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How Important is Technology Capital?
Measurement and Theory
Marek Kapicka∗
Department of Economics, UC Santa Barbara
March 27, 2008, ver. 0.4.2
Abstract
I construct the measures of technology capital and country openness, as
introduced by McGrattan and Prescott (2007a), for the US economy and the
rest of the world for 1976-2005. The key assumption behind the estimation
is that the net rates of return on both tangible and technology capital are
equalized within a country.
The ratio of technology capital and tangible capital is estimated to be about
one fourth for both the US economy and for the rest of the world. US economy
is also found to be less open than the rest of the world, although the difference
has been diminishing, especially before 1990.
I use the estimates to show that the neoclassical growth model with technology capital successfully explains movements in foreign direct investment in the
last thirty years. While the welfare losses from totally closing US economy are
found to be only about 0.13% of consumption, the welfare gains from totally
opening US economy are much larger: they are equal to 6.9% of consumption.
∗
Comments welcome at [email protected].
1
1. INTRODUCTION
Recently, McGrattan and Prescott (2007a) have extended the neoclassical growth
model by introducing the concept of technology capital. The theoretical framework
incorporates two seemingly incompatible features: technology capital is costlessly
replicated at all locations at which a firm operates, but at the same time the aggregate production function exhibits constant returns to scale in relevant inputs and
a standard general competitive analysis applies. Depending on country openness,
technology capital can be also replicated abroad through foreign direct investment.
The model thus offers a promising new mechanism through which (more or less open)
economies interact.
However, in order to develop sensible quantitative implications of the model, one
needs to have estimates of two of the main ingredients of the model: the stock of
technology capital, an the degree of openness of an economy. Both of them are
directly unobservable in the data. In this paper I provide an estimation procedure
that uncovers both from the data, and estimate them for the US economy and the
rest of the world for the 1976-2005 period. I then use the estimates to evaluate the
performance of a neoclassical growth model with technology capital. I also estimate
the welfare gains from totally closing US economy and from opening US economy
further.
The estimation of the stock of technology capital and the degree of openness relies
on two main ingredients. First, it requires an identifying assumption. I use the
assumption that the net rates of return from investments to both technology capital
and to the tangible capital are equalized within each country. This assumption is quite
weak, as it is not required that the net rates of return are equalized across countries.
The estimates are therefore consistent with a large set of models. For instance, models
with any type of international trade frictions, or models with imperfect consumption
2
insurance within each country will be consistent with the estimates.1 The second
ingredient in the estimation follows from the theory of aggregate production function.
The theory implies that both tangible capital inputs and labor inputs of all firms
within a given country are related to the country-wide inputs in the same proportion,
which is a function of the stock of technology capital of all countries, and of country
openness.
The US technology capital is estimated to be 25% of the US tangible capital in
1976, and then slightly decreases to 23% in 2005. For the rest of the world, the ratio
is about 2% lower than the US ratio. US economy is found to be less open than
the rest of the world in the whole period. While the openness of the US economy
increases over the whole period, the openness of the rest of the world decreases until
about 1985, and increases after that. Although no immediate effect of NAFTA can
be seen in the estimates, there is a gradual increase in the US openness between
1994 and 2000, which can possibly be attributed to NAFTA. Even then, the effect of
NAFTA on US openness appears to be small.
I find that the neoclassical growth model with technology capital successfully captures the movements in the foreign direct investment in the US and abroad. I compute
the welfare gains from either totally closing or totally opening US economy. While
the welfare losses from totally closing US economy are found to be very small (0.13%
of consumption, in consumption equivalents), the welfare gains from totally opening
US economy are large (6.9% of consumption, in consumption equivalents). This indicates that, despite the increase in openness after 1976, the US economy is still closer
to being totally closed, rather than totally open.
1
The idea that theory can be used to obtain estimates of an unobserved quantity has been
used frequently in a business cycle analysis, starting with the estimation of Solow residuals in the
aggregate production function (Prescott 1986). For instance, Ambler and Paquet (1994) estimate
the stock of physical capital and stochastic depreciation shocks, Burnside, Eichenbaum and Rebelo
(1993) estimate the work effort, and Ingram, Kocherlakota and Savin (1997) estimate nonmarket
hours worked.
