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Appropriate Technology and Growth Susanto Basu; David N. Weil The Quarterly Journal of Economics, Vol. 113, No. 4. (Nov., 1998), pp. 10251054.
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APPROPRIATE TECHNOLOGY AND GROWTH*
We model growth and technology transfer in a world where technologies are specific to particular combinations of inputs. Unlike the usual specification, our model does not imply that a n improvement in one technique for producing a given good improves all other techniques for producing that good. Technology improvements diffuse slowly across countries, although knowledge spreads instantaneously and there are no technology adoption costs. However, even with "Ah" production, our model implies conditional convergence. This model, with appropriate technology and technology diffusion, has more realistic predictions for convergence and growth than either the standard neoclassical model or simple endogenousgrowth models.
Do all countries in the world use the same technology? Many would view even the posing of this question as absurd. 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 airconditioned 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. But this stance raises a new problem when one considers technological change: do technology improvements that raise the productivity of combines in America also improve the productivity of farmers in India? The answer obviously seems to be No. However, standard models of economic growth, which index technology by a single coefficient that is independent of factor proportions, would say Yes. In these models, technology improvements in the United States should immediately improve total factor productivity in Indiawhich seems counterfactual. To escape this problem, standard models often assume that technological improvements are countryspecific. But since there are many examples of technology transfer within countries, why should the flow of technology stop at national boundaries? We provide a new way out of this quandary by focusing on the
* We thank James Feyrer for outstanding research assistance. We are grateful to Olivier Blanchard, Charles Jones, Michael Kremer, two anonymous referees, and seminar participants a t Brown University, Columbia University, Harvard University, Hebrew University, the National Bureau of Economic Research, Stanford University, and the University of Haifa for comments. Basu thanks the National Science Foundation for financial support, and gratefully acknowledges a National Fellowship a t the Hoover Institution and a Research Fellowship from the Alfred P. Sloan Foundation.
o 1998 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology. The Quarterly Journal of Economics, November 1998
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issue of appropriate technology. We believe it is reasonable to model technological advances as benefiting certain types of technologies and not others. For example, an advance in transportation technology in Japan may take the form of a refinement of the newest maglev train. Such an advance may have very few spillovers to the technology of the transportation sector in Bangladesh, which relies in large part on bicycles and bullock carts. As a convenient shortcut for modeling "appropriateness," we index technologies by capital intensity, where in the definition of capital we include both human and physical capital. Each technology is thus appropriate for one and only one capitallabor ratio. We also model technological improvements as expanding the production possibilities frontier for a given capitallabor ratio. This is reasonable if one thinks of technological improvements as taking place because of learningbydoing. It may also be a reasonable reducedform model of the outcome of investment in R&D. So in our model technology transfer is not immediate because countries take time to achieve a level of development that can take advantage of the progress being made by the technology 1eaders.l Of course, we are not the first to consider this issue. A large literature examines barriers to the transfer of technology across countries. Parente and Prescott [I9941 present a model in which there is a world "best practice" technology and different countries erect barriers that raise the cost of adopting a higher level of technology The qualityladder model of Segerstrom, Anant, and Dinapoulos [I9901 simply assumes that a fixed amount of time must elapse before hightechnology"Northern" innovations spread to the laborintensive "South."Grossman and Helpman [1991, Ch. 1 1 and Barro and SalaiMartin [I9971 present models where 1 technology diffusion is slow because imitation is c o ~ t l y . ~ We view the mechanism presented in this paper as comple1. The concept of appropriate technology that we use is related to that of Schumacher [1973], although there are significant differences between the two. Schumacher critiqued development policies that stressed large, capitalintensive projects as a means of technology transfer. He argued that in poor countries capitalintensive processes would be unproductive due to lack of marketing and financial infrastructure, inappropriate inputs, and untrained workers. The intermediate technology movement Schumacher founded attempts to create new technologies and to locate and transfer existing technologies for smallscale, lowcapitalintensive rural production. The model that we present here shares the property that capitalintensive technologies are inappropriate for poor countries, but in our model there is never a problem of countries using technologies that do not match their level of development. 2. Yet another possibility is that new technology is embodied in capital goods: in order to achieve stateoftheart technology, a country needs to import stateoftheart equipment. If there are borrowing constraints or adjustment costs,
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mentary to these other models of impediments to technology transfer. The existing literature focuses on the impediments to the transfer of a specific technology. We, by contrast, assume that all technology 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.3 Within any given country, our model of technological progress is similar to that of "localised learning by doing," as introduced by Atkinson and Stiglitz [19691.4They examine a model in which a firm (or economy) learns over time to improve the productivity of the particular mix of capital and labor that it is currently using. The model of learningbydoing that we present is simply a less extreme version of the AtkinsonStiglitz model: firms improve the productivity not only of the specific capitallabor mix that they are using, but also of similar techniques. When we consider a world with many countries, we assume that technological improvements made in one country are immediately available to all. Our model of technical progress combines properties of the endogenousgrowth and neoclassical growth models. As in endogenousgrowth models, the longterm growth rate in our model depends on the saving rate (which is the only exogenous variable in our model). However, as in the neoclassical growth model, countries that differ in their saving rates may share a common growth rate in steady state and differ only in their levels of output. This result follows from the fact that spillovers are usually not symmetric in our appropriatetechnology model: a country that is the technology leader benefits less from its followers than they benefit from it. Since a follower country can use the technology of the leading country only if it has a sufficiently high level of development, the relation between growth and saving can be highly nonlinear. Over a range of parameters, changes in the saving rate just change the steadystate level of income relative to the income of the technology leader. But outside that range, changes in the saving rate affect the growth rate. This result suggests that empirical tests of
investment rates will be bounded away from infinity, and technology transfer will proceed a t a finite rate [Lee 1995; Mazumdar 19961. 3. Benhabib and Rustichini [I9931 present a similar model, focusing mostly on welfare issues. 4. See also Stiglitz [1987]. Lucas [I9931 presents a similar model, in which learning by doing applies only to a specific factor mix and output good.
