Factor shares

In exogenous growth models, the production function can be represented by:^[1]

${\displaystyle Y=F(K,L)\,}$

with Y total production, K capital, and L labor. So a representative agent will attempt to maximize a profit function:^[2]

${\displaystyle \pi =max_{\{K,L\}}F(K,L)*P-(rK+wL)\,}$

where ${\displaystyle rK+wL\,}$ 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:

${\displaystyle w=D_{L}[F(K,L)]\,}$ and ${\displaystyle r=D_{K}[F(K,L)]\,}$.

Now we can write the expenditure allocated to labor as ${\displaystyle wL=D_{L}[F(K,L)]*L\,}$

and to capital as ${\displaystyle rK=D_{K}[F(K,L)]*K\,}$

So the factor share devoted to labor is:

${\displaystyle wL/Y=D_{L}[F(K,L)]*L/F(K,L)\,}$

and the factor share devoted to capital is:

${\displaystyle rK/Y=D_{K}[F(K,L)]*K/F(K,L)\,}$