3
The importance of foreign direct investment and openness for welfare has long been
recognized by the economic literature.2 However, their quantitative importance has
been addressed only recently. Most notably, Ramondo (2007) analyzes and estimates
an Eaton and Kortum (2002) type model with technology diffusion across countries.3
While the source of welfare gains in her model is similar to this model, both approaches differ along several dimensions. First, the diffusion of technology in her
model is exogenously given, rather than being endogenous like here. Thus, there is
no interaction between the amount of technology diffusion in one country and openness of other countries. Second, her model is static, while this model is dynamic and
one can analyze the transitional adjustment to a change in country openness. On
the other hand, her model can be easily integrated with modern theories of international trade and one can jointly analyze gains from openness and gains from trade, as
in Rodriguez-Clare (2007) and Ramondo and Rodriguez-Clare (2008). However, the
main advantage of the approach in this paper is, in my view, that it is fully integrated
with the neoclassical growth model. Therefore, all the available knowledge about the
neoclassical growth model and its ability to match the data can be put to a new use.
Next section introduces the technology capital. It also reviews the aggregation
theory of McGrattan and Prescott, and highlights the elements of the aggregation
that are important for the estimation that follows in Section 3. Section 4 describes
the data used in the estimation, and provides the estimates. Section 5 then analyzes
the neoclassical growth model with technology capital, evaluates its performance and
provides results of the numerical simulations. Section 6 concludes.
2
See e.g. Horstmann and Markusen (1989) for an early analysis.
See also Burstein and Monge-Naranjo (2006) where the gains from openness are derived from
reallocation of managerial talent across countries, and quantitatively evaluated.
3
4
2. TECHNOLOGY CAPITAL
There is I countries in the world. Country i has a population Ni . In each country
there is a large number of locations where production can take place. The measure
of locations is, without loss of generality, taken to be equal to Ni . In each location,
at most one firm from each country can operate. All firms in country i, domestic and
foreign, share the same total factor productivity Ai .
There are two types of capital. Tangible capital is location and firm specific and
depreciates at rate δK . Technology capital is firm specific but not location specific:
investment in technology capital increases production abilities in all locations, both
domestic and foreign, at which the firm operates. The technology capital depreciates
at rate δM .
The technology capital allows the firm to expand production by effectively multiplying number of plants that can be operated at each location. A domestic firm
with M units of technology capital, k units of tangible capital and l units of labor
produces in each location in country i
¶(1−φ)
µ
k α l (1−α)
yi = Ai M ( ) ( )
,
M
M
¡
¢(1−φ)
= Ai M φ kα l(1−α)
,
α, φ ∈ (0, 1),
since k and l is evenly spread over all plants at a given location. Similarly, it is efficient to spread all the available labor and tangible capital evenly across all locations.
Therefore, the production of a domestic firm in country i that has technology capital
M and uses K units of tangible capital and L units of labor in total is4
4
¡
¢(1−φ)
Yii = Fii (Ni , M, K, L) = Ai (MNi )φ K α L(1−α)
.
In what follows, a subscript on a variable indicates the country where production takes place,
while superscript indicates country of origin.
5
Production of firms from a foreign country is determined similarly but depends,
in addition, on the openness of country i. The degree of openness is measured by a
parameter ωi ∈ [0, 1] that determines the fraction of foreign technology capital that
is permitted to enter the domestic country. A firm from country j 6= i produces a
total output in country i given by
¡
¢(1−φ)
Yij = Fij (Ni , M, K, L) = Ai (ωi MNi )φ K α L(1−α)
.
As in McGrattan and Prescott, a country i is called totally open if ωi = 1 and
totally closed if ωi = 0.
The aggregate production function maximizes the output of a country i given the
population Ni , aggregate capital and labor inputs Ki and Li , and aggregate stock of
technology capital in all countries M = (M 1 , ...M I ) :
Fi (Ni , M, Ki , Li ) = max
j j
X j
Fi (Ni , M j , Kij , Lji )
{Ki ,Li }
(1)
j
P j
subject to the resource constraint on tangible capital
j Ki ≤ Ki and on labor
P j
j Li ≤ Li . The maximization problem yields the aggregate production function
that exhibits constant returns to scale in M, Ki and Li and is given by
Fi (Ni , M, Ki , Li ) = Ai Niφ (M i + ωi
X
α(1−φ) (1−α)(1−φ)
M j )φ Ki
Li
.
(2)
j6=i
The solution to (1) requires that the factors of production, as well as the production
6
of all firms, are related to the country aggregates in the same proportions:
where Yi =
P
j
Yij = vij Yi ,
(3)
Kij = vij Ki ,
(4)
Lji = vij Li ,
(5)
Yij is the total production of country i. The proportion factors {vij }i,j
are given by
vii =
vij =
Mi
P
M i + ωi M j
(6)
j6=i
j
Mi
ωi M
P
,
+ ωi M j
j 6= i.
