# Protection with static collusion.

I. 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.

II. 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 maximum price that category k consumers are willing to pay for a unit, replace the weak inequalities in (3) with strict equalities, and recall that p [bar] = ([p.sub.i] + [p.sub.j])/2. This is stated as (5) [Mathematical Expression Omitted] for k = h, l; i, j = d, f and i [is not equal to] j, where [Mathematical Expression Omitted] is the highest price that category k consumers will pay firm i when firm j has posted [p.sub.j]. Since expected net surplus is zero, it is the highest price at which each category would purchase a unit. Because surplus is - [s.sup.k] if a category k consumer samples but does not purchase, these consumers would prefer a surplus of - [s.sup.k]/2 from making purchase. Thus expected surplus is zero, but realized surplus will be negative for half of the category k = h, l consumers when there is price dispersion. Since price dispersion affects the consumers' incentive to search, the price that firm j posts directly affects the price that firm i can post. Furthermore, [s.sup.h] > [s.sup.l] implies [Mathematical Expression Omitted].

Suppose now that there is sufficient dispersion to induce category k = h, l consumers that draw the higher priced firm i initially to sample again. This requires [p.sub.i] - [p.sub.j] > [s.sup.k]. Suppose also that expected net surplus is zero, so that [Theta] = [p.sub.j] + [3s.sup.k] 3s.sup.k]/2 in (4). Since the consumer knows that (s)he will purchase from firm j (that is, pj is the expected price), the highest price at which sampling and purchase will occur is (6) [Mathematical Expression Omitted] for k = h, l and j = d, f. A comparison of (5) and (6) reveals that the expectation of drawing more samples lowers the choke price for consumers: (7) [Mathematical Expression Omitted] for k = h, l; i, j = d, f and i [is not equal to] j.

When consumers enter the market to sample a firm, they calculate an endogenous reservation price at which they would buy the product. The reservation price is the highest price at which they would purchase a unit of the good, given the prices known by the consumers to be posted by the duopolists. Denoting the reservation price by [r.sup.k] for k = h, l, a consumer equilibrium is generated by (8) [Mathematical Expression Omitted] for i, j = d, f; i [is not equal to] j and [p.sub.i] > [p.sub.j].

Proof. If a random sample by a consumer yields [p.sub.j], (s)he will purchase a unit of the good as long as [Mathematical Expression Omitted] for k = h, l. If (s)he draws [Mathematical Expression Omitted] the other firm will be sampled as long as there is a gain from doing so. The other firm can be sampled at a cost of [s.sup.k], and its product purchase at a total expenditure of p.sub.j] + [s.sup.k]. Thus, there will be a gain from additional as long as [p.sup.j + s.sup.k] < [p.sub.i], in which case category k consumers will only purchase from the lower price duopolist irrespective of which firm is sampled initially. This implies that category k consumers would be willing to pay [r.sup.k] = [p.sub.j] + [s.sup.k] if that price were drawn initially. On the other hand, [Mathematical Expression Omitted] results in category k consumers purchasing from the first firm sampled.

A unit of the good is purchased by each consumer as long as [Mathematical Expression Omitted] for i = d, f. In order to further explore the relationship among prices, search costs, choke (maximum) prices, and goods, the duopolists' behavior must be explicitly portrayed.

III. The Duopolists

The production of the homogenous good is characterized by constant marginal costs. These are denoted by [c.sub.i] for i = d, f where [Mathematical Expression Omitted] for k = h, l in R +. Both duopolists post prices in order to maximize expected profits, which are denoted by [Pi].sub.i] for i = d, f. Expected profits are the difference between posted price and marginal cost multiplied by expected sales. Denoting the latter by [q.sub.i] we can state (9) [Pi].sub.i] = ([p.sub.i] - [c.sub.i])[q.sub.i] for i = d, f.

In choosing a profit maximizing price, recall that each duopolist has complete and almost perfect information.(9) Information is almost perfect because the firms post prices simultaneously. Since the duopolists post prices in stage one and consumers search in stage two, the duopolists are von Stackelberg leaders with respect to consumers.

In maximizing (9), the firms post prices with ex post flexibility. In markets where there are a small number of interdependent producers, and in which consumers are imperfectly informed about prices, it is reasonable to believe that producers are better informed about the prices posted by rivals than are consumers.(10) Because of this, each duopolist has the opportunity to reply to the price posted by its rival before consumer search takes place in the next stage. This is the basis for static collusion on the part of the duopolists. It requires zero menu costs in the posting of prices.(11) That is presented in this paper as a limiting case, in which prices can be revised with an arbitrarily short time lag.(12)

Denoting by [p [bar].sub.j] the price that firm i expects firm j to post (with [p [bar].sub.i] correspondingly defined), ex post price flexibility is depicted as (10) [Mathematical Expression Omitted] for i, j = d, f and i [is not equal to] j. If firm j deviates from the price that firm i expects it to post (with firm i posting the price that firm j expects i to post), firm i can prevent sales from taking place at the deviation price by quickly changing its own posted price. That is, it can preclude unanticipated undercutting. Hence imperfect consumer information limits ( in this case completely) the ability of a firm to gain at the expense of its rival by not acting collusively. As a result unanticipated prices are not observed. That is, [p.sub.j] = [p [bar].sub.j] for j = d, f.

In choosing its expected profits maximizing price, each duopolist can adopt an aggressive or a soft pricing strategy. These strategies are termed [s.sup.a] and [s.sup.s], respectively. They are defined as (11) [Mathematical Expression Omitted] and (12) [Mathematical Expression Omitted] where [q [bar].sub.j] is the level of sales that firm i expects firm j to make. Recall that [q.sub.i] is the sales that firm i expects to make. Under an aggressive pricing strategy, it is senseless to post a price below the price that a rival is expected to make, unless it is sufficiently below to induce those consumers that draw the rival initially to sample again. Thus expected sales are part of the definition of the strategies. With the soft pricing strategy, a firm is willing to allow itself to be undercut by its rival. Because the firms have complete and almost perfect information and price with ex post flexibility, as in (10), expected prices equal posted prices and each firm makes the level of sales that its rival expects it to make. That is, [p.sub.j] = [p [bar].sub.j] and [q.sub.j] = [q [bar].sub.j] for j = d, f.

If firm j does post a disequilibrium (unexpected) price for j = d, f, it is necessary to specify firm i's best reply for i = d, f and i [is not equal to] j. Recalling that each firm has complete and almost perfect information, and prices with ex post flexibility, this reply will depend upon the cost advantage (if any) that firm j is known by i to have. These replies are (13) [Mathematical Expression Omitted] for [c.sub.i] - [c.sub.j] [is less than or equal to] [s.sup.l]; i, j = d, f and i [is not equal to] j. (14) [Mathematical Expression Omitted] for [s.sup.h] [is greater than or equal to] [c.sub.i] - [c.sub.j] > [s.sup.l]; i, j = d, f and i [is not equal to] j. (15) [Mathematical Expression Omitted] for [s.sup.h] [is greater than or equal to] [c.sub.j] - [c.sub.i] > [s.sup.l]; i, j = d, f and i [is not equal to] j.

