Discussion Paper No. 940

CAPITAL HETEROGENEITY
AS A SOURCE OF
COMPARATIVE ADVANTAGE:
PUTTY-CLAY TECHNOLOGYIN A RICARDIAN MODEL

Hirokazu Ishise

June 2015

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Capital Heterogeneity as a Source of Comparative
Advantage: Putty-Clay Technology in a Ricardian
Model
Hirokazu Ishise∗
June 23, 2015
(First version: February 19, 2013)

Abstract
I consider how heterogeneity in capital goods aﬀects international trade patterns,
and I show a novel source of comparative advantage: the magnitude of capital goods
heterogeneity. Capital goods are heterogeneous in their vintage and productivity, anddue to capacity constraints, only productive capital goods are activated in the equilibrium. Through this selection, the distribution of capital goods determines the industrylevel productivity: industry-level productivity is higher in an industry with relatively
larger variation in capital goods, and hence in a perfectly competitive two-country,
two-good, two-factor equilibrium, the industry has Ricardian comparative advantage.
An extension of the model, including fixed trade cost, describes a sorting situationin which the most productive production units (which are generally newer vintage)export, the moderately productive units serve the domestic market, and the least productive units (older) do not operate.

∗

Institute of Social and Economic Research, Osaka University. Address: 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan; Phone: +81-6-6879-8581; Email to: ishise@iser.osaka-u.ac.jp; I would like to thankcomments by two anonymous referees, Marianne Baxter, Jonathan Eaton, Taiji Furusawa, Simon Gilchrist,
François Gourio, Samuel Kortum, Noriaki Matsushima, Ray Riezman, and seminar/conference participants
at Kansai, Kobe, Kyoto, Osaka (Econ and ISER), Tohoku, Yokohama National, Spring 2013 Midwest International Trade meeting, Spring 2013 JEA meeting, and 2014 Asian Meeting of the Econometric Society.
This work is supported by JSPS KAKENHI Grant Number 26885043. All errors are mine.

1Keywords: Ricardian trade model; Putty-clay technology; Vintage capital; Capacity
utilization rate; Sorting in destination.
JEL Codes: F11, D24, E22, E23.

1

Introduction

While investment drives several aspects of aggregate economy, trade models frequently abstract investment decisions regarding capital goods by considering an endowment economy.
In models that include investment decision, capital goods are commonly assumed to be homogeneous. However, an important empirical aspect of capital goods is heterogeneity inproductivity. Some capital goods are more productive than others within an industry.
This paper considers the role of investment and productivity heterogeneity of capital
goods in the pattern of international trade. Specifically, I introduce putty-clay production
technology in the style of Gilchrist and Williams (2000, 2004, 2005) into an international
trade model considered by Baxter (1992), and discover that the magnitude of heterogeneity
in capital goods is a source of Ricardian comparative advantage.
Baxter (1992) includes endogenous capital accumulation, intertemporal optimization andneoclassical production function in the classical “2 by 2 by 2” (two-country, two-good, andtwo-factor) international trade model. She shows that in the steady-state of the dynamic
economy, the pattern of comparative advantage is described as a Ricardian model, compared to the classical endowment “2 by 2 by 2” model that implies a Heckscher-Ohlin tradepattern. Intuitively, since returns to investment are determined by the user cost, which isultimately determined by the subjective discount factor and the capital depreciation rate,
the amount of capital is fully adjusted so that the returns to investment are equalized acrossindustries. As a result, the economy eﬀectively has only one input (labor) as in the Ricardianmodel. The relative price of goods under autarky is determined by the ratio of productivities
across industries, and this relative price predicts the pattern of specialization that occurs
when countries are engaged in trade. Baxter’s result indicates the importance of explicitly2considering investment decisions when capital goods exist. However, she does not consider
heterogeneity in capital goods.
Accumulated empirical evidence shows that even in a narrowly defined industry, thequality (i.e., productivity and capital-intensity) of capital goods diﬀers significantly acrossfirms (Goolsbee, 2004; Foster et al., 2008). A large diﬀerence in capital goods presents
even within firms (e.g., Goolsbee and Gross, 2000).1 Theoretically, a part of productivity
diﬀerence is caused by the capacity constraint and the inflexibility (e.g., irreversibility) ofcapital goods (Cooper and Haltiwanger, 2006). Without capacity constraint, production
occurs only for the most productive capital good. Besides, inflexibility is important. If
capital goods could be instantaneously adjusted at no additional cost, firms would adjust
the quality of capital goods according to their economic environment.
One source of heterogeneity in capital goods is vintage. New capital goods tend be
more productive than older ones, thanks to technological progress. For example, using
U.S. manufacturing plant data, Jensen et al. (2001) show that new entrants in recent years
are significantly more productive than past entrants in their entry year. Another sourceof heterogeneity is idiosyncratic productivity variations among capital goods of the same
vintage. For example, among the same make and model of equipment (industrial, mining,
and farming machines, and oﬃce and transportation-related equipment), some machines
malfunction, which requires a series of inspections and repairs, and eﬀectively renders themless productive than other machines of the make and model.2 Since industrial machines suchas assembly lines, supercomputers, transportation equipment and farm equipment, havebecome more and more complex and consist of many parts and/or computerized functions, a1

Goolsbee and Gross (2000) use data from airline companies. Obviously, capital goods (aircrafts) diﬀer
in their quality (e.g., capacity and fuel eﬃciency) even within the same airline company.
2
Typical examples are transportation, construction and farm equipment (e.g., vehicles, dump tracks andcombine harversters). This type of troubles is suﬃciently common such that in the U.S., lemon laws (atthe federal level, the Magnuson-Moss Warranty Act) protect consumers by ensuring appropriate recourse
for compensation in the transaction of consumer vehicles and some other goods. However, the applicationof lemon laws for business vehicles is limited. For farming, a similar problem of new “lemon” machines
is an important issue. Mechanical deficiencies lower the productivity of machines. Moreover, during theinspection and repairs, the machines cannot be operated, and hence eﬀective productivity is much lower.
The application of lemon laws to farming equipment depends on individual states.

3

malfunction in one part of the machinery often leads to a breakdown of the entire machinery(“O-ring” theory: Kremer, 1993; Jones, 2011). Such variation leads to ex post productivity
variations among ex ante same capital goods.
Putty-clay production technology is attractive for analyzing heterogeneity in capital
goods since the approach incorporates vintage, capacity constraint, investment irreversibility,and endogenous utilization decision. Gilchrist and Williams (2000, 2005) introduce puttyclay technology into a closed-economy single-industry business cycle model. Capital goods(called “machines”) have distinct vintage and idiosyncratic productivity. The machine-level
labor-productivity depends on endogenously chosen capital-intensity and an exogenously determined idiosyncratic productivity component. This idiosyncratic productivity component
is determined after the determination of capital-intensity. Once a machine is built, it is impossible to change its capital-intensity or to revert to investment goods. Moreover, capacity
is constrained: the operational choice is to allocate one worker to each machine or to keep
the machine idle.3
Under this heterogeneous capital setting, the aggregate production function in the steadystate is represented as a standard Cobb-Douglas function, but its Solow residual is determinedin part by the capital heterogeneity. With machine heterogeneity, less productive machines
are not profitable, and hence not used in the equilibrium. Gilchrist and Williams (2005) showthat a temporary increase in the idiosyncratic productivity variation provides an economywide productivity benefit through the optimal reallocation of variable factors, which is labor,
across machines. That is, a change in the variance of idiosyncratic productivity component
leads to a change in the aggregate productivity.
3

Johansen (1972, chap. 9), Fuss (1977), Lasserre (1985) and Salvanes and Tveteras (2004) empiricallyconfirm these key characterizations of putty-clay production technology using various micro-level data. In
the context of the transportation, construction and farm equipment examples in Footnote 2, putty-clay works
as follows. First, before installation or adoption, a firm can choose the size of, for example, dump tracks, but
the size cannot be changed afterwards. The choice of the size determines capital-intensity because a dump
track requires only one operator at a time. Some dump trucks experience a series of mechanical troubles(“lemons”), and possibly cannot be used. Productivity over a certain period of time becomes low or zerofor these lemons. Production machines experiencing mechanical troubles also may produce products that do
not meet quality standards. This also leads to low idiosyncratic productivity.

