Increasing Returns, Imperfect Markets, And Trade Theory

8/13/2019 Increasing Returns, Imperfect Markets, And Trade Theory

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Chapter 7INCREASING RETURNS IMPERFECT MARKETS

ND TR DE T H E O R YE L H A N A N H E L P M A NTel Aviv University

ontent s1 . I n t r o d u c t i o n 3 2 62 . T y p e s o f e c o n o m i e s o f s c a le 3 2 73. T y p e s o f c o m p e t i t i o n 3 3 04 . H o m o g e n e o u s p r o d u c t s 3 3 25 . I n t e r n a t i o n a l r e t u r n s t o s c a l e 3 3 76 . N a t i o n a l r e t u r n s t o s c a l e 3 4 17 . L i m i t e d e n t r y a n d m a r k e t s e g m e n t a t i o n 3 4 88. D i f f e r e n t i a t e d p r o d u c t s 3 5 5R e f e r e n c e s 3 6 3

This chap ter is a revised version of Work ing Paper no. 18-82, T he Foe rder Inst i tute of Econom icResearch, Tel Aviv Universi ty. As with much of m y previous work, the prese nt one has also benefi tedfrom discussions at the Tel Aviv Workshop on International Economics. I would l ike to thankespecial ly Eitan Berglas for his ma ny wise comm ents on a writ ten draft and in oral discussions. Thisversion also benefi ted from the comments of Alan Deardorff, Wilfred Ethier, James Markusen, JamesMelvin, Arvind Panagarlya and Lars Svensson, as well as from discussions at the conference of theHandbook s authors at Princeton Universi ty. Finally, I would l ike to thank the Foerder Inst i tute forEconomic Research for financial support .Handbook Of International Economics, vol. 1, Edited by R. W. Jones and P.B. Kenen Elsevier Science Publisher s B.V ., 1984

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1 Introduct ion

The effects of increasing returns to scale and noncompeti tive behavior on interna-tional trade have been discussed for many years. They have a bearing on a host oftrade problems such as explanations of trade patterns, gains from trade, commer-cial poficy, transnational corporations and direct foreign investment. This surveyconcentrates on two major issues: explanations of trade patterns and gains fromtrade. 1

Early writers on this subject whose approach was grounded in the classicaltradition were mainly concerned with gains from trade and other welfare effects.Thus Marshall (1879, p. 13) discussed terms of trade effects, arguing that withincreasing returns to scale (economies of scale) a country may improve its termsof trade by expanding demand for its imports, while Graham (1923) argued thateconomies of scale may cause a country to lose from trade, concluding that in thiscase a tariff is beneficial. Later, with the development of the neoclassical tradetheory, the pattern of trade became of major concern. Thus Ohlin (1933) pointedout that economies of scale serve as one explanation of foreign trade patterns,while other post 1933 writers emphasized the role of monopolistic competition indifferentiated products. In the preface to the English edition of his famous book,Haberler (1936) wrote:

it seems to me that the theory of international trade, as outlined in thefollowing pages, requires further development, in two main directions. Thetheory of imperfect competition and the theory of short-run oscillations(business cycle theory) must be applied to the problems of internationaltrade. It will soon be possible to do this in a systematic way, since muchprogress has been made in both fields in recent years. With regard to the firstof these questions, there is the literature which centers around the twooutstanding books, Monopolistic ompetitionby Professor E. Chamberlin andImperfect ompetition by Mrs. Joan Robinson. In the second field wherefurther development is required, it is not easy to refer to a body of acceptedtheory.

The hope that was expressed by Haberler was probably shared in the thirties byother trade experts as well. Indeed, there exist early attempts to extend tradetheory on the basis of Chamberlin's work [see, for example, Beach (1936) andLovasy (1941)]. However, only recently has this goal been achieved.

In what follows I survey the theory of international trade in the presence ofeconomies of scale and monopolistic (not monopsonistic) competition, with an

1Some issues which are not dealt with in this surveyare surveyed n Caves (1974).326


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Ch 7: Increas ing Re turns 3 7emphasis on predictions of trade patterns and gains from trade. I will take aunified approach to the subject trying to bring out the common logic in much ofthe seemingly unrelated parts of the literature. The main ingredient of this logic isthat the allocation of productive resources is guided in every country by thereward level of every sector s employed combination of factors of production.

In order to obtain an idea what these sectoral rewards are, consider twoexamples. First, suppose there is pure competition, no joint production, andconstant returns to scale. Then the commodity price, which is equated to marginalcosts of production, is the reward to a combination of factors of production thatproduce a single unit at minimum costs. Without impediments to trade, thereward levels-which equal prices-are the same in every country, and tradepatterns are predicted by the standard theories. Secondly, suppose there ismonopolistic competition and constant returns to scale. Suppose also that thereexists one producer of each good in every country. Since a monopolist equatesmarginal costs to marginal revenue, marginal revenue is the sectoral reward to thecost minimizing combination of factors of production that produce a single unitat minimum costs. If in a trading equilibrium marginal revenues are the same inevery country, identical reward levels guide the allocation of productive resourcesin every country and we can again predict trade patterns by means of standardtheories. If they are not, other avenues have to be explored. Whether marginalrevenues are the same depends on the nature of the monopolistic competition.

After discussing types of economies of scale in Section 2 and types ofcompetition in Section 3, I present in ~Section 4 a general model of trade inhomogeneous products when there are variable returns to scale and free entry intoindustries. This model covers many of the cases discussed in the literature.International returns to scale are explored in Section 5 while national returns toscale are explored in Section 6. In Section 7 the assumption of variable returns toscale is abandoned and cases of monopolistic competition with limited entry andmarket segmentation are considered. In the final section, Section 8, I discussdifferentiated products.

2 T y p e s o f e c o n o m i e s o f s c a leThe role of economies of scale (increasing returns to scale) in international tradecannot be dealt with unless their nature is specified, because the behavioralassumptions which are appropriate for firms depend on them. Consequently, theresulting market structure and equilibrium allocations depend on the underlyingeconomies of scale. The importance of this point emerged in discussions ofinternational trade with increasing returns to scale, starting with the debatebetween Knight and Graham [see Graham (1923), Knight (1924), Graham (1925)and Knight (1925)]. Knight s view was that Graham s analysis of the possible


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328 E. ttelpmanl o s s e s f r o m t r a d e i s v a l i d i f t h e e c o n o m i e s o f s c a l e a r e e x t e r n a l t o t h e f i r m a n di n t e r n a l t o t h e i n d u s t r y , b u t t h a t i t is w r o n g i f th e e c o n o m i e s o f s c a le a r e i n t e r n a lt o t h e f ir m . 2 W e w i ll c o m e b a c k t o t h e G r a h a m - K n i g h t d e b a t e i n S e c t io n 4. I t h a sb e e n b r o u g h t u p a t th i s p o i n t o n l y in o r d e r t o a r g u e t h e r e l e v a n c e o f a l t e rn a t i v es p e c i f i c a t i o n s ,

V a r i a b l e r e t u r n s t o s c a l e w h i c h a r e i n t e r n a l t o t h e f i r m a r e d e f i n e d a s f o l l o w s .L e t f v ) b e t h e f ir m s q u a s i - c o n c a v e p r o d u c t i o n f u n c t i o n , w h e r e v is a v e c t o r o fi n p u t s . T h e n f ( . ) e x h i b i t s e c o n o m i e s o f s c a le ( in c r e a s i n g re t u r n s t o s c a le ) a t v i ff o r ) t > 1 , b u t s u f f ic i e n t ly c l o s e t o o n e , f ? w ) > X f ( v ) . N a m e l y , a s m a ll p r o p o r -t i o n a l i n c r e a s e i n a ll f a c t o r i n p u t s i n c r e a s e s o u t p u t m o r e t h a n p r o p o r t i o n a t e l y .S i m i la r ly , t h e r e a r e d i s e c o n o m i e s o f s c a le ( d e c r e a s i n g r e t u r n s t o s c a le ) if f ( X v ) 1 . I w i l l u s e t h e m e a s u r e 0 ( . ) i n t h e f o l l o w i n ga n a l ys i s. C l e a r l y , 0 ( - ) < 1 m e a n s t h a t m a r g i n a l c o s t s e x c e e d a v e r a g e c o s t s o r t h a ta v e r a g e c o s t s a r e i n c r e a s i n g w i t h o u t p u t . E x p l a n a t i o n s o f e c o n o m i e s o f s c a l ew h i c h a r e i n t e r n a l to t h e f ir m a r e b a s e d o n e c o n o m i e s o f i n t e r n a l o r g a n i z a t i o na n d s p e c i a l i z a t i o n [s e e M a r s h a l l ( 1 9 2 0 , B o o k I V , c h . I X ) a n d S t i g l e r ( 1 95 1 ) ], o ni n d i v i s i b il i ti e s [ se e O h l i n ( 1 9 3 3 , p . 5 2 )] a n d o n t h e e x i s t e n c e o f f i x e d c o s ts .

E c o n o m i e s o r d i s e c o n o m i e s o f s c a l e w h i c h a r e e x t e r n a l t o t h e f i r m b u t i n t e r n a lt o t h e i n d u s t r y a r e u s u a l l y r e p r e s e n t e d b y a p r o d u c t i o n f u n c t i o n o f t h e f o r mx = j ~( v, X ) , w h e r e x i s t h e o u t p u t l e v e l o f t h e s i n g l e fi rm , v i s i t s v e c t o r o f i n p u t sa n d X is t h e i n d u s t r y s l e v e l o f o u t p u t [ se e J o n e s ( 19 6 8) ]. T h e f u n c t i o n j ? (. ) i sa s s u m e d t o b e q u a s i -c o n c a v e a n d p o s i ti v e ly l in e a r h o m o g e n e o u s i n v . T h i s m e a n st h a t f r o m t h e p o i n t o f v i e w o f a s i n g le f ir m w h i c h c o n s i d e r s t h e in d u s t r y s o u t p u t

2Th is view was su pported by H aberler (1936, pp. 204-20 8). See Caves (1960, pp. 169-17 6) for aclear description of the debate. The co ncepts of internal and external econo mies of scale areextensivelydiscussed in M arshall (1920, Bo ok IV, chs. IX-X I).


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Ch. 7: IncreasingReturns 329l e v el a s in v a r i a n t t o i ts d e c i si o n s , t h e p r o d u c t i o n p r o c e s s e x h i b i t s c o n s t a n t r e t u r n st o s c a l e . A s s u m i n g t h a t a l l f i r m s i n t h e i n d u s t r y h a v e i d e n t i c a l p r o d u c t i o nf u n c ti o n s , t h e i n d u s t r y 's o u t p u t le v el is X = ~ = 1 x k = ~ = l f ( v k, X ) w h e r e n ist h e n u m b e r o f f ir m s i n t h e i n d u s t r y a n d k i s a n i n d e x o f f i rm s . S i n c e j T (. ) isp o s i t iv e l y l i n e a r h o m o g e n e o u s in v , t h e n a s s u m i n g t h a t a l l fi rm s p a y t h e s a m ef a c t o r p r ic e s a n d t h a t f a c t o r i n p u t s a r e c o s t m i n i m i z i n g , w e g e t X = ) 7 ( ~ = l v k , X )o r X = f V , X ) , w h e r e V is t h e v e c t o r o f i n p u ts e m p l o y e d b y t h e i n d u s tr y . H e n c e ,i f a ll f ir m s a r e i d e n t i c a l a n d f a c t o r m a r k e t s a r e c o m p e t i t iv e , t h e i m p l i c i t r e la t i o n -s h i p b e t w e e n t h e i n d u s t r y ' s i n p u t v e c t o r a n d o u t p u t l e v el i s g i v e n b y : 3

X = f V , X ) . ( 2 . 2 )A s s u m e t h a t ( 2 . 2 ) c a n b e i n v e r t e d t o y i e l d a s o l u t i o n :

X = F V ) , ( 2 . 3 )w h e r e F ( . ) is a f u n c t io n , t h e n w i t h c o m p e t i t i v e f a c t o r m a r k e t s t h e i n d u s t r yo p e r a t e s a s i f F ( . ) i s i ts i m p l i c it p r o d u c t i o n f u n c t i o n .