Replies (13) and (14) pertain to the soft pricing strategy. With these replies, the firms cannot expect to earn negative profits if the rival does not change its posted price. That is, [p.sub.i] [is greater than or equal to] [c.sub.i] for i = d, f. Of course, [Mathematical Expression Omitted]. That is, a reply is not credible if a firm cannot expect positive sales at that price. In (13), firm i will not permit j to post a price that induces any consumers that draw i initially to sample j. It maintains this price (or revises according to (13) if j posts some other disequilibrium price ) until j posts [p [bar].sub.j] for j = d, f and i [is not equal to] j. Firm j will revise its price quickly, as there is no gain to posting disequilibrium (unexpected) price. Because of ex post price flexibility, firm j cannot gain a temporary advantage by undercutting firm i. Since there isn't a significant cost advantage for either firm, their expected sales are identical. In the present case, these are ([n.sup.l] + [n.sup.h])/2. That is, neither firm has the cost advantage relative to consumer search costs to impose an uneven split of the market on its rival. In (14), firm j does have such a cost advantage. This allows firm j to post a price that induces all low cost consumers that draw i initially to sample again. (For instance, j could post [p.sub.j]: [p.sub.j] + [s.sup.l] < [c.sub.i].) Thus firm j's expected sales exceed those of i. Firm j's expected sales are [n.sup.l] + [n.sup.h]/2 and firm i's are [n.sup.h]/2. Hence, firm i's best reply to a disequilibrium price by j takes this into account.

The reply to a disequilibrium price when firm i is acting aggressively is depicted by (15). Firm i is known by both firms to have a cost advantage that enables it to capture all of the low cost consumers. If firm j posts a disequilibrium price, firm i replies by posting a price sufficiently below j's marginal cost of production to induce all low cost consumers that draw j to sample again. It maintains this price until [p.sub.j] = [p [bar].sub.j], which must occur very quickly.

These replies to disequilibrium prices by either of the duopolists ensure that such prices are not observed. Thus [q.sub.i] = [q [bar].sub.i] for i = d, f.

A strategy will be considered feasible if can be successfully implemented. That is, if a firm can price aggressively without its rival revising its posted price through its ex post price flexibility and precluding the former's expansion of its expected sales, then the aggressive strategy is feasible for the former firm. Otherwise, it is not. Strategy sets are defined by [S.sub.i] for i = d, f.

Free Trade

In the free trade duopoly equilibrium, it is assumed that [s.sup.h] [is greater than or equal to] [c.sub.d] - [c.sub.f] > [s.sup.l]. That is, the foreign firm has a cost advantage that enables it to capture all of the home country's low cost consumers. The producer equilibrium that emerges is depicted as (16) [p.sub.i] [is greater than or equal to] [c.sub.i] : [q.sub.i] > 0 for i = d, f. (17) [S.sub.d] = {S.sup.s}. (18) [S.sub.f] = {S.sup.a], [S.sup.s]} (19) [Pi].sub.f] (S.sup.a]) > [Pi].sub.f] (S.sup.f]). (20) [P.sub.f] = [P.sub.d] - [s.sup.l] - [Delta] for [Delta] [Epsilon] (0, [s.sup.h] - [s.sup.l]] (21) [Mathematical Expression Omitted] (22) 2 [s.sup.l] - 2 [s.sup.h] - [Theta] [is less than or equal to] 0.

Proof. Inequality (16) simply requires nonnegative profits for production to occur. Since a firm will not produce if it cannot cover its costs, its expected sales are positive only when (16) holds. Because of its cost disadvantage relative to the search costs of the low cost consumers, the home duopolist can only price softly. This is apparent from (14). The foreign firm, however, can price aggressively, as (15) reveals. It can, of course, also price softly. It can always respond to a disequilibrium price by the home firm according to (13). The home firm will not try to prevent soft pricing by the foreign firm as its expected sales would be ([n.sup.h] + [n.sup.l])/2 rather than [n.sup.h]/2. Thus, the firm's strategy sets are given by (17) and (18). Inequalities (19) states that the foreign firm will prefer to price aggressively when it is Given that the foreign firm can capture all of the low cost consumers because of its production cost advantage, ex post price flexibility, and response to a disequilibrium price, the home firm knows that it can expect to sell to only [n.sup.h]/2 consumers. It naturally wants to do so at the highest possible price. The foreign firm, knowing that it has the advantage to sell on [n.sup.l] + [n.sup.h]/2 consumers, also wants to do so at the highest possible price. Both firms know that their posted prices must be such that (3) holds, and their combines profits are maximized when it holds with equality. This is because high cost consumers will draw only once. They also know that that the foreign firm needs to preserve the incentive for low cost consumers that draw the home firm to sample again, and that the foreign firm will enforce this through (15) The static collusive equilibrium (focal equilibrium) that meets the price dispersion and profit maximization requirements has the foreign firm posting its price according to (20) and the home firm pricing according to (21). Equation (21) is derived from setting setting p [bar] = [Theta] - [s.sup.h] = ([p.sub.d] + [p.sub.d] - [s.sup.l] - [Theta]) /2. The foreign firm prefers that the home firm posts [p.sub.d]. Recognizing that the foreign firm will sell to [n.sup.l] + [n.sup.h]/2 consumers, the home firm prefers that the foreign firm posts [p.sub.d] - [s.sup.l] - [Theta]. Finally, (22) is derived by substituting (21) into (20), and then substituting (20) into (4). It indicates that the expected surplus of low cost consumers is positive, thereby ensuring that they will draw at least one sample, and will purchase a unit of the good. (They will sample only once if they draw the foreign firm on the first try.)

Protection with a Low Tariff

Suppose now that the government seeks to protect its constituent duopolist by imposing a tariff on the foreign firm's sales in the home market. This raises the cost of producing for and selling in the home market for the foreign firm. This is expressed as (23) [Mathematical Expression Omitted] where t denotes the tariff. In order for the tariff to change the duopoly equilibrium, it must be that [Mathematical Expression Omitted]. This removes the ability of the foreign firm to capture all of the low cost consumers. The tariff distorted producer equilibrium becomes. (24) [p.sub.d] [is greater than or equal to] [c.sub.d] > 0. (25) [Mathematical Expression Omitted] (26) [S.sub.i] = {[S.sup.s]} for i = d, f. (27) [Mathematical Expression Omitted] (28) ([Theta] - [s.sup.h] - [c.sub.d] ([n.sup.h] + [n.sup.l])/2 > ([Theta] - [s.sup.l] - [c.sub.d] [n.sup./2

Proof. Inequalities (24) and (25) are the nonnegative profits constraints. Because the home country's tariff has reduced the cost advantage of the foreign firm relative to the search costs of the low cost consumers, both firms can now only price softly. This is apparent from (13). Thus strategy sets are given by (26). Each firm knows that the absence of a production cost advantage (relative to search costs), the existence of ex post price flexibility, and the response to a disequilibrium price means that each firm cannot expect to sell to more than half of the market. That is, expected sales must be ([n.sup.h] + [n.sup.l]/2. This partition of the market can most profitably occur at the price at which high cost consumers expect zero surplus from purchasing a unit of the good. (See discussion of (28) below). In this equilibrium, high cost consumers expect and receive zero surplus, as [p.sub.f] = p [bar]. Inequality (3) reveals that consumers will sample only once in this circumstance. Equation (5) discloses that consumers will pay more than p [bar] when there is price dispersion. That is, when expected surplus is zero, but realized surplus for half of category [kappa] consumers is negative. However, this requires one duopolist to post a lower price than the other. Since each firm can impose equal expected sales on the other through (13), neither will be willing to post a lower price than the other to sell to ([n.sup.h] + [n.sup.l])/2 consumers. Thus the collusive equilibrium prices are given by (27). Inequality (28) states that the market is covered. That is, all consumers will buy a unit of the good in stage two. The high cost firm finds it more profitable to sell to ([n.sup.l] + [n.sup.h])/2 consumers at the price at which high cost consumers expect zero surplus rather than the price at which low cost consumers expect zero surplus (recall (3)), thereby excluding high cost consumers from the market.(14,15) If (28) holds for the high cost firm, it also holds for the low cost foreign firm.(16)

