# Factor shares and Cobb-Douglas Factor shares

Factor shares are how much income a factor of production (for example capital or workers) receive as a proportion of income.

Suppose we have the following simple economy which produces only beer:

• $100 of income being produced by the economy • 10 workers in the economy • 10 units of capital (could be machines for bottling etc) • the wage rate is$7 per worker
• the rental rate of capital is $3 per machine The factor share is defined as: (total factor payments ÷ total income of the economy). This means that the factor share for workers is as follows:$7 * 10 ÷ $100 =$70/$100 = .7 = 70% this means that the factor share for workers (labor) is 70%. The factor share for capital can be worked out as follows:$3 * 10 ÷ $100 =$30/\$100 = .3 = 30%

This means that the factor share for capital is 30%.

Factor share for Cobb Douglas production function

This section is a little bit more advanced and will assume that you are familiar with some calculus.

Consider the following Cobb Douglas production function:

Y = L^a *K^(1-a)     0 < a < 1

The factor share for L (labor) is defined:

w * L ÷ Y

The factor share for K (capital) is defined:

r * K ÷ Y

The factor shares are solved by maximising the profit of the cobb douglas production function such that:

Profit = L^a * K^(1-a) - w*L - R*k

dprofit/dl = a*L^(a-1) * K^(1-a) - w = 0 -> MPL = w

dprofit/dk = (1-a)*L^(a) * K^(1-a-1) - w = 0 -> MPK = r

(MPL is marginal product of labor and MPK is marginal product of capital)

From our definition of factor share we get that the factor share of income is:

MPL * L ÷ Y

->  a*[L^(a-1) * K^(1-a) * L] ÷ L^a *K^(1-a)

The part in the [] becomes Y which cancels out with the Y on the denominator since Y = L^a *K^(1-a)

Thus we are just left with as our factor share for labor

For capital we have:

MPK * K ÷ Y

->  (1-a)*[L^(a) * K^(1-a-1) * K] ÷ L^a *K^(1-a)

The part in the [] becomes Y again, which cancels out with the Y on the denominator since Y = L^a *K^(1-a)

Thus we are just left with (1-a) as our factor share for labor