Assessing Dsge Models with Capital Accumulation and Indeterminacy

Vadim Khramov

doi:10.5089/9781475502350.001.A001

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Assessing Dsge Models with Capital Accumulation and Indeterminacy

Author: Mr. Vadim Khramov

Contributor Notes

Author’s E-Mail Address: vkhramov@imf.org

Publication Date:: 01 Mar 2012

eISBN:: 9781475502350

Language:: English

Keywords:: WP; monetary policy; interest rate; monetary policy authority; inflation dynamics; New Keynesian model; estimate monetary policy; steady-state inflation; monetary policy influence

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The simulated results of this paper show that New Keynesian DSGE models with capital accumulation can generate substantial persistencies in the dynamics of the main economic variables, due to the stock nature of capital. Empirical estimates on U.S. data from 1960:I to 2008:I show the response of monetary policy to inflation was almost twice lower than traditionally considered, as capital accumulation creates an additional channel of influence through real interest rates in the production sector. Versions of the model with indeterminacy empirically outperform determinate versions. This paper allows for the reconsideration of previous findings and has significant monetary policy implications.

Abstract

Introduction

Dynamic stochastic general equilibrium (DSGE) models with nominal rigidities (“New Keynesian” models) became very popular in the analysis of monetary policy in the last decades. Canonical versions of these models are well-studied and estimated. As these models are micro-founded, their estimation using the Bayesian approach, which allows us to specify prior distributions for parameters, became very popular. As multiple equilibria can arise in these models under a wide set of parameters, only is a limited set of econometric estimation methods can be applied.

Traditionally, papers estimated canonical models only under determinacy, ruling out possibilities for indeterminacy. The standard result is that determinacy arises mainly under active monetary policy rules (the nominal interest rate increases more than one-to-one with inflation) in New Keynesian DSGE models, while passive monetary policy (the nominal interest rate increases less than one-to-one with inflation) leads to indeterminacy (see Kerr and King (1996), Rotemberg and Woodford (1998), and Christiano and Gust (1999)). Clarida, Gali and Gertler (2000) estimate monetary policy rules in the U.S. for different periods of time and show that U.S. monetary policy was passive during the pre-Volcker period (1960-1979) and active into the Volcker-Greenspan period (1979-1996). Lubik and Schorfheide (2004) were the first to estimate a standard New Keynesian model allowing for indeterminacy.

In contrast to canonical New Keynesian models, where the interest rate affects output only through the consumption and saving decisions of households, in this paper the investment decisions of firms are added. The no-arbitrage condition between the real return on bonds and the real return on capital implies that the capital rental rate increases when monetary policy responds to higher inflation by increasing the interest rate. This response increases the cost of renting capital, leading to cost-push inflation. Dupor (2001), Carlstrom and Fuerst (2005), Kurozumi and Zandweghe (2008), Kurozumi (2006), Huang and Meng (2007), and Xiao (2008) study properties of models with capital accumulation and show that this crucial feature changes the stability structure and dynamics of the models and makes indeterminacy likely.

This paper studies New Keynesian DSGE models with capital accumulation, different Taylor rules, and the potential for indeterminacy. While the stability properties of models with capital accumulation are well-analyzed¹, there appear to be no papers that simulate and estimate these models allowing indeterminacy to occur. This paper is offered to fill this gap. The analysis is distinguished from the conventional New Keynesian studies in three ways. First, different versions of New Keynesian models with capital accumulation are simulated, and their dynamic properties are discussed. Second, the models are estimated on U.S. data from 1960:I to 2008:I. Following Lubik and Schorfheide (2004), passive monetary policy (consistent with indeterminacy) is assumed a priori for the pre-Volcker period and active monetary policy (consistent with determinacy) for the post-1982 period. Using state-space decomposition and the Kalman filter, the overall likelihood of the model is maximized taking into account prior distributions of the parameters, and inferences are made with a likelihood-based approach by adopting the Metropolis-Hastings techniques. Third, the estimated models are compared using the Bayesian approach. While some explanations of the results are consistent with the recent findings of Mavroeidis (2010), the empirical estimates of this paper differ from the results of Lubik and Schorfheide (2004) and Clarida, Gali and Gertler (2000).

