Addendum to: New Trade Models, Same Old Gains?

Transcription

1 Addendum to: New Trade Models, Same Old Gains? Costas Arkolakis Yale and NBER Arnaud Costinot MIT and NBER September 5, 200 Andrés Rodríguez-Clare Penn State and NBER Abstract This addendum provides generalizations of Proposition for the cases of multiple sectors and tradable intermediate goods discussed in Section 5 of our main paper.

2 Extension (I): Multiple Sectors. Assumptions Preferences, Technology, and Market Structure. It is standard to interpret models with Dixit-Stiglitz preferences, such as the one presented in Section 3 of our main paper, as onesector models with a continuum of varieties. Under this interpretation, our model can be extended to multiple sectors, s = ; :::; S, by assuming that the representative agent has a two-tier utility function, with the upper-tier being Cobb-Douglas, with consumption shares s 0, and the lower-tier being Dixit-Stiglitz with elasticity of substitution s >. Under this assumption, the consumer price index in country j formally becomes P j = Q S s= P s j s, () where P s j = R!2 s p s j(!) s d! s is the Dixit-Stiglitz price index associated with varieties from sector s. For each sector s, primitive assumptions on technology and market structure are as described in Section 3 of our main paper. Superscripts s denote all sector-level variables. Macro-level Restrictions. In this extension we use the following counterparts of R-R3: R(MS) For any country j and any sector s, w j L s j + s j w j N s j F s j = P n i= Xs ji. In the one-sector case R states that P n i= X = P n i= X ji. This is equivalent to w j L j + j w j N j F j = P n i= X ji since w j L j + j w j N j F j = P n i= X by country j s representative agent s budget constraint. R(MS) is simply the sector-level counterpart of the previous expression. R2 (MS) For any country j and any sector s, s j = R s j with 2 [0; ]. Compared to R2 in the one-sector case, R2(MS) states that aggregate pro ts are a constant share of revenues in each sector, but also that the share of pro ts is common across sectors. R3 (MS) The import demand system is such that for any sector s, any importer j, and any pair of exporters i 6= j and i 0 " si ln X s =X s ln s i 0 j. 6= j, " si j = " s < 0 if i = i 0 and zero otherwise, with Note that R3(MS) allows the trade elasticities " s to vary across sectors.

3 .2 Welfare Evaluation Under the previous assumptions, Proposition generalizes to: Proposition (MS) Suppose that R-R3(MS) hold. Then the change in real income associated with any foreign shock in country j can be computed as W c j = Q S b s s j s= jj /"s, under perfect competition and monopolistic competition with restricted entry, and W c j = Q. s S s= ^s bl s j jj j /"s, under monopolistic competition with free entry. Proof. Like in our main paper, we consider separately the cases of perfect and monopolistic competition and use labor in country j as our numeraire, w j =. For expositional purposes, we describe in detail the steps of the proof that are distinct from those in Appendix A of our main paper and omit others. Case : Perfect competition By the same logic as in Appendix A (Step ), we have d ln Y j = 0. Combining this observation with Equation () and the de nition of W j, we obtain d ln W j = P S s= s jd ln P s j. Following the exact same reasoning as in Appendix A (Steps 2-4), one can easily check that R3(MS) implies d ln P s j = d ln s jj " s. Combining the two previous expressions and integrating (Step 5 in Appendix A), we get cw j = Q S b s s j s= jj /"s, which completes our proof under perfect competition. Case 2: Monopolistic Competition Using the same logic as in Appendix A (Step ), we rst show that d ln Y j = 0. Under free entry we know that Y j = L j, which immediately implies d ln Y j = 0. Under restricted entry, we know from the budget constraint of the representative agent in country j that Y j = L j + P S s= s j. 2

