A Unified Model of International Business Cycles and Trade

Transcription

1 A Unified Model of International Business Cycles and Trade Saroj Bhattarai U of Texas at Austin Konstantin Kucheryavyy U of Tokyo February Abstract We present a general competive open economy business cycle model wh capal accumulation trade in intermediate goods production externalies in the intermediate and final goods sectors and iceberg trade costs Our main theoretical result shows that models developed in the modern international trade lerature that feature comparative advantage monopolistic competion and cost of entry and firm heterogeney and cost of exporting are isomorphic in terms of aggregate equilibrium to versions of this competive dynamic model under appropriate restrictions on the externalies In particular the restrictions apply on the overall scale of externalies the spl of externalies between the different factors of production and the identy of the sectors wh production externalies Our quantative exercise assesses whether various restricted versions of the general model in forms they are typically considered in the lerature are able to resolve the well-known aggregate empirical puzzles in international business cycle models Our theoretical result on isomorphism between models then provides insights on why they fail to do so in many instances We thus provide a unified theoretical and quantative treatment of the international business cycles and trade leratures in a general dynamic framework Key words: International business cycles; Dynamic trade models; Production externalies; Monopolistic competion; Fixed costs JEL classifications: F2; F4; F44; F32 We thank Andres Rodriguez-Clare for helpful comments This version: February 208 Department of Economics Universy of Texas at Austin; sarojbhattarai@austinutexasedu Graduate School of Economics Universy of Tokyo; kucher@eu-tokyoacjp

2 Introduction Are margins identified in the modern international trade lerature important for international business cycle dynamics? Do features such as heterogeneous comparative advantage monopolistic competion wh sunk cost of entry or heterogeneous firms wh fixed cost of exporting combined wh costly trade matter eher qualatively or quantatively for the transmission of aggregate shocks in a dynamic open economy business cycle model? Do these margins change the aggregate predictions one gets from using a neoclassical competive open economy business cycle model? If so is by enabling a better f between the data and the model by resolving some well-known empirical puzzles such as the high correlation of output compared to consumption and the high volatily of real exchange rates? We provide a unified model of international business cycles and trade that can address these important questions We provide such a unified treatment in steps First we formulate a general dynamic open economy model where all sectors are competive but there are externalies in the production function In particular in this model there is trade in the intermediate goods sector which uses labor and capal for production and features factor externalies Trade is costly which we model using iceberg costs as is standard in the trade lerature The intermediate goods including imported ones are combined into a final good using a production function that features external economies of scale The aggregator we consider is a standard Armington type The final good is used in the production of both the investment and consumption good The production function for investment good addionally uses labor input as well Our set-up is general in assumptions on international trade in assets and we consider all three standard cases: financial autarky bond economy and a complete markets economy Second we formulate general dynamic versions of three classic international trade models: Eaton-Kortum Krugman and Melz models 2 In terms of intertemporal linkages apart from trade in assets internationally the dynamic Eaton-Kortum model features capal accumulation while the dynamic Krugman and Melz models feature a law of motion of differentiated varieties driven by firm entry and ex After formulating the general competive dynamic model wh production externali- As is somewhat obvious from this description the canonical open economy real business cycle model as presented in Heathcote and Perri 2002) for instance is nested in this version as a case wh no iceberg trade costs no externalies and no labor as input in the production of the investment good 2 We explain in detail later in the paper why our set-ups are more general than similar models in the lerature and precisely how we generalize them Here we simply want to point out that these generalizations are in fact key to establish complete isomorphisms between the unified competive model and the three trade models 2

3 ties and the three dynamic trade models we then derive the main theoretical result We show that in terms of aggregate implications after appropriate re-labeling of variables and parameters the three dynamic trade models are isomorphic to the general competive dynamic model 3 This isomorphism holds even though the three dynamic trade models have very different micro-foundations Let us now discuss more what our main theoretical result implies In terms of relabeling of variables the result is based on the similary between the law of motion of physical capal in the general competive model wh the law of motion of the differentiated varieties in the dynamic Krugman and Melz models In terms of re-labeling of parameters the isomorphism between the general competive model and the Eaton- Kortum model is very direct This re-labeling of parameters is more interesting and involved for the dynamic Krugman and Melz models The dynamic Krugman model has monopolistic competion and sunk cost of entry For the isomorphism between the unified model and the Krugman model we need that the share of capal in the production function be governed by the elasticy of substution across varieties; the total scale of externalies in the intermediate goods sector be governed by a parameter determining love for variety and individual factor externalies; and that the spl in factor externalies be governed by the elasticy of substution across varieties the love for variety parameter and the individual factor externalies in the production function 4 Finally the dynamic Melz model compared to the dynamic Krugman model addionally features heterogeneous productivies across firms and a fixed cost of serving different markets For the isomorphism between the unified model and the Melz model we need that the share of capal in the production function be governed by the elasticy of substution across varieties as well as the Pareto parameter governing dispersion in productivies; the total scale of externalies in the intermediate goods sector be governed by a parameter determining externaly in the fixed cost of serving markets and individual factor externalies; and that the spl in factor externalies be governed by the elasticy of substution across varieties the externaly in fixed cost of serving markets and the individual factor externalies in the production function Moreover a key distinction of the dynamic Melz model is that also addionally features external economies of scale in the aggregate good sector where intermediate goods are combined Given the theoretical result we then undertake a quantative exercise We consider 3 More precisely we show that the equilibrium system of equations that governs aggregate dynamics are the same across these variants 4 The spl is essentially a free parameterization because of this general structure of the dynamic Krugman model 3

