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Technology Assimilation and Growth
Tsz-Nga WongDepartment of Economics
Washington University at St. Louis, USA
Chong K. YipDepartment of Economics
The Chinese University of Hong Kong, Hong Kong
January 29, 2009
Abstract
This paper examines the effect of technology adoption on economic devel-opment in a standard overlapping generations model. The key to successfultechnology adoption is a country’s ability to technology assimilation. If tech-nology assimilation is weak, then wages are not sufficient to support capitalused by the advanced capital-intensive technology. As a result, future pro-duction declines and a poverty trap is formed. If technology assimilationis strong, then converging to the steady state of the rich is almost a sureoutcome through technology adoption.Key Words: technology assimilation, elasticity of substitution, poverty trap.JEL Classification Codes: O110, O330, O410.
Address for Correspondence: Chong K. Yip, Department of Economics, The Chi-nese University of Hong Kong, Shatin, Hong Kong. Tel: (852) 26098187, Fax: (852)26035805, Email: [email protected].
1 Introduction
An important question in macroeconomics is why there are large disparities in per
capita income across countries. A common answer offered by economists is the
differences in production technologies and knowledge.1 In standard models of eco-
nomic growth, productivity is usually indexed by a single coefficient independent of
factor inputs. Unless the improvements are country-specific, technological improve-
ment in one country should then also improve productivity of the others. Given
the vast examples of technological transfer across countries, how can we avoid the
counterfactual prediction of standard growth models?
Existing literature offers alternative explanations and here we highlight two
strands. One well-developed strand of literature emphasizes that technological
transfer across countries is subject to different barriers of adoption. For instance,
Parente and Prescott (1994) models the barrier to technology transfer as a cost of its
adoption. Chen et al (2002) specifies the barriers of technology adoption in terms of
search and entry frictions in the labor market. Other interesting work in this area
includes the quality-ladder model of Segerstrom et al (1990), the costly-imitation
models of technology diffusion by Grossman and Helpman (1991) and Barro and
Sala-i-Martin (1997), the monopoly model of rent-seeking by Parente and Prescott
(1999) and the industrial transformation model by Wang and Xie (2004). Allowing
for free and instant technology transfer across countries, another popular strand of
literature abandons the assumption that technological improvement affects all parts
of the production structure to the same extent. Instead it focuses on the "appro-
priateness" of technology and the main idea is that "countries with different factor
endowments should choose different technologies." [Caselli and Coleman (2006),
p.501] The origin of the appropriate-technology literature goes back to the work
of Atkinson and Stiglitz on the "localised technology." Productivity improvement
is based on learning by doing on the firm’s current production technique which is
specified by a given capital-labor ratio. Recently, Basu and Weil (1998) extends
the Atkinson-Stiglitz model by considering that technological improvement are not
only for a specific combination of capital and labor but also of similar techniques.2
1See Prescott (1998) for a discussion. Empirically, it is found that cross-country differences intotal factor productivity are significant (e.g., Hall and Jones 1999).
2According to Basu and Weil (1998), it is in the neighborhood of γ of the specific capital-laborratio.
1
The North-South model of Acemoglu and Zilibotti (2001) differs from Atkinson and
Stiglitz (1969) and Basu and Weil (1998) by focusing on skilled and unskilled labor
instead of capital intensities. More importantly, technological improvement is an
outcome of optimizing behavior of firms and the results of productivity differences
followed from the fact that there is a technology-skill mismatch between the North
and the South.3
Our paper belongs to the latter strand of literature on appropriate technology.
The main motivation of our work is the well-documented phenomenon of technology
adoption. Consider the following quotation from the classic development textbook
of Kindleberger (1956):
"...whether countries at early stages of development, ..., should take advantage of
the modern technology developed by advanced countries, ..., or whether they should
devise a technology of their own or use production methods which are obsolete in
countries abroad." (p. 249)
We believe that the answer depends on the ability of technology assimilation of the
developing countries. Also, as Lin (2007) observes:
"... enterprises in developing countries can fully utilise the industrial and techno-
logical gap with developed countries and acquire industrial and technological inno-
vations that are consistent with their new comparative advantage through learning
and borrowing from developed countries, especially from those countries whose stage
of development is higher than but not far away from theirs." (p.32)
Two examples are cited by Lin in his Marshall lecture. The first one was the expe-
rience of the continental countries in Western Europe who adopted the technology
from the United Kingdom in the nineteenth century. But the same industrial strat-
egy did not work for the poorer countries in Eastern Europe. The second example
was the post-World War II development of the four East Asian NIEs (Hong Kong,
Singapore, South Korea and Taiwan). They borrowed technology from Japan in-
stead of USA and Western Europe and were successful. In this paper, we propose
the concept of technology assimilation to help to understand the above examples.4
3In the words of Acemoglu and Zilibotti (2001), it is "because unskilled workers in the Southperform some of the tasks performed by skilled worker in the North." (p.566)
4Lin (2007) discusses these examples in terms of different development policies pursued bythe governments: the comparative advantage-defying strategy versus the comparative advantage-following strategy. Matching our results with his discussion, it seems that technology assimilationis an important condition for a successful comparative advantage-following policy.
2
Also, recall the farmer example considered in Basu and Weil (1998):
"In India, fields are harvested by bands of sweating workers, bending to use their
scythes. In the United States one farmer does the same work, riding in an air-
conditioned combine. Yet an economist might argue that the two countries do have
access to the same technology and simply choose different combinations of inputs
(points along an isoquant) due to differences in factor prices." (p.1025)
Then Basu and Weil (1998) proposes an explanation by agreeing that "all technol-
ogy is freely available and instantly transferred. But a country may nonetheless
refrain from using a new technology until it reaches a level of development at which
this technology would be appropriate to its needs." (p. 1027) We instead explain
the farmer example in terms of technology assimilation, i.e., Indian farmers are un-
able to assimilate the combine technology used by the US farmer.5 Moreover, our
story of technology assimilation corroborates the empirical work by Los and Tim-
mer (2005) on the Basu-Weil (BW) model of appropriate technology. According to
Los and Timmer (2005), "[a]n important assumption in the BW model is that new
knowledge generated in one country is immediately available to all other countries."
