Two-fator trade model with monopolisti ompetition S.Kihko, S.Kokovin, E.Zhelobodko, August 17, 2012 NRU HSE, NSU, Sobolev Institute of mathematis, Russia (To be ompleted) Abstrat We develop a two-fator trade model with monopolisti ompetition, a variable elastiity of substitution and trade ost. We show that the ountry endowed with more apital is a net exporter of manufatured good, its apital prie is bigger and rm's output is smaller. The relative added value in manufaturing is smaller in the apital-endowed ountry. Total trade value inreases with trade liberalization. Last, dumping or reverse dumping may aim depending on onsumers' altitude towards variety love and ountries relative apital endowment. 1 Introdution Classi questions in trade theory inlude number of rms in dierent ountries and their size, speialization of ountries, total trade value and et. How dereasing of trade bariers and globalization aet them? It's one of the main interests in many studies (Helpman and Krugman, 1987; Brakman and Heijdra, 2004). Another topi is dumping (Brander and Krugman, 1983; Greenhut et al., 1987). What leads rms to use prie disrimination? New trade theories typially use CES preferenes whih yield a well-dened set of preditions regarding rms' priing poliies aross ountries. However, most of them are not onsistent with empirial evidene. First, mark-ups vary with market size (Syverson, 2007). Seond, rms prie disriminate aross destinations, using either dumping or reverse dumping (Martin, 2009; Manova and Zhang, 2009; Bernard et al., 2007). Last, rms set up in ountries endowed with a large amount of human or physial apital harge pries higher than rms loated in poorly endowed ountries (Shott, 2004; Hummels and Klenow, 2005; Hallak, 2006; Hallak and Shott, 2008). The aim of this paper is to obviate some of these drawbaks by studying trade ows and rms' priing 1
in a monopolisti ompetition setting with non-homotheti preferenes. To be preise, instead of using the CES or any other spei preferenes, we use general additive preferenes. To ahieve our goal, we onsider a trade model with two asymmetri ountries in whih prodution fators are owned by dierent agents, i.e. workers and apital-owners. Unlike the model developed by Krugman and Helpman (1987) and others who assume that the relative endowment of labor and apital is the same in both ountries, we follow Heksher and Ohlin and onsider a setting in whih the shares of labor and apital dier between ountries. Our model may thus be viewed as a ombination of new and standard trade theories. By ombining the two main types of asymmetries used in the literature with non-homotheti preferenes, we are able to derive a new set of preditions that agree with the above-mentioned empirial literature. Our main results may be summarized as follows. Elastiity of inverse demand (or, the same, inverse elastiity of substitution of varietes) an inrease/derease with individual onsumption and all markets an be lassied respetively as imitating prie dereasing/inreasing ompetition. Under inreasing (dereasing) elastiity of inverse demand the produt prie dereases (inreases) with the number of rms. 1 It ould take plae when total endowment of apital or share of apital in the ountry inrease. In both ases the number of rms remains proportional to gross apital (as in traditional models) and individual onsumption of eah variety dereases. However, this dereasing is outweighed by inreasing number of varieties and onsumer's utility inreases. Another topi is the eet of the trade liberalization whih means dereasing trade ost. Lowing trade ost makes prie of any imported variety inreasing in ase of prie inreasing ompetition (the hange being ambiguous in the opposite ase), whereas prie of any domesti variety dereases (inreases) under prie dereasing (inreasing) ompetition. Total trade value inreases with trade liberalization. We show that rms' priing behavior may exhibit (reiproal or unilateral) dumping or (reiproal or unilateral) reverse dumping, depending on the nature of the utility and the relative sizes of the trading partners. Under small asymmetry of ountries in apital endowment both ountries pratie dumping or reverse-dumping depend on prie dereasing or inreasing ompetition. In ase big asymmetry prie dereasing (inreasing) ompetition yields dumping (reverse-dumping) used by smaller ountry and reverse-dumping (dumping) used by bigger ountry. Trade equlibrium displays that apital prie is less in ountry with relative advantage in apital in ontrast to Puger (2004) and ountry with bigger endowment of apital is net exporter of manufatured varieties. These results are the similar to Helpman and Krugman (1987). One of trade pattern of a model with CES-preferenes (Helpman and Krugman, 1987) presents relative GDP 1 Anti-ompetitive eet is not unnatural. It is known in IO that shrinking demand dereases the output, but the impat on prie is ambiguous, depending upon the elastiity of the demand. more ompetitors. A monopolisti ompetitor experiene the same eets when faing 2
