30. Equilibrium Unemployment Theory

30.1 Set-up

• Firms offer wage \(w\) in each period to the employed workers until terminal (unemployment) date.
• Terminal date \(t\) is drawn from an exponential distribution with parameter \(\chi\), i.e. \(t\sim\exp(\chi)\) with the p.d.f. \(f(t)=\chi e^{-\chi t}\).
  • The probability of termination prior to \(t\) is just the c.d.f. of the exponential distribution, i.e. \(F(t)=1-e^{-\chi t}\).
  • Define the hazard rate at \(t\) by \(\dfrac{f(t)}{1-F(t)}=\dfrac{\chi e^{-\chi t}}{1-(1-e^{-\chi t})}=\chi\), which is exactly \(\chi\).
  • The hazard rate is not an unconditional probability of losing job at \(t\). Instead, it is a conditional probability of losing job at \(t\) conditioning on the job has not be lost prior to \(t\).
  • So, the hazard rate is the relevant probability to think about when standing in the shoes of date \(t\) employed worker. But when calculating at the beginning, the worker would use unconditional probability, which is the p.d.f., to evaluate the expected value.
• Discount rate for the value is \(r\).

30.2 One-time compensation case

We assume that after termination, the worker gets a termination payoff of \(\bar V\) (already discounted back to the termination date) as the one-time unemployment compensation, and cannot find another job again in the future. In this simple case, we can directly write the value function for the worker at date zero as

$$ \begin{aligned} V&=\int_0^{\infty}\underbrace{e^{-rt}}_{\text{discount}}\underbrace{e^{-\chi t}}_{\text{not terminated}}\underbrace{w}_{\text{period return}}dt+\int_0^{\infty}\underbrace{e^{-rt}}_{\text{discount}}\underbrace{\chi e^{-\chi t}}_{\text{terminated at }t}\underbrace{\bar V}_{\text{payoff}}dt\\ &=w\int_0^{\infty}e^{-(r+\chi)t}dt+\chi\bar V\int_0^{\infty}e^{-(r+\chi)t}dt=\frac{w+\chi\bar V}{r+\chi}\\ \Rightarrow\ rV&=w+\chi(\bar V-V) \end{aligned} \tag{30.1} $$

30.3 Pissarides model: allow for job search when unemployed

30.3.1 Additional set-up details

• We assume that the unemployed worker receives zero compensation but can freely search for the next job.
• The only input for production is labor; the productivity is \(Z\).
• Each individual makes a discrete labor supply decision, i.e. either work to produce \(Z\) amount of good and suffers the disutility of labor \(\gamma\), or not work to receive 0 payoff.

30.3.2 Matching function

The matching function \(m(U,V)\) has two argument: \(U\) is the number of unemployed workers in the economy, and \(V\) is the number of vacancies. Assume that \(m(U,V)\) is constant return to scale, i.e. h.o.d. 1 in \((U,V)\) (there is an empirical literature on whether the matching function is h.o.d. 1; the results are mixed, but the literature cannot reject this assumption), and assume that \(m(U,V)\) is increasing in both arguments.

Job finding rate on the worker's side: denote the vacancy to unemployment ratio by \(\theta\). The rate at which a worker finds a job is

$$ \frac{\text{number of matches}}{\text{total unemployed workers}}=\frac{m(U,V)}{U}=m\left(1,\frac{V}{U}\right)=m(1,\theta)\equiv f(\theta) $$

where \(f\) is an increasing function due to the monotonicity of \(m(U,V)\) in \(V\).

Job filling rate on the firm's side:

$$ \frac{\text{total matches}}{\text{total postings}}=\frac{m(U,V)}{V}=m\left(\frac{U}{V},1\right)=m(\theta^{-1},1)\equiv\mu(\theta) $$

where \(\mu(\theta)\) is decreasing in \(\theta\) because \(m(\theta^{-1},1)\) is increasing in \(\theta^{-1}\), and \(\mu(\theta)=m(\theta^{-1},1)=\dfrac{m(1,\theta)}{\theta}=\dfrac{f(\theta)}{\theta}\). Inada conditions: \(f(0)=0\), \(\lim_{\theta\to\infty}\mu(\theta)=0\), \(\mu(0)=\infty\), \(\lim_{\theta\to\infty}f(\theta)=\infty\) (which ensure the interiority of solutions).