(p.519) Following the decomposition technique based on data envelopment analy-
sis developed by Kumar and Russell (2002), they conclude that "for the original
BW model to have empirical relevance, the assumption of immediate spillovers of
appropriate technology needs to be relaxed. Assimilation of technologies new to
a country is a costly and slow process." (Los and Timmer, 2005, p.529) Thus, we
believe that understanding the role of technology assimilation is a challenging and
important task of the appropriate technology literature.
Our paper contributes to the literature of appropriate technology by explicitly
modelling the mechanism of technology assimilation. Specifically, the appropriate-
ness of technology depends on the ability of assimilation of the backward country.
In order to understand how technology assimilation affects the development process,
we build an adaptability function in normalized constant-elasticity-of-substitution
(CES) form for its measurement.6 Following Basu and Weil (1998), we index tech-
5Our concept of technology assimilation can be regarded as a formalization of the idea of theintermediate technology proposed by Schumacher (1973). According to Schumacher, capital-intensive processes are unproductive in developing countries because they are "lack of marketingand financial infrastructure, inappropriate inputs, and untrained workers." (Basu and Weil, 1998,p.1026) In our understanding, the lack of these factors is the cause for the failure of technologyassimilation.
6For an introduction of the normalized CES function, see Klump and de la Grandville (2000).
3
nology in terms of the capital-labor ratio. As a result, the degree of technology
assimilation is measured by the elasticity of substitution between the domestic and
foreign capital intensities in the adaptability function. Different from the existing
work such as Basu and Weil (1998) and Acemoglu and Zilibotti (2001), we do not
just look at the two extreme cases: appropriate technology versus inappropriate
technology.7 Our adaptability function highlights the interaction of technologies
between the advanced country (the North) and the backward country (the South).
It provides a much wider spectrum of appropriate technology and fills in the in-
termediate cases that are overlooked in the literature. An interesting but realistic
prediction from our model is that the lower the degree of technological assimilation,
the more likely that borrowing technology would lead to a poverty trap so that
the South is worse off. This may help to understand why developing countries
sometimes refuse to import advanced foreign production technologies.
The organization of the paper is as follows. The autarkic model is developed in
the next section so as to highlight the structure of the economy and the character-
istics of the adaptability function for technology assimilation. Section 3 consider
the extreme case where technology assimilation is absent so that poverty trap is
an outcome from technology borrowing. Section 4 studies the full version of the
North-South model of technology assimilation. Finally, there is a fifth concluding
section.
2 The Autarkic Model
To sketch how firms make the technology choice, we first introduce the notion of
"production technique" F̃ , thereby developing the notion of "assimilated technol-
ogy" F (local production function). Given a collection of production techniques, we
have a collection of assimilated technologies, amongst which firms can choose the
most profitable one based on the factor prices. This collection of profitable tech-
nologies then forms the "global production function."8 Figure 1 provides a graphical
7For instance, Basu and Weil (1998) assumes that technology is appropriate for a country ifit is in the neigborhood of magnitude γ of its capita-labor ratio. On the other hand, Acemogluand Zilibotti (2001) assumes that the North is skill labor abundant and there is no interaction ofthe skill ratios between the North and the South. As a result, the technology developed from theNorth is always skill intensive which is inappropriate for the South who is abundant in unskilledlabor.
8As all the premises of the Welfare theorems are satisfied, we are able to use the "efficient"technology to characterize the profitable technology at the competitive equilibrium.
4
representation of these concepts.9
2.1 Technique, Technology, and Production Function
To be more specific, for a given production technique, output in our economy is
produced using two factors inputs: capital K and labor N . Following Jones (2005),
a particular production technique i is defined by two parameters, ai and bi, so that
the output Y can be written as10
Y = F̃ (aiK, biN). (1)
It is assumed that F̃ is a constant-returns-to-scale (CRTS) function. Rewriting F̃
in per capita terms, we have
y ≡ Y/N = biF̃ (aiK
biN, 1) ≡ biF̃ (
aibik, 1). (2)
Define ki ≡ bi/ai, then the expression becomes11
y = biF̃ (k
ki, 1). (3)
With the normalization that F̃ (1, 1) = 1, we have k = ki and hence y = bi = b so
that the production technique is indexed by ki and bi instead. Given our focus is
on technology assimilation, we assume F̃ to take the Leontief form:
y = bimin(k
ki, 1).
Thus, output is at its highest level when k = ki. If k < ki, firms have idle labor
so that the production technique is not "appropriate." As a result, firms would try
their best to make use of it by looking for or adapting a "better" technique. Let
σ be a measure of technology assimilation that captures how well firms can utilize
their idle labor given a production technique. In other words, countries can be char-
acterized and compared according to their ability of technology assimilation, i.e.,
the parameter σ. To this end, the normalized constant-elasticity-of-substitution
9In Figure 1, the production "technique" is the Leontief kink; while the "assimilated technol-ogy" is the CES curve developed from the kink point. The "(global) production function" is theenvelope of all the CES curves.10Jones (2005) calls F̃ below "the local production function associated with technique i."11Although strictly speaking ki is not capital intensity, our concept of technological appropri-
ateness is closely related to the capital intensity k. To facilitate model solving, we simply followBasu and Weil (1998) to index technology by the capital-labor ratio.
5
(CES) function introduced by Klump and de la Grandville (2000) provides us with
the perfect specification. Thus, we assume the "assimilated" output level is given
by the following adaptability function Q:12
Y = F (aiK, biN, σ) ≡ biQ(K/ki, N, σ) ≡ biAhα (K/ki)
σ−1σ + (1− α)N
σ−1σ
i σσ−1
.
