Capital Accumulation and International Trade

67 Pages·2017·1.03 MB·English

Capital Accumulation and
∗
International Trade
Fernando Alvarez
University of Chicago and NBER
September 18, 2017
Abstract
Capital accumulation is introduced into a version of Eaton-Kortum model of inter-
national trade, imposing period by period balanced trade. The effects of tariff changes
on world steady states and transition dynamics are studied. A calibrated version of
the model is used to assess the short- and long-run gains from a world-wide elimination
of trade tariffs. The determinants and importance of convergence in world-wide capi-
tal as well as convergence on the relative capitals and incomes are analyzed. Positive
and normative comparisons with an analogous static model are conducted, as well as
comparisons steady state welfare comparisons vs full dynamic gains.
JEL Classification Numbers: E5
Key Words: Capital Accumulation, International Trade, Dynamics, Convergence, Tar-
iffs.
∗Thetitleofthispaperandsomeofitsmaterialistakenfroma2009draftcirculated(butneverpublished)
with Robert E. Lucas, Jr. Bob has contributed fundamentally to some of the main ideas and results of this
2009 paper but he felt that his lack of involvement in the current, quite different paper should be reflected
in his withdrawal from the title. I thank Bob for his ideas, encouragement and generosity. Of course, all
the errors and shortcomings are, quite literally, only mine. I thank Lorenzo Caliendo, Jaume Ventura, and
Jonathan Eaton for excellent comments. Finally, I thank Francisca Sara-Zaror for her research assistance.
Corresponding author: Fernando Alvarez, 1126 E. 59th Street, Chicago, IL 60637, phone [phone],
[email protected] .
1 Introduction
1
Capital accumulation is introduced into the Eaton and Kortum (2002) model of trade,
2
modified as in Alvarez and Lucas (2007). The Eaton-Kortum model has a single primary
3
input: non-tradeable labor. In Alvarez and Lucas (2007) labor was interpreted as “equipped
4
labor” and identified the income of this factor with value added. This broadened view
5
enabled to calibrate the model realistically to U.S. national income and product data, but
6
did not give a framework for analyzing any genuine dynamics. In this paper I add physical
7
capital as a second primary input, also assumed non-tradable, and add investment goods
8
as a second final good, along with consumption. International trade occurs in continuum
9
of intermediate goods, modeled as in Alvarez and Lucas (2007). Intertemporal preferences
10
and a law of motion for capital are taken from standard growth theory. It is assumed that
11
countries must balance trade continuously, i.e. no borrowing or lending is allowed. The
12
combined model describes the equilibrium of a world of n economies evolving over time as a
13
system of autonomous differential equations.
14
Section 2 considers capital accumulation into a single, autarchic economy, with the in-
15
dustrial structure and commodity space of the Eaton-Kortum and Alvarez-Lucas setup,
16
obtaining the one-sector model of economic growth that is the basis for the analysis to fol-
17
low. Section 3 sets up the notation for a world economy of n such economies and discuss
18
the general mathematical structure of the theory. Sections 5-7 study income “convergence”
19
in a world of different economies, under the assumption of costless trade. In this context,
20
the existence and uniqueness of equilibrium steady states can be established quite easily and
21
generally, as done in in Section 4. Steady states have a very similar nature as equilibrium for
22
a static economy without capital. Section 5 examines the dynamics of a small, open economy
23
when the rest of the world is at stead state. A small open economy has dynamics similar, but
24
not identical, to the ones from the one sector closed economy growth model. The speed of
25
convergenceoftheonecountrygrowthmodelofSection2, asitiswellknown, iscontrolledby
26
a race between intertemporal willingness to substitute consumption and the elasticity of the
27
1
marginal productivity of capital with respect to an additional unit of capital. Interestingly,
28
the elasticity of the marginal productivity of capital is higher for the small open economy,
29
since an additional unit of capital expand production, as in the closed economy but this
30
expansion has the additional effect of depressing the prices of the products the small open
31
economy sells. As a consequence, the small open economy has a higher speed of convergence
32
than the one sector closed economy growth model. Section 6 examines the stability of a
33
n-country steady state in a symmetric world. The speed and nature of adjustment to steady
34
state in a world of n countries, whose state is the vector on the n capital stocks for each
35
country, is dictated by two considerations, i.e. it is dictated by two distinct eigenvalues. One
36
eigenvalue is the same as in the standard one country closed economy growth model –which
37
is natural since then entire world is a closed economy– and controls the convergence of the
38
average world-wide capital to its steady state. The other eigenvalue is the same as in the
39
small open economy –which is natural since in a world with countries with different capital
40
levels there are further incentive for international trade– and controls the convergence of
41
each country level of capital relative to the average world-wide capital.