30.3.3 Bellman equations for the workers

The worker's problem is defined by the following parameters: discount rate \(r\), constant hazard rate \(\chi\), job finding rate \(f(\theta)\), equilibrium wage \(w^{\star}\), disutility of work \(\gamma\) (assume \(w^{\star}\ge\gamma\)).

The unemployment Bellman equation (the integrals are similar to §30.2, since the job finding date follows an exponential distribution with p.d.f. \(f(\theta)e^{-f(\theta)t}\)):

$$ rV^u=f(\theta)\underbrace{\big(V^e(w^{\star})-V^u\big)}_{\text{gain from switch}} \tag{30.2} $$

The employment Bellman equation:

$$ rV^e(w)=w-\gamma+\chi\underbrace{\big(V^u-V^e(w)\big)}_{\text{gain from switch}} \tag{30.3} $$

30.3.4 Bellman equations for the firms

There is no capital in this model. Firms hire employed workers and convert them into output. Two conditions: (1) firms use CRS production technology using only labor with each labor producing \(Z\), and assume \(Z>\gamma\); (2) there is a cost \(c\) for maintaining each job vacancy for each period.

First, we require the equilibrium value of a vacancy is zero, i.e. \(V^v=0\) (free entry condition). If this did not hold, since firms cannot lose money in equilibrium, it must be that \(V^v>0\). However, in that case the firms would like to post an infinite number of vacancies, which contradicts the equilibrium condition.

The Bellman equation for a firm with a worker (job) (using \(V^v=0\)):

$$ rV^j(w)=Z-w-\chi V^j(w) \tag{30.4} $$

The Bellman equation for a firm without a worker (vacancy):

$$ rV^v=-c+\mu(\theta)\big(V^j(w^{\star})-V^v\big) \tag{30.5} $$

30.3.5 Five equations with six unknowns

(30.2), (30.3), (30.4), (30.5) plus the free entry condition

$$ V^v=0 \tag{30.6} $$

are five equations; six unknowns \(V^u,V^e(w),V^j(w),V^v,w^{\star},f(\theta)\) (we did not include \(\mu(\theta)\) because \(\mu(\theta)=f(\theta)/\theta\) is known once \(f(\theta)\) is known). So, we are short of one more equation to solve for the system.

30.3.6 Equilibrium wage setting and Nash bargaining solution

In this employment relationship, the worker is willing to work for a range of wages \(w\in[\gamma,Z]\) and the firm is willing to pay for a range of wages \(w\in[\gamma,Z]\). To characterize the exact solution we will use the Nash bargaining solution: the equilibrium wage is the one that maximizes a geometric average of the surplus of each party, i.e.

$$ w^{\star}\in\arg\max_w\big(V^e(w)-V^u\big)^{\phi}\big(V^j(w)-0\big)^{1-\phi} $$

where \(\phi\in(0,1)\) is the worker's bargaining power (in the limit as \(\phi\) goes to 1 the firm will have to pay the worker a wage \(\phi=Z\); in the limit as \(\phi\) goes to zero the worker has no bargaining power, so the equilibrium wage will go to \(\gamma+rV^u\)). Taking the derivative of this expression to find \(w^{\star}\), we have that

$$ w^{\star}=\phi Z+(1-\phi)(\gamma+rV^u) \tag{30.7} $$

30.3.7 Solve for equilibrium with the system of six equations and six unknowns

Using (30.2), (30.3), (30.7) and (30.4), we have (denote the joint surplus \(V^s\equiv V^e(w^{\star})+V^j(w^{\star})-V^u\))

