Protection with static collusion.

• Author: Hartigan, James C.

1. Introduction

   Recent developments in doupoly pricing by Stahl [17] and Bhaskar [4] have disclosed that collusion can occur in a single period game when firms can revise prices rapidly. This phenomenon may be termed static collusion. Previous game theoretic of collusion have required infinite period games of complete information, or finite (but greater than one) period games of incomplete information.(1)

   The present paper argues that when static collusion takes place, there is a new justification for a tariff. This occur in the context of a model in which consumers observe prices at a cost, and the industry structure is a duopoly with a home and a foreign firm. Because costly search induces imperfect information about prices on the part of consumers, the ability to rapidly revise posted prices on the part of the duopolists leads to the appropriation of a substantial amount of consumer surplus. The static collusion takes place under any trade policy. Thus a home tariff that shifts a duopoly equilibrium to the advantage of the home firm will be demonstrated to entail, in the case of a nonprohibitive tariff, no consumption cost from protection. Because this tariff generates revenue for the home government and greater profits for the home firm, it will be welfare improving to the home country.

   The concept of profit shifting as a motive for commercial policy in imperfectly competitive markets was pioneered by Brander an Spencer [6].(2) In a model where home and foreign duopolists are Nash-Cournot competitors, a tariff can shift profits from the foreign to the home firm by inducing the foreign firm to play less aggresively. In the present paper, the tariff shifts profits within the collusive outcome by precluding more aggressive play by the foreign firm.(3) Furthermore, Davidson [8] had disclosed that the level of a tariff can affect the incentives to collude in a quantity setting supergame with Nash-Cournot punishments. In the present model, the tariff affects the collusive outcome that can be supported. It does not, however, affect the incentives to collude.

   The paper assumes that consumers are distinguished by their costs of search. There are two categories of consumers: high and low search costs. The low cost consumers gather information more efficiently that their high cost counterparts. This can be due to differences is ability, education, transportation, or proximity to retail facilities.(4)

   Firm behavior falls within the purview of imperfectly competitive models of international trade. The duopolists are endowed with complete and almost perfect information. They post prices simultaneously so that expected profits are maximized. The firms are assumed to price with ex post flexibility. That is, they post prices with zero menu costs. They can instantaneously revise a posted price if their rival tries to expand its expected sales beyond the collusive outcome. This revision precludes an undercutting strategy from being effective. Thus, the firms engage in static collusion.

   The tariff is set by the home government prior to the beginning of play. The firms post prices simultaneously in stage one and the consumers search and purchase in stage two. That is, the firms price as von Stackelberg leaders with respect to consumers. The level of the tariff affects the collusive equilibrium. In particular, the foreign firm is assumed to have a sufficiently large cost advantage for it to capture all of the low cost consumers in free trade. That is, the free trade equilibrium is characterized by price dispersion. The foreign firm will post a price that is sufficiently below that of the home firm that low cost consumers who sample the home firm initially will then sample the foreign firm. (Note, however, that this still entails collusion.) A low tariff imposed by the home government will induce a degenerate price distribution. A high nonprohibitive tariff will enable the home duopolist to capture all of the low cost consumers. The price distribution will be identical to that of free trade. Hence, there will not be any impact on home consumers, and the home firm will gain.

   Because obtaining price quotes is costly for the home consumers, they must devise a sampling strategy. They are endowed with knowledge of the first two moments of the distribution of prices, and the number of firms in the industry. They are assumed to sample sequentially, and to do so randomly and without replacement.(5) Drawing one price quote enables them to determine what price each firm has posted. However, they still must pay the sampling cost in order to contact the other firm, even though they have determined the price it has posted.

   The model is solved in reverse for subgame perfection. Hence, the consumer equilibrium is discussed. next. Following that, the duopoly equilibrium under free trade, a low tariff, and a high tariff is disclosed. A conclusion follows.

2. The Consumers

   Consumers are homogeneous in their tastes for the good. being characterized by a common parameter [Theta]. However, they are distinguished by their costs of search. There are [n.sup.l] low cost consumers that obtain price quotes at a cost of [s.sup.l] where [n.sup.l] [Epsilon] R +. Then [n.sup.l] high cost consumers obtain a price quote at a cost of [s.sup.h], where [n.sup.h] [Epsilon] R +. Furthermore, [s.sup.h] > in R +. The low cost consumers, through education and/or access to technology, are more efficient in their search activities.

   Each consumer maximizes the expected net surplus from consumption of a unit of the good subject to an expenditure constraint.(6) Expected expenditures is the sum of expected purchase price and the expected number of samples drawn times the costs of sampling. (1) [e.sup.k] = [p.sup.ek] + [a.sup.k] [s.sup.k] for k = h,l, where [p.sup.ek] is the price that the consumer of category k expects to pay, and [[Alpha.sup.ak] is the number of samples this consumer expects to draw to purchase being made. Letting [u.sup.k] denote expected net surplus for category k consumers permits the following definition: (2) [u.sup.k] = [Theta] - [p.sup.ek] - [[Alpha].sup.k] [s.sup.k] for k = h, l.

   Each consumer is endowed with the knowledge of the first two moments of the price distribution and the size of the industry. They do not know which (if any) firm has posted the lower price. However, their knowledge does permit them to infer the posted prices prior to sampling. If a consumer learns that firm j = d, f has posted the lower price by contacting i = d, f and i [is not equal to] j, (s)he must still incur [s.sup.k] = h, l in order to contact firm j, where d(f) denotes the home (foreign) duopolist. The consumer sample randomly without replacement, and do so sequentially.(7,8)

   Determining the producer equilibrium requires specification of the maximum price that each category of consumer will pay for a unit of the good. This depends upon the number of samples [[Alpha].sup.k] that each category of consumers expects to take, which, in turn, depends upon the dispersion of posted prices. For a distribution in which [p.sub.i] - [p.sub.j.] [is less than or equal to] [s.sup.k] for k = h, l, consumers expect to sample only once. Suppose that [p.sub.i] [is greater than or equal to] [p.sub.j]. If firm i is drawn initially, there isn't any gain from sampling j. This is because the total expenditure of doing so ([p.sub.j] + [s.sup.k] is at least as great as the initial price drawn. Thus [[Alpha].sup.k] = 1. Letting p [bar] denote the average posted price (which is also the expected price here) permits a necessary condition for drawing a sample to be (3) [Theta] [is greater than or equal to] p [bar] + [s.sup.k] if [p.sub.i] - [p.sub.j] [is less than or equal to] [s.sup.k] for k = h, l.

   If [p.sub.i] - [p.sub.j] > [s.sup.k] for i, j = d, f, i [is not equal to] j, and k = h, l, category k consumers expect to sample 3/2 times. That is, [[Alpha].sub.k] = 3/2. In this case, [e.sup.k] = [p.sub.j] + [3s.sup.k] /2 as a consumer drawing [p.sup.i] initially will sample again. Category k consumers will only buy from firm j = d, f. That is, [p.sub.j] is the price they expect to pay. The only question is the number of times a sample must be drawn before the lower price firm is contacted (recall that prior to drawing a sample, they do not know which firm has posted the lower price, but they do know the value of this lower price, due to their knowledge of the price distribution). The condition for sampling in this case is (4) [Theta] [is greater than or equal to] [p.sub.j] + [3s.sup.k]/2 if [p.sub.i] - [p.sub.j] > [s.sup.k] for k = h, l.

   To determine the...