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Strategic trade policy for exporting industries: more general results in the oligopolistic case.

1. Introduction

SINCE the seminal contribution of Spencer and Brander (1983), the scope for stategic trade policy has been extensively investigated in the literature (see, e.g., the surveys by Dixit 1987; Krugman 1988). However, most of the theoretical work has been carried out in the duopoly case and/or within specific models imposing strong assumptions on functional forms (linear demand, constant marginal costs and so forth). From the work of, e.g. Eaton and Grossman (1986) and Markusen and Venables (1988) we know that conclusions about the optimal strategic trade policy might not carry over when assumptions about the nature of the strategic interaction, entry conditions, functional forms, etc. are altered. That is to say, the robustness of the conclusions from the literature on strategic trade policy is largely an open question. The purpose of this paper is to explore the robustness of conclusions about the direction (taxes or subsidies) of optimal trade policy in broad classes of models, incorporating scale economies, product differentiation, minimum restrictions on functional forms, a general number of home and foreign firms as well as price and quantity competition. The paper also examines to what extent optimal trade policy is affected by the presence of home consumption of both domestic and foreign goods.

Markusen and Venables (1988) have examined strategic trade policy in a wide range of cases for exporting industries, taking into consideration home consumption and competition from imports. They compare cases when markets are segmented versus integrated between countries. They examine both industries with free and restricted entry.(1) However, to accomplish this task they restrict their analysis to homogeneous products, linear demand curves, constant marginal costs, and Cournot competition.

The present paper establishes sufficient conditions for drawing definite conclusions about the direction of an optimal trade policy in the oligopolistic case with only weak assumptions about the properties of the profit functions of the firms. Only symmetric equilibria with a given number of firms are considered. Price competition is shown to provide an unambiguous case for an export tax, when we impose mild regularity conditions on the profit functions. No clear cut results can be obtained in the case of quantity competition. However, a simple rule of thumb is derived which determines the direction of an optimal trade policy in terms of the total number of firms in the industry, the relative number of domestic firms and what one might call 'the relative strength of strategic substitution'. From this rule of thumb the impact of scale economies and product differentiation can be derived. Section 3 applies the rule of thumb to determine the cases for taxes or subsidies in the quantity setting case within the widely used model of product differentiation developed by Spence (1976) and Dixit--Stiglitz (1977). The analysis shows how scale economies (in a particular sense) favour the case for an export subsidy, while product differentiation tends to support the case for an export tax. The analysis also reveals that the case for a tax on exports is stronger with a high share of domestic firms and a large number of firms in the industry. Section 4 shows that the presence of home consumption will tend to pull an optimal policy towards a subsidy to domestic producers.

Dixit (1984) has considered the importance of the relative number of home firms in the case with quantity competition and homogeneous products. However, the importance of scale economies, product differentiation and the total number of firms in the industry have not been considered in the literature to my knowledge.

2. The policy problem for an exporting industry

The government's policy choice is formulated in a two stage game framework. The government is able to commit itself to a policy action at the first stage of the game. In the second stage of the game, the firms choose simultaneously their actions subject to the policy determined at the first stage.(2) The industry investigated is characterized by imperfect competition in differentiated products. The policy problem is viewed from the point of view of the home government, which can levy a tax or a subsidy on the domestic producers. The domestic producers sell only to the world market, i.e. the discussion is limited to purely exporting industries. The consequences of home consumption are considered in Section 4. We will only examine cases where the government is unable to influence the number of firms in the industry. More precisely, I will assume there is a fixed number of foreign and domestic firms.

The welfare contribution from a purely exporting industry is equal to total pre-tax profits

[MATHEMATICAL EXPRESSION OMITTED] where [[ps].sup.i]([S, bar above]; t) is firm i's profit given the strategy vector [S, bar above] of the firms in this industry. [N.sub.A] is the set of domestic producers. t is a per unit output tax, which is assumed to be zero initially.