(4)
In per capita terms, we have
y(k/ki, σ) = biq(k/ki, σ) = biAhα (k/ki)
σ−1σ + (1− α)
i σσ−1
. (5)
Before proceeding further, let us take a closer look at the adaptability function
q and understand some of its properties. First of all, when there is no assimilation,
we expect Q to take the Leontief form, i.e., Q = K/ki, so that there exists a
lower bound for q such that k/ki ≤ q(k/ki, σ).13 Secondly, it is expected that
more capital and higher σ improve technology assimilation so that ∂q/∂k > 0 and
∂q/∂σ > 0. Thirdly, when k = ki, we should have y = bi = b so that q(1, σ) = 1.
This in turn implies that A = 1. Fourthly, when k = 0, we should get y = 0 so
that q(0, σ) = 0. This in turn implies that σ cannot exceed unity. Finally, with
maximum adaptability σ = 1, q takes the Cobb-Douglas form and we have
q(k/ki, 1) = (k/ki)α.
We interpret it is the case that firms can perform perfect technology assimilation,
so they can produce at the highest output level where y = bi = b. Equivalently we
have
biq(k/ki, 1) = b or b/bi = (k/ki)α.
Thus the intensive local production function with a given "technology" ki or simply
the production technology ki is
y(k/ki, σ) = biq(k/ki, σ) = bihα (k/ki)
σ−1σ + (1− α)
i σσ−1
. (6)
The corresponding isoquants are depicted in Figure 1.
Next, we want to characterize the collection of efficient technology, which is the
technology ki maximizing y given k. Let ρ be the output ratio of technology ki
12The derivation details of the normalized CES adaptability function are given in the Appendix6.1.13In this case, the production technique and the assimilated output function coincide and is
studied in Jones (2005).
6
over technology k:
ρ(k, ki, σ) ≡y(k/ki, σ)
y(1, σ)=
hα (k/ki)
σ−1σ + (1− α)
i σσ−1
(k/ki)α.
Direct differentiation with respect to ki yields
dρ
dki=
α (1− α)hα (k/ki)
σ−1σ + (1− α)
i 1σ−1
(k/ki)1+α
h(k/ki)
σ−1σ − 1
i> 0 iff ki < k.
So output is at its highest level whenever ki = k. Hence an efficient technology
given capital per labor k is simply the technology k. With the exception for the
case where σ = 1, adopting the "right" technique (technology k) is always more
efficient than assimilating a "wrong" technique (technology other k) for a firm. It
thus hints that when firms are induced to use a "wrong" technique, a poverty trap
can result. Next, the convex hull of efficient technologies is the "global production
function". As the number of techniques available increases, the global production
function f can be approximated by a Cobb-Douglas production function, i.e.,14
y = f(k) = zkα, (7)
where z ≡ biki−α is a constant that measures the "productivity". The global
production function is an envelope of all the local production functions of technology
ki.15
Turning to the preference side, we consider a discrete-time economy populated
by an infinite sequence of two-period lived, overlapping generations. At each date
a set of young agents is born and all agents are risk neutral, and care only about old
age consumption. Hence all young period income is saved and invested in capital
formation. Each agent is endowed with one unit of labor when young, which is
supplied inelastically and earns a wage income w. Agents are retired when old and
agents other than the initial old have no endowment of capital or the final good at
any date. We assume that the initial old of a country is each endowed with K0
units of capital.
14See Jones (2005) for the technical details. Such a production function is called the globalproduction function.15The isoquant associated with the global production function is given in Figure 1.
7
2.2 The Autarkic Steady State
We assume that there are competitive rental markets for capital and labor and
that factors of production are immobile across the border. There is no population
growth and capital depreciates completely in production each period. Standard
marginal productivity conditions yield
r = fk(k) = αzkα−1, (8)
w = f(k)− kfk(k) = (1− α)zkα. (9)
Equilibrium requires that savings equal investment in each period:
kt+1 = w(kt). (10)
The unique fixed point given by (10) is
k∗A = w(k∗A) or k∗A = [(1− α)z]1
1−α . (11)
According to (9), w(kt) is increasing and concave. In addition, we have limk→0w0(k) <
1 and limk→∞(kt+1/kt) > 1 so that the fixed point is globally stable. Thus the law-
of-motion equation (10) implies that the convergence toward the fixed point k∗Amust be monotone. We depict the fixed point in Figure 2.
We are now ready to characterize the steady state in autarky. In order to have
the capital market equilibrium k∗A to be the steady state, we must make sure that
k∗A is also an equilibrium for the production technique ki = k∗A, i.e., firms have no
incentive to switch to other production techniques. With the production technology
ki, the corresponding wage level is given by
w(k, ki, σ) = (1− α)bihα (k/ki)
σ−1σ + (1− α)
i 1σ−1
.
Since ki = k = k∗A at the fixed point, we have
w(k∗A, k∗A, σ) = (1− α)bi = (1− α)z (k∗A)
α
so that the same equilibrium is obtained. Thus we conclude that k∗A is a steady
state at which both the level of capital stock as well as the production technique do
not change. We also note that k∗A is independent of the adaptability parameter σ.
This is an implication of the fact that the global production function is an envelope
of all local production functions or technologies, i.e., they all tangent at the fixed
point.
8
3 Imitating Foreign Techniques
In this section, we build a two-country model by introducing a foreign country whose
production technique is much more advanced than that used by the domestic firms.
Denote the foreign technique by k̄, then we have k̄ > k∗A. In order to highlight
the effect of technology assimilation, we assume that domestic firms are unable to
assimilate the foreign technique in the current section, i.e., the adaptability function
of domestic firms is Leontief.16 As a result, if a domestic firm decide to imitate the
foreign production technique, its production function becomes
Y = z̄k̄αmin
µK
k̄,N
¶, (12)
where z̄ is a measure of foreign productivity and z < z̄. Of course, domestic firms
always have the option to use their domestic techniques to produce, i.e., follow the
domestic global production technique:
Y = zKαN1−α. (13)
Without the consideration of technology assimilation, the decision of imitation of
domestic firms depends purely on the underlying production costs. To be specific,
the unit cost function of using domestic production technique is
c =rαw1−α
C, (14)
where C ≡ zαα (1− α)1−α. On the other hand, the unit cost function of imitating
the foreign production technique is
c̄ =rk̄ + w
z̄k̄α. (15)
We further note that at the steady-state equilibrium, the zero profit condition must
hold for firms so that min(c, c̄) = 1.