(7)
j6=i
Equation (4) shows that the proportion factor vij determines the amount of capital
invested in country i by firms from country j, i.e. the foreign direct investment from
country j to country j. Similarly, equation (3) shows that the same proportion factor
determines the sales of foreign firms. As follows from (6) and (7), the proportion
factor depends on the technology capital in all countries, as well as on the openness
of the domestic country.5
Rates of Return
It follows from the aggregation procedure that the capital output ratio is the same
for all firms operating in country i. Hence the net rate of return from all investments
5
Equation (4) can be related to a similar equation (18) in Ramondo (2007). In her model, the
fixed costs tij play a role similar to the role of country openness here, and the variable Γij plays a
role similar to the stock of technology capital here.
7
in tangible capital in country i is equalized and is equal to
riK
= (1 −
τip )
¶
µ
Yi
α(1 − φ)
− δK ,
Ki
(8)
where τip is the tax rate on profits in country i. The net rate of return from investment
in technology capital riM is given by
Yii +
riM = φ
P
Yji
j6=i
Mi
− δM .
(9)
3. THE ESTIMATION PROCEDURE
In this section I develop a procedure that estimates the stock of technology capital,
and the degree of openness. I will assume that there are only two countries in the
world, United States US and the rest of the world RW.6
The key identifying assumption is that the net rates of return on tangible and
technology capital in all countries are equalized:
M
K
M
Assumption 1 rUKS = rUS
and rRW
= rRW
.
US
, KURW
Since the tangible capital stocks of both domestic and foreign firms KUS
S ,
RW
US
KRW
and KRW
are all observed, one can use them to compute the proportion factors
US
RW
, YRW
, and
vU S and vRW from equation (4).7 The production of domestic firms YUS
US
of foreign firms YURW
S , YRW can be eliminated from the rate of return formula (9)
6
The rest of the world will occasionaly be called a country.
US
RW
For the case of two countries the notation is simplified by writing vU S = vU
S and vRW = vRW .
RW
US
One then obtains vU S = 1 − vU S and vRW = 1 − vRW .
7
8
using equation (3):
vUS YU S + (1 − vRW )YRW
− δM ,
M US
vRW YRW + (1 − vU S )YU S
=φ
− δM .
M RW
rUMS = φ
M
rRW
(10)
(11)
Because the investment in technology capital is expensed rather than treated as an
investment, country’s production Y is not equal to its Gross Domestic Product Ŷ .
Rather, GDP is lower than the actual production:
US
ŶUS = YUS − XM
,
(12)
RW
,
ŶRW = YRW − XM
(13)
US
RW
and XM
is the investment in technology capital by US and RW firms.
where XM
Since the investment in technology capital is not observed in the data, YU S and YRW
are not directly observed as well.
Substituting the identities (12) and (13) for YUS and YRW in (10) and (11) and
using Assumption 1, one can express the implied US stock of technology capital as
follows:
MtUS = φ
US
RW
vU S,t (ŶUS,t + XM,t
) + (1 − vRW,t )(ŶRW,t + XM,t
)
(1 −
p
τUS
)(α(1
−
ŶU S,t +X U S
φ) KU S,tM,t
,
(14)
− δK ) + δM
Similarly, the stock of technology capital of the rest of the world satisfies:
MtRW = φ
RW
US
vRW,t (ŶRW,t + XM,t
) + (1 − vUS,t )(ŶU S,t + XM,t
)
(1 −
p
τRW
)(α(1
−
ŶRW,t +X RW
φ) KRW,tM,t
.
(15)
− δK ) + δM
US
RW
The investment in technology capital XM
and XM
can be recovered simultane-
9
ously with technology capital by using its law of motion:
US
US
= XM,t
+ (1 − δM )MtUS
Mt+1
RW
RW
= XM,t
+ (1 − δM )MtRW .
Mt+1
(16)
(17)
Equations (14), (15), (16) and (17) form a system of first order difference equations
in the technology capital MtU S and MtRW . Given some initial or terminal condition on
technology capital, one can solve these difference equations for the time series of technology capital in both countries. After the stock of technology capital is computed,
the equations (6) and (7) can be inverted to compute the openness parameters:
1 − vUS M U S
,
vU S M RW
1 − vRW M RW
=
.
vRW M US
ωUS =
ωRW
Note that there are alternative ways of estimating the stock of technology capiRW
US
RW
tal. If firm level production YUS
, YUS
, YRW
and YURW
S is observed directly, then the
proportion factors vU S and vRW can be computed from (3) rather than from (4). SimUS
RW
RW
ilarly, if firm level employment LRW
US , LU S , LRW and LU S is observed, vUS and vRW
can be computed from (5). Although data availability may lead one to prefer one
estimation procedure over the other, all three approaches are theoretically equivalent
and should in principle yield the same results.
