Countercyclical unemployment benefits: a general equilibrium analysis of transition dynamics

Abstract

We analyze the general equilibrium effects of countercyclical unemployment benefit policies. Our heterogenous-agent model features costly job search with imperfect insurance of unemployment risk and individual savings. Our model predicts: (1) the additional unemployment under a countercyclical policy relative to that under an acyclical policy to be a superlinear function of the aggregate shock?s size, (2) a higher unemployment rate sensitivity to UI policy changes when individual savings are relatively low. Our estimates of the effects of UI policy changes are based on transition dynamics following a large, unanticipated increase in the unemployment rate.

Algorithm

1.1 Algorithm for stationary equilibrium

In the precrisis period $t<0$, the UI payments are a a piece-wise linear function of k, $y_1^{\star }(k) = \min \left( a + b (k-{\underline{k}}, {\bar{y}}) \right)$, which increases linearly with slope b for individuals with $k \le k_H = {\underline{k}}+\frac{{\bar{y}}-a}{b}$. Given these parameters $a, b, {\bar{y}}$, we consider a stationary equilibrium which can be summarized by a coupled HJB-KF equation system. For $k \in [{\underline{k}}, \infty )$,

$$\begin{aligned}&\rho v_1(k) = \max _{c_1 s} u(c_1) + \left( y_1^{\star }(k) + (r^{\star } -\delta )k - c_1 \right) \dot{v}_1(k) + \left( v_2(k) - v_1(k)\right) \lambda _2(s) - \psi (s) \end{aligned}$$

(A1)

$$\begin{aligned}&\rho v_2(k) = \max _{c_2} u(c_2) + \left( (1-\theta ^{\star }) w^{\star } + (r^{\star } -\delta )k - c_2 \right) \dot{v}_2(k) + \left( v_1(k) - v_2(k)\right) \lambda _1 \end{aligned}$$

(A2)

$$\begin{aligned}&0 = - \frac{d}{d k} \left( g_1(k) \left( y_1^{\star }(k) + (r^{\star } -\delta )k - c_1^{\star } \right) \right) - g_1(k) \lambda _2(s^{\star }) + g_2(k) \lambda _1 \end{aligned}$$

(A3)

$$\begin{aligned}&0 = - \frac{d}{d k} \left( g_2(k) \left( (1-\theta ^{\star }) w^{\star } + (r^{\star } -\delta )k - c_2^{\star }\right) \right) - g_2(k) \lambda _1 + g_1(k) \lambda _2(s^{\star }) \end{aligned}$$

(A4)

where $c^{\star }_1, s^{\star }$ are such that the right hand side of (A1) attains maximum, $c^{\star }_2$ is such that the right hand side of (A2) attains maximum, more specifically,

$$\begin{aligned}&c^{\star }_1(k) = ({\dot{v}}_1(k))^{-1/\gamma }\\&c^{\star }_2(k) = ({\dot{v}}_2(k))^{-1/\gamma }\\&s^{\star }(k) = \left( \frac{v_2(k)- v_1(k)}{\phi }\right) ^{1/\kappa } \end{aligned}$$

and $r^{\star }, w^{\star }, \theta ^{\star }$ satisfy market clearing conditions

$$\begin{aligned}&r^{\star } = \alpha \left( K^{\star }/L^{\star } \right) ^{\alpha -1} \end{aligned}$$

(A5)

$$\begin{aligned}&w^{\star } = (1-\alpha ) \left( K^{\star } /L^{\star } \right) ^{\alpha } \end{aligned}$$

(A6)

$$\begin{aligned}&\theta ^{\star } w^{\star } L^{\star }= \int _{{\underline{k}}}^{\infty } g_1(k)y_1^{\star } dk \end{aligned}$$

(A7)

with $K^{\star } = \int _{{\underline{k}}}^{\infty } k(g_1(k)+ g_2(k)) dk$ and $L^{\star } = \int _{{\underline{k}}}^{\infty } g_2(k) dk$.