Define x ≡ α(1−α)k̄
wr. Then using (14) and (15), we have
c ≥ c̄⇐⇒ 0 ≥ (1− α) (z/z̄)x− x1−α + α (z/z̄) ≡ Γ(x). (16)
Thus the comparison between the unit costs depends on the sign of Γ. To take
a closer look at Γ(x), we first note that Γ(x) is a U-shape convex function. Also,
16We would relax this assumption in the next section by restoring the normalized CES adapt-ability fundtion. Then we are able to examine the effect of technology assimilation.
9
Γ(0) > 0, Γ(∞) > 0 and Γ(xmin) < 0. So we can conclude that Γ(x) = 0 has two
roots. We provide the graphical representation of Γ(x) in Figure 3. The following
proposition then summarizes our findings of technological imitation.
Proposition 1 Let x1 and x2 be the two roots of Γ(x) = 0 and assume that x1 < x2.
Then we have
1. When w/r = (1− α) k̄x1/α ≡ (w/r)1 or w/r = (1− α) k̄x2/α ≡ (w/r)2, firmsare indifferent between domestic and foreign production techniques.
2. When (w/r)1 < w/r < (w/r)2, firms imitate foreign production technique.
3. When w/r < (w/r)1 or w/r > (w/r)2, firms continue to use domestic production
technique.
According to the proposition, it is clear that foreign production technique is
imitated if the wage-rental ratio falls within an appropriate range. We provide
a graphical representation in terms of the firms’ isocost curves in Figure 4. The
intuition is that since the "adoptable" foreign production technique is fixed at the
capital-labor ratio k̄, the factor prices have to be "appropriate" (i.e., cannot be too
high or too low) in order to have k̄ to be cost efficient for domestic firms. Otherwise,
it does not pay the domestic firms to imitate such a technique. For the "normal"
case where k < k̄, then in order to be profitable to adopt k̄, the rental-wage ratio
has to be low enough to justify the imitation of such a capital-intensive technique,
i.e., w/r > (w/r)1.17
We are now ready to characterize the steady state when the foreign production
technique is introduced. For foreign firms, without loss of generality, we simply
assume the steady state is given at the capital-labor ratio k̄. Since the production
structure of the two countries are identical, the steady-state level of foreign capital-
labor ratio is
k̄t = k̄ = [(1− α) z̄]1
1−α . (17)
Next, we recall the law-of-motion equation (10) that characterizes capital accumu-
lation in the domestic economy:
kt+1 = w(kt).
17The logic of course goes both ways. Even if the domestic technology is superior to the foreigntechnique (k > k̄), it still pays domestic firms to adopt a less superior foreign technique as longas such a technique is much less costly [w/r < (w/r)2].
10
Depending on the production technique being used, the wage function will be dif-
ferent. Using the information given in the proposition on technological imitation
and letting kj = k̄xj (j = 1, 2), we have
= (1− α) zkαt for kt < k1 or kt > k2,
w(kt) = (1− α) zkα1 for k1 ≤ kt < k̄,
= (1− α) zkα2 for k̄ < kt ≤ k2.
The graphical representation of w(kt) is given in Figure 5. The stepwise wage
function w(kt) is a consequence of the Leontief specification of the adaptability
function. If we relax the Leontief specification of zero adaptability, a smoother
wage function can be obtained.18 In the presence of this stepwise wage function,
multiple steady states emerge. Finally, recall that k̄ is the foreign steady state, it
must also be a steady state for domestic firms using the same foreign technique. The
intuition is straightforward. If k = k̄, then there is no need for any adaptability
consideration and domestic firms can just directly use the foreign technique for
production. We summarize our findings in the following proposition.
Proposition 2 In the presence of an advanced foreign production technique k̄ >
k∗A, then multiple steady states exist, namely k̄, k∗L and k∗H, where k
∗L < k∗A < k̄ <
k∗H. The first steady state k̄ is unstable, while the other two (k∗L and k
∗H) are stable.
We illustrate these three steady states in Figure 6. Since k∗L is a stable steady
state, our finding implies a "strong" form of poverty trap. Starting from the au-
tarkic situation and in the presence of foreign production technique, both capital
intensity and wage are lower in the new steady state. The flat portion of the
wage function implies that wage would not increase as capital accumulates. This
is because the increase in capital stock will be used to support the increase in the
number of domestic firms adopting the capital-intensive foreign technique.19 Such
18This extension is studied in the next section.19Such a coexistence of firms using backward domestic technique and firms using advanced
foreign technique has long been noted in the economic development literature; for instance, seeNelson (1968). To compute the share of domestic firms that adopt foreign technique at the steadystate, denoted by h, we note that
k∗L = hk̄ + (1− h)k1
or
h =k∗L − k1k̄ − k1
.
11
a technological adoption creates a vicious circle of low wage and low investment so
that a poverty trap below the autarkic steady state is formed. Notice that the
conclusion of this "strong" form of poverty trap is valid in transition too. Since the
law-of-motion equation (10) yields a monotonic converging process, importing the
foreign technique at any point of time for any kt < k̄ will leads to a poverty trap
at the steady-state equilibrium. Thus, introducing an advanced but "inappropri-
ate" production technique can hurt the domestic economy. Of course, we should
emphasize that our conclusion on the inappropriateness of foreign technique in this
section is based on the assumption that domestic firms are unable to assimilate it.
In the absence of technology assimilation, the mismatch between the capital-scarce
domestic economy and the capital-intensive foreign technique cannot be alleviated
which leads to a poverty trap. Finally, in the presence of an advanced foreign
technique, notice that the autarkic equilibrium is no longer a steady state. To
show this, first note that if the equilibrium technique at k∗A is f , then we must have
(w/r)A =(1− α) k∗A
α.
