflation forecast based (IFB) rules that we refer to as IFB1 rules—replaces the second term in equation (14) with a model-consistent measure of inflation expectations. Specifically, IFB1 rules can be written in the general form:

\begin{matrix} {\tilde{r}}_{t} = r^{*} + {\tilde{E}}_{t} {w_{π} ({π 4}_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} & (15) \end{matrix}

where

\begin{matrix} {\tilde{r}}_{t} = r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} & (16) \end{matrix}

Here ${\tilde{r}}_{t}$ is the monetary authority’s ex ante measure of the real interest rate on which aggregate demand and unemployment depend; E_tπ4_t+4 denotes the public’s expectation at time t of the inflation rate over the year ahead; and ${\tilde{E}}_{t} {| Ω_{t}}$ denotes a model-consistent forecast at time t based on the authorities’ information set Ω_t, which includes information about the model along with the observed values of the inflation rate through quarter t and all other economic variables through quarter t—1.

Note that the IFB1 rule is specified in the form of a rule for real interest rate adjustment. Although monetary policy operates by setting the nominal interest rate, in our model (and most others) the extent to which monetary policy adjustment stimulates or restrains aggregate demand and employment depends on the real interest rate. It would thus make no sense to propose that policy be guided by a nominal interest rate rule that could not be explicitly translated into an economically reasonable rule for the real interest rate.

The IFB1 rule involves a forward-looking measure of the real interest rate. A third class of rules that has received attention in the literature, which we refer to as IFB2 rules and make a number of references to in parallel with our discussion of IFB1 rules, is defined through a simple modification of the bracketed expression in equation (15) in which the period-t inflation rate is replaced by an inflation forecast.

By focusing on the deviation from target of the authorities’ inflation forecast, inflation–forecast-based rules have the appealing feature of inducing the authorities to condition their interest rate settings on current information about the determinants of future inflation, given their information/assumptions about the structure of the model.³³ As we will demonstrate, the conditioning of monetary policy reactions on forward-looking inflation forecasts, rather than backward-looking inflation measures, appears to be critical to stability in moderately nonlinear models in which inflation expectations have a model-consistent component and policymakers confront an historically normal degree of uncertainty about the NAIRU.

B. The Policy Objective Function

The literature on optimal policy rules has traditionally relied on quadratic loss functions that are separably additive in the deviation of inflation from target, the unemployment (or output) gap, and sometimes also the change in the nominal interest rate; see, for example, Rudebusch and Svensson (1999) and Wieland (1998). To remain consistent with this literature, we adopt an objective function in which the period-t loss has the following general form

\begin{matrix} L_{t} = {(π_{t} - π^{T A R})}^{2} + θ {[u_{t} - (u_{t}^{*} - β)]}^{2} + ν {(r s_{t} - r s_{t - 1})}^{2} & (17) \end{matrix}

where θ, β, and ν are parameters and u_t^* is the DNAIRU (deterministic NAIRU). For β = 0 this corresponds to the specification that it has been popular to use in recent simulation studies of policy rules. More generally, it also allows us, somewhat in the spirit of Barro and Gordon (1983a, 1983b) and Rogoff (1985), to consider cases in which the authorities’ preferences with regard to unemployment are not symmetric around the DNAIRU but center on an unemployment rate below the DNAIRU (i.e., cases with β > 0), and to note how our simulation results are affected by credibility issues in these cases.³⁴

C. The Monetary Authority’s Behavior

The monetary authority adjusts a short-term nominal interest rate period by period in accordance with a prespecified simple policy rule. We assume that its action in quarter t is timed to come soon after the announcement of the observed inflation rate for quarter t, when the period-t values of other macroeconomic variables have not yet been observed.

The monetary authority is assumed to have full information about the structure of the model and the ex ante distributions of the exogenous shocks. After observing the period-t inflation rate, the central bank is assumed to update its estimates of the DNAIRU and NAIRU (i.e., resolve the Kalman filter problem defined by equations (6)-(10)) and set the period-t interest rate based on an information set Ω_t that includes: the complete specification of the true model, including the process that generates the DNAIRU and NAIRU as well as the bounds on the DNAIRU; the history of all observable variables (including the survey measures of inflation expectations) through period t—1, along with the inflation rate for period t; and the probability distributions (but not the realizations) of the shocks for period t and all future periods.

Under these strong informational assumptions, the central bank knows that the exogenous shocks have independent normal distributions with zero means, and it also knows the standard deviations of the shocks (which we calibrated to reflect the unexplained variances of the dependent variables during the historical periods over which the model equations were estimated). For purposes of implementing its policy rule, it needs to solve for the expected rate of inflation that defines the level of the real interest rate in equation (16), which requires it to solve its forward-looking macro model for the expected future time paths of all the endogenous variables, since inflation expectations have a forward-looking model-consistent component. We make the assumption that the central bank follows a certainty equivalence procedure in solving the model—in particular, that it assumes that all future shocks will be equal to their expected values of zero.³⁵ The forecasting rule that it uses to project the path of the DNAIRU (a bounded random walk) is described in Appendix I. In solving the model, the central bank determines, inter alia, a set of projections for the entire future time paths of both its policy instrument and the rate of inflation.

D. The Stochastic Simulation Experiments

Our first set of stochastic simulation experiments is oriented toward identifying the optimal calibrations of IFB1 rules, based on a grid search, under different model variants and parameterizations of the loss function. Additional simulations focus on evaluating the performances of selected calibrations of other classes of simple rules.

With regard to the first objective, we simulated the performance of the economy for a range of reaction function weights (ω_π, ω_u), with ω_π running over a grid from 0.1 to 2.0 in intervals of 0.1 and ω_u running from 0.0 to 2.0 in intervals of 0.1. Focusing first on the base–case model, we computed the value of the loss function under several parameterizations in order to evaluate how the monetary authorities’ preferences would influence the optimal parameters in the IFB1 rule. We then considered several alternative variants of the model in order to see how specific modeling assumptions influence the optimal calibration of the parameters in the reaction function. In each case the hypothetical path of the economy was simulated 64 times over a horizon of 100 quarters, starting in a position with the inflation rate at its target (specifically, 2.5) and the unemployment rate at the long-run NAIRU (specifically, 6.0), and using a common set of the 64 different random drawings of the timepaths of the various shocks that enter the model. This generated 6,400 observations for evaluating the cumulative (undiscounted) loss in each case, and provided information both on the optimal calibration of the rule (conditional on the loss function parameters, particular model variant, and the grid over which we searched), its implications for a range of performance indicators in addition to the cumulative loss, and the sensitivity of the cumulative loss to the calibration of the rule.

Table 4.

Optimal calibrations of IFB1 rules. (The case of a convex Phillips curve, historically-normal NAIRU uncertainty, and 12-quarter contracts.) 1/

^¹Inflation expectations are described by equation (11).

^²Loss function is L_t = {π_t - π^TAR)² + θ[u_t - (u^*_t - β)]² + ν(rs_t-rs_t-1)₂.

^³Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 4.

Optimal calibrations of IFB1 rules. (The case of a convex Phillips curve, historically-normal NAIRU uncertainty, and 12-quarter contracts.) 1/

	Loss Function Parameters 2/			Optimal Weights 3/
	θ	β	ν	ω_u	ω_π
1.	0	Irrelevant	0.5	0.2	0.8
2.	1	0	0.5	0.3	0.8
3.	1	1	0.5	0.4	0.8
4.	1	2	0.5	0.4	0.7
5.	2	0	0.5	0.4	0.7
6.	2	1	0.5	0.5	0.7
7.	2	2	0.5	0.6	0.7
8.	1	1	0	1.5	1.0

^¹Inflation expectations are described by equation (11).

^²Loss function is L_t = {π_t - π^TAR)² + θ[u_t - (u^*_t - β)]² + ν(rs_t-rs_t-1)₂.

^³Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 4.

Optimal calibrations of IFB1 rules. (The case of a convex Phillips curve, historically-normal NAIRU uncertainty, and 12-quarter contracts.) 1/

	Loss Function Parameters 2/			Optimal Weights 3/
	θ	β	ν	ω_u	ω_π
1.	0	Irrelevant	0.5	0.2	0.8
2.	1	0	0.5	0.3	0.8
3.	1	1	0.5	0.4	0.8
4.	1	2	0.5	0.4	0.7
5.	2	0	0.5	0.4	0.7
6.	2	1	0.5	0.5	0.7
7.	2	2	0.5	0.6	0.7
8.	1	1	0	1.5	1.0

^¹Inflation expectations are described by equation (11).

^²Loss function is L_t = {π_t - π^TAR)² + θ[u_t - (u^*_t - β)]² + ν(rs_t-rs_t-1)₂.

^³Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

IV. Simulation Results

A. Optimal Calibrations oflFBl Rules

This section describes the optimal calibrations of IFB1 rules, as defined by equations (15) and (16), under different model variants and for a range of loss-function parameters.

We first focus on the case in which the behavior of inflation expectations follows our preferred model, as specified in equation (11). Table 4 reports optimal calibrations of IFB1 rules under base-case assumptions about the shape of the Phillips curve, the degree of NAIRU uncertainty, and the length of wage-price contracts. The first seven rows consider combinations of three different settings of θ (the loss attached to unemployment variance relative to inflation variance) and three different settings of β (the strength of the short-run temptation to push unemployment below the DNAIRU) when a positive loss is attached to interest rate variability (ν = 0.5).³⁶ Several points may be noted.

Table 5.

Optimal calibrations of IFB1 rules under a backward-and-forward-looking components model of inflation expectations.

(The case of a convex Phillips curve, historically-normal NAIRU uncertainty, and 12-quarter contracts.) 1/

^¹Inflation expectations are described by equation (11a).

^²Loss function is L_t = {π_t - π^TAR)² + θ[u_t - (u^*_t - β)]² + ν(rs_t-rs_t-1)₂.

^³Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 5.

Optimal calibrations of IFB1 rules under a backward-and-forward-looking components model of inflation expectations.

(The case of a convex Phillips curve, historically-normal NAIRU uncertainty, and 12-quarter contracts.) 1/

	Loss Function Parameters 2/			Optimal Calibrations 3/
	θ	β	ν	ω_u	ω_π
1.	0	Irrelevant	0.5	0.2	0.8
2.	1	0	0.5	0.3	0.8
3.	1	1	0.5	0.4	0.8
4.	1	2	0.5	0.4	0.8
5.	2	0	0.5	0.4	0.8
6.	2	1	0.5	0.5	0.7
7.	2	2	0.5	0.6	0.7
8.	1	1	0	1.8	1.1

^¹Inflation expectations are described by equation (11a).

^²Loss function is L_t = {π_t - π^TAR)² + θ[u_t - (u^*_t - β)]² + ν(rs_t-rs_t-1)₂.

^³Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 5.

Optimal calibrations of IFB1 rules under a backward-and-forward-looking components model of inflation expectations.

(The case of a convex Phillips curve, historically-normal NAIRU uncertainty, and 12-quarter contracts.) 1/

	Loss Function Parameters 2/			Optimal Calibrations 3/
	θ	β	ν	ω_u	ω_π
1.	0	Irrelevant	0.5	0.2	0.8
2.	1	0	0.5	0.3	0.8
3.	1	1	0.5	0.4	0.8
4.	1	2	0.5	0.4	0.8
5.	2	0	0.5	0.4	0.8
6.	2	1	0.5	0.5	0.7
7.	2	2	0.5	0.6	0.7
8.	1	1	0	1.8	1.1

^¹Inflation expectations are described by equation (11a).

^²Loss function is L_t = {π_t - π^TAR)² + θ[u_t - (u^*_t - β)]² + ν(rs_t-rs_t-1)₂.

^³Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

First, even for cases in which no loss is attached to unemployment variance (top row), the optimal calibration of the IFB1 rule places a positive weight on the unemployment gap. Thus, in setting the nominal interest rate relative to a model-consistent measure of expected inflation, authorities who condition their interest rate settings on information about both inflation and unemployment can achieve a more desirable path for future inflation than authorities who ignore information about unemployment. Second, as the relative loss attached to unemployment variance increases, so do the relative weights on unemployment in the optimal calibrations of these rules (compare, e.g., rows 1, 2, and 5). Third, as β increases and the “target” unemployment rate (u_*—β) declines, the optimal relative weight on unemployment increases (compare rows 2, 3, and 4 and rows 5, 6, and 7). We regard these results as intuitively very plausible and likely to prove fairly robust across both models and different classes of simple policy rules. It may be noted, however, that when we double the relative loss associated with interest rate variability by raising the setting of ν from 0.5 to 1.0, the optimal calibration of ω_u declines in all cases and rounds to 0.0 in cases with θ = 0.

A fourth result is that the optimal weights on inflation and unemployment are inversely related to the loss attached to interest rate variability; compare rows 3 and 8. The lower is the loss associated with interest rate variability, other things equal, the more aggressive are the optimal responses to unemployment gaps and deviations of inflation from target.

As a check on the sensitivity of these results to our assumption about the dynamics of inflation expectations, Table 5 reports results comparable to those in Table 4 in all respects except for the assumption about inflation expectations. In Table 5, inflation expectations are assumed to reflect the traditional forward-and-backward looking components model defined by equation (11a). It may be seen that the four points noted about Table 4 are equally evident in Table 5.

Table 6 characterizes the sensitivity of the optimal calibrations to the degree of NAIRU uncertainty and alternative assumptions about the model. Each of the five panels corresponds to a particular choice of the loss function parameters. Within each panel, the top row corresponds to the base-case results reported in Table 4.

Table 6.

Sensitivity of optimal calibrations of IFB1 rules to different assumptions.

^{^1/}The convex Phillips curves are described by equations (1) and (2). The linear Phillips curves are described by equations (la) and (2a).

^{^2/}F denotes partially forward-looking expectations as characterized by equation (11). B denotes completely backward-looking expectations defined by equation (11b).

^{^3/}Normal level of NAIRU uncertainty reflects sample period variances of ε^π, ε^π^x, and ε^u^x in equations (1), (2), and (6). Low (high) NAIRU uncertainty corresponds to variances of ε^π, ε^π^x, and ε^u^x that are half (twice) as large as in the normal case.

^{^4/}Length of standard price and wage contracts measured in calendar quarters; value of N in equation (3).

^{^5/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 6.

Sensitivity of optimal calibrations of IFB1 rules to different assumptions.