What will be the welfare consequences of a unilateral introduction of a small export tax? That is, what is the sign of

[MATHEMATICAL EXPRESSION OMITTED] where N is the set of all firms operating in this market. Notice that the welfare contribution from the tax-revenue creates no net welfare increase, as it is completely offset by the exact same reduction in profits and is therefore a pure transfer. Some of the elements on the right hand side of eq. (2); that is [delta][[pi].sup.i]/[delta][S.sub.i], will be zero according to the Envelope theorem.

The changes in strategies can be obtained by totally differentiating the set of first order conditions which identifies the firms' optimal strategy choices:

[MATHEMATICAL EXPRESSION OMITTED] where [N.sub.B] is the set of foreign firms. The notation is as follows

[MATHEMATICAL EXPRESSION OMITTED] As is well known, nothing of interest can be said about the solutions to the set of eqs. (3) and (4) without further restrictions on the profit functions, except in the duopoly case (cf. Dixit 1986). The duopoly case has been extensively examined by Eaton and Grossman (1986), and will not be discussed here.

In what follows I will stick to the symmetric case. In the symmetric case equation (2) can be rewritten

[MATHEMATICAL EXPRESSION OMITTED] where [n.sub.A] and [n.sub.B] denote the number of domestic and foreign firms. [[pi].sub.y] represents

[MATHEMATICAL EXPRESSION OMITTED] d[S.sub.A] is the strategy change of domestic firms, whereas d[S.sub.B] is the strategy change of foreign firms. In order to determine the direction of an optimal trade policy, the task is to sign the expression inside the square brackets.

Let me define


[MATHEMATICAL EXPRESSION OMITTED] Using these definitions, (3) and (4) can be simplified to

[MATHEMATICAL EXPRESSION OMITTED] [[pi].sub.xt] is defined as [[delta].sup.2][[pi].sup.i]/([delta][S.sub.i] [delta]t), i [element of] [S.sub.A]. Solving this set of equations gives the result



It is standard practice in oligopoly theory to assume that the system of (3) and (4) is stable (see e.g. Dixit 1986). As shown in the Appendix, stability implies that [absolute value of [pi].sub.xx] > ([n.sub.A] + [n.sub.B] - 1)[absolute value of [pi].sub.xy]. Clearly, stability implies that [absolute value of [pi].sub.xx] > ([n.sub.B] - 1) [absolute value of [pi].sub.xy], and [absolute value of [pi].sub.xx] > [absolute value of [pi].sub.xy]. Consequently, the denominator in (11) and (12) is positive. Notice that [[pi].sub.xx] is negative at a profit maximizing point. Hence, d[S.sub.A]/dt has the sign of [[pi].sub.xt]. The sign of d[S.sub.B]/dt depends in addition on [[pi].sub.xy].

Let me now focus on competition in prices and quantities. It is easy to show that [[pi].sub.xt] is negative if quantities are the strategy variables. Price competition implies that [[pi].sub.xt] is positive.

A general discussion of the sign of [[pi].sub.xy] has been presented by Bulow et al. (1985). They discussed how various primitives of standard models of imperfect competition determine whether goods are strategic complements or substitutes. The analysis showed that [[pi].sub.xy] could be either positive (what they called 'strategic complements') or negative 'strategic substitutes' for both price and quantity competition. It follows that no general conclusion can be drawn for the welfare problem. In what follows I will investigate cases with common assumptions about the sign of [[pi].sub.xy].

In the theories of imperfect competition, the standard (implicit) assumption is that price competition goes together with 'strategic complements' (defined as [[pi].sub.xy] > 0).(3) With strategic complements, both terms in the square bracket in (6) are positive. Hence, with price competition and strategic complements there is an unambiguous case for an export tax.(4) The same conclusion clearly holds if [[pi].sub.xy] is zero.

