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 <r*, the graph shows that nr sF(, 1),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