Substituting this wage-rental ratio into the definition of x, we have
xA = (z/z̄)1
1−α . (18)
According to (16), we have
Γ(xA) = (1− α) (z/z̄)xA − x1−αA + α (z/z̄) = (1− α) (z/z̄)h(z/z̄)
11−α − 1
i< 0
so that c > c̄. This leads to a contradiction and we conclude that f cannot be the
equilibrium production technique. Consequently, k∗A cannot be the steady state
when an advanced foreign technique is available to domestic firms.20
Before closing the section, we note that kj (j = 1, 2) are increasing functions of
the productivity level z/z̄. If the productivity gap between the two countries is
widened, then this wage-rental ratio decreases so that k∗L becomes lower relative to
the autarkic steady state k∗A. Consequently, the availability of a more productive or
advanced technique can be more harmful to a developing country if her adaptability
is low. The intuition is not difficult to understand. The larger the productivity gap
between the two countries is, the more serious the mismatch between the capital-
scarce domestic economy and the capital-intensive foreign technique becomes. In20This conclusion is robust to the CES specification of the adaptability function.
12
order to be profitable to imitate a more productive technique, firms must face a
higher return of capital so that wage has to stop growing earlier (or at a lower
level).
4 A Model of Technology Assimilation
In this section, we relax the assumption that domestic firms are unable to assimilate
the foreign technique. Consequently, the effect of technological adoption depends
crucially on the firms’ ability to assimilate the foreign technique. As already
discussed in the autarkic section, we specify the adaptability function for a given
foreign technique k̄ to be a normalized CES function:
Y = z̄k̄αhα¡K/k̄
¢σ−1σ + (1− α)N
σ−1σ
i σσ−1
,
or in per capita terms
y = z̄k̄αhα¡k/k̄
¢σ−1σ + (1− α)
i σσ−1
. (19)
Notice that when the adaptability parameter σ tends to one, we get the foreign
global production function as the assimilation is perfect. Of course, when σ tends
to zero, we get back the Leontief function which is the zero-adaptability case studied
in the previous section.
If firms continue to use domestic production technique, then the unit cost func-
tion remains unchanged:
c =rαw1−α
C,
where C ≡ zαα (1− α)1−α. On the other hand, the unit cost function of adopting
the foreign production technique takes a CES form:
c̄ =
hασ¡rk̄¢1−σ
+ (1− α)σ w1−σi 11−σ
z̄k̄α. (20)
As before, at the steady-state equilibrium, the zero profit condition must hold for
firms so that min(c, c̄) = 1.
Define χ ≡h
α(1−α)k̄
wr
i1−σ. Then using (14) and (20), we have
c ≥ c̄⇐⇒ 0 ≥ (1− α)Zχ− χ1−α + αZ ≡ Λ(χ),
where Z ≡ (z/z̄)1−σ < 1. Thus the comparison between the unit costs depends
on the sign of Λ(χ). To study Λ(χ), we first note that it is a U-shape convex
13
function. Also, Λ(0) > 0, Λ(∞) > 0 and Λ(χmin) < 0. So we can conclude that
Λ(χ) = 0 has two roots. We provide the graphical representation of Λ(χ) in Figure
7. The following proposition then summarizes our findings about firms’ choice of
techniques based on unit costs comparison.21
Proposition 3 Let χ1 and χ2 be the two roots of Λ(χ) = 0 and assume that χ1 <
χ2. Then we have
1. When w/r = [(1− α) /α] (χ1)1
1−σ k̄ ≡ (w/r)1 or w/r = [(1− α) /α] (χ2)1
1−σ k̄ ≡(w/r)2, firms are indifferent between domestic and foreign production techniques.
2. When (w/r)1 < w/r < (w/r)2, firms adopt the foreign production technique.
3. When w/r < (w/r)1 or w/r > (w/r)2, firms continue to use the domestic
production technique.
Similar to the previous section, the foreign production technique is adopted if
the wage-rental ratio falls within an appropriate range. We provide a graphical
representation in terms of the firms’ isocost curves in Figure 8. The intuition
is similar to the case where domestic firms have zero adaptability to the foreign
technique. For the case where k < k̄, in order to be profitable to adopt the foreign
technique k̄, the rental-wage ratio has to be low enough to justify the adoption of
such a capital-intensive technique, i.e., w/r > (w/r)1. Allowing for firms’ ability
to assimilate foreign techniques so far has not made a difference.
4.1 Steady State Analysis
We now characterize the steady state and recall that the law-of-motion equation of
capital accumulation is given by (10):
kt+1 = wt.
Depending on the production technique being used, the wage function will be differ-
ent. First of all, we let kj ≡ k̄¡χj¢ 11−σ (j = 1, 2). To characterize the wage function,
note that when kj = k̄, we have χ̄ = 1 and wr= (1−α)k̄
α. Since Λ(χ̄) = Z − 1 < 0,
we get χ1 < χ̄ < χ2. Next, let k̄j ≡ k̄¡χj¢ σ1−σ , then we have
χ1 < χ̄ =⇒ k1 < k̄1 < k̄,
χ̄ < χ2 =⇒ k̄ < k̄2 < k2.
21Its proof is given in the Appendix 6.2.
14
Then we can rephrase the above proposition in terms of k instead of the wage-rental
ratio w/r :
Corollary 4 Let kj ≡ k̄¡χj¢ 11−σ (j = 1, 2) and k̄j ≡ k̄
¡χj¢ σ1−σ . Then we have
1. When k1 ≤ k ≤ k̄1 or k̄2 ≤ k ≤ k2, firms are indifferent between domestic and
foreign production techniques.