Let us turn to cases with quantity competition. In models with quantity competition 'strategic substitutes' is the standard assumption.(5) With strategic substitutes (i.e. [[pi].sub.xy] < 0), we can not sign the expression inside the square brackets in eq. (6) without further restrictions. Inserting (11) and (12) into (6) we get

[MATHEMATICAL EXPRESSION OMITTED] Collecting terms inside the square brackets, it follows that dW/dt will be positive if(6) [[pi].sub.xx] / [[pi].sub.xy] > n - 1 / vn - 1 (14) where n is the total number of firms and v is the relative number of home firms.(7) From (14) some rules of thumbs for the direction of an optimal trade policy emerge: (i) relatively 'weak strategic substitutes', and (ii) a high proportion of domestic firms in an industry, will tend to favour an export tax. Vice versa for an export subsidy. The importance of the relative number of home firms was pointed out by Dixit (1984) in the case of homogeneous products.

As marginal cost becomes a declining function of scale, [[pi].sub.xx] will be less negative, while [[pi].sub.xy] is not affected. Consequently, marginal costs decreasing with scale will in general favour the case for an export subsidy. Product differentiation, on the other hand, will weaken the strategic interaction between firms. One might expect that as the products in the industry become more differentiated, [[pi].sub.xx]/[[pi].sub.xy] will increase and thereby strengthen the case for an export tax. In the next section I will examine how scale economies and product differentiation affect the left hand side of (14) using a specific model of product differentiation.

The left hand side of (14) will in general depend on the total number of firms in the industry (but not on the relative number of home firms). Consequently, we are not able to draw definite conclusions about the impact on trade policy of changes in the total number of firms, without further restrictions on functional forms etc. In the next section I will pursue this question within a particular model of product differentiation.

3. Product differentiation and scale economies

This section will present an analysis of a special model with quantity competition in order to see how the question of whether to tax or subsidize an exporting industry, can be referred back to the size of a few interesting parameters (the absolute number of firms, the relative number of home firms, the elasticity of scale in production and the degree of homogeneity of the products). Product differentiation is modeled in the way developed by Spence (1976) and Dixit--Stiglitz (1977). This model allows a simple parametric representation of the degree of product differentiation. Since the model is extensively discussed elsewhere, only a brief presentation will be given below.

The functional form of the utility function in the Spence--Dixit--Stiglitz model imply that the inverse demand functions for the differentiated product ([x.sub.i]) can be written

[MATHEMATICAL EXPRESSION OMITTED] where [p.sub.i] is the price of the product. V is total expenditure by the (representative) consumer, on the set of differentiated products (N). V is assumed to be fixed.(8) [gamma] is defined by(9)

[MATHEMATICAL EXPRESSION OMITTED] It follows that the profit of firm 'i' is

[MATHEMATICAL EXPRESSION OMITTED] where C([x.sub.i]) represents total costs. Hence

[MATHEMATICAL EXPRESSION OMITTED] where c([x.sub.i]) = C'([x.sub.i]). Consequently

[MATHEMATICAL EXPRESSION OMITTED] In the symmetric case this expression simplifies to


[MATHEMATICAL EXPRESSION OMITTED] Hence, in the symmetric case

[MATHEMATICAL EXPRESSION OMITTED] A symmetric Nash--Cournot equilibrium in the Spence--Dixit--Stiglitz model implies that p = c(x)/[rho](1 - 1/n) (23) An important parameter in what follows is what one could call 'the scale elasticity of marginal cost' defined as e [equivalent] -d ln c(x)/d ln x.(10) Hence c'(x) = -ec(x)/x (24) It is evident that fixed costs do not affect e.

In a symmetric equilibrium p = V/(nx). Using this expression and (23) and (24)

[MATHEMATICAL EXPRESSION OMITTED] Inserting (20), (22) and (25) into (14) it follows that dW/dt > 0 if and only if (n - 1)(n + 2[rho] - n[rho] - ne)/[rho](n - 2) > n - 1/vn - 1 (26) Hence, there is a case for an export tax if

[MATHEMATICAL EXPRESSION OMITTED] and vice versa.(11) The following findings can be seen from (27): in industries with (i) more homogeneous products and (ii) a higher scale elasticity of marginal costs there will be a stronger case for an export subsidy. (iii) More firms and (iv) a higher share of domestic firms will favour an export tax.

