Theory Appendx for Market Penetraton Costs and the New Consumers Margn n Internatonal Trade Costas Arkolaks y Yale Unversty, Federal Reserve Bank of Mnneapols, and NBER October 00 Abstract Ths s an onlne theoretcal appendx supplementng the paper Market Penetraton Costs and the New Consumers Margn n Internatonal Trade. Contents I Theoretcal Results 3 General Marketng Technology 3. The Setup....................................... 3. Entry and Frm Sze................................. 4.3 Growth of Sales.................................... 5.4 Appendx to the Secton............................... 8 I thank Jonathan Eaton for varous suggestons. I am also extremely grateful to Treb Allen and Olga Tmoshenko for excellent research assstance. The vews expressed heren are those of the author and not necessarly those of the Federal Reserve Bank of Mnneapols or the Federal Reserve System. All remanng errors are mne. y Department of Economcs, Yale Unversty. Emal: costas.arkolaks@yale.edu.
Idosyncratc Shocks to Entry Costs 0. The Setup....................................... 0. Entry and Aggregate Sales...............................3 Exporters to Indvdual Destnatons and Sales n France..............4 Normalzed Exportng Intensty........................... 3.5 Appendx to the Secton............................... 5 3 Free Entry 7 3. The Setup....................................... 7 3. Solvng for the Equlbrum............................. 9 4 Departng from the CES Aggregator Assumpton: The Lnear Demand Case 4. The Setup....................................... 4. Dstrbuton of Sales................................. 3 4.3 Entry and Aggregate Sales.............................. 4 4.4 Appendx to the Secton............................... 7 5 Market Penetraton Costs: Alternatve Interpretatons 30 5. Persuasve Advertsement.............................. 30 5. Random Evaluaton.................................. 3 II Robustness 37 6 Dstrbuton of Sales 37 7 Normalzed Average Sales 4 8 The Parameters a; and Normalzed Frm Entry 43 9 Export Growth and Marketng Convexty 45 0 Export Growth n the US-France,-Germany,-Mexco cases 47 Trade Costs 50
Part I Theoretcal Results General Marketng Technology In ths secton we derve a set of results for a marketng technology that s more general than the one presented n Arkolaks (008). The purpose s to llustrate a general set of condtons under whch you can have an equlbrum behavor that resembles the one consdered n that paper. However, we wll abstract from general equlbrum consderatons and focus on the behavor of the ndvdual rm.. The Setup We wll retan the same notaton as n the man text of the paper. The entry cost f (n ) nto a market depends on the fracton of the consumers n [0; ] reached. The followng assumptons are made concernng f (n ) (also f (0) 0 s an auxlary assumpton): Assumpton f 0 (n ) > 0 8n [0; ] Assumpton f 00 (n ) > 0 8n [0; ] Assumpton 3 lm n % f 0 (n ). These assumptons are sats ed by the functon presented n Arkolaks (008), but also hold for other functons, e.g. f (n) n. Note that assumptons and mply the exstence of one-to-one ncreasng functons f () and f 0 Followng the setup n the man paper, whch uses a Dxt-Stgltz demand functon, and usng the optmal prcng rules of the rm, the pro ts for a rm wth productvty reachng n fracton of consumers n market are gven by ~ w () n P (). X {z } varable pro ts f (n ) {z }. () entry costs where ~ ( ) s the constant markup, s the elastcty of substtuton among varetes of the good, s the ceberg transportaton cost, w s the wage n country (the exporter), X 3
s the market sze of the (mportng) country and nally P s the Dxt-Stgltz prce ndex. For an nteror soluton the FOC wth respect to n s: Notce that f ~ w P X f 0 (n ) f n (0; ). () ~ w P X f 0 (0), (3) then n 0 so that an nteror soluton does not exst for such s. Addtonally, Assumpton 3 guarantees there does not exst a [0; ) such that n ;.e. no rm wll reach all consumers, regardless of how great ts productvty.. Entry and Frm Sze Entry We can solve for n by nvertng equaton (). Notce that Assumptons and 3 for the functon f and the nverse functon theorem mply that f 0 x tends toward n nty. Snce >, (~ w ) ncreasng n. P (x) ncreases and approaches as ) s w X s ncreasng n, so f 0 (~ P From the above dscusson t drectly follows that there exsts a threshold productvty such that: n () f 0 f (0)! 0 f ; X (4a) n () 0 f (0; ], (4b) where s de ned by: ~ w P X f 0 (0), f 0 (0) X (~w ) P. (5) Sales The sales of a rm wth productvty are: y () n () ~ w P X. (6) 4
