Incorporating Modern Trade Theory into CGE Models

Transcription

1 Incorporatng Modern Trade Theory nto G Models ddy ekkers Unversty of ern Joseph Francos Unversty of ern & R (London) STRT: We propose a way to ncorporate the four workhorse models n the modern trade lterature nto computable general equlbrum models (Gs). We show that the ther- Krugman monopolstc competton model, the Meltz rm heterogenety model and the aton and Kortum model can be de ned as an rmngton model wth generalzed margnal costs, generalzed trade costs and a demand externalty. s already known n the lterature n both the ther-krugman model and the Meltz model generalzed margnal costs are a functon of the amount of factor nput bundles. In the Meltz model generalzed margnal costs are also a functon of the prce of the factor nput bundles. Lower factor prces rase the number of rms that can enter the market pro tably (extensve margn), reducng generalzed margnal costs of a representatve rm. For the same reason the Meltz model features a demand externalty: n a larger market more rms can enter. We mplement the d erent models n a G settng wth multple sectors, ntermedate lnkes, non-homothetc preferences and detaled data on trade costs. We nd the largest welfare e ects from trade cost reductons n the Meltz model. Keywords: Frm Heterogenety, G Model, Demand xternalty JL codes: F2, F4 prntdate: July 3, 25 ddress for correspondence: ddy ekkers, Johannes Kepler Unversty Lnz, Department of conomcs, ltenbergerstraße 69, - 44 Lnz, USTRI. emal: eddybekkersgmal.com

2 Incorporatng Modern Trade Theory nto G Models STRT: We propose a way to ncorporate the four workhorse models n the modern trade lterature nto computable general equlbrum models (Gs). We show that the ther- Krugman monopolstc competton model, the Meltz rm heterogenety model and the aton and Kortum model can be de ned as an rmngton model wth generalzed margnal costs, generalzed trade costs and a demand externalty. s already known n the lterature n both the ther-krugman model and the Meltz model generalzed margnal costs are a functon of the amount of factor nput bundles. In the Meltz model generalzed margnal costs are also a functon of the prce of the factor nput bundles. Lower factor prces rase the number of rms that can enter the market pro tably (extensve margn), reducng generalzed margnal costs of a representatve rm. For the same reason the Meltz model features a demand externalty: n a larger market more rms can enter. We mplement the d erent models n a G settng wth multple sectors, ntermedate lnkes, non-homothetc preferences and detaled data on trade costs. We nd the largest welfare e ects from trade cost reductons n the Meltz model. Keywords: Frm Heterogenety, G Model, Demand xternalty JL codes: F2, F4 Introducton There s a lvely debate n the recent trade lterature about the value added of rm heterogenety n trade models. rkolaks, et al. (22) show that the welfare gans from trade can be expressed wth two su cent statstcs, the domestc spendng share and the trade elastcty. Ths holds n the rmngton model, the Rcardan aton-kortum model, the equal rms monopolstc competton ther-krugman model and the rm heterogenety Meltz model. The only d erence s the nterpretaton of the trade elastcty. In rmngton and ther-krugman the trade elastcty s determned by the substtuton elastcty between varetes, whereas n aton-kortum and Meltz t s determned by productvty dsperson. Meltz and Reddng (23) nstead show that trade cost reductons generate larger welfare gans n the Meltz rm heterogenety model than n the equvalent model wth homogeneous rms, the ther-krugman model. Frm heterogenety has not been ncorporated n a comprehensve way n multsector G models. Most mportant work n ths respect s alstrer (22), who have ncluded rm heterogenety n one sector n a G model wth other sectors characterzed by an rmngton setup. llowng for rm heterogenety n all sectors mght be useful for varous reasons. Frst, t can shed lght on the dscusson about the value added of rm heterogenety n trade models by explorng the d erences n modellng outcomes wth other models. Second, varous realstc mcroeconomc features can be modelled lke the dstncton of welfare e ects nto an ntensve

3 and extensve margn e ect. Thrd, G models contan a large degree of sectoral detal, but are sometmes somewhat outdated n terms of modellng setup. Wth the ncorporaton of rm heterogenety n all sectors, ths drawback would dsappear. In ths paper we map out a parsmonous representaton of rm heterogenety enablng ncorporaton n multsector G models. In partcular, we show that both the ther-krugman and the Meltz model can be de ned as an rmngton model by generalzng the expressons for ceberg trade costs and for margnal costs and by allowng for a demand externalty n the Meltz model. In ther-krugman generalzed margnal costs are a functon of the number of nput bundles leadng to so-called varety scalng (Francos (23)). Varety scalng also props up n the Meltz model, but on top of that generalzed margnal costs are also a functon of the prce of nput bundles. The reason s that the extensve margn relatve and the compostonal margn are a ected by the prce of nput bundles. Wth a lower prce of nput bundles more rms can sell pro tably to the d erent destnaton markets generatng a postve e ect through the extensve margn (more varetes) and a negatve e ect through the compostonal margn (lower avere productvty because of the survval of the least productve rms as well). For the same reason there s a demand externalty n the Meltz model: n a larger market wth a hgher prce ndex more rms can survve, rasng the extensve margn relatve to the compostonal margn. Generalzed ceberg trade costs are a functon of xed and ceberg trade costs and of tar s. We show theoretcally that the ther-krugman model s a specal verson of the Meltz model f the rm sze dstrbuton becomes granular. Granularty corresponds wth ade elastcty n Meltz equal to the substtuton elastcty mnus one. The reason s that under granularty the destnaton-varyng component of the extensve margn cancels out anst the compostonal margn leavng only the ntensve margn and the number of entrants-component of the extensve margn, the two channels also operatve n ther-krugman. We mplement the parsmonous representaton of the d erent models n the multsector, multcountry, multfactor G model GT featurng ntermedate lnkes on non-homothetc preferences based on a detaled consstent dataset on output, trade ows, tar s and transport servces. Followng? we decompose changes n trade ows n response to polcy shocks nto an ntensve margn, an extensve margn and a compostonal margn. It s shown wth smulatons that the destnaton-spec c component of the extensve margn relatve to the compostonal margn rses when the rm sze dstrbuton becomes less granular In lne wth ths ndng we show that the welfare gans from reductons n trade costs are largest n the Meltz model and 2

