Exchange-rate pass-through and market structure

Vol. 64 (1996), No. 2, pp. 129-154 Journal of Economics Zeitschrift fDr National~konomie

�9 Springer-Verlag 1996 - Printed in Austria

Exchange-rate Pass-through and Market Structure

Alan Kirman and Louis Phlips

Received October 2, 1995; revised version received July 24, 1996

We consider a situation in which n firms located in market 1 and m firms located in market 2 each sell a commodity which is homogeneous within each market but may differ between markets. All firms sell on both markets. Each market has its own currency. The market demand functions differ. We give some basic results on the effects of exchange-rate changes and then show the following. When these markets are independent on the cost side (constant marginal costs) and demands are linear, a reduction in the number of firms (which might result from a merger) in market 1 increases the pass-through (of an appreciation of currency 2) in market 1 and decreases the pass-through in market 2. A similar occurrence in market 2 has the opposite effect. We give conditions under which, with identical economies of scope linking the markets, the sign of the price changes will be reversed when the number of foreign firms is small enough compared to the number of local firms. However, such sign reversals cannot occur in the two markets simultaneously.

Keywords: exchange rates, pass-through, market structure.

JEL classification: D43, Fi2, F31, LI3.

1 Introduction

Empirical evidence (see Feinberg, 1986; Fisher, 1989a; Feinberg and Kaplan, 1992) suggests that the pass-through of exchange-rate changes (into export prices) is smaller when the exporting country is more concentrated. In particular, more concentrated industries lower their export- price markup more than less concentrated industries, when there is an appreciation of their domestic market 's currency. The econometric re- suits obtained by Feinberg (1986, 1989, 1991) also indicate that domestic prices react more to exchange-rate changes in industries with a larger import share. These findings are in accordance with the theo- retical results on the relationship between exchange-rate changes and domestic prices in oligopolistic "home" markets obtained by Dornbusch (1987) and Fisher (1989b).

130 A. Kirman and L. Phlips

We shall emphasize the feedback between markets which are effectively separated and we shall look at industries with an oligopolistic structure. In looking at imperfect competition we follow the spirit of Krugman's (1987) "pricing-to-market" approach, though, since he has a monopolistically competitive model, strategic interaction is not di- rectly considered. Of course there are many other explanations for the failure of price changes to reflect exchange-rate changes such as the difference between exchange-rate movements which are perceived to be permanent and those which are perceived to be temporary (see Froot and Klemperer, 1989). We focus here on the consequences of strategic interaction in a one-shot game.

This paper presents a model related to those of Hens et al. (1991) and Hens (1994) which allows one to analyze the effects of an exchange-rate change on the prices in two oligopolistic markets which are separated but in which the same firms compete. Some of the firms produce in one market or country and the rest in the other. However, they all sell in both markets. The separation of the two markets may be a result of legal or technical problems. The recent case in the UK of the sale of the same compact disks at very different prices in London and NewYork is a good example of this. In this case copyright laws prevent any arbitrage between the two markets. Alternatively, the nature of the product produced by the firms may be such as to prevent effective arbitrage. The important feature to note is that, depending on the nature of the factor causing the separation of the markets, firms' cost structures may be very different. If an identical product can be sold on either of the two markets but its short life makes it difficult to arbitrage, an international newspaper in one language for example, then the cost of production should just be the cost of the sum of the quantities produced for each country (see the studies by Huertas-Rubio, 1996; Gosh and Wolf, 1994, on the price of the Economist). If, at the other extreme, the product for one market is completely different from that for the other, the cross derivatives of the cost function with respect to the quantities of the goods produced for each country will be zero. However, there are many intermediate cases in which changes in the production for one country have some impact on the costs of production for the other. Right-hand drive cars cannot be substituted for left-hand drive cars, so arbitrage is impossible. Yet an increase in the production of one will have an effect on the cost of the many components common to both variants. Thus, when economies of "scope" are allowed for, these do not reduce to economies of scale exploited by exporting, since the products may differ between markets. In our previous example, economies of scope may arise from the impact of exports of a particular type of car on the cost of purchase of parts which are used in both the domestically sold

Exchange-rate Pass-through and Market Structure 131

and the exported cars, although their production technology is different. Diseconomies of scope may result from the necessity of setting up a specific production unit for the production of the exported cars. There may thus be varying levels of economies or diseconomies of scope. Our model provides a framework within which to analyze these possibilities.

To simplify matters we assume that firms sell identical products in the same market but products may be differentiated between markets. Thus for a food product, health and hygiene requirements may differ between countries but within a country the products of the different firms are perfect substitutes. There are many goods which are very close substitutes, which sell at similar prices in each country but whose prices differ substantially between countries. Colour films and officially approved tennis balls are obvious examples. The assumption of perfect substitutability could, of course, be relaxed, but it would considerably complicate the analysis.

Our model lies between that of Hens et al. (1991) and Hens (1994). In the first of these papers the problem is of two firms each producing in one country and selling their products both in their own country and that of the other. Thus they play a duopoly game in both markets. Unless special restrictions are put on cost and demand functions, exchange-rate changes can lead to many different consequences. Stan- dard effects, with the prices of the finn located in the depreciation country falling, can follow although the prices might not change as much as the exchange-rate change ("incomplete pass-through") or pos- sibly might change more ("excessive pass-through"). Prices might also move in the same direction as exchange-rate changes ("surprising pass- through"), for one producer, the price of exports of the firm in the depreciation country increasing. If this "wrong" effect occurs in both markets one refers to a "perverse pass-through."

