Optimal Monetary Policy with Government-Provided Unemployment Benefits

Appendix A: Model Evaluation

In this section, the baseline model is judged against quarterly U.S. data to see how well it accounts for empirically relevant labor market dynamics. I compare the model’s business cycle implications for inflation, output, employment, market tightness, real wage, unemployment rate, and vacancies with those in the post-war US data from 1964:1 to 2014:1. Panel A in Table 6 reports the standard deviation relative to GDP, autocorrelation, and the correlation with GDP of each variable, along with the correlation between unemployment and vacancy postings. The relative standard deviation and correlation with GDP of inflation rate are taken from Gali (2010) (Table 1), and its autocorrelation is based on Fuhrer (2010). The business cycle statistics for GDP, employment, real wage, market tightness, unemployment, and vacancies, along with the correlation between vacancies and unemployment rate (the Beveridge curve) are taken from Chugh et al. (2018), which are in line with what Gertler and Trigari (2009) and Gertler, Huckfeldt, and Trigari (2020) report.

Before turning to discuss the result, note that in presenting the dynamics of the decentralized economy, I assumed that the monetary authority sets the nominal interest rate according to:

where R, π, and Y are the steady-state rates of nominal interest, inflation, and the steady state of output, respectively. The coefficients in the interest rate rule are set to the conventional values α _π = 1.5 and α _Y = 0.05. Φ _t is an exogenous policy shifter that follows the following AR(1) process

where ε t Φ is an i.i.d. innovation with mean zero and standard deviation σ ^Φ.

Panel B of Table 6 reports unconditional moments from the model for the baseline calibration discussed above. Panels C and D report unconditional moments when, respectively, the per-vacancy posting cost is assumed to be 0.25 and 0.75 (and, as a result, lower values for the unemployment benefits Ω), while keeping all other structural parameters fixed at their baseline settings.

Panel B shows that, under baseline calibration, the model performs quite well in reproducing key business cycle facts, specifically in reproducing the cyclical behavior of key labor market variables.

It succeeds in generating empirically relevant large fluctuations in unemployment, market tightness, and vacancies, and small fluctuations in real wages. In addition, the relative volatility of inflation is the same as in the data. Moreover, the model is able to generate a Beveridge curve. The contemporaneous correlation between vacancies and unemployment conditional on both monetary and TFP shocks is −0.61 versus −0.89 in the data.^[12]

Panels C and D confirm what is discussed in Section 4 regarding how a low value of κ, that results in a high value for Ω, is needed to generate high volatility in unemployment and market tightness and a low volatility in wages. A mere rise in κ, that leads to a fall in unemployment benefits Ω, all else equal, leads to a substantial deterioration in the model’s performance in matching the empirically relevant labor market dynamics. More specifically, increasing κ from its baseline 0.025 to 0.25 (that is accompanied by a fall in Ω from 0.977 to 0.767) causes the (relative) volatilities of unemployment, market tightness, and vacancies to enormously decline from 5.7, 13.8, and 9.3, to 1.7, 4, and 2.7, respectively. At the same time, the relative volatility of wages significantly increases from 0.27 to 0.76. These changes are much more significant in the case of κ = 0.75, as panel D clearly documents. Therefore, in order for the model to be empirically relevant for a clear optimal monetary policy recommendation, the government-provided unemployment benefits, as a result of a low κ, must be high.

Figures 5 and 6 display the dynamic responses of eight macro variables (output, inflation, nominal interest rate, unemployment, vacancies, market tightness, employment, and wages) to, respectively, exogenous positive monetary policy and technology shocks. In line with VAR-based estimates of impulse responses to a monetary policy shock reported by Gali (2010) and Christiano et al. (2016), Figure 5 shows that in response to an expansionary monetary shock both output and employment significantly rise, and the nominal interest rate falls on impact. The figure also shows that the response of inflation is muted, and the real wage rises only modestly, and specifically, its increase is substantially smaller than the increase in employment, all in line with the facts. Moreover, the model implies that unemployment falls, vacancy postings rise, and as a result labor market tightness increases, all in line with the data.

Figure 5:

Responses to a monetary policy shock: baseline calibration. x-axis in quarters; y-axis for inflation and nominal interest in annual percentage points, for unemployment rate in percentage points in the panel that is plotted alone, but in percent in the panel with market tightness and vacancies, and for all other variables in percent.

