ISSN 1178-2293 (Online)

University of Otago Economics Discussion Papers

No. 1002

May 2010

Capital Accumulation, Non-traded Goods and International Macroeconomic Dynamics

with Heterogeneous Firms*

Daniel Farhat

Address for correspondence:

Daniel Farhat Department of Economics University of Otago P O Box 56 Dunedin 9054 NEW ZEALAND E-mail: dan.farhat@otago.ac.nz Telephone: +(64) 3 479-8645 Fax: +(64) 3 479-8174 __________ *Special thanks to Jang-Ting Guo, Roger Farmer, R. Robert Russell, Richard Suen, Victoria Umanskaya for their invaluable comments. All errors are my own. For additional comments or suggestions, I can be reached at dan.farhat@otago.ac.nz

Abstract

This paper examines international business cycle transmission within a two-country dynamicstochastic general equilibrium model featuring an endogenously determined trade pattern. In con-trast to existing literature, this model distinguishes between non-traded final goods and traded inputs.The model incorporates capital into the production of final goods and shows that shocks to final goodsproduction are important in replicating the empirical regularities of imports, exports, the real ex-change rate and their relationship to GDP. Endogenously determined labour supply and high assetmarket frictions are incorporated into the model to improve the model’s ability to replicate labourmarket statistics and international co-movement.Keywords: International Real Business Cycles, Non-traded Final Goods, Imperfect Competition,International Trade.JEL: F12, F21, F41.

1 Introduction

Many would agree that trade and financial linkagesare important to the transmission of business cy-cles across countries. However, past internationalreal business cycle (IRBC) models featuring tradein goods and assets have had difficulty generatingrealistic international co-movement1. Poor inter-national co-movement tends to result from the freeflow of assets across countries in particular. Whenone country experiences a positive productivityshock, asset trade leads to an inflow of produc-tive capital from the rest of the world, strengthen-ing the economic expansion at home but inducingrecessions abroad. These models2, which extendmicro-founded business cycle frameworks to includeinternational trade while taking the trade patternas exogenously given, require asset market rigiditiesto produce appropriate business cycle transmission.

An emerging class3 of IRBC models do just theopposite: adapt new4 models of international tradeto measure macroeconomic dynamics. In theseframeworks, the trade pattern is endogenously de-termined and fluctuates with the business cycle.One such model, proposed by Ghironi and Melitz(2005) [GM], is capable of replicating a varietyof empirical regularities, including deviations inpurchasing power parity, persistent trade deficits,the Harrod-Balassa-Samuelson5 effect and an envi-ronment in which only the most productive firmsbecome exporters6. Calibrated results suggest thatchanges in the trade pattern over time enhancebusiness cycle persistence and induce internationalco-movement. While the model produces an arrayof desirable results, it oversimplifies the structureof production by omitting capital.

When capital is omitted from the framework, “in-vestment” is defined only as new firm construction.Investment in the data, however, includes plant andequipment purchases for both new firm creation andexisting firm expansion. Without capital, compar-ing “investment” in the model and investment inthe data is erroneous. In this paper, I will includecapital accumulation into framework proposed byGhironi and Melitz (2005) in such as way as toleave the underlying mechanism of trade undis-turbed. By including capital in the production ofnon-traded final goods, I can improve the definitionof “investment” as well as explore an alternative

source of business cycle fluctuations: productivityshocks to final goods production7.

I show that the model with capital can generatemore realistic relationships between imports, ex-ports, and GDP when business cycles are driven bythe same process as in the original GM framework:shocks to the production of intermediate goods.The inclusion of capital improves the model’s fitfor these international variables, but does so atthe expense of international co-movement8. Whenthere is a positive productivity shock, changes inthe capital stock prolong economic expansions andstrengthen the flow of investment from abroad.Further, an increase in capital prices leads to anincrease in the domestic price level and a fall in theexchange rate. The inflow of investment funds fromabroad coupled with declines in the exchange rateworsen international correlations. This effect per-sists even when asset market frictions are present.Productivity shocks to intermediate goods produc-tion alone are not enough to replicate business cycleco-movement.

When shocks to the production of final goods arepresent, the model can produce positive correlationof output, consumption and investment. Further,the model can more closely match the data by gen-erating increased volatility for imports, exports andthe real exchange rate since shocks to final goodsproduction directly affect the demand for interme-diate inputs. Fluctuations in the exchange ratealong with changes in the trade pattern in responseto shocks to final goods production generate theseimproved results. Although helpful, productivityshocks to final goods production alone are not suf-ficient to generate high output volatility. A modelwith shocks to both intermediate and final goodsproduction is proposed. The improved results sug-gest both sources of business cycle fluctuations arenecessary to produce international co-movementadequately.

Because the original GM framework assumes in-elastic labour supply, I further extend the modelwith capital by adding a developed labour market.As in Farhat (2009), output volatility more closelymatches the data for relatively high elasticities oflabour supply. In the presence of shocks to finalgoods production, however, employment becomesnegatively correlated across countries. With lower

1As measured by international correlations of output, consumption, investment and employment.2Mendoza (1991) and Backus, Kehoe, and Kydland (1992, 1994) are seminal in this strand of literature.3Examples of models in this class include Head (2002); Cook (2002) and Ghironi and Melitz (2005).4Such as Helpman and Krugman (1987).5The Harrod-Balassa-Samuelson effect states that prices are generally higher in wealthier (or more productive) countries.6Which is consistent with firm-level data.7Shocks to final goods production behave as ‘demand shocks’ for intermediate goods in the model.8As in the older strand of IRBC literature, trade in assets leads to capital inflow, inducing negative business cycle

correlations between countries.

1

labour supply elasticities9, output volatility is sim-ilar to the benchmark framework without capitaland positive employment correlation is achieved.

In summary, including capital in the productionof final goods induces improved relationships be-tween imports, exports and GDP when businesscycles are driven by shocks to intermediate goodsproduction, but results in negative business cy-cle correlation. Shocks to final goods production,which act as demand shocks for intermediate in-puts, is capable of generating positive internationalcorrelation of consumption, output and investmentalong with increased volatility of the internationalmarket variables. Adding fluctuations to employ-ment increase output volatility, but may result innegative correlation of employment if labour supplyelasticity assumed to be high.

The remainder of the chapter is as follows. I out-line recent literature on international business cycletransmission and the set of stylized facts the modelintends to reproduce. The model is then describedand a series of numerical experiments are presented.I end with a short conclusion.

2 Literature Review and Styl-ized Facts

2.1 Literature Review

Starting in the mid-1980’s, researchers such asDellas (1986); Stockman and Svensson (1987) andCantor and Mark (1988) began extending the newlydeveloped real business cycle models of Kydlandand Prescott (1982) and Long and Plosser (1983) tomultiple countries (coined International Real Busi-ness Cycle models or IRBC models). The mainempirical findings of these early studies suggestedthat the main sources of business cycle transmissionwere likely correlated productivity shocks10, tradelinkages (in inputs11) and international sales of as-sets12. Backus et al. (1992) and Mendoza (1991) ap-plied calibration techniques to two-country DSGEmodels featuring complete financial markets (whichfacilitate international risk sharing), correlated pro-ductivity shocks, frictionless trade and capital ad-justment costs to find how accurately a simulatedmodel can reflect the real economy. Unfortunatelytheir answer was not much.

Many inconsistencies between the simulated results

and the real economy were derived by Backus, Ke-hoe and Kydland and persist in the face of severalexperiments. Unlike the real economy, simulationresults often suggest consumption among countrieshas high correlation while output has negative orlow correlation (dubbed the “consumption-output”or C-O anomaly). This result occurs from usingfinancial markets to pool risk13. To a somewhatlesser extent, international correlations betweeninvestment, savings and employment were also dif-ficult the match. Resolving the C-O anomaly, how-ever, became the most sought after target for thenext wave of research.

Continual advancements in the research programwere made as economists tried to resolve the mys-tery. Some researchers attempt to augment theway financial markets are modeled to achieve betterresults. Baxter and Crucini (1995) and Kollmann(1996) include incomplete asset markets to try to re-tard the effect of risk pooling. Heathcote and Perri(2002) and Kose and Yi (2001, 2002, 2006) close fi-nancial markets completely and compare the resultsto those that occur in complete markets. Kehoe andPerri (2002) look not at exogenously constrainedfinancial markets, but endogenously constrainedfinancial markets by including imperfectly enforce-able lending agreements. Olivero (2004) modelsnon-competitive lenders which reduces the effectsof risk-pooling via counter-cyclical mark-ups forloans. Some notion of asset market friction seemsnecessary in this class of models to prevent capitalexodus and improve output correlation across coun-tries. Other researchers have found, however, thatchanging non-financial components can improve in-ternational co-movement as well. Stockman andTesar (1995) introduce a non-traded goods sectorto allow for high correlation of traded goods, butnot necessarily for non-traded goods.

The source of business cycle fluctuations is alsoimportant in matching international data with anIRBC model. Stockman and Tesar (1995) and Wen(2002) illustrate the importance of demand-sideshocks to international co-movement by incorporat-ing persistent demand shocks into an IRBC frame-work. The intuition is that business cycles driven bytechnology shocks, assuming perfect risk pooling,induce capital exodus out of less productive coun-tries (lowering output correlation across countries)while demand shocks force the international market(trade in goods) to work as a medium for positive

9Fiorito and Zanella (2008) note that labour supply elasticities are empirically lower at the individual level than at theaggregate level.

