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Chapter 9

N E O C L A S S I C A L G R O W T H T H E O R Y

ROBERT M. SOLOW

Massachusetts Institute of Technology, Department of Economics, E52-383B,

Cambridge, MA 02139, USA

Contents

Abstract 638 Keywords 638 1. Introduction 639 2. The Harrod-Domar model 640 3. The basic one-sector model 641 4. Completing the model 642 5. The behaviorist tradition 643 6. The optimizing tradition 646 7. Comparing the models 648 8. The Ramsey problem 649 9. Exogenous technological progress 650 10. The role of labor-augmentation 651 11. Increasing returns to scale 652 12. Human capital 653 13. Natural resources 655 14. Endogenous population growth and endogenous technological progress in

the neoclassical framework 657 15. Convergence 659 16. Overlapping generations 660 17. Open questions 663 References 665

Handbook of Macroeconomics, Volume 1, Edited by J.B. Taylor and M. WoodJbrd © 1999 Elsevier Science B.V. All rights reserved

637

638 R.M. Solow

Abstract

This chapter is an exposition, rather than a survey, of the one-sector neoclassical growth model. It describes how the model is constructed as a simplified description of the real side of a growing capitalist economy that happens to be free of fluctuations in aggregate demand. Once that is done, the emphasis is on the versatility of the model, in the sense that it can easily be adapted, without much complication, to allow for the analysis of important issues that are excluded from the basic model.

Among the issues treated are: increasing returns to scale (but not to capital alone), human capital, renewable and non-renewable natural resources, endogenous population growth and technological progress. In each case, the purpose is to show how the model can be minimally extended to allow incorporation of something new, without making the analysis excessively complex.

Toward the end, there is a brief exposition of the standard overlapping-generations model, to show how it admits qualitative behavior generally absent from the original model.

The chapter concludes with brief mention of some continuing research questions within the framework of the simple model.

Keywords

growth, technological progress, neoclassical model

JEL classification: 04, E1

Ch. 9: Neoclassical Growth Theory 639

1. Introduction

As part of macroeconomics, growth theory functions as the study of the undisturbed evolution of potential (or normal capacity) output. The force of "undisturbed" in this context is the maintained assumption that the goods and labor markets clear, i.e., that labor and capital are always fully or normally utilized (or, at the very minimum, that the degree of utilization does not vary). The scope of specifically "neoclassical" growth theory is harder to state, because it is a matter of judgment or convention how much more of the neoclassical general equilibrium apparatus to incorporate in a model of undisturbed growth. As in most of macroeconomics, modeling strategy in growth theory tends to be weighted away from generality and toward simplicity, because the usual intention is to compare model with data at an early stage.

Simplicity does not mean rigidity. On the contrary, it will emerge from this review that the neoclassical growth model is extraordinarily versatile. Like one of those handy rotary power tools that can do any of a dozen jobs if only the right attachment is snapped on, the simple neoclassical model can be extended to encompass increasing and decreasing returns to scale, natural resources, human capital, endogenous population growth and endogenous technological change all without major alteration in the character of the model.

In this survey, the completely aggregated one-sector model will be the main focus of attention. Models with several sectors (agriculture and industry, consumption goods and capital goods) have attracted attention from time to time, but they tend to raise different issues. To discuss them would break continuity. The main loss from this limitation is that the important literature on open-economy aspects of growth theory has to be ignored. [The main reference is Grossman and Helpman (1991) and the work they stimulated.] Apart from the underlying restriction to "equilibrium growth" (meaning, in practice, the full utilization already mentioned), the most important neoclassical attribute is the assumption of diminishing returns to capital and labor. Here "capital" means (the services of) the stock of accumulated real output in the strictest one-good case, or the complex of stocks of all accumulatable factors of production, including human capital and produced knowledge, when they are explicitly present. The further assumption of constant returns to scale is typically neoclassical, no doubt, but it is not needed in some unmistakably neoclassical approaches to growth theory.

The text will be neutral as between the ultra-strong neoclassical assumption that the economy traces out the intertemporal utility-maximizing program for a single immortal representative consumer (or a number of identical such consumers) and the weaker assumption that saving and investment are merely common-sense functions of observables like income and factor returns. The long-run implications tend to be rather similar anyway.

Much of growth theory, neoclassical or otherwise, is about the structural character- istics of steady states and about their asymptotic stability (i.e., whether equilibrium paths from arbitrary initial conditions tend to a steady state). The precise definition of a steady state may differ from model to model. Most often it is an evolution

640 R.M. Solow

along which output and the stock of capital grow at the same constant rate. It would be possible to pay much more attention to non-steady-state behavior, by computer simulation if necessary. The importance of steady states in growth theory has both theoretical and empirical roots. Most growth models have at least one stable steady state; it is a natural object of attention. Moreover, ever since Kaldor's catalogue of "stylized facts" [Kaldor (1961)], it has generally, if casually, been accepted that advanced industrial economies are close to their steady-state configurations, at least in the absence of major exogenous shocks. The current vogue for large international cross-section regressions, with national rates of growth as dependent variables, was stimulated by the availability of the immensely valuable Summers-Heston (1991) collection of real national-accounts data for many countries over a fairly long interval of time. The results of all those regressions are neither impressively robust nor clearly causally interpretable. Some of them do suggest, however, that the advanced industrial (OECD) economies may be converging to appropriate steady states.

There is nothing in growth theory to require that the steady-state configuration be given once and for all. The usefulness of the theory only requires that large changes in the determinants of steady states occur infrequently enough that the model can do meaningful work in the meanwhile. Then the steady state will shift from time to time whenever there are major technological revolutions, demographic changes, or variations in the willingness to save and invest. These determinants of behavior have an endogenous side, no doubt, but even when established relationships are taken into account there will remain shocks that are too deep or too unpredictable to be endogenized. No economy is a close approximation to Laplace's clockwork universe, in which knowledge of initial positions and velocities is supposed to determine the whole future.

2. The Harrod-Domar model

This survey is not intended as a history of thought, It is worth saying, however, that neoclassical growth theory arose as a reaction to the Harrod-Domar models of the 1940s and 1950s [Harrod (1939), Domar (1946)]. (Although their names are always linked, the two versions have significant differences. Harrod is much more concerned with sometimes unclear thoughts about entrepreneurial investment decisions in a growing economy. Domar's more straightforward treatment links up more naturally with recent ideas.)

Suppose that efficient production with the aggregate technology requires a constant ratio of capital to (net) output, say Y = vK. Suppose also that net saving and investment are always a fixed fraction of net output, say I = d K / d t = s Y . Then, in order for the utilization rate of capital to stay constant, capital and output have to grow at the proportional rate st;. Since labor input is proportional to output, employment would then grow at the same rate. I f labor productivity were increasing at the rate m, the growth rate of employment would be sv - m. Let the growth of the labor force, governed


mainly by demography, be n. Then the persistence of any sort of equilibrium requires that sv = m + n; if so > m + n there would be intensifying labor shortage, limiting the use of capital, while if so < m + n there would be increasing unemployment.

But the equilibrium condition so = m + n is a relation among four parameters that are treated, within the Harrod-Domar model, as essentially independently determined constants: s characterizes the economy's propensity to save and invest, v its technology, n its demography, and m its tendency to innovate. There is no reason for them to satisfy any particular equation. An economy evolving according to Harrod-Domar rules would be expected to alternate long periods of intensifying labor shortage and long periods of increasing unemployment. But this is an unsatisfactory picture of 20th century capitalism. The Harrod-Domar model also tempted many - though not its authors - to the mechanical belief that a doubling of the saving-investment quota (s) would double the long-term growth rate of a developing or developed economy. Experience has suggested that this is far too optimistic. There was a theoretical gap to be filled. The natural step is to turn at least one of the four basic parameters into an equilibrating variable.