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endogenous growth that attempt to estimate simple savinggrowth relations might be severely misspecified. Our paper is organized as follows. In Section I we present a simple onecountry model of growth in which technology advances through learningbydoing and where the level of technology is specific to a particular degree of capital intensity. In Section I1 we consider the same model in a world with two countries, in which technology moves freely. We examine the conditions under which the two countries will have different growth rates in steady state, and the conditions under which they will have different income levels but the same growth rate. We also examine the conditions under which the "follower country" can have higher consumption than the leader. Section I11 examines a world with more than two countries, and demonstrates the possibility that "convergence clubs" of countries with similar growth rates will arise. Section IV extends the twocountry model to allow for perfect and imperfect capital mobility. We show that our qualitative conclusions survive under imperfect capital mobility, and even under perfect capital mobility if there are countryspecific differences in productivity. Section V discusses empirical implications of the model. Our results lead us to reappraise the interpretation of the "conditional convergence" finding, which has been taken as evidence against the endogenousgrowth model. Such an interpretation is not necessarily true in a world with technology transfer. For example, our model generally displays conditional convergence, even though the growth rate is endogen~us.~ Section VI concludes.
Although we focus on technology spillovers between countries, we first develop our model in the context of economic growth in a onecountry world. In order to highlight the novel features of our model of appropriate technology, we keep the structure very simple. For example, rather than modeling preferences explicitly, we assume a fixed saving rate. (Of course, for certain parameter values and demographic assumptions, a constant saving rate may be optimal.) As we show below, for analyzing output growth (but not
5 . Barro and SalaiMartin [I9971 and Howitt [I9971 make the same observation in discussing models of technology diffusion.
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consumption) differences in saving rates between countries are isomorphic to crosscountry differences in productivity levels, of the kind documented by Islam [19951; Caselli, Esquivel, and Lefort [19961; Klenow and RodriguezClare [19971; and Hall and Jones [19991. These productivity differences may be due to variations in naturalresource endowments or political institutions. Thus, the saving rate in the model below should be interpreted broadly, as a metaphor for a more general set of factors that affect output per worker. We begin by considering a model in which there is no technological progress. Assuming that the firmlevel production function is CobbDouglas and has constant returns to scale in capital and labor, the aggregate production function for per capita output in the economy is
where a is the share of capital in output. Equation (1) embodies the idea that there are externalities to capital. This idea has been discussed extensively in the growth l i t e r a t ~ r eFor simplicity and ease of comparison to this previous .~ literature, we assume that A*(K) = BK1".I The parameter B can be thought of as representing the components of technology, broadly defined, that do not change over time and are not subject to imitation from abroad. Examples are natural resources, culture, and institutions. (In Section IV we consider the case where B varies across countries.) Thus, for this model the production function is just Y = BK, as in Rebelo [19911. We now alter the model by assuming that the technology for producing output with a given capitallabor ratio K can itself change over time. Let A(K,t) be the level of technology for capitallabor ratio K at time t. Thus, the production function for this amended model is given by
As noted, the distinguishing feature of our model is that we allow the level of technology, A(K,t), to change over time. We now interpret A*(K) as the maximum level of technology at the given value of capital per worker. This specification captures the idea
6. See, e.g., Romer [I9861 and Rebelo [19911. De Long and Summers [1991, 19931 argue that there are positive spillovers to investment in capital equipment. 7. Alternatively, we could derive our aggregate model by simply assuming that the firmlevel marginal product of capital is constant, as in Rebelo [19911.
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that there are always new productive techniques to be developed, but after a point there is no new technique that uses a given level of capital per worker; that is, the oxcart can only be improved so much.8 However, we also assume that A" increases with K, which captures the idea that technologies have increasingly high potential at higher levels of development: the maglev train can be improved more than the oxcart. From now on, it will generally be convenient to discuss quantities in logarithms. We henceforth use lowercase letters to represent natural logs of their uppercase counterparts; e.g., k = In (K). We now specify the process by which technology improves. We assume, for reasons we do not model explicitly, that producing at some level of capital per worker raises the level of technology appropriate for capitallabor ratios within a neighborhood of a country's current capitallabor ratio (henceforth called the "capital stock"). Specifically, we assume that the growth rate of the technology appropriate for some (log) capital stock j is positive if the log distance between the country's current capital stock k and capital stockj is less than some positive parameter y:
=
0
otherwise.
Note that the change in the level of technology is a function of the log of the capital stock. P is a parameter of our model. We naturally assume that P > 0. The initial capital stock is ko. We assume that, beyond a certain range, technologies were completely undeveloped at the beginning of time; i.e., we assume that there exists some x > ko such that A(j,O) = 0 ' j > x. We also i l assume that A(ko,O)> 0. Equation (3) implies that a country uniformly improves technologies that are related to the technology it is currently using. We use the uniform distribution for simplicity, but one could use others. For example, a distribution that shifts mass from the tails to the center would capture the idea that the usefulness of spillovers is proportional to the distance between a technology and the technique currently being used. Our qualitative results would not be affected by this change.
8. Young [I9931 presents historical evidence supporting the proposition that the maximum effect of learning by doing is bounded for any given technology.