10Dellas (1986).11Dellas (1986); Canova and Dellas (1993).12Cantor and Mark (1988); Stockman and Svensson (1987).13Countries that receive positive productivity shocks also receive a flood of investment from the rest of the world. This

leads to ‘recessions’ in other countries and capital is transmitted to the most productive nations via asset markets.

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business cyclical synchronization14 (improving out-put correlation across countries). However, whendemand shocks are not persistent, investment iscrowded out to increase consumption temporarilyin the affected country, resulting in low or negativeinvestment correlation across countries.

Although IRBC models featuring demand shocksand asset market frictions can replicate some of theempirical regularities of international co-movement,the vast majority of these models fail to explicitlyfocus on the functioning of the international mar-ket in transmitting business cycles. Focus on thedynamic properties imports, exports, the terms oftrade (TOT) and the real exchange rate is explic-itly done within a somewhat separate but relatedstrand of research. Backus et al. (1994); Mendoza(1995); Zimmermann (1999) and Cunat and Maf-fezzoli (2004) focus on constructing IRBC modelsthat can replicate the data associated with thesevariables, but generally reach the same conclusionsas the models that focus predominately on interna-tional co-movement: many refinements are requiredto the model to replicate the full gambit of stylizedfacts.

The source of the poor fit of IRBC models in gen-eral may be related to the ‘microfoundations’ thatthe models are build upon. All of the models de-scribed so far apply micro-founded macro modelsto the open economy. The evolution and determi-nation of the trade pattern has been exogenous. Asmall but separate sect of IRBC literature has de-veloped within the rubric of ‘international trade’, inwhich micro-founded trade models are adapted formacroeconomic analysis. These models, which arestructured to reflect modern perceptions of inter-national trade15, endogenize firm entry then allowfirms to choose whether or not to export. The tradepattern is thus endogenous and sources of businesscycle transmission are twofold: through new firmconstruction and through changes in export de-cisions. Head (2002); Cook (2002); Alessandriaand Choi (2004) and Ghironi and Melitz (2005)incorporate endogenous firm entry and export deci-sions in monopolistically competitive input marketsand succeeds in reducing the C-O anomaly as wellmatching many features of international data16.

Of these new models, the Ghironi and Melitz (2005)has shown to be particularly effective in studyinginternational co-movement. The original Ghironiand Melitz model, however, is still quite under-developed when compared to the other branch ofIRBC research and resent developments in closed-economy real business cycles. Labor is supplied in-elastically to simplify the models already complexstructure. The drawback of this feature is a lackof employment dynamics in the model. Since laboris used in the production of traded goods, to con-struct new firms and to pay export costs, changesin employment can dramatically affect the tradepattern. Further, the model distorts the definitionof “investment” typically used in real business cy-cle models by omitting capital accumulation. Sinceinvestment exodus has been shown to be a centralcause of low output correlation across countries,and since the presence of capital can prolong theeffects of business cycle movements, incorporatingcapital into the model is certainly not a trivialexercise. This paper attempts to relax these struc-tural assumptions and include a notion of demandshocks and asset market frictions, which have al-ready been shown to be useful in replicating thestylized facts seen in the data. In the process, par-ticular attention will be paid to the data associatedwith the medium of business cycle transmission:the international market.

2.2 Stylized Facts

When evaluating the success of an internationalreal business cycle model, many researchers referto the econometric analysis performed by Backuset al. (1992). This analysis looks at quarterly datafrom 1954q1 to 1983q7 for domestic (U.S.) businesscycle statistics and transformed17 quarterly datafrom 1970q1 to 1986q4 for international18 busi-ness cycle correlations. Stock and Watson (2005)note that both domestic business cycle volatilityand the correlation of business cycle movementsbetween the U.S. and several European countrieshave changed since the 1980s. U.S. business cy-cles have become more moderate and less synchro-nized with Europe19. Because nearly 20 years ofadditional data are now available and the interrela-

14Building on this idea, researchers have also explored the value of indeterminacy and animal spirits in IRBC models(Guo and Sturzenegger (1998); Xaio (2002, 2003, 2004); Fukuda (2004)). These models are currently at the forefront of theIRBC literature.

15Specifically, ‘new trade theory’ as developed by Helpman and Krugman (1987). These trade models feature non-competitive markets incorporating intra-industry trade.

16such as persistent deviations in purchasing power parity, persistent trade deficits, the Harrod-Balassa-Samuelson effectand the Harberger-Larsen-Metzler effect

17Deflated quarterly data is transformed into comparable terms by comparing the 1985 annual average to the Penn WorldTable 1985 value.

18Business cycle correlations are between the U.S. and a “European Aggregate” which consists of Austria, Finland, France,Germany, Italy, Switzerland and the United Kingdom.

19According to Stock and Watson (2005), this is due to the reduction in “common shocks”, i.e. oil price shocks andcommodity price shocks

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tionships between business cycle fluctuations havepotentially changed, it is worthwhile to generateupdated stylized facts before testing the fit of mymodel.

I organize the data on international co-movementinto three categories: domestic business cycle statis-tics, international correlations and internationalmarket statistics. Table 1, which is available in theappendix, reports domestic business cycle statisticsfor the United States, 1957q1-2007q1. As is com-monly found, consumption, investment and em-ployment are pro-cyclical. Investment is 3.41 timesmore volatile than GDP, while both consumptionand employment are less volatile than GDP (0.75and 0.61 times as volatile respectively).

Table 2, also available in the appendix, reportsinternational business cycle co-movement betweenthe U.S. and Europe, 1970-200420. The degree ofbusiness cycle co-movement, however, proves to betime-dependant. In the original analysis, Backus,Kehoe and Kydland find that output is more cor-related across countries than consumption (withcorrelation coefficients 0.70 and 0.46 respectively).Many researchers have difficulty constructing amodel that can replicate this relationship (the C-Oanomaly). Table 2 confirms that this relationshipis much weaker, as predicted by Stock and Wat-son (2005), when recent data is considered. Figure1 shows how the correlation coefficients describedin table 2 change over time by looking at 5-yearand 10-year rolling window estimates. When look-ing at the 10-year rolling window for the Backus,Kehoe and Kydland sample (1970 - 1985), outputcorrelation is significantly higher than consumptioncorrelation. However, consumption is more corre-lated than output in later years (1985 - 2004). Ingeneral, we can say that output correlation andconsumption correlation are “close” and generallypositive.

A rich set of statistics pertaining to the interna-tional market are also available in the appendix.These measures include imports (IM), exports(EX), the trade balance as a fraction of output(Net Exports/Output = NX/Y) the terms of trade(TOT, measured as the ratio of import prices toexport prices) and the real exchange rate index(Q) and are reported for the U.S. in table 3. Thedata suggests high volatility of quantities (importsand exports which are 3.31 and 3.74 times morevolatile than GDP) as well as prices (the terms oftrade and the real exchange rate, which are 1.69and 3.37 times more volatile than GDP). Exports

and imports are both pro-cyclical, net exports arecounter-cyclical and prices are generally a-cyclical.Exports tend to be more volatile than imports, andless correlated with output. This set of data willbe used to test how well the model captures thebehavior of trade linkages, an important source ofbusiness cycle transmission across countries, overthe business cycle.

3 The Model

The theoretical model is heavily influenced byGhironi and Melitz (2005). I will briefly outline theparts of the original GM framework that I augmentwhen adding capital accumulation21. The modelconsiders two countries, home and foreign (whichis denoted by *). Both countries are large and as-sumed to be structurally identical. That in mind, Iconstruct the framework for one country (the homecountry) knowing that a symmetric framework ex-ists for the foreign counterpart.

3.1 The Consumer’s Problem

Denote the universe of intermediate input varietiesas Ω. At any time, t, a subset Ωt ⊆ Ω are actuallyproduced by firms. Ωt contains both domesticallyproduced and imported goods. Further, it need notbe the case that Ωt = Ωs for t 6= s nor Ωt = Ω∗t . Acomposite intermediate good, Mt, is produced usingDixit-Stiglitz technology:

Mt =(∫

ω∈Ωt

mt(ω)(a−1)/adω

)a/(a−1)

where mt(ω) denotes the quantity of variety ω usedin the production of the composite, and a denotesthe elasticity of substitution across varieties. Thecomposite intermediate good is then combined withcapital, Kt, to produce consumption using a simpleCobb-Douglas production process:

Ct = AtMbtK

1−bt

At denotes total factor productivity at time, t. AtAt = b = 1, the model reverts to the structuredescribed by Ghironi and Melitz (2005).

It is straightforward to construct demand equationsand price indices for mt,Mt,Kt and Ct. DenotePXt as the price index for good X. I construct thefollowing conditional demand equations by solvinga straight-forward cost-minimization problem22:

mt(ω) = Mt(Pt(ω)/PMt)−a

Mt = Ct

At(PMt

PKt

1−bb )b−1

Kt = Ct

At(PMt

PKt

1−bb )b

20Unlike Backus et al. (1992), the annual data from the Penn World Tables are considered directly.21A detailed work-up of the model is available upon request.22The production function for Ct is essentially a multi-step CES production function. First, I find the lowest-cost way to

produce one unit of the intermediate index, M′t . Then, I solve a cost-minimization problem for Ct over M

′t and Kt.