2. When k̄1 < k < k̄2, firms adopt the foreign technique.
3. When k < k1 or k > k2, firms continue to use the domestic technique.
In the first case of the above corollary, when firms are indifferent between domes-
tic and foreign production techniques, we expect some firms use domestic technique
while others adopt foreign technique. Let h be the share of firms using the foreign
technique f̄ , then we have
k = hk̄j + (1− h)kj or h =k − kjk̄j − kj
(21)
for k being in between kj and k̄j. This then implies that whenever k is in between
kj and k̄j, the wage function is flat since the increase in capital stock will be used to
support the increase in the number of domestic firms adopting the capital-intensive
foreign technique. As a result, wage would not increase as capital accumulates.
Thus, we are ready to specify the wage function as follows:
= w(kt) for kt < k1, (22)
= w (k1) = w̄¡k̄1¢for k1 ≤ kt ≤ k̄1,
wt = w̄(kt) for k̄1 < kt < k̄2,
= w (k2) = w̄¡k̄2¢for k̄2 ≤ kt ≤ k2,
= w(kt) for kt > k2,
where w(kt) = (1− α) zkαt and w̄(kt) = (1− α) z̄k̄αhα¡kt/k̄
¢σ−1σ + (1− α)
i 1σ−1.
The graphical representation of wt is given in Figure 9.
To characterize the steady states given the wage function (22), we first consider
the case where the foreign technique is the equilibrium production technology, i.e.,
when k̄1 ≤ kt ≤ k̄2. Then (10) implies the following equivalent relation:
k = w̄(k)⇐⇒ S(χ) ≡ (1− α)χ− χ1−σ + α = 0
15
where χ = (k/k̄)(1−σ)/σ.22 Since S00(χ) > 0, S(0) = α and S(∞) =∞, so S(χ) = 0at most have two roots.23 In addition, k = k̄ or χ = 1 is one of the roots and we
denote the other root by k̃. The local stability condition of the roots is given by
w0(k) < 1⇐⇒ S0(χ) > 0. (23)
For k = k̄ or χ = 1, then we have Λ(1) < 0 so that c > c̄. Thus k̄ is a steady
state for the case where f̄ is the equilibrium technique. For the root k = k̃ and
hence χ = χ̃, in order to have f̄ to be the equilibrium technique, we need Λ(χ̃) < 0.
Using the fact that S(χ̃) = 0, we have
Λ(χ̃) = χ̃1−σ¡Z − χ̃σ−α
¢< 0⇐⇒ z
z̄<
Ãk̃
k̄
!1−α/σ. (24)
We summarize our steady-state characterization for the case where f̄ is the equi-
librium technique as follows:
Lemma 5 When f̄ is the equilibrium production technique, then we have at most
two steady states, k̄ and k̃. For the latter to be a steady state, we need zz̄<³
k̃k̄
´1−α/σ. Under dual steady states, if σ > α, then k̄ > k̃ and w0(k̄) < 1 < w0(k̃).
If σ < α, then k̄ < k̃ and w0(k̄) > 1 > w0(k̃).
To avoid being taxonomic, let us focus on the "plausible" case of our steady-
state characterization where k ≤ k̄. For the region where k1 ≤ kt < k̄1, the wage
function is a flat line so at most we have a single steady state. It turns out that the
existence of the steady state in this case depends on (24). Such a steady state, if
it exists, consists of a mix of firms using production techniques f and f̄ . We state
our characterization of the steady state for k1 ≤ kt < k̄1 in the following lemma
with its proof being given in the Appendix 6.3:
Lemma 6 There is a unique steady state, k∗m, in the region£k1, k̄1
¤if and only if
zz̄<³k̃k̄
´1−α/σ.
We are ready to complete our steady state characterization for the general case
of technology assimilation. For kt ≤ k̄, then we have either three steady states
22It is easy to show that χ ≡h
α(1−α)k̄
wr
i1−σ= (k/k̄)(1−σ)/σ.
23For the extreme case of σ = 0 studied in the last section, then k = k̄ or χ = 1 is the only rootof S(χ) = 0. Thus, we have a unique steady state where f̄ is the equilibrium technique.
16
or a unique steady state. Whenever a unique steady state exists, it must be
k̄ and is stable. In this case all firms adopt the foreign production technique.
When multiple steady states emerge, it is possible to have some firms still using the
domestic technique. However, the stability property of the steady states depends
on the parameterization of the model. We summarize our findings in the following
proposition.
Proposition 7 If zz̄<³k̃k̄
´1−α/σ, then multiple steady states exist, namely k̄, k̃
and k∗m, where k∗m < min
³k̃, k̄
´. If σ > α, then k∗m is stable and the production
techniques are mixed, i.e., some firms using f while others use f̄ . At either k̃
(unstable) or k̄ (stable), all firms adopt foreign technique. The stability property
of k̃ and k̄ will be reversed if σ < α.
On the other hand, if zz̄>³k̃k̄
´1−α/σ, we have a unique stable steady state k̄ where
all firms adopt the foreign production technique.
We illustrate the multiple-steady-state cases in Figures 10 and 11, while Figure
12 shows the case of a unique steady state.
4.2 Poverty Traps
What is the main difference between the general case studied here and the Leontief
case of the previous section? The answer lies on the possibility of poverty traps.
In the Leontief case where technology assimilation is absent, poverty trap is a must
outcome when the domestic economy is exposed to an advanced production tech-
nique. However, if technology assimilation is possible (though may not be perfect),
then we need not have poverty traps as steady states. For instance, in the case
where we have a unique steady state as shown in Figure 12, domestic firms are
able to adapt foreign technique successfully so that we can achieve a steady state
with higher output level given by k̄. On the other hand, according to Figure 11, if
the adaptability is restricted so that σ < α, then we are again trapped in poverty
at the steady state k∗m. The mixed case is given in Figure 10 where the presence
of the poverty trap depends on the locations of the autarkic steady state k∗A and
the threshold k̃. We summarize our findings on poverty traps in the following
proposition.24
24Its proof is given in the Appendix 6.4.
17
Proposition 8 A poverty-trap steady state emerges if and only if zz̄<³k̃k̄
´1−α.