Returning to the consumer decision rules reveals an important implication of protection. With the producer equilibrium under the low tariff, [p.sub.d] = [p.sub.f] = p [bar]. Expected and realized surplus for high cost consumers is exactly zero, as p [bar] = [Theta] - [s.sup.h]. Note that [Mathematical Expression Omitted] for i = d, f from (5). However, high cost consumers pay an expected price under free trade which is denoted by [p.sup.e] and is expressed as (29) [p.sup.e] = ([p.sub.d] + [p.sub.f])/2. Substituting from (20) and (21) into (29) reveals [p.sup.e] = [Theta] - [s.sup.h]. That is, the expected price equals the mean price for the high cost consumers. Assuming that the consumers are risk neutral, reveals that protection does not have an impact on the expected surplus of the high cost consumers. Of course, realized surplus can be higher or lower under free trade, because there is price dispersion.

Because the high cost consumers sample only once in each equilibrium, the impact of protection on them be assessed through purchase price. However, low cost consumers expect to draw one sample under protection and 3/2 samples in free trade. Thus, the impact of protection on them must be assessed through expected expenditure. Under protection, expected expenditure is [Theta] - [s.sup.h] + [s.sup.l]. Since the low cost consumers will only buy from the foreign firm in free trade, the only issue is the number of samples that must be drawn before the foreign firm is located. Thus their expected expenditure under free trade is obtained by adding [3s.sup.l]/2 to (20), after substituting from (21). This yields [Theta] - [s.sup.h] = [s.sup.l] - [Delta]/2. Since expenditure under protection is [Theta] - [s.sup.h] + [s.sup.l], they are trivially harmed by protection.(17) Although the low cost consumers reduce their expected expenditure on search, this saving of expenditure is trivially exceeded by the higher price that these consumer pay under protection with a low tariff.

To assess the values of t for which the indicated shift in the duopoly equilibrium occurs, recall that [Mathematical Expression Omitted] in free trade. To obtain the protection equilibrium, [Mathematical Expression Omitted]. The smallest tariff that effects this change in the relationship between the marginal cost differences of the firms and the search costs of the low cost consumers is [Mathematical Expression Omitted]. The largest tariff that includes the protection equilibrium occurs with the home firm having a cost advantage over the foreign firm in selling to the home market. That is, [c.sub.f] + t > [c.sub.d]. This tariff must not be so large that low cost consumers that draw the foreign firm are induced to sample again. Thus [Mathematical Expression Omitted] The largest value of t that shifts the free trade equilibrium to the protection equilibrium is t [bar], where [Mathematical Expression Omitted]. Hence, any t [Mathematical Expression Omitted] generates the shift in the duopoly equilibrium discussed above.

The impact of the low tariff on the expected surplus of the home consumers is quite surprising. For any [Mathematical Expression Omitted], the tariff does not effect the cost advantage of the foreign firm relative to the search costs of the low cost consumers. Thus expected surplus for each category of consumer is identical to that of free trade. This is zero for the high cost consumers and [Mathematical Expression Omitted] for the low cost consumers. For t [Mathematical Expression Omitted], high cost consumers are not affected, but low cost consumers incur a decline of expected surplus to [s.sup.h] - [s.sup.l]. That is, relative to free trade, the low cost consumer will expect to pay ([s.sup.l] + [Delta])/2 more in purchase price with a low tariff. However, the low cost consumer expects to save [s.sup.l] /2 in search costs under that low tariff. Thus, the low tariff essentially entails a zero consumption cost of protection.

To consider the effect of the low tariff on home welfare, let (30) [Mathematical Expression Omitted] where [w.sub.d] is home welfare and m is the level of home imports. In the decision to apply a low tariff, the change in home welfare relative to free trade [Delta] [w.sub.d] is the relevant calculation. Since [u.sup.h] is always zero under static collusion, [n.sup.h] [u.sup.h] is always zero, Because t is zero in free trade, tariff revenue is also zero. Thus the change in home welfare from applying a low tariff is (31) [Mathematical Expression Omitted] The first expression on the right hand side of (31) is the increase in the home duopolist's profits from selling to an additional [n.sup.l] /2 consumers under the tariff. The second expression is the tariff revenue, where the tariff that maximizes the revenue for ([n.sup.h] + n.sup.l]) /2 sales by the foreign duopolist is the highest tariff that will induce this equilibrium. The third term is the loss in expected surplus for the low cost consumers resulting from not being able to buy from the foreign firm at a lower price as in free trade. The fourth term arises from the lower price the home firm posts in the degenerate price distribution under static collusion with a low tariff relative to the price it posts under the dispersed price distribution with free trade. (Note that the home firm sells to [n.sup.h] /2 high cost consumers in each equilibrium.) The sign of this expression is ambiguous. However, it is apparent that the home producer gains if the [Mathematical Expression Omitted]. This will occur as long as the frequency of low cost consumers in the home economy is reasonably large. Inequality (19) in the free trade equilibrium, which states that the foreign duopolist prefers to lower its price to capture the low cost consumer, suggests that this is the case. Thus (31) can be expected to hold, since tariff revenue exceeds the trivial loss in expected surplus of the low cost consumers. A low tariff can therefore raise home welfare when a home and a foreign duopolist engage in static collusion.

Protection with a High Tariff

Suppose that a nonprohibitive tariff t : t > t [bar] is imposed on foreign sales in the home market. This results in [s.sup.h] [is not greater than or equal to] [c.sub.f] + t - [c.sub.d] > [s.sup.l]. This provides the home firm with the cost advantage necessary to capture all of the low cost consumers under ex post price flexibility with complete and almost perfect information. The tariff induces the following (17') [Mathematical Expression Omitted] (18') [Mathematical Expression Omitted]. (19') [Mathematical Expression Omitted]. (20') [Mathematical Expression Omitted]. (21') [P.sub.d] = [P.sub.f] - [s.sup.l] = [Delta] for [Delta] [Epsilon] (0, [s.sup.h] - [s.sup.l]). Inequalities (16) and (22) continue to hold as in free trade. The proof is identical to that given earlier.

Because all low cost consumers buy from the home firm at [P.sub.d], and expect to draw 3/2 samples, there is no effect on their surplus when a high tariff is compared to free trade. That is, their expected expenditure is the same under both policies. Similarly, the high cost consumer are evenly distributed between the two firms, and their expected expenditure is identical under free trade and a high tariff.

Referring to equations (20), (20'), (21), and (21') reveals that the expected surplus levels are identical for each category of consumers under free trade and a high tariff. Since there is no consumption cost to a high tariff under static collusion, the entire welfare depends upon the change in the home duopolist's profits relative to free trade, and the tariff revenue. This is expressed as (31') [Mathematical Expression Omitted] where t [Epsilon] (t [bar], [t.sub.p]) in R + with [t.sub.p] being the prohibitive tariff.(18) The first term in (31') is the home government's tariff revenue. The second term is the loss in the home duopolist's profits from selling to the [n.sup.h] /2 high cost consumers at a lower price. The third term is the gain the home duopolist's profits from selling to an additional [n.sup.l] consumers relative to free trade. Since t > t [bar] = [c.sub.d] - [c.sub.f] + [s.sup.l] and [Delta] is close to zero, [tn.sup.h] /2 > ([s.sup.l] + [Delta]) [n.sup.h] /4 if [c.sub.d] > [c.sub.f]. Thus a high tariff will improve home welfare if the home duopolist has higher (constant) marginal costs than the foreign duopolist, and the duopolists engage in static collusion.