Hens (1994) shows that equally arbitrary changes can occur when both markets are purely competitive. The model presented in this paper considers the intermediate case where there are n finns in one country and m in the other. As might be expected, given the result in the two polar cases, unless special restrictions are imposed on cost and demand functions one cannot expect to obtain clear analytical results as to the consequences of an exchange-rate change on the movement of prices. However, imposing such restrictions does allow us to produce results and the question then is how plausible are these restrictions. We restrict our attention to the oligopolistic case though, for certain classes of cost and demand functions, limit results as the number of firms goes to infinity can be obtained (see Martin and Phlips, 1993; Herguera, 1993).

Studying the consequences of a merger and the resultant Cournot- Nash equilibrium is complicated, as Salant et al. (1983) pointed out.


In cooperative games merging amounts to forming an indissoluble coalition but in noncooperative games such as oligopolies the situation is more complicated. Does a merger mean that the new firm can now use the production facilities of the old in which case decreasing returns to scale, for example, will make the splitting of production between the two finns desirable and hence will produce a new player with a new production technology? We take the simple position here that a merger corresponds to the disappearance of a firm and thus the firm that absorbed the firm that is no longer active still acts like one of the original agents. Thus one might more properly talk of an "increase in concentration."

The model is presented in Sect. 2. A comparative statics analysis is then carried out, starting with the standard assumption of constant marginal costs (Sect. 3). The impact of strategic complementarity and substitutability on individual quantity responses in the two markets to exchange-rate changes is examined in some detail. Then the effect of mergers on the extent of the pass-through is traced out. Section 4 allows for economies and diseconomies of scope linking the two markets. When the firms benefit from identical economies of scope, the sign of the price changes may be reversed: the price may go up in the market whose currency appreciates, or the price may go down in the market whose currency depreciates, depending on the relative number of firms located in the two markets. Under the assumptions made, the two market prices may go up or the two market prices may go down. But the perverse situation where the price goes up in the country that appreciates and goes down in the country that depreciates cannot occur.

2 The Model

As we have said, there are two markets for a commodity which is homogeneous within each of the markets, market 1 and market 2, separated by barriers other than tariffs and transportation costs. However, the product may differ between markets. Market structure is oligopolistic in each: there are n firms located in market 1, selling (xil, xi2) each (i = 1 . . . . . n); there are m firms located in market 2, selling (Zkl, Zk2) each (k = 1 . . . . . m).

The profit function of a firm located in market 1, expressed in market 1 currency, is

l I i = P l ( X ) x i l -~- e p 2 ( Z ) x i 2 - ci (X/l, x i 2 ) , (1)

where


X = Xil q- Zkl ~-- X1 --~ Z l , (2.1) i=1 k= l

n m

Z = ~ xi2 "1- ~ Zk2 = X2 -}- Z 2 . ( 2 . 2 )

i=1 k=l

Xil represents the sales of firm i in market 1 and xi2 represents its sales in market 2. And similarly for Zkl and Zk2. The inverse market demand functions are p l ( X ) and p2(Z), respectively. The exchange rate, e, is the worth in market 1 currency of the currency used in market 2. The cost functions a r e ci(xi l , xi2) and Ck(Zkl, Zk2), with marginal costs c], c 2, c~, and c 2. Superscripts denote derivatives with respect to the first, respectively second, argument. Firms located in market 2 have profit functions

Ilk = pl (X)Zkl -1- ep2(Z)zk2 -- eck(Zkl, Zk2) , (3)

also expressed in market 1 currency. We assume that the demand, cost, and profit functions satisfy the

following restrictions.

AI" The inverse demand functions p j ( X ) and p j (Z ) , j = 1, 2, are continuous for all X > 0, Z > 0.

A2: For each market there exists )) > 0 and 2 > 0 such that Pl (X) = 0 for all X > X, p2(Z) = 0 for all Z > 2 and p l ( X ) > 0 for X < J ( , p 2 ( Z ) > 0 f o r Z < 2 .

A3: pj(O) = /Sj < c~ ( j = 1,2) and for all X and Z such that 0 < X < J~ and 0 < Z < 2, respectively.

A4: p l ( X ) and p2(Z) have a continuous second derivative pj' with

p' l(X) < 0 and p'2(Z) < 0 for all X and Z. A5: The cost functions Ci(Xil,Xi2 ) and ck(zkl, Zk2) are defined and

continuous for all output levels xil > O, xi2 > O, zkl > O, and Zk2 >_ O.

A6: c1(0, 0) _> 0, ck(0, 0) >_ 0. A7: ci and Ck have continuous first and second partial derivatives for

all Xil, Xi2, Zkl, Zk2 >__ O. 2 A8: c], c 2, c~, c k > 0 for all xil , xi2, Zkl, zk2 > O.

A9: For all Xil, xi2, zkl, zk2 > 0, X < J~, and Z < 2. I'Ii(xil , Xi2, X, Z) and Ilk(Zkl, Zk2, X, Z) are concave in xi and zi, respectively.

Let us derive the first-order conditions of a Cournot-Nash equilib-


rium. Differentiating I-I i and FI k, the marginal profits are

l'I] (Xil, xi2, X, Z) -~- p~(X)Xil d- p l ( X ) - c](xi l , xi2) , (4.1)

l--I2(Xil, Xi2 , X , Z) -~- e(p~(Z)xi2 -~- p2 (Z) ) -- c2(xil , xi2) (4.2)

for firms i and

1 t 1 I-Ik(Zkl Zk2, X , Z) = -~- P l ( X ) - Zk2) ec k (Zkl, (4.3) , Pl (X)zk l ,

2 2 , = ck(zkl, Zk2)) (4.4) e(p2(Z)zk2 + p2(Z) - FIk(Zkl Zk2, X , Z)

for firms k. The system of 2(n + m) equations

II{(Xi*l,Xi2, X * , Z * ) = 0 , j = 1,2; i = 1 . . . . . n ; (5)

FI~(z~I, z~2, X*, Z*) = 0, j = 1, 2; k = 1 . . . . . m ,

describes the first-order equil ibrium conditions, where asterisks denote equil ibrium values o f the relevant variables, l These conditions allow one to divide (4.2) and (4.3) by e. A change in the exchange rate can therefore be interpreted alternatively as a rotation o f a f irm's foreign marginal revenue curve or as a change in the opposite direction o f its marginal cost.