4. THE ESTIMATES
I focus on the 1976-2005 time period. The data for real US tangible capital stock
KUUSS are taken from the National Economic Accounts Fixed Assets Table. In the
10
benchmark estimates, the rest of the world definition includes OECD countries.8
RW
Hence, KRW
is is constructed as a sum of real capital stock for these countries, and
the data are taken from the AMECO database. The data are converted into 1990 US
dollars using Purchasing Power Parity.
US
The tangible capital stock of US firms abroad KRW
is measured by the amount of
US-owned private assets owned abroad from the International Investment Position of
RW
the United States. Similarly, the tangible capital stock of foreign firms in the US KUS
is measured by foreign-owned private assets owned in the US from the International
Investment Position of the United States. Current cost valuation is used for both
assets.
The real US Gross Domestic Product ŶU S is taken from the National Income and
Product Accounts. To be consistent with the measure of US capital stock, US GDP
is adjusted by imputing the flow of services from consumer durables and from government capital. The adjustment procedure is standard.9 The real Gross Domestic
Product for the rest of the world ŶRW is constructed as a sum of real GDP for all
the countries that are included in the definition of the rest of the world capital stock,
and converted into 1990 US dollars using Purchasing Power Parity. The data are
taken from the Groningen Growth and Development Center (GGDC) Total Economy
database.
The values of profit tax rate τUp S is taken to be equal to the ratio of total taxes
on corporate income to corporate profits, and is taken from the National Income and
p
Product Accounts. The value of τRW
is assumed to be equal to τUp S .
The definition of the rest of the world is an important factor in the estimation. On
one hand, one would like to take the rest of the world as large as possible. The cost
of doing so is that by aggregating foreign countries into one, it is implicitly assumed
8
Czech Republic, Hungary, Slovak Republic and Poland are excluded because of limited data
availability.
9
See Cooley and Prescott (1995).
11
that i) all the foreign countries included in the definition are totally open toward
each other and ii) they have the same degree of openness toward USA. Especially
the first assumption becomes increasingly unrealistic as the number of foreign countries included in the definition increases. Since foreign direct investment is mostly
concentrated among OECD countries, I select only those countries in the benchmark
estimation. However, I will also provide alternative estimates where the rest of the
world includes a larger set of countries to see how the definition of the rest of the
world affects the results.
Benchmark Estimates
I set the tangible capital share in GDP α to be equal to 0.326, and the depreciation
rate of tangible capital δK to be equal to 5.6% annually. Both numbers are computed
as an average over the 1976-2005 period. I follow McGrattan and Prescott (2007b) by
setting the depreciation rate of technology capital to be 8% annually.10 The remaining
parameter to be chosen is the degree of decreasing returns in each location φ. I set
φ in such a way that the average ratio of investment in technology capital to the
production of US economy is 7%, which is somewhat higher than the ratio of US.
expenditures on R&D and advertisement to GDP. The resulting value of φ is 0.075.
The terminal condition I use in solving the difference equations (14), (15), (16)
and (17) is that the investment in technology capital in 2005 in both countries is the
same as the investment in technology capital in 2004. Since this assumption is highly
arbitrary, I will later experiment with alternative terminal conditions.
Figure 1 plots the ratio of technology capital and tangible capital for both countries.
The ratio of US technology capital and US tangible capital is approximately 0.25
initially, and then decreases to 0.23 in 2005. In the rest of the world, the ratio is
10
The argument why technology capital depreciates faster than the tangible capital is that it includes R&D investment, and BEA estimates that R&D investment depreciates at rate 15% annually.
12
USA
Rest of the World
0.25
0.2
0.15
1975
1980
1985
1990
year
1995
2000
2005
Fig. 1. Ratio of Technology Capital and Tangible Capital
about 2% lower in all periods. US investment in technology capital is found to be
highly procyclical (at annual frequency), and less volatile than investment in tangible
capital.