4

I also construct the associated price indices:

PMt =(∫

ω∈ΩDt

Pt(ω)1−adω

)1/(1−a)

PCt =P bMtP

1−bKt

At(1− b)1−bbb

Next, I turn to the household’s problem faced by therepresentative agent. It is assumed that there existsa representative household that chooses to work,consume and save. The household is endowed with1 unit of time that it can divide between labour, lt,and leisure. labour earns the real wage, wt. Thehousehold can choose to save by purchasing bonds,Bt+1, which each cost one unit of consumption, butyield (1 + rt+1) units in the next period. Both do-mestic and foreign bonds (B∗) are available to thehousehold and are subject to a quadratic transac-tion cost which is rebated, lump-sum (Γt), to thehousehold. The household can also purchase sharesof a mutual fund, xt+1, which entitle the owner toa fraction of the profits of the producing firms. Ar-bitrage prices the shares at the firms expected dis-counted value (Nhtvt for new shares, Ndtvt for oldshares). Households are obligated to hold bondsand stocks for only one period before they are resoldin asset markets. Finally, the households have theoption of saving real wealth in the form of savingsaccounts, St+1. Savings are used to construct cap-ital in the next period and earn a real return ofRt

23. The consumer’s real period budget constraintis then given by:

Ct +Bt+1 +QtB∗t+1 + n12 (B2

t+1)+n22 (QtB2

∗t+1) + (Nhtvt)xt+1 + St+1

=wtlt + (1 + rt)Bt +Qt(1 + r∗t )B∗t + Γt+xt(Ndtvt +Ndtdt) +RtSt

(1)

where n2 ≥ n1 > 0 are scale parameters on for-eign and domestic bond adjustment costs, Γt =n12 (B2

t+1) + n22 (QtB2

∗t+1) in equilibrium and Qt isthe real exchange rate.

The instantaneous utility function for the agentis assumed to have the form:

Ut(Ct, lt) = logCt −Hl1+λt

1 + λ

The agent’s maximization problem is then to choosea sequence:

Cs, ls, Bs+1, B∗s+1,xs+1, Ss+1∞s=t

to maximize the discounted sum of expected life-time utility, Et

∑∞s=t β

s−tUt(Ct, lt) subject to thebudget constraint in equation (1).The first-order

conditions for the consumer’s problem generatethree Euler equations and an equation that guideslabour supply when H 6= 0. The Euler equation fordomestic bonds is:

C−1t (1 + n1Bt+1) = (1 + rt+1)βEtC−1

t+1

Similarly, the Euler equation for foreign bonds is:

QtC−1t (1 + n2B∗t+1) = (1 + r∗t+1)βEtQt+1C

−1t+1

The Euler equation for stocks is:

vt = β(1− δ)Et(Ct+1

Ct)−1(dt+1 + vt+1)

Notice that repetitive forward substitution of theEuler equation for stocks generates the expectedvalue of the firm’s lifetime profit stream. Finally,the Euler equation for savings is:

C−1t = βEtRt+1C

−1t+1

In the original Ghironi and Melitz (2005) frame-work, labour was supplied inelastically (H = 0,therefore lt = 1).When endogenous labour supply isconsidered, H = 1 and the condition which guideslabour supply is:

lt = C−1/λt w

1/λt

The parameter λ represents the elasticity of laboursupply.

3.2 The Firm’s Problem

I now consider the problem faced by a typical in-termediate good producer in the home country. Itis assumed that there exists an unbounded mass offirms in the economy that may begin production atany time. These firms are monopolistically compet-itive producers of unique varieties who produce fordomestic and foreign markets separately 24. Giventhe demand for their products derived in the previ-ous section, the firm’s:

Step 1: Decide whether or not to enter.Step 2: Upon entry, choose how much output toproduce for each market and what prices to set.Step 3: Die or exit at the end of the period.

In practice, firms will first derive the solutions tostep 2 and step 3 and form estimates on their futureprofit streams before choosing whether or not it’sworth it to enter the industry. Prior to entry, thefirm calculates average (expected) profit. Each firmj in the home country produces output accordingto the production function:

yjt = ZtzjLjt23All savings instruments are denoted with the time subscript in which they yield a return.24Markets are segmented – firm’s practice third-degree price discrimination.

5

where yjt denotes the quantity of output firm jproduces, Zt denotes an economy-wide technol-ogy variable, zj is a productivity draw specificto firm j and Ljtis the quantity of labour firm jhires in the production of output. The firm willchoose how much to produce for the domestic mar-ket (mdjt) and how much to produce for export(mxjt = m∗mjt). Exports are subject to an icebergcost, τ ≥ 1. Thus, total output for the firm is givenby yjt = mdjt + τmxjt.

The firm’s expenses include a wage bill, a start-up cost paid in terms of labour, and an exporterfee (also paid in terms of labour). The wage bill issimply wtLjt. Start-up costs are a fixed cost paidonce during the period the firm begins production.The ”production” of entry is given byFEt = ZtLEjtwhich suggests a total fixed cost of wtFEt/Zt. Ex-porter fees are paid each period the firm chooses toexport (mxjt > 0). The production of the exporterfee is given as FXt = ZtLXjt. Thus, the per-periodexporter cost is wtFXt/Zt. Dropping the start-upcost, the firm’s real per-period profit maximizationproblem is to choose domestic and exported output,cxjt, cdjt to maximize profits, ΠjtPCt = Pdjt

PCtmdjt +

εtPxjt

PCtmxjt − wt

Ztzj(mdjt + τmxjt) − Ixt wtFXt

Zt,where

Ixt = 1 if cxjt > 0, else Ixt = 0. εt denotes the nom-inal exchange rate. It is important to note that Pxjtis measured in terms of foreign currency. For anyvariable, X, we denote real prices as ρX = PX/PCand substitute the demand functions from the pre-vious section into the firm’s problem (as is standardwith monopolistic competition) to generate the firstorder conditions for the firm25:

mdjt = bCtρa−1Mt ( a

a−1wt

Ztzj)−a

mxjt = bC∗t ρ∗a−1Mt ( a

a−1wt

Ztzj

τQt

)−a

where Qt = εtP∗Ct/PCt is the real exchange rate.

For simplicity, it is assumed that εt = 1. We thenuse the demand equations to find the optimal realprices charged by the firm:

ρdjt = aa−1

wt

Ztzj

ρxjt = aa−1

wt

Ztzj

τQt

The prices suggest a constant markup over marginalcost (wt/Ztzj).

Profits for the firm can be divided into “domes-tic production profits” and “export profits”. Theyare given as:

ddjt = baCtρ

a−1Mt ρ

1−adjt

dxjt =baC∗t ρ∗a−1Mt Qtρ

1−axjt −

wtFXt

Ztif cxjt > 0

0 if cxjt = 0

For a continuum of potential firms, there existssome productivity draw, zxt, such that the firm

which receives that draw upon entering at time tearns zero profits whether it exports or not (theexporter cutoff). For this firm, b

aC∗t ρ∗a−1Mt Qtρ

1−axjt =

wtFXt

Zt.

Knowing the profits for any given productivitydraw, potential firms decide whether or not toenter. Their productivity draw is not assured priorto entry, so firms form an expectation of their prof-its (they calculate an “average” profit from entry).To do this, we employ the “special averages” de-veloped by Melitz (2003). First, we suppose thatthe productivity draws follow a Pareto distribu-tion with CDF G(z) = 1 − (zmin/zj)k and PDFg(z) = kzkminz

−k−1j where k denotes the shape pa-

rameter of the distribution. Next, define:

z1−ad =

∫∞zmin

z1−aj g(z)dz

z1−axt =

∫∞zxt

z1−aj

g(z)1−G(zxt)dz

Finally, substitute these definitions in construct-ing average prices: ρ1−a

dt =∫∞zmin

ρ1−adjt g(z)dz =

( aa−1

wt

Ztzd)1−a and ρ1−a

xt =∫∞zxt

ρ1−axjt

g(z)1−G(zxt)dz =

( aa−1

wt

Ztzxt

τQt

)1−a. We define Ndt as the number offirms that produce for the domestic market and Nxtas the number of firms that produce for export. Themodel is recast in terms of average prices:

ρdt = ( aa−1

wt

Ztzd)

ρxt = ( aa−1

wt

Ztzxt

τQt

)ρMt = (Ndtρ1−b

dt +N∗xtρ∗1−bxt )1/(1−b)

ddt = baCtρ

a−1Mt ρ

1−adt

dxt = baC∗t ρ∗a−1Mt Qtρ

1−axt − wtFXt

Zt

Completing the integral for the special productiv-ity averages suggests that zD = ( k

k+1−a )1/(a−1)zmin

and zxt = ( kk+1−a )1/(a−1)zxt where k > a − 1 for

boundedness. Knowing that all existing firms pro-duce for the domestic market, and a fraction ofthose become exporters, the expected per-periodprofit for a potential firm, on average, is:

dt = ddt + (1−G(zxt))dxt

There are two important features guiding firm cre-ation and destruction. The first is a lag in produc-tion. A firm that enters in period t starts producingat period t+ 1. The entering firm, however, is stillcounted as a firm in period t. The total numberof firms that exist at period t, Nht, is given by thenumber of producing firms that already exist, Ndt,plus the number of new firms, Net. Second, firmsare subject to an exogenous exit shock. The num-ber of firms that “survive” to produce in period t+1is given by Ndt+1 = (1−δ)Nht = (1−δ)(Ndt+Net).