		Model Characteristics			Optimal Calibraltons5/
Phillips		Inflation	NAIRU	Contract
Curve ^1/		Expectations ^2/	Uncertainty ^3/	Length ^4/	ω_u	ω_π
1. Base Case: (θ, β, ν)= (1, 1,0.5)
	Convex	F	Normal	12	0.4	0.8
	Convex	F	Low	12	0.8	0.8
	Convex	F	High	12	0.0	0.7
	Linear	F	Normal	12	0.0	0.8
	Convex	F	Normal	4	1.1	0.9
	Linear	F	Normal	4	0.7	0.9
	Convex	B	Normal	12	0.0	0.9
2. No Unemployment Loss and Preference for u= DNAIRU: (θ, β, ν) = (0,0,0.5)
	Convex	F	Normal	12	0.2	0.8
	Convex	F	Low	12	0.6	0.9
	Convex	F	High	12	0.0	0.8
	Linear	F	Normal	12	0.0	0.9
	Convex	F	Normal	4	0.9	0.9
	Linear	F	Normal	4	0.6	0.9
	Convex	B	Normal	12	0.2	1.0
3. High Unemployment Loss and Preference for u = DNAIRU: (θ, β, ν) = (2,0,0.5)
	Convex	F	Normal	12	0.4	0.7
	Convex	F	Low	12	0.8	0.8
	Convex	F	High	12	0.1	0.7
	Linear	F	Normal	12	0.1	0.8
	Convex	F	Normal	4	0.9	0.8
	Linear	F	Normal	4	0.7	0.8
	Convex	B	Normal	12	0.0	0.9
4. High Unemployment Loss and Strong Preference for u < DNAIRU: (θ, β, ν) = (2, 2, 0,5)
	Convex	F	Normal	12	0.6	0.7
	Convex	F	Low	12	1.0	0.7
	Convex	F	High	12	0.1	0.6
	Linear	F	Normal	12	0.1	0.8
	Convex	F	Normal	4	1.1	0.8
	Linear	F	Normal	4	0.7	0.8
	Convex	B	Normal	12	0.2	0.8
5. No Loss on Interest Rate Volatility: (0, B, v) = (1, 1, 0)
	Convex	F	Normal	12	1.5	1.0

^{^1/}The convex Phillips curves are described by equations (1) and (2). The linear Phillips curves are described by equations (la) and (2a).

^{^2/}F denotes partially forward-looking expectations as characterized by equation (11). B denotes completely backward-looking expectations defined by equation (11b).

^{^4/}Length of standard price and wage contracts measured in calendar quarters; value of N in equation (3).

^{^5/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 6.

Sensitivity of optimal calibrations of IFB1 rules to different assumptions.

		Model Characteristics			Optimal Calibraltons5/
Phillips		Inflation	NAIRU	Contract
Curve ^1/		Expectations ^2/	Uncertainty ^3/	Length ^4/	ω_u	ω_π
1. Base Case: (θ, β, ν)= (1, 1,0.5)
	Convex	F	Normal	12	0.4	0.8
	Convex	F	Low	12	0.8	0.8
	Convex	F	High	12	0.0	0.7
	Linear	F	Normal	12	0.0	0.8
	Convex	F	Normal	4	1.1	0.9
	Linear	F	Normal	4	0.7	0.9
	Convex	B	Normal	12	0.0	0.9
2. No Unemployment Loss and Preference for u= DNAIRU: (θ, β, ν) = (0,0,0.5)
	Convex	F	Normal	12	0.2	0.8
	Convex	F	Low	12	0.6	0.9
	Convex	F	High	12	0.0	0.8
	Linear	F	Normal	12	0.0	0.9
	Convex	F	Normal	4	0.9	0.9
	Linear	F	Normal	4	0.6	0.9
	Convex	B	Normal	12	0.2	1.0
3. High Unemployment Loss and Preference for u = DNAIRU: (θ, β, ν) = (2,0,0.5)
	Convex	F	Normal	12	0.4	0.7
	Convex	F	Low	12	0.8	0.8
	Convex	F	High	12	0.1	0.7
	Linear	F	Normal	12	0.1	0.8
	Convex	F	Normal	4	0.9	0.8
	Linear	F	Normal	4	0.7	0.8
	Convex	B	Normal	12	0.0	0.9
4. High Unemployment Loss and Strong Preference for u < DNAIRU: (θ, β, ν) = (2, 2, 0,5)
	Convex	F	Normal	12	0.6	0.7
	Convex	F	Low	12	1.0	0.7
	Convex	F	High	12	0.1	0.6
	Linear	F	Normal	12	0.1	0.8
	Convex	F	Normal	4	1.1	0.8
	Linear	F	Normal	4	0.7	0.8
	Convex	B	Normal	12	0.2	0.8
5. No Loss on Interest Rate Volatility: (0, B, v) = (1, 1, 0)
	Convex	F	Normal	12	1.5	1.0

^{^1/}The convex Phillips curves are described by equations (1) and (2). The linear Phillips curves are described by equations (la) and (2a).

^{^2/}F denotes partially forward-looking expectations as characterized by equation (11). B denotes completely backward-looking expectations defined by equation (11b).

^{^4/}Length of standard price and wage contracts measured in calendar quarters; value of N in equation (3).

^{^5/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

We noted above that the optimal weight on the unemployment gap depends on the relative loss attached to unemployment variance. Here we add an intuitive result on how the optimal reaction function parameters vary with the degree of NAIRU uncertainty. In particular, rows 2 and 3 report results for the cases in which the magnitudes of NAIRU uncertainty are, respectively, half as much and twice as great as the base-case level. These results confirm that the optimal relative weight on the unemployment gap is inversely related to the degree of NAIRU uncertainty.

Another finding is that the optimal weights increase (implying more aggressive policy reactions) as the length of standard wage and price contracts shortens and thereby reduces the degree of inertia in the backward-looking component of inflation expectations. This can be seen by comparing rows 1 and 5 or rows 4 and 6. Consistently, for cases in which market participants are assumed to have completely backward-looking expectations, the optimal calibrations of the IFB1 rule involve weaker policy responses to unemployment gaps than for analogous cases with partially forward-looking expectations; compare rows 1 and 7.

We expected to also find that, other things equal, the optimal policy reaction is more aggressive in model variants with convex Phillips curves than in model variants with linear approximations to the same Phillips curves.³⁷ This is simply because the greater the degree of Phillips-curve convexity, the higher is the inflation and/or unemployment variance that tends to be generated by shocks to the economy, other things equal. The results in Table 6 support these priors insofar as the optimal weights on unemployment are higher in the cases with convex Phillips curves than they are in the cases with linear Phillips curves (compare rows 1 and 4 and rows 5 and 6). In reflecting on this result, it may be noted, in addition, that the differences in weights across the two models is moderated by two factors: first, the forward-looking IFB1 rules, which implicitly take account of the nonlinearities in the model, are highly successful in avoiding large boom and bust cycles; and second, our convex Phillips curves are approximately linear in the region of the NAIRU.

Table 7 reports a number of relevant performance characteristics associated with a selected subset of the optimally-calibrated IFB1 rules shown in Table 6. The performance characteristics include the cumulative undiscounted losses (over 6,400 simulated observations, with a scale factor); the average rates of inflation and unemployment; the standard deviations of the inflation and unemployment rates; and the standard deviations of both the level of, and the change in, the nominal interest rate (the policy instrument). Several points may be noted.

Table 7.

Performance characteristics of optimally-calibrated IFB1 rules 1/.

^{^1/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 7.

Performance characteristics of optimally-calibrated IFB1 rules 1/.

Model Characteristics				Performance Characteristics
										Standard
					Inflation Outcomes		Unemployment Outcomes		Standard	Deviation of
									Deviation of	Change in
Phillips	Inflation	NAIRU	Contract	Cumulative		Standard		Standard	Nominal	Nominal
Curve	Expectations	Uncertainty	Length	Loss	Mean	Deviation	Mean	Deviation	Interest Rate	Interest Rate
(θ, β, ν) = (1, 1, 0.5)
Convex	F	Normal	12	6.90	2.65	2.17	6.03	0.54	2.17	1.06
Convex	F	Low	12	2.71	2.57	1.13	6.01	0.36	1.20	0.58
Linear	F	Normal	12	6.80	2.51	2.17	6.00	0.55	2.17	1.04
Convex	F	Normal	4	8.46	2.71	2.44	6.04	0.59	3.01	1.22
Linear	F	Normal	4	8.31	2.51	2.44	6.00	0.60	3.03	1.20
Convex	B	Normal	12	7.13	2.66	2.04	6.03	0.54	2.90	1.52
(θ, β, ν) = (2, 0, 0.5)
Convex	F	Normal	12	6.02	2.64	2.17	6.02	0.52	2.04	1.01
Convex	B	Normal	12	6.26	2.66	2.04	6.03	0.54	2.90	1.52
(θ, β, ν) = (2, 2, 0.5)
Convex	F	Normal	12	14.20	2.64	2.17	6.02	0.52	2.04	1.02
Convex	B	Normal	12	14.52	2.65	2.04	6.03	0.52	2.77	1.47

^{^1/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 7.

Performance characteristics of optimally-calibrated IFB1 rules 1/.

Model Characteristics				Performance Characteristics
										Standard
					Inflation Outcomes		Unemployment Outcomes		Standard	Deviation of
									Deviation of	Change in
Phillips	Inflation	NAIRU	Contract	Cumulative		Standard		Standard	Nominal	Nominal
Curve	Expectations	Uncertainty	Length	Loss	Mean	Deviation	Mean	Deviation	Interest Rate	Interest Rate
(θ, β, ν) = (1, 1, 0.5)
Convex	F	Normal	12	6.90	2.65	2.17	6.03	0.54	2.17	1.06
Convex	F	Low	12	2.71	2.57	1.13	6.01	0.36	1.20	0.58
Linear	F	Normal	12	6.80	2.51	2.17	6.00	0.55	2.17	1.04
Convex	F	Normal	4	8.46	2.71	2.44	6.04	0.59	3.01	1.22
Linear	F	Normal	4	8.31	2.51	2.44	6.00	0.60	3.03	1.20
Convex	B	Normal	12	7.13	2.66	2.04	6.03	0.54	2.90	1.52
(θ, β, ν) = (2, 0, 0.5)
Convex	F	Normal	12	6.02	2.64	2.17	6.02	0.52	2.04	1.01
Convex	B	Normal	12	6.26	2.66	2.04	6.03	0.54	2.90	1.52
(θ, β, ν) = (2, 2, 0.5)
Convex	F	Normal	12	14.20	2.64	2.17	6.02	0.52	2.04	1.02
Convex	B	Normal	12	14.52	2.65	2.04	6.03	0.52	2.77	1.47

^{^1/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} .$

First, for the linear model variants, the simple policy rules succeed in hitting the inflation target to a very close approximation (i.e., with an error averaging less than 015 percentage points over 6400 observations) and achieving an average unemployment rate equal to the long-run DNAIRU (i.e., the center of the band within which the DNAIRU randomly walks).

Second, as noted in Section II, for model variants with convex Phillips curves (and simulations that start with inflation on target and the unemployment rate at the long-run DNAIRU), it is infeasible to hit the inflation target on average in a stochastic environment without generating an average unemployment rate above the long-run DNAIRU. For these cases, the policy rule calibrations that are optimal under the quadratic loss function—which trades off deviations of inflation from its target against deviations of unemployment from an implicit target at or below the DNAIRU—result in both above-target average inflation rates and above-DNAIRU average unemployment rates.

Third, the realized means and standard deviations of inflation and unemployment are essentially independent of the loss-function parameter β, other things equal; compare, for example, the results in the second and third panels of Table 7. Thus, under the assumptions that the macro model is well defined and known to the authorities, that inflation expectations are either backward looking or model consistent, and that the authorities are motivated to optimize over a long horizon, there may be some basis for taking comfort in the traditional assumption that adherence to a well-performing policy rule would be fully credible. As noted earlier, however, except in completely linear models such an assumption does not imply that the prospect of achieving the announced inflation target is fully credible.

Fourth, for all cases shown in Table 7—and more generally, for most cases with historically normal (or low) levels of NAIRU uncertainty—the optimal calibrations of the IFB1 rules succeed in keeping average inflation within .25 percentage points of target and average unemployment within .05 percentage points of the long-run DNAIRU. Moreover, the associated standard deviations of inflation, unemployment, and the nominal interest rate might also be regarded as attractively small. This highly attractive performance is an interesting result in light of Schaling’s (1998) argument that for models with convex Phillips curves, optimal policy reaction functions are nonlinear in the observable state variables. In particular, our simulation results provide the additional perspective that when policymakers are assumed to have complete information about the structure of the model, linear IFB rules that embody model-consistent measures of real interest rates and thereby implicitly take account of the nonlinearities in the model may provide acceptably-close approximations to “optimal rules.” By contrast, as is emphasized in the next section, complete information about the structure of the model would have little value if policymakers were committed to follow a conventional Taylor rule.

A fifth result, evident from simulation results not reported in this paper, is that the performances of IFB 1 rules are slightly dominated by the performances of IFB2 rules, which we have described verbally in Section III.A above and characterize formally by equation (19) in Section IV.B below. But the shift from a Taylor rule to an IFB1 rule achieves a much larger gain in macroeconomic stability than the shift from an IFB1 rule to an IFB2 rule.

While the results reported in this section suggest that simple rules are capable of performing very well as monetary policy guidelines when macro models are well defined, such a conclusion does not necessarily extend to the model-uncertain environments in which monetary policy is actually conducted. At best, such inference would rest on a presumption that the monetary authorities can identify a rule calibration that comes close to achieving the performance of the optimal calibration. Moreover, a point that we find even more alarming is that economists seem to have a poor track record at identifying rules with good stabilizing properties. In particular, as the next section illustrates, a number of rules that economists have chosen to advocate on the basis of their performances in linear models of the U.S. economy perform very poorly in models with moderate nonlinearities, particularly when policymakers tend to make serially-correlated errors in estimating the NAIRU.

B. Perspectives on Several of the Simple Rules Proposed in the Literature

This section focuses on the stabilizing properties of several types of simple rules that have been proposed in the literature, and compares their performances with the performance of an optimally-calibrated IFB1 rule. The additional rules on which we focus are: (i) the conventional Taylor rule advocated by Taylor (1993,1999a), (ii) an inflation-forecast-based rule with interest rate smoothing, as estimated for the United States by Clarida, Gali, and Gertler (1998), (iii) an optimally-calibrated IFB2 rule, as analyzed previously in Isard and Laxton (1998) and Isard, Laxton, and Eliasson (1998), and (iv) a first-difference rule for the interest rate, as proposed by Levin, Wieland, and Williams (1999).

