閤 A Contribution to the theory of Economic growth OR。 Robert m. solow The Quarterly Journal of Economics, Vol. 70, No. 1.(Feb, 1956), pp. 65-94 Stable url: http://inksistor.org/sici?sic0033-5533%028195602%02970%3a1%03c65%03aacttto%03e2.0.co%3b2-m The Quarterly Journal of Economics is currently published by The MIT Press Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyouhaveobtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the jsTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that ap on the screen or printed page of such transmission STOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support @jstor. org Thu May3l1102:492007
A Contribution to the Theory of Economic Growth Robert M. Solow The Quarterly Journal of Economics, Vol. 70, No. 1. (Feb., 1956), pp. 65-94. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28195602%2970%3A1%3C65%3AACTTTO%3E2.0.CO%3B2-M The Quarterly Journal of Economics is currently published by The MIT Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/mitpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Thu May 31 11:02:49 2007
A CONTRIBUTION TO THE THEORY OF ECONOMIC GROWTH By roBert M I. Introduction, 65.-Il. A model of long-run growth, 66.- Ill. Possible growth patterns, 68.-IV. Examples, 73- V. Behavior of interest and wag rates, 78.-VI. Extensions, 85.-VII. Qualifications, 91 I. INtRODUCtION All theory depends on assumptions which are not quite true. make the inevitable simplifying assumptions in such a way that the inal results are not very sensitive. A"crucial"assumption is one on which the conclusions do depend sensitively, and it is important that crucial assumptions be reasonably realistic. When the results of a theory seem to flow specifically from a special crucial assumption then if the assumption is dubious, the results are suspect. I wish to argue that something like this is true of the Harrod Domar model of economic growth. The characteristic and powerful conclusion of the Harrod-Domar line of thought is that even for the long run the economic system is at best balanced on a knife-edge of quilibrium growth. Were the magnitudes of the key parameters the savings ratio, the capital-output ratio, the rate of increase of the labor force-to slip ever so slightly from dead center, the conse- quence would be either growing unemployment or prolonged inflation. In Harrod, s terms the critical question of balance boils down to a comparison between the natural rate of growth which depends, in the absence of technological change, on the increase of the labor force, and the warranted rate of growth which depends on the saving and invest- ing habits of households and firms But this fundamental opposition of warranted and natural rates turns out in the end to flow from the crucial assumption that produc- tion takes place under conditions of fixed proportions possibility of substituting labor for capital in production. If this assumption is abandoned, the knife-edge notion of unstable balance seems to go with it. Indeed it is hardly surprising that such a gross 1. Thus transport costs were merely a negligible complication to Ricardian trade theory, but a vital characteristic of reality to von Thunen
QUARTERLY JOURNAL OF ECONOMICS rigidity in one part of the system should entail lack of flexibility in A remarkable characteristic of the Harrod-Domar model is that it consistently studies long-run problems with the usual short-run tools. One usually thinks of the long run as the domain of the neo- classical analysis, the land of the margin. Instead Harrod and Domar alk of the long run in terms of the multiplier, the accelerator, "the capital coefficient. The bulk of this paper is devoted to a model of ong-run growth which accepts all the Harrod-Domar assumptions except that of fixed proportions. Instead I suppose that the single composite commodity is produced by labor and capital under the standard neoclassical conditions. The adaptation of the system to an exogenously given rate of increase of the labor force is worked out in ome detail to see if the Harrod instability appears. The price-wage interest reactions play an important role in this neoclassical adjust ment process, so they are analyzed too. Then some of the other rigid assumptions are relaxed slightly to see what qualitative changes result: neutral technological change is allowed, and an interest-elastic savings schedule. Finally the consequences of certain more"Keynes- ian''relations and rigidities are briefly considered IL. A MODEL OF LONG-RUN GROWTH There is only one commodity, output as a whole, whose rate of production is designated Y((). Thus we can speak unambiguously of the community's real income. Part of each instant,'s output consumed and the rest is saved and invested. The fraction of output saved is a constant s, so that the rate of saving is sr(. The com munity's stock of capital K(t)takes the form of an accumulation of is then just th increase of this capital stock dK/dt or K, so we have the basic identity at every instant of time Output is produced with the help of two factors of production capital and labor, whose rate of input is L(t). Technological possi- bilities are represented by a production function Y= F(K, L) Output is to be understood as net output after making good the depre- ciation of capital. About production all we will say at the moment is