25It is assumed that firms do not observe the impact of price setting on the aggregate price level when solving the firm’smaximization problem. Also, note that the definition of PCt implies Mt = bCtρ

−1Mt.

6

Therefore, the expected value of the firm’s lifetimeprofit stream, vt, is given by:

vt = Et

∞∑s=t+1

[1− δ]s−t[βs−t(

CsCt

)−1

]ds

where[βs−t(Cs

Ct)−1]

is the stochastic discount fac-tor to be discussed shortly and (1− δ) is the firm’ssurvival probability. Firms continue to enter aslong as the discounted value of their profit streamexceeds the cost of entry. Therefore, the entry cut-off is determined by vt = wtFEt/Zt.

Finally, there are two important characteristics ofthe firm’s problem with capital accumulation thatare of note. First, the indexed price of interme-diate goods (which equals the aggregate price ofconsumption in the model without capital, but notin the model with capital) affects the equilibriumquantity produced by any individual firm. Recall:

mdjt = bCtρa−1Mt ( a

a−1wt

Ztzj)−a

mxjt = bC∗t ρ∗a−1Mt ( a

a−1wt

Ztzj

τQt

)−a

Second, the domestic and export profits are stronglyrelated to the level of consumption. Recall:

ddjt = baCtρ

a−1Mt ρ

1−adjt

dxjt =baC∗t ρ∗a−1Mt Qtρ

1−axjt −

wtFXt

Ztif cxjt > 0

0 if cxjt = 0

Increasing the volatility of consumption or thevolatility of the aggregate price of labour serviceswill result in increased volatility of firm profitsand thus more volatility in the decision to export.Changes in export decisions affect the size of theexport sectors as well as the trade pattern, whichinfluences the transmission of business cycle fluctu-ations across countries.

3.3 Market Clearing and VariableDefinitions

The market clearing conditions and definitionsneeded to solve the model are available in theappendix. The central differences from the orig-inal Ghironi and Melitz (2005) model when capitalaccumulation is added to the framework are theexistence of a market for capital, the redefinitionof investment and the redefinition of the aggregateprice index.

In the capital market, savings today become thefoundation for tomorrow’s capital stock (St+1 =Kt+1). In equilibrium, the gross return to savings(RtSt = RtKt) should be equal to total paymentsto capital plus the return of non-depreciated capital(ρKtKt+(1−δK)Kt). This suggests the equilibriumprice of capital is given by:

ρKt = Rt − 1 + δK

Capital demand for the production of consumptionin period t is given by:

Kt =CtAt

(ρMt

ρKt

1− bb

)b

while supply of capital in the next period is deter-mined by the Euler equation:

C−1t = βEtRt+1C

−1t+1

Capital market clearing suggests St+1 − RtSt =Kt+1−(ρKtKt+(1−δK)Kt) = Kt+1−(1−δK)Kt−ρKtKt. I impose mutual fund market clearing andlump-sum transfers of bond adjustment costs togenerate the balance of payments condition:

2(1 + rt+1)Bt + 2(1 + r∗t+1)QtB∗t=(Ct −QtC∗t ) + (Netvt −QtN∗etv∗t )+2(Bt+1 +QtB∗t+1)− (wtlt −Qtw∗t l∗t )−(Ndtddt −QtN∗dtd∗dt)− (Nxtdxt −QtN∗xtd∗xt)+(Kt+1 −QtK∗t+1)− (1− δK)(Kt −QtK∗t )−(ρKtKt −Qtρ∗KtK∗t )

Investment is defined as expenditures on new firmentry plus the purchases of new capital:

It = Netvt +Kt+1 − (1− δK)Kt

Additional definitions are also required. FollowingGhironi and Melitz (2005), income (GDP) is definedas:

Yt = wtlt + ρKtKt +Ndtddt

Further, total imports is constructed using the aver-age revenue foreign exporters earn from their salesabroad:

IMt = bN∗xtCtρa−1Mt ρ

∗1−axt

Total exports is constructed using the averagerevenue domestic exporters earn from their salesabroad:

EXt = bQtNxtC∗t ρ∗Mtρ

1−axt

Terms of trade is defines as the ratio of importprices to export prices:

TOTt =ρ∗xtQtρxt

In the model, the price of consumption (PCt) ismeasured as a welfare-based price index followingFeenstra (2003). It is thus important to trans-form this welfare-based index into one which closermatches the price index calculated in the data. Todo so, we redefine the price index for aggregate con-sumption:

1 =[Ndtρ1−a

dt +N∗xtρ∗1−axt ]b/(1−a)ρ1−b

Kt

At(1− b)1−bbb

7

If we assume that all prices are, on average, ρ =Pt/PCt, we construct:

PCt

Pt=

[Ndt +N∗xt]b/(1−a)

At(1− b)1−bbb

Any variable measured in terms of real consump-tion, Xt, is adjusted to this index: Xt = PCtXt

Pt.

Further, since the real exchange rate is con-structed using the welfare-based price indices (Qt =P ∗Ct/PCt), we construct an adjusted real exchangerate:

Q = P ∗t /Pt = Qt[A∗tAt

][Ndt +N∗xtN∗dt +Nxt

]b/(1−a)

4 Numerical Experiments

Many of the model’s parameter values are cali-brated to match Ghironi and Melitz (2005) and arelisted in table 4. The value of the depreciation rateof capital is a new parameter to the model and isset to 2.5% quarterly and 10% annually (a stan-dard in the literature). The other model parame-ters, (b, n1, n2, λ, and the parameters of the shockprocesses, will be thought of as “free parameters”and will be calibrated under different experimentsdiscussed shortly. The model is solved numericallyusing the brute force algorithm developed by Uhlig(1999)26.

Simulations are performed in order to generatea set of statistics to compare to the data. In eachexperiment, a 200-period model is simulated 200times by drawing a random vector of innovations.During each simulation, I make the appropriateprice adjustments as described above, then applythe HP filter27. Summary statistics (volatility, cor-relation with output, etc.) are calculated for eachsimulation and I report the average statistics acrosssimulations in the results tables.

4.1 The Benchmark Model

To match the original Ghironi and Melitz (2005)framework as closely as possible, I set n1 = 0.0025,

b = 1 and omit endogenous labour supply fromthe model. Since the inclusion of capital accumu-lation may lead to a highly persistent and extremecapital exodus if bond markets are fluid, I limitthe amount of international asset flow by imposinghigh adjustment costs for foreign bonds (n2 = 1).I calibrate the technology shocks to intermediategoods production as a near-unit-root processes withcorrelated innovations28:

[ZtZ∗t

]=

[0.99 0

0 0.99

] [Zt−1

Z∗t−1

]+[ξZtξ∗Zt

]σξZt

= σξ∗Zt= σξ = 0.00852

ρξZtξ∗Zt= ρξZξ∗Z

= 0.258

with covariance of (ξZt, ξ∗Zt) 0.0019. Technologyshocks are calibrated as such in all the followingexperiments. I set At = 1 for all t. Results fromsimulations are reported in column (2) of table5. The model fails to predict high correlation ofinvestment across countries and high volatility ofimports, exports, the trade balance, the terms oftrade and the real exchange rate. Further, themodel over-predicts the correlation of exports, theterms of trade and the real exchange rate with GDP.

The failure to capture the appropriate investmentdynamics is due to how investment is defined. Inthe model without capital, investment is only newfirm construction. When there is a positive pro-ductivity shock to intermediate goods productionin the home country, each individual firm in thatcountry earns increased profits, leading to a surgeof new firm construction and an increase in home’sinterest rate. The increase in home’s interest rateinduces foreign agents to adjust their investmentportfolio in favor of home bonds (via the Euler re-lationship). Although high bond adjustment costshave limited the extent to which foreign investorssubstitute, there is still an exodus of investmentdollars out of the foreign country. This causes firmattrition in the foreign economy and a net reductionin foreign real investment29.

26The algorithm used in experimentation is described as follows. After calibrated parameters are chosen, I restructurethe linearized equation system: (Et[FXt+1 +GXt+HXt−1 +MZt] = 0;Zt = NZt−1 +et, where X is a vector of variables,Z is a vector of stochastic processes guided by a VAR and e is a vector of innovations). Using Uhlig’s brute force method ofundetermined coefficients, I find a policy rule: (Xt = PXt−1 +QZt). In each experiment, a 200-period model is simulated200 times by drawing a random vector of innovations.

27Since the model exhibits a high degree of persistence, we take out the low-frequency trend using the HP filter.28Backus et al. (1992) calibrate technology shocks as a VAR process with spill-overs:[

ZtZ∗t

]=

[0.906 0.0880.088 0.906

][Zt−1

Z∗t−1

]+

[ξZtξ∗Zt

]σξZt

= σξ∗Zt

= σξ = 0.00852

ρξZtξ∗Zt

= ρξZξ∗Z

= 0.258

Baxter and Crucini (1995) note that this specification is not statistically different from a near-unit-root process withoutspill-overs.

