One weakness of Coate and Loury (1993a)’s model is that wages are not determined in a competitive labor market, but are fixed exogenously. Because affirmative action policies change the profitability of hiring workers from different groups, this is not an innocuous assumption. Moreover, workers from the discriminated group face a more favorable task assignment rule, but, conditional on the signal, receive the same wages as before, therefore affirmative action can only be a benefit to them.
Moro and Norman (2003a) study the effect affirmative action policies in the general equilibrium setting analyzed in Section 4.1, where wages are determined endogenously by firms engaged in Bertrand competition for workers. Their analysis confirms the perverse
incentive effects of government-mandated policies found by Coate and Loury (1993a). More- over, it finds perverse effects on equilibrium wages and proves that in some circumstances affirmative action may hurt its intended beneficiaries.
The affirmative action constraint is the same as that assumed in Section 6.2.1, that is, employers are forced to hire the same proportion of workers from both groups in the complex task (and, residually, in the simple task). Employers therefore solve the following problem (assuming for simplicity that groups have identical size):
max ˜ 𝜃𝐵,˜𝜃𝑊 𝑦 (𝐶, 𝑆) = max ˜ 𝜃𝐵,˜𝜃𝑊 𝑦 ⎛ ⎝ ∑ 𝑗=𝐵,𝑊 𝜋𝑗 [ 1 − 𝐹𝑞(˜𝜃𝑗) ] , ∑ 𝑗=𝐵,𝑊 [ 𝜋𝑗𝐹𝑞(˜𝜃𝑗) + (1 − 𝜋𝑗)𝐹𝑢(˜𝜃𝑗) ] ⎞ ⎠ s.t. 𝜋𝐵𝐹𝑞(˜𝜃𝐵) + (1 − 𝜋𝐵)𝐹𝑢(˜𝜃𝐵) = 𝜋𝑊𝐹𝑞(˜𝜃𝑊) + (1 − 𝜋𝑊)𝐹𝑢(˜𝜃𝑊)
Denote ˆ𝜃𝑗(𝝅), 𝑗 = 𝐵, 𝑊 the optimal group-specific cutoff rules that solve this problem
for a given vector 𝝅 = (𝜋𝐵, 𝜋𝑊). Employers assign all workers with signal above such
thresholds to the complex task, and all other workers to the simple task. Observe that from the constraint, it follows directly that if 𝜋𝐵 < 𝜋𝑊 then ˆ𝜃𝐵(𝝅) > ˆ𝜃𝑊(𝝅). The direct
(partial-equilibrium) effect of the policy on the task assignment rule is to force employers to lower the task assignment threshold for the discriminated group, and to raise the threshold for the dominant group. It can be proved that the equilibrium wages are:
ˆ 𝑤𝑗(𝜃; 𝜋) = { 𝑝(ˆ𝜃𝑗(𝝅), 𝜋𝑗)𝑥𝑞( ˆ𝐶, ˆ𝑆) for 𝜃 < ˆ𝜃𝑗(𝝅) 𝑝 (𝜃, 𝜋𝑗) 𝑥𝑞( ˆ𝐶, ˆ𝑆) for 𝜃 ≥ ˆ𝜃𝑗(𝝅) (52)
where ˆ𝐶, ˆ𝑆 are the optimal inputs of the production function computed from the opti- mization problem satisfying the affirmative action constraint, 𝑥𝑞 and 𝑥𝑢 are the marginal
products of workers in the complex and simple task, and 𝑝 (𝜃, 𝜋𝑗) is the probability that a
worker with signal 𝜃 is qualified, given by (4). This result says that the wage is a continu- ous function of the signal, that workers in the complex task are paid exactly their marginal products, and that workers in the simple task are paid the wage of the marginal worker. In the simple task, workers are therefore paid above the marginal product if they belong to the dominant group, and below their marginal product if they belong to the discriminated group. Figure6 illustrates the equilibrium wages under the assumption 𝜋𝐵< 𝜋𝑊.
