Production Sharing and Business Cycle Co-Movements

with Heterogeneous Firms

Andrei Zlate�

Boston College

April 28, 2008

Abstract

In this paper I examine whether the co-movement of business cycles can be explained by the

�rms�decision to relocate the production of intermediate goods across borders through vertical FDI.

First, using data from Mexico�s maquiladora industry, I show that o¤shore production is pro-cyclical

with the U.S. manufacturing output, although the timing of the maquiladora�s response di¤ers at

the intensive and the extensive margins: hours worked respond immediately, whereas the number

of o¤shore plants responds with a lag of three quarters. Second, I build a dynamic stochastic

general equilibrium (DSGE) model with heterogeneous �rms and �rm entry dynamics in a two-

country framework (North and South). The model examines the ability of three internationalization

strategies of multinational �rms - vertical FDI, horizontal FDI, and endogenous exports - to act

as transmission channels of aggregate shocks across countries. I derive an asymmetric steady

state in which the relatively lower wage in South acts as an incentive for the �rms in North to

relocate production o¤shore. A positive technology shock in North encourages �rm entry, triggers

the appreciation of relative wages and causes more �rms to relocate production o¤shore. Higher

demand for labor in North (due to �rm entry) and simultaneously in South (due to o¤shoring)

generates positive co-movement of wages and aggregate incomes. Relative to the benchmark models

with either endogenous exports or horizontal FDI, o¤shoring through vertical FDI increases the co-

movement of output and decreases the co-movement of consumption, results which come closer to

the empirical correlations reported in Backus, Kehoe, Kydland (1992).

JEL classi�cation: E32, F21, F23, F41, F42

Keywords: o¤shoring, vertical FDI, horizontal FDI, exports, business cycle co-movements,

heterogeneous �rms, �rm entry, relative wages

�Department of Economics, Boston College, MA 02467, USA; zlate@bc.edu, http://www2.bc.edu/~zlate.

1

1 Introduction

In this paper I examine whether the �rms�decision to relocate the production of intermediate goods

across borders can explain the co-movement of aggregate incomes across countries. Firms often follow

strategies that involve the fragmentation of production chain and the establishment of foreign a¢ liates

at locations with relatively lower labor costs that serve as export platforms, an action which I refer

to as "o¤shoring through vertical FDI" (foreign direct investment) in this paper.1 Unlike production

under horizontal FDI - which implies that foreign a¢ liates produce mainly for the local market of

the host country - this type of vertically-integrated production involves foreign a¢ liates that add

value towards the �nal goods purchased by consumers in the multinationals�country of origin or in

third countries. Therefore, in my model, vertical FDI and trade are complements: The creation of

production-based a¢ liates o¤shore generates trade �ows, as multinationals and their foreign a¢ liates

exchange intermediate goods with each other.

Recent literature documents empirically that country pairs that exchange intermediate goods more

intensely �exchanges which often take place between o¤shore a¢ liates and the headquarters of multi-

national �rms engaged in vertical FDI �exhibit higher correlations of aggregate incomes (Burstein,

Kurtz, Tesar, 2007). Trade data also shows that, despite notable variation across sectors and indus-

tries, the extent of cross-border production under vertical FDI in manufacturing has been particularly

large within the NAFTA region, Central and South America. In 2005, as much as 32, 50 and 26

percent of the manufacturing sales of U.S. a¢ liates from Canada, Mexico and Latin America as a

whole were directed towards their U.S. parent �rms, a pattern which re�ects the key economic role of

production sharing through vertical FDI in the region (Bureau of Economic Analysis, 2007).2

I build a dynamic stochastic general equilibrium (DSGE) model that features endogenous o¤-

shoring, heterogeneous �rms and �rm entry dynamics in a two-country framework (North and South).

Firms in each economy are monopolistically competitive, and each �rm produces a di¤erent variety of

�nal goods using either domestic or foreign inputs. Firms also face sunk entry costs, period-by-period

�xed o¤shoring costs, and iceberg trade costs. The sunk entry costs re�ect the regulation of starting

1 I attach di¤erent meanings to "o¤shoring" and "outsourcing." The former concerns the integration of multinational

�rms across borders through vertical or/and horizontal FDI. In contrast to o¤shoring, outsourcing takes place when a

�rm purchases intermediate inputs or services from an una¢ liated supplier - either at home or abroad - rather than

producing it in house. See Helpman (2006) for an ample discussion of the related literature.2 In contrast, considerably smaller shares of the sales of U.S. a¢ liates in Europe and Asia-Paci�c were directed towards

their U.S. parent multinationals (3 and 5 percent in 2005).

2

a business, whereas the �xed costs of producing o¤shore represent headquarters activities and the

maintenance of production plants in the foreign country. Nevertheless, the iceberg trade costs re�ect

transportation, insurance and trade barriers that a¤ect the imports of intermediate goods produced by

o¤shore a¢ liates and sold to their parent �rms in the country of origin. Thus, when deciding where to

locate production (domestically vs. o¤shore), �rms balance the incentive of lower labor costs against

the �xed and iceberg trade costs associated with o¤shore production.

I derive an asymmetric steady state in which di¤erences in �rm entry regulation (i.e. the cost

of starting a business) a¤ect labor demand and generate wage di¤erences across countries. Indeed,

empirical evidence shows large variation in �rm entry costs across countries: The cost of starting a

business is three times higher in Mexico than in the U.S. or Canada; it is six times higher in Hungary

than in the U.K. (World Bank, 2007, see Table 4). In my model, I set �rm entry costs to be higher

in South; consequently, labor demand and the e¤ective wage are lower in South. Therefore, only

�rms originating in North have the incentive to relocate production across the border; all �rms that

originate and serve the South market produce domestically.

The mechanism that generates the co-movement of aggregate incomes through endogenous o¤-

shoring in my model works as follows. Firms choose endogenously to locate production either at home

or overseas. A positive technology shock in the North economy encourages �rm entry. In turn, �rm

entry causes wages to rise faster than productivity, and leads to an appreciation of the cost of e¤ective

labor in North relative to South. Subsequently, the more productive �rms originating in North choose

to relocate production overseas. Higher demand for labor in North (due to �rm entry) and higher

demand for labor in South (due to o¤shoring) generate positive co-movement of wages and aggregate

incomes.

Throughout this paper I use the �terms of labor�as a measurement of the relative cost of e¤ective

labor between North and South. I de�ne the terms of labor as the ratio between the South and North

real costs of e¤ective labor expressed in units of the North consumption basket, TOLt =w�t =Z

�t

wt=ZtQt,

where the cost of e¤ective labor is the ratio between the real wage and aggregate productivity, i.e. wtZt

in the North economy. Over the business cycle, �rm entry leads to an increase of the relative cost

of e¤ective labor (i.e. the terms of labor appreciate/decrease), which in turn causes more �rms from

North to relocate production o¤shore.

Firm entry and �rm heterogeneity are two necessary ingredients of my model. In steady state,

cross-country di¤erences in the regulation of �rm entry are translated into di¤erences in real e¤ective

wages across countries, and therefore provide an incentive for some of the �rms in North to relocate

3

production to South, where the cost of e¤ective labor is relatively lower. Over the business cycle,

�rm entry triggered by a positive technology shock in the home economy drives the appreciation of

the domestic e¤ective wage relative to South, and thus causes more �rms to relocate production to

o¤shore.3

Firm heterogeneity �i.e. some �rms are more productive than others �is also necessary to generate

endogenous o¤shoring in my model. If �rms were identical and o¤shoring costs negligible, all �rms

would prefer to produce intermediate goods at the location with cheaper e¤ective labor. However, with

�rm heterogeneity, only some of the �rms can a¤ord the �xed costs of o¤shoring, and their share in the

number of total �rms varies over the business cycle. The relocation of production by home �rms a¤ects

wages in South in the same way that �rm entry a¤ects wages in North �namely, it induces upward

pressure on wages due to higher demand for labor �a mechanism which then generates co-movement

of aggregate incomes.4

Given the increased complexity of multinational �rms�operations across borders (Helpman, 2006),

the traditional classi�cation of FDI into vertical and horizontal types has become increasingly di¢ cult.

Therefore, in addition to o¤shore production through vertical FDI (i.e. motivated by lower production

costs in the foreign economy), my model nests the benchmark models with endogenous exports a la

Ghironi and Melitz (2005) as well as multinational production through horizontal FDI (i.e. motivated

not by low production cost, but by access to the local market) a la Contessi (2006). Thus, the model

examines the ability of three di¤erent internationalization strategies of multinational �rms to act as

channels through which aggregate shocks are transmitted across economies. The analytical moments

of my model show that endogenous o¤shoring through vertical FDI increases the co-movement of

national incomes and decreases the co-movement of consumption relative to the benchmark models

with endogenous exports a la Ghironi and Melitz (2005) and horizontal FDI a la Contessi (2006),

results which are more in line with the corresponding empirical correlations.

This paper is organized as follows. Section 2 provides an empirical analysis of the cyclicality

of o¤shoring to Mexico�s maquiladora sector relative to the U.S. manufacturing output. Section 3

introduces a DSGE model of o¤shoring with heterogeneous �rms and �rm entry dynamics. Section

3Without entry, the cost of e¤ective labor in North would not appreciate relative to South after a positive technology

shock, as the increase of home wage relative to its steady state level would be just as large as the increase in aggregate

productivity every period; �rms would have no incentive to relocate production overseas.4Firm heterogeneity is not only a useful feature of my model, but also represents a well-established empirical regularity.

Firm productivity varies within and across sectors, and only the more productive �rms import intermediate goods from

overseas (Kasahara and Rodriguez, 2005; Yasar and Morrison Paul, 2006).

