I used excel to construct a visualization of the Cobb-Douglas production function (explicitly presented in the graph titles).
The production function expresses the output of any given firm as a function of two inputs (labor and capital) and parameters (alpha and beta). When the sum of alpha and beta equals 1, it can be shown that they represent labor’s and capital’s share of output, respectively .
This condition also means that the firm is operating with constant returns to scale. When a firm expands its inputs by a certain percent, output increases by the same amount.
We can plot each amount of labor, capital, and output in x-y-z space if we specify alpha and beta.
We do this for labor and capital ranging from 1 to 100 and alpha=beta=.5.The result is the Cobb-Douglas production surface with capital and labor each comprising 50% of the input.
Notice that the lines that separate the differently colored areas are equally spaced. This is a property of increasing returns to scale.
When labor and capital expand, the level of utility rises proportionately at a constant rate.
It is also useful to view the surface from above.
Those L-shaped curves are called isoquants or rectangular hyperbolas. They represent the different combinations of labor and capital that produce the same (“iso”) quantity of output (“quant”). For example, L=4 and K=4, L=16 and K=1, and L=1 and K=16 all produce O=4 level of output. The L-shaped curve simply connects these points for Q=4. As the curves move northwest, they plot the combinations for higher values of output.
A particular firm’s efficient combination depends on the price of its inputs, but that’s for another day.