The economics of poverty traps in our model is as follows. For σ < α so
that k̃ > k̄, then the poverty-trap condition is satisfied automatically. What this
is implying is that when technology assimilation is limited (σ is relatively low),
then domestic firms are unable to utilize the advanced foreign technique so that
its adoption generates a poverty trap.25 Thus, to avoid poverty traps, we need to
have a minimal level of technological assimilation which is given by the condition
σ > α. In this case, the unstable steady state k̃ serves as a capital threshold
beyond which an economy can take off. Domestic firms are induced by lower unit
costs to adopt the foreign technique. But if adaptability is low while capital is
not sufficient, then wages paid by firms are not enough to support the next-period
capital for using the advanced foreign technique. This lowers future production
and capital accumulation so that a poverty trap is eventually formed. On the other
hand, if domestic firms have a good adaptability of production techniques, then the
labor productivity will be high enough to support a sufficiently high wage for future
capital accumulation so that the convergence to the rich steady state k̄ can occur.
To further analyze the relation between poverty traps and technology assimila-
tion, we take a closer look at the multiple-steady-state and poverty-trap conditions.
We first rewrite the multiple-steady-state condition in terms of χ:
z
z̄<
Ãk̃
k̄
!1−α/σ⇐⇒ z
z̄< χ̃
σ−α1−σ (25)
where χ̃ = (k̃/k̄)(1−σ)/σ. To study the effect of technology assimilation, we differ-
entiate its borderline case, i.e., z/z̄ = χ̃σ−α1−σ :
d (z/z̄)
dσ=
1
1− σ
∙1− α
1− σχ̃
σ−α1−σ ln χ̃+ (σ − α) χ̃
σ−α1−σ −1
dχ̃
dσ
¸.
To compute dχ̃/dσ, we recall the fact that S(χ̃) = 0 and totally differentiate it to
getdχ̃
dσ= − χ̃
1−σ ln χ̃
S0(χ̃)
<
>0 for α
<
>σ
where χ̃ < 1 and S0(χ̃) < (>)0 since k̃ is unstable (stable) when α < (>)σ. Thus
we get
(σ − α)dχ̃
dσ< 0 =⇒ d (z/z̄)
dσ< 0.
25Again our Leontief case in the last section is an extreme example.
18
Similarly, we rewrite the poverty-trap condition in terms of χ:
z
z̄<
Ãk̃
k̄
!1−α⇐⇒ z
z̄< χ̃
σ(1−α)1−σ . (26)
But χ̃ < 1 so that χ̃σ(1−α) < χ̃σ−α, we can conclude that the locus of the borderline
poverty-trap condition (z/z̄ = χ̃σ(1−α)1−σ ) must lie below the locus of the borderline
multiple-steady-state condition (z/z̄ = χ̃σ−α1−σ ) in the (σ,Z) space. We illustrate
these loci in Figure 13. We denote the poverty-trap condition (z/z̄ = χ̃σ(1−α)1−σ ) as
the PT locus and the multiple-steady-state condition (z/z̄ = χ̃σ−α1−σ ) as the ME locus.
Figure 13 delivers our main message crystal clear. For any given technological gap
across countries (i.e., given z/z̄), if technology assimilation is mature (i.e., σ is high
enough), then adopting foreign techniques can be beneficial since higher production
can be achieved. However, if domestic firms are unable to assimilate the advanced
techniques when σ is small, then a poverty trap emerges and the presence of foreign
techniques worsens the economy from autarky. On the other hand, given the
technological adaptability of a country, the larger its productivity gap (or lower
z/z̄), the more likely it will be fallen into a poverty trap. This finding is intuitive
since a larger productivity gap implying that it is harder for a country to adjust to
the advanced foreign technique.
Finally, Figure 13 can shed light on the issue that whether a developing country
should import an advance foreign production technique at a given point in time.
The answer lies on the structure of the economy. Specifically, it depends on both the
productivity gap and the country’s adaptability to new technology. If the structure
implies that the country is located below the PT locus, then it is preferred to be
isolated in autarky at any point in time. The logic is based on the accumulation
equation (10) which describes a monotonic increasing process of capital stock toward
its steady state. Given that the autarkic steady state is below the capital threshold
(because k̃ > k̄ > k∗A), it implies that capital stock at any moment in time in
transition is below the threshold, i.e., k̃ > k̄ > kt. The law of motion then takes us
toward the poverty trap in the presence of the foreign technique at any point in time.
On the other hand, if its structure is located above the ME locus, importing the
foreign technique is the right policy at any time. This is because, in the presence
of foreign technique, k̄ is the only steady state toward which the economy will
converge. The last scenario is that if the location of the country is between the two
19
loci in Figure 13, then the timing of importing an advance production technique is
important. If the country is already in its autarkic steady state, then it converges
toward the "rich" steady state k̄ so that opening is the right choice. But this
conclusion does not hold for all time in transition. It is possible that at time τ
when the foreign technique is introduced, we have k∗A > k̃ > kτ so that the capital
threshold has not been reached. If this is the case, then the economy will converge
toward the poverty trap despite the fact that we have k̃ < k∗A. This implies that
there exists a time threshold for capital accumulation in transition beyond which
would make opening the right policy to follow. Opening too early would not be a
good matter which gives us an "infant-industry" favor from the trade literature.
5 Concluding Remarks
This paper studies the mechanism of technology assimilation and examines its effects
on economic development. In the absence of technology assimilation, imitating
an advanced foreign technology is inappropriate in the sense that the backward
country will achieve a steady state where both its capital intensity and wage are
lower than in autarky. This is due to the fact that the increase in capital from
higher production is directed to support the increase in the number of domestic
firms adopting the new capital-intensive technique so that a vicious cycle of low
wage and low investment is created. This then leads to a poverty trap below the
autarkic steady state. However, if technology assimilation is allowed, then adopting
an advance foreign technology can be a successful policy for development. We find
that the key to successful technology adoption is a country’s ability to technology
assimilation. Using a CES adaptability function where the degree of technology
assimilation is measured by the elasticity of substitution, we provide a condition in
terms of a minimum level of the elasticity so as to avoid poverty traps in technology
adoption. The main message is that if technology assimilation is not mature (given
by a lower elasticity of substitution of the CES adaptability function), then wages
are not sufficient to support capital for the advanced technology. As a result, future
production declines and a poverty trap is formed.