of riher in apital ountry less than one despite to bigger output of manufaturing setor. Should this non-trivial pattern remain true under our generalized assumption? We show that value added per rm and rm's size are less in ountry with advantage in apital. Finally, we apply our setting to trade theory and unover several new properties hindered by the CES, suh as dumping and reverse dumping. We also derive general properties of the trade pattern. So, the relative value added in manufaturing is smaller in the bigger endowment of apital ountry as well as rm's size in terms of value of output. 2 Closed eonomy Before analyzing our main objet of interest - trade, we are explaining our main onepts and main impats of labor and apital on equilibria on the simpler objet - losed eonomy. Our goal is to study eets of market size and ownership struture on equilibrium without inuene of trade ost. One big ountry an be interpret as integrated equilibrium in Helpman meaning. 2.1 Model To simplify aggregating demands of apital owners and workers, we assume two setors alled (traditionally) manufaturing and agriulture, the latter being used as numerarie. Manufaturing inludes one dierentiated good and agriulture inludes one homogeneous good. The two prodution fators are alled labor and apital though there an be alternative interpretations: skilled and unskilled labor, et. Consumer side inludes workers and apital owners. Among a big number L of idential workers, eah oers one unit of labor inelastially and similarly among a big number K of apital owners, eah oers one unit of apital. The dierentiated good is represented by ontinuum of varieties indexed by i [0, N], where N is the mass of varieties. We interpret L + K as exogenous ountry size (population size) and N as an endogenous industry size. Innite-dimensional onsumption vetor is x = (x i ) i [0, N] where x i is the individual onsumption of variety i. We follow Ottaviano et al. (2002) and assume quasi-linear preferenes of onsumers (same for workers and apital owners). Absene inome eet is drawbak of using quasi-linear preferenes but it neessary to avoid dierenes in onsumers' expenditure. Utility maximization has the form: ˆ N ˆ N U = V ( u(x i )di) + A max, s.t. p i x i di + Ap a E 0 x,a 0 Here p = (p i ) i [0, N] is the prie vetor, p a is the prie of agriulture good, E is inome (dened below dierently for workers and apital owners), u( ) is an elementary utility funtion, V ( ) is an upper-layer utility 3
funtion. Both utility funtions are thrie ontinuously dierentiable, stritly inreasing (at least at some zone of equilibria [0, ˇx)) and stritly onave with u(0) = 0, V (0) = 0. Unlike Dixit and Stiglitz (1977) and Behrens and Murata (2007) we do not assume spei form of funtion u( ). Critial role in the analysis of the equilibrium variables is an Arrow-Pratt measure of onavity is r u (z) = u (z)z u (z). On the other hand, r u is the elastiity of the inverse-demand funtion for variety i. Also r u (z) an be treated as relative love for variety (further, RLV). Note that for CES-funtion (u(z) = z ρ ) r u (z) is a onstant - r u (z) = 1 ρ but for other funtions it may inrease or derease. The rst-order ondition for the onsumer's problem imply the inverse demand funtion for variety i: ˆ N p(x i, λ) = V ( u(x j )dj)u (x i ) = u (x i )/λ. 0 1 where λ = V ( N > 0 denotes an analogue of the Lagrange multiplier of the budget onstraint for 0 u(x i)di) sub-optimization problem with manufaturing only (unlike real budget multiplier equal to 1). It is interpreted as the marginal utility of expenditure for manufaturing or the intensity of ompetition in manufaturing. On prodution side, agriulture setor produe homogeneous good with marginal ost of one unit of labor, perfet ompetition and onstant return to sale, so prie p a 1. Manufaturing setor presents idential rms. Eah rm pereives urrent λ and funtion p(.,.) as given and faes xed ost of one unit of apital and marginal ost amounting to units of labor. Its prodution ost is C(y) = π +wy, where π is the prie of apital (interest rate) and y is output. Labor is intersetorally mobile, that lead us to same wages in both setors, normalized without loss of generality to w = 1. Then total prodution ost of output y beomes C(y) = π + y. Then inome of worker (wage) is E = 1. Eah apital owner spends E = from apital and K denotes total supply of apital. K 0 π idi K whih is individual inome Eah rm produes one unique variety (eah is produed by single rm). The number of rms N is big enough to ignore impat of eah rm on the market and any i-th produer's program is: (L + K)p(x i, λ)x i (L + K)x i π i max x i. Sine all produers are idential (we assume anave prot funtion and week border onditions), there is unique 4
symmetri solution of produer's program. We drop index i and, using the rst order ondition, haraterize the (symmetri) prot-maximizing prie as p = 1 r u (x) (1) and mark-up as M = p p = r u (x) (0, 1). (2) Seond order ondition for produer's program an be derived as r u (x) < 2, where r u (x) = xu (x) u (x) denotes onavity of u (see Apendix A). For symmetri outome, apital owners inome beomes E = π, and sine eah rm uses one unit of apital, the number of rms at equilibrium should beome N = K. (3) Equilibrium. The (symmetri long-run) equilibrium is the bundle (p, x, λ, π ) satisfying onsumers' and produers' maximization onditions and zero-prot ondition in manufaturing setor (L + K)p x (L + K)x π = 0. Labor market learing for manufaturing is unneeded beause agriultural setor uses the same labor, whereas total labor balane follows from the budget onstraint. We an rearrange the main equilibrium equations in the onvenient form dropping index : V (Ku(x))u (x) = 1 r u (x) (4) π = (L + K)(p )x. (5) Proposition 1. To solve for equilibrium, one an nd onsumption x from (4), then prie p from (1), then apital prie (interest) π from (5), number of rms always being onstant as (3). Summarize, equilibrium is determined as system solution inluded (1), (3), (4), (5). 5