42
The models analyzed in Section 2 to Section 6 are selected to reveal the structure of
43
the equilibrium dynamics in the clearest possible way, free of unnecessary complications.
44
For calibration and policy simulations trade costs (i.e. tariffs) and differential content of
45
tradeable inputs in the production of investment goods relative to consumption goods are
46
incorporated. The first element is required to study a trade liberalization. The second
47
element, a feature of actual economies, is important to obtain a realistic estimate of the long-
48
run response to reductions in trade costs, since this feature implies a permanent reduction
49
in the relative price of investment to consumption goods. This version of the model, its
50
calibration, as well as positive and normative exercises are in Section 7. In particular, I study
51
the path of adjustment for the whole world after a trade liberalization –i.e. the dynamics
52
for an entire world starting at a steady state calibrated to the observed heterogeneous tariffs
53
and GDP size, and whose tariffs are permanently reduced to zero for all countries. Following
54
2
the theoretical decomposition of the dynamics, I study this liberalization first in a model
55
with all countries are assumed to be identical, and later in one with a realistic level of
56
heterogeneity in tariffs and country size. The welfare implications in the dynamic model
57
that properly takes the transition path into account is compared with an analogous static
58
modelwithoutcapitalaccumulation, andalsocomparedwiththesteadystateofthedynamic
59
model. For the average country, the steady state comparisons vastly exaggerates the welfare
60
gains relative to the welfare gains properly computed taking the transition into account.
61
Instead, the static model underestimate the welfare gains relative to the dynamic model, but
62
they are much closer. This is true even though the prediction for the volume of trade (i.e.
63
trade to GDP) is almost identical between the steady state comparison and the static model
64
without capital accumulation. Furthermore, the pattern of welfare gains, as a function of the
65
pre-trade liberalization tariff of each country, are similar for the static and dynamic model
66
that takes the transition into account. Yet these patterns and magnitudes are different if one
67
compares the welfare gains of each country as a function of the pre-trade liberalization tariff
68
across steady states. As a summary, the dynamic model implies larger property measured
69
welfare gains than the static one, and, as in many related setups, the steady state welfare
70
comparisons prove to be misleading.
71
The analytical results that extend the treatment of Section 2-Section 6 to the more
72
realistic setup can be found in Appendix A to Appendix D. Among these results are the
73
effects of the average size of each country and of the trade costs on the speed of convergence
74
on the relative capitals –see Proposition 6, the effect of the higher tradable component
75
on the production of capital goods relative to consumption goods on the transitions and
76
steady states –see Proposition 4 and Proposition 7, and the behavior the relative price of
77
consumption to tradeables goods and the relative price of investment to consumption goods
78
during the transition –see again Proposition 7. Conclusions are offered in Section 8.
79
3
1.1 Related research
80
This paper is a contribution to the research on integrated models of trade and growth dating
81
back at least to Stiglitz (1970) and including Chen (1992), Ventura (1997), Atkeson and
82
Kehoe (2000), Acemoglu and Ventura (2002), Bajona and Kehoe (2010), Caliendo (2011)
83
and Eaton et al. (2016). With the exceptions of Acemoglu and Ventura (2002) and Eaton
84
et al. (2016), discussed below, all of these earlier papers combine versions of the two-factor
85
Heckscher-Ohlintypemodelwithendogenouscapitalaccumulation. Withacommontechnol-
86
ogy and preferences, capital-labor ratios in different countries will converge to common levels
87
under autarky. The goal of these analyses is to characterize the way that trade—assumed
88
to be costless and continuously balanced— affects the steady state and the transition paths
89
of the world economy. As it is well known, factor price equalization can occur in these
90
models without equalization in capital-labor ratios, and the nature of the dynamics depends
91
critically on whether or not the world economy is in the “cone of diversification” (the region
92
of factor price equalization). Ventura (1997) is based on assumptions that ensure that the
93
cone of diversification is the entire non-negative orthant. Chen (1992), Atkeson and Kehoe
94
(2000), Bajona and Kehoe (2010) and Caliendo (2011) adopt weaker assumptions so they
95
can examine behavior in and outside of the cone and the transitions between. These studies
96
differ in many details with different results, but in instances where long-run behavior can
97
be characterized, the finding is that factor prices are ultimately equalized but capital-labor
98
ratiosneednotbe. Atagenerallevel, thesemodelslinkscapitalaccumulationandtradewith
99
structural transformation, i.e. they link the composition of output in sectors with different
100
capital intensity with the overall level of capital per worker.
