At some point when studying economics, you’ve likely studied production functions, which detail how firms use inputs like labor and capital to produce goods and services. You’ve also studied aggregate production functions that detail inputs versus outputs on a national scale. We can thus model shifts and changes using these production functions.
The Cobb-Douglas model consists of production functions that represent how labour and capital translate as inputs into real GDP. The general form of the production function is as follows:
Y = AKᵅNᵝ
Cobb-Douglas Production Function
Whereas:
Y is the total quantity of output produced (total output).N is the quantity of labour (either measured in # of workers or hours).A is the level of technology (or productivity; generally a measure of efficiency).K is the quantity of capital.α (alpha) and β (beta) are positive constants (not fixed, but rather parameters) that represent the output elasticities of labour and capital, meaning how much output changes when labor or capital changes. These values typically sum to 1 for reasons we’ll see later.Since α and β sum to 1, α + β = 1, then β =1 — α. So, we can simplify our equation to: Y = A(Kᵅ)(N¹⁻ᵅ).
We can first test whether there are constant returns to scale by multiplying Y by a fixed amount, let’s say x:
xY = A(xKᵅ)(xN¹⁻ᵅ) = (A)(xᵅ)(Kᵅ)(x¹⁻ᵅ)(N¹⁻ᵅ) = (xᵅ)(x¹⁻ᵅ)(A)(Kᵅ)(N¹⁻ᵅ)
Note that the back half of the expanded equation is what we started with — A(Kᵅ)(N¹⁻ᵅ). Thus, since (xᵅ)(x¹⁻ᵅ) cancel down to x. So, the Cobb-Douglas does have constant returns to scale.
For our second calculation, we’ll look at the marginal product of labor (MPN) and marginal product of capital (MPK).
The marginal product of capital must equal the first partial derivative of the production function since doing so gives you the increase in output given a one-unit increase in the input. For sake of simplicity, we’ll assume that technology = 1, so we can proceed with the general production function Y = (Kᵅ)(N¹⁻ᵅ).
MPN = dY/dN = d(Kᵅ)(N¹⁻ᵅ)/dN = (1-α)N⁻ᵅKᵅ
So, the marginal unit of labor gets paid (1-α)N⁻ᵅKᵅ. Assuming we have N units of labor, total payments can be described as follows:
MPN(N) = ((1-α)N⁻ᵅKᵅ)N
Since total income in the economy is real GDP, in turn given by our overall production function, the total share of total income paid to labor is:
1 – α = (((1-α)N⁻ᵅKᵅ)N)/Y
So, basically, the share of income paid to labour equals the total payment to laborers divided by the real GDP. If total payments is $100 and GDP is $200, then 1 – α = .5 and α = .5, and so on.
Output Per Worker in the Cobb-Douglas Production Function
Output per worker is the focus of many undergrad applications of the Cobb-Douglas, so we’ll give that some attention here. Output per worker is simply defined as Y/N, or Y (being total output, or real GDP) divided by N (the quantity of laborers). We in turn simply divide the right side of the production function by N and can then simplify.
Y = (Kᵅ)(N¹⁻ᵅ)Y/N = ((Kᵅ)(N¹⁻ᵅ))/NY/N = (Kᵅ)(N⁻ᵅ)Y/N = Kᵅ/NᵅY/N = (K/N)ᵅ
So, the output per worker is equivalent to the capital per worker (K/N) raised to alpha. That’s a pretty easy (thankfully!) formula to remember.
Output Per Worker in the Cobb-Douglas Production Function