AutorCespedes, Daniel Villalobos


The Theory of Production Growth by Cobb and Douglas (1928) is an attempt to explain the relation between capital, labor, and production of values. These authors criticized "the progressive refinement ... in the measurement of the volume of physical production in" United States manufacturing, concluding that such relation "is purely fortuitous" and arguing that it is a "reductio ad absurdum" (Cobb & Douglas, 1928, pp. 139, 160). The start of their endeavor to "deal with" and "to throw some light upon" it is based on the following questions:

* Can we estimate, within limits, whether this increase in production was purely fortuitous, whether it was primarily caused by technique, and the degree, if any, to which it responded to changes in the quantity of labor or capital?

* May it be possible to determine, [...], the relative influence upon production of labor as compared with capital?

* As the proportion of labor to capital changed from year to year, may it be possible to deduce the relative amount added to the total physical product by each unit of labor and capital and what is more important still by the final units of labor and capital in the respective years?

* Can we measure the probable slopes of the curves of incremental product which are imputed to labor and capital and thus give greater definiteness to what is at present purely a hypothesis with no quantitative values attached?

* May we shed light upon the question as to whether or not the processes of distribution are modeled at all closely upon those of the production of value? (Cobb & Douglas, 1928, pp. 139-140)

So, it is the purpose of this research to offer a method of analysis to discover new results and to provide some answers to those questions, focusing on developing a "method of attack" according to the following Cobb and Douglas' suggestion and challenge:

We should (1) be prepared to devise formulas which will not necessarily be based upon constant relative 'contributions' of each factor to the total product but which will allow for variations from year to year, and (2) will eliminate so far as possible the time element from the process. (Cobb & Douglas, 1928, p. 165)

The previous suggestion and challenge are the core points in this research's attempt to find answers to those questions. But before doing it, in the first section, we will work with some remarkable authors like Swan (1956), Solow (1956) and (1957), Arrow et al. (1961), and recently Piketty (2014a), all of which perform amazing efforts in giving answers to those questions. But no endeavor of these authors satisfies Cobb and Douglas's suggestion, challenge, and questions. By working with Cobb and Douglas's production function, part two of this research will offer new formulas which measure (a), "the elasticity of the product with respect to small changes in labor alone" (Cobb & Douglas, 1928, pp. 155-156), and thus the slope of production function. Cobb and Douglas's data is useful in revealing results shown in tables and figures. The third section deals with further assumptions to analyze the process of distribution of value. From there, the elasticity of distribution (p) shows as an original result by developing another new formula that measures the elasticity of resource composition (p). To evaluate these findings, we will use Cobb and Douglas's data.

We conclude that "the processes of distribution are modeled at all closely upon those of the production of value" (Cobb & Douglas, 1928, pp. 139-140, 161). Neither production growth nor distribution of production are purely fortuitous; they might be caused by technique and the degree to which it responds to changes in the quantity of labor or capital (Cobb & Douglas, 1928, pp. 139-140) and business cycle, wars, and government policies. Those processes reveal the convergence and divergence phenomenon in economic growth. Finally, conclusions and possible refinements to the theory of economic growth will be stated followed by the bibliography.

Progressive Refinement in Economic Growth Theory

Cobb and Douglas's theory on production growth suggests that it is valid as a result of the United States manufacturing analysis during 1899-1922 (Cobb & Douglas, 1928, pp. 159-164). Nevertheless, in "A Program for Further Work" these authors advice: "It should be made clear that we do not claim to have actually solved the law of production, but merely that we have made an approximation to it and suggested a method of attack" (Cobb & Douglas, 1928, pp. 164-165). Since Cobb and Douglas, "progressive refinement" on the theory represents remarkable efforts in finding new answers to those questions. Even though Harrod (1939) did not have the intention of answering those questions, he proposed:

a new method of approach [to] the study of the operations of the forces maintaining a trend of increase [because] even in a condition of growth, which generally speaking is steady, it is not to be supposed that all the component individuals are expanding at the same rate. (pp. 15-16) He insisted on it in his "second essay in dynamic theory" (Harrod, 1960, p. 281). Cobb and Douglas's questions imply that resources' relative contributions to production growth must converge with comparative distributions of value. Harrod (1960) assumes "constant income distribution," but:

