Academic journal article Economic Quarterly - Federal Reserve Bank of Richmond

# Algebraic Production Functions and Their Uses before Cobb-Douglas

Academic journal article Economic Quarterly - Federal Reserve Bank of Richmond

# Algebraic Production Functions and Their Uses before Cobb-Douglas

## Article excerpt

Fundamental to economic analysis is the idea of a production function. It and its allied concept, the utility function, form the twin pillars of neoclassical economics. Written

the production function relates total product P to the labor L, capital C, land T (terrain), and other inputs that combine to produce it. The function expresses a technological relationship. It describes the maximum output obtainable, at the existing state of technological knowledge, from given amounts of factor inputs. Put differently, a production function is simply a set of recipes or techniques for combining inputs to produce output. Only efficient techniques qualify for inclusion in the function, however, namely those yielding maximum output from any given combination of inputs.

Production functions apply at the level of the individual firm and the macro economy at large. At the micro level, economists use production functions to generate cost functions and input demand schedules for the firm. The famous profit-maximizing conditions of optimal factor hire derive from such microeconomic functions. At the level of the macro economy, analysts use aggregate production functions to explain the determination of factor income shares and to specify the relative contributions of technological progress and expansion of factor supplies to economic growth.

The foregoing applications are well known. Not so well known, however, is the early history of the concept. Textbooks and survey articles largely ignore an extensive body of eighteenth and nineteenth century work on production functions. Instead, they typically start with the famous two-factor Cobb-Douglas version

That version dates from 1927 when University of Chicago economist Paul Douglas, on a sabbatical at Amherst, asked mathematics professor Charles W. Cobb to suggest an equation describing the relationship among the time series on manufacturing output, labor input, and capital input that Douglas had assembled for the period 1889-1922.

The resulting equation

exhibited constant returns to scale, assumed unchanged technology, and omitted land and raw material inputs. With its exponents k and 1- k summing to one, the function seemed to embody the entire marginal productivity theory of distribution. The exponents constitute the output elasticities with respect to labor and capital. These elasticities, in competitive equilibrium where inputs are paid their marginal products, represent factor income shares that just add up to unity and so exhaust the national product as the theory contends.

The function also seemed to resolve the puzzling empirical constancy of the relative shares. How could those shares remain unchanged in the face of secular changes in the labor force and the capital stock? The function supplied an answer. Increases in the quantity of one factor drive down its marginal productivity and hence its real price. That price falls in the same proportion as the increase in quantity so that the factor's income share stays constant. The resulting share terms k and 1- k are fixed and independent of the variables P, L, and C. It follows that even massive changes in those variables and their ratios would leave the shares unchanged.

From Cobb-Douglas, textbooks and surveys then proceed to the more exotic CES, or constant elasticity of substitution, function They observe that the CES function includes Cobb-Douglas as a special case when the elasticity, or flexibility, with which capital can be substituted for labor or vice versa approaches unity.

Finally, the texts arrive at functions that allow for technological change. The simplest of these is the Tinbergen-Solow equation. It prefixes a residual term e^sup rt^ to the simple Cobb-Douglas function to obtain

This term captures the contribution of exogenous technological progress, occurring at trend rate r over time t, to economic growth. Should new inventions and innovations fail to materialize exogenously like manna from heaven, however, more complex functions are available to handle endogenous technical change. …

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