4 how these different versions of the dynamic trade models as they are often used in the lerature whout the generalizations we propose lead to differential aggregate implications in terms of business cycle moments Our point of comparison is wh the standard open economy business cycle model that has no externalies at all We show that these models are not able to resolve some key empirical puzzles related to cross-country output consumption and real exchange rate behavior that plague the standard business cycle model We provide an interpretation based on our theoretical results: standard formulations of these models and the calibration lead to very small production externalies which are in turn tightly restricted in terms of spls across factors This then leads to aggregate second moments very similar to the standard competive business cycle moment wh no externalies Our paper is related to several strands of the lerature In formulating a dynamic international business cycle model that incorporates the margins of the model international trade lerature we are clearly building on seminal trade contributions of Krugman 980) Eaton and Kortum 2002) and Melz 2003) In particular Ghironi and Melz 2005) Alessandria and Choi 2007) Fattal Jaef and Lopez 204) and Eaton et al 206) also develop dynamic models similar to ours and assess how important international trade features are for aggregate dynamics and business cycle moments Our main theoretical contribution is to develop a general competive model wh production externalies that is isomorphic to various versions of such dynamic models Moreover our theoretical results are then useful to understand the quantative findings of Alessandria and Choi 2007) and Fattal Jaef and Lopez 204) that firm heterogeney and costs of entry and exporting does not matter for aggregate dynamics Our paper is also related to the closed economy lerature Benhabib and Farmer 994) in a well-known paper introduced production externalies to the standard neoclassical business cycle model to increase the propagation of aggregate productivy shocks and in particular to generate the possibily of multiple bounded equilibria Moreover the closed economy set-up in Bilbiie et al 202) also links firm dynamics and entry in a model wh monopolistic competion and sunk cost of entry thus similar to the closed economy dynamic version of the Krugman model we develop in this paper) to the capal stock dynamics and investment in the standard competive business cycle model Our general model provides a similar interpretation as well while addionally showing precisely theoretically how a competive open economy set-up wh different levels and types of production externaly is in fact isomorphic to various versions of monopolistic competion models wh firm heterogeney and costs of entry and exporting 4

5 2 Unified Model of Trade and Business Cycle We present a dynamic stochastic general equilibrium model wh multiple countries and international trade Time is discrete The world consists of countries 5 Each country has four production sectors: intermediate final aggregate consumption investment Intermediate goods are produced from capal and labor Final aggregate is assembled from intermediate goods Consumption good is produced directly from the final aggregate Investment good is produced from the final aggregate and labor All markets are perfectly competive Labor is perfectly mobile between the sectors where is used Technology of production of intermediate goods and final aggregates features external economies of scale Only intermediate goods can be traded Trade is subject to iceberg trade costs Financial markets structure can be one of the three standard alternatives: financial autarky bond economy or complete financial marketswe now describe the model in detail 2 Intermediate Goods and International Trade Denote by X the total output of intermediate goods in country i at time t Technology of production of intermediate goods is given by X = S X K α XK X Lα XL X where α XK 0 α XL 0 and α XK + α XL = Here K X and L X are the total amounts of country i s capal and labor used in production of intermediates and S X = Θ Xi Z X K ψ XK X Lψ XL X is aggregate productivy The aggregate productivy consists of two parts: exogenous productivy Θ Xi Z X and endogenous productivy K ψ XK X Lψ XL X wh ψ XK 0 and 0 The endogenous productivy part captures external economies of scale in ψ XL the production of intermediates and is taken by firms as given The term Θ Xi in the exogenous productivy part is a normalization constant that is introduced for convenience to later show isomorphisms between the current setup and dynamic versions of Eaton- Kortum Krugman and Melz models Let P X denote the price of country i s intermediate good in period t Let W and R be the wage and capal rental rate in country i in period t Perfect competion in the 5 In all our quantative exercises we focus on the case of = 2 But there is nothing that prevents us from formulating the theoretical framework wh any number of countries Moreover following the modern quantative trade lerature we prefer to set the environment in terms of a general 5

6 production of intermediates implies that K X = α XK P X X R and L X = α XL P X X W Also P X = R α XK W α XL Θ Xi Z X K ψ XK X Lψ XL X where Θ Xi α α XK XK αα XL XL Θ Xi Intermediate goods are the only traded goods and trade in these goods is costly Trade costs are of the iceberg nature: in order to deliver one un of intermediate good to country n country i needs to ship τ nit uns of this good To guarantee absence of arbrage in the transportation of goods we require that trade costs satisfy the triangle inequaly: τ nkt τ kit τ nit for any n k and i This implies that the price of country i s intermediate good sold in country n is given by τ nit P X 22 Final Aggregates and Consumption Goods Denote by Y nt the total output of final aggregate in country n at time t and by P Ynt the price of this aggregate Final aggregate is produced by combining intermediate goods sourced from different counties: Y nt = S Ynt [ i= ] ω ni X nit where ω ni 0 and > 0 Here X nit is the amount of intermediate good that country n buys from country i and ) PYnt Y ψy nt S Ynt = Θ Yn Z Ynt W nt is productivy As in the production of intermediates productivy in the production of W nt the final aggregate has two components: exogenous component Θ Yn Z Ynt and endogenous component wh ψ Y 0 The endogenous component of S Ynt ) PYnt Y ψy nt captures external economies of scale in the production of the final aggregate and is taken by firms as given The term Θ Yn is a normalization constant introduced for convenience Perfect competion in the production of the final aggregate implies that the price of 6

