Shapiro-Stiglitz theory

Carl Shapiro

Joseph E. Stiglitz

The Shapiro-Stiglitz theory is an economic theory of wages and unemployment . She explains why wages cannot clear the labor market and why there is involuntary unemployment in equilibrium . It is based on the idea of asymmetrical information in the employment relationship between employee and employer , more precisely the inability of the employer to monitor the employee adequately and free of charge. The Shapiro-Stiglitz theorem was developed by Carl Shapiro and Joseph E. Stiglitz .

overview

A wage above the market clearing level leads to unemployment ${\displaystyle \textstyle w_{G}}$

The Shirking approach is based on the fact that monitoring of employees is not possible for cost reasons, but only random checks of performance. When there is full employment and a worker is laid off for inadequate work performance, he will immediately find a new job. Since shirking lowers a company's productivity, it increases wages to reduce shirking. If all companies do this, there will be unemployment. In equilibrium, all companies will pay a wage that is higher than the market-clearing wage, making job loss more costly for workers. Unemployment serves as an incentive for workers to be very committed. An unemployed person cannot make credible that he would work effectively for a wage less than the efficiency wage . Unemployment is involuntary.

model

The model explains the incentive effect of unemployment on the effort of workers in a company.

Workers

There is a fixed number of identical workers. All have a utility function of form , which means that they derive a benefit from their wages, which they receive and from which they finance their consumption, and a disuse from the work effort . A worker can choose between minimal effort ( ) and positive effort ( ). If he is unemployed, he receives unemployment benefit, (and ). A worker is likely to lose his job even if he has worked hard. There is also an interest rate of . ${\displaystyle \textstyle N}$ ${\displaystyle \textstyle U(w,e)=w-e}$ ${\displaystyle \textstyle w}$ ${\displaystyle \textstyle e}$ ${\displaystyle \textstyle e=0}$ ${\displaystyle \textstyle e>0}$ ${\displaystyle \textstyle {\overline {w}}}$ ${\displaystyle \textstyle e=0}$ ${\displaystyle \textstyle b}$ ${\displaystyle \textstyle r>0}$

A worker is faced with the decision to choose his level of exertion, which is by assumption a discrete choice. If he opts for the high level of effort, he receives the wages and working so long, until it a result of exogenous factors dismissal comes. If he chooses the low level of exertion (shirking) he is likely to be spotted doing it and then released. ${\displaystyle \textstyle w}$ ${\displaystyle \textstyle q}$

There are:

• ${\displaystyle \textstyle V_{E}^{N}}$: total expected benefit of a non-shirker
• ${\displaystyle \textstyle V_{E}^{S}}$: overall expected benefit of a shirker
• ${\displaystyle \textstyle V_{U}}$: total expected benefit of an unemployed person

The following applies to a Shirker:

${\displaystyle {\begin{aligned}r\cdot V_{E}^{S}=w+(b+q)\cdot (V_{U}-V_{E}^{S})\end{aligned}}}$ and thus ${\displaystyle {\begin{aligned}V_{E}^{S}={\frac {w+(b+q)\cdot V_{U}}{r+b+q}}\end{aligned}}}$

and for a non-shirker:

${\displaystyle {\begin{aligned}r\cdot V_{E}^{N}=w-e+b\cdot (V_{U}-V_{E}^{N})\end{aligned}}}$ and thus ${\displaystyle {\begin{aligned}V_{E}^{N}={\frac {(w-e)+b\cdot V_{U}}{r+b}}\end{aligned}}}$

A worker will only choose to work when applies. This is the non-shirking condition (NSC) . It follows: ${\displaystyle \textstyle V_{E}^{N}\geq V_{E}^{S}}$

${\displaystyle \textstyle w\geq r\cdot V_{U}+(r+b+q)\cdot {\frac {e}{q}}={\hat {w}}}$

${\displaystyle \textstyle {\hat {w}}}$ describes the critical wage at which the worker just decides to work.

${\displaystyle \textstyle {\hat {w}}}$ must be higher, the

• higher the effort required
• the worker's benefit is greater when he is unemployed
• it is less likely to be discovered
• the interest is higher
• the exogenous termination rate is higher

Alternatively, the NSC can be represented in the form . Thus the meaning of unemployment becomes clear: Without unemployment, a shirking worker would incur no costs if he is discovered and dismissed, as he finds a new job immediately after his dismissal. Because then it would be , and the NPC could never be met. ${\displaystyle \textstyle q\cdot (V_{E}^{S}-V_{U})\geq e}$ ${\displaystyle \textstyle V_{U}=V_{E}^{S}}$

employer

There are identical firms that all have a form production function , which leads to an aggregate form production function . describes the effective workforce of the company . Assume that a non-acting worker is an effective worker. ${\displaystyle \textstyle M}$ ${\displaystyle \textstyle i=1,...,M}$ ${\displaystyle \textstyle Q_{i}=f(L_{i})}$ ${\displaystyle \textstyle Q=F(L)}$ ${\displaystyle \textstyle L_{i}}$ ${\displaystyle \textstyle i}$

The wage bundle that every company offers consists of a wage and an unemployment benefit . It is optimal for any company to lay off shirking workers as this is the only punishment for them. A wage cut would only induce the employee to shirk. ${\displaystyle \textstyle w}$ ${\displaystyle \textstyle {\overline {w}}}$

