The Rebitzer and Taylor Efficiency Wage Model

Since the monopsony model proved to be an unlikely explanation for the results described thus far, economists have looked to other models. Bhaskar and To considered the monopsonistic competition model where firms have monopsony power due to non-wage differentials between jobs.[3] In other words, since different people prefer different types of occupations, a certain amount of monopsony power is possible even for a small firm that employs from a large labor market. Rebitzer and Tayor consider the efficiency wage model to explain some of these findings.[8] In standard labor market analysis, the wage is normally set by the market forces. In the efficiency wage model, the wage is set above the market wage due to the added benefits of paying a higher wage. It is based on the assumption that the productivity of the workers hired in a firm is a function of their wage. Paying the efficiency wage means that the firm will attract more productive workers, have less job turnover, and lower recruiting costs. The intuition behind the following analysis is as follows,

Higher Wage$$\Rightarrow$$   Higher Cost of Losing Job$$\Rightarrow$$   Lower Monitoring Cost$$\Rightarrow$$   Increase Employment$$.$$

To begin this model, consider a firm that maximizes a standard profit function, $\pi(w,l).$⁶ To accomplish this, the firm determines optimal labor and wage by taking the partial derivatives of $\pi$ with respect to $l$ and $w$ and setting them equal to zero. Rearranging the terms of these optimizations yields the following equation:

$$\frac{d l}{d w} = \frac{-\partial^2 \pi / \partial w \partial l}{\partial^2 \pi / \partial l^2}.$$ (1)

This derivative will normally have a negative sign, indicating that an increase in the wage will have to be accompanied by a decrease in employment. Thus $d l / d w < 0$. We will show that in some cases (besides monopsony) the sign on this derivative could be positive, thus explaining the positive employment effects of a minimum wage. We will start with the assumption that there are a large number of small firms and each of these firms must monitor their employees to make sure they are being productive and not shirking. The cost of monitoring the employees increases with the size of their workforce. Define an output function for the firm, $f(w,l) = g(l)$ if and only if workers are not shirking. Otherwise, $f(w,l) = 0,$ because a firm cannot be productive if its employees are not putting a sufficient amount of effort, $e$, into their jobs. Now, consider a worker at this firm. He or she must decide to work and provide an effort level of at least $e$, or to shirk, but risk getting caught and dismissed from the job. The decision would surely not only depend on the wage of the present job, but also the utility he would gain from working elsewhere. To continue the analysis, we must first define several new variables.

$$\begin{array}{llll} V^N &=& \textrm{The level of utility g... ...e level of utility gained from not working.}\\ \end{array}$$

The precise definitions of these utility equations can be found in Appendix B. These three equations make intuitive sense once one breaks them down into several parts. The expected flow of utility from not shirking, $V^N$, is first a function of the wage minus the effort that must be put into the job. It is also a function of the recursive term which accounts for the utility from working in the future, given the possibility of quitting and discounted by $(1+r)$. The last term comes from the possibility that a worker will quit the firm and resort to the next best alternative. The expected lifetime utility of a shirker, $V^S$, is derived from their wage, the same recursive term as above, but this time is reduced by $(1-D)$, or the probability of not getting caught. Finally the last term reflects the utility from leaving the firm, either by getting caught or quitting, and then moving into the next best alternative. The utility gained from persuing an alternative way of life, $V^A$, is a function of the $\bar{w}$ term, plus a discounted recursive term that reflects the probability of attaining another job versus remaining unemployed. The goal of the firm, by ensuring that their workers are being productive and not shirking, is to set a wage such that $V^N = V^S.$ They want to make sure that the utility that an employee gets from working hard, which is a function of their wage, is just high enough to be greater than the utility from shirking. Setting equation 13 equal to equation 14, a firm determines the lowest possible wage it can pay to assure that its workers provide optimal effort. This reduces to the non-shirking wage, $w_{ns}$, such that:

$$w_{ns} = \bar{w} + e + \frac{e(r+s+q)}{D(1-q)}.$$ (2)

Substituting $w_{ns}$, the efficiency wage, into the production function, we obtain the profit function, $\pi(w,l) = f(w,l) - wl = g(l) - w_{ns} l.$ Taking the derivative of profits with respect to labor, the resulting equation is simply the difference between marginal product and marginal cost since $g(l$) is equal to total product and $w_{ns}l$ is equal to total cost. This is a normal result one would expect for a profit maximizing firm. To determine $l'(w)$, take the reciprocal of the derivative of equation 2 with respect to $l$, yielding:

$$l'(w) = -\frac{D^2(1-q)}{e(r+s+q)D'(l)}>0.$$ (3)