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Define R as the ratio of technology to its maximum level:
Note that this definition ofR implies that the production function can be written as Y = RBK. The ratio R is determined by a country's growth rate. If a country grows rapidly, it will have relatively little time to improve the technology at any given capital stock. The intuition is that the faster the growth rate the less time a country will be "in rangenwithin distance yof a given capital stock. Thus, R is a negative function of the growth rate. This result is a general featureindeed, the essenceof a model of localized learningbydoing. Localized learningbydoing implies that the longer one uses a given technology the more efficiently one can use it. Thus, fast growth necessarily goes hand in hand with lower efficiency relative to potential. For example, Young [I9921 finds that Singapore had high rates of investment and low average TFP growth; he argues that the two outcomes are related, because Singapore continually changed technologies of production without reaping the benefits of learningby doing for any one technology. Our assumptions about how technology improves can best be understood by examining Figure I, which shows the relative level of technology, R, as a function of the (log of the) capital stock at two different points in time. We assume that the capital stock is growing at a constant rate; we show below that this is true in the steady state. The top panel shows an economy in which capital is &owing slowly, and the bottom panel an economy in which capital is growing rapidly. The two figures are, of course, qualitatively similar. At time 1the economy has capital stock kl, a corresponding ratio of technology to its maximum level of R*, and is improving technologies between k1  y and kl + y. Technologies appropriate fork < kl  y are at the maximum level ofR that they will attain, which we call R. (Note that E < 1: no technology is perfected before it ceases to be improved.) The value ofR for each technology being improved is proportional to the length of time that the corresponding level of capital has spent "within range" of the economy's actual capital stock. Thus, as we expect, R is close to R for levels of capital near k1  y, R = 0 for all k > k1 + y, and R is between 0 and R at all intermediate levels of capital, including the economy's actual capital stock kl. At time 2 the whole figure shifts to the right. Note that the level of R at the economy's actual
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A. Slow Capital Growth
klY
kl
k,
k,+y
log capital
B. Rapid Capital Growth
R
log capital
FIGURE I Relative Technology as a Function of Capital
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capital stock remains constant at R*: we show below that this is another property of the steady state. Now note the differences between the fast and slowgrowing economies. In the fastgrowing economy shown in the bottom panel, k2 is naturally farther to the right of kl than in the slowgrowing economy. But since the fastgrowing economy spends less time "within range" of any given level of capital, it has both a lower steadystate technology ratio, R*, and a lower R. Thus, there is a penalty for fast growth in our model that is not present in the usual Rebelo model. We now show that the steady state of the onecountry model is characterized by a constant growth rate of capital and output and a constant ratio of the existing level of technology to the maximum level of technology. We derive R'k, the steadystate level of R, as follows. Consider some level of capital, k, which will be reached at time t by an economy that is growing in steady state at rate g . We are interested in deriving A(k,t). First, note that the technology used to produce at capital level k will begin improving when the economy's capital stock reaches the level k  y; this takes place at time t  ylg. Since technologies start at a level of zero, we know thatA(k,t  ylg) = 0. Equation (3) for technology change can be rewritten in the following form, which makes it clear that the gap between the maximum level of technology and its current level is subject to exponential decay:
Integrating this equation between t and t  ylg and using the boundary condition A(k,t  ylg) = 0, we find the steadystate A(k,t),which we write simply as just A(k):
Hence, in steady state,
As we claimed above, fastergrowing economies (higher g ) have lower steadystate values ofR. In order to close the model, we have to make an assumption about saving behavior. We assume simply that the saving rate is
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constant:
where s is the exogenous saving rate and 6 is the rate of depreciation. (For simplicity, we assume that there is no population growth.) Thus, it is always true that
The growth rate of output is
In the steady state R is constant at some level R'k, and thus the steadystate growth rate of output is9 (8) g =sR*B

6.
Figure I1 shows the determination of steadystate growth and technology in the singlecountry case as the solution to equations (6) and (8). A higher rate of technology improvement (larger P) rotates the R(g) curve clockwise, simultaneously increasing both the growth rate and the level of technology. A higher saving rate will rotate the g(R) line counterclockwise, thus increasing the growth rate of output but lowering the technology ratio in the new steady state.1° Since the steadystate growth rate is unique, it clearly depends only on the parameters of the model and not on initial conditions. There is hysteresis, however, in the sense that a country's initial conditions (the level of the capital stock and the
9. Note that in equation (3) we assumed that technologies improve for capital stocks that are a constant log distance from the current level of capital. Assuming instead that the zone in which learningbydoing takes place differs from the current capital stock in terms of levels rather than logs would imply that the growth rate asymptotically goes to zero. To see why this result must be true, note that a t a constant growth rate the length of time during which the economy is "in range" of a given capital stock would fall steadily over time, implying that R would fall. But ifR falls, the growth rate must fall as well. 10. The dynamic analysis of the model is complicated by the fact that equation (8) holds only in steady state. As we npted, outside the steady state the growth rate of output, g , depends on both KIK and RIR. We conjecture, however, that one can establish stability as follows. Note that if we interpret g as the growth rate of capital rather than output, equation (8) holds even out of steady state. Thus, an increase in the saving rate, for example, rotates the g(R) line counterclockwise. The immediate effect of a savings increase is to leave R unchanged and move g to the value dictated by the new g(R) line. However, this new level ofg iq higher than the steadystate level, since the faster growth rate of capital makes R negative. Therefore, the economy slides along the new g(R) line toward the new steady state. Unfortunately, one cannot easily take the analysis further, because the R ( g ) function, equation (6), also holds only in the steady state.
APPROPRIATE TECHNOLOGY AND GROWTH
FIGURE I1 Determination of the Steady State for a Single Country
initial level of technology) will be reflected in its level of output at any point in time.ll
The only difference between the one and twocountry models is in the technology growth equation. Allowing for more than one country creates the possibility that technology at a given capital ratio may be simultaneously improved by spillovers from multiple countries. Letting i index countries, we thus have
(9)
A(j,t) = p(A*(j)  A(j,t))
2 I(ki y <j < hi + y),
i
11. In the model of Atkinson and Stiglitz [1969], by contrast, there can be hysteresis in growth rates. Our model does not have this property because factor proportions do not vary in response to the available technology (so the capitallabor ratio never gets "stuck) and learning is not completely localized (so the cost, in terms of productivity, of moving to a new capitallabor ratio is never too high).