Because the producers are colluding under any trade policy, they are extracting surplus from the consumers irrespective of whether the home government protects its constituent duopolist. All that the tariff does is affect the ability of either of the duopolists to offer a lower price to the low consumers. Because there are no menu costs and each firm is completely and almost perfectly informed, static collusion takes place in any case.(19)

IV. Conclusion

This paper introduced a new argument for protection. It is based upon recent advances in the study of oligopoly pricing which discloses that collusion in a duopoly can take in a static context. Contrary to the classical model of international trade, there are essentially no consumption costs to protection when a domestic and a foreign duopolist engage in static collusion. Thus protection can raise the welfare of the home country. It will rise even if the home government prohibits trade. This is unlike the classical optimum tariff argument, because of the absence of the consumption cost, and because the welfare gain arises from imperfect consumer information. (1)For a very clear introduction to this topic, see Tirole [18]. (2)For articles addressing the profit shifting motive for trade policy, see the papers in Krugman [1], and the references cited therein. Brander's [5] article in this volume reviews this literature. A more recent article of note concerning this this topic is Spencer [16]. (3)I am grateful to the anonymous referee for calling my attention to this distinction. (4)There has been some consideration given to trade policy in which consumers were nonhomogeneous. In Fieleke [9] and Clark [7], consumers were distinguished by their level of income. In Benson and Hartigan [1; 2; 3], Hatzipanayatou and Heffley [10], and Porter [14], consumers were distinguished by their location. (5)An alternative sampling for consumers is to draw a sample of a fixed size. (6)Manning and Morgan [12] have explored the consistency of utility maximization and consumer search. (7)See Morgan and Manning [13] for an elaboration on search strategies. (8)Prices have commitment value to consumers. Firms do not know if they are the first or second sampled, and they do not know if they are with a high or a low cost consumer each time they are visited. Furthermore, price setting is not secret. (9)The foreign producers' production and marketing decisions for its internal market are independent of that for its exports. Hence, the two markets are segmented. (10)If producers are endowed with the same information as consumers, which is the first two moments of the price distribution and the number of firms, they will always know the price posted by their rival without incurring any search costs. This is because they must know their own posted price. Hence, determination of the rival's price follows immediately. (11)Even if price lists are printed, a sale price can immediately be announced. Thus prices can be instantaneously revised, and can be done so without limit. (12)In the standard one shot Nash-Bertrand game, a firm that is undercut by its rival's price loses all of its sales. In the present model, an undercut firm can preclude any loss of sales by a price revision. Of course, a slower response time will affect the incentives to collude. (13)Because the foreign firm wants to post the highest possible price that preserves the incentives of the low cost consumers that draw the home firm initially to sample again, [Delta] will be a very small real number. (14)A well known result from price dispersion models for a single good is that a nondegenerate equilibrium distribution of prices may emerge when firms have equal costs of production. In that literature, firms are Nash-Bertrand strategies. That is, they select a profit maximizing price with the assumption that the existing distribution (or the expected distribution when they move simultaneously) of prices remain fixed. In these models, the number of firms is "large." The number of firms and consumers typically increases without bound as the expected customers that sample each firm is fixed. In that context, firms may not be aware of their interdependence. In this paper, they are aware of that interdependence and of an information advantage relative to consumers. This induces a degenerate price distribution. See Rob [15] for an excellent introduction to this topic. (15)This implies that the low cost firm also finds it more profitable to sell to ([n.sup.h] + [n.sup.l]) /2 consumers. (16)Note that this equilibrium corresponds to that which would occur under monopoly. That is, a monopolist would post p = [Theta] - [s.sup.h]. Contrasting this result with the free trade equilibrium reveals that under static collusion, one duopolist (the one with higher production costs) will post a price greater than the monopoly price. (17)Recall from footnote 9 that [Delta] is very small. (18)With a prohibitive tariff, there is no tariff revenue. The home duopolist sells to [n.sup.h] + [n.sup.l] consumers at [p.sub.d] = [Theta] - [s.sup.h]. There is essentially no consumption cost in prohibiting trade. Thus, the effect of home welfare of prohibiting trade is ([Theta] - [s.sup.h] - [c.sub.d]) ([n.sup.l] + [n.sup.h] /2) - ([s.sup.l] + [Delta]) [n.sup.h] /4. That is, it depends entirely on the gain to the home duopolist, and this is very likely to be positive. (19)It is not the interest of the foreign producer to punish the home producer for being protected by its government by not colluding, as the game lasts for one period and the tariff is set exogenously. Furthermore, there is no retaliation by the foreign government. Recall from the introduction that the tariff affects the collusive outcome that can be supported, not the incentives to collude.

References

[1]Benson, Bruce L. and James C. Hartigan, "Tariffs Which Lower Price in the Restricting Country: An Analysis of Spatial Markets." Journal of International Economics, August 1983, 117-33. [2]-- and --, "Tariffs and Quotas in a Spatial Duopoly." Southern Economic Journal, April 1984, 965-78. [3]-- and --, "Tariffs and Location Specific Income Distribution." Regional Science and Urban Economics, May 1987, 223-43. [4]Bhaskar, V., "Quick Responses in Duopoly Ensure Monopoly Pricing." Economics Letters, February 1989, 103-107. [5]Brander, James A. "Rationales for Strategic Trade and Industrial Policy," in Strategic Trade Policy and the New International Economics, edited by Paul Krugman. Cambridge: MIT Press, 1986, pp. 23-46. [6]-- and Barbara J. Spencer, "Export Subsidies and International Market Share Rivalry." Journal of International Economics, February 1985, 83-100. [7]Clark, Don P., "How Regressive are United States Distortions of International Trade?" National Tax Journal, September 1982, 215-21. [8]Davidson, Carl, "Cartel Stability and Tariff Policy." Journal of International Economics, November 1984, 219-37. [9]Fieleke, Norman S., "The Incidence of the U.S. Tariff Structure on Consumption." Public Policy, Fall 1971, 629-52. [10]Hatzipanayotou, Panos and Dennis Heffley, "Tariff Protection in an Open Spatial Economy." Journal of Regional Science, February 1991, 1-15. [11]Krugman, Paul, ed. Strategic Trade Policy and the New International Economics. Cambridge: MIT Press, 1986. [12]Manning, Richard and Peter Morgan, "Search and Consumer Theory." Review of Economic Studies, April 1982, 203-16. [13]Morgan, Peter and Richard Manning, "Optimal Search." Econometrica, July 1985, 923-44. [14]Porter, Robert H., "Tariff Policies in a Small Open Spatial Economy." Canadian Journal of Economics, May 1984, 270-82. [15]Rob, Rafael, "Equilibrium Price Distributions." Review of Economic Studies, July 1985, 487-504. [16]Spencer, Barbara J., "Capital Subsidies and Counterveiling Duties in Oligopolistic Industries." Journal of International Economics, August 1988, 45-69. [17]Stahl, Dale, "Revocable Pricing Can Yield Collusive Outcomes." Economics Letters, February 1986, 87-90. [18]Tirole, Jean. The Theory of Industrial Organization. Cambridge: MIT Press, 1988.

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.

