A Generalized Solution of the Monopolistic Competition Model with Heterogeneous Firms and a Linear Demand (Melitz-Ottaviano)

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1 A Generalzed Soluton of the Monopolstc Competton Model th Heterogeneous Frms and a near Demand (Meltz-Ottavano) Costas Arkolaks y Yale Unversty Frst verson: Aprl 008 Ths verson: Octoer 008 Astract Ths note solves the monopolstc competton model th lnear demand for multple countres, ut thout an outsde sector. The purpose s to compare the model s predctons regardng rm sales and ther dstruton th those arsng from a model n hch consumers are characterzed y symmetrc CES preferences. Introducton In ths note, e solve a verson of the monopolstc competton model th heterogeneous rms and lnear demand (Meltz Ottavano (008)). We extend the model to a mult-country settng, and allo for ncome e ects y dspensng th the outsde sector. The man results of Meltz-Ottavano stll hold; n partcular, rm mark-ups depend on rm sze. Hoever, e hghlght a numer of ne results for the model. Frst, the dstruton of sales of small rms s more skeed than the smple Pareto dstruton, hch s hat the constant elastcty of susttuton (CES) model ould mply. Ths result s smlar to Arkolaks (008). Second, hen compared to the CES frameork, the model lacks an addtonal degree of freedom n order to match oth lateral aggregate trade across countres and the dstruton of sales. Fnally, the equlrum numer of entrants s a functon of the populaton and other constants, ut not of lateral trade costs. Ths result s the same as the one found y Arkolaks, Demdova, Kleno and Rodrguez-Clare (008) for the CES model. Many thanks to Turkmen Goksel and Ina Smonovska for ther comments. Alex Torgovtsky provded excellent research assstance. All remanng errors are mne. y Department of Economcs, Yale Unversty. Emal: costas.arkolaks@yale.edu.

2 Solvng for the FOCs Assume a measure of dentcal consumers, here each one of them s endoed th unt of laor and does not value lesure. Preferences of a representatve consumer over a contnuum of products! are gven y U q c (!) d! (q c (!)) d! q c (!) d! here ; ; are all postve and q c (!) s the quantty consumed. The consumer maxmzes ths utlty functon suect to the udget constrant q c (!)p(!)d! ; here s the unt age and p(!) s the prce of good!. The FOCs of the aove prolem yeld (8q c (!) > 0) : p (!) q c (!) q c (!) d!: () here s the agrangan multpler. Also, from (), e can derve: q c (!) p (!) q c (!) d! : () et represent consumed varetes, and let M e the measure of ths set. De nng: q c : q c (!)d!; p : p(!)d!; M M and ntegratng () over all! yelds: p q c M q c ) q c p + M : Follong (), demand for varety! s: q (!) p (!) q c (!) d! : R R De nng q : q c M q c (!)d! M q(!)d! (y de nton of q(!)) and susttutng the expresson for q c nto () yelds:

3 q (!) ( p (!) M q c ) ) q (!) p (!) M q ) q (!) p (!) M p + M ) q (!) + M p (!) + M p + M : It follos that q (!) 0 exactly hen p( ) p : + M p + M + M p + M : The pro t maxmzaton prolem of the rm th productvty dra s max p () q () q() max p () + M hch mples the FOC q () p () + M p + M + M p () + M p ; + M + M p () + M p + M + 0 ) p () q () : (3) The FOCs also mply that: + M + M p + M + p () ) p + p () () y de nton of p. Recall that the margnal rm must satsfy p ) p( ) p hch yelds: + p () ) 3

4 Susttutng () nto (3) yelds: + p () (5) q () (6) Also () yelds the follong mark-up: () (7) Usng (5) and (6), total revenue s: p () q () +! (8) and pro ts are: ()! (9) 3 Equlrum Assume the Pareto dstruton of productvtes such that the rms pay an entry cost and dra ther productvtes from G ( ) We assume that the rms pay an entry cost f e to enter n the market. After payng the cost they learn ther productvty. et G e the dstruton of productvty condtonal on entry, so G (). Then expected

5 pro ts, condtonal on entry, are ()dg () ()( ) d ( ) d ( ) " # + d ( ) ( ) + + ( + )( ) + ( + )( ) + ( ) + ( + )( ) ( + )( ) ( ) ( + )( + ) Therefore the free entry condton mples ( G( )) ( ) f e ( + )( + ), ( ) f e ( + )( + ),( ) + f e ( + )( + ), () + + : (0) f e ( + )( + ) aor market clearng, hch n ths case s equvalent to the consumers spendng ther entre laor ncome and s also equvalent to laor market clearng mples r()mdg () M p + p M( ) M ( ) M ( ) + : ( + )( ) dg ()! d 5