4 rse when the rm sze dstrbuton moves away from granularty. ostnot and Rodrguez-lare (23) compare the welfare e ects of trade and trade lberalzaton n the d erent trade models n d erent setups. They show that the expresson for the prce ndex n the most general model, the rm heterogenety model, nests the expressons n the rmngton and ther-krugman model. Ther exposton s d erent n several respects. Frst, they concentrate on welfare and thus only derve an expresson for the prce ndex. Second, they do not wrte the d erent models as specal versons of an rmngton economy wth generalzed margnal costs, generalzed trade costs and a demand sde externalty. Thrd, they use exact hat algebra to derve ther results on the welfare e ects of trade lberalzaton. 2 Model 2. General Setup onsder an economy wth J countres. There are three groups of ents wth demand for goods n sector r, prvate households p, government g and rms f. The group of ents n country has demand q representatve goods q d; wth S preferences over quanttes of domestc and mported and q m;. We omt sector r subscrpts as well as the dervaton of demand for sector r goods and take ths demand as our startng pont: q e d q d; + e m q m; () Quanttes of mported varetes from the three d erent groups of ents n turn can be wrtten as a functon of quanttes from d erent sources : q m 2fp;g;fg q m; 6 (q ) (2) e s s a demand sde externalty playng a role n the rm heterogenety verson of the model. The demand externalty s dentcal for the d erent groups of ents. The reason s that upon payng xed export costs for a destnaton country rms can serve all three groups of ents n the destnaton country and the zero cuto pro t condton s thus formulated over all three groups together. The externalty s source-spec c wth the source domestc or mportng, s d; m.?. Dervatons and expressons for sectoral demand for the three groups of ents can be found n? and also n 3

5 The reason s that we want to allow for d erent destnaton-spec c taxes for mported goods and domestc goods. Demand for q s; can be wrtten as: q s; e s ta s; p s q (3) ta s; s a group-mporter spec c mport tar, expressed n power terms. For domestc goods s d, equaton (3) s the nal equaton but for mported goods, mport demand conssts of demand for goods from d erent sources, q : q p p m q m (4) p as: s the prce of the representatve good traded from to. and p m; are respectvely the prce ndces correspondng to q ;e and q m; and de ned tad; p d e d p m (p ) 6 + tam; p m e m (5) (6) p ta t c te p Z + ptr (7) p d c p Z (8) The prce of the representatve good, p, n equaton (7) s equal to cf-prce calculated as the sum of the prce of nput bundles n the exportng country, p Z, tmes the export subsdy appled to the fob-prce plus the prce of transport servces p tr dvded by ansport servces technology shfter, multpled by generalzed margnal costs n the exportng country, c, generalzed ceberg trade costs t and blateral ad valorem tar s, ta, both expressed n power terms. Frms spend a xed quantty share of sales on transport servces. Techncally, the cfquantty traded o cf s a Leontef functon of the quanty n fob-terms o fob and transport servces tr. The mplcaton s that transport servces work as a per unt trade cost and appear thus as an addtve term to the fob prce te p Z. 4

6 The rmngton model, the Krugman/ther model and the Meltz model can all be seen as specal versons of the above structure, dependng upon how the demand externalty e s n equaton (??), generalzed ceberg trade costs t, and generalzed margnal cost c n equaton (7) are spec ed. 2.2 rmngton conomy erfectly compettve rms n country produce homogeneous country varetes wth margnal cost b. So, nput bundles can be transformed nto output x accordng to x b. Wth margnal cost prcng the prce of output n country, p x, s gven by, px b p Z. Frms face ceberg trade cost. There s no demand externalty n the rmngton economy, so e s. Therefore, the rmngton economy s characterzed by equatons ()-(8) wth the followng expressons for c, t and e s : c (9) t () e s () 2.3 ther-krugman conomy In the ther-krugman economy, preferences are characterzed by love for varety over varetes produced n d erent countres. Utlty q (output) o () of varetes 2 shpped from all exporters : can thus be de ned over physcal quanttes q Z 2 o () d (2) The correspondng prce ndex s de ned over the prces of physcal quanttes of the varetes, p o (): Z p ;o () d 2 (3) Frms n country produce wth an dentcal ncreasng returns to scale technology wth xed cost a and margnal cost b mplyng that each rm produces a unque varety. Increasng returns n combnaton wth love for varety mples also that a larger number of nput bundles 5

7 leads to a more than proportonal ncrease n utlty snce the number of varetes s larger. To capture ths externalty, generalzed margnal costs c are fallng n the number of varetes N and thus n the amount of nput bundles. mployng the expressons for markup prcng, the free entry condton and factor market closure, c can be expressed as follows: 2 c N (4) Wth N J N r p Z ta g ta te b + ptr p Z N r p Z ta g ta a (5) The number of varetes N does not ncrease proportonally wth the amount of nput bundles. N s calculated by combnng factor market equlbrum and the free entry condton. Snce transport servces are sourced employng separate nput bundles, they have to be subtracted n calculatng the demand for nput bundles from a spec c country and sector. So an ncrease n transport costs leads to less labor demand for gven zero-pro t-revenues. s a resut hgher transport costs rase the number of varetes for a gven number of nput bundles. 3 Representatve output x can be transformed nto q accountng for the ceberg trade costs.there s no demand externalty n the ther-krugman economy, so we have: t (6) e s (7) So, the ther/krugman economy s characterzed by equatons ()-(8) wth c, t and e as de ned n equatons (4)-(7): 2.4 Meltz conomy In the Meltz economy preferences are lke n ther/krugman characterzed by love for varety over varetes produced by d erent rms from d erent countres as n equaton (2)-(3). Goods are produced by rms wth heterogeneous productvty. To start producng, rms can draw a productvty parameter ' from a dstrbuton G (') after payng a sunk entry cost en. The 2 Dervatons n ppendx 3 n ncrease n transport costs rases nput bundle demand also through the demand for transport servces, but n the transport sector we assume perfect competton so there s no number of rms externalty. 6

