The fair wage-effort hypothesis. (Akerlof and Yellen, 1990.) Suppose there are a large number of firms, N, each with profits given by F (eL)−wL, F (•) > 0, F (•) < 0.="" l="" is="" the="" number="" of="" workers="" the="" firm="" hires,="" w="" is="" the="" wage="" it="" pays,="" and="" e="" is="" workers’="" effort.="" effort="" is="" given="" by="" e="">∗,1], where w∗ is the “fair wage”; that is, if workers are paid less than the fair wage, they reduce their effort in proportion to the shortfall. Assume that there are L workers who are willing to work at any positive wage.

(a) If a firm can hire workers at any wage, what value (or range of values) of w minimizes the cost per unit of effective labor, w/e? For the remainder of the problem, assume that if the firm is indifferent over a range of possible wages, it pays the highest value in this range.

(b) Suppose w∗ is given by  where u is the unemployment rate and w is the average wage paid by the firms in the economy. Assume b > 0 and a/b <>

(i) Given your answer to part (a ) (and the assumption about what firms pay in cases of indifference), what wage does the representative firm pay if it can choose w freely (taking w and u as given)?

(ii) Under what conditions does the equilibrium involve positive unemployment and no constraints on firms’ choice of w? (Hint: In this case, equilibrium requires that the representative firm, taking was given, wishes to pay w.) What is the unemployment rate in this case?

(iii) Under what conditions is there full employment?

(c) Suppose the representative firm’s production function is modified to be  where L₁ and L₂ are the numbers of high productivity and low-productivity workers the firm hires. Assume that , where w∗ i is the fair wage for type-i workers.

given by  is the average wage paid to workers of type i, and u_i is their unemployment rate. Finally, assume there are L workers of each type.

(i) Explain why, given your answer to part (a) (and the assumption about what firms pay in cases of indifference), neither type of worker will be paid less than the fair wage for that type.

(ii) Explain why w1 will exceed w2 by a factor of A

(iii) In equilibrium, is there unemployment among high-productivity workers? Explain. (Hint: If u1 is positive, firms are unconstrained in their choice of w1.)

(iv) In equilibrium, is there unemployment among low-productivity workers? Explain.

Sol:

(a) The firm obtains e units of effective labor for a wage cost of w. Thus the cost to the firm of one unit of effective labor is w/e.

See the figure at right, which plots the cost of a unit of effective labor, w/e, as a function of the firm&#39;s wage relative to the fair wage, w/w*. We can see that any wage such that w/w* £ 1 or w £ w*, minimizes the cost of effective labor.

(b) (i) Given the assumption that the firm pays the highest wage in the range described in part (a), it chooses w = w*; that is, in order to minimize the cost per unit of effective labor, the firm chooses to pay the fair wage. Thus

(3) w = w +a - bu.

(b) (ii) Assume that there is positive unemployment. If this is the case, the firm is unconstrained in its choice of the wage and therefore pays the fair wage, nothing more. Since this is true for all firms, the average wage, w, must equal w. Thus we have

(4) w = w + a - bu.