4

4 shows the impulse responses and the second moments generated by the model with vertical FDI

relative to the benchmark models with endogenous exports a la Ghironi and Melitz (2005) and o¤shore

production through horizontal FDI a la Contessi (2006). Section 5 includes an extension of the model

with vertical FDI which allows for elastic labor supply, and explores the ability of the model to

reproduce the empirical moments of o¤shoring at the intensive and extensive margins. Section 6

concludes.

2 Measuring the cyclicality of o¤shoring

Although the link between trade �ows and business cycle co-movements has received large attention

in empirical literature, there is still little known about the �uctuation of o¤shore production over the

business cycle. Therefore, in this section I explore the cyclicality of o¤shore production to Mexico�s

maquialdora sector relative to �uctuations in the U.S. manufacturing output and in the U.S.-Mexico

relative manufacturing wage.

2.1 Trade in intermediate goods and business cycle correlations

Figure 1. Share of intermediates in bilateral trade vs. GDP correlations. Source:

Burstein, Kurtz, Tesar (2007).

Recent literature documents the link between trade in intermediate goods and the co-movement of

aggregate incomes, which is one of the stylized facts that my theoretical model attempts to replicate.

For instance, Burstein, Kurtz, Tesar (2007) show that a positive and signi�cant link exists between

(1) the volume of intermediate goods exported by o¤shore a¢ liates back to headquarters, measured as

5

the share of a¢ liate exports in the total exports of the host economy to their country of origin in 2003,

and (2) the GDP correlation between the pairs of countries involved, measured over 1985-2003 (Figure

1).5 This result suggests that the cross-country interdependencies generated by the involvement in

production sharing are associated with a closer synchronization of business cycle �uctuations across

the countries involved.

2.2 The cyclicality of o¤shoring with quarterly data

Next I explore the cyclicality of o¤shore production using data on Mexico�s maquiladora and the U.S.

manufacturing industries. In particular, I explore the link between (1) variables describing Mexico�

maquiladora sector: value added, hours worked, and number of plants, and (2) variables describing the

U.S. manufacturing business cycle: industrial production, and the ratio of U.S. to Mexico�s nominal

hourly wage in manufacturing expressed in the same currency. The data is provided by Mexico�s

National Institute of Statistics (real value added, hours worked, hourly nominal wages and number of

plants in Mexico�s maquiladora sector), the Board of Governors of the Federal Reserve System (real

industrial output in U.S. manufacturing) and the U.S. Bureau of Labor Statistics (hourly nominal

wages in U.S. manufacturing).6

The maquiladora industry in Mexico constitutes a good example to study the pattern of o¤shoring

in U.S. manufacturing, due to several reasons. First, it represents a classic example of o¤shoring under

vertical FDI: Plants that operate under Mexico�s maquiladora program import inputs, process them

and ship them back to the country of origin. On the goods�return to the country of origin, tari¤s are

paid only on the value added by the plant in Mexico (Gruben, 2001). Second, although not all plants

in Mexico�s maquiladora sector are owned by U.S. �rms, most of maquiladora�s value added - roughly

90 percent - is exported to the U.S. (Burstein, Kurtz, Tesar, 2007).

I employ the Baxter-King bandpass �lter to the variables expressed in natural logs in order to

�lter out �uctuations with periodicity lower than 18 months and greater than eight years. The re-

sulting bandpassed series are plotted in Figure 2(a). The three charts on the left in Figure 2(a) plot

Mexico�s maquiladora value added (real), hours worked and number of plants together with the U.S.

manufacturing output (industrial production, real) at quarterly frequency between Q1-1993 and Q4-

2003. Similarly, the three charts on the right in Figure 2(a) plot Mexico�s indicators together with the

relative U.S.-Mexico nominal hourly wage in manufacturing.7

5The OLS coe¢ cient is positive (0.65) and signi�cant at the 1 percent level.6Data on Mexico�s maquiladora sector is provided at monthly frequency. I aggregate it into quarters.7The data on Mexico�s maquiladora sector was provided by the Instituto Nacional de Estadistica, Geogra�a e Infor-

6

The visual inspection of the data suggests that o¤shoring to Mexico - measured as value added,

hours worked and number of o¤shore plants - is pro-cylical with the U.S. manufacturing output. More,

Mexico�s value added is notably more volatile than the U.S. manufacturing output. The charts also

suggest a delayed response of o¤shoring at the extensive margin (i.e. the number of plants in Mexico�s

maquiladora sector) to �uctuations of industrial production in U.S. manufacturing.

Figure 2(a). The cyclicality of o¤shoring to Mexico�s maquiladora sector relative to (1) U.S.

manufacturing ouput (left) and (2) the U.S.-Mexico relative wage (right)

matica (INEGI) Mexico. Industrial production in the U.S. manufacturing sector at monthly frequency was provided by

the U.S. Board of Governors of the Federal Reserve System.

7

2.2.1 Correlations with lags and leads of U.S. output

In order to account for the possibility of delayed responses, I compute the correlation of each of

Mexico�s maquiladora indicators (value added, hours worked, numbers of plants) with various lags

and leads of the U.S. business cycle indicators (industrial output and the U.S.-Mexico relative wage).

Figure 2(b). Correlations between Mexico�s mauiladora indicators and various lags and leads of (1)

the U.S. manufacturing output (left) and (2) the U.S.-Mexico relative wage (right)

The charts on the left in Figure 2(b) show that o¤shoring to Mexico�s maquiladora industry is pro-

cyclical with U.S. manufacturing output, although the timing of maquiladora�s response di¤ers at the

intensive and the extensive margins. On one hand, correlations show a positive immediate response

8

of maquiladora�s hours worked to �uctuations of U.S. industrial production in manufacturing. On

the other hand, value added responds with a lag of one or two quarters, and the number of plants

in Mexico�s maquiladora sector (i.e. o¤shoring at the extensive margin) responds with a lag of three

quarters to the U.S. manufacturing output. The correlation of the number of maquiladora plants with

the contemporaneous U.S. manufacturing output is less than 0.5, whereas the correlation with U.S.

manufacturing output lagged by three quarters is positive and exceeds 0.7.

I interpret the correlations between Mexico�s maquiladora indicators and the U.S.-Mexico relative

manufacturing wage ratio as evidence in favor of the cross-country transmission mechanism of business

cycles that I build in my model. The charts on the right in Figure 2(b) show that the correlations

of maquiladora value added, hours worked and number of plants with the lagged relative wage ratio

are positive, i.e. past increases in manufacturing wage of the U.S. relative to Mexico are followed by

the relocation of production from the U.S. to Mexico at both the intensive and the extensive margins.

However, the correlations of each of the maquiladora indicators with the contemporaneous U.S.-Mexico

wage ratio are negative, a result which shows that the relocation of production to Mexico is associated

with a contemporaneous increase in Mexico�s wage, and therefore a decrease in the relative wage ratio.

2.2.2 OLS analysis

I use the following econometric speci�cation to explore the link between the time series described in

Figure 2(a):

ln(YMEX;t) = �+ � ln(OutputUS;t) + dNAFTA;t + "t;

ln(YMEX;t) = �+ � ln(WageUS;tWageMEX;t

) + dNAFTA;t + "t;

where the dependent variable YMEX;t represents alternatively Mexico�s manufacturing real value

added, hours worked and the number of plants; the dummy variable dNAFTA;t controls for the impact

of NAFTA membership on correlations. I expect o¤shore value added, hours worked and number of

plants to be pro-cyclical with the U.S. manufacturing output, and negatively related to the U.S.-Mexico

relative wage in manufacturing.8 I estimate the equations in both levels (to obtain the elasticities of

levels) and di¤erences (for elasticities of growth rates).

8The results should be interpreted as correlations; they do not provide any insight on the causality between variables.

9

Table 1. U.S.-Mexico co-movement in manufacturing, OLS estimates

Equations in levels

Dependent Variables

Explanatory + ln(V AMEX;t) ln(HoursMEX;t) ln(PlantsMEX;t)

ln(OutputUS) 1.425*** (0.223) 2.854*** (0.301) 1.450*** (0.349)

ln( WUSWMEX

) -0.045 (0.074) 0.002 (0.127) -0.155 (0.095)

Equations in di¤erences

Dependent Variables

Explanatory + dln(V AMEX;t) dln(HoursMEX;t) dln(PlantsMEX;t)

dln(OutputUS) 1.152*** (0.295) 1.976*** (0.397) -0.141 (0.381)

dln( WUSWMEX

) 0.152* (0.070) 0.051 (0.107) -0.015 (0.079)

The results in Table 1 show that Mexico�s maquiladora value added, hours worked and the number

of plants are indeed pro-cyclical with U.S. manufacturing output. In the equations in levels, when each

of Mexico�s value added, hours worked and number of plants alternatively represents the dependent

variable, the coe¢ cients on U.S. output are positive and statistically signi�cant at the 1 percent level.

The equations in levels also highlight the larger volatility of Mexico�s maquiladora sector relative to

U.S. manufacturing. The coe¢ cient estimates represent elasticities and are larger than unit, i.e. a

1 percent increase in U.S. manufacturing output is associated with a more than 1 percent increase

in Mexico�s maquiladora sector. The coe¢ cient estimates on the relative wage as the explanatory

variable are not statistically signi�cant.

In the equations in di¤erences with either Mexico�s value added or hours worked as the dependent

variable, the coe¢ cients on U.S. output are positive and statistically signi�cant at the 1 percent level.

The sign and magnitude of coe¢ cients show that faster growth in U.S. manufacturing is associated with

accelerated growth in Mexico�s maquiladora value added and hours worked. With one exception (value

added as the dependent variable), the coe¢ cient estimates on the relative wage as the explanatory

variable are not statistically signi�cant.