The proof of this result first argues that wages must be continuous, otherwise one employer could exploit the discontinuity and increase profit by offering a slightly higher wage to workers that are cheaper near the discontinuity, and zero to workers that are more expensive. Second, note that there is a difference between quantity of workers in the complex task and their labor input, because not all workers employed in the complex
Group 𝐵 Group 𝑊 - 6 𝜃 𝑤 𝑥𝑢( ˆ𝐶, ˆ𝑆) ˆ 𝜃𝐵 6 - 𝑤 𝑥𝑢( ˆ𝐶, ˆ𝑆) ˆ𝜃𝑊 𝑥𝑞( ˆ𝐶, ˆ𝑆)𝑝(𝜃, 𝜋𝐵) 𝑥𝑞( ˆ𝐶, ˆ𝑆)𝑝(ˆ𝜃𝑊, 𝜋𝑊) 𝑥𝑞( ˆ𝐶, ˆ𝑆)𝑝(𝜃, 𝜋𝑊) 𝑥𝑞( ˆ𝐶, ˆ𝑆)𝑝(ˆ𝜃𝐵, 𝜋𝐵) 𝜃
Figure 6: Equilibrium wage schedules under affirmative action in Moro and Norman (2003) task are productive. If workers in the complex task were not paid their expected marginal product, then employers could generate a profitable deviation that exploits the difference between quantity of workers and quantity of effective inputs.18 But because of continuity, this implies that workers in the simple task are paid above or below the marginal product depending on their group identity. It is not difficult to show from the first order condition of the task assignment problem that the average pay of all workers in the simple task (from both groups) is exactly the marginal product 𝑥𝑢( ˆ𝐶, ˆ𝑆).
Incentives to invest for group 𝑗 are: 𝐼𝑗(𝝅) = ∫ 𝜃 ˆ 𝑤𝑗(𝜃)𝑓 𝑞(𝜃) − ∫ 𝜃 ˆ 𝑤𝑗(𝜃)𝑓𝑢(𝜃), 𝑗 = 𝐵, 𝑊 (53)
and the equilibria are characterized by the solution to the system of fixed-point equations 𝜋𝑗 = 𝐺(𝐼𝑗(𝝅)), 𝑗 = 𝐵, 𝑊 where as usual 𝐺 is the CDF of the cost of human capital
investment. Any symmetric equilibrium of the model without the policy trivially satisfies the affirmative action constraint and therefore is also an equilibrium under affirmative action.
The full equilibrium effects of affirmative action are indeterminate. While it is possible that imposing affirmative action completely eliminates asymmetric equilibria, it is also possible for asymmetric equilibria to exist that satisfy the quota imposed by the policy for reasons similar to those illustrated by the patronizing equilibria derived in Section6.2.2. A proof may be derived by construction fixing fundamentals 𝑦, 𝑓𝑞 and 𝑓𝑢, and looking for a
cost of investment distribution 𝐺 that satisfies the equilibrium conditions under affirmative action. Note that if 𝜋𝐵 = 0, and 0 < 𝜋𝑊 < 1, then from (52) and (53) it must be
that 𝐼𝐵(0,𝜋𝑊) = 0 < 𝐼𝑊(0, 𝜋𝑊) (all group-𝐵 workers are offered zero wage, equivalent
to their productivity in the complex task but some are employed in the complex task to satisfy the affirmative action constraint). But then since 𝐼𝑗(⋅) is continuous and initially
increasing near 𝜋𝐵= 0, one can find 𝜋𝐵 > 0 such that 0 < 𝜋𝐵 < 𝜋𝑊 < 1 and, at the same
time, 0 < 𝐼𝐵(𝜋𝐵,𝜋𝑊) < 𝐼𝑊(𝜋𝐵, 𝜋𝑊). Hence one can find a strictly increasing CDF 𝐺 such
that 𝐺(0) > 0, 𝐺(𝐼𝐵(𝜋𝐵,𝜋𝑊)) = 𝜋𝐵, and 𝐺 (𝐼𝑊(𝜋𝐵, 𝜋𝑊)) = 𝜋𝑊 so that (𝜋𝐵, 𝜋𝑊) is an
equilibrium of the model.
In general, comparing outcomes with and without the policy is difficult because outcomes depend on the equilibrium selection. It is possible to show that the policy may have negative welfare effects for its intended beneficiaries. The negative direct effects on the discriminated group’s wages are evident from Figure 6. The picture however hides the full equilibrium effects because factor ratios will change in equilibrium. Unless such factor ratios do not change significantly, expected earnings for group-𝐵 decrease. Note also from the figure that the direct effect of the policy is to increase incentives to invest for the discriminated group. This tends to moderate the negative wage effects, but unless this effect is significant, workers in the discriminated groups are made worse-off by the policy.
The wage determination in this model is specific to the modeling assumptions made regarding production and information technologies. In this simplified setting, a slightly more complex policy that combines affirmative action employment quota and racial equality of average wages in each task would be effective in inducing symmetric equilibria. It is not clear, however, whether such a policy would be easily implementable in a more complex environment.19 Nevertheless, the model is useful to illustrate that affirmative action policies have non trivial general equilibrium effects.