Journal of Economcs and Econometrcs Vol. 55, No.2, 202 pp. 78-93 SSN 2032-9652 E-SSN 2032-9660 Structural Estmaton of Varety Gans from Trade ntegraton n a Heterogeneous Frms Framework VCTOR RVAS ABSTRACT n ths artcle we develop a smple analytcally solvable model of heterogeneous frms. The heterogeneous frm framework presented n ths paper s partcularly sutable for the structural estmaton of varety gans from trade ntegraton, as all structural equatons for emprcal estmatons can be drectly derved from the theoretcal model. JEL Classfcaton: C68, F2, F4, F5, F7, R2, R3, R23. Keywords: Varety gans, structural estmaton, trade ntegraton, heterogeneous frms. Department of Economc Hstory, Houghton Street, London WC2A 2AE, Unted Kngdom. E-mal: v.rvas@lse.ac.uk. The author s grateful to two anonymous referees and to the Edtor of the Journal for many useful comments and suggestons on an earler draft of the paper. The artcle represents only the author s vew, the usual dsclamer apples.
Journal of Economcs and Econometrcs Vol. 55, No. 2, 202 pp. 78-93 NTRODUCTON We consder an economy wth two factors of producton: labour, L, and captal, K. These two producton factors are suppled nelastcally and are moble between sectors mplyng that n the long run wages are equalsed across sectors. Factors are used for producton of goods and servces by a contnuum of ndustres. Wthn ndustres goods are produced by a large number of heterogeneous frms, N. 2 CONSUMPTON As usual, the representatve consumer's utlty s an ncreasng functon of consumed goods produced by one of ndustres, each of whch supples a large number of horzontally dfferentated varetes,, of good. For smplcty, we assume that the upper ter of utlty, U, determnng consumpton of each ndustry's output takes the Cobb- Douglas form wth the fracton of ncome spent on ndustry 's good equal to : U lnq d 0 () where the quantty consumed, Q, s a consumpton ndex defned over the contnuum of horzontally dfferentated varetes,, of good. 2 The lower ter of utlty functon determnng the consumpton of good takes the CES form: N Q q d 0 (2) where q s consumer demand for varety, N s the number of avalable varetes of good and s the constant elastcty of We use captal Latn letters for varables referrng to ndustry (sector), small Latn letters for varables referrng to ndvdual varetes (frms) and Greek letters for denotng parameters of the model. 2 We use the terms "good", "sector", and "ndustry", as synonyms whle term "varety" s reserved for horzontally dfferentated products wthn an ndustry.
Vctor Rvas 8 substtuton between varetes wth. The utlty maxmsaton under ncome constrant, Y, yelds the demand for ndvdual varetes, q : q Y p P P (3) where P s the dual prce ndex defned over prces of ndvdual varetes, p : N P p d 0 (4) 3 PRODUCTON A contnuum of ndustres wth 0, use both factors (labour and captal) for producng goods and servces. Factor ntensty vares across ndustres. The ndex ranks ndustres by ther relatve factor ntenstes L / K, ndustres wth hgher are more labour ntensve. n equaton (5) the labour ntensty n sector s captured by parameter. t s hgher n sectors whch use more ntensvely labour, that s for sectors whch are ranked wth a hgher. Producton process nvolves two types of costs: producton costs and, for those frms whch enter market, also market entry costs. The producton costs are frm-specfc - they vary wth frm productvty,, wth 0,. n contrast, market entry costs are ndustry specfc and have two components: varable market transacton cost, V and fxed market transacton cost, F. 3 3 Gven that all frms share the same ndustry-specfc market entry costs, we use captal letters n order to dstngush them from the frm-specfc producton costs ndexed.
82 Journal of Economcs and Econometrcs Vol. 55, No. 2. n order to avod undue complexty, analogously to the functonal form of the utlty functon we assume that the cost functon, tc, takes the Cobb-Douglas form: 4 q tc W R V F (5) where W s wage rate for labour, R s captal rental rate and s elastcty of substtuton between labour and captal. Constant returns n the technology for formng the composte nput ndcates that the sum of the share parameters,, equals to one. The presence of fxed market transacton cost, F, mples that n equlbrum each frm chooses to produce a unque varety, all of whch are consumed due to the love of varetes of consumers. Proft maxmsaton mples that the equlbrum output prce s equal to a constant mark-up over margnal cost: p W R V where revenue, r /. Wth prcng rule (6), frm's equlbrum, s proportonal to frm's productvty, : (6) P r Y W R V (7) Accordng to equaton (7), gven productvty, frm revenue, r, s ncreasng n the expendture share,, allocated to ndustry,, ncreasng n aggregate consumer expendture (whch equals aggregate ncome, Y ), ncreasng n the ndustry prce ndex, P, whch measures the degree of competton n market, and ncreasng n, whch s an nverse measure of the sze of the mark-up over margnal cost. Frm 4 Despte ths smplfyng assumpton, our analyss generalses to any homothetc cost functon for whch the rato of margnal cost to average cost s a functon of output alone.