I t i s n o w s t r a i g h t f o r w a r d t o s h o w t h a t t h e i n d u s t r y ' s i m p l i c i t p r o d u c t i o nf u n c t i o n e x h i b i ts d e c r e a s i n g r e t u r n s t o s c a l e w h e n t h e e l a s t ic i t y o f f ( . ) w i t hr e s p e c t t o X is n e g a ti v e , t h a t i t e x h i b i ts c o n s t a n t r e t u r n s t o s c a le w h e n t h ise l a s t i c i ty i s z e ro , a n d i n c r e a s i n g r e t u r n s t o s c a l e w h e n t h is e l a s t i c i t y i s p o s i t i v e b u ts m a l l e r t h a n o n e . I t is a s s u m e d t h a t t h e e l a st ic i t y o f t h e f i rm ' s p r o d u c t i o nf u n c t i o n w i t h r e s p e c t to t h e i n d u s t r y ' s o u t p u t is sm a l l e r t h a n o n e i n o r d e r t o a v o i dl a n d o f C o c k a i g n e p h e n o m e n a . 4

E x p l a n a t i o n s o f e x t e r n a l e c o n o m i e s - e c o n o m i e s o f sc a le w h i c h a r e e x t e r n a l t ot h e f i r m b u t i n t e r n a l t o t h e i n d u s t r y ( I w i ll n o t d i sc u s s d i s e c o n o m i e s o f s c al e o ft h i s t y p e ) - r e s t o n t h e a r g u m e n t t h a t a l a r g e r i n d u s t r y t a k e s b e t t e r a d v a n t a g e o fw i t h i n - i n d u s t r y s p e c i a l i z a t i o n ( t h e d i v i s io n o f l a b o r i s l i m i t e d b y t h e e x t e n t o f t h em a r k e t , a n d s o is p r o b a b l y t h e d i v i si o n o f o t h e r f a c t o r s o f p r o d u c t i o n ) , a s w e l l a sb e t t e r a d v a n t a g e o f c o n g l o m e r a t i o n , i n d i v i s i b i l i t i e s , a n d p u b l i c i n t e r m e d i a t e i n -p u t s s u c h a s r o a d s [ s ee M a r s h a l l ( 1 9 2 0, B o o k I V , c hs . X - X I ) , O h l i n ( 19 3 3 , p . 5 3 )a n d S t ig l e r ( 19 5 1 )] . T h e p r o c e s s b y m e a n s o f w h i c h h i g h e r o u t p u t l e v el s l e a d t om o r e i n t r a - i n d u s t r y s p e c i a l i z a t i o n i s n o t s p e l l e d o u t e x p l i c i t l y . E a r l y w r i t e r s o nt h e s u b j e c t w e r e a w a r e o f t h e d i f fi cu l ti e s i n h e r e n t i n t h e c o n c e p t o f e x t e r n a le c o n o m i e s o f s c al e [ se e C h i p m a n ( 1 9 6 5 ) f o r a s u m m a r y o f t h e v i e w s o n t h is m a t t e ra n d C h i p m a n ( 1 9 7 0 ) f o r a g e n e r a l e q u i l i b r i u m a n a l y s is o f e x t e r n a l e c o n o m i e s o fsca le ] .

3These assumptions are really stronger than required, bu t they facilitate the exposition.4Using eq. (2.2), the elasticity of F(X v) w ith respect to ?~ evaluated at X = I is c alculated to be[ 1 - e V, X )]- ~, where e(-) l s the elasticity of f(- ) w ith respect to X. T he conclusion discussed in thetext followsfrom here.


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330 E HelpmanIt has, nevertheless, remained a concept of major use in the theory of interna-tional trade.

It is clear from our discussion that the specification of what constitutes anindustry for the definition of external economies is of major importance. Most ofthe literature has used a national basis for this purpose. Namely, it assumed that afirm operating in a particular country derives cost savings f rom the expansion ofoutput of that country s industry. This view was criticized by Ethier (1979) whosuggested tha t in a world economy which is integrated via international trade, theinternational basis is more appropriate for the definition of external economies.According to this view, a firm derives cost savings from the expansion of worldoutput of its product [see also Viner (1937, p. 480)] due to within-industryspecialization which is diffused throughout the world via intra-industry trade inintermediate inputs and stages of production. However, economies of scale whicharise from conglomeration or public intermediate inputs (such as roads) seem tobe country specific, and due to transportation costs within-industry specializationin some stages of production may also be country specific. It seems, therefore,that the size of a domestic industry plays a role in the determination of externaleconomies of scale. But it is also clear that some sources of these scale economiesaffect more than a single industry, which implies cross-industry spillovers ofexternal economies of scale. Cross industry spillovers are not discussed in detailin this chapter [see, however, Anderson (1936), Haberler (1936, p. 208), Manningand Macmillan (1979), Chang (1981), and Herberg, Kemp and Tawada (1982)].In what follows I confine attention to the type of economies of scale that havebeen discussed in the formal part of this section. The reader should, however, beaware of the fact that these concepts are rather limited. Firstly, the suggestedformulat ion is restricted to output generating economies of scale. From the verbaldiscussion it is clear that this need not be the case. In each case the precisespecification should be derived from more basic microeconomic structures. Sec-ondly, a firm may be able to save costs when other sectors expand. I havementioned earlier some of the reasons for such effects. Thirdly, pecuniary externaleffects which are derived from other industries when output expands as well asfrom domestic factor markets might play a role [see Viner (1937, p. 481)]. Thesespillover effects can conceptually also be derived from more basic microeconomicstructures. However, we will proceed with what is available.

3 Types of compet it ionThe literature on international trade under increasing returns to scale employs avariety of assumptions concerning firms behavior. These assumptions play animportant role in the determination of trade patterns and welfare effects. In


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C h . 7 . I n c r e a s i n g R e t u r n s 331addition it is necessary to specify whether there is free entry, because the easewith which firms can enter significantly affects the degree of competition in anindustry.Broadly speaking, there are three types of assumptions about firms' behaviorthat have been employed. When possible, it has been assumed that firms behavepurely competitively. Namely, that firms take prices of inputs and outputs asgiven, and that they choose the input- output combination that maximizes profits.The consequence is, of course, marginal cost pricing. This pricing procedure isviable if the resulting profits are non-negative, which means that perceivedmarginal costs exceed perceived average costs, or-using eq. (2.1)-that at theresulting input- output combination there do not exist economies of scale. If theprocess of production is characterized by global economies of scale, the competi-tive assumption is inappropriate. Indeed, this assumption has been employedexclusively in cases in which the economies (or diseconomies) of scale are externalto the firm but internal to the industry, in which case the single firm operatesunder perceived constant returns to scale.

The second behavioral assumption that has been employed is that of pricecompetition, which is associated with the name of Bertrand [see Modigliani(1958)]. Under this assumption a firm takes as given the prices that are chargedby its competitors and it chooses a price for its product so as to maximize profits.Clearly, this assumption is appropriate when the firm faces a downward slopingdemand curve for its product, but it can also be employed on other occasions.5The third assumption in the broad classification is that of quantity competition,which is associated with the name of Cournot [see Modigliani (1958)]. In theinternational trade literature it has been used mainly for industries which producea homogeneous product. Under this assumption a firm observes the quantities ofthe product offered for sale by its competitors. It assumes that variations in itsown sales will not affect the sales of its competitors. Then it calculates theresponse of the price to changes in its sales and it chooses a profit maximizinglevel of sales.

Existing models can be classified along several lines, generating a multi-dimen-sional matrix. However, not all of the entries in this matrix have been investi-gated; some of them may even be meaningless. The three categories of behavioralassumptions that have been described above represent a single axis of the matrix.Other axes which are relevant to the literature include the following:

(a) Entry: It can be assumed that there are barriers to entry so that the numberof firms in an industry is predetermined, or, alternatively, that there is free entry.The second assumption legds to long-run equilibria with normal profit levels(usually assumed to be zero).

5I will come back to this point at a later stage.


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33 E Helpman( b ) P r o d u c t ty p e : A n i n d u s t r y s p r o d u c t m a y b e a h o m o g e n e o u s g o o d o r it m a y

b e a d i f f e r e n t ia t e d p r o d u c t w i t h m a n y p o s s i b l e v ar ie t ie s . I n t h e s e c o n d c a s e a na d d i t i o n a l d i m e n s i o n o f c o m p e t i t i o n is a d d e d ; c o m p e t i t i o n i n p r o d u c t t y p e .( c) M a r k e t t y p e: M a r k e t s o f d i f fe r e n t c o u n t r i e s f o r t h e s a m e p r o d u c t m a y b ei n t e g r a t e d o r s e g r e g a t e d . I n p a r t i c u l a r , i t m a k e s a s i g n i f i c a n t d i f f e r e n c e w h e t h e r af i rm r e c o g n i z e s th i s i n t e r d e p e n d e n c e o r i t i g n o r e s i t.

T h i s c l a s si f ic a t io n is i n t h e d o m a i n o f c o m p e t i t i o n t y p e s . I f w e a d d t h e p o s s i b let y p e s o f e c o n o m i e s o f s c a le ( in t e r n a l o r e x t e r n a l ), t h e n u m b e r o f c a se s is d o u b l e d .T h e r e i s , o f c o u r s e , n o i n t e n t i o n t o g o o v e r a l l t h e s e c a s e s i n t h i s s u r v e y . I w i l ls u r v e y t h e m a j o r c o m b i n a t i o n s t h a t h a v e b e e n d i s c u ss e d in t h e l it e r a t u r e a n d Iw i l l p r e s e n t t h e m a i n c o n c l u s i o n s t h a t h a v e b e e n r e a c h e d i n o t h e r c a s e s .

4 Hom ogeneous productsI n t e r n a t i o n a l t r a d e i n h o m o g e n e o u s p r o d u c t s h a s r e c ei v e d m o s t o f t h e a t t e n t io ni n t h e t h e o r e t i c a l l i t e r a tu r e . I n t h i s s e c t i o n I d e s c r i b e a m o d e l o f t r a d e i nh o m o g e n e o u s p r o d u c t s w h i c h is g e n e ra l e n o u g h t o c o v e r th e c a s es o f n a t io n a l a n di n t e r n a t i o n a l r e t u r n s t o s c al e, a n d I p r o v e a g a i n s f r o m t r a d e t h e o r e m f o r th em o d e l .

M a n y o f t h e im p o r t a n t i s su e s t h a t h a v e a b e a r i n g o n t h e p r o b l e m s d i s c u ss e d int hi s s e c ti o n w e r e d is c u ss e d i n c o n n e c t i o n w i th t h e G r a h a m - K n i g h t d e b a t e.G r a h a m ( 19 2 3 ) c o n s tr u c t e d a n u m e r i c a l e x a m p l e ( o r s o h e b e li e v e d ) w h i c h s h o w st h a t w h e n a c o u n t r y h a s a s e c t o r w i t h i n c r e a s i n g r e t u r n s t o s c a l e a n d a s e c t o r w i t hd e c r e a s in g r e t u rn s t o s c al e i t m a y l o s e f r o m t r ad e . G r a h a m s a r g u m e n t c a n b e ,r e s t a t e d a s f o ll o w s . S u p p o s e t h e r e is a s i n gl e f a c t o r o f p r o d u c t i o n , s a y l a b o r , a n de q u a l p r i c e s o f b o t h g o o d s . A l s o s u p p o s e t h a t a s a r e s u lt o f f o r e ig n t r a d e ac o u n t r y s h i f ts l a b o r f r o m t h e in c r e a s i n g r e t u r n s t o s c a le i n d u s t r y t o t h e d e c r e a s i n gr e t u r n s t o s c a le i n d u s t r y . T h e n o u t p u t p e r m a n f a ll s i n b o t h i n d u s t ri e s , t h e r e b yr e d u c i n g g r o s s d o m e s t i c p r o d u c t a t c o n s t a n t c o m m o d i t y p ri ce s . T h i s le a d s t o aw e l f a r e l o s s .