4 Combining this observation with R(MS), we get Y j = P S s= Rs j. Since s j = R s j for all s = ; :::; S by R2(MS), we thus have Y j = L j + Y j. Totally di erentiating the previous expression, we get dy j = 0 and thus d ln Y j = 0. Since d ln Y j = 0 under monopolistic competition, the same reasoning as under perfect competition implies d ln W j = P S s= s jd ln P s j. Following the exact same reasoning as in Appendix A (Steps 2-4), one can also easily check that R3(MS) implies d ln P s j = d ln s jj " s + d ln N s j " s. Using the same logic as in Appendix A (Step 5), let us now show that d ln N s j = d ln L s j, under free entry, and d ln N s j = 0, under restricted entry. Under free entry, we know that s j = N s j F s j, which implies d ln s j = d ln N s j. By R(MS) and free entry, we know that d ln L s j = d ln R s j. By R2(MS), we also know that d ln R s j = d ln s j. Combining the previous series of equations, we obtain d ln N s j = d ln L s j. Finally, under restricted entry, N s j = N s j immediately implies d ln N s j QED = 0. The last part of the proof is the same as under perfect competition and omitted. 2 Extension (II): Tradable Intermediate Goods 2. Assumptions Preferences, Technology, and Market Structure. The primitive assumptions on preferences and market structure are the same as in Section 3 of our main paper. In terms of technology, however, we now allow goods! 2 to be used in the production of other goods. Formally, we assume that all goods can be aggregated into a unique intermediate good using the same Dixit-Stiglitz aggregator as for nal consumption. Thus P i now represents both the consumer price index in country i and the price of intermediate goods in this country. In this extension, the cost function for each good! is given by C i (w; P ; q; t;!) = P n j= [c (w i ; P i ; t j ;!) q j + f (w i ; P i ; w j ; P j ; t j ;!) I(q j > 0)], 3

5 where P fp i g is the vector of intermediate good prices. In line with the previous literature we further assume that constant marginal costs and xed exporting costs can be written as c (w i ; P i ; t j ;!) w i P i (!) t, f (w i ; P i ; w j ; P j ; t j ;!) h (w i P i ; w j P i ) (!) m (t), with 2 [0; ] governing the share of intermediate goods in variable and xed production costs. Similarly, we assume that xed entry costs (if any) are given by wi P i F i, with 2 [0; ] governing the share of intermediate goods in entry costs. Macro-level Restrictions. In this extension our rst two macro-level restrictions, R(TI) and R2(TI), are exactly the same as in Section 3 of our main paper. R3(TI) still requires the import demand system to be such that for any importer j and any pair of exporters i 6= j and i 0 6= j, " i j = " < 0 if i = i 0, and zero otherwise. The only di erence with Section 3 of our main paper is that the import demand system now refers to the mapping from (w; P ; N; ) into X, and, so the partial elasticities " i j also hold xed the price of intermediate goods, P. 2.2 Welfare Evaluation Under the previous assumptions, Proposition generalizes to: Proposition (TI) Suppose that R-R3(TI) hold. Then the change in real income associated with any foreign shock in country j can be computed as W c j = b /" jj, under perfect competition; cw j = b " /[" ( )( +)] jj, under monopolistic competition with restricted entry; and W c j = " /[" ( )( +)+( )] b, under monopolistic competition with free entry. jj Proof. Like in the previous proof, we consider separately the cases of perfect and monopolistic competition and use labor in country j as our numeraire, w j =. For expositional purposes, we again describe in detail the steps of the proof that are distinct from those in Appendix A of our main paper and omit others. Throughout this proof we let c i w i P i and c c i. Case : Perfect competition This generalization of the de nition of the import demand system re ects the fact that there are now two inputs in production, labor and the aggregate intermediate goods, with prices given by w and P, respectively. 4

6 By the same logic as in Appendix A (Step ), we have d ln Y j = 0, which implies d ln W j = d ln P j. (2) Similarly, by the same logic as in Appendix A (Step 2), small changes in the consumer price index satisfy P n i= d ln c. (3) Finally, by the same logic as in Appendix A (Step 3), small changes in expenditure shares satisfy d ln d ln jj = + i i jj d ln c + P n i 0 6=i jj d ln c i 0 j j + jj j d ln cjj, where is given by the same expression as in Appendix A. Compared to Appendix A, the main di erence is that we now have d ln c jj 6= 0. Combining Equations (3) and (4), we obtain (4) P P n i= d ln d ln n jj i 0 6=i;j jj d ln ci 0 j + j + jj j d ln cjj. + i i jj (5) Following the same logic as in Appendix A (Step 4), it is easy to check that Equation (4) and R3(TI) imply + i i jj = ", for all i 6= j, and = jj for all i 0 6= i; j. Combining this observation with Equation (5), we get jj " + P n i= j + jj j d ln cjj. (6) " Using the de nition of in Appendix A, it is easy to check that j = P i 0 6=j j jj = P i 0 6=j jj, which implies and j + j jj = + i i jj + X i 0 6=i;j jj = ". Together with Equation (6), the previous expression further implies jj " + d ln c jj. (7) 5