THE THEORY OF ECONOMIC GROWTH that it shows constant returns to scale. Hence the production func- tion is homogeneous of first degree. This amounts to assuming that there is no scarce nonaugmentable resource like land. Constant returns to scale seems the natural assumption to make in a theory of growth. The scarce-land case would lead to decreasing returns to scale in capital and labor and the model would become more Inserting()in(1)we get K=sF(K,L) This is one equation in two unknowns. One way to close the system would be to add a demand-for-labor equation: marginal physical productivity of labor equals real wage rate; and a supply-of-labor equation, The latter could take the general form of making labor supply a function of the real wage or more classically of putting the real wage equal to a conventional subsistence level. In any case there would be three equations in the three unknowns K, L, real wage Instead we proceed more in the spirit of the Harrod model esult of exogenous population growth the labor force increases at a constant relative rate n. In the absence of technological change n is Harrod' s natural rate of growth. Thus (4) L(o= Loe In (3)L stands for total employment; in(4)L stands for the available supply of labor. By identifying the two we are assuming that ful employment is perpetually maintained. When we insert(4)in( 3) K= 8F(K, Loe") re have the basic equation which determines the time path of capital accumulation that must be followed if all available labor is to be Iternatively(4)can be looked at as a supply curve of labor. It says that the exponentially growing labor force is offered for employ- ment completely inelastically. The labor supply curve is a vertical 2. See, for example, Haavelmo: A Study in the Theory of Economic Evolutin (Amsterdam, 1954), pp 9-11. Not all "underdeveloped"countries are areas of land shortage. Ethiopia is a counterexample. One can imagine the theory as applying as long as arable land can be hacked out of the wilderness at essentially onstant cost
QUARTERLY JOURNAL OF ECONOMICS line which shifts to the right in time as the labor force grows according to(4). Then the real wage rate adjusts so that all available labor is employed, and the marginal productivity equation determines the wage rate which will actually rule. 3 In summary, (5)is a differential equation in the K ) Its solution gives the only time profile of the community capital stock which will fully employ the available labor. Once we know the time path of capital stock and that of the labor force, we can compute from the production function the corresponding time path of real output. The marginal productivity equation determines the time path of the real wage rate. There is also involved an assumption of full employment of the available stock of capital. At any point of time the pre-existing stock of capital(the result of previous accumula tion)is inelastically supplied. Hence there is a similar marginal productivity equation for capital which determines the real rental per unit of time for the services of capital stock. The process can be viewed in this way: at any moment of time the available labor supply is given by(4)and the available stock of capital is also a datum. Since the real return to factors will adjust to bring about full employment of labor and capital we can use the production function(2)to find the current rate of output. Then the propensity to save tells us how much of net output will be saved and invested. Hence we know the net accumulation of capital during the current period. Added to the already accumulated stock this gives the capital available for the next period, and the whole process can be repeated III. POSSIBLE GROWTH PATTERNS To see if there is always a capital accumulation path consistent with any rate of growth of the labor force, we must study the differen tial equation (5)for the qualitative nature of its solutions. Naturally canlt hope to find the exact solution of the production fur are surprisingly easy to isolate, even graphically in braig nction we without specifying the exact shape 'o do so we introduce a new variabler& the ratio of capital to labor. Hence we have K= rL rLoe". Differentiating with respect to time we get K= Loe"r+ nrLoe 3. The complete set of three equations consists of(3), (4) and F(K, D-w
THE THEORY OF ECONOMIC GROWTH lnstitute this in(5) (r nr)Loe"= SF(K, Lo But because of constant returns to scale we can divide both variables in F by L= Loe"provided we multiply F by the same factor. Thus (r +nrloe"= sLoe"F and dividing out the common factor we arrive finally at 7=8F(r,1)一nr Here we have a differential equation involving the eapital-labor ratio alone This fundamental equation can be reached somewhat less formally. Sincer =- the relative rate of change of ris the difference between the relative rates of change of K and L. That is rK L Now first of all L n. Secondly K= sF(K, L). Making these sub- stitutions F(K,L) 一nr, K Now divide L out of F as before, note thatL 1 get(6)aga The function F(r, 1)appearing in(6)is easy to interpret. It is the total product curve as varying amounts r of capital are employed with one unit of labor. Alternatively it gives output per worker as change of the capital-Iak worker. Thus(6) states that the rate of a function of capital or ratio is the difference of two terms, one representing the increment of capital and one the increment of labor When r=0, the capital-labor ratio is a constant, and the capital stock must be expanding at the same rate as the labor force, namely n
