31. Unemployment and Job Searching in Real Business Cycle Model

In this section, we will bridge the ideas of the job searching in Pissarides model discussed in subsection 30.3 and of the real business cycle model. We will be solving both the decentralized competitive equilibrium and the social planner's problem, and then we will compare these two results.

31.1 Set-up

• Discrete periods \(t=0,1,2,\ldots\); state \(s_t\); history \(s^t=(s_0,\ldots,s_t)\), successor \(s^{t+1}>s^t\).
• The period \(t\) consumption \(C_t(s^t)\), period \(t+1\) labor supply \(N_{t+1}(s^t)\), and capital \(K_{t+1}(s^t)\) are all functions of history \(s^t\).
  • Labor supply breaks down to two parts: production labor \(L_t(s^t)\); recruiting labor (e.g. HR) \(V_t(s^t)\); \(N_t(s^{t-1})=L_t(s^t)+V_t(s^t)\).
• Probability distribution \(\Pi_t(s^t)\in[0,1]\).
• Utility function: \(u(C_t(s^t),N_t(s^{t-1}))=\ln C_t(s^t)-\gamma N_t(s^{t-1})\).
• Production function CRS, Cobb-Douglas: \(F(K_t(s^{t-1}),Z_t(s^t)L_t(s^t))=[K_t(s^{t-1})]^{\alpha}[Z_t(s^t)L_t(s^t)]^{1-\alpha}\), where \(Z_t(s^t)\) is the productivity of production labor.
• Hazard rate \(\chi\); discount factor \(\beta\); capital depreciation rate \(\delta\).
• Matching function \(m(U,V)\) has two argument: \(U\) is the number of unemployed workers, \(V\) is the number of labor employed for recruitment; CRS (h.o.d. 1), increasing in both arguments.

31.2 Social planner's problem

31.2.1 The problem

$$ \max_{\{C_t(s^t),V_t(s^t),L_t(s^t),K_{t+1}(s^t)\}}\sum_{t=0}^{\infty}\sum_{s^t}\beta^t\Pi_t(s^t)\big(\ln C_t(s^t)-\gamma N_t(s^{t-1})\big) $$

subject to: resource constraint \(K_{t+1}(s^t)-(1-\delta)K_t(s^{t-1})+C_t(s^t)=[K_t(s^{t-1})]^{\alpha}[Z_t(s^t)L_t(s^t)]^{1-\alpha}\); law of motion of employment \(N_{t+1}(s^t)=(1-\chi)N_t(s^{t-1})+m(U_t(s^{t-1}),V_t(s^t))\); labor split \(N_t(s^{t-1})=L_t(s^t)+V_t(s^t)\); total population \(N_t(s^{t-1})+U_t(s^{t-1})=1\); initial \(N_0,K_0\) given.

31.2.2 First-order conditions and the two Euler equations

(31.1) and (31.4) give the first Euler equation (the usual capital Euler):

$$ \frac{1}{C_t(s^t)}=\beta\mathbb{E}\left[\frac{F_K(K_{t+1}(s^t),Z_{t+1}(s^{t+1})L_{t+1}(s^{t+1}))+(1-\delta)}{C_{t+1}(s^{t+1})}\,\Big|\,s^t\right] \tag{31.6} $$

(31.1), (31.5) and (31.3) give the second Euler equation (the inter-temporal indifference condition for HR staff recruiting):

$$ \frac{1}{C_t(s^t)}=\beta m_V(1-N_t(s^t),V_t(s^t))\,\mathbb{E}\left[\frac{(1-\chi)-m_U(1-N_{t+1}(s^t),V_{t+1}(s^{t+1}))+m_V(1-N_{t+1}(s^t),V_{t+1}(s^{t+1}))F_{L,t+1}(s^{t+1})}{C_{t+1}(s^{t+1})m_V(1-N_{t+1}(s^t),V_{t+1}(s^{t+1}))}-\gamma\,\Big|\,s^t\right] \tag{31.7} $$

31.2.5 Balanced growth path with no uncertainty

Assume that the productivity series is deterministic with trend, i.e. \(Z_t(s^t)=Z_0(1+g)^t\). One can prove the economy has a balanced growth path where output, consumption, investment and capital all grow at \(1+g\). Detrend \(k_t=\dfrac{K_t}{(1+g)^t}\), \(c_t=\dfrac{C_t}{(1+g)^t}\), then the first Euler equation (31.6) becomes

$$ \frac{1}{c_t}=\beta\frac{F_K(k_t,Z_0(N_t-V_t))+(1-\delta)}{c_{t+1}} \tag{31.8} $$