Substtutng nto (5), we can rewrte the sales of the rms as: 0 ~ w C ~ w y () f 0 B @ P X A P! f 0 f 0 (0) X f 0 (0) h (7a). (7b) Gven Assumptons -3, h has the followng propertes: a) Sales tend to 0 as!. Ths can be seen from (7b) ; notng that h h () 0 snce f 0 (f 0 (0)) 0. Snce f 0 () s ncreasng and contnuous, h s ncreasng and contnuous n, so the result s mmedate. Also, more productve rms sell more. b) Sales tend to as!. Notce that goes to n nty as goes to n nty. From Assumpton 3, f 0 f 0 (0) goes to as goes to n nty, so the result s mmedate from (7b)..3 Growth of Sales A more subtle queston s how sales of rms of d erent productvtes respond d erentally to changes n the trade costs, whch can be wrtten purely as a functon of changes n. Spec cally, do smaller rms ncrease ther sales more than larger rms when trade costs declne? In what follows, we wll show that they do, provded certan reasonable condtons are sats ed. Frst note that as varable trade costs fall equaton (5) mples that falls;.e. less productve rms can trade. Takng logs of (7b), we have that: For smplcty, we focus on the e ect of a change n on sales.! ln y ln + ln f 0 f 0 (0) + ( ) ln + ln f 0 (0), (8) so that @ ln y @ ln @ ln f f 0 0 (0) @ ln ( ). (9) 5
Consder. Usng the chan rule and de nng x f 0 (0) : @ ln f 0 @ ln (x) @ ln f 0 (x) @ ln x @ ln x @ ln (0) @f 0 (x) x ( ). () @x f 0 (x) By the nverse functon theorem, @f 0 @x (x) f 00 (f (x)). () Hence, @ ln f 0 @ ln (x) x ( ). (3) f 00 (f 0 (x)) f 0 (x) From Assumpton, f 00 () > 0 and f 0 () > 0 f, so by (9), @ ln y < 0,.e. a fall n @ ln (such as from a fall n transportaton costs) results n an ncrease n sales. How does such an e ect change wth rm sze? Snce x s monotoncally ncreasng n, smaller rms (.e. rms wth lower productvty ) sales wll ncrease more to a fall n than larger rms sales f and only f: Note that: f 00 (f 0 (x)) f 0 (x) @ x @x f 00 (f 0 (x)) f 0 (x) < 0. (4) @ x @x f 00 (f 0 (x)) f 0 (x) x (f 000 (f 0 (x)) f 0 (x) f 0 0 (x) + f 00 (f 0 (x)) f 0 0 (x)) [(f 00 (f 0 (x)) f 0 (x))] (5a) f 00 (f 0 (x)) f 0 (x) x (f 000 (f 0 (x)) f 0 (x) f 0 0 (x) + f 00 (f 0 (x)) f 0 0 (x)) [(f 00 (f 0 (x)) f 0 (x))]. (5b) Hence x f 000 f 0 (x) f 0 @ x < 0, (6a) @x f 00 (f 0 (x)) f 0 (x) (x) f 0 0 (x) + f 00 f 0 (x) f 0 0 (x) > f 00 f 0 (x) f 0 (x),(6b) f xf 0 0 (f 0 (x)) (x) f 00 (f 0 (x)) + >. (6c) f 0 (x) Usng equaton () equaton (6c) smpl es to: f 000 (f 0 (x)) x [f 00 (f 0 (x))] + >, (7) f 0 (x) f 00 (f (x)) 6
whch s the condton so that the growth s hgher for rms wth smaller ntal sze. Note that f 0 (x)f 00 (f (x)) > 0 snce f 0 (x) > 0 and f 00 (f (x)) > 0. Hence, a su cent condton for (7) to be sats ed s: x f 000 (f 0 (x)) [f 00 (f 0 (x))] >, (8a) @f 00 (f 0 (x)) x @x f 00 (f 0 (x)) >, (8b) @ ln f 00 (f (x)) @ ln x >, (8c).e. the elastcty of the second dervatve of the cost functon wth respect to productvty of the rm needs to be bgger than one. Notce that ths condton requres f 000 > 0. Notce also that so that the above condton can be wrtten as Usng (9) we can rewrte condton n equaton (7) as n f 0 (x) ) f 0 (n) x, (9) @ ln f 00 (n) @ ln f 0 (n) > f 000 f 0 [f 00 ] >. (0) f 0 (n) f 000 (n) [f 00 (n)] + f 0 (n) nf 00 (n) >. () In the appendx of ths secton we show that an dentcal condton s requred f we are tryng to explan asymmetres n growth wth a convex producton cost functon (n a perfect competton envronment). A somewhat smlar condton s necessary and su cent for growth of younger rms n the Jovanovc (98) model. We wll now llustrate some examples of partcular functons. The functon that s used n the man text of Arkolaks (008) s f (n) ( n) so that f 000 f 0 [f 00 ] ( ) ( n) ( n) + ( n) Notce that for the functon n, usng condton (7) we have >. f 0 (n) f 000 (n) [f 00 (n)] + f 0 (n) nf 00 (n) n ( ) ( ) n 3 n [ ( ) n ] + n ( ) n ( ) ( ) + ( ). 7
It can be ver ed by assumng f (n) n a nsde the framework of Arkolaks (008) that ths marketng functon mples that, locally, all rms grow at the same rate. However, f n s bounded, due to e.g. saturaton, the small rms (the ones wth n < ) wll grow faster when trade costs declne even wth that assumpton..4 Appendx to the Secton I now consder the condton for whch a producton cost functon can generate asymmetres n the growth rates of rms of d erent sze. Assume that c (q) s an ncreasng and convex cost functon (so that the producton functon s ncreasng and concave). s the productvty of the rm and p s the compettve prce of the good. Assume that a rm takes prces as gven and solves: max pq q c (q) ) p c 0 (q) c 0 (p) q. ) The nverse functon theorem mples c 0 (p) 0 c 00 (c 0 (p)). The growth rate as a functon of a reducton n prce (e.g. trade lberalzaton) s gven by @ ln c 0 (p) @ ln (p) p c 0 (p) c 00 (c 0 (p)) c0 (q) qc 00 (q). To see how ths growth rate changes wth we take the dervatve of the negatve of the above expresson wth respect to q (snce q and are ) c 00 (q) qc 00 (q) c 0 (q) [c 00 (q) + qc 000 (q)] [qc 00 (q)] < 0 ) qc 0 (q) c 000 (q) c 00 (q) qc 00 (q) + c0 (q) c 00 (q) c 00 (q) qc 00 (q) > ) c 0 (q) c 000 (q) [c 00 (q)] + c0 (q) qc 00 (q) >. 8
Ths s the same exact condton as for marketng cost functon, equaton (), but now n terms of quanttes and cost functons. 9