6 Ths gves the age tmes the multpler as a functon of productvty cuto : Susttutng () nto (0): ( + )( ) : () M f e ( + )( + ) ( + )( ) M f e ( ) + ( + )Mf e, ( ) M M( + )f e ) + ( + )( + ) + ( ) () ( + )f e hch s the expresson for n terms of prmtves and the equlrum numer of rms. In other ords the numer of entrants s and out of them (+)f e a fracton survves. Thus, the Krugman e ect of the numer of rms ( ) enterng the market s lnear n the populaton of the market s stll here, as far as t concerns the numer of rms that try to enter the market. Hoever, ho many operate ll depend on ho changes th. No to determne the equlrum numer of rms,recall that for 6 0 e have: Frst, note that, usng () and (), + M p + N, ( + M) + M p, M : (3) p ( + )( ) M ( + ) M M( + )f e () 6

7 and p : p()dg () + So usng () and (5) n (3) yelds ( ) d ( ) + d (5) M + ( + ) f e ( + ) M + ( + ) The HS s a lne th slope, the RHS s convex n M s.t. at M0 t s negatve, and as M tends to n nty, t also goes to n nty. Thus, the to curves must ntersect and an equlrum exsts. Gven that e nd M e can also recover from. The case 0 By the de nton of a and usng equaton () then e have a ( + ) M No usng the equaton hch susttutng n (). (6) and of course M ( + ) a M ( ) ( + )f e ) ( ) ( + )f e ) a (+) (7) ( + )( + )f e ( + ) a a ( + )( + )f e (+) (8) 7

8 hch mples that total output for the case that 0 s gven only y laor ncome snce! M p () q () d M ( ) + d! M + M a, ( ) + M ( ) + here e used (), (6) n the last equalty. Notce, that even though average productvty ncreases th also the numer of rms s not as large so n the end GDP depends lnearly n.. Dscusson of results for 0 There are not so many d erent thngs related to the equlrum expressons (7), (8). Whle enters ; N and enters N n the same ay as n the CES model, a d erence can e seen th respect to the relaton and. In partcular, th lnear demand hgher mples hgher. Hoever, hgher productvty does not change the relatonshp of aggregate GDP th snce t s also lnear (gven that the numer of rms s not grong as fast as ncreases). Regardng the dstruton of sales e have p () q () + Usng a ; ( + ) a M and from the Pareto dstruton that e have ( Pr)! p () q () ( Pr) 8

9 and hen e dvde y average sales p () q () dg () ( ) e get the percentle sales normalzed y mean sales y y ( Pr) + + hch has the desrale property that s ndependent of market sze. Fnally, notce that trade shares are gven y P J ( ) J ( ) (+) (+) P J ( ) J ) (9) ( ) J P J. Usng the near Demand for quanttatve analyss Notce that the only coe cent a ectng the dstruton of sales s the curvature of the Pareto,. Ths fact hghlghts one of the man draacks of usng the smple lnear demand model for quanttatve analyss n trade: the same parameter ll e the only one appearng n the gravty equatons, not allong enough degrees of freedom to oth match the oserved dstruton of rm sales and the total trade among countres (see the appendx n the orgnal Meltz- Ottavano paper for the dervaton of total trade os as a functon of trade costs and ages and also expresson (9)). On the contrary, the model th CES demand features that addtonal degree of freedom snce oth and the elastcty of demand,, appear n the expresson for the dstruton of sales. Thus, a generalzaton of the CES frameork such as the frameork n Arkolaks (008) allos oth for a mcro-foundaton of the devatons from CES demand (so that there s a theoretcal underpnnng for a non-strctly CES demand structure) and a quanttatve successful frameork for predctng trade os across countres. Of course the model th lnear demand may e extended n the An addtonal feature of the CES model th market spec c costs of entry s that t can acheve the log-lnear assocaton eteen numer of entrants n a market and the sze of the market. Ths emprcal pattern may not e replcated n a straghtforard ay n the lnear demand model thout market spec c costs of entry. Goksel (008) consders the predctons of the lnear demand model for lateral trade os under these d erent spec catons. 9

