Maggi - Strategic Trade Policies With Endogenous Mode of Competition

American Economic Association

Strategic Trade Policies with Endogenous Mode of CompetitionAuthor(s): Giovanni MaggiSource: The American Economic Review, Vol. 86, No. 1 (Mar., 1996), pp. 237-258Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/2118265Accessed: 17/11/2009 08:55

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* Department of Economics, Princeton University, Princeton, NJ 08544. I gratefully acknowledge financial support from the Alfred P. Sloan Foundation and the Cen- ter for Economic Policy Research (CEPR) at Stanford. I am grateful to Robert W. Staiger for his constant encour- agement and help. I thank Dirk Bergemann, Avinash Dixit, Mario Epelbaum, Gene Grossman, Thomas Hellmann, Kenneth Judd, Paul Milgrom, Andr6s Rodrfguez-Clare, Peter Thompson, Ernst Ludwig von Thadden, and seminar participants at Stanford University and the University of Chicago for helpful discussions and comments. I also thank an anonymous referee, whose comments led to substantial improvement of the paper.

Strategic Trade Policies with Endogenous Mode of Competition

By GIOVANNI MAGGI *

This paper develops a model of capacity-price competition in which the equilibrium outcome ranges from the Bertrand to the Cournot outcome as capacity constraints become more important. This model is employed to reexamine aspects of strategic- trade-policy theory and, in particular, the theory's well-known sensitivity to the mode of oligopolistic competition. Among other things, the analysis identifies a simple single-rate policy, namely, capacity subsidies, which can increase the home country's income regardless of the mode of competition. This suggests that the presence of critical informational constraints need not diminish governments' incentives to distort the international competition. (JEL D43, F13, L13)

Since its very inception, the theory of strategic trade policy has attracted much attention and much criticism. The "skeptical" view of strategic trade theory is well represented by the following statement, taken from a recent arti- cle of Paul Krugman (1993 p. 363):

After several years of theoretical and empirical investigation, it has become clear that the strategic trade argument, while ingenious, is probably of minor real importance. Theoretical work has shown that the appropriate strategic trade policy is highly sensitive to details of market structure that governments are unlikely to get right, while efforts to quantify the gains from rent-snatching suggest small payoffs.

Two lines of criticism can be detected in the above quote: one empirical and one theoreti-

cal. Focusing on the empirical evidence first, the results in this area, while negative, are far from conclusive. The impact of export policies on national incomes has been evaluated using "calibration" models, rather than estimation procedures that employ real data. Most trade economists interpret these quantification ex- ercises with caution, allowing for a wide mar- gin of error. Overall it is safe to say that the jury is still out for the empirical evaluation of trade policies.

A potentially more damaging criticism raised against strategic trade models is of theoretical nature. It is pointed out that their conclusions rely heavily on the assumption that governments have complete information about the markets targeted for intervention. This criticism is all the more relevant because the predictions of the theory are very sensitive to the particularities of the markets. One key sensitivity of the strategic-trade models concerns the mode of competition: if firms compete in quantities, the optimal policy tends to be a subsidy to the home firm (James A. Brander and Barbara J. Spencer, 1985), but if firms are price-setters the optimal policy tends to be a tax (Jonathan Eaton and Gene M. Grossman, 1986). If a government lacks information about the exact nature of competition, trade policies based on the wrong beliefs can be harmful for the home country: for example, an export subsidy reduces the home country's welfare if firms compete in prices.

237

238 THE AMERICAN ECONOMIC REVIEW MARCH 1996

These theoretical findings-the policy sensitivity and the informational requirements that this sensitivity implies-have generally been interpreted as striking a devastating blow to strategic-trade theory, as either a normative or a positive theory. Thus far, however, the theoretical literature has fallen short of a rig- orous examination of the implications of this form of incomplete information on trade policy issues. Arguably, an important step toward a better understanding of this issue is to introduce explicitly the policymakers' informational constraints in the trade-policy models, in order to address the question of whether governments are still driven to intervene in the international competition even when they do not know the nature of the firms' oligopolistic conduct.'

The present paper takes a step in this direction. The analysis is in three stages. First, a new model of oligopoly is developed, in which the mode of competition is traced back to a structural parameter, whose level determines whether the outcome more resembles price or quantity competition. This model is then employed to analyze strategic trade issues in a complete-information environment. The final step is to examine trade policies when the government is uncertain about the mode of competition.

The mode of competition, rather than being identified with the variable chosen by the firms (price or quantity), is determined endoge- nously by the importance of capacity constraints. Unlike more traditional models of capacity-constrained price competition (such as David M. Kreps and Jose A. Scheinkman, 1983), capacity commitment is not modeled as an inflexible constraint: firms can produce in excess of capacity, but at a higher marginal cost; the additional cost of producing above capacity is the parameter that captures the importance of capacity constraints. Firms set capacities at the first stage and then compete in prices given the cost curves generated by their

choice of capacity. The model predicts that the equilibrium outcome ranges from the Bertrand to the Cournot outcome as capacity constraints become more important. In this perspective, the mode of competition is identified with the extent to which firms can mitigate price competition through their capacity choices, and it becomes a continuous variable rather than a binary variable, thus capturing intermediate situations between the pure Bertrand and Cournot cases.

The model can also be contrasted with the conjectural-variations model, which is a common way to parameterize oligopolistic conduct.2 The similarity between the two models lies in the fact that the mode of competition is ultimately indexed by a single parameter, and the equilibrium prices and quantities range from the Bertrand to the Cournot outcome as this parameter varies. The model proposed here, however, has a dual advantage over the conjectural-variations model: first, the mode of competition is indexed by a structural parameter which is (at least in principle) observable, whereas the conjectural-variation parameter is not even observable in principle; and second, it is based on game-theoretical foundations, which are lacking in the conjectural-variations model.

When oligopolistic competition is modeled in the way described above, new insights arise about the role of strategic trade policies. One interesting point is the following. Capacities are strategic substitutes, yet it is not always desirable to induce an expansion of the home firm's capacity: this is true only if capacity constraints are important enough, but when capacity constraints are less important it is optimal to induce a contraction of the home firm's capacity (and output). This result can be understood intuitively in the following way. In this model, capacity is an imperfect commitment device that helps the firm to limit production and sustain higher prices. When this commitment device is effective enough, it is optimal to encourage a capacity expansion by the home firm in order to induce a contraction of the foreign firm. However, if capacity is not

' Larry D. Qiu (1994), Maggi (1992), and Lael Brainard and David Martimort (1992) examine strategic trade models in which governments lack information about cost or demand parameters but do not address issues of uncertainty about the mode of competition.

2 See Carl Shapiro (1989) for a concise exposition of the conjectural-variations model.

VOL. 86 NO. 1 MAGGI: STRATEGIC TRADE POLICIES 239

a very effective commitment device, it is desirable to tax the home firm's output, in order to provide the home firm with some additional commitment ability.

The last step of the analysis is to reexamine the "infonnational" criticism of the strategic- trade theory. As already pointed out, output policies are potentially harmful for the home country if the mode of competition is un- known. However the analysis identifies a simple single-rate policy, namely, subsidies to investment in capacity, which are (weakly) beneficial to the home country regardless of the mode of competition: a small capacity subsidy is strictly beneficial if capacity constraints are important enough, and it is neutral if capacity constraints are less important. More- over, this is true regardless of the specific values of demand and cost parameters. This result suggests that investment policies are less sensitive than output policies to the specific characteristics of the market: governments may have incentives to distort the international competition even when they lack information about all the relevant parameters of the targeted industry.3

A capacity subsidy can be interpreted as any policy that encourages expansion of the domestic firms' productive base. Examples in- clude incentives for investment in new plants and machinery, either by direct subsidies or by indirect means such as low-interest loans or tax credits. Gary C. Hufbauer and Joanna S. Erb (1984) report anecdotal evidence of such policies in several countries. For example, they report Alan K. Wolff's ( 1983, pp. 7-8) description of "industrial-targeting" policies pursued by the Japanese government in industries like computers, microelectronics, robot- ics, machine tools, and aerospace: "... with the help of government grants and loans, and the cooperation of the banking system, large new investments are made in advanced production equipment...."

One possible criticism to the approach fol- lowed in this paper is that it abstracts from the

political-economy motivations that often lie behind trade policies. One might argue that the reason governments intervene in spite of all their informational constraints is that in reality the motivations behind trade policies are chiefly "political." The insights of this paper however should be viewed as complementary, not in contrast, to the political-economy ex- planations of trade policies: the point of the paper is to show that the presence of stringent informational constraints need not reduce the governments' economic incentives to distort the firms' competition, and this is potentially important for the interpretation of real trade policies.

. Another paper that examines strategic trade policies under "endogenous" mode of competition is Didier Laussel (1992), who em- ploys Paul D. Klemperer and Margaret A. Meyer's (1989) model of supply-function competition; Laussel does not consider informational issues. Since the supply-function model can be viewed as an alternative way to endogenize price versus quantity competition, one might wonder whether the supply-function model could be helpful in examining the issue of uncertainty about the mode of competition. It should be noted first that, in the supply- function model, single-rate subsidies/taxes (which affect the level of the firm's marginal cost) have no impact on the equilibrium prices and quantities: making the home firm more or less aggressive does not per se influence the equilibrium outcome. Only nonlinear subsidies/taxes (which affect the slope of the marginal-cost curve) can affect the equilibrium outcome. Hence the supply-function model does not seem suitable for examining the "traditional" sensitivity issue of single- rate subsidies versus single-rate taxes (or, in other words, the Brander-Spencer vs. the Eaton-Grossman policy), which is the focus of the present paper: the Brander-Spencer and Eaton-Grossman cases are not encompassed by the Laussel model, while they constitute the benchmarks of the present model.

