Equilibrium Unemployment as a Worker Discipline Device1984

Involuntary unemployment is a feature of labor markets. Why don't wages fall to clear the market? Imperfect monitoring in this model requires unemployment on equilibrium. Thus, the threat of firing a worker can be seen as disciplinary. In this paper, the punishment for being fired is endogenous as it depends on the equilibrium rate of unemployment. Workers have perfect information about hobs, and unemployed people prefer to work at wage \(w<w_m\) (market) over being jobless. This model gives some interesting implications. First, unemployment benefits lead to an increase in the unemployment rate. This is not due to the explanation that it induces laziness or not enough incentives to get a job, rather than in \((w,L)\) space where \(L\) is the employment level, the NSC (no shirking condition) and demand for labor curves intersect at a point. Introducing unemployment benefits means being unemployed is less painful, so firms have to pay higher efficiency wages, and the NSC curve shifts upwards. The movement in the demand curve to find a new equilibrium leads to lower employment level, and thus higher unemployment. Secondly, wages adjust slowly to shocks, since a decreased demand for labor will lead to lower wages and higher unemployment. Finally, the market equilibrium is not Pareto optimal. Improvements are shown.

Let us begin with the model. There are \(N\) identical workers with infinite life (something hopefully coming soon to me) who dislike effort but enjoy consumption. Utility is \(U(w,e)\) where \(e\) is effort. \(U(\cdot)\) is separable and workers are risk neutral for now. We can express \(U=w-e\) with either \(e=0\) or \(e\in \mathbb{R}^{+}\). When worker is unemployed they get benefits \(\bar{w}\) and provide effort \(e=0\). There is a probability \(b\) of exogenously losing one's job. Workers maximize expected present discounted value of utility with discount rate \(r \in \mathbb{R}^{+}\). \[ \max W = \mathbb{E} \int_{0}^{\infty}u[w(t),e(t)] \exp(-rt)dt \] We suppose they cannot borrow or lend. The model is set in continuous time. Workers only choose the level of \(e\), which is discrete. At normal \(e\), that is, when they do not shirk, workers get wage \(w\) and remain employed until an exogenous shock (can't be death) causes separation. If they shirk, there is a probability \(q\) that they het caught, after which they are fired and enter the unemployment pool. The endogenous job acquisition rate determines the expected length of unemployment. The expected lifetime utilities are named \(V_E^S\) and \(V_E^N\) for shirker and non shirker \[ \begin{align} rV_E^S = \underbrace{w+(b+q)(V_u-V_E^S)}_{\text{shirker}} && && rV_E^N = \underbrace{w-e+b(V_u-V_E^N)}_{\text{non-shirker}} \end{align} \] are the asset (Bellman) equations, where \(V_u\) is the value of being unemployed. Workers have states, either employed (shirking or non-shirking) or unemployed. Each state has these lifetime values attached to them. These Bellman equations state that the value of the current state equals current utility plus the expected gain or loss from changing states. Since the model is in continuous time, \(b\) and \(q\) are Poisson transition rates. Let us solve for both (I will only show one) expected lifetime utilities: \[\begin{split}rV_E^S & = w+bV_u-bV_E^S+qV_u-qV_E^S \\ rV_E^S+qV_E^S+bV_E^S& =w+bV_u+qV_u \\ r+q+b(V_E^S) & = w+ V_u(b+q) \end{split} \] And similarly we also obtain \[ \begin{align} V_E^S = \frac{w+ V_u(b+q)}{r+q+b}&& && V_E^N = \frac{(w-e)+ bV_u}{r+b} \end{align} \] The worker does not shirk \(\iff V_E^N \geqslant V_E^S\). This is what we call the No Shirk Condition (NSC) i.e. \[ \begin{align} w \geqslant \frac{rV_u + (r+b+q)e}{q} \equiv \hat{w} && \text{or} && q(V_E^S-V_u) \geqslant e \end{align} \] Unless there is a penalty associated with unemployment everybody will shirk. So NSC cannot be satisfied (i.e. \(V_E^N = V_E^S\)) if we can get immediate employment after job loss. So the critical wage \(\hat{w}\) is higher the higher the \(e\), \(V_u\), \(b\), and \(r\), and the lower the \(q\).