II. 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 maximum price that category k consumers are willing to pay for a unit, replace the weak inequalities in (3) with strict equalities, and recall that p [bar] = ([p.sub.i] + [p.sub.j])/2. This is stated as (5) [Mathematical Expression Omitted] for k = h, l; i, j = d, f and i [is not equal to] j, where [Mathematical Expression Omitted] is the highest price that category k consumers will pay firm i when firm j has posted [p.sub.j]. Since expected net surplus is zero, it is the highest price at which each category would purchase a unit. Because surplus is - [s.sup.k] if a category k consumer samples but does not purchase, these consumers would prefer a surplus of - [s.sup.k]/2 from making purchase. Thus expected surplus is zero, but realized surplus will be negative for half of the category k = h, l consumers when there is price dispersion. Since price dispersion affects the consumers' incentive to search, the price that firm j posts directly affects the price that firm i can post. Furthermore, [s.sup.h] > [s.sup.l] implies [Mathematical Expression Omitted].

Suppose now that there is sufficient dispersion to induce category k = h, l consumers that draw the higher priced firm i initially to sample again. This requires [p.sub.i] - [p.sub.j] > [s.sup.k]. Suppose also that expected net surplus is zero, so that [Theta] = [p.sub.j] + [3s.sup.k] 3s.sup.k]/2 in (4). Since the consumer knows that (s)he will purchase from firm j (that is, pj is the expected price), the highest price at which sampling and purchase will occur is (6) [Mathematical Expression Omitted] for k = h, l and j = d, f. A comparison of (5) and (6) reveals that the expectation of drawing more samples lowers the choke price for consumers: (7) [Mathematical Expression Omitted] for k = h, l; i, j = d, f and i [is not equal to] j.

When consumers enter the market to sample a firm, they calculate an endogenous reservation price at which they would buy the product. The reservation price is the highest price at which they would purchase a unit of the good, given the prices known by the consumers to be posted by the duopolists. Denoting the reservation price by [r.sup.k] for k = h, l, a consumer equilibrium is generated by (8) [Mathematical Expression Omitted] for i, j = d, f; i [is not equal to] j and [p.sub.i] > [p.sub.j].

Proof. If a random sample by a consumer yields [p.sub.j], (s)he will purchase a unit of the good as long as [Mathematical Expression Omitted] for k = h, l. If (s)he draws [Mathematical Expression Omitted] the other firm will be sampled as long as there is a gain from doing so. The other firm can be sampled at a cost of [s.sup.k], and its product purchase at a total expenditure of p.sub.j] + [s.sup.k]. Thus, there will be a gain from additional as long as [p.sup.j + s.sup.k] < [p.sub.i], in which case category k consumers will only purchase from the lower price duopolist irrespective of which firm is sampled initially. This implies that category k consumers would be willing to pay [r.sup.k] = [p.sub.j] + [s.sup.k] if that price were drawn initially. On the other hand, [Mathematical Expression Omitted] results in category k consumers purchasing from the first firm sampled.

A unit of the good is purchased by each consumer as long as [Mathematical Expression Omitted] for i = d, f. In order to further explore the relationship among prices, search costs, choke (maximum) prices, and goods, the duopolists' behavior must be explicitly portrayed.

III. The Duopolists

The production of the homogenous good is characterized by constant marginal costs. These are denoted by [c.sub.i] for i = d, f where [Mathematical Expression Omitted] for k = h, l in R +. Both duopolists post prices in order to maximize expected profits, which are denoted by [Pi].sub.i] for i = d, f. Expected profits are the difference between posted price and marginal cost multiplied by expected sales. Denoting the latter by [q.sub.i] we can state (9) [Pi].sub.i] = ([p.sub.i] - [c.sub.i])[q.sub.i] for i = d, f.

In choosing a profit maximizing price, recall that each duopolist has complete and almost perfect information.(9) Information is almost perfect because the firms post prices simultaneously. Since the duopolists post prices in stage one and consumers search in stage two, the duopolists are von Stackelberg leaders with respect to consumers.

In maximizing (9), the firms post prices with ex post flexibility. In markets where there are a small number of interdependent producers, and in which consumers are imperfectly informed about prices, it is reasonable to believe that producers are better informed about the prices posted by rivals than are consumers.(10) Because of this, each duopolist has the opportunity to reply to the price posted by its rival before consumer search takes place in the next stage. This is the basis for static collusion on the part of the duopolists. It requires zero menu costs in the posting of prices.(11) That is presented in this paper as a limiting case, in which prices can be revised with an arbitrarily short time lag.(12)

Denoting by [p [bar].sub.j] the price that firm i expects firm j to post (with [p [bar].sub.i] correspondingly defined), ex post price flexibility is depicted as (10) [Mathematical Expression Omitted] for i, j = d, f and i [is not equal to] j. If firm j deviates from the price that firm i expects it to post (with firm i posting the price that firm j expects i to post), firm i can prevent sales from taking place at the deviation price by quickly changing its own posted price. That is, it can preclude unanticipated undercutting. Hence imperfect consumer information limits ( in this case completely) the ability of a firm to gain at the expense of its rival by not acting collusively. As a result unanticipated prices are not observed. That is, [p.sub.j] = [p [bar].sub.j] for j = d, f.

In choosing its expected profits maximizing price, each duopolist can adopt an aggressive or a soft pricing strategy. These strategies are termed [s.sup.a] and [s.sup.s], respectively. They are defined as (11) [Mathematical Expression Omitted] and (12) [Mathematical Expression Omitted] where [q [bar].sub.j] is the level of sales that firm i expects firm j to make. Recall that [q.sub.i] is the sales that firm i expects to make. Under an aggressive pricing strategy, it is senseless to post a price below the price that a rival is expected to make, unless it is sufficiently below to induce those consumers that draw the rival initially to sample again. Thus expected sales are part of the definition of the strategies. With the soft pricing strategy, a firm is willing to allow itself to be undercut by its rival. Because the firms have complete and almost perfect information and price with ex post flexibility, as in (10), expected prices equal posted prices and each firm makes the level of sales that its rival expects it to make. That is, [p.sub.j] = [p [bar].sub.j] and [q.sub.j] = [q [bar].sub.j] for j = d, f.

If firm j does post a disequilibrium (unexpected) price for j = d, f, it is necessary to specify firm i's best reply for i = d, f and i [is not equal to] j. Recalling that each firm has complete and almost perfect information, and prices with ex post flexibility, this reply will depend upon the cost advantage (if any) that firm j is known by i to have. These replies are (13) [Mathematical Expression Omitted] for [c.sub.i] - [c.sub.j] [is less than or equal to] [s.sup.l]; i, j = d, f and i [is not equal to] j. (14) [Mathematical Expression Omitted] for [s.sup.h] [is greater than or equal to] [c.sub.i] - [c.sub.j] > [s.sup.l]; i, j = d, f and i [is not equal to] j. (15) [Mathematical Expression Omitted] for [s.sup.h] [is greater than or equal to] [c.sub.j] - [c.sub.i] > [s.sup.l]; i, j = d, f and i [is not equal to] j.