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where I( ) is the indicator function. The assumption implicit in equation (9) is that when two countries are simultaneously within range of a given capitallabor ratio, the associated technology improves twice as much per unit of time as it would if it were being improved by only one country.12 Technology improvements can also spill over from one country to another even if they are not simultaneously improving the same technique: for example, a "follower country" benefits from technology improvements made by a "leader country" in the past, because it inherits a positive level of technology and thus starts from a higher level than did the leader. In a multicountry world we need to keep careful track of the history of technology changes; this is why A is also a function of time. There are two kinds of steady states in the twocountry model: ones in which the two countries have the same growth rate of output, and ones in which they do not. Which type of steady state is observed will depend on the parameters of the model, but not on initial conditions. We first consider the steady state in which the two countries have the same growth rate. Let g be the common growth rate, sl and s2 the saving rates for the two countries, and R; and R; the steadystate ratios of A(k)lA"(k) for the two countries. Without loss of generality, assume that s l > s2. We show below that if both countries grow at the same rate, then the country with the higher saving rate must have higher capital intensity of production. The common growth rate for the two countries in steady state is given by the following equations:
and
Country 1has a higher saving rate than Country 2. If they are to have a common growth rate, then Country 1must have a lower level of technology at each capital stock. This can happen only if Country 1is always "in the lead;" i.e., if at all times Country 1has a higher level of capital per worker than does Country 2.
12. We always examine a world with a fixed number of countries. If the number of countries is variable, one obviously needs to normalize the effect of each country on world technology by some scale variable to avoid nonsensical results.
APPROPRIATE TECHNOLOGY AND GROWTH
I
I
I
rn
log capital stock
k1Y
k 2
kl
k2+Y
kl+Y
FIGURE
I11
The TwoCountry Model
Intuitively, Country 1 "passes over" a given capital ratio and leaves an improved level of technology for Country 2 to use, allowing Country 2 to grow a t g despite its lower saving rate. Define d as the log distance between the capital stocks of the two countries in steady state: d = k l  k2.Figure I11 illustrates the relation between the two countries' capital stocks. Technologies with capital intensity greater than kl + y are not affected by either country. Technologies with capital intensity between k2 + y and kl + y are affected only by Country 1. Technologies with capital intensity between kl  y and k2 + y are affected by both countries. Technologies between k l  y and k2 are affected only by Country 2. (Country 2 also improves technologies from k2 to k2 y, but these technologies are irrelevant since they will never again be used by either country.) Thus, the two countries have steadystate ratios of current to maximum technologies ofy3
13. These equations are derived by integrating equation (91, analogously to the derivation of equation (6). Consider a level of capital, k , which will be reached by Country 1a t time t and by Country 2 a t time t + d l g . Between time t  ylg and time t  ( y  d)lg, only Country 1 will influence technology a t h , and so the gap between A " ( k ) and A ( h ) will decay a t rate P. From time t  ( y  d)lg to time t , the gap between A * ( k ) and A ( k ) will decay a t rate 2P, since both countries will be influencing technology a t k . The overall decline in In ( A * ( k ) A ( k ) )will thus
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Combining the four equations (lo), ( l l ) , (12), and (13) yields solutions for the four endogenous variables, R;, R;, d, andg. We are now in a position to say when the steady state will be characterized by a single common growth rate for the two countries. Specifically, there will be a common growth rate if the implied value of d from the equations above is less than y (see Figure 111). If this condition is violated, then (12) and (13) are no longer correct. Instead, RT and R; are given by
and
Note that equation (14) for R; is the same as equation (6) in the onecountry case. The reason is that if the two countries grow at different rates, then in the steady state Country 1(which by virtue of its higher saving rate grows faster) does not receive any benefit from Country 2. Country 2, on the other hand, receives the maximum benefit from Country 1 (2ylgl) as well as the benefit from its own technology improvements (ylg2).14 this steady In state the two countries will have different growth rates, given by equations analogous to (8) in the onecountry case:
and
As we noted above, our model can readily accommodate crosscountry differences in productivity levels, which are naturally modeled as differences in the parameter B across countries. If B is allowed to differ across countries, then equations (lo),( l l ) , (16), and (17) would be replaced with equations of the form, gi =
be p(dlg) + 2P((y  d ) / g ) = P((2y  d)lg). This can be rearranged to yield equation (12). 14. In terms of Figure 111, if d > y, then Country 1's "region of influence" passes over a given k before Country 2 reaches that level of k . Hence that technology receives a full 2y/g, benefit from Country 1 (both the leading and trailing regions of influence pass over k ) as well as the usual y/gz benefit from the leading region of Country 2.