Analyzing the cyclicality of o¤shoring represents a necessary step which provides empirical ground-

ing for the o¤shoring-based transmission channel of business cycles that I develop in the theoretical

section below.

10

3 Model of o¤shoring with heterogeneous �rms

The model describes three main activities in each economy.

(1) Households supply labor inelastically, consume an aggregate consumption basket, and �nance

�rm entry.

(2) Firms (monopolistically-competitive, heterogeneous in productivity) produce �nal goods for the

domestic market using intermediate goods under two possible strategies: (a) use only domestic inputs;

or (b) use a mix of intermediate goods produced both domestically and o¤shore through vertical FDI,

i.e. relocate the production of some inputs to o¤shore locations where labor is cheaper.

(3) Some of the �rms also produce �nal goods for the foreign market under two possible strategies:

(a) produce domestically and export the �nal good, like in Ghironi and Melitz (2005); or (b) use a mix

of intermediate goods produced both domestically and o¤shore through horizontal FDI as in Contessi

(2006).

Figure 3. Internationalization strategies of multinational �rms

Figure 3 describes the various production strategies available to �rms that serve the domestic and

foreign markets. The di¤erence between production under vertical and horizontal FDI is that the

former strategy is adopted by �rms serving the domestic market, is motivated by the relatively lower

labor costs abroad, and involves the relocation of production o¤shore simultaneously with shutting

down the domestic facility. In contrast, production under horizontal FDI is motivated by access to the

o¤shore market, and involves the simultaneous production of the same variety of intermediate goods

11

both at home (for the domestic market) and o¤shore (for the foreign market).

The following sections describe the detailed model while focussing on the North economy. The

equations for South are similar unless indicated otherwise. Variables in South are denoted with a star

superscript.

3.1 Households

Households in each country maximize the expected lifetime utility and provide labor inelastically,

subject to the budget constraint:

maxfBt; xtg

"Et

1Xs=t

�s�tC1� s

1�

#;

s:t: Bt+1 + evt (Nt +NE;t)xt+1 + Ct = (1 + rt)Bt + (edt + evt)Ntxt + wtL;

where � 2 (0; 1) is the subjective discount factor, Ct is aggregate consumption, and > 0 is the

inverse of the inter-temporal elasticity of substitution.

The representative household in North purchases two types of assets. Every period t, the household

buys the risk-free bond Bt+1 denominated in units of the consumption basket Ct. It also purchases

xt+1 shares in a mutual fund of �rms in North that include: (i) Nt �rms that already produce at time

t, either domestically or o¤shore, and (ii) NE;t new �rms that enter the market in period t. Each

share is worth its market value evt, which equals the net present value of the expected stream of future

pro�ts of the average �rm, measured in units of the North consumption basket.

The household enters every period t with real bond holdings Bt and mutual fund share holdings

xt whose market value is evtNt. It receives interest rtBt on bond holdings, and dividend income edtNt

on the mutual fund stocks equal to the pro�t of the average �rm times the number of existing home

�rms in period t, in proportion with its stock holdings xt.9 Finally, the representative household also

receives labor income equal to the real wage wt. I set L = 1, i.e. labor is supplied inelastically.

9 In this version of the model (complete �nancial autarky), stocks in the mutual fund and bonds are not traded across

countries; hence, the equilibrium conditions for stock and bond holdings are xt = xt+1 = 1; Bt = Bt+1 = 0.

12

The �rst-order conditions generate the Euler equations for bonds and stocks:10

C� t = � (1 + rt+1)Et

hC� t+1

i; (1)

evt = �(1� �)Et�Ct+1Ct

�� (edt+1 + evt+1): (2)

Households in North consume the consumption basket Ct, which includes varieties of �nal goods

produced by North �rms (! 2 NNt ) using a mix of domestic and foreign inputs, through either

domestic or o¤shore production under vertical FDI. In addition, the consumption basket Ct also

includes �nal good varieties produced by South �rms (! 2 SNt ) and which are either imported or

produced in North under horizontal FDI:

Ct =

266666664zV;tZzmin

yD;t(!)��1� d!

| {z }Domestic production

+

1ZzV;t

yV;t(!)��1� d!

| {z }O¤shore production

+

1Zz�H;t

y�H;t(!)��1� d!

| {z }SNt

377777775

���1

;

where � > 1 is the symmetric elasticity of substitution across goods. Let pt(!) denote the price

of each variety of �nal goods yt(!); then the price index of the home consumption basket is Pt =�Rpt(!)

1��d!� 11�� for ! 2 NNt [ SNt , and the demand for each variety of �nal goods ! is yt(!) =

(pt(!)=Pt)�� Ct. Using the home consumption basket Ct as the numeraire good and de�ning the real

price of variety ! as �t(!) = pt(!)=Pt, I write the consumption-based price index as:

1 =

�Z�t(!)

1��d!

� 11��

; ! 2 NNt [ SNt : (3)

3.2 Firms serving the domestic market: domestic production vs. vertical FDI

The North economy includes a continuum of monopolistically competitive �rms that produce �nal

goods. Each �rm produces a di¤erent variety of �nal goods - therefore, the �rm-speci�c labor pro-

ductivity z also serves as an index for the existing varieties on the support interval [zmin;1). As

shown in the expression for the North consumption basket Ct above, �rms with productivity below

10As described below, �rms - including the new entrants - encounter a random exit shock with probability � at the end

of every period. Moreover, new entrants at time t do not produce and do not obtain any pro�t until period t+ 1. Thus,

the law of motion for the number of producing �rms is Nt+1 = (1� �)(Nt+NE;t). The representative household at time

t purchases stocks in a mutual fund of Nt +NE;t �rms, among which some will experience the random exit shock at the

end of period t.

13

the endogenous cuto¤ zV;t produce their varieties of �nal goods using exclusively domestic inputs,

whereas those above the cuto¤ use foreign inputs.

Thus, each �rm producing for the domestic market can choose one of two possible strategies:

(1) Under domestic production, the home �rm with idiosyncratic labor productivity z employs an

amount of labor lt and obtains output:

yD;t(z) = Ztzlt;

where Zt is the aggregate productivity of labor in North;

(2) Under vertical FDI, the same �rm uses a combination of domestic and foreign inputs produced

with labor from North and South, lt and l�t ; respectively. Following Antras and Helpman (2004),

output of every variety z of �nal goods is a Cobb-Douglas function of the inputs:

yV;t(z) =

�Ztzlt�

�� �Z�t zl�t1� �

�1��:

Under di¤erent calibrations of the model, I vary the share of foreign inputs contributed by o¤shore

a¢ liates. The smaller � is, the more intensive the production process is in inputs produced by o¤shore

a¢ liates. Setting � = 0 implies that vertical FDI �rms use exclusively foreign inputs in the production

of �nal good varieties.11 At the other extreme, � = 1 shuts down o¤shore production under vertical

FDI as a transmission channel for aggregate shocks.

Given the demand for �nal goods produced domestically, yD;t(z) = �D;t(z)��Ct, and the demand

for �nal goods produced under vertical FDI, yV;t(z) = �V;t(z)��Ct, �rms set prices by solving the

following pro�t-maximizing problem:12

maxf�D;t(z)g

dD(z) = �D;t(z)yD;t(z)�wtZtz

yD;t(z);

maxf�V;t(z)g

dV (z) = �V;t(z)yV;t(z)��wtZtz

����w�tQtZ�t z

�1��yV;t(z)� fV

�wtZt

���w�tQtZ�t

�1��:

The cost of producing one unit of output either domestically or under vertical FDI varies according

to di¤erences in labor productivity z across �rms and di¤erences in the cost of e¤ective labor across

countries. Given the domestic and o¤shore real wages wt and w�t , the unit cost of producing interme-

11Using l�Hôpital�s rule, lim�!0

�1�

��= lim

�!1�1=� = lim

�!1e(1=�) ln � = e

lim�!1

[(1=�) ln �]= e

lim�!1

(1=�)= e0 = 1:

12Appendix 1 shows the derivations of the aggregate price index and the demand functions of intermediate goods

produced domestically and o¤shore.

14

diate goods domestically is wtZtz, and the unit cost of producing o¤shore is

�wtZtz

�� ��w�tQtZ�t z

�1��.13 The

real exchange rate Qt =P �t "tPt

is de�ned as the ratio of consumer price indexes expressed in North and

South in the same currency, where "t is the nominal exchange rate.

In addition to the unit production costs, �rms producing under vertical FDI also incur the period-

by-period �xed o¤shoring costs equal to a mix of fV units of e¤ective labor from North and South,

costs which re�ect headquarter operations as well as the building and maintenance of the production

facility o¤shore.14 They also face an iceberg trade cost (� > 1) which applies to the value of inputs

produced by o¤shore a¢ liates and incorporated in the �nal good varieties consumed in North.15 The

iceberg trade costs represent transportation costs, insurance and fees, as well as language barriers and

di¤erences in the legal systems, as discussed in Anderson and Wincoop (2004).