2.2 Impat of market size: apital and labor supply From now on we study only equilibrium values of variables (p, x, λ, π). This is done through total dierentiation of our equilibrium equations, allowed by the impliit funtion theorem (the funtions involved are twie dierentiable). Consumption. We use main equation (4) for inverse demand and dierentiate it w.r.t. K to get x K = V (Ku(x))u(x)u (x) (1 r u (x)) KV (Ku(x)) (u (x)) 2 (1 r u (x)) + V (Ku(x))u (x)(2 r u (x)) < 0, sine V > 0, V < 0, u > 0, u < 0, r u < 1, r u < 2. Thus, individual onsumption of eah variety dereases with the supply of apital K, being independent of labor supply L (see (4)). One an understand that independene of labor suply is the artifat of the assumption of onstant marginal ost. Dereasing individual onsumption w.r.t. mass of varieties is onsistent with the assumption that agents onsume all available varieties. Individual onsumption of eah variety dereases w.r.t. mass of varieties beause eah onsumer onsumes bigger mass of varieties distributing her inome between larger mass of varieties. Prie. Now using behavior of individual onsumption w.r.t. apital endowment we study rms' priing poliy. Thus, we analyze the behavior of equilibrium prie (1) as a funtion of K: p K = r u(x) x (1 r u (x)) 2 K We observe that behavior of equilibrium prie as funtion of K depends on r u. When r u (x) is an inreasing funtion - under inreasing RLV - the equilibrium prie dereases with apital K, thereby under growing ompetition (N = K) market mimis pro-ompetitive eet. When r u (x) is dereasing funtion - under dereasing RLV - the equilibrium prie inreases with apital K that orresponds to anti-ompetitive eet. Further, using (2) we note that: (i) behavior of equilibrium mark-up is the same as the pries; (ii) both prie and mark-up are independent on number of workers L. As before (ii) is the artifat of the assumption of onstant marginal ost. Eonomi intuition suggests that prie dereases with number of ompetitors. But as desribe above inreasing/dereasing prie w.r.t. number of ompetitors depends on the elastiity of the inverse demand. So, inreasing equilibrium prie with number of ompetitiors is native if inverse demand urve is too onvex. Interest rate. Now we study the behavior of apital prie (5) π. Dierentiating (in K) equation π = (L + K) xr u(x) 1 r u (x) 6
and rearranging we get ( π K = Kr u(x) 1 r u (x) x K x K x K + ( L K + 1)(2 r u (x)) ). 1 r u (x) The sign here is not lear but one an note that it depends on omparison between (the inverse of) elastiity of equilibrium onsumption and labor/apital ratio multiplied by some demand harateristi. Intuition suggests that the sign should be negative, the prie of apital should derease with its supply, whih is true when gross parentheses term is positive. However, formally it is not so lear. As soon as x is a funtion only of K, not L, we see from the above that for any given u, K one an found suiently big labor endowment of eonomy that the parentheses term is positive and our intuitive onlusion holds true. I.e., under suiently big population interest dereases with apital supply, that makes sense. Is the same true for smaller population or not is not lear so far, we do not have ounter-intuitive examples. Now we study impat of labor supply on apital prie by dierentiating equation (5) w.r.t. L π L = xr u(x) 1 r u (x) > 0 So, apital prie unambiguously inreases with the number of workers. Individual onsumption doesn't depend on number of workers. It means that rm size (equilibrium rm output) inreases with number of workers. Consequently, apital prie inreases with labor suply too. Welfare. Equilibrium utility of eah worker does not depend on labor supply, beause onsumption of manufaturing good x does not hange and inome does not hange. By ontrast, utility of eah apital owner inreases with labor supply, beause onsumption of x does not hange, inome π inreases and additional inome an be spent on agriulture. When r u > 0, equilibrium utility of eah worker as funtion of apital is lear: variety inreases, prie dereases and bigger range of variety beomes available for smaller pries, so, under inreasing RLV, worker's utility inreases in apital supply (by the revealed-preferene argument). Similar question for apital owners is more involved. In ase pro-ompetitve eet and inreasing apital prie with apital supply utility of apital owners' inrease. But behavior of apital owners' utility ould be ontraditory in more general ase: U(K) = V (Ku(x(K))) + A(K) = V (Ku(x(K))) + π(k) Kp(x(K))x(K) By the envelope theorem, we an ignore indiret eets (x(k)) on the optimal utility, so, dierentiating 7
utility w.r.t. K we get U K = V ( )u( ) + π p px Kx K x x K, with unlear answer. Intuition suggests that inreasing as well as dereasing apitalists' utility is native behavior. As we show above equilibrium apital prie as well as produt prie behave dier w.r.t. apital supply. So, market an reets dierent eets, in ase pro-ompetitive eet and dereasing apital prie apitalists' utility an dereases or inreases. The same take plae in ase anti-ompetitive eet and inreasing apital prie. 3 International trade In previous setion we study losed eonomy model that help us understand main interest of our researh - trade patterns in following model. We assume that the world eonomy has similar preferenes and tehnologies but inludes two ountries named Home and Foreign. Agriultural good requires zero trade ost. By ontrast, τ > 1 is the ieberg-type trade ost for manufature good. There is a big number L = s a L + (1 s a )L of idential workers and K = sk + (1 s)k of idential apital owners. Here s a and (1 s a ) denote the shares of workers in Home and Foreign ountries respetively. Similarly, s and (1 s) are the shares of apital owners in these ountries. We assume that Home ountry has bigger supply of apital, i.e. s > 1 2. Let x ij be the individual onsumption