Figure 2 plots the openness parameters ωU S and ωRW . Overall, the magnitude of
the openness parameters shows that both the US economy and the rest of the world
are rather closed than open. The fraction of foreign technology capital permitted in
any of the two countries never exceeds 6%. The rest of the world is more open than
USA throughout the whole period, despite the fact that the difference in openness
between both economies has been diminishing. While the US economy has increased
its openness more or less steadily throughout the whole period (from 0.0054 to 0.0283),
the openness of the rest of the world has been declining until about 1985. The degree
of openness of the US economy is uncorrelated with US production, although it is
13
positively correlated with US GDP.
0.06
0.05
0.04
0.03
0.02
0.01
USA
Rest of the World
0
1975
1980
1985
1990
year
1995
2000
2005
Fig. 2. Openness Parameters
Interestingly, there is no spike in the degree of openness of the US economy in
or immediately after 1994 when NAFTA came into effect. This may be so because
NAFTA investment restrictions have been lifted only gradually. This is consistent
with the estimates: before 1994 the degree of openness was constant for several years,
but started to increase gradually after 1994. But even in that case the effect of
NAFTA on openness appears to be small, at least compared to the secular trends in
openness.
The US production has been growing on average at 2.98% per year which is slightly
lower than the growth rate of GDP. The total factor productivity has been growing
at a rate of 0.88% per year. In contrast, the average growth rate of the total factor
14
productivity one would incorrectly measure by ignoring technology capital has been
growing at a rate of 1.03% per year. Thus, the contribution of the total factor productivity, why still of first order importance, is somewhat diminished if one properly
accounts for the technology capital.
Alternative Estimates
To examine the robustness of the results, I have considered alternative assumptions
about i) the definition of the rest of the world, ii) the way proportion factors vUS and
vRW were computed, iii) the average ratio of investment to technology capital and
production, and iv) the terminal condition that was used in solving the difference
equations (14), (15), (16) and (17).
Definition of the Rest of the World I have redefined the rest of the world to
include a larger number of foreign countries for which all data were available.11 I use
the World Bank estimates of Nehru and Dhareshwar (1993) for capital stock data12 ,
and the GGDC Total Economy database for GDP data.
Figure 3 shows that, while the alternative estimates of the US technology capital
to tangible capital stock ratio are almost identical to the benchmark estimates, the
estimates for the rest of the world are now higher by about 1%.13 The alternative estimates of the openness parameters are plotted in Figure 4. The openness parameters
for the US economy are slightly lower, but the time profile is practically the same as
before. For the rest of the world, the estimates are almost the same as before. Thus,
the effect of the definition of the rest of the world is rather small.
11
The following countries were added: Argentina, Bolivia, Brazil, Chile, China, Colombia, Costa
Rica, Cyprus, Dominican Republic, Ecuador, Guatemala, Jamaica, Malta, Peru, Singapore, Taiwan,
Uruguay and Venezuela.
12
Since the World Bank time series end in 1990, I extend the series by assuming that capital-output
ratio in 1991-2005 is the same as capital-output ratio in 1990.
13
Naturally, since the stock of tangible capital in the rest of the world is now higher (by about
25% on average), the stock of technology capital in the rest of the world is also higher.
15
0.28
0.27
US, benchmark
RW, benchmark
US, alt RW definition
RW, alt RW definition
US, alt proportion factor
RW, alt proportion factor
0.26
0.25
0.24
0.23
0.22
0.21
0.2
1975
1980
1985
1990
year
1995
2000
2005
Fig. 3. Ratio of Technology Capital and Tangible Capital, Alternative Estimates
Computation of Proportion Factors As mentioned previously, if one observes
the production of domestic and foreign firms separately, then the proportion factors
vU S and vRW can be computed from equation (3) rather than from (4). The production of domestic and foreign firms is not observed directly in the data, but one
can infer it from the data on Foreign Direct Investment Income, which is available in
the U.S. International Transactions Accounts. The production of US firms abroad is
related to the Foreign Direct Investment Income F DIU S : The foreign direct investUS
S
US
ment income of US firms from abroad equals dividends YRW
− WRW LURW
− IRW
plus
16
0.09
0.08
0.07
US, benchmark
RW, benchmark
US, alt RW definition
RW, alt RW definition
US, alt proportion factor
RW, alt proportion factor
US, alt investment ratio
RW, alt investment ratio
0.06
0.05
0.04
0.03
0.02
0.01
0
1975
1980
1985
1990
year
1995
2000
2005
Fig. 4. Openness Parameters, Alternative Estimates
US
US
reinvested earnings IRW
− δK KRW
, or
p
US
US
)(YRW
− WRW LUS
F DIU S = (1 − τUS
RW − δK KRW )
p
US
US
)[(φ + α(1 − φ))YRW
− δK KRW
],
= (1 − τUS
(18)
S
by using
where the second equality follows from elimination of labor income WRW LURW
the first order condition in labor demand. Similarly, the production of firms from the
rest of the world in the US can be obtained from their Foreign Direct Investment
Income F DIRW :
p
RW
)[(φ + α(1 − φ))YURW
F DIRW = (1 − τRW
S − δK KUS ].