Now $l'(w)$, or equivalently, $dl/dw$, is the relationship that we have been trying to find from the beginning. It is going to be positive because all of its terms are positive. Thus the no shirking condition is upward sloping. In order to show that the profit maximizing condition also holds when wages and labor rise, a graphical representation is necessary. Consider the efficiency wage model in figure 4. Figure 4 is actually just the competitive labor market model that was shown in figure 2, but also included is the no-shirking condition and an isoprofit curve. The no-shirking curve represents the lowest wage that would guarantee high effort. Notice that the no-shirking curve and $MC_L$ are upward sloping reflecting the increasing cost of monitoring larger numbers of workers and the result in equation 3. The isoprofit curve represents combinations of wages and quantity of labor where profits are equal. The firm continues to set $MRP_L = MC_L$ to determine $L_0$, but similar to monopsonists, they do not have to pay a wage equal to the marginal revenue product. With the inclusion of the no-shirking constraint, before the minimum wage is in place the firm hires $L_0$ workers at a wage of $w_0$ because this is where the isoprofit curve is just tangent to the no shirking condition. Moving away from this point will either result in lower profits if the wage is above $w_0$ or a violation of the no-shirking requirement if the wage is below $w_0$. Now, consider a binding wage floor $w_{min}$. With the wage in place, $MC_L = w_{min}.$ Since initially, $MC(L_0)$ was above $w_{min}$, instating a minimum wage actually lowers the marginal cost of hiring an additional employee. This results in a firm hiring additional workers and employment rises to $L_1$. As shown in figure 4, since the wage must be high enough to satisfy the no-shirking condition, the firm must now be on a new (lower) isoprofit curve, but they are at virtually the same level of profits as before. After the minimum wage increase, firm profits do fall in the short term. For the long run analysis, we will extend the profit function as follows,

$$\pi = pg(l) - wl - R.$$ (4)

Where R represents the normal profit of the firm, or the cost of using one's entrepreneurial abilities in the production of a good. Therefore $\pi$ is truly the economic profit of the firm. To determine the wage, as was shown before, take the derivative of profits with respect to labor and set it equal to zero. Also set equation 4 equal to zero because in the long run, firms should be making zero economic profit. This simplifies to,

$$p(AP_L - MP_L) = \frac{R}{l} - lw'(l).$$ (5)

We find that setting these two equations equal yields an equation involving the difference between the average product of labor and the marginal product of labor. Consider figure 5 which shows their relationship to each other and to marginal and average cost. We will now consider several cases in the long run.

• If $R$ is small, the right hand side of equation 5 will be negative implying that $MP_L > AP_L$. This implies that the firm is on the downward sloping portion of their long-run average total cost curve. It has been shown that following a minimum wage increase, employment will rise in the short run. Once profits fall, firms will be forced to leave the industry, causing total employment to fall back to the original level. However, at the remaining firms in the market, employment per firm has increased. Since the firms are on an increasing portion of their $AP_L$ curves, industry output will have risen if only a relatively small number of firms exited, and price therefore must fall. If profits are still negative, more firms will exit. However, in the end, though per firm employment has risen, industry employment is lower.
• If $R$ is large, the right hand side of equation 5 will be positive implying that $MP_L < AP_L$. This implies that the firm is on the upward sloping portion of their long-run average total cost curve. So following the minimum wage increase, employment will rise in the short run. Once profits fall, firms will be forced to leave the industry, causing total employment to fall back to the original level. However, at the remaining firms in the market, employment is higher per firm. Since the firms are on a decreasing portion of their $AP_L$ curves, industry output will have fallen, and price therefore must rise. Profit will be positive which will cause firms to enter until the zero profit equilibrium is attained. Thus, industry employment rises above the previous equilibrium level in the long run.
• If $\frac{R}{l} - lw'(l) = 0$, then $MP_L = AP_L$, and firms must be operating at the minimum of their long run average cost curve. An increase in the minimum wage here will cause employment to rise in the short run and profits will fall. As firms exit the industry, industry employment falls back to its original level though per firm employment has actually risen.

This model has clearly shown that via the efficiency wage model, one can show that an increase in the minimum wage will have positive employment effects in the short run and ambiguous effects in the long run. However, if firms are initially operating at the minimum of their long run average cost curve, long run industry employment remains unchanged.