One of the key findings of our simulation analysis, supported by the stability analysis presented in Appendix II, is that in a world in which inflation expectations have a forward–looking model-consistent component, monetary policy guided by a myopic rule that incorporates a backward-looking measure of the real interest rate, such as a conventional Taylor rule, can be destabilizing in our moderately nonlinear model. A second key finding is that rules with high degrees of interest rate smoothing, such as the forward-looking rule estimated by Clarida, Gali, and Gertler (CGG) and the first-difference rule proposed by Levin, Wieland, and Williams (LWW), can also lead to instability in our moderately nonlinear model.

The specification of conventional Taylor rules has been described by equation (14) above. The CGG rules that we consider can be written as:

\begin{matrix} r s_{t} = (1 - ρ) c + ρ r s_{t - 1} + (1 - ρ) {\tilde{E}}_{t} {β π 4_{t + 4} + γ ({\bar{u}}_{t} - u_{t}) | Ω_{t}} & (18) \end{matrix}

where (ρ, β, γ) are the parameters to be chosen.³⁸ Note that in CGG rules the interest rate is adjusted in reaction to an inflation forecast rather than a backward-looking measure of inflationary pressures as embodied in the conventional Taylor rule.³⁹

The IFB2 rule that we consider here has the following form:

\begin{matrix} {\tilde{r}}_{t} = r^{*} + {\tilde{E}}_{t} {w_{π} (π_{t + 3} - π^{T A R}) + w_{u} ({\bar{u}}_{t} - u_{t}) | Ω_{t}} & (19) \end{matrix}

where

\begin{matrix} {\tilde{r}}_{t} = r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} & (20) \end{matrix}

As in the IFB1 rule, ${\tilde{r}}_{t}$ is the monetary authority’s ex ante measure of the real interest rate on which aggregate demand and unemployment depend; E_tπ4_t+4 denotes the public’s expectations at time t of the inflation rate over the year ahead; and ${\tilde{E}}_{t} {| Ω_{t}}$ denotes a model-consistent forecast at time t based on the authorities’ information set Ω_t, which includes information about the model along with the observed values of the inflation rate through quarter t and all other economic variables through quarter t—1.⁴⁰ However, in the IFB2 rule the real interest rate is adjusted in response to a forecast of the annualized inflation rate three quarters ahead rather than the contemporaneous year-on-year inflation measure that is used in the conventional Taylor rule and IFB1 rule. While the assumption of a three-quarter-ahead inflation forecast is somewhat arbitrary, some experimentation suggested that such a specification was capable of producing reasonable macroeconomic stability, and of offsetting unanticipated shocks to inflation within a horizon of two or three years.⁴¹ Nevertheless, the issue may deserve more consideration in future work.⁴²

There are three potentially important differences between the CGG rule and the IFB2 rule. First, the IFB2 rule embodies information about the real interest rate on which aggregate demand depends. Consequently, with prespecified reaction-function parameters, the behavior of nominal interest rates under the IFB2 rule appears to be more sensitive to assumptions about the manner in which inflation expectations are formed by the private sector. Second, the CGG rule employs a fourth-quarter-ahead forecast of the year-on–year inflation rate while the IFB2 rule employs a third-quarter-ahead forecast of quarterly inflation (measured at an annual rate). Third, the IFB2 rule does not incorporate interest rate smoothing.

CGG provide estimates of the “Volcker-Greenspan” calibrations that best fit the post-October 1979 and post-1982 data for the United States; these estimates, in rounded numbers, are (ρ, β, γ) = (0.7, 2.0, 0.1) and (ρ, β, γ) = (0.8, 1.6, 0.9), respectively. CGG implicitly suggest that the “Volcker-Greenspan” calibrations have attractive stabilizing properties, and that a CGG rule with these calibrations could have avoided the stagflation that occurred in the late 1960s and 1970s. We test this conjecture by using our stochastic simulation framework to explore how well their rule would work in our model under the Volcker-Greenspan calibrations and a historically-normal degree of ex ante NAIRU uncertainty.

The final rule that we consider is a first-difference rule for the nominal interest rate, as explored by Levin, Wieland, and Williams (1999). This LWW rule can be written as:

\begin{matrix} r s_{t} - \begin{matrix} r s_{t - 1} = w_{π} \sum_{i = 0}^{N - 1} (π_{t - i} - π^{T A R}) / N | w_{u} ({\bar{u}}_{t} - u_{t}) \end{matrix} & (21) \end{matrix}

where we consider both N = 12 and N = 4.

Table 8 summarizes the performance characteristics of the selected policy rules in the context of our base-case model and base-case loss function parameters. Among other things, the base-case model embodies an historically normal degree of NAIRU uncertainty and recognizes that such uncertainty creates a tendency for policymakers to make serially-correlated errors in estimating the unemployment gap. In this context, both the conventional Taylor rule and the LWW first-difference rule are too myopic to satisfy the stability conditions for our moderately nonlinear model in which inflation expectations have a model-consistent component.

Table 8.

Performance characteristics of selected rules. (The case of a convex Phillips curve, historically-normal NAIRU uncertainty, and 12-quarter contracts under loss function parameterization (θ, β, ν) = (1, 1, 0.5).)

^1/Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\tilde{u}}_{t} - u_{t}) | Ω_{t}} .$

^{^2/}Reaction function is $r s_{t} = (1 - ρ) c + ρ r s_{t - 1} + (1 - ρ) {\bar{E}}_{t} {β π 4_{t + 4} + y ({\bar{u}}_{t + 1} - u_{t + 1}) | Ω_{t}} .$

^{^3/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π_{t + 3} - π^{T A R}) + w_{u} ({\tilde{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 8.

		Performance Characteristics
								Standard
			Inflation Outcomes		Unemployment Outcomes		Standard	Deviation of
							Deviation of	Change in
		Cumulative		Standard		Standard	Nominal	Nominal
Rule		Loss	Mean	Deviation	Mean	Deviation	Interest Rate	Interest Rate
Conventional Taylor rules		too myopic to satisfy stability conditions (see text)
IFB1 rulel/
	(wu, wpi) = (0.4,0.8)	6.90	2.65	2.17	6.03	0.54	2.17	1.06
CGG rules 2/		based on cases that do not explode (see text)
	(P, B, Y) = (0.8,1.6,0.9)	8.13	2.90	2.33	5.99	0.53	1.74	0.30
	(P, B, Y)= (0.7,2,0.1)	6.98	2.74	2.24	6.02	0.56	1.85	0.45
IFB2 rule 3/
	(wu, wpi) = (0.0, 1.5)	6.32	2.57	2.06	6.02	0.50	1.93	l.12
LWW first-difference rule		too myopic to satisfy stability conditions (see text)

^1/Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\tilde{u}}_{t} - u_{t}) | Ω_{t}} .$

^{^2/}Reaction function is $r s_{t} = (1 - ρ) c + ρ r s_{t - 1} + (1 - ρ) {\bar{E}}_{t} {β π 4_{t + 4} + y ({\bar{u}}_{t + 1} - u_{t + 1}) | Ω_{t}} .$

^{^3/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π_{t + 3} - π^{T A R}) + w_{u} ({\tilde{u}}_{t} - u_{t}) | Ω_{t}} .$

Table 8.

		Performance Characteristics
								Standard
			Inflation Outcomes		Unemployment Outcomes		Standard	Deviation of
							Deviation of	Change in
		Cumulative		Standard		Standard	Nominal	Nominal
Rule		Loss	Mean	Deviation	Mean	Deviation	Interest Rate	Interest Rate
Conventional Taylor rules		too myopic to satisfy stability conditions (see text)
IFB1 rulel/
	(wu, wpi) = (0.4,0.8)	6.90	2.65	2.17	6.03	0.54	2.17	1.06
CGG rules 2/		based on cases that do not explode (see text)
	(P, B, Y) = (0.8,1.6,0.9)	8.13	2.90	2.33	5.99	0.53	1.74	0.30
	(P, B, Y)= (0.7,2,0.1)	6.98	2.74	2.24	6.02	0.56	1.85	0.45
IFB2 rule 3/
	(wu, wpi) = (0.0, 1.5)	6.32	2.57	2.06	6.02	0.50	1.93	l.12
LWW first-difference rule		too myopic to satisfy stability conditions (see text)

^1/Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π 4_{t} - π^{T A R}) + w_{u} ({\tilde{u}}_{t} - u_{t}) | Ω_{t}} .$

^{^2/}Reaction function is $r s_{t} = (1 - ρ) c + ρ r s_{t - 1} + (1 - ρ) {\bar{E}}_{t} {β π 4_{t + 4} + y ({\bar{u}}_{t + 1} - u_{t + 1}) | Ω_{t}} .$

^{^3/}Reaction function is $r s_{t} - {\tilde{E}}_{t} {E_{t} π 4_{t + 4} | Ω_{t}} = r^{*} + {\tilde{E}}_{t} {w_{π} (π_{t + 3} - π^{T A R}) + w_{u} ({\tilde{u}}_{t} - u_{t}) | Ω_{t}} .$

Some intuition for these results is provided by the following points. First, the stability conditions depend essentially on the inflationary consequences of excess demand. Second, when an economy gets into a region of significant overheating, the inflationary consequences of a marginal increase in excess demand can be significantly greater with a convex Phillips curve than with a linear Phillips curve. Third, serially-correlated errors in estimating unemployment gaps increase the likelihood of getting into states with significant excess demand. Fourth, when inflationary expectations have a forward-looking model-consistent component and policy reactions are not sufficiently forward looking, attempts to achieve a target average rate of inflation can put the economy through boom and bust cycles where the busts are more pronounced than the booms. And fifth, as an alternative and more extreme outcome, reliance on backward-looking or sluggish policy rules can leave policymakers so far behind “shifts in the curve” that they fail to provide an anchor for inflation expectations.

Appendix II provides an extended discussion of the stability properties of Taylor rules and LWW rules. It also demonstrates that LWW rules bear a close resemblance to price level targeting.

Instability was encountered in some of the simulations with CGG rules. Although as Table 8 indicates, reasonably good performances are delivered, on average, for those cases of random shock drawings in which the simulations do not explode, it may be noted that the first calibration of the CGG rule failed to prevent instability in 8 out of 64 simulations, and the second in 22 of 128 simulations.

These results for CGG rules, along with the analysis of LWW rules, emphasize that it would be very dangerous to constrain policymakers to always exercise gradualism in adjusting interest rates. It may be quite appropriate to associate losses with interest rate variability and to take these losses into consideration when calibrating the strength of policy reactions to estimated unemployment gaps and deviations of inflation from target. But to adhere mechanically to either an LWW rule or a CGG rule with a high degree of interest rate smoothing would be a recipe for disaster.⁴³

By contrast, the optimally-calibrated IFB1 and IFB2 rules succeed in preventing instability in all 64 simulations and deliver relatively attractive outcomes for the means and variances of inflation and unemployment. By incorporating model-consistent measures of inflation expectations and the real interest rate, which implicitly takes account of the nonlinearities in macroeconomic behavior, these rules are highly successful in avoiding boom and bust cycles within our well-defined macro model, and in delivering average rates of inflation and unemployment respectively close to the inflation target and the long-run DNAIRU.

V. Conclusions

The various simulation results reported in this paper illustrate the prospective dangers of adopting a simple policy rule as an automatic pilot for monetary policy. These dangers are reflected, among other places, in the observation that economists seem to have a poor track record in identifying rules with good stabilizing properties. In particular, several prominent types of simple monetary policy rules—rules that have been shown to perform well in linear models of the U.S. economy and that have accordingly received prominent attention in the economics literature—are too myopic to deliver macroeconomic stability in our moderately nonlinear model in which inflation expectations have a forward-looking model-consistent component.

While recognizing that simple policy rules should not be followed mechanically, many economists argue that the adoption of simple rules as guidelines can be helpful for communication, accountability, and credibility. This is reflected, for example, in the widespread attention that inflation targeting strategies have received during the 1990s. Moreover, even for advocates of discretionary monetary policies, simulation experiments and analytic studies of the properties of simple rules can provide valuable insights. In this context, two key messages of this paper are that it is very important for policymakers to calibrate their nominal interest rate adjustments on the basis of forward-looking measures of real interest rates, and that it is also important to be aware that excessive caution (interest rate smoothing) in policy reactions can be much more costly in a nonlinear world than is apparent from simulation experiments with linear models. These messages are especially relevant for a world characterized by NAIRU uncertainty in which policymakers tend to make serially–correlated errors in estimating the unemployment gap. Our analysis suggests that relying heavily on guidance from a conventional Taylor rule would lead to a repeat of the policy errors of the 1970s, independently of how the Taylor rule was calibrated.

A third message of the paper is that the propensity of economists to analyze the properties of simple policy rules within the confines of linear models is difficult to defend as a research strategy. In linear models, bad policy rules affect the variances of unemployment and inflation, but not the means: bad rules do not have first-order welfare consequences. By contrast, nonlinear models recognize that policy rules that allow economies to overheat significantly can leave policy “behind shifts in the curve,” with consequences that are much more dire than simply increasing the variances of unemployment and inflation. While the nonlinearity on which this paper has focused is a moderate one associated with convex Phillips curves that are highly linear in the region of the DNAIRU (deterministic NAIRU), in reality policymakers also confront other possibly-important nonlinearities, such as nonlinearities in the response of inflation expectations (policy credibility) to the authorities’ track record in hitting announced inflation targets (including asymmetry in the speeds with which credibility can be lost and regained), asymmetric hysteresis in the dynamics of unemployment, and floors on nominal interest rates.

Appendix I. Optimal Forecasting Rules for Bounded Random Walks

As summarized by equation (6), our simulations assume that the DNAIRU follows abounded random walk centered at 6 percent, with a floor and ceiling of 4 percent and 8 percent, respectively. The central bank is assumed to know the process that generates the DNAIRU, including the variance of the random walk and its upper and lower bounds. In each period the central bank updates its estimates of the historical path of the DNAIRU, based on knowledge of the structural model, the ex ante distributions of the exogenous shocks, and the history of all observable variables. For purposes of implementing its IFB1 or IFB2 rules, it needs to solve its forward-looking macro model, which, among other things, requires it to forecast the timepath of the DNAIRU.

The optimal forecasts for a bounded random walk depend on the upper and lower bounds, the variance of the random walk process, and the most recently observed or estimated value of the time series. Figure A1 shows optimal forecasts over horizons of 100 periods, based on different estimates of the initial value of the DNAIRU, a standard deviation of 0.12, and the lower and upper bounds of 4 percent and 8 percent. As can be seen in the figure, the optimal forecasts revert very gradually toward the long-run DNAIRU of 6 percent. Note, however, that the expected speed of convergence is positively related to the distance between the estimated initial value of the DNAIRU and its long-run value.