The second-order conditions imply

f! * * ! a i l = P l ( X )Xil -}- 2 p l ( X * ) - c~l(x*l,xi*z) < 0 , (6.1)

! l * * ai2 = e ( p 2 ( Z )xi2 + 2p~(Z*)) _ ci22,tx i*~ , x* 2) < 0 , (6.2) ! ! * *

akl P l ( X )Zkl + 2 p ~ ( X * ) H �9 = - ec k (Zkl, Z~2 ) < 0 , (6.3)

it , , 2p~(Z*) 22 �9 �9 ak2 = e(p2 (Z )zk2 + - ck (Zkl, Zk2)) < 0 . (6.4)

We now introduce the fol lowing derivatives:

! l * * bil = p~ (X )Xil + p!l(X*) , (7.1)

!' * * " Z * ) , bi2 = p 2 ( Z )xi2 -1- p2 I, (7.2)

1 Equilibrium exists under assumptions A1-A9 (Friedman, 1977). These assumptions are satisfied i f p j < 0, pj' < 0 for both countries and c~ ~, c~ > 0 for all i and k. They may still hold under economies of scale and if pj' > 0 provided that p~ is large enough.


I I * * I bkl = Pl (X )Zkl + Pl(X*) ,

" * *

bk2 = P2 (Z )zk2 +

(7.3) (7.4)

An interesting interpretation was suggested by Bulow et al. (1985). Think of bil a s O (O lI i/OXi* 1)/Oxj* 1 where j represents firms other than i selling in market 1: bil represents the change in the marginal profitabil- ity to firm i of being more aggressive (selling more) when competitor j becomes more aggressive (sells more) in market 1. Note that bil takes

* * for a l l j ~ i a n d k , the same value with respect to Xjl ( j ~ i) and Zkl bk~ takes the same value with respect to Z~l (I r k) and x/* 1 for all l r k and i, and similarly for bi2 and bk2. When bil > O, firm i regards its product as a "strategic complement" to the product of its competitors in market 1. When bil < O, firm i regards its product as a "strategic substitute" to the product of its competitors in market 1. These signs clearly depend on the shape of the market demand functions within each market and the cost functions of the two firms, and may therefore differ between markets even when the commodity is homogeneous. Since demand in one market is independent of the price in the other market, there is no room for strategic substitutability or complementarity between markets. In the special case where demands are linear (p~ = pI~2 = 0 ) ,

then bil = bkl = p~I(X*) < 0 and bi2 = bk2 = p~(Z*) < 0, so that strategic complementarity is eliminated in both markets.

Under strategic complementarity in market 1, firm i reacts by also selling more when firm j adopts a more aggressive sales policy and sells more in that market. Strategic complementarity thus implies quantitative\.reactions going in the same direction. In the case of strategic substitutability, an increase in j ' s sales makes i sell less, so that aggregate sales in market 1 change less. Strategic substitutability thus implies smaller prk~e reactions. One should not be surprised, therefore, to find incomplete pass-through in a linear world (with strategic substitutability only).

Exchange-rate Pass-through

In order to do comparative statics to examine the consequences of an exchange-rate change, we need to impose some additional structure on our model. To do this we will follow Dixit (1986) and assume that the model is stable with respect to an adjustment process. Here we will do this for each market taken separately and later we will extend this to consider both markets together. A natural adjustment process is to suppose that a firm will increase its sales if it obtains a positive


marginal profit from so doing. For market 1, the adjustment process is

3Oil = I-I](Xll . . . . . Xnl, Zll . . . . . Zml, e ) , (8.1)

Zkl ~- 1 l-'lk(Xll . . . . . Xnl, Zll . . . . . Zml, e ) . (8.2)

Linearizing around the equilibrium point x ~ l , . . . . . . , x ~ n , z~ l , , Zml,* i.e., taking a first-order Taylor expansion, one obtains

L1] all bll ...... blllFxlx lj X21 = b21 a21 b21 " ' " b21 x21 X~l

Zml [_bml brnl bml "'" aml_l lZml Z* 1

(9)

Therefore

F nWm ] [ n + m b j l ] (-1)n+m[j~=l(ajl-bjl) 1 - I - Z : j = l ajl---bjl > 0 . (10)

Sufficient conditions are obtained by requiring that the coefficient matrix in (9) has the "dominant diagonal" property

lajll > (n + m - 1)lbjll (11)

for j = 1 . . . . . n + m. In conjunction with the second-order conditions,

] I - I ( a j l - - + - - l " j=l j=l ajl -- bjl A

and similarly for market 2. We have

a i l =FI ] 1 = b i l + P l l ( X * ) < 0 for a l l i ,

a k l = 1 7 1 1 = b k 1 + p r I ( X * ) < 0 for a l l k ,

a necessary condition for stability. Another necessary condition is that the determinant of the coefficient matrix in (9) should have the same sign as ( -1 ) n+m. This determinant can be written as


this implies

a j l + (n + m - 1 ) b j l < 0

a j l - (n + m - 1 ) b j l < 0

i f b j l > O ,

if b j l < 0 . (12)

These inequalities in turn imply

a j l - b j l < 0 , j = l . . . . . n + m . (13)