Given the North-South technological gap, if the South’s ability to technology
assimilation is good, then converging to the steady state equilibrium of the North is
almost a sure outcome through technology adoption. For a lower ability of assim-
20
ilation where multiple steady states emerge, then economic development through
technology adoption requires a minimum level of capital. Finally, if the ability
of assimilation is poor, then imitating foreign technology leads to a poverty trap.
The policy implication from our paper is clear. If the ability of technology as-
similation of a country is not strong, then a big-push type of development policy
is called for so as to make sure that the country has enough capital to start off.
If technology assimilation is almost absent, then conventional policies like the big
push or subsidizing R&D for technology adoption may be futile which is in contrast
to the existing literature of economic development. For instance, Rustichini and
Schmitz (1991) argue that imitation plays an important role to developing countries
so that it is optimal to subsidize imitation in their model. Our paper provides an
alternative mechanism to generate a different result.26 We emphasize that whether
imitation can be successful depends crucially on a country’s ability of technology
assimilation.26The conventional wisdom of subsidizing technology-advancing activities has also been chal-
lenged by others. For example, from the perspective of political economy, Boldrin and Levine(2004) show that policies of forstering innovations and their adoption like patents may not beoptimal due to rent-seeking behavior of the private and public sectors.
21
6 Appendix
6.1 The Derivation of the Adaptability Function
Following Klump and de la Grandville (2000), we first specify the three arbitrarily
chosen baseline variables of the normalized CES function: a capital-labor ratio ki,
the per capita production yi = f(ki) = zki, and the marginal rate of substitution
mi = [f(ki)− kif0(ki)] /f
0(ki) = (1− α) ki/α. The normalized capital-labor ratio
will be taken to be the production technique ki that being assimilated. As a result,
the family of the normalized CES functions mimics the family of the adaptability
functions ordered by their ability to assimilate advanced production techniques. In
summary, the normalized CES function becomes
f(k, σ) = A (σ)ha (σ) (k/ki)
σ−1σ + (1− a (σ))
i σσ−1
(27)
where
A (σ) = yi
"k1/σi +mi
ki +mi
# σσ−1
, (28)
a (σ) =k1/σi
k1/σi +mi
. (29)
Substituting (28) and (29) into (27), we get
f(k, σ) = Ahα (k/ki)
σ−1σ + (1− α)
i σσ−1
where A = yi. This is the per capita adaptability function q(k/ki, σ) given by (6)
in the text.
6.2 Proof of Proposition 3
Since χ1 and χ2 are the two roots of Λ(χ) = 0 and χ1 < χ2, we have
1. When χ < χ1 or χ > χ2, firms continue to use domestic production technique
because c < c̄.
2. When χ1 < χ < χ2, firms adopt foreign production technique because c > c̄.
3. When χ = χ1 or χ = χ2, firms are indifferent between domestic and foreign
production techniques because c = c̄.
Using the definition of χ and rewrite the above findings in terms of w/r, the results
follow.
22
6.3 Proof of Lemma 6
In the region of£k1, k̄1
¤, if the steady state exists, then the wage function has to
satisfy the following inequalities:
k1 ≤ w (k1) = w̄¡k̄1¢≤ k̄1. (30)
The first inequality can be rewritten as
k1 ≤ (1− α) zkα1 or k1 ≤ [(1− α) z]1/(1−α) = k∗A.
But we have already shown that in the presence of f̄ no firms will be using f in the
autarkic steady state. But k1 is the maximum level of k when firms are using f ,
thus we must have k∗A > k1. For the second inequality, we get
w̄¡k̄1¢≤ k̄1 ⇐⇒ S(χ(k̄1)) ≥ 0. (31)
Now recall (24). If it holds, then both k̄ and k̃ are steady states for k̄1 < kt < k̄2.
So we have min³k̄, k̃
´> k̄1 or min
hχ¡k̄¢, χ³k̃´i
> χ¡k̄1¢. Recall that S(χ)
is U-shaped and S(χ¡k̄¢) = S
³χ³k̃´´
= 0, we get S¡χ¡k̄1¢¢
> 0 and (30) is
satisfied.
If (24) does not hold, then only k̄ is the steady state for k̄1 < kt < k̄2. Since
f̄ is not the equilibrium technique at k = k̃, we then have k̄ > k̄1 > k̃ and hence
S¡χ¡k̄1¢¢
< 0. The inequality (31) is violated and so there is no steady state in
the region of£k1, k̄1
¤.
Likewise, using the same procedure, we can show that for the range£k̄2, k2
¤, if
(24) holds, we get a unique steady state. Otherwise, no steady state exists when
(24) fails.
6.4 Proof of Proposition 8
It is easy to see that a poverty trap emerges when k∗A < min³k̃, k̄
´. For k̃ > k̄,
then we must have k̃ > k̄ > k∗A since k̄ is the foreign steady state. For the case
where k̃ < k̄, then k̃ is unstable so that poverty traps require k̃ > k∗A. But it is
easy to show that k̃ > k∗A ⇐⇒ zz̄<³k̃k̄
´1−αand the result follows.
23
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25
production techniques (σ < 0)
production technology (σ < 1)
production function (σ = 1)
K
N0
Figure 1: The Production Concepts
•
•
0
w(k)
k1k 2k
Figure 6: Multiple Steady States under Imitation
•
•
••
••
•
•
*Lk *
Ak *Hk
1+tk
tk
45°
Λ(χ)
χ0 • •
•
χ1 χ2
•Zα
⎟⎠⎞⎜
⎝⎛ −
αα 1
11Z
Z •
Figure 7: Unit Costs Comparison under Adaptation
χmin