4

Introducing this putty-clay technology into an international model considered by Baxter(1992), I find a novel source of comparative advantage: the heterogeneity of capital goods. In
the steady-state of the dynamic “2 by 2 by 2” economy, trade pattern and gains from trade
are generally described by Ricardian comparative advantage, as in Baxter (1992). Under
autarky, diﬀerences in the magnitude of machine-level productivity heterogeneity acrossindustries lead to diﬀerences in industry-level productivity. Larger machine-level variation
provides an industry-level productivity benefit through the optimal allocation of labor acrossmachines since less productive machines are not used in the equilibrium. Hence, an industry
with larger idiosyncratic variation has higher industry-level productivity than an industry
with less variation, and the price of the good is lower than the price of the other good.
Moreover, in the industry-level aggregation, the production is represented as a Cobb-Douglas
function, and the contribution of heterogeneity appears in a part of the Solow residual. By
considering costless trade equilibrium, since the diﬀerence in the relative price under autarky
determines the pattern of specialization, a country specializes in an industry with relatively
larger capital goods heterogeneity. In this sense, trade is based on the technology-driven(i.e., Ricardian) comparative advantage, and the industry-level Solow residual is useful topredict trade pattern.
As an extension, I present the model with a fixed trade cost, and show that the model
describes a situation in which the most productive (typically newer) machines export, moderately productive machines serve the domestic market, and the least productive (older)machines do not operate. This sorting implication is closely related to recent empiricalliterature findings (e.g., Bernard and Jensen, 1997, 1999; Bernard et al., 2003; Tomiura,2007). While a popular explanation of this type of sorting developed by Melitz (2003) appeals to productivity heterogeneity, fixed costs of operation and exporting, and monopolisticcompetition, my model oﬀers a fundamentally diﬀerent mechanism of sorting based on the
combination of productivity heterogeneity, fixed exporting cost, and capacity constraint.
Essentially, since each machine faces capacity constraints, a machine is used for its most5

profitable operation. If the price of a good is higher in the foreign country, but delivering
the good to the foreign country requires a fixed trade cost, exporting is profitable only for themost productive machines. Moreover, the model provides a prediction regarding the sortingand vintage. When technology is gradually improving and capital-embodied, a newer machine tends to be more productive and hence a newer machine is more likely to be used forexporting.
Even with the introduction of the fixed trade cost, the overall trade pattern is described
as a Ricardian model. In general, a country produces more on the product with comparative advantage. The gains from trade are hence the combined and interactive results of
the reallocation across industries (a reallocation that itself arises by exploiting comparativeadvantage), and reallocation across machines within an industry.
This paper oﬀers new insights into three distinct research programs: (1) the source ofcomparative advantage, (2) firm-level trade, and (3) implications of putty-clay technology
and micro-production based theories of aggregate productivity.
The source of comparative advantage has been the central subject of international tradestudies (e.g., Feenstra, 2004). Traditionally, the comparative advantage results from a country’s industry-level technology and aggregate factor endowment.4 Emerging literature considers diﬀerences in the distribution of skill across labor as a source of comparative advantage(Grossman and Maggi, 2000; Antràs et al., 2006; Bougheas and Riezman, 2007; Ohnsorge
and Trefler, 2007). In particular, these studies underscore that a country with a concentrated
skill distribution (e.g., Germany) has comparative advantage in an industry requiring many
moderately skilled workers (e.g., precision machinery), while a country with sparse skill distribution (e.g., Italy) has a comparative advantage in an industry requiring a few outstanding
experts (e.g., fashion design). While this logic is intuitive and appealing, another important
aspect of heterogeneous distribution is selection mechanism, which my paper also emphasizes. Moreover, the distribution in my model is endogenously determined in the sense that
4

Costinot (2009) organizes traditional theories of comparative advantage, and shows conditions of technology and factor endowment to emerge comparative advantages.

6

capital-intensity is optimally chosen based on the knowledge of distribution of idiosyncratic
productivity variation.
Second, my model adds a new view to micro-level trade literature. The sorting implication
in my extended model is closely related to Melitz (2003), and an analysis regarding the gains
from trade in this extended model resembles the way Bernard et al. (2007) applied Melitz(2003) to two industries. In the context of the sorting, the vast majority of the model
is a version of Melitz (2003) in which monopolistic competition is an essential feature in
addition to the underlying assumption, namely, the love of variety.5 A potential problem
is that the assumption of the love of variety is too critical in the welfare analysis, andthe literature attempts to organize the contribution of monopolistic competition and its
parameterization to the welfare implications of firm-level trade models (c.f., Arkolakis et al.,2010, 2012; Behrens and Murata, 2011). An alternative view to sorting helps clarify the
relationship between sorting in destination and gains from trade. In addition, my modeloﬀers several additional empirical predictions regarding the exporting status and vintage ofcapital goods: a firm with newer vintage machines is more likely to be an exporter; among
exporters, export intensity is higher for the firm with newer vintage machines; and a firm
starts exporting after an installation of new and productive machines. These predictions will
be empirically pursued in future research.
Finally, the paper contributes to the literature of putty-clay technology and microproduction based theories of aggregate productivity. After Johansen (1959) introduced
putty-clay technology, researchers intensively studied it from the 1960s to 80s mainly inthe context of growth and investment models.6 The original idea is that in the aggregate
production function, ex ante capital-intensity is freely chosen (“putty” or Cobb-Douglas
function), but cannot be changed ex post (“clay” or Leontief function). Three papers fromthat era are closely related to my paper: Inada (1966) considers a two-sector Solow model5

A notable exception is a model by Bernard et al. (2003), who introduce Bertrand competition into amodel of Eaton and Kortum (2002).
6
See footnote 1 of Gilchrist and Williams (2000).

7

with putty-clay technology; Grossman (1983), a trade application, considers the implicationsof a putty-clay production (without investment decision) to compare the specific-factor andHeckscher-Ohlin trade models; and Eaton (1979), another international application, examines the implication of exchange rate change on the allocation of factors. These classicalpapers assume exogenous investment decisions. A modern revival of putty-clay technology, with endogeneous investment decision, includes Atkeson and Kehoe (1999), Gilchrist
and Williams (2000, 2004, 2005), Wei (2003), Birchenall (2004), Gourio (2011) and Choiand Rios-Rull (2012). In addition, the version of putty-clay in my paper, which relies on
Gilchrist and Williams (2005), introduces heterogeneity in productivity across machines, andanalyzes the industry-level productivity. This micro-founded analysis of aggregate productivity has its roots in a classical analysis by Houthakker (1955–56).7 These papers focus on
implications of business cycle, labor market or asset price in dynamic single-industry closedeconomy settings. None of the recent papers features multi-industry and/or multi-country
implications.
The rest of the paper is organized as follows. Section 2 is the main section of the paper,
which first describes the investment decision problem, and then introduces this investment
decision problem into a classical “2 by 2 by 2” economy with endogenous capital accumulationand intertemporal decision. Section 3 is an extension of the baseline model, in which fixed
trade cost is included, and shows the implication of the sorting in destination. Section 4
concludes.

2

Model

The heart of the model is the investment decision problem and its steady-state implications.
The problem extends Gilchrist and Williams (2005) by including two industries. The investment decision involves an explicit dynamic stochastic problem, while the focus of the paper
7

Related to Houthakker (1955–56) but distinct from studies based on putty-clay technology, Lagos (2006)
also incorporates a Leontief micro-structure based on Houthakker (1955–56) with a search-and-matchingmodel for explaining measured total factor productivity.

8

is its steady-state. After describing the investment problem, I combine the problem into
the “2 by 2 by 2” economy with endogenous capital accumulation. Following the traditionof trade literature, I first consider a relative price of two goods under autarky, and then I
analyze an open-economy equilibrium without trade cost. There are two countries, two final
goods, and two factors of production. Final good A is used just for consumption, while final
good B is used for both consumption and investment. Factors of production are capitaland labor. Importantly, capital is endogenously accumulated. Time is discrete, and denotedby t = 0, ..., ∞. Countries are indexed by subscript i = 1, 2, and industries are distinct bysuperscript ι = A, B.

2.1

Dynamic problem

The investment decision model is stated with explicit treatment of uncertainty and dynamics.
I first state the problem of the households, and then consider the investment problem.