Therefore ( - 1) n+m r-rn+m ,. l l j = l ~ajl - b j l ) > 0 and (10) implies

n+m bj 1 f l l = l + ~ -- > 0 .

j=l a j l -- b j l (14)

Similar conditions apply to market 2. Since ail - bil = p t 1 - - C] 1 , i = 1 . . . . . n, and akl -- bkl = p~ -- e c 11, k = 1 . . . . . m, condition (13) implies that the slope of the inverse market demand curve (expressed in home currency) is smaller than the slope of the marginal cost curve. When, as in the next section, marginal cost is constant, condition (14) simplifies to

tl ( P l "~X /~1 : 1 + n + in + \-~1 j > 0 , (14.1)

which restricts complementarity (occurring when p~ is positive). Note also that condition (14) ceases to be restrictive when, in addition, market demand is linear (p~' = P"2 = 0), since then

t~1 = f12 = 1 + n + m , (14.2)

strategic complementarity being eliminated altogether as noticed above. We are now ready for comparative statics, and we first obtain the

following result which is standard in the (static) literature on this problem.

P r o p o s i t i o n 1: Under assumptions A1-A9 an appreciation of the currency of market 2 decreases p2 and increases Pl, when markets are independent.

Proof" Let Yi = X - X i l , Yk = X - - Z k l , gi = Z - - x i 2 , gk = Z - Zk2.


Total differentiation of the first-order conditions (5) gives

aildXil -kbi l dyl = 0 ,

akl dzkl + b k l dyk = S k l d e ,

ai2dxi2-l-bi2 dgi --~Si2 de ,

ak2 dZk2+bk2 dg~ = 0 ,

(15.1)

(15.2)

(15.3)

(15.4)

where 3kl = - l " [ l e and ~i2 ~-- - l " [ 2 e - I n particular, ~kl -~- C~ > 0 is the cost increase (in market 1 currency) resulting from an appreciation of market 2 currency for firms located in market 2. Similarly, 6i2 = - (1 / e )c2 i < 0 is the cost reduction (in market 2 currency) resulting from an appreciation of market 2 currency for firms located in market 1.

Rewrite (15.1) as ail dxil + bil ( d X - dXi l ) = 0 o r

bil dXil -~ - - d X = 0 .

ail - bil

Summing over i,

dX1 + -- dX = 0 . all -- bil

(16.1)

Rewrite (15.2) as akl dZkl + b k l (dX --dZkl) = ~ k l de or

bkl 8kl d Z k l + d X -

akl - - bk l akl - -bk l d e .

Summing over k,

m m

dZ1 _k [k__~ 1 bk..__l ] dX = [k__~l 8kl ] d e . akl -- bkl akl -- bkl d

(16.2)

The sum of (16.1) and (16.2) in turn gives

dX ---- -- de , = akl - bk~

(17)

so that

dyil bil [ k ~ ~kl ] d e fll(ai-1 - - b i l ) = akl ~ bkl

(18)


and

= [ 8~ bk~ [~-~ ~k~l ] ] d e dzkl L(akl -- bkl) /~1 (akl -- bkl) L~= 1 akl -- bkl 3d "

(19)

A similar aggregation procedure applied to market 2 gives

1 [/__~1 ~i2 ] d e , (20) dZ = -~2 ai2 ~ bi2

[ ' i2 be2 (~'~=1 'i2 ) ] d e , (21) dxi2 : (ai2 ~-bi2) /~2(ai2 -- bi2) ai2~bi2

bk2 [~'~--1 'i2 ]de" (22) dZk2 = fl2(a S - bk2) ai2 _ bi 2

Since ~kl > 0 and 3i2 < 0, we have

dX dZ - - < 0 and > 0 . [] (23) de de

The sum appearing in dX is over the m firms located in market 2, whereas the sum appearing in dZ is over the n firms located in market 1: the price change in a market is due to the quantity response of the foreign firms to the exchange rate shift. This response is related to the fact that an appreciation of market 2 currency amounts to a cost increase for foreign firms selling in market 1 (reflected in 8~1) and a cost decrease for firms i selling in market 2 (reflected in ~i2). These cost effects are weighted by the degree of aggressiveness encountered in the foreign market (1~ill in market 1, 1/f12 in market 2). The sign of the response, however, is independent of the sign of bil, bkl, bi2, and bk2, that is, of the strategic nature of the commodity, since fll > 0 and f12 > O.

3 Constant Marginal Costs

To obtain some first insights into our problem, we start with the standard assumption that all firms have constant marginal costs of production but which can differ for the products in each of the two markets. This is an extreme case in which there are no economies or diseconomies of scale: C]1 = C22 = c l l = Ck22 -m- O; there are no economies or d i seconomies


of scope between markets: c~2 = c2il = Ck12 = Ck21 = 0, for all i and k. Consequently the two markets are independent on the cost side, in the sense that quantities sold in one market do not affect the marginal cost of quantities sold in the other market, and each market's equilibrium can be determined separately.

We now look at individual quantity responses with this restriction and obtain the following proposition.

Proposition 2: Under assumptions A1-A9, and constant marginal costs individual quantity responses to exchange-rate changes depend on the strategic effect on marginal profits. In the case of strategic complementarity in both markets, all firms increase sales in the market whose currency appreciates and reduce sales in the other market. In the case of strategic substitutability in both markets, firms located in the market whose currency appreciates sell less in both markets, whereas finns located in the other market sell more in both markets. If there is strategic complementarity in the market whose currency appreciates and strategic substitutability in the other market, then firms located in the former sell more at home and less abroad, while firms located in the latter sell more in both markets. In the opposite case, the reverse is true.