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siRTBi  6 for i = 1, 2. Note that for analyzing g and R, crosscountry differences in B are completely isomorphic to the differences in s on which we focus, since only the product sB matters for our results. (However,consumption will differ depending on whether it is B or s that varies across countries.) One natural question that might arise is whether the follower country should "convexify"use some of its capital stock to set up a hightech enclave with a high capitallabor ratio that can use cuttingedge technologies, and leave the rest of the country with a lower capitallabor ratio. In the steady state the follower country will never wish to convexify: since the technology being developed by the leader is farther fromA*(k) than the technology used by the follower, a convex combination of the two will leave the follower with worse average technology.15
A. Comparative Steady States In this subsection we examine the effect of changes in saving rates on the steady states of the twocountry model. Changing saving in a single country will change the distance between the two countries' capital stocks in steady state, and also change their common growth rate. Changes in saving can also move the steady state from being one with common growth rates to being one in which the countries have different growth rates. Figure IV shows the growth rates of Countries 1 and 2 as functions of the saving rate in Country 1, holding constant the saving rate in Country 2 (we no longer assume that sl > sz).16 For a sufficiently low saving rate in Country 1, the steady state will be such that Country 1grows more slowly than Country 2. Increases in the saving rate in Country 1 will increase the growth rate in Country 1, but leave Country 2's growth rate unaffected. At a critical level of saving sl the steady state switches to one in which
15. The reason for this result is that, from the point of view of the follower country a t any point in time, the A(k,t) function is concave in k. With more than two countries, however, this condition is not guaranteed. In particular, one country might find itself between two large groups of countries, both of which have improved their technology to a high level. In that case, the country in between may want to take advantage of this nonconvexity inA by splitting into two sections: one with higherthanaverage k and the other with lowerthanaverage k. We rule out this behavior by assumption. One way of justifying it is to hypothesize that there are strong sectoral complementarities within a country, so that all sectors have to improve together to adopt a better technology. 16. All figures and simulations use parameter values of y = 1,B = 1 , 6 = 0.05, and p = 0.01/n, where n is the number of countries.
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0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Saving Rate in Country 1
FIGURE
IV
Growth as a Function of Saving in the TwoCountry Model
The saving rate in Country 2 is 0.2.
the two countries grow at the same rate, and the distance d between their capital stocks is less than y.17 At this point, further increases in s l have two effects. First, for a given RT, increases in saving increase the growth rate. Second, and partially offsetting, however, the higher growth rate pushes the two countries closer together, decreasing Thus, the overall effect of increases in sl is to raise the common growth rate of the two countries and to narrow d, the distance between them. If sl increases beyond a critical value &, the two countries again grow at different rates in steady state, with Country 1 now growing faster. Note that, as s l rises above q, dglldsl increases. The logic for this result is as follows. For saving rates below q, Country 1 is benefiting from Country 2's presence in the world,
RT.
17. A referee pointed out that qualitatively different steady states often require large differences in saving rates. As we noted above, crosscountry differences in productivity levels would have the same effect as differences in saving rates. Thus, Figure IV and similar exercises below can also be interpreted as exploring the effects of changes in productivity resulting, e.g., from changes in political institutions.
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and Country 1's growth rate is higher than it would be if it were in autarky. As sl rises toward q, however, this benefit diminishes gradually, since Country 1receives less and less spillover benefit from Country 2. This reduced spillover depresses dglldsl. Once its saving rate is beyond q, Country 1is effectively in a onecountry world, and its growth rate responds to its saving rate as if Country 2 did not exist. An important effect to note is that when sl increases beyond G, The growth rate of Country 2 falls. The rationale for this result is as follows. At the point where d = y, Country 2 is getting the maximum benefit from the spillover from Country 1(and Country 1 is getting no benefit from Country 2). As the growth rate of Country 1increases further, it will spend less time improving the technology at any given capital ratio, and thus Country 2 will inherit technology that has been improved less than it would have been if Country 1were growing more slowly.
B. Relative Consumption Consider two countries that are growing at the same rate. When one country raises its saving rate, it both raises its level of income relative to the other country, and also raises their joint growth rate. The effect on relative consumption in the two countries is ambiguous, however. By saving more, a country uses productive technology that is less mature, and also consumes a smaller fraction of its incomefactors that can offset the positive effect of a higher capital stock on output. Our previous analysis indicates that there is a positive externality from the saving of at least the leading country in the twocountry model. In steady states where both countries grow at the same rate there are positive spillovers from saving in both countries. This situation can lead each country to try to free ride off the other, by being a follower and inheriting the leader's improved technology. An analogy is bicycleracing, where one wishes to follow rather than to lead in order to conserve energy and still maintain speed by using the slipstream of the leader. Figure V explores this issue of relative consumption using our baseline set of parameters. The vertical axis measures the log of C1/C2, SO positive values indicate that Country 1 has higher relative consumption. We consider consumption in Country 1 relative to consumption in Country 2, as a function of the saving rate in Country 2. We plot this function for three different saving rates in Country 1. Obviously, if the two countries have equal
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Country 1 = 0.15
saving rate in Country 1 = 0.25
saving rate in
0.05 0.05
1
I
I
0.15 0.2 0.25 0.3 0.35
0.1
saving rate in Country 2
FIGURE V Relative Consumption in the TwoCountry Model
saving rates, they will have equal consumption. For a low saving rate in Country 1(0.15), relative consumption in Country 1is a negative function of Country 2's saving. In other words, if Country 2 saves more than Country 1, it will have higher relative consumption. For a high saving rate in Country 1 (0.25), the opposite holds: Country 2 will have higher relative consumption if it saves less than Country 1. The intuition for this result is straightforward. If Country 1 has a'low saving rate, it has already substantially improved the technology it leaves Country 2 by spending a long time "within range" of each capitallabor ratio. Thus, Country 2 has little to gain by keeping its own saving rate low and trying to improve the technology further; it is better off by increasing savings and moving to a higher capitallabor ratio. But the logic is just the reverse when Country 1 has a high saving rate; in that case Country 2 can do better (relatively) by having a low saving rate itself, improving its technology further and securing higher consumption. A natural scenario to consider is the case where saving is set
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taking into account only relative consumption. It is clear from Figure V that one Nash equilibrium in this case will be the saving rate s*. If both countries save at this level, then neither can raise its relative consumption by changing its saving rate slightly.ls
One of the themes in the recent growth literature has been the possibility that countries will endogenously clump into discrete "convergence clubs."19 In Durlauf and Johnson [I9951 this clumping takes place because nonconvexities in the aggregate production function create multiple locally stable steady states. In our model, the possibility of convergence clubs arises because of spillovers between countries. Countries with small differences in saving rates will tend to have the same growth rate, since the lowersaving country will benefit from spillovers from higher saving one. But if countries differ by too much in their saving rates, then they will grow at different rates. Thus, a world comprising numerous countries that have slightly different saving rates might be expected to break up into a number of discrete clumps, within each of which countries would differ only in their levels of income, while clumps would differ from each other in their growth rates. To explore this possibility, we examine a world with three countries, in which equations analogous to (10)(13) can be solved analytically. Figure VI shows how the steady state configuration of growth rates depends on saving rates in the three countries. The figure is generated by holding constant the saving rate in a single country (Country 3, with a saving rate of 0.20), and varying the saving rates in the other two countries. For each pair of saving rates in Countries 1 and 2, we indicate the configuration of the world economy. There are four possibilities: first, each country can grow at its own rate; second, there can be a convergence club of the two lowsaving countries, with the highestsaving country growing at its own, faster rate; third there can be a convergence club of the two highsaving countries, with the lowsaving country growing at a slower rate; and finally, there can be a single worldwide
18. For the parameters that we use, s* is approximately 0.204. 19. Quah [1993, 19961 argues that countries are converging into two such clubs, one where the poor are getting poorer and the other where the rich are getting richer.