Similarly, the second Euler equation (31.9), the law of motion of capital (31.10), and the law of motion of employment (31.11) can all be detrended and rewritten. At the steady state \(k_{t+1}=k_t=k\), \(c_t=c\), \(N_t=N\), \(V_t=V\), we get a system of five steady state equations: (31.12) from (31.9) pins down \(\frac{1-N}{V}\); (31.13) from (31.11) \(\chi N=m(1-N,V)\) pins down \(N,V\); (31.14) from (31.8) pins down \(k\) and then (31.10) pins down \(c\). Therefore, output \(Y_t\), consumption \(C_t\), investment \(I_t\) and capital \(K_t\) are all growing at \(1+g\) in steady state, which is a balanced growth path.

31.2.6 Calibration and log-linearization

Same as in sections 24 and 25, we can use observed (or assumed) parameters in steady state to calibrate and do log-linearization around the steady state in the same fashion (but with more variables). Denote \(\phi_t=(\hat k_t,\hat c_t,\hat s_t,\hat N_t,\hat V_t)'\) where the hatted variables are defined as the difference between the log detrended variable and the log steady state detrended variable, then \(\phi_{t+1}=\mathbf{A}\phi_t\) where \(\mathbf{A}\) is a \(5\times5\) matrix.

31.3 Decentralized problem: competitive equilibrium

31.3.1 Complete market assumption and always-want-to-work condition

In the decentralized problem, consumers maximize their utility, firms maximize their profits, labor and capital are owned by firms, firms are owned by consumers, and finally both goods market and labor market clear. We need to impose the complete market condition: assume that there is an exogenously pinned down sequence of state prices \(\{q_0^t(s^t)\}\).

We would also impose the condition that consumers always want to work when offered a job. This can be true if we imagine that each consumer lives in a household of many (infinitely) such households (members). In each household, the matriarch or patriarch (the family leader) requires that all the family members go to work, and the leader allocate consumption equally among household members regardless of their employment status. All individuals obey the order of the leader, so they always want to work (to kill the incentive problem due to the disutility of working).

Matching: on the worker's side, the rate at which a worker finds a job is \(f(\theta)=m(1,\theta)\); on the firm's side, the number of workers attracted to the firm by each recruiter is \(\mu(\theta)=m(\theta^{-1},1)=\dfrac{f(\theta)}{\theta}\).

31.3.2 Definition of decentralized competitive equilibrium

A decentralized competitive equilibrium is the sequence of endogenous variables \(\{C_t(s^t),L_t(s^t),V_t(s^t),N_{t+1}(s^t),K_{t+1}(s^t),W_t(s^t),q_0^t(s^t),\theta_t(s^t)\}_{t=0}^{\infty}\), where the recruiter to unemployment ratio \(\theta_t(s^t)=\dfrac{V_t(s^t)}{1-N_t(s^{t-1})}\), such that:

• Household problem: given \(\{W_t(s^t),q_0^t(s^t),\theta_t(s^t)\}\), initial \(N_0,A_0\), each household chooses \(\{C_t(s^t),N_{t+1}(s^t)\}\) to solve

$$ \max\sum_{t=0}^{\infty}\sum_{s^t}\beta^t\Pi_t(s^t)\big(\ln C_t(s^t)-\gamma N_t(s^{t-1})\big) $$

$$ \text{s.t.}\ A_0=\sum_{t=0}^{\infty}\sum_{s^t}q_0^t(s^t)\big(C_t(s^t)-W_t(s^t)N_t(s^{t-1})\big) \tag{31.15} $$

$$ N_{t+1}(s^t)=(1-\chi)N_t(s^{t-1})+f(\theta_t(s^t))(1-N_t(s^{t-1})) \tag{31.16} $$

• Firm problem: given \(\{W_t(s^t),q_0^t(s^t),\theta_t(s^t)\}\), initial \(N_0,K_0\), each firm chooses \(\{L_t,V_t,N_{t+1},K_{t+1}\}\) to solve

$$ \max\sum_{t=0}^{\infty}\sum_{s^t}q_0^t(s^t)\Big([K_t(s^{t-1})]^{\alpha}[Z_t(s^t)L_t(s^t)]^{1-\alpha}+(1-\delta)K_t(s^{t-1})-K_{t+1}(s^t)-W_t(s^t)N_t(s^{t-1})\Big) \tag{31.17} $$

$$ \text{s.t.}\ N_t(s^{t-1})=V_t(s^t)+L_t(s^t)\ \text{(31.18)},\quad N_{t+1}(s^t)=(1-\chi)N_t(s^{t-1})+\mu(\theta_t(s^t))V_t(s^t)\ \text{(31.19)} $$

where \(L_t(s^t)\) is the production labor, \(V_t(s^t)\) is the recruiting labor, and \(N_t(s^{t-1})\) is the total labor. Note that if \(L_t(s^t)=N_t(s^{t-1})\) (no recruiting), then this problem becomes exactly the same as a standard Neoclassical growth model.