Hence, the optimal prices per unit of output produced either domestically or o¤shore are:16

�D;t(z) =�

� � 1wtZtz

;

�V;t(z) =�

� � 1

�wtZtz

����w�tQtZ�t z

�1��:

The resulting pro�ts from domestic and o¤shore production, both expressed in units of the North

consumption basket Ct, are:17

dD;t(z) =1

��D;t(z)

1��Ct;

dV;t(z) =1

��V;t(z)

1��Ct � fV�wtZt

���w�tQtZ�t

�1��:

To summarize the above, the structure of production costs depends on the �rm-speci�c productivity

factor z, the cost of e¤ective labor in North and South (wtZt andw�tQtZ�t

, respectively), the �xed costs

13The latter is translated into units of the consumption basket in North. The real wage wt = Wt=Pt in North is

expressed in units of the domestic consumption basket; the o¤shore real wage w�t = W�t =P

�t is expressed in units of the

consumption basket in South.14The mix of fV units of e¤ective labor is equivalent to fV

�wtZt

�� �w�tQtZ�t

�1��units of the consumption basket in

North.15The assumption that the iceberg trade costs apply only to the inputs produced o¤shore is a normalization. Domestic

inputs that undergo o¤shore processing and then are re-imported in the country of origin, although exempt from tari¤s,

are still subject to transportation and insurance costs. However, I assume that the iceberg trade costs applying do

domestic intermediates are less than those applying to imported o¤shore intermediates.16Appendix 2 shows the derivation of the optimal price formulas for intermediate goods produced by the

monopolistically-competitive �rms domestically and o¤shore.17Given the demand yD(z) = �D(z)��C and optimal price �D(z) = �

��1wZz

charged for the intermediate good variety

z produced domestically, the pro�t formula is derived as dD(z) =�

���1 � 1

�wZzyD(z) =

1��D(z)yD(z): The pro�t from

producing through vertical FDI is derived in a similar fashion.

15

of o¤shoring fV as well as the iceberg trade costs � . Firms that produce intermediate goods o¤shore

bene�t from the relatively lower cost of e¤ective labor, but incur �xed o¤shoring costs every period

and iceberg trade costs when shipping intermediates for �nal assembly back in North. Hence, when

deciding the location of production facilities, �rms compare the pro�ts they would obtain from domestic

production dD;t(z) to those obtained from producing intermediates o¤shore dV;t(z), a feature of the

model which generates endogenous o¤shoring.

Every period t, �rms with idiosyncratic productivity z above a certain cuto¤ zV;t �nd it pro�table

to locate the production of intermediate inputs o¤shore. Given the di¤erence between the costs of

e¤ective labor in North and South, the relatively more productive �rms (z > zV;t) obtain pro�ts that

are large enough to cover the �xed o¤shoring cost fV and the iceberg trade cost � of shipping the

inputs back home. On the contrary, �rms with relatively low labor productivity (z < zV;t) are unable

to take advantage of the lower cost of e¤ective labor in South, since their pro�ts would fall short of

covering the �xed and the iceberg trade costs.

As a particular case, the �rm with labor productivity equal to the cuto¤ zV;t is indi¤erent between

producing domestically or o¤shore: After subtracting the o¤shoring and trade costs, the pro�ts it

would obtain from domestic and o¤shore production are equal. The equality of pro�ts allows to solve

for the endogenous productivity cuto¤ zV;t that governs the location decision of production under

vertical FDI:

zV;t = fz j dD;t(zV;t) = dV;t(zV;t)g :

Existence of equilibrium. Below I show that the existence of equilibrium productivity cuto¤ zV;t

requires cross-country asymmetries with respect to the cost of e¤ective labor. In other words, �rms

need the incentive of a lower cost of e¤ective labor in order to relocate production o¤shore through

vertical FDI.

Setting � = 0; I re-write the pro�t functions for domestic and o¤shore production through vertical

FDI, respectively, as dD;t(z) = Bt

�wtZt

�1��z��1 and dV;t(z) = Bt

��w�tQtZ�t

�1��z��1; where � > 1 and

Bt � 1�

��1��

�1��Ct is a measure of market size: Figure 3(b) plots pro�ts as functions of z��1. The

vertical intercepts are the annualized value of the sunk entry cost (for dD;t(z)) and the annualized

value of the sunk entry cost plus the �xed o¤shoring costs (for dV;t(z)). The existence of equilibrium

cuto¤ zV;t requires:

(1) dV;t(z) to be steeper than dD;t(z); and

(2) zmin < zV;t must hold every period:

16

Figure 3(b). Existence of equilibrium productivity cuto¤ zV ; t

The �rst condition implies that e¤ective wages in South must be low enough (TOLt < 1) in order

to o¤set the iceberg trade costs (� > 1):

�w�tQtZ�t

<wtZtz

() �TOLt < 1:

The second condition, discussed in Appendix 4, implies:

dD;t(zmin) < �fEwtZt+ fV

w�tQtZ�t

;

where � = 1��(1��)�(1��) : The inequality shows that the pro�t obtained from domestic production by the

�rm with the minimum productivity zmin is less than the per-period value of the sunk entry cost plus

the �xed o¤shoring cost. In other words, the �rm that obtains zero pro�t from domestic production

would make negative pro�ts if it engages in o¤shore production, i.e. the �rm with productivity zmin

does not produce in either country.

3.3 Firms serving the foreign market: exports vs. horizontal FDI

Firms in the North economy also have the option to serve the foreign market through one of two

possible strategies:

(1) Under endogenous exports, �rms in North use only domestic inputs in the production of �nal

goods which they export to the South market;

17

(2) Under horizontal FDI, �rms use a combination of domestic and foreign inputs and sell the

resulting �nal goods to consumers in the South market. The production function for both strategies

can be summarized as:

yH;t(z) =

�Ztzlt�

�� �Z�t zl�t1� �

�1��;

where a smaller � corresponds to a larger content of inputs produced by o¤shore a¢ liates and that is

incorporated in the �nal goods sold to consumers in the host market. Thus, my model of o¤shoring

nests both Ghironi and Melitz (2005) and Contessi (2006). On one hand, setting � = 1 implies that

�rms serve the South market exclusively through endogenous exports, as in Ghironi and Melitz (2005).

On the other hand, setting � = 0 implies that �rms produce exclusively through their o¤shore a¢ liates

in order to serve to South market, like in Contessi (2006).

Both exports and horizontal FDI imply a �xed cost equal to a mix of fH units of e¤ective labor

in North and South. In addition, exports also require the iceberg trade cost �� which applies to the

imported content of �nal goods consumed in South.

Unlike o¤shore production through vertical FDI, which is motivated by lower labor costs, o¤shore

production through horizontal FDI is motivated by market access, since it allows �rms to avoid the

trade costs associated with inputs that are produced in the host economy rather than imported. Under

production through horizontal FDI, �rms continue to produce in their country of origin (in order to

serve the home market) at the same time with producing the same variety of �nal goods at the o¤shore

location (in order to serve the market of the host economy), .

Following the approach described above, pro�t-maximizing �rms from North serving the southern

market set optimal prices for output that is either exported or produced under horizontal FDI:

�H;t(z) =�

� � 1

���wtQ

�1t

Ztz

�� �w�tZ�t z

�1��:

Every period t, �rms with idiosyncratic productivity z above a certain cuto¤ zH;t �nd it pro�table

to serve the foreign market (South) - at the same time with serving their domestic market (North)

- through either exports or horizontal FDI. Their pro�ts are large enough to cover the �xed cost of

serving the foreign market fH . In addition to that, �rms save the iceberg trade cost when using inputs

produced locally through horizontal FDI in the country which they serve (i.e. � = 0). The pro�t

function is:

dH;t(z) =1

��H;t(z)

1��C�tQt � fH�wtZt

�� �w�tQtZ�t

�1��:

The option to serve the foreign market (in addition to serving the domestic one) is endogenous:

18

the productivity cuto¤ zH;t is time-variant:

zH;t = inf fz j dH;t(zV;t) > 0g :

3.4 Firm averages

Firms serving the domestic market: Following the approach in Melitz (2003), I de�ne two av-

erage productivity levels that correspond to: (1) �rms producing �nal goods using entirely domestic

components (average productivity level ezD;t), and (2) �rms producing �nal goods through verticalFDI using inputs produced both domestically and through o¤shore a¢ liates (average productivity

level ezV;t). Then I solve the model in terms of two representative North �rms: (1) the average home�rm producing domestically, and (2) the average home �rm producing o¤shore through vertical FDI.

Figure 4. Average labor productivities for �rms serving the

domestic market: domestic production (ezD; t) and o¤shoreproduction through vertical FDI (ezV ; t)

As shown in Figure 4, in every period t there are ND;t home �rms producing domestically with

average �rm-speci�c labor productivity ezD;t, and NV;t �rms producing o¤shore through vertical FDI

with average �rm-speci�c labor productivity ezV;t.18 Since the �rm-speci�c labor productivities z

are random draws from a common distribution G(z) with density g(z) and support on the interval

18One should distinguish between ezV;t and zV;t. The former is the average �rm-speci�c labor productivity of �rmsproducing o¤shore, whereas the latter is the labor productivity cuto¤ on support interval [zmin;1); all �rms above cuto¤

zV;t establish production facilities o¤shore.