2.1.1

Households

In each country i, there is a stand-in household. Households inelastically supply homogeneous
labor Li .8 Households consume both good A and B. Let cιi,t denote consumption of goods
ι in country i at period t. The utility function is additively separable across time, and theperiod-utility is homothetic and identical across countries:

E0

∞
∑

B
β t u(cAi,t , ci,t ),

(1)

t=0

where β is the subjective discount factor of the household. For simplicity of calculation, I
use log-utility:

B
A
B
u(cA
i,t , ci,t ) = (1 − ϵ) ln ci,t + ϵ ln ci,t .
8(2)Including a leisure-labor choice is straightforward, and it does not change any implications. See Appendix.

9The second good is used for consumption and investment. The intertemporal decision between period t and t + s is summarized by the ratio of the marginal utilities of the second
goods at t and t + s (i.e., the “stochastic discount factor” in the macro-finance terminology):∂ui,t+s /∂cBi,t+s
mi,t,t+s = β.
∂ui,t /∂cB
i,t
s

2.1.2

(3)

Investment problem

Production takes place in a unit called “machine.” In each period and for each industry, a setof investment opportunities to create new machines becomes available. Households determine
capital-intensity of machines, and the number (quantity) of machines to be installed. At thetime of investment, technology is constant-returns-to-scale, and the size of each machine isnormalized to employ a maximum of one unit of labor. Once capital-intensity is determined,
the investment is irreversible, and there is no substitutability between capital and labor.
Investment is subject to idiosyncratic variation in productivity. After the determinationof the capital-intensity, the productivity is randomly drawn from a known log-normal distribution, which is industry- and country-specific. The productivity variation is a source ofheterogeneity across machines within a vintage of capital goods. Each machine in a vintage
is distinguished by an index ξ. The log-productivity of a machine ξ of industry ι in country
i installed at period t is:)( ιιln θi,t(ξ) ∼ N ln θi,t
− 0.5σiι 2 , σiι 2 .

(4)

ι
by the adjustment of −0.5σiι 2 of the log-normal disThe mean productivity is E(θiι (ξ)) = θi,t

tribution. Log-normal is empirically relevant (Campbell, 1998) and theoretically convenient(Gilchrist and Williams, 2000, 2005).
The capital-intensity of a newly installed machine is the same within an industry and aιunits of investment
country, since the investment project is ex ante identical. By installing ki,t

10

ι α
goods, the capital’s contribution to labor-productivity is ki,t
. The parameter α ∈ (0, 1)

directly controls marginal product of capital.9 At period t + s, a machine installed at periodιt (vintage t) can produce yi,t+s(ξ) units of goods ι if one unit of labor is assigned:

ι
ι
ι α ι
ι
yi,t+s(ξ) = θi,t(ξ)ki,tli,t+s (ξ), where li,t+s(ξ) ∈ [0, 1].

(5)

If a machine is not used at a period, it remains idle at no cost.
Each period after the initial production (which takes place at t + 1), machines have aconstant probability δ of breaking down regardless of vintage, productivity, capital-intensity,
industry or country.
The optimization problem at the time of capital-intensity decision (t) is:max∞ι , lι
ki,t{ i,t+s (ξ)∈[0,1]}

s=1

ι−pkt ki,t
+ Et

[∞
∑

mi,t,t+s (1 − δ)

s−1

(

ι α
− wt+spi,t+s θi,t (ξ)ki,t]) ι
li,t+s (ξ) ,

s=1

(6)
where pki,t is the price of the investment goods (which is pB
i,t under the current specification),mi,t,t+s is the stochastic discount factor, and the expectation operator Et takes over the time
t idiosyncratic productivity diﬀerence and all the other values at and after period t + 1. The
capital-intensity is chosen to maximize expected profits conditional on the known distribution
but not on the realization of the idiosyncratic productivity. This optimization problem
includes decisions about each of the diﬀerent machines (ξ), and directly solving the problem
is diﬃcult. However, the problem is simplified by introducing a variable summarizing a
cut-oﬀ decision.
ι αι, while the cost is(ξ)ki,tAt period t + s, the revenue from operating a machine is pιi,t+s θi,t

wi,t+s . By optimization, the decision follows a cut-oﬀ rule: one unit of labor is assigned if itι αι
≥ wi,t+s ), and no labor is assigned if it is not profitable. Let fΘ(ξ)ki,tis profitable (pιi,t+s θi,t

9

It is straightforward to make the share parameter α diﬀers across industries, but the assumption of thecommon share parameter is useful to highlight the role of level and variance of productivity.

11ι
denote the probability distribution function of θi,t(ξ). As shown in Appendix, the expectedlabor-use at period t + s for a machine built at period t is as:(1 − δ)s−1∫ ∞wi,t+spιkι αi,t+s i,tιιιfΘ (θi,t(ξ)|wi,t+s , pιi,t+s , θi,t)dθi,t
(ξ)

(
( ι
))
=(1 − δ)s−1 1 − Φ zi,t,t+s,(7)where Φ(.) is the cumulative distribution function of the standard normal distribution, andιzi,t,t+s
≡

1
ln
σiι

(

wi,t+sιι ι αpi,t+s θi,tki,t)
+ 0.5σiι .

(8)Among machines built at period t, less productive machines are kept idle, while more proι
ductive machines are in operation. The variable zi,t,t+s
summarizes the cutoﬀ, which ingeneral depends not only on the time t + s but also the vintage s. As shown in Appendix,
the expected output is also expressed by Φ(.) as:

∫ ∞
s−1 
(1 − δ)wi,t+spιkι αi,t+s i,t



( ι
) ι
ι
ι
ι α
fΘ θi,t(ξ)|wi,t+s , pιi,t+s , θi,tdθi,t (ξ)θi,t(ξ)ki,t(( ι)) ι ι α=(1 − δ)s−1 1 − Φ zi,t,t+s− σiι θi,tki,t .

(9)( ι)
The term 1 − Φ zi,t,t+s− σiι is a capacity utilization rate, i.e., the ratio of output produced
from the capital goods installed at t to the level of maximum output using the capital goods.10
Using (7) and (9), at period t + s the expected net income of a surviving machine build at t
is as:))(( ι)) ι ι α(( ι.
ki,t − wi,t+s 1 − Φ zi,t,t+s− σiι θi,tpιi,t+s 1 − Φ zi,t,t+s

10

(10)

The inside of Φ(.) is z − σ, not z. This adjustment comes from the use of truncated expectation formula.

12ιThe first derivative of this equation with respect to zi,t,t+sis:( ι) ι ι α( ι)
0 = −pιi,t+s ϕ zi,t,t+s− σiι θi,tki,t + wi,t+s ϕ zi,t,t+s,(11)where ϕ(.) is the probability density function of the standard normal distribution. Using thefunctional form of ϕ(.), a rearrangement of this expression (11) leads to equation (8). Thismeans that the capital-intensity decision problem (6) involving a machine-level problem canbe rewritten in an industry-level problem using zi,t,t+s as an additional variable:max∞ι ,{z ιki,ti,t,t+s }s=1

ι+ Et
− pki,t ki,t[ ∞∑mi,t,t+s (1 − δ)s−1

]
( ι
(
( ι
))
(
(
)))
α
ι
ι
ι
ki,t
− wi,t+s 1 − Φ zi,t,t+s.
× pi,t+s 1 − Φ zi,t,t+s − σiι θi,t
s=1

(12)

ι
The first order condition with respect to zi,t,t+s
is (11), or equivalently, (8). Depending on
ιthe prices, ki,t
could be zero. When the investment is positive, the first order condition isι 11
the first derivative with respect to ki,t
, and hence:

0 = −pki,t + αEt

[∞
∑

]
(
))
ι
ι ι α−1.
mi,t,t+s (1 − δ)s−1 pιi,t+s 1 − Φ zi,t,t+s− σiι θi,t
ki,t
(

(13)

s=1

Once a machine is installed, each machine earns positive profits if it operates. An equilibrium
requires expected profits to be zero. Such a condition, a free-entry, implies that:ι
pki,t ki,t
=Et

[ ∞
∑

mi,t,t+s (1 − δ)s−1

]
(
(
)))
))
( ι
(
( ι
ι α
ι
ι
ki,t
− wi,t+s 1 − Φ zi,t,t+s.
× pi,t+s 1 − Φ zi,t,t+s − σiι θi,t
s=1

(14)

This free-entry condition, together with the resource constraints presented below, determinesι.
the total number of newly installed machines in each industry, qi,t11ιιPrecisely, a change in ki,tchanges future decisions of zi,t,t+s. By the envelope condition (11), thisindirect eﬀect does not appear in (13).