Idosyncratc Shocks to Entry Costs Ths secton dscusses the mplcatons of addng random shocks to entry costs n a framework of monopolstc competton and heterogeneous rms. The man ndng s that a random shock does not alter a number of the man predctons of the monopolstc competton model when the dstrbuton of productvtes s Pareto.. The Setup Consumer s demand n country s Constant Elastcty of Substtuton (CES) wth elastcty and wth prce ndex P and rms produce n a monopolstcally compettve envronment wth constant margnal costs of producton. There s a measure of J rms n country. Frms pay a cost to reach foregn consumers that depends on the fracton of consumers reached n where the spec caton of that cost follows the dervatons n the man paper (Arkolaks (008)). We assume that the rm has a separate random shock n ts entry cost to each market, a shock whch s..d. across rms. Thus, a rm s xed cost of entry to country s represented as the product of a certan cost F, a random component f, and a component that depends on the fracton of consumers reached, n. The probablty dstrbuton of f s the same across destnaton markets so that f G f. We wll denote by M (f ) the measure of rms from wth a shock f that enter n country. Each rm n country receves a productvty draw G ;, where t s assumed that G ; s a Pareto dstrbuton and where productvty determnes the margnal cost of producton. There s an ceberg transportaton cost for a rm n country to sell to country. These assumptons mply that the pro ts of a gven rm! from country n country chargng a prce p and reachng a fracton of consumers n and wth random entry shock f (!) and productvty are gven by P ( (!) ; f (!)) n p X w (!) n p P X F f (!) ( n ), where w s the wage n country (the exporter) and X s the market sze of the (mportng) 0
country. Maxmzaton of ths expresson wth respect to p gves us ( ) p P X + w (!) p P X 0 ) p w. Substtutng the optmal prce to the pro t functon results n ( (!) ; f (!)) n w (!) P X F f (!) ( n ). Optmzng wth respect to n gves the entry cuto rule for market for rms wth entry shock f, and optmal n, 0 (f ) B @ n (; f ) ( F f (!) w ) 6 4 P (( X F f (!) C A ) w ) X P 3 7 5, (a). (3) Note that the threshold productvty depends on the realzaton of the random xed cost shock f ; to emphasze ths fact, n what follows we refer to as (f ). We can also calculate the sales of a rm n market, y ( (!) ; f (!)) as a functon of (f ): y ( (!) ; f (!)) 0 B @ 6 4 (( F f (!) Notce that sales are ncreasng n productvty. F f (!) ) w ) X P 3 C 7 5 C A w (!) P 4 (!) (!) (f ) (f ) X! ( ) 3 5. (4). Entry and Aggregate Sales We can calculate the average sales of rms from country that sell to some country. Snce G ; s Pareto dstrbuted, the probablty densty for all the rms wth a common entry
shock f condtonal on sellng to country can be wrtten as () sales are then gven by Z Z X y ( (!) ; f (!)) (f) (f ) ddg f ( (f )). Average Z Z F f (f) " ( )! 3 4 5 (f ) (f ) (f ) ddg f # F ( ) E (f), (5) where Z E (f) fdg f. Hence, average sales depend only on the expected value of f ; they are una ected by all other characterstcs of the dstrbuton. In the appendx for ths secton we show that the number of rms s related to market sze of the destnaton market by M + X ( ) F E (f). (6) The mmedate concluson from these two expressons s that the magntude of aggregate blateral entry and average sales of these entrants are una ected by random entry costs (Notce that specfyng a blateral xed cost F, as n Chaney (008), nstead of F wll leave the results ntact)..3 Exporters to Indvdual Destnatons and Sales n France A systematc ndng n French trade data of Eaton, Kortum, and Kramarz (00) s that rms that sell to a greater number of destnatons have larger domestc sales on average. The theory outlned above s consstent wth ths ndng snce more productve rms export to more destnatons and sell more n each destnaton. Formally, consder the domestc sales of all rms n country that are su cently productve to export to country ;.e. for each possble cost f, consder all rms wth productvty at least (f ). As above, assume that the productvty dstrbuton s Pareto, so that the probablty densty functon condtonal on
exportng to country for a gven f s (; f ) sales condtonal on sellng to market are 0 Z Z Z F f @ (f ) ( (f )) (f ) Z Z F + " (f ) f (f ) (f ) (f ) (f ). Thus, average domestc! A # dg f dg f. (f ) ddg fdg f Ths expresson nvolves both the entry thresholds for country and country, and these thresholds depend on the entry shocks f and f. In the appendx of ths secton we show that the rato (f ) (f ) s gven by M f M f ; where M s the measure of rms n country exportng to country, so that the expresson above equals to Z Z F ( + ) F ( + ) X " M M " M f " M M M M M M M M #, M # E (f) # f dg f dg f. f where we used the expresson for average sales gven by (5). If M > M (.e. f more French rms produce domestcally than export to country ), those rms that export to country wll have hgher than average sales domestcally. Notce that ths expresson s not n uenced by dosyncratc entry shocks at all..4 Normalzed Exportng Intensty De ne the normalzed export ntensty for a gven rm as the rato of ts relatve export sales to country to ts relatve domestc sales. Normalzed export ntensty for a rm wth productvty 3