This assumption is simply carried over from the welfare optimum of static economics, where the income distribution is taken as given. Ultimately, we should be able to accommodate a steadily changing distribution of income in dynamic theory; this would (or might) entail a steadily changing rate of growth. (p. 282) Swan's Basic Formula

Swan (1956) aggregated the rate of saving to measure the annual addition to the capital stock as a ratio of saving to output: d/K=sY. Assuming that capital and labor are paid relative contribution to production growth, Swan (1956) affirmed, "The relative shares of total profits and total wages in income are constants, given by the production elasticities [alpha] and [beta] (p. 335) where ([alpha], [beta]) measures, correspondingly, the relative contributions of capital (K) and labor (L) to production (Y) growth; furthermore, the ratios (Y/K, Y/L) measure the average product of capital and labor with which he arrives at the relative shares. Swan (1956) distinguishes between resources' relative contribution to production growth and the relative distribution of value (Villalobos, 2020). He makes it clear that the share of resources on production growth depends on its contribution; "the forces of perfect competition drive the rate of profit or interest [r] and the (real) wage rate [w] into equality with the marginal productivities of capital and labor, derived from the production function" (Swan, 1956, p. 335).

Swan's basic formula of production growth surges from the definition of marginal products; for labor [[partial derivative]Y/[partial derivative]L=[beta] Y/L [therefore] [y.sub.L]=[beta]n] and for capital [[partial derivative]Y/[partial derivative]K=[alpha] Y/K [therefore] [y.sub.K]] = [alpha]k] where (k, n) represent the relative rates of capital and labor increment, so that the relative rates of capital and labor contribution to production growth define the relative rate of production growth (Swan, 1956):

y = ([y.sub.K], [y.sub.L]) = ([alpha]k, [beta]n) (1)

which in terms of income is equals to:

y = (sr, nw) (2)

Solow's Fundamental Equation

Solow assumes that 1. There is only one commodity, output as a whole, whose rate of production is designate Y(t), where (t) represents time, in each instant of production. 2. Part of each instant's output is consumed and the rest is saved and invested. 3. The fraction of output saved is a constant s, so that the rate of saving is sY(t). 4. The community'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, so we have the basic identity at every instant of time: [??] = sY, as found in Swan (1956). 5. Output is produced with capital and labor, whose rate of input is L(i) = [L.sub.0] [e.sup.nt]. Solow's production function is Y = F(K, L) where "constant returns to scale seems the natural assumption to make in a theory of growth" Solow (1956, pp. 66-67). Hence, dK=sY, so that:

k = s Y/K (3)

Inserting Y=F(K,L) into the previous equation will result in:

[??] = sF(1, 1/r)-n (4)

By the derivative of the ratio capital/labor (r=K/L)such that (r=k-n) it states that the relative rate of growth of resource composition (r) is the difference of the relative rate of growth in capital (k) and in labor (n). Inserting equation (4) into this definition results:

f =sF(l,l/r)-n (5)

This is Solow's fundamental equation in terms of relative rate of change on resource composition; it is found by multiplying both sides of equation (5) by (r) and "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 a function of capital per worker [...] [It] 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" (1956, p. 69).

Equation (5) can be simplified in terms of the average capital contribution to production growth ([[bar.y].sub.K].

[??] = s[[bar.y].sub.K]--n (6)

If the relative contribution of capital to production growth is [dY/dK=a Y/K], from equation (3) we obtain Y/K = k/s, and thus, [[partial derivative]Y/[partial derivative]K = [alpha] k/s] [therefore] [y/k = [alpha] k/s K/Yj. Due to (k/s K/Y=l), then [y.sub.K] = [alpha]k. From equation (6), ([??] = k-n) and let [[bar.y].sub.K] = F(1, 1/r), and then, k-n = s[[bar.y].sub.K] [therefore] k = [[bar.y].sub.K]K and [y.sub.K] = [alpha]s[[bar.y].sub.K]. Labor relative contribution to production growth is [[partial derivative]Y/[partial derivative]L = [??] Y/L [therefore]...

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