8 dstrbuton of ntal productvtes s areto wth a shape parameter and a sze parameter : G (') ' (8) hgher reduces the dsperson of the productvty dstrbuton and a hgher rases all ntal productvty draws proportonally. We mpose > to guarantee that expected revenues are nte. The productvty of rms stays xed and rms face a xed death probablty n each perod. Frms ether decde to start producng for at least one of the markets or leave the market mmedately. In equlbrum there s a steady state of entry and ext wth a steady number of entrants drawng a productvty parameter, mplyng that the productvty dstrbuton of producng rms s constant. Frms produce wth an ncreasng returns to scale technology wth margnal cost equal to '. Frms pay xed costs f for each market n whch they sell. The xed costs are pad partly n nput bundles of the source country and partly n bundles of the destnaton country accordng to a obb Douglas spec caton wth a fracton pad n source country nput bundles. Upon payng the xed entry costs for a destnaton market, rms can sell goods to all three groups of ents. Snce preferences are characterzed by love for varety and producton occurs wth ncreasng returns to scale, an ncrease n the number of nput bundles leads to a more than proportonal ncrease n utlty. To account for ths externalty, representatve output s lke n the ther/krugman economy de ned as varety scaled output. Snce productvty s heterogeneous, varety scaled output s also a ected by nput costs. Followng? changes n costs lead to an adustment n output along three margns, an ntensve margn, an extensve margn and a compostonal margn. Lower costs lead to more sales of rms already n the market, the ntensve margn. Ths s a prce e ect and hence does not a ect varety scaled output. Lower costs also rases the mass of rms that can produce pro tably, the extensve margn. Ths leads to a rse n varety scaled output. nd nally, lower costs reduces the avere productvty of rms n the market, as more rms can survve, the compostonal margn. Ths margn also a ects varety scaled output. ccountng for the latter two margns, 7

9 generalzed margnal costs c can be wrtten as: m c N + p ( ) 2 Z en (9) The number of entrants N s a functon of nput bundles and transport costs: N J N r ( e' ) p Z ta g ta te + ptr p Z N r ( e' ) p Z ta g ta en (2) The expresson for N s almost dentcal to the expresson for N n the ther-krugman model n equaton (5) except for d erences n parameters. m s a functon of and and an addtonal converson parameter for later use set equal to : m (+) + + (2) x can be transformed nto q accountng for generalzed ceberg trade costs, whch are a functon of ceberg trade costs, xed trade costs f, mport tar s c and the cf prce te p Z + ptr. Iceberg and xed trade costs a ect the transformaton n the same way through the extensve and compostonal margn as the prce of nput bundles p Z margnal costs. 4 We get the followng expresson for generalzed ceberg trade costs: a ect generalzed t + te p Z + ptr + + ta + + ( ) 2 f + ( ) 2 (22) The four terms between brackets represent the e ects of the cf-prce, tar s, and ceberg and xed trade costs through the extensve and compostonal margn on convertng fob varety scaled output nto cf varety scaled output. Iceberg trade costs also have a drect e ect through the ntensve margn, represented by the last term outsde of the brackets. Fnally, the demand externalty does play a role under rm heterogenety, an drven by the extensve and compostonal margn. The followng expresson can be derved for the 4 ro ts are calculated dvdng revenues nclusve of tar s by tar s, r cq f. ostnot and +ta Rodrguez-lare call ths demand shftng. The alternatve would be cost shftng wth pro ts calculated as r c ( + ta) q f. Ths makes t mpossble to nd an expresson for the mass of rms as a functon of market sze, a problem also occurng n the ther/krugman model. 8

10 demand externalty e : e s ;e ta s; ta s; p Z + ( ) 2 (23) s expendture by n country. oth larger prce ndexes, larger market szes and lower group-spec c tar s for the d erent groups of ents rase the extensve margn relatve to the compostonal margn and thus reduce the prce ndex and rase utlty q. lower prce of nput bundles p Z n the destnaton country also rases utlty, as t rases welfare through the extensve margn relatve to the compostonal margn. The Meltz economy s characterzed by equatons (??)-(7) wth the expressons c, t and e gven n equatons (9)-(23). 2.5 Nestng From the expressons n the prevous 3 subsectons t follows drectly that Krugman/ther s a specal case of Meltz up to a constant and rmngton s a specal case of both. Meltz can be converted nto an ther/krugman model by settng equal to, the sze parameter of the productvty dstrbuton equal to the nverse of margnal cost b, sunk entry costs tmes the death probablty en equal to the xed cost a and the converson parameter n equaton (2) as follows: +2 ( + ) (24) mples that the demand externalty e s s. It can be easly ver ed that the expressons for c and t n equatons (9)-(22) become equal to the prce of the representatve good n the ther/krugman economy n equatons (4)-(6). ther/krugman can be converted nto rmngton by settng the margnal cost parameter c equal to b and thus droppng the varety scalng. The ntuton for why mples that Meltz leads to Krugman/ther s the followng. s ponted out above a change n trade costs generates a change n trade ows along three margns, an ntensve margn of already exportng rms, an extensve margn representng an ncrease n the mass of varetes and a compostonal margn representng the change 9

11 n avere productvty of rms exportng. If trade costs fall, trade rses wth an elastcty of along the ntensve margn and wth an elastcty along the extensve margn. It falls along the compostonal margn wth an elastcty. So, f, the extensve and compostonal margn cancel out and only the ntensve margn remans. Therefore, the model wth heterogeneous rms works out dentcally as a model wth homogeneous rms. The converson factor n movng from Meltz to ther/krugman s necessary. Wthout ths converson factor utlty would become n nte. The reason s that that avere productvty would become n nte. Stll, when approaches would mply the e ect of changes n trade costs wll be dentcal to the e ect n an ther/krugman economy. Therefore, when studyng the e ect of polcy changes we can apply the converson factor wthout any consequences. 3 Margn Decomposton of Trade n Meltz Model Total trade ows can be wrtten as: V N er N G ' Z r (') g (') d' (25) ' Log d erentatng equaton (25) on the RHS and LHS wrt to the endogenous varables gves: d ln V d ln N +N G ' Z ' d ln r (') r (') ln G ' r ( e') g (') d'+ ln ' d ln ' r ' r ( e') The rst term represents the extensve margn, M, the second term the ntensve margn, IM, and the thrd term the compostonal margn, M. To elaborate on these expressons, we rst log d erentate the expresson for ' n equaton (.7): ' ta te p Z + ptr f p p Z ta ;e (26)

12 c' cp + cp Z + + \ dta + c + te p Z + ptr p s qs; ( ) d + d d fs;p;fg p s qs; + c f (27) We can elaborate on the extensve margn, employng the expresson for N and N n equatons (.9)-(.2) and the expresson for c ' n equaton (??): M d ln N + c ' + d N cp ( ) cp Z p s qs; fs;p;fg + dta p s qs; ( ) d + d \ te p Z + ptr ta [ s; c c f + d N (28) We can elaborate on the ntensve margn, IM, employng the expresson for r n equatons (.3)-(.4) and summng over the three ncome groups: IM G ' Z ' ( ) c + dta + d ln ta te p Z + ptr ' \ te p Z + ptr + p s qs; p s qs; fs;p;fg (') and p (') ;e ( ) d + d + ta [ s; (29) Fnally, we can express the compostonal margn, M, as follows, usng the dstrbuton functon of the areto dstrbuton n equaton (8) and the expresson for r (') n equaton (.3): M c ' + ( ) ' c cp Z + ( ) cp Z + dta + ( ) c + ( ) p s qs; p s qs; fs;p;fg ( ) d + d ta [ s; \ te p Z + ptr + f c (3)