It follows from the NSC that all firms will provide the lowest possible unemployment benefit. An increase in increases and thus increases the critical wage . A higher unemployment benefit costs companies twice: on the one hand directly because of the higher , on the other hand indirectly through the higher wages that the company then has to pay. ${\displaystyle \textstyle {\overline {w}}}$ ${\displaystyle \textstyle V_{U}}$ ${\displaystyle \textstyle w}$ ${\displaystyle \textstyle {\overline {w}}}$ ${\displaystyle \textstyle w}$

In addition, the company pays a wage to induce high work exertion. ${\displaystyle \textstyle w={\hat {w}}}$

balance

The total expected benefit of an unemployed person is given by

${\displaystyle \textstyle r\cdot V_{U}={\overline {w}}+a(V_{E}-V_{U})}$,

where the percentage of the unemployed who find employment and the expected total utility of an employed worker (equilibrium applies ). ${\displaystyle \textstyle a}$ ${\displaystyle \textstyle V_{E}}$ ${\displaystyle \textstyle V_{E}=V_{E}^{N}}$

It follows:

${\displaystyle \textstyle r\cdot V_{E}={\frac {(w-e)(a+r)+{\overline {w}}\cdot b}{a+b+r}}}$

${\displaystyle \textstyle r\cdot V_{U}={\frac {(w-e)\cdot a+{\overline {w}}\cdot (b+r)}{a+b+r}}}$

If you put the expression from in the NSC, you get ${\displaystyle \textstyle V_{U}}$

${\displaystyle \textstyle w\geq {\overline {w}}+e+e(a+b+r)/q}$.

This leads to the new realization that the higher the proportion of unemployed people who are newly employed, the higher the critical wage.

The number of employees who lose their jobs is . The number of unemployed people finding new employment is . In the state of equilibrium these agree: ${\displaystyle \textstyle b\cdot L}$ ${\displaystyle \textstyle a(N-L)}$

${\displaystyle \textstyle b\cdot L=a(N-L)}$.

If you insert the expression for in the NSC, you get: ${\displaystyle \textstyle a}$

${\displaystyle \textstyle w\geq e+{\overline {w}}+{\frac {e}{q}}({\frac {b\cdot N}{N-L}}+r)=e+{\overline {w}}+{\frac {e}{q}}\cdot ({\frac {b}{u}}+r)={\hat {w}}}$,

where represents the unemployment rate. ${\displaystyle \textstyle u={\frac {(N-L)}{N}}}$

Unemployment in balance

This shows that it is not possible to prevent workers from shirken when there is full employment ( ) and every dismissed worker would immediately find a new job ( ). Every company takes it for granted and knows that it has to offer at least a wage of . How many workers the company hires for each wage indicates the company's labor demand. ${\displaystyle \textstyle L=N}$ ${\displaystyle \textstyle a=+\infty }$ ${\displaystyle \textstyle a}$ ${\displaystyle \textstyle {\hat {w}}}$

The balance is at the intersection of the NSC and the company's labor demand.

For : ${\displaystyle \textstyle {\overline {w}}=0}$

${\displaystyle \textstyle F'(L)=e+{\frac {e}{q}}\cdot (b\cdot N/(N-L)+r)}$

In equilibrium , the company has no incentive to further increase wages, since the workers already choose the high level of exertion. Lowering wages would induce shirking. ${\displaystyle \textstyle E}$

Welfare

Unemployment that exists in equilibrium is inefficient. Every company hires too few workers. The private cost of hiring an additional worker is while the social cost is, which is less. On the other hand, companies that hire an additional worker do not notice the effect . Hiring an additional worker reduces unemployment. This leads to a negative externality of overemployment. However, the first effect dominates, which means that there is too much unemployment in equilibrium and that it is therefore not Pareto-efficient . ${\displaystyle \textstyle w}$ ${\displaystyle \textstyle e}$ ${\displaystyle \textstyle V_{U}}$

Alternative methods

• Deposit : Every worker leaves a deposit when starting work in a new company. If the worker is caught shirking and fired, he loses bail. The problem here is that the worker must first of all have enough own resources to pay the deposit. A more serious problem, however, is that the company would now have an incentive to accuse a Shirking worker, then fire him, get bail and replace him with a new bail worker.

• Seniority remuneration: Here, the worker is initially paid a wage that is below his marginal product and later above it. An employee therefore has no incentive to shirk, as he would lose his future higher wages if he were laid off. However, here too there is a problematic incentive for a company to lay off workers who receive wages above their productivity and replace them with younger workers.

• Further costs of a layoff : In the model, unemployment serves to generate costs for laid-off workers. However, if other costs, such as the cost of finding a new job, moving costs or the costs that arise when company-specific human capital is lost, are sufficiently high, they alone offer an incentive not to work - even with full employment.

• Heterogeneous workforce : The assumption that all workers are identical ensures that they will not carry any stigma in the event of a layoff . If this assumption is relaxed, a worker will endeavor not to be laid off for shirking, as his next potential employer would suspect that he will also work in the new company again. If this risk exists and the costs are sufficient, this can act as an incentive to show a high level of labor input despite high employment.

See also

• Efficiency wage

literature

• Salop, SC (1979): A Model of the Natural Rate of Unemployment . In: American Economic Review, 69, pp. 117-125
• Shapiro, C. and J. E. Stiglitz (1984): Equilibrium Unemployment as a Worker Disciplin Device , American Economic Review , Volume 74, pp. 433-444.
• Yellen, JL (1984): Efficiency Wage Models of Unemployment , American Economic Association, Volume 74, pp. 200-205