and entry shocks f and f s gven by y ( (!) ; f (!)) X y ( (!) ; f (!)) X ( w ) n P X + F E(f) ( w ) n P X + F E(f) f f ( (f )) (f ) M M Notce that we used the relatonshp (a) and the fact that (proven n the appendx of ths secton). Usng ths fact agan, y ( (!) ; f (!)) X y ( (!) ; f (!)) X M M ( (f ))! ( (f )) ( (f )) M M! ( (f )) (f ) (f ) ( (f )). s equal to M M f f ( (f )) Notce that the dstrbuton of f f s by assumpton ndependent of. Notce also that whereas a gven f mples a d erent (f ) for each country, the dstrbuton of (f ) s always Pareto. Ths analyss means that the dstrbutons of both f f and ( (f )) ( (f )) are ndependent of the destnaton country. Of course, for d erent spec caton of the dstrbutons of f s, the levels of f f, could be d erent for a gven percentle ( (f )) ( (f )) of sales (but stll the same across each destnaton country, for a gven percentle wthn the country). The calbraton of Arkolaks (008) s based on choosng a certan percentle of sales for each country, and lookng at the varatons of normalzed exportng ntensty as the number of entrants, M, changes for each country. The above mply that choosng d erent dstrbutons for f s should only mnorly a ect the calbrated values of obtaned n Arkolaks (008): The calbraton s heavly based on varatons of normalzed exportng ntensty (and normalzed average sales, see above) across s, caused entrely by changes n M M, to dentfy (and ~ ). 4. f f Thus, the fact that the
level of f f ( (f )) wll be d erent for d erent dstrbutons of f s means that lttle on that calbraton hnges on the spec caton of the dstrbuton of f s..5 Appendx to the Secton We start by computng the market shares of rms from country n country. We use the measure of rms that would enter n market f they had shock f, M (f ) J (b ) (f ). (7) Usng equaton (4) that gves the sales of an ndvdual rm, we have that the market share of country to country are gven by R M (f ) R " f (f ) ( (f )) ( (f )) P R M (f ) R " f (f ) ( (f )) ( (f )) # ddg ( (f )) f # ( (f )) ddg f R J (b ) (f ) dgf P R J (b ) (f ) dgf J (b ) ( w ) ( E f ) P J (b ) ( w ) E (f ( ) ). cancellng out the expectatons and lettng b 8 for the rest of our analyss we have J ( w ) P J ( w ). (8) The total sales of country to country can be represented as X X ) Notce that these dervatons mply that the total number of avalable varetes s gven by N X N X + X ( ) F R f dg f + X ( ) F R f dg f 5
M M X X ) + The prce ndex n market s: X Z Z M (f ) p () n (; f ) X " w + P X ( ) F E (f) (f ) ddg f ( ). (9) # Z M (f ) (f ) dgf. where we have substtuted for n (; f ) from equaton (3) and for p () usng the constant markup choce of the rm. Notce that the total measure of entrants s R M (f ) dg f. Replacng for the measure of rms wth entry cost f that would enter n market ; M (f ) J (b ) (f ) (equaton (7)) we have P X Z J (b ) w c where c. + ( ) Replacng wth the de nton of (f ), equaton (a) we get 4 P w! F X 3 5 Z F c X f (f ) dgf ; dg f P J (b ) w w and usng the cuto rule from equaton (a) and the formula for the market shares, equaton (8), we have: (f ) Z F c X Notce that usng (9) we have that (f ) J (b ) M f R f Usng ths last equaton we get that (f ) (f ) dg f J (b ) E (f) M M dg f (f ). (f ). f f. (30) 6
3 Free Entry In ths secton we show that the model n Arkolaks (008) wth a predetermned number of potental entrants (followng Chaney (008)) gves dentcal solutons to a model where there s a free entry of new rms n each country (followng Meltz (003)). The only d erence of the free entry setup wth the Chaney (008) setup s that all the pro ts are accrued to labor n order to pay the xed costs of entry. 3. The Setup All the assumptons except the one regardng entry of rms s as n the man paper. Consumer preferences are Dxt-Stgltz wth an elastcty. A rm n country wth productvty that s reachng fracton n of the consumers of country and chargng prce p earns pro t P () n p w L w P w L n p w w L ( n ) + +, where s the ceberg transportaton cost, w s the wage n country (the exporter), w s the wage rate and L the populaton of country, and nally P s the Dxt-Stgltz prce ndex. The rm s optmal choce of p, whch s ndependent of n, s determned by the rst order condton Gven ths choce of p, the rm s pro ts are () n p ~ w, ~ ( ) : ~ w P w L w w L ( n ) + : + The rst order condton for the rm s market penetraton choce s then gven by n 0 B @ ~ P w L w w L C A : (3) Let supf : () 0g. Clearly, a rm wth productvty wll choose n 0. The rst order condton for market access thus mples L P w w (~ w ) ; w L 7