13 ddng up the three margns, we can express the overall margn thus as follows: d ln V T M M + IM + M cp ( ) \ te p Z + ptr cf + cp Z + d N + dta p s qs; fs;p;fg 4 Implementaton n GT GMK p s qs; c ( ) d + d ta [ s; (3) We mplement the Meltz structure wth demand and supply sde externaltes and generalzed ceberg trade costs n the GT model programmed n GMK. We outlne for each of the three topcs rst the blocks added to the GMK code and then how the exstng code s adusted. Then we dscuss parameterzaton n GMK to contnue ths secton wth a dscusson of how to move between the d erent models employng closure swaps. We nsh ths secton wth a dscusson of the margn decomposton n GMK. In the mplementaton we assume that all xed exportng costs are pad n the source country,.e.. 4. Supply-Sde xternalty The supply-sde externalty n the ther-krugman and Meltz model can be gathered by log d erentatng respectvely equatons (4) and (9): bc bc N c (32) N d + + ( ) 2 cp (33) In GMK we model respectvely the ther-krugman and Meltz supply-sde externalty as follows: oscaleek(; r) ekscale(; r) [( )] nne(; r) (34) oscalem(; r) mscale(; r) [( )] nne(; r) ( + ) ( ) 2 [ps(; r) pfactwld] (35) 2

14 We de ate the prce change term ps (; r) n the calculaton of the externalty by the numerare pfactwld, such that a change n all prces does not change the sze of the externalty and s neutral. To move between the d erent supply-sde externaltes we add the followng addtonal equaton: oscaleekm(; r) ekscale(; r) + emscale(; r) sext(; r) (36) We use the same varable for the relatve change n the number of rms n the ther- Krugman model and n the number of entrants n the Meltz model, nne (; r), snce the two are dentcal. Ths becomes clear by log d erentatng equaton (.8) or equvalently equaton (.2). In GMK notaton we get: nneh(; r) V OM (; r) s V OM (; r) qo (; r) J (V MD (; r; t) V IW S (; r; t)) t V OM (; r) V MD (; r; s) (pcf (; r; s) + qxs (; r; s) J (V MD (; r; t) V IW S (; r; t)) t V W D (; r; s) (ps (; r) + ao (; r) tx (; r) tx (; r; s)) V IW S (; r; s) V IW S (; r; s) V W D (; r; s) ptrans (; r; s)) V IW S (; r; s) V IW S (; r; s) + s J V OM (; r) (V MD (; r; t) V IW S (; r; t)) t (pcf (; r; s) + qxs (; r; s) (ps (; r) + ao (; r))) nne (; r) (37) So the expresson for the number of varetes contans addtonal terms, re ectng the sze of transport servces and export subsdes to all destnaton partners. Moreover, we have to take nto account that the varety scalng term has to be appled to the cf-prce, so nclusve of transport costs, for the nternatonal prce and quantty. Therefore, we have to wrte the ceberg trade costs technology shfter ams (; r; s) as a functon of the supply-sde externalty. We cannot nclude the supply-sde externalty before the transport sector s added, snce we would have to multply all terms by F OSHR (; r; s) whch would be destnaton spec c. Snce the domestcally sold goods do not feature transport costs, but do bene t from varety scalng, the varety scalng term also a ects domestc prces and quanttes,.e. ppd, pgd and pfd and qpd, qgd and qfd. 3

15 4.2 Demand-Sde xternalty To model the demand-sde externalty, we add a block to the model calculatng the demand-sde externalty and we adust the prce and quantty expressons for domestc and mported goods for the three groups of ents, prvate households, governments and rms. Frst, we dscuss the addtonal block for the demand-sde externalty. Log d erentatng the theoretcal expresson for the externalty n equaton (23) gves: be s fs;p;fg ta s; ta s; ta s; ta s; + d [ ta s; + + d ( ) 2 [ ta s; (38), Multplyng the numerator and denomnator of the coe cent by p s we can rewrte equaton (38) as follows: be s p s qs; p s qs; fs;p;fg + d [ ta s; + + d ( ) 2 [ (39) ta s; To nd the equvalent expresson n GT notaton, we observe that p s qs; represents the expendtures of group f; p; g on source s d; m, V; S; G; M. So, equaton (39) can be wrtten n GMK notaton as follows wth s m; d: dscales (; r) + (prceds (; r) pfactwld) (valueds (; r) ( ) pfactwld) ( + ) ( ) 2 tarffds (; r) (4) Wth prceds (; r) the prce ndex term of the externalty n sector n country r for source s d; m, valueds (; r) the value term and tarffds (; r) the tar term and de ned for s m as (the expressons for s d are smlar): prcedm(; r) SHRI M [pp(; r)] + SHRIGM [pg(; r)] + sum(; ROD_OMM; SHRIF M(; ; r)) [pf(; ; r)]) (4) 4

16 nd: valuedm(; r) SHRI M [pp(; r) + qp (; r)] + SHRIGM [pg(; r) + qg (; r)] + sum(; ROD_OMM; SHRIF M(; ; r)) [pf(; ; r) + qf (; ; r)]) (42) nd: tarffdm(; r) SHRI M tpm(; r) + SHRIGM tgm(; r) + sum(; ROD_OMM; SHRIF M(; ; r)) tfm(; ; r)) (43) pp, pg, and pf are the relatve prce changes for prvate households, government and rms and qp, qg, and qf the quantty equvalents. SHRI M (; r) s de ned as: SHRI M (; r) V I M (; r) V IM (; r) (44) Wth V IM (; r) the sum of mport demand at market prces: V IM(; r) V I M(; r) + V IGM(; r) + sum(; ROD_OMM; V IF M(; ; r)) (45) SHRIGM (; r) and SHRIF M (; ; r) are de ned smlarly. s for the supply-sde externalty, we de ate the prce and value changes (based on prce changes) n the calculaton of the externalty by the numerare, such that a change n all prces does not change the externalty. To determne how the expressons for domestc and mporter demand and prce for the three groups of ents n the GT model change, we de ne the domestc and mporter prce, nclusve of the externalty and the ent-spec c tax, ep s;, as follows: ep s; tas; p s; e s (46) Log d erentatng both equaton (46) and the rewrtten expresson for demand n equaton (3) 5