Proof." For individual sales, we find

Oxi 1 OZk 1 - - < 0 and - - < 0 (24) Oe 0e

for all i and k in the case of strategic complementarity (bi l > 0 and bkl > 0). However, in the case of strategic substitutability

OXil OZkl - - > 0 and - - < 0 . (25) 0e 0e

Indeed, since, from Proposition 1, dX/de < 0, the positive quantity response of the firms located in market 1, which take advantage of the local price increase, must be over-compensated by a negative response of the foreign firms whose currency appreciates. This negative response results from (19) and the fact that

i=l--~-e --[- - - < 0 or - - - > k=l 0e k=l 0e i=1 0e


can be written as

n bi 1 n bi 1

i=1 all -- bi l .= a i l - bil

In market 2, OXi2 OZk2

- - > 0 and - - > 0 (26) Oe Oe

when there is strategic complementarity, and

OXi2 OZk2 - - > 0 and - - < 0 (27) 0e 0e

when there is strategic substitutability. The negative quantity response of the firms located in the market whose currency appreciates is over- compensated by the increased sales, in market 2, of the foreign firms.

[]

It is clear that it is not possible to derive a general result about the incomplete pass-through of exchange-rate changes into prices in oligopolistic markets. This can be seen from an examination of the elasticities of prices with respect to e which are

e l - - f l l P l h = l

6 2 - f12P2 i=1

~kl

akl -- bkl

(~i2

ai2 - b i 2

> 0 , (28.1)

< 0 , (28.2)

respectively. However, when market demands are linear, we do have the following simple result.

Proposition 2a: With constant marginal costs and linear market demands, the pass-through is incomplete in both directions.

Proof." In this case these elasticities simplify to

) Z l q - m < < 1 el - - l + n + m ' , p l l + n + m

142 A. Kirrnan and L. Phlips

and 1 (p~ ) n

t'2X2+n > > - 1 e2-- l + n + m \ p 2 l + n + m

using (14.2). Consequently

0 < e l < 1 and - 1 < e 2 < 0 . [] (29)

Mergers

In this section, we adopt the even stronger assumptions of constant marginal costs and linear demands. This puts us in the framework adopted by Salant et al. (1983): merged firms are treated as one firm under the control of a particular player in a noncooperative game. We shall also adopt their simplifying assumption that all firms have equal marginal costs (~), since nothing essential is lost in doing so once marginal costs are constant. 2 Note that the linearity of market demands implies that all firms regard their product as a "strategic substitute" for their competitors' in both markets.

We first consider a merger of two firms in market 1 and prove the following.

Proposition 3: Assume marginal costs are constant and demands are linear. A merger in market 1 increases the pass-through of an appreciation of currency 2 in market 1, where the post-merger equilibrium price increases more than the pre-merger price, and decreases the pass- through in market 2, where the post-merger equilibrium price decreases less than the pre-merger price, when the products are strategic substitutes in two separate markets.

Proof: To see the effect on the pass-through, in market 1, of an appreciation of the currency of country 2, we use Eq. (17), which becomes

de ~ ~=l a~l - bkl (30)

where the superscript M indicates post-merger values. We have, from

2 If marginal costs were different the merged firm would use the production facilities of the firm with lower marginal costs.


(14.2), f i~ = n + m < i l l . The sum between brackets remains un-

changed since 3~1 = c 1 and a~l - b~l = P'I are constant under the assumptions made. /~M < fil implies that the degree of aggressiveness is increased in market 1 (1~ill M > 1~ill), so that X M decreases more than X: the post-merger price pM increases more than Pl as market 2 currency appreciates.

How does the same merger (in market 1) affect the pass-through in market 2? Equation (20) becomes

d Z M 1 n - 1 ~]i2 > 0

i=1 ai2 7 bi2 / (31)

H e r e ~M2 = tiM < f12, while the sum between brackets is reduced.

Recall that ai2 - - b i 2 = p ' and ~i2 = --c2/e for all i We thus have 2

dZ - i (n~2/e'~ dZ M __ _ _ - I ( ( n - 1 ) ~ 2 / e )

d ~ - - - l + n + m \ p; 1 > d---~ n + m P'2 (32)

since this can be written as n/ (n - 1) > (1 + n + m) / (n + m) which is always true. Z M increases less than Z and pM decreases less than P2- 2

[]

We next consider the effects of a merger of two firms in market 2 and obtain

Proposition 4: When marginal costs are constant and equal across firms and demands are linear a merger in market 2 decreases the pass-through of an appreciation of currency 2 in market 1, where the post-merger equilibrium price increases less than the pre-merger price, and increases the pass-through in market 2, where the post-merger equilibrium price decreases more than the pre-merger price, when the products are strategic substitutes in two separate markets.

Proof." Using Eqs. (17) and (20) again, when the currency of market 2 appreciates, a similar argument gives

dX dX M - - < - - < 0 de de


since m/(m - 1) > (1 + n + m)/(n + m). In market 1, X M decreases less than X and p~ increases less than Pa. However, in market 2, in which the merger occurs, Z M increases more, simply because of the reduction in fiN. []

In terms of the empirical results summarized in the Introduction, Propositions 3 and 4 are compatible with the finding that the pass- through into export prices is smaller when the exporting industry is more concentrated, in the sense that such industries lower their export price markup more than less concentrated industries, when there is an appreciation of their domestic currency.

What about the indication that domestic prices react more to exchange-rate changes in industries with a larger import share? This would be compatible with our Propositions 3 and 4, to the extent that mergers occurring in a market increase the import share of that market. Under the assumptions adopted from Salant et al. (1983), such an increase in import shares indeed occurs. In post-merger equilibrium, each surviving firm sells more in each of our two markets since the total number of firms is reduced. Each surviving firm also sells the same quantity as the others in each market, since the equilibrium is sym- metric in each market. However, the two merged firms contract their aggregate output after the merger, for any given output of the other firms, because they internalize the inframarginal loss that they impart to each other. Consequently, X M < X*, Z M > Z~, and X M < X~ according to (2.1), when two firms merge in market 1, and the post- merger import share z M / x M is larger than the pre-merger share. And similarly if the merger occurs in the other market.

