• 10 August 2009

A Formal Model of Passive Discrimination

Jonah Gelbach & Lesley Wexler & Jonathan Klick

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In this Editorial, we present a basic, one-period microeconomic model in which equilibrium occurs in both perfectly competitive labor markets and goods markets. This piece is a companion to our earlier Legal Workshop Editorial, Passive Discrimination, which was posted on June 22, 2009. Because all hypothesized workers are equally productive, and because no market power exists and no state mandate supports segregation, models like this one typically cannot sustain a deliberately segregated equilibrium. However, when workers’ preferences for an amenity good are correlated with worker types, segregated equilibria are possible, and possibly even unique. Prejudiced firms can use compensation plans that combine cash wages and fringe benefits in an effort to hire only the favored types of workers.

We first discuss some basic background details of the model, focusing on the production technology, competitive labor and goods markets, and the (assumed) systematic differences in preferences of two worker types: Deltas and Omegas. Then we introduce the possibility of compensation plans that involve both cash wages and fringe benefits, considering first the case where firms face the same amenity price as do their workers. In such cases, a no-fringe-benefits equilibrium exists. This equilibrium is integrated. However, at least one cash-and-fringe compensation plan1 allows prejudiced firms to hire only Deltas in equilibrium. The resulting equilibrium is segregated, in the sense that at least some firms may (a) deliberately avoid hiring Omegas, and (b) stay in business. Interestingly, the Omegas’ utility is no lower in this equilibrium than it would be in the cash wage-only equilibrium. This result follows because of the competitive nature of the labor market, which ensures that other firms will hire Omegas and pay them their marginal product.2 If discrimination is regarded as socially or individually harmful in ways that do not show up in consumption-based utility, then Omegas may still be worse off in a segregated equilibrium.

We also consider the case where firms have a price advantage in purchasing amenities, perhaps because of economies of scale. In this case, no integrated equilibrium exists. Any equilibrium necessarily involves cash-and-fringe compensation plans used to pay Deltas and a cash-only plan used to pay Omegas. Banning fringe benefits would (a) eliminate segregation, (b) reduce Deltas’ utility compared to the segregated equilibrium, and (c) have no impact on the Omegas’ utility compared to the segregated equilibrium. We note that this condition for equilibrium would hold even when employers harbor no animus toward Omegas. If discrimination is regarded as socially or individually harmful in ways that do not show up in consumption-based utility, then Omegas’ welfare may be improved by a policy of banning fringe benefits.

Model Details

Suppose two types of workers, Deltas and Omegas, exist. Every member of each group is identical to all other members of that group in terms of tastes. Preferences of Deltas and Omegas are given by

(1) UΔ = xα b1-α – c × W                                                    

(2) UΩ = x – c × W

where x is the number of units of a generic consumption good a person consumes, b is the number of units of an amenity good (mnemonically, think of this good as “beer”), c is the disutility of working, W equals one if the person works for pay and zero otherwise, and α is a preference parameter for Deltas and lies between zero and one.3

            We assume that all jobs in the economy involve the same type of labor, all workers have one unit of labor to offer, and all workers are equally productive. Firms can hire as many workers as they like. If hired by a firm, each worker can produce Q units of the generic good with her one unit of labor. We normalize the price of a unit of the generic consumption good to be one,4 and we assume that the price of the amenity good is fixed at level pb.

A.     Optimal Consumption and Labor Supply Decisions

Let YΔ be the income a Delta worker receives if she works; assuming for simplicity that she has zero income otherwise, her income equals W × YΔ. Her utility-maximizing choice of consumption in terms of x and b will then be given by whatever choice of (x, b) maximizes UΔ subject to the constraint that

(3) x + pb × b = W × YΔ

Equation (3) is known as the budget constraint. Each unit of x costs one unit of income, and each unit of b costs pb, so altogether the Delta’s total expenditure is x plus pb × b. In our static, one-period model, individuals lack a reason to save any income, and thus people will want to spend all their income. Similarly, individuals lack an opportunity to borrow, so people’s spending will be limited to their income.