K n i g h t ( 1 9 2 4) a c c u se d G r a h a m o f n o t d i s ti n g u is h i ng b e t w e e n i n t e rn a l a n de x t e r n a l e c o n o m i e s . I f t h e e c o n o m i e s o f s c a le a r e i n t e r n a l t o t h e f i r m t h e r e c a n b en o c o m p e t i t i o n a n d o n e h a s t o d e a l e x p li c it ly w i t h m o n o p o l y . G r a h a m ( 19 2 5)d e n i e d t h e n e e d t o d i s ti n g u i sh b e t w e e n i n t e r n a l a n d e x t e r n a l e c o n o m i e s o f s ca le .H a b e r l e r ( 19 3 6 , p . 2 0 4 ) a n d V i n e r (1 9 3 7 , p . 4 7 3 ) a g r e e d w i t h K n i g h t s p o s i t i o n o nt h i s m a t t e r . V i n e r a l s o p o i n t e d o u t t h a t G r a h a m c o n f u s e d a v e r a g e a n d m a r g i n a lc o s ts i n h is p r ic i n g r ul es a n d t h a t e x t e r n a l e c o n o m i e s m a y d e p e n d o n w o r l do u t p u t r a t h e r t h a n n a t i o n a l o u t p u t , i n w h i c h c a s e G r a h a m s a r g u m e n t i s si gn if i-c a n t l y w e a k e n e d .

F i r s t c o n s i d e r t h e c a s e o f e x t e r n a l e c o n o m i e s , a n d r e c a ll t h a t t h e f i r m sp r o d u c t i o n f u n c t i o n i s a s s u m e d t o b e o f t h e f o r m x = ) ?( v, X ) , w h e r e X i s t h er e l e v a n t i n d u s t r y s ( n a t i o n a l o r i n t e r n a t i o n a l ) o u t p u t l e v e l. T h e l i t e r a t u r e u n d e r


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Ch. 7: Increasing Returns 333s u r v ey d ea l s m a i n l y w i t h t h e f o ll o w i n g mu l t ip l ic a t iv e l y s ep a r ab l e f o r m o f ] ( . ) :

] v, X) = g X) f v) , (4 .1 )wh ere f ( . ) i s i ncreas ing , s t r ic t l y quas i - concav e , and p os i t i ve ly l i near homo ge-n eo u s . Th e f u n c t i o n g X) has an e l as t i c i t y e X) s ma l l e r th an o n e . I f g ( - ) i s anincreas ing func t ion o f X , t here a re economies o f sca l e , and i f i t i s a decreas ingf u n c t i o n t h e r e a r e d i s eco n o m i es o f s cal e.

The spec i f i ca t i on o f t he p roduct ion func t ion i n (4 .1 ) has a conven ien t i n t e rp re-t a ti o n . Th e f u n c t i o n f ( . ) , w h i ch h as t h e s t an d a r d p r o p e r t ie s , c an b e co n s i d e r ed a sr ep r e s en t i n g f ac t o r a l v a l u e ad d ed . Th i s v a l u e ad d ed i s au g men t ed b y t h e p r o d u c -t i v i t y o f t he sca l e e f f ec t , wh ich i s r ep resen t ed by g ( . ) , t o y i e ld t o t a l ou tpu t . I t i sc l ear f rom th i s spec i f i ca t i on t ha t i f c w) r ep r e s en t s u n i t co s t s o f f ac t o r a l v a l u ead d ed [ th e u n i t co s t f u n c t i o n a s s o c i a t ed w i th f ( v ) ] t h en co m p e t i t iv e fi rms i n t h ei n d u s t r y en g ag e i n marginalcos t p r i c ing accord ing to p = c w)/g X), wh ere p i st h e p r o d u c t p r i ce [c w)/g X) i s t he f i nn s m arg ina l cos t func t ion ] . P u t d i f fe r -ent ly :

pg X) = c w), (4 .2)w h e r e pg X) i s t h e r ew ar d p e r u n i t f a c t o r a l v a l u e ad d ed . I t i s n o w s t r a i g h t f o r -w a r d t o s h o w t h a t c w)X/g X) i s th e i n d u s t r y s co s t f u n c t i o n s o t h a t c w)/g X)r ep r e s en ts i ts av e r ag e co s t s . 6 H en ce , t h e i n d u s t r y p r i ce s i t s o u t p u t a cco r d i n g t oaverage cos t s .Le t g ( . ) b e i n c r ea si n g i n X , t h en t h e i n d u s t r y s i mp l i c it p r o d u c t i o n f u n c t i o nF ( V ) ex h i b it s i n c r eas i n g re t u r n s t o s ca le . N o w ch a n g e t h e a s s u m p t i o n an d a s s u m et h a t F v) i s a f i rm s p r o d u c t i o n f u n c t i o n , s o t h a t t h e s ca l e eco n o m i es a r e i n te r n a lto t he f irm. B u t assum e a l so t ha t t here i s f r ee access t o t h i s t echno logy , f r ee en t ry(and ex i t ) i n to t he i ndus t ry , and p r i ce compet i t i on h l a Ber t r and . I f an en t e r ingf i r m co n j ec t u r e s t h a t b y ch a r g i n g t h e mar k e t p r i ce i t c an g e t an y mar k e t s h a r e i tdes i res , t ha t b y charg ing a h igher p r i ce i t wi l l ge t a zero ma rke t share , and bycharg ing a l ower p r i ce i t wi l l ge t t he en t i r e marke t , t hen i n t he r esu l t i ngequ i l i b r ium there wi l l be a s i ng l e f i rm in t he i ndus t ry and i t wi l l charge a p r i ceequal t o average cos t s [ see Kemp (1969 , p . 155) and Grossman (1981) ] . In t h i scase t he p r i c ing e qua t ion (4 .2 ) is a l so app l icab l e . D esp i t e t here be ing a s ing le f irmin t he i ndus t ry , entry co m p e t i t i o n f o r ce s t h e mo n o p o l i s t t o en g ag e i n av e r ag e co s t

6F rom (4.1) t he i ndus t ry ' s i mp l i c it p roduc t i on func t ion i s F ( V ) -= g - l [ f ( V) ], whe re ~ 1 (. ) i s t hei nve rse o f g ( ) , de f i ne d by ~ (X ) -= X / g X ) . He nc e , i f Cf w , Z ) i s the cos t func t ion assoc ia ted wi thf v ) and CF(W, X) i s the cos t func t ion assoc ia ted wi th F(V ), the two a re re la ted by C e(w , X)---Cflw, ~ (X) ] . S i nce f ( - ) i s pos i t ive l y l i ne ar homoge ne ous , Cf[w, ~( x)] --- c w ) ~ x ) , i mp l y i ng CF w X )=- c w)X/g x).


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3 3 4 E . H e l p m a np r i c in g . 7 I t s e e m s , th e r e f o r e , t h a t G r a h a m ( 1 9 2 5 ) h a d a p o i n t w h e n h e a r g u e d t h a th e d o e s n o t c a r e w h e t h e r t h e e c o n o m i e s o f s c a le a r e i n t e r n a l o r e x t e r n a l . I t is a ls oc l e a r t h a t i n o u r c a s e s a v e r a g e c o s t p r i c i n g i s t h e p r o p e r s p e c i f i c a t i o n , a n d I w i l la d o p t i t i n t h e f o l l o w i n g p r e s e n t a t i o n .

I n o r d e r t o r e p r e s e n t th e e q u i li b r iu m c o n d i t i o n s c o n s i d e r a m a n y s e c to re c o n o m y , w i t h f ~ (V i ), i = l , 2 , . . . , m , b e i n g t h e f a c to r a l v a l u e a d d e d f u n c t i o n ins e c t o r i. G i v e n t h e a g g r e ga t e e n d o w m e n t o f f a c t o r s o f p r o d u c t i o n o n e c a n d e f in et h e e c o n o m y s t r a n s f o r m a t i o n s u r f a ce b e t w e e n f a c t o r a l v a lu e a d d e d l ev e ls . 8 T h i ss u r f ac e h a s t h e u s u a l p r o p e r t i es o f a t r a n s f o r m a t i o n s u r fa c e d e r iv e d f r o m c o n s t a n tr e t u r n s t o s c a l e t e c h n o l o g i e s . E q u i l i b r i u m f a c t o r a l v a l u e a d d e d l e v e l s w i l l b e o nt h i s s u r f a c e d u e t o c o s t m i n i m i z a t i o n . I n a d d i t i o n , d u e t o t h e a v e r a g e c o s t p r ic i n gr u l e ( 4 . 2 ) t h e e c o n o m y s f a c t o r a l v a l u e a d d e d l ev e ls w i ll b e c h o s e n a t ap o i n t o f t a n g e n c y b e t w e e n t h is s u r f a c e a n d a h y p e r p l a n e w i t h w e i gh t s[ p l g l X 1 ) , p 2 g 2 X 2 ) , . . . , p m g , , , X , , , ) ] . T h e e c o n o m y s G D P c an , t h e r ef o re , b e r e p -r e s e n t e d b y t h e f u n c t i o n G D P [ p l g l X 1 ) , p z g 2 X z ) . .. . , p m g , , X m ) ; V ] , w h e r eG D P ( . ) h a s t h e u s u a l p r o p e r t i e s o f a r e s t r i c t e d p r o f i t f u n c t i o n w h i c h i s d e r i v e df r o m c o n s t a n t r e t u r n s t o s c al e te c h n o l o g i e s . 9 I t i s p o s i t iv e l y l i n e a r h o m o g e n e o u si n t h e f ir s t m a r g u m e n t s , i t i s p o s i t i v e l y l i n e a r h o m o g e n e o u s i n V , i t s p a r t i a ld e r i v a t i v e w i t h r e s p e c t t o P i g i S i ) e q u a l s t h e f a c t o r a l v a l u e a d d e d i n s e c t o r i , 1a n d i t s d e r iv a t i v e w i t h r e s p e c t to V e q u a l s t h e c o m p e t i t i v e r e w a r d t o f a c t o r l. I na d d i t i o n , i t i s c o n v e x i n t h e f ir s t m a r g u m e n t s a n d c o n c a v e i n V [ se e V a r i a n( 1 9 7 8 , c h . 1 )]. S in c e G D P ( . ) i s p o s i t i v e l y l i n e a r h o m o g e n e o u s in V a n d t h ec o m p e t i t iv e r e w a r d o f f a c t o r l e q u a l s 0 G D P / 0 V t , f a c t o r p a y m e n t s e x h a u s t th ee n t i r e G D P , a n d t h e r e a r e n o p u r e p r o f i t s . I n w h a t f o l l o w s I w i l l u s e t h e n o t a t i o n

7 T h i s d i s c u s s i o n w a s c o n f i n e d t o t h e c a s e o f i n c r e a s i n g r e t u r n s t o s c a le . I f t h e r e a r e d e c r e a s i n gr e t u r n s t o s c a l e , f re e e n t r y a n d f r e e a c c e s s t o t h e t e c h n o l o g y l e a d t o i n f i n i t e l y m a n y f i r m s w h i c ho p e r a t e a t a n i n f i n i t e s i m a l l e v e l a n d t h e i n d u s t r y ' s i m p l i c i t p r o d u c t i o n f u n c t i o n e x h i b i t s c o n s t a n tr e t u r n s t o s c a l e .S L e t Z = ( Z 1 , Z 2 . . . . Z m ) b e a v e c t o r o f f a c t o r a l v a l u e a d d e d l e v e ls . T h e n , g i v e n a n e n d o w m e n tv e c t o r V , t h i s t r a n s f o r m a t i o n s u r f a c e is d e f i n e d a s fo l l o w s :

T V ) = { Z I V i > _ O , i = I , 2 . . . . . m , s u c h t h a t ~ v i < _ V , a n di = lZ t = f i V i ) , a n d t ] ~ : i > _ o , i = l , 2 . . . . . m s u c h t h a t ~ ( z i < Vi = 1 \a n d f i ( ( z i ) _> Z ~ , i = 1 , 2 . . . . . m , w i t h s t r i c t i n e q u a l i t y h o l d i n g f o r s o m e i /

S e e a l s o H e r b e r g a n d K e m p ( 1 9 6 9 ) a n d I n o u e ( 1 9 8 1 ) .9 B y d e f i n i t i o n , G D P ( q , V ) -= m a x q . Z , s .t . Z ~ T V ) . I n o u r c a s e q i = P i g i X i ) .1 F o r e c o n o m i e s i n w h i c h t h e n u m b e r o f g o o d s i s la r g e r t h a n t h e n u m b e r o f f a c t o r s o f p ro d u c t i o n ,s u c h a s i n t h e R i c a r d i a n c a s e, t h e G D P f u n c t i o n m a y n o t b e d i f f e r e n ti a b l e w i t h r e s p e c t t o t h e f i rs t r na r g u m e n t s . I n t h i s c a se it s g r a d i e n t w i t h r e s p e c t t o t h e s e a r g u m e n t s s h o u l d b e i n t e r p r e t e d a s a s e t.