Figure Al.

Optimal forecast trajectories of a bounded random walk.

Sources: Each line in the figure is derived from taking the average values over 500 simulations of a bounded random walk with a standard deviation of 0.12, a lower bound of 4.0 percent and an upper bound of 8.0 percent.

In solving for the optimal forecast paths for the DNAIRU, we rely on numerical derivations. In particular, the paths shown in Figure A1 were constructed by averaging over 500 outcomes for each initial estimated value of the DNAIRU. For purposes of conducting the stochastic simulation experiments reported in the text, we created a grid of candidate optimal forecasts, corresponding to a grid of initial values that varied between 4 percent and 8 percent in increments of 001 percent. We then assumed that the central bank’s forecast, in period t, of the path of the DNAIRU from period t to t + 100 corresponded to the optimal path associated with its period-t estimate of the DNAIRU for period t—1.

The bounded random walk has a number of advantages over other stochastic processes that might be assumed in modeling the DNAIRU. First, it allows for quasi-permanent, or highly persistent, shifts in the underlying DNAIRU. Second, it allows us to differentiate between long-run concepts like the natural rate of unemployment, which is usually presumed to be fairly stable over time, and short-run concepts like the DNAIRU, which potentially is considerably more variable as a reflection of mismatches resulting from stochastic variability in the offer curves of workers and firms. For purposes of simplification, we have based our analysis on the assumption of a constant expected long-run DNAIRU of 6 percent, but it would be relatively straightforward in principle to vary the expected long-run DNAIRU as a function of unemployment compensation schemes, demographics, and so forth.

Appendix II. Stability Analysis of Taylor Rules and LWW Rules

Research during the last few years has suggested that conventional Taylor rules, when appropriately calibrated, have reasonably attractive stabilization properties within a range of different macro models. Along similar lines, Levin, Wieland, and Williams (LWW: 1999) have recently shown that simple rules linking the change in the interest rate to the variables that enter conventional Taylor rules have desirable properties in four different macro models.

Such favorable impressions of Taylor rules and LWW rules have not gone unchallenged, however. Christiano and Gust (CG: 1999) have been quick to point out that the stabilization properties of these rules have been evaluated almost exclusively in sticky-price IS-LM models with similar structures. CG demonstrate that in a limited-participation-rate model developed by Christiano, Eichenbaum and Evans (1998), Taylor rules or LWW rules may be dangerous, especially if the rules place too high a weight on output (or unemployment) relative to inflation.

This appendix argues that Taylor rules and LWW rules are also likely to be highly destabilizing in plausible specifications of sticky-price IS-LM models. In particular we emphasize that in contrast to their favorable performances in linearized versions of such models where overheating does not have first-order welfare consequences, Taylor rules and LWW rules tend to perform very poorly in nonlinear rational-expectations models in which myopic or sluggish policy responses can fail to provide an anchor for inflation expectations.

For each of the two classes of rules, we start by reporting the Blanchard-Kahn (1980) saddle-point stability conditions for a linear forward-looking model developed by Fuhrer and Moore (FM: 1995a, 1995b).⁴⁴ We show that both classes of rules produce saddle–point stability over an enormous range of parameter values. We then go on to argue that the stability of these rules in linear sticky-price IS-LM models with rational expectations breaks down in models that do not presume global linearity.⁴⁵ In particular, we show that in our model with moderate nonlinearities, such rules can give rise to extreme instabilities in inflation expectations and would risk a repeat of the monetary policy errors of the 1970s. As a corollary, the assumption that such myopic policy rules would be fully credible in these models is untenable, especially in cases where the monetary authorities place a high weight on output (or unemployment) relative to inflation.

A. Conventional Taylor Rules Generalized for Interest Rate Smoothing

Figure B1 reports the combinations of parameter settings that lead to unique and indeterminate solution paths in the Fuhrer-Moore (1995b) model under a conventional Taylor rule that has been generalized to allow for interest rate smoothing. This rule can be written as:

Figure B1.

Regions of uniqueness and indeterminacy (generalized Taylor rule in Fuhrer-Moore model). Policy reaction function: rs_t = ρrs_t-1 + (1—ρ)[ω_π(π4_t) + ω_y(y_t)].

\begin{matrix} r s_{t} = ρ r s_{t - 1} + (1 - ρ) [w_{π} (π 4_{t}) + w_{y} (y_{t})] & (B .1) \end{matrix}

where: rs_t is the nominal interest rate setting at time t; π4_t is the average inflation rate over the previous four quarters; y_t represents the output gap in the Fuhrer-Moore model; and ρ, ω_π, ω_y are parameters.⁴⁶ Note that the interest rate reaction function has been coded so that the parameters ω_π and ω_y represent asymptotic long-run responses of interest rates to the year-over-year inflation rate and the output gap.⁴⁷

A striking feature of Figure B1 is that for a very wide range of parameter values—and independently of the speed with which monetary policy reacts to inflation and output gaps (i.e., independently of ρ)—the model has a stable and unique solution. Indeed, the stability properties of the generalized Taylor rule in this linear rational expectations IS-LM model are extremely simple. The only condition necessary for stability and uniqueness is that the long-run response of the interest rate to year-over-year inflation must be greater than one. Provided this condition is met, even a Taylor rule that reacts much more aggressively to output than to inflation will provide an anchor for inflation expectations in the Fuhrer-Moore model.

What is it that explains the “excessive stability” generated by Taylor rules in these sticky–price linear rational expectations models? What gives rise to stable macroeconomic behavior even when the monetary authorities respond in a very myopic way to inflation developments, or place an extremely high weight on real objectives relative to inflation objectives? Two assumptions appear to be critical here. The first is the assumption that the economy can be characterized by global linearity.⁴⁸ The second is the premise that no matter how myopic policy responses are in the short run, the private sector forms its expectations under the assumption that the monetary policy rule will be adhered to forever.

For the nonlinear model considered in this paper, which is cast in terms of unemployment gaps rather than output gaps, even moderately myopic policy rules like the conventional Taylor rule can result in explosive behavior if the economy is subjected to a significant degree of overheating. This reflects a combination of factors. Recall, first, that even moderate convexity in the Phillips curve implies that at some point the short-run unemployment–inflation tradeoff must worsen considerably when unemployment falls significantly below the NAIRU, and beyond this point a further marginal easing of monetary policy results mainly in inflation with only a very small incremental reduction in unemployment. Second, to the extent that policymakers tend to make serially correlated errors in estimating unemployment and output gaps, as reflected in our model, the probability of experiencing a significant degree of overheating is heightened. Third, when inflation expectations have a model-consistent component and rational agents possess information about the policy rule and the nonlinear nature of the expansionary effects of monetary policy, attempting to adhere to a conventional Taylor rule with a high weight on imprecise measures of unemployment gaps relative to a backward-looking measure of inflation could be conducive to wide swings or explosiveness in inflation expectations.

To sharpen quantitative perspectives, and to emphasize as well that in the region of small unemployment gaps our nonlinear model exhibits similar behavior to models that impose global linearity, consider the following. From equation (1), the direct effects of unemployment gaps (u^*—u) on the annualized inflation rate in the nonlinear model are given by the functional form γ(u^*—u)/(u—Φ), where we also assume Φ= u^*—4. To simplify, define g = (u^*—u) such that the direct effects of the gap on inflation can be written as F (g) = γ g/(4—g). Differentiating this last term with respect to g leads to the expression F’(g) = [γ(4—g) + γg]/(4—g)² and shows how the slope of the Phillips curve will depend on the initial state of excess demand conditions in the labor market. For example, if we linearize the Phillips curve at g = 0, the point where there is zero labor market tightness, and also substitute in our empirical estimate of γ = 3.2, we can see that the slope of the Phillips curve is 0.8. This estimate would suggest that a surprise of 0.1 percentage point in labor market tightness would have a direct effect on the contemporaneous annualized inflation rate of 0.08 percentage point, and hence would raise the year-on-year inflation rate by 0.02 percentage points. Such an order of magnitude of the direct impact effect of marginal changes in labor market tightness on year-on-year inflation is consistent with numerous studies of the U.S. inflation process based on linear models that suggest that the direct impact effects on inflation of changes in unemployment or output gaps are extremely small. Indeed, such findings have led some commentators in U.S. policymaking circles to sometimes describe the in-quarter and 1 to 2-quarters-ahead year-on-year inflation rate as “predetermined,” or as effectively independent of small surprises in the degree of labor market tightness.

Table B1.

Slope of Phillips curve and impact effects of surprises in the unemployment gap.

^{^1/}Response of year-on-year inflation to an unanticipated shift of 0.1 percentage point in the unemployment gap.

Table B1.

Slope of Phillips curve and impact effects of surprises in the unemployment gap.

Initial	Slope of	Impact Effect of
Unemployment	Phillips	Surprise in the
Gap	Curve	Unemployment Gap ^1/
g = u^ - u*	F’[g]	0.1 F’[g]/4
-3.00	0.26	0.01
-2.00	0.36	0.01
-1.00	0.51	0.01
0.00	0.80	0.02
+ 1.00	1.42	0.04
+ 1.6	2.22	0.06
+2.00	3.20	0.08
+3.00	12.80	0.32

^{^1/}Response of year-on-year inflation to an unanticipated shift of 0.1 percentage point in the unemployment gap.

Table B1.

Slope of Phillips curve and impact effects of surprises in the unemployment gap.

Initial	Slope of	Impact Effect of
Unemployment	Phillips	Surprise in the
Gap	Curve	Unemployment Gap ^1/
g = u^ - u*	F’[g]	0.1 F’[g]/4
-3.00	0.26	0.01
-2.00	0.36	0.01
-1.00	0.51	0.01
0.00	0.80	0.02
+ 1.00	1.42	0.04
+ 1.6	2.22	0.06
+2.00	3.20	0.08
+3.00	12.80	0.32

^{^1/}Response of year-on-year inflation to an unanticipated shift of 0.1 percentage point in the unemployment gap.

Although our nonlinear model encompasses this prediction in the neighborhood of zero excess demand, it also suggests that the direct impact effects of a marginal increase in labor market tightness can be much more significant when there is already substantial tightness in the labor market. To illustrate, Table B1 shows the slope of the Phillips curve, and the direct impact effects on the year-on-year inflation rate of a 0.1 percentage point change in the unemployment gap, at various initial levels of the unemployment gap. Note that the direct impact effects accelerate as the unemployment gap widens.

To appreciate how easily the inflation process can become explosive in our nonlinear model when policy follows a conventional Taylor rule, it needs to be recognized that the full effects on inflation of shocks to the unemployment gap can be much larger than the direct effects. This reflects the fact that inflation expectations have a model-consistent component and depend on the entire timepath of the unemployment gap that rational agents come to expect, given their information about the nature of the policy rule. When our Phillips curve is linearized around the NAIRU, our model satisfies the Blanchard-Kahn conditions for a Taylor rule calibrated with the weights originally suggested by Taylor (1993): ω_π = 1.5, ω_u = 1.0, and ρ = 0.⁴⁹ However, if we linearize the Phillips curve around a point of excess demand at which the unemployment gap exceeds 1.6 percentage points, the Blanchard-Kahn conditions are no longer satisfied. In this case, the Taylor rule is sufficiently myopic in terms of responding to inflationary pressures that monetary policy fails to provide an anchor for inflation expectations and the solutions of the model become explosive.

Figure B2.

Regions of uniqueness and explosiveness (generalized Taylor rule in the nonlinear model). Policy reaction function: rs_t = ρrs_t-1 + (1—ρ)[ω_π(π4_t) + ω_u(g_t)].

As is evident in Figure B1, one of the striking features of the stability conditions for the Fuhrer-Moore model is that they appear to be independent of the degree of interest rate smoothing. This points to a general problem with linear models of the inflation process, which imply that slow monetary policy responses to information about future inflation developments only have second-order welfare consequences. Figure B2 reports the regions of stability for our nonlinear model in terms of both the slope of the Phillips curve and the degree of interest rate smoothing. For ρ = 0, the critical slope of the Phillips curve, about 2.2, corresponds to an unemployment rate of 1.6 percentage points below the NAIRU, as noted in Table B1. Notice that in the nonlinear model, the region of stability shrinks further if interest rate smoothing is imposed on an already myopic policy rule.

B. LWW Interest Rate Change Rules

Figure B3 reports the regions of stability for the class of interest rate change rules suggested by Levin, Wieland, and Williams (LWW: 1999). In this case, the general form of the reaction function is:

Figure B3.

Regions of uniqueness and explosiveness (LWW rule in the Fuhrer-Moore model). Policy reaction function: rs_t = rs_t-1 + ω_π π4_t) + ω_y(y_t).

\begin{matrix} r s_{t} = r s_{t - 1} + (w_{π} (π n_{t}) + w_{y} (y_{t})) & (B .2) \end{matrix}

where π n_t is an n-quarter moving average of inflation measured over the previous n quarters. The top and middle panels of Figure B3 consider the two optimal rule parameterizations reported by Levin, Wieland and Williams (1999), where n is equal to 4 quarters and 12 quarters; the longer lag structure on inflation was found to be optimal in a linearized version of the FRB-U.S. model, while the shorter lag structure was found to be optimal in the other linear models that they included in their study. In this case again, even where there is extreme interest rate smoothing and monetary policy responds to very backward-looking measures of inflation, the linear model is stable for an incredibly wide range of weights on inflation and output. The lower panel of Figure B3 considers an even more extreme case of myopic reaction functions, where the reaction function now depends on a six-year moving average of past inflation. Here there is some evidence of instability in the model; but in contrast to the type of results found by Cristiano and Gust (1999), in this case explosiveness can arise from setting too low a weight on output.

The LWW rule has extremely poor stabilizing properties in the nonlinear model developed in this paper. First, for the model that was estimated the rule is so myopic and backward–looking that it fails to provide an anchor for inflation expectations. Second, even if one recalibrates the model to reduce the effects of overheating very substantially, an optimal parameterization of the LWW rule still gives rise to significant boom and bust cycles.