132.1.3

Industry aggregation

Total labor-use in the industry ι is the sum of labor requirements of all existing machines:

ι
li,t
=

∞
∑

(
( ι
)) ι
(1 − δ)s−1 1 − Φ zi,t−s,tqi,t−s(15)

s=1

and the total production of industry ι is:

ι
yi,t
=

∞
∑

(
( ι
)) ι
α ι
ι
(1 − δ)s−1 1 − Φ zi,t−s,t− σiι θi,t−ski,t−sqi,t−s .

(16)

s=1

The total gross investment for industry ι at period t is the product of the number of machinesι ιand their capital-intensity, ki,tqi,t . In each period, δ fraction of capital goods is exogenously

broken. If a researcher uses a standard perpetual inventory method to calculate capital stockιof industry ι, the calculated industry-level capital stock Ki,tfollows the usual transition

expression:ιιι ι
Ki,t+1= (1 − δ)Ki,t+ ki,tqi,t .

2.1.4

(17)

Resource constraints and other conditions

The first goods are pure consumption goods, while the second goods are consumption andinvestment goods. The resource constraints are:
N
∑

A
yi,t
=

N
∑

cA
i,t ,

i=1

i=1

N
∑

N
∑
(

i=1

B
=
yi,t

)
B B
A A
cB
i,t + ki,t qi,t + ki,t qi,t ,

(18)

(19)

i=1

A
B
li,t
+ li,t= Li,t .

(20)

There is one country (N = 1) under autarky, and two (N = 1, 2) under costless trade. Since
the second goods are used as investment goods, the price of the investment goods is that of14

the second goods:

pki,t = pB
i,t .

(21)

Under costless trade, pιt = pι1,t = pι2,t . I also impose a period-by-period trade balance:
( B
)
A
A
B
B
A A
B B
pA
t (yit − cit ) + pt yit − cit − kit qit − kit qit = 0.

(22)

A competitive equilibrium of the economy is defined as usual: sequences of prices andquantities that are consistent with the household’s optimization, the investment decision,
the resource constraints, and conditions on prices. For costless trade equilibrium, the tradebalance is also required.

2.2

Steady-state implications of zero-growth economy

In the rest of the paper, the analysis focuses on the steady-state/balanced-growth equilibrium
in which all the variables stay constant, and the distribution of machines is stationary. In
particular, the analysis starts from a case without trend growth. The mean productivity isιconstant over time E(θiι (ξ)) = θi,t
= θiι . Under this assumption, I can show many results

analytically.

2.2.1

The investment problem

The variables in the steady-state are expressed without time subscripts, for example, kitι = kiι .
The stochastic discount factor mi,t,t+s defined by (3) is replaced by β s . Let R denote thesteady-state user cost:R=1− 1 + δ.
β

15

(23)

The variable of the cut-oﬀ decision in general depends both on time and vintage. In thesteady-state, the right-hand-side of (8) depends on neither, and hence ziι is constant within
industry and country. The utilization decision condition (11) is as:

0 = −pιi ϕ (ziι − σ ι ) θiι kiι α + wi ϕ (ziι ) .

(24)

Similarly, equations (13) and (14) are as:α ιpi (1 − Φ (ziι − σiι )) θiι kiι α ,
R
α 1
k ι
p i ki =w (1 − Φ (ziι )) .
1−αR
pki kiι =

(25)
(26)

Combining (24), (25) and (26) leads to the following equation, which pins down ziι :ϕ (ziι − σiι ) / (1 − Φ (ziι − σiι ))1−α=.
ϕ (ziι ) / (1 − Φ (ziι ))(27)

Lemma 1. Equation (27) has a unique solution, and dziι /dσiι > 1. (Gilchrist and Williams,
2005)
The proof is given by Gilchrist and Williams (2005).12 Lemma 1 implies that ziι depends
only on α and σiι . It does not depend on the mean productivity θiι , the real wage wi , or the
prices of the good pιi and investment goods pki . Since Gilchrist and Williams (2005) analyze asingle-industry economy, the finding that ziι does not depend on the prices is not surprising.
However, equation (27) shows that even with a two-industry extension, the steady-state
operation decision does not depend on the prices.

12

The basic idea of the proof is to calculate the upper limit (which is unity) and lower limit (which iszero) of the right-hand-side, and to show monotonicity of the right-hand-side with respect to z. The sign ofthe derivative is determined by the implicit function theorem.

16

The aggregation equations (15) and (16) are as:1liι = (1 − Φ (ziι )) qiι ,δ1yiι = (1 − Φ (ziι − σiι )) θiι kiι α qiι .
δ

(28)
(29)

By combining (25), (26), (28), and (29):(1 − α)pιi yiι = wi liι .

(30)that is, labor’s income share is constant. This condition motivates Cobb and Douglas (1928)
to discover the famous functional form. Hence, in the steady-state, this putty-clay microstructure is observationally consistent with industry-level Cobb-Douglas technology.
Moreover, from (17), the implied industry capital stock is as:
1
Kiι = kiι qiι ,δ

(31)

and combining this with (28) and (29) leads to an expression of the industry-level production
function as:1yiι = (1 − Φ (ziι − σiι )) θiι kiι α qiιδ()1−α ι α1= (1 − Φ (ziι − σiι )) θiι kiι α liι δ (1 − Φ (ziι ))−1qi
δ
1 − Φ(ziι − σiι ) ι α ι 1−α=θiιKi l i .
(1 − Φ(ziι ))1−α(32)

In the steady-state the industry-level aggregate production function is exactly Cobb-Douglasform. Its Solow residual has two components: the part coming from the mean productivity,
and the part coming from the magnitude of heterogeneity.13 A straightforward calculation
13

This aggregate representation of Cobb-Douglas production function based on the putty-clay microstructure is available in the steady-state of a single-good setting. Gilchrist and Williams (2000, 2005) do
not explicitly present this representation, although they study the eﬀect of a short-run change in σ on the17

shows that the Solow residual is increasing in σiι .
Lemma 2. The Solow residual in the steady-state is increasing in the parameter σiι .
Proof. For notational simplicity, both superscripts and subscripts are dropped. By taking
the first derivative:())−ϕ(z − σ)dz
1 − Φ(z − σ)dz
− 1 + (1 − α)ϕ(z)dσ(1 − Φ(z))1−α dσ(1 − Φ(z))2−α(())
1 − Φ(z − σ)−ϕ(z − σ)
dz
ϕ(z)
dz
=θ
− 1 + (1 − α)(1 − Φ(z)) dσ(1 − Φ(z))1−α (1 − Φ(z − σ)) dσ(())
1 − Φ(z − σ)−ϕ(z − σ)dzϕ(z − σ)dz=θ
−1 +(1 − Φ(z − σ)) dσ(1 − Φ(z))1−α (1 − Φ(z − σ)) dσϕ(z − σ)=θ.
(1 − Φ(z))1−α∂Solow=θ∂σ(

From the second to third line, equation (27) is used. Since all the terms in the last line arepositive, the first derivative is positive.
The intuition behind this result is that large heterogeneity in productivity across machinesprovides an industry-wide productivity benefit through the optimal allocation of labor acrossmachines. In this model, less productive machines are not used. Large σ means that the
distribution of machines (with respect to labor-productivity) is flat; there are many highly
productive machines and many unproductive machines. The unproductive machines are
not suﬃciently profitable to cover the operational cost (wage), so they are kept idle. The
industry-level productivity is high since the production mainly takes place with productive
machines. On the contrary, small σ means the productivity distribution is concentrated;
there are many “so-so” productive machines. Most of the machines are in operation, andthe industry-wide productivity gain through the selection is small. In an extreme case, if all
machines are the same, the industry-level productivity is the same as that of the individualmachines. The industry has no productivity gains from the selection.
aggregate productivity.