Substtutng ths back nto (3) yelds n ()! as the rm s optmal market access, condtonal on enterng market. The rm s pro ts from market are () 4 w w L! 3 5 " ( ) + ~ w P + # : w L Frms have to pay a xed entry cost, f e, n order to enter the market and receve a productvty draw. The productvty of a new entrant s assumed to be a Pareto random varable wth shape parameter > entrant s productvty s therefore G () and locaton parameter b. The dstrbuton functon of a new b. A rm that receves a productvty draw lower than mmedately exts the market. In equlbrum, free entry mples that expected pro ts must be zero: X Z X Z " X ww X ww X ww w w L L ( L ) # " + L 4 w P + ( ) + ( ) ( ) ( ) ( ) + ( ) ( ) w L ( ) # ( ) w f e b ( ) + ( ) ( ) d ( ) d w f e, b ( ) ( ) ( ) ( ), 3 ( ) ( + ) 5 ( ) ( ) w f e : (3) b ( ) 8
3. Solvng for the Equlbrum The equlbrum number of rms producng n country, N, s determned by the followng labor market clearng condton: " X N ww L ( ) + + X Z N w w L ( ) + X Z + N ( ) ww L N X + w L w ( ) ( ) ( ) + + ( ) + # ( ) ( ) + ( ) + f e ( ) + N G ( ) ( ) d ( ) d L, 3 ( ) 4 5 ( ) ( + ) ( ) ( ) + N f e G ( ) " X w L ( N ) + w ( ) + ( ) X # w L N w ( ) L ( ) ; (33) where the last equvalence follows after substtutng the free entry condton, (3). We wll use the fact that marketng spendng s a constant fracton, m, of sales (see the paper for detals). Total ncome s composed of ) ncome other than xed costs, ) ncome from xed costs from exportng actvtes, ) ncome from exportng actvtes of foregn countres n country, and the sum of all these equals total spendng: ( m + m) X " ww L + ( ) m X " w w L + " L + X w w X ww L N ( ) ( ) X ( ) # ( ) # ( ) w w # N ( ) ( ) + N ( ) ( ) N ( ) ( ), L ( N ) ( ) ; 9
whch, substtutng nto (33) and usng (3), yelds 3 X N ww L ( ) 4 5 ( ) ( + ) ( ) ( ) + N f e G ( ) " # ( ) X + N L, ( ) + f e ( + ) N + N b ( ) N f e L b ; ( ) + ww L ( ) ( ) f e L b, ( ) whch yelds the measure of operatng rms N but also the number of equlbrum entrants N ( ) b. Notce that the share of ncome that goes to labor for the producton of the entry cost, ( ) (), equals exactly the share of total ncome that goes to pro ts n the model wth no free entry n the man paper n Arkolaks (008). All the remanng equatons of the model are dentcal to the ones of the model wth no free entry. 0
4 Departng from the CES Aggregator Assumpton: The Lnear Demand Case In ths secton, we solve a verson of the monopolstc competton model wth heterogeneous rms and lnear demand (Meltz and Ottavano (008)) and characterze rm entry and the dstrbuton of rm sales n ndvdual exportng destnatons. We show that whereas the lnear demand (a model that features departures from the CES aggregator and non-homothetc demand) has some qualtatve propertes that are algned wth the data, quanttatvely t s not able to match the rm entry patterns and the sze dstrbuton of rms. 4. The Setup Assume a measure L of dentcal consumers n each country, where each one of them s endowed wth unt of labor and does not value lesure. Preferences of a representatve consumer over a contnuum of products! are gven by Z U q c (!) d! Z q c (!) d! Z q c (!) d! where ; ; are all postve and q c (!) s the quantty consumed. The consumer maxmzes ths utlty functon subect to the budget constrant Z q c (!)p (!)d! w ; where w s the unt wage and p (!) s the prce of good! n country. The FOCs of the above problem yeld (8q c (!) > 0) : Z p (!) q c (!) q c (!) d!: (34) where s the Lagrangan multplers. Also, we can derve: q c (!) Z! p (!) q c (!) d! : (35) Let represent consumed varetes n country, and let M c set. De nng: q c M c Z q c (!)d!; p M c Z p (!)d!;! be the measure of ths
and ntegratng (34) over all! yelds: L s: p q c M c q c ) q c p : + M c Followng (35), demand for varety! for a country wth a contnuum of consumers of measure L Z! p (!) q c (!) d! : We wll consder a symmetrc equlbrum where all the rms from source country wth productvty choose the same equlbrum varables. It follows that q () 0 exactly when for some. market : p ( ) p + M c p + M c + M c p : (36) + M c Frm maxmzes revenues mnus producton and shppng cost n each () max p () q () q ();p () L max p () p () + M c w The above problem mples the FOC L + M c w q () L p () + L M c p + M c L p () + L M c p. + M c L + M c L p () + L M c p + M c + w L 0 ) q () L p () w (37) The FOCs also mply that: L p w 0 ) p ( ) w. From the FOC we also have L + M c + L M c p + M c + w L L p L w ) L + M c + L M c p + M c L w
whch can be wrtten as L L + M c + L M c p w L + + M c p () ) p () p w +, (38) and therefore usng (39) and (38) the quantty s gven by q () L p w. (39) From ths pont on we wll assume the Pareto dstrbuton of productvtes of rms so that G () (b ). The probablty densty condtonal on s gven by and from now on we also mantan the assumpton that whch mples that 4. Dstrbuton of Sales (), (40) + 0, p ( ) ) The sales of the rm can be wrtten as w w p () q () + " L p () q () 4 w. (4) L w w And usng equaton (4) n equaton (4) we have p () q () w L 4 ( ) w L 4 3 # w. w ). (4)!