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Ch. 7: IncreasingReturns 335G D P [ p g ( X ) , V ], w h e r e p g ( X ) i s i n t e r p r e t e d a s t h e v e c t o r [ p l g l ( X 1 ) ,P 2 g 2 ( X 2 ) . . . . . P m g m ( X , ,) ] O b s e r v e t h a t t h e e m p l o y e d s p e c i fi c a t io n a d m i t s a l s os e c t o r s w i t h c o n s t a n t r e t u r n s t o s c a l e . S u c h s e c t o r s a r e r e p r e s e n t e d b y c o n s t a n tg ( . ) f u n c ti o n s . In p a r t ic u l a r , o n e m a y c h o o s e g i ( X / ) = 1 f o r a c o n s t a n t r e t u r n s t os c a l e s e c t o r .

I n t h e r e m a i n i n g p a r t o f t h is s e c t i o n I s p e c if y t h e e q u i l i b r i u m c o n d i t i o n s o f at r a d i n g w o r l d i n w h i c h t h e e c o n o m i e s o f s c al e a r e n a t i o n a l i n s o m e s e c t o r s,i n t e r n a t i o n a l i n o t h e r s o r h a v e a m o r e g e n e r a l f o r m , a n d I p r o v e a g a i n s f r o mt r a d e t h e o r e m . F o r s o m e p u r p o s e s w e w i l l a l s o n e e d a u t a r k y e q u i l i b r i u m c o n d i -t io n s . H o w e v e r , t h e s e w i ll n o t b e f o r m a l i z e d i n o r d e r t o s a v e s p a c e ; t h e i r n a t u r ew i l l b e c o m e c l e a r f r o m t h e t r a d i n g e q u i l i b r i u m c o n d i t i o n s .

L e t s u p e r s c ri p t j d e n o t e f u n c t io n s a n d v a r ia b l es o f c o u n t r y j . F o r e x a m p l e ,G D W ( . ) is c o u n t r y j s G D P f u n c ti o n , w h i c h d e p e n d s o n i ts t e c hn o lo g ie s o fp r o d u c t i o n o f f a c t o r a l v a l u e a d d e d . L e t D i ( p , I ) b e c o u n t r y j s vector o f d e m a n df u n c t i o n s . T h e n t h e f o l l o w i n g a r e e q u i l i b r i u m c o n d i t i o n s :

~ . , D J ( p , G D P J [ p g J ( X J ) , v J ] } = X j , ( 4 . 3 )J J

X j = G D P J [ p g J ( X J ) , W ] , f o r a l l j , ( 4 . 4 )w j = G D P ~ [ p g J ( . ~ J ) , V J ] , f o r a l l j . ( 4 . 5 )

C o n d i t i o n ( 4 .3 ) i s th e e q u i f i b r i u m c o n d i t i o n i n c o m m o d i t y m a r k e t s , w h e r e X j ist h e o u t p u t v e c t o r i n c o u n t r y j . A l l g o o d s a re t ra d e d a n d t h e r e a re n o i m p e d i m e n t st o t r a d e . H e n c e , p i s th e v e c t o r o f c o m m o d i t y p r ic e s i n e v e r y c o u n t r y . C o u n t r ie sa r e a l lo w e d t o d i ff e r i n d e m a n d p a t t e r n s , t e c h n o l o g i e s [ i n c lu d i n g t h e g ( . ) f u n c -t io n s] a n d f a c t o r e n d o w m e n t s .

C o n d i t i o n ( 4 .4 ) s a y s th a t a c o u n t r y s o u t p u t v e c t o r e q u a l s i ts s u p p l y . T h es u p p l y e q u a ls t h e g r a d i e n t ( G D P p ) o f t h e G D P f u n c t i o n w i t h r e s p e ct to p . l lC o n d i t i o n ( 4 .5 ) r e p r e s e n t s e q u i l ib r i u m f a c t o r p ri c e s. T h e s e e q u a l m a r g i n a l p r o d -u c t v a lu e s o r t h e g r a d ie n t ( G D P v ) o f t h e G D P f u n c t i o n w i th r e s p e c t t o th e f a c t o re n d o w m e n t v e c t or .I t r e m a i n s t o e x p l a i n t h e X J t h a t a p p e a r s i n g ( . ) . T h i s v e c t o r r e p r e s e n t s t h er e l e v a n t t y p e o f re t u r n s t o s ca le . I f i n s e c t o r i t h e r e t u r n s t o s c a l e a r e n a t i o n a l a n ds e c t o r s p e c if ic , t h e n ~ J = X / , i . e. o n l y t h e n a t i o n a l o u t p u t l e v e l o f s e c t o r i a f f e c tsp r o d u c t i v i t y i n s e c t o r i. I f, o n t h e o t h e r h a n d , t h e r e t u r n s t o s c a le in s e c t o r i a r ei n t e r n a t i o n a l a n d s e c t o r s p e c if ic , t h e n .~ .J = Y .gX ~ , i. e. w o r l d o u t p u t o f s e c t o r ia f fe c t s p r o d u c t i v i t y i n s e c t o r i. B y a n a p p r o p r i a t e s p e c i f ic a t i o n o f t h e X , J s o n eo b t a i n s e v e r y d e s i r e d m i x t u r e o f n a t i o n a l a n d i n t e r n a t i o n a l r e t u r n s t o s c a le .

l tFrom the properties of the GD P function,we have: OGDP/api = g i [ 3 G D P / O P i g i ) ] = g i f i = X i .If the GD P function is no t differentiable, he gradient is a set and (4.4) should be interpreted asX GDP/(-).


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336 E. HelpmanM o r e o v e r , f o r p r e s e n t p u r p o s e s X / m a y b e a v e c t o r w h i c h r e p r e s e n t s i n t e rs e c t o r a ls p i l l o v e r e f f e c ts o f e c o n o m i e s o f s c a l e [i n th i s c a s e g / ( . ) i s a f u n c t i o n o f a v e c t o r ],o r i r r e v e r s i b l e e c o n o m i e s o f s c a l e a s d i s c u s s e d i n M a r s h a l l ( 1 9 2 0 , a p p e n d i x H )a n d N e g i s h i ( 1 9 7 2, c h . 5) . S o m e s e c t o r s m a y a l s o e x h i b i t c o n s t a n t r e t u r n s t o s c al e .T h i s s p e c i f i c a t i o n t o g e t h e r w i t h ( 4 . 3 ) - ( 4 . 5 ) p r o v i d e a c o m p l e t e d e s c r i p t i o n o fe q u i l i b r i u m c o n d i ti o n s .

I t r e m a i n s t o f o r m u l a t e t h e g a in s f r o m t r a d e t h e o r e m . L e t X A j b e t h e a u t a r k yo u t p u t v e c t o r in c o u n t r y j . I n t h e a b s e n c e o f t r a d e o n l y n a t i o n a l o u t p u t l ev e lsa f f e c t p r o d u c t i v i t y i n s e c t o rs w i t h v a r i a b l e r e t u r n s t o s c a le , s o t h a t i n a u t a r k yg / ( . ) d e p e n d s on X A j [ h e r e I a l l o w g / . ) t o b e a f u n c t i o n o f t h e e n t i re o u t p u tv e c t o r ] .Gains f rom t rade theorem 1T h e f o l l o w i n g is a s u f f ic i en t c o n d i t i o n f o r g a i n s f r o m t r a d e :

P i [ gi g+ )- gi xAO] LA/>- 0i =

w h e r e L j is a u t a r k y f a c t o r a l v a lu e a d d e d i n c o u n t r y j s s e c t o r i. 12T h i s c o n d i t i o n i s n e c e s s a r i l y sa t is f i e d i f ~ J i s a sc a l a r w h i c h r e p r e s e n t s t h e p o s t

t r a d e o u t p u t l ev e l o f s e c to r i i n c o u n t r y j o r t h e w o r l d s p o s t t r a d e o u t p u t l ev e l o f12The proof of this the orem is as follows. Let e/ p, u) be country j s minimum expenditurefunction (associated with its u tility function from wh ich the vector of dem ands DJ p , I ) has beenderived). Then:

eJ p, u A J)


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Ch. 7: IncreasingReturns 337industry i and:(a) for g / . ) strictly increasing (economies of scale) X/>_ x/A/;

(b) for g/J(.) strictly decreasing (diseconomies of scale) X//< X/A/.The interpretation of conditions (a) and (b) is that a country gains from trade if(i) as a result of trade it expands output of industries with national economies ofscale and it contracts output of industries with national diseconomies of scale,and if (ii) the post trade world output of industries with international economiesof scale is larger than the country s output of these industries in autarky and thepost trade world output of industries with international diseconomies of scale issmaller than the country s output of these industries in autarky. No restrictionsare imposed on industries with constant returns to scale. This is a generalizationof the gains from trade theorem proved by Kemp and Negishi (1970), who dealtonly with the case of national economies of scale 13 Negishi (1972, ch. 5) provedtha t irreversible nat ional economies of scale assure gains from trade, whereirreversibility means that a contract ion of output does not reduce productivi ty [ifX 0 initially, then g i x / ) = gi(X/) for all X/< X/0]. His case is covered by mysufficient condit ion as is the case of irreversible international economies of scaleand cases of intersectoral interdependencies.

I have provided a sufficient condition for gains from trade. This conditionmakes clear the point that international economies of scale are more conducive togains from trade than national economies of scale. Moreover, the smaller thecountry compared to the rest of the world, the more it stands to gain from tradedue to international economies of scale (because the more likely it is that wor ldoutput of these industr ies will exceed significantly its pre-trade output levels). Bythe same token international diseconomies of scale are more harmful to gainsfrom trade than national diseconomies of scale. If economies of scale areirreversible, then a country always gains from trade. The sufficient conditiondescribed in the theorem states simply that trade does not reduce the economy saverage productivity (as measured at the autarky factor allocation).

5 I n t e r n a t i o n a l r e t u r n s t o s c a l e

The relevance of international economies of scale for trade problems was pointedout by Viner (1937, p. 480) in his evaluation of the Graham-Knight debate.However, only in Ethier (1979) has this concept been seriously explored. Ethierexpressed the view that international economies of scale are more important thannational economies of scale. He developed a two sector model; one operatingunder constant returns to scale and the other (manufacturing) operating under

13Eaton and Panagariya 1979) provide a stronger version of the Kemp-Negishi heorem. However,their global analysis,which should have relied on line integrals, is incomplete.