18

The role of the selection in determining industry-level productivity becomes clear byconsidering the capacity utilization rate. As shown in (9) the term 1 − Φ(z − σ) is thecapacity utilization rate of the industry. Note that Φ(.) is a monotonically increasing function
because it is the cumulative distribution function of the standard normal distribution. Sincedz/dσ > 1 (by Lemma 1) and Φ(.) is monotonically increasing, the capacity utilization rate
1 − Φ(z − σ) is monotonically decreasing in σ. In other words, the more productivity varies,
the more severe the selection. Together with the previous result,Lemma 3. In the steady-state there is a one-to-one relationship between the capacity utilization rate 1 − Φ(ziι − σiι ) and the parameter σiι . The Solow residual is larger if the capacity
utilization rate is lower.
The statement appears to contradict a standard short-run argument that the productivity(calculated without adjusting capacity utilization rate) is high if the capacity utilization rate
is high (e.g., Greenwood et al., 1988). The statement here is about the steady-state withheterogeneous capital goods. Productivity is high if the steady-state capacity utilization rate
is low, because a low capacity utilization results from a severe selection.
In summary, a diﬀerence in the productivity across capital goods is a source of theindustry-level productivity through the selection.
2.2.2

Autarky

I first consider the implications of the putty-clay technology under autarky. For notational
simplicity, country subscript i will be dropped. The focus is the relative price of two industries
pB /pA . From equations (25) and (26), the relative price is:1−Φ(z A −σ A )pθ (1−Φ(zA ))1−α=B) .
pA θB 1−Φ(zBB−σ1−α(1−Φ(z ))
B

A

(33)

The relative price is the inverse of the ratio of the Solow residuals. If σ A = σ B and θA = θB ,then z A = z B and the relative price is 1. If σ A = σ B but θA ̸= θB , then the relative price19

depends only on the mean productivities, as in the classical Ricardian model. If σ A ̸= σ B ,since the Solow residual is increasing in σ,
Proposition 1. Under autarky, the relative price pB /pA depends only on the capital shareparameter α, the levels θι and variations σ ι of productivities. It does not depend on the
demand conditions. The relative price pB /pA is monotonically increasing in σ A , and monotonically decreasing in σ B .
Under autarky, the relative price is determined by technology. Intuitively, since theamount of capital is adjustable, and is determined by the user cost, labor is eﬀectively the
only input used to determine the relative supply. However, labor is homogeneous, and hence
wage is adjusted to be consistent with relative technology. Consequently, the relative priceis purely determined by technology. This logic is the same as that used for an economy with
the neoclassical production technology (Baxter, 1992).
Adding to the mean productivity, the variance plays a role in the putty-clay setting. Thiseﬀect occurs because, contrary to standard neoclassical production function, productivity isheterogeneous across machines. Less productive machines are not used since they are notprofitable. Due to this industry-level selection, productivity is higher if the distribution isflatter. The parameter σ ι determines the variability of productivity distribution, and hence
the relative price is lower for a sector with higher σ ι .
In this model, however, not just exogenous distribution but also endogenous investment
plays an important role in determining labor productivity. In particular, under autarky,
capital-intensity decision actually attenuates the selection mechanism. A rearrangement of(26) leads to:
1 − Φ(z B )
kB
=
.
kA
1 − Φ(z A )(34)

It is straightforward to obtain the first derivative with respect to σ ι , for example, with respect

20

to σ B :B

d kkA
−ϕ(z B ) dz B
=
< 0.
dσ B
1 − Φ(z A ) dσ B

(35)

An industry with higher variation has lower capital-intensity. The diﬀerence in the laborproductivities across industries is not as large as the diﬀerence in the exogenous productivity
component because in this model the labor-productivity is the product of the underlying
idiosyncratic productivity θiι (ξ) and the capital-intensity term kiι α . The endogenous natureof a capital-intensity decision decreases the diﬀerence between labor-productivities acrossindustries. In the industry with large σ a machine with low labor-productivity will not be
used and hence it is “risky” to make a large capital-intensity investment for an individualmachine. Instead, investment is made mainly in increasing the number of machines (q ι ). On
contrary, in the industry with σ, most of the machines will be used, so that investment isdirected toward developing the labor-productivity of each machine.
The capital-intensity of each machine k ι is not the same as the industry-level capital-laborratio K ι /lι . From equations (26), (28), and (31):
1 ι ι
kq
Kι
α w 1
δ
=
=
.
ι
ι
l
l
1 − α R pk

(36)

The right-hand-side does not depend on industry ι, and hence industry-level capital-laborratios are the same across industries. The relative capital-labor ratio is not correlated withthe relative price. Accordingly, the industry-level capital-labor ratio does not contain any
useful information for predicting the comparative advantage. The important variable istechnology.

2.2.3

Two-country equilibrium without trade cost

Given the autarky relative price, it is straightforward to determine the pattern of specialization when two countries are engaged in costless trade. Two countries (country 1 and 2) are21

distinguished by subscript i = 1, 2. Without trade cost, the price must be equalized acrossA
B
B
countries and hence pA
i = p and pi = p . Suppose that country 2 possesses comparativeA
B
A
advantage in producing good B, i.e., pB
2 /p2 < p1 /p1 . Country 2 then tends to specialize ingood B productions. If the price is so low (due to cheaper imported goods), then even very
productive machines of industry A in country 2 would earn negative profits.14 In this case,the industry-level investment, labor and output are zero: y2A = k2A = q2A = l2A = 0, that is,country 2 completely specializes.

/pA
pB
1 1
/pA
pB
2 2

0

yB
2,autarkyyB
2(2 producing only B)

+y B
yB
2
1
(both producing only B)

Figure 1: World supply of good B
Given this structure, the rest of the equilibrium properties are parallel to the textbook
Ricardian model (e.g., Feenstra, 2004, Chapter 1). The relative price depends on the demandside. Figure 1 illustrates the world supply of good B, which is a typical “stair-step” pattern.
The vertical axis is the relative price. If the world relative price, pB /pA , turns out to besmaller than the autarky relative price in country 2, both countries are fully specialized in
production of good A, and supply of good B is zero. If the relative price with trade is thesame as the autarky relative price in country 2, then country 2 produces both goods, while
country 1 completely specializes in the production of good A. Instead, if the relative priceis larger than the autarky relative price in country 1, both countries produce good B. If theworld relative price is in-between, country 1 specializes in good A and country 2 specializes

14

Formally, for given prices (26) satisfies only if kiι = 0. See Appendix for details.

22good B, and as shown in the Appendix, the relative price is given by:

B

p
=
pA

(

αδ
ϵ + R−αδ1−ϵ)1−α1−Φ(z1A −σ1A )A 1−Φ(z A ) 1−α
θ1 (
1 )
,
B 1−Φ(z2B −σ2B )θ21−α(1−Φ(z2B ))(37)

The results are summarized as:
Proposition 2. The world supply curve in the steady-state has a “stair-step” pattern. The
relative price under the complete specialization is determined by three diﬀerent terms that
individually capture (1) demand condition, (2) mean productivities of the distributions, and(3) relative dispersions of the productivities.
Other properties are also Ricardian. For example, trade improves welfare for both countries, although, in general, the wages are not equalized across countries. In this model,
welfare is precisely calculated by the household utility. In the Appendix, I show that (1)trade is welfare improving for both countries, and (2) the wage diﬀerence across countries
captures the diﬀerence in the utility. Moreover, under the complete specialization, the wage
diﬀerence across countries depends on demand for two goods.

2.2.4

Comparison

The model resembles the standard “2 by 2 by 1” Ricardian trade model, and the “2 by 2 by
2” economy with endogenous capital accumulation model of Baxter (1992).
If the production of each industry is given by a linear-in-labor technology, the model is
identical to the textbook “2 by 2 by 1” Ricardian trade model (e.g., Feenstra, 2004). In
particular, suppose that the industry-level production function is as:yiι = θiι liι ,(38)then the model is that of the classical Ricardian model.15 In this case, the relative price
15

Notice that I follow macro-tradition in that the productivity θiι expresses marginal product of labor23

under the complete specialization is as:
pB
ϵ θ1A
=
.
pA
1 − ϵ θ2B

(39)

The first term captures the demand condition, while the second term captures the supply
condition. In the classical Ricardian model, capital accumulation is abstracted so that
there is no demand associated with investment. The demand is entirely determined byconsumption share parameter in the utility function, ϵ.
The second term captures the eﬀect of technological diﬀerence. If θ1A is much higher than
θ2B , then it is less costly to produce good A than good B, so that the world price of good A
is lower than the world price of good B.
If the industry-level production technology is as:yiι =θiι Kiι α liι 1−α ,(40)where Kiι is total capital stock in the industry, and capital accumulation follows the standard
law of motion:ιι
Ki,t+1= (1 − δ)Ki,t+ investmentιi,t ,(41)

then, this is a case analyzed by Baxter (1992), and the relative price under complete specialization is as:
pB
=
pA

(

αδ
ϵ + R−αδ
1−ϵ

)1−α

θ1A
.
θ2B

(42)

The first term again captures the eﬀect of the demand, while the second term captures thetechnology. Contrary to the classical Ricardian model, the demand term reflects not only
consumption demand but also the demand coming from investment motive, which is αδ/(R−rather than the marginal labor requirement as is common in trade literature.