The form of preferences we have assumed for Deltas implies that their optimal consumption choice is to spend the fraction α of income on the generic consumption good and the remainder on the amenity. Thus, optimal consumption levels for a Delta are given by

(4) x* = α × YΔ     and      b* = (1 – α) × YΔ / pb

if she works for pay, and zero otherwise.5 If the Delta works, then her utility will be

(5) UΔ,work = [αα(1 - α)1-αpbα-1] × YΔ – c

(6) = vα(pb) × YΔ - c

where the first term on the right hand side of (5) is the result of plugging in the optimal consumption levels in (4) to the utility function in (1). The function vα(pb) collects the part of UΔ,work that varies with the preference parameter and the amenity price. The important thing to notice is that this function, and thus the highest possible value of utility, UΔ,work, is a decreasing function of the amenity price.

The above assumptions imply that if a Delta forgoes work, then her utility will be zero.6 Thus, she will work for pay if and only if the right-hand side of (6) is non-negative. In other words, inducing Deltas to work requires that firms pay them at least enough income, YΔ, to allow them to realize c units of consumption utility.

Next, consider the simpler case of Omegas. Let YΩ be an Omega’s income if she works for pay, and again assume zero income for nonworkers. Omegas have very simple consumption plans based on (2): they consume all their income by purchasing the generic good and spend nothing on the amenity good. Since each unit of the generic good costs one unit of income, this means that Omegas will consume YΩ units of the generic good. Utility for working Omegas is thus

(7) UΩ,work = YΩ – c

As with Deltas, Omegas will work if and only if this utility level is at least zero, since that is their utility if they do not work. Thus, an Omega worker will be willing to work if and only if she is paid at least YΩ. We assume that each worker’s productivity, Q, is greater than the maximum of (YΩ – c) and (vα(pb)× YΔ – c). This assumption ensures that the labor market equilibria described below exist.

B.     Labor Market Equilibrium with Fringe Benefits

We allow for firms to pay both fringe benefits and cash wages. Firms will provide compensation by giving fringe-receiving workers some number of units of the amenity good. We assume that workers cannot resell these fringe benefits.7 Since Omegas place no value on the amenity good, intuition suggests that employers should be able to design a combination of wages and amenities compensation that will (a) attract Deltas, and (b) repel Omegas. This intuition is correct, as we now show.

1.   Firms Face Amenity Price pb

Consider a firm that wishes to hire only Deltas. Suppose this firm offers a cash-wage amount, Yf, less than the labor disutility, c, together with some positive number, bf, of units of the amenity good. We refer to this compensation plan as F = (Yf, bf). Omegas place no value on the amenity goods, and we have assumed away the possibility of resale. Therefore, the value of this compensation to an Omega is simply the Yf units of the generic good that the Omega can purchase with the cash wage. Since Yf  < c by assumption, an Omega would rather have zero income than work for this compensation plan.

Will compensation plan F attract Deltas? Suppose that bf  = b* from equation (4). This will cost the firm pb × b* = (1 – α) × Q. Suppose the firm combines this fringe-benefit level with the cash wage Yf = α × Q. This compensation plan costs the firm exactly Q dollars, which is the break-even worker cost that allows firms to produce in competitive equilibrium as explained above. Thus, this compensation plan is feasible from the firm’s point of view. The compensation plan also allows a Delta to attain exactly the consumption bundle she would have chosen for herself had she been paid Q dollars in cash and nothing in fringe benefits. Since we assumed previously that α and the amenity price pb were such that Deltas would choose to work when offered the wage Q, they will also choose to work when offered the compensation plan F just described.

We conclude that there exists a segregated equilibrium in which some firms—those that screen out Omegas, which we call screening firms—offer compensation plan F only, and other firms offer a cash wage of Q together with no fringe benefits. In this equilibrium: (a) only Deltas work for the screening firms, (b) Omegas work only for cash-only firms, and (c) some Deltas may work for cash-only firms.8 In fact, when Deltas strictly prefer to work when offered a no-fringe wage of Q, firms can design a variety of compensation plans that generate segregated equilibria with fringe benefits.9 It can be shown that all equilibria require that the cost of a screening firm’s compensation plan equal Q. Any firm that pays less than that will face competition for its workers, as before. Finally, we note when no prejudiced firm owners exist, both the segregated and the integrated equilibria described above continue to exist, with firm owners being indifferent between them. We have thus shown that when firms face the same amenity price as consumers, they can design a compensation plan that both repels Omegas and attracts Deltas and allows the firm to stay in business in competitive equilibrium.