Notice that $w^{\star } = w(r^{\star }) = (1-\alpha ) (\frac{r^{\star }}{\alpha })^{\alpha /(\alpha -1)}$ while $\theta ^{\star }$ depends on both the interest rate and the distribution functions.

Our computational approach is adopted from [5] (hereafter referred to as “AHLLM”). In AHLLM, individuals make decisions solely on consumption and their employment state is determined by an exogenous two-state continuous-time Markov chain. Consequently, the employment rate in the stationary equilibrium is a constant $L= \frac{\lambda _{1, 2}}{\lambda _{1, 2}+\lambda _{2, 1}}$ where $\lambda _{1,2}(\lambda _{2,1})$ is the constant intensity rate of transition from the unemployment(employment) state to the employment(unemployment) state. For a each given interest rate r, the total capital of demand $K_d(r) = L(\frac{r}{\alpha })^{1/(\alpha -1)}$ can be computed directly and a total capital of supply $K_s(r) = \int _{{\underline{k}}}^{\infty } k(g^r_1(k)+ g^r_2(k)) dk$ can be obtained by solving a system of HJB-KF equations. Here, $g^r_1$ and $g^r_2$ are solutions of KF equations whose coefficients depend on the solution of the HJB equation given the interest rate r and the corresponding $w= w(r)$.

In AHLLM, a fixed-point algorithm is employed to find the interest rate r such that $K_s(r) = K_d(r)$, indicating a stationary equilibrium. In each iteration of the fixed point algorithm, AHLLM uses a variate of finite difference method, "upwind-scheme", to solve HJB equations. The solution of HJB equations are then used to generate coefficients in KF equations, whose solution is obtained by solving a linear equation system. The value of r will be adjusted after each iteration based on whether there is an excess demand or an excess supply of total capital.

In our case, we aim to find a pair $(r^{\star }, \theta ^{\star })$ such that not only does $K_s(r^{\star }) = K_d(r^{\star })$ hold but also (A7) holds. We decompose this task into a two-layer fixed-point algorithm. The algorithmic details are outlined in Algorithm 1, where, in the second loop of the fixed point algorithm, we obtain L and $\theta$ such that (A7) holds for each $r^{(n)}$ in the first loop. In the first loop, we iterate over $r^{(n)}$ until $K_s(r^{\star }) = K_d(r^{\star })$ holds.

Remark 2

In the algorithm, all value functions $v_i^{\star }$ and density functions $g^{\star }_i$ are represented by $2\times I$ matrices, where I is the number of mesh points on interval $[{\underline{k}}, {\bar{k}}]$ for a large ${{\bar{k}}}$. The $\Vert \cdot \Vert$ norm is $\Vert \cdot \Vert _{\sup }$ on $2\times I$ matrices.

Remark 3

For fixed $r^{(n)}, \theta ^{(n)}$, the solution of (A2,A1) is obtained using "upwind-scheme", as in AHLLM. For detailed information, please refer to the Online appendix of AHLLM.

Algorithm 1

Algorithm for stationary equilibrium

1.2 Algorithm for time dependent equilibrium

Although we have infinite horizon in our problem, we do not start from the steady state due to the unanticipated shock at time $t = 0$. Moreover, with a time-dependent UI payments $y_{1t}$, the equilibrium solution for our model is in form of transition dynamics. As in AHLLM, for the convenience of numerical computation, we assign a large time T as the terminal time and assume that with this long enough time T, the economy has converged to the same steady state as in the precrisis period. We use the time-dependent equilibrium of this finite horizon problem to approximate the transition dynamics of the original infinite horizon problem, which can be summarized by a coupled HJB-KF equation system. For $k \in [{\underline{k}}, \infty ), t\in [0, T]$,

$$\begin{aligned} \rho v_1(k,t)&= \max _{c_1, s} u(c_1) + \left( y_{1t}(k) + (r_t -\delta )k - c_1 \right) \partial _{k} v_1(k,t) \nonumber \\&\quad + \left( v_2(k,t) - v_1(k,t)\right) \lambda _2(s) - \psi (s) + \partial _t v_1(k,t) \end{aligned}$$