$$ \frac{V^e(w^{\star})-V^u}{\phi}=\frac{V^j(w^{\star})}{1-\phi}=V^e(w^{\star})+V^j(w^{\star})-V^u\equiv V^s \tag{30.11} $$

which means that the worker's gain from bargaining is the fraction \(\phi\) of the joint surplus \(V^s\), and the firm's share is \(1-\phi\) of the joint surplus \(V^s\). Furthermore,

$$ rV^s=Z-\gamma-\chi V^s-\phi f(\theta)V^s\ \Rightarrow\ V^s=\frac{Z-\gamma}{r+\chi+\phi f(\theta)} $$

Using (30.5) and \(V^v=0\), we have \(c=\mu(\theta)V^j(w^{\star})=\mu(\theta)(1-\phi)V^s\) (30.12)–(30.13). Combining with the above (and using \(f(\theta)/\mu(\theta)=\theta\)), we obtain

$$ \frac{r+\chi}{\mu(\theta)}=(1-\phi)\frac{Z-\gamma}{c}-\phi\theta \tag{30.14} $$

Note that the LHS of (30.14) is increasing in \(\theta\), always positive, and by Inada condition can take any value between \((0,\infty)\); and the RHS of (30.14) is clearly decreasing in \(\theta\), positive for small \(\theta\) and negative for large \(\theta\). So, there is one and only one intersection of the LHS and RHS of (30.14), which uniquely pins down the equilibrium vacancy to unemployment ratio \(\theta\). Once \(\theta\) is pinned down, all other five variables are all pinned down by \(\theta\), and the model's equilibrium is solved.

Equilibrium wage: by (30.7), (30.14) and \(rV^u=\dfrac{\phi\theta c}{1-\phi}\), the equilibrium wage \(w^{\star}\) is determined by

$$ w^{\star}=\phi(Z+c\theta)+(1-\phi)\gamma \tag{30.19} $$

30.3.8 Steady state unemployment rate

We can characterize the steady state unemployment rate by imposing the dynamic equal flow-in and flow-out condition. We first normalize the total population to 1, i.e. \(U+N=1\) (the total employed worker \(N\) and total unemployed workers \(U\) add up to 1). Then, the unemployment rate is \(\dfrac{U}{U+N}=U\). The employed workers lose their jobs at rate \(\chi\) in each period, and the unemployed workers find jobs at rate \(f(\theta)\) in each period. So, in steady state, the number of people who newly find their job should equal the number of people who newly lose their job in each period, i.e.

$$ U\cdot f(\theta)=(1-U)\cdot\chi\ \Rightarrow\ U^{\star}=\frac{\chi}{f(\theta)+\chi} \tag{30.20} $$

which is uniquely pinned down by \(\theta\).

30.3.9 Relationship between steady state and equilibrium

• Steady state must be an equilibrium.
• An equilibrium is not necessarily in the steady state.
• So, steady state is only a special case of equilibrium.

So, suppose we start with a non-steady state unemployment rate \(U(0)\ne U^{\star}\), the \(\theta\) is constant along the way of convergence back to \(U^{\star}\) as long as the economy is in equilibrium along that path. So, using the flow-in and flow-out argument, we can write down the dynamics of convergence to steady state:

$$ \dot U(t)=(1-U(t))\chi-U(t)f(\theta) \tag{30.21} $$

Notice that \(0=(1-U^{\star})\chi-U^{\star}f(\theta)\), so solving the differential equation (30.21) gives

$$ U(t)=U^{\star}+(U(0)-U^{\star})e^{-(\chi+f(\theta))t} \tag{30.23} $$

Figure 17 (Relationship between Steady State and Equilibrium, paraphrased): vertical axis \(V\), horizontal axis \(U\). The black line represents the uniquely pinned down equilibrium \(\theta=V/U\) (the set of equilibrium points, all with the same ratio \(\theta\)); the red curved line is the set of steady state points \(m(U,V)=\chi(1-U)\). They intersect at point \(P\) = the unique steady state. If we start at \(U(0)\ne U^{\star}\) (point \(Q\)), since the economy maintains equilibrium \(\theta\), the system converges back to \(P\) along the black line.