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338 E. Helpmani n t e r n a t i o n a l l y s e c t o r s p e c if ic i n c r e a s i n g r e t u r n s t o s c a le . T h e n h e s h o w e d t h a t f o rf ix e d b u d g e t s h a r e d e m a n d p a t t e rn s a n d a n e x p o n e n t i a l g ( . ) f u n c t i o n in m a n u f a c -t u r in g , t h e r e e x is ts a u n i q u e e q u i l i b r iu m i n w h i c h t r a d i n g t a k e s p l a c e a c c o r d i n g t os t a n d a r d t h e o r i e s o f c o m p a r a t i v e a d v a n t a g e . 14 A l t h o u g h h e d i d n o t s t a t e e x p l i c it l ya g a i n s f r o m t r a d e t h e o r e m , h i s d i s c u s s i o n s g i v e t h e i m p r e s s i o n t h a t c o u n t r i e s d og a i n f r o m t r a d e .

F i r s t c o n s i d e r th e p a t t e r n o f t ra d e . S u p p o s e t h a t e v e r y c o u n t r y h a s t h e s a m eg i ( ) f u n c t i o n i n s e c t o r i. I f al l r e t u r n s t o s c a le a r e i n t e r n a t i o n a l a n d s e c t o rs p e c if ic ( w e a d m i t b o t h e c o n o m i e s a n d d i s e c o n o m i e s o f s c al e) , a s a s s u m e d b yE t h i e r ( 1 9 79 ) , t h e n ~ J = ~ k X ~ a n d t h e r e w a r d t o f a c t o r a l v a l u e a d d e d i n s e c t o r i ,p i g i ~ k x i k ) , i s t h e s a m e i n e v e r y c o u n t r y . H o w e v e r , f o r p r e d i c t i o n s o f t r a d ep a t t e r n s w h a t i s r e q u i r e d is t o h a v e t h e s a m e X J in e v e r y c o u n t r y . T h i s m e a n s t h a tw e c a n a l l o w X [ t o b e a v e c t o r w h i c h r e p r e s e n t s a l so i n t e r s e c t o r a l e ff e ct s, w i t h~ k X ~k b e i n g i t s i t h c o o r d i n a t e . I n t h is c a s e i t c a n b e e a s i ly s h o w n t h a t w h e nf a c t o r a l v a l u e a d d e d i s p r o d u c e d b y m e a n s o f R i c a r d i a n t e c h n o l o g i e s w i t h a s i ng l ef a c t o r o f p r o d u c t i o n ( l a b o r ) th e p a t t e r n o f s p e c i a l iz a t i o n i s p r e d i c t e d a c c o r d i n g t oc o m p a r a t i v e c o s t s , j u s t a s i n t h e c l a s s ic a l m o d e l o f t r a d e [ s e e e q s . ( 4 .3 ) a n d( 4. 4) ]. 15 I f i n s t e a d o f t h e R i c a r d i a n t e c h n o l o g y f a c t o r a l v a l u e a d d e d i s p r o d u c e db y m e a n s o f H e c k s c h e r - O h l i n t y p e t e c h n o l o g ie s w h i c h a r e i d e n ti c a l ac r os sc o u n t r i e s a n d p r e f e r e n c e s a r e h o m o t h e t i c a n d i d e n t i c a l ac r o s s c o u n t r i e s , t h e n d u et o t h e f a c t t h a t t h e r e w a r d t o f a c t o r a l v a l u e a d d e d i s t h e s a m e i n e v e r y c o u n t r y (i ne a c h s e c t o r ) r e l a t i v e o u t p u t l e v e l s a r e d e t e r m i n e d b y r e l a t i v e f a c t o r e n d o w m e n t sw h i l e r e l a ti v e c o n s u m p t i o n l e v e ls a r e t h e s a m e i n e v e r y c o u n t r y . H e n c e , t h ep a t t e r n o f t r a d e i s p re d i c t e d b y t h e H e c k s c h e r - O h l i n t h e o r y [s ee eq s. ( 4. 3) a n d( 4 .4 ) ]. I n a d d i t i o n , d u e t o ( 4 .5 ) , f a c t o r p r i c e e q u a l i z a t i o n o b t a i n s i f V b e l o n g s t ot h e c o n e o f d i v e r s i f ic a t i o n ( s e e C h a p t e r 3 ).

I t i s c l e a r f r o m t h i s d i s c u s s i o n t h a t t h e k e y t o o u r a b i l i t y t o p r e d i c t t r a d ep a t t e r n s f o r e c o n o m i e s w i t h i n t e r n a t i o n a l l y v a r i a b l e r e t u r n s t o s c a le li es in t h ef a c t t h a t i n a t r a d i n g e q u i l i b r i u m t h e r e a l r e w a r d t o f a c t o r a l v a l u e a d d e d i s t h es a m e i n e v e r y c o u n t r y s e c t o r b y s e c to r . T h i s i s n o t t h e c a s e w i t h n a t i o n a l r e t u r n st o s c a le . I t i s a ls o c le a r t h a t d u e p r e c i s e ly to t h e s a m e k e y f e a t u r e a n d t h e a s s u m e dh o m o t h e t i c i t y o f p r e fe r e n c e s, u n d e r i n t e r n a t i o n a l r e t u r n s t o s c a le t h e p a t t e r n o ft r a d e d o e s n o t d e p e n d o n c o u n t r y siz e. O n t h e o t h e r h a n d , s in c e a u t a r k y s c al ep r o d u c t i v i t y l e v e l s g l , g2 , . . . , gm) d e p e n d o n a u t a r k y o u t p u t l ev e ls , p r e - t r a d e

14Ethier (1979) has dev eloped n that pap er a new analytical construct, the allocation curve, n orderto analyze the problem s at han d.15In this c a s e T J v ) iS the no n-neg ativepart o f a hyperplane andQoPJ[pg X) ,VJ]=-max E p igi X iJ )Z i , s . t .Z ~T J W ) ,

i ~ l

implies com er values of Zi s except for spe cial com binationsOf pig i , i 1,2 ,. . . , rn.


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Ch. 7: Increasing Re turns 339r e la t iv e c o m m o d i t y p ri c es o r f a c t o r r e w a r d s s e rv e a s b i a s e d p r e d i c t o r s o f t r a d ep a t t e r n s . T a k e , f o r e x a m p l e , t h e c a s e o f t w o c o u n t r i e s w h i c h p r o d u c e t w o g o o d sb y m e a n s o f t w o f a c t o r s o f p r o d u c t i o n w i t h id e n t i c a l te c h n o l o g ie s . O n e s e c to r is ac o n s t a n t r e t u r n s t o s c a l e s e c t o r w h i l e t h e o t h e r i s s u b j e c t t o s e c t o r s p e c i f i ci n t e r n a t i o n a l l y in c r e a s i n g r e t u r n s t o s ca le . S u p p o s e a l so t h a t t h e c o u n t r i e s h a v ei d e n t i c a l h o m o t h e t i c p r e f e r e n c e s a n d t h e s a m e r e l a t i v e f a c t o r e n d o w m e n t s . T h e na c c o r d i n g t o o u r p r e d i c t i o n t h e s e c o u n t r i e s w i l l n o t t r a d e w i t h e a c h o t h e r .H o w e v e r , i f o n e c o u n t r y i s la r g e r t h a n t h e o t h e r , t h e l a r g e r c o u n t r y w i ll h a v e i na u t a r k y a l o w e r r e l a ti v e p ri c e o f t h e g o o d t h a t i s p r o d u c e d w i t h i n c r e a s i n g re t u r n st o s c al e. U n d e r t h e s e c i r c u m s t a n c e s p r e - t r a d e c o m m o d i t y p ri c es a n d f a c t o rr e w a r d s d o n o t s e r v e a s r e li a b le p r e d i c t o rs o f t h e p a t t e r n o f t r a d e .

T h e i d e a t h a t t h e r e a r e i n t e r n a t i o n a l e x te r n a li t ie s w h e n t r a d e i s a ll o w e d t o t a k ep l a c e b u t i t d o e s n o t t a k e p l a c e a n d n o s u c h e x t er n a l it i es e x is t i n a u t a r k y , s o u n d sl ik e a b i z a rr e o n e . H o w e v e r , t h e a b o v e d i s c u s s i o n s h o u l d b e i n t e r p r e t e d a s d e a l i n gw i t h p r e d i c t i o n s o f i n t e r s e c t o r a l t r a d e p a t t e r n s . I f o n e i n t e r p r e t s t h e i n t e r n a t i o n a le c o n o m i e s o f s c al e as b e i n g o b t a i n e d v i a i n t r a - i n d u s t r y t r a d e ( d e s p i t e t h e a b s e n c eo f a n e x p l i c i t m o d e l i n g o f i t s e x is t e n c e ), t h e p a r a d o x i s r e s o l v e d .

N o w c o n s id e r t h e u n iq u e n e s s p ro b l e m . I t h a s b e e n k n o w n f o r a l o n g t i m e t h a tw i t h n a t i o n a l l y i n c r e a s i n g r e t u r n s t o s c a l e th e r e e x i s t m u l t i p l e e q u i l i b r i a [ se eC h i p m a n ( 1 96 5) ]. I n t h e c a s e a n a l y z e d b y E t h i e r ( 19 7 9 ) t h i s d i f fi c u l t y d o e s n o te x i s t, b u t i t r e m a i n s a p r o b l e m i n o t h e r c a s e s , j u s t a s i t i s a p r o b l e m i n a c l o s e de c o n o m y . T o s e e t h i s p o i n t a s w e l l a s s o m e w e l f a r e i m p l i c a t i o n s , t a k e t h ef o l l o w i n g e x a m p l e . T h e r e a r e t w o g o o d s a n d a s i n g le fa c t o r o f p r o d u c t i o n , s a y ,l a b o r . O n e s e c to r p r o d u c e s w i t h s e c to r s p ec if ic i n t e r n a t i o n a l l y i n c r e a s in g r e t u r n st o s c a le w h i l e th e o t h e r p r o d u c e s w i t h c o n s t a n t r e t u r n s t o s ca le . I n p a r t i c u l a r :

f i L i ) = L i , i = 1 , 2 , L 1 + L 2 = L ,g l (X 1 ) - X ; , O < e < l , g 2 (~ 2 ) - 1 .

P r e f e r e n c e s a r e o f t h e C E S f o r m w i t h e q u a l w e i g h t s o n b o t h g o o d s a n d a ne l a s t i c i ty o f s u b s t i t u t i o n o > e - 1 > 1 . I f p s t a n d s f o r t h e r e l a t i v e p r ic e o f t h e f i rs tg o o d , t h i s i m p l i e s t h a t i t s b u d g e t s h a r e is [ 1 + p O - 1 ] - l , a n d t h e b u d g e t s h a r ed e c l in e s i n p . L e t g o o d t w o s e r v e a s n u m e r a i r e . T h e n i n a u t a r k y t h e w a g e r a tee q u a l s o n e a n d a c c o r d i n g t o ( 4 . 2 ) :

p = x f 5 . 1 )I n a d d i t i o n , a u t a r k y v e r s i o n s o f ( 4 . 3 ) - ( 4 . 4 ) i m p l y t h a t t h e b u d g e t s h a r e o f g o o do n e s h o u l d e q u a l p X t / L = X ~ - e / L , w h e r e t h e e q u a l i t y i s o b t a i n e d b y m e a n s o f(5 .1 ) . Hence ,

[ l + p - l ] - l = x ~ - T L , (5 .2 )


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3 4 0

Fig u re 5 .1 .