It does not seem to be widely recognized that interest rate change rules such as equation B.2 are exactly equivalent to targeting a trend change in the price level when ω_y = 0, and result in approximate price level targeting for small values of ω_y. To see this, consider a simple case in which the interest rate change depends solely on the quarterly change in the price level (P) expressed at an annual rate:

\begin{matrix} r s_{t} = r s_{t - 1} + w_{π} π_{t} & (\begin{matrix} B .3 \end{matrix}) \end{matrix}

where π_t = 4(P_t—P_t-1). As initial conditions, assume that inflation is on target and the real interest rate is at its equilibrium value (i.e., in period 0, rs_o = rs^* and π_o = π^* = π^e, where ^* denotes equilibrium).

Now assume that a demand or supply shock raises the inflation rate in period 1 to some arbitrary value π₁. It is interesting, and perhaps even surprising, that monetary policy governed by equation B.3 would attempt to move the price level back to the original baseline path. This will be the case, for example, if long-run neutrality holds (as LWW claim for each of the models they consider), because long-run neutrality implies that the real interest rate must return back to its initial value. But if the real interest rate returns back to control, the nominal interest rate must also eventually return back to control in some period T since, by assumption, the rule is successful in moving inflation back to its initial level of π^*.

If we now sum equation B.3 between periods 1 and T we obtain

\begin{matrix} r s_{t} - r s_{0} = w_{π} \sum_{i = 1}^{T} π_{i} & (B .4) \end{matrix}

So rs_T—rs₀ = 0 implies

\begin{matrix} \sum_{i = 1}^{T} π_{i} = 0. & (B .5) \end{matrix}

Thus, under the assumption that long-run neutrality holds, a policy rule in the form of equation B.3 essentially amounts to a price-level targeting rule, since any shock that generates positive inflation must be offset at some point by negative inflation rates. This result obviously carries over to cases in which the contemporaneous inflation rate in equation B.3 is replaced by some finite moving average lag structure on past inflation; and even when the rule is extended to include a term in the output gap, as in the general form of LWW rules described by equation B.2, it continues to bear a close resemblance to price-level targeting. Thus, it should not be surprising that such myopic LWW rules can generate extremely poor business cycle properties in models with strong inflation persistence and convexity in the Phillips curve.

Acknowledgments

The views expressed are those of the authors and do not necessarily reflect the views of the International Monetary Fund. We thank Jeffrey Fuhrer, Lars Svensson, Robert Tetlow, and Volker Wieland for useful discussions and Susanna Mursula and Sarma Jayanthi for extensive research assistance.

This assertion can be confirmed using the methods developed in Rotemberg and Woodford (1998) and imposing our assumption that subsidies are used to eliminate the output and employement distortions arising from monopolistic competition. Even when the variances of shocks are small, aproximate solutions yield incorrect welfare rankings in some models. For example, Kim and Kim (1999) show that in a model of international risk sharing a standard approximation implies that welfare is lower with a complete market than with autarky.

However, if prices are fixed by staggered contracts instead of by one-period contracts (or by synchronized multiperiod contracts), results depend crucially on whether wages are fixed by contracts or are flexible as shown by Erceg, Henderson, and Levin (forthcoming).

Even the fully optimal policy under complete information cannot attain the Pareto-optimal equilibrium if both wages and prices are fixed by staggered contracts as shown by Erceg, Henderson, and Levin (forthcoming).

This set includes Ireland (1997), Goodfriend and King (1997), Rotemberg and Woodford (1998), Henderson and Kim (1999), King and Wolman (1999), and Rotemberg and Woodford (1999).

This set includes Corsetti and Pesenti (1998), which is based on a perfect foresight model, and Obstfeld and Rogoff (1998) Devereux and Engel (1998), and the paper by Engel in this collection which are based on stochastic models.

That is, we assume for simplicity that there are no factors of production other than labour and no fixed costs. Kim (1998) shows that our formulation can be viewed as a model with capital in which the marginal adjustment cost for the first unit of net investment approaches infinity. Kim (1997) explores the implications of allowing for fixed costs.

These equal shares exhaust aggregate profits:

\int_{0}^{1} Γ_{s} d h = \int_{0}^{1} (Y_{f, s} - W_{s} L_{f, s}) d f

That is, we assume that u_s = log U_s ∼ N (O, 2σ_u²), v_s = log V_s ∼ N (0, 2σ_v²), and z_s = log Z_s ∼ N (0, 2σ_z²).

As is well known, given the form that we have assumed for the utility of consumption, as ρ → 1, the utility of consumption approaches c = ln C.

Assumptions about the paths of government spending and taxes have implications for which monetary policies are feasible and for the effects of different feasible monetary policies. For an up to date discussion of the interaction between monetary and fiscal policy and citations of other recent contributions see Canzoneri, Cumby, and Diba (1998).

We assume a monetary policy reaction function that implies that the expected rate of inflation, the solution for inflation in the model with flexible wages and prices when all shocks take on their mean values, is equal to zero. The analysis could be modified to allow for a nonzero expected rate of inflation. If the expected rate of rate of inflation were positive, the expected government deficit would have to be positive.

The properties of log normal distributions used in this paper are summarized in Appendix A.

Analogous logic applies in the case with price contracts and flexible wages. That is, the optimal rule with price contracts is the rule that keeps prices constant with completely flexible prices and wages. As we show in Appendix B, outcomes with price contracts and flexible wages are the same as the outcomes with wage and price contracts for all variables except the nominal wage. Therefore, the optimal rule with wage and price contracts is the same as the optimal rule with price contracts and flexible wages.

This result was obtained by Bean (1983).

This result was obtained by Koenig (1996).

Of course, the solution for W can also be obtained using the wage equation.

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Notes

Losses associated with interest rate variability are reflected in the policy objective function, however, and thereby influence the optimal calibration of the policy reaction function.

Attention to the credibility problem was stimulated by Kydland and Prescott (1977) and Calvo (1978).

Barro and Gorden (1983a, 1983b), Backus and Driffill (1985), Canzoneri (1985), and Rogoff (1985), among others, were instrumental in advancing the analysis of these issues.

The arguments are further articulated in Flood and Isard (1990), which corrects a technical error identified by Lohmann (1990).

The terminology emanated from Persson and Tabellini (1990).

Recent examples of such papers were included in the programs of the NBER Conference on Monetary Policy Rules (January 15-17, 1998), the Federal Reserve Bank of San Francisco Conference on Central Bank Inflation Targeting (March 6-7, 1998), the Riksbank-IIES Conference on Monetary Policy Rules (June 12-13, 1998), the 1998 Symposium on Computational Economics at Cambridge University (June 29-July 1, 1998), and the Reserve Bank of New Zealand Conference on Monetary Policy Under Uncertainty (June 29-July 3, 1998). Earlier contributions to the inflation targeting literature include the conference volumes Leiderman and Svensson (1995), Haldane (1995), Federal Reserve Bank of Kansas City (1996), and Lowe (1997).

A crucial prerequisite for exercising discretion intelligently, of course, is that the monetary authorities must understand the time-consistency issue and continuously evaluate the extent to which their behavior may be affecting the credibility of their announced objectives.

In this context, Svensson argues that the strategy of inflation targeting can be viewed as a way of committing to minimize a particular loss function by adopting a rule (first-order condition) that involves target variables or forecasts of target variables and by implementing communication practices that allow the public to evaluate the monetary authority’s performance and hold it accountable.

For example, Wieland (1998), Rudebusch and Svensson (1999), Svensson (1999a).

For example, McCallum and Nelson (1988, 1999), Henderson and Kim (1999).

For example, McCallum (1988), Levin, Wieland, and Williams (1999).

Data uncertainties, which also have implications for the optimal strength of policy reactions, are not addressed in this paper; see Orphanides (1998) for a recent exploration of this topic. On the general importance of changing the strength of policy responses when the (perceived) macro model changes, see Amano, Coletti, and Macklem (1998).

See McCallum and Nelson (1988, 1999) and Henderson and Kim (1999) for studies of monetary policy rules in models with optimizing agents rather than postulated Phillips curves.

See, for example, the Symposium comprised of Blanchard and Katz (1997), Galbraith (1997), Gordon (1997), Rogerson (1997), Staiger, Stock, and Watson (1997), and Stiglitz (1997), as published in the Winter 1997 issue of the Journal of Economic Perspectives.

Staiger, Stock, and Watson (1997).

For example, Galbraith (1997).

Other recent analyses of the implications of NAIRU uncertainty (or output gap uncertainty) include Wieland (1998) and Smets (1998), who use simple linear models with backward-looking expectations to demonstrate that uncertainty about the NAIRU (or about the Phillips-curve parameters on which NAIRU estimates depend) provides a motive for cautious policy reactions.

Isard and Laxton (1998), Isard, Laxton, and Eliasson (1998).

Williams (1999) finds that even in models with hundreds of state variables, parsimonious specifications of simple policy rules appear to be very effective in achieving stabilization objectives. See also Rudebusch and Svensson (1999).

The model contains important backward- and forward-looking components, as derived from the bargaining framework in Fuhrer and Moore (1995a, 1995b), but the functional form is less restrictive and is more consistent with empirical evidence that suggests that there is a small weight on the “rational” or forwardlooking component of the US inflation process—for example, see Fuhrer (1997).

The base-case model variant assumes N = 12, but to explore the sensitivity of the results to the length of contracts we have also conducted simulations with N = 4.

In equations (1) and (2), the estimated value of γ is 3.20. The estimation and stochastic simulations are based on the assumption that (Φ_t—max[0, u^*_t—4], and u^*_t is always strictly greater than 4 in the actual and hypothetical data we address.

For discussions of the potential pitfalls associated with conventional tests for asymmetries in the Phillips curve, see Clark, Laxton, and Rose (1996) and Laxton, Rose, and Tambakis (1999).

The latter would be implied by constant adherence to a given policy rule. Note that equation (7) is relevant for interpreting history and updating estimates of the NAIRU, but that apart from assuming stationarity, we do not require specific assumptions about the distribution of the ε^u_t terms, which are not drawn directly in the simulation analysis.

Kuttner (1992, 1994) has applied this idea to measuring potential output. In using information about the error terms in each of the two Phillips curves, our procedure for estimating the NAIRU and DNAIRU essentially gives equal weight to the data on the CPI and the CPI excluding food and energy.

The simulations set Φ_t—u^*_t—4, so the convex term in the unemployment rate in equations (1) and (2), based on the estimates reported in Table’1, can be expressed as 3.20F (g), where g = u^*—u. The linear approximations in equations (la) and (2a) replace F (g) with [F’(0)](u^* - u) = (3.20/4)(u^* - u) = 0.80(u^* - u).

Estimates of equations based on the Michigan survey measures of inflation expectations suggested a weight of 6 on the model consistent component, but there was significant evidence of residual autocorrelation in the estimated equations.

Fuhrer and Moore (1995b) argue that longer-term interest rates are more relevant for explaining aggregate demand and unemployment. The implications of such an alternative representation of the monetary transmission mechanism might be interesting to explore as an extension of the analysis in this paper.

See Laxton, Rose, and Tambakis (1999) for details on the estimation.

The qualitative results and main conclusions of this paper do not hinge on the precise nature of the unemployment dynamics, although they clearly depend on a positive response of unemployment to the real interest rate, as well as on the existence of both lags in the response of unemployment to policy actions and a persistent component of unemployment. It might be interesting, in future work, to consider modifications of the model in which the response of unemployment to the interest rate was forward looking. It might also be interesting to treat the parameters of the unemployment equation as an additional element of uncertainty—along with the level of the NAIRU—that policymakers take into account when choosing the “optimal calibration” for a policy rule.

Interest in this formulation received considerable impetus from Taylor (1993), who defined his rule in terms of the output gap. Recent studies of the performance of Taylor rules can be found, for example, in Levin, Wieland, and Williams (1999) and Taylor (1999a).

Appendix II discusses the stability conditions for models that are based on linear and nonlinear Phillips curves and explains why the conventional Taylor rule ensures stability in models with linear Phillips curves but does not ensure stability in nonlinear models of the inflation process.

Different types of IFB rules have been shown to deliver reasonable economic performances over a wide range of disturbances; see, for example, Amano, Coletti, and Macklem (1998), Haldane and Batini (1999), and Rudebusch and Svensson (1999).

The β parameter also provides a basis for analyzing the pros and cons of central bank transparency. As Faust and Svensson (1998) emphasize, transparency about the central bank’s objectives, by improving the accuracy of the public’s information about β (or alternatively, about the difference between u^*_t—β and the long-run DNAIRU, u_*), can make it possible for the public to distinguish more accurately between the intended components of macroeconomic outcomes and the central bank’s control errors, thereby making the central bank’s reputation and credibility more sensitive to its actions.

The stochastic simulations were performed using a robust and efficient Newton-Raphson simulation algorithm that is now available in portable TROLL—for a discussion of the properties of this algorithm see Juillard and others (1998).

The setting ν = 0.5 corresponds to the base-case value used by Rudebusch and Svensson (1999). Note also that the setting of β is irrelevant when θ = 0.

Recall the discussion in Section III.A on how the linear Phillips curves are calibrated.

Clarida, Gali, and Gertler (1998) consider specifications based, alternatively, on output gaps and unemployment gaps. The CGG rule is derived by combining the following two equations

r s_{t}^{*} = r^{*} + {\tilde{E}}_{t} {(β - 1) (π 4_{t + 1} - π^{T A R}) + γ ({\bar{u}}_{t} - u_{t}) | Ω_{t}}

r s_{t} = (1 - ρ) r s_{t}^{*} + ρ r s_{t - 1}

where rs^*_t represents a target nominal interest rate. Thus, the constant term in equation (18) can be decomposed into c = r^* - (β - 1)π^TAR.

Clarida, Gali, and Gertler (1998) also suggest that it is more consistent with actual Fed behavior for the interest rate reaction function to depend upon the one or two-quarter ahead forecast of the unemployment gap rather than the contemporaneous unemployment gap. It would be interesting to further explore the stabilizing properties of CGG rules to see if they change significantly when the rule is based on forecasts of the unemployment gap rather than contemporaneous measures.

It may be noted here that the literature has distinguished between IFB rules that embody rule-consistent inflation forecasts, as does our IFB2 rule, and IFB rules defined in terms of constant-interest-rate inflation forecasts.

We did not undertake an extensive search for the optimal inflation forecast horizon but note that three quarters is roughly half the time that is generally believed to be required for interest rates to have their full effects on the economy. By comparison, Clarida, Gali, and Gertler (1998) use a four-quarter-ahead inflation forecast, while Haldane and Batini (1999) and Rudebusch and Svensson (1999) explore the performances of IFB rules with a range of forecast horizons.