More specifically, with no scale dependence of marginal costs and no product differentiation, the bordercase between taxes and subsidies is when the number of domestic firms is equal to half the total number of firms. This result was obtained by Dixit (1984). With some degree of product differentiation in the industry, there will be a case for an export tax even with a lower proportion of domestic firms. On the other hand, with marginal costs decreasing with scale, there might be a case for export subsidies even with a higher proportion of domestic firms. In particular, if e > 1, it is always welfare increasing to subsidize the domestic producers.

In fig. 1 I have plotted the border lines between taxing and subsidizing regions in {[rho], v}-space. As is evident from (27), these borderlines will be functions of 'the scale elasticity of marginal costs' (e) and the total number of firms (n). Notice that there is a case for an export subsidy northwest in Fig. 1, and a case for an export tax in the south-eastern region.


4. Optimal policy with home consumption

So far, this paper has been concerned with trade policy for purely exporting industries. This section will examine to what extent the trade policy analysis presented above is affected by the presence of home consumption. To focus ideas, I will limit the analysis to cases where markets are globally integrated in the sense that consumers at home and abroad face the same price for each product.(12)

The (representative) domestic consumer is assumed to have an indirect utility function v([p.sub.1],..., [p.sub.n], ??, m), where the [p.sub.i]'s are the consumer prices, as above. m is total income. Imposing a small tax on the domestically produced goods will give the welfare effects(13)

[MATHEMATICAL EXPRESSION OMITTED] Let us normalize the marginal utility of income at unity. Then Roy's identity implies that

[MATHEMATICAL EXPRESSION OMITTED] where [h.sub.i] is the domestic consumption of good i. Starting from a zero tax situation, it follows that (28) can be rewritten

[MATHEMATICAL EXPRESSION OMITTED] The last term captures the income changes. These income changes are the sum of changes in profits and government tax-revenues. This sum of profits and tax-revenues was referred to as pre-tax profits in the previous sections. If we let [sigma]?? [[pi].sup.i] denote this sum (as above), (30) can be rewritten

[MATHEMATICAL EXPRESSION OMITTED] The last term on the right hand side of this equation was examined in Sections 2 and 3. The question addressed in this section is the sign and magnitude of the first term, the changes in consumer surplus. In particular, I will discuss to what extent changes in consumer surplus reinforces or counteracts the changes in domestic producer surplus, as discussed in the previous sections.

In the case of price competition, I showed above that both [dp.sub.A]/dt and [dp.sub.B]/dt are positive under standard assumptions.(14) (The subscripts A and B refer to the consumer prices of products produced at home and abroad.) It follows that the first term on the right hand side of (30) is negative. Hence, with price competition, the case for imposing a tax on the domestic producers is weaker the larger is the domestic consumption of the differentiated product.

The case with quantity competition requires a more detailed argument. To keep matters as transparent as possible, I will follow the literature and disregard the income effects (at home and abroad) in the demand functions for the differentiated products. The inverse world demand functions for the differentiated products can be written [p.sub.i] = [p.sub.i]([x.sub.1],..., [x.sub.n]). Hence

[MATHEMATICAL EXPRESSION OMITTED] Define [p.sub.x] = [delta][p.sub.i]/[delta][x.sub.i] and [p.sub.y] [equivalent] [delta][p.sub.i]/[delta][x.sub.j] when i [not equal to] j.(15) Notice that downward sloping demand curves imply that [p.sub.x] < 0, while substitute products mean [p.sub.y] < 0. For the domestically produced goods, (32) can be rewritten (assuming symmetry)