Wth the use of the Pareto dstrbuton assumpton we get a stark predcton about the dstrbuton of sales. In partcular, usng the Pareto dstrbuton together wth (4) we have that Pr. Thus, usng equaton (4), the sales of a rm at a gven percentle Pr are gven by y ( Pr) L 4 h ( Pr). (43) Combnng (43) and the expresson for average sales, X, equaton (45) (derved rght below) mples that the dstrbuton of sales normalzed by average sales s y ( Pr) X ( Pr) (+), (44) whch s ndependent of mportng or exportng country characterstcs ust as n the model wth CES demand. However, one addtonal feature that should be ponted out s that as Pr! the relatve sales of the largest exporters do not go to n nty but rather to a constant number. Thus, arbtrary large ncreases n do not translate to arbtrarly large ncreases n relatve sales as t would happen n the CES model. The reason for ths result s the fact that the lnear demand s asymptotcally nelastc and not elastc as the CES demand. Nevertheless, when! ; y! 0 whch allows the model to generate rms wth tny sales as n the Arkolaks (008) model. 4.3 Entry and Aggregate Sales Average sales are gven by ntegratng the above expresson over the condtonal pdf, equaton (40), Z X w L 4 w L 4 L 4 d + ( + ). (45) ( + ) 4
Notce that the last lne mples that average sales per rm, are not source country spec c. The number of rms from source country sellng to country s M J b, and therefore total sales are gven X J b L 4 ( + ), (46) J can be determned usng free entry and labor market clearng condtons. In the appendx of ths secton we show that n equlbrum. J L ( + ) f e, (47) We wll start rst by de nng the market share of country to country : X P X where we can use (46) and (47) to wrte the relatonshp as P L (+)f e L (+)f e b ( ) L 4 b ( ) L 4 ; (+) (+) L b (w ) P L b (w ). (48) To examne the relatonshp between market share and number of rms, use the dentty w L X ) w L M L ( + ) ) M w (+). (49) From expresson (4) ths equaton can be re-expressed as M (+). Therefore, we would expect to see more rms n a destnaton f the average producer there s more productve. 5
Notce that usng the results of secton (4.4) and equaton (4) we can wrte w X + w b P J J b w b ( w ) (w ) + L (+)f e b + (+) + ( + ) f e ( + ) ( + ) ) (+) ) L ) (+) : (50) Replacng ths expresson nto the equaton (49) we see that the model mples more rms (normalzed by market share) n markets wth hgher b and L condtonal on, h (+) (+) M + b (+) L (+)f e(+) (+) : (5) Arkolaks (008) usng the French data by Eaton, Kortum, and Kramarz (00) reports that normalzed entry s postvely related to both populaton and ncome per capta of the market. In the relatonshp above, b (whch s postvely related to ncome per capta, w ) and L postvely a ect normalzed entry, M. Ths means that qualtatvely the model has the ablty to match the stylzed fact that normalzed entry s ncreasng n w and L. However, usng the relatonshp (48) to replace for, we then have " # h (+) M X (w ) (+) L b (w ) Ths relatonshp mples the followng: (+) + (+)f e(+) (+). (5) a) The relatonshp of normalzed entry wth respect to populaton s weakly postve snce the populaton of country s n the summaton term and ts changes are lkely to a ect the sum only by a lttle. b) The relatonshp of normalzed entry wth respect to ncome per capta s postve snce the term outsde the summaton has the correct coe cent. Notce that nsde the summaton there are the terms b and w whch are both related to country ncome, but n opposte ways. Snce both terms are n the summaton they are lkely to a ect entry less than the term outsde the summaton. Smonovska (009) also assumes a non-homothetc demand system and ponts out a smlar ndng. 6
Smlar dervatons can be used to express average sales as X w L M h b w b (+) w L (+) + L (+)f e(+) (L ) (+) (+) ( + ) h (+) + (+)f e(+) (+). (53) The data of Eaton, Kortum, and Kramarz (00) also ndcate a postve relatonshp between average sales of French rms n a market wth populaton and ncome per capta of the market. The postve relatonshp between average sales and populaton seems dubous for the same reason as for the case of normalzed entry. The relatonshp of average sales wth ncome per capta seems more lkely to be sats ed. 4.4 Appendx to the Secton Aggregate Entry In partcular, budget constrant (whch s equvalent to labor market clearng) mples that w L X w L X w L X J J J b b b Z Z p ()q () () d ) L 4 w L 4 ( ) w! w () d ). (54) + Usng equaton (4) pro ts can be wrtten as () L 4 w! w L () 4 ( ) w w L w w ) w. (55) 7
Thus, expected pro ts are X Z () () d X Z ( ) X X w Therefore the free entry condton mples b X ( ) w b X ( ) ()( ) d L 4 ( ) L ( ) 4 L ( ) 4 ( ) + + L w ( ) ( ) now replacng the above equaton nsde (54) w L X b L v J ( ) 4 w ( ) X b L w L J 4 w ( ) J Z w ( ) + + ( + )( + ) w w f e ) + f e. + + ) ) + d L ( + ) f e (56) Thus, the number of entrants s ndependent of varable trade costs and trade n general. Trade Balance To complete the descrpton of the model we dscuss the mplementaton of the trade balance condton. The budget constrant of the representatve consumers mples. w L X J b {z } measure of entrants from av.sales n Usng equaton (45) we can wrte ths expresson as w L X J b L 4 ( + ). (57) We can also ask where s the ncome of the consumers derved from. Ths ncome s derved from pro ts (whch due to free entry equal entry costs) and producton costs. Notce that both 8