Svensson (1999b) suggests that a two-year horizon might be preferable on conceptual grounds, reflecting his notion (perhaps inferred from models with substantial inflation inertia) that inflation expectations at shorter horizons have significant predetermined components. This suggestion seems to reflect a preference for rules that (approximately) correspond to the first-order conditions from policy optimization problems, along with the notion that such first-order conditions boil down to simpler expressions of the relationship between the interest rate and an inflation forecast when the inflation forecast is not largely predetermined.

In a separate forthcoming paper we argue that the high estimates of ρ (and associated high t-statistics) that are obtained when CGG rules are fitted to historical data are probably reflections of specification error.

We chose this model because it was more easily accessible than the other models considered by Levin, Wieland, and Williams (1999). We are indebted to Jeffrey Fuhrer for taking the time to help us replicate some of his earlier results. The results reported in this appendix have been derived from the parameter estimates reported in Fuhrer and Moore (1995b).

Our line of argument in this appendix is broadly similar to the one presented in Christiano and Gust (1999), who show not only that poorly parameterized simple rules can give rise to poor simulation properties in their particular model, but also that choosing the parameters of rules on the basis of one particular class of models can give rise to indeterminacy or explosiveness in other models.

It is convenient here to follow Taylor (1993) in defining the rule in terms of the output gap rather than the unemployment gap. For notational convenience we have dropped the constant term in the equation by assuming that the equilibrium real interest rate and long-run inflation target are zero.

For example, the long-run effects of a permanent unitary change in the output gap is equal to the short-run effect, (1—ρ) w_y, divided by (1—ρ).

Under the global linearity assumption, the estimated slope of the Phillips curve (based on post war U.S. data) suggests that unemployment gaps or output gaps have small effects on the inflation process. These small effects imply that it can be very costly, in the context of these models, to reduce inflation once high inflation expectations have become entrenched. The estimated slope also means that for given inflation expectations, the marginal effect on inflation of an increase in excess demand is small, even when the level of excess demand is high.

Taylor (1993) suggested a weight of 0.5 on the output gap, which we translate into a weight of 1.0 on the unemployment gap under the Okun’s Law assumption that the unemployment gap varies approximately half as widely as the output gap over the business cycle.

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Comment

BY JO ANNA GRAY

The Henderson and Kim paper is part of a larger research agenda concerned with the evaluation of monetary policy rules. Among the issues addressed in the paper are several that involve aspects of the choice of microfoundations in modeling macroeconomic behavior. This discussion notes three in particular:

(i) The appropriate choice of a welfare criterion, with emphasis on the choice between the objective function of a policy maker and the utility of a representative agent.
(ii) The error that may be introduced by using approximate solutions of models rather than exact solutions that permit exact utility calculations.
(iii) The justification provided for the non-neutrality of money.

Direct evaluation of the utility of economic agents rather than indirect evaluation through the objective function of a policy maker is surely a direction worth exploring. In general, however, one cannot expect the convenience of an economy in which welfare is adequately characterized by the utility of a single group of homogeneous agents (the household sector in the Henderson and Kim paper). The introduction of multiple groups with independent economic interests (laborers, entrepreneurs, retirees, etc.) means that the choice of a policy rule will typically have distributional consequences. It follows that a single representation of utility will generally be inadequate as a metric for policy evaluation, and that retreat to a social welfare function with ad hoc attributes is not an unlikely outcome.

Another emphasis of the paper is exact utility calculation rather than approximate solution. In the paper’s introduction, the authors suggest that reliance on approximation rather than exact solutions may produce erroneous welfare conclusions. However, the valued-added of exact utility calculation is not actually addressed in the analysis of the paper. In other work the authors have analyzed the choice of monetary policy rules using approximate solutions and so they are in a position to compare the outcomes of the two approaches. This is an exercise they promise to undertake in a future paper.

Perhaps the most ambitious modeling choice in the paper is the characterization of both firms and households as monopolistic competitors, a decision with non-trivial implications for the complexity of the paper’s analysis. The motivation for the choice is to provide improved microfoundations for the wage and price rigidities assumed in the paper. But the case is blurred by several considerations. One would normally incorporate microfoundations into a model of price stickiness in order to insure that the conclusions drawn are robust with respect to the additional assumptions needed to explain the existence of price stickiness. However, the authors assume, rather than derive, one-period fixed nominal wage and price contracts. In addition, they leave unaddressed the question of whether the paper’s conclusions would be any different if perfectly competitive workers and firms, rather than monopolistically competitive entities, were assumed. The reader is left to wonder if the advantages of the more sophisticated modeling approach, once they are articulated and evaluated, will outweigh the disadvantages of the more cumbersome analytics.

The decision to model firms and households as monopolistic competitors leads the authors to a second choice that is of independent interest. The authors use subsidies to eliminate the distorting effects of monopoly on resource allocation. In doing so, they also eliminate the possibility that optimal policy may involve a trade-off between stabilization objectives and other economic objectives such as Pareto-efficient resource allocation. It is natural to ask if something interesting is lost here, particularly given that exact utility calculations may introduce an interaction between stabilization outcomes and resource allocation.

Finally, does it make sense to characterize individual households as monopsonistic suppliers of labor, and to offer their monopsonistic behavior as the underlying reason for nominal wage contracts? The assumption that monopolistic competition characterizes the structure of the goods market has become widely acceptable, and one might wish to extend the assumption to the labor market in cases in which workers act as a collective (as in the case of unions, for example). The case for characterizing the behavior of individual households in this way is less obvious, and the authors may wish to offer more of their thinking on the matter in future work.

Comment

BY LARS E. O. SVENSSON

Isard, Laxton and Eliasson (ILE) have written a fine and impressive paper, with much content. It presents an estimated empirical model of the U.S. economy, with a nonlinear Phillips curve (with both forward- and backward-looking elements) and an unobservable time-varying natural unemployment rate. Stochastic simulations and stability analysis are undertaken with alternative simple “instrument rules,” that is, rules specifying the instrument as a given reaction function of directly observable or constructed, synthetic variables. Furthermore, since the natural unemployment rate is unobservable, instrument rules that involve responses to the unemployment gap (the deviation between unemployment and the natural rate) must rely on the policy-makers’ estimate of the time-vary ing natural rate.

A number of interesting detailed results are presented. The main result of the paper, as I interpret it, can be expressed as: “Beware of simple instrument rules (especially conventional linear and backward-looking ones) as an automatic pilot for the economy.” Such rules may even result in instability. So-called “forecast-based” rules, where the instrument responds to the deviation of model-consistent inflation forecasts from the inflation target, perform better than the backward-looking instrument rules when the instrument responds to the deviation between current and lagged inflation and the inflation target (in addition to responding to the estimated unemployment gap).

My discussion will focus on an intuitive explanation of why linear backward-looking reaction functions will be inferior in a non-linear model, and why a reaction function, where the instrument responds to an inflation forecast, is likely to perform better, but is not optimal.

1. Instrument Rules and Targeting Rules

First, however, I would like to raise the issue about how to handle practical monetary policy, given the paper’s warning about the possible instability of conventional instrument rules. The paper states that it is very important for policymakers to calibrate their nominal interest rate adjustments on the basis of forward-looking measures of real interest rates. This can be interpreted as the paper advocating modified instrument rules rather than the conventional backward-looking instrument rules. However, in the real world, instrument rules are never applied mechanically. For several reasons, they are, at best, used as guidelines and benchmarks, which may illuminate the monetary policy decision but never be a substitute for a forward-looking decision framework for monetary policy. One of these reasons is the lack of a commitment mechanism, by which the central bank could commit itself to a given instrument rule. Another is the manifest inefficiency of any simple rule, and the strong incentives to deviate from it, since it relies on much less information than is efficient, including extra-model information that motivate judgemental adjustments.

Instead, as argued in Svensson (1999a, 1999b), I believe that the so-called “targeting rules,” which involve a commitment to minimize a given loss function or to fulfill some (approximate first-order) condition for (forecasts of) the target variables, but allow the optimization to be done under discretion, is a more fruitful and realistic formalization of real-world monetary policy. In particular, as discussed in Svensson (1999b), I believe that “forecast targeting” (meaning selecting an instrument path such that resulting forecasts of inflation and the output gap minimizes an intertemporal loss function) rather than a commitment to a simple instrument rule is the best way of maintaining price stability. Furthermore, I believe that a generalization from “mean” forecast targeting to “distribution” forecast targeting is a practical way of handling both nonlinearity and model uncertainty in monetary policy.

2. A Simple Model with a Nonlinear Phillips Curve

Let me now illustrate what inflation targeting, interpreted as a commitment to minimize a particular loss function, implies in a model with a nonlinear Phillips curve, and how the resulting equilibrium can be used to illuminate the inferiority of linear backward-looking instrument rules and the somewhat better performance of a forecast-based instrument rule. I choose a simple model which can be seen as a simple variant of the more elaborate model of lLE.

Assume that the aggregate supply is given by the simple accelerationist Phillips curve

\begin{matrix} π_{t + 1} = π_{t} + f ({\tilde{u}}_{t}) + ε_{t + 1}^{π}, & (2.1) \end{matrix}

where π_t is inflation in quarter t,

{\tilde{u}}_{t} \equiv u_{t} - u_{t}^{*}

is the unemployment gap (where u_t is the unemployment rate and u_t^* is the natural unemployment rate), and ε^π_t is N (0,σ²_π), a normally distributed exogenous shock with zero mean and variance σ²_π. The natural unemployment rate is a random walk,

u_{t + 1}^{*} = u_{t}^{*} + ε_{t + 1}^{u *},

where ε^u*_t is N (0,σ²_u*).

Nonlinearity enters in the Phillips curve via the nonlinear function f, which fulfills

\begin{matrix} \begin{matrix} f^{'} < 0, & f (0) = 0, \end{matrix} & f^{''} \geq 0. \end{matrix}

Several functional forms can be used. ILE use the form

f ({\tilde{u}}_{t}) = - γ \frac{{\tilde{u}}_{t}}{{\tilde{u}}_{t} - 4} .

Schaling (1998) instead uses

f ({\tilde{u}}_{t}) = - \frac{γ {\tilde{u}}_{t}}{1 - ϕ γ {\tilde{u}}_{t}},

where φ ≥ 0 is used as an index of convexity. For convenience, I choose a simple quadratic function that allows an (approximate) analytical solution, namely

\begin{matrix} f ({\bar{u}}_{t}) = {\begin{matrix} \begin{matrix} - γ {\tilde{u}}_{t} + ϕ {\tilde{u}}_{t}^{2} & {\tilde{u}}_{t} \leq γ / 2 ϕ \end{matrix} \\ \begin{matrix} - γ^{2} / 4 ϕ & {\tilde{u}}_{t} \leq γ / 2 ϕ \end{matrix}, \end{matrix}} & (2.2) \end{matrix}

for the parameters φ ≥ 0 and γ > 0. This function is continuous and differentiable. For φ > 0, it is decreasing and convex for ${\tilde{u}}_{t} \leq γ / 2 ϕ$ and constant for ${\tilde{u}}_{t} > γ / 2 ϕ .$ . For φ = 0, it is linear, $f ({\tilde{u}}_{t}) = - γ {\tilde{u}}_{t} .$ . Thus, φ can be interpreted as an index of convexity and nonlinearity.

Aggregate demand is taken to be a linear function in terms of the unemployment gap,

\begin{matrix} {\tilde{u}}_{t + 1} = η_{u} {\tilde{u}}_{t} + η_{r} (i_{t} - π_{t + 1 | t} - \bar{r}) + ε_{t + 1}^{\bar{u}}, & (2.3) \end{matrix}

where i_t is a short nominal interest rate (denoted rs_t in ILE) and the central bank’s instrument, X_t+τ|t ≡ E_tx_t+τ for any variable x denotes the expectation of x_t+τ conditional on information available in quarter t, $\bar{r} > 0$ is the “natural” real interest rate, ε^u_t+1 is $N (0, σ_{\bar{u}}^{2}),$ , and parameters η_u and η_r are positive. The natural real interest rate is the constant real interest rate that, in the absence of shocks, would result in a constant zero unemployment gap.

Assume an intertemporal loss function for the central bank,

E_{t} \sum_{τ = 0}^{\infty} Δ^{τ} L_{t + τ},

where 0 < δ < 1 is a discount factor and the period loss, L_t, is given by the period loss function

L_{t} = \frac{1}{2} [{(π_{t} - π^{*})}^{2} + θ {\tilde{u}}_{t}^{2}],

where π^* is the inflation target (denoted π^TAR in ILE). (I have simplified the period loss function relative to ILE by setting β = ν = 0.)

Let me simplify further by setting θ = 0 and consider “strict” inflation targeting,

L_{t} = \frac{1}{2} {(π_{t} - π^{*})}^{2},

leaving the case of “flexible” inflation targeting, θ > 0, as an extension. We note from (2.1) that π_t and π_t+1 are predetermined with respect to quarter t. By (2.3), the instrument i_t affects ${\tilde{u}}_{t + 1},$ , which, in turn, affects π_t+2 (and later inflation rates). Since there is no cost to instrument adjustment (since ν = 0), it is clear that i_t should be set so as to minimize

\begin{matrix} E_{t} Δ^{2} L_{t + 2} & (2.4) \end{matrix}

(since i_t+τ for τ ≥ 1 can be freely used to minimize E_t+τδ^τ+2L_t+τ+2).

Since ${\tilde{u}}_{t + 1}$ by (2.3) is linear in i_t we realize from (2.1) and (2.4) that the first-order condition for an optimum can be written

\begin{matrix} 0 = E_{t} [(π_{t + 2} - π^{*}) f^{'} ({\tilde{u}}_{t + 1})] = E_{t} {[π_{t + 1} + f ({\tilde{u}}_{t + 1}) + ε_{t + 2}^{π} - π^{*}] f^{'} ({\tilde{u}}_{t + 1})} = (π_{t + 2 | t} - π^{*}) E_{t} f^{'} ({\tilde{u}}_{t + 1}) + {C O v}_{t} [f ({\tilde{u}}_{t + 1}), f^{'} ({\tilde{u}}_{t + 1})], & (2.5) \end{matrix}

(where I have used that E_t[x, y] = E_t[x]E_t[y] + COV_t[x, y]). Furthermore, assuming that negligible probability mass falls in the interval ${\tilde{u}}_{t + 1} > γ / 2 ϕ,$ , we have

C o v_{t} [f ({\bar{u}}_{t + 1}), f^{'} ({\tilde{u}}_{t + 1})] = 2 ϕ {C o v}_{t} [f ({\tilde{u}}_{t + 1}), {\tilde{u}}_{t + 1}] .