(A8)

$$\begin{aligned} \rho v_2(k,t)&= \max _{c_2} u(c_2) + \left( (1-\theta _t) w_t + (r_t -\delta )k - c_2 \right) \partial _k v_2(k,t) \nonumber \\&\quad + \left( v_1(k,t) - v_2(k,t)\right) \lambda _1+ \partial _t v_2(k,t) \end{aligned}$$

(A9)

$$\begin{aligned} \partial _t g_1(k, t)&= -\partial _k \left( g_1(k,t) \left( y_{1t}(k) + (r_t -\delta )k - c_1(k,t) \right) \right) \nonumber \\&\quad - g_1(k,t) \lambda _2(s(k,t)) + g_2(k, t) \lambda _1 \end{aligned}$$

(A10)

$$\begin{aligned} \partial _t g_2(k,t)&= -\partial _k \left( g_2(k,t) \left( (1-\theta _t) w_t + (r_t-\delta )k - c_2(k,t) \right) \right) \nonumber \\&\quad - g_2(k,t) \lambda _1 + g_1(k,t) \lambda _2(s(k,t)) \end{aligned}$$

(A11)

where in (A10, A11)

$$\begin{aligned}&c_1(k,t) = (\partial _k v_1(k,t))^{-1/\gamma }\\&c_2(k,t) = (\partial _k v_2(k,t))^{-1/\gamma }\\&s(k,t) = \left( \frac{v_2(k,t)- v_1(k,t)}{\phi }\right) ^{1/\kappa } \end{aligned}$$

and $r_t, w_t, y_{1t}, \theta _t$ satisfy

$$\begin{aligned}&r_t = \alpha \left( K_t/L_t \right) ^{\alpha -1}\\&w_t = (1-\alpha ) \left( K_t/L_t\right) ^{\alpha }\\&y_{1t}(k) = \min \left( a + b(k-{\underline{k}}), {\bar{y}} \right) + \eta (U_t - U^{\star })= \min \left( a + b(k-{\underline{k}}), {\bar{y}} \right) + \eta (L^{\star }-L_t)\\&\theta _t w_t L_t= \int _{{\underline{k}}}^{\infty } g_{1}(k,t)y_{1t}(k) dk \end{aligned}$$

with given employment rate $L^{\star }$ in steady state, $K_t = \int _{{\underline{k}}}^{\infty } k(g_1(k,t)+ g_2(k,t)) dk$ and $L_t = \int _{{\underline{k}}}^{\infty } g_2(k,t) dk$.

The terminal conditions are given by $v_2(k,T) = v_2^{\star }(k), v_1(k, T) = v_1^{\star }(k)$ where $v_2^{\star }(k), v_1^{\star }(k)$ are the solutions of the stationary equilibrium (A1,A2,A3,A4). The initial distributions $g_1(k, 0), g_2(k,0)$ are given by (7) and $g_2(\cdot , 0) = g_1^{\star }(\cdot , 0)+ g_2^{\star }(\cdot , 0) - g_1(\cdot , 0)$. This indicates a uniform job loss for all k and guarantees a drop of employment rate from $L^{\star }$ to an exogenous constant $L_0$.

With the above initial and terminal conditions, Algorithm 2 outlines the procedure for solving (A8,A9, A10, A11). In each iteration the value functions $v_i(\cdot , t)$ is derived backward in time given the terminal condition and the density function $g_i(\cdot , t)$ is derived forward in time given the initial condition. A detailed description is also available in the Online appendix of AHLLM. Paths of $(K_t, L_t, \theta _t)$ are updated simultaneously in each iteration based on the density functions $g_i(\cdot , t)$. The algorithm converges to the equilibrium when the maximum change in paths of $(K_t, L_t, \theta _t)$ between two consecutive iterations is sufficiently small.

Algorithm 2

Algorithm for time-dependent equilibrium