pro ts and producton costs are pad by domestc rms to domestc consumers and pro ts + producton costs total sales of domestc rms. Therefore, we can also wrte the trade balance n the followng form: w L X X J b {z } measure of entrants n J b L 4 av.sales n ( + ). (58) As a result of the assumptons of the model, equatons (57) and (58) mply that trade s balanced. 9
5 Market Penetraton Costs: Alternatve Interpretatons In ths secton we dscuss alternatve hypotheses for the nformatve advertsng theory presented n Arkolaks (008). The dscusson s focused on developng mathematcal somorphsms to that framework. 5. Persuasve Advertsement The purpose of ths subsecton s to show that there exst an somorphsm of the model of Arkolaks (008), where rms pay a cost to reach more consumers, wth a model where rms pay a market penetraton cost to ncrease ther sales per consumer. The problem of the consumer s Z max x(!) Z s.t. u (!) x (!) d! p (!) x (!) w + where p (!) represents the prce of good!, x (!) the quantty demanded by the representatve consumer, y s ncome per capta and ( demand per consumer s where x (!) u (!) p (!) Z P! ) >. Frst order condtons gve that the P (w + ), u (!) p (!) d!. (59) There s a measure of L consumers. Lookng at a symmetrc equlbrum where all the rms wth the same productvty face the same optmzaton problem we can wrte the problem of a rm wth productvty as thus max u p u;p P (w + ) L uw p (w + ) L wg (u ()), (60) P P Z M u () p () () d, (6) where () s the probablty densty of rms condtonal on operatng ( > ) and M s the measure of operatng rms. 30
FOC gve us: wth respect to p () wth respect to u () p () w w, (w + ) L P g0 (u ()). Proposton Assume that the market penetraton cost functon s g (u ()) ( u) + +. Then the problem de ned above s somorphc to the one of Arkolaks (008) where the market penetraton cost ncreases as a functon of the number of consumers reached. Proof: Frst notce that the cuto of producton, that determnes the number of operatng rms M; s gven by ( ) (w) (w+)l P In the case that the market penetraton cost s a functon of the fracton of consumers reached (I denote all equlbrum varables n ths case wth a tlde). ~ ( ~w) ( ~w+~)l ~P. Total labor requred for market penetraton costs s Z ( u ()) l m () d, where for the case that the market penetraton cost s a functon of the fracton of consumers reached, ~n (), Labor demand for producton s Z ~ ( ~n ()) lm Z l p u () w Z ~ lp ~n () ~w () d. p () (w + ) L () d, P ~p () ( ~w + ~) L () d, ~P 3
respectvely and wth Z ~P M ~ ~n () ~p () ~ () d. We also have that the e ectve demand for a rm s y () u () L p () (w + ), P where for the case that the market penetraton cost s a functon of the fracton of consumers reached we have ~y () ~n () L ~p () ( ~w + ~). ~P Fnally, for the case of market penetraton n terms of fracton of consumers reached we have ~x () ~p () ( ~w + ~). ~P Now, de ne the followng varables ~n () u () ; P P ~, y () ~y (), x () ~x (), p () ~p (), M M ~, l m () ~ l m (), l p () ~ l p (), w ~w; ~. Then, assumng Constant Elastcty of Substtuton utlty functon wth the same elastcty of substtuton parameters, the same productvty dstrbuton for rms as well as the same technology for producng the goods, the models of market penetraton by reachng more consumers or sellng more per consumer are somorphc. 5. Random Evaluaton We now construct a case that s partally somorphc to Arkolaks (008). We assume that consumers have a random evaluaton of each good whle the rm has a constant margnal cost to reach addtonal consumers. Therefore, there are constant returns to scale n the marketng technology but decreasng revenues accrued from addtonal consumers. For smplcty we assume that there s a measure of consumers n the market. The notaton s as n the prevous subsecton. Let the demand of a consumer for a spec c good be x (!) a (!) + y p (!), y w + P 3
where a (!) s an d shock that each consumers gets for each good! that s randomly drawn from an dentcal Pareto dstrbuton wth support (0; a] and > a Pr [A < aa], a. In partcular, for > 0. In the cases where > 0 the probablty densty functon s well de ned n the nterval (0; a], In fact, the mean s gven by Pr [A aa] a a +. Z a 0 a a a da Z a a 0 a da + a, whch means that t s well de ned for > 0. To prove that Pr [A a] s a pdf we have Z a a 0 a da a a 0 (f > 0). a The cases that we wll analyze correspond to the theory of Arkolaks (008) for [0; ] as we wll llustrate below. Frm problem We assume that the rm has to pay a xed cost f to reach each ndvdual consumer (the problem can be easly extended to the case that Arkolaks consders where the cost s not lnear but convex n the number of consumers). The pro ts of a rm that charges a prce p and reaches n fracton of the consumers are Z a p a + y p ( n) a P a (a) da {z } sales Z a w a + y p ( n) a P a (a) da {z } producton cost nf {z} marketng cost The rst term s the total sales to n fracton of the consumers (to calculate the lower bound of the dstrbuton ( n) a that corresponds to reachng n fracton of the total populaton smply set n Pr [A > aa] and use the cdf). The second term corresponds to the producton and shppng costs (wth margnal cost w, where w s the wage and s the ceberg transportaton 33