of eah variety made in ountry i and onsumed in ountry j, p ij is the prie of x ij. Preferenes of onsumers are the same as before and inlude onsumption of goods produed in Home and Foreign ountries. Consumer's programs for Home and Foreign ountry are: [ ˆ NH ˆ NH max V ( u(x HH +N F i )di + X,A 0 N H ] ˆ NH u(x F i H )di) + A, s.t. 0 ˆ NH p HH i x HH +N F i di + N H p F H i x F i H di + A E [ ˆ NH ˆ NH max V ( u(x HF +N F i )di + X,A 0 N H ] ˆ NH u(x F i F )di) + A, s.t. 0 ˆ NH p HF i x HF +N F i di + N H p F F i x F i F di + A E Here N H and N F denote number of rms in Home and Foreign ountry, and of varieties in Foreign are numbered starting from N H. As before, for workers E = 1 and apital owners' inomes in H and F are interests E = π H, E = π F, respetively. We use, as in previous setion, the marginal utility of expenditure for manufaturing goods, but now these 8
magnitudes an dier in H and F, being denoted as λ H, λ F, respetively. The rst-order ondition for the onsumer's optimization imply the inverse demand funtions of any variety k: p(x HH k k ) λ H, λ H ) = u (x HH, p(x F H k ) = u (x F H k ) λ H λ H = V ( N H 0 u(x HH k 1 )di + N H +N F N H u(x F H k )di), p(x F F k k ) λ F, λ H ) = u (x F F, p(x HF k ) = u (x HF k ) λ F, λ F = V ( N H 0 u(x HF i 1 )di + N H +N F N H u(x F F i )di). Prodution side is the same as in previous setion. Sine rms are idential, we symmetrized outputs are q H i = q H, q F i = q F. Total demand (output) q H of Home rm and output q F of Foreign rm are: q H (x) (sk + s a L)x HH + ((1 s)k + (1 s a )L)x HF, q F ((1 s)k + (1 s a )L)x F F + (sk + s a L)x F H. Produers of Home and Foreign rms pereive λ H, λ F as given and maximize prots as (p HH (x HH, λ H ) )(sk + s a L)x HH + (p HF (x HF, λ F ) τ)((1 s)k + (1 s a )L)x HF π H max x HH, x HF, (p F F (x F F, λ F ) )((1 s)k + (1 s a )L)x F F + (p F H (x F H, λ H ) τ)(sk + s a L)x F H π F max x F F, x F H, where π H and π F are apital pries in Home and Foreign ountry. Using the rst order ondition we haraterize the (symmetri) prot-maximizing pries as p HH = 1 r u (x HH ), pf H = τ 1 r u (x F H ) 9
p F F = 1 r u (x F F ), phf = τ 1 r u (x HF ), Under seond order ondition for produer's problem we get the same as in the previos setion ondition for prot onavity: r u (x) < 2. Capital (as well as labor) is immobile among ountries, so, as previously, the masses of rms are predetermined by its apital balane: N H = sk, N F = (1 s)k. Equilibrium. Trade equilibrium is the bundle (x HH, x F H, x HF, x F F, p HH, p F H, p HF, p F F, N H, N F, π H, π F, λ H, λ satisfying onsumers' and produers' maximization onditions, apital balane and zero-prot ondition (p HH (x HH ) )(sk + s a L)x HH + (p HF (x HF ) τ)((1 s)k + (1 s a )L)x HF = π H, (6) (p F F (x F F ) )((1 s)k + (1 s a )L)x F F + (p F H (x F H ) τ)(sk + s a L)x F H = π F. (7) This denition does not present onsumption of agriultural good and related trade expliitly, but has in mind that it satises the balane of payments. We an rearrange the equilibrium equations in terms of onsumptions only as follows (see Appendix B for details). Lemma 1. The equilibrium onsumption bundle (x HH, x F H, x HF, x F F ) is the solution to the system u (x HH ) u (x F H ) = 1 τ 1 r u(x F H ) 1 r u (x HH ) V [ sku(x HH ) + (1 s) Ku(x F H ) ] u (x HH ) = 1 r u (x HH ) u (x F F ) u (x HF ) = 1 τ 1 r u(x HF ) 1 r u (x F F ) V [ sku(x HF ) + (1 s) Ku(x F F ) ] u (x F F ) = 1 r u (x F F ) This system ontains two independent systems of two equations eah. First system (rst and seond equation) determines the individual onsumption for home ountry (x HH, x F H ). Seond system determines (x HF, x F F ). 10
Analyzing this system bring us to the following onlusions on omparisons among onsumptions of Home and Foreign varieties. Proposition 2. (i) there is not more than one solution (x HH, x F H, x HF, x F F ) of the equilibrium system. (ii) individual onsumption of any domestially produed variety is higher than the onsumption of any imported variety, i.e. (x HH > x F H, x F F > x HF ). (iii) onsumption of a domesti variety is smaller in the ountry with higher endowment of apital (x F F > x HH ). (iv) There exists suh ritial value ŝ (0.5, 1] of apital share s of Home, suh that orderings of individual onsumptions satisfy: x F F > x HF > x HH > x F H when s > ŝ (very asymmetri ountries), x F F > x HH > x HF > x F H when s < ŝ (similar ountries). Proof see in Appendix B. Commenting, we explain that one an nd some value s as the root (proven to be unique) of equation x HF ( s) = x F H ( s) and s an be lower or higher than 1 but anyway ŝ = min{1, s}. 3.1 Impat of fators supply and osts Further, we study omparative statis of the equilibrium onsumptions, pries, et. by totally dierentiating w.r.t. exogenous parameters K, s, s a,, τ (or diretly observing monotoniity of funtions). First, we nd what happens to individual onsumptions (sizes of eah purhase). Proposition 3. The equilibrium onsumptions respond to exogenous shoks as follows: (i) individual onsumption x ii of any domesti variety inreases whereas onsumption x ij of any imported variety dereases w.r.t. transport ost τ. (ii) both individual onsumptions x ii,x ji in ountry i derease: (1) w.r.t. its share (s for Home and (1 s) for Foreign) of world apital, (2) w.r.t. total endowment of world apital K, (3) w.r.t. marginal ost. (iii) all individual onsumptions x HH,x HF,x F F,x F H are independent from the world endowment of labor L, and from ountries' shares of workers s a, (1 s a ). Proof of proposition 3 see in Appendix B. Now we analyze the behavior of equilibrium pries (p HH, p F H, p HF, p F F ) as a funtion of K. We drop 11