Exploiting a theorem of Rubinstein (1976),¹ we have

{C o v}_{t} [f ({\tilde{u}}_{t + 1}), {\tilde{u}}_{t + 1}] = E_{t} f^{'} ({\tilde{u}}_{t + 1}) σ_{\bar{u}}^{2} .

Using this in (2.5), I get the first-order condition

\begin{matrix} π_{t + 2 | t} = π^{*} - 2 ϕ σ_{\bar{u}}^{2} . & (2.6) \end{matrix}

Thus, under strict inflation targeting with a convex Phillips curve, it is optimal to undershoot the inflation target, on average. Average inflation, the unconditional mean of inflation, will fulfill $\begin{matrix} E [π_{t}] = π^{*} - 2 ϕ σ_{\bar{u}}^{2} . \end{matrix}$

In order to determine the optimal setting of i_t, I need to solve (2.6) for $\begin{matrix} {\tilde{u}}_{t + 1 | t} \end{matrix} .$ We have

\begin{matrix} π_{t + 2 | t} \equiv π_{t + 1 | t} + E_{t} f ({\tilde{u}}_{t + 1}) . & (2.7) \end{matrix}

We note, in passing, that by taking the unconditional mean of (2.7), we have $\begin{matrix} E [f ({\bar{u}}_{t})] = 0. \end{matrix}$ Since $f ({\tilde{u}}_{t})$ is convex and f(0) = 0, it follows (as in ILE) that the average unemployment gap will be positive, $E [{\tilde{u}}_{t}] > 0.$ We can directly infer from (2.3) that the average real interest rate, r_t ≡ i_t—π_t+1|t, must exceed the natural real interest rate, $E [r_{t}] > \bar{r} .$

In order to solve for ${\tilde{u}}_{t + 1 | t},$ assume that the variance $σ_{\bar{u}}^{2}$ is sufficiently small to warrant the second-order Taylor approximation

\begin{matrix} E_{t} f ({\tilde{u}}_{t + 1}) = E_{t} f ({\tilde{u}}_{t + 1 | t} + ε_{t + 1}^{\bar{u}}) = f ({\bar{u}}_{t + 1 | t}) + \frac{1}{2} f^{''} ({\bar{u}}_{t + 1 | t}) σ_{\bar{u}}^{2} = f ({\tilde{u}}_{t + 1 | t}) + ϕ σ_{\bar{u}}^{2} . & (2.8) \end{matrix}

Combining (2.2) and (2.7)-(2.8) leads to a second-order equation for ${\tilde{u}}_{t + 1 | t} .$ The solution for the relevant root can be written

\begin{matrix} {\tilde{u}}_{t + 1 | t} = g (π_{t + 1 | t} - π^{*}), & (2.9) \end{matrix}

where

g (π_{t + 1 | t} - π^{*}) \equiv {\begin{matrix} \frac{y}{2 ϕ} (1 - \sqrt{1 - \frac{4 ϕ}{y^{2}} (π_{t + 1 | t} - π^{*} + 3 ϕ σ_{\bar{u}}^{2}}), ϕ > 0 \\ \frac{1}{y} (π_{t + 1 | t} - π^{*}), ϕ = 0 \end{matrix}}

Combining the expectation in quarter t of (2.3) with (2.9) results in the optimal reaction function,

i_{t} \begin{matrix} \equiv \bar{r} + π_{t + 1 | t} + \frac{1}{η_{r}} g (π_{t + 1 | t} - π^{*}) - \frac{η_{u}}{η_{r}} {\tilde{u}}_{t} & (2.10) \end{matrix}

\begin{matrix} \equiv \bar{r} + π_{t} + f ({\tilde{u}}_{t}) + \frac{1}{η_{r}} g (π_{t} - π^{*} + f ({\tilde{u}}_{t})) - \frac{η_{u}}{η_{r}} {\tilde{u}}_{t} . & (2.11) \end{matrix}

The reaction function can be expressed in terms of π_t+1|t and ${\tilde{u}}_{t}$ as in (2.10). Alternatively, since the predetermined π_t+1|t fulfills

\begin{matrix} π_{t + 1 | t} \equiv π_{t} + f ({\tilde{u}}_{t}) & (2.12) \end{matrix}

(recall that ${\tilde{u}}_{t}$ is observable, since I simplify by assuming that u^*_t is observable), it can be expressed in terms of π_t and ${\tilde{u}}_{t}$ as in (2.11).

3. Comparing Reaction Functions

Thus, under strict inflation targeting, the endogenous reaction function in the equivalent forms (2. 10) or (2. 11) will result. The reaction function (2.11) is nonlinear in π_t+1|t and linear in ${\tilde{u}}_{t} .$ The reaction function (2.11) is nonlinear in both π_t and ${\tilde{u}}_{t} .$ . We note that it is optimal to respond to ${\tilde{u}}_{t},$ even under strict inflation targeting with no weight on ${\tilde{u}}_{t}$ in the loss function, since, as emphasized in Svensson (1999a), it is generally optimal to respond to the determinants of the target variable (s) rather than just the target variable (s) themselves.

We can now compare these optimal reaction functions to a Taylor-type rule,

\begin{matrix} i_{t} = \bar{r} + π_{t + 4 | t}^{4} + w_{π} (π_{t} - π^{*}) - w_{u} {\tilde{u}}_{t}, & (3.1) \end{matrix}

and the two forecast-based instrument rules examined by ILE,

\begin{matrix} i_{t} = \bar{r} + π_{t + 4 | t}^{4} + w_{π} (π_{t}^{4} - π^{*}) - w_{u} {\tilde{u}}_{t} \end{matrix}

which is denoted IFB1 (inflation-forecast-based rule 1), and

\begin{matrix} i_{t} = \bar{r} + π_{t + 4 | t}^{4} + w_{π} (π_{t + 3 | t} - π^{*}) - w_{u} {\tilde{u}}_{t} \end{matrix}

which is denoted IFB2. Here $π_{t}^{4} = \frac{1}{4} \sum_{τ = 0}^{3} π_{t - τ}$ denotes 4-quarter inflation. Both IFB rules are somewhat simplified by the assumption that u^*_t and hence ${\tilde{u}}_{t}$ are observable.

We see that the Taylor-type rule, as a function of π₁ and ${\tilde{u}}_{t},$ is quite different from the optimal reaction function (2.11), since the former is linear in both arguments whereas the latter is nonlinear. Thus, it is quite intuitive that, with a nonlinear Phillips curve, the linear Taylor rule is inferior.

Furthermore, for IFB1, the term $π_{t + 4 | t}^{4} = \frac{1}{4} \sum_{τ = 0}^{3} π_{t + 4 - τ | t}$ enters. For IFB2, the term π_t+3|t also enters. Since the terms $π_{t + 4 - τ | t} = π_{t + 3 - τ | t} + E_{t} f ({\tilde{u}}_{t + 3 - τ | t})$ for τ = 0,...,3, are nonlinear functions of ${\tilde{u}}_{t},$ , this means that the instrument becomes a nonlinear function of ${\tilde{u}}_{t} .$ . The resulting nonlinear function is, of course, not equal to the optimal reaction function (2.10) or (2.11), so it will not be optimal. Still, it may be closer to the optimal reaction function than the backward-looking Taylor-type rule (3.1). This seems to be the reason why IFB1 and IFB2 perform better than the backward-looking linear rule.

Furthermore, note that IFB1 and IFB2 are not reaction functions that are functions of predetermined variables only. Instead, they make the instrument a function of an endogenous model-consistent inflation forecast, which depends on the instrument rule itself and requires the solution of the whole model to be determined. Therefore, IFB1 and IFB2 are actually examples of quite complex equilibrium conditions.² For this reason and others discussed in Svensson (1999a), I remain sceptical about their usefulness in practical monetary policy.

4. An Instrument Rule Involving an Optimal Response to an Inflation Forecast

Suppose, however, that we would insist on applying an instrument rule involving a response to an inflation forecast. What could we do within the present model? First, consider the two-period inflation forecast π_t+2|t as a function Π_t+2|t(i_t) of the instrument, i_t, and the state of the economy in periods t, π_t and ${\tilde{u}}_{t} .$ . This function is by (2.7) defined by

π_{t + 2 | t} = π_{t + 2 | t} (i_{t}) \equiv π_{t + 1 | t} + f [η_{u} {\bar{u}}_{t} + η_{r} (i_{t} - π_{t + 1 | t})] + ϕ σ_{\bar{u}}^{2},

where we recall that π_t+1|t is given by (2.12) and where I have used the approximation (2.8). Now, we can, of course, consider a first-order Taylor approximation to this forecast around the interest rate i_t-1 in the previous quarter,

π_{t + 2 | t} = Π_{t + 2 | t} (i_{t - 1}) + \frac{\partial Π_{t + 2 | t} (i_{t} - 1)}{\partial i} Δ i_{t},

where Δi_t ≡ i_t—i_t-1. Combining this with the first-order condition (2.6) and solving for Δi_t leads to

Δ i_{t} = \frac{1}{- \frac{\partial _{t + 2 | t} (i_{t} - 1)}{\partial i}} [Π_{t + 2 | t} (i_{t} - 1) - (π^{*} - 2 ϕ σ_{\bar{u}}^{2})] . (4.2)

Here, we get an optimal instrument rule, which says that the optimal adjustment of the interest rate, Δi_t, should be proportional to the deviation between the unchanged–interest- rate inflation forecasts Π_t+2|t(i_t-1) and the adjusted inflation target, $π^{*} - 2 ϕ σ_{\bar{u}}^{2} .$ Furthermore, the response coefficient is given by

\frac{1}{- \frac{\partial Π_{t + 2} (i_{t} - 1)}{\partial i}} = \frac{1}{- η_{r} f^{'} [η_{u} {\tilde{u}}_{t} + η_{r} (i_{t - 1} - π_{t + 1 | t} - \bar{r})]}

\frac{1}{η_{r} {y - 2 ϕ [η_{u} \tilde{u_{t}} + η_{r} (i_{t - 1} - π_{t + 1 | t} - \bar{r})]}} .

Several comments are in order. First, the inflation forecast is not the model-consistent inflation forecast for the endogenous interest rate but the unchanged-interest-rate forecast (that is, for i_t = i_t-1). Second, the instrument rule involves the change in the interest rate, not the level. Third, an adjusted inflation target should be applied (when φ > 0 and the Phillips curve is nonlinear). Fourth, the response coefficient is not constant but state-dependent (when φ > 0). The response coefficient is the reciprocal of the slope of the Phillips curve for an unchanged-interest-rate unemployment-gap forecast for period t + 1, given by $n_{u} \tilde{u_{t}} + n_{r} (i_{t - 1} - π_{t + 1 | t}) .$ Finally, even if the response coefficient is statedependent, the instrument rule is only an approximation, since it follows from a first-order approximation of a nonlinear function.

Clearly, the optimal instrument rule (4.2) is, in several respects, quite different from the IFB rules discussed by ILE. In a linear model (when φ = 0), the problem of the adjusted inflation target and the state-dependent response coefficient would disappear. Then, the optimal instrument adjustment is proportional to the deviation between an unchanged–interest- rate inflation forecast and the inflation target. Thus, it seems more intuitive that any response should be to the unchanged-interest-rate forecast than to a model-consistent forecast.

Notes

The theorem says that, if x and 3; are bivariate normal, under some mild regularity conditions,

C o v [f (x), y] = E [f^{'} (x)] C o v [x, y] .

Rudebusch and Svensson (1999, Appendix) demonstrate the complexity of this instrument rule.

References

Rubinstein, Mark. (1976). “The Valuation of Uncertain Income Streams and the Pricing of Options.” Bell Journal of Economics 7, 407–425.
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Rudebusch, Glenn D., and Lars E. O. Svensson. (1999). “Policy Rules for Inflation Targeting.” in Taylor (1999), forthcoming.
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Schaling, Eric. (1999). “The Nonlinear Phillips Curve and Inflation Forecast Targeting—Symmetric versus Asymmetric Monetary Policy Rules.” Working Paper.
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Svensson, Lars E. O. (1999a). “Inflation Targeting as a Monetary Policy Rule.” Journal of Monetary Economics 43, forthcoming.
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Svensson, Lars E. O. and (1999b) “Price Stability as a Target for Monetary Policy: Defining and Maintaining Price Stability.” Presented at Deutsche Bundesbank’s conference on The Monetary Transmission Process: Recent Developments and Lessons for Europe, March 26-27, 1999. http://www.iies.su.se/leosven/.
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Taylor, John B. (ed.). (1999). Chicago University Press, forthcoming. pp. 31–32.
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Comment

BY BENNETT T. MCCALLUM

Participating in this affair has helped me to recognize that my own concern for robustness in policy rule design was in part due to Bob Flood’s influence. When we were together at Virginia, I would get excited about a particular analytical policy result. But every few weeks Bob would say to me: “Well, yes, Ben, but what if you make the following small change in the model?” And, of course, doing so would overturn my result entirely. So I gradually came to understand that it was not so exciting.

Peter Isard gave me the assignment of doing whatever I wanted with regard to the two papers in this session. Since I was having enough trouble in getting my own paper pulled together, I took the easy way out and decided to discuss only one—the Isard, Laxton, Eliasson paper—with apologies to Henderson and Kim.

Their paper is really a very ambitious and rather complex one. I am extremely supportive of their general approach, and of the aspect of their approach that treats the NAIRU as nonobservable and that treats output as observable only with a one-quarter lag. I have been arguing in favor of these kinds of information assumptions for several years, and more generally for the strategy of looking for simple policy rules that are reasonably effective in a variety of models, rather than looking for the “optimal policy” in some particular model. So, to repeat, I am entirely supportive of their general approach.

When I say that the paper is complex, what I have in mind is that there are several features of the model that are “nonstandard.” Most prominent among these is the nonlinearity in the Phillips curve relation. Two others that I have noted are that expectational behavior is not straightforward rational expectations, but sort of a hybrid, and that the IS function, which they call the unemployment rate equation, is entirely backward looking. With respect to this latter function I have to observe, parenthetically, that this specification is not very nice theoretically, and even so doesn’t work very well empirically. The largest t-ratio on the three lagged real interest rates in their Table 3 is 1.8. This is somewhat important, because the effects of monetary policy on both real and nominal variables in their model is transmitted entirely via these three parameters. If they were zeros, the model would just fall apart and policy wouldn’t do anything to anything.