2.   Firms Face Amenity Price pbf < pb

Next, we consider the case when firms have a price advantage relative to consumers in purchasing the amenity, so that the per-unit price for the amenity that firms must pay is pbf  < pb. An example is group purchase of insurance plans, but many other examples exist. In equilibrium, this price advantage means that Deltas must always be paid a compensation plan that involves fringe benefits. The reason is simple: any firm that pays a Delta only in cash is providing less than the maximum possible utility that can be provided at that cost. Another firm could come along and offer to pay the Delta slightly less in cash together with some fringe benefits. Because firms acquire the fringe benefits more cheaply than Deltas, the second firm could provide greater utility to the Delta than the first, while paying less to do so. The Delta would switch jobs and the second firm would earn a greater profit than the first. Thus no competitive equilibrium exists in which any Delta is paid only in cash.

In fact, firms’ advantage in purchasing the amenity means that Deltas will want their employers to purchase all units of the amenity that the Deltas consume. As in the models above, equilibrium requires that firms pay Q for each worker, whether Delta or Omega; if a firm paid less than that, our familiar compensation-competition story would apply. Thus, Deltas will be paid a cash-and-fringe compensation plan that costs Q dollars, while Omegas will once again be paid Q dollars in cash. To find the utility level for Deltas in equilibrium, we need only act as if the Deltas themselves faced the firms’ amenity price, pbf, rather than the higher price of pb. Thus, we simply plug pbf into equation (6) above, yielding

(8) UΔ,f = vα(pbf) × Q – c

Now, it is easy to show that when pbf  < pb, it must be true that

(9) vα(pbf) > vα(pb)

and this implies that

(10) UΔ,f > UΔ,cash only = vα(pb) × Q – c

We have thus shown that when firms have an amenity-price advantage relative to consumers, Deltas’ utility is strictly greater in the with-fringe equilibrium than it would be if fringe benefits were banned. This equilibrium, which is unique under the argument above, is segregated. However, since Omegas continue to receive cash compensation in the amount of Q dollars, their equilibrium utility is unaffected by the existence of fringe compensation: banning fringe compensation plans would not increase Omegas’ utility. A fringe ban in the presence of an amenity-price advantage on the part of firms would simply cause deadweight loss by forcing Deltas to purchase amenities at an unnecessarily high price, while giving nothing extra to Omegas. Notice that if firms must provide only one compensation plan, workers will be segregated in equilibrium even if no employers harbor animus toward Omegas: the fact that employers have a cost advantage in providing the amenity, while worker type is perfectly correlated with amenity preference, ensures full separation of workers. If firms can offer compensation menus, however, then unprejudiced employers can avoid segregation by offering workers their choice of plan F or all-cash compensation of Q dollars.

Conclusion and Extensions

When firms possess no cost advantages for amenities, multiple equilibria exist, including a nonsegregated one in which employers pay every worker in wages only. When firms possess an amenity cost advantage and cannot offer workers a choice between compensation plans, a unique equilibrium exists, and it is segregated. Interestingly, Omegas are just as well-off economically; they would not benefit economically from eliminating fringe-induced segregation. However, Deltas are strictly better off in the segregated equilibrium than in the no-fringe equilibrium when firms have a cost advantage. Hence, banning segregation-inducing fringe compensation would (a) eliminate segregation, (b) not improve the economic welfare of Omegas, and (c) economically harm Deltas.

Some firms may possess market power in either the labor or product markets, which would allow economically harmful discrimination to persist. Fringe-generated segregation might be especially troublesome in such market-power cases because employers could use it to skirt easily monitored disparate-treatment proscriptions. Banning fringe benefits would still harm Deltas if firms have an amenity-price advantage, though with market power, such a ban could also help Omegas. Fringe-based discrimination could be prevented in this model without harming Deltas by mandating that firms offering fringe benefits also offer a cash-only compensation plan whose wage/salary equals the cost to the firm of the cash-and-fringe compensation plan.10

It is, of course, also possible that segregation is undesirable in its own right because stereotypes break down in integrated workplaces, because individuals incur psychic costs in experiencing discrimination even if they voluntarily sort themselves into segregated workplaces, or because the individuals that do not sort themselves out may incur psychic costs in a segregated workplace.11 In such situations, reducing the net pecuniary compensation received by Deltas might be worthwhile in order to bring about an integrated economy. Dealing with either of these extensions would markedly change the welfare implications of the segregated result, and we do not mean to discount the relevance of either case. However, in terms of consumption utility, segregation induced by fringe compensation does not harm the group that is “segregated against,” given perfect competition.dingbat



Copyright © 2009 University of Chicago Law Review.

