Addendum to: Importing SkillBiased Technology


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1 Addendum to: Importng SkllBased Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our man paper.
2 Alternatve quanttatve trade models In ths secton we embed captalskll complementarty nto an alternatve quanttatve trade model featurng monopolstc competton and heterogeneous rms. We rst analyze the case wth restrcted entry, n whch the mass of actve rms n each country s exogenously determned, and rms productvtes are drawn from a Pareto dstrbuton, as n the Chaney (2008) model. We show how to solve for real wages and the skll premum as a functon of domestc varables n ths settng. We note, however, that the expressons lnkng changes n the skll premum to changes n domestc expendture shares d er from those n the Rcardan model developed n the paper, and that changes n real wages now also depend on changes n the domestc trade balance. Fnally, we allow for endogenous entry as n Meltz (2003) and show that n ths case domestc expendture shares and the domestc trade balance are no longer su cent statstcs for the mpact of trade on the skll premum. We assume that there s a contnuum of producers n each country and sector, each producng a d erentated good. Producers operate the constant returns to scale technology detaled n the paper, y ('; j) = A 'b c;, where b c; b 3; b 4; s an nput bundle, b 3; and b 4; are de ned as n the paper, ' s the producer s dosyncratc productvty, and we ndex producers by ther productvty and ther sector. In addton to producton costs, we assume that ntermedate producers from country and sector j must ncur an ceberg trade cost n and a xed cost f n to sell n country n. The xed cost s denomnated n unts of the composte producton bundle b c;. Ths mples that producers n sector j must pay a xed cost c f n to sell n country n, where c s the unt cost of the composte producton bundle and s de ned n the paper. The dstrbuton of productvtes s gven by G ('; j) n sector j n each country. Productvtes are dstrbuted Pareto wth shape parameter and support [; ]: G ('; j) = '.. Restrcted entry We wll rst assume that the mass of rms operatng n each sector n each country M s determned exogenously. Frm s problem. The pro t maxmzng prce of a rm wth productvty ', operatng n A
3 sector j, and sellng from country to country n s p n ('; j) = n n c n A '. Varable pro ts, & n ('; j), for ths rm are & n ('; j) = ' n n n [ n ] n n c A P n n Y n. If the rm s actve n market n, ts demand for the composte bundle s b c;n ('; j) = n p n ('; j) q n ('; j) + f n. n c The productvty threshold ' n below whch rms decde not to serve the market s gven by & n (' n ; j) = c f n. Aggregates. Aggregatng across rms, countres, and sectors, we obtan total demand of the nput bundle to be used n producton Bc;n P = X X c n j n n ('; j) P n Y n n where h M s the mass of rms from country operatng n sector j and where n ('; j) = R. p ' n n ('; j) q n ('; j) M dg ('; j) [P n Y n ] s the share of country n expendtures n sector j spent on goods produced n country. Total demand for the nput bundle to cover xed costs s gven by B F;c = X n X j f n M [ From the de nton of the partcpaton cuto, we obtan n ('; j) P n Y n c = Aggregate demand for the nput bundle s G (' n ; j)] n + n f n M [ G (' n ; j)] B c B F;c + B P c;n = X n X n + n n P n Y n. n c j A2