[MATHEMATICAL EXPRESSION OMITTED] while for the foreign goods

[MATHEMATICAL EXPRESSION OMITTED] Section 2 established that standard assumptions ensure that [dx.sub.A]/dt > 0, while [dx.sub.B]/dt < 0. Consequently, the signs of [dp.sub.A]/dt and [dp.sub.B]/dt are not evident. In appendix 2, I have shown that stability of the system (as stated in Appendix 1) is sufficient to ensure that [dp.sub.A]/dt is positive. The same is true for [dp.sub.B]/dt if the products are not too differentiated, as one might expect.(16) If the products are not too differentiated, a tax on domestic producers will have an unambiguous negative impact on domestic consumer surplus. That is, a subsidy to the domestic producers will create an even larger welfare gain in the presence of home consumption, if the products are not too differentiated. Appendix 2 shows that for industries with highly differentiated products, stability is not sufficient to rule out the possibility that an export subsidy will cause a price increase for the foreign products (i.e. to sign [dp.sub.B]/dt). It follows that changes in consumer surplus might call for an export tax in this case.(17) Hence, stability is not sufficient to sign the overall effect on consumer suplus of a tax on domestic producers, if there is a significant consumption of foreign goods and the products are highly differentiated.

To summarize; both in the case of price and quantity competition the consumer surplus will tend to decline if a tax is imposed on the domestic producers, and increase with a subsidy. Consequently, the welfare gains from taxing the domestic producers in the price setting case will be reduced in the presence of domestic consumption. In the quantity setting case with not too differentiated products, both consumer surplus and pre-tax profits will increase by a subsidy to the domestic producers. However, as the degree of product differentiation increases in the quantity setting case, the welfare gains from a subsidy will tend to be more ambiguous--and possibly negative--both in terms of consumer surplus and pre-tax profits.

5. Summary

This paper presents a simple rule of thumb to determine whether to tax or subsidize (purely) exporting firms operating in a symmetric, quantity setting industry. The rule is expressed in terms of some general properties of the profit function and the number of foreign and domestic firms. This rule of thumb reveals how the case for an export subsidy is strengthened as marginal cost becomes a declining function of scale. A larger fraction of domestic producers and more product differentiation tend to pull the argument in the opposite direction. The rule is applied to the Spence--Dixit--Stiglitz model of product differentiation. It is shown how the rule can be redefined in this model, in terms of a few interesting parameters of the model (the elasticity of substitution in demand, 'the scale elasticity of marginal costs', the relative number of domestic firms and the total number of firms in the industry). In a symmetric, price setting industry only mild regularity conditions on the profit function are required to prove the case for an export tax.

The presence of home consumption tends to strengthen the case for a subsidy to domestic producers. Standard regularity conditions ensure that in almost all cases, domestic consumer surplus will increase with a subsidy to domestic producers. Consequently, the argument for taxing the domestic producers with price competition, is weaker in the presence of home consumption. In the case of quantity competition home consumption reinforces the case in favour of a subsidy to the domestic producers. However, the analysis with quantity competition revealed that the positive effect of a subsidy on both pre-tax profits and consumer surplus will be smaller (or even negative), as the degree of product differentiation increases.

The kind of trade policy analysis presented in this paper has been criticized for being too narrow. Dixit and Grossman (1986) examined how optimal trade policy is affected if there are several imperfectly competitive industries in the economy using the same resources. With several industries, subsidizing one industry is similar to imposing a tax on the other industries, competing for the same resources. The point is that even if a partial analysis of individual industries identify cases for export subsidies (or taxes), a more comprehensive welfare analysis would have to consider the relative merits of strategic trade policies in the different industries affected by the policy. This issue is a specific example of the more general point that the analysis presented above has abstracted from distortions in the rest of the economy. The presence of such distortions (e.g. subsidies and taxes, imperfect competition etc.) might interfere with the effects and conclusions identified above. In particular, the argument in favour of an export tax is stronger than indicated in the simple surplus analysis presented in this paper, when the marginal opportunity cost of government revenue exceeds unity.

Another criticism raised against this kind of analysis, is that it abstracts from retaliation from foreign governments when the home government imposes an aggressive trade policy. However, if the governments act in a static and non-cooperative way, the possibility of retaliation will not alter the direction of an optimal trade policy, but it might reduce the potential gains from an optimal trade policy (see Brander and Spencer 1985; or Eaton and Grossman 1986, for a formal analysis). The possibilities for retaliation could be more important when the governments are involved in a repeated game. The analysis of such a dynamic game between governments (and/or producers), with a formal study of optimal retaliation etc., is left for future research.