The paper is organized as follows. In Section I the oligopoly model is developed. Section II examines optimal trade policies under complete information. Section Im examines trade policies under incomplete information. Section IV contains concluding remarks.

3 This result is consistent with Kyle Bagwell and Robert W. Staiger's (1994) analysis of strategic subsidies to R&D investment. They find that the optimal R&D policy is independent of the mode of competition.


I. Preliminaries: Price versus Quantity Competition

The Cournot model of quantity competition has been the subject of considerable criticism in the theory of industrial organization. If taken literally, the Cournot model assumes that firms dump their production on the market and that an auctioneer determines the price that clears the market. In most industries there is nothing that resembles an auctioneer, and firms use prices as a strategic variable. The Cournot model has been recently reinterpreted as the reduced form of a richer game, in which firms can commit to capacity levels prior to setting prices (Kreps and Scheinkman, 1983) .4 The common assumption of these models is that the cost of producing beyond capacity is infinite; that is, capacity imposes a strict constraint on production. These capacity- constrained models, however, present a few unsatisfactory features, aside from the dubious realism of the assumption of infinite cost of production beyond capacity. First, they predict that for certain regions of capacities the price subgame has only a mixed-strategy equilibrium. This equilibrium does not possess the "no-regret" property: after prices are realized, a firm will generally like to change its price; if one believes that prices can be changed much more quickly than capacities, this prediction is problematic. Furthermore, the prediction about the emergence of Bertrand versus Cournot outcomes is of an all-or-nothing nature: the equilibrium is Bertrand or Cournot depending on whether or not the industry is characterized by capacity constraints; these models do not capture situations that lie between these two extremes.

Rigid capacity constraints are not the only way to formalize the notion of quantity competition. As Jean Tirole (1988 pp. 217-18) observes, " [i] n most cases, firms do not face

rigid capacity constraints. ... More generally, what we mean by quantity competition is re- ally a choice of scale that determines the firm's cost functions and thus determines the conditions of price competition." Here I follow this more "flexible" approach, modeling capacity as an imperfect commitment device. This modeling approach will generate price subgames with unique pure-strategy equilibria and will yield richer predictions as to the de- terminants of the mode of competition, while at the same time simplifying the mathematical structure considerably. Consider two firms that produce (symmetrically) differentiated products, and assume that demand is linear for each product: q' = D(pi,p-) = a - b1p' + b2p-', i = 1,2, b, > b2 > 0.5 On the cost side, assume that each firm can produce at constant marginal cost c up to the capacity level k and incurs an additional cost 0 for each unit pro- duced in excess of k. The "short-run" marginal cost is thus given by

c for qi k MCi9

c + 0 for qi > k.

The "short run" total cost is

T cq' forq' k TC9Q

1 ck' + 9(qi - ki') for qi > kI'. The unit cost of capacity is assumed to be

constant and is denoted by c0, so that the total cost of capacity is given by TCi = cok'. The "long-run" marginal cost is therefore given by c + c0. For the firm to have an incentive to invest in capacity, the long-run cost must be lower than the cost of producing in excess of capacity; therefore, I will assume c0 ' 0 throughout the paper.

This cost structure admits several interpre- tations. One is that the firm can produce k units of the good "in house," and if it wants to produce more than k it has to incur a higher cost

4 For the Cournot outcome to obtain, several assumptions must be satisfied: efficient rationing, perfectly substitutable products, and symmetric costs. If any of these assumptions does not hold, the Cournot outcome may not obtain. See for example Carl Davidson and Raymond Deneckere (1986) for the extension of the Kreps- Scheinkman model to different rationing rules.

' I will point out how results can be generalized to nonlinear demand functions in later footnotes. In order to economize on space and algebra, I will generally use the notation D(p', p-') rather than the explicit linear notation.

VOL 86 NO. I MAGGI: STRATEGIC TRADE POLICIES 241

(e.g., because it has to purchase inputs from outside the firm, or it has to subcontract part of the production process). An alternative interpretation is that 0 may represent the cost of overtime work; under this interpretation, the value of 0 would be influenced by such factors as the degree of unionization of the industry: the presence of a union may raise the cost of overtime work, thus making capacity constraints more inflexible. In all cases, k represents the "maximum efficient scale" of production, and the parameter 0 captures the importance of capacity constraints in the industry.6 I assume that 0 has the same value for the home and the foreign firm, but the model can be easily extended to allow for differences in the importance of capacity constraints across countries.

The game consists of two stages. In the first stage, firms simultaneously choose capacities. In the second stage, after observing both capacities, firms simultaneously choose prices, then produce to satisfy demand. Firm i maximizes total profits, given by xr' = p'D(p', p-i) - TC9 - TC. If the firm produces at capacity (q' _ k'), which will always be the case in equilibrium, the profit function reduces to 7r' = (pi c - co)D(p, pi).

It will be useful to define a "Bertrand" benchmark and a "Cournot" benchmark for this game. Since the long-run per-unit cost is given by c + co, the relevant Bertrand benchmark is the one-shot game in which firms have constant marginal cost c + co and compete in prices, while the Cournot benchmark is the one-shot game in which firms have constant marginal cost c + co and compete in quantities.

The mechanism through which capacity commitment,can mitigate price competition is

intuitive. By choosing scale k, a firm generates a disincentive for itself to produce more than k at the following stage, and this commitment device is used strategically to sustain higher prices. The parameter 0 measures the effectiveness of this commitment device, thereby in- dexing the mode of competition: I will show that as 0 increases the outcome of the game (prices, quantities and profits) moves from the Bertrand benchmark to the Cournot benchmark. The Bertrand outcome obtains for 0 = co, and the Cournot outcome obtains for 0 higher than some critical level 9C.

This result is broadly consistent with the results of Klemperer and Meyer (1989) and Kenneth L. Judd (1989), in spite of the very different structure of these models: both models find that a critical factor in determining the nature of competition lies in the convexity of the cost curve, with higher convexity pushing toward the Cournot outcome. In the model proposed here, the total cost of production is piecewise linear but globally convex, with a higher 0 implying higher convexity: here too, higher convexity of the cost curve pushes the equilibrium closer to the Cournot outcome. In this sense, the present model, in spite of its static and very stylized structure, seems to capture some essential elements of more fully fledged models of oligopolistic competition.

A. The Price Subgame

The first step is to derive the subgame reaction function of each firm given the installed capacities. First define: r'(p-'; x) argmaxp(p - x)D(p, p'), that is, the Bertrand reaction function when marginal cost is constant at x. Also let: pi = 'V(p-'; ki) be defined implicitly by D(p', p -) = k', which represents the price combinations such that the demand for firm i is constant at k'; in other terms, i(p`; ki) is an isoquantity curve.7 It is easy to verify that &4Vlap` > ar'/jp-F. The following lemma derives the subgame reaction function, denoted by pi = R'(p-'; ki) (see Figure 1).

6A similar cost structure is postulated in Avinash Dixit's (1980) model of entry barriers. Dixit's model dif- fers from the present one not only because one firm (the incumbent) enjoys a first-mover advantage, but because firms compete in quantities, not in prices, after the investment stage. Another key difference is that Dixit assumes 0 = co: investment in capacity has the character of a pure sunk cost, whereas in this paper investment has a cost- saving function. Staiger and Frank A. Wolak (1992), in the context of a capacity-constrained game with demand uncertainty, make a similar assumption of "flexible" capacity constraints: after the realization of demand, firms can add to the initial capacity but at a higher cost.

'The closed-form expressions for r' and 4'1 are r'(p-'; x) = (a + b2p-')/2b, + x/2 and VD(p-'; k') = (a + b2p-' -k')Ib,.


pz R1(p2;k1

r1(p2; C) / 1(p2; k1)

r2c +0)

p

FIGURE 1. THE SUBGAME REACTION FUNCTION

LEMMA 1:

R'(p'; k') = r'(p'; c) where

bJ'(p-; kl) < r'(p-'; c) (branch 1)

R'(p'; k') = 1(p-'; k') where

ri(p -; C) s 1,(p; k') ' r'(p1; c + 0)

(branch 2)

R (p'; k') = r' (p; c + 0) where

JV(p-'; k') > r'(p'-; c + 0) (branch 3).

The proof of Lemma 1 is intuitive. Given p, firm i chooses the optimal point along its residual demand curve. This choice can be described as picking the quantity level at which firm i's residual marginal revenue curve, say MR j, crosses its marginal cost curve MCi. If p' is low, so that 4V (p'; k') < r'(p'; c), MR rcrosses MCi where MCi = c, so the best- response price coincides with the Bertrand best-response price for constant marginal cost c, namely, r'(p'-; c). This explains "branch 1" of R'. If p-' is intermediate, so that r'(p'-; c) s 1?'(p'; k') c r'(p-'; c + 0), MR i crosses MCi at its vertical segment, where qi = k'; thus, firm i's best-response price is the one that keeps q' constant at k', namely,

VJ (p ` i; k'). This explains "branch 2" of R'. Finally, if p-' is high, so that 4iV(p` ; k') > r'(p'; c + 0), MR' crosses MCi where MCi = c + 0; hence the best-response price coincides with the Bertrand best-response price for constant marginal cost c + 0, that is, r'(p-i; c + 0). This explains "branch 3" of R'.