Every one of those four parameters has its obvious endogenous side, and models of economic growth have been built that endogenize them. Within the neoclassical framework, most attention has been paid to treating the capital intensity of production and the rate of saving and investment as variables determined by normal economic processes. The tradition of relating population growth to economic development goes back a lot further; and labor-force participation clearly has both economic and sociological determinants. The case of technological progress is interesting. Most neoclassical growth theory has treated it as exogenous. Some authors, e.g., Fellner (1961) and von Weizsficker (1966), have discussed the possibility that internal economic factors might influence the factor-saving bias of innovations. It is then no distance at all to the hypothesis that the voltune of innovation should be sensitive to economic incentives. This was certainly widely understood. But there was little or no formal theorizing about the rate of endogenous technological progress until the question was taken up in the 1980s. The original references are Lucas (1988) and Romer (1986), but there is now a vast literature.

3. The basic one-sector model

The model economy has a single produced good ("output") whose production per unit time is Y( t ) . The available technology allows output to be produced from current inputs of labor, L( t ) , and the services of a stock of"capital" that consists of previously accumulated and partially depreciated quantities of the good itself, according to the production function Y = F ( K , L). (Time indexes will be suppressed when not needed.) The production function exhibits (strictly) diminishing returns to capital and labor separately, and constant returns to scale. (More will be said about this later.)

642 R.M. Solow

Constant returns to scale allows the reduction

Y : F ( K , L) : L F ( K / L , 1) : L F ( k , 1) = L f ( k ) ,

and thus finally that y =f(k), where y is output per unit of labor input, i.e., productivity, and k is the ratio of capital to labor input, i.e., capital intensity. From diminishing returns, f ( k ) is increasing and strictly concave. The usual further assumptions (Inada conditions) about f (k ) eliminate uninteresting possibilities: f (0) ~> 0, f ' (0 ) = cx~, f ' ( c~) = 0. These are overly strong: the idea is that the marginal product of capital should be large at very low capital intensity and small at very large capital intensity. [No more than continuity and piecewise differentiability is required of f ( . ) , but nothing is lost by assuming it to be at least twice continuously differentiable, so strict diminishing returns means that f l ' ( k ) < 0.]

The universal assumption in growth theory is that each instant's depreciation is just proportional to that instant's stock of capital, say D=dK. This is known to be empirically inaccurate, but it is the only assumption that makes depreciation independent of the details of the history of past gross investment. The convenience is too great to give up.

Since this point is usually glossed over, it is worth a moment here. A much more general description is that there is a non-increasing survivorship function j (a ) , with j(0) = 1 and j(A)=0. (A may be infinite.) The interpretation is that j ( a ) is the fraction of any investment that survives to age a. Then if I ( t ) is gross investment,

K ( t ) = f A I ( t -- a ) j (a ) da. Now differentiation with respect to time and one integration by parts with respect to a leads to

~0 'A K ' : I ( t ) - I ( t - a) d(a) da, (3.1)

where d ( a ) = - j ' ( a ) is the rate of depreciation at age a. So net investment at time t depends on the whole stream of gross investments over an interval equal to the maximum possible lifetime of capital. It can be checked that only exponential survivorship, j ( a ) = e -da, simplifies to K ' = I ( t ) - d K ( t ) . This assumption will be maintained for analytical convenience. The more complicated formula could easily be adapted to computer simulation.

4. Completing the model

At each instant, current output has to be allocated to current consumption or gross investment: Y= C + I . It follows that

K ' = Y - d K - C = F ( K , L ) - d K - C. (4.1)

If the labor force is exogenous and fully employed, L(t) is a given function of time. (In this completely aggregated context, the clearing of the markets for goods and


labor amounts to the equality of saving and investment at full employment.) Then any systematic relationship that determines C(t) as a function of K( t ) and t converts Equation (4.1) into an ordinary differential equation that can be integrated to determine the future path of the economy, given L(t) and the initial value of K. Suppose L(t) = e nt. Then simple transformations convert Equation (4.1) into autonomous per capita terms:

k / - f ( k ) ( d + n ) k c, (4.2)

where, of course, c = C/L. The last component to be filled in is a rule that determines consumption per

capita. Here there are two distinct strategies plus some intermediate cases. The simplest possibility, as mentioned earlier, is just to introduce a plausible consumption function with some empirical support. This was the earliest device [Solow (1956), Swan (1956)]. The other extreme, now more common, is to imagine the economy to be populated by a single immortal representative household that optimizes its consumption plans over infinite time in the sort of institutional environment that will translate its wishes into actual resource allocation at every instant. The origins are in Ramsey (1928), Cass (1965) and Koopmans (1965), but there is a large contemporary literature on this basis. For excellent surveys with further references, see Barro and Sala-i-Martin (1995), Blanchard and Fischer (1989, Ch. 2), and D. Romer (1996, Chs. 1, 2).

5. The behaviorist tradition

The two simplest examples in the "behaviorist" tradition are (a) saving-investment is a given fraction of income-output, and (b) saving-investment is a given fraction (which may be unity) of non-wage income, however the distribution of income between wages and profit or interest is determined in the society at hand. The case where different fractions of wage and non-wage income are saved amounts to a mixture of (a) and (b) and does not need to be examined separately. [Complications arise if the correct distinction is between "workers" and "capitalists" instead of wages and non-wages, because workers who save must obviously become partial capitalists. See Samuelson and Modigliani (1966), and also Bertola (t994).] In all this, an important role is played by the maintained assumption that investment always equals saving at full utilization.

Under the first of these hypotheses, (4.2) becomes

k' = f ( k ) - (d + n) k - (1 - s ) f ( k ) = s f ( k ) - (d + n)k, (5.1)

where of course s is the fraction of output saved and invested. The conditions imposed o n f ( k ) imply that the right-hand side (RHS) of Equation (5.1) is positive for small k

644 R.M. Solow

(a+d+n)k///

Fig. 1.

because f'(O) > (d + n)/s, first increasing and then decreasing because f " ( k ) < 0, and eventually becomes and remains negative because i f ( k ) becomes and remains very small. It follows that there is a unique k* >0 such that U ( t ) > 0 when k(t)<k*, U(t )=0 at k*, and U(t)< 0 when k(t)>k*. Thus k* is the globally asymptotically stable rest point for k (leaving aside the origin which may be an unstable rest point [if f (0 )=0] . The phase diagram, Figure 1, drawn for the case f ( 0 ) = 0 , makes this clear.)

The properties of k* will be discussed later. For now it is enough to note that, starting from any initial capital intensity, the model moves monotonically to a predetermined capital intensity defined from Equation (5.1) by s f ( k * ) - ( d + n ) k *= O. [Note that k* precisely validates the Harrod-Domar condition because f ( k ) / k corresponds precisely to v, now a variable. The depreciation rate appears only because Equation (5.1) makes gross saving proportional to gross output instead of net saving proportional to net output.] When the economy has reached the stationary capital intensity k*, the stock of capital is growing at the same rate as the labor force - n - and, by constant returns to scale, so is output. The only sustainable growth rate is the exogenously given n, and productivity is constant. A reasonable model of growth must obviously go beyond this.

The second hypothesis mentioned earlier, that saving-investment is proportional to non-wage income, requires a theory of the distribution of income between wages and profits. The usual presumption is the perfectly competitive one: profit (per capita) is k S ( k ) because f t (k ) is the marginal product of capital. Some further generality is almost costlessly available: if the economy is characterized by a constant degree of monopoly in the goods market and monopsony in the labor market, then profit per capita will be proportional to kf~(k) with a factor of proportionality greater than one. If sk is the fraction of profits saved and invested (or the product of that fraction and the monopoly-monopsony factor) Equation (4.2) can be replaced by

U = skkf ' (k) - (d + n) k. (5.2)

d+n

Neoclassical Growth Theory

(

k ~-

Ch. 9:

k)

Fig. 2.