Vctor Rvas 83 revenue s decreasng n own prce and hence own (margnal) producton costs, mc W R /. Under the equlbrum prcng rule (6), frm profts,, equal frm revenue, r, scaled by the elastcty of substtuton,, mnus fxed market transacton costs, F : r F V Accordng to equaton (8), frm profts,, are ncreasng n frm revenue, r, and decreasng n fxed market transacton cost, F. 4 FRM PRODUCTVTY A frm drawng productvty operates n market f ts revenue, 0 r, covers at least the fxed market transacton cost, F, that s,. Ths defnes a zero-proft productvty cut-off, ndustry : r V F (8), for Only those frms drawng productvty equal to or above operate n market. Substtutng frm revenue (7) nto zero proft condton (9) yelds the zero-proft productvty cut-off, : (9) F V W R V Y P (0) where Y Y s the expendture share on sector 's goods, MC s margnal cost of the frm margnal frm whch makes zero proft from enterng market, and V are varable market transacton costs. n order to make the model operatonal we need to assume a specfc functonal form of the productvty dstrbuton functon. n order to avod undue complexty, we assume that each of the M potental
84 Journal of Economcs and Econometrcs Vol. 55, No. 2. entrants nto market receve ther frm-specfc productvty draw from a Pareto dstrbuton wth probablty densty, g : g b and cumulatve dstrbuton, G : G b () (2) where b s the mnmum productvty ( [, b )) and s shape parameter ( ). Parameter can be nterpreted as an nverse measure of frm heterogenety n sector. Sectors wth lower are more heterogeneous n sense that more output s concentrated among the most productve frms. Choosng the unts of measurement such that the mnmum productvty, b, equals to unty, the probablty densty functon, g, and the cumulatve dstrbuton functon, G, can be rewrtten as follows: g G where G (3) (4) s the probablty that frm,, wll enter market,, whch equals to the fracton of operatng frms, N, over all frms, M, n sector. Equatons (3) and (4) allow us to derve explct expressons for margnal productvty, average productvty of frms enterng market and for average productvty of frms not enterng market. Frst, nvertng equaton (4) we can express the zero-proft productvty cut-off,, as a functon of frms enterng market :
Vctor Rvas 85 N M (5) accordng to whch zero-proft productvty cut-off s decreasng n the fracton of frms enterng market and frm heterogenety,. More frms can enter market only f the margnal productvty,, decreases wth respect to productvty of potental entrants. Second, from the cumulatve dstrbuton functon (4) we can derve the CES weghted average productvty,, for all frms operatng n market wth productvty. g d G whch smplfes to (6) (7) The second term on the rght hand sde,, s a constant capturng the hgher productvty of entrants. 5 Gven that the mnmum productvty, b, equals to one and b,. Fnally, from the cumulatve dstrbuton functon (4) we can also fnd the CES weghted average productvty,, for all frms not enterng market wth productvty. The dervaton of average productvty for frms not enterng market s analogous to (7). 5 Under the estmated parameter values t takes a value between.25 and.75 mplyng that the average productvty of frms, N, enterng market s 25% to 75% hgher than the cut-off productvty of a margnal frm, whch earns zero profts from enterng market.
86 Journal of Economcs and Econometrcs Vol. 55, No. 2. 5 GENERAL EQULBRUM From the demand functon (3), the optmal prcng rule (6), zero-proft productvty cut-off (0) and average frm productvty (7) we can derve the aggregate market supply functon, X, of N frms operatng n sector : p X P Y M W R F V sze of sector producton and transacton costs (8) where p s a constant. 6 Accordng to equaton (8), sector 's output s ncreasng n the sze of sector and declnng n costs assocated wth producton and market transactons. The aggregate output s ncreasng n the output prce, P, for sector 's goods wth an ncreasng rate, whch s the largest elastcty among the rght hand sde explanatory varables wth respect to output. A hgher prce ndex ndcates a less ferce competton n sector, whch attracts new frms and expands producton of the ncumbent frms. Sectoral output s also ncreasng n the sector 's expendture share, Y Y wth an ncreasng rate.7 Fnally, the aggregate sector 's output s ncreasng n the number of potental entrants (frms) wth a constant rate over the entre nterval. As expected, the costs assocated wth producton and market transactons, MC, V and F, have a negatve mpact on frm producton and hence on the aggregate output. Ceters parbus, the mpact of margnal cost on sectoral output s larger than the mpact of market transacton costs, V and F, because value of the elastcty of margnal cost,, s larger than the elastcty of fxed and varable market transacton costs. 6 p. 7 Ths mples that a hgher demand for sector 's goods (measured n expendture share) ncreases the aggregate producton at an ncreasng rate.
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