Exercise for Unemployment

Question 1

In a competitive labor market, shirking on the job can be a problem. In this market for labor services, the demand for labor is expressed as:

$$W_d=\frac{270,000,000}{L^{1.5}}$$

Where $W$ is wage rate (dollars per hour) and $L$ is number of employed per unit of time. The no-shirking constraint is expressed as:

$$NSC=\frac{L^2}{1\times 10^9}$$

Where $NSC$ is the minimum wage workers need not to shirk, and $L$ is the employment rate. Assume that the labor force $L^{\ast }=150,000$. Determine the following:

1. The level of unemployment that would result when firms pay the efficiency wage.
2. The market clearing wage.
3. The efficiency wage.

Answer

1

Since it is the efficiency wage rate, so that $NSC=W_d$, that is,

$$\frac{L^2}{10^9}=\frac{27\times 10^7}{L^{1.5}}$$

After calculation, $L_e=95585$, then we use the existing labor force to subtract it: $L^{\ast }-L^e=54415$, which is also the Involuntary Unemployment.

2

When market clearing, it is that all labor force participate in the work. That is

$$w_d=\frac{27\times 10^7}{(150,000{)}^{1.5}}=4.648$$

3

$$w_e=\frac{27\times 10^7}{(95585{)}^{1.5}}=9.14>>w^{\ast }$$

Question 2

Consider the following explanation of why wages can be above the competitive equilibrium prediction, as suggested by Shapiro and Stiglitz (1984). Assume a risk neutral worker with utility $u(w,e)=w-e$, where effort $e$ is binary $e=\{0,1\}$. The firm obtains a product $\alpha m$, where $\alpha >0$ and where $m$ represents the number of workers choosing an effort $e=1$. The firm can monitor only one worker, each with equal probability, paying each worker salary $w$ if he exerts a positive effort $e=1$, but zero otherwise. There are $n>2$ workers in the firm.

1. For a given salary $w$, under which conditions does each worker choose effort $e=1$?
2. Set up the firm’s profit maximization problem, find the profit-maximizing salary $w$ and the number of workers $n$ hired.
3. If monitoring was perfect, how do your results in Q 1 and Q 2 change?