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.06
.30
Saving Rate in Country 1
/
all three countries grow at the same rate convergence club of two richest countries convergence club of two poorest countries each country grows at a different rate
FIGURE
VI Convergence Clubs in the ThreeCountry Model
The saving rate in Country 3 is 0.2.
convergence club. For the parameters we examine, all four cases are present. In Figure VII we consider the experiment of varying a single country's saving rate, holding constant the saving rates of the
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: '
growth rate in Country 1
growth rate in
  
,,.A':
 ..  
_____
/
Country 2
Saving Rate in Country 1
FIGURE II
V Growth as a Function of Saving in the ThreeCountry Model
The saving rate in Country 2 is 0.07 and in Country 3 is 0.30.
other two countries in the world. Specifically, we hold the saving rate of Country 2 constant at 0.07 and the saving rate of Country 3 constant at 0.30, and vary the saving rate of Country 1 between 0.06 and 0.22. When Country 1has a sufficiently low saving rate, Country 1 and Country 2 have a common growth rate, while Country 3 grows faster. As Country 1's saving rate rises, the convergence club of Countries 1 and 2 breaks up, and all three countries grow at different rates. As Country 1's saving rate rises even further, Countries 1 and 3 form a convergence club with a common growth rate, while Country 2 grows more slowly. Note the relation between saving and growth rates in Country 1: at the point where Country 1breaks out of its convergence club with Country 2, the effect of saving changes on the growth rate rises. When Country 1joins a convergence club with Country 3, the size of the effect of saving on growth falls. For the parameters that we use, the derivative of Country 1's growth rate with respect to its saving rate is 0.05 just before it leaves the lower convergence club, rises from 0.20 to 0.25 over the period when it is in its own "convergence club," and is 0.09 immediately after it
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joins the upper convergence club. We discuss this feature of the model in Section IV. Also worthy of note is the effect of increases in saving in Country 1on the growth rate of Country 2. When the two countries have similar saving rates, increasing saving in Country 1raises the growth rate of output in Country 2. As saving increases in Country 1, and as the lower "convergence club" breaks up, increases in saving in Country 1harm Country 2 not only in terms of its position relative to the rest of the world, but in absolute terms as well. Finally, note that qualitative differences in the configurations of the three countries (i.e., transitions between convergence clubs), happen with much smaller differences in saving rates than was the case in the twocountry case. This result is particularly striking when one recalls that the saving rate in our model also captures crosscountry differences in productivity due to factors such as differences in political institutions, which recent research shows can be important.20Thus, even in a threecountry world, relatively small changes in economic policy and legal organization can lead to significant differences in the world steady state. We conjecture that the exogenous changes required to move countries between convergence clubs are even smaller in more realistic environments where each country is a smaller fraction of the world. IV. CAPITAL OBILITY M So far, we have assumed that capital is not mobile across borders. However, as Lucas [I9881 argues, ideally one wants to derive the result that crosscountry capital flows are small rather than simply assuming that capital cannot move across countries. We thus investigate the implications of both perfect and imperfect capital. mobility in our model. For simplicity, the discussion assumes that there are only two countries in the world, but many of our results generalize to the case of more than two c ~ u n t r i e s . ~ ~
20. For example, Hall and Jones [I9991 find that even countries with relatively similar levels of income per worker and capitallabor ratios often have quite different levels of total factor productivitye.g., India's TFP level is about 2.5 times as large as China's. They attribute these gaps to differences in "social infrastructure." 21. In a world with capital mobility, we have to distinguish between GDP and GNP. We assume that saving is a fixed fraction of GDP, an assumption which can be justified in a twoperiod overlappinggenerations model where all saving is done by young workers, whose only income is (domestic) labor income.