Though baseline New Keynesian models have become very popular in the analysis of monetary policy, many authors show that these models are unable to generate enough persistence in inflation and output. Fuhrer and Moore (1995) show that a sticky wage model can generate persistence in the price level but not in the inflation rate. Chari, Kehoe and McGrattan (2000) point out that models with nominal rigidities do not generate enough persistence in output following a monetary shock. The simulated results of this paper show that models with capital accumulation can generate substantial persistencies among the major economic variables, as the stock nature of capital adds persistency to the dynamics of all other variables in the models.

Later research attempts to determine whether there were switches in monetary policy or structural changes in the fundamental parameters of the economy. Sims and Zha (2006) use a vector autoregression (VAR) approach to estimate a multivariate regime-switching model for U.S. monetary policy. They find that the main changes were in the monetary policy rules. Smets and Wouters (2003) and Smets and Wouters (2007) find most of the structural parameters are stable over those two periods. The biggest difference concerns the variances of the structural shocks. The main drawback of the VAR approach is that, due to rational expectations, agents can anticipate changes in parameters of the economy, leading to inconsistent estimates. One method to overcome this problem is to estimate a fully-specified DSGE model that can be re-solved for alternative policy rules. Lubik and Schorfheide (2004) exogenously split data into two sets and show that U.S. monetary policy during the post-1982 period was consistent with determinacy, whereas during the pre-Volcker policy was not. Schorfheide (2005) estimates a basic New Keynesian monetary DSGE model, in which monetary policy follows a regime switching process, and confirms the switch in monetary policy between the pre-Volcker and post-Volker periods.

There are identification issues with estimation of forward-looking Markov-switching rational expectations models. Beyer and Farmer (2007) argue that it is not always possible to decide whether the data are generated from determinate or indeterminate models. Farmer, Waggoner and Zha (2008) provide a set of necessary and sufficient conditions for determinacy in a class of forward looking Markov-switching rational expectations models. Mavroeidis’s (2010) model shows that policy before Volcker led to indeterminacy, however, the model is not accurately identifiable using data after 1979.

The paper is structured as follows. In Section 1, a New Keynesian DSGE model with capital accumulation and different monetary policy rules is derived. In Section 2, different versions of this model are simulated and their dynamic properties are analyzed. In Section 3, the model is fitted to quarterly U.S. data on output, inflation, nominal interest rates, consumption, and capital from 1960:I to 2008:I and the estimation methodology and prior distributions of the parameters are discussed. The empirical results are presented in Section 4. Estimated models are compared in Section 5. The last section contains concluding remarks.

1. Model

Following Yun (1998), Carlstrom and Fuerst (2005), and Kurozumi and Zandweghe (2008), a New Keynesian DSGE model with sticky prices and capital accumulation in discrete time is constructed. The economy consists of a large number of households, monopolistically competitive firms, and a monetary authority that changes the nominal interest rate in response to inflation and output.

1.1. Households

Households seek to maximize their expected life-time utility function:

\begin{matrix} (1) & E_{t} \end{matrix} Σ_{t = 0}^{\infty} β^{t} U (C_{t}, \frac{M_{t + 1}}{P_{t}}, 1 - L_{t})

where E_t is the conditional expectations operator on the information set available at date t, β is the discount factor, C_t is consumption, M_t+1 is nominal money holdings and the beginning of the period (t+1), P_t is a price level, $\frac{M_{t + 1}}{P_{t}}$ is real money balances², and (1 - L_t) is leisure.

The utility function is separable in leisure and takes the following functional form:

(2) U (C_{t}, \frac{M_{t + 1}}{P_{t}}, 1 - L_{t}) = \frac{{C_{t}}^{1 - σ}}{1 - σ} + θ \ln \frac{M_{t + 1}}{P_{t}} + Φ L_{t}

At the beginning of each period t, a household has M_t cash balances and B_t-1 nominal bonds. A household starts period t by trading bonds and receiving a lump-sum monetary transfer T_t from the government. A household receives interest payments on bonds B_t-1 with gross interest rate R_t-1 and spends money on new bonds B_t. A household also receives real factor payments from the labor market w_tL_t and capital market [r_t + (1 - δ)]K_t, receives firm’s profits Π_t, and spends money on next period capital K_t+1 and current consumption C_t at current prices P_t. Each household chooses C_t, M_t+1, and L_t to maximize (1) subject to the sequence of intertemporal budget constraints:

\begin{matrix} (3) & M_{t + 1} \end{matrix} + B_{t} + P_{t} C_{t} + P_{t} K_{t + 1} = M_{t} + T_{t} + B_{t - 1} R_{t - 1} + P_{t} {w_{t} L_{t} + [r_{t} + (1 - δ)] K_{t}} + Π_{t}