7 the final aggregate is given by P Ynt = [ i= ω ni τ nit P X Θ Yn Z Ynt PYnt Y nt W nt ] ) ψy and country n s share of expendure on country i s intermediate good is given by λ nit = ω ni τ nit P X l= ω nl τ nlt P Xlt Final aggregate in country n is used directly as the consumption good in this country as well as in the production process of the investment good which is described next 23 Investment Goods Denote by I the total output of the investment good in country i in period t and by P I the price of this good Investment good is produced from labor and the final aggregate: I = Θ Ii Z I L β I I Y β I I where 0 β I Here L I and Y I are the total amounts of labor and final aggregate used in the production of the investment good and the term Θ Ii Z I is an exogenous productivy The term Θ Ii is a normalization constant introduced for convenience Perfect competion in the production of the investment good implies L I = β I P I I W Y I = β I ) P II P Y Also P I = Wβ I P β I Y Θ Ii Z I where Θ Ii β β I I β I ) β I Θ Ii 7

8 24 Households Households in country i maximize expected sum of discounted utilies E 0 t=0 βt U C L ) by choosing consumption C supply of labor L investment I and holdings of financial assets if allowed) The law of motion of capal is given by K it+ = δ) K + I Depending on the financial markets structure households face different budget constraints Below we consider three standard alternatives for financial markets: financial autarky bond economy and complete markets 24 Financial Autarky In the case of financial autarky there is no international trade in financial assets Households in country i face the following budget constraint: P Y C + P I I = W L + R K First-order condions for the household s optimization problem imply: { } PY P I = βe t U C it+ L it+ ) [R P Yit+ U C L ) it+ + δ) P Iit+ ] ) U 2 C L ) U C L ) = W P Y 2) where U ) and U 2 ) are derivatives of the utily function wh respect to consumption and labor correspondingly 242 Bond Economy We consider a variant of the bond economy where each country issues a non-state-contingent bond denominated in consumption uns of this country Every country chooses holdings of bonds of all countries Holdings of country i s bond by country n are denoted by B nit 8

9 Household s budget constraint is given by P Y C + P I I + P Ylt B ilt + b ) adj 2 B2 ilt l= = W L + R K + P Ylt + r lt ) B ilt + T B l= where r lt is period-t return on country-l s bond and T B b adj 2 P Ylt Bilt 2 l= is the bond fee rebate taken as given by households Here b adj is the adjustment cost of bond holdings which is introduced to ensure stationary First-order condions in the case of the bond economy imply condions ) and 2) plus an addional set of Euler equations: P Ylt U C L ) P Y for l = + badj B ilt ) = βet { } U C it+ L it+ ) P P Ylt+ + r lt ) Yit+ International trade in bonds allows unbalanced trade in intermediate goods Define trade balance as the value of net exports of intermediate goods: TB P X X P Y Y Define country i s current account as the change in this country s financial assets posion: 6 CA P Ylt B ilt B ilt ) l= 6 Using markets clearing condions described later) can be shown that trade balance and current account can also be wrten as TB = W L + R K P Y C P I I CA = TB + l= r lt P Ylt B ilt 9

10 243 Complete Financial Markets To introduce the household s budget constraint in the case of complete markets we employ notation for the states of the world in period t denoted by s t and history of states in period t denoted by s t We consider a variant of complete markets where in each state wh history s t countries trade a complete set of state-contingent nominal bonds denominated in the numeraire currency Let B it+ s t s t+ ) denote the amount of the nominal bond wh return in state s t+ that country i acquires in the state wh history s t Assuming that there are no costs of trading currency or securies between countries we can denote by Q t s t s t+ ) the international price of this bond in the state wh history s t Country i s budget constraint is given by where P Y s t ) C s t ) + P I s t ) I s t ) + A s t ) = W s t ) L s t ) + R s t ) K s t ) + B s t ) A s t ) s t+ Q t s t s t+ ) Bit+ s t s t+ ) is country i s net foreign assets posion in period t First-order condions in the case of complete markets imply condions ) and 2) wh the state-dependent notation added to them) plus an addional set of condions: Q t s t ) π t+ s t+ ) P Y s t ) s t+ = β π t s t ) P Yit+ s t+ ) U Cit+ s t+ ) L it+ s t+ )) U C s t ) L s t )) P Y s t ) P Yjt s t ) = κ U C s t ) L s t )) ij ) U Cjt s t ) L jt s t for each i and j ) where π t s t ) is the probabily of history s t occurring in period t and U Ci0 s 0 ) L i0 s 0 )) /P Yi0 s 0 ) ) κ ij U Cj0 s 0 ) L j0 s 0 ) ) /P Yj0 s 0 ) 0