QUARTERLY JOURNAL OF ECONOMICS (The warranted rate of growth, warranted by the appropriate real rate of return to capital, equals the natural rate. In Figure I, the ray through the origin with slope n represents the function nr. The other curve is the function 8F(r, 1).It is here drawn to pass through the origin and convex upward: no output unless both inputs are positive and diminishing marginal productivity of capital uld be the case, for example, with the Cobb-Douglas function. At the point of intersection nr= sF(, 1)and r=0. If the capital-labor ratio r* should ever be established, it will be maintained, and capital and labor will grow thenceforward in proportion. By constant returns to FIGURE I scale, real output will also grow at the same relative rate n, and out put per head of labor force will be constant. But if r r*, how will the capital-labor ratio develop over time? To the right of the intersection point, when r>r", nr >8F(r, 1)and from(6)we see that r will decrease toward r*. Conversely if initially r 0, and r will increase toward r*. Thus the equilibrium value r* is stable. Whatever the al value of the capital-labor ratio, the system will develop toward a state of balanced growth at the natural rate. The time path of apital and output will not be exactly exponential except asymptote- cally. 4 If the initial capital librium ratio 4. There is an exception to this. If K=0, r=0 and the system can't ge started with no and hence no accumulation. But this
THE THEORY OF ECONOMIC GROWTH capital and output will grow at a faster pace than the labor force unt the equilibrium ratio is approached. If the initial ratio is above the equilibrium value, capital and output will grow more slowly than the labor force. The growth of output is always intermediate between those of labor and capital f course the strong stability shown in Figure I is not inevitable The steady adjustment of capital and output to a state of balanced growth comes about because of the way I have drawn the produc- ivity curve F(r, 1). Many other configurations are a priori possible For example in Figure II there are three intersection points. Inspec- FIGURE tion will show that ri and ra are stable, r? is not. Depending on the initially observed capital-labor ratio, the system will develop either to balanced growth at capital-labor ratio r or ra. In either case labor supply, capital stock and real output will asymptotically expand at rate n, but around r there is less capital than around r3, hence the level of output per head will be lower in the former case than in the latter. The relevant balanced growth equilibrium is at ri for an initial ratio anywhere en0 and ra, it is at r3 for any initial ratio greater than r2. The ratio r? is itself an equilibrium growth ratio, but an unstable one; any accidental disturbance will be magnified over time. Figure Ii has been drawn so that production is possible without capital; hence the origin is not an equilibrium"growth"configurati Even Figure iI does not exhaust the possibilities. It is possible rium is unstable: the slightest windfall capital accumulation will start the
QUARTERLY JOURNAL OF ECONOMICS that no balanced growth equilibrium might exist. 5 Any nondecreasing function F(, 1)can be converted into a constant returns to scale production function simply by multiplying it by L; the reader can nstruct a wide variety of such curves and examine the resulting solutions to(6). In Figure III are shown two possibilities, together F( 2(,) FIGURE III with a ray nr. Both have diminishing marginal productivity through out, and one lies wholly above mr while the other lies wholly below. The first system is so productive and saves so much that perpetual full employment will increase the capital-labor ratio (and also the output per head)beyond all limits; capital and income both increase 5. This seems to contradict a theorem in R. M. Solow and P. A Sa Balanced Growth under Constant Returns to Scale, "Econometrica, XXI (1953) 12-24, but the contradiction is only apparent. It was there assumed that every commodity had positive marginal productivity in the production of each com- modity. Here capital cannot be used to produce labor. 6. The equation of the first might be s FI(, 1)=nr +yr, that of the second P2(r,1)=
THE THEORY OF ECONOMIC GROWTH more rapidly than the labor supply. The second system is so unpro- ductive that the full employment path leads only to forever diminish- ing income per capita. Since net investment is always positive and labor supply is increasing, aggregate income can only rise The basic conclusion of this analysis is that, when production takes place under the usual neoclassical conditions of variable pro- portions and constant returns to scale, no simple opposition between natural and warranted rates of growth is possible. There may not be -in fact in the case of the Cobb-Douglas function there never can be-any knife-edge The system can adjust to any given rate of growth of the labor force, and eventually approach a state of steady proportional expansion IV, EXAMPLES In this section I propose very briefly to work out three three simple choices of the shape of the production function for whic it is possible to solve the basic differential equation( 6)explicitly. Example 1: Fixed Proportions. This is the Harrod-Domar case It takes a units of capital to produce a unit of output and b units of labor. Thus a is an acceleration coefficient. Of course, a unit of output can be produced with more capital and / or labor than this (the isoquants are right-angled corners); the first bottleneck to be eached limits the rate of output. This can be expressed in the form (2)by saying Y= F(K, L)= min K L where"min(.. ""means the smaller of the numbers in parentheses. The basic differential equation(6)becomes =mi(1 Evidently for very small r we must have<-, so that in this range a2b,1.e,r 2 b, the equa- tion becomes r=h-nr. It is easier to see how this works graphi ally. In Figure Iv the function s min(r, a)is epresented by a