E HelpmanE

p = X 1 5 . 1 )

1 / l - E )L

5 . 2 )

l

A n e x a m p l e o f m u l t ip l e e q u i li b r ia a n d l o s s es f r o m t r a d e i n t h e p r e s e n c e o fi n t e r n a t i o n a l e c o n o m i e s o f s c a le .

where the left-hand side represents the first commodity s budget share. The curvesdescribed by (5.1) and (5.2) are drawn in Figure 5.1. For sufficiently large valuesof L the two curves have several points in common [observe that condition (5.1)does not depend on L], these represent autarky equilibria. Two equilibrium pointsof interest are points A and B. Assuming that L is large enough, the existence ofthese points can be seen as follows. Point A exists because the curve described by(5.1) is asymptotic to the horizontal axis while the curve described by (5.2) cutsthis axis at X = L 1/-~ . Given the existence of A, the existence of B can beverified by observing that for X1 sufficiently small the curve described by (5.1) liesabove the curve described by (5.2). 16 Hence, we have identified two autarkyequilibrium points. These can also be interpreted as equilibrium points of atrading world which consists of two identical countries, each one of size L / 2 , withX1 being world output of the first commodity.

Apart from demonstrating the possibility of multiple equilibria, this examplealso brings out another impor tant feature of models with internationally increas-ing returns to scale; under these circumstances an economy may lose from freetrade. To see this, consider two identical economies, each one in an equilibrium atpoint B in Figure 5.1 prior to trade.-Now suppose that these economies engage intrade. Trading equilibria can be described by moving out the curve that intersects

1 6 T h is r e l a t i o n s h i p h o l d s i f a n d o n l y i f e o > 1 . I n E t h i e r ( 1 9 7 9) o = 1 , i m p l y i n g e o < 1 . H e n c e , i nE t h i e r s c a s e p o i n t B d o e s n o t e x i st .


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C h 7 : In crea s in g Re tu rn s 341t h e h o r i z o n t a l a x i s . T h e n e w e q u i l i b r i u m c o r r e s p o n d i n g t o p o i n t B w i l l b e t o t h el e f t o f B o n t h e c u r v e d e s c r i b e d b y ( 5 .1 ) , s a y B ' . I f th i s is t h e r e s u l t i n g e q u i l i b r i u mp o i n t , b o t h c o u n t r i e s l o s e f r o m t r a d e , b e c a u s e t h e i r i n c o m e h a s n o t c h a n g e d a n dt h e y p a y h i g h e r p r i c e s f o r t h e fi rs t c o m m o d i t y . A s a r e s u lt o f t r a d e w o r l d o u t p u to f g o o d o n e i s l o w e r t h a n e a c h c o u n t r y ' s a u t a r k y o u t p u t l e ve l ( w h i c h m e a n s t h a tt h e s u f f i c i e n t c o n d i t i o n o f o u r g a i n s f r o m t r a d e t h e o r e m i s n o t s a t i s f i e d ) . T h e r ew i ll b e g a i n s f r o m t r a d e i f t h e n e w e q u i l i b ri u m i s a t A ' , w h i c h c o r r e s p o n d s t o A .

I h a v e d e m o n s t r a t e d t h e p o s s i b il i ty o f lo s s es f r o m t r a d e . T h e r e r e m a i n , h o w -e v e r , o p e n q u e s t i o n s w h i c h h a v e t o b e a n s w e r e d b e f o r e t h e r e l e v a n c e o f t h i sp o s s i b i li t y c a n b e e v a l u a t e d . T h e s e h a v e t o d o w i t h d y n a m i c a d j u s t m e n t p r o c e s s e sw h i c h s h o u l d h e l p d e t e r m i n e b o t h a u t a r k y a n d t r a d i n g e q u il ib r ia .

6 N a t i o n a l r e tu r n s t o s c a l e

T h e c a s e o f n a t i o n a l r e t u r n s t o s c al e h a s b e e n i n v e s t i g a te d m o r e t h a n o t h e rs . I tw a s d i s c u s s e d b y M a r s h a l l ( 1 8 79 ) i n r e l a t i o n t o t e r m s o f t r a d e e f fe c ts , 17 b yG r a h a m ( 1 92 3 ) i n r e l a t i o n t o g a in s f r o m t r a d e , b y O h l i n ( 19 3 3 ) in r e l a ti o n t ot r a d e p a t t e r n s , a n d b y L e r n e r ( 1 9 32 ) in r e l a t i o n t o ef f ic i e n t w o r l d p r o d u c t i o n .L a t e r w r i te r s h a v e e x p a n d e d o n s o m e o f t h e s e is su e s.

A s f a r a s t h e t r a d e p a t t e r n i s c o n c e r n e d , i t h a s b e e n a r g u e d b y O h l i n ( 1 9 3 3 , p .5 4 ) th a t e c o n o m i e s o f s c al e p e r s e c a u s e i n t e rn a t i o n a l t r a d e . T h i s s h o u l d b ei n t e rp r e t e d t o m e a n t h a t i n t h e a b s e n c e o f o th e r c a u s es o f t r a d e - s u c h a sd i f f e r e n c e s i n p r e f e r e n c e s , r e l a t i v e p r o d u c t i v i t y l e v e l s , a n d r e l a t i v e f a c t o r e n d o w -m e n t s - t h e r e w i ll b e f o r e i g n t r a d e d u e t o t h e e x i st e n c e o f e c o n o m i e s o f sc a le . T h i sw a s i n d e e d s h o w n t o b e t ru e b y M a t t h e w s ( 1 9 4 9 - 5 0 ) a n d l a t e r b y M e l v in( 1 96 9 ). 18 ( I w i l l c o m e b a c k t o t h i s p o i n t w h e n d i s c u s s i n g a s p e c if i c e x a m p l e . )A p a r t f r o m c a u s i n g i n t e r n a t i o n a l t r a d e , n a t io n a l l y i n c r e a s in g r e t u r n s t o s c a le a rec a p a b l e o f a s s i g n i n g t r a d e p a t t e r n s o n t h e b a s i s o f re l a t i v e c o u n t r y s iz e. T h i s i s an e w f e a tu r e . R e m e m b e r t h a t in t h e p r e s e n c e o f i n t e r n a t i o n a l l y v a r i a b l e r e t u r n s t os c a l e t h e r e c o u l d b e n o i n t e r s e c t o r a l t r a d e b e t w e e n i d e n t i c a l e c o n o m i e s a n dr e l at i v e c o u n t r y s i z e p l a y e d n o r o l e in t h e d e t e r m i n a t i o n o f t h e t r a d e p a t t e r n .

17,,... an inc rease in Germ any's dem and for Eng lish clo th may to an extent dev elop the facilitieswhich England has for produ cing cloth as to cause a grea t and perm anent fall in the value of cloth inEngland .. . an increase in G ermany's demand for English cloth may cause her to obtain an import ofEnglish cloth increased in a g rea ter ra t io th a n is her export of linen to England [M arshall (1879, p.13)]. Italics in original.lSMelvin (1969) concen trates his a nalysis on com plete specialization and misleads the read er tobelieve that severe restrictions on preferences are required to assure the existence of an equilibrium.W hen incom plete specialization is not excluded by assum ption (as it sho uld not be ), his restrictionsare not required (see his discussion on p p. 392-39 3 an d esp ecially footnote 8).


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342 E. HelpmanW i t h n a t i o n a l l y s e c t o r s p e ci fi c v a r i a b l e r e t u r n s t o s c a le t h e r e w a r d t o f a c t o r a l

v a l u e a d d e d i n s e c t o r i , p ig i X~J) , d e p e n d s o n t h e c o u n t r y ' s o u t p u t l ev e l i n t hi ss e c to r . A s a re s u lt , c o u n t r i e s w h i c h h a v e d i f f e r e n t f a c t o r e n d o w m e n t s h a v ed i f f e re n t f a c t o r p r i c e s [s e e ( 4. 5) ] e v e n if re l a t iv e f a c t o r e n d o w m e n t s a r e t h e s a m e ,t h e r e b y i n d u c i n g f a c t o r m o v e m e n t s . 19 B u t i f f a c t o r s d o n o t m o v e , t h e d i f f er e n c e si n f a c t o r r e w a r d s w i ll a f fe c t t h e t r a d i n g c o u n t r i e s ' c r o s s - i n d u s t r y o r d e r i n g o fr e l a t iv e c o s t s , a n d t h e r e f o r e a l s o t h e p a t t e r n o f s p e c i a l i za t i o n .

F o r c o n c r e t e n e ss , c o n s i d e r a w o r l d w h i c h c o n s i s ts o f t w o e c o n o m i e s . T h e r e a r et w o g o o d s ; g o o d o n e i s p r o d u c e d w i t h n a t i o n a l l y i n c r e a si n g r e t u r n s t o s c a le a n dg o o d t w o is p r o d u c e d w i t h c o n s t a n t r e t u r n s t o s ca le . T h e p r o d u c t i o n f u n c t i o n s a r et h e s a m e i n b o t h c o u n t r i e s , g l S l ) = X ~, 0 < e < l , a n d V 2 = ~ V 1, ~ > 1 [ t h ea s s u m p t i o n o f a c o n s t a n t e l a st i c it y g l ( ' ) f u n c t i o n i s n o t t r iv ia l] . T h e l a st a s s u m p -t i o n m e a n s t h a t t h e s e c o n d c o u n t r y i s l a r g e r t h a n t h e fi rs t, b u t t h a t i t h a s t h e s a m ec o m p o s i t i o n o f f a c to r s o f p r o d u c t i o n . A s s u m e a l s o t h a t b o t h c o u n t r i es h a v e t h es a m e h o m o t h e t i c p r e f e r e n c es . I n t h i s c a s e t h e r e e x i st s a n e q u i l i b r i u m i n w h i c h t h el a rg e r c o u n t r y ( c o u n t r y 2 ) e x p o r t s t h e c o m m o d i t y p r o d u c e d w i t h i n c re a s in gr e t u r n s t o s c a le [ se e M a r k u s e n a n d M e l v i n ( 19 8 1 )] .

T h e p r o o f p r o c e e d s a s fo l l o w s. F o r a f i x ed o u t p u t r a t io X 2 / X ~ , t h e M R T= - O X 2 / g X 1 ) o f th e l a rg e r c o u n t r y i s sm a l l e r t h a n t h e M R T o f th e s m a l l er

c o u n t r y , b e c a u s e t h e s ca l e e ff e ct m a k e s m a r g i n a l r e s o u r c e r e q u i r e m e n t s f o r t h ep r o d u c t i o n o f g o o d o n e l o w e r in t h e l a rg e r c o u n t r y . S i nc e u n d e r t h e c u r r e n ts p e c if i ca t io n o u t p u t i s d e t e r m i n e d b y p = M R T / ( 1 - e ) , w h e r e p = P l / P 2 [ s e eH e r b e r g a n d K e m p ( 19 69 )] , t h i s m e a n s t h a t w h e n t h e a u t a r k y e q u i l ib r i u m i su n i q u e , t h e h o m o t h e t i c i t y o f p r e f e r e n c e s a s s u re s t h a t i n a u t a r k y t h e r e l a t iv e p r ic ep i s s m a l l e r i n t h e l a r g e r c o u n t r y . 2 T h e l a s t p o i n t i s d e m o n s t r a t e d i n F i g u r e 6 . 1.P o i n t A r e p r e s e n t s t h e a u t a r k y e q u i l i b r i u m p o i n t o f th e s m a l l e r c o u n t r y . A t t h i s

19Melvin's (1969) discussion of factor mob ility is inaccurate, becau se he lets relativefactor rewardsdetermine factor m ovements. Ho we ver, with econom ies of scale, a higher relative return is notnecessarily associated with a higher absolute return.2The most technical step in the pro of consists of showing that fo r the sam e X 2 / X 1 ratio MRT issmaller in the larger country. Since p = M R T/ (1 - e), this can be proved by show ing that (4.4) impliesa lower p when V expands proportionately and 8 --- X 2 / X 1 is kept constant. In the current two sectoreconomy, with g l X1) = S ~ and g2(X2 ) --- 1, (4.4) can b e written as:X~ ~=G I q ,1 ;XV); 8XI=G 2 q ,1 ;hV ) ,

where q = pX~.If there exists a single factor of production G .) does no t exist, but then q does not change with 2,,implying that X1 increases and p declines when ~ rises. If there exist at least two factors of productionand the transformation curve of factoral value added is strictly concave, GI( ) increases in q and G 2(')decreases in q. Since GI( ) and G2(') are positively linear homogeneous in V, it is straightforward tocalculate from the first two equations that in this case an increase in 2, raises X1 and reduces q. Thisimplies by means o f the last equation tha t p declines, proving the required result. Ma rkusen andMelvin (1981) provide a com plicated proof for the two factor case.