There are \(M\) firms indexed by \(i= 1, \cdots, M\). Then, \(\forall \: i \Rightarrow\) the production function is \(Q_i=f(L_i)\) with the aggregate function \(Q=f(L)\). Firm \(i\)'s labor force is \(L_i\). Non-shirk worker contributes 1 unit of labor, and the shirker 0. We assume \(F^{\prime}(N)>e\), that is, that full employment is efficient. All firms find optimal to fire shirkers. Monitoring technology \(q\) is exogenous. Firms pay smallest \(\bar{w}\) allowed, since increasing it increases \(V_u\), which implies an increase in wage \(w\). So firm is hit by costs in both unemployment benefits and actual wage. Firms pay the minimum to induce effort \(w=\bar{w}\) to meet NSC. Its labor demand is determined by setting the marginal product of labor equal to the cost of an additional employee. \[ \begin{align} \underbrace{f^{\prime}(L) = \hat{w}}_{\bar{w}=0} && && \underbrace{f^{\prime}(L) =w+ \frac{\bar{w}b}{a+r}}_{\bar{w}>0} \end{align} \] The authors then give the heuristics for equilibriym. High wages imply workers value jobs because the bread is crazy and there is low unemployment (because of low demand for labor). So firm can't reduce \(w\) without people sjhirking. If wage is low they workers want to shirk, as working \(\approx\) unemployment, as there is high employment. Firms have to raise \(w\) to induce less shirk. Let us turn to \(rV_u = \bar{w}+a(V_E-V_u)\) where \(a\) is job acquisition rate and \(V_E\) is expected utility of employment, which is equal to \(V_E^N\) in equilibrium, which occurs when firms find optimal to offer the going wage and not another. \[ \begin{align} rV_E = \frac{(w-e)(a+r)+\bar{w}b}{a+b+r} && && rV_u = \frac{(w-e)a+\bar{w}(b+r)}{a+b+r} \end{align} \] We then substitute \(V_u\) to NSC and get the aggregate NSC. \[ w \geqslant \bar{w}+e+ \frac{e(a+b+r)}{q}\] Critical wage for not shirking is larger the higher the \(e\), \(b\), \(r\), \(\bar{w}\), \(a\), and the smaller the lower the \(q\). \(\tfrac{1}{a}\) is the expected unemployment duration. The longer the duration, the longer the punishment. In steady state, the rate of acquisition \(a\) is related to other variables. The flow into unemployment pool is \(bL\), and the flow out is \(a(N-L)\). Since they must be equal we have the following \[ \begin{split} a=\frac{bL}{N-L} \Rightarrow w & \geqslant e + \bar{w}+ \frac{e}{q} \left( \frac{bN}{N-L}+r\right) \\ & = e + \bar{w} \left( \frac{e}{q} \right) \left( \frac{b}{u+r} \right) \equiv \hat{w}\end{split} \] where \(u = \frac{N-L}{N}\) is the unemployment rate. No shirking cannot happen with full employment. Equilibirum occurs when aggregate demand for labor (how many workers hired at \(\bar{w}\) taking \(a\) as given) and NSC intersect. For \(\bar{w}\) we have \[ F^{\prime}(L) = e + \left( \frac{e}{q} \right) \left( \frac{bN}{N-L}+r \right) \] This is an equilibrium because raising wages is senseless, as there is appropriate effort and firms are getting all the labor they want. From the worker POV unemployment is involuntary. Indeed, the unemployed would be happy to work for equilibrium wage \(w^{*}\) or lower, but they cannot promise not to shirk at such a wage level. In \((w,L)\) space, shifts in NSC curve (\(\uparrow b\), \(\downarrow q\)) decrease incentive to exert effort. The labor demand curve is unaffected, and so the effect of shift is is \(\uparrow w\) and \(\downarrow L\), which implies \(\uparrow u\).