19

[zmin;1), I write the domestic and o¤shore average productivities respectively as:

ezD;t =24 1

G(zV;t)

zV;tZzmin

z��1g(z)dz

351��1

;

ezV;t =264 1

1�G(zV;t)

1ZzV;t

z��1g(z)dz

3751��1

:

After de�ning the two average productivities, I re-write the model in terms of two representative

�rms, one producing domestically and the other producing o¤shore through vertical FDI. I start by

re-writting the prices charged by each of the two representative �rms as:

e�D;t = �

� � 1wt

Z etzD;t ; (4)

e�V;t = �

� � 1

�wt

ZtezV;t���

�w�tQtZ�t ezV;t

�1��: (5)

The corresponding pro�ts are:

edD;t = 1

�(e�D;t)1�� Ct; (6)

edV;t = 1

�(e�V;t)1�� Ct � fV �wt

Zt

���w�tQtZ�t

�1��: (7)

Due to the wage asymmetry across countries, �rms originating in South do not engage in vertical

FDI, i.e. they do not use intermediates produced o¤shore - in the high-wage North economy - when

serving their domestic market. Their average price and pro�ts are:

e��D;t = �

� � 1w�t

Z�t ezD;t� ; (8)

ed�D;t = 1

�

�e��D;t�1�� C�t : (9)

Firms serving the foreign market: As in Ghironi and Melitz (2005), the pro�t-maximizing

representative �rms in North and South that serve the foreign market - through either exports or

horizontal FDI - charge the following prices:

e�H;t = �

� � 1

���wtQ

�1t

ZtezH;t�� �

w�tZ�t ezH;t

�1��; (10)

e��H;t = �

� � 1

�w�tQtZ�t ez�H;t

!�� wt

Ztez�H;t!1���

: (11)

20

The corresponding average pro�ts are:

edH;t = 1

�(e�H;t)1�� C�tQt � fH �wtZt

�� �w�tQtZ�t

�1��; (12)

ed�H;t = 1

�

�e��H;t�1�� CtQ�1t � f�H�w�tZ�t

��� �wtQ�1tZt

�1���: (13)

Price indexes: The consumption price index in North include the average price of the �nal good

produced domestically, the average price of the good produced by domestic �rms under vertical FDI,

and the average price of the good supplied by South �rms through either exports or local production

under horizontal FDI, respectively:

1 = ND;t (e�D;t)1�� +NV;t (e�V;t)1�� +N�H;t

�e��H;t�1�� (14)

In contrast to �rms originating in the North economy, none of the �rms in South has an incentive

to establish o¤shore production facilities in North, where labor is relatively more expensive. Therefore,

none of the �rms in the South economy produces intermediates through o¤shore a¢ liates under vertical

FDI. The consumption price index in South includes the average price of goods supplied by southern

�rms through domestic production and the price of goods provided by northern �rms through either

exports or horizontal FDI:

1 = N�D;t

�e��D;t�1�� +NH;t (e�H;t)1�� (15)

Total pro�ts: The total pro�t of the average �rm originating in North includes pro�ts from

domestic production, vertical FDI and horizontal FDI:

Ntedt = ND;t

edD;t +NV;tedV;t +NH;t

edH;t: (16)

In contrast to northern �rms, none of the �rms in South engages in vertical FDI:

N�D;tedt = N�

D;tedD;t +N�

H;ted�H;t: (17)

3.5 Pareto-distributed �rm productivity

Firms serving the domestic market: I solve the model in terms of two representative �rms -

producing either domestically or through vertical FDI, respectively - under the assumption that �rm-

speci�c labor productivity z is Pareto distributed over the support interval [zmin;1), with p.d.f.

g(z) = kzmin=zk+1 and c.d.f. G(z) = 1 � (zmin=z)k as in Melitz (2003). Parameter k re�ects the

dispersion of the productivity draws: A relatively larger k implies a smaller dispersion and a higher

concentration of productivities z towards the lower productivity bound zmin.

21

Under the Pareto assumption, I re-write the average productivity levels of the home �rms producing

domestically and o¤shore as follows:19

ezD;t = �zminzV;t

24zk�(��1)V;t � zk�(��1)min

zkV;t � zkmin

35 1��1

; (18)

ezV;t = �zV;t; (19)

where � �h

kk�(��1)

i 1��1, zV;t = zmin(Nt=NV;t)

(1=k) and k > � � 1.20, 21

Since �rms originating in South serve their domestic market exclusively through domestic produc-

tion, the average productivity of southern �rms is constant:

ez�D;t = �z�min: (20)

Finally, using that the �rm at the productivity cuto¤ zV;t is indi¤erent about the location of its

production plants - since it obtains equal pro�ts from domestic and o¤shore production, dD;t(zV;t) =

dV;t(zV;t), I derive the link between the pro�ts of the two average �rms as follows:22

edV;t = k

k � (� � 1)

�zV;tezD;t

���1 edD;t + � � 1k � (� � 1)fV

�wtZt

���w�tQtZ�t

�1��: (21)

Firms serving the foreign market: Under the assumption of Pareto-distributed productivity, I

re-write the average productivity levels of the �rms serving the foreign market through either exports

or horizontal FDI as follows:

ezH;t = �zmin

�Nt

NH;t

�1=k; (22)

ez�H;t = �z�min

N�D;t

N�H;t

!1=k: (23)

Using that the �rm at the productivity cuto¤ zH;t obtains zero pro�ts from serving the foreign

market through either exports or horizontal FDI, dH;t(zH;t) = 0, I derive the following pro�t cuto¤s:

edH;t = � � 1k � (� � 1)fH

�wtZt

�� �w�tQtZ�t

�1��; (24)

ed�H;t = � � 1k � (� � 1)f

�H

�w�tZ�t

��� �wtQ�1tZt

�1���: (25)

19See Appendix 3.20 I derive the shares of home �rms producing domestically and o¤shore, respectively, as ND;t

Nt= G (zV;t) and

NV;tNt

=

1 � G (zV;t), where the total number of �rms in every period is Nt = ND;t + NV;t. Then I use the expression for the

Pareto c.d.f. G(zV;t) to derive the productivity cuto¤ zV;t from the share of o¤shoring �rms, zV;t = zmin(Nt=NV;t)(1=k):21The condition that k > � � 1 ensures that the variance of �rm size is �nite, since the average productivities of the

domestic-based and outsourcing �rms are, respectively, ezV;t = h kk�(��1)

i 1��1

zV;t.22See Appendix 4.

22

3.6 Firm entry and exit

Firm entry and exit take place every period. Following Gironi and Melitz (2005), �rm entry requires

a sunk entry cost � i.e. which re�ects the cost of starting a business � equal to fE units of home

e¤ective labor.23 Potential entrants do not become aware of their idiosyncratic labor productivity z

until after they enter the market. After paying the sunk entry cost, each �rm is randomly assigned

an idiosyncratic labor productivity z drawn independently from a common distribution G(z) with

support on the interval [zmin;1), which the �rm keeps for the entire duration of its life.

Potential entrants are forward looking and correctly anticipate their expected pro�ts edt as wellas the probability � of receiving an exit-inducing shock every period. Hence, potential entrants do

anticipate their expected post-entry value given by the present discounted value of their expected

stream of pro�ts: evt = Et

1Xs=t+1

[�(1� �)]s�t�CsCt

�� eds:Therefore, �rm entry takes place until the value of the average �rm equals the sunk entry cost:

evt = fEwtZt: (26)

Each of the NE;t �rms entering at time t does not produce until period t+ 1, which introduces a

one period time-to-build into the model. Irrespective of their productivity, �rms - including the new

entrants - are also subject to a random exit shock that occurs with probability � at the end of every

period after production has taken place. Thus, the law of motion for the number of producing �rms

is:

Nt+1 = (1� �)(Nt +NE;t): (27)

3.7 Aggregate demand and value added

In order to quantify the business cycle co-movements generated by my model, I use value added as a

measure of the aggregate incomes of the two countries. Thus, the model takes into account the fact

that inputs produced in South and used in the production of the �nal good consumed in North are

part of the value added of the South economy, and therefore must not be double-counted in the value

added of North. Under �nancial autarky on the market for both bonds and stocks (i.e. Bt+1 = Bt = 0

and xt+1 = xt = 1 in equilibrium), value added is equal to the wage bill plus the total amount of stock

23Or fEwt=Zt units of the consumption basket in North.

23

dividends derived by households in each country:

Yt = wtL+Ntedt:

Finally, aggregate accounting under �nancial autarky implies that households spend their income

from labor and stock holdings on consumption and investment in new �rms:

Ct +NE;tevt = wtL+Ntedt: (28)

3.8 Balanced trade

Under �nancial autarky, the real exchange rate Qt is pinned down by the balanced current account

condition for North (which re�ects the corresponding balance for South):

(1� �)NV;t (e�V;t)1�� Ct| {z }(1) VFDI imports

+ ��N�H;t

�e��H;t�1�� Ct| {z }(2) HFDI imports

+ (1� ��)N�H;ted�H;tQt| {z }

(3) HFDI pro�ts*

= �NH;t (e�H;t)1�� C�tQt| {z }(4) HFDI exports

+ (1� �)NV;tedV;t| {z }

(5) VFDI pro�ts

+ (1� �)NH;tedH;t| {z }

(6) HFDI pro�ts

:

The equations shows that the sum of (1) North�s imports of inputs manufactured by o¤shore a¢ liates

under vertical FDI, (2) North�s imports of inputs used by South �rms for local production under

horizontal FDI, and (3) the repatriated pro�ts of South �rms active in North through horizontal FDI

must equal the sum of (4) North�s exports of inputs used by North �rms serving the South market

through horizontal FDI, (5) and (6), i.e. the repatriated pro�ts of North �rms obtained from vertical

and horizontal FDI, respectively.

3.9 Model summary and asymmetries between North and South

As shown in Table 2, the model for the North economy is summarized by 16 equations in 16 endogenous

variables: Nt, ND;t, NV;t, NH;t, NE;t, edt, edD;t, edV;t, edH;t, ezD;t, ezV;t, ezH;t, evt, rt, wt and Ct, plus theequation describing aggregate productivity Zt. Firms in the South economy produce domestically

for the domestic market, and serve the foreign market through either exports or local production

under horizontal FDI, but have no incentive to engage in vertical FDI since e¤ective labor in North

is relatively more expensive. Therefore, the South economy is described by only 11 equations in 11

endogenous variables, i.e. there are no counterparts for Nt, NV;t, edV;t, ezD;t and ezV;t. In particular,since no �rms from South produce o¤shore, the average labor productivity of the Southern �rms is a

constant, ez�D = �z�min, where � =h

kk�(��1)

i 1��1

: Finally, I use the pricing and pro�t formulas de�ned

by equations (6) - (15) and summarized in Table 3.