The first order conditions for the household’s maximization problem are the following:

\begin{matrix} (4) & \frac{U_{C}}{U_{L}} \end{matrix} = - \frac{1}{w}

\begin{matrix} (5) & U_{C} \end{matrix} (t) = β E_{t} {\begin{matrix} U_{C} (t + 1) \end{matrix} [r_{t + 1} + (1 - δ)]}

\begin{matrix} (6) & \frac{U_{C} (t)}{P_{t}} \end{matrix} = β R_{t} E_{t} (\begin{matrix} \frac{U_{C} (t + 1)}{P_{t + 1}} \end{matrix})

\begin{matrix} (7) & \frac{U_{m} (t)}{U_{C} (t)} \end{matrix} = \frac{R_{t} - 1}{R_{t}}

Equation (4) is a standard consumption-labor condition. Equation (5) is the Euler equation of consumption dynamics. Equation (6) is the Fisher equation that connects inflation and interest rates. Equation (7) is a money demand equation.

1.2. Firms

Firms are monopolistic competitors in the intermediate good market. The final output Y_t is produced from intermediate goods y_t (i) with Dixit-Stiglitz (1977) technology:

\begin{matrix} (8) & Y_{t} \end{matrix} = {\int_{0}^{1} [y_{t} {(i)}^{\frac{η - 1}{η}}] ⅆ i}^{\frac{η}{η - 1}}

The corresponding demand for an intermediate good i possesses constant price elasticityη:

\begin{matrix} (9) & {\begin{matrix} y \end{matrix}}_{t} \end{matrix} (i) = \begin{matrix} Y_{t} \end{matrix} {(\frac{\begin{matrix} P_{t} (i) \end{matrix}}{P_{t}})}^{- η}

where P_t (i) is the price of the intermediate good and P_t is the price of the final good.

The production function of each firm exhibits constant returns to scale:

\begin{matrix} (10) & f (K, L) \end{matrix} = K^{α} L^{1 - α}

The first order conditions for the cost minimization problem are the following:

\begin{matrix} (11) & r_{t} = z_{t} f_{K} (K_{t}, L_{t}) \end{matrix}

\begin{matrix} (12) & w_{t} = z_{t} f_{L} (K_{t}, L_{t}) \end{matrix}

where z_t is the marginal cost of production (see Appendix 1 for details).

With the Cobb-Douglas production function (10) the first order conditions take the form:

\begin{matrix} (13) & r_{t} \end{matrix} = α z_{t} Y_{t} / K_{t}

\begin{matrix} (14) & w_{t} \end{matrix} = (1 - α) z_{t} {(Y_{t} / K_{t})}^{\frac{- α}{(1 - α)}}

The Calvo (1983) staggered pricing model is used, assuming that each period a fraction (1-ν) of firms gets a signal to set a new price. Therefore, each firm maximizes the sum of discounted profits taking into account the probability of changing its price. The optimization problem of a firm takes the form:

\begin{matrix} (15) & E_{t} \end{matrix} Σ_{j = t}^{\infty} {(\frac{ν}{Π_{i = 0}^{j} R_{i}})}^{j} [{(\frac{P_{t} (i)}{P_{t}})}^{- η} Y_{t} (\frac{P_{t} (i)}{P_{t}} - z_{t})] \to \max_{P_{t} (i)}

The profit maximization conditions give a log-linearized New Keynesian Phillips Curve³ of the form:

\begin{matrix} (16) & {\begin{matrix} \hat{π} \end{matrix}}_{t} \end{matrix} = β E_{t} \begin{matrix} {\begin{matrix} \hat{π} \end{matrix}}_{t + 1} \end{matrix} + λ \begin{matrix} {\begin{matrix} \hat{z} \end{matrix}}_{t} \end{matrix}

where ${\hat{π}}_{t}$ is inflation and $λ = \frac{(1 - ν) (1 - β ν)}{ν}$ is the real marginal cost elasticity of inflation.