The equilibrium of the price subgame is given by the intersection of the two subgame reaction functions. One can check that the slope 0R'1/p -i lies between 0 and 1 for any pair of capacities. This implies that for any pair of capacities the subgame admits a unique pure-strategy equilibrium.8

It may be interesting to relate this result to the well-known fact that pure-strategy equilibria (in prices) often fail to exist in Bertrand- competition games with homogeneous goods and rigid capacity constraints (e.g., Kreps and Scheinkman, 1983). The nonexistence problem is caused by two features of these models. One is product homogeneity, which implies a discontinuity in the residual demand function. The other is that the production cost is infinite for output in excess of capacity. This implies that consumers must be rationed for certain ranges of prices, and the demand "spillovers" from one firm to the other may imply profit functions that are not quasi-concave. In the model proposed here, neither of those features is present, since I assume differentiated goods and firms do not have to ration consumers (since the marginal cost of production is finite for any production level); hence, profit functions are continuous and quasi-concave, and a unique pure-strategy equilibrium exists.9

x For nonlinear demand functions, Lemma I and the uniqueness of a pure-strategy equilibrium in the subgame still hold as long as: (i) products are substitutes and own effects dominate cross effects in demand, so that the slope of 1D is between 0 and 1; (ii) the Bertrand reaction function (for constant marginal cost) is upward-sloping and "stable," meaning that its slope is also between 0 and 1; (iii) if p-i is increased, firm i's optimal point on its residual demand curve entails a higher price and a higher quantity. One can show that this implies 09V/0p-' > Or'/0p`.

9 When products are differentiated and capacity constraints are rigid, price subgames may or may not have pure-strategy equilibria. A model with this structure is examined by James W. Friedman (1988). Friedman shows that in this case profit functions may or may not be quasi- concave: if the demand spillover (generated when one firm

VOL. 86 NO. I MAGGI: STRATEGIC TRADE POLICIES 243

B. The Full Game

I now solve for the subgame-perfect equilibria of the whole game. I first examine the case in which firms have symmetric demand and cost functions and then generalize the results to the case of asymmetric firms. Let {pb(x), qb(x)1 and {pc(x) qc(x)1 denote, respectively, the Bertrand and Cournot price and quantity when both firms have constant marginal cost x. As already argued, the relevant benchmarks for this game are the Bertrand and Cournot prices and quantities when x = c + co. I assume that output is positive at the benchmark Bertrand and Cournot equilibria. Given the linear-demand specification, this is ensured if c + co < a/(b - b2).

It is first useful to compare the Bertrand and Cournot benchmarks in price space. The Bertrand price pb(c + co) is given by the intersection of the two curves r (p2; C + co) and r2(pl;c + co) (point B in Figure 2). The Cournot price pc(c + co) can be identified graphically if one first defines a new curve. Suppose firm i chooses the optimal price pair taking the rival's quantity as fixed, say, at k-i. This entails choosing the point of tangency between firm i's highest isoprofit curve and the rival's isoquantity 4i(pi, k').l0 As k' changes, this tangency point traces a curve in price space, which I denote by 0i(p i; c + co) (this curve can be seen in Figure 3, together with another curve to be defined later). The

A .

curve C can be interpreted as representing "Cournot conjectures" in price space." The Cournot price pair is given by the intersection of O' and C2 or, equivalently, the point at which each firm's isoprofit is tangent to the rival's isoquantity (point C in Figure 2).12 It

should be noted that the curves r' and C' are in general not parallel. With linear demand, one can check that Orl/Op' < Ci/Opi.13

C. Symmetric Equilibria

For expositional purposes I first look for symmetric equilibria. Asymmetric equilibria, which can arise for certain parameter values, are discussed in the next subsection. The next proposition establishes that the game admits a unique symmetric equilibrium. As 0 increases, this equilibrium moves from the Bertrand point B to the Cournot point C. In particular, the Bertrand outcome obtains when 0 = co, and the Cournot outcome obtains when 0 is higher than a critical level OC, defined implicitly by pb(c + OC) = pc(c + co) (it can be shown that this equality defines a unique OC). For intermediate levels of 0, the symmetric equilibrium price is given by the Bertrand price for marginal cost (c + 0), that is, pb(c + 0). This result is recorded in the next proposition, whose proof can be found in the Ap- pendix, along with all other proofs of the paper.

PROPOSITION 1: If firms are symmetric, the game admits a unique symmetric subgame- perfect equilibrium. This equilibrium entails

[p b(C + 0) if co 0 < OC

1 pC(c + co) if 0 oc

k= D(p, p)

qb(C + 0) if co- 0 < 0C

- qc(c + co) if 0 oc = [p - (c + co)]k.

Proposition 1 suggests that the parameter 0, which measures the importance of capacity

rations its customers) is small enough relative to the degree of product differentiation, the quasi-concavity of profit functions is preserved, and the price subgame may admit pure-strategy equilibria.

'?When I use the term "isoprofit" without further specification I am referring to the "long-run" profit function, 7r' = (p' - c - c)D'(p', p-').

" The closed-form expression for C' is C'(p-'; c + co) = [bb2p-i + ab1 + (c + co)(b 2 - b2)]1/(2b2 - b2).

12 With linear demand, the Bertrand price is always lower than the Cournot price. With nonlinear demand, the Bertrand price is lower than the Cournot price under fairly

general conditions; a sufficient condition, for example, is that the Bertrand and Cournot prices are unique, as shown in Xavier Vives (1985).

'3 With linear demand, the condition for r' to be flatter than e' is b2 < 2b1/1F, which is always the case, since b2 < b.


p2

r (p2; C + C.) ,)1(p2; k') /

1 1 2(p' k2

'r /

r2(pl; C + C.)

FIGURE 2. THE BERTRAND AND COURNOT BENCHMARKS

constraints, indexes in a natural way the mode of competition, from the case of pure price competition (Bertrand) when H = co, to that of pure quantity competition (Cournot) when

0 c 14

The intuition for this result is the following. In the absence of capacity constraints, the equilibrium is given by the Bertrand price, at which each firm's marginal benefit from lowering the price ( and increasing its market share) equals the marginal cost of expanding production. The Cournot price cannot be sustained as an equilibrium, because at this price a firm's marginal benefit from price-cutting outweighs the marginal cost of expanding production. Now suppose that 0 is high enough and capacities are set at the Cournot level. Now the marginal benefit from price-cutting is outweighed by the marginal cost of expanding production beyond capacity; therefore, the Cournot outcome can be sustained as an equilibrium. In general, the highest price that can be sustained is the one at which the marginal benefit from price-cutting equals the marginal cost of expanding output beyond capacity, c + 0. But this is equivalent to saying that firms can sustain the same price as if they were com- peting 'a la Bertrand with constant marginal

cost c + 0, that is, pb(c + 0), as long as this price does not exceed the Cournot price.

In sum, the choice of capacity is an imperfect commitment to limit production and raise prices. The higher is 0, the more effective is this commitment device, and the higher is the price that firms can sustain.

D. Asymmetric Equilibria

For a certain parameter range, the game also admits asymmetric equilibria, as well as the symmetric one just characterized. To identify these additional equilibria, I need to introduce some more notation. I will use r'(0) as a shorthand for r'(p'-; c + 0). For a given 0, consider the point of tangency between firm i's highest isoprofit curve and the curve r-i(0). As 0 changes, r-'(0) is shifted, and this tangency point traces a curve in price space. I denote this curve by S'(p'; c + c(). This curve is linear, it lies between r'(c0) and Ci, and it can be shown that its slope is intermediate between the slopes of r'(co) and ci (see Figure 3). Also, letp5 denote the (symmetric) price at which S' crosses the diagonal, and let OS (< c) be the value of 0 such that pb(c + 0) = pS. The game admits asymmetric equilibria when 0 lies in the interval (OS, OC). If firms are symmetric, for 0 E (O', Qc) the set of equilibrium prices is given by

7 {(p', p2): p' = r1(0), Sc p'

'r-(0) pi c ci p -ir-^-i

i I lor i= 2}

The structure of this set is illustrated in Figure 4. Note that the only symmetric point in F is pb(c + 0).16 It turns out, however, that

14 Proposition 1 holds also with nonlinear demand, under the assumptions spelled out in footnote 8.

1 The closed-form expression for S3 is S'(p-i; c + cO) = 2b,b2p- Y(4b 2 - b2) + ab, + (c + co)(2b2 - b 2)2. If demand is not linear, 3i still lies between r' and C, provided the assumptions of footnote 8 are met, but the slopes of these curves may be ranked differently. The qualitative results of the model are not affected, however.