Anyhow, to get back to the main argument, it’s quite notable that in their model the original Taylor rule (which responds to lagged variables) performs very badly indeed, leading to explosive behavior. Also performing very badly are variants utilized by Clarida, Gali, and Gertler (1997) and by Levin, Wieland, and Williams (1998). This finding that is presented in the paper contrasts very sharply with results reported by several authors at the NBER Conference organized by John Taylor and held exactly a year ago in Florida. These results are summarized in a recent paper by Taylor (1999), which argues that Taylor-style rules performed very well in a robust sense. So this finding is quite significant, I would say.

My main reservation about the paper is that it is not clear to me which of the nonstandard features are responsible for the very different behavior in their model and in the bundle of models presented by several authors at the NBER conference. Their exposition emphasizes the nonlinearity in the Phillips curves, but it is not entirely obvious that it is not another one of the nonstandard features that is crucial. Maybe that information is present in the paper, but I couldn’t find it. So I would like to see results that shut down the nonlinearity and keep all of the other nonstandard features. And, I would also be interested in seeing how the results, especially those in the Appendix, would fare if nominal income growth were used as the target variable instead of inflation and the output gap—perhaps an expected value of nominal income growth. I don’t really want to suggest that they report more results, but possibly somewhat different ones that make for a slightly clearer comparison.

I’ll conclude by noting that the results in their Appendix are quite closely related to issues discussed in the paper that I prepared for this Festschrift—issues concerning regions of instability, unique solutions, and indeterminacy in these models. In that regard, I was surprised to see the left hand regions in their Figure B1 labeled as indeterminacy. I would have thought one would get explosions there from Taylor’s discussion and from Peter’s discussion. But additional investigation indicates that their result is correct, subject to considerations raised in Section II of my paper.

Anyhow, none of these comments represents any major complaint. Basically, I think this is an extremely interesting and valuable paper.

References

Clarida, R., and J. Gali, and M. Gertler. (1998). “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory.” NBER Working Paper No. 6442, March.
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Levin, A., V. Wieland, and J. Williams. (1999). “Robustness of Simple Monetary Policy Rules Under Model Uncertainty,” in J. B. Taylor (ed.), Monetary Policy Rules. Chicago: University of Chicago Press.
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Taylor, J. B. (1999). “The Robustness and Efficiency of Monetary Policy Rules as Guidelines for Interest Rate Setting by the European Central Bank.” Journal of Monetary Economics 43, forthcoming.
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General Discussion

Charles Engel contrasted the Henderson-Kim paper with work that he had done with Michael Devereux, noting that he didn’t understand how Henderson and Kim had treated the monopolistic inefficiency or their rationale for doing so. It wasn’t clear to him what the Pigouvian taxes and subsidies ought to be. This was essentially a Jensen’s inequality issue. Prices could be set to get the level of output to the competitive level, but this wouldn’t get consumption or the expected utility of consumption to the competitive level. He and Devereux had chosen to leave the monopolistic inefficiencies in their model when evaluating policy rules, and he wasn’t clear about the role that inefficiencies played in the Henderson-Kim analysis.

Robert Hodrick pointed out that it was difficult to explain inflation inertia in models in which price-setting behavior was characterized in terms of one-period-ahead contracts. This seemed to be a problem for both the Henderson-Kim and Engel papers, since considerable inflation inertia was clearly present in the data, as revealed, for example, in the estimation results presented in the Isard-Laxton-Eliasson paper.

Kim and Henderson responded to several questions that Gray had raised. In justifying the hassles of constructing a model that could be used to make exact utility calculations, Kim cited another paper in which reliance on approximate calculations had generated an incorrect ordering of the utilities associated with having complete and incomplete markets. In explaining what they felt was gained by introducing monopolistic competition among firms and households, they expressed the view that it was important to be able to rationalize price- and wage-setting behavior, but that the introduction of monopolistic competition didn’t have major effects on the conclusions from their analysis.

In responding to Engel’s request for clarification, Henderson explained that they had introduced subsidies that were not time varying, and that served to remove the monopolistic competition distortions for agents who made decisions under full current information. This essentially eliminated the distortions from the first-order conditions of price- and wagesetters who had full current information. Whether it completely eliminated the effects of the distortions from the general equilibrium was a deeper question, given that some agents had to set prices and wages without full current information. Henderson seemed to feel that by knocking the markup parameter out of all the equilibrium conditions, the taxes and subsidies did what they were intended to do. Assaf Razin suggested that this would be done perfectly if the taxes and subsidies were shock specific.

In responding to Hodrick, Henderson agreed with the empirical evidence on the importance of inflation inertia and indicated that he and Kim wouldn’t be taking their model to the data any time soon. Henderson also applauded Svensson’s approach of optimizing within classes of rules, indicating that despite the model specificity of the results, this approach can sometimes be very helpful in clarifying thinking, for example, about why certain ranges of parameter values are good or bad.

Laxton responded to McCallum’s request for clarification on why the stabilizing properties of various classes of rules were so much different in the Isard-Laxton-Eliasson (ILE) model than in other models in which they had been evaluated. He emphasized that the other models tended to be linear with a lot of inflation persistence and very small effects of the output gap on the inflation process. Under these assumptions, bad policy rules could generate relatively high variances of output and inflation but did not lead to explosive behavior. This was illustrated by the first chart in Appendix II of the ILE paper, which focused on the stabilizing properties of a Taylor rule in the Fuhrer-Moore model, where the rule had been generalized to allow for interest rate smoothing. Laxton emphasized that the region of stability, defined in terms of the long-run responses of the interest rate to output gaps and inflation, essentially included all rule calibrations in which the long-run response of the nominal interest rate to inflation was greater than one. Moreover, this region of stability was completely unaffected by the lags in the response of monetary policy: even with extremely slow responses of monetary policy to inflation and the output gap, stability under a generalized Taylor rule essentially depended only on having a positive long-run response of the real interest rate to an excess demand shock.

Laxton noted that work at the Federal Reserve by Andrew Levin, Volker Wieland, and John Williams (LWW) had looked at several different linear rational-expectations models and found that a first-difference rule was superior to the Taylor rule. The ILE analysis had found that explosive behavior was hard to generate in the Fuhrer-Moore model, even under the extremely myopic policy behavior that resulted when the LWW rule was combined with an assumption that inflation expectations could be described by a three-year or six-year moving average of past inflation. By contrast, in the nonlinear ILE model, if excess demand in the economy became quite high and policy was responding to a backward-looking measure of inflation, while inflation expectations were forward looking and conditioned (in the simulation analysis) on the implicit assumption that the monetary authority was committed to its backward-looking rule, a situation could develop in which the nominal interest rate did not move sufficiently to make the real interest rate rise in response to the excess demand. This would be a situation in which the Phillips curve was shifting and monetary policy was always behind shifts in the curve. In response to a follow-up question from McCallum, Laxton noted that while the stability analysis reported in Appendix II was based on the assumption that inflation expectations were only partly model-consistent, the performances of Taylor rules and LWW rules in the ILE model became even more unstable when full weight was placed on the model-consistent expectations term.

A final set of questions from the audience was posed by Michael Dooley, who wondered who would actually want to use a simple policy rule and for what purpose. Why would central banks want to rely on simple rules when they could base their policy decisions on sophisticated models? And what was the logic of using simple rules as a communication device; how could credibility be increased if central banks explained their actions in terms of simple rules when they were actually basing policy on more sophisticated analysis?

Several of the authors and discussants responded. Henderson started by noting that there has been a long tradition of asking whether optimal policy can be closely approximated by a simple rule. He also emphasized that there is considerable disagreement in policy circles about how to describe the policy process to the public; while some people believe the public is quite sophisticated and can handle complicated descriptions, others recommend simple descriptions. In addition, he pointed to the fact that the Federal Reserve staff receives ongoing requests from the Board of Governors to analyze how simple monetary policy rules would guide the economy and to explain why the economy would behave in this way or that under these rules.

In a separate response to Dooley’s questions, Svensson suggested that the targeting rule approach provided an answer, assuming that the authorities could clearly define the objectives of policy. The inflation targeting framework was designed to be explicit about the policy loss function and to have simple procedures so that the public could monitor whether the central bank was actually optimizing that particular loss function. McCallum disagreed with Svensson on the importance of a targeting rule approach and the associated emphasis on communication. Citing the Bundesbank as an example, he argued that a central bank’s credibility depended on what it did, not on what it said.

Estimated equations: 1/
	$π_{t} = \underset{(9.39)}{\begin{matrix} 3.20 (u_{t}^{*} - u_{t}) / (u_{t} - Φ_{t}) \end{matrix}} + \underset{(6.97)}{0.41 {\bar{π}}_{t}^{e}} + (1 - 0.41) π_{t - 1}$
	$π_{t}^{x} = \underset{(9.39)}{\begin{matrix} 3.20 (u_{t}^{*} - u_{t}) / (u_{t} - Φ_{t}) \end{matrix}} + \underset{(9.59)}{0.64 {\bar{π}}_{t}^{e}} + (1 - 0.64) π_{t - 1}^{x}$
	${\bar{u}}_{t} = \underset{(0.30)}{0.20} + u_{t}^{*}$
	$u_{t}^{} = u_{t - 1}^{} + ε_{t}^{u^{*}}$
Variables:
π_t	Percent change in consumer price index, quarterly at annual rate.
π_t^x	Percent change in CPI excluding food and energy.
${\bar{π}}_{t}^{e}$	Expected inflation rate based on equation (3) with N = 12.
φ_t	Time-varying “wall parameter” see text.
u_t^*	Unobserved DNAIRU as estimated from the Kalman filter.
${\bar{u}}_{t}$	Estimated NAIRU.

Estimated equations: 1/
	$π_{t} = \underset{(9.39)}{\begin{matrix} 3.20 (u_{t}^{*} - u_{t}) / (u_{t} - Φ_{t}) \end{matrix}} + \underset{(6.97)}{0.41 {\bar{π}}_{t}^{e}} + (1 - 0.41) π_{t - 1}$
	$π_{t}^{x} = \underset{(9.39)}{\begin{matrix} 3.20 (u_{t}^{*} - u_{t}) / (u_{t} - Φ_{t}) \end{matrix}} + \underset{(9.59)}{0.64 {\bar{π}}_{t}^{e}} + (1 - 0.64) π_{t - 1}^{x}$
	${\bar{u}}_{t} = \underset{(0.30)}{0.20} + u_{t}^{*}$
	$u_{t}^{} = u_{t - 1}^{} + ε_{t}^{u^{*}}$
Variables:
π_t	Percent change in consumer price index, quarterly at annual rate.
π_t^x	Percent change in CPI excluding food and energy.
${\bar{π}}_{t}^{e}$	Expected inflation rate based on equation (3) with N = 12.
φ_t	Time-varying “wall parameter” see text.
u_t^*	Unobserved DNAIRU as estimated from the Kalman filter.
${\bar{u}}_{t}$	Estimated NAIRU.

Unemployment rate equation
$u_{t} = c_{t} + \sum_{i = 1}^{2} η_{i} u_{t - i} + \sum_{i = 1}^{3} Φ_{i} r_{t - i} + ε_{t}^{u}$
$c_{t} = c_{t - 1} + ε_{t}^{c}$
Real federal funds rate
$r_{t} = r s_{t} - E_{t} {π 4}_{t} + 4$
u_t unemployment rate
rs_t federal funds rate
E_tπ 4_t+4 expected inflation over the next year from the Michigan survey
c_t time-varying parameter estimated using Kalmanfiltering methods
Estimation period: 1968:Q1 to 1998:Q2
Coefficient	Estimate	t-ratio
η₁	1.027	11.6
η₂	-0.258	2.9
φ₁	0.010	0.6
φ₂	0.034	1.8
φ₃	0.023	1.2
Residual variance	0.0285
Log likelihood	-28.96

Unemployment rate equation
$u_{t} = c_{t} + \sum_{i = 1}^{2} η_{i} u_{t - i} + \sum_{i = 1}^{3} Φ_{i} r_{t - i} + ε_{t}^{u}$
$c_{t} = c_{t - 1} + ε_{t}^{c}$
Real federal funds rate
$r_{t} = r s_{t} - E_{t} {π 4}_{t} + 4$
u_t unemployment rate
rs_t federal funds rate
E_tπ 4_t+4 expected inflation over the next year from the Michigan survey
c_t time-varying parameter estimated using Kalmanfiltering methods
Estimation period: 1968:Q1 to 1998:Q2
Coefficient	Estimate	t-ratio
η₁	1.027	11.6
η₂	-0.258	2.9
φ₁	0.010	0.6
φ₂	0.034	1.8
φ₃	0.023	1.2
Residual variance	0.0285
Log likelihood	-28.96

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Abstract

Abstract

1. Introduction

2. The Model

2.1. Firms

2.2. Households

2.3. Government

3. Flexible Wages and Prices

3.1. Solution

3.2. Discussion

4. Wage Contracts and Flexible Prices

4.1. Solution

4.2. Expected Loss

4.3. Optimal Policy

4.4. Output Gap Stabilization

4.5. Nominal Income Stabilization and Related Hybrid Rules

4.6. Price Level Stabilization

4.7. Output Stabilization

4.8. Money Supply Stabilization

5. Wage and Price Contracts

5.1. Solution

5.2. Optimal Policy and Output Gap Stabilization

5.3. Simple Policy Rules

6. Conclusions

Appendix A

Appendix B

Acknowledgments

Simple Monetary Policy Rules Under Model Uncertainty

Abstract

I. Introduction and Overview

II. A Model of the Unemployment-Inflation Process

A. The Short-Run Phillips Curves

B. The Dynamics of Inflation Expectations

C. The Dynamics of the Unemployment Rate

III. The Policy Rules and Stochastic Simulation Framework

A. The Simple Policy Rules

B. The Policy Objective Function

C. The Monetary Authority’s Behavior

D. The Stochastic Simulation Experiments

IV. Simulation Results

A. Optimal Calibrations oflFBl Rules

B. Perspectives on Several of the Simple Rules Proposed in the Literature

V. Conclusions

Appendix I. Optimal Forecasting Rules for Bounded Random Walks

Appendix II. Stability Analysis of Taylor Rules and LWW Rules

A. Conventional Taylor Rules Generalized for Interest Rate Smoothing

B. LWW Interest Rate Change Rules

Acknowledgments

References

Notes

References

1. Instrument Rules and Targeting Rules

2. A Simple Model with a Nonlinear Phillips Curve

3. Comparing Reaction Functions

4. An Instrument Rule Involving an Optimal Response to an Inflation Forecast

Notes

References

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