Figure 6:

Responses to a technology shock: baseline calibration. x-axis in quarters; y-axis for inflation and nominal interest in annual percentage points, for unemployment rate in percentage points in the panel that is plotted alone, but in percent in the panel with market tightness and vacancies, and for all other variables in percent.

In response to a one percent increase in z _t , and in line with the empirical impulse responses in Christiano, Eichenbaum, and Trabandt (2016), Figure 2 shows that inflation sharply falls and output rises and the increase in employment is far larger than the rise in wages. Labor market tightness again rises due to more vacancies posted and unemployment falling. It is important to note that in the VAR-based impulse response functions presented in Christiano, Eichenbaum, and Trabandt (2016) unemployment initially rises and after 5–6 quarters it starts to fall. However, this is not the case here, and unemployment declines sharply on impact, as is the case in the standard search and matching framework.

These results are in line with Christiano, Eichenbaum, and Trabandt (2016), who show that their estimated New Keynesian model accounts well for key business cycle properties of macroeconomic aggregates, including labor market variables. They do so in a medium-scale DSGE framework with frictional labor markets, adjustment costs in capital formation, capacity utilization costs, habit formation in preferences, Calvo sticky price frictions, a Taylor rule for monetary policy, and with neutral and investment-specific technology shocks and monetary policy shocks. In their baseline model real wages are determined by alternating-offer bargaining (AOB). However, they also consider the case that wages are determined by Nash bargaining and show that this version of their model also succeeds in accounting for the key features of the estimated impulse response functions. This section shows that the Canonical New Keynesian framework with matching frictions and Nash bargaining, and in the absence of many of those factors considered by Christiano, Eichenbaum, and Trabandt (2016), is capable of accounting for key business cycle properties of labor market variables as long as vacancy posting costs are low (and therefore, the unemployment benefits are high enough).

Appendix B: Efficient Allocations

The social planner problem in the case of government-provided unemployment benefits is to maximize.

subject to the law of motion for the employment

the fraction of searching workers

and the goods market resource constraint

Denote by λ t 1 , λ t 2 , and λ t 3 the Lagrange multipliers on these three constraints, respectively, the Lagrangian problem is

L = E 0 ∑ t = 0 ∞ β t u c t + λ t 1 n t − 1 − ρ n t − 1 − m s t , v t + λ t 2 s t − 1 + 1 − ρ n t − 1 + λ t 3 z t n t − c t − g t − κ v t

The first-order conditions with respect to c _t , v _t , n _t , and s _t are thus

Eliminating λ t 3 between conditions (52) and (53) gives

Now using (52) and (55) we get

Plugging (56) and (57) into (54) we have

Given Cobb-Douglas matching m s t , v t = ζ s t α v t 1 − α , we have m s s t + 1 , v t + 1 = ζ α θ t 1 − α and m v s t + 1 , v t + 1 = ζ 1 − α θ t − α . Plugging these values into (58) we will get (39).

Appendix C: Measuring Welfare Costs

Let the Ramsey-optimal policy and the inflation targeting policy be denoted by r and t, respectively. The conditional welfare related to the time-invariant equilibrium implied by the Ramsey-optimal policy evaluated at c t is denoted by V 0 r and is defined as

The conditional welfare related to regime t is denoted by V 0 t and is defined as

Let Θ be the welfare cost of adopting policy t instead of the Ramsey-optimal policy. Θ measures the fraction of the Ramsey-optimal policy consumption process that a household would be willing to give up to be as well off under price stability policy as under the Ramsey-optimal policy. More precisely

Given the utility form (34), expression (61) can be written as

Solving the above expression for Θ we get

In equilibrium, V 0 r = V r x 0 , σ ε and V 0 t = V t x 0 , σ ε where x ₀ is the initial state vector, and σ _ɛ is a parameter scaling the standard deviation of the exogenous shocks (for details see Schmitt-Grohé and Uribe 2007). Therefore, we can write the conditional welfare cost Θ as follows

Now, a second-order approximation of ∇ x 0 , σ ε around x 0 = x , σ ε = 0 , where x stands for the deterministic Ramsey steady state of the state vector yields

where V σ ε σ ε r x , 0 and V σ ε σ ε t x , 0 are the second derivatives of the policy functions with respect to σ _ɛ evaluated at x 0 = x , σ ε = 0 .