645

The analysis is not very different from that of Equation (5.1). Indeed if F(K,L) is Cobb-Douglas with elasticities b and 1 -b , so that kfl(k)=by, Equations (5.1) and (5.2) coincide, with s replaced by skb. More generally, the conditions imposed on f (k) do not quite pin down the behavior of kfr(k), though they help. For instance, kf'(k) <f(k) as long as the marginal product of labor is positive; so the fact that in Figure 2 the graph of sf(k) eventually falls below the ray (n + d)k, irrespective of s, implies that the RHS of Equation (5.2) becomes and remains negative for large k.

The other Inada condition is more complicated. Obviously f ( 0 ) = 0 implies that kH(k) goes to zero at the origin. Now the derivative ofkf '(k) i s f ( k ) + kf"(k) <f'(k), so the behavior of the phase diagram for small k depends on the second derivative of f . If the elasticity of substitution between K and L exceeds (or equals) unity near the origin, the slope of kJV(k) is indeed large there, and the phase diagram for Equation (5.2) looks like the one drawn for Equation (5.1). Even if the elasticity of substitution is less than one, the RHS of Equation (5.2) will be positive for small k unless the curvature o f f is extreme, given the Inada conditions.

These are only details. It follows from Equation (5.2) that any steady state, apart from the trivial possibility at k = 0, must satisfy s J ' ( k *) =d +n. But then the Inada conditions and concavity imply that there is one and only one nontrivial steady state, and the RHS of Equation (5.2) is positive to its left and negative to its right. In every essential respect, Equation (5.2) will behave qualitatively like Equation (5.1). Starting anywhere, k(t) will approach a unique non-zero steady state k* defined by skf~(k*)=n+d. The steady-state implications of Equations (5.1) and (5.2) are qualitatively identical. The sustainable growth rate is still just the rate of growth of the labor supply.

Playing with the example f ( k )= (c/b)(1- e -bk) will show that it is easy to find specifications - this one is bounded - for which kH(k) decreases for some values of k (here for k > 1/b). This makes at most a trivial difference in the qualitative behavior

646 R.M. Solow

of the solution of Equation (5.2). For some parameter choices the origin is the only steady state; for the rest there is one and only one non-zero steady state, and it is an attractor. So nothing special happens. In discrete time, however, the qualitative possibilities are diverse and complex. The discrete analogue of Equation (5.2) can easily exhibit periodic or chaotic dynamics (and even more so if there is saving from wages). It is not clear how much "practical" macroeconomic significance one should attach to this possibility; but it is surely worth study. For an excellent treatment, see B6hm and Kaas (1997).

Since k f ' ( k ) / f ( k )=e(k ) , the elasticity of f ( . ) with respect to k, the RHS of Equation (5.2) could be translated as s k e ( k ) f ( k ) - ( n + d ) k . As this suggests, a wide variety of assumptions about (market-clearing) saving and investment can be incorporated in the model if Equation (5.1) is generalized to

k' (t) = s ( k ) f (k) - (n + d) k. (5.3)

For example, suppose s(k) is zero for an initial interval of low values of k and y, and thereafter rises fairly steeply toward the standard value s. This pattern might correspond to a subsistence level of per capita income, below which no saving takes place. The modified phase diagram now has two non-zero steady-state values of k, the larger of which is as before. The smaller steady state is now unstable, in the sense that a small upward perturbation will launch a trajectory toward the stable steady state, while a small downward perturbation will begin a path leading to k =y = 0. This is a sort of low-equilibrium trap; similar variations can be arranged by making n a function of, say, the wage rate, and thus of k. The details are straightforward.

6. The optimizing tradition

These formulations all allocate current output between consumption and investment according to a more or less mechanical rule. The rule usually has an economic interpretation, and possibly some robust empirical validity, but it lacks "microfoundations". The current fashion is to derive the consumption-investment decision from the decentralized behavior of intertemporal-utility-maximizing households and perfectly competitive profit-maximizing firms. This is not without cost. The economy has to be populated by a fixed number of identical immortal households, each endowed with perfect foresight over the infinite future. No market imperfections can be allowed on the side of firms. The firms have access to a perfect rental market for capital goods; thus they can afford to maximize profits instant by instant. For expository purposes, nothing is lost by assuming there to be just one household and one firm, both price- takers in the markets for labor, goods, loans and the renting of capital.

The firm's behavior is easy to characterize because it can afford to be myopic under these assumptions. To unclutter the notation, take d = 0. The market provides a real wage w(t) (in terms of the produced commodity) and a real (own) rate of interest i(t).


It is always profit-maximizing for the firm to hire labor and rent capital up to the point where

i ( t ) = S ( k ( t ) ) , (6.1a)

w ( t ) = f ( k ( t ) ) - k f ' ( k ( t ) ) , (6. lb)

the two RHS expressions being the marginal products of capital and labor. [To allow for depreciation, just subtract d from the RHS of Equation (6.1a).]

As before, the size of the household grows like e nt, and each member supplies one unit of labor per unit time, without disutility. (This last simplifying assumption is dispensable.) The household's preferences are expressed by an instantaneous utility function u(c( t ) ) , where c( t ) is the flow of consumption per person, and a discount rate for utility, denoted by r. The household's objective at time 0 is the maximization of

f o °~ e r t u (c ( t ) ) e nt d t = U. (6.2)

(The term e ~t can be omitted, defining a slightly different but basically similar optimization problem for the household or clan.) The maximizing c( t ) must, of course, satisfy a lifetime budget constraint that needs to be spelled out.

Let J ( t ) = Jo t i ( t ) dt so that e -J(t) is the appropriate factor for discounting output from time t back to time zero. The household's intertemporal budget constraint requires that the present value (at t = 0) of its infinite-horizon consumption program should not exceed the sum of its initial wealth and the present value of its future wage earnings. In per-capita terms this says

ff e J(O c( t ) e nt d t = ko + e J(t) w ( t ) e nt dt , (6.3)

where non-satiation is taken for granted, so the budget constraint holds with equality, and k0 is real wealth at t = 0.

Maximization of Equation (6.2) subject to condition (6.3) is standard after introdnction of a Lagrange multiplier, and leads to tile classical (Ramsey) first-order condition

IC~Ff ~C ~ 1 c'(t) = i ( t ) - r.

u'(c) c (6.4)

The first fraction is the (absolute) elasticity of the marginal utility of consumption. So the optimizing household has increasing, stationary, or decreasing consumption according as the current real interest rate (real return on saving) exceeds, equals, or falls short of the utility discount rate. For a given discrepancy, say a positive one, consumption per head will grow faster the less elastic the marginal utility of consumption.

648 R.M. Solow

C ~

C O

( ( k ) : n

I I

ko k * k

Fig. 3.

In the ubiquitous special case of constant elasticity, i.e., if u ( c ) = (c 1 - h _ 1)/(1 - h), Equation (6.3) becomes

l_c , ( t ) _ i ( t ) - r _ f ' ( k ( t ) ) - r (6.3a)

c h h

by Equation (6.1a). Under these rules of the game, the trajectory of the economy is determined by

Equation (6.4) or, for concreteness, Equations (6.3a) and (4.2), reproduced here with d=O as

k ' ( t ) = f ( k ( t ) ) - n k ( t ) - c ( t ) . (6.5)

The phase diagram in c and k is as shown in Figure 3. c ' ( t ) = 0 along the vertical line k = k * defined b y f t ( k * ) = r , with e increasing to the left and decreasing to the right. U = 0 along the locus defined by c =f (k ) - n k , with k decreasing above the curve and decreasing below it. Under the normal assumption that r > n [otherwise Equation (6.2) is unbounded along feasible paths] the intersection of the two loci defines a unique steady state:

f ' ( k * ) = r , c* =f(k*) - nk*. (6.4a,b)

7. Comparing the models

This steady state is exactly like the steady state defined by a "behaviorist" model: capital per head and output per head are both constant, so capital and output grow


at the same rate as the labor force, namely n. In the steady state, the ratio of saving and investment to output is a constant, nk*/f(k*). The steady-state investment rate is higher the higher k* turns out to be, and thus, from Equation (6.4a) the lower is r. So far as steady-state behavior is concerned, choosing a value for s is just like choosing a value for r, a higher s corresponding to a lower r.

