not reflect abflity differences. Second, higher-wage industries have higher capital-labor ratios, which suggests that they need more skilled workers. Third, workers in higher-wage industries have higher measured ability (in terms of education, experience, and so on); thus it seems likely that they have higher unmeasured ability. Finally, the same patterns of interindustry earnings differences occur, although less strongly, among self-employed workers.

The hypothesis that estimated interindustry wage differences reflect unmeasured ability cannot easily account for all of the findings about these differences, however. First, quantitative attempts to estimate how much of the differences can plausibly be due to unmeasured abihty generally leave a substantial portion of the differences unaccounted for (see, for example, Katz and Summers, 1989). Second, the unmeasured-ability hypothesis cannot readily explain Gibbons and Katzs findings about the wage cuts of displaced workers. Third, the estimated wage premia are higher in industries where profits are higher; this is not what the unmeasured-ability hypothesis naturally predicts. Finally, industries that pay higher wages generally do so in all occupations, from janitors to managers; it is not clear that unmeasured ability differences should be so strongly related across occupations. Thus, although the view that interindustry wage differences reflect unmeasured ability is troubling for rent-based explanations of those differences, it does not definitively refute them.

The second aspect of this literatures findings that is not easily accounted for by efficiency-wage theories concerns the characteristics of industries that pay high wages. As described above, higher-wage industries tend to have higher capital-labor ratios, more educated and experienced workers, and higher profits. In addition, they have larger establishments, and larger fractions of male and of unionized workers (Dickens and Katz, 1987b). No single efficiency-wage theory predicts aU of these patterns. As a result, authors who believe that the estimated interindustry wage differences reflect rents tend to resort to complicated explanations of them. Dickens and Katz and Krueger and Summers, for example, appeal to a combination of efficiency-wage theories based on imperfect monitoring, efficiency-wage theories based on workers perceptions of fairness, and worker power in wage determination.

In sum, the literature on interindustry wage differences has identified an interesting set of regularities that differ greatly from what simple theories of the labor market predict. The reasons for those regularities, however, have not yet been convincingly identified.

Problems

10.1. Union wage premia and efficiency wages. (Summers, 1988.) Consider the efficiency-wage model analyzed in equaflons (10.12)-(10.19). Suppose, however, that fraction f of workers belong to unions that are able to obtain

a wage that exceeds the nonunion wage by proportion p. Thus, = (1+, where Wu and w„ denote wages in the union and nonunion sectors; and the average wage, w, is given by fwu + (1 - Own- Nonunion employers continue to set their wages freely; thus (by the same reasoning used to derive [10.15] in the text), w„ = (1 - bu)w„ /(1 - p).

(a) Find the equilibrium unemployment rate in terms of p, b, f, and p. ib) Suppose = f = 0.15.

(i) What Is the equilibrium unemployment rate if /3 - 0.06 and b - 17 By what proportion is the cost of effective labor higher in the union sector than in the nonunion sector?

(ll) Repeat part (i) for the case of p = 0.03 and b = 0.5.

10.2. Efficiency wages and bargaining. Summers (1988, p. 386) states "In an efficiency wage environment, hrms that are forced to pay their workers premium wages suffer only second-order losses. In almost any plausible bargaining framework, this makes it easier for workers to extract concessions." This problem asks you to investigate this claim.

Consider a firm with profits given by 77 = (eL)"/a wL.O < a < 1, and a union witb objective ftmction U = w - x, where x is an index of its workers outside opportunities. Assume that the firm and the union bargain over the wage, and that the firm then chooses I taking w as given.

(a) Suppose that e = 1, so that efficiency-wage considerations are absent.

(i) What value of I does the firm choose given w7 What is the resulting level of profits?

(ii) Suppose that the firm and the union choose w to maximize Utt, where 0 <y < a indexes the unions power in the bargaining (this is known as the Nash bargaining solution). What level of w do they choose?

(ni) What is d(\nw)ldy at 7 = 0?

(b) Suppose that e is given by equation (10.12) in the text: e = [(w - x)/x] for w > X, where 0 < /3 < 1.

(/) What value of I does the firm choose given w? What is the resulting level of profits?

(ti) Suppose that the firm and the union choose w to maximize UTT-T.O < < a. What level of w do they choose? (Hint: for the case of p =0, your answer should simplify to your answer in part [a][ii].)

(in) What is d(lnw)lSy at = 0? Is this elasticity higher with efficiency wages than without, as Summers argues? Is the impact of bargaining on the wage qualitatively different with efficiency wages, as Summers implies?

10.3 Describe how each of the following affect equilibrium employment and the wage in the Shapiro-Stiglitz model:

(a) An increase in workers discount rate, p.

(b) An increase in the job breakup rate, b.

ic) A positive multiplicative shock to the production function (that is, suppose the production function is AF(L), and consider an increase in A).

id) An increase in the size of the labor force, I.

10.4 Suppose that in the Shapiro-Stiglitz model, unemployed workers are hired according to how long they have been unemployed rather than at random; specifically, suppose that workers who have been unemployed the longest are hired first.

(a) Consider a steady state where there is no shirking. Derive an expression for how long it takes a worker who becomes unemployed to get a job as a function of b, L, N, and L

(b) Let Vu be the value of being a worker who is newly unemployed. Derive an expression for Vu as a function of the time it takes to gel a job, workers discount rate (p), and the value of being employed (Vt).

(c) Using your answers to parts ( ) and (b), find the no-shirking condition for this version of the model.

(d) How, if at all, does the assumption that the longer-term unemployed get priority affect the equilibrium unemployment rate?

10.5 The fair wage-efforl hypothesis. (Akerlof and Yellen, 1990.) Suppose there is a large number of firms, N, each with prohts given by F{eL) - wL,F{-) > 0, f "(•) < 0. i is the number of workers the firm hires, w is the wage it pays, and e is workers effort. Effort is given by e = minlw/w*, 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 I 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 yields the highest profits? Eor 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 vv* is given by w = w + a ~bu,b > 0, where is the unemployment rate and W is the average wage paid by the Arms in this economy.

(i) Given your answer to part (a), what wage does the representative firm pay if it can choose w freely (taking w and 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 W as given, wishes to pay W.) What is the unemployment rate in this case?

(Hi) Under what conditions is there full employment?

(c) Suppose the representative firms production function is modified to be F(ALi + U),A > 1, where L] and Li are the numbers of high-