1.3. Monetary policy rules

Monetary policy reacts to inflation and output with interest rate smoothing:

\begin{matrix} (17) & R_{t} \end{matrix} = {(\begin{matrix} R_{t - 1} \end{matrix})}^{ρ_{R}} {[R {(\frac{\begin{matrix} E_{t} π_{t + k} \end{matrix}}{π})}^{Ψ_{π}} {(\frac{Y_{t}}{Y})}^{Ψ_{π}}]}^{(1 - ρ_{R})}

where R,π, Y are the steady-state values of the interest rate, inflation, and output, respectively. Parameters ψ_π and ψ_Y are the elasticities of the interest rate with respect to inflation and output, respectively. Two basic specifications of the monetary policy rule are considered: with response to current inflation (k=0) and future expected inflation (k=1). Interest rate smoothing is introduced with the autocorrelation coefficient ρ_R. In this framework, the monetary policy is active if the nominal interest rate increases more than one-to-one with inflation (ψ_π>1), otherwise, it is passive (ψ_π< 1). Also, a model with ψ_Y=0 would give a standard model with capital in discrete time as in Carlstrom and Fuerst (2005)—an analog of the Dupor (2001) continuous time model.

1.4. Dynamics of the model

The dynamics of the model are represented by a system of first order conditions log-linearized around the steady state for households and firms (18-23), the monetary policy rule (24), shocks of preferences and marginal cost (25-26) (see Appendix 2 for details):

\begin{matrix} (18) & {\begin{matrix} \hat{R} \end{matrix}}_{t} \end{matrix} - E_{t} {\begin{matrix} \hat{π} \end{matrix}}_{t + 1} = σ (E_{t} {\begin{matrix} \hat{C} \end{matrix}}_{t + 1} - {\begin{matrix} \hat{C} \end{matrix}}_{t} - ɛ_{g, t})

\begin{matrix} (19) & {\begin{matrix} \hat{R} \end{matrix}}_{t} \end{matrix} - E_{t} {\begin{matrix} \hat{π} \end{matrix}}_{t + 1} = [1 - β (1 - δ)] (E_{t} {\begin{matrix} \hat{z} \end{matrix}}_{t + 1} + E_{t} {\begin{matrix} \hat{Y} \end{matrix}}_{t + 1} - {\begin{matrix} \hat{K} \end{matrix}}_{t + 1})

(20) σ {\hat{C}}_{t} = {\hat{z}}_{t} + \frac{α}{1 - α} ({\hat{K}}_{t} - {\hat{Y}}_{t})

\begin{matrix} (21) & {\hat{K}}_{t + 1} \end{matrix} = (1 - δ) {\hat{K}}_{t} + δ {\hat{I}}_{t}

\begin{matrix} (22) & {\hat{Y}}_{t} \end{matrix} = s_{C} \begin{matrix} {\hat{C}}_{t} \end{matrix} + s_{I} \begin{matrix} {\hat{I}}_{t} \end{matrix}

\begin{matrix} (23) & {\hat{π}}_{t} \end{matrix} = β E_{t} {\hat{π}}_{t + 1} + λ {\hat{z}}_{t} + ɛ_{z, t}

\begin{matrix} (24) & {\hat{R}}_{t} \end{matrix} = ρ_{R} \begin{matrix} {\hat{R}}_{t - 1} \end{matrix} + (1 - ρ_{R}) (Ψ_{π} E_{t} \begin{matrix} {\hat{π}}_{t + k} \end{matrix} + Ψ_{Y} \begin{matrix} {\hat{Y}}_{t} \end{matrix}) + ɛ_{R, t}

\begin{matrix} (25) & ɛ_{g, t} \end{matrix} = ρ_{g} \begin{matrix} ɛ_{g, t - 1} \end{matrix} + ν_{g, t}, ν_{g, t} is iid (0, σ_{g}^{2})

\begin{matrix} (26) & ɛ_{z, t} \end{matrix} = ρ_{z} \begin{matrix} ɛ_{z, t - 1} \end{matrix} + ν_{z, t}, ν_{z, t} is iid (0, σ_{z}^{2})

