How to Calculate Output From a Production Function
A production function gives the output a firm can make from its inputs. The general form Q = f(L, K) becomes computable once written as Cobb-Douglas: Q = A × L^α × K^β.
The Production Function formula
Calculator
Enter technology, labor, capital and the two exponents to get output, marginal products and returns to scale.
Total factor productivity, the multiplier that scales the whole function.
Units of the labor input the firm employs this period.
Machines, buildings and equipment in use alongside that labor.
Labor's output elasticity. An exponent of 0.5 is a square root.
Capital's output elasticity. Add it to α to read returns to scale.
These inputs and this technology produce 500 units of output.
- Average product of labor (APL)
- 20
- Marginal product of labor (MPL)
- 10
- Marginal product of capital (MPK)
- 2.5
- Returns to scale
- Constant returns to scale
- Output if both inputs double
- 1,000
Output per worker is 20 units.
One more worker adds about 10 units, and the exponent on labor is what holds MPL below the average.
One more unit of capital adds about 2.5 units of output.
The exponents sum to 1, and that sum alone settles what happens when every input grows together.
Doubling labor and capital together takes output to 1,000 units, which is the returns-to-scale answer in numbers.
How to calculate Production Function, step by step
- 1List the inputs and the technology term. L is the quantity of labor, K is the quantity of capital, and A is the technology or total factor productivity term that scales the whole function.
- 2Pick a functional form. Q = f(L, K) is a shell that says output depends on inputs without saying how. The Cobb-Douglas form Q = A × L^α × K^β is the specification that lets you put numbers in.
- 3Raise each input to its exponent. Work out L^α and K^β separately. An exponent of 0.5 is a square root, so those two terms are easy to check by hand.
- 4Multiply the pieces together. Total output Q = A × L^α × K^β. Multiply the two input terms, then scale the result by A.
- 5Read the marginal and average products. Average product of labor is Q ÷ L. Marginal product of labor is α × Q ÷ L, and marginal product of capital is β × Q ÷ K.
- 6Add the exponents for returns to scale. If α + β = 1 returns to scale are constant, if the sum is above 1 they are increasing, and if it is below 1 they are decreasing.
Worked example: Production Function
A firm produces with Q = 10 × L^0.5 × K^0.5 and uses 25 workers and 100 units of capital. The square root of 25 is 5 and the square root of 100 is 10, so Q = 10 × 5 × 10 = 500 units. Average product of labor = 500 ÷ 25 = 20 units per worker, while MPL = 0.5 × 500 ÷ 25 = 10 units and MPK = 0.5 × 500 ÷ 100 = 2.5 units. The exponents sum to 1, so returns to scale are constant, and doubling both inputs to 50 workers and 200 units of capital lifts output to 1,000 units, exactly twice as much.
Production Function questions
What is the difference between a production function and a cost function?
A production function maps input quantities to output in physical units, such as workers and machines to bags of coffee. A cost function maps output back to dollars, and it is derived from the production function once input prices are known.
Why does the marginal product of labor equal α × Q ÷ L?
Differentiating Q = A × L^α × K^β with respect to L gives α × A × L^(α − 1) × K^β, which is the same thing as α × Q ÷ L. That is also why the labor exponent doubles as labor's share of output in the Cobb-Douglas case.
Do the two exponents have to add up to 1?
No. Their sum is exactly what tells you the returns to scale, so it can sit above or below 1. A sum of 1 means doubling every input doubles output, and each input can still show diminishing marginal returns as long as its own exponent lies between 0 and 1.
Get AP Econ exam tips in your inbox
Occasional emails with study tips, new interactive graphs, and exam-season reminders. Free, no spam.
No spam. Unsubscribe anytime. Read our privacy policy.
Keep track of what you have studied
A free EconLearn account adds progress tracking, your quiz history, and achievements. Studying here is free either way, and there is nothing to pay for as a student.
Create a free accountAlready have one? Sign in
Last updated