While this paper has explored the direction of optimal trade policy under fairly general conditions, the importance of (optimal) strategic trade policy remains an open question. For instance, I have shown that as the total number of firms in the industry increases, the case for an export tax tends to be stronger. But one might think that as the total number of firms in an industry becomes very large, the size of the optimal export tax, as well as the associated welfare gains, will be negligible. A formal examination of the importance of strategic trade policy remains an unsettled research topic.

Research Department, Statistics Norway, P.O. Box 8131 Dep., N-0033 Oslo, Norway


The author gratefully acknowledges comments from K. Moene, V. Norman, H. Vennemo, and two anonymous referees. The usual disclaimer applies. This paper is a revised and extended version of essay 3 in my Ph.D. thesis.

(1)Eaton and Grossman (1986) have provided in some respects a more general, but informal, analysis of the changes which occur when the possibility of free entry and exit is incorporated into an analysis of optimal trade policy. Cowan (1989) has considered the optimal combination of trade and competition policies in situations where the government directly can determine the number of home firms.

(2)Greunspecht (1988) has considered the reverse case, when the government determines whether to tax or subsidize after the firms have decided their stategies (prices).

(3)Strategic complementarity is equivalent to assuming upward sloping reaction functions in the duopoly case. In situations with price competition, Bulow et al. (1985) showed that constant marginal costs imply that goods i and j are strategic complements if demand for product i becomes more inelastic if the price of good j increases. If prices tend to increase with market share, this will pull the situation (further) towards strategic complements. Similarly, if the producer of the competing good has increasing marginal costs, strategic complementarity is more likely. With linear demand and increasing marginal costs, the goods are always strategic complements in the price setting case, as stated by Bulow et al. (1985, p. 501).

(4)I am assuming that the goods in the industry are substitutes; i.e.


(5)'Strategic subsitutes' is equivalent to downward sloping reaction functions in the duopoly case. Once more the reader is referred to Bulow et al. (1985, pp. 499--501) for a discussion of the required assumptions on preferences and technology for this property to hold in the quantity setting case. They show that a sufficient condition for strategic substitutes in the homogeneous product case, is that the slope of the marginal revenue curve is larger than the slope of the industry demand curve. This property always holds with linear demand curves.

(6)We stick to the case of the goods in this industry being (ordinary) substitutes. In the case of quantity competition this implies that [[pi].sub.y] < 0. Notice that dW/dt is always negative if there is only a single domestic firm.

(7)I.e. v [equivalent] [n.sub.A]/n.

(8)This assumption greatly facilitates the analysis. A log-linear aggregation function between the differentiated product and the other goods will imply this property, see e.g. Dixit and Stiglitz (1977, section II). More generally, we would expect the total expenditure on the differentiated product to be a declining function of the aggregate price index for the differentiated product. See Dixit and Stiglitz (1977, section I) for an examination of the complications which emerge in this more general case.

(9)In the Spence--Dixit--Stiglitz model the elasticity of substitution in demand between the different variants is given by [sigma] = 1/(1 - [rho]). That is to say, [rho] close to zero implies that the products are highly differentiated, while [rho] equal to unity implies that the products are homogeneous.

(10)When there is (locally) a constant elasticity of scale (in the traditional sense) termed [epsilon], e = [epsilon]([epsilon] - 1).

(11)Using (20), (22), and (25), we can check whether the stability condition

[MATHEMATICAL EXPRESSION OMITTED] put any bounds on permissible parameter values. After some manipulations we can show that

[MATHEMATICAL EXPRESSION OMITTED] which is positive for all parameter values.

(12)This is the same definition as given by Eaton and Grossman (1986, pp. 399--400). Markusen and Venables (1988, p. 303) have a somewhat different definition of an integrated market. They define an integrated market as a situation where arbitrage ensures the equality of producer prices across markets.