Finally, we note that if a firm located in market 1 takes over a firm located in market 2, the effect on dX/de and on dZ/de is the same, under the assumptions made, as if the merger had been between firms located in market 2 (and vice versa). All that matters is the reduction in the number of players.

4 Economies and Diseconomies of Scope Across Markets

We now take account of the fact that each firm' s cost function may have nonzero cross partial derivatives: selling abroad may lead to economies or diseconomies of scope, for the sort of reasons that we explained earlier. Economies of scope contribute positively to marginal profits, whereas diseconomies do the opposite. Since


a2ci c~2, n , 02c~ ~2 17] 2 _ _ _ - - _ .ec k , OXi 10Xi2 OZkl OZk2

i - i /21 __ 02Ci __ C/21 ' 1-121 - - 0 2 C k . ec21 ,

Oxi2OXil OZk2OZkl

there are d iseconomies o f scope across markets i f C] 2 > 0, C k 12 > 0,

C/21 > 0, and c 21 > 0. There are economies o f scope across markets when these derivatives are negative.

4.1 Exchange-ra te Pass- through

Total differentiation o f Eqs. (5) gives

ail dxil + bil dyi - c]2 dxi2 -~- 0 ,

akl dZkl + bkl dyk -- ec 12 dzk2 = 3kl de ,

ai2 dx i2 @ bi2 dgi - c 21 dXil = (~i2 de ,

ak2 dZk2 + bk2 dgk -- ec 21 dZkl = 0 .

These equations can be rewritten as

( bil 4 2 dx i l q- \ a i l - bil ) d X

, a i l - - bil

( bkl ) d X ecl2

dZkl q- kak~ -- bkl akl -- bkl

" be2 ) c21

dxi2 -~- ( a i 2 - bi2 d Z ai2 - bi2

( bk2 "] ec 21

dz~2 + kak2--_-bk2 / d Z ak2 -- b~2

- - d x i 2 = 0 , (33.1)

~kl dzk2 -- de , (33.2)

akl -- bkl

~ i 2 - - d x i l - - - - de , (33.3)

ai2 -- hi2

dZkl = 0 . (33.4)

To solve this system for dX and dZ we have to simplify matters. We suppose that all firms are alike, in the fol lowing sense. First we assume that

B I : c] 2 = c 12k = c 12 and c/21 = c 21 = c 21 for all i and k, and thus

e42 = 4 2 = c 12, c~ = e 4 1 = c 2', and c 1 2 = ec 21


The last assumption can be justified if we do local analysis of a situation in which the exchange rate is normalized to e = 1. Increased sales in market 2 have the same effect on the marginal cost of sales in market 1 for firms located in market 1 as for firms located in market 2, and similarly for increased sales in market 1.

Further, we assume

B2: b i l = b k l = b l and bi2 = bk2 -= b2, that is, all firms selling in a particular market consider their product either as a strategic substitute or as a strategic complement.

B3: a i l --=. a k l = a l and a i2 = ak2 = a2, that is, the effect of increased sales on marginal profit is the same for all firms selling in a particular market.

B4: 3kl = 31 and 3i2 = 3 2. The cost disadvantage (advantage) resulting from an appreciation (depreciation) of home currency is the same for all firms in a given market.

Summing (33.1)-(33.4) over i and k under these assumptions, we obtain

and

[ Y -aft] I d X z ] = [ ede - 0 ~ de ]

dX 1 de

where

a2 + (n + m - 1)b2. al + (n + m - 1)bl 0 ~ = , y = ;

a2 -- b2 al - bl c 12 c 21

- - - - ; 0 = - - ; f al -- bl a2 - b2

mS1 n32 e - ; ~ = - - - - ~ ; A = oey - Off .

a l - b l a2 - 2

To interpret (34) we need stability conditions. We can use the stability conditions for each market derived separately in Sect. 2. In addition we now need to take the two markets together and consider the profit adjustment process described by the system of 2(n + m) differential


equations

Xil = 1-I~(Xil, Yi , Xi2, e ) ,

x i2 = I I ~ ( x i 2 , g i , X i l , e ) ,

Zkl 1 e ) , = I-Ik(Zkl, Yk, Zk2,

2 , e) . Zk2 = l-lk (Zk2, gk , Zkl (35)

Linearizing a r o u n d t h e equilibrium (x~l . . . Zml*, X12" . �9 . Zm2),* w e o b -

t a i n

- X l l -

X21

Zml

XI2

X22

- ~m2-

)<

al bl bl . . . bl --C12 0 0 . . . 0 b~ a l b~ . . . b~ 0 --C12 0 . . . 0

bl b~ bl . . . a l 0 0 0 . . . - c 1 2

-c21 0 0 . . . 0 a2 b2 b2 . . . b2

0 --C21 0 . . . 0 b2 a2 b2 . . � 9 b2

0 0 0 . . . - - C 2 1 b2 b2 b2 . . . a2

X l l - Xl l

x21 - X~l

:g

Zml -- Zml

X12 - - X~2

X22 - - X22

Zm2 -- Zm2

X

(36)

The 2(n + m) x 2(n + m) coefficient matrix satisfies a necessary condition for stability that the trace be negative�9 We will add a condition which is sufficient for uniqueness and stability and therefore justifies the use of comparative statics, namely we shall make the assumption

B5: The coefficient matrix (36) has the "dominant diagonal" property,


that is,

]aa[> (n4-m-1)lb11+[c121, l a2 l> (n4-m-1)lb21q-lc211. (37)

This, together with al < 0 and a2 < 0, implies ot > 0, y > 0, a l - b l < 0 , a 2 - b 2 < 0 , and

a l - - h i < C 12, a2 -- b2 < c 21 �9 (38)

The decrease in marginal profit, corrected for strategic complementarity or substitutability, must remain larger than the cost decrease due to economies of scope.

