Rent shrinking.

• Author: Alexeev, Michael

1. Introduction

   Economic models of such diverse activities as rent seeking, patent races, wars, and election campaign spending often employ the assumption that the competitors are chasing a prize of a fixed size.(1) In many circumstances, however, the size of the prize that goes to the winner is a decreasing function of the competitive expenditures of the losers. In a patent race, for example, the value of the patent to the winning firm will often be lower if other firms have been actively competing. The research expenditures of the close substitutes provided by the losing firms makes the patent less valuable to the winning firm. Anticipating this behavior, firms may have less incentive to engage in research and development when the actions of their competitors lower the value of the patent. A second example involves the market for corporate control. A takeover competition can lower the value of the firm that is the prize if the firm's current management responds to the takeover threat with a poison pill. The value of the firm may be lowered even if the takeover attempt is unsuccessful, i.e., if the current management wins the contest.

   This article examines situations where the size of a prize to the winner falls as the expenditures of competing firms increase, a phenomenon we call "rent shrinking." A commonly used model of rent seeking is modified to incorporate rent shrinking. Rent shrinking lowers the rent seeking expenditures of the competing firms.(2) Collusive behavior, which acts to reduce the number of competitors, is more likely to be profitable with rent shrinking. By reducing the number of competitors, collusion increases the prize available to the colluding firms, ceteris paribus. This represents an additional benefit to collusion available in a rent shrinking environment.

   In some circumstances, the shrinking of the rent to the winning firm that arises from opponents' expenditures is mirrored by the consideration that the losing firms' expenditures were not pure losses. For example, losers in a patent race may still be positioned to capture some rent, if they can successfully innovate around the patent. Such effects can be incorporated into the rent shrinking model, by assuming that losing firms are reimbursed for some percentage of their expenditures. This approach also serves to unify the standard rent seeking game with the rent shrinking game. If none of the expenditures of losing firms are reimbursed, the game is one of pure rent shrinking. If the entire expenditures of losers are reimbursed, the standard rent seeking game emerges. As the share of losers' expenditures that are returned goes from 0 to 1, the game moves from pure rent shrinking to one of pure rent seeking, and an individual firm's expenditures monotonically increase.

   The basic rent seeking and rent shrinking models are presented in section II, where it is shown that rent shrinking reduces the expenditures of firms relative to rent seeking. Section III examines the potential for collusive behavior by firms. Section IV includes the possibility that a fraction of the expenditures of losing firms may be returned, an approach that generalizes the rent seeking and rent shrinking games; collusive behavior is also examined in this more general framework. Section V presents conclusions.

2. The Model

   The standard rent seeking game consists of n risk neutral firms competing for a fixed rent of size [Pi].(3) Let [x.sub.i] [greater than or equal to] 0 represent the rent seeking expenditures of firm i, i = l, . . ., n. The probability that firm i wins the prize, [p.sub.i], is given by firm i's share of total rent seeking expenditures: [p.sub.i] = [x.sub.i]/([x.sub.i] + [summation of][x.sub.j]where j[not equal to]i). Firm i chooses expenditures [x.sub.i] to maximize its expected utility [U.sub.i] = ([p.sub.i] [center dot] [Pi]) - [x.sub.i]. At the symmetric Nash equilibrium, firm i's rent seeking expenditures are [Mathematical Expression Omitted], and its expected utility is [Mathematical Expression Omitted]. Total rent seeking expenditures are (n - 1)[Pi]/n. As the number of competing firms n gets large, total rent seeking expenditures approach the size of the rent [Pi], though an individual firm's rent seeking expenditures fall as n rises.(4)

   The standard rent seeking game can be modified to incorporate rent shrinking. Let the original prize remain fixed at size [Pi]. If firm i, i = 1, . . ., n, is the winner, however, firm i only gets to enjoy a...