24αδ). The second term, capturing the technology, is the same as the classical Ricardian model.
This is the finding by Baxter (1992), that is, since the capital is endogenously determined,
technology, not the endowment, is the determinant of the comparative advantage and therelative price under costless trade equilibrium.
Table 1: Comparison of the models
Technology

AutarkyComplete specialization

Linear-in-labor

pB
θiA
i
=
pA
θiB
i

Cobb-Douglas

pB
θiA
i
= B
pA
θi
i

ϵ θ1A
pB
=
pA
1 − ϵ θ2B
(
)1−α
αδ
B
ϵ
+
p
θ1A
R−αδ
=
pA
1−ϵ
θ2B

Putty-clay

1−Φ(ziA −σiA )A 1−Φ(z A ) 1−α
B
θ (
pi
i )
= iB 1−Φ(zB −σB )
A
i
i
pi
θi
1−α
(1−Φ(ziB ))

B

p
=
pA

(

αδ
ϵ + R−αδ

1−ϵ

)1−α

1−Φ(z1A −σ1A )A 1−Φ(z A ) 1−αθ1 (1 )B 1−Φ(z2B −σ2B )θ21−α(1−Φ(z2B ))Notes: i = 1, 2 is country, ι = A, B is industry, ϵ is consumption share of good B in the period-utility,
R = 1/β − 1 + δ, β is subjective discount factor, δ is depreciation rate, α is capital’s share in the
production function, and z is utilization rate (see equation (27)). Complete specialization is a case inwhich country 1 specializes in production of good A, and country 2 specializes in production of good B.

The relative price under costless trade of the putty-clay model is (37), and Table 1
compares these three models. As is clear from Table 1, the implication of these models is
Ricardian. The pattern of comparative advantage is determined by the relative price underautarky, and it is fully captured by the industry-level Solow residuals. In the case of the
putty-clay technology, not only the mean level of productivity (θ) but also the magnitude of
the heterogeneity (σ) contributes to the measured Solow residual.
The relative price under costless trade with perfect specialization is determined by bothdemand and supply conditions. When we compare the linear-in-labor and Cobb-Douglas
production technologies, the diﬀerence is driven by the presence of investment. In the caseof the Cobb-Douglas production technology, technology is homogeneous within country andindustry. The mean productivities are the key indicators capturing the autarky relative
price and the pattern of trade. In the case of the putty-clay technology, selection across25

heterogeneous capital goods additionally aﬀects supply.

2.3

The balanced growth economy

By including capital-embodied technological progress, the model provides additional predictions regarding productivity variation, industry-level productivity, and the vintage of capital
goods. Suppose that technology is improving, and it is embodied in the capital goods; specifι
ically, the mean productivity has a common trend growth, E(θiι (ξ)) = θi,t
= (1 + g)t(1−α) θiι .

Let variables without time subscripts denote their values on the balanced growth path. The
analysis in Section 2.2 is a special case in which g = 0. When g > 0, the calculations are
not as simple as in the case of g = 0. In particular, with trend growth of technology, an oldmachine is generally less productive than a new machine. Accordingly, the cutoﬀ depends
on the vintage of the capital. Since newer machines are in general more productive thanolder machines, the ratio of operation of an “s + 1 periods old” machine is lower than thatof an “s periods old” machine.
ιThe cutoﬀ variable zis
depends on the vintage of the machine:

ι
zi,s
=

1
ln
σiι

(

wi
ι ι ιαp i θ i ki)
+ 0.5σiι +sln(1 + g).
σiι

(43)

Since g > 0, zi,s+1 > zi,s . A higher z means a larger fraction of machines is below theoperational cutoﬀ.
ιAs shown in the Appendix, the condition corresponding to (27) for g > 0 involves zi,sfor s = 1, 2, .... Gilchrist and Williams (2005) show that even when g > 0, the existenceιι
is uniquely
is always ensured. Moreover, they show that under a mild condition zi,sof zi,sdetermined.
ι
is determined, the industry-level output can be expressed as a generalized
Once zi,s26

version of (32):
(∞
∑
s=1

yiι = θiι (g + δ)α

(
( ι
))
(1 + g)−s (1 − δ)s−1 1 − Φ zi,s
− σiι
(∞
∑

(
( ι ))(1 − δ)s−1 1 − Φ zi,s)Kiι α liι 1−α .

)1−α

(44)

s=1
ι
Since zi,sdepends on σiι , the Solow residual depends on σiι .

Figure 2 illustrates the relationship between σ, vintage, and capacity utilization rateι(1 − Φ(zi,s− σiι )).16 If σ is large, the selection works even immediately after installation,that is, some machines are not used even if they are new. As time passes, the utilization
rate gradually declines because these machines are relatively less productive than newer
machines. This relationship between vintage and selection holds regardless of the value ofσ, while the pattern is diﬀerent. If σ is small, productivity variation among machines inthe same vintage is small. Most of the machines are initially used, and the utilization rate
sharply drops once the vintage of machines becomes outdated.
1

1.05

0.7
0.6
0.5
0.4
0.3

8.5

1

8

0.95

7.5

0.9

Solow residual

Avg. vintage (left scale)
Solow residual (right scale)
Average vintage in years

0.8

Capacity utilization rate9

σ=0.05σ=0.20σ=0.35

0.9

0.2
0.1
0

0

5

10

15

20

25

30

35

7
0.05

40

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.85
0.5

Sigma

Vintage in years

Figure 3: Sigma, average vintage, and Solow
residual

Figure 2: Sigma, vintage, and capacity utilization rates

16

The calibration strategy basically follows Gilchrist and Williams (2000, 2005). One period of the model
is quarter. Parameter values are as follows. The capital-share in the production is 0.3 (α = 0.3). The annual
depreciation rate is 8% (δ = 1 − (1 − 0.08)1/4 ). The annual trend growth rate is 2% (g = (1 + 0.02)1/4 − 1).
The subjective discount factor is determined so that the implied annual real interest rate is 4% if g = 0(β = (1/1.04)1/4 ). The share of leisure in the utility function is 0.3 (ψ = 0.3) when I include labor-leisurechoice. The mean productivity is set to 1 (θ = 1).

27

One important factor implicit in Figure 2 is the exogenous retirement by δ. As time
passes, more machines break. The total mass of old machines is not large, however. In
Figure 3, the dotted-line with left-axis-scale shows the average vintage of operating machines
for diﬀerent values of σ. When σ is small, most of the machines are active until a certain
age. Due to the exogenous retirement, the average vintage is about 8.5 years if σ = 0.05.
As σ increases, the selection contributes to the industry-level productivity. Less productive
machines are not used even in the initial period, and as time passes, more and more machines
are replaced by younger machines. The average vintage is even smaller than the case of asmall σ.
The solid-line with right-axis scale of Figure 3 illustrates the relationship between σ andthe industry-level Solow residual calculated by (44). When σ is small, the selection does notcontribute to the industry-level productivity. If σ is large, the industry-level productivity isobserved to be large, thanks to the selection.
In summary, if σ is large, the selection eliminates less productive machines from operation.
The operating machines are young and more productive, and hence the average vintage ofactive machines is small. An industry using young machines tends on average to be moreproductive than other industries.
The rest of the analysis is basically the same as the case of g = 0. If an industry has
higher σ than another, the productivity is higher. Observationally, an industry is moreproductive if the industry uses young machines on average and/or the capacity utilization
rate of industry is low. If productivity is high, the relative price of the good is low underautarky. This relative price is an indicator of trade patterns when countries are engaged in
trade.