Replies (13) and (14) pertain to the soft pricing strategy. With these replies, the firms cannot expect to earn negative profits if the rival does not change its posted price. That is, [p.sub.i] [is greater than or equal to] [c.sub.i] for i = d, f. Of course, [Mathematical Expression Omitted]. That is, a reply is not credible if a firm cannot expect positive sales at that price. In (13), firm i will not permit j to post a price that induces any consumers that draw i initially to sample j. It maintains this price (or revises according to (13) if j posts some other disequilibrium price ) until j posts [p [bar].sub.j] for j = d, f and i [is not equal to] j. Firm j will revise its price quickly, as there is no gain to posting disequilibrium (unexpected) price. Because of ex post price flexibility, firm j cannot gain a temporary advantage by undercutting firm i. Since there isn't a significant cost advantage for either firm, their expected sales are identical. In the present case, these are ([n.sup.l] + [n.sup.h])/2. That is, neither firm has the cost advantage relative to consumer search costs to impose an uneven split of the market on its rival. In (14), firm j does have such a cost advantage. This allows firm j to post a price that induces all low cost consumers that draw i initially to sample again. (For instance, j could post [p.sub.j]: [p.sub.j] + [s.sup.l] < [c.sub.i].) Thus firm j's expected sales exceed those of i. Firm j's expected sales are [n.sup.l] + [n.sup.h]/2 and firm i's are [n.sup.h]/2. Hence, firm i's best reply to a disequilibrium price by j takes this into account.

The reply to a disequilibrium price when firm i is acting aggressively is depicted by (15). Firm i is known by both firms to have a cost advantage that enables it to capture all of the low cost consumers. If firm j posts a disequilibrium price, firm i replies by posting a price sufficiently below j's marginal cost of production to induce all low cost consumers that draw j to sample again. It maintains this price until [p.sub.j] = [p [bar].sub.j], which must occur very quickly.

These replies to disequilibrium prices by either of the duopolists ensure that such prices are not observed. Thus [q.sub.i] = [q [bar].sub.i] for i = d, f.

A strategy will be considered feasible if can be successfully implemented. That is, if a firm can price aggressively without its rival revising its posted price through its ex post price flexibility and precluding the former's expansion of its expected sales, then the aggressive strategy is feasible for the former firm. Otherwise, it is not. Strategy sets are defined by [S.sub.i] for i = d, f.

Free Trade

In the free trade duopoly equilibrium, it is assumed that [s.sup.h] [is greater than or equal to] [c.sub.d] - [c.sub.f] > [s.sup.l]. That is, the foreign firm has a cost advantage that enables it to capture all of the home country's low cost consumers. The producer equilibrium that emerges is depicted as (16) [p.sub.i] [is greater than or equal to] [c.sub.i] : [q.sub.i] > 0 for i = d, f. (17) [S.sub.d] = {S.sup.s}. (18) [S.sub.f] = {S.sup.a], [S.sup.s]} (19) [Pi].sub.f] (S.sup.a]) > [Pi].sub.f] (S.sup.f]). (20) [P.sub.f] = [P.sub.d] - [s.sup.l] - [Delta] for [Delta] [Epsilon] (0, [s.sup.h] - [s.sup.l]] (21) [Mathematical Expression Omitted] (22) 2 [s.sup.l] - 2 [s.sup.h] - [Theta] [is less than or equal to] 0.

Proof. Inequality (16) simply requires nonnegative profits for production to occur. Since a firm will not produce if it cannot cover its costs, its expected sales are positive only when (16) holds. Because of its cost disadvantage relative to the search costs of the low cost consumers, the home duopolist can only price softly. This is apparent from (14). The foreign firm, however, can price aggressively, as (15) reveals. It can, of course, also price softly. It can always respond to a disequilibrium price by the home firm according to (13). The home firm will not try to prevent soft pricing by the foreign firm as its expected sales would be ([n.sup.h] + [n.sup.l])/2 rather than [n.sup.h]/2. Thus, the firm's strategy sets are given by (17) and (18). Inequalities (19) states that the foreign firm will prefer to price aggressively when it is Given that the foreign firm can capture all of the low cost consumers because of its production cost advantage, ex post price flexibility, and response to a disequilibrium price, the home firm knows that it can expect to sell to only [n.sup.h]/2 consumers. It naturally wants to do so at the highest possible price. The foreign firm, knowing that it has the advantage to sell on [n.sup.l] + [n.sup.h]/2 consumers, also wants to do so at the highest possible price. Both firms know that their posted prices must be such that (3) holds, and their combines profits are maximized when it holds with equality. This is because high cost consumers will draw only once. They also know that that the foreign firm needs to preserve the incentive for low cost consumers that draw the home firm to sample again, and that the foreign firm will enforce this through (15) The static collusive equilibrium (focal equilibrium) that meets the price dispersion and profit maximization requirements has the foreign firm posting its price according to (20) and the home firm pricing according to (21). Equation (21) is derived from setting setting p [bar] = [Theta] - [s.sup.h] = ([p.sub.d] + [p.sub.d] - [s.sup.l] - [Theta]) /2. The foreign firm prefers that the home firm posts [p.sub.d]. Recognizing that the foreign firm will sell to [n.sup.l] + [n.sup.h]/2 consumers, the home firm prefers that the foreign firm posts [p.sub.d] - [s.sup.l] - [Theta]. Finally, (22) is derived by substituting (21) into (20), and then substituting (20) into (4). It indicates that the expected surplus of low cost consumers is positive, thereby ensuring that they will draw at least one sample, and will purchase a unit of the good. (They will sample only once if they draw the foreign firm on the first try.)

Protection with a Low Tariff

Suppose now that the government seeks to protect its constituent duopolist by imposing a tariff on the foreign firm's sales in the home market. This raises the cost of producing for and selling in the home market for the foreign firm. This is expressed as (23) [Mathematical Expression Omitted] where t denotes the tariff. In order for the tariff to change the duopoly equilibrium, it must be that [Mathematical Expression Omitted]. This removes the ability of the foreign firm to capture all of the low cost consumers. The tariff distorted producer equilibrium becomes. (24) [p.sub.d] [is greater than or equal to] [c.sub.d] > 0. (25) [Mathematical Expression Omitted] (26) [S.sub.i] = {[S.sup.s]} for i = d, f. (27) [Mathematical Expression Omitted] (28) ([Theta] - [s.sup.h] - [c.sub.d] ([n.sup.h] + [n.sup.l])/2 > ([Theta] - [s.sup.l] - [c.sub.d] [n.sup./2

Proof. Inequalities (24) and (25) are the nonnegative profits constraints. Because the home country's tariff has reduced the cost advantage of the foreign firm relative to the search costs of the low cost consumers, both firms can now only price softly. This is apparent from (13). Thus strategy sets are given by (26). Each firm knows that the absence of a production cost advantage (relative to search costs), the existence of ex post price flexibility, and the response to a disequilibrium price means that each firm cannot expect to sell to more than half of the market. That is, expected sales must be ([n.sup.h] + [n.sup.l]/2. This partition of the market can most profitably occur at the price at which high cost consumers expect zero surplus from purchasing a unit of the good. (See discussion of (28) below). In this equilibrium, high cost consumers expect and receive zero surplus, as [p.sub.f] = p [bar]. Inequality (3) reveals that consumers will sample only once in this circumstance. Equation (5) discloses that consumers will pay more than p [bar] when there is price dispersion. That is, when expected surplus is zero, but realized surplus for half of category [kappa] consumers is negative. However, this requires one duopolist to post a lower price than the other. Since each firm can impose equal expected sales on the other through (13), neither will be willing to post a lower price than the other to sell to ([n.sup.h] + [n.sup.l])/2 consumers. Thus the collusive equilibrium prices are given by (27). Inequality (28) states that the market is covered. That is, all consumers will buy a unit of the good in stage two. The high cost firm finds it more profitable to sell to ([n.sup.l] + [n.sup.h])/2 consumers at the price at which high cost consumers expect zero surplus rather than the price at which low cost consumers expect zero surplus (recall (3)), thereby excluding high cost consumers from the market.(14,15) If (28) holds for the high cost firm, it also holds for the low cost foreign firm.(16)

Returning to the consumer decision rules reveals an important implication of protection. With the producer equilibrium under the low tariff, [p.sub.d] = [p.sub.f] = p [bar]. Expected and realized surplus for high cost consumers is exactly zero, as p [bar] = [Theta] - [s.sup.h]. Note that [Mathematical Expression Omitted] for i = d, f from (5). However, high cost consumers pay an expected price under free trade which is denoted by [p.sup.e] and is expressed as (29) [p.sup.e] = ([p.sub.d] + [p.sub.f])/2. Substituting from (20) and (21) into (29) reveals [p.sup.e] = [Theta] - [s.sup.h]. That is, the expected price equals the mean price for the high cost consumers. Assuming that the consumers are risk neutral, reveals that protection does not have an impact on the expected surplus of the high cost consumers. Of course, realized surplus can be higher or lower under free trade, because there is price dispersion.