indies of ountry, sine the behavior of pries repeats the behavior in the losed eonomy: p ii K = r u(x ii ) (1 r u (x ii )) 2 x ii K, p ij K = τ r u(x ij ) x ij (1 r u (x ij )) 2 K In the ase of inreasing (dereasing) RLV, eah equilibrium prie dereases (inreases) with K, thereby market displays pro-ompetitive (anti-ompetitive) eet. As before, equilibrium mark-ups behave as pries. As well as onsumptions, both equilibrium prie and mark-up are independent from the number of workers L and from the share of workers in eah ountry. Proposition 4. The equilibrium pries respond to exogenous shoks as follows: (i) dereasing transport ost τ makes prie p ij of any imported variety dereasing when RLV dereases (the hange being ambiguous in the opposite ase), whereas prie p ii of any domesti variety dereases (inreases) under inreasing (dereasing) RLV. (ii) growing total world apital K makes all pries p ii,p ji of domesti and imported goods dereasing (inreasing) under inreasing (dereasing) RLV. (iii) growing ountry share (s for Home, (1 s) for Foreign) of world apital makes pries p ii,p ji of domesti and imported goods in this ountry dereasing (inreasing) under inreasing (dereasing) RLV. (Proof see in Appendix B). Of ourse, downward shoks push all these magnitudes oppositely to upward shoks. Another obvious observation is that the ase of CES preferenes is the borderline between inreasing and dereasing RLV, so, prie eets are absent (unrealistially). To interpret both above propositions, suppose, for example, inreasing world apital K. The rst onsequene is that number of rms in the world inreases proportionally beause one rm requires 1 unit of K. This invites eah onsumer to spread her budget over a larger range of varieties, thereby eah onsumption (size of a purhase) dereases. However, the impat on pries an be two-fold. When the inverse demand elastiity is an inreasing funtion (inreasing RLV, r > 0), then this derease onsumption pushes this elastiity down. It means that the demand elastiity (equal to the elastiity of substitution) goes up. Stronger substitution means stronger ompetition among varieties, smaller markup and pries. The same logi under dereasing RLV (r < 0) makes pries inreasing in spite of more rms. 2 Similar response to apital inrease works oppositely in Home and Foreign when the apital share s of Home inreases. Naturally, in r u > 0 ase pries p HH and p F H in Home derease but the apital share of Foreign dereases, so pries p HF and p F F in it go up. 2 Analogously, as everybody knows from IO theory, a downward shift of the demand urve an indue positive or negative prie eet depending on elastiity, the same remains true in monopolisti ompetition. 12
Probably, suh prie eets ould explain (manufaturing) prie dierentials between the developed ountries and less developed ountries when apital was less mobile than nowadays (it remains immobile in our model). More atual interpretation is human apital or skilled workers supply as the meaning of K parameter. Then our prie eets an be pronouned as follows. The developed ountries should have heaper high-teh goods than less developed ountries. Again, suh tendeny is not neessarilry observed in reality. The reason is, probably, the notieable dierentials in wages between North and South, the dierentials assumed out from our model by our wage-equalizing hypotheses, still dominating in the literature. 3.2 Dumping, relative fator pries, trade ows and Home-market eet Now we study the dumping eet. We say that ountry i's rm uses dumping in trade when its domesti prie exeeds its export prie divided by trade oeient p(x ii ) > p(xij ). τ Proposition 5. There exists suh ritial value ŝ (0.5, 1] of apital share s of Home, suh that governs two possible prie orderings, yielding two dierent dumping eets: (i) under small asymmetry, i.e., s < ŝ, (implying x F F > x HH > x HF > x F H ), inreasing RLV yields dumping priing pratied by eah ountry: p(x F F ) > p(x HH ) > p(xhf ) τ > p(xf H ), τ whereas dereasing RLV anti-ompetitive yields reverse-dumping used by eah ountry: p(x F F ) < p(x HH ) < p(xhf ) τ < p(xf H ) ; τ (ii) under big asymmetry, i.e., s > ŝ, (implying x F F > x HF > x HH > x F H ), inreasing RLV yields dumping used by smaller ountry and reverse-dumping used by bigger ountry: p(x F F ) > p(xhf ) τ > p(x HH ) > p(xf H ), τ whereas dereasing RLV yields dumping used by bigger ountry and reverse-dumping used by smaller ountry: p(x F F ) < p(xhf ) τ < p(x HH ) < p(xf H ). τ (Proof see in Appendix C). 13
Hene, when ountries are not too dissimilar, rms adopt the same priing behavior as they do in the ase of idential ountries. However, one H and F beome suiently dissimilar, both domesti and foreign rms hoose opposite priing behavior. The hoie between dumping or reverse dumping depends on the nature of preferenes, while the dierene in size explains why rms loated in the bigger and smaller ountries may adopt similar or dierent priing behaviors. To sum up, the priing pattern hosen by rms ritially depends on onsumers' preferenes in a way that vastly diers from what we know from the CES-ase whih exhibits no market segmentation poliies. Our next goal is to study the impat of dierene in apital among ountries, in the spirit of Heksher- Ohlin topis, but under monopolisti ompetition instead of perfetly substitutable goods. To separate this eet from impats from heterogeneity in population per se, we onsider the same populations in both ountries: sk + s a L = (1 s)k + (1 s a )L, the share of apital being larger in Home ountry s > 1 2. Further we use term larger ountry implying ountry H that has bigger share of apital. The value exported from Home ountry e H equals e H = sk((1 s)k + (1 s a )L)p HF x HF (8) and export from Foreign ountry e F is e F = (1 s)k(sk + s a L)p F H x F H (9) Agriultural setor serving as a dumper, these two need not be equal and we an nd who is the main exporter in manufaturing. Studying expressions (8)-(9) and (6)-(7) for value exported and interest we get Proposition 6. The ountry with relative advantage in apital (Home) has bigger value of export in manufaturing and its apital prie is bigger, i.e. e H > e F, π H > π F. Proof is in Appendix D. This eet ould be understanding as some eet with avor of Home market eet. 14