101
In contrast, the present paper is based on the Eaton and Kortum (2002) model, in the
102
particular form described in A/L. The dynamic analysis carried to completion in Section 6
103
and Section 7 of this paper is conducted under assumptions similar to used used in the
104
dynamic Heckscher-Ohlin models: overall technologies and preferences are symmetric across
105
countries, there are no trade costs, and current accounts are always balanced. But here
106
4
there is a continuum of traded intermediate goods, and countries differ in their technologies
107
for producing any particular one of them. There are gains from trade even when countries
108
have identical capital-labor ratios, and factor prices will not be exactly equalized except
109
when capital-labor ratios are. The unique steady state of this economy is locally stable and
110
factor prices are equalized at and only at the steady state. Away from the steady state
111
the transition dynamics are governed by two negative characteristic roots, one describing
112
the approach of the world average stock to its long-run level and the other describing the
113
approach of any individual country’s capital stock to the world average.
114
In this paper, and in most of those in the Heckscher-Ohlin tradition, there is no engine
115
of growth and capital stocks converge to constant levels. The models can easily be modified
116
to incorporate common, constant, exogenous rates of population growth or technological
117
change or both. An exception is Ventura (1997), which follows Jones and Manuelli (1990) in
118
assuming a marginal product of capital bounded away from zero. The world economy does
119
nothaveasteadystateandgrowthinalleconomiesremainsstrictlypositive. Asymptotically,
120
growth rates converge to zero.
121
Acemoglu and Ventura (2002) study of capital accumulation in world of many countries
122
linked through trade also departs from the Heckscher-Ohlin tradition. In their model, trade
123
is motivated by Armington (1969) preferences, in which consumption in any one country
124
is an aggregate over a continuum of country-specific intermediate goods. At a country-
125
level this aspect of their formulation is essentially interchangeable with Eaton and Kortum’s
126
formulation, and hence with my model. Nonetheless, they study capital accumulation in a
127
world consisting of many “AK” economies, in which goods production is linear in a single,
128
capital input –although this is later relaxed. In their equilibrium, the relative capital stocks
129
and income levels in any two countries remain constant over time, even if they are identical
130
in all other respects.1 Yet my characterizations have a very similar flavor, especially so for
131
the small open economy case.
132
1There are other small differences, such as absence of intermediate goods, log utility vs isoelastic utiity,
the modeling of tariffs and trade cost, and whether the set of goods is endogenous.
5
Finally, Eaton et al. (2016) consider a general multi-sector version of the Eaton and
133
Kortum (2002) model with capital accumulation. A key difference between with the model
134
in Eaton et al. and the one in this paper, is that they allow for unrestricted borrowing
135
and lending across countries.2 The results of both papers are complementary, since they are
136
arguably extreme versions of the degree of feasible intertemporal trade in actual economies.
137
Additionally while Eaton et al. (2016) adapt the methods in Kehoe et al. (forthcoming) to
138
numerically solve for exact equilibrium path of a multi-country world, this paper focuses
139
on an analytical characterization of local equilibrium dynamics. A recent paper with a
140
similar approach is Ravikumar et al. (2017), which numerically computes transitions with
141
and without balanced trade in a closely related model. Overall, leaving aside the modeling
142
of a realistic degree of intertemporal trade, the extreme assumption of continuous balanced
143
trade imposed in this paper has the disadvantage that important issues such as the dynamics
144
effects of “global imbalances”, and the degree of the Feldstein-Horioka puzzle –as addressed
145
for instance in Eaton et al. (2016), Eaton et al. (2016), and Kehoe et al. (forthcoming)–
146
cannot be considered.
147
2 Growth in a closed economy
148
The first step is to introduce capital goods into the closed economy described in Section 2
149
of Alvarez and Lucas (2007) (abbreviate as A/L). In this paper, as in the earlier one, there
150
is a single, non-traded final good and a continuum of tradeable intermediate goods, each
151
produced using a common, Cobb-Douglas production function but with different intercepts.