29The reduction in foreign investment is reduced after applying the real exchange rate to transform foreign denominationsinto domestic terms. The positive productivity shock to home intermediate good production leads to a fall in the price of

8

The failure of the model to appropriately repro-duce features of the international market suggeststhat international trade in goods is failing to actas the appropriate mechanism for the transmis-sion of business cycles across countries. Inspectionof the impulse response functions for the endoge-nous variables of the model suggest the relationshipbetween the international market statistics and rel-atively docile endogenous variables are responsiblefor the low volatilities of imports, exports, thetrade balance and international prices. Aggregateproductivity shocks in the benchmark model leadto increases in foreign consumption, the size of thedomestic export sector, export prices and the realexchange rate. Each of these are less volatile thandomestic income, which results in low volatility ofexports relative to GDP. A similar story holds forimports. Since prices in the model are marked-up over marginal costs, and marginal costs in themodel are less volatile than GDP, both the termsof trade and the real exchange rate are predicted tohave low volatility.

4.2 Capital Accumulation

I now turn to the model with capital in the produc-tion of final consumption to improve the definitionof investment. There are two important charac-teristics of the model worth noting when capitalis used in the production of domestic consumables.First, since capital is not used to produce the tradedintermediate inputs, the two countries are in effecttrading labour services as in the original frame-work. Including capital in this fashion allows meto bypass a potentially complicated discussion ofchanges in relative factor intensities. Second, sincethe production of final consumption entails its ownproduction process, I can introduce an additionalsource of business cycle fluctuations, At. While Atacts as a technology shock to the domestic pro-duction of final consumption, it acts as an inputdemand shock to intermediate goods producers30.

To proceed with experimentation, I first calibratethe parameter b so that the steady state capital-to-output ratio matches the average value for the US(approximately 3.1031). I then analyze the modelwith aggregate shocks to intermediate goods pro-

duction only, followed by the model with shocksto the production of composite consumption. Bothmodels fail to reproduce sufficiently volatile output,so as an aside I analyze a version of the model withendogenous labour supply.

4.2.1 Capital Accumulation and Shocksto the Production of IntermediateGoods

I start with a model whose only deviation fromthe original framework is the significance of capitalin the production of final consumption. Labouris supplied inelastically, At = 1 for all t, and theunderlying process driving business cycles is shocksto the production of intermediate inputs. As de-scribed above, the costs associated with purchasingforeign bonds is high (n2 = 1) to limit the amountof investment exodus that can occur with openbond markets. The simulation results are reportedin column (3) of table 5.

At first glance, the model with capital results inslightly improved volatilities of imports and exportsas well as improved correlations between imports,exports and GDP. The model produces worse es-timates of output and investment volatility andreduced correlation of consumption and investmentacross countries. Inspection of the impulse responsefunctions link the worsening results of the simula-tion to changes in international prices once capitalis incorporated into the model.

When there is a positive productivity shock tothe production of input goods in the home coun-try, consumption increases with an expansion offirm output. The production of intermediate goodsbecomes more profitable and new firms enter themarket. More workers are demanded to expandoutput and to build new firms, leading to an in-crease in the domestic wage pushing up domesticincome. The equilibrium price of capital (ρKt)rises via the Euler equation for savings, leading tofurther increases in income. The home country ex-periences a state of expansion.

Home consumers demand more intermediate goods,leading to an increase in the demand for imports. Inthe foreign country, the export sector expands and

domestic goods and expansion of the foreign export sector (both of which push down the domestic price level in spite ofrising import prices). Increases in the home export sector are not sufficient to outweigh increases in the prices of domesti-cally produced goods in the foreign country; the foreign price level rises. Changes in aggregate prices brought about by theproductivity shock cause the exchange rate to rise.

30Stockman and Tesar (1995) and Wen (2002) show that demand shocks are important to international business cyclesynchronization. Although the formulation I adopt here does not impose demand shocks on behalf of consumers, demandshocks for intermediate inputs are potentially equally important.

31Annual data for the US, 1990-2007, from the Bureau of Economic Analysis for real total fixed assets plus consumerdurables (Table 9.1) and real GDP (Table 1.1.6) are used to construct this average. MATLAB has difficulty numericallydetermining b in the model, so it is calibrated by hand. For quarterly simulations, b = 0.88 corresponds to a capital-outputratio of 3.09. For annual simulations, b = 0.57 corresponds to a capital-output ratio of 2.84. When labour supply isendogenous, b = 0.60 is used for annual calibrations which corresponds to a capital-output ratio of 2.77.

9

demand for labour increases, pushing up foreignwages and income. Although open asset marketslead to firm attrition and a reduction in labour de-mand for new firm construction, the model predictswages to rise in equilibrium implying the effect ofan expanding export sector on employment out-weighs the effect of firm attrition. Capital pricesin the foreign country are also higher via the Eulerequations for savings and home bonds.

Although wages rise in both countries, the pro-ductivity shock in the home country pushes downthe marginal cost of labour while the marginal costof labour abroad rises. This results in a fall inthe price of domestically produced goods in thehome country and a rise in the price of goods pro-duced abroad. An expansion of the export sectorin both countries occurs to meet increased demandand export prices both at home and abroad rise.Changes in the number of exporting firms cou-pled with changes in domestic and export pricesresult in a fall in the intermediate good price in-dex (ρMt, ρ

∗Mt) for both countries. The fall in ρMt

results primarily from a decline in domestic pricescoupled with an increase in the number of foreignexporters. The fall in ρ∗Mt seems to result onlyfrom the increase in home’s export sector and is ofsmaller magnitude than the change in ρ∗Mt. Theinput price ratio (ρKt/ρMt) rises in both countriesinducing households to use relatively more labourservices to produce final consumption. In equi-librium, households actually reduce their capitalholdings32 in favor of new firm investment. Totalinvestment in the home country is higher due tonew firm construction although investment in cap-ital has fallen. Investment in the foreign country,however, is lower due to a fall in both new firmconstruction and capital purchases.

The impulse response functions for a positive shockto domestic intermediate goods production showincreases in both foreign and domestic consump-tion denoted in national terms. The simulationresults, however, predict negative correlation whenconsumption is measured in comparable terms dueto changes in the real exchange rate. In the modelwithout capital, the real exchange rate increaseswhen there is a positive productivity shock33. Inthe model with capital, changes in capital pricesgreatly influence changes in the real exchange rate.

Percentage changes in the real exchange rate canbe written as:

Qt = b(ρ∗Mt − ρMt) + (1− b)(ρ∗Kt − ρKt)

Because declines in the price index for intermedi-ate goods are severe in the home country, (ρ∗Mt −ρMt > 0), the real exchange rate is pushed up.Increases in capital prices are also severe in thehome country (ρ∗Kt − ρKt < 0), which pushes thereal exchange rate down. Changes in the price forcapital in the home country is sufficiently strong(ρKt > b

(1−b) (ρ∗Mt − ρMt) + ρ∗Kt) to induce in asmall fall in the real exchange rate when there isa positive productivity shock to intermediate goodsproduction. The correlation between output andthe real exchange rate is therefore negative in simu-lation, which corresponds more closely to the data.However when foreign variables have small positivechanges, converting them to domestic terms usingthe reduced real exchange rate results in a nega-tive movement. Consumption in the model, whichincreases in both countries, becomes negatively cor-related after converting the foreign measure to com-parable terms34. Output in the foreign country,which changes only slightly in real terms, declinesafter applying the exchange rate.

4.2.2 Capital Accumulation Shocks to theProduction of Aggregate Consump-tion

By developing a production process for final goodsthat includes both capital and labour services, itis possible to extend the model by incorporatingshocks to final goods production. Shocks of thisnature can also be thought of as demand shocksfor intermediate inputs. To analyze versions of themodel with this feature, I will allow the final goodstechnology variables (At, A∗t ) to follow a VAR pro-cess. Since At is also related to technology, I use thesame near-unit root shock process as was used forthe shocks to the production of intermediate goods:[

AtA∗t

]=

[0.99 0

0 0.99

] [At−1

A∗t−1

]+[ξAtξ∗At

]σξAt

= σξ∗At= σξA

= 0.00852ρξAtξ∗At

= ρξAξ∗A= 0.258

As an intermediate step, I allow only At to followa stochastic process while Zt = 1 for all t. Theresults for this calibration are reported in column

32Although less capital is purchased, investment in the model (Kt+1 − (1 − δK)Kt + Netvt) still increases due to newfirm construction.

33A positive productivity shock leads to a fall in the price for domestically produced goods and an increase in the variety ofimported goods, which offsets increases in import prices. In the foreign country, only an increase in the number of importedgoods offsets increases in the prices of all goods. Although prices in both countries are likely to decrease, prices in the homecountry would decrease more than the foreign country, leading to a rise in the real exchange rate. This conclusion is basedon impulse response functions for PCt/Pt, which decline with a positive productivity shock, suggesting that changes in thenumber of firms are more influential than changes in the prices of intermediate goods.

34The home export sector expands as the real exchange rate falls, leading to an increase in foreign consumption. Thisincrease in consumption, however, is not enough to overpower the fall in the exchange rate after converting foreign variablesinto comparable terms.

