A Model of Imperfect Competition under Adverse Selection

This paper develops and analyzes a model of imperfect competition under adverse selection with private values among a finite set of heterogeneous principals (henceforth firms) for (a large number of) heterogeneous agents (henceforth workers). Firms differ in the technology that translates the effort of a worker into revenues for the firm, and we assume that firms are ordered by a single crossing property between effort and the type of technology (better firms have a higher marginal revenue from effort). In turn, workers differ in ability, which determines their disutility of effort. Firms compete with each other by offering menus consisting of wage-effort pairs (contracts), one for each type of worker. Alternative, one can think of firms offering effort-utility pairs for each type. After observing the menus, workers self-select by choosing the best contract for them. Although we cast the model as a labor market, it is straightforward to change the notation and think about it in terms of firms that produce goods of different qualities for different consumers. Instead of a revenue function there will be a cost function, instead of a wage a payment from consumers to firms, instead of worker's ability a consumer's value for quality, and instead of disutility of effort a utility for quality. The analysis reveals that instead of the standard efficiency vs. information rents trade off, the relevant one when there is imperfect competition among firms under adverse selection is efficiency vs. information rents plus market coverage trade off, since changing the menu offered not only affects efficiency and the information rents given to the workers hired but also affects the measure of workers targeted by a firm. We show that a pure strategy Nash equilibrium (PSNE) exists in this modified game (which is in turn a PSNE of the original game), and that in equilibrium the market segments into contiguous intervals, with each firm hiring only workers whose types belong to a given interval, with better firms targeting better interval of worker types and thus having better workforce composition. In equilibrium, the worst firm (in the single-crossing order defined above) distorts effort provision upward for all the types it serves, while the best distort it downward. All other firms exhibit both types of distortions: downward distortions for the lower types they serve, and upward distortions beyond an interior efficient type. Interestingly, the equilibrium effort function exhibits jumps at transition points between firms. In the firms-customers interpretation, this asserts that quality provided in equilibrium on the entire spectrum of the market will exhibit gaps, a potentially testable implication. Regarding curvature properties of the equilibrium menus, we show that it exhibits `quantity discounts' in the following sense: the wage per unit of effort is decreasing in the amount of effort induced in each worker hired by the firm (a similar interpretation holds for the other applications mentioned above regarding firms and customers).