(13)That is, we consider a tax on all domestic production, and not only a pure export tax. A pure export tax is not feasible if foreign and domestic consumers are to face the same price on every product, as assumed above. If we allow foreign and domestic consumer prices to differ in the presence of differentiated products, the analysis becomes significantly more complex as we would have to consider four different prices (consumer prices at home and abroad for products produced domestically and abroad), even if we assume symmetric producers. This is considered beyond the scope of the present paper. An analysis of special cases with homogeneous products (with only two prices) have been provided by Eaton and Grossman (1986) and Markusen and Venables (1988).

(14)In particular, assuming strategic complements.

(15)This notation for [p.sub.y] (implicitly) assumes symmetry.

(16)There is no general, clear cut definition of what properties of demand is altered by changes in the degree of product differentiation. In the present context we associate more differentiation with a lower value for the ratio

[MATHEMATICAL EXPRESSION OMITTED] For homogeneous products this ratio is unity, while more product differentiation pulls this ratio towards zero. This property is consistent with the definition of more product differentiation in the Spence--Dixit--Stiglitz model, given in footnote 11.

(17)It is interesting that in this sense the effects on consumer surplus have a tendency to move in parallel with the effects on pre-tax profits.

(18)See e.g. Seade (1980) for a discussion.

(19)A formal proof is given by Murata (1977); theorem 20, p. 23 and theorem 8, p. 88.


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Stability of equilibrium

The following analysis of local stability is widely applied in the literature, despite its well known limitations as a model of the dynamics of the agents strategic behavior.(18) Assume that the firms increase their strategy variable, if this will have a (marginal) positive effect on profit:

[MATHEMATICAL EXPRESSION OMITTED] where [S.sup.i] denotes the time derviative of the strategy choice for firm i. The [[alpha].sub.i]'s are parameters representing the adjustment speeds which are assumed to be positive. Let us investigate the question of local stability around the equilibrium. Linearizing around the equilibrium point


[MATHEMATICAL EXPRESSION OMITTED] It is well known that the following property is a sufficient (but not necessary) condition for the stability of this system(19)

[MATHEMATICAL EXPRESSION OMITTED] In the symmetric case this implies that |[[pi].sub.xx]| > ([n.sub.A] + [n.sub.B] - 1) |[[pi].sub.xy]| (38)


Price effects in the quantity setting case

This appendix will show that stability is sufficient to prove that d[p.sub.A]/dt > 0 (in the quantity setting game). The analysis presented here furthermore reveals that stability is only sufficient to prove that d[p.sub.B]/dt > 0 if the products are fairly homogeneous, in a sense which will be made precise below.

From (33) we have that

[MATHEMATICAL EXPRESSION OMITTED] Both [p.sub.x] and d[x.sub.A]/dt are negative. Hence, the problem is to sign the expression inside the square brackets. For homogeneous products [p.sub.y]/[p.sub.x] is unity, while product differentiation will pull this ratio towards zero. A sufficient condition ensuring d[p.sub.A]/dt > 0 is therefore to prove that the absolute value of (d[x.sub.B]/dt)/(d[x.sub.A]/dt) is smaller than [n.sub.A]/[n.sub.B].

Equations (11) and (12) imply that

[MATHEMATICAL EXPRESSION OMITTED] Stability implies that [[pi].sub.xx]/[[pi].sub.xy] > [n.sub.A] + [n.sub.B] - 1 (cf. appendix 1). Hence

[MATHEMATICAL EXPRESSION OMITTED] where the last inequality follows since [n.sub.A] + [n.sub.B] > 1. This proves that stability is a sufficient condition to ensure that d[p.sub.A]/dt is negative.

Let us turn to signing d[p.sub.B]/dt. From (33) we have that

[MATHEMATICAL EXPRESSION OMITTED] The sign of the first term inside the square brackets has been examined above, and signed positive. The point here is that as the products become more and more differentiated, the ratio [p.sub.y]/[p.sub.x] will approach zero. As this ratio gets sufficiently close to zero, the last term inside the square brackets will dominate. This term is definitely negative. Consequently, as the degree of product differentiation increases, d[p.sub.B]/dt will change sign from positive to negative.
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Author:Klette, Tor Jakob
Publication:Oxford Economic Papers
Date:Apr 1, 1994
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