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 a 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