Sine population being deomposed into workers and apitalists, we seek some disproportional eet in the following monetary form. We use the value added in manufaturing (M i, i = H, M) as the mesure of the industry size. Proposition 7. The trade equilibrium displays: 1) the ountry with advantage in apital (Home) has disproportionally lower added value per rm M H N H < M F N F, 3, i.e. where M H is added volue in manufaturing in Home, M F is added volue in manufaturing in Foreign; 2) rm's output loated in Home is less than rm output in Foreign under assumptions u (x) > 0andr u (x) < q H < q F, where y H is rm output in Home, y F is rm output in in Foreign, 3) total trade value inreases with trade liberalization. (proof in Appendix E). 4 Conlusion Thus, for a two-fator two-setor losed monopolistially ompetitive eonomy we have found that under inreasing RLV (dereasing elastiity of substitution) - the equilibrium prie dereases with apital supply, under dereasing RLV the eet is opposite. Moreover, under suiently big population, interest rate dereases with apital supply. These basi tendenies display in two-ountries two-fator trade model through several important regularities: - omparative statis in both fators supply, prodution and transport osts shows, among other things, that under inreasing (dereasing) RLV equilibrium pries of domesti and imported varieties derease (inrease) with the share of the ountry's apital; - dumping and reverse-dumping may our, and suh patterns of priing behavior depend both of the ountries' apital advantage and elastiity of substitution, CES ase being the borderline between the opposite eets; - when populations are equal: 1) the ountry better endowed with apital is a net exporter of manufaturing 15
goods, has lower value added per rm, bigger apital prie, smaller size of rm loated in this ountry; 2) total trade value inreases with trade liberalization. These ndings suggest agglomeration eets in a similar model of eonomi geography with mobile apital. Referenes [1] Behrens, K. and Murata Y. General equilibrium models of monopolisti ompetition: a new approah, Journal of Eonomi Theory 136(1), 2007, 776-87 [2] Brakman, S. and B.J. Heijdra The Monopolisti Competition Revolution in Retrospet. Cambridge: Cambridge University Press, 2004 [3] Dixit, A.K. and J.E. Stiglitz Monopolisti ompetition and optimum produt diversity, Amerian Eonomi Review 67, 1977, 297-308 [4] Krugman, P. Inreasing returns, monopolisti ompetition, and international trade, Journal of international eonomis 9, 1979, 151-75. [5] Ottaviano G.I.P., Tabuhi T., Thisse J.-F. Agglomeration and trade revisited. International Eonomi Review, 43, 2002, 409-36. [6] Baldwin R., Forslid R., Martin P., Ottaviano G.I.P., and Robert-Nioud F. Eonomi Geography and Publi Poliy. Prineton, NJ: Prineton University Press, 2003 [7] Ottaviano, Gianmaro I. P. Monopolisti ompetition, trade, and endogenous spatial utuations, Regional Siene and Urban Eonomis, Elsevier, vol. 31(1), 2001, pages 51-77 [8] Brander, J. and P.R. Krugman A `reiproal dumping' model of international trade. Journal of International Eonomis 15, 1983, 313-321. [9] Greenhut, M.L., H. Ohta and J. Sailors Reverse dumping: a form of spatial prie disrimination, Journal of Industrial Eonomis, XXXIV, 1987, 167-181. [10] Dinopoulos E., Fujiwara K., and Shimomura K. International Trade and Volume Patterns under Quasilinear Preferenesrode, Review of Development Eonomis, 15(1), 2011, 154167 [11] Puger M. (2004) A simple, analytially solvable, Chamberlinian agglomeration model, Regional Siene and Urban Eonomis 34, 565 573 16
[12] Helpman E., Krugman P. R. Market Struture and Foreign Trade: Inreasing Returns, Imperfet Competition, and the International Eonomy, MIT Press, 1987 [13] Davis D.R. Eonomi Geography and Regional Prodution Struture: An Empirial Investigation, Harvard University, Federal Reserve Bank of New York, and NBER, 1999 [14] Martin P., Rogers C.A. Industrial loation and publi infrastruture, Journal of International Eonomis, 39, 1995. [15] Syverson C. Pries, spatial ompetition, and heterogeneous produers: an empirial test. Journal of Industrial Eonomis LV, 2007, 197-222. [16] Bernard A. B., Jensen J.B., Redding S., Shott P.K. Firms in International Trade, CEP Disussion Paper No 795, 2007 [17] Manova K., Zhang Z. Quality Heterogeneity aross Firms and Export Destinations, NBER Working Paper No. 15342, 2009 [18] Hummels D. and Klenow P. J. The Variety and Quality of a Nation's Exports, Amerian Eonomi Review, 95, 2005, 704-723 [19] Shott P. K. Aroos-produt versus within produt speialization in international trade, The Quarterly Journal of Eonomis, 2004, 647-678 [20] Hallak J. G. Produt quality and the diretion of trade, Journal of International Eonomis 68(1), 2006, 238-265 [21] Hallak J. G. and Shott P. Estimating Cross-Country Dierenes in Produt Quality, Yale Shool of Management, mimeograph, 2008. 5 Appendix. 5.1 Appendix A. The produer's program of produer i is ˆ N (L + K)V ( u(x j )dj)u (x i )x i (L + K)x i π i max. 0 x i First order ondition of produer's program: 17