10

(4) of table 5. Although shocks to final goodsproduction alone are not sufficient to capture allthe dynamics of the real economy, the behaviorof many economic variables are dramatically im-proved. Imposing shocks of this nature results inincreased volatility of investment, increased inter-national correlation, increased volatility of importsand exports, increased volatility of terms of trade,and increased volatility of the exchange rate index.

Investment volatility increases since input demandshocks have a more direct impact on capital thanthe productivity shocks of input producers. A pos-itive productivity shock to the production of finalgoods will directly increase the marginal productof capital along with the marginal product of in-termediate ‘labour services’. The higher marginalproduct of capital increases the demand for capitalin the current period which leads to an immediateincrease in capital prices, since the current capitalstock is predetermined. Further, since the shockis persistent, consumers expect future returns tosavings to be higher and supply more savings tocapital markets, expanding future capital holdings.The part of investment that is held as capital dra-matically increases, pushing up investment volatil-ity.

Inspection of the impulse response functions sug-gests that a positive productivity shock to finalgoods production results in firm attrition in thehome country (Net declines) because the returnto capital makes savings relatively more profitableas an investment tool than new firm constructionin the current period. Further, since a positiveproductivity shock to final goods production leadsto an immediate expansion of consumption, thereis higher demand for input products and thus in-creased demand for labour services in intermediategoods production. Intermediate producers expandproduction and demand more workers, causing thereal wage to rise. With higher wages, it becomesmore expensive to pay the fixed entry cost associ-ated with starting a firm. Although the discountedvalue of existing firms (vt) rises since each exist-ing firm now earns more profits, this increase ismore than offset by the decline in firm entry. Thepart of investment that is comprised by new firmconstruction (Netvt) decreases, but only slightly.There is an overall increase in investment resultingfrom capital purchases. Higher labour income, cou-pled with an increased price of capital, leads to anincrease in aggregate income. In the home country,aggregate output, investment and consumption allincrease.

Because capital earns a higher return, there is amovement away from issuing domestic bonds, pur-chasing foreign bonds and building new firms to-wards capital accumulation in the home country.A reduction in the supply of home bonds results inlower bond issuance and higher bond returns. Areduction in home’s demand for foreign assets leadsto a fall in foreign bond purchases and a reductionin the foreign interest rate. As the return to foreignbonds falls, foreign agents adjust their investmentportfolios in favor of capital and new firm construc-tion. There is a wave of firm entry in the foreigncountry coupled with rising capital prices, leadingto higher foreign investment. As home’s demandfor imports expands, the pressure in the foreignexport sector to expand production coupled withan increase in labour demand for new firm con-struction leads to higher foreign wages. Aggregateforeign income rises as does foreign consumption.The two countries exhibit synchronous business cy-cle fluctuations.

Final goods production shocks have a more di-rect effect on aggregate prices, resulting in a highervolatility of the exchange rate index. We can definethe real exchange rate as:

Qt =P ∗CtPCt

= [AtA∗t

][P ∗bMtP

∗(1−b)Kt

P bMtP(1−b)Kt

]

The effect of a movement in At on Qt is strong andpositive35. Sharp increases in the real exchange rateincrease the profitability of exporting in the homecountry and a fall in the profitability of exportingin the foreign country. There is an expansion ofthe home export sector and a contraction of theforeign export sector. In addition, increases in con-sumption in both countries lead to higher demandfor intermediate inputs and thus higher prices forboth imported and exported goods. The volatil-ity of the terms of trade rises in part due to anincrease in intermediate good price volatility aswell as increases in the volatility of the real ex-change rate. Although intermediate goods pricesrise, the expansion of the domestic export sectorand the contraction of the foreign export sectorinduce home’s exports to sharply increase whilehome’s imports change negligibly. This, however,implies a pro-cyclical trade balance.

Changes in aggregate prices are not trivial. Thefactor used to transform welfare-based variables tovariables that more closely match the data (PCt

Pt)

sharply declines in the home country in response35This is confirmed in the impulse response function for Qt for a 1 standard deviation shock to At. A positive

productivity shock to final goods production results in an increase in the real interest rate, which suggests At >b(PMt − P ∗Mt) + (1− b)(PKt − P ∗Kt).

36 PCt

Pt=

[Ndt+N∗xt]

b/(1−a)

At(1−b)1−bbb. A positive shock to At coupled with a net increase in Ndt +N∗

xt lead to a sharp fall in PCt

Pt.

11

to a positive shock to final goods production36.All home variables subject to the price adjust-ment are slightly reduced. Home country variablesthat exhibit small or marginal increases, such asconsumption and imports, can even show declinesafter applying the price adjustment. Foreign vari-ables subject to the price adjustment are also sub-ject to changes in the real exchange rate37. Sincethe real exchange rate is rising, the negative effectof changes in the price adjustment factor is morethan made up for. Foreign variables are seeminglymore responsive to shocks in the home country thanhome variables due to changes in the exchange rate.

Timing is an important factor when connecting theintuition gleamed from the impulse response func-tions to the simulation results. From the impulseresponse functions, consumption and imports seemcontemporaneously negatively correlated with GDPafter applying the price adjustment while exportsand investment seem contemporaneously positivelycorrelated with GDP. Over time, after a positiveproductivity shock, GDP rises before returning tothe steady state. Consumption and imports followthe same pattern, they rise then gradually returnto the steady state. Imports and exports, however,continuously fall to the steady state after the initialshock. Consumption and imports are positivelycorrelated with GDP when considering the entiretransition period, while movements of investmentand exports along with the trade balance becomenegatively correlated with GDP. These results areconsistent with predictions made in the simulationresults.

The central benefits of productivity shocks to fi-nal goods production are an increase in investmentvolatility, high international correlation of output,investment and consumption, high volatility of ex-ports and imports, high volatility of internationalprices. This stochastic feature can not, however,capture correctly the volatility of output and con-sumption. Further, changes in aggregate prices,changes in the exchange rate and timing inducecounter-cyclical investment and exports. Since pro-ductivity shocks to intermediate goods productionhad many successes on these dimensions, I now turnto the model with both sources of shocks.

The model in which both At and Zt follow a nearunit root process is reported in column (5) of table538. The model captures output and consumptionvolatility as well as the benchmark model, but alsosucceeds in generating highly correlated consump-tion, output and investment across countries, higher

volatility of imports and exports, higher volatilityof the real exchange rate and low correlation be-tween international prices and GDP. These featuresare not captured by the model with capital accu-mulation and productivity shocks to intermediategoods production alone. The volatility of the termsof trade is improved, but still falls short of unity.The model over-predicts the volatility of investmentand slightly under-predicts correlation between ex-ports, imports and GDP as well as investment andGDP.

When capital is used in the production of non-traded final goods, the benchmark framework is un-able to reproduce a fully functioning internationalmarket. The mechanism that correlates businesscycles across countries becomes distorted when themodel is driven by productivity shocks to interme-diate goods production alone. When we subjectfinal goods production to productivity shocks, wecan improve the functioning of the internationalmarket. These sorts of productivity shocks can beinterpreted as demand shocks for intermediate in-puts. A model in which business cycles are drivenby both sorts of shocks can capture a larger num-ber of stylized facts more accurately while allowingus to represent investment in a manner that isconsistent with the rest of the real business cycleliterature.

4.2.3 Endogenous Labour Supply

As in Farhat (2009), I try to push the model furtherby incorporating endogenously determined laboursupply. I calibrate the elasticity of labour supplyto be consistent with empirical measurements atthe macro level (λ = 1) and impose strong coststo adjusting foreign bond holdings (n2 = 1). Iallow for capital accumulation as in the previoussections39. The results from the model with in-termediate goods shocks as the source of businesscycle fluctuations, the model with shocks to finalgoods production and the model with both typesof shocks are reported in columns (6), (7) and (8)in table 6 respectively. The incorporation of thelabour-leisure decision increases output volatilityand captures pro-cyclical employment and positivecorrelation of employment across countries. Fur-ther, some of the successes that were achieved inthe previous sections (such as increased volatility ofimports and exports, more correlated consumption,higher volatilities of international prices) can stillbe achieved.

When business cycles are driven by shocks to theproduction of intermediate goods, both the im-

37In addition to converting foreign welfare-based variables to measures that more closely match the data, the real exchangerate must also be used to make the variables comparable to home measures.

38Although there is underlying correlation between Zt and Z∗t , it is assumed that Zt and At are uncorrelated.39See Farhat (2009) for the model with endogenous labour supply without capital accumulation

12

pulse response functions and many of the simu-lation statistics are strikingly similar between themodel with capital accumulation and endogenouslabour supply (table 6) and the model with onlycapital accumulation (table 5). The model with amore complete labour market, however, can achievea higher estimate of output volatility, can gen-erate pro-cyclical employment and can reproducepositive correlation of employment across coun-tries. In the model with endogenously determinedlabour supply, a positive productivity shock leadsto firm entry as well as firm expansion, resultingin increased labour demand. Increased compositeconsumption from both firm entry (more variety)and expansion (more goods produced) in the homecountry compels households to supply less labour.The increase in labour demand resulting from thepositive productivity shock offsets the decline inlabour supply, resulting in more employment andhigher wages in equilibrium. The boost in employ-ment further increases aggregate output, leading tothe higher output volatility seen in the results table.