17 gves: qd s; [ ;e dep s; ta [ s; + d p s; dep s; + q d;e be s (47) be s (48) The equvalent expressons n GT for domestc government goods s gven by: qgd(; s) SUD() [pg(; s) pgd(; s)] + qg(; s) Dextd(; s) (49) pgd(; s) tgd(; s) + pm(; s) Dextd(; s) (5) pgm(; s) tgm(; s) + pm(; s) Dextm(; s) (5) wth qgd and qg the domestc and total government demand; pgd, pgm and pg, the domestc, mported and overall prce of government consumpton; tgd and tgm the tax on domestc and mported government consumpton; pm and pm the domestc and mport prce of goods; and Dextd the domestc demand externalty. So we model the demand externalty as a technology shfter to domestc and mported demand. 4.3 Generalzed Iceberg Trade osts The generalzed ceberg trade costs are equal to the normal ceberg trade costs n the rmngton, ther-krugman and aton-kortum model. Only n the Meltz model the two are dstnct and generalzed ceberg trade costs are de ned n equaton (22). Log d erentatng ths equaton gves: ct + + \ te p Z + ptr c ( + ) ( ) 2 dta + + ( ) 2 c f (52) In the GT model (wth all varables expressed n relatve change terms) blateral ad-valorem tar s dta consst of mport tar s tm and tms and the ceberg trade costs c consst of an ceberg-trade-costs-lke technology shfter ams. Tar s are pad based on the marked-up prces, whereas ceberg trade costs and the transport margn operate on the physcal quanttes and are thus based on costs. s a result, the coe cent on tar s n generalzed trade costs s d erent. Snce both the generalzed ceberg trade costs t and the generalzed margnal costs c are 6

18 appled on the cf-prce, we endogenze the ceberg-trade-cost-lke technology shfter ams (; r; s) as a functon of the supply-sde externalty sext (; r) and generalzed ceberg trade costs. In GMK notaton we get n the ther-krugman and Meltz model respectvely: gentcekh(; r; s) sext(; r) + tc(; r; s) gentcek(; r; s) (53) gentcmh(; r; s) sext(; r) ( + ) ( ) 2 (tm(; s) + tms(; r; s)) + tc(; r; s) fex (; r; s) + pcf (; r; s) gentcm (; r; s) (54) ( ) We shft between the ther-krugman and Meltz model wth the followng equaton: gentcekm(; r; s) gentcek(; r; s) + gentcm(; r; s) + ams(; r; s) (55) We add the varable tc to the model, whch represents normal ceberg trade cost n the ther- Krugman and Meltz spec caton of the model. Snce ams (; r; s) s a technology-shfter and a postve shock to ams represents a reducton n ceberg trade costs n the standard model, we add ams n the above equaton nstead of subtractng t. The exstng code of the model does not have to be adusted to account for Meltz-generalzed trade costs and only requres a closure swap. 4.4 arameterzaton We need values for the parameters and. We de ne the trade elastcty as e, the degree of granularty as gran and the tar elastcty e: e (56) We can express and as a functon of e and gran as follow: gran e (57) e gran (58) gran approachng means that the model s approachng granularty/ther-krugman. Under granularty the tar elastcty would be equal to +. 7

19 4.5 Movng between D erent Models wth losure Swaps We move between the d erent models usng closure swaps and employng d erent parameter values. Frst we dscuss closure swaps. The baselne model wth the addtonal blocks and wthout closure swaps mples the rmngton model. We move from rmngton to ther-krugman by turnng on the ther-krugman supply-sde externalty and by endogenzng ceberg trade costs. We move from rmngton to Meltz by turnng on the Meltz supply-sde and demand-sde externaltes and by endogenzng ceberg trade costs. y swappng oscaleekm wth sext n equaton (36) and nneh wth nne n equaton (37) we turn on the supply-sde externalty and by swappng oscaleek wth ekscale or oscalem wth mscale n respectvely equatons (34)-(35) we turn respectvely the ther-krugman and Meltz supply-sde externalty on. To turn on the Meltz demand-sde externalty, we swap dscale2d wth Dextd n the followng equaton: dscale2d(; r) dscaled(; r) Dextd(; r) (59) Fnally, to model generalzed trade costs n ther-krugman or Meltz, ams (; r; s) s swapped wth gentcekm (; r; s) n equaton (55). y swappng gentcekh wth gentcek or gentcmh wth gentcm n respectvely equatons (53)-(54) we choose for respectvely ther-krugman and Meltz generalzed ceberg trade costs. We set and as spec ed n equatons (57)-(58). For the rmngton and ther-krugman verson of model we set gran equal to mplyng that the substtuton elastcty s equal to the tar elastcty. 4.6 Margn Decomposton To calculate the three margns n GMK, we rewrte equatons (27)-(3) n GMK notaton as follows: psstarh (; r; s) [ps(; r) + ao (; r) pfactwld] + + (tm (; s) + tms (; r; s)) + pcf (; r; s) + tc (; r; s) + fex (; r; s) prceds(; s) valueds (; s) + tarffds (; s) psstar (; r; s) 8

20 The extensve margn s gven by: extm (; r; s) psstar(; r; s) + nne(; r) nd the ntensve margn s de ned by: ntm (; r; s) ( ) (tc (; r; s) + tm (; s) + tms (; r; s) + pcf (; r; s)) + ( ) prceds(; s) + valueds (; s) tarffds (; s) The compostonal margn can be expressed as: compm (; r; s) ( ) psstar (; r; s) nd nally the overall e ect can be wrtten as: d ln V T M M + IM + M (ps (; r) + ao (; r) pfactwld) + nne(; r) + (tm (; s) + tms (; r; s)) (tc (; r; s) + pcf (; r; s)) fex (; r; s) + prceds(; s) + valueds (; s) tarffds (; s) (6) Wth prceds, valueds and tarffds de ned as n equatons (4)-(43), except for the fact that values are expressed employng ents prces nstead of market prces. 5 oncludng Remarks We have shown that both the ther-krugman monopolstc competton model and the Meltz rm heterogenety model can be de ned as an rmngton representatve ent model. Ths representaton of these two models also makes clear that the Meltz model generates the same equlbrum outcome as the ther-krugman model when the rm sze dstrbuton s granular and spec c values are chosen for parameters values lke trade costs and sunk entry costs. Ths representaton s n partcular useful for mplementaton of the Meltz rm heterogenety model n multsector G models. 9