11 By dropping the state-dependent notation the above condions can be wrten compactly as P Y C + P I I + A = W L + R K + B { } PY A = βe t U C it+ L it+ ) B P Yit+ U C L ) it+ P Y U = κ C L ) P ij ) for each i and j Yjt U Cjt L jt As in the case of the bond economy trade balance is defined as net exports of intermediate goods and current account is defined as the change in net foreign assets posion: TB = P X X P Y Y CA = A A it 25 Market Clearing Condions The labor market clearing condion is not standard and is given by W L X + W L I = W L + atb for i = where a 0 is a constant When a = 0 we have a standard labor market clearing condion The ad-hoc term atb is introduced to later show isomorphism wh the Melz model for which a > 0 The rest of the market clearing condions for the economy are of a standard form Capal is used only in the production of intermediate goods: K X = K for i = demand for intermediate goods is equal to supply: τ nit M nit = X for i = n= and final aggregate is used in consumption and production of the investment good: C + Y I = Y for i =

12 In the case of the bond economy and complete markets we also have the sets of bonds market clearing condions which are given by for the bond economy and by for complete markets B nit = 0 for i = n= A = 0 i= For convenience the full set of equilibrium condions is provided in Appendix A 3 Generalized Versions of the Standard Trade Models In this section we present the key elements of generalized versions of the workhorse international trade models: Eaton-Kortum Krugman and Melz The focus of this section is to present the elements of these models that differ from their standard exposions Thus the presentation oms all the derivations which are provided in Appendix B 3 Generalized Version of the Eaton-Kortum Model Household s problem is identical to the one in the unified model Also as in the unified model the production side consists of intermediate final consumption and investment goods All markets are perfectly competive The intermediate goods sector consists of a continuum of goods indexed by ν [0 ] The production technology of good ν in country i in period t is given by x ν) = S X z ν) K X ν) α XK L X ν) α XL where K X ν) and L X ν) are capal and labor used in production of good ν z ν) is the efficiency of production of good ν and S X K ψ XK X Lψ XL X is aggregate productivy that depends on the total amount of capal K X and labor L X used in production of all intermediate goods in country i in period t Aggregate productivy S X captures external economies of scale in the production of intermediates and is taken by firms as given Efficiencies z ν) are drawn from the Fréchet distribution wh parameters Z X 2

13 and θ Its cumulative distribution function is given by Prob [z ν) z] = e z/z X) θ Intermediate goods can be traded Trade is costly and is subject to iceberg trade costs denoted by τ nit Intermediate goods are combined into the non-tradeable final aggregate: [ Y = 0 x ν) ] dν Producers of the final aggregate in country i buy each good ν from the cheapest source The price of the final aggregate is given by P Y = γ EK l= ) θ + where γ EK Γ is a constant and θ τ ilt P Xlt ) θ ) θ P X R α XK W α XL Θ X Z X K ψ 3) XK X Lψ XL X wh Θ X α α XK XK αα XL XL Price P X can be interpreted as the price of the output of intermediates in country i in period t The final aggregate is used for consumption and investment Technologies of production of the consumption and investment goods are the same as in the unified model The complete set of equilibrium condions for the generalized Eaton-Kortum model is provided in Appendix B 32 Generalized Version of the Krugman Model 32 Production of Varieties International Trade and Final Aggregate Each country i produces a unique set of varieties Ω which is endogenously determined in every period t Let M be the measure of this set All varieties can be internationally traded Let p nit ν) denote the price of variety ν Ω produced by country i and sold in country n Assuming iceberg trade costs and no arbrage in international trade we have that p nit ν) = τ nit p iit ν) 3

14 Countries use varieties to produce non-traded final aggregates Technology of production of the final aggregate in country n is given by the nested CES production function: Y nt = i= [ [ ] M φ x nit ν) dν ν Ω ] η η η η 4) where x nit ν) is the amount of variety ν Ω that country n buys from country i in period t The nested CES structure implies that the elasticy of substution between varieties produced in one country given by is different from the elasticy of substution between varieties produced in different countries given by η The term M φ introduces correction for the love-of-variety effect which is the only source of externalies in the standard Krugman model wh CES preferences Assuming perfect competion in the production of the final aggregate we get the usual CES demand: x nit ν) = M )φ ) [ P nit = M φ ) P Ynt = ) pnit ν) ) η Pnit Y nt 5) P nit P Ynt ] 6) p nit ν) dν ν Ω ) P η η nit 7) i Production of variety ν Ω requires only labor and is given by x ν) = S X l ν) where l ν) is the amount of labor used in production of variety ν and S X Z X M φ XM L φ XL X is the aggregate productivy in production of varieties The aggregate productivy S X consists of two parts: exogenous productivy Z X and endogenous productivy M φ XM L φ XL X where L X is the total amount of labor allocated to the production of varieties in country i in period t The endogenous part of the aggregate productivy is an addional source of external economies of scale on top of the love-of-variety effect) and is taken by firms as given Producers of varieties ν are engaged in monopolistic competion Hence the price of variety ν Ω is p nit ν) = τnitw S X 4

