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

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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

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.

\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

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

Topics

Countries

Series

About

Resources

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