28

3

Fixed trade cost and sorting implication

This section presents an extension of the model. The extended model describes a sorting
situation in which the most productive machines export, the moderately productive machinesserve the domestic market, and the least productive machines do not operate. Since newer
vintage machines are in general more productive, these machines are more likely to be used
for exporting. The gains from trade in the extended model are the combined and interactive
results of the reallocation across industries by exploiting the comparative advantage, and of
the reallocation across machines within an industry.
In Section 2, good A is a pure consumption good, while good B is used both for consumption and investment. To simplify the analysis with fixed trade cost, this section starts froma modified version of the model. Both goods are intermediate inputs used for producing thenon-tradeable final goods. The production of the final goods is given by an equally-weighted
B
Cobb-Douglas production function, ln Yi,t = 0.5 ln cA
i,t + 0.5 ln ci,t . The producers of the fiA
B B
nal goods face perfect competition and their profits are Pi,t Yi,t − pAi,t ci,t − pi,t ci,t . The final
A A
B B
goods are used for the consumption and investment, Yi,t = Ci,t + ki,tqi,t + ki,tqi,t . Finally,

the consumer’s period utility is given by the log of consumption and leisure, u(Ci,t , Li,t ) =ψ ln Ci,t + (1 − ψ) ln(1 − Li,t ).

3.1

Autarky and costless trade

The implications of the closed economy or costless trade are almost identical to the previous
model; as I discussed in the previous section, demand condition plays a minimal role underautarky and costless trade equilibria. The relative price under autarky is determined bytechnology. For example, in the closed-economy of g = 0, the relative price across industries
is given by (33).
Under costless trade, the relative price possibly depends on demand condition. If bothcountries specialize in producing one of the two goods, the relative price is given by a modified

29version of (37). Specifically, the first term, which captures relative demand for two goods,
becomes unity due to the symmetry assumption of the two goods.

3.2

Investment decision

I introduce a trade cost via fixed labor cost. Namely, to deliver the output yiι (ξ) of one
machine to the foreign country, the machine requires 1 + τ units of domestic labor insteadof one unit of labor for the domestic delivery. Owing to this fixed trade cost, price is notequalized across countries.
In the rest of this section, I will focus on a symmetric situation in which one country
possesses comparative advantage in one industry, while the other country possesses advantage in the other industry. Moreover, the source of the comparative advantage is purely fromthe variation term. The mean productivities are assumed to be the same across countries
and industries, and variance is symmetric: σ1A = σ2B > σ1B = σ2A . Under these assumptions,
it is suﬃcient to analyze two types of industries: an industry with comparative advantage(exporting industry) and one without comparative advantage (non-exporting industry). Instead of using country subscripts and industry superscripts, I use superscripts f , d, o, andn, where f is exported values and foreign prices of exporting industry, d is domesticallyconsumed values and domestic prices of exporting industry, o is exporting industry, and n is
non-exporting industry. By considering a case in which country 1 has comparative advantage
in good A, for example, in country 1, capital-intensity in industry A is attached with o as
k o . Similarly, y o represents the total production of the exporting industry, and it is sum of
the production for domestic supplies (y d ) and foreign supplies (y f ), y o = y d + y f . The priceof the exporting good in the home country is pd , and the price of the exporting good in theforeign country is pf . Let ρ be the ratio of the two minus one, pf = (1 + ρ)pd .
One might think that ρ = τ , (i.e., pf = (1 + τ )pd ), as is typical in iceberg trade costmodels. However, this is not true. The reason is that if ρ ≥ τ , all the machines are used
for exporting, and domestic supply is zero. This domestic supply shortfall pushes up the30domestic price, and shrinks price ratio ρ. Hence, an open-economy equilibrium is consistent
with ρ < τ .
When ρ < τ , the profits from exporting and domestic sales are diﬀerent, and machines
are sorted by their productivities θo (ξ). If a machine is suﬃciently productive to satisfy

pft+s θto (ξ)kto α − (1 + τ )wt+s > pdt+s θto (ξ)kto α − wt+s ,(45)then the machine is used for exporting operation. If a machine is not suﬃciently productive
to satisfy the inequality but covers operational costs (i.e., wages), it is still used for domestic
sales.
Net income by exporting
Net income by domestic sales

p f θ(ξ) koα - (1+τ)w →Net income← p d θ(ξ) koα - w

0-w-(1+τ)w
←

Idle

→←

→←
Domestic
Productivity (θ(ξ))

Exporting→Figure 4: Trade cost, operation decision, and sorting
Figure 4 depicts this sorting situation. The horizontal axis shows the productivity ofeach machine, and the vertical axis shows the net income from operation. The solid-line
shows net income from domestic operation. The machine operates if net income is greaterthan zero, and is idled if net income is negative. The dotted-line is net income if a machineis used for exporting operation. The net income is negative even at the domestic operational31

cutoﬀ, owing to trade cost. However, if a machine is highly productive, it covers trade costs
and earns larger profits from exports than domestic sales. This sorting pattern is closely
related to findings in micro-level empirical trade literature (e.g., Bernard and Jensen, 1999).
The investment decision problem takes this sorting into account:max− Pt kto + Et
o
kt

(∫

×

mt,t+s (1 − δ)s−1

s=1

(

∞

τ wt+soαρt+s pdt+s ktτ wt+soαρt+s pdt+s kt

∫
+

[ ∞
∑

wt+s
oα
pd
t+s ktpft+s θto (ξ)kto α − (1 + τ )wt+s)fΘ (θto (ξ)|wt+s , pdt+s , pft+s , θto )dθto (ξ)

)]
( d o
)
f
oα
o
d
o
o
pt+s θt (ξ)kt − wt+s fΘ (θt (ξ)|wt+s , pt+s , pt+s , θt )dθt (ξ) .

(46)

For each vintage s, there are two cut-oﬀs, zsd for domestic operation, and zsf for exporting.
As shown in the Appendix, on the balanced growth path, two cutoﬀs are linked by zsf =zsd + σ1o ln(τ /ρ). In the case of g = 0, two cutoﬀs z d and z f are summarized by the following
equation, which is an extension of (27):1−α=ρ ϕ(z f − σ o )τ (1 − Φ(z f )) + 1 − Φ(z d ).
τ ϕ(z f ) ρ (1 − Φ(z f − σ o )) + 1 − Φ(z d − σ o )(47)Note that ρ is an endogenous variable, which ultimately depends on z f . Hence, theequilibrium is a fixed point for solving ρ, z d and z f , which are consistent with (47) and othersteady-state equilibrium conditions. If g > 0, a condition extending (47) includes zsd and zsf(see the Appendix), but the equilibrium is similarly calculated.
I solve the equilibrium numerically.17 The mean productivities are θo = θn = θiι = 1 for
all i and ι. Gilchrist and Williams (2005) set σ = 0.2 for their baseline analysis. Here I
set σ o (= σ1A = σ2B ) = 0.35, and σ n (= σ1B = σ2A ) = 0.15. From the results obtained in theprevious sections, country 1 possesses comparative and absolute advantages for producing
good A, and country 2 possesses comparative and absolute advantages for producing good17

See footnote 16 for other parameter values.

32

B. For trade cost, I examine the range of τ between 0 and 0.2. Under this parameterization,
it turns out that τ = 0.2 gives results almost identical to those of autarky equilibrium.

3.3

Vintage, productivity, and operating status

Figure 5 shows the distribution of machines of an exporting industry. The vertical axis shows
quantity of machines, while the horizontal axis shows the labor-productivity of machines,
which is the product of idiosyncratic productivity (θ(ξ)), the contribution of trend growth(1 + g)−s , and capital-intensity (k α ). The distribution of newly installed machines (the solidline) is on the right of the distributions of 10-year and 20-year-old machines indicated by,respectively, the broken and dotted lines. New (small s) machines are, in general, moreproductive than old (large s) machines. The quantity of machine of old machine is small,reflecting the exogenous retirement.
New machines are more productive, and hence these are more likely to be in operation.
Furthermore, among new machines, top-tier machines are used for exporting. Old machines
are not only few due to retirement but also less productive due to embodied vintage technology. This sorting for diﬀerent vintages is also illustrated by Figure 6, which shows therelationship between vintage and operational status. The figure shows that among new
machines, approximately the top 5% are used for exporting, the middle 70% are used for domestic supply, and the bottom 25% are idle. The fractions of exporting and domestic supply
decrease as machines age. Among 20-year-old machines, the top 35% are in operation, butalmost none of these machines are used for exporting.18
From these figures, the model predicts that (1) more productive machines are more likely
used for exporting, (2) newer machines are more likely used for exporting, (3) if an oldmachine is used for exporting, the machine was initially extremely productive.
In the light of the literature, (1) is routinely described using a type of Melitz (2003) model.
However, the baseline Melitz (2003) model is silent about the relationship between opera18

These numbers depend on parameters, but qualitative implications are robust under reasonable calibrations.