Equation (18) is the Euler equation for the household’s dynamic optimization problem with a preference shock ε_{g, t}, which follows an AR(1) process with an autocorrelation coefficient of ρ_g (Equation 25). Equation (19) is the Fisher relation between the nominal interest rate, expected future inflation, and real interest rate, where the latter is determined in the production sector. Equation (20) is the wage-equilibrium relation of the log-linearized equations (4) and (14). Equation (21) is the capital accumulation relation with a depreciation rate δ. Equation (22) is the division of the steady-state output between consumption and investment with shares s_C and s_I, respectively. Equation (23) is a New Keynesian Phillips Curve derived from the Calvo staggered-pricing model with a marginal cost shock ε_{z, t}, which follows an AR(1) process with an autocorrelation coefficient of ρ_z (Equation 26). Equation (24) is the log-linearized monetary policy rule (17) with an interest rate shock ε_{R, t}. As the money supply is endogenous and the Ricardian equivalence holds in this model, the hidden government budget constraint and the equation for the evolution of government debt are implicitly satisfied.

Straightforward re-arrangements of the variables ẑ_t and ${\hat{I}}_{t}$ in the model give a system of variables ${\hat{C}}_{t}, {\hat{R}}_{t}, {\hat{π}}_{t}, {\hat{Y}}_{t}, {\hat{K}}_{t}, ɛ_{g, t}, ɛ_{z, t} :$

\begin{matrix} (27) & {\hat{C}}_{t} = E_{t} {\hat{C}}_{t + 1} - \frac{1}{σ} [{\hat{R}}_{t} - E_{t} {\hat{π}}_{t + 1}] + ɛ_{g, t} \end{matrix}

\begin{matrix} (28) & {\hat{R}}_{t} - E_{t} {\hat{π}}_{t + 1} = [1 - β (1 - δ)] (σ E_{t} ({\hat{C}}_{t + 1}) + \frac{1}{1 - α} [E_{t} {\hat{Y}}_{t + 1} - E_{t} {\hat{K}}_{t + 1}]) \end{matrix}

\begin{matrix} (29) & {\hat{K}}_{t + 1} \end{matrix} = (1 - δ) {\hat{K}}_{t} + \frac{δ}{s_{I}} ({\hat{Y}}_{t} - (1 - s_{I})) {\hat{C}}_{t})

\begin{matrix} (30) & {\hat{π}}_{t} \end{matrix} = β E {t \hat{π}}_{t + 1} + λ (σ \begin{matrix} {\hat{C}}_{t} \end{matrix} - \frac{α}{1 - α} ({\hat{K}}_{t} - {\hat{Y}}_{t})) + ɛ_{z, t}

\begin{matrix} (31) & {\hat{R}}_{t} \end{matrix} = ρ_{R} \begin{matrix} {\hat{R}}_{t - 1} \end{matrix} + (1 - ρ_{R}) (Ψ_{π} E_{t} \begin{matrix} {\hat{π}}_{t + k} + Ψ_{Y} \end{matrix} \begin{matrix} {\hat{Y}}_{t} \end{matrix}) + ɛ_{R, t}, ɛ_{R, t} i s i i d (0, σ_{R}^{2})

\begin{matrix} (32) & ɛ_{g, t} \end{matrix} = ρ_{g} \begin{matrix} ɛ_{g, t - 1} \end{matrix} + ν_{g, t}, ν_{g, t} is iid (0, σ_{g}^{2})

\begin{matrix} (33) & ɛ_{z, t} \end{matrix} = ρ_{z} \begin{matrix} ɛ_{z, t - 1} \end{matrix} + ν_{z, t}, ν_{z, t} is iid (0, σ_{z}^{2})

Adding capital accumulation to the model makes real interest rate connected to the marginal product of capital, this is contained in the Fisher equation (28). Also, output does not equal consumption in the absence of capital, as in Lubik and Schorfheide (2004), but is split between consumption and investment in this model. This is incorporated in the New Keynesian Phillips curve equation (30) through the output equation (22). By including investment, this model has the capital accumulation equation (29), which influences interest rates through the equations (26) and (28).

To compare, the canonical New Keynesian model can be presented by a system of three equations: the IS equation, the Phillips curve, and a monetary policy rule, similar to equations (27), (30), and (31), respectively. In that way, the interest rate affects output only through the consumption-savings decision of the household and not through the production sector.