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C h . 7 . I n c r e a s i n g R e t u r n sgood 2

343

Figure 6.1.g o o d

A comparison between autarky equilibria of two countries which differ only insize,

p o i n t t h e s l o p e o f t h e i n d i f f e r e n c e c u r v e is (1 - e ) - 1 t i m e s l a r g e r t h a n t h e s l o p e o ft h e t r a n s f o r m a t i o n c u r v e , i .e . M R S - - M R T / ( 1 - e ). 21 I f w e d r a w a r a y t h r o u g ht h e o r ig i n a n d p o i n t A , w e o b t a i n p o i n t B o n t h e l a r g e r c o u n t r y s t r a n s f o r m a t i o nc u r v e . A t B t h e s l o p e o f t h e i n d i f f e r e n c e c u r v e i s t h e s a m e a s a t A , b u t t h e s l o p e o ft h e t r a n s f o r m a t i o n c u r v e i s s m a l le r , i .e ., a t B w e h a v e M R S > M R T / ( 1 - e ).H o w e v e r , u n l e s s t h e r e i s a c o r n e r e q u i l i b r i u m i n t h e l a r g e r c o u n t r y t h e r e i s a p o i n to f ta n g e n c y b e t w e e n a n i n d i ff e r e n c e c u r v e a n d t h e t r a n s f o r m a t i o n c u r v e ( M R S =M R T ) t o th e r ig h t o f B , i m p l y i n g b y c o n t i n u i t y t h a t b e t w e e n t h a t p o i n t a n d Bt h e r e is a p o i n t , s a y A , a t w h i c h M R S = M R T / ( 1 - e ). P o i n t A i s t h e l a rg e rc o u n t r y s e q u i l i b r iu m p o i n t . S i n c e a t A t h e c o n s u m p t i o n o f g o o d 1 is r e l a ti v e l yh i g h e r t h a n a t A , M R S i s s m a l l e r a t A t h a n a t A . I f w e h a v e a c o r n e r s o l u ti o n ,M R S i n t h e l a r g e r c o u n t r y i s a l s o h i g h e r t h a n a t A . T h i s p r o v e s t h a t w h e n t h ee q u i l i b r iu m i s u n i q u e t h e a u t a r k y r e l a t iv e p r ic e o f g o o d o n e i s l o w e r i n t h e l a r g e rc o u n t r y .

T a k i n g a d v a n t a g e o f t h e f a c t t h a t i n a u t a r k y t h e r e l a t iv e p r ic e o f g o o d o n e i sl o w e r i n t h e l a r g e r c o u n t r y , d r a w e a c h c o u n t r y s o f fe r c u r v e t o f i n d o u t t h a t t h e r ee x i s ts a t l e a s t o n e i n t e r s e c t i o n p o i n t o f t h e o f f e r c u r v e s a t w h i c h t h e l a r g e r c o u n t r ye x p o r t s t h e c o m m o d i t y t h a t is p r o d u c e d u n d e r i n c r e a s in g r e t u r n s t o s c al e. O n e o f

2tThere are two opposing forces which shape the transformation curve. The factoral value addedtransformation curve makes the comm odity transformation curve concave to the origin while the scaleeffect in the first industry m akes it convex to the origin. W ith the em ployed g(- ) function, the firstfactor dominates near the ax is of the increasing returns to scale go od while the second factordominates near the other axis [see Herberg and Kemp (1969)]. Starting with Tinbergen (1945), itbecame a comm on mistake to draw this curve with reversed concavity-convexity properties.


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3 4 4 E . H e l p m a nt h e s e e q u i l i b r i u m p o i n t s i s a l s o s t a b l e a c c o r d i n g t o t h e K e m p ( 1 9 6 9 , c h . 8 )a d j u s t m e n t p r o ce s s, in w h i c h p r o d u c e r s b e a r t h e b u r d e n o f a d j u s tm e n t . C o n -s u m e r s a r e a l w a y s o n t h e ir d e m a n d c u rv e s, b u t p r o d u c e r s m a y b e o f f t h e ir s u p p l yc u r v e s . W h e n e v e r p r o d u c t i o n t a k e s p l a c e o f f a s u p p l y c u r v e , o u t p u t is a d j u s t e d t oa p p r o a c h t h e s u p p l y c u r v e [ se e M a r k u s e n a n d M e l v i n (1 9 81 )] . F o r t h e c a s e o f as i n g l e f a c t o r o f p r o d u c t i o n t h e s t a b i l i t y p r o p e r t y o f t h e a b o v e m e n t i o n e d e q u i -l i b ri u m p o i n t w a s a ls o c o n f ir m e d f o r th e a d j u s t m e n t m e c h a n i s m u n d e r w h i c ho u t p u t i s e x p a n d e d w h e n e v e r t h e d e m a n d p r i c e e x c e e d s t h e s u p p l y p r i c e a n do u t p u t i s c o n t r a c t e d w h e n e v e r t h e s u p p l y p r i c e e x c e e d s t h e d e m a n d p r i c e [ s e eE t h i e r ( 19 8 2a )] . A l t h o u g h t h e i m p l ic a t io n s o f a d j u s t m e n t m e c h a n i s m s h a v e b e e ne x p l o r e d f o r e c o n o m i e s w i t h n a t i o n a l l y in c r e a s i n g r e t u r n s t o s c a le m o r e t h a n f o re c o n o m i e s w i t h i n t e r n a t i o n a l l y in c r e a s i n g r e t u r n s t o s c al e, it s e e m s t h a t w e h a v en o t y e t r e a c h e d a s a t i s f a c t o r y s t a te o f k n o w l e d g e i n t h is a r e a . C o m p e t i n gm e c h a n i s m s ( s uc h a s K e m p s a n d E t h i e r s ) c a n n o t c l a i m t o o m u c h i n t e r m s o fw e l l b a s e d o p e r a t i n g a s s u m p t io n s , b u t i f w e d o n o t r e ly o n t h e m , w e h a v e n o w a yo f c o m p a r i n g t h e m a n y p o s s i b le e q u i li b ri a i n o r d e r t o m a k e a j u d g m e n t a b o u tw h i c h a r e t h e m o r e l i k e ly to o c c u r . M o r e w o r k i n t h is a r e a i s c e r ta i n l y r e q u i r e d .

W e h a v e s e e n t h a t t h e r e e x is t e q u i l i b r ia i n w h i c h t h e l a rg e r c o u n t r y e x p o r t s t h ec o m m o d i t y t h a t i s b e i n g p r o d u c e d w i t h i n c re a s in g r e t u r n s t o s c a le . H o w e v e r , e v e ni f b o t h c o u n t r i e s a r e o f t h e s a m e s i z e , t r a d e w i l l e x i s t . T o s e e t h i s p o s s i b i l i t y a sw e l l as s o m e w e l f a r e e f fe c ts , f u r t h e r s p e c i a l iz e t h e e x a m p l e a n d a s s u m e t h a t t h e r ei s a s in g l e f a c t o r o f p r o d u c t i o n , s a y l a b o r L , w i t h f~ (L ~ ) = L i . A l s o a s s u m e t h a te a c h c o u n t r y s p e n d s t he s a m e f ix e d b u d g e t s h a r e a o n g o o d o n e . S u p p o s eL 1 = L 2 = L . T h e n f o r a < ] / 2 t h e f o l l o w i n g i s a n e q u i l ib r i u m :

L 1 = 2 a L , L 2 = 0,L ~ = ( 1 - 2 a ) L , L 2 = L ,X 1 = 2 a L ) 1 / 1 - ~ ) , X 2 = O ,X 1 = ( 1 - 2 a ) L , X 2 = L ,W = 1 W 2 = 1

p = 2 a L ) - ~ / 1 - ~ ) .

H e r e c o u n t r y 2 s p e c ia l iz e s i n g o o d 2 w h i le c o u n t r y 1 is i n c o m p l e t e l y s p ec i a li z e d .C l e a r l y , th e r o l e s o f e a c h c o u n t r y c a n b e r e v e r s e d , w h i c h d e m o n s t r a t e s t h a t t h e r ea r e m u l t i p l e e q u i l ib r i a , a p o s s i b i l i t y t h a t w a s p o i n t e d o u t b y O h l i n ( 1 9 3 3 , p . 5 5 ).D u e t o t h e f a c t t h a t a < 1 / 2 , t h e d e m a n d f o r g o o d 1 is lo w e n o u g h t o e n a b l e as in g l e u n s p e c i a l i z e d c o u n t r y t o s u p p l y t h e m a r k e t f o r t h a t g o o d . 22

2 2 I f the sec ond industry was to produc e under decreasing returns to scale with g 2 X 2 ) = X ~ , lJ < O,then, as shown in Panagariya (1981), each country would have produced at least some of good 2.


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Ch 7: Increas ing Re turn s 3 4 5I n a u t a r k y L A j = a L , L A j = (1 - a ) L , X A j = (o /L ) x / (1 -e ) , X A j = (1 - a ) L , w A j

= 1 a n d p A j = a L ) - ~ / 1 -0 . T h i s i s a l s o a t r a d i n g e q u i l i b r i u m w i t h o u t a c t i v et r a d e , b u t i t is u n s t a b l e f o r s o m e a d j u s t m e n t m e c h a n i s m s [ se e E t h i e r ( 1 9 82 a )] .S i n c e p A j > p a n d t h e w a g e r a te e q u a l s o n e i n a u t a r k y a n d u n d e r f r ee tr a d e , e a c hc o u n t r y g a i n s f r o m t r a d e i n t h e a b o v e d e s c r ib e d e q u i l ib r i u m . I n a d d i t io n , i n t h ea c t iv e t ra d i n g e q u i l i b r iu m o u t p u t i s o n t h e w o r l d s t r a n s f o r m a t i o n c u r v e . O b s e r v e ,h o w e v e r , t h a t d e s p i t e t h e r e b e i n g g a i n s f r o m t r a d e t h e c o n d i t i o n s o f t h e g a i n sf r o m t r a d e t h e o r e m d u e t o K e m p a n d N e g i s h i (1 9 7 0) a r e n o t s a ti sf ie d . R e c a l l th a ta c c o r d i n g t o t h a t t h e o r e m a s u ff ic ie n t c o n d i t i o n f o r a c o u n t r y w i t h t h e e x a m p l e sd a t a t o g a in f r o m t r a d e i s th a t a s a r es u l t o f t r a d e i t e x p a n d s p r o d u c t i o n o f th eg o o d p r o d u c e d w i t h n a t i o n a l l y in c r e a s i n g r e t u r n s t o s c al e. I n o u r e x a m p l e b o t hc o u n t r ie s g a i n , b u t o n e o f t h e m r e d u c e s t o z er o i ts o u t p u t o f th e g o o d p r o d u c e dw i t h n a t i o n a l l y i n c r e a s i n g r e t u r n s t o s c a l e . T h i s i m p l i e s t h a t t h e i r s u f f i c i e n tc o n d i t i o n s f o r g a i n s f r o m t r a d e a r e q u i t e r e s t r i ct i v e . 23 I n f a c t , t h e g e n e r a ls u f fi c ie n t c o n d i t i o n f o r g a i n s f r o m t r a d e p r e s e n t e d i n S e c t i o n 4 i n t h e g a i n s f r o mt r a d e t h e o r e m i s a l s o n o t s a t is f ie d f o r th e c o u n t r y t h a t s p e c i a l i z e s i n t h e s e c o n dc o m m o d i t y , w h i c h m e a n s t h a t t h a t c o n d i t i o n i s a ls o q u i te r e s tr ic t iv e .