As income and consumption rise, the demand forimported goods rises inducing an expansion of theforeign export sector. This boosts foreign labourdemand. Increases in the return to home bondsleads to less firm construction in the foreign coun-try (as investment portfolios adjust) which reduceslabour demand. The contradicting effects of ex-port expansion and firm attrition make deducingchanges in labour demand difficult. Slight increasesin the home country’s export sector in response tothe improved productivity of labour lead to mildincreases in foreign consumption40, causing a reduc-tion in foreign labour supply. The model predictsan overall fall in foreign employment.

Although home employment rises and foreign em-ployment falls contemporaneously, implying nega-tive international correlation, the lagged responseto a positive productivity shock matter in gener-ating the positive employment correlation seen inthe simulation results. After the initial positiveshock, employment in the home country graduallyfalls back to the steady state. In the foreign coun-try, employment continues to fall after the initialshock for several periods41. As both the foreign anddomestic employment series move in the same direc-tion for several periods (in addition to employmentchanges in the foreign country being quite small),the simulation statistics report a positive corre-

lation. The analysis of output, consumption andinvestment correlations across countries, as well asthe other variables reported after applying the ap-propriate price adjustments follow from the modelwith capital and inelastic labour supply describedabove.

When business cycles are driven by shocks to theproduction of final consumption, major differencesbetween the model with only capital accumulationand the model with both capital accumulation andcomplete labour markets lie in the dynamic re-sponse to employment. A positive shock to finalgoods production in the home country leads to animmediate increase in the demand for intermediategoods. Employment increases, then gradually re-turns to its steady state. With shocks to final goodsproduction, there is no prolonged increase in thequantity or variety of goods that are exported tothe foreign country. Although foreign consumptionimmediately increases42, it then smoothly returnsto the steady state. Foreign labour supply im-mediately falls, then returns to its steady state,resulting in the dynamic negative correlation be-tween domestic and foreign employment observedin the simulation results.

The model with both shock processes achieves thesame successes as it had in the model with inelas-tic labour supply. A noticeable drawback is thenegative international correlation of employment.The effect of final goods production shocks on thevariety of products is dominate, resulting in a dy-namic negative correlation of employment acrosscountries.When comparing international correlations betweenthe model with developed labour markets (column(8) of table 6) and the model with inelastic laboursupply (column (5) of table 5) we notice an in-verted relationship between output volatility andbusiness cycle synchronization. When we incorpo-rate endogenous labour supply, any productivityshock results in a dramatic increase in output inthe home country. This triggers a larger degree ofinvestment exodus out of the foreign country andreduces the international correlations of consump-tion, output and investment. To test the impli-cations of this result, I experiment with a lowerlabour supply elasticity43 to mitigate changes inemployment when shocks occur. The results arereported in columns (9), (10) and (11) of table 6for the three different specifications of the shock

40This is due to changes in the variety of goods available to foreign customers.41Since the home country is expanding, increased varieties and quantities of exports to the foreign country result in

increasing foreign consumption over time which results in persistent declines in foreign labour supply. Firm attrition andexpansion of the foreign export sector continue to keep changes in labour demand marginal. Employment falls and wagesrise in the foreign country for several periods after the initial shock.

42This is due to increased investment in capital in response to the shock, as described in the previous section.43λ = 5 corresponds to a labour supply elasticity of 0.2, which is a common estimate for men according to Fiorito and

Zanella (2008)).

13

processes. The results for output volatility and theinternational correlations are as expected. Out-put volatility declines since labour supply is lessresponsive to economic shocks. The reduction inoutput volatility increases the international cor-relation of output, consumption, investment andemployment since there is less investment exodusfrom the foreign country in response to a positiveproductivity shock. Reductions in labour supplyelasticity increase the volatility of imports, exportsand international prices relative to GDP since GDPvolatility decreased. A better statistical fit for allthe other endogenous variables save output volatil-ity is achieved with a lower labour supply elasticity.

5 Conclusion

Without any notion of capital, the definition of “in-vestment” in the original Ghironi and Melitz (2005)model is inconsistent with the real business cycleliterature. Although new firm construction is animportant component, changes in plant and equip-ment by existing firms is included in the statisticalmeasure of investment. By including capital intothe production of non-traded goods, more reason-able comparisons between the data and the model’sresults can be made.

At first, the inclusion of capital into the modelimproves the correlation between imports, exportsand GDP when foreign bonds purchases are costlyand economic fluctuations are driven by productiv-ity shocks to intermediate goods production. Thepresence of capital, however, results in negative in-ternational correlations of output, investment andconsumption and inadequately captures the volatil-ity of the international market. Increases in theprice of capital when there are positive productiv-ity shocks leads to decreases in the real interestrate which induces statistical declines in foreignvariables after transforming them into comparableterms.

When productivity shocks to final goods are incor-porated into the model, dramatic improvements ininternational correlations along with higher volatil-

ities of international market variables are achieved.Positive shocks to final goods production directlyincrease the exchange rate, making positive inter-national correlations of output, consumption andinvestment statistically achievable. Since shocks tofinal goods production also acts as demand shocksfor intermediate goods, there is increased volatilityof trade measures and international prices. Fur-ther, the relationships between the exchange rate,terms of trade and GDP that are seen in the dataare captured by the model. This suggests thatproductivity shocks to both final (non-traded) andintermediate (traded) goods are essential to re-producing the mechanism that transmits businesscycles across countries. Although results are muchimproved when shocks to final goods productionare added to the model, they are far from ideal.Improving the fit further by estimating and re-calibrating the VAR processes for both shocks tointermediate and final goods production is left forfuture work.

When endogenous labour supply is incorporatedinto the model, increased output volatility alongwith pro-cyclical employment and positive corre-lation of employment across countries is achievedwhen business cycles are driven by shocks to inter-mediate goods production (productivity shocks tofirms that demand labour). When shocks to finalgoods production are added, the model continues toproduce improved international co-movement andreplicates the behaviour of international marketvariables. To keep employment positively corre-lated across countries, low labour supply elasticitiesmust be considered which results in reduced outputvolatility. Correcting this attribute of the modelcontinues to be a mystery44 and is left for futurework.

6 Acknowledgements

Special thanks to Jang-Ting Guo, Roger Farmer,R. Robert Russell, Richard Suen, Victoria Uman-skaya for their invaluable comments. All errors aremy own. For additional comments or suggestions, Ican be reached at dan.farhat@otago.ac.nz.

44Neither the preferences described by Greenwood, Hercowitz, and Huffman (1988), which produce labour supply equa-tions without income effects, nor indivisible labour as described by Hansen (1985) seem to improve output volatility withoutdistorting the other results.

14

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16

A Appendix

Table 1:U.S. Business Cycle Statistics (1957q1-2007q1)Volatility (Std. Dev.) Correlation with GDPσY (%) 1.54 ρY,Y 1.00σC/σY 0.75 ρC,Y 0.88σI/σY 3.41 ρI,Y 0.90σL/σY 0.61 ρL,Y 0.81

Table 1: U.S. quarterly data (1957q1-2007q1) for output (Y), consumption (C) and investment (I) is extracted fromthe International Financial Statistics maintained by the International Monetary fund. Data for investment includes grossfixed capital formation plus changes in inventories. Labor data is generated from civilian employment measures from theOECD.Stat database. Logs of deflated measures are de-trended using the HP filter.

Table 2:U.S. - European Co-movement (1970-2007)

CorrelationρY,Y ∗ 0.56ρC,C∗ 0.45ρI,I∗ 0.37ρL,L∗ 0.39

Table 2: Annual data for output (Y), consumption (C) and investment (I) for the U.S. and a ”European Aggregate”consisting of Austria, Finland, France, Germany, Italy, Switzerland and the United Kingdom are taken from the Penn WorldTables maintained by Heston, Summers, and Aten (2006). Annual labor (L) data is taken from the OECD.Stat databaseand omits Finland due to missing data. Data availability for Germany restricts the sample to 1970-2004. Logs for deflatedvariables are de-trended using the HP filter.

Table 3:U.S. International Market Statistics (1957q1-2007q1)Volatility (Std. Dev.) Correlation with GDPσIM/σY 3.31 ρIM,Y 0.6σEX/σY 3.74 ρEX,Y 0.28σNX/Y 0.4 ρNX/Y,Y -0.36σTOT /σY 1.69 ρTOT,Y 0.07σQ/σY 3.37 ρQ,Y -0.18

Table 3: U.S. quarterly data (1957q1-2007q1) for imports (IM), exports (EX) the trade balance relative to output(NX/Y), the terms of trade (TOT) (index) and the real exchange rate (Q) (index) is extracted from the InternationalFinancial Statistics database maintained by the World Bank. Logs of deflated measures are de-trended using the HP filter,with exception to the trade balance which is de-trended in levels.

17

1970 1975 1980 1985 1990 1995 2000 2005−1

−0.8

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GDPConsumptionInvestmentEmployment

1970 1975 1980 1985 1990 1995 2000 2005−1

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Figure 1: U.S. - European Co-movement (1970-2004) - Rolling Window

Figure 1: Annual data for output (Y), consumption (C) and investment (I) for the U.S. and a ”European Aggregate”consisting of Austria, Finland, France, Germany, Italy, Switzerland and the United Kingdom are taken from the PennWorld Tables maintained by Heston et al. (2006). Annual labour (L) data is taken from the OECD.Stat database. Dataavailability for Germany restricts the sample to 1970-2004. Logs for deflated variables are de-trended using the HP filter.At any time, t, the figure plots the international correlations of GDP, consumption, and investment using data from theproceeding j years, where j=5 and 10 respectively.