17
(19)
US
Inverting the equations (18) and (19) yields the estimates of YRW
and YURW
S .
While using the income data to compute the proportion factors is theoretically
equivalent to using the capital stock data, there are reasons why it may be inferior
in practice. In particular, as argued by Prescott and McGrattan (2007b), the foreign
direct investment income includes return on investment on intangible location specific
capital. In contrast, no such problems affect the data on tangible capital stock. Using
capital stock data to compute the proportion factors is therefore a better alternative.
Nevertheless, it is interesting to obtain the alternative estimates to see how they differ
from the benchmark ones.
Figure 3 shows that the estimates of the technology capital to tangible capital
ratio are very close to the benchmark estimates for the US economy, and about 1%
lower for the rest of the world. The alternative estimates of the openness parameters
are shown in Figure 4. They are substantially more volatile than the benchmark
estimates, presumably because of the fluctuations in the return from investments in
intangible capital. They are slightly higher on average for the rest of the world, but
have about the same magnitude for the US economy. Overall, despite the problems
in using income data to compute the proportion factors, the alternative estimates
confirm the results that were obtained in the benchmark estimation.
Ratio of Investment to Technology Capital and Production I have experimented with alternative values for the average ratio of investment in technology
capital and production. I have increased this ratio to be 9% rather than 7%. As a
result, the ratio of technology capital and tangible capital has increased by about 7%
for both US economy and for the rest of the world. The change in country openness
was negligible. The results were similar when the ratio has been 5% rather than 7%:
the technology and tangible capital stock ratio has decreased by about 7%, while the
openness parameters were almost unchanged.
18
Terminal Condition I have also experimented with several sensible alternatives.
First, I have assumed that the ratio of investment in technology capital to output
in 2005 is equal to its long run average of 7%. Second, I have assumed that the
investment in technology capital in 2005 is 20% higher (lower) than the investment in
2004. In all cases, the effects on the aggregate estimates of capital stock and openness
parameters were negligible.
5. THEORY
I will now use the estimates of technology capital and country openness to evaluate
the performance of a neoclassical growth model with technology capital. I will also
estimate the welfare loss from totally closing US economy in 1976, welfare gain from
totally opening US economy in 1976, and welfare gains from NAFTA.
The agents in country i ∈ {US, RW } evaluate sequences of consumption according
to the following utility function:
³ ´1−θ
Cit
∞
X
Nit
1 t
(
) Nit
,
1
+
ρ
1
−
θ
t=0
θ > 0,
where ρ is the discount rate, Nit is the population in country i and
(20)
Cit
Nit
is consumption
per person in country i.
Agents face a tax rate on profits (net of depreciation) τitP . The proceeds from
taxation are rebated back as a lump sum transfer. The budget constraint is
i
= Wit Nit + (1 − τitP )RitK Kit + τitP δK Kit + RitM Mti
Cit + Ki,t+1 + Mt+1
+ (1 − δK )Kit + (1 − δM )Mti + Tit + NXit ,
(21)
where Wit is the wage rate, RitK and RitM are the gross rates of return on tangible and
19
technology capital, Tit is lump sum transfer, and NXit is net export.14 Each agent
supplies one unit of labor inelastically, and so the aggregate labor supply in country
i is Nit . The production of country i is thus given by (2) with L = N, i.e. by
1−α(1−φ)
Yit = Ait Nit
α(1−φ)
(Mti + ωit Mt−i )φ Kit
.
Consumption, tangible capital and technology capital are all required to be nonnegative. Moreover, it is not possible to convert technology capital back to consumption
goods and so the investment in technology capital is required to be nonnegative as
well.
i
The competitive equilibrium consists of allocations {Cit , Yit , Ki,t+1 , Mt+1
, NXit }
and prices {Wit , RitK , RitM } such that, given the initial capital stocks Ki0 , M0i and the
exogenous sequences {Nit , Ait , ωit , τitP }, i) households in each country maximize (20)
subject to (21) taking prices as given, ii) prices are given by Wit = (1 − α)(1 − φ) NYitit ,
Y−it
Yit
RitK = α(1 − φ) K
and RitM = φ M i +ωYit M −i + φω−it M −i +ω
it
t
it
t
t
i
−it Mt
, iii) the government
budget is balanced each period, and iv) markets clear.