Out-of-steady-state behavior differs in the two schemes. In the usual way, it is shown that the steady state or singular point (c*, k*) is a saddle-point for the differential equations (4.2) and (6.3a). History provides only one initial condition, namely k0. If the initial value for c is chosen anywhere but on the saddle path, the resulting trajectory is easily shown to be non-optimal for the household (or else ultimately infeasible). The appropriate path for this economy is thus defined by the saddle path, which leads asymptotically to the steady state already discussed. Of course the saving-investment rate is not constant along that path, although it converges to the appropriate constant value (from above if k0 <k*) as the economy nears the steady state. A clear and detailed exposition can be found in D. Romer (1996, Ch. 2).

Evidently it does not matter much which approach is taken, at least for steady-state analysis. For illustrative purposes, the "behaviorist" approach is usually simpler. The comparative analysis of steady states goes similarly in the two cases, through variation in n and s, or in n and r, with analogous results. When a parameter changes, the new "initial" k is always given by history; the allocation of output between current consumption and investment is allowed to jump. In the optimizing approach, the jump is always to the appropriate arm of the saddle path.

Out-of-steady-state behavior is, of course, sensitive to the underlying assumption about household behavior. One example will illustrate the point. Start in steady state and suppose that war or natural disaster destroys a substantial fraction of the capital stock, but the population is unchanged. It has been found [King and Rebelo (1993)] that the return to the (neighborhood of the) initial steady state takes a very long time if the "behaviorist" saving-investment rate is fixed at its original value; the optimizing version gets back to its steady state much sooner. This is because the optimizing path will respond to the loss of capital by saving and investing more than in the steady state. In a common-sense way, the same effect could be achieved behavioristically by presuming that any economy will respond to the destruction of part of its capital stock by temporarily increasing its rate of investment.

8. The Ramsey problem

The optimizing model just described originated with Ramsey (1928) and was further developed by Cass (1965) and Koopmans (1965). They regarded it, however, as a story about centralized economic planning. In that version, Equation (6.2) is a social welfare indicator. A well-meaning planner seeks to choose c(t) so as to maximize Equation (6.2), subject only to the technologically determined constraint (4.2) and the initial stock of capital. [In that context, Ramsey thought that discounting future utilities

650 R.M. Solow

was inadmissible. He got around the unboundedness of Equation (4.2) when r = 0 by assuming that u(.) had a least upper bound B and then minimizing the undiscounted integral of B-u (c ( t ) ) , either omitting the factor e nt on principle or assuming n = 0. The undiscounted case can sometimes be dealt with, despite the unbounded integral, by introducing a more general criterion of optimality known as "overtaking". For this see von Weizs/icker (1965).]

Then straightforward appeal to the Euler equation of the calculus of variations or to the Maximum Principle leads precisely to the conditions (6.4) (or 6.3a) and (6.5) given above. A transversality condition rules out trajectories other than the saddle path. The competitive trajectory is thus the same as the planner's optimal trajectory. One can say either that the solution to the planning problem can be used to calculate the solution to the competitive outcome, or that the competitive rules offer a way to decentralize the planning problem. Lest this seem too easy, it should be remembered that the competitive solution simply presumes either that the household have perfect foresight out to infinity or that all the markets, for every value of t, are open and able to clear at time zero. That strikes many workers in this field as natural and some others as gratuitous.

9. Exogenous technological progress

These models eventuate in a steady state in which y, k and c are constant, i.e., aggregate output and capital are growing at the same rate as employment and the standard of living is stationary. That is not what models of growth are supposed to be about. Within the neoclassical framework, this emergency is met by postulating that there is exogenous technological progress. The extensive and intensive production functions are written as F(K, L; t) andf(k; t), so the dependence on calendar time represents the level of technology available at that moment.

So general an assumption is an analytical dead end. The behaviorist version of the model can only be dealt with by simulation; the optimizing version leads to a complicated, non-autonomous Euler equation. The standard simplifying assumption is that technological progress is "purely labor-augmenting" so that the extensive production function can be written in the form Y(t) = F(K(t) , A(t)L(t)). Technological progress operates as if it just multiplied the actual labor input by an increasing (usually exponential) function of time. The quantity A(t)L(t) is referred to as "labor in efficiency units" or "effective labor". It will be shown below that this apparently gratuitous assumption is not quite as arbitrary as it sounds.

I f y(t) is now redefined as Y(t ) /A( t )L( t )= Y(t)/eate nt= Y(t) /e (a+n~t, and similarly for k and c, the basic differential equation (5.1) of the behaviorist model is replaced by

k' = s f (k) - (a + n + d)k , (9.1)

the only change from Equation (5.1) being that the rate of growth of employment in efficiency units replaces the rate of growth in natural units. [(One can write f ( k )


rather than f ( k ; t) because the time-dependence is completely absorbed by the new version of k.] Under the standard assumptions about f ( . ) there is once again a unique non-trivial steady-state value k*, defined as the non-zero root o f s f ( k * ) = (a + n + d)k* .

This steady state attracts every path of the model starting from arbitrary k0 > 0. The difference is that in this steady state aggregate capital, output and consumption

are all proportional to e (a+n)t SO that capital, output and consumption per person in natural units are all growing at the exponential rate a, to be thought of as the growth rate of productivity. This growth rate is obviously independent of s. The effect of a sustained step increase in s, starting from a steady state, is a temporary increase in the aggregate and productivity growth rates that starts to taper off immediately. Eventually the new path approaches its own steady state, growing at the same old rate, but proportionally higher than the old one. [There is a possibility of over investment, if f~(k*) < a + n + d, in which case higher s increases output but decreases consumption. This will be elaborated later, in connection with the Diamond overlapping-generations model.]

The situation is slightly more complicated in the optimizing version of the model because the argument of c(t) must continue to be current consumption per person in natural units, i.e. consumption per effective unit of labor multiplied by e nt. This does not change the structure of the model in any important way. The details can be found in D. Romer (1996, Ch. 2).

It goes without saying that the introduction of exogenous technical progress achieves a steady state with increasing productivity, but does not in any way explain it. Recent attempts to model explicitly the generation of A(t) fall under the heading of "endogenous growth models" discussed in the original papers by Lucas (1988) and EM. Romer (1990), and in the textbooks of Barro and Sala-i-Marfin (1995) and D. Romer (1996). A few remarks about endogenizing aspects of technical progress within the neoclassical framework are deferred until later.

10. The role of labor-augmentation

The question remains: what is the role of the assumption that exogenous technical progress is purely labor-augmenting? It is clear that either version of the model, and especially easily with the behaviorist version, can be solved numerically without any such assumption. It is just a matter of integrating the differential equation k I = s f ( k ; t) - (n + d) k. The deeper role of labor-augmentation has to do with the importance - in theory and in practice - attached to steady states. It can be shown that purely labor-augmenting technical progress is the only kind that is compatible with the existence of a steady-state trajectory for the model. This observation was due originally to Uzawa (1961). Since the proof is not easily accessible, a compact version is given here.