N o w t a k e t h e c a s e a > 1 / 2 . I n t h is c a s e t h e d e m a n d f o r g o o d o n e i s t o o l ar g ef o r a s in g le u n s p e c ia l i z e d c o u n t r y t o s u p p l y t h e d e m a n d e d q u a n t it ie s . I n t h ef o l lo w i n g a s y m m e t r i c a l e q u i l i b r iu m o n e c o u n t r y s p e c ia l iz e s i n g o o d o n e w h i l e th eo t h e r s p e c i a li z e s i n g o o d t w o :

L x = L , = O ,L =0 L

= L 1 1- X? = 0x l = 0 - - L ,W 1 = O r / 1 - 0 ~ ) , W2 = 1 ,

a L - ~ / ( 1 - e ) .P = l - aI n t h is e q u i l i b r i u m t h e f ir st c o u n t r y s r e a l w a g e r a t e i s h i g h e r t h a n i n a u t a r k y i nt e r m s o f e v e r y g o o d [ w 1 = a / 1 - a ) > w A1 = 1 f o r a > 1 / 2 a n d w l / p = L ~ / 0 - 0 >w A 1 / p A1 = a L ) ~ / O - O ] , w h i l e t h e s e c o n d c o u n t r y s r e a l w a g e r a t e i s a s h i g h a s i na u t a r k y i n t e r m s o f g o o d t w o b u t m a y b e h i g h e r o r l o w e r t h a n i n a u t a r k y i n t e r m so f g o o d o n e [ w 2 = 1 = w A2 a n d w 2 / p = a - l ( 1 - a ) L ~ / 1 -~ ), w A 2 / p A 2 =

23Eaton and Panagariya (1979) prov ide weaker conditions. They require exp ansion of output inindustries with a degree of economies of scale which exceeds some reference degree and contraction ofoutput in ind ustries with a lower degree of econom ies of sca le. Ho wev er, their sutticient conditions arealso not satisfied in ou r exam ple.


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346 E. Helpmana ~ / ~ 1 0 L ~ A I - ) ] . F o r a s u f f i c ie n t l y c l o s e to o n e t h e c o u n t r y t h a t s p e c ia l iz e s i ng o o d t w o h a s a w a g e ra t e w h i c h i s l o w e r i n t e r m s o f g o o d o n e t h a n i n a u t a r k y a n di t l o s e s f r o m t r a d e .T h i s e x a m p l e s h o w s t h a t i d e n t i c al co u n t r i e s m a y e n g a g e i n t r a d e , it s h o w s t h a tb o t h c o u n t ri e s m a y g a i n f r o m t r a d e o r o n e o f t h e m m a y lo s e, a n d i t sh o w s t h a tg a i n s fr o m t r a d e d e p e n d o n t h e s t re n g t h o f d e m a n d f o r th e g o o d t h a t i s p r o d u c e du n d e r n a t i o n a l l y i n c r e as i n g r e t u r n s t o s c a le . It c o n f i r m s G r a h a m s a r g u m e n t t h a ta c o u n t r y m a y l o s e f r o m t r a d e [s ee G r a h a m ( 19 23 )] . V i n e r ( 19 3 7 ) a t t r i b u t e s as i m i l a r a r g u m e n t t o N i c h o l s o n ( 1 89 7 ). 24 I t a l s o s h o w s t h a t t h e r e n e e d n o t b ef a c t o r p r i c e e q u a l i z a t i o n e v e n w h e n t h e t r a d i n g p a r t n e r s a r e i d e n t i c a l . I n t h ee x a m p l e t h e r e i s f a c t o r p r i c e e q u a l i z a t i o n i n a u t a r k y w h i l e t r a d e b r i n g s a b o u tu n e q u a l f a c t o r p r i ce s [ O h l i n (1 9 3 3, p . 1 0 6 ) p o i n t s o u t t h e u n e q u a l i z i n g e f f e c t o ft r a d e o n f a c t o r p r i c e s i n t h e p r e s e n c e o f e c o n o m i c s o f s c a l e ] . T h i s s h o w s a g a i nt h a t w i t h e c o n o m i e s o f s c a le n e i th e r c o m m o d i t y p r i ce s n o r f a c t o r r ew a r d s s e rv e a sr e l ia b l e p r e d ic t o r s o f t h e p a t t e r n o f t r a d e . W h e n t h e c o u n t r i e s a r e o f u n e q u a l s iz et h e r e i s a n e q u i l i b r i u m i n w h i c h t h e l a r g e r c o u n t r y e x p o r t s t h e g o o d p r o d u c e dw i t h e c o n o m i e s o f s c a le .

A p a r t f r o m t h e is s u es d i s c u s s ed s o f a r, c o m p a r a t i v e s t a t ic s r es p o n s e s o f t h ep r o d u c t i o n s t r u c t u r e o f e c o n o m i e s w i t h n a t i o n a l l y i n c r e as i n g r e t u r n s t o s c a l e h a v eb e e n i n v e s t i g a te d . I n t h is c o n t e x t f a c t o r a l v a l u e a d d e d i s a s s u m e d t o b e p r o d u c e db y m e a n s o f H e c k s c h e r - O h l i n - t y p e t e c h n o l o g i es r e s u l ti n g i n t h e f o l l o w i n g e q u i-l i b r i u m c o n d i t i o n s in p r o d u c t i o n ( t h e c o u n t r y s p e ci fi c s u p e r s c ri p t j i s d r o p p e d a tth i s s t age ) :

? , = c , ( w ) / g i ( X i ) , fo r a l l i , (6 .1 )[ a l i ( w ) / g i ( g i ) ] X i = V t , fo r a l l l , (6 .2 )i = 1

w h e r e i t i s a s s u m e d t h a t t h e r e t u r n s t o s c a l e a r e s e c t o r s p e c if ic .25 Eq . (6 .1) i s ju st ar e p r o d u c t i o n o f t h e p r i c i n g e q u a t i o n ( 4 . 2 ) . E q . ( 6 . 2 ) r e p r e s e n t s e q u i l i b r i u m i nf a c t o r m a r k e t s . T h e c o e f f i c i e n t a t i ( w ) i s th e d e m a n d f o r f a c t o r I p e r - u n i t f a c t o r a lv a l u e a d d e d i n s e c t o r i [ = O c i ( w ) / O w t ] . D i v i d e d b y g i ( X i ) t h i s c o e f f i c i e n tr e p r e s e n t s d e m a n d f o r f a c t o r l p e r - u n i t o u t p u t i n s e c to r i. S e e n in t h i s f o r m , i t i sc l e a r t h a t c h a n g e s i n o u t p u t l e v el s a f f ec t b o t h p r o d u c t i v i t y a n d c o s ts . T h e r e f o r ew e s h o u l d n o t e x p e c t t h e s t a n d a r d r e s u l t s - s u c h a s t h e S t o l p e r - S a m u e l s o n a n dt h e R y b c z y n s k i t h e o r e m s - t o r e m a i n i n t a c t i n t h e p r e s en t f r a m e w o r k .

24For an analysisof gains from trade with decreasing returns to scale in one in dustry, wh ich wasGrah am s original assumption, see Pan agariya 1981) and Ethier (1982a):25See Jones (1968), K em p (1969), Panagariya (1980), and Ino ue (1981). Jones (1968) does notassume homo theticity.Chan g (1981) and Herberg, Kem p and Taw ada (1982) discussalso intersectoralspillovers.


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Ch. 7 : Inc reas ing Re turns 3 4 7T a k e , f o r e x a m p l e , t h e S t o l p e r - S a m u e l s o n t h e o r e m i n t h e t w o - c o m m o d i t y

t w o - f a c t o r c as e . I t s t a t e s t h a t i n a n i n c o m p l e t e l y s p e c ia l iz e d e c o n o m y a n i n c re a s ei n t h e r e l a t i v e p r i c e o f a g o o d i n c r e a s e s t h e r e a l r e w a r d o f t h e f a c t o r t h a t i s b e i n gu s e d r e l a ti v e ly i n t e n s i v e l y i n t h e p r o d u c t i o n o f t h is g o o d . T h e s t a n d a r d p r o o fr e li es o n t h e f a c t t h a t t h e r e l a t i o n s h i p b e t w e e n c o m m o d i t y p r ic e s a n d f a c t o rr e w a r d s i s i n d e p e n d e n t o f o u t p u t l ev e ls [ th i s i s i n d e e d t h e c as e w h e n ( X / ) = 1 ;s e e (6 .1 )]. H o w e v e r , i n t h e p r e s e n t c o n t e x t c h a n g e s i n o u t p u t l e v e ls a f fe c t b o t ha b s o l u t e a n d r e l a t i v e c o s t s [ se e (6 .1 )] , w h i c h m e a n s t h a t t h e e f f e c t o f a r e l a t iv ep r i c e i n c r e a s e o n r e a l f a c t o r r e w a r d s d e p e n d s o n t h e f e e d b a c k o f o u t p u t c h a n g e s( t h a t a r e c a u s e d b y t h e r e l a t i v e p r i c e c h a n g e ) o n s e c t o r a l c o s t s tr u c t u r e s . B ylog a r i th m ic d i f fe re n t i a t ion o f (6 .1 ) an d (6.2) g iven i = 1 , 2 , l = 1 , 2 , c a l l ing th e f i r s tf a c t o r o f p r o d u c t i o n l a b o r ( w i t h w = w ) , a n d t h e s e c o n d f a c t o r o f p r o d u c t i o nc a p i t a l ( w i t h w2 = r ) , w e o b t a i n :

0L 2 OK ^ , (6 .3)[ 1 - -e X )~ k L i 1 - - E 2 ))K L 2 ] 1 ) L ) ~ L 11 - e l ) ~ t K l 1 - -e 2 ) )k K 2 X 2 = \* x I q - 1 ~ - /~ ) - - ~ K ] (6 .4 )

w he re 0 jr i s t he co s t s ha re o f f a c to r j i n in du s t r y i , Xj~ i s t he a l loca t ive sha r e o ff a c t o r j i n i n d u s t r y i , 8L = ~ k L l O g l 6 1 -1 - ~ k L 2 O K 2 O 2 ~ K = ~ k K I O L I 6 I q - ~ k K 2 0 L 2 0 2 O i =e l a s t i c i ty o f s u b s t i t u t i o n b e t w e e n l a b o r a n d c a p i t a l i n s e c t o r i , e ; i s t h e e l a s t i c i ty o f

( X , ) , a n d h a t s i n d i c a t e p r o p o r t i o n a l r a t e s o f ch a n g e . A s s u m i n g t h a t s e c t o r 1 isr e l at iv e l y l a b o r i n t e n s i v e , th e d e t e r m i n a n t o f t h e m a t r i x o n t h e l e f t - h a n d s i d e o f(6 .3) , 0 , i s posi t ive , i .e . 101 > 0 a n d s o i s t h e d e t e r m i n a n t o f t h e m a t r i x o n t h el e f t- h a n d s id e o f ( 6.4 ), w h i c h e q u a ls ( I - e l ) ( 1 - e 2 ) l X l , w h e r e IX l = X L l X x = -X K I ~ K L 2 .U s i n g ( 6.3 ) a n d ( 6.4 ), w e c a n c a l c u l a te t h e li n k b e t w e e n c o m m o d i t y p r ic e s a n df a c t o r r e w a r d s w h i c h t a k e s i n t o a c c o u n t b o t h t h e d i r e c t a n d i n d i r e c t ( t h r o u g ho u t p u t c h a n g e s ) e f f e c t s . T h i s c a l c u l a t i o n y i e l d s :

w h e r e(6 .5 )

o = O K , + Ie (X K j L + X L j K )~ = I ) , l ( 1 - e , ) i = 1 , 2 ; j = l , 2 ; j i .


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48 E HelpmanA typical element of the matrix 0 represents the effect of a factor price increaseon a v e r a g e costs of production, given constant factor endowments [see Jones(1968)]. It takes in

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Increasing Returns, Imperfect Markets, And Trade Theory

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