Because the high cost consumers sample only once in each equilibrium, the impact of protection on them be assessed through purchase price. However, low cost consumers expect to draw one sample under protection and 3/2 samples in free trade. Thus, the impact of protection on them must be assessed through expected expenditure. Under protection, expected expenditure is [Theta] - [s.sup.h] + [s.sup.l]. Since the low cost consumers will only buy from the foreign firm in free trade, the only issue is the number of samples that must be drawn before the foreign firm is located. Thus their expected expenditure under free trade is obtained by adding [3s.sup.l]/2 to (20), after substituting from (21). This yields [Theta] - [s.sup.h] = [s.sup.l] - [Delta]/2. Since expenditure under protection is [Theta] - [s.sup.h] + [s.sup.l], they are trivially harmed by protection.(17) Although the low cost consumers reduce their expected expenditure on search, this saving of expenditure is trivially exceeded by the higher price that these consumer pay under protection with a low tariff.

To assess the values of t for which the indicated shift in the duopoly equilibrium occurs, recall that [Mathematical Expression Omitted] in free trade. To obtain the protection equilibrium, [Mathematical Expression Omitted]. The smallest tariff that effects this change in the relationship between the marginal cost differences of the firms and the search costs of the low cost consumers is [Mathematical Expression Omitted]. The largest tariff that includes the protection equilibrium occurs with the home firm having a cost advantage over the foreign firm in selling to the home market. That is, [c.sub.f] + t > [c.sub.d]. This tariff must not be so large that low cost consumers that draw the foreign firm are induced to sample again. Thus [Mathematical Expression Omitted] The largest value of t that shifts the free trade equilibrium to the protection equilibrium is t [bar], where [Mathematical Expression Omitted]. Hence, any t [Mathematical Expression Omitted] generates the shift in the duopoly equilibrium discussed above.

The impact of the low tariff on the expected surplus of the home consumers is quite surprising. For any [Mathematical Expression Omitted], the tariff does not effect the cost advantage of the foreign firm relative to the search costs of the low cost consumers. Thus expected surplus for each category of consumer is identical to that of free trade. This is zero for the high cost consumers and [Mathematical Expression Omitted] for the low cost consumers. For t [Mathematical Expression Omitted], high cost consumers are not affected, but low cost consumers incur a decline of expected surplus to [s.sup.h] - [s.sup.l]. That is, relative to free trade, the low cost consumer will expect to pay ([s.sup.l] + [Delta])/2 more in purchase price with a low tariff. However, the low cost consumer expects to save [s.sup.l] /2 in search costs under that low tariff. Thus, the low tariff essentially entails a zero consumption cost of protection.

To consider the effect of the low tariff on home welfare, let (30) [Mathematical Expression Omitted] where [w.sub.d] is home welfare and m is the level of home imports. In the decision to apply a low tariff, the change in home welfare relative to free trade [Delta] [w.sub.d] is the relevant calculation. Since [u.sup.h] is always zero under static collusion, [n.sup.h] [u.sup.h] is always zero, Because t is zero in free trade, tariff revenue is also zero. Thus the change in home welfare from applying a low tariff is (31) [Mathematical Expression Omitted] The first expression on the right hand side of (31) is the increase in the home duopolist's profits from selling to an additional [n.sup.l] /2 consumers under the tariff. The second expression is the tariff revenue, where the tariff that maximizes the revenue for ([n.sup.h] + n.sup.l]) /2 sales by the foreign duopolist is the highest tariff that will induce this equilibrium. The third term is the loss in expected surplus for the low cost consumers resulting from not being able to buy from the foreign firm at a lower price as in free trade. The fourth term arises from the lower price the home firm posts in the degenerate price distribution under static collusion with a low tariff relative to the price it posts under the dispersed price distribution with free trade. (Note that the home firm sells to [n.sup.h] /2 high cost consumers in each equilibrium.) The sign of this expression is ambiguous. However, it is apparent that the home producer gains if the [Mathematical Expression Omitted]. This will occur as long as the frequency of low cost consumers in the home economy is reasonably large. Inequality (19) in the free trade equilibrium, which states that the foreign duopolist prefers to lower its price to capture the low cost consumer, suggests that this is the case. Thus (31) can be expected to hold, since tariff revenue exceeds the trivial loss in expected surplus of the low cost consumers. A low tariff can therefore raise home welfare when a home and a foreign duopolist engage in static collusion.

Protection with a High Tariff

Suppose that a nonprohibitive tariff t : t > t [bar] is imposed on foreign sales in the home market. This results in [s.sup.h] [is not greater than or equal to] [c.sub.f] + t - [c.sub.d] > [s.sup.l]. This provides the home firm with the cost advantage necessary to capture all of the low cost consumers under ex post price flexibility with complete and almost perfect information. The tariff induces the following (17') [Mathematical Expression Omitted] (18') [Mathematical Expression Omitted]. (19') [Mathematical Expression Omitted]. (20') [Mathematical Expression Omitted]. (21') [P.sub.d] = [P.sub.f] - [s.sup.l] = [Delta] for [Delta] [Epsilon] (0, [s.sup.h] - [s.sup.l]). Inequalities (16) and (22) continue to hold as in free trade. The proof is identical to that given earlier.

Because all low cost consumers buy from the home firm at [P.sub.d], and expect to draw 3/2 samples, there is no effect on their surplus when a high tariff is compared to free trade. That is, their expected expenditure is the same under both policies. Similarly, the high cost consumer are evenly distributed between the two firms, and their expected expenditure is identical under free trade and a high tariff.

Referring to equations (20), (20'), (21), and (21') reveals that the expected surplus levels are identical for each category of consumers under free trade and a high tariff. Since there is no consumption cost to a high tariff under static collusion, the entire welfare depends upon the change in the home duopolist's profits relative to free trade, and the tariff revenue. This is expressed as (31') [Mathematical Expression Omitted] where t [Epsilon] (t [bar], [t.sub.p]) in R + with [t.sub.p] being the prohibitive tariff.(18) The first term in (31') is the home government's tariff revenue. The second term is the loss in the home duopolist's profits from selling to the [n.sup.h] /2 high cost consumers at a lower price. The third term is the gain the home duopolist's profits from selling to an additional [n.sup.l] consumers relative to free trade. Since t > t [bar] = [c.sub.d] - [c.sub.f] + [s.sup.l] and [Delta] is close to zero, [tn.sup.h] /2 > ([s.sup.l] + [Delta]) [n.sup.h] /4 if [c.sub.d] > [c.sub.f]. Thus a high tariff will improve home welfare if the home duopolist has higher (constant) marginal costs than the foreign duopolist, and the duopolists engage in static collusion.

