Production function is a mathematical function that links the quantity of output with the input used in the production process. Usually, economists assume capital and labor are the only production inputs.
The function represents the output limit that producers can produce from any possible input combination. The function is useful to determine how much output they should produce, given the price of a product. And, to achieve that output, what input combinations they should use.
Next, economists distinguish the short run and the long run. In the short run, one factor of production is considered fixed, usually capital. Meanwhile, in the long run, all factors of production are variable.
The general equation for the production function is Q = F (K, L), where Q is the quantity of output, L is labor, and K is capital.
Let’s take a simple production function as an example.
Q = K + L
From the equation, the company can produce an output of 5 units using 2 units of capital and 3 units of labor. When labor is increased to 5 units, output increases to 7 units (2 units increase). Furthermore, when capital is increased to 5 units, output increases by 8 units (3 units increases).
From that function, output increases proportionally with each additional input unit. In economics, we call it a constant return scale.
The next is the Cobb-Douglas function.
Y = AKα Lβ
Where A represents the total factor productivity (TFP), which measures factors outside of labor and capital that contribute to changes in output. Economists refer to it as a factor in technological improvement.
Meanwhile, alpha (α) refers to the output elasticity of capital. And, beta (β) represents the output elasticity of labor. The sum of alpha and beta is equal to 1 if it is constant returns to scale, less than 1 if it is decreasing returns to scale, and more than 1 if it is increasing returns to scale.
We can approximate the formula concerning its growth over time as:
∆Y/Y = α * ∆K/K + β * ∆L/L + ∆A/A
- ∆Y/Y = Growth rate in output
- ∆K/K = Growth rate of capital
- ∆L/L = Growth rate of labor
- ∆A/A = TFP growth
For example, if α = 0.5, an increase in capital use of 2% will cause about a 1% increase in output.