6 The reason a point like Q in Figure 4 constitutes an equilibrium can be explained heuristically as follows. The capacity levels that sustain Q as equilibrium prices are such that 4V and 12 trace through Q; call these capacities kQ and kQ. Given kQ and kQ, prices Q clearly constitute an


p

r'p; c + Co) S'(p2 c + c0)

lC(p2; C + CO)

l l l ___ ?~~~~(D2 (p; k2 1) _ 2(p1; k2")

/ - ~~~~(2(p1;k2)

I r2(p'; c + 6") -- - - - - - - - - r~~2(pl; C ? 9')

2p;C + Co)

p

FIGURE 3. THE CURVES C AND 3

most of the asymmetric equilibria are not robust to a small perturbation of the model, namely, an approximation of the marginal-cost function with a close-by smooth function. The only asymmetric equilibria that survive this perturbation are points z' and z2 (see Figure 4), where z' is the intersection between r"(0) and s Moreover, these points constitute equilibria only when 0 lies in the interval ()V, C), where Ov denotes the value of 0 such that point z'I lies on c& (one can check that OS < Ov < Oc). When H > C)V, the points z' and z2 fall outside F; hence, they are no longer equilibria of the game. The next proposition describes the set of

robust equilibria of the game (bold characters denote vectors).

PROPOSITION 2: If firms are symmetric, the robust equilibrium prices are given by:

(i) the symmetric equilibrium: p = min { pb(c + 0), pc(c + co)), and (ii) p = zi, i = 1, 2, if E (s, Ov]. At any equilibrium, capacities are given by k = D(p).

I will now generalize the previous result to allow for asymmetric costs or demand functions, but first I need to define one more curve. Let R'(p -') denote the lower envelope of the curves r'(0) and Ci: Ri(p-) = min{r'(p-'; c' + 0), C'(p-; c' + c')} No- tice that, if firms are symmetric, the intersection between R' (p2) and R2 (pl) is given by the symmetric equilibrium, min { pb (C + 0), pc ( c + co) ). The next proposition describes the set of robust equilibria allowing for asymmetric firms. In it, the definitions of point z' and the set F are analogous to the definitions in the case of symmetric firms. The proof follows logic similar to that of Proposition 2 and is omitted.

PROPOSITION 3: The robust equilibrium prices are given by:

(i) the intersection of R'(p2) and R2(p'), and

(ii) z' if z' e F (i = 1, 2).

At any equilibrium, capacities are given by k = D(p).

E. Comparative Statics

Before turning to trade policy issues, it is worth pointing out some comparative-statics implications of the model that may be of interest for oligopoly theory. The first one concerns the impact of the parameter 0 on equilibrium prices, quantities, and profits. As 0 increases, prices and profits increase (and quantities decrease) for both firms, unless 0 is sufficiently high, in which case further

equilibrium of the subgame. The question is: why are kQ and kQ equilibrium capacities? (i) Given kQ, firm 1 would like to implement the Stackelberg point on firm 2's subgame reaction function, R2(p'; k2) - R2, by suitable choice of k'. Since RQ has a kink at Q, and firm l's isoprofit is tangent to R2 at that kink, firm l's Stackelberg point is given by point Q; hence choosing kQ is optimal. (ii) Given kQ, firm 2's Stackelberg point lies somewhere on R'(p2; kQ) above Q; however, no point above r2(0) can constitute an equilibrium (this follows directly from Lemma 1). Thus firm 2 cannot do better than inducing equilibrium prices Q, by setting capacity k .

17 There is no simple intuition for this result, but the reason an equilibrium like Q in Figure 4 is not robust is that this equilibrium relies heavily on the presence of a kink in firm 2's subgame reaction function R2. If the marginal cost curve is approximated with a smooth function, R2 also becomes smooth, and the equilibrium breaks down.


p2

, / I2

, / |() p1

FIGURE 4. THE ASYMMETRIC EQ2uILIBRIA

changes of 0 have no impact. When there are multiple equilibria, this statement holds for each equilibrium. In sum, the model predicts that when 0 is higher ( i.e., when capacity constraints are more important), the outcome tends to be less competitive. This prediction of the model is potentially testable, to the extent that 0 can be quantified. This is in sharp contrast with the conjectural-variations model, in which the parameter that indexes oligopolistic conduct is not even observable in principle ( let alone its problematic interpretation).

The second interesting implication of the model concerns the impact of the cost of capacity co,. The existing models of capacity- constrained price competition suggest, in Jean Tirole's ( 1988 p. 217) words, that "the Cournot-outcome results are more likely to hold when the investment cost c,, is high." I The idea is that, when capacity constraints are rigid, a high cost of capacity co generates a large discrepancy between first-period (ex

ante) costs and second-period (ex post) costs; hence it induces a higher willingness to "dump" existing capacities (charge market- clearing prices). The present model suggests a qualification to this view. Focusing for sim- plicity on the symmetric equilibrium price, p = min { pb(c + 0), pc(c + co)}, forgiven 0 an increase in co can only increase the divergence between the equilibrium price and the Cournot price (and introduces no divergence if 0 is high enough).'9 The intuition is that here capacity is only an imperfect commitment device: if co is high, so that there is a large discrepancy between ex ante and ex post costs, firms are more likely to produce beyond capacity at the second stage, and hence are less willing to dump the existing capacities.

II. Strategic Trade Policy under Complete Information

Suppose a home firm and a foreign firm compete on a third market, and assume that the firms' products are consumed only in the third country, so that there are no consumption effects for the exporting countries.20 The firms are assumed to have identical costs of production and capacity.

The model focuses on unilateral trade policies, but the key insights of the analysis should carry over to the case of bilateral intervention. The home government is allowed to use two policy instruments: output-per-unit subsidies, 21 which affect the production cost c, and apacity-per-unit subsidies, which affect the cost of capacity co. The government's objec- tive is given by the domestic firm's net profits, that is, its total profits minus the subsidy pay-

Ix In this statement, Tirole refers to the class of two- period games in which firms choose (rigid) capacities in the first period and prices in the second period. In Kreps and Scheinkman (1983), the equilibrium outcome coincides with the Cournot outcome regardless of co, but as soon as one introduces asymmetries in the firms' costs, rationing rules different from the "efficient" one, or product differentiation, the equilibrium may not coincide with the Coumot outcome if c( is relatively small, whereas the Coumot outcome obtains if co is high enough.

19 To see this, note that for 0 < 9 an increase in co does not affect the equilibrium, pb(c + 0), while it increases the Coumot price, pc(c + co). For 0 > 9c, the equilibrium is pc(c + co); hence an increase in co introduces no divergence.

20 This assumption, which is equivalent to assuming that markets are segmented, is made to isolate the profit- shifting motive for trade policy from issues of terms of trade and correction of monopoly distortions. The insights offered by the model carry over to situations where other motives for trade policies are also present.

21 1 will generally speak of subsidies, with the understanding that a negative subsidy represents a tax.


ments (whether positive or negative) from the government.

The trade-policy game consists of three stages: in the first stage the home government commits to a policy scheme, which is observed by both firms; in the second stage firms simultaneously choose capacities; and in the third stage firms simultaneously choose prices. What is essential for the analysis is that the government has superior commitment ability relative to the domestic firm. This assumption is adequate for countries that have stable and credible governments, but in other situations this may not be a good description of the trade- policy game. The examination of alternative timing assumptions is potentially interesting, but beyond the scope of the present paper.

Before proceeding, I need to clarify the methodology of the analysis to follow. Ex- amining the impact of unilateral trade policies amounts essentially to a comparative-statics exercise on the equilibria of the oligopoly game: one needs to understand how changes in the home firm's production cost or capacity cost affect the outcome of the game, and in particular the home firm's net profits. In the last section I showed that when 0 E (Os, Ov) absent trade policy, the game admits two asymmetric robust equilibria as well as a symmetric one. In order to evaluate the effects of trade policies, I make two assumptions: (i) absent trade policies (i.e., when firms have symmetric costs) firms focus on the symmetric equilibrium. (ii) small changes in parameters have small effects on the equilibrium selected by the firms. Formally speaking, if ai denotes the vector of relevant parameters and Pe(a) is the equilibrium price correspondence, the firms' equilibrium selection pe(a) E Pe(a) is restricted to be continuous in a .22,23

Together, these assumptions make it possible to examine the impact of subsidies and

taxes even for the multiple-equilibrium range of parameters. For example, the assumptions imply that, for 0 E (OS, Ov), absent trade policy, firms focus on the point of intersection between r'(p2; c + 0) and r2(p'; c + 0); and if an output subsidy s is given to the home firm, firms focus on the point of intersection between r1(p2; c - s + 0) and r2(pI; c + 0).

I now turn to the analysis of strategic trade policies. The existing strategic-trade literature has pointed out that output subsidies tend to be optimal when firms compete in quantities (if these are strategic substitutes), and output taxes tend to be optimal when they compete in prices (if these are strategic complements). In the present model capacities are strategic substitutes (as will be seen shortly),' while the price subgame is one of strategic complements. This might suggest that there is scope for capacity policies as distinct from output policies, and perhaps that the optimal trade policy might entail both an output tax and a capacity subsidy. In contrast with this intuition, I will show that: (i) capacity subsidies are redundant if output subsidies are set opti- mally (thus there is no loss of generality in restricting attention to output policies); and (ii) the optimal output subsidy is negative for low values of 9, 0 for an intermediate value of 9, and positive for high values of 9, equalling the Eaton-Grossman (negative) subsidy when 0 = co, and the Brander-Spencer subsidy when 0 is high enough.