4 Note that our mod ed market structure, relatve to the perfectly compettve market structure n our man paper, does not alter how total payments to the nput bundle, c B c, are splt among nputs. In addton, note that aggregate pro ts are: = X X n P n Y n c B c n j = X X n n P n Y n n j n Fnally, we can wrte the prce ndex n country for goods n sector j as " X Z P = n ' n n c n A n ' # n M n dg ('; j), and usng the de nton of the cuto, we can wrte the expendture share as c n = A n n f n n n n P n M ; (A) P n Y n wth + n n n (n ). We now characterze the steadystate equlbrum n the world economy... SteadyState Equlbrum We now de ne and characterze the steady state equlbrum for the world economy. In dong so, we show how aggregate quanttes and prces can be determned before solvng for product level varables. A steadystate equlbrum for the aggregate varables n the world economy conssts of a set of prces fv ; w ; r ; s g 2I, fp b; ; p b2; ; p b3; ; p b4; ; c g 2I, fp (S) ; P (M) ; P (E)g 2I, aggregate quanttes fk (S) ; K (E) ; X (M) ; X (S)g 2I, fc (M) ; C (S)g 2I, fy (M) ; Y (S); Y (E)g 2I ; partcpaton cuto s f' n g ;n2i ; j2j and trade shares f n g ;n2i ; j2j such that, gven factor supples, fh ; L g 2I and fm g ;n2i ; j2j, technologes, fa (S) ; A (M) ; A (E) g 2I, and net exports, fnx g 2I ; n each country, the followng are sats ed:. Household s maxmze utlty subject to ther budget constrants: The household s optmalty condtons n steady state are gven by the Euler equatons (6) and (7) n the man paper, and the ntratemporal consumpton equaton (8) n the man paper. A3
5 The budget constrant s now wrtten as, (w L + s H + v K (S) + r K (E) + ) ( + nx ) = P (E) (E) K (E) (A2) +P (M) C (M) +P (S) [C (S) + (S) K (S)] h where aggregate pro ts are gven by: = n PI n n= n ('; j) P n Y n. As n the man paper, nx denotes net exports as a share of GDP, whch we take as a parameter. 2. Cost mnmzaton by producers of ntermedate goods: Cost mnmzaton mples equatons (20) (24) n the man paper. 3. Cost mnmzaton by producers of nal goods: The sectoral prce ndces for nal goods are gven by P = X n n c n ' n M n A n ' +! n (A3) and trade shares between any par of countres are gven by equaton (A). 4. Aggregate factor market clearng: Equatons (26) (3) n the man paper are now mod ed as, and, v K (S) = c B c, (A4) w L = ( ) p b2; w c B c, (A5) r K (E) = ( ) ( ) pb; s H = ( ) ( ) ( ) r pb2; c B c, p b; pb; pb2; s p b; c B c, (A6) (A7) P (S) X (S) = " ( ) c B c, (A8) P (M) X (M) = ( " ) ( ) c B c, (A9) where c B c = X n X j n + n n P n Y n. (A0) n c 5. Aggregate goods markets clear n each country: These are gven by equatons (33) (35) n the man paper. A4
6 6. The partcpaton cuto s are gven by: ' n = n c A " c f n [ n ] n n n P n n Y n..2 Solvng n Terms of Domestc Expendture Shares # n. (A) In ths secton we show how to solve for all aggregate domestc varables as functons of domestc expendture shares,, domestc productvtes, A, domestc factor supples fh ; L g and fm g, and net exports nx. To better understand the d erence wth the Rcardan model, we use equaton (A) to wrte aggregate prce ndces as functons of domestc expendture shares, " c P = ('; j) A n n f P Y n n # : (A2) Note that sectoral prce ndces P are functons of sectoral expendtures, P Y, snce these determne the partcpaton cuto s and hence the dstrbuton of prces n each sector. Snce sectoral expendtures are not constant across equlbra, we cannot solve for the model s prces ndependently of the quanttes, as we dd n the Rcardan model. Instead, we can use the followng algorthm to solve for all aggregate domestc varables as a functon of domestc trade shares. Start by guessng expendtures n each sector P Y. From equatons (A5) and (A7) we obtan s w H L = ( ) p b;. (A3) The 3 prce ndex equatons (A2), together wth equaton (A3), the Euler equatons (6) and (7) and the cost mnmzaton equatons (20) (24) make a system of equatons. Together wth a choce of numerare these equatons can be used to solve for the 2 domestc prces. Gven prces, we can solve for quanttes as follows. Frst, solve for K (E) and K (S) usng equatons (A4), (A6), and (A7). Second, addng equatons (A4) (A7), we solve for c B c as c B c = v K (S) + w L + r K (E) + s H. Thrd, usng equatons (A8) and (A9), we obtan ntermedate nputs X (M) and X (S). Fourth, from equaton (8) n the paper and equaton (A2), we can solve for the consumpton levels C (S) and C (M). Fnally, from the market clearng equatons (33) (35) we obtan A5