ˆ N (L + K)V ( u(x j )dj)(u (x i )x i + u (x i )) (L + K) = 0 0 ˆ N V ( u(x j )dj)(u (x i )x i + u (x i )) = 0 ˆ N ( u V ( u(x j )dj)u ) (x i )x i (x i ) 0 u + 1 = (x i ) p(x i ) (1 r u (x)) = p(x i ) = Seond order ondition of produer's program: 1 r u (x). 5.2 Appendix B. ˆ N (L + K)V ( u(x j )dj)(u (x i ) + u (x i )x i + u (x i )) < 0 0 ( ) u (x i ) 2 + u (x i )x i u < 0 (2 r u (x)) > 0 r u (x) < 2 (x i ) Proof of Proposition 2. In equilibrium solutions of onsumer's and produer's program is equal. Than V [ sku(x HH ) + (1 s) Ku(x F H ) ] u (x HH ) = 1 r u (x HH ) V [ sku(x HH ) + (1 s) Ku(x F H ) ] u (x F H ) = τ 1 r u (x F H ) V [ sku(x HF ) + (1 s) Ku(x F F ) ] u (x F F ) = 1 r u (x F F ) V [ sku(x HF ) + (1 s) Ku(x F F ) ] u (x HF ) = Dividing equations mutually we get system desribed in Proposition A: τ 1 r u (x HF ) u (x HH ) u (x F H ) = 1 τ 1 r u(x F H ) 1 r u (x HH ) 18
V [ sku(x HH ) + (1 s) Ku(x F H ) ] u (x HH ) = 1 r u (x HH ) u (x F F ) u (x HF ) = 1 τ 1 r u(x HF ) 1 r u (x F F ) V [ sku(x HF ) + (1 s) Ku(x F F ) ] u (x F F ) = Proof of Proposition 3 and 4. Rewrite equation 1 r u (x F F ) as follow u (x HH ) u (x F H ) = 1 τ 1 r u(x F H ) 1 r u (x HH ) u (x HH )(1 r u (x HH )) u (x F H )(1 r u (x F H )) = 1 τ Left part of equation is ratio of marginal inome: u (x)(1 r u (x)) = u (x) + xu (x) = (xu (x)) = (xp). Well-known that marginal inome dereases with output. Right part of equation less then one ( 1 τ < 1). Then in home ountry individual onsumption of loal varieties bigger than individual onsumption of foreign varieties (x HH > x F H ). The same ondition for foreign ountry gives x F F > x HF. Let assume following funtions: 1. Funtion y(x, τ) dened from equation u (x) u (y(x, τ)) = 1 τ 1 r u(y(x, τ)) ; 1 r u (x) where x = x HH, y = x F H or x = x F F, y = x HF. 2. Funtion G(x, s, K, τ) dened as G(x, s, K, τ) = V (sku(x) + (1 s) Ku(y(x, τ)))u (x)(1 r u (x)). we study the behavior y as funtion of x. First we dierentiated funtion r u (x) w.r.t. x: r u(x) = ( ) xu (x) u = (u (x) + xu (x))u (x) x(u (x)) 2 (x) (u (x)) 2 = 19
= (u (x) + xu (x))u (x) x(u (x)) 2 (u (x)) 2 = xu (x) xu (x) xu (x)u (x)x u (x)u (x)x + 1 x ( xu ) (x) 2 u = (x) = r u(x) x r u(x)r u (x) x + r2 u(x) x = r u(x) x (1 + r u(x) r u (x)) Now we dierentiated funtion y w.r.t. x: u (x)(1 r u (x)) u (x)r u(x) = 1 τ (u (y)(1 r u (y)) u (y)r u(y)) y x ( xu u ) (x) (x) xu (x) (1 r u(x)) r u(x) = 1 τ u (y)( yu (y) yu (y) (1 r u(y)) r u(y)) y x u (x)r u (x) x (1 r u (x) + 1 + r u (x) r u (x)) = 1 τ u (y)r u (y) y (1 r u (y) + 1 + r u (y) r u (y)) y x u (x)r u (x) x (2 r u (x)) = 1 τ u (y)r u (y) y (2 r u (y)) y x y x = τ u (x)r u (x)(2 r u (x))y u (y)r u (y)(2 r u (y))x = τ u (x)(2 r u (x)) u (y)(2 r u (y)) > 0 We found positive dependene between onsumptions of loal varieties and foreign varieties in eah ountry. The behavior y w.r.t. transport ost desribes by derivative y τ : u (x)(1 r u (x)) = (u (y)(1 r u (y)) u (y)r u(y)) y τ u (x)(1 r u (x)) = (u (y)(1 r u (y)) u (y) r u(y) (1 + r u (y) r u (y))) y y τ u (x)(1 r u (x)) = u (y) y ( r u(y)(1 r u (y)) r u (y)(1 + r u (y) r u (y))) y τ u (x)(1 r u (x)) = u (y) y ( r u(y) + (r u (y)) 2 r u (y) (r u (y)) 2 + r u (y)r u (y))) y τ 20
u (x)(1 r u (x)) = u (y) y ( 2r u(y) + r u (y)r u (y))) y τ u (x)(1 r u (x)) = u (y)r u (y) (r u (y) 2) y y τ u (x)(1 r u (x)) = u (y)(2 r u (y)) y τ y τ = u (x)(1 r u (x)) u (y)(2 r u (y)) < 0 Now we turn to study funtion G(x, s, K, τ). Easy nd that argument of funtion V (sku(x)+(1 s) Ku(y(x, τ))) stritly inreases with individual onsumption of loal varietes x, share of apital in ountry s, total endowment of apital K and stritly dereases with transport ost τ. Then funtion V (sku(x) + (1 s) Ku(y(x, τ))) stritly dereases with x, s, K and striktly inreases with τ. As we found before u (x)(1 r u (x)) stritly dereases with x. Consequently funtion G(x, s, K, τ) stritly dereases with s, x and K and stritly inreases with τ. Sine funtion G(x, s, K, τ) stritly monotone with x there is at most one solution of equation G(x, s, K, τ) = Moreover, denition x of this equation stritly dereases with s, and K and stritly inreases with τ. So, we nd out that: (i) onsumption of loal varieties (x HH, x F F ) dereases with s, and K and stritly inreases with τ; (ii) onsumption of export varieties (x F H, x HF ) dereases with s,, K and τ. The denition of equation G(x, 1 s, K, τ) =. is individual onsumption of varieties in Foreign. Sine funtion G(x, s, K, τ) stritly dereases with s, x and K and stritly inreases with τ onsumption of loal varieties less in ountry with bigger share of apital (x F F > x HH ). The same reasons brings us to expression x HF > x F H. Consequently there are only two types of sorting individual onsumptions: x F F > x HF > x HH > x F H x F F > x HH > x HF > x F H, 21
QED. 5.3 Appendix C. Proof of proposition 5. (i) Consider the sorting of individual onsumption x F F > x HH > x HF > x F H In ase pro-ompetitive behavior if x > y then So, p(x) = 1 r u (x) > 1 r u (y) = p(y) Consequently, 1 r u (x F F ) > 1 r u (x HH ) > 1 r u (x HF ) > 1 r u (x F H ) p(x F F ) > p(x HH ) > p(xhf ) τ > p(xf H ) τ In ase anti-ompetitive behavior if x > y then So, p(x) = 1 r u (x) < 1 r u (y) = p(y) Consequently, 1 r u (x F F ) < 1 r u (x HH ) < 1 r u (x HF ) < 1 r u (x F H ) p(x F F ) < p(x HH ) < p(xhf ) τ < p(xf H ) τ Proof of seond piont of Proposition is the same, QED. 5.4 Appendix D. Proof of proposition 6. Sine the ountry size is equals total export valume for home e H and foreign e F ountry are 22