Now authors turn to welfare. The unemployment equilibrium is shown to not be Pareto optimal. They begin with a planned economy reminiscing of the lower-stage alluded to in the Gotha program, where workers own the means of production, but exist nevertheless in this environment. The planner must solve \[ \begin{split} \max_{w, \bar{w}, L} & \quad (w-e)L+\bar{w}(N-L) \\ \mathrm{s.t} & \quad \bar{w} \in \mathbb{R}^{+} \\ \mathrm{s.t} & \quad wL+\bar{w}(N-L) \leqslant F(L) \\ \mathrm{s.t} & \quad w \geqslant e + \bar{w} + \left( \frac{e}{q} \right) \left( \frac{bN}{N-L}+r \right) \end{split} \] Remember that \(U(w,e) = w-e \Rightarrow \frac{\partial U}{\partial w}=1 > 0\) so an extra dollar always increases utility. Because this utility is increasing in consumption and workers are risk neutral means we can increase \(w\) or \(\bar{w}\) to induce no slack in production, so we end up with \(F(L)-wL-\bar{w}(N-L)=0\). NSC also binds. Thus, we can write the problem as \[ \begin{split} \max_{w, \bar{w}, L} & \quad (w-e)L+\bar{w}(N-L) \\ \mathrm{s.t} & \quad F(L) -wL-\bar{w}(N-L) = 0 \\ \mathrm{s.t} & \quad w - e - \bar{w} - \left( \frac{e}{q} \right) \left( \frac{bN}{N-L}+r \right) = 0 \end{split} \] So the Lagrangian is \[ \begin{split} \mathscr{L} = & \: (w-e)L+\bar{w}(N-L) \\ + & \: \lambda\left[ w - e - \bar{w} - \left( \frac{e}{q} \right) \left( \frac{bN}{N-L}+r \right) \right] \\ + & \: \mu[F(L) -wL-\bar{w}(N-L)] \end{split}\] Taking the FOC we obtain \[ \begin{align} \frac{\partial \mathscr{L} }{\partial w} = L+\lambda-\mu L =0 && && \frac{\partial \mathscr{L} }{\partial \bar{w}}=(N-L)-\lambda-\mu(N-L) \end{align} \] So we get \(\lambda=L(\mu-1)\). We substitute it in \(\frac{\partial \mathscr{L} }{\partial \bar{w}} = N(1-\mu)\). Remember that Lagrange multipliers on \(\geqslant 0\) constraints must be non-negative. So \(\lambda \in \mathbb{R}^{+}\). Then \(\mu > 1\) because we have \(\tfrac{\lambda}{L}=\mu-1\). Since \(\lambda >0 \Rightarrow \mu-1 >0 \Rightarrow \mu >1 \). So \(\forall \: \bar{w} \in \mathbb{R}^{+} \Longrightarrow\) any smaller \(\bar{w}\) is better since \(\tfrac{\partial \mathscr{L} }{\partial \bar{w}} < 0\). So optimum is the corner solution \(\bar{w}=0\). With this we can re-write again \[ \begin{split} \max_{w, L} & \quad (w-e)L \\ \mathrm{s.t} & \quad wL\leqslant F(L) \\ \mathrm{s.t} & \quad w \geqslant e + \left( \frac{e}{q} \right) \left( \frac{bN}{N-L}+r \right) \end{split} \] In \((w,L)\) space iso-utility curves are rectangular hyperboles. With \(F^{\prime}>e\) they are steeper than average product locus. So optimum is the intersection of NSC and \(w = \tfrac{F(L)}{L}\) (wages equal average product of labor). However, market equilibrium is at a different intersection, between NSC and \(w= F^{\prime}(L)\). In CRS the market equilibrium is the optimum because \(F(L)= F^{\prime}(L)L\).

There are several relaxations made. Firstly, with endogenous monitoring (employers choose \(q\)), there is a trade-off between stricter monitoring and higher wages as disciplinary methods. Firms' intense monitoring is not optimal. Firms believe that they can only control shirking with higher monitoring, but unemployment is also there as a disciplinary measure. This saves on monitoring costs socially, gains that would offset ant loss from reduced employment. So a public policy would be like a tax on monitoring, which increases welfare. This is one of the reasons I find KelleyLaneSchonholzerAER2024 so stupid. Of course, the context is slightly different. Secondly, under risk aversion \(\bar{w}=0\) is clearly not optimal, but all firms will offer this anyway, which is a possible justification for minimum unemployment benefit levels. In Mexico, for example, there is no unemployment benefits. There is also no legal requirement. Finally, with endogenous turnover rate \(b\) the rate of hiring out of the unemployment pool is affected, and thus so is \(V_u\). This also affects other firms' NSC. So employment packages from firms are not optimal. A policy intervention would discourage turnover to make unemployment more costly to shirkers.

Finally we turn to less corpo-dystopian enforcement mechanisms. Firstly there is the suggestion of performance bonds. The problem with this and other such rewards is that firms have incentives to claim shirking, since there is no real way to discipline firms. A possible solution is firm reputation as good employers serving as a disclosed signal, but if the reputation mechanism is not perfect it will be contaminated by the unemployment one. Secondly, other costs (search costs, moving, friends at work) may be enough to induce effort. Finally, the authors argue that even with reputational costs (i.e. removing homogenous workers assumptions to recover stigma of firing or unemployment) there would exist unemployment as disciplinary device.

Very good and simple paper. One of the most emancipatory works in economic research, with postmodern qualities. Of course, it all remains within market ontology.