Here I illustrate the effects of small output subsidies/taxes; the optimal trade policy is fully characterized in Proposition 4. The reader can refer to Figure 5, where ri(0) is used as shorthand for r1(p-1, c + 0). Also recall the definition of the curve 31 (drawn in Figure 3) and of Os, the value of 0 such that r1(0) intersects g1 on the diagonal. Consider first the case 0 < OS. Absent trade policy, the equilibrium is given by the intersection between r'(0) and r2(9). An output tax shifts r1 (0) to the right, thus moving the equilibrium point upward along r 2(9). Since the laissez- faire equilibrium point lies on the left of the curve g the home firm's net profits increase upward along r2(9). Hence an output tax is beneficial.

Next consider the case 0 = Ss. In the absence of trade policy, the equilibrium point

22 Notice that a continuous selection from a correspondence exists whenever the correspondence is lower- hemicontinuous. In my model, the (robust) equilibrium correspondence is continuous in the underlying parameters; hence a continuous selection exists.

23 An equivalent way of phrasing this restriction is with reference to the players' expectations: if I expect you to play pe when parameters are et, my expectation about your action can change only slightly if et changes slightly.


ph(c + 0) is also the tangency point between the home net isoprofit and r2(0); hence, an output subsidy or an output tax would result in lower net profits for the home country. A laissez-faire approach is optimal.

Suppose now 0 lies in the interval (9S, Oc). In the absence of trade policies the symmetric equilibrium is again given by the intersection between r1 (0) and r2(0), but now to the right of the curve S1; hence an output subsidy, which shifts r1 (0) toward the left, is desirable.

Finally, suppose 0 > 9c. The no-intervention equilibrium is given by the intersection of C1 and C2, that is, the Cournot point. A positive output subsidy shifts C1 to the left, so equilibrium prices move down along C2. But home net profits increase in this direction; hence, an output subsidy is again desirable.

The reason why capacity policies are redundant is that output policies turn out to be sufficient to implement the optimal price pair. Thus a capacity policy cannot do any better.

Proposition 4 characterizes the optimal complete-information trade policy as a function of 0. Let s * (0) denote the optimal output subsidy, 5BS the Brander-Spencer subsidy when firms play Cournot and have constant marginal cost c + c0, and 5EG the Eaton- Grossman (negative) subsidy when firms play Bertrand and have constant marginal cost c + CO.

PROPOSITION 4: (i) Output policies can do everything that capacity policies can. (ii) The optimal output policy entails

s*(9) < O for co 0 < Os

s*(O) = O for 0 =S

s*(O) > 0 for 0 > S Furthermore, s*(9) = 5EG for 0 = co and s*(9) = sBSfor 0 2 O?

Proposition 4 establishes that output taxes are called for when capacity constraints are not very important, and output subsidies are called for when capacity commitment is effective enough. When capacity constraints are of intermediate strength, a laissez-faire policy is optimal. The textbook results for pure price

competition and pure quantity competition are special cases of Proposition 4: when 0 = co, the optimal tax is equal to the Eaton-Grossman tax; when 0 2 OC, the optimal subsidy is equal to the Brander-Spencer optimal subsidy.24

Notice that the result of Proposition 4 bears a superficial resemblance to the result that obtains from a conjectural-variation model (see e.g., Eaton and Grossman, 1986). In that model, whether the government subsidizes or taxes the home firm, or does not intervene at all, is determined by the conjectural-variation parameter: the subsidy is at a maximum under "Cournot conjectures," is zero under "consistent conjectures," and is negative under "Bertrand conjectures." In my model of imperfect quantity commitment, the strength of the commitment (0) determines the optimal strategic distortion of incentives. A stronger (weaker) capacity commitment yields the same result as a small (large) conjectured variation of the rival's output. The case of "consistent conjectures" corresponds to the case: 0 = O9.

The model suggests new insights about the role of strategic distortions of the firms' incentives. The existing literature (in particular, Jeremy I. Bulow, John D. Geanakoplos and Klemperer [1985] and Drew Fudenberg and Tirole [1984]) has identified a simple principle to predict the nature of the optimal incentive distortion: a more (less) aggressive behavior is called for when the firms' choice variables are strategic substitutes (complements). I will refer to this as the "BGK-FT principle" of strategic distortions.

Can the BGK-FT principle help predict the nature of optimal trade policies in this setting? The answer is no. For example, in this model, capacities are strategic substitutes, yet it is not

24When H lies in (9S, 9V), so that multiple (robust) equilibria arise, Proposition 4 uses the assumption that the equilibrium selection is continuous in the trade-policy parameter (and symmetric, absent trade policy). While this continuity assumption seems quite natural for small subsidies/taxes, it might be considered more problematic for large policy changes. If one remains agnostic about the effects of large policy changes in the presence of multiple equilibria, one should weaken Proposition 4 for the interval H E (9S, 9V) to read "a small output subsidy is beneficial to the home country."


p2 A.