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A. Perfect Capital Mobility In the model we have developed, the private net marginal product of capital in country i is ri = &,Bi  6. In a world without capital mobility, the rate of return to capital may differ across countries if they have different values ofR or B . First, assume that countries have the same productivity parameter B (and common values of all the other technological parameters), and differ only in their saving rates. Then, since trailing countries always have a higher ratio of technology to its maximum level, they always have a higher marginal product of capital. Thus, in a world with perfect capital mobility and no country fixed effects, one would expect to see capital flowing from rich to poor c ~ u n t r i e s . ~ ~ In such a world it is immediately apparent that per capita output and its growth rate also equalize. If the two countries would have been in the same convergence club absent capital mobility, we can show that their common growth rate rises when capital becomes mobile internationally. To do so, we note that we can conceptualize the case of capital mobility as the case where two countries have identical saving rates, and think about autarky as the case where national saving rates can differ. Formally, take the twocountry model of Section 11, and let sl = S + E and s2 = S  E . We can study the impact of capital mobility between (similar) countries on their common growth rate by taking the derivative d g l d ~ and evaluating it at E = 0. We find that the derivative is zero. (We check the secondorder condition for a maximum, and find that it is satisfied.) Thus, we conclude that allowing capital mobility between two (slightly) dissimilar countries will increase their common rate of growth. With capital mobility the longrun level of output per capita in the following country will be higher than it would have been absent capital mobility (at least in the case where the two countries initially grew at the same rate). Now consider a world where two countries have the same saving rate but different values of B. Then, as long as they are members of the same convergence club (i.e., have the same growth rate), we can prove a striking result. Countries can have different
22. The fact that trailing countries have better technology, and thus higher marginal products of capital, changes the prediction of the basic Ah model that marginal products are independent of capitallabor ratios. Thus, our model introduces a new force leading to capital mobility that does not exist in the Ah model.
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capitallabor ratios, and different values of R, but will have identical marginal products of capital. Thus, even with perfect capital mobility, capital will not flow from the rich country, which has a lower R, to the poor country, which has a higher R! The proof of this proposition is straightforward. It is easy to see that differences in B create effects that go in opposite directions. Raising B has the direct effect of raising the marginal product of capital, but it also leads to faster growth and thus a lower value of R, which lowers the marginal product of capital. Surprisingly, if two countries have the same growth rate and saving rate, these two effects just offset one another. To see this result, compare equations (10) and (11)in Section 11, assuming that countries have the same saving rate but different values of B. It is clear that the two countries must have identical values of the product RB. But since the marginal product of capital in country i is just d i B i  6, it follows that they have the same marginal product.
B. Imperfect Capital Mobility
We have shown that even a model with full capital mobility is consistent with crosscountry differences in technology, and can give rise to the leaderfollower pattern that is present in our autarky world. But full capital mobility is, of course, an extreme case. A more empirically relevant model is one in which capital flows between countries only in response to sufficiently large interestrate differentials. Assume again that two countries have identical values of B, but different saving rates. Let the maximum interestrate differential between Countries 1and 2 be p. Assume that Country 1has a higher saving rate than Country 2. If capital is flowing between the two countries, then the gap between their steadystate technology levels is
In this case the technology levels will be closer than they would otherwise have been, as will the levels of output per capita (assuming that the two countries grow at the same rate in the steady state with capital mobility). As in the case of full capital mobility, partial capital mobility will tend to raise the joint growth rate of the two countries. Of course, if the interestrate difference between the two countries in the autarky steady state was less
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than p, then introducing limited capital mobility will not change our previous results. Note that limited capital mobility need not rule out the existence of multiple "convergence clubs." If p is large enough to allow sufficiently large differences in R (as given by equations (14) and (15) in the twocountry case), then capital mobility is consistent with countries growing at different rates even in the steady state. This result may seem surprising, until one recalls the basic "Ah" structure of our model. Countries differ in their marginal products of capital because they have different technology levels, but not because marginal products are driven down by capital accumulation at a given level of technology. Thus, different longrun growth rates of output do not imply infinitely large differences in the marginal product of capital.
A. CrossCountry Growth Regressions There is now a large literature using crosscountry growth regressions to test for "conditional c~nvergence."~~ we disHere cuss what such regressions would show if the world behaves as our model. Like most of this literature our discussion focuses on the model with no capital mobility. The standard growth regression uses a country's average growth rate of output as the dependent variable and its saving rate and initial income as the righthandside variables.24 Mankiw, Romer, and Weil [I9921 (hereinafter MRW) find a positive effect of saving on growth and a negative effect of initial income on growth. Both of these results are consistent with the hypothesis that each country is an independent economy that is well described by the Solow growth model. Using the framework of the Solow model, MRW interpret the coefficient on the saving rate as a measure of the output elasticity of capital. They find that this elasticity is significantly smaller than one, which is the minimum value that would give endogenous growth, and argue that the size of the elasticity they estimate is consistent with the rate of conditional convergence implied by the coefficient on initial income. Further, they find that the coefficient on the initial level of income is
23. See, e.g., Mankiw, Romer, and Weil [I9921 and Barro and SalaiMartin [1991,1992,19951. 24. These regressions also include countryspecific population growth rates, but these are all zero in our model.
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negative, which is inconsistent with standard endogenous growth models. Thus, they argue that the evidence supports an extended version of the Solow model, where "capital" encompasses both human and physical capital. We argue that such tests are not dispositive in a world with technology diffusion. Data generated from our model would display a pattern similar to the one that MRW found, but the policy implications of our model are quite different from those of the Solow model. Note first that countries in our model display conditional convergence, since two countries with the same saving rate will eventually converge to the same level of income. The reason is that at each point in its development the poor country is able to use more mature technology than is the rich one, so it grows faster. Barro and SalaiMartin [I9971 obtain a similar result in a model with costly imitation. In their model, convergence is driven by the assumption that there are diminishing returns to imitation. It is also clear that in our world the saving rate enters a crosscountry growth regression with a positive sign. If two countries have the same level of income, the highersaving country grows faster. But the estimated coefficient on the saving rate will typically not reveal the Ak structure of production present in our model. Since highersaving countries will typically have higher capital stocks, and thus less mature levels of technology, crosscountry regressions will tend to show an output elasticity of capital smaller than 1,which is its true value for any given technology, From the point of view of policy, it will not be apparent from the usual crosscountry regressions that policies can affect longrun growth. Thus, although the work of MRW and others has cast doubt on the plausibility of closedeconomyAk models, we believe that our model, which is based on a similar production function, is quite consistent with the accepted empirical findings. Sharper tests are needed to discriminate between a world where convergence takes place because of diminishing returns, and one where convergence is due to technology diffusion.
