Factor shares explained

In macroeconomics, factor shares are the share of production given to the factors of production, usually capital and labor. This concept uses the methods and fits into the framework of neoclassical economics.

Derivation

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

Y = F(K,L)

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

\pi = max _{\{K,L\} } F(K,L)*P - (rK + wL)\,

where

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:

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

Now we can write the expenditure allocated to labor as

wL = D _{L[F(K,L)]*L}

and to capital as

rK = D _{K[F(K,L)]*K}

So the factor share devoted to labor is:

wL/Y = D _{L[F(K,L)]*L/F(K,L)}

and the factor share devoted to capital is:

rK/Y = D _{K[F(K,L)]*K/F(K,L)}

Notes and References

1. Web site: What is the Solow Growth Model?. Corporate Finance Institute. 22 October 2021.