; / ~ ~~~~~~2(o)O

B. Si r) (0)(

~~~~~~C2

/a/~~~~~ B~~~~~ /~~~~

L p 1

p2

C. Ci ri (0)

/ 1/ / r~~~~~2(o)

/~~~~~~~~ /~~~~ ._--- B/ S c

FIGURE 5. THE EFFECTS OF SMALL OUTPUT SUBSIDIES/ TAXES: A) THE CASE H < HS; B) THE CASE H C (HS, HC);

C) THE CASE H > HC

always desirable to induce an expansion of the home firm's capacity: if 0 < O' it is optimal to induce a contraction of the home firm's capacity (and output). The reason the BGK-FT principle is of little guidance for understanding optimal trade policies in this setting is the following. Since the principle is based on the assumption that firms play a one-shot game, it can be applied only to the reduced-form game in capacities; hence, it can yield predictions only about capacity policies when output policies are not available. To draw out the rela- tionship between the present model and the BGK-FT principle, it is helpful to focus first on an amended version of the model, in which governments are allowed to use only capacity policies. This game is also interesting in its own right, since export subsidies (but not capacity subsidies) are prohibited by GATT.25

A. Capacity Policies Only

In this section I analyze the effects of capacity policies when output policies are not available, explain why capacity policies are less effective than output policies (even though capacities are equal to outputs in equilibrium), and discuss the logic of optimal trade policies in this setting. Let ori (ki, k-i) denote firm i's reduced-form profit function in terms of capacities. In order to relate the results to the BGK-FT principle, one needs to sign the externality effect 07ri /Ok - and the slope of the capacity reaction function 0k1/0k-1.

A complete characterization of these signs for all capacity levels turns out to be complex, but to make the important points it suffices to characterize them around the (laissez-faire) symmetric equilibrium point, k .

LEMMA 2: Around the laissez-faire equilibrium point, capacities are conventional and Wtrntfpaio V,h.Vtitfuft.V 26

25 In spite of the GATT ban, however, it is important to understand the governments' incentives to use export policies, for at least two of reasons: (i) the GATT law is not directly enforceable, and hence it is important to analyze the incentives of governments to violate the agreements; (ii) governments are often able to offer subsidy-like policies that fall in the gray area of the GATT law.

26 If the reduced-form profit functions or the capacity


(i) (O1rICik )Ike c 0 (ii) (0k0/0k`C)Ike c 0.

The intuition for result (i) is very simple: an increase in a firm's capacity gives this firm an incentive to expand its production and lower its price in the second period, thereby harming the rival firm. The intuition for result (ii) is more subtle. For 0 > OC, firms can sustain the Cournot outcome, and the intuition is the same as for why quantities are strategic substitutes in the Cournot model. For 0 < OC, ke = qb(c + 0) andpe = pb(c + 0). If a firm increases its capacity above qb(c + 0), this gives both firms incentives to lower prices below pb(c + 0) in the subgame (both prices face downward pressure because they are strategic complements). Anticipating this, the rival firm is induced to reduce its capacity in order to counter this downward pressure on prices.

Having established that capacities are both conventional and strategic substitutes, I now consider the effects of capacity policies. The BGK-FT principle, applied to this game, implies that a small capacity subsidy, which reduces the home firm's cost of capacity, is beneficial for the home country, provided c0 affects the equilibrium capacities. The interesting point here is that, for 0 < OC, small changes in c0 do not affect the (symmetric) equilibrium ke = qb(c + 0); hence, small capacity subsidies are neutral. On the other hand, for 0 2 OC a small capacity subsidy does affect the equilibrium and is beneficial to the home country.

Of course, these results for the capacity- policy-only game are entirely consistent with the BGK-FT principle. However, the logic that governs strategic trade policies when output policies are available is very different from the BGK-FT logic, and it requires a different line of intuition.

The key point is to understand why output policies can do more than capacity policies. Capacity policies can affect only the capacity reaction function of the home firm, not that of the foreign firm. Output policies, on the other

hand, affect the second-period incentives of the home firm, and this feeds back to affect the capacity reaction functions of both the home firm and the foreign firm.

In particular, taxing the home firm's output reduces both firms' incentives to build capacity,27 and for low levels of 0 this is beneficial (to both firms). This explains why for low levels of 0 it is optimal to induce a contraction of the home firm's capacity (through an output tax), even though capacities are strategic substitutes.

The logic of strategic trade policies when both capacity and output policies are available admits a very simple intuition. The government wants to induce less (more) aggressive behavior when 0 is low (high), and the intuition is the following. Capacity-setting is used by the firms as a commitment to limit production, and the parameter 0 measures the effectiveness of this commitment device. When this commitment device is not very effective, an output tax by the home government provides the home firm with some additional commitment ability, inducing a further contraction of capacity. However, if capacity performs its commitment function effectively enough, the government finds it optimal to induce an expansion of the home firm and a contraction of the foreign firm. In sum, whether the home firm should be encouraged to expand or to contract can be understood not by looking at whether capacities are strategic substitutes or complements, but rather by looking at the effectiveness of capacity as a commitment device.

Another insight provided by the model, which the BGK-FT principle could not suggest, is the following. A priori, one might expect the optimal policy to entail an output tax coupled with a capacity subsidy, since prices are strategic complements and capacities are strategic substitutes; in other words, one might

reaction functions are not differentiable, these signs apply to both the left derivative and the right derivative.

27 To see this, start from the equilibrium of the nonin- tervention game and introduce an output tax, which increases the home cost of production. If capacities were unchanged, both firms would charge higher prices in the subgame and would produce below capacity. Anticipating this, however, both firms will reduce their capacities in the first period, to avoid capacity idleness.


expect the two policies to be complementary. However, the model yields quite different results. For example, if 0 > Oc the two policies are perfectly substitutable, both serving the purpose of inducing the home firm to increase its capacity and output (recall that for 0 > Oc the purpose of trade policy is to shift the curve C (p2; c + co) leftward; since the parameters c and co affect this curve only through their sum, controlling one is equivalent to controlling the other).28

Before turning to issues of uncertainty, it is worth pointing out an implication of the model concerning the value of incumbency, or capacity leadership. Suppose that firm 1 (the "incumbent") builds capacity at t = 1, firm 2 (the "entrant") chooses its capacity at t = 2, and firms simultaneously choose prices at t = 3. Further suppose that firms have symmetric costs. I do not intend to analyze fully this leadership version of the model here, but one interesting result can be easily shown: when 0 is low (namely, when 0 : 9S) the leader's optimal capacity level is just the simultaneous- move equilibrium capacity qb(c + 0); hence, incumbency has no value.29 To understand

why incumbency has no value, recall the traditional argument for why a Stackelberg leader wants to commit to a higher quantity than the Cournot level: increasing production slightly beyond the Cournot level entails a first-order benefit due to the contraction of the rival's output, and a second-order loss because the leader is moving away from its own reaction function. In the capacity-price model, when 0 is low the loss from expanding capacity is first- order, because firms are at a "kinky" maximum, and when 0 < 9S it actually outweighs the benefit.30 Incumbency becomes valuable only as 0 becomes high, that is, if capacity constraints are important enough.

This allows me to make a further point about the role of trade policies. In most strategic-trade models, the government intervention accomplishes exactly what the home firm would accomplish were it in the position of a Stackelberg leader. In the present model, if 0 is low, trade policy can accomplish more than capacity leadership, and the reason is that (as pointed out before) an output policy can affect both the home and the foreign capacity reaction function. In contrast, a capacity leader cannot influence its rival's reaction function.

III. Trade Policy under Uncertainty About the Mode of Competition

The previous section has pointed out that the optimal trade policy depends critically on the mode of competition, determined by the importance of capacity constraints. In practice, a government is likely to have limited information about the technological characteristics of the targeted industry, and consequently it might not know whether firms behave more

28 This suggests a point about the role of strategic cost- reducing investment. Within a simple Cournot model, the BGK-FT principle implies that a firm has an incentive to "overinvest" in cost-reducing activities, provided quantities are strategic substitutes. As Tirole (1988) notes, this is less clear when "quantity" competition is interpreted as capacity-price competition. Suppose activity A reduces the production cost, c, and activity A2 reduces the capacity cost, c0. In this case, "we would predict strong strategic investment ... when this investment reduces the cost of accumulating capacity. In contrast, a firm may be less ea- ger to reduce production costs and trigger tough price competition ..." (Tirole, 1988 pp. 327-28). In other words, Tirole's conjecture is that a firm might want to "overinvest" in A2, but perhaps "underinvest" in A 1. The analysis of the last section does not support this conjecture. Suppose that H is high, so that firms behave like Coumot competitors, and that the two activities Al and A2 have independent and convex costs. Then a firm will want to overinvest both in Al and in A2. Once again, one must be careful in extending the BGK-FT principle to the capacity-price game.

29To see this, consider the (unique and symmetric) simultaneous-move equilibrium capacity, ke = qb(c + 9), and check whether firm 1 would like to commit to a different level of capacity, taking into account firm 2's optimal response. Since the equilibrium prices cannot lie outside the region UB ={p' < r'(9), p2 ? r2(9)}, the best

firm 1 can possibly do is to implement the point of tangency between its highest isoprofit curve and region UBO. But, since H ? 9', firm l's isoprofit curves are flatter than r2(0); hence, such point of tangency is given by pb(c + 9), the simultaneous-move equilibrium. To implement pb(c + 0), it suffices for firm 1 to pick capacity qb(c + 9); hence, the outcome of the leadership game is the same as that of the simultaneous-move game.

30 Notice that, if the marginal-cost function is approximated with a smooth function, the leader's capacity will be approximately equal to the simultaneous-move equilibrium capacity, and the value of incumbency will be of second-order magnitude.