2. Model simulations

The model (27)-(33) exhibits different types of dynamics depending on its parameter values (see Carlstrom and Fuerst (2005), Sosunov and Khramov (2008), Kurozumi and Zandweghe (2008), Kurozumi (2006), Huang and Meng (2007), and Xiao (2008)). Under a wide set of parameters, the model is determinate if the monetary authority implements an active monetary policy (Ψ_π> 1), and the model is indeterminate if the monetary policy is passive (Ψ_π< 1). Therefore, two major versions of the model (27)-(33), with active and passive monetary policies, are simulated.⁴ For the baseline calibration most of the parameter values are the same as prior means used by Lubik and Schorfheide (2004); the rest of the parameters are calibrated according to stylized facts (Table 6). The Matlab-based computer package Dynare was used to calculate theoretical moments for the endogenous variables of the model. The simulation results of the two versions of the model with current-looking passive and active monetary policy rules and corresponding moments of consumption, interest rate, inflation, output, and capital are presented in Tables 1-4.

The version of the model with passive monetary policy demonstrates substantially higher volatility of interest rate and inflation compared to the version with active monetary policy (Table 1). This can be explained by the existence of indeterminate equilibria. As long as the monetary policy authority is unable to respond sufficiently to changes in inflation by raising interest rates substantially, the volatility of inflation and, therefore, nominal interest rates, is higher. Both models reproduce a similar volatility of capital to that of U.S. data with lower volatilities of consumption, interest rate, and inflation (Table 5).

The crucial differences between the two versions of the model arise from the variance decomposition of shocks, correlation matrices of endogenous variables, and impulse response functions (IRFs). First, the preferences (demand) shock is the main drivers of volatility in the version of the model with passive monetary policy, explaining more than 90 percent of the volatility of endogenous variables (Table 2). In contrast, the marginal cost (supply) shock explains more than 50 percent of variance in the model with active monetary policy. These results are similar to the findings of Smets and Wouters (2007), who show that “demand” shocks can explain a substantial share of the variance in output in the general version of a New Keynesian model.

The two versions of the model demonstrate different correlations among the major variables (Table 3). In the version of the model with active monetary policy, the nominal interest rate plays the role of the active monetary policy instrument and is negatively correlated with output and capital. These theoretical results are consistent with the stabilizing role of a nominal interest rate in the economy. In contrast, in the version of the model with passive monetary policy, the nominal interest rate is positively correlated with output, capital, and consumption. Passive monetary policy is unable to respond sufficiently to shocks, indeterminacy and additional shock propagation. A response of the monetary authority to supply and demand shocks leads to co-movements in the dynamics of the interest rate and real variables as changes in the nominal interest rate are not enough to diminish the effect of shocks and reverse the dynamics of the economy. Therefore, the version of the model with passive monetary policy demonstrates substantially higher volatility among the economic variables, which is consistent with U.S. data for the pre-Volcker period (1960:I to 1979:II). Theoretical IRFs support this intuition (Appendixes 3-4).

As stated in the Introduction, while the baseline “New Keynesian” models became very popular in the analysis of monetary policy, many papers show that these models are unable to generate enough persistencies in inflation and output (see Chari, Kehoe, and McGrattan (2000), Fuhrer and Moore (1995), Fernandez-Villaverde and Rubio-Ramirez (2004)). First-order autocorrelation coefficients for consumption, interest rate, output, and capital are more than 0.85 in the U.S. data (Appendix 6). Most of the New Keynesian models fail to replicate even half of these correlation levels (see Rubio-Ramirez and Rabanal (2005)). In contrast, the simulated results of this paper show that models with capital accumulation can, in fact, generate substantial persistencies (Table 4). The autocorrelation coefficients for consumption and capital are more than 0.9 in both active and passive monetary policy rule versions of the model. The autocorrelations of nominal variables, such as inflation and the nominal interest rate, are also very high due to the stock nature of capital, which adds persistency to the dynamics of all other variables. The autocorrelation coefficients for consumption and capital are very high, representing substantial consumption smoothing and slow adjustment of capital stock. In the version with passive monetary policy, the interest rate and output autocorrelation coefficients are higher than in the version with active monetary policy, again, due to the fact that passive monetary policy is unable to adjust the interest rate sufficiently to control shock propagation.

Table 1.

Simulation results of the model. Theoretical moments.