21 References rkolaks,.,. ostnot and. Rodrguez-lare (22). New Trade Models, Same Old Gans? mercan conomc Revew 2(): alstrer,., R. Hllberry and T. Rutherford (2). Structural stmaton and Soluton of Internatonal Trade Model wth Heterogeneous Frms. Journal of Intermatonal conomcs 83(): ernard, ndrew., Stephen J. Reddng and eter K. Schott (27). omparatve dvante and Heterogeneous Frms. Revew of conomc Studes 74, pp ostnot, rnaud and ndres Rodrguez-lare (23). Trade Theory wth Numbers: Quantfyng the onsequences of Globalzaton. Francos, Joseph, Mram Manchn and Wll Martn (23). Market Structure n G Models of Open conomes. Meltz, Marc J. and Stephen J. Reddng (23). Frm Heterogenety and ggregate Welfare. Mmeo rnceton Unversty 2

22 ppendx ther/krugman conomy The goal of ths secton s to derve the expressons for c and t n the man text n equatons (4)-(6). We follow the dual exposton as t s clearer. gents of group fs; p; fg wth g government, p prvate sector and f rms n country have S preferences over physcal quanttes o () of varetes from d erent countres. The quantty and prce ndex are de ned n equatons (2)-(3). Demand for a varety shpped from to and sold to group s equal to: o () p () (.) Varetes are produced by dentcal rms wth an ncreasng returns to scale technology wth xed cost a and margnal cost b, mplyng that each rm produces a unque varety. s rms are dentcal, can be dropped n the remander. Frms face ceberg trade costs, blateral export taxes te, blateral mport tar s ta, and group spec c mport tar s. Moreover, there s ansport sector wth rms havng to spend a xed quantty share of sales on transport servces. Techncally, the cf-quantty traded o cf s a Leontef functon of the quanty n fob-terms o fob and transport servces tr : o cf mn o fob ; atr tr (.2) ro ts are therefore gven by: p o ta p o p o te p Z + ptr ta p o te p Z + ptr ta o o (.3) Ths expresson for pro t mples the followng markup prcng rule: p o ta te b p Z + ptr (.4) p o s the cf prce of physcal output o before the group-spec c mport tar s appled. Frms do not face destnaton spec c xed costs and can enter all markets upon payng the 2

23 xed costs a. ro ts from sales to all markets are thus equal to: p o o ta a p Z (.5) s a next step, N s de ned as the mass of varetes produced n country. N s dentcal for all destnatons by absence of destnaton spec c xed costs. It follows from the followng labor market equlbrum: o + a N (.6) To rewrte ths expresson, we rst rewrte the expresson for o usng the markup equaton (.4): o p o o + p Z ta p o o p Z ta te b + ptr p Z (.7) Usng equatons (.5) and (.7), we can solve for N from equaton (.6) as follows: N J N r p Z ta g ta te b + ptr p Z a N r p Z ta g ta (.8) The prce of the representatve good p s de ned such that substtutng p n the expresson for the prce ndex de ned over representatve prces n equaton (5) generates the expresson for the prce ndex n the ther/krugman economy as de ned n equaton (3). Ths gves: p Z 2 p o () d (.9) Gven that all rms are dentcal and all varetes N are exported to all destnatons, equaton (.9) can be rewrtten as: Substtutng equaton (.4) for p o leads to: p N p o (.) p ta N te b p Z + ptr (.) So the externalty s appled after expendtures on the transport sector have been ncurred. We 22

24 can wrte generalzed margnal costs c thus as follows wth N as de ned n equaton (.8): c N ppendx ppendx. Meltz conomy Demand and roducton Lke n the ther/krugman economy the goal of ths secton s to derve the expressons for generalzed margnal costs c, generalzed ceberg trade costs t and the demand externalty e n the Meltz economy n equatons (9)-(23) and to derve the demand externalty. gents of group n country have the same S preferences over varetes from d erent countres as n the ther/krugman economy. The quantty and prce ndex are thus gven by equatons (2)-(3) and demand for physcal quanttes o () of a varety by equaton (.). In contrast to the ther/krugman economy goods are produced by rms wth heterogeneous productvty. Frms can sell both n domestc and foregn markets and have to pay xed costs f to sell n each market. The xed costs are pad n wes of both countres wth accordng to a obb Douglas spec caton a fracton pad n domestc nput bundles. The xed costs are destnaton-spec c, but not ent-spec c. So a rm pays the xed costs only once for sales to all three groups of ents. xportng rms also face ceberg trade costs, blateral tar s ta, ent-spec c tar s ta g, export taxes te. Moreover, there s ansport sector wth rms havng to spend a xed quantty share of sales on transport servces as n the ther-krugman model wth the cf-quantty traded o cf by: de ned as n equaton (.2). ro ts are therefore gven po o ta ta po o p Z te + ptr ta p Z p o o te p Z + ptr o ' o ' (.) We assume that productvty ' operates both on the costs of producton and on the transport sector. Ths means that more productve rms also need less transport servces, an assumpton also made for ceberg trade costs. If productvty would only operate on the cost of producton, the model would become ntractable n a multcountry, multsector settng. 5 5 In lne wth the GT model we de ne p o as the prce before group spec c mport tar s are pad. 23

25 ach rm produces a unque varety, so we can dentfy demand for varety by the productvty ' of the rm producng ths varety. Demand o (') and revenues r (') of a rm wth productvty ' producng n and sellng n are equal to: o (') r (') po (') po (') (.2) (.3) Maxmzng pro ts mples the followng markup prcng rule: p o (') ta te p + ptr Z ' (.4) Substtutng equaton (.4) back nto equaton (.) shows that pro ts for sales to destnaton market are equal to: (') po (') ta f p p Z (.5) So we add up the revenues for sales to the three groups of ents. ppendx.2 ntry and xt ntry and ext are lke n Meltz (23),.e. rms can draw a productvty parameter ' from a dstrbuton G (') after payng a sunk entry cost en. The productvty of rms stays xed and rms face a xed death probablty n each perod. Frms ether decde to start producng for at least one of the markets or leave the market mmedately. In equlbrum there s a steady state of entry and ext wth a steady number of entrants N drawng a productvty parameter, mplyng that the productvty dstrbuton of producng rms s constant. Denotng ' as the cuto productvty, only rms wth a productvty ' ' from country sell n market. ppendx.3 Free ntry and Zero uto ro t ondtons qulbrum s de ned wth a zero cuto pro t condton (Z) and a free entry condton (F). ccordng to the zero cuto pro t condton rms from country wth cuto productvty ' 24