like price-setters or like quantity-setters. As the "informational" criticism of strategic- trade theory points out, if the home government does not observe the value of 0, trade policies can be harmful if based on the wrong beliefs about 0. Consider an output tax: the net profits of the home firm will increase for low values of 0, but for high values of 0 the tax will be harmful. The same reasoning applies, reversed, to an output subsidy. In sum, any output policy can result in a reduction of the home country's income if it is based on the wrong beliefs about 0.

The point of this section is to identify a simple single-rate policy that (weakly) increases the home country's income regardless of the mode of competition. In particular, a small single-rate capacity subsidy increases the home country's income for high values of 0, leaving it unaffected for low values of 0. This is true for any values of the demand and cost parameters; thus a capacity subsidy can be an at- tractive form of intervention even for a government that has no information about the spe- cifics of the targeted market.

This result is a straightforward consequence of the analysis in Section 1I-A. There I argued that a small capacity subsidy is strictly beneficial if 0 2 9C, encouraging an expansion of the home firm's capacity and a contraction of the foreign firm's capacity. For 0 < OC, on the other hand, the (symmetric) equilibrium capacities and prices are given by (pb(c + 0), qb(c + 0)). This equilibrium is not affected by the capacity subsidy; hence, net profits are also unaffected. The following proposition summarizes the result.

PROPOSITION 5: A small single-rate capacity subsidy (weakly) increases the home country's income, independently of the demand and cost parameters of the model. In particular, it is strictly beneficial for 0 2 OC and neutral for 0 < Oc.

The result that capacity subsidies are (weakly) beneficial regardless of the mode of competition is consistent, in a broad sense, with Bagwell and Staiger's (1994) analysis of R&D subsidies. They consider purely cost- reducing investments, and one of their findings is that the optimal investment policy is inde-

pendent of the mode of competition. In the present model, investment in capacity is also a form of cost reduction, although it takes a different form: by building capacity k the firm lowers the cost of producing the first k units of output, while in Bagwell and Staiger the firm can reduce the marginal cost uniformly by investing in R&D. Both results suggest that investment policies are considerably less sensitive than output policies to the nature of product-market competition. In my model, this happens because capacity subsidies are more selective than output subsidies: they induce an expansion of the home firm only when capacity constraints are important and a more aggressive behavior is beneficial, not when it is harmful.

It should be emphasized that the notion of capacity subsidy adopted here requires that the subsidy affect co without affecting the short- run marginal cost c + 0. The most natural interpretation of this kind of policy is in terms of subsidies to investment in new plants and machinery (in the form of direct subsidies, subsidized loans, or tax credits), but one can think of other policies that have the effect of lowering the long-run cost of production, without affecting the short-run marginal cost. For example, if the number of workers can be interpreted as the firm's "capacity," subsidies tied to the wages paid by the firm (e.g., a reduction in the firm's share of social-security payments) can be interpreted as a capacity subsidy.3'

A natural extension of the model is to allow the government to design more sophisticated, incentive-compatible trade policies. In my working paper (Maggi, 1995), the government is allowed to design menus of two-part

3'To see this, suppose that labor (L) is the only factor of production and that the firm can change its price more quickly than it can change L. Let wo denote the cost of labor per unit of output, and let q = f(L) be the production function. Producing more than fiL) requires subcontract- ing production, which costs H > wo per unit of output. Finally, suppose workers must be paid in full even if they 'Are partially idle, that is, if q < fiL). In this case, the number of workers L can be thought of as the firm's "capacity," with a unit cost of wo. The marginal cost of production is zero if q ' f(L), and H if q > f(L). In this setting. any policy that reduces the base wage wo can be interpreted as a "capacity" subsidy.


policies, composed of an output subsidy/tax and a lump-sum transfer, which can induce the home firm to reveal its information about 0. This extension of the model strengthens the conclusions of the previous analysis: not only does the informational constraint not diminish the scope for strategic trade intervention, but it may even augment the policy distortion relative to the complete-information case. More specifically, if firms behave like quantity- setters the home firm gets subsidized to the same extent as under complete information, and if firms behave as price-setters the home firm gets taxed to the same or to a larger extent than under complete information. However, the informational requirements for this kind of policies are stringent: the distribution of the key parameter has to be common knowledge, and the government must have precise information about all the cost and demand parameters. This may help explain why such policy menus are rarely observed in practice, and why governments seem to prefer simpler trade policies.

IV. Conclusion

Precise information about the nature of oligopolistic conduct in international markets is crucial to determine the "appropriate" trade interventions, but governments are unlikely to have this information. The analysis of this paper has suggested that, in spite of such critical informational constraints, individual governments may still be driven to intervene in the international competition. In particular, I have argued that there exist simple single-rate policies, namely, subsidies to the firms' productive capacity, which can increase the home country's income without requiring any information about the particularities of the market.

In view of the analysis, the "'informational" criticism of the strategic-trade-policy theory appears less important than previously envi- sioned. It must be emphasized that this does not amount to a more "activist" view of international trade; in fact, the results of the model should be interpreted in the opposite way: they should be taken as a warning that informational constraints are not likely to re- move the individual governments' economic incentives to engage in export policies. There-

fore, they strengthen-not weaken-the case for international institutions like the GATT, which are intended to promote cooperative agreements among governments.

APPENDIX

PROOF OF PROPOSITION 1: I will focus separately on the two cases. 1. Case 0 < OC.-First notice that, given

capacities k' = k' = qb(c + 9), Lemma 1 ensures that the unique equilibrium of the subgame is given byp' = p2 = pb(C + 0). Next I will focus on capacity choices. The first step is to show that, for any given k-', firm i chooses k' so that in the subgame equilibrium it will produce at capacity: k' = q= D(pi, p'). In other words, the equilibrium prices must lie on "branch 2" of firm i's subgame reaction function R'. I will argue by contradiction. Suppose first that k' > q': then, the equilibrium prices must lie on "branch 1" of firm i's reaction function; but then firm i can slightly reduce k' without affecting the equilibrium prices, and incurring lower costs. Now suppose that k' < q': then, the equilibrium must lie on "branch 3" of firm i's reaction function; hence, firm i can increase k' without affecting prices, and saving costs. Therefore, it must be k' = q .

Next I formulate firm i's capacity-choice problem given k-'. Firm i's problem can be described as selecting the price pair that maximizes profits (p' - c - co)D(', p -') subject to two constraints: that the price pair lies on the rival's price reaction function R -'(p ; k-) and that it lies in the band between r'(p -'; c) and r (p -; c + 9). These constraints can be thought of as "implementability" constraints: firm i can induce, by suitable choice of k', all price pairs that satisfy these two constraints (and only those price pairs); this is an immediate consequence of Lemma 1. The optimal level of k' is then the one that implements the optimal price pair, namely, k' = D(pi, pi). Formally, the best-reply capacity is determined as k (k- ) =D(p (k- ), p-'(k )) where (p'(k), p-'(k)) satisfies

max(p' - c - co)D(p',p-) P'p


p2

r1 (p2; co) r1(p2; c+ )

c 2)p;+

//c (p;C +c)

L r~~~~~~~~~2(pl; C + Co)

p

FIGURE Al. SYMMETRIC EQUILIBRIUM, 9 < HC

subject to

(Al) P = R (p'; ki)

(A2) r'(p ; c) c pi c r'(p'; c + 9) Notice that, if one lets Q (k-') denote the set of price pairs that satisfy (Al) and (A2), firm i's problem can be viewed graphically as picking the point of tangency between its highest isoprofit curve and Q'(k- ). Next I will check that, when the rival's capacity is qh(c + 0), firm i's optimal capacity is also qh(c + 0). First note that the set Qi(qb(c + 0)) is given by a segment of the line J-'(p'; qh(c + 9)), whose highest point is (ph(c + 0), pb(c + 0)) (bold segment in Figure Al). Since the point (ph(C + 0), pb(C + 0)) lies lower on the diagonal than the Coumot point C, firm i's isoprofit at this point is flatter than ( J(p'; k-'); therefore the point (ph(C + 9), pb(c + 0)) is firm i's optimal choice, and the capacity level that implements this point is given by k1 = D(ph(C + 0), ph(C + 0)) = qb(c + 0). It is easy to show that this is also the unique symmetric equilibrium.

2. Case 0 99 . -Let kc qc(c + c) and pC pc(c + co). Following the same logic as in the previous case, one can establish that (a) given capacities (kc, kC) the unique equilibrium of the subgame is (nC- nC)- and (b) for

any k-', firm i sets k' so that qi = k' in the subgame equilibrium. Next, I will check that firm i's best response to k-' = kc is given by kc (see Figure A2). Firm i chooses the best price pair on R (p; kc) such that r'(p-'; c) p' c r'(p-'; c + 0). Noting that "branch 2" of R-'(p'; k) is given by a segment of 41Y`(p'; kC) and contains the Cournot point C (this follows from 0 < 9c), and since C is the point of tangency between firm i's highest isoprofit and (J-', it follows that C is firm i's optimal price pair, and that the optimal capacity is given by kC. The same reasoning applies for the other firm. That (kc, kC) is the unique symmetric equilibrium readily follows from the fact that the Cournot equilibrium is unique.

PROOF OF PROPOSITION 2: Here I present a sketch of the proof; the

complete proof can be found in my working paper (Maggi, 1995). The proof is in two parts: derivation of the equilibrium set and analysis of the perturbed game.

Starting with the original game, the first step is to establish two necessary conditions for an equilibrium. Defining B0 as the region southwest of the r'(0) curves [pi c r'(0), i = 1, 2] and Z as the region southwest of the ci curves [pi C', i = 1, 2], the first step is to show that equilibrium prices must satisfy the following two conditions: (N I ) pC E B0 and (N2) pC E z. Next, three cases must be analyzed separately:

1. Case 0 c 9s. Let po denote the vector (ph(c + 0), p0(c + 0)). It is already known from Proposition 1 that pb, the intersection between r'(0) and r-'(0), is an equilibrium. One can further show that there can be no other equilibria than p^(c + 9).

2. Case 9 E (9S, 9c). Since firms are symmetric, one can focus on the region p1 l p 2 the same arguments apply to the region p' 1 p2, with labels reversed. The first step here is to show that any point in F is an equilibrium. Consider an arbitrary point Q in F (see Figure 4). The capacity levels that sustain Q as equilibrium prices are such that the isoquantities 1' and J2 trace through Q. Call these capacities k' and k , the corresponding isoquantities JQ and 4JQ, and the implied subgame reaction functions RQ and R Q. First notice that, given k' and k , the firms' subgame