e H = sk((1 s)k + (1 s a )L)p HF x HF e F = (1 s)k(sk + s a L)p F H x F H Consider funtion v(x) = px. We study monotoniity of this funtion: v(x) x = x (px) = ( ) x x 1 r u (x) = 1 r u(x) + xr u(x) (1 r u (x)) 2 = 1 r u(x) + r u (x)(1 + r u (x) r u (x)) (1 r u (x)) 2 = = 1 + r2 u(x) r u (x)r u (x) (1 r u (x)) 2 > 1 + r2 u(x) 2r u (x) (1 r u (x)) 2 = (1 r u(x)) 2 (1 r u (x)) 2 = > 0 We found that total ountry export stritly inreases with x. Sine we have x HF > x F H total export of home ountry bigger than total export of foreign ountry: e H > e F Capital pries in home π H and foreign π F ountries are π H = (sk + s a L)((p HH )x HH + (p HF τ)x HF ) π F = (sk + s a L)((p F F )x F F + (p F H τ)x F H ) π H π F = (sk + s a L)((p HH )x HH + (p HF τ)x HF (p F F )x F F (p F H τ)x F H ) = ( x HH r u (x HH ) = (sk + s a L) 1 r u (x HH ) τ xf H r u (x F H ( ) x F F 1 r u (x F H ) r u (x F F ) 1 r u (x F F ) τ xhf r u (x HF )) ) 1 r u (x HF ) Consider funtion z(x) = xru(x) 1 r u(x) τ y(x,τ)ru(y(x,τ)) 1 r u(y(x,τ)). z x = (r u(x) + xr u(x))(1 r u (x)) + r u(x)xr u (x) (1 r u (x)) 2 τ (r u(y) + yr u(y))(1 r u (y)) + r u(y)yr u (y) y (1 r u (y)) 2 x = 23
= r u(x) ru(x) 2 + xr u(x) (1 r u (x)) 2 τ r u(y) ru(y) 2 + xr u(y) y (1 r u (y)) 2 x = = r u(x) ru(x) 2 + r u (x)(1 + r u (x) r u (x)) (1 r u (x)) 2 τ r u(y) ru(y) 2 + r u (y)(1 + r u (y) r u (y)) y (1 r u (y)) 2 x = = r u (x) 2 r u (x) (1 r u (x)) 2 τr u(y) 2 r u (y) (1 r u (y)) 2 y x = r u(x) 2 r u (x) (1 r u (x)) 2 τ 2 r u (y) 2 r u (y) (1 r u (y)) 2 u (x)(2 r u (x)) u (y)(2 r u (y)) = = r u (x) 2 r u (x) (1 r u (x)) 2 τ 2 r u (y) u (x)(2 r u (x)) u (y)(1 r u (y)) 2 = 2 r u (x) (1 r u (x)) 2 (r u (x) τ 2 r u(y)u (x)(1 r u (x)) 2 ) u (y)(1 r u (y)) 2 = = 2 r u (x) ( (1 r u (x)) 2 r u (x) τ 2 r u(y)u ( (x) u ) ) (y) 2 u (y) τu = 2 r u (x) (x) (1 r u (x)) 2 (r u (x) r u(y)u (x)(u (y)) 2 ) u (y)(u (x)) 2 = = 2 r u (x) ( (1 r u (x)) 2 r u (x) yr u(y)r u (x)u ) (y) xr u (y)u = 2 r u (x) ( ) (x) (1 r u (x)) 2 r u(x) 1 yu (y) xu < 0 (x) Sine funtion z(x) stritly dereasing and x F F > x HH we have QED. π H π F = (sk + s a L)(z(x HH ) z(x F F )) > 0 π H > π F, 5.5 Appendix E. Proof of Proposition 7. If added volue (M H ) in manufaturing divided by number of rms in Home (bigger endowment with apital ountry) less then ratio in Foreign (M F ) following equation should take plae: M H N H < M F N F M H = sk [ x HH (sk + s a L) + τx HF ((1 s)k + (1 s a )L) + π H] 24
M H N H = sk [ x HH (sk + s a L) + τx HF ((1 s)k + (1 s a )L) + π H] sk = (sk + s a L)(x HH + τx HF ) + π H = = /use zero profit ondition/ = (sk + s a L) [ p HH x HH + p HF x HF ] = [ x HH = (sk + s a L) 1 r u (x HH ) + x HF ] 1 r u (x HF ) [ = (sk + s a L) x HH 1 r u (x HH ) + x HF 1 r u (x HF ) ] M F N F [ = (sk + s x F F al) 1 r u (x F F ) + x F ] H 1 r u (x F H ) M H N H M F N F [ = (sk + s al) x HH 1 r u (x HH ) + x HF 1 r u (x HF ) ] [ (sk + s a L) x F F 1 r u (x F F ) + x F H 1 r u (x F H ) ] Let t 1 (x) = [ x HH = (sk + s a L) 1 r u (x HH ) x F H 1 r u (x F H ) x 1 r u(x) + y(x,τ) 1 r u(y(x,τ)). We diferentiated w.r.t. x: ( x F F 1 r u (x F F ) x HF )] 1 r u (x HF ) t 1 (x) x = 1 r u(x) + xr u(x) (1 r u (x)) 2 + 1 r u(y(x, τ)) + yr u(y(x, τ)) y (1 r u (y(x, τ))) 2 x = = 1 r u(x) + r u (x)(1 + r u (x) r u (x)) (1 r u (x)) 2 + 1 r u(y) + r u (y)(1 + r u (y) r u (y)) y (1 r u (y)) 2 x = = 1 + r2 u(x) r u (x)r u (x) (1 r u (x)) 2 + 1 + r2 u(y) r u (y)r u (y) y (1 r u (y)) 2 x = = 1 2r u(x) + ru(x) 2 + 2r u (x) r u (x)r u (x) (1 r u (x)) 2 + 1 2r u(y) + ru(y) 2 + 2r u (y) r u (y)r u (y) y (1 r u (y)) 2 x = = (1 r u(x)) 2 + r u (x)(2 r u (x)) (1 r u (x)) 2 + (1 r u(y)) 2 + r u (y)(2 r u (y)) y (1 r u (y)) 2 x > 0, Sine 2 > r u (x) and y x > 0. So, we use x HH < x F F and get t 1 (x HH ) < t 1 (x F F ) as far as funtion t 1 (x) is stritly inrease. It brings us 25
to following inequation M H N H M F N F = (sk + s al) [ t 1 (x HH ) t 1 (x F F ) ] < 0 M H N H < M F N F Finally M H N H < M F N F. Firm's output in Home: q H = x HH (sk + s a L) + τx HF ((1 s)k + (1 s a )L) = (sk + s a L)(x HH + τx HF ). q F = (sk + s a L) ( x F F + τx F H). q H q F = (sk + s a L) ( x HH τx F H ( x F F τx HF )). Let t 2 (x) = x τy(x, τ). We diferentiated w.r.t. x: t 2 (x) x = 1 τ y x = 1 τ u (x)(2 r u (x)) u (y)(2 r u (y)) = 1 u (y)(1 r u (y)) u (x)(1 r u (x)) u (x)(2 r u (x)) u (y)(2 r u (y)) = = 1 u (y)(1 + yu (y) u (y) ) ) u (x)(2 r u (x)) u (x)(1 + xu (x) u (y)(2 r u (y)) = 1 (u (y) + yu (y)) (x)) u (x)(2 r u (x)) (u (x) + xu u (y)(2 r u (y)) = 1 (yu (y)) u (x)(2 r u (x)) (xu (x)) u (y)(2 r u (y)) u (x) Sine x > y we have (yu (y)) > (xu (x)). Consider funtion u (x)(2 r u (x)), it inreases when (u (x)(2 r u (x))) = u (x)(3 r u (x)) > 0 So, under assumptions u (x) > 0 and r u (x) < 3 we get t 2 (x) x < 0. It brings us to follows 26
q H q F = (sk + s a L) ( x HH τx F H ( x F F τx HF )) = (sk + s a L)(t 2 (x HH ) t 2 (x F F )) < 0 q H < q F, sine x HH < x F F. It nothing but rm's output in Home less then in Foreign. Total trade value is T = x HF p HF ((1 s)k + (1 s a )L) + x F H p F H (sk + s a L) = (sk + s a L)(x HF p HF + x F H p F H ) = ( x HF = (sk + s a L) 1 r u (x HF ) + x F ) H 1 r u (x F H ) ( T 1 τ = (sk + s ru (x HF ) + x HF r u(x HF ) x HF al) (1 r u (x HF )) 2 τ + 1 r u(x F H ) + x F H r u(x F H ) x F ) H (1 r u (x F H )) 2 = τ ( 1 + r 2 = (sk + s a L) u (x HF ) r u (x HF )r u (x HF ) x HF (1 r u (x HF )) 2 τ + 1 + r2 u(x F H ) r u (x F H )r u (x F H )) x F ) H (1 r u (x F H )) 2 = τ ( (1 ru (x HF )) 2 + r u (x HF )(2 r u (x HF )) x HF = (sk+s a L) (1 r u (x HF )) 2 τ + (1 r u(x F H )) 2 + r u (x F H )(2 r u (x F H )) x F ) H (1 r u (x F H )) 2 < 0, τ Sine 2 > r u (x), r u (x) > 0 and xij τ < 0, QED. 27