Because the producers are colluding under any trade policy, they are extracting surplus from the consumers irrespective of whether the home government protects its constituent duopolist. All that the tariff does is affect the ability of either of the duopolists to offer a lower price to the low consumers. Because there are no menu costs and each firm is completely and almost perfectly informed, static collusion takes place in any case.(19)

IV. Conclusion

This paper introduced a new argument for protection. It is based upon recent advances in the study of oligopoly pricing which discloses that collusion in a duopoly can take in a static context. Contrary to the classical model of international trade, there are essentially no consumption costs to protection when a domestic and a foreign duopolist engage in static collusion. Thus protection can raise the welfare of the home country. It will rise even if the home government prohibits trade. This is unlike the classical optimum tariff argument, because of the absence of the consumption cost, and because the welfare gain arises from imperfect consumer information. (1)For a very clear introduction to this topic, see Tirole [18]. (2)For articles addressing the profit shifting motive for trade policy, see the papers in Krugman [1], and the references cited therein. Brander's [5] article in this volume reviews this literature. A more recent article of note concerning this this topic is Spencer [16]. (3)I am grateful to the anonymous referee for calling my attention to this distinction. (4)There has been some consideration given to trade policy in which consumers were nonhomogeneous. In Fieleke [9] and Clark [7], consumers were distinguished by their level of income. In Benson and Hartigan [1; 2; 3], Hatzipanayatou and Heffley [10], and Porter [14], consumers were distinguished by their location. (5)An alternative sampling for consumers is to draw a sample of a fixed size. (6)Manning and Morgan [12] have explored the consistency of utility maximization and consumer search. (7)See Morgan and Manning [13] for an elaboration on search strategies. (8)Prices have commitment value to consumers. Firms do not know if they are the first or second sampled, and they do not know if they are with a high or a low cost consumer each time they are visited. Furthermore, price setting is not secret. (9)The foreign producers' production and marketing decisions for its internal market are independent of that for its exports. Hence, the two markets are segmented. (10)If producers are endowed with the same information as consumers, which is the first two moments of the price distribution and the number of firms, they will always know the price posted by their rival without incurring any search costs. This is because they must know their own posted price. Hence, determination of the rival's price follows immediately. (11)Even if price lists are printed, a sale price can immediately be announced. Thus prices can be instantaneously revised, and can be done so without limit. (12)In the standard one shot Nash-Bertrand game, a firm that is undercut by its rival's price loses all of its sales. In the present model, an undercut firm can preclude any loss of sales by a price revision. Of course, a slower response time will affect the incentives to collude. (13)Because the foreign firm wants to post the highest possible price that preserves the incentives of the low cost consumers that draw the home firm initially to sample again, [Delta] will be a very small real number. (14)A well known result from price dispersion models for a single good is that a nondegenerate equilibrium distribution of prices may emerge when firms have equal costs of production. In that literature, firms are Nash-Bertrand strategies. That is, they select a profit maximizing price with the assumption that the existing distribution (or the expected distribution when they move simultaneously) of prices remain fixed. In these models, the number of firms is "large." The number of firms and consumers typically increases without bound as the expected customers that sample each firm is fixed. In that context, firms may not be aware of their interdependence. In this paper, they are aware of that interdependence and of an information advantage relative to consumers. This induces a degenerate price distribution. See Rob [15] for an excellent introduction to this topic. (15)This implies that the low cost firm also finds it more profitable to sell to ([n.sup.h] + [n.sup.l]) /2 consumers. (16)Note that this equilibrium corresponds to that which would occur under monopoly. That is, a monopolist would post p = [Theta] - [s.sup.h]. Contrasting this result with the free trade equilibrium reveals that under static collusion, one duopolist (the one with higher production costs) will post a price greater than the monopoly price. (17)Recall from footnote 9 that [Delta] is very small. (18)With a prohibitive tariff, there is no tariff revenue. The home duopolist sells to [n.sup.h] + [n.sup.l] consumers at [p.sub.d] = [Theta] - [s.sup.h]. There is essentially no consumption cost in prohibiting trade. Thus, the effect of home welfare of prohibiting trade is ([Theta] - [s.sup.h] - [c.sub.d]) ([n.sup.l] + [n.sup.h] /2) - ([s.sup.l] + [Delta]) [n.sup.h] /4. That is, it depends entirely on the gain to the home duopolist, and this is very likely to be positive. (19)It is not the interest of the foreign producer to punish the home producer for being protected by its government by not colluding, as the game lasts for one period and the tariff is set exogenously. Furthermore, there is no retaliation by the foreign government. Recall from the introduction that the tariff affects the collusive outcome that can be supported, not the incentives to collude.

References

[1]Benson, Bruce L. and James C. Hartigan, "Tariffs Which Lower Price in the Restricting Country: An Analysis of Spatial Markets." Journal of International Economics, August 1983, 117-33. [2]-- and --, "Tariffs and Quotas in a Spatial Duopoly." Southern Economic Journal, April 1984, 965-78. [3]-- and --, "Tariffs and Location Specific Income Distribution." Regional Science and Urban Economics, May 1987, 223-43. [4]Bhaskar, V., "Quick Responses in Duopoly Ensure Monopoly Pricing." Economics Letters, February 1989, 103-107. [5]Brander, James A. "Rationales for Strategic Trade and Industrial Policy," in Strategic Trade Policy and the New International Economics, edited by Paul Krugman. Cambridge: MIT Press, 1986, pp. 23-46. [6]-- and Barbara J. Spencer, "Export Subsidies and International Market Share Rivalry." Journal of International Economics, February 1985, 83-100. [7]Clark, Don P., "How Regressive are United States Distortions of International Trade?" National Tax Journal, September 1982, 215-21. [8]Davidson, Carl, "Cartel Stability and Tariff Policy." Journal of International Economics, November 1984, 219-37. [9]Fieleke, Norman S., "The Incidence of the U.S. Tariff Structure on Consumption." Public Policy, Fall 1971, 629-52. [10]Hatzipanayotou, Panos and Dennis Heffley, "Tariff Protection in an Open Spatial Economy." Journal of Regional Science, February 1991, 1-15. [11]Krugman, Paul, ed. Strategic Trade Policy and the New International Economics. Cambridge: MIT Press, 1986. [12]Manning, Richard and Peter Morgan, "Search and Consumer Theory." Review of Economic Studies, April 1982, 203-16. [13]Morgan, Peter and Richard Manning, "Optimal Search." Econometrica, July 1985, 923-44. [14]Porter, Robert H., "Tariff Policies in a Small Open Spatial Economy." Canadian Journal of Economics, May 1984, 270-82. [15]Rob, Rafael, "Equilibrium Price Distributions." Review of Economic Studies, July 1985, 487-504. [16]Spencer, Barbara J., "Capital Subsidies and Counterveiling Duties in Oligopolistic Industries." Journal of International Economics, August 1988, 45-69. [17]Stahl, Dale, "Revocable Pricing Can Yield Collusive Outcomes." Economics Letters, February 1986, 87-90. [18]Tirole, Jean. The Theory of Industrial Organization. Cambridge: MIT Press, 1988.

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Author: | Hartigan, James C. |
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Publication: | Southern Economic Journal |

Date: | Apr 1, 1992 |

Words: | 7307 |

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