The Cobb-Douglas function has many applications in economics; from being a well-behaved preference in microeconomics to a production function in macroeconomics. It is named after Paul Douglas, an American Congressmen who was researching labour and capital shares and asked Charles Cobb, a mathematician, for help in formulating this into a function. In this article we will explore its use as a production function.
In its simplicity, a CD (Cobb-Douglas) function is just a function. A function, in mathematical jargon, transforms an input into a single output: it is a one-to-one mapping. For example Y=2X is a simple function. X is the independent variable and Y is the dependent variable, because Y is determined by whatever the value of X is. If we say that X is 2 then Y has to be 4. Because we have been given an input (X=2) and we have a single output (Y=4) then Y=2X can be described as a function. We can make our function more complex by adding different variables, or operators. We could make our function Y=2X + 5, or Y=XZ so Y is now a function of two variables.
A Cobb-Douglas Function takes the form of Q=KαLβ where Q=output, K=capital, L=labour, and alpha and beta are used to represent input shares of capital and labour respectively. In this form we have used CD as a production function. If we are given values for K,α,L,β (our independent variables) then we can calculate a value for output, our dependent variable. Note: sometimes Y is used instead of Q, this effectively the same thing since in economics output has to equal input so Q and Y can be used interchangeably.
It makes sense that this is a production function; the amount of product a firm can make depends on the amount of capital it uses (this will include things like raw materials alongside the more conventional machines and factories) and the amount of labour which utilises the labour to produce goods. Alpha is simply the percentage of capital I use in my production process, whilst beta is the percentage of labour used. If I spend £50 on capital and £50 on labour then the total cost of my inputs is £100 and alpha and beta both equal 0.5 (50%).
Returns to Scale
The returns to scale from my production is simply how much output I get from the amount of inputs I use. If I double the number of inputs, what will happen to the amount of output? Intuitively, we would assume that output would double. If this were the case then we have a production function which exhibits constant returns to scale. Simply put this means that if I double the inputs I double the output. If we double the inputs but get more than double the output, we say we have increasing returns to scale. This is a bit more unusual but could happen when there are economies of scale to be had. Finally, if I double inputs but get less than double output we have the situation of decreasing returns to scale.
If given a CD with input shares we can calculate what will happen if we double output:
Q = K0.2L1
If we multiply each side by a and expand, we can find out if we have increasing, decreasing or constant returns to scale:
aQ = (aK)0.2(aL)1
aQ = a0.2*a1*K0.2*L1
But we know that K0.2*L1 equals Q
aQ = a0.2*a1*Q
Now, using the power of indices, we can add 0.2 and 1:
aQ = a1.2*Q
We know that any positive value of a to the power of 1.2 is going to be greater than a, therefore this function must exhibit increasing returns to scale.
Increasing Returns to Scale: α + β > 1
Constant Returns to Scale: α + β = 1
Decreasing Returns to Scale: α + β < 1