10 future to other forms of demand that ncorporate lnear demand as a sucase. In addton, the lnear demand frameork s successful n terms of delverng varale markups somethng that a model ased on CES s not easy to get. 5 Mult-Country Model We can construct a mult-country verson of the model U q c (!) d! q c (!) d! q c (!) d!! here ; ; are all postve and q c (!) s the quantty consumed. The consumer maxmzes ths utlty functon suect to the udget constrant q c (!)p (!)d! ; here s the unt age and p(!) s the prce of good!. The FOCs of the aove prolem yeld (8q c (!) > 0) : p (!) q c (!) q c (!) d!: (0) here s the agrangan multplers. Also, e can derve: q c (!)! p (!) q c (!) d! : () et represent consumed varetes, and let M e the measure of ths set. De nng: q c : q M (!)d!; c p : p (!)d!; M and ntegratng () over all! yelds: p q c M q c ) q c p + M : Smonovska (008) proposes a model th non-homothetc preferences that features varale mark-ups, allong her to capture a postve relatonshp eteen prces of tradeales and per-capta ncome levels across countres. She also tests the quanttatve predctons of the model analyzed n ths note regardng prces. 0

11 Follong (), demand for varety! for a country th a contnuum of consumers of measure s:! p (!) q c (!) d! : We ll consder a symmetrc equlrum here all the rms from source country th productvty choose the same equlrum varales. It follos that q () 0 exactly hen p ( ) p : + M p + M + M p + M : () The pro t maxmzaton prolem of the rm th productvty dra s () max p () q () q ();p (); max p () + M q () p () + M p + M + N p () + M p + here e have replaced for the ceerg transportaton costs and the producton functon for a rm n the cost functon of producng and shppng the good aroad. The aove prolem mples the FOC + M p () + M p + + M 0 ) p () q () : (3) The FOCs also mply that: p 0 ) p ( ) >From the FOC e also have + M p + + N + M p ) + M p + M + M and thus + N p () + M p + + M p ()

12 + M p + + M + M p () ) p () p + and therefore usng (5) and () the quantty s gven y q () p The sales of the rm usng the aove to equatons are p () q () and pro ts are: () + p () q () ( ) 5. Total sales "!!! () ( ) We have that p ( p ) + M + N and lookng at the case 0 p ( ) ) and thus usng equaton (6) can e rtten as p () q () ( )! ) () (5) (6)!#! (7)

13 and thus the average sales are p q J fe(+)(+) X XXX P. P ( + ) ( + ). ( + ) J fe(+)(+) J J ( ) J f e ( + )J ( ) + d ( + ) P N XXXX Notce that the last lne mples that average sales per rm, are not source country spec c. The numer of rms from source sellng to country s and therefore total sales are gven T J N J ( + ) here J can e determned usng free entry and laor market clearng. In partcular udget constrant (hch s equvalent to laor market clearng) mples that X X X J J J p ()q () dg () ( )! dg () + (8) 3

14 pro ts are gven y X ()dg () X ( ) X ( ) X X ()( ) d ( ) ( ) Therefore the free entry condton mples X ( ) ( ) ( ) + + X ( ) ( ) ( ) no replacng the aove equaton nsde (8) X v J ( ) ( ) X J ( ) ( + )( + ) ( ) + + f e ) + f e + + J (9) ( + ) f e Thus, the numer of entrans s ndependent of tar s and trade n general. 6 Blography Costas Arkolaks, 008, Market Penetraton Costs and the Ne Consumers Margn, NBER orkng paper. Costas Arkolaks, 008, Endogenous Varety and the Gans from Entry, The Amercan Economc Reve Papers and Proceedngs, 98 (), -50 Turkmen Goksel, 008, Income, Trade Barrers, and Internatonal Trade, Unversty of Mnnesota, unpulshed mmeo. Marc Meltz and Ganmarco I. P. Ottavano, 008, Market Sze, Trade, and Productvty, Reve of Economc Studes, 75 (), Ina Smonovska, 008, Income D erences and Prces of Tradeales, Unversty of Mnnesota, unpulshed mmeo. ) + d d

15 6. Appendx p (!) (!) q (!) Whch mples that for! and! 0 e have p (!) (!) q (!) p (!) (!) q (! 0 ) d! 0 q (!) X p (!) (!) q (!) p (! 0 ) (! 0 ) q (! 0 ) q (! 0 ) d! 0 X ) p (! 0 ) (! 0 ) p (! 0 ) d! 0 d! 0 ) X (! 0 ) d!0 de ne Q X P X X q (! 0 ) d! 0 p (! 0 ) d! 0 (! 0 ) d!0 p (!) (!) P Q q (!) p (!) (!) q (!) Q P 5