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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 Wed May 9 12:13:11 2007 A CONTRIBUTIOX TO T H E THEORY OF ECOKOIIIC GROWTH I . Introduction, 65. - 11. A model of long-run growth. 66. -111. Possible growth patterns. 68. - IT7. Examples. 73. -V. Behavior of interest and wage rates. 78. -VI. Extensions, 85. -VII. Qualifications. 91. I. IR'TRODUCTIOS All theory depends on assumptions which are not quite true. That is what makes it theory. The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final 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. 1 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 equilibrium 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. There is no 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. 65 6 6 QUARTERLY JOGR-VAL OF ECOSO-IIICS rigidity in one part of the system should entail lack of flexibility in another. 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 talk 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 long-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 some 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. 11. A MODEL O F LONG-RUNGROWTH There is only one commodity, output as a whole, whose rate of production is designated Y( t ) . Thus we can speak unambiguously of the community's real income. Part of each instant's output is consumed and the rest is saved and invested. The fraction of output saved is a constant s, so that the rate of saving is sY(t). The com- munity's stock of capital K( t ) takes the form of an accumulation of the composite commodity. Net investment is then just the rate of increase of this capital stock dK/dt or K, SO we have the basic identity a t every instant of time: (1) K = sY. 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 Output is to be understood as net output after making good the depre- ciation of capital. About production all we will say a t the moment is THE THEORY OF ECONOAIZC GROWTH 67 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 R i~a rd i an .~ Inserting (2) in (1) we get 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. As a result 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: 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 full employment is perpetually maintained. When we insert (4) in (3) to get ( 5 ) K' = SF (K , L~~ " ~ ) we have the basic equation which determines the time path of capital accumulation that must be followed if all availabIe labor is to be employed. Alternatively (4) can be looked a t as a supply curve of labor. I t 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 Theor?) o f Economzc Evolution (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 a t essentially constant cost. 68 QL'ARTERLY 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 mill actually rule.3 In summary, (5) is a differential equation in the single variable K( t ) . I ts solution gives the only time profile of the community's 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: a t 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. 111. POSSIBLE PATTERNSGROWTH To see if there is al~vaysa 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 without specifying the exact shape of the production function we can't hope to find the exact solubion. But certain broad properties are surprisingly easy to isolate, even graphically. KTo do so we introduce a new variable r =-, the ratio of capital L to labor. Hence we have K = r L Differentiating with= r ~ ~ e " ~ . raspect to time we get K = I,,,ent; + nrLoent. aF(K,L)3. The complete set of three equations consists of (3), (4) and -------- W. aL THE THEORY OF ECONOMIC GROWTH Substitute this in (5): But because of constant returns to scale we can divide both variables in F by L = Loentprovided we multiply F by the same factor. Thus and dividing out the common factor we arrive finally a t ( 6 ) r = sF(r , l ) - nr. Here we have a differential equation involving the capital-labor ratio alone. This fundamental equation can be reached somewhat less Kformally. Since r = -, the relative rate of change of r isthedifference L between the relative rates of change of K and L. That is: LKow first of all - = IL. Secondly K = sF (K ,L ) . Making these sub- L stitutions: sF (K ,L ) r = r ------ - nr. K L 1Kow divide L out of F as before, note that -= -, and we get (6) again. K r The function F( r , l ) appearing in ( 6 )is easy to interpret. I t 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 a function of capital per worker. Thus (6) states that the rate of change of the capital-labor ratio is the difference of two terms, one representing the increment of capital and one the increment of labor. When i = 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. 