To begin with, it is worth noting that labor-augmenting technical progress is often described as "Harrod-neutral" because Roy Harrod first observed its particular

652 R.M. Solow

significance for steady states. We have defined a steady state as a growth path characterized by a constant ratio of capital to output. For a well-behaved f ( k ; t) it is clear that constancy of the average product o f capital is equivalent to constancy of the marginal product o f capital. So a steady state might just as well be characterized as a path with constant marginal product o f capital. In the same way, because y/k is monotone in k, one can express k as a function of k/y and t and therefore y = f ( k ; t) =f(k(k /y; t); t) = g(z; t), where z stands for k/y.

Now a straightforward calculation leads to dy/dk = gz/(g +zgz). The require- ment that dy/dk be independent o f time for given k/y says that the RHS of this equation is independent o f t and therefore equal to a function of z alone, say c(z). Thus one can write gz(z; t )=c(z)[g(z; t )+zgz(z; t)], and finally that gz(z; t)/g(z; t ) = c (z ) / (1 -zc (z ) ) . The RItS depends only on z, and thus also the LHS, which is d(lng)/dz. Integrating this last, one sees that lng(z; t) must be the sum of a function of t and a function of z, so that g(z; t) = y = A(t)h(z). Finally z = k/y = h -1(y/A), whence k/A = (y/A) h -1 (y/A) =j (y /A) and y/A =j-I (k/A). This is exactly the purely-labor-augmenting form: Y = F(K, AL) means Y = ALF(K/AL, 1) or y/A = f (k/A).

The assumption that technical progress is purely labor-augmenting is thus just as arbitrary as the desire that a steady-state trajectory should be admissible. That property brings along the further simplifications.

11. Increasing returns to scale

There is an almost exactly analogous, and less well understood, way of dealing with increasing returns to scale. Leaving the model in extensive variables, one sees that the equation K'(t) = sF[K(t) ,A(t)L(t)] can be integrated numerically for any sort of scale economies. Trouble arises only when one looks for steady-state trajectories, as a simple example shows. Suppose F is homogeneous of degree h in K and AL. I f K and AL are growing at the same exponential rate g, Y =F(K,AL) must be growing at the rate gh. Unless h = 1, steady state trajectories are ruled out.

There is a simple way to restore that possibility. Let h be a positive number not equal to 1 and suppose the production function F[K,AL h] is homogeneous of degree 1 in K and AL h. Production exhibits increasing returns to scale in K and L if h > 1: doubling K and L will more than double AL h and thus more than double output, though F is generally not homogeneous of any degree in K and L. (Obviously, i f F is Cobb- Douglas with exponents adding to more than 1, it can always be written in this special form.) But now, if A grows at the exponential rate a and L at the rate n, it is clearly possible to have a steady state with Y and K growing at rate g = a + nh. (The same goes for h < 1, but the case of increasing returns to scale is what attracts attention.) It is an interesting property of such a steady state that productivity, i.e., output per unit o f labor in natural units, Y/L, grows at the rate g - n = a + ( h - 1)n. Thus the model with increasing returns to scale predicts that countries with faster growth of the labor force


will have faster growth rates of productivity, other things equal. This seems empirically doubtful.

This discussion speaks only to the existence of a steady state with non-constant returns to scale. More is true. Within the behaviorist model, the steady state just described is a global attractor (apart from the trivial trap at the origin). To see this it is only necessary to redefine y as Y/AL h and k similarly. The standard calculation then shows that k' = s f (k ) - (a + hn + d) k, with a unique stable steady state at k*, defined as the unique non-zero root of the equation s f (k*) = (a + hn + d) k*. Note that, with h > 1, a higher n goes along with a smaller k* but a higher productivity growth rate.

The appropriate conclusion is that the neoclassical model can easily accommodate increasing returns to scale, as long as there are diminishing returns to capital and augmented labor separately. Exactly as in the case of exogenous technical progress, a special functional form is needed only to guarantee the possibility of steady-state growth.

The optimizing version of the model requires more revision, because competition is no longer a viable market form under increasing returns to scale; but this difficulty is not special to growth theory.

12. Human capital

Ever since the path-breaking work of T.W. Schultz (1961) and Gary Becker (1975) it has been understood that improvement in the quality of labor through education, training, better health, etc., could be an important factor in economic growth, and, more specifically, could be analogized as a stock of"human capital". Empirical growth accounting has tried to give effect to this insight in various ways, despite the obvious measurement difficulties. (For lack of data it is often necessary to use a current flow of schooling as a surrogate for the appropriate stock.) See for just a few of many examples, Denison (1985) and Collins and Bosworth (1996). The U.S. Bureau of Labor Statistics, in its own growth-accounting exercises, weights hours worked with relative wage rates, and other techniques have been tried. These considerations began to play a central role in theory with the advent of endogenous growth theory following Romer and Lucas, for which references have already been given. Here there is need only for a sketch of the way human capital fits into the basic neo-classical model. Corresponding empirical calibration can be found in Mankiw, Romer and Weil (1992) and Islam (1995).

Let H(t ) be a scalar index of the stock of human capital, however defined, and assume as usual that the flow of services is simply proportional to the stock. Then the extensive production function can be written as Y = F(K, H, L). I f there is exogenous technical progress, L can be replaced by AL as before. Assume that F exhibits constant returns to scale in its three arguments. (If desired, increasing returns to scale can be accommodated via the device described in the preceding section.) Then the intensive productivity function is y = F(k, h, 1) = f ( k , h). In the endogenous-growth literature, it

k t= 0

I~" /Y "-~ 'h:

R.M. Solow 654

0

k Fig. 4.

is more usual to start with the assumption that Y = F(K, HL), so that HL is interpreted as quality-adjusted labor input. The really important difference is that it is then assumed that F is homogeneous of degree 1 in the two arguments K and HL. Obviously this implies that there are constant returns to K and H, the two accumulatable inputs, taken by themselves. This is a very powerful assumption, not innocent at all.

Within the neo-classical framework, the next step is a specification of the rules according to which K and H are accumulated. Simple symmetry suggests the assumption that fractions sx and sH of output are invested (gross) in physical and human capital. (This is undoubtedly too crude; a few qualifications will be considered later.) Under these simple assumptions, the model is described by two equations:

k' = s ~ f ( k , h ) - ( a + n + d x ) k , h' = s H f ( k , h ) - ( a + n + d H ) h . (12.1)

As usual, a + n is the rate of growth of the (raw) labor supply in efficiency units and dx and dH are the rates of depreciation of physical and human capital.

Under assumptions on f( . , . ) analogous to those usually made on f( . ) , there is just one non-trivial steady state, at the intersection in the (h, k) plane of the curves defined by setting the LHS of Equation (12.1) equal to zero. In the Cobb-Douglas case If(k, h) = kbh c, b + c < 1] the phase diagram is easily calculated to look like the accompanying Figure 4. With more effort it can be shown, quite generally, that the locus of stationary k intersects the locus of stationary h from below; since both curves emanate from the origin, the qualitative picture must be as in Figure 4. Thus the steady state at (h*,k*) is stable. [It is obvious from Equation (12.1) that k*/h* =sx/sH if the depreciation rates are equal; otherwise the formula is only slightly more complicated.] Thus, starting from any initial conditions, K, H and Y eventually grow at the same rate, a + n. This model with human capital is exactly analogous to the model without it.

But this model is unsatisfactory in at least two ways. For one thing, the production of human capital is probably not fruitfully thought of, even at this level of abstraction, as a simple diversion of part of aggregate output. It is not clear how to model the production


of human capital. The standard line taken in endogenous-growth theory has problems of its own. (It simply assumes, entirely gratuitously, that the rate of growth of human capital depends on the level of effort devoted to it.) Nothing further will be said here about this issue. The second deficiency is that, if investment in physical capital and in human capital are alternative uses of aggregate output, the choice between them deserves to be modeled in some less mechanical way than fixed shares.