15 the bilateral price index is P nit = τ nit P X where P X W Z X M φ XM+φ and the share of expendure of country n on country i s varieties is λ nit = τ nitp X ) η l= τ nltp Xlt ) η L φ 8) XL X Similarly to the generalized Eaton-Kortum model P X can be interpreted as the price of the output of varieties in country i in period t Let X denote the value of total output of varieties in country i in period t and D denote the average prof of country i s producers of varieties Ω We have X = W L X D = X M 322 Entry and Ex of Producers of Varieties In order to enter the economy producer of a variety in country i in period t needs to pay sunk cost equal to Wβ I P β I Y where 0 β I and Θ Ii Z I is an exogenous cost shifter Θ Ii Z I V Paying this sunk cost involves hiring β I uns of labor and using β I ) V uns W P Y of the final aggregate where V is the value of a variety in country i in period t 7 In every period t each country has an unbounded mass of prospective entrants firms) into the production of varieties Entry into the economy is free and therefore the value of a variety is equal to the sunk cost of entry: V = Wβ I P β I Y Θ Ii Z I Firms entering in period t start producing in the next period At the end of each period t an exogenous fraction δ of the total mass of firms ie a fraction δ of M ) exs The 7 In Appendix B2 we derive the sunk cost by introducing an R&D sector and specifying an invention process for new varieties Labor and final aggregate needed to pay the sunk cost of entry are interpreted as the production factors used in the R&D sector for the invention of varieties 5

16 probabily of ex is the same for all firms regardless of their age Since ex occurs at the end of a period any firm that entered into the economy produces for at least one period Let M I denote the number of producers of varieties that enter into the country i s economy in period t Given the described process of entry and ex of firms the law of motion of varieties is M it+ = δ) M + M I 9) All producers of varieties are owned by households We turn next to their problem 323 Households Similarly to the unified model households in country i maximize expected sum of discounted utilies E 0 t=0 βt U C L ) by choosing consumption C supply of labor L the number of new varieties M I and holdings of financial assets if allowed) Constraints faced by the households are the budget constraint and the law of motion of varieties given by 9) The specification of the budget constraint depends on the financial markets structure In the case of financial autarky the budget constraint is given by P Y C + V M I = W L + D M The left-hand side of this expression contains household s expendure in period t: the household spends s budget on consumption and entry of new firms The right-hand side of this expression contains household s income in period t: consists of labor income and profs of firms In the case of the bond economy and complete markets the budget constraints can be wrten in a similar way by adding the expendure and income from financial assets in the same manner as is done in the unified model 324 Markets Clearing Condions All market clearing condions are standard Labor is used for production and invention of varieties: demand for varieties is equal to supply: L X + L I = L λ nit P Ynt Y nt = X n= 6

17 and final aggregate is used for consumption and invention of varieties: C + Y I = Y The complete set of equilibrium condions for the generalized Krugman model is provided in Appendix B2 33 Generalized Version of the Melz Model 33 Production of Varieties International Trade and Final Aggregate In every period t country i can produce any of the varieties from an endogenously determined set of varieties Ω wh measure M All varieties from the set Ω can be internationally traded but not all of them are available in a particular country n The subset of country-i s varieties available in country n is denoted by Ω nit wh Ω nit Ω ) and s measure is denoted by M nit Importantly only a subset Ω iit of the whole set of varieties Ω is available in the domestic market i and generally some varieties from Ω are not available in any country ie some varieties from Ω are not produced in period t) Subsets of varieties Ω nit are endogenously determined In order to sell in the country-n s market a country-i s producer of a variety has to pay two types of costs: the usual per-un iceberg trade costs τ nit and fixed cost Φ nit > 0 which are paid in terms of the country-n s labor The fixed cost Φ nit is an endogenous object Its formal definion is introduced later As in the Krugman model countries combine varieties to produce non-traded final aggregates: Y nt = i= [ x nit ν) ν Ω nit ] dν η η Differently from the Krugman model there is no correction for the love-of-variety effect because in the type of the Melz model presented here the love-of-variety effect induces an externaly only through the interaction wh the fixed costs of serving markets which is discussed below) Perfect competion in the production of the final aggregate implies the usual expressions for the CES demand that are almost the same as the corresponding expressions 5)-7) in the Krugman model except for there is no term correcting for the love of variety Production technology of variety ν Ω is given by x ν) = S X z i ν) l ν) where η η 7

18 l ν) is the amount of labor used in production of ν z i ν) is the efficiency of production of ν and S X Z X M φ XM L φ XL X is the aggregate productivy in production of varieties wh L X being the total amount of labor used in the production of varieties in country i As in the Krugman model S X features external economies of scale and is taken by firms as given Monopolistic competion in the production of varieties implies that the price of variety ν Ω nit is given by p nit ν) = 332 Entry and Ex of Producers of Varieties τ nit W S X z i ν) This part of the Melz model is almost the same as the corresponding part of the Krugman model wh one important difference that upon entry producer of a new variety gets an idiosyncratic draw of efficiency of production z i ν) from the Pareto distribution given by s cumulative distribution function: G i z) Prob [z i ν) z] = zmini As in the Krugman model the expected value of entry before drawing the efficiency of production) is denoted by V The sunk cost of entry is equal to Wβ I P β I Y and in Θ Ii Z I equilibrium is equalized wh the expected value of entry The number of producers of varieties entering into the country i s economy in period t is denoted by M I The law of motion of varieties is M it+ = δ) M + M I Since the probabily of ex is the same for all varieties ν Ω the distribution of efficiencies of production of varieties ν Ω in any period t is given by G i z) Under the assumption that efficiencies of production of varieties are distributed Pareto we can derive that the set of country-i s varieties available in country n is given by where z nit is given by Ω nit = z mini z nit { } ν Ω zi ν) z nit ) θ = θ + θ z X nit W nt Φ nit M wh X nit being the total value of varieties that country n buys from i in period t ) θ 8