Properties of Equilibrium
Balanced Growth Path It is assumed that the total factor productivity Ait and
population Nit fluctuate in a finite number of initial periods and grow at a constant
rate γ and η afterwards. Similarly, the tax rate on profits τitP and the openness
parameter wit is assumed to be constant after a finite number of periods. The economy
then converges to a balanced growth path where consumption per person
per person
14
Yit
,
Nit
technology capital per person
Net export satisfies N Xit + N X−it = 0.
20
Mti
,
Nit
Cit
,
Nit
output
and tangible capital per person
Kit
Nit
all grow at a common rate g, given by
1
g = [(1 + γ)(1 + η)φ ] (1−α)(1−φ) − 1.
Equality of Rates of Return Depending on whether the nonnegativity constraints on investment in technology capital bind, three possibilities can arise in any
given period. In the first case, investment in technology capital is strictly positive in
both countries. The net rates of return from all investments are then equalized:
K
M
RitM − δ M = RitK − δ K = R−it
− δ K = R−it
− δM .
In the second case, investment in technology capital is zero in country i but strictly
positive in country −i. Then the net rates of return from investments in tangible
capital and from investment in technology capital in country i are still equalized, and
they are greater than the net rate of return from investment in technology capital in
country −i:
K
M
RitM − δ M = RitK − δ K = R−it
− δ K > R−it
− δM .
In the third case, both investments in technology capital are zero. Then the net
rates of return from investment in technology capital is smaller in both countries:
K
M
− δ K > R−it
− δM .
RitM − δ M < RitK − δ K = R−it
The Results
In the benchmark scenario, the openness parameters are assumed to be equal to
the estimated values in the first thirty periods (corresponding to years 1976-2005)
and constant after that. The tax rates on profits are equal to their values used in the
estimation for the first thirty period and are constant after that as well. The model
21
total factor productivity equals the Solow Residual. The common long run growth
rate of total factor productivity γ is taken to be the average growth rate of US total
factor productivity in the 1976-2005 time period. Finally, the country population is
equal to the US and rest of the world population in the first thirty years, and grows
at a rate equal to the the average growth rate of US population after that.
The discount rate ρ is chosen in such a way that the net rate of return in steady
state equals 4.29%, which is the average net rate of return in the US economy between
1976-2005. The coefficient of relative risk aversion θ is set equal to one. The remaining
parameters φ, α, δ M , δ K are the same as the ones used in the estimation.
Figure 5 plots the (detrended) US production in the data and in the model. It
shows that the model is able to replicate the fluctuations in US output. Figure 6
shows that the same can be said about fluctuations in both US investments abroad
and foreign investments in the US. The model successfully captures the decline in US
investments abroad until about 1985 and its rise thereafter. Both the initial decline
and subsequent rise can be explained by changes in the openness of the rest of the
world towards USA.15
Although the model is successful in explaining movements in foreign direct investment, it is not very successful in explaining movements in US net exports. US net
exports in the data are negative and small. US net exports predicted by the model
are negative and large. The inability to explain net exports is not surprising since
the model assumes that there are no trade frictions between both countries.
15
It is worth noting that, to some extent, the success of the model is to be expected because
the openness parameters and technology capital stocks were estimated using one of the equilibrium
condition of the model, namely the equality of the net rates of return within country. Naturally, the
model provides more restrictions which determine its success in explaining the data.
22
5000
4900
1990 USD, detrended
4800
4700
4600
4500
4400
data
benchmark model
4300
1975
1980
1985
1990
year
1995
2000
2005
Fig. 5. US Output
Gains from Current US Openness
The welfare loss from forever totally closing US economy in 1976 turns out to be
very small: they are equal to 0.13% of consumption (in consumption equivalents).
When both US economy and the rest of the world are totally closed in 1976, the
welfare losses are larger, but still small: they are equal to 0.36% of consumption.
Figure 7 illustrates the results by plotting US production in all three cases.