p2

r (p; C + 0) rI(p2; C + Co)

4c l (p 2; kc) /o1o

B / o// 1 / r2(p1; c + 0) | ,r1_ ~~~2(pl; kc)

,:B / r2(p~~~r1; c + co)

p1

FIGURE A2. SYMMETRIC EQUILIBRIUM, 0 > Oc

reaction functions cross at Q; hence Q constitutes equilibrium prices for the subgame. Next, check that k' and k2 constitute equilibrium capacities. Focus on firm 1 first: given kQ, and the subgame reaction function RQ that it induces, firm 1 seeks to implement the Stackelberg point on RQ by suitable choice of k1. Noting that RQ is kinked at Q and that firm l's isoprofit is tangent to R4 at the kink, it follows that Q constitutes the Stackelberg point for firm 1; hence, choosing kQ is optimal. Next focus on firm 2: given R4, firm 2's Stackelberg point lies somewhere on RQ above Q; however no point above r2(0) can be implemented (this follows directly from Lemma 1). Thus, firm 2 cannot do better than implementing point Q; hence choosing k 2is optimal. One can further show that no point outside F can be an equilibrium or, equivalently, that the conditions p2 = r2(o), Sl p l rX(9),p2 C C2 and p X C, are all necessary for an equilibrium.

3. Case 9 : Oc.-From Proposition 1, it is known that the Cournot point (i.e., the intersection of ci and C-i) is an equilibrium. It is fairly easy to verify that there can be no other equilibrium for this range of 9.

I will now turn to the issue of robustness of equilibria for the range 0 E (9S, Oc). Consider the following sequence of smooth functions converging to the step function MCi:

MCi (q1; k') [c qi k

fn(q) ki < q< ki + l/n

C + 0 qi'2 k'+ lln

wherefn(q') is a monotonic, smooth function satisfyingfn(k') = c andfn(k' + l/n) = c + 0. The structure of the argument is as follows: first I show that for n large enough there can be no equilibria outside small neighborhoods of p(c + 9), z1, and z2; then I argue that an equilibrium exists in each of these neighborhoods, provided 0 E (Os, 9v] If 0 E (9v OC) z' and z2 fall outside F, and the only robust equilibrium is the symmetric one.

Let R (p-'; ki) denote firm i's subgame reaction function when MCI (qi; k') is the marginal cost curve. R' (p -; ki) is monotonic and smooth, and it converges to R'(p -; k') as n co. Let -'(p1; k') denote the part of R' that connects the two linear branches r'(O) and r'(9).

First notice that for any finite n, an equilibrium need not entail q' = k'. It can be shown, however, that Vn must trace through the equilibrium point; that is, it must be p' = 4(D (p i; k') at equilibrium. Let k' = on,(p1, p-1) denote the value of k' such that JV, traces through (pi, p-i)* Note that (p' converges to D' as n oo. Let Vrn(p', p'; ki) denote firm i's profits when MCq is the marginal-cost curve, and let qrn(p1 p i) _ irn(pi p i; (pn(p1i pi))* If one defines an iso-7r' curve as the locus of points where 7r1(p1, p-i) is constant, firm i's problem can be described as picking the point of tangency between firm i's highest iso-7rn curve and its rival's subgame reaction function Rn-, subject to the constraint r'(p'; c) ? pi r'(p1; c + 9)

Focusing on the regionp l c p2, the key step is to show that for n large enough there are no equilibria outside small neighborhoods of Z2 and p', denoted respectively by N6(z2) and N6(p'). Suppose by contradiction that there is an equilibrium point V outside N6(z1) and N6(p) . There are four possible cases. (i) Sup- pose V is in F. Since R 2 is smooth and coincides with r2(0) at V, it must be that R 2 has the same slope as r2(0) at V. But the iso-7rn


curve at V is steeper than r2(0) for n large enough; hence firm 1 has an incentive to deviate and choose a lower price on R', a contradiction. (ii) Suppose V is in the interior of B0. The curve V l must trace through V, and the iso-ir' curve at V is steeper than V; hence firm 2 has an incentive to deviate and choose a higher point on I, a contradiction. The re- maining possibilities are (iii) V lies to the left of g', and (iv) V is outside B0. These two possibilities are easily contradicted, leaving only the possible equilibria in N, (z') and Np(p'). The final step of the proof, which is not reported here, is to show that there exists an equilibrium in each of the three neighborhoods N6(p') and N6(z') (i = 1, 2) if 0 E (OS, Ov). The result then follows by noting that, as n approaches infinity, these neighborhoods collapse, respectively, to the points pa and z' (i = 1, 2).

PROOF OF PROPOSITION 4: I will first derive the optimal output policy

and then argue that capacity policies are redundant.

1. Case 0 < OS.-Since in equilibrium it must be thatp2 < r2(O) (from condition (Ni ) in the proof of Proposition 2), the government can do no better than implementing the point of tangency between the home firm's highest net isoprofit curve and r2(0) or, equivalently, the point of intersection between r2(0) and S (say, point p *). To implement point p * it suffices to shift r1 (0) outward, just enough to cross r 2(0) at p *. This can be accomplished by an appropriate output tax. Using Proposi- tion 3 one can show that, under such output tax, p * is the unique equilibrium of the firms' game.

For the extreme case 0 = c0, recall that the Eaton-Grossman equilibrium point is given by the tangency between the home firn's highest net isoprofit curve and the curve r2(cO); hence, the government's optimal point coincides with the Eaton-Grossman point, and the output tax that implements it is just the Eaton-Grossman tax.

2. Case 0 = 0s.-By the same logic as in the previous case, the government can do no better than implementing the point of intersection between r2 (0) and S1. But this coincides with the laissez-faire equilibrium, so no intervention is required.

3. Case 0 E ('S, 9v). The point of intersection between A' and r2(0) is the government's preferred price pair, by the same logic as in cases 1 and 2. Since by assumption, absent trade policies, firms focus on the intersection of r' (0) and r2(0), and since the equilibrium selection is continuous in the trade policy parameter, the preferred price pair can be implemented by shifting r'(0) inward, in such a way that it crosses r2 (0) at the preferred price pair. This can be accomplished by offer- ing the home firm an appropriate output subsidy.

4. Case 0 E (0V Oc).-Recalling the definition of 9v from Section I-D, in the case r2 (0) and S' intersect each other above C2. From conditions (Ni ) and (N2) (see the proof of Proposition 2), the government can do no better than implementing the point of tangency between the home firm's highest net isoprofit curve and the curve min{r2(9), C2}; such a point is given by the intersection between r2(0) and C2. By the same logic as in the previous case, the preferred point can be implemented by shifting r' (9) through an appropriate output subsidy.

5. Case 0 :- Oc.-Absent intervention, the equilibrium is given by the Cournot point, that is, the intersection of C' and C2. Since in equilibrium it must be p2 < Ce2, the government can do no better than implementing the point of tangency between the home isoprofit curve and C2. This is the Brander-Spencer equilibrium price pair. The government can implement this point as the unique equilibrium of the firms' game by shifting C' inward, through an appropriate output subsidy (equal to the Brander-Spencer subsidy).

That capacity policies are redundant follows from the logic of the above argument: an output policy always exists that implements the optimal price pair. Therefore a capacity policy cannot do any better.

PROOF OF LEMMA 2: The proof of part (i) is straightforward and

left to the reader. For part (ii), to analyze firm l's optimal response to a change in firm 2's capacity, it is convenient once again to work in price space.

1. Case 0 < 9s.-The (laissez-faire) equilibrium capacities k' are such that the iso-


quantities 4" and 4)2 trace through the equilibrium point pb (c + 0)). An increase in k2 from its equilibrium level shifts 2 downward, say, to 42D. This amounts to lowering "branch 2" of firm 2's subgame reaction function. Firm l's optimal response to this change is to select the most profitable, imple- mentable point on firm 2's new subgame reaction function. Such point is given by the intersection between 4'22 and r'(0). Since firm l's output decreases moving down along r'(0), firm I's optimal capacity decreases as well. Now consider a decrease in k2 starting from the equilibrium level. This shifts 4)2 Up_ ward, say, to 4'2 1. It is easy to see that, after this change in k2, firm l's optimal price pair is still given by pb(c + 0), and its optimal capacity does not change. I conclude that, for this range of 0, capacities are (weak) strategic substitutes around the equilibrium.

2. Case 0 E (OS, Oc).-The logic is the same as in the previous case, except that now firm l's isoprofits are steeper than r2 (0) around the equilibrium pointpb(c + 9). When k2 is increased, firm l's optimal change in k' is identical to that in the previous case. On the other hand, when k2 is decreased and 4)2 iS shifted upward, firm l's optimal price pair moves leftward along r 2(6); since firm l's output increases in this direction, firm l's optimal capacity increases as well. For this range of 0, capacities are (strict) strategic substitutes around the equilibrium.

3. Case 0 2 Sc.-For this range of 0, firm l's isoprofit curve through the equilibrium point pb(c + 0) is tangent to 4'2. When k2 increases, 4)2 iS shifted downward. In response to this change, firm l's optimal price pair moves down along C'. Since firm l's output decreases moving down along C , firm I's optimal capacity decreases as well. On the other hand, if k2 is decreased, 2 is shifted upward, and firm l's optimal point moves up along C'; hence firm I's optimal capacity increases. Thus, for this range of 0, capacities are (strict) strategic substitutes around the equilibrium.

REFERENCES

Bagwell, Kyle and Staiger, Robert W. "The Sen- sitivity of Strategic and Corrective R&D Policy in Oligopolistic Industries." Journal

of International Economics, February 1994, 36(1-2), pp. 133-50.

Brainard, Lael and Martimort, David. "Strategic Trade Policy with Ignorant Policy- Makers." National Bureau of Economic Research (Cambridge, MA) Working Paper No. 4069, 1992.

Brander, James A. and Spencer, Barbara J. "Ex- port Subsidies and International Market Share Rivalry." Journal of International Economics, February 1985, 18(1-2), pp. 83-100.

Bulow, Jeremy I.; Geanakoplos, John D. and Klemperer, Paul D. "Multimarket Oligopoly: Strategic Substitutes and Complements." Journal of Political Economy, June 1985, 93(3), pp. 488-511.

Davidson, Carl and Deneckere, Raymond. "Long-Term Competition in Capacity, Short-Run Competition in Price, and the Cournot Model." Rand Journal of Eco- nomics, Autumn 1986, 17(3), pp. 404- 15.

Dixit, Avinash. "The Role of Investment in Entry-Deterrence." Economic Journal, March 1980, 90(357), pp. 95-106.

Eaton, Jonathan and Grossman, Gene M. "Op- timal Trade and Industrial Policy under Oli- gopoly." Quarterly Journal of Economics, May 1986, 51(2), pp. 383-406.

Friedman, James W. "On the Strategic Impor- tance of Prices versus Quantities." Rand Journal of Economics, Winter 1988, 19(4), pp. 607-22.

Fudenberg, Drew and Tirole, Jean. "The Fat-Cat Effect, the Puppy-Dog Ploy, and the Lean- and-Hungry Look." American Economic Review, May 1984 (Papers and Proceed- ings), 74(2), pp. 361-68.

Hufbauer, Gary C. and Erb, Joanna S. Subsidies in international trade. Cambridge, MA: MIT Press, 1984.

Judd, Ke'nneth L. "Cournot vs. Bertrand: A Dy- namic Resolution." Mimeo, Hoover Insti- tution, Stanford University, 1989.

Klemperer, Paul D. and Meyer, Margaret A. "Supply Function Equilibria in Oligopoly under Uncertain

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