24

Table 2. Model summary

Euler equation, bonds C� t = � (1 + rt+1)Et

hC� t+1

iC�� t = �

�1 + r�t+1

�Et

hC�� t+1

iEuler equation, stocks evt = �(1� �)Et

�Ct+1Ct

�� (edt+1 + evt+1)ev�t = �(1� �)Et

�C�t+1C�t

�� (ed�D;t+1 + ev�t+1)

Free entry evt = fEwtZtev�t = f�Ew

�t

Z�t

Rule of motion, # �rms Nt+1 = (1� �)(Nt +NE;t)

N�D;t+1 = (1� �)(N�

D;t +N�E;t)

Aggregate accounting Ct +NE;tevt = wtL+Ntedt

C�t +N�E;tev�t = w�tL

� +N�D;ted�D;t

Consumption price index 1 = ND;t (e�D;t)1�� +NV;t (e�V;t)1�� +N�H;t

�e��H;t�1��1 = N�

D;t

�e��D;t�1�� +NH;t (e�H;t)1��Total pro�ts Nt

edt = ND;tedD;t +NV;t

edV;t +NH;tedH;t

N�D;tedt = N�

D;tedD;t +N�

H;ted�H;t

Number of �rms (Home) Nt = ND;t +NV;t

VFDI pro�ts link (Home) edV;t = kk�(��1)

�zV;tezD;t���1 edD;t + ��1

k�(��1)fV�wtZt

�� �w�tQtZ�t

�1��HFDI pro�ts link edH;t = ��1

k�(��1)fH�wtZt

�� �w�tQtZ�t

�1��ed�H;t = ��1

k�(��1)f�H

�w�tZ�t

��� �wtQ�1tZt

�1���Dom. productivity (Home) ezD;t = �zminzV;t

�zk�(��1)V;t �zk�(��1)min

zkV;t�zkmin

� 1��1

VFDI productivity (Home) ezV;t = �zmin

�NtNV;t

�1=kHFDI productivity ezH;t = �zmin

�NtNH;t

�1=kez�H;t = �z�min

�N�D;t

N�H;t

�1=kBalanced trade (1� �)NV;t (e�V;t)1�� Ct| {z }

VFDI imports

+ ��N�H;t

�e��H;t�1�� Ct| {z }HFDI imports

+ (1� ��)N�H;ted�H;tQt| {z }

HFDI pro�ts*

=

= �NH;t (e�H;t)1�� C�tQt| {z }Exports

+ (1� �)NV;tedV;t| {z }

VFDI pro�ts

+ (1� �)NH;tedH;t| {z }

HFDI pro�ts

25

Table 3. Pricing and pro�t formulas

Production Prices Pro�ts

Domestic e�D;t = ���1

wtZ etzD;t edD;t = 1

� (e�D;t)1�� Cte��D;t = ���1

w�tZ�t ezD;t� ed�D;t = 1

�

�e��D;t�1�� C�tVFDI e�V;t = �

��1

�wt

ZtezV;t�� �

�w�tQtZ�t ezV;t

�1�� edV;t = 1� (e�V;t)1�� Ct � fV �wtZt�� �w�tQtZ�t

�1��HFDI e�H;t = �

��1

���

wtQ�1t

ZtezH;t�� �

w�tZ�t ezH;t

�1�� edH;t = 1� (e�H;t)1�� C�tQt � fH �wtZt�� �w�tQtZ�t

�1��e��H;t = �

��1

��w�tQtZ�t ez�H;t

��� �wt

Ztez�H;t�1��� ed�H;t = 1

�

�e��H;t�1�� CtQ�1t � f�H�w�tZ�t

��� �wtQ�1tZt

�1���4 Results

4.1 Steady state

Intuitively, �rms from North have an incentive to relocate input production to South only if the

di¤erence in the cost of e¤ective labor is large enough to cover both the �xed costs of o¤shoring and

the iceberg trade costs of shipping the inputs produced o¤shore back home. In line with this intuition,

in this section I demonstrate analytically the need for an asymmetric steady state with respect to

wages.

Under � = 0; the steady state solutions for all variables of my model can be obtained analytically

after solving for the labor productivity of the average �rm that produces intermediate goods o¤shore,ezV . The solution is described by the hyperbola:�1ez1��V + �2ez�kV = �3;

with parameters �2 > 0 and �3 > 0 for all plausible calibrations, k > 0 and � > 1.

In Figure 5, I plot the hyperbola described above for the cases of �1 > 0 (squares) and �1 < 0

(diamonds).24 The �gure shows that the steady-state solution for ezV lies within the support interval[zmin;1), with zmin � 1, if and only if:

�1 = z��1min

�k

k � (� � 1)

�2 (�TOL)��1

1� (�TOL)��1> 0;

where the terms of labor, TOL = w�=Z�

w=Z Q, represent a measure of the relative cost of e¤ective labor

24ezV is on the horizontal axis, �3 is on the vertical axis.

26

between the two countries.

Figure 5. Steady state solution for ezVThe resulting condition, �TOL < 1, can be satis�ed in two di¤erent ways:

(a) TOL = 1 and � < 1;

(b) � > 1 and TOL < 1.

With method (a), the model would have a symmetric steady state in which real e¤ective wages in

North and South are identical, i.e. TOL = 1. In absence of cross-country di¤erences in wages, �rms

would be motivated to relocate production o¤shore by the existence of an �iceberg subsidy,� � < 1;

that reduces the marginal cost of intermediate goods manufactured o¤shore and shipped home.

With method (b), the model would have an asymmetric steady state in which the cost of e¤ective

labor is relatively higher in North than in South, i.e. TOL < 1. The cross-country di¤erence in the

cost of e¤ective labor acts as an incentive for �rms originating in North to relocate production to

South. The di¤erence in the cost of e¤ective labor must be large enough to cover the �xed o¤shoring

cost and the iceberg trade cost � > 1.

I follow method (b): I impose f�E > fE , i.e. the cost of starting a business is higher in South than

in North, to obtain an asymmetric steady of e¤ective wages. The calibration is supported empirically

by the data on cross-country di¤erences in the cost of starting a business described in Table 4.

27

4.2 Calibration

I set the cost of starting a business to be 2 times larger in South than in North (f�E = 2fE), where I

assume fE = 1 without loss of generality. As shown in Table 4, this calibration re�ects the considerable

variation in the cost of starting a business across countries: the monetary cost is 3 times higher in

Mexico than in the U.S. or Canada; it is 6 times higher in Hungary than in the U.K.

Table 4. Firm entry costs, selected economies

(Source: World Bank, Doing Business, 2007)

The asymetric sunk entry costs - together with the �xed o¤shoring cost fV = 0:002, the trade

iceberg cost � = 1:3, and the �xed cost of serving the foreign market fH = 0:006 - generate TOL = 0:74:

In steady state, the cost of e¤ective labor in South �de�ned as real wage over aggregate productivity

�is 74 percent the value of the corresponding measure in North.

Interpreting periods as quarters, I set the subjective rate of time discount � = 0:99 and the

coe¢ cient of relative risk aversion � = 2. Following Ghironi and Melitz (2005), I set the probability of

�rm exit � = 0:025 to match annual 10 percent job destruction in the U.S. Following Bernard, Eaton,

Jensen and Kortum (2003), I also set the intra-temporal elasticity of substitution � = 3:8, calibrated

after U.S. plant and macro trade data.

Finally, I set the Pareto distribution coe¢ cient k = 5:5 and the lower bound of the support interval

of idiosyncratic �rm productivities zmin = 1 without loss of generality. In future work, I aim to �ne-

tune the calibration of k, fV and fH in order to match empirical patterns such as the fraction of �rms

that use imported inputs in production (14 percent, as documented by Bernard, Jensen, Redding and

28

Schott, 2007), and the fraction of �rms that serve the foreign market (21 percent thorugh exports, as

documented by Bernard, Eaton, Jensen and Kortum, 2003).

4.3 Impulse responses

I log-linearize the model around the steady state and compute the impulse responses to a transitory

1 percent increase in aggregate productivity in the North economy assuming logZt+1 = � logZt + ut

with � = 0:9.

Figure 6. Endogenous o¤shoring with Ghironi and Melitz (2005) as benchmark: impulse responses to a

transitory 1 percent technology shock

Figure 6 shows the impulse responses of the model with both exports and endogenous o¤shoring

29

through vertical FDI (solid line, � = 0; � = 1), and contrasts them with the impulse responses of the

benchmark model with endogenous exports a la Ghironi and Melitz (2005) (dotted line, � = 1; � = 1).

For each variable, the horizontal axis illustrates quarters after the initial shock, and the vertical axis

shows the percentage deviations from the original steady state in each quarter.

On impact, the 1 percent increase in aggregate labor productivity in North generates an equal

increase in the real wage wt. The increase in demand for inputs produced in both North and South

also generates a jump of real wages in South w�t and of the terms of labor TOLt: Since the increase of

aggregate productivity in North is not replicated in South, on impact there is excess demand for units

of e¤ective labor in South. Therefore, the rise of South wages above aggregate productivity deters

some of the North �rms from producing o¤shore, due to: (1) the increase in the unit cost of producing

o¤shore, and (2) the increase in the �xed cost of relocation, both sensitive to the cost of e¤ective labor

in South. Hence, the number of North �rms that operate o¤shore a¢ liates in South (NV;t) drops on

impact.

Over the business cycle, the increase in aggregate labor productivity and higher expected pro�ts

in North stimulate �rm entry (Nt increases). Firm entry generates higher labor demand in North, and

thus leads to an appreciation of the cost of e¤ective labor in North relative to South (as shown by

the decline of TOLt below the initial steady state). Following the appreciation of the terms of labor,

more �rms from North have an incentive to relocate production to South. Hence, after the initial drop

triggered by the spike in South real wages, the number of �rms from North that relocate production

to South (NV;t) rises above the original steady state in the medium run.

