Factor shares

Derivation

In exogenous growth models, the production function can be represented by:

Y = F ( K , L )

with Y total production, K capital, and L labor. So a representative agent will attempt to maximize a profit function:

π = m a x { K , L } F ( K , L ) ∗ P − r K − w L

where r K + w L is the cost to the firm, r the rental rate of capital, w the wage rate for labor, and P is the price of the output.

As in microeconomics supply and demand models, first-order conditions that the derivative of this function with respect to capital and labor will be zero at the functions maximum. Thus (assuming P = 1) we can calculate the wages and the rental rate of capital:

w = D L [ F ( K , L ) ] and r = D K [ F ( K , L ) ] .

Now we can write the expenditure allocated to labor as w L = D L [ F ( K , L ) ] ∗ L

and to capital as r K = D K [ F ( K , L ) ] ∗ K

So the factor share devoted to labor is:

w L / Y = D L [ F ( K , L ) ] ∗ L / F ( K , L )

and the factor share devoted to capital is:

r K / Y = D K [ F ( K , L ) ] ∗ K / F ( K , L )