70 QUARTERLY JOURNAL OF ECONOMICS (The warranted rate of growth, warrantedby 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 sF(r,l). I t is here drawn to pass through the origin and convex upward: no output unless both inputs are positive, and diminishing marginal productivity of capital, as would be the case, for example, with the Cobb-Douglas function. At the point of intersection nr = sF(r,l) and ;= 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 r * FIGURE I scale, real output will also grow a t 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 > sF(r,l) and from (6) we see that r will decrease toward r*. Conversely if initially r < r*, the graph shows that nr < sF(r,l),;> 0, and r will increase toward r*. Thus the equilibrium value r* is stable. Whatever the initial value of the capital-labor ratio, the system will develop toward a state of balanced growth a t the natural rate. The time path of capital and output will not be exactly exponential except asymptoti- ally.^ If the initial capital stock is below the equilibrium ratio, 4. There is an exception to this. If K = 0, T = 0 and the system can't get started; with no capital there is no output and hence no accumulation. But this 71 THE THEORY OF ECONOMIC GROWTH capital and output will grow a t a faster pace than the labor force until 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. Of 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- tivity curve F(r,l). Many other configurations are a priori possible. For example in Figure I1 there are three intersection points. Inspec- r nr s ~ ( r ,1) r r3 FIGURE I1 tion will show that rl and r3 are stable, rz is not. Depending on the initially observed capital-labor ratio, the system will develop either to balanced growth a t capital-labor ratio rl or r3. In either case labor supply, capital stock and real output will asymptotically expand a t rate n, but around rl 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 a t rl for an initial ratio anywhere between 0 and rz, i t is a t r3 for any initial ratio greater than rz. The ratio rz is itself an equilibrium growth ratio, but an unstable one; any accidental disturbance will be magnified over time. Figure I1has been drawn so that production is possible without capital; hence the origin is not an equilibrium "growth" configuration. Even Figure I1 does not exhaust the possibilities. I t is possible equilibrium is unstable: the slightest windfall capital accumulation will start the system off toward r*. 72 QUAR T E R L Y J OU RN A L OF ECONOll l ICS that no balanced growth equilibrium might exist.6 Any nondecreasing function F(r , l ) can be converted into a constant returns to scale production function simply by multiplying i t by L; the reader can construct a wide variety of such curves and examine the resulting solutions to (6). In Figure I11 are shown two possibilities, together with a ray nr. Both have diminishing marginal productivity through- out, and one lies wholly above nr while the other lies wholly below.6 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. Samuelson: "Balanced Growth under Constant Returns to Scale," Econometrica, XXI (1953), 412-21, 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,Fl(r,l) = nr + .\l -, the equa- a a b b S tion becomes 1: = -- nr. I t is easier to see how this works graphi- b r 1 cally. In Figure IV the function s min (; x)is represented by a 7 74 QUARTERLY JOURNAL OF ECONOMICS Sbroken line: the ray from the origin with slope -until r reaches the a value 2 , and then a horizontal lineat height 5 . In the Harrod model b b - S is the warranted rate of growth. a . r " f 6b FIGURE IV There are now three possibilities: S (a) nl > -, the natural rate exceeds the warranted rate. I t can a be seen from Figure IV that nlr is always greater than s min so that r always decreases. Suppose the initial value of the a s capital-labor ratio is r , > -, then r = - - nlr , whose solution is b b 1 = (7. - :)e-"' +-S . Thus r decreases toward -S which is nlb n ~ b 75 THE THEORY OF ECONOMIC GROWTH a in turn less than -. At an easily calculable point of time t l , r reaches b - a a ( t - $ 1 . From then on 7: = r , whose solutionis r = -e b ' b Since -s < nl, r will decrease toward zero. At time tl , when r = -a a b the labor supply and capital stock are in balance. From then on as the capital-labor ratio decreases labor becomes redundant, and the extent of the redundancy grows. The amount of unemployment can be calculated from the fact that K = rL,ent remembering that, when K K capital is the bottleneck factor, output is -and employment is b -. a a S(b) n2= -,the warranted and natural rates are equal. If initially a a a r > -so that labor is the bottleneck, then r decreases to -and staysb b there. If initially r < a -, then r remains constant over time, in a sort b of neutral equilibrium. Capital stock and labor supply grow a t a common rate nz; whatever percentage