19 333 Fixed Costs of Serving Markets and the Rest of the Model At this point we need to introduce the formal definion of the fixed costs of serving market n by firms from market i Φ nit Let L Fnt be the total amount of country n s labor that is used to pay the fixed costs of serving s market We assume that [ ] Φ nit M θ φ FM L δ φ δ FL Fnt Fnit 0) [ ] where F nit is an exogenous part of the fixed costs M θ φ FM L δ φ δ FL Fnt is an endogenous part of the fixed costs that is taken by firms as given and δ θ [ ] Under the assumption that θ > we have that δ > 0 The term M θ φ FM L δ φ δ FL Fnt corrects for the externaly that arises due to interaction of scale and love-of-variety effects The intuion is the following If the market is served by a small set of large firms then is cheaper to serve this market because average costs for each of the firms are lower This is the scale effect The scale effect goes against the love-of-variety effect: consumers of the final good gain from access to a larger set of varieties The trade-off between these two effects is captured by δ When δ = 0 the two effects just offset each other Given larger θ implies larger δ High θ implies lower variance of Pareto productivies When variance of productivies is low all firms look similar And so they eher all enter the market or none of them enters Conversely low θ implies higher variance of Pareto productivies which allows for the scale effect to kick in Under the assumption 0) on the form on fixed costs of serving markets we can derive that the bilateral price index is P nit = τ nit P X where P X = W z mini Z X M φ XM+φ FM L φ XL X ) is interpreted as the price of the output of varieties in country i in period t The price of the final aggregate is θ P Ynt = θ + ) +φ FL P Ynt Y nt W nt ) φfl [ i= F δnit τ nitp X ) θξ ] θξ 9

20 where ξ η ) θ + and the share of expendure of country n on country i s varieties is F δ nit τ nitp X ) θξ λ nit = l= F δ nlt τ nltp Xlt ) θξ The value of total output of varieties in country i in period t is X = W L X and total average profs of country i s producers of varieties are D = θ X M The total amount of country n s labor used to serve s market is which can also be wrten as L Fnt = θ + θ PYntY nt W nt L Fnt = δp YntY nt X nt L Xnt If trade is balanced then P Ynt Y nt = X nt and so L Fnt = δl Xnt The household s problem is identical to the one in the Krugman model Labor market clearing condion is different from the corresponding condion in the Krugman model involves labor used for serving markets L F : L X + L F + L I = L The other condions are the same as in the Krugman model: λ nit P Ynt Y nt = X n= C + Y I = Y The complete set of equilibrium condions for the generalized Melz model is pro- 20

21 vided in Appendix B3 4 Theoretical Results In this section we first describe relation to relevant models in the lerature of the the unified model of Section 2 henceforth simply the unified model ) and the generalized models of international trade described in Section 3 After that we establish isomorphisms between the unified model and the models of Section 3 And then we explore quantatively the abily of the unified model to match business cycle moments observed in the data 4 Relation to Models in the Lerature The standard international real business cycle model as is described for example in Heathcote and Perri 2002) can be obtained as a special case of the unified model by shutting down externalies removing iceberg trade costs and requiring that capal investment uses the final aggregate only ie does not use labor) Formally the unified model gives the standard international real business cycle model if we set ψ XK = ψ XL = ψ Y = 0 Θ Xi = Θ Yi = Θ Ii = Z Y = Z I = β I = 0 and τ nit = A straightforward extension of the standard static version of the Eaton-Kortum model to a dynamic version wh intertemporal investment decisions along the lines of for example Eaton et al 206) can be obtained from the generalized Eaton-Kortum model of Section 3 by shutting down externalies and requiring that capal investment uses the final aggregate only Formally this is achieved by setting ψ XK = ψ XL = 0 and β I = 0 A dynamic version of the standard Krugman model which can be obtained by for example a straightforward extension of Bilbiie et al 202) to a multi-country environment can be obtained from the generalized Krugman model of Section 3 by removing correction for the love of variety shutting down external economies of scale in production of varieties and requiring that producers of varieties pay entry costs in terms of labor only Formally this is achieved by setting φ = φ XM = φ XL = 0 and β I = There are no direct analogs in the existing lerature of the generalized Melz model of Section 3 There are two important differences of the generalized Melz model wh the dynamic versions of the Melz model described in for example Ghironi and Melz 2005) Alessandria and Choi 2007) and Fattal Jaef and Lopez 204) First fixed costs of serving markets in the generalized Melz model are payed in terms of the destinationcountry labor while in the existing dynamic Melz models the fixed costs are paid in 2