• Market clearing: goods market \([K_t(s^{t-1})]^{\alpha}[Z_t(s^t)L_t(s^t)]^{1-\alpha}=C_t(s^t)+K_{t+1}(s^t)-(1-\delta)K_t(s^{t-1})\) (31.20); labor market clearing is already implicitly imposed in (31.15), (31.16), (31.18), (31.19), so we only need consistency between the definition of \(\theta_t(s^t)\) on the worker's side and the actual recruiter to unemployment ratio on the firm's side, i.e. \(\theta_t(s^t)=\dfrac{V_t(s^t)}{1-N_t(s^{t-1})}\).

31.3.3 Property of equilibrium wage

The household has no complete control over labor supply (the leader decides it subject to the law of motion of employment), so the condition that the expected wage should be equal to the expected marginal rate of substitution is gone. Instead, to guarantee that the workers would always want to work given the offer, we need the condition that the expected wage should be greater or equal to the expected marginal rate of substitution (workers always want to work). Also, only part of the employed labor is used for productive purpose, so the expected wage should be lower than the expected marginal product of labor. Thus, the expected marginal product of labor would be higher than the expected marginal rate of substitution. This distortion of household's intra-temporal indifference condition is caused by the labor market searching friction.

31.3.4 Two Euler equations for the decentralized equilibrium

Combine the f.o.c. for the household's problem and the firm's problem to get the first Euler equation (the same as the SP's (31.6)):

$$ \frac{1}{C_t(s^t)}=\beta\mathbb{E}\left[\frac{F_K(K_{t+1}(s^t),Z_{t+1}(s^{t+1})L_{t+1}(s^{t+1}))+(1-\delta)}{C_{t+1}(s^{t+1})}\,\Big|\,s^t\right] \tag{31.23} $$

The second Euler equation (the inter-temporal indifference condition for HR staff recruiting):

$$ \frac{F_{L,t}(s^t)}{C_t(s^t)}=\beta\mu(\theta_t(s^t))\,\mathbb{E}\left[\frac{F_{L,t+1}(s^{t+1})\left(1+\frac{1-\chi}{\mu(\theta_{t+1}(s^{t+1}))}\right)-W_{t+1}(s^{t+1})}{C_{t+1}(s^{t+1})}-\gamma\,\Big|\,s^t\right] \tag{31.24} $$

31.3.5 Equilibrium wage setting and Nash bargaining solution

Now we can pin down the sequence of wages by Nash bargaining. Again, similar as in subsection 30.3.6, we can write down the full set of Bellman equations, and the equilibrium wage is the one that maximizes a geometric average of the surplus of both parties, and the result is

$$ W_t(s^t)=\phi F_{L,t}(s^t)\big(1+\theta_t(s^t)\big)+(1-\phi)\gamma C_t(s^t) \tag{31.25} $$

Again, similarly, if the matching function is Cobb-Douglas, i.e. \(m(U,V)=\bar m U^{\eta}V^{1-\eta}\), then \(\phi=\eta\) implies efficiency of decentralized equilibrium in the sense that the decentralized equilibrium allocation solves the planner's problem (Hosios condition).

31.3.6 System of 5 equations

Now we have obtained a system of five equations for the decentralized equilibrium: (1) the first Euler equation of the household (capital, indifference condition for investment), i.e. (31.6)/(31.23); (2) the second Euler equation of the household (indifference condition for recruiting recruiters), i.e. (31.24); (3) law of motion of employment \(N_{t+1}=(1-\chi)N_t+f(\theta_t)(1-N_t)\); (4) law of motion of capital \(K_{t+1}=(1-\delta)K_t+[K_t]^{\alpha}[Z_t L_t]^{1-\alpha}-C_t\); (5) equilibrium wages from Nash bargaining (31.25). Five unknowns \(C_t,K_{t+1},N_{t+1},L_t,W_t\) (with \(Z_t\) exogenous). The steady state is a balanced growth path, and log-linearization is done the same fashion as before.

In the world with \(Z_t(s^t)=Z_0(1+g)^t\), we can alternatively have the wage sequence satisfy

$$ W_t=W_0(1+g)^t \tag{31.26} $$