One alternative is to treat human capital exactly as physical capital is treated in the optimizing-competitive version of the neo-classical model. Two common-sense considerations speak against that option. The market for human capital is surely as far from competitive as any other; and reverting to infinite-horizon intertemporal optimization on the part of identical individuals is not very attractive either.

It is possible to find alternatives that give some economic structure to the allocation of investment resources without going all the way to full intertemporal optimization. For example, if in fact one unit of output can be transformed into either one unit of physical or one unit of human capital, market forces might be expected to keep the rates of return on the two types of investment close to one another as long as both are occurring. This implies, given equal depreciation rates, that f l ( k , h)=.f2(k, h) at every instant. The condition f12 > 0 is sufficient (but by no means necessary) for the implicit function theorem to give k as a function of h. I f F(K, H, L) is Cobb-Douglas, k is proportional to h; the same is true for a wider class of production functions including all CES functions.

The simplest expedient is to combine this with something like k ~= s f ( k ,h ) - (a + n + d) k with h replaced by h(k). Then physical investment is a fraction of output, and human-capital investment is determined by the equal-rate-of-return condition. In the Cobb-Douglas case, this amounts to the one-capital-good model with a Cobb- Douglas exponent equal to the sum of the original exponents for k and h. It happens that this set-up reproduces exactly the empirical results of Mankiw, Romer and Weil (1992), with the original exponents for k and h each estimated to be about 0.3.

A more symmetric but more complicated version is to postulate that aggregate investment is a fraction of output, with the total allocated between physical and human capital so as to maintain equal rates of return. With depreciation rates put equal for simplicity, this reduces to the equation k ~ + H = sf(k, h) - (a + n + d)(k + h), together with h = h(k). The Cobb-Douglas case is, as usual, especially easy. But the main purpose of these examples is only to show that the neoclassical model can accommodate a role for human capital, with frameworks ranging from rules of thumb to full optimization.

13. Natura l resources

There is a large literature on the economics of renewable and nonrenewable resources, some of it dealing with the implications of resource scarcity for economic growth. [An early treatise is Dasgupta and Heal (1979). See also the Handbook of Natural

656 R.M. Solow

Resource and Energy Economics, edited by Kneese and Sweeney (1989) for a more recent survey.] This is too large and distant a topic to be discussed fully here, but there is room for a sketch of the way natural resources fit into the neoclassical growth- theoretic framework.

The case of renewable natural resources is simplest. Some renewable resources, like sunlight or wind, can be thought of as providing a technology for converting capital and labor (and a small amount of materials) into usable energy. They require no conceptual change in the aggregate production function. More interesting are those renewable resources - like fish stocks and forests - that can be exploited indefinitely, but whose maximal sustainable yield is bounded.

Suppose the production function is Y = F ( K , R , e(g+n)t), with constant returns to scale, where R is the input of a renewable natural resource (assumed constant at a sustainable level) and the third input is labor in efficiency units. I f a constant fraction of gross output is saved and invested, the full-utilization dynamics are K ~ = sF(K , R, e (g+n)t) - dK, where R is remembered to be constant. For simplicity, take F to be Cobb-Douglas with elasticities a, b and 1 - a - b for K, R and L respectively.

The model then looks very much like the standard neoclassical case with decreasing returns to scale. It is straightforward to calculate that the only possible exponential path for K and Y has them both growing at the rate h = (1 - a - b)(g + n)/(1 - a). If intensive variables are defined by y = Ye ht and k = K e -ht, the usual calculations show that this steady state is stable. In it, output per person in natural units is growing at the rate h - n = [(1 - a - b ) g - bn]/(1 - a ) . For this to be positive, g must exceed b n / ( 1 - a - b). This inequality is obviously easier to satisfy the less important an input R is, in the sense of having a smaller Cobb-Douglas elasticity, i.e., a smaller competitive share.

I f the resource in question is nonrenewable, the situation is quite different. In the notation above, R/> 0 stands for the rate of depletion of a fixed initial stock So given at t = 0. Thus the stock remaining at any time t > 0 is S(t) and S(t) = ft ~ R(u) du, assuming eventual exhaustion, so that R( t )=-S~( t ) . Along any non-strange trajectory for this economy, R(t) must tend to zero. Even if F(K,O, A L ) = 0, it is possible in principle for enough capital formation and technological progress to sustain growth. But this has not been felt to be an interesting question to pursue. It depends so much on the magic of technological progress that both plausibility and intellectual interest suffer.

The literature has focused on two other questions. First, taking L to be constant, and without technological progress, when is a constant level of consumption per person sustainable indefinitely, through capital accumulation alone? The answer is: if the asymptotic elasticity of substitution between K and R exceeds 1, or equals 1 and the elasticity of output with respect to capital exceeds that with respect to R. For representative references, see Solow (1974), Dasgupta and Heal (1979), and Hartwick (1977). Second, and more interesting, how might such an economy evolve if there is a "backstop" technology in which dependence on nonrenewable resources is replaced by dependence on renewable resources available at constant cost (which may decrease


through time as technology improves). In pursuing these trails, capital investment can be governed either by intertemporal optimization or by rule of thumb. The depletion of nonrenewable resources is usually governed by "Hotelling's rule" that stocks of a resource will rationally be held only if they appreciate in value at a rate equal to the return on reproducible capital; in the notation above, this provides one differential equation: dFR/dt=FRFK. The other comes from any model of capital investment.

14. Endogenous population growth and endogenous technological progress in the neoclassical framework

Making population growth and technological progress endogenous is one of the hallmarks of the "New" growth theory [see Barro and Sata-i-Martin (1995) for references]. Needless to say, one way of endogenizing population growth goes back to Malthus and other classical authors, and has been adapted to the neoclassical-growth framework from the very beginning. There was also a small literature on endogenous technical progress in the general neoclassical framework, e.g., Fellner (1961) and von Weizs~icker (1966), but it was concerned with the likely incidence of teclmical change, i.e., its labor-saving or capital-saving character, and not with its pace. However the same simple device used in the case of population can also be used in the case of technology. It is outlined briefly here for completeness.

The Malthusian model can be simplified to say just that the rate of population (labor-force) growth is an increasing function of the real wage; and there is at any time a subsistence wage - perhaps slowly changing - at which the population is stationary. In the neoclassical model, the real wage is itself an increasing function of the capital intensity (k), so the subsistence wage translates into a value k0 that separates falling population from growing population. There is no change in the derivation of the standard differential equation, except that the rate of growth of employment is now n(k), an increasing function of k vanishing at k0. One might wish to entertain the further hypothesis that there is a higher real wage, occurring at a higher capital intensity kl such that n(k) is decreasing for k > kl, and may fall to zero or even beyond.

Technical progress can be handled in the same way. Imagine that some unspecified decision process makes the rate of (labor-augmenting) technological progress depend on the price configuration in the economy, and therefore on k. (The plausibility of this sort of assumption will be discussed briefly below.) In effect we add the equation Al(t) = a(k)A(t) to the model. The remaining calculations are as before, and they lead to the equation

k' = s f ( k ) - ( d + n ( k ) + a ( k ) ) k , (14.1)

where it is to be remembered that k= K / AL stands for capital per unit of labor in efficiency units.

658

(d+a(k)+h(k))k s~f(k)

~ k l ~ ~ k 2 k 3

R.M. Solow

Fig. 5.