Thus, the initial jump of the South real wage, caused by the higher demand for existing o¤shore

varieties on impact (i.e. o¤shoring at the intensive margin), is followed by an additional increase once

more �rms from North start re-locating production to South over the business cycle (i.e. o¤shoring at

the extensive margin).

In comparison with Ghironi and Melitz (2005), the model with vertical FDI generates a larger

increase in real wages in South, which is generated by the relocation of production by �rms originating

in North.25 Hence, higher demand for labor in North due to �rm entry and higher demand for labor in

South due to o¤shoring through vertical FDI generate a co-movement of wages and aggregate incomes

which is notably larger than in Ghironi and Melitz (2005).

25Over the business cycle, the terms of labor in the model with vertical FDI appreciate by less (i.e. TOL decreases by

less) than in Ghironi and Melitz (2005) due to the stronger increase in real wages in South caused by o¤shoring.

30

4.4 Second moments

I simulate the model and calculate the cross-country correlations of national income and consumption

in North and South using the shocks to aggregate productivities Zt and Z�t as the only sources of

business cycle �uctuations, as in Backus, Kehoe, and Kydland (1992):

24 Zt

Z�t

35 =24 �Z �ZZ�

�Z�Z �Z�

3524 Zt�1

Z�t�1

35+24 �Zt

�Z�

t

35 ;where �Z = �Z� = 0:906 and �ZZ� = �Z�Z = 0:088. The variances of innovations are 0.73% and the

covariance is 0.19%.

Before computing the co-movement of national incomes and consumption across countries, I de�ate

home and foreign variables by the average price indices in each country in order to eliminate the variety

e¤ect generated by di¤erences in the number of varieties of �nal goods available in each country. For

instance, I de�ate the value added of North as YR;t = PtYt= ePt, where Pt = N1��1t

ePt. As discussed inGhironi and Melitz (2005), the empirical price de�ators are best represented by the average price indexePt rather than the welfare-based price index Pt, as the latter includes the welfare e¤ect generated bythe availability of Nt product varieties.

Table 5 shows the e¤ect of vertical FDI on the cross-country co-movement of national incomes and

consumption. I take as benchmarks the model with exogenous exports a la Ghironi and Melitz (2005)

and the model with local production through horizontal FDI a la Contessi (2006), both of which are

nested by my model.

Table 5. Business Cycle Co-movements

Correlations Calibration YR; Y�R CR; C

�R

Domestic production + VFDI + exports � = 0; � = 1 0.50 0.76

Only domestic production + exports (GM 2005) � = 1; � = 1 0.23 0.87

Domestic production + VFDI + HFDI � = 0; � = 0 0.45 0.71

Only domestic production + HFDI (Contessi 2006) � = 1; � = 0 0.28 0.94

Data (BKK 1992) 0.70 0.46

The results show that endogenous o¤shoring through vertical FDI increases the co-movement of

output and decreases the co-movement of consumption relative to the benchmark models with either

endogenous exports or horizontal FDI. Although the rank of correlations is inverted (i.e. the co-

31

movement is still greater for consumption than for national income), the results are closer to the

corresponding empirical correlations reported in Backus, Kehoe, Kydland (1992).

5 Extended model: elastic labor supply

In this section I extend my benchmark model of o¤shoring by introducing elastic labor supply. Then

I explore the ability of the extended model to reproduce the correlation of the maquiladora indicators

with U.S. output measured at various lags and leads. In particular, I am interested in the ability of the

model to reproduce the time pattern of o¤shoring at the intensive and extensive margins, as described

by the data in Figure 2(b) and summarized in Figure 8 (�rst panel).

Figure 8. The co-movement between U.S. manufacturing (at lags and leads) and

Mexico�s maquiladora manufacturing: the intensive vs. extensive margins

The �rst panel in Figure 8 shows the correlation between Mexico�s maquiladora indicators (i.e.

value added, hours worked, and plants) and various lags and leads of U.S. industrial output in manu-

facturing, measured in real terms at quarterly frequency. It shows that, among the three maquiladora

32

indicators, the number of o¤shore plants (i.e. o¤shoring at the extensive margin) has the lowest con-

temporaneous correlation with U.S. output. Whereas the correlation of the number of plants with

contemporaneous U.S. manufacturing output is below 0.5, the correlation with U.S. output lagged by

three quarters exceeds 0.75. Unlike the number of plants, the maquiladora value added and hours

worked respond faster to �uctuations in U.S. manufacturing output (i.e. hours worked respond im-

mediately, and value added responds with a lag of one or two quarters). The pattern suggests that

the relocation of production o¤shore across the business cycle takes place �rst at the intensive margin

(the number of hours worked per �rm reacts �rst), followed by the relocation at the extensive margin

(the number of plants reacts with a lag of three quarters).

In the extended model I introduce elastic labor supply: The representative household consumes

the basket of goods Ct and supplies hours of work Lt in a competitive labor market each period t, at

the going real wage wt. The household maximizes the expected intertemporal utility:

maxfCt; Bt; xtg

"Et

1Xs=t

�s�t

lnCs � �

L1+1= s

1 + 1=

!#;

s:t: Bt+1 + evt (Nt +NE;t)xt+1 + Ct = (1 + rt)Bt + (edt + evt)Ntxt + wtLt;

where � > 0 is the weight of disutility from labor in the period utility function, and � 0 is the

Frisch elasticity of labor supply to wages and the intertemporal elasticity of substitution in labor

supply. Following King, Plosser and Rebello (1988) and the discussions in Campbell (1994) and Bilbie

et al. (2006), log utility for consumption is required to obtain constant steady state labor supply

(balanced growth) in a model with utility additively separable over consumption and hours.

In order to account for elastic labor supply in the model, I add the �rst order condition for hours

worked in North and South to the equations in Table 2. I also adjust the aggregate accounting

equations to allow for variation in hours worked, as shown in Table 6.

Table 6. Extended model, elastic labor supply

Euler equation, labor � (Lt)1 = wtC

�1t

� (Lt)1 � = w�tC

��1t

Aggregate accounting Ct +NE;tevt = wtLt +Ntedt

C�t +N�E;tev�t = w�tL

�t +N

�D;ted�D;t

In my model, the value added by �rms originating in North that produce through vertical FDI in

South is the theoretical counterpart of the empirical measure of value added in Mexico�s maquiladora

33

sector:

V At = (1� �)NV;t (e�V;t)1�� Ct:I also de�ate value added by the average CPI in the North economy in order to eliminate the variety

e¤ect, i.e. V AR;t = Pt (V At) = ePt, where Pt = N1

1��t

ePt.The theoretical counterpart of the number of hours worked in Mexico�s maquiladora sector is L�V;t:

The variable represents hours worked by the South workers employed in the NV;t o¤shore plants of

�rms originating in North that produce under vertical FDI in the South economy. Their labor is

associated with both �xed and marginal costs activities:

L�V;t = NV;t

26664 fV

(Zt)� (Z�t )

1��| {z }Fixed cost labor

+(1� �) (e�V;t)�� Ct

Z�t ezV;t| {z }Variable cost labor

37775 :I calibrate the extended model to allow for endogenous o¤shoring through vertical FDI (� = 0)

while nesting the model with endogenous exports (� = 1) a la Ghironi and Melitz (2005). I derive an

asymmetric steady state using di¤erences in �rm entry regulations (f�E = 4fE) in order to generate

wage di¤erences across countries. As in Bilbie et al. (2006), I set � = 0:924271 in order to obtain

the steady-state level of hours worked being equal to unit irrespective of the value of , i.e. L =n1�

h1� r

�(1+�)

io 1+ . I also set the elasticity of labor supply to wages in North at = 2, and consider

di¤erent values for the elasticity of labor supply in South, i.e. low elasticity when = 2, and high

elasticity when = 10.

Figure 8 contrasts the empirical second moments (�rst panel) with the corresponding analytical

moments. The second panel shows the correlations of maquiladora�s value added, hours worked, and

number of plants generated by the model with inelastic labor supply. Although the model replicates

the rank of contemporaneous correlations between the maquiladora indicators and U.S. output, the

magnitudes and signs are not in line with the empirical moments. In the model, the number of plants

has a negative correlation with contemporaneous U.S. manufacturing output, rather than positive as

in the data. Moreover, the rank of correlations of value added and number of plants with lags and leads

of U.S. output departs from the empirical moments: In the model, the correlations with lagged U.S.

output are too small, and the correlations with lead values of U.S. output are too large (particularly

for o¤shore value added).

The third and fourth panels show the simulated moments generated by the model with elastic

labor supply when the elasticity of labor supply to wage in the South economy is either low ( = 2) or

34

high ( = 10) respectively. In particular, the model with high elasticity of labor supply in the South

is partially successful in reproducing the empirical moments of o¤shoring. The number of o¤shore

plants (i.e. o¤shoring at the extensive margin) responds to �uctuations in U.S. manufacturing output

with a lag of three quarters, a result which is in line with the empirical moments. The magnitudes are

also respected, i.e. the correlation of the number of plants with U.S. output lagged by three quarters

is 0.6 in the model and 0.75 in the data.

However, the model with elastic labor supply does not reproduce the immediate response of hours

worked shown by the data (i.e. o¤shoring at the intensive margin). Instead, the correlations between

hours worked and U.S. output follow the pattern of value added and the number of plants, variables

which respond with a lag to �uctuations in U.S. output. Therefore, in further work I aim to disconnect

the responses of o¤shoring at the intensive and extensive margins to �uctuations in U.S. manufacturing

output.