26 can ust make zero pro t from sales n country : po (') ta f p p Z (.6) Snce the xed costs are destnaton-spec c and not group-spec c there s only one Z for each source-destnaton par and thus also only one cuto productvty level '. Usng equatons (.3)-(.5) the Z can be wrtten as follows: ' ta te p Z + ptr f p p Z ta ;e (.7) The free entry condton (F) equalzes the expected pro ts before entry wth the sunk entry costs: G ' ( e' ) en p Z (.8) e' s a measure of avere productvty and de ned as: e' Z ' ' g (') d' G ' (.9) Usng r (' ) r (' 2) ' ' 2 wrtten as: and the Z n equaton (.6), the F n equaton (.8) can be G ' p p Z f e' ' The dstrbuton of ntal productvtes G (') s areto: en p Z (.) G (') ' (.) wth the shape parameter and the sze parameter. We mpose > to guarantee that expected revenues are nte. Wth a areto dstrbuton e' s proportonal to ' : e' + ' (.2) 25

27 Substtutng equatons (.)-(.2) nto the fe, equaton (.), gves: ' p p Z f + en p Z (.3) ppendx.4 Dervng The omponents of The Representatve rce To derve the t, c, and e s such that general-setup model s equvalent to Meltz model, we rst rewrte the expresson for the prce ndex n the general setup-model by substtutng equatons (6)-(7) nto equaton (5): tad; c p Z e d + 6 ta m; ta t c p Z e m (.4) Hence, the prce ndex can also be wrtten as follows: p ta s; e s ta s; ta t c p Z e s (.5) wth s d f and s m f 6. Next, we derve the expresson for the representatve prce p ta s; e s followng from the Meltzsetup as derved n ths appendx. We can wrte the representatve prce as follows from equaton (3): p ta s; e s Z 2 p o () d (.6) p ta s; e s s the representatve prce ncludng the demand externalty. The representatve prce n equaton (.6) can be rede ned as an ntegral over productvtes of the producng rms as follows: p ta s; e s Z ' N p ;o (') g (') d' G ' (.7) Usng equatons (.4) and (.9) the representatve prce n equaton (.7) can be rewrtten as a functon of avere productvtes: p ta s; e s N ta ta s; te p Z + ptr e' (.8) 26

28 The mass of varetes sold from country to country, N s related to the mass of entrants N and the cuto productvty ' by the followng steady state condton: N G ' N ' N (.9) The steady state of entry and ext mples that N can be wrtten as a functon of the number of nput bundles : N J N te + p tr p Z ptr te + en p Z r ( e' ) p Z ta g ta (.2) Usng equatons (.2), (.9) and (.2), the representatve prce n equaton (.8) can be wrtten as: p ta s; e s ( + ) J N te + p tr p Z r ( e' ) te + ptr p Z ta g p Z ta en ta ta s; (.2) te p Z + ptr + ' The nal step s to substtute the Z solved for ' n equaton (.7) nto equaton (.2) generatng the followng expresson: p ta s; e s m J N te + p tr p Z r ( e' ) te + ptr p Z ta g p Z ta te p Z + ptr p + en ta (.22) + + ( ) ( f m s de ned n equaton (2) n the man text. From equaton (.22) we can easly determne the source-spec c component, c, the blateral component, ta t, and the destnaton spec c component, e, n: p ta s; e s tas; ta t c p Z e s (.23) 27

29 The source spec c component n equaton (.22) s equal to: m c N + p ( ) 2 Z en (.24) The parwse component n equaton (.22) s gven by: t ta te p Z + ptr te p + ptr Z (ta ) (ta f ) + ( ) 2 (.25) Rearrangng leads to the expresson for t n the man text, equaton (22): t + te p Z + ptr + ( ta +) ( ) 2 f + ( ) 2 (.26) Fnally, the destnaton spec c terms n equaton (.22) represents the demand externalty, gvng: e s ;e ta s; ta s; p Z + ( ) 2 (.27) So we have shown that the general setup-expresson for the prce ndex n equaton (.5) employng expressons for c n equaton (9), t n equaton (22) and e s n equaton (23) follows from a Meltz structure and s thus equvalent to a Meltz structure. To prove equvalence between the general setup and the Meltz setup, we also have to show that the general setupexpresson for demand n equatons (3)-(4) wth expressons for prces n equatons (5)-(8) s equvalent to the expresson for demand followng from the Meltz structure. The general setupexpresson mpled by equatons (3)-(8) s equal to: q e s (ta ) (ta f ) + ( ) 2 c p Z e s 2fp;g;fg ta s; (.28) 28

30 Substtutng the expressons for t, c and e s leads to: q e s m ta t L en ;e ta s; p Z + + ( ) 2 p ta s; + ( ) 2 2fp;g;fg ta s; (.29) Next we show that the expresson for quantty q e s nclusve of the demand-sde externalty startng from the Meltz-setup s dentcal to the expresson n equaton (.29). We can wrte the quantty startng from the Meltz-setup as follows: q e s Z 2 o () d (.3) Rede nng quantty n equaton (.3) as an ntegral over the productvty of producng rms gves: R q e s N ' o (') g (') d' G ' (.3) Substtutng the expresson for q (') n equaton (.2), representatve quantty n equaton (.3) can be wrtten as a functon of avere productvty: q e s N o ( e' ) (.32) The next step s to use q (' ) q (' 2 ) ' ' 2 and equaton (.2) to wrte q as a functon of cuto : quantty q ' q e s N o ' + The Z n equaton (.6) can be employed to express cuto quantty o (.33) ' as follows: o ' ( ) f ' p p Z p Z (.34) Substtutng equaton (.34) and also the expressons for N and N n equatons (.9)- 29