The big change is that the last term in Equation (14.1) is no longer a ray from the origin, and may not behave simply at all. It will start at the origin. For small k, it will no doubt be dominated by the Malthusian decline in population and will therefore be negative. One does not expect rapid technological progress in poor economies. For larger values of k, n(k) is positive and so, presumably, is a(k). Thus the last term of Equation (14.1) rises into positive values; one expects it to intersect sf(k) from below. Eventually - the "demographic transition" - n(k) diminishes back to zero or even becomes negative. We have no such confident intuition about a(k). On the whole, the most advanced economies seem to have faster growth of total factor productivity, but within limits.

Figure 5 shows one possible phase diagram, without allowing for any bizarre patterns. The steady state at the origin is unstable. The next one to the right is at least locally stable, and might be regarded as a "poverty trap". The third steady state is again unstable; in the diagram it is followed by yet another stable steady state with a finite basin of attraction. Depending on the behavior of a(k), there might be further intersections. For a story rather like this one, see Azariadis and Drazen (1990). There are many other ideas that lead to a multiplicity of steady states.

The interesting aspect of this version of the model is that k is output per worker in efficiency units. At any steady state k*, output per worker in natural units is growing at the rate a(k*). It is clear from the diagram that a change in s, for instance, will shift k* and thus the steady-state growth rate of productivity. It will also shift n(k*) and this is a second way in which the aggregate growth rate is affected. So this is a neoclassical model whose growth rate is endogenous.

The question is whether the relation A ~ ~ a(k)A has any plausibility. The Malthusian analogue L / = n(k)L has a claim to verisimilitude. Birth and death rates are likely to

Ch. 9.. Neoclassical Growth Theory 659

depend on income per head; more to the point, births and deaths might be expected to be proportional to the numbers at risk, and therefore to the size of the population. One has no such confidence when it comes to technical change. Within the general spirit of the neoclassical model, something like a(k) seems reasonable; k is the natural state variable, determining the relevant prices. But the competitive market form seems an inappropriate vehicle for studying the incentive to innovate. And why should increments to productive knowledge be proportional to the stock of existing knowledge? No answer will be given here, and there may be no good answer.

The relevant conclusion is that population growth and technological progress can in principle be endogenized within the framework of the neoclassical growth model; the hard problem is to find an intuitively and empirically satisfying story about the growth of productive technology.

15. Convergence

The simplest form of the neoclassical growth model has a single, globally stable steady state; if the model economy is far from its steady state, it will move rapidly toward it, slowing down as it gets closer. Given the availability of the Summers-Heston cross- country data set of comparable time series for basic national-product aggregates, it is tempting to use this generalization as a test of the neoclassical growth model: over any common interval, poorer countries should grow faster than rich ones ( inper capita terms). This thought has given rise to a vast empirical literature. Useful surveys are Barro and Sala-i-Martin (1995) and Sala-i-Martin (1996), both of which give many further references. The empirical findings are too varied to be usefully discussed here, but see chapter 4 by Durlauf and Quah in this volume, and also chapter 10.

Sala-i-Martin distinguishes between/j-convergence and a-convergence. The first is essentially the statement given above; it occurs when poor countries tend to grow faster than rich ones. On the other hand, a-convergence occurs within a group of countries when the variance of their per capita GDP levels tends to get smaller as time goes on. Clearly/j-convergence is a necessary condition for a-convergence; it is not quite sufficient, however, though one would normally expect/J-convergence to lead eventually to a-convergence.

Something can be said about the speed of convergence if the neoclassical model holds. Let gt, r stand for the economy's per capita growth rate over the interval from t to t + T , meaning that gt, r = T-11og[y(t+T)/y(t)] . Then linearizing the neoclassical model near its steady state yields an equation of the form

gt, r = const. - T-l(1 - e -/~r) logy(t). (15.1)

Obviously gt,0 =/Jlogy. Moreover, in the Cobb-Douglas case with f ( k ) = k b, it tma~s out that/3 = (1 - b)(d + n + a). Another way to put this is that the solution to the basic differential equation, near the steady state at k*, is approximately

k(t) - k* ~ e-b(a+n+")t(k0 - k*). (15.2)

660 R.M. Solow

Since b is conventionally thought to be near 0.3, this relation can be used to make /3-convergence into a tighter test of the neoclassical model. [It usually turns out that b must be considerably larger than that to make the model fit; this has led to the thought that human capital should be included in k, in which case the magnitudes become quite plausible. On this see Mankiw, Romer and Weil (1992).]

One difficulty with all this is that different countries do not have a common steady state. In the simplest model, the steady-state configuration depends at least on the population growth rate (n) and the saving-investment rate (s) or the utility parameters that govern s in the optimizing version of the model. One might even be permitted to wonder if countries at different levels of development really have effective access to a common world technology and its rate of progress; "backwardness" may not be quite the same thing as "low income". In that case, an adequate treatment of convergence across countries depends on the ability to control for all the determinants of the steady- state configuration. The empirical literature consists largely of attempts to deal with this complex problem. On this, see again chapter 10 in this Handbook.

The natural interim conclusion is that the simple neoclassical model accounts moderately well for the data on conditional convergence, at least once one allows for the likelihood that there are complex differences in the determination of steady states in economies at different stages of development. The main discrepancy has to do with the speed of convergence. This is perhaps not surprising: actual investment paths will follow neither optimizing rules nor simple ratios to real output.

Outside the simplest neoclassical growth model, there may even be multiple steady states, and this clearly renders the question of/3-convergence even more complicated. This possibility leads naturally to the notion of club-convergence: subsets of "similar" countries may exhibit/3-convergence within such subsets but not between them. Thus the states of the United States may exhibit convergence, and also the member countries of OECD, but not larger groupings. This is discussed in Galor (1996). See also Azariadis and Drazen (1990) for a model with this property.

16. Overlapping generations

Literal microfoundations for the optimizing version of the standard neoclassical model usually call for identical, immortal households who plan to infinity. An equivalent - and equally limiting - assumption involves a family dynasty of successive generations with finite lives, each of which fully internalizes the preferences of all succeeding generations. An alternative model avoids some restrictiveness by populating the economy with short-lived overlapping generations, each of which cares only about its own consumption and leaves no bequests. The simplest, and still standard, version involves two-period lives, so that two generations - young and old - coexist in each period.

As the previous sentence suggests, overlapping-generations models are written in discrete time, although this is not absolutely necessary [Blanchard and Fischer (1989),


p. 115]. There is a large literature beginning with Samuelson (1958) [anticipated in part by Allais (1947)]. An excellent exposition is to be found in D. Romer (1996, Ch. 2), and a full treatment in Azariadis (1993), where further references can be fotmd. The OG model has uses in macroeconomic theory generally [for instance Grandmont (1985), Hahn and Solow (1996)], but here attention is restricted to its use in growth theory, beginning with Diamond (1965).

There is only one produced good, with the production function Yt = F ( K t , A t N t ) as usual. In the standard notation, we can write Yt =f (k t ) , where y and k are output and capital per unit o f labor in efficiency traits. In each period, then, the competitively determined interest rate is rt =f~(kt ) and the wage in terms of the single good is Atwt = A t ( f ( k t ) - ktf~(kt)). Note that wt is the wage per efficiency unit o f labor; a person working in period t earns Atwt.

Nt families are born at the beginning of period t and die at the end of period t + 1. Set Nt = (1 + n) t so the total population in period t is (1 + n) t 1 + (1 + n) t. Each family supplies one unit o f labor inelastically when it is young, earns the going (real) wage Atwt, chooses how much to spend on the single good for current consumption etl, earns the going rate of return rt+l on its savings ( A t w t - c t l ) , and spends all o f its wealth on consumption when old, so that ct2 = (1 + rt+l)(Atwt - Ctl). Note that savings in period t are invested in period t + 1.