7 total producton n each sector. Use these equatons to verfy the guess..2 Endogenous Entry We now examne the case when entry s endogenous. We assume that n every perod there s an unbounded mass of potental entrants that can pay a sunk cost f e; to enter sector j and produce a d erentated good. Producers de each perod wth probablty M 0. In an equlbrum wth entry, the entry cost must equal dscounted expected pro ts. Ths mples: where c f e; = = X n M =M n n P n Y n n are total revenues n sector j. These two equatons determne the mass of rms n each sector M n ths settng. Note that, snce M depends on revenues accrung from each destnaton, n P n Y n, we can no longer solve for the equlbrum as a functon of domestc varables only even f there s trade balance (but not sectorbysector) for each country. 2 D erences n factor ntenstes across sectors In ths secton we extend the model to allow for heterogeneous producton functons across sectors. In partcular, we allow for the parameters the parameters f ; ; ; ; " ; ; g j2j to be sector spec c: y (!; j) = A z (!; j) b 3; b 4; wth b ; = b 2; = h = k ( E )= + ( ) = h ( =( )= ), h = l ( )= + ( ) = ( b =( )= ), b 3; = k S b 2, b 4; = x " " S x M, where we have dropped countryspec c subscrpts,, from the producton functon parameters to facltate exposton. The unt cost of producton for supplyng the domestc A6
8 market of a producer wth productvty A z (!; j) = s now sector spec c, c = p b 3; p b4; ( ), where p b; = p b2; = h r + ( ) s, h w + ( ) p b; p b3; = v p b2; ( ), p b4; = P (S) " P (M) " " ( " ) "., Aggregate factor market clearng condtons are now wrtten as, v K (S) = X j w L = X j ( pb2; ) w r K (E) = X j pb; ( ) ( ) r pb2; p b; s H = X j pb; ( ) ( ) ( ) s pb2; p b; and ntermedate nput market clearng s wrtten as P (S) X (S) = X j P (M) X (M) = X j " ( ) ( " ) ( ) where P n n P n Y n denotes total revenue accrung to all country producers n sector j. We now express the above condtons n terms of the factor content of trade. De ne by NX F the unts of factor F that are emboded n county s net exports, NX F = X j F!, (A4) A7
9 where F denotes the utlzaton of factor F n country and sector j, and where! s the rato of county s net exports n sector j to country s total revenue n sector j, P n! = [ n n Y n n P Y ] P n. (A5) n P n Y n By equatons (A4) and (A5), we have p F NX F = F X n6= [ n P n Y n n P Y ], (A6) where F h denotes the share of sector j revenue pad to factor F. By equaton (A6) and P = n6= n, we express equlbrum n the ntermedate nput markets as P (S) = P (M) = and n the factor markets as P j " ( ) P Y X (S) P j ( " ) ( ) P Y X (M) NX X (S) NX X(M) (A7) (A8) v = X j, P Y K (S) NX K(S) (A9) w = X j ( pb2; ) P Y, w L NX L (A20) r = X pb; ( ) ( ) (A2) r j pb2;!, P Y K (E) NX K(E) p b; s = X pb; ( ) ( ) ( ) (A22) s j pb2; P Y!, p b; H NX H A8
10 Aggregate prces are: P = c. A (A23) Equatons (6) (9) ; (33) (35) ; and (A7) (A23) yeld a system of equatons n fw ; s ; v ; r g I =, fp g I =;j2j, fy g I =;j2j, fk (E) ; K (S)g I =, fc (M) ; C (S)g I =, and fx (M) ; X (S)g I = that characterzes the steadystate equlbrum and that depends only on domestc expendture shares by sector f g j2j, sectoral domestc productvtes fa g j2j, the factor content of trade for all factors NX F f ; ; ; ; " ; ; ; g j2j., and the parameters F 2F A9
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