Table 4: Calibrated Parameters.Parameter Valueβ Discount factor 0.99, 0.96δ Probability of death shock 0.025, 0.10δk Capital depreciation rate 0.025, 0.10τ Iceberg costs associated with trade 1.3a Elasticity of substitution between inputs 3.8k Shape parameter on Pareto distribution 3.4zmin Lower bound on Pareto distribution 1zD ”Special” average productivity draw zmin( k

k+1−a )1/(a−1)

FEt Fixed cost of entry 1FXt Fixed exporter costs 0.235(FEt)(

1−β(1−δ)β(1−δ) )

18

Table 5: Simulation Results: Capital Accumulation(1) (2) (3) (4) (5)

Domestic StatisticsVolatilityσY (%) 1.54 0.97 0.76 0.15 0.77σC/σY 0.75 0.61 0.50 1.78 0.57σI/σY 3.41 3.11 2.19 43.45 8.12σL/σY 0.61 — — — —Correlation with GDPρC,Y 0.88 0.99 0.99 0.48 0.86ρI,Y 0.90 0.98 0.59 -0.17 0.14ρL,Y 0.81 — — — —

International Co-movementCorrelationρY,Y∗ 0.56 0.28 -0.15 0.67 0.45ρC,C∗ 0.45 0.34 -0.27 0.31 0.29ρI,I∗ 0.37 0.07 -0.75 0.77 0.72ρL,L∗ 0.39 — — — —

International Market StatisticsVolatilityσIM/σY 3.31 0.58 0.68 8.87 1.70σEX/σY 3.74 0.57 0.71 11.73 2.18σNX/Y 0.40 0.02 0.12 0.56 0.58σTOT /σY 1.69 0.12 0.21 4.22 0.77σQ/σY 3.37 0.10 0.14 3.19 0.58Correlation with GDPρIM,Y 0.60 0.82 0.79 0.14 0.31ρEX,Y 0.28 0.80 0.40 0.10 0.13ρNX/Y,Y -0.36 -0.22 -0.34 -0.002 -0.06ρTOT,Y 0.07 0.55 0.56 -0.03 0.15ρQ,Y -0.18 0.41 -0.55 0.17 -0.12

(1) Data(2) Benchmark,(Kt = 0), shocks to Zt only(3) Capital Accumulation, shocks to Zt only(4) Capital Accumulation, shocks to At only(5) Capital Accumulation, shocks to Zt and At

19

Table 6: Simulation Results: Capital Accumulation with Endogenous Labour Supply.(1) (6) (7) (8) (9) (10) (11)

Domestic StatisticsVolatilityσY (%) 1.54 1.05 0.20 1.08 0.83 0.15 0.86σC/σY 0.75 0.44 1.11 0.47 0.48 1.55 0.55σI/σY 3.41 2.36 30.86 6.32 2.26 40.81 7.48σL/σY 0.61 0.38 0.88 0.41 0.12 0.29 0.13Correlation with GDPρC,Y 0.88 0.98 -0.05 0.86 0.99 0.30 0.86ρI,Y 0.90 0.60 0.12 0.28 0.60 -0.06 0.18ρL,Y 0.81 0.98 0.69 0.94 0.99 0.36 0.91

International Co-movementCorrelationρY,Y∗ 0.56 -0.05 0.87 0.42 -0.11 0.72 0.46ρC,C∗ 0.45 -0.26 0.10 0.07 -0.32 0.21 0.18ρI,I∗ 0.37 -0.59 0.48 0.41 -0.73 0.69 0.64ρL,L∗ 0.39 0.33 -0.31 -0.12 0.42 0.01 0.14International Market StatisticsVolatilityσIM/σY 3.31 0.64 5.70 1.24 0.69 7.95 1.53σEX/σY 3.74 0.66 7.42 1.54 0.70 10.45 1.89σNX/Y 0.40 0.17 0.53 0.57 0.13 0.54 0.56σTOT /σY 1.69 0.20 2.64 0.52 0.21 3.72 0.67σQ/σY 3.37 0.13 2.01 0.38 0.14 2.80 0.51Correlation with GDPρIM,Y 0.60 0.76 -0.50 0.28 0.78 -0.10 0.33ρEX,Y 0.28 0.35 0.70 0.28 0.36 0.34 0.17ρNX/Y,Y -0.36 -0.32 0.62 0.03 -0.35 0.24 -0.05ρTOT,Y 0.07 0.56 -0.62 0.07 0.57 -0.27 0.14ρQ,Y -0.18 -0.54 0.68 -0.04 -0.54 0.38 -0.06

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−0.

010

0.01

0.02

0.03

I

050

100

−50510

x 10

−4

L

050

100

−50510

x 10

−3

IM

050

100

−5051015

x 10

−3

EX

050

100

−1

−0.

50

0.51

x 10

−3

NX

/Y

050

100

−50510

x 10

−3

Q,T

OT

Q TO

T

050

100

−50510

x 10

−3

K

050

100

−50510

x 10

−3

PK

050

100

−50510

x 10

−3

PM

050

100

−505

x 10

−5

B,B

∗

050

100

−0.

010

0.01

0.02

0.03

r

050

100

0

0.51

1.5

x 10

−3

Y

050

100

−1012

x 10

−3

C

050

100

0

0.01

0.02

0.03

I

050

100

−505

x 10

−3

˜IM

050

100

−50510

x 10

−3

EX

050

100

−10−505

x 10

−3

PC

t

Pt

050

100

0

0.51

1.5

x 10

−3

Q

Fig

ure

6:Im

puls

eR

espo

nse

Func

tion

s-

Cap

ital

Acc

umul

atio

n,E

ndog

enou

sL

abou

rSu

pply

(λ=

1),

Shoc

ksto

Fin

alG

oods

Pro

duct

ion

25

050

100

−202468

x 10

−3

ND

Hom

eF

orei

gn

050

100

−0.

010

0.01

0.02

0.03

0.04

NE

050

100

012345x

10−

3N

X

050

100

−1012

x 10

−3

W/Z

050

100

−50510

x 10

−3

Y

050

100

02468x

10−

3C

050

100

−0.

010

0.01

0.02

I

050

100

−50510

x 10

−4

L

050

100

012345x

10−

3IM

050

100

0246x

10−

3E

X

050

100

−4

−2024

x 10

−4

NX

/Y

050

100

−20246

x 10

−3

Q,T

OT

Q TO

T

050

100

−202468

x 10

−3

K

050

100

−202468

x 10

−3

PK

050

100

−10−505

x 10

−4

PM

050

100

−8

−6

−4

−202

x 10

−5

B,B

∗

050

100

−0.

010

0.01

0.02

0.03

r

050

100

−202468

x 10

−3

Y

050

100

−20246

x 10

−3

C

050

100

−0.

02

−0.

010

0.01

0.02

I

050

100

012345x

10−

3˜

IM

050

100

012345x

10−

3E

X

050

100

−2.

5

−2

−1.

5

−1

−0.

50x

10−

3P

Ct

Pt

050

100

−2

−1.

5

−1

−0.

50x

10−

3Q

Fig

ure

7:Im

puls

eR

espo

nse

Func

tion

s-

Cap

ital

Acc

umul

atio

n,E

ndog

enou

sL

abou

rSu

pply

(λ=

5),

Shoc

ksto

Inte

rmed

iate

Goo

dsP

rodu

ctio

n

26

050

100

−1

−0.

50

0.51

x 10

−3

ND

Hom

eF

orei

gn

050

100

−0.

01

−0.

0050

0.00

5

0.01

NE

050

100

−0.

01

−0.

0050

0.00

5

0.01

NX

050

100

−50510

x 10

−3

W/Z

050

100

−50510

x 10

−3

Y

050

100

−50510

x 10

−3

C

050

100

−0.

010

0.01

0.02

0.03

I

050

100

−1012

x 10

−4

L

050

100

−50510

x 10

−3

IM

050

100

−5051015

x 10

−3

EX

050

100

−505

x 10

−4

NX

/Y

050

100

−50510

x 10

−3

Q,T

OT

Q TO

T

050

100

−50510

x 10

−3

K

050

100

−202468

x 10

−3

PK

050

100

−50510

x 10

−3

PM

050

100

−4

−2024

x 10

−5

B,B

∗

050

100

−0.

010

0.01

0.02

r

050

100

0

0.51

1.5

x 10

−3

Y

050

100

−1012

x 10

−3

C

050

100

0

0.00

5

0.01

0.01

5

0.02

I

050

100

−4

−20246

x 10

−3

˜IM

050

100

−50510

x 10

−3

EX

050

100

−10−505

x 10

−3

PC

t

Pt

050

100

0

0.51

1.5

x 10

−3

Q

Fig

ure

8:Im

puls

eR

espo

nse

Func

tion

s-

Cap

ital

Acc

umul

atio

n,E

ndog

enou

sL

abou

rSu

pply

(λ=

5),

Shoc

ksto

Fin

alG

oods

Pro

duct

ion

27