As with other versions of growth theory, it is usual to give each household the same time-additive utility function u (c t l )+(1 + i ) lu(ct2). It is then straightforward to write down the first-order condition for choice of Ctl and c~2. It is U'(Ct2)/U'(Ctl) = (1 +i ) / (1 +rt+l); together with the family's intertemporal budget constraint it determines etl and ct2, and therefore the family's savings in period t as a function of rt+l and Atwt.

In the ever-popular special case that u(x) = (1 - m)- lx I m (SO that m is the absolute elasticity of the marginal utility function), it follows directly that the young family saves a fraction s(r) of its wage income, where

(1 + r) (1-m)/m s(r) = (1 + r)( 1-m)/m + (1 + i) 1/m (16.1)

and the obvious time subscripts are omitted. The formulas for young and old consumption follow immediately.

Now assume that At -- (1 + a) t as usual. Since the savings of the young finance capital investment in the next period, we have Kt+l = s(rt+l)AtwtNt. Remembering that k is KlAN, we find that

kt+l = (1 + n) l(1 + a)-ls(rt+l)wt. (16.2)

Substitution of rt+l =f~(kt+l) and wt = f ( k t ) - k t f t ( k t ) leaves a first-order difference equation for kt.

In simple special cases, the difference equation is very well behaved. For instance (see D. Romer 1996 for details and exposition), i f f ( - ) is Cobb-Douglas and u(.) is

662

kt+l

J

k 0 k I k2 k3 k~ kt Fig. 6.

R.M. Solow

logarithmic, the difference equation takes the form kt+l = const.kt b and the situation is as in Figure 6. (Note that logarithmic utility implies that the young save a constant fraction o f their earnings, so this case is exactly like the standard neoclassical model.) There is one and only one stationary state for k, and it is an attractor for any initial conditions k0 > 0. Exactly as in the standard model, k* decreases when n or a increases and also when i increases (in which case s naturally decreases). Since k =K/AN, the steady-state rate of growth of output is a + n and the growth rate of labor productivity is a, both independent o f i (or s).

There are, however, other possibilities, and they can arise under apparently "normal" assumptions about utility and production. Some o f these possibilities allow for a multiplicity of steady states, alternately stable and unstable. This kind of configuration can arise just as easily in the standard model when the saving rate is a function of k. They are no surprise.

The novel possibility is illustrated in Figure 7. The curve defined by Equation (16.2) in the (kt, k t + l ) plane may bend back, so that in some intervals - in the diagram, when kt is between km and kM - kt is compatible with several values of kt+t. The difference equation can take more than one path from such a kt. This is the situation that gives rise to so-called "sunspot" paths. See Cass and Shell (1983), Woodford (1991), and an extensive treatment in Farmer (1993).

The mechanism is roughly this. Suppose sl(r) < 0. Then a young household at t ime t that expects a low value of rt+l will save a lot and help to bring about a low value of r. I f it had expected a high value of r next period, it would have saved only a little and helped to bring about a high value of r. The possibility exists that the household may condition its behavior on some totally extraneous phenomenon (the "sunspot" cycle) in such a way that its behavior validates the implicit prediction and thus confirms the significance of the fundamentally irrelevant signal. In this particular model, the sunspot phenomenon seems to require that saving be highly sensitive to the interest

kt+l

Neoclassical Growth Theory Ch. 9: 663

k m k M k t Fig. 7.

rate, and in the "wrong" direction at that. This goes against empirical findings, so that indeterminacy of this kind may not be central to growth theory, even if it is significant for short-run macroeconomic fluctuations.

17. Open questions

This survey has stayed close to basics and has not attempted anything like a complete catalogue of results in growth theory within the neoclassical framework. In that spirit, it seems appropriate to end with a short list of research directions that are currently being pursued, or seem worth pursuing.

The role of human capital needs clarification, in both theoretical and empirical terms. Human capital is widely agreed to be an important factor in economic growth. Maybe more to the point, it seems to offer a way to reconcile the apparent facts of convergence with the model. One difficulty is that the measurement of human capital is insecure. See Judson (1996) and Klenow and Rodriguez-Clare (1998). School enrollment data are fairly widely available, but they clearly represent a flow, not a stock. Direct measurement of the stock runs into deep uncertainty about depreciation and obsolescence, and about the equivalence of schooling and investment in human capital. Mention has already been made of the use of relative wages as indicators of relative human capital; the well-known Mincer regressions can also be used, as in Klenow and Rodriguez-Clare (1998). (Better measurement might throw some light on the way human capital should enter the production function: as a labor-augmentation factor, as a separate factor of production, or in some other way. On this, as on several

664 R.M. Solow

other matters, the distinction between "neoclassical" and "endogenous" growth theory seems to be artificial.)

It was suggested in the text above that there is no mechanical obstacle to the endogenization of technical change within the neoclassical model. But the analytical devices mentioned by example were all too mechanical. The modeling of technological progress should be rethought, and made more empirical using whatever insights come from micro-studies of the research-and-development process. It seems pretty clear that the endogenous-growth literature has been excessively generous in simply assuming a connection between the level of innovative effort and the rate of growth of the index of technology. It is impossible to know what further empirical work and theoretical modeling will suggest about the nature of that connection. But it is a central task to find out.

One of the earliest stories about endogenous technical change was Arrow's model of "learning by doing" (Arrow 1962), which is well within the neoclassical framework. It, too, was rather mechanical, with automatic productivity increase as simple fall-out from gross investment. Many economists have found the basic idea to be plausible; it is a source of technical change that is entirely independent of R&D. But very little econometric work has taken off from learning-by-doing, and there seems to have been no attempt to test it. Recently there have been renewed efforts to elaborate and improve the underlying idea [Young (1993), Solow (1997)]. The next step should probably be empirical.

Very similarly, the notion that technical change has to be "embodied" in new investment in order to be effective seems instantly plausible [Solow (1960), Jorgenson (1966)]. For many years, however, it proved to be impossible to verify its importance in macroeconomic time series data. Just recently there has been a revival of interest in this question. Lau (1992) may have isolated a significant embodiment-effect in a combined time-series analysis involving several advanced and newly-industrialized countries. And Wolff (1996) claims to have found in the embodiment-effect an explanation of the productivity slowdown that occurred almost worldwide in the early 1970s. Hulten (1992), on the other hand, came to different conclusions using different methods and data. The interesting possibility of using changes in the relative prices of investment goods and consumption goods to isolate the embodiment effect has opened up new vistas. Greenwood, Hercowitz and Krusell (1997) is a pioneering reference. See also the survey by Hercowitz (1998) and an (as yet) unpublished paper by Greenwood and Jovanovic (1997). Some new theory might help develop this work further.

A powerful embodiment effect (and the same could be said about learning by doing) will strengthen the connection between short-run macroeconomic fluctuations and long-run growth. As things stand now, the only effect of business cycles on the growth path comes through the "initial" value of the stock of capital. These more sophisticated mechanisms would also link growth to cycle through the level of achieved technology. There are no doubt other ways in which better integration of growth theory and business-cycle theory would improve both of them.

A last issue that needs exploring is the matter of increasing returns to scale. It was shown earlier that the neoclassical model can easily accommodate increasing (or,


for that matter, decreasing) returns to scale, just as a matter o f model ing production. The important question lies elsewhere. The ubiquity o f increasing returns to scale implies the ubiquity o f imperfect competit ion as a market form. There is plenty o f microeconomic theory to link imperfect competi t ion with investment and perhaps with innovative activity. The systematic relation - i f any - between imperfect competi t ion and growth has not been fully explored. What there is has come mostly through the endogenous-growth literature [Aghion and Howitt (1992, 1998), Romer (1990)), and there it has been an appendage to specialized models o f the R&D process. Imperfect competi t ion is finding its way slowly into general macroeconomics. Growth theory should not be far behind.

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