7 By de nition of c jj and our choice of numeraire, we know that d ln c jj = ( small changes in the consumer price index satisfy ) d ln P j. Thus d ln jj ". Combining the previous expression with Equation (2), we get d ln W j = d ln jj ". The rest of the proof is the same as in Appendix A (Step 5). Case 2: Monopolistic Competition Using the same logic as in Appendix A (Step ), we rst show that d ln Y j = d ln R j = 0. Note that compared to Appendix A, the rst of these two equalities is no longer a trivial implication of R(TI): whereas total revenues are still R j = P n i= X ji, the total expenditure of the representative agent in country j is now Y j 6= P n i= X since total imports also include expenditures on intermediate goods by rms from country j. Let us start with the case of free entry. Under free entry we know that Y j = L j, which immediately implies d ln Y j = 0. By R(TI), R2(TI), and our Cobb-Douglas assumptions, we also know that total payments to labor are ( ) R j + P j N j F j, which must be equal to L j. Since free entry and R2(TI) imply that P j N j F j = R j, we then have L j = [ ( ) + ] R j, hence d ln R j = 0 as well. Let us now turn to the case of restricted entry. Under restricted entry, R(TI), R2(TI), and our Cobb-Douglas assumptions imply that total payments to labor are ( ) R j, which must be equal to L j. This immediately implies d ln R j = 0. By R2(TI) and the budget constraint of the representative agent in country j, we also know that Y j = L j + R j. Since d ln R j = 0, this implies that d ln Y j = 0. Like in Appendix A (Step ), d ln Y j = 0 immediately implies d ln W j = d ln P j. (8) The next part of the proof follows closely Steps 2 through 4 in Appendix A. Compared to Appendix A, the main di erence is that, like under perfect competition before, we now have d ln c j 6= 0. Using ( ) P j c h h (c j ;c j ) R j i =( ) together with the fact that d ln Rj = 6

8 0, and following the same logic as in Appendix A (Step 2), small changes in the consumer price index satisfy P n i= (d ln + d ln c i ) (9) j d ln + ln h (c i ; c j ) d ln c ln c i ln h (c i ; c j ln c j d ln c j + d ln N i, where and j are given by the same expressions as in Appendix A. Similarly, by the same logic as in Appendix A (Step 3), small changes in expenditure shares satisfy d ln d ln jj = (d ln + d ln c i d ln c j ) ln h (c i ; c j ) + (d ln c i d ln c j ln c i + d ln + jj d ln jj + d ln N i d ln N j. Combining the previous expression with Equation (9) and noting ln h (c i ;c j ln c i ln h (c i ;c j ln c j =, by the assumption that h() is homogeneous of degree, we then get P n i= + j + d ln d ln jj jj d ln jj () d ln c j + d ln N j. Following the same logic as in Appendix A (Step 4), it is easy to check that Equation (0) and R3(TI) imply = " for all i. Combining this observation with Equation (5), we obtain d ln jj + d ln N j " + " d ln c j. " ( ) By de nition of c j and our choice of numeraire, we know that d ln c j = ( small changes in the consumer price index satisfy ) d ln P j. Thus d ln jj + d ln N j (2) " " ( ) +. 7

9 Finally, by the same logic as in Appendix A (Step 5), we must have d ln N j = ( ) d ln P j under free entry (since d ln R j = 0); and d ln N j = 0 under restricted entry (since N j = N j ). Combining these observations with Equations (8) and (2), we obtain d ln jj " ( )( " d ln W j =, under monopolistic competition with restricted entry, d ln d ln W j = jj " ( )( " +)+( ), under monopolistic competition with free entry. The last part of the proof is the same as under perfect competition. QED 8