B. Implications for ((Miracles"
Much of the recent growth literature has investigated the subject of "miracles:" countries that make a rapid transition from poverty to relative affluence, seemingly without taking extraordi
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nary measures.25 We discuss the implications of our model for such miracles, under the assumption that all countries in the world belong to a single convergence club and thus grow at the same rate in the steady state.26 Consider the different effects of an increase in the saving rate (which in our model is a metaphor for a variety of improvements in economic policy), depending on whether the country is toward the front or the back of the pack. An increase in saving on the part of the leading country has a relatively small effect on its income, since an increase in the capitallabor ratio leads the country to produce with significantly less mature technology.A poor country raising its saving rate, by contrast, will experience a period of rapid growth as it moves quickly to the front of the pack, taking advantage of the relatively mature technology at the rear. Thus, our model predicts that miracles should occur through a process of "catchup:" countries that can make rapid gains in per capita income through small changes in policy must be poor, not rich. This prediction is certainly consistent with the facts of miracles: no rich country has experienced the kind of sustained rapid per capita income growth displayed by countries like South Korea, Hong Kong, and Taiwan. Why is rapid catchup called a miracle? The reason is that the technologies we estimate from the data suggest that such events .~~ are i m p r ~ b a b l eSuppose that the world is in a stochastic steady state where countries have small, stationary fluctuations in their saving rates. Estimates of the elasticity of growth with respect to saving (which in a model with a neoclassical production function also imply the output elasticity of capital) will be averages across tKe rich and poor countries.28 our model, however, this parameIn ter is not equal across countries. In particular, in rich countries the effect of saving on growth will be small for the reasons we discussed in the previous section. Thus, the large effect of saving on growth in a poor country will appear miraculous: treating countries symmetrically would not uncover the statedependent
25. See Lucas' [I9931 discussion and comparison of the growth histories of South Korea and the Philippines. Parente and Prescott [I9931 document some of the stylized facts. 26. Parente and Prescott [I9931 argue that there is no tendency for the distribution of country incomes to spread out over time, which seems to argue in favor of a common growth rate. However, Quah I19961 comes to the opposite .. conclusion. 27. These events are also quite infrequent in the data: Quah [I9961 finds that the probability of going from the tenth percentile in world income to anywhere above the ninetieth percentile in 60 years is about 5 percent. 28. They will actually be weighted averages, where the weights depend on the relative variances of countries' saving rates.
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effects of saving on output that make it easier for poor countries to become rich than for rich countries to become richereZ9 Our model also has an interesting implication: it predicts that miracles may become easier as more countries experience them. If one country increases saving, it improves the level of technology available to other poor countries. This then makes it easier for these countries to have miracles of their own (in the sense that they require smaller increases in their saving rates to achieve the same increase in relative income). Of course, the poor countries are better off whether or not they increase savings, since they now inherit more mature technology at each level of capital.30
Consider a world with two countries: a wealthy country with a relatively low saving rate (e.g.,the United States) and a poorer, fastergrowing country with a higher saving rate (e.g., Japan). According to the Solow model, Japan will be growing faster than the United States because it is farther below its steadystate level of output. This model predicts that Japan will pass the United States, and that Japanese growth will slow down as it reaches its steady state. Eventually the ratio of output per capita in the two countries will stabilize. The fact that Japan is passing it by will have no effect on the growth rate of output in the United States. Endogenousgrowth models also predict that Japan will pass the United States and that this overtaking will have no effect on growth in the United States. But these models do not predict any subsequent slowing in Japan's growth: over time the gap between the levels of per capita output in the two countries is supposed to grow arbitrarily large. The predictions of the model presented in this paper seem more reasonable than those of either the Solow or endogenous29. Recall from our discussion of Figure VII that the effect of a country's saving rate on even its steadystate growth rate can vary a great deal, depending on its position in the world economy. Differences in the transitional dynamics following a change in the saving rate are naturally even more pronounced. 30. This prediction depends on our assumption that all countries in the world are in fact members of a single convergence club. If there are two (or more) such clubs and miracles take the form of leaving the lower club to join a higher one, then poor countries that experience miracles worsen the situation of the countries they leave behind. First, as we discuss in Section 111, they will lower growth rates for the countries left behind. Second, since they grow a t a faster rate, they do not improve the technologies they use as much as they otherwise would. The first effect lowers the welfare of countries left behind, and the second makes it harder for the remaining lowgrowth countries to create miracles of their own.
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growth models. Our model predicts, first, that the two countries will eventually reach a steady state in which the ratio of their per capita incomes is constant (assuming that the gap in saving rates between the two countries is not too large). Second, the fact that Japan passes it in terms of income per capita will be good for the United States: growth will accelerate as Japan takes over the burden of technology leadership. In the steady state, the common rate of growth in the two countries will be higher than growth in the United States when it was the technology leader, but lower than growth in Japan when it was the technology follower. Our result that being overtaken by Japan is good for the United States is one that finds little support in the popular press, although it is more widely accepted by economists; see, for example, Krugrnan [19961. In order to highlight the implications of our appropriatetechnology model for relative income and growth rates across countries, we have kept the model as simple as possible. In particular, we have used a simple learningbydoing model to describe the process of technological advancement, rather than explicitly modeling expenditures on research and development. Similarly, we have assumed that technology is instantly and freely mobile across countries, rather than modeling technology transfer as requiring time or expenditures. And finally, we have taken saving rates to be exogenous and constant, rather than modeling them as the outcome of optimization. All three of these areas hold the potential for interesting extensions of the model that we present.
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