Factor-Intensive Production Function

A pervasive concept in economic thinking is factor intensity, the ratio of capital to labor in production. The neoclassical growth model of Solow concerns the determination of the long-run capital/labor ratio and how it sets the wage and the interest rate and consumption per worker (Solow, R. M. (1956, February). A contribution to the thory of economic growth. Quarterly Journal of Economics, 70 (1), 65–94). The Heckscher-Ohlin approach to international trade posits a key role for the capital/labor ratio and how it differs between sectors in the economy. Assuming constant returns to scale, Solow models production by the factor-intensive production function: output per worker is a nondecreasing, concave function of the capital/labor ratio. Epi-multiplication of the factor-intensive production function recovers the original production function. The conjugate of the factor-intensive production function is the factor-price frontier, the combinations of capital cost and labor cost such that the minimum cost of production is one. The conjugate evaluated at the capital cost is the negative of the labor cost. In combination the factor-intensive production function and the factor-price frontier are valuable for economic applications. Below an application of this relationship is the golden rule of saving, the saving rate that maximizes long-run per capita consumption in the neoclassical one-sector growth model (Phelps, E. (1961, September). The golden rule of accumulation: A fable for growthmen. American Economic Review, 51 (4), 638–643). Choosing the optimum saving rate reduces to the evaluation of the conjugate of the factor-intensive production function. Per capita consumption is the negative of the conjugate evaluated at the rate of population growth, and the steady-state capital/labor ratio is a subgradient. For a competitive economy in market equilibrium, that the conjugate is the factor-price frontier links the golden rule of saving to the real interest rate and the real wage. The golden rule of saving is achieved if the real interest rate equals the rate of population growth and the real wage equals per capita consumption. Saving equals profit.