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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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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
:
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