Assuming that all the changes in US openness after 1994 are associated with
NAFTA, one can compute the implications of not joining NAFTA. The welfare costs
of not joining NAFTA are naturally even smaller than the welfare costs of totally
closing US economy, and are equal to 0.03% of consumption. There is, however, a
significant effect on the quantity of foreign direct investment in the US. About a third
23
800
700
1990 USD, detrended
600
500
400
300
200
Foreign Investment in US, data
Foreign Investment in US, model
US Investment Abroad, data
US Investment Abroad, model
100
0
1975
1980
1985
1990
year
1995
2000
2005
Fig. 6. Foreign Direct Investment
of foreign direct investment after 1994 can be attributed to NAFTA.
These results reinforce the view that the US economy still appears to be very much
closed: not much has been gained from the current level of openness and not much
would be lost from closing them completely. Next subsection investigates whether
there are potential gains from opening the US economy further.
Gains from Opening US Economy Further
If the US economy opens totally, it is no longer efficient to invest in technology
capital in the US. Rather, the US economy imports all its technology capital from
abroad. The steady state gain in terms of output is 9.35%. The gain in measured
productivity is even larger: it is about 12% in all periods. The welfare gain from
24
4900
4800
1990 USD, detrended
4700
4600
4500
4400
Benchmark
US totally closed
US and RW totally closed
4300
1975
1980
1985
1990
year
1995
2000
2005
Fig. 7. US Output, Alternative Openness Assumptions
totally opening US economy turn out to be large as well: It is equal to 6.91% of
consumption. Figure 7 illustrates the gains in terms of US production.16
The fact that there is no investment in US technology capital allows for an immediate increase in consumption in both countries. At the same time, total openness of
US economy increases the rate of return on investment in technology capital in the
rest of the world. In response, rest of the world increases its investment in technology
capital. Increases in foreign technology capital is more significant than decreases in
US technology capital and so consumption grows faster over the transition.
16
The magnitudes of both welfare losses from totally closing and welfare gains of US economy
from totally opening are similar to the ones reported by Ramondo (2007) where the welfare losses
from closing US economy are 0.00% and the welfare gains from opening are 8%. See her Table 9,
case II.
25
6. CONCLUSIONS
This paper has two goals. First, it estimates the stock of technology capital, country
openness in the US and in the rest of the world. Second, using the estimates, it
evaluates the performance of a neoclassical growth model with technology capital,
and quantifies the gains from country openness.
I identify the time series of technology capital and country openness by assuming
that the net rates of return on both types of capital are equalized within each country.
I estimate that the technology capital is about one fourth of the stock of tangible
capital stock for both the US economy and for the rest of the world. The openness of
US economy has been increasing over time, but is smaller than the openness in the
rest of the world. These estimates are found to be robust to alternative assumptions
about the definition of the rest of the world, the way proportion factors are computed,
average size of the investment in technology capital and the terminal condition on
investment in technology capital.
I find that the neoclassical growth model with technology capital performs well in
explaining the movements in output and foreign direct investment between 1976 and
2005. I also find that the losses from totally closing both economies are small. On
the other hand, the gains from opening US economy totally are much larger.
Several assumptions are likely critical in the estimation of country openness, technology capital stock, and the welfare gains from opening an economy. First, the
estimation procedure relies on an assumption that the rest of the world countries
are mutually totally open. While this assumption seems to be the only alternative
if one wants to restrict attention to a tractable two country model, it may result in
estimating an implausibly large stock of the world technology capital. If this is the
case, the actual welfare gain from totally opening US economy may be smaller than
the one computed in this paper.
26
But how sensitive are the estimates and the welfare calculations to this assumption? This question is not easy to answer in the current two country framework,
because a relaxation of this assumption will simultaneously change the estimates of
the technology capital stock, country openness, as well as the theoretical predictions
of the model. Clearly, a multicountry model is needed to find out how important this
assumption is.
Another reason why the estimates of welfare gains from opening the economy further might be too large is that total openness may be impossible to achieve. The
implicit assumption in the welfare calculation is that the degree of openness is related to government policies and one thus computes the welfare gains from moving
toward the best government policy. But maybe no government policy can achieve
total openness because there are other limitations on the flow of foreign direct investment like physical distance (see the evidence in Ramondo (2007)). If that is the case,
one should really compare the current degree of openness with a degree of openness
that is achievable by the best government policy. Estimation of the upper bound on
country openness is left for future research.
On the other hand, the welfare gains from opening less developed countries or
countries with smaller population are likely to be much larger than the welfare gains
from opening US economy. In both cases foreign technology capital will, at least
potentially, play much larger role in domestic production than in the case of US
economy. In this sense, studying US economy versus rest of the world probably gives
us a lower bound on potential gains from openness across the world.
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