33

8
7

×10-3
←

1 exporting

Idle

→←

Domestic→←

Exporting→
Proportion of machines

Quantity of machine

0.8

6
5
New
10-year-old
20-year-old

4
3
2

domestic

0.6

0.4
idle
0.2

1
0

0

0

0.5

1

1.5
2
2.5
Productivity of machine

3

3.5

4

5

10
Vitange (in years)

15

20

Figure 6: Vintage and and operation status

Figure 5: Distribution of machines

tional status and vintage. My model oﬀers new insights regarding vintage and operational
status.
In this model, a unit of production is a machine. An empirical problem is how to map
the model to data. A simple approach is to consider this production unit as a firm (orplant). Then, the model’s prediction regarding operational status of a machine is regarded
as operational status of the firm. Another, and probably more reasonable, approach is tothink that firm is a collection of multiple machines.
A simple way to include a firm in the model is to follow a three-step procedure. First,
in each period, households determine the total investment of capital-intensity and numberof machines. Second, each firm draws from a lottery, which specifies the number of machines to be installed for each firm. Third, idiosyncratic productivity is assigned for eachmachine. If financial and labor markets are perfect, the investment decision of the households is purely determined by the stochastic discount factor, and is independent from theinter-firm allocations. Then, the aggregate properties of the equilibrium are the same as theprevious analysis. In this world, some firm have many machines, while other firms have only
a few machines. Some firms have new and productive machines, other firms have old but
productive machines, and still other firms have old and unproductive machines.
This modified model gives not only standard but also several new predictions regarding34

the firm characteristics and exporting status. The model predicts that the standard relationship between productivity and operation status: (4) more productive firms are likely
exporters, and (5) export intensity is higher for more productive firms. In addition, themodel provides three new predictions with respect to the vintage of machines and exporting status: (6) a firm with newer vintage machines is more likely to be an exporter, (7)among exporters, export intensity is higher for the firm with newer vintage machines, and(8) a firm starts exporting after an installation of new and productive machines.19 Future
investigations to test these predictions are expected.

3.4

Welfare eﬀects of trade cost reduction

There are two possible types of equilibrium. One is that both countries completely specialize.
This equilibrium is realized if trade cost is small. In particular, if trade cost is zero, thesituation is then exactly the same as costless trade equilibrium. The other equilibrium exists
when both countries produce both goods. This is the case if trade cost is large. When trade
cost is extremely large, the model is eﬀectively the same as autarky. As trade cost decreases,
highly productive machines start to export (as described by Figure 4). However, since thetotal number of such productive machines is small, the supply of exporting goods is smaller
than the demand. This imbalance drives up the price of the goods, and stimulates domestic
production. That is, a country lacking comparative advantage in this good continues to
produce.
Figure 7 shows the realized equilibrium values for diﬀerent fixed trade cost τ . In thepanels, the horizontal axis represents values of τ , while the vertical axis represents othervariables. If τ = 0, the economy is the same as the costless trade. Similarly, if τ is large, theeconomy is eﬀectively the same as autarky.
The top-left panel of Figure 7 shows the utilization ratios of newly installed machines(s = 1) for various operations (ratio of activated, domestic, and exporting operation to total
19

I extend appreciatation to one of the referees for encouraging me to include capital-embodied technology
progress, and suggesting implications (6) and (7).

35

8

1
0.8

6

0.6
4
ratio f
d
ratio
o
ratio
ratio n

0.4
0.2
0

0

0.05

0.1

2

0.15

0

0.2

o

k
kn

0

0.05

0.1

12.2040.01
0.008

12.202

0.006

12.2

0.004

12.1980.002

12.196

0

0

0.05

0.1

0.15

0.2

utility

o

q
qn

0.15

0.2

12.194

0

0.05

0.1

0.15

0.2

Figure 7: Eﬀects of change in trade costs
existing machines of the vintage). Under autarky (τ = 0.2), no machines are used for exporting operation (ratiof , the dash-dotted-line). As trade cost decreases, more machines areused for exporting, and fewer are used for domestic supply (ratiod , the solid-line). Capacity
utilization rate of this industry (ratioo (= ratiof + ratiod , the dotted-line) increases as τ
decreases until a critical value, and then silightly decreases. This critical value divides thetwo types of equilibria, whether n-industry is active or not. The utilization ratio of machines
used in n-industry (ration ) is high (approximately 0.999) when the industry is active.
The top-right panel of Figure 7 shows the capital-intensity of o- and n-industry, and thebottom-left panel shows the quantities of machines. Starting from autarky (right-end), thecapital-intensity of o-industry (k o , the solid-line) gradually increases as trade cost decreases,
and then decreases. As trade cost decreases, more resources are available for o-industry.
A rapid increase in the machines used for exporting is accompanied by a rapid increase ininvestment in o-industry. The capital-intensity of n-industry (k n , the dashed-line) remains

36

nearly flat until the critical value. Under autarky, the capital-intensity is higher in n-industry
than in o-industry, as discussed in Section 2.
The quantity of machines of n-industry (q n ) gradually decreases as trade cost decreases.
Since more foreign products are supplied, the production of n-goods in this country decreases. This internal supply adjustment occurs through the number of machines since
capital-intensity of this industry remains unchanged. In this phase, a source of aggregate
welfare gain is this inter-industry reallocation.
Once the inter-industry reallocation ends, and two countries completely specialize, theselection among machines is the only source of industry-level productivity gain. As trade
cost decreases, capacity utilization rate decreases capital-intensity decreases and the numberof machines increases.
A reduction in trade cost definitely increases welfare, but how it changes depends on thespecialization phases. As shown in the bottom-right panel of Figure 7, the utility monotonically increases as trade cost decreases. The change in the phase changes the slope of thecurve. In particular, the welfare gain is largest in the timing of large increase in exporting.
In summary, a reduction of trade cost has two diﬀerent types of impacts: inter-industry
reallocation and intra-industry reallocation. Starting from autarky, a reduction in trade costcauses inter-industry reallocation. More machines are created in the comparative advantage
industry. The pattern is further amplified by an increase in the capital-intensity of thisindustry. At the same time, relatively less productive machines are also activated in theindustry with comparative advantage. Once countries completely specialize, less productive
machines fall idle. This change of the phase leads to a nonlinear impact of trade cost
reduction on welfare. Since nonlinearity is not considered in models analyzed by Arkolakis
et al. (2012), their finding that gains from trade are unexpectedly low in the standard trademodels might, indeed, be explained by the role of nonlinearlity. Exploring this possibility is
an important future research topic.

37

4Conclusion

This paper considers the role of investment and heterogeneity in productivity across capital
goods on the pattern of international trade. Specifically, I introduce putty-clay production
technology in the style of Gilchrist and Williams (2000, 2005) into an international trademodel considered by Baxter (1992).
Motivated by several micro-studies, I assume capital goods are heterogeneous due toidiosyncratic productivity variation and vintage. In a two-country, two-good, two-factor
model, trade pattern is generally described by Ricardian comparative advantage. However,I find a new source of comparative advantage: the magnitude of capital goods heterogeneity.
A country has comparative advantage in an industry with relatively larger capital goods variation. Accordingly, the model predicts that an industry is more likely to have comparativeadvantage if the industry uses on average new machines, the industry’s capital utilization
rate is low, and capital goods in the industry has large productivity variation.
With fixed trade cost, the model describes a sorting situation in which the most productive machines (which are generally newer vintage) export, the moderately productive
machines serve the domestic market, and the least productive machines (old) do not operate. The model gives novel predictions regarding vintage and exporting status: a firm withnewer vintage machines is more likely to be an exporter, among exporters, export intensity is higher for the firm with newer vintage machines, and a firm starts exporting after
an installation of new and productive machines. The gains from trade are the combined
and interactive results of the reallocation across industries by exploiting the comparativeadvantage, and of the reallocation across production units within an industry. As a result,a change in the trade cost has nonlinear implications on the change in the welfare.
The steady-state implications of the investment decision and the aggregation results ofthe putty-clay model are suﬃciently simple to incorporate a Ricardian model, although theunderlying investment decision is suﬃciently rich to capture the essential features of capital
goods heterogeneity. This paper presents an extension including fixed cost, and there are38many possibilities to extend the model. One direction is to apply the framework to humancapital investment instead of physical capital investment. Future research is expected.

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43