Jonah Gelbach is Associate Professor of Economics at University of Arizona.
Jonathan Klick is Professor of Law at University of Pennsylvania Law School.
Lesley Wexler is Assistant Professor of Law at Florida State University College of Law.

This Editorial is a companion to the following previous Legal Workshop Editorial: Jonah Gelbach, Jonathan Klick & Lesley Wexler, Passive Discrimination, LEGAL WORKSHOP (U. CHI. L. REV. June 22, 2009).

The following is a Response by Richard Epstein to this series of Editorials: Richard A. Epstein, Protect Us, Lord, from Title VII: A Response to Gelbach, Klick, and Wexler, LEGAL WORKSHOP (U. CHI. L. REV., June 22, 2009).

  1. Typically, infinitely many compensation plans exist.
  2. If participating in an economy with a segregated workforce itself bothers these workers, then their overall welfare will be reduced in any segregated equilibrium. Where this fact is relevant, we will note it below. However, the equilibria themselves do not depend on the existence of such a phenomenon.
  3. The function f(x, b) = (x^α)(b^(1 – α)) is an example of a type of preferences known as Cobb-Douglas. This is a very standard form to assume for preferences, because it leads to the result that the consumer spends the fraction α of her income on good x and the rest on good b. See Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory 55 (Oxford 1995). Nothing important about our results hinges on assuming this type of preferences; we do so for expositional ease.
  4. In an equilibrium model like this one, generality is not lost in making such a normalization because only relative prices matter. As a result, we simply choose to measure the currency in convenient units.
  5. To see why equation (4) holds, first consider optimal consumption on the generic good. The Delta spends the fraction α of her income Y{Δ} on this good, and each unit of x costs one dollar, since the price of x is one by assumption. Thus she will buy α × Y{Δ} units of this good. She will spend the remainder of her income, (1 – α) × Y{Δ}, on the amenity good. Its price is p{b}, so we must have p{b} × b* = (1 – α) × Y{Δ}, and dividing by p{b} yields the expression in the text.
  6. This is another normalization: it does not affect any result but simply involves choosing a convenient basis for measurement.
  7. This assumption is stronger than necessary. We just need the cost of resale to be sufficiently positive (in other words, transactions costs are nonzero).
  8. For simplicity, we assume that neither type of worker is unemployed in equilibrium; free entry by firms is sufficient for this result. In addition, and also for simplicity, we assume (a) that there are fewer prejudiced firms than there are Deltas, or (b) that prejudiced firm owners would prefer to operate with an Omega than to go out of business. Assumption (a) ensures that in segregated equilibria, all prejudiced employers hire only Deltas. Assumption (b) ensures that Omegas will be employed in a segregated equilibrium even if Deltas are rationed in such an equilibrium. It would be straightforward to derive assumption (a) as a result of a slightly more general model that required capital for production, with capital having positive opportunity cost. In such a model, prejudiced employers who are unable to hire Deltas would exit the industry, choosing to do something else with their costly capital. Since the return on capital in this industry would rise, other nonprejudiced capital owners would then enter, and these employers would be willing to hire Omegas. This entry would continue until the industry had no more unemployed Omegas, at which point we would have an equilibrium like the one described in the text. These sorts of assumptions and arguments are conventional in the study of how perfect competition interacts with employers’ taste-based preferences over worker types.
  9. In each of these equilibria, screening firms offer a fringe level b{f} that is more than zero and not more than b*, with the cash wage then equaling Q minus p{b} × b{f}.
  10. It is important to note that the proper mandate would involve the cost of the cash-and-fringe compensation plan to the firm, not its value to Deltas. Mandating the latter would have the effect of raising the cost of employing Omegas relative to Deltas, even though each type of worker is equally productive.
  11. See Devah Pager and Hana Shepherd, The Sociology of Discrimination: Racial Discrimination in Employment, Housing, Credit, and Consumer Markets, 34 Annual Rev Sociology 181, 183 (2008) (discussing costs such as depression, anxiety, and other negative health outcomes, as well as diminished effort or performance in the workplace).

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