cost). Smplfyng the expresson we get Z a p p w p w y p P + + w a + y p ( n) a 0 B @ a++ (a) P a da nf ) a h ++ ( n) a (a) C A nf ) y p P + + a+ ( n) (++) nf. Notce that the soluton of the optmal prcng problem s a constant markup such that p w. The choce of reachng an addtonal fracton of the consumers can be represented as the dervatve wth respect to n y w P a ( n) (++) f, whch mples 0 + n @ f A. (6) ( y w ) a P Notce that more productve rms reach more people (but no rms optmally reaches all of them). Fnally, notce that rms sell only f f ( y w) a P ( ) ) (63) 0 @ f ( y w) a P A ( ). (64) Ths cuto rule s exactly the same as n Arkolaks (f we normalze a ). Usng (6) ths cuto rule mples that n + ) ( n) +. and n, We can now replace n the expresson for sales gven the optmal choces of the rm for p y () p w f + + y p P + + a 34 ( n) (++) + ( )!. (65)
Marketng costs as a fracton of total sales Notce that varable pro ts for each rm are gven by V () f + + Also average varable pro ts for operatng rms are V f + + f + + Z + + + ( )!. + ( )! ( )!. ( ) + + On the other hand we have that entry costs 0 n () f @ " # + A f, and that average entry costs f 0 f @ " # + A ( ) +!. ( ) + + Ths means that entry costs are a constant fracton of overall pro ts snce f f ++ + + ( )+ + ( )+ ++ + + ( )+ <, + ( )+ as long as whch, of course, holds. < +, The somorphsm We now dscuss the somorphsm that the sales equaton (65) wth the correspondng one n Arkolaks (008) exhbts. We consder the smple case where 0. The results follows by drect comparson of equaton (65) to equaton (4) n Arkolaks (008) and notng that the two equatons are the same f +. Cases wth > 0 yeld the same results to the ones below by smply rede nng notaton. We dstngush the followng two cases. 35
case a) If > ths case corresponds to the case (0; ) ; ( ; ) for the theory n Arkolaks (008). In ths case hgher a mples hgher densty. In the lmt where ( )! + we have! and also +! 0 for any > whch e ectvely means that y () f as n Meltz (003). How should we nterpret the result? If there s so much homogenety of tastes that all the mass s concentrated n a then we e ectvely have the Meltz model. case b) If 0 < < then the densty decreases for hgher a so that there many people that are not so fond of each good. Ths case corresponds to the case that (; ) so that ( ; 0). The lmt! 0 corresponds to the case!. 36
Part II Robustness In ths secton we llustrate some robustness of the results wth respect to the parameters of the model. 6 Dstrbuton of Sales Fgure : Dstrbuton of sales relatve to mean sales n model ( :5, ~ :65) and n the French data (for small, medum, and larger exportng destnatons) 37
Fgure : Dstrbuton of sales relatve to mean sales n model (, ~ :65) and n the French data (for small, medum, and larger exportng destnatons) 38
Fgure 3: Dstrbuton of sales relatve to mean sales n model ( :95, ~ :9) and n the French data (for small, medum, and larger exportng destnatons) 39
Fgure 4: Dstrbuton of sales relatve to mean sales n model ( :95, ~ :5) and n the French data (for small, medum, and larger exportng destnatons) 40
7 Normalzed Average Sales Fgure 5: Normalzed average sales n the French data and the model ( :5; ~ :65) 4
Fgure 6: Normalzed average sales n the French data and the model ( ; ~ :65) 4
8 The Parameters a; and Normalzed Frm Entry Fgure 7: Normalzed Entry (M ) n the data and the reestmated model (by assumng 0 and reestmatng usng the relatonshp between average sales and y,l ) 43
Fgure 8: Normalzed Entry (M ) n the data and the reestmated model (by assumng 0 and reestmatng usng the relatonshp between average sales and y,l ) 44
9 Export Growth and Marketng Convexty Rato of total mports n 998 00 to 99 93 64 3 6 8 4 Endogenous cost 3 4 5 6 7 8 9 0 Decles of prevously traded goods Fgure 9: Growth by decle of prevously traded goods, data and model ( :5, ~ :65) 45
Rato of total mports n 998 00 to 99 93 64 3 6 8 4 Endogenous cost 3 4 5 6 7 8 9 0 Decles of prevously traded goods Fgure 0: Growth by decle of prevously traded goods, data and model ( ; ~ :65) 46
0 Export Growth n the US-France,-Germany,-Mexco cases Fgure : Growth by decle of prevously traded goods, data for US-France and model calbrated to the US-France case under the two parameterzatons. 47
Fgure : Growth by decle of prevously traded goods, data for US-Germany and model calbrated to the US-Germany case under the two parameterzatons. 48
Fgure 3: Growth by decle of prevously traded goods, data for the Unted Kngdom and model calbrated to the US-Unted Kngdom case under the two parameterzatons. 49
Trade Costs Fgure 4: Trade costs changes n the US-Mexco NAFTA lberalzaton for prevously traded goods at the Harmonzed System 6-dgt level categorzed by ntal trade. Restrcton to manufacturng Standard Industral Class caton sectors. 50
Fgure 5: Trade costs changes n the US-Mexco NAFTA lberalzaton for prevously traded goods at the Standard Industral Class caton 4-dgt level categorzed by ntal trade. Restrcton to manufacturng Standard Industral Class caton sectors. References Arkolaks, C. (008): Market Penetraton Costs and the New Consumers Margn n Internatonal Trade, NBER workng paper 44. Chaney, T. (008): Dstorted Gravty: The Intensve and Extensve Margns of Internatonal Trade, The Amercan Economc Revew, 98(4), 707 7. Eaton, J., S. Kortum, and F. Kramarz (00): An Anatomy of Internatonal Trade: Evdence from French Frms, NBER Workng Paper 460. Jovanovc, B. (98): Selecton and the Evoluton of Industry, Econometrca, 50(3), 649 670. Meltz, M. J. (003): The Impact of Trade on Intra-Industry Reallocatons and Aggregate Industry Productvty, Econometrca, 7(6), 695 75. Meltz, M. J., and G. I. P. Ottavano (008): Market Sze, Trade, and Productvty, The Revew of Economc Studes, 75(), 95 36. 5
Smonovska, I. (009): Income D erences and Prces of Tradables, Manuscrpt, Unversty of Calforna, Davs. 5