Now we can give the following result,

Proposition 5: Under assumptions B 1-B5, an appreciation of the currency of market 2 increases Pl and decreases P2 if exporting leads to diseconomies of scope.

Proof." From

A = 1

X (al -- bl)(a2 - b2)

• [(a2 + (n 4- m - 1)b2)(al 4- (n 4- m - 1)bl) - c 2 1 c 12]

and (37) it follows that A > 0. Indeed, a2 + (b2 4- c 21) < 0 implies - ( a 2 + be) > c 21 and al + (bl + c 12) < 0 i m p l i e s - ( a l + bl) > c 12.

We are now in a position to determine the signs of the changes in quantities sold in each country and hence of the changes in prices using Eqs. (34). We have

dX 1

de A ( a ~ + / ~ ) < 0 and hence Pl increases if C 12 > 0 . (39.1)

This follows since from the diagonal dominance condition (37) we have al - bl < 0 and a 2 - - b2 < 0 since al < 0 and a2 < 0.

Now oe can be written as oe = ( a 2 q- ( n q- m - 1 ) b 2 ) / ( a 2 - b2) > 0. In addition, e < 0 since 81 > 0 and fl < 0 if c 12 < 0. Furthermore,

> 0 since 82 < 0. This together with A > 0 means that

dX - - < 0 and hence pl increases if c 12 < 0 . de


Note also that with a similar argument and using the fact that g > O, s < O , t / > O , a n d O > O i f c 12 > O, we have

dZ 1

de A (0s + FO) > 0 and hence P2 decreases if c 21 > 0 . (39.2)

[]

The coefficient s reflects the fact that an appreciation of country 2 currency represents a cost increase for firms from country 2 when exporting to market 1, whereas ~ reflects the cost reduction for finns from country 1 when exporting to market 2. Note that both s and t/ appear in dX and dZ: both affect total sales in each market. /3 and 0 are proportional to the economies or diseconomies of scope resulting from exports to the other market, o~ and F reflect the aggregate degree of aggressiveness due to strategic substitutability or complementarity and correspond to the coefficients 132 and /31 defined in (14). When c 1 1 = c 2 2 = 0 a n d p i ~ = p ' z = 0 , o t = F = l + n + m .

The standard conclusion that dX/de < 0 and dZ/de > 0 reappears when exports lead to diseconomies of scope (for all firms). It is the end result of a series of reactions. First, the appreciation creates cost (dis)advantages which make market 2 firms export less and market 1 firms export more to the other market. However, this is only a first explanation of the reduction in X. Because of the diseconomies, increased exporting increases the cost of production for local producers in market 1. This is another reason for reducing their local sales. Re- duced exports to market 1 decrease the cost of production of market 2 firms, which therefore increase their local sales. The effects on local sales, in this second round, are reinforced, in a third round, when the commodities are strategic complements: the reduction of local sales by firms i further reduces the exports by firms k; the increase in local sales by firms k is reinforced by a strategic increase in the exports of firms i. Strategic substitutability implies reactions in the opposite direction, which are not strong enough to prevent X from decreasing and Z from increasing.

These changes in X and Z may go in the opposite direction when all firms benefit from economies of scope. This sign reversal clearly depends on the impact of these economies on local sales, which in turn depends (all firms being alike) on the number of local firms. The role of n and rn in this respect will be analyzed in the next section.


4.2 Market Structure

We now examine the impact on exchange-rate pass-through of changes in the number of firms. Changes in n or m affect the equilibrium quantities X* and Z*, which in turn affect the parameters appearing in dX/de and dZ/de. Given the comparative statics approach followed, it would be an impossible task to trace out these effects. What can be done is to make the parameters, other than n and m, constant with respect to X* and Z*. To that effect, we suppose that the demand functions are linear (bl < 0, b2 < 0) and that the second derivatives of the cost functions are constant. As a consequence, al - bl = pl 1 - c 11 and a2 -- b2 = p~ - C 22 2 are constant. These are, of course, restrictive assumptions. In a world with economies or diseconomies of scale and scope, they include, though, the assumptions of linearity and constant marginal costs made in Sect. 3: effects of changes in market concentration are reduced to effects of changes in the number of players.

Of course, the plausibility of these assumptions depends very much on the technologies of the firms. If each firm prior to a merger had increasing costs but, with constant second derivatives, once the merger is made and the firm can utilize two plants, the second derivative of its cost will change. However, in the case which is most interesting, that is, where there are decreasing marginal costs, the firm will concentrate its entire production in one of the plants and the second derivative of its costs will therefore by assumption be unchanged.

Notice, however, that the sign of the pass-through may be reversed in the case of economies of scope. We now examine how such sign reversals are related to differences in market structure and obtain

Proposition 6: If all finns benefit from the same economies of scope and the commodities are strategic substitutes it cannot happen that as the result of an appreciation of currency 2, Pl decreases and P2 increases simultaneously.

Proof." Notice first that the sign of the determinant in Eqs. (39) does not change with n or m. The sign of dX/de depends on the sum de + f i t and the sign of dZ/de on the sum 0e + yT/. We have

and

ore < O, f i t / > O, if c 12 < 0 ;

de < 0, Y~7 > 0, if C 21 < 0 .

dX/de turns positive when f i t /dominates ore in absolute value. How-


ever, ot increases with both m and n, while ~ increases with n and [el increases with m. Similarly, dZ/de turns negative when 0e dominates g ~ in absolute value. Here, y increases with m or n. In both markets, the sign of the effect of an appreciation therefore depends on the ratio m/n.