22 terms of the source-country labor Second there are non-zero fixed costs of serving the domestic markets in the generalized Melz model while in the existing dynamic Melz models there are no fixed costs of serving domestic markets The presence of such costs in the generalized Melz model creates a suation when in every period there are some firms that neher produce nor ex These firms have too low efficiency of production to overcome fixed costs of serving markets but had high enough efficiency of production to enter the economy at some point In the existing dynamic Melz models all firms that enter the economy produce for at least the domestic market Quantatively the effects of the differences in these assumptions are small in the environment wh two symmetric countries which is tradionally the focus of the international business cycle lerature and which is studied in the quantative part of the current paper) At the same time the assumptions about fixed costs of serving markets made in the generalized Melz model here allow us to establish isomorphism wh the unified model 8 If we shut down external economies of scale in production of varieties and in the fixed costs of serving markets by setting φ XM = φ XL = 0 φ FM = θ and φ FL = δ) and if we require that the sunk costs of entry into the economy are paid in terms of labor only by setting β I = ) then the only essential differences between the generalized Melz model and the existing dynamic versions of the Melz model will be the differences in the assumptions about fixed costs of serving markets described in the previous paragraph In the rest of this paper for brevy when there is no risk of creating confusion we refer to the generalized international trade models of Section 3 simply as the Eaton-Kortum model the Krugman model and the Melz model 42 Isomorphisms The key result in establishing the link between the unified model and the models of Section 3 is the following lemma: Lemma By an appropriate relabeling of variables and parameters price of country i s output of varieties in the Eaton-Kortum Krugman and Melz models given correspondingly by 8 The generalized Melz model can be considered as an extension to a dynamic environment of the static version of the Melz model described in Kucheryavyy et al 207) who make the same assumptions about fixed costs of serving markets as in the current paper These assumptions allow Kucheryavyy et al 207) to establish isomorphism of the static version of the Melz model to a static version of the Eaton-Kortum model wh external economies of scale 22

23 expressions 3) 8) and ) can be wrten as P X = R α XK W α XL Θ Xi Z X K ψ 2) XK X Lψ XL X which is the price of country i s intermediates in the unified model There is nothing to prove in the case of the Eaton-Kortum model: the price of output of varieties in the Eaton-Kortum model given by 3) is identical to the price in expression 2) The formal proofs of the above lemma for the Krugman and Melz models are provided in Appendices B2 and B3 Here we provide an informal discussion of the mapping between prices 8) and ) in the Krugman and Melz models and price 2) in the unified model In both the Krugman and Melz models we need to relabel variables D as R and M as K X to obtain expression 2) Informally the average firms prof in country i and the measure of country i s varieties in the Krugman and Melz models play the role of correspondingly return on capal in country i and the stock of country i s capal in the unified model In addion to this the amount of labor used in production of varieties in the Melz model L X needs to be multiplied by ) in order for to map θ to the amount of labor used in production of intermediates in the unified model also denoted by L X This adjustment to L X in the Melz model has to be done because in the Melz model differently from the other models there is an extra use of the total labor available in the economy: to pay fixed costs of serving markets And is the sum of the labor used in production of varieties and the labor used to pay fixed costs of serving markets that in the Melz model corresponds to the labor used in production of intermediates in the unified model Turning to parameter mappings for the Krugman model we need to set α XK α XL ψ XK φ XM + φ ψ XL φ XL + ) and Θ Xi ) And for the Melz model we need to set α XK θ α XL θ ψ XK φ XM + φ FM θ ψ XL φ XL + θ and Θ Xi θ ) θ ) θ θ ) φxl z θ mini Lemma leads to our main theoretical result formulated in the next proposion 23

24 Proposion By an appropriate relabeling of variables and parameters in the Eaton-Kortum Krugman and Melz models we can wre the equilibrium system of equations in each of these models in a form identical to the equilibrium system of equations in the unified model Thus these models are isomorphic to each other in their aggregate predictions This proposion says that up to relabeling the Eaton-Kortum Krugman and Melz models are essentially the same despe having very different micro-foundations In particular under certain parameterizations these models are identical to a standard international business cycle model extended to allow for external economies of scale in production and iceberg trade costs We now provide a discussion of the isomorphism Both in the Krugman and Melz models the exponents of R and W which are the Cobb-Douglas shares in production of intermediates in the unified model) are pinned down by the fundamental parameters of these models: elasticy of substution between varieties produced in one country and in the case of the Melz model also by the shape of Pareto distribution θ In addion to determining exponents of R and W parameters and θ drive other economic outcomes in the standard Krugman and Melz models Several modeling assumptions made in the generalized versions of the Krugman and Melz models in the current paper allow us to break the links between parameters and θ and economic outcomes different from the exponents of R and W in expression 2) Let us first consider the Krugman model In order to understand our modeling assumptions for the generalized Krugman model helps to understand how the standard Krugman model more precisely a dynamic version of the standard Krugman model) relates to the generalized Krugman model The standard Krugman model can be obtained as a special case of the generalized Krugman model by making several parameter restrictions First we need to assume that in the production technology for the final aggregate given by expression 4) the elasticy of substution between varieties produced in different countries is equal to the elasticy of substution between varieties produced in one country ie assume that η = ) Second we need to remove the correction for the love-of-variety effect in the production technology for the final aggregate by setting φ = Third we need to shut down external economies of scale in production of varieties by setting φ XM = φ XL = 0 As should be clear from these parameter restrictions the standard Krugman model has trade elasticy equal to ) More importantly viewed through the lenses of the unified model the standard Krugman model implies that the strength of economies of scale in production of intermediates is given by ψ XK = for capal and by ψ XL = for labor Thus in the standard Krug 24

