Capital in the Twenty-First Century

by Thomas Piketty

The case of an elasticity of substitution exactly equal to one corresponds to the so-called Cobb-Douglas production function, named for the economists Charles Cobb and Paul Douglas, who first proposed it in 1928. With a Cobb-Douglas production function, no matter what happens, and in particular no matter what quantities of capital and labor are available, the capital share of income is always equal to the fixed coefficient α, which can be taken as a purely technological parameter.17 For example, if α = 30 percent, then no matter what the capital/income ratio is, income from capital will account for 30 percent of national income (and income from labor for 70 percent). If the savings rate and growth rate are such that the long-term capital/income ratio β = s / g corresponds to six years of national income, then the rate of return on capital will be 5 percent, so that the capital share of income will be 30 percent. If the long-term capital stock is only three years of national income, then the return on capital will rise to 10 percent. And if the savings and growth rates are such that the capital stock represents ten years of national income, then the return on capital will fall to 3 percent. In all cases, the capital share of income will be 30 percent.