30.3.10 Convergence to steady state

The convergence is very fast. Consider the half-time convergence \(U(t)-U^{\star}=\tfrac12(U(0)-U^{\star})\), which needs \(e^{-(\chi+f(\theta))t}=\tfrac12\Rightarrow t=\dfrac{\ln2}{\chi+f(\theta)}\). Suppose we let \(\chi+f(\theta)\approx\tfrac13\), then \(t\approx2\), which means that it takes two months (period length set as month) for the economy to converge half-way back to the steady state.

30.3.11 Comparative statics

• Suppose we experience a positive MIT shock (unexpected one-time shock) in productivity, i.e. \(Z\uparrow\):
  • For change in \(\theta\) (by (30.14)): the LHS remains unaffected by \(Z\) while the RHS goes up; to maintain the equality, we must have \(\theta\) go up.
  • For change in \(w^{\star}\) (by (30.19)): both \(Z\) and \(\theta\) go up, and other parameters are unaffected, so equilibrium wage goes up. The wage increases by increased \(Z\) is due to higher productivity; the wage increases by increased \(\theta\) is intuitively due to the higher threatening power of the worker since higher \(\theta\) means easier job finding and harder job recruiting.
  • Figure 18 (Comparative Statics for Steady State Unemployment Rate: Change in Productivity, paraphrased): increase in \(Z\) leads to new equilibrium \(\theta'>\theta\) (blue line); since the system attains equilibrium, the system immediately jumps from point \(P\) to point \(Q\); then the system gradually converges to new steady state point \(M\) along the blue line from point \(Q\); the new steady state unemployment rate \((U^{\star})'
• Suppose we experience an increase in matching technology, i.e. \(m'(U,V)>m(U,V)\) for the same pair of \((U,V)\):
  • For change in \(\theta\) (by (30.14)): the RHS remains unaffected by change in matching technology while the LHS goes down (\(\mu\uparrow\)); to maintain the equality, we must have \(\theta\) go up.
  • For change in \(w^{\star}\) (by (30.19)): since \(\theta\) goes up, equilibrium wage goes up.
  • Figure 19 (Comparative Statics for Steady State Unemployment Rate: Change in Matching Technology, paraphrased): increase in matching technology leads to new equilibrium \(\theta'>\theta\) (blue straight line); the set of steady state points curve also shifts inward because \(m'(U,V)>m(U,V)\) implies that for the same \(V\), we have a lower \(U\) to maintain the equality \(m(U,V)=(1-U)\chi\); the system jumps from \(P\) to \(Q\) and converges to new steady state \(M\) along the blue line; \((U^{\star})'
  • It seems obvious that with the advent of internet, the matching technology has improved significantly. However, empirically, we don't observe significant drop in steady state unemployment rate, which might because: the matching technology improvement due to internet is offset of the higher volume of inappropriate application, so the overall matching technology is not improved; or it is offset of the more picky selection standard of firms.

30.3.12 The efficiency of equilibrium

To consider the efficiency of equilibrium characterized by (30.14), we should compare it against the socially optimal outcome, which is obtained by solving the social planner's problem that maximizes the social surplus:

$$ \max_{\{U(t),V(t)\}}\int_0^{\infty}e^{-rt}\big[(1-U(t))(Z-\gamma)-cV(t)\big]dt\quad\text{s.t.}\quad \dot U(t)=\chi(1-U(t))-m(U(t),V(t)) $$

Therefore, the equilibrium is efficient when the coefficients in matching function and bargaining function are the same. So, generally suppose the matching function (Cobb-Douglas form) is \(m(U,V)=\bar m U^{\eta}V^{1-\eta}\). Then, only when the worker's bargaining power \(\phi=\eta\) will we have the efficient equilibrium outcome (Hosios condition).