6 Conclusion

In this paper, I aim to rationalize the link between o¤shoring and the co-movement of business cycles in

a DSGE model in which trade in intermediate goods and o¤shore production under vertical FDI play a

special role. The model is successful in replicating the stylized facts described in the empirical section:

Over the business cycle, an economic boom and the appreciation of wages in North are followed by

the relocation of production o¤shore to the country with lower labor costs. In turn, higher demand

for labor in South exercises upwards pressure on wages. Higher demand for labor in North (due to

�rm entry) and simultaneously in South (due to o¤shoring) results into the positive co-movement of

wages and national incomes.

Empirical data from Mexico�s maquiladora industry also suggests that the timing of relocation

di¤ers at the intensive and extensive margins: Hours worked in the maquiladora sector respond im-

mediately to �uctuations in U.S. manufacturing output, whereas the number of plants responds with

a lag of three quarters. My model with elastic labor supply is partially successful in replicating this

pattern. The model does generate a delayed response in the number of o¤shore plants; the sign and

magnitude of correlations with lagged North output are in line with the empirical moments. How-

ever, hours worked in the o¤shore sector also respond with a lag �therefore, in future work, I aim to

disconnect the timing of relocation at the intensive and extensive margins.

35

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in a Ricardian Model with a Continuum of Goods," The American Economic Review, 67(5),

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37

A Appendix

A.1 The price index of Ct and the demand for intermediate goods

Under � = 0, households in North minimize total expenditure subject to the aggregator constraint:

minfyD;t(z); yV;t(z)g

PtCt =

zV;tZzmin

pD;t(z)yD;t(z)dz +

1ZzV;t

pV;t(z)yV;t(z)dz;

s:t: Ct =

26666664zV;tZzmin

yD;t(z)��1� dz

| {z }Produced Domestically

+

1ZzV;t

yV;t(z)��1� dz

| {z }Produced Offshore

37777775

���1

:

Given the �rst-order conditions:

pD;t(z) = �C1�t yD;t(z)

� 1� ;

pV;t(z) = �C1�t yO;t(z)

� 1� ;

the ammount of total expenditures can be re-written as:

PtCt =

zV;tZzmin

pD;t(z)yD;t(z)dz +

1ZzV;t

pV;t(z)yV;t(z)dz =

= �tC1�t

264zV;tZzmin

yD;t(z)��1� dz +

1ZzV;t

yV;t(z)��1� dz

375| {z }

C��1�

t

=

= �tCt:

The resulting identity �t = Pt leads to the price index and the demand formulas for intermediate

goods produced domestically and o¤shore:

Pt =

264zV;tZzmin

pD;t(z)1��dz +

1ZzV;t

pV;t(z)1��dz

3751

1��

;

yD;t(z) =

�pD;t(z)

Pt

���Ct = �D;t(z)

��Ct;

yV;t(z) =

�pV;t(z)

Pt

���Ct = �V;t(z)

��Ct:

38

A.2 Optimal pricing of intermediate goods produced domestically and o¤shore

The pro�t maximization problem for the home �rm producing intermediate good variety z domestically

is:

maxf�D;t(z)g

�D;t(z)yD;t(z)�wtZtz

yD;t(z):

Given the demand function yD;t(z) = �D;t(z)��Ct, the �rst-order condition leads to the pricing formula:

yD;t(z) + �D;t(z)@yD;t(z)

@�D;t(z)� wtZtz

@yD;t(z)

@�D;t(z)= 0) �D;t(z) =

�

� � 1wtZtz

:

The home �rm that produces o¤shore the intermediate good variety z solves the following pro�t

maximizing problem:

maxf�V;tg

�V;t(z)yV;t(z)��wtZtz

����w�tZ�t z

Qt

�1��yD;t(z)� fV

�wtZt

���w�tZ�tQt

�1��Given the demand function yV;t(z) = �V;t(z)

��Ct, the �rst-order condition generates the pricing for-

mula:

�V;t(z) =�

� � 1

�wtZtz

����w�tZ�t z

Qt

�1��:

A.3 Implications of �rm heterogeneity

My model implies that the more productive �rms use imported intermediate goods in their production

process. Moreover, following Melitz (2003), the more productive �rms have lower marginal costs,

charge lower prices, obtain larger revenues and pro�ts. Given two �rms with �rm-speci�c productivity

factors z2 > z1,y(z2)

y(z1)=

�z2z1

��> 1;

r(z2)

r(z1)=

�z2z1

���1> 1;

i.e., the more productive �rm has larger output and revenue, where output and revenue are given

by yt(z) = [�t(z)]�� Vt and rt(z) = �t(z)yt(z), respectively.

Both implications of �rm heterogeneity are in line with the results of empirical literature. This

property matches the stylized fact that �rms using imported components are more productive and

larger than non-importing �rms, as documented in Kurz (2006), Kasahara and Lapham (2006), Ra-

manarayanan (2006).

39

A.4 Existence of equilibrium productivity cuto¤ zV;t

As discussed in the text, two conditions must hold every period in order to ensure for the existence of

equilibrium productivity cuto¤ zV;t : (1) dV;t(z) is steeper than dD;t(z); and (2) zmin < zV;t. The �rst

condition implies that the e¤ective wage in South must be low enough relative to teh e¤ective wage

in North (TOLt < 1) in order to o¤set the iceberg trade cost (� > 1):

�w�tQtZ�t

<wtZtz

() �TOLt < 1:

The second condition, zmin < zV;t, requires that:

Slope(dV;t(z)) <�fE

wtZt+ fV

w�tQtZ�t

z��1min

;

z��1min

1

�

��

� � 1wtZt

�1��Ct < �fE

wtZt+ fV

w�tQtZ�t

;

1

�

��

� � 1wt

Ztzmin

�1��Ct < �fE

wtZt+ fV

w�tQtZ�t

;

dD;t(zmin) < �fEwtZt+ fV

w�tQtZ�t

;

where � = 1��(1��)�(1��) : The last inequality shows that the pro�t obtained from domestic production by

the �rm with the minimum productivity zmin must be smaller than the sum of the per-period value of

the sunk entry cost and the �xed cost of o¤shoring. In other words, the �rm that obtains zero pro�t

from domestic production would make negative pro�ts if it engages in o¤shore production, i.e. the

�rm with productivity zmin does not produce in either country.

A.5 Average productivities under the Pareto distribution

The average productivity of home �rms that produce o¤shore is:

ezV;t =264 1

1�G(zV;t)

1ZzV;t

z��1g(z)dz

3751��1

=

264� zV;tzmin

�k 1ZzV;t

z��1kzkminzk+1

dz

3751��1

=

=

"�zV;tzmin

�k kzkmink � (� � 1)z

��1�kV;t

# 1��1

=

�k

k � (� � 1)

� 1��1

zV;t =

= �zV;t;

where � �h

kk�(��1)

i 1��1

:

40

The average productivity of home �rms producing domestically is:

ezD;t =24 1

G(zV;t)

zV;tZzmin

z��1g(z)dz

351��1

=

24 zkV;t

zkV;t � zkmin

zV;tZzmin

z��1kzkminzk+1

dz

351��1

=

=

"zkV;t

zkV;t � zkminkzkmin

� � k � 1

�z��1�kV;t � z��1�kmin

�# 1��1

=

=

24 k

k � (� � 1)(zminzV;t)

k

zkmin � zkV;t

0@ 1

zk�(��1)V;t

� 1

zk�(��1)min

1A35 1��1

=

= �zminzV;t

24zk�(��1)V;t � zk�(��1)min

zkV;t � zkmin

35 1��1

:

A.6 Link between average pro�ts with �xed costs fOw�tQtZ�t

Under � = 0, the average pro�t of the home �rms producing domestically is:

edD;t = dD;t(ezD;t) = 1

�

��

� � 1wt

ZtezD;t�1��

Ct =1

�

��

� � 1wtZt

�1��Ctez��1D;t =

=1

�

��

� � 1wtZt

�1��Ct (�zminzV;t)

��1

24zk�(��1)V;t � zk�(��1)min

zkV;t � zkmin

35 ==1

�

��

� � 1wt

ZtzO;t

�1��Ct| {z }

dD;t(zV;t)

(�zmin)��1

24zk�(��1)V;t � zk�(��1)min

zkV;t � zkmin

35 =

= dD;t(zV;t) (�zmin)��1

24zk�(��1)V;t � zk�(��1)min

zkV;t � zkmin

35 :Similalry, the average pro�t of the home �rm producing o¤shore is:

edV;t = dV;t(ezV;t) = 1

�

��

�

� � 1w�tQtZ�t ezV;t

�1��Ct � fV

w�tQtZ�t

=

=1

�

��

�

� � 1w�tQtZ�t

�1��Ctez��1V;t � fV

w�tQtZ�t

=

=

(1

�

��

�

� � 1w�tQtZ�t zV;t

�1��Ct � fV

w�tQtZ�t

)| {z }

dV;t(zV;t)

���1 +����1 � 1

�fVw�tQtZ�t

=

= dV;t(zV;t)���1 +

� � 1k � (� � 1)fV

w�tQtZ�t

:

The home �rm with productivity equal to the cuto¤ zV;t is indi¤erent between locating its pro-

duction plants domestically or o¤shore. Hence, the identity of pro�ts at the productivity cuto¤,

41

dD;t(zV;t) = dV;t(zV;t), and the two expressions above allow me to derive the following link between

the pro�ts to the two representative �rms in the home economy:

edV;t = � 1

�zmin

���1 24zk�(��1)V;t � zk�(��1)min

zkV;t � zkmin

35�1 edD;t| {z }=dD;t(zV;t)

���1 +� � 1

k � (� � 1)fVw�tQtZ�t

=

= z1��min

24zk�(��1)V;t � zk�(��1)min

zkV;t � zkmin

35�1 edD;t + � � 1k � (� � 1)fV

w�tQtZ�t

=

=k

k � (� � 1)

�zV;tezD;t

���1 edD;t + � � 1k � (� � 1)fV

w�tQtZ�t

:

42