31 (.2) nto equaton (.33) leads to: q e s ( + ) ( ) en f + ' p p Z p Z (.35) Fnally, the Z solved for ' n equaton (??) can be substtuted nto equaton (.35) and after several rearrangngs, we get the same expresson as the general setup-expresson n equaton (.29). ppendx.5 Summary of qulbrum quatons and Implementaton n GMS s a check on the correctness of the expressons, we show n GMS that a soluton of the model n a settng wth countres generates the same soluton usng the ntal equlbrum condtons of the Meltz rm heterogenety model as usng the sngle equlbrum condton. We work wth a verson of the model wthout ntermedate lnkes. The nput bundle and ts prce p Z wll be equal to respectvely factor nput bundies L and ts prce w. Imposng the general equlbrum condton that output w L s equal to the value of exports to all destnaton countres, leads to: w L (t c w ) w L (.36) J k (t kc k w k ) k We have used n equaton (.36) that the absence of tar s and trade mbalances mples that demand s equal to w L. Substtutng the expressons for t and c n the Meltz economy n equatons (22)-(9) and abstractng from transport servces and export taxes gves: w L en L w (+ + ) ( ) + f ( ) J k en k L k w (+ + ) ( ) k k + k f ( ) k k w L (.37) Wth J equatons (.37) the model can be solved for J unknown w. We use populaton for the number of workers and tted trade costs from the gravty regressons on dstance for the bggest countres n terms of populaton from the sample, the countres angladesh, razl, hna, Indonesa, Inda, Ngera, akstan, Russa and US. For the model wth the full set of equatons we use the followng condtons: the expresson 3

32 for the prce ndex followng from equaton (.2); the expresson for the number of varetes followng from equatons (.9) and (.2); a demand equaton; an expresson for cuto revenues followng from equaton (.3); a markup prcng expresson n equaton (.4); and a zero cuto pro t condton n equaton (.6). The free entry condton s substtuted n both the expresson for the number of varetes and the demand equaton. Ths gves the followng set of equatons. ( e ) N p Z ' N + p ' (.38) en N + r ' (.39) (.4) r ' p ' ( e ) (.4) p ' p Z ' (.42) r ' f p p Z (.43) GMS code avalable upon request shows that both representatons of the model generate exactly dentcal outcomes for the prce of nput bundles when dentcal parameters and data for populaton and trade costs are used. s parameter values we used the same values as n the man text wth 3:8, 3:4 and :5. The sngle equaton code solves the baselne n 3 teratons n GMS, whereas the code wth all equatons requres 398 teratons. Wth countres ths s stll a relatvely fast process, but wth more than countres t s lkely to encounter problems n solvng the model. ppendx aton and Kortum conomy The man structure of the aton and Kortum economy s descrbed n the man text. Gven the Frechet dstrbuton of productvtes ' n equaton (Frechet) the prce p of a good sold from country to s als Frechet dstrbuted: G (p) exp T (( + ta ) p Z ) p (.) 3

33 The realsed prce of varety n country s the mnmum prce of all potental supplers: p () mn fp () ; ::; p J ()g (.2) Therefore, the dstrbuton of prces n country s gven by: 6 G (p) JY ( G (p)) exp p (.3) Wth de ned as: T (( + ta ) p Z ) The probablty that country delvers a good to country s equal to: (.4) T (( + ta ) p Z ) J T k (( + ta k ) k p Zk ) k (.5) It can be shown that the prce dstrbuton of goods bought from country n country s equal to the general dstrbuton of prces n country, G ('). Ths mples that avere expendture n country does not vary by source as ponted out by aton and Kortum, snce expendture can only vary by source because of prce d erences. Ths mplcaton thus also holds for quantty and thus the quantty sold from to s gven by: q e T (( + ta ) p Z ) q J e T k (( + ta k ) k p Zk ) k Fnally, the prce ndex follows from calculatng the expected prce and substtutng the result nto the expresson for the prce ndex correspondng to utlty n equaton (??): ek k T k (( + ta k ) k p Zk ) We can rewrte ths prce ndex as n the expresson for the prce ndex wth a represenatve 6 The probablty that a prce n country s smaller than p s equal to mnus the probablty that none of the supplers has a prce smaller than p. 32

34 rm n equaton (5): e e e e ek k T (( + ta ) p Zk ) ek T (( + ta ) p Z ) ( +) (( + ta ) p Z ) k ( ek T ) k (( + ta ) p Z ) ( + ) (( + ta ) p Z ) ( + ta ) (( + ta ) ) + p + Z ek T (p Z ) (.6) Hence, from equaton (.6) we have for c and t : c + p ek T t (( + ta ) ) + To determne e, we use the rst equalty sgn n (.6): e + + Hence, usng e e, we have for e : e + Substtutng p ( + ta ) t c p Z nto the demand equaton for the general model (omttng 33

35 the rmngton shfter) gves: q q p q (( + ta ) t c p Z ) J k (( + ta ) ) J ( + ta ) (( + ta ) ) + p + Z ( + ta k ) (( + ta k ) k ) + p + Z k k (ek T ) p (( + ta k ) k ) ek T k p Z k k ek T pz ek T k (pzk ) Ths expresson s not dentcal to the expresson for the probablty that goods are sourced from country n equaton (.5). In aton and Kortum the quantty share of goods sourced from country s dentcal to the probablty that goods are sourced from country, snce avere prces are dentcal. Ths s not the case n the generalzed model, where the quantty share s equal to: q q p J p p For the expendture share, the expresson generated from the expressons for c and t are correct n the aton and Kortum model: p q p J k p J p ( + ta ) (( + ta ) ) + p + Z ( + ta k ) (( + ta k ) k ) + p + Z k k ekt (( + ta ) ) p J ek T k (( + ta k ) k ) p Z k k ek T pz ek T k (pzk ) Hence, we can use the representatve rm representaton of the aton and Kortum model f we 34

36 solve the model usng closures wth expendtures (or expendture shares) and the prce ndex. xpendture closures typcally have the form that sales from an exporter has to be equal to the mport shares tmes spendng n the tradng partners. If we work wth quanttes, the representaton s not correct. 35

37 Supplementary ppendces of Dervatons quaton (24) To convert Meltz nto ther/krugman the followng should hold: m ek Substtutng the expressons for ek and m n equatons (??) and (2) leads to the followng expresson for : (+) + + (+) ( + ) + + quaton (??) D erentatng equaton (25) on the RHS and LHS wrt to the endogenous varables gves: dv dn r + N G + N G ' Z ' dn r + N G g ' + V d' G ' Z dr (') g (') d' N ' G ' g ' r (') g (') d' d' G ' ' Z ' dr (') g (') d' N G ' ' r ' g ' d' r ' g ' d'