For market 1, we find (ore + fl~) > 0 if

- - < . ( 4 0 )

n a2 + (n + m -- 1)b2

In market 2, a sign reversal occurs, that is, dZ/de < 0 or (0e + y~) < 0, if

_m > [a1-1-(n-l-m-l)b,](c_~) (41) n c 21 "

It remains to show that a sign reversal cannot occur in both markets simultaneously. In other words, that (40) and (41) cannot be satisfied simultaneously. If they were, then

C12C21 > (al q- (n + m -- 1)bl)(a2 q- (n + m - 1)b2)

which is not compatible with the dominant diagonal property (37). Whatever the number of firms in the two markets, it cannot happen that pl decreases and P2 increases, when all firms are identical in the sense defined above. []

It is thus clear that if we wish to obtain the perverse result that Pl decreases and p2 increases one has to look for differences in production technologies. 3

The economic rationale of the result for market 1 is as follows. Increased exporting to market 2, as a result of the appreciation of currency 2, reduces the cost of production for firms located in market 1, which therefore increase their local sales. Strategic substitutability (bl < 0) reduces imports into market 1. Nevertheless, total sales increase in market 1 if the number of local firms (n) is large enough compared to the number of foreign firms (m). In the aggregate, the effect of economies of scope then dominates the strategic effect.

The critical ratio m/n decreases with Icml and c2: the smaller the economies of scope or the relative cost disadvantage of foreign firms, the larger n must be compared to m. Mergers in market 2 make a sign reversal in market 1 more likely.

3 This conclusion corresponds to proposition 6 of Hens et al. (1991) where the case m = n = 1 is considered.


Note that (40) does not imply that market 2 must be more concentrated than market 1. With c 2 = c 1, m must be smaller than n only if IC121 < la2 + (n + m -- 1)b21.

The second part of the result can be explained as follows. Reduced exporting to market 1, as a result of the appreciation of their currency, increases the cost of production of firms located in market 2. They reduce their local sales, which increases imports into market 2 through the strategic substitution effect. Nevertheless, total sales decrease in market 2 if the number of local firms (m) is large enough compared to the number of foreign firms. In the aggregate, the effect of economies of scope then dominates the strategic effect.

The smaller the economies of scope resulting from exports to market 1, and the larger the cost disadvantage for firms located in market 2 resulting from the appreciation of their currency, the larger the number of these firms must be for a sign reversal to occur. Now mergers in market 1 make this reversal more likely. With c 2 = c 1, m must be larger than n only i f [C211 < [al + (n § m -- 1)b1[.

We have noted so far that mergers in market 2 make a sign reversal more likely in market 1 and that mergers in market 1 make such a reversal more likely in market 2. We can now add that, in the event of a sign reversal, these mergers accentuate the pass-through (in the "wrong" direction). Thus we have

Proposition 7: The pass-through in the market where the price change, resulting from an appreciation, goes in the "wrong" direction, is accentuated by a reduction in the number of players in the other market.

Proof" Reductions in n or m make the determinant appearing in (39) smaller. A smaller m gives a smaller ]otel, so that dXM/de > dX/de > 0, where the superscript M again designates post-merger values. A smaller n makes yO smaller, with the result that dZM/de < dZ/de < O.

[]

What about the impact of mergers on the extent of the pass-through, when prices move in the "correct" direction? In the case of constant marginal costs (3.2), clearcut impacts could be detected. With economies or diseconomies of scope, this does not seem possible in the framework of our model: the extent of the pass-through may be reduced as well as accentuated.

One thing is clear: in the case of strategic substitutability, a country's antimerger policy is likely to stabilize prices in that country after a revaluation of its currency by (a) reducing the danger of "perverse"


price increases (Proposition 6) and (b) reducing the "normal" decrease of import prices - at least in the absence of diseconomies of scope (Proposition 4). If the other countries also adopt an antimerger policy, this further reduces the danger of a perverse price increase after a revaluation (Proposition 7) but tends to accentuate the normal price decrease (Proposition 3). On balance, antitrust policies tend to make sure that a revaluation leads to decreasing import prices, as predicted by the standard literature. However, what remains to be examined is whether the restrictions that we have imposed are more or less likely, themselves to lead to increased concentration.

6 Conclusion

We have analyzed the consequence of an exchange-rate change for the prices charged and quantities sold by oligopolistic firms selling in two markets which are effectively separated. Since the cost structures of firms may exhibit economies of scope and scale, the interdependence between markets may lead to "perverse" responses and indeed this is what should be expected given the results of Hens et al. (1991) and Hens (1994). However, when each of a number of simple restrictions such as linear demand or identical firms are imposed, the direction of changes is predictable. In addition, the model used here allows us to study the effect of mergers on exchange-rate pass-through and with appropriate restrictions to predict the way in which such mergers modify exchange- rate pass-through.

Acknowledgements

We are grateful to Patrick Sinnesael for correcting an error in an earlier draft, to Thorsten Hens for many helpful discussions, and to two anonymous referees for useful comments.

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Addresses of authors: Alan Kirman, Groupement de Recherche en Econo- mie Quantitative d'Aix-Marseille, Ecole des Hautes Etudes en Sciences So- ciales, 2 rue de la Charitt, F-13002 Marseille, France; - Louis Phlips, Depart- ment of Economics, European University Institute, Badia Fiesolana, 1-50016 S. Domenico di Fiesole (FI), Italy.

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Exchange-rate pass-through and market structure