25 man model there is a tight link between the elasticy of substution between varieties in production of the final aggregate trade elasticy and the strength of economies of scale In the generalized Krugman model trade elasticy is given by η) Thus by assuming that η = we break the link between parameter and trade elasticy Also by introducing correction for the love-of-variety effect in the generalized Krugman model by assuming that φ = we break the tight link between parameter and the strength of economies of scale for capal We can get any desired value of parameter ψ XK by varying φ However correction for the love-of-variety effect does not break the link between parameter and the strength of economies of scale for labor To break this last link we directly introduce external economies of scale in production of varieties in the generalized Krugman model wh the strength of these economies of scale given by parameters φ XL for labor and φ XM for the number of varieties By varying φ XL we can get any desired level of the strength of economies of scale for labor in production of intermediates in the unified model Varying parameter φ XM achieves the same effect as varying parameter φ and the reasons we simultaneously introduce parameters φ XM and φ are to highlight the role of the love-of-variety effect and to have symmetry in the production technology of varieties Let us now turn to the Melz model Again let us understand how a dynamic version of the standard Melz model relates to the generalized Melz model First the same as in the generalized Krugman model in order to obtain the standard Melz model from the generalized Melz model we need to set η = Second we need to remove correction for the externaly that arises due to interaction of scale and love-of-variety effects in the presence of the fixed costs of serving markets This involves setting φ FM = θ and φ FL = δ Third we need to shut down external economies of scale in production of varieties by setting φ XM = φ XL = 0 43 Quantative Results 43 Calibration In the quantative section we focus on the world economy that consist of two symmetric countries We consider the following preferences: U C L ) = [ C µ γ L ) µ] γ Calibration of the parameters for the unified Krugman and Melz models is presented in Table The calibration for the Krugman model implies that in the isomorphic unified 25

26 model α XK = 026 ψ XK = 0095 ψ XL = 026 and ψ Y = 0 For the Melz model the isomorphic unified model has the following parameter values: α XK = 025 ψ XK = 0 ψ XL = 025 and ψ Y = 0075 Common Parameters Productivy Process Parameters β = 099 γ = 2 δ = 0025 µ = 034 τ nit = 57 ω ni = 05 β I = 0 Z I = Θ Xi = Θ Yi = Θ Ii = Z Y = [ ] log ZXt ) = log Z X2t ) [ εxt [ ] [ ] ρx ρ X2 log ZXt ) + ρ X2 ρ X22 log Z X2t ) ] [ 0 2 ] ) 0 X X2 X2 X2 2 ε X2t [ εxt ε X2t wh ρ X = ρ X22 = 097 ρ X2 = ρ X2 = 0025 X = X2 = X2 = X2 = 028 ] Unified Model = 2 α XK = 036 ψ XK = ψ XL = 0 ψ Y = 0 a = 0 Krugman = + /α XK η = 2 φ = φ XK = φ XL = 0 Melz θ = /α XK = 33 η = 2 φ XK = φ XL = 0 φ FK = θ φ FL = δ Table : Moments for Benchmark Calibration 432 Results Moments for the benchmark calibration are presented in Table 2 Column provides moments calcuated from the data These moments are taken from Heathcote and Perri 2002) and Alessandria and Choi 2007) Columns present results for the standard international business cycles model wh iceberg trade costs IRBC) columns and present results for versions of Krugman and Melz models that feature sunk costs of entry into the economy paid in terms of the final aggregate only Comparing outcomes of the three models wh moments in the data we see that Krugman and Melz models perform no better than the standard IRBC model: Krugman and Melz perform well or even worse) and fail in the same moments where the standard IRBC model performs well or fails 26

27 Financial Autarky Bond Economy Complete Markets Moment Data IRBC Krug Mel IRBC Krug Mel IRBC Krug Mel ) 2) 3) 4) 5) 6) 7) 8) 9) 0) Corr GDP GDP 2 ) Corr C C 2 ) Corr I I 2 ) ) C Corr P Y2 /P Y C 2 TB Corr GDP GDP ) CA Corr GDP GDP TB Corr P ) Y2 GDP P Y CA Corr P ) Y2 GDP P Y ) X2 Corr GDP P Y ) X2 Corr GDP P Y Var P Y2 /P Y ) Var GDP ) TB Var GDP) X2 Var P Y ) X2 Var P Y ) Conclusion Table 2: Moments for Benchmark Calibration How important are margins identified in the modern international trade lerature in explaining aggregate business cycle dynamics? We provide a unified theoretical and quantative treatment of the international business cycles and trade leratures in a general dynamic framework to answer this question We present a general competive open economy business cycle model wh capal accumulation production externalies trade in intermediate goods and iceberg trade costs Our main theoretical result shows that models developed in the modern inernational trade lerature that feature comparative advantage monopolistic competion and cost of entry and firm heterogeney and cost of exporting are isomorphic in terms of aggregate equilibrium to versions of this competive dynamic model under appropriate restrictions on the externalies In particular the restrictions apply on the overall scale of externalies the spl of externalies between the different factors of production and the identy of the sectors wh production exter- 27