152
The intercept for any given tradeable good is given by x−θ, where θ > 0 and where x is a
153
random variable, independently drawn for each good from an exponential distribution with
154
parameter λ.3 Each good is identified by its productivity level, referring to “good x.” If q(x)
155
2Thereareotherdifferences. Eatonetal.(2016)modelhasdifferenttypesofcapitalineachcountry, and
a form of adjustment cost on investment. They don’t consider tariffs, but allow trade costs.
3That is, the random variable y =x−θ has a Frechet distribution with parameters (λ,θ).
6
unitsofeachgoodxareproducedatdatet, defineq tobetheSpence-Dixit-Stiglitzaggregate
156
(cid:20)(cid:90) ∞ (cid:21)η/(η−1)
q = λe−λxq(x)1−1/ηdx . (1)
0
This aggregate has two uses: as an input in the production of the non-tradeable final good,
157
y, and as an input in the production of each tradeable intermediate good x, q (x). Any
158 m
individual tradeable x affects production only through its effect on the aggregate q.
159
In the closed economy we have
160
(cid:90) ∞
q ≡ λe−λxq (x)dx (2)
m m
0
and
161
q = q +q . (3)
m f
Let capital per person be k = K/L. The law of motion for capital is assumed to be
162
dk
= z −δk, (4)
dt
where z is gross investment.
163
Each person has one unit of labor, of which s is employed in the production of the final
164 f
good and s (x) is employed in the production of good x. Define
165 m
(cid:90) ∞
s = λe−λxs (x)dx. (5)
m m
0
It is assumed that
166
s +s = 1. (6)
f m
The production functions for the final good involves labor, capital, and intermediate goods:
167
168
y = (cid:0)s1−ϕkϕ(cid:1)αq1−α. (7)
f f f
7
The final good can be used as consumption or investment,
169
y = c+z. (8)
Appendix A considers the case where investment and consumption are not perfect substi-
170
tutes. In particular, investment and consumption are allowed to have a different share of
171
intermediate goods.4
172
The production functions for each intermediate good x involve the same three inputs:
173
q(x) = x−θ(cid:2)(s (x))1−ϕk (x)ϕ(cid:3)β[q (x)]1−β. (9)
m m m
The assumption that the relative share parameter ϕ is the same in all these technologies
174
implies that
175
k k (x)
f m
k = = (10)
s s (x)
f m
for all x, and we can write
176
y = (s kϕ)αq1−α, (11)
f f
and
177
q(x) = x−θ[s (x)(k(x))ϕ]β[q (x)]1−β. (12)
m m
Capital is treated as owned by households and rented to firms. Then we can think of
178
a household at date t as endowed with kϕ units of a composite “equipped labor”, rented to
179
firms at a competitive price w. In the temporary equilibrium established at each date t, the
180
pricesp,p (x),andp ofthefinalgood, ofeachtradeablex, andofthetradeableaggregate,
181 m m
are all determined as multiples of w by the cost-minimizing behavior of competitive firms.
182
4This differential share will be important for the model with international trade, since it is assumed that
intermediaries are tradeable.
8
In particular, the price of the tradeable aggregate is
183
p = χ λ−θ/βw, (13)
m m
and the price of the final good is
184
p = χwαp 1−α, (14)
m
where the parameters χ and χ depend on the technology parameters α, and β.
185 m
The intertemporal decisions in this economy are made entirely by households. Everyone
186
has the same preferences:
187
(cid:90) ∞ c1−σ
e−ρt u(c(t)) dt, where u(c) = (15)
1−σ
0
with σ > 0.
188
In each period households sell kϕ units of equipped labor at price w, and buy y units
189
of the goods at price p. Thus in the absence of taxes and transfers,
190
w
c+z = kϕ. (16)
p
must hold for all t. The equilibrium relative price w/p depends only on technological param-
191
eters. The problem for the household is to maximize equation (15) by choice of paths for
192
consumption c, investment z, and capital k, subject to equation (4) and (16), and given the
193
initial capital holdings k(0).
194
The solution of this problem must satisfy the familiar Euler equation
195
(cid:20) (cid:21)
1dc 1 w
= ϕkϕ−1 −(ρ+δ) . (17)
c dt σ p
9