Fundamentals of Business Economics and Management · Utility, Wealth, Production

Production Function and Laws of Returns Explained

Q: What is the production function in simple words?

It is a rule that links the quantity of inputs to the maximum output you can produce, given the technology. It is usually written Q = f(L, K). A better technology changes the function itself.

Q: What is the difference between returns to scale and returns to a factor?

Returns to a factor (law of variable proportions) is a short-run idea where only one input changes and others stay fixed. Returns to scale is a long-run idea where all inputs change in the same proportion.

Q: What are the three stages of the production function?

In Stage I, AP rises up to its maximum. In Stage II, AP and MP fall but MP stays positive, and TP keeps rising to its maximum. In Stage III, MP is negative and TP falls. Producers operate in Stage II.

Q: Why does diminishing return set in?

The fixed factor gets used too heavily. Each extra unit of the variable factor has less of the fixed factor to work with. So the extra output per unit starts to fall.

Updated 10 October 2026 · Fact-checked

A production function shows the maximum output you can get from given inputs. The laws of returns describe how output changes when inputs change. The law of variable proportions applies in the short run, when one input varies. Returns to scale apply in the long run, when all inputs change together.

Understand Production Function and Laws of Returns

A production function is a technical relationship between inputs and output. It is written as Q = f(L, K, ...), where Q is output, L is labour and K is capital. It shows the maximum output possible from each combination of inputs, given the technology.

Economists split time into two periods. In the short run, at least one input is fixed, such as land or machinery. Output changes only by changing variable inputs like labour. In the long run, all inputs can be changed, so the scale of production can change.

The law of variable proportions (law of returns to a factor) applies in the short run. You add more units of a variable input to a fixed input. At first, output rises faster and faster. Then it rises, but more slowly. Finally it stops rising and may fall. Think of a small shop with one counter: the first few extra workers help a lot, but too many workers get in each other's way.

This law has three stages. In Stage I, marginal product (MP) rises, then falls, while average product (AP) rises until it meets MP. In Stage II, both AP and MP fall but MP stays positive, so total product (TP) keeps rising until MP is zero. In Stage III, MP is negative and TP falls. A rational producer works in Stage II.

Returns to scale apply in the long run. Here all inputs rise by the same proportion. If output rises by a larger proportion, there are increasing returns to scale. If by the same proportion, constant returns. If by a smaller proportion, decreasing returns. The key difference: returns to a factor change one input; returns to scale change all inputs together.

Key formulas to remember

Production function: Q = f(L, K, ...); Output depends on inputs and technology. Short run: some inputs fixed. Long run: all variable.
Average product: AP = TP ÷ units of variable input; Output per unit of the variable factor.
Marginal product: MP = change in TP ÷ change in variable input = TPn − TPn−1; Extra output from one more unit of the variable input.
Total product from MP: TP = Σ MP; TP is the running total of all MP values.
Relation of AP and MP: MP > AP: AP rises. MP = AP: AP is at maximum. MP < AP: AP falls.; MP cuts AP at AP's highest point.
Three stages of variable proportions: Stage I: from start to where AP is maximum. Stage II: from max AP to where MP = 0. Stage III: MP < 0.; TP is maximum where MP = 0.
Returns to scale: Increasing: output rises by a larger %. Constant: same %. Decreasing: smaller %.; Compare % rise in output with % rise in all inputs.

How to solve Production Function and Laws of Returns questions

Use this method for table-based or concept questions on production and returns.

1Decide the time period. One input fixed means short run and law of variable proportions. All inputs changing means long run and returns to scale.
2For a table question, find MP as the difference between successive TP values. Find AP as TP divided by units of input.
3Mark where MP is highest, where AP is highest (MP equals AP there), and where MP becomes zero or negative.
4Assign stages: Stage I up to maximum AP, Stage II from there to MP = 0, Stage III after MP turns negative.
5For returns to scale, work out the % rise in inputs and the % rise in output. Compare them.
6Check the answer options against the definitions and remove those that mix up the two laws.

Quickest way: Three-point scan for stage and scale questions

When to use it: Use it for any MCQ that gives a table of TP, AP or MP, or a scale-up statement.

TP maximum means MP = 0. This marks the end of Stage II.
AP maximum means MP = AP. This marks the end of Stage I.
For scale questions, just compare the multipliers: inputs doubled, output more than doubled means increasing returns.
If the question says one factor is fixed, rule out every returns to scale option at once.

Common mistakes in Production Function and Laws of Returns

Treating returns to scale and the law of variable proportions as the same thing.
Both use the words increasing and diminishing returns.
Fix: Ask: is one input changing, or all? One input is variable proportions (short run). All inputs is returns to scale (long run).
Saying TP falls when MP starts to fall.
Students confuse a falling rate with a falling total.
Fix: TP keeps rising as long as MP is positive. TP falls only when MP is negative.
Placing the end of Stage I where MP is highest.
Students link Stage I with increasing returns only.
Fix: Stage I ends where AP is highest, which is where MP equals AP. MP peaks earlier.
Choosing Stage I or Stage III as the rational stage.
Students look for the stage with the highest AP or the largest TP.
Fix: A producer works in Stage II. In Stage III more input reduces output, and in Stage I the fixed factor is underused.
Calling a doubling of one input with output less than doubled 'decreasing returns to scale'.
The scale idea is applied without checking what changed.
Fix: If only one input was changed, it is diminishing returns to a factor, not returns to scale.

Worked examples

Example 1

A farm has a fixed plot. Total product with 1, 2, 3, 4, 5 workers is 10, 30, 48, 56, 56 quintals. Find MP and AP at each level. Where does Stage I end, at which number of workers is MP zero, and which range of workers is Stage II?

Show the solution

MP of 1st worker = 10.
MP of 2nd = 30 − 10 = 20.
MP of 3rd = 48 − 30 = 18.
MP of 4th = 56 − 48 = 8.
MP of 5th = 56 − 56 = 0.
AP = TP ÷ workers: 10 ÷ 1 = 10, 30 ÷ 2 = 15, 48 ÷ 3 = 16, 56 ÷ 4 = 14, 56 ÷ 5 = 11.2.
MP is highest (20) at the 2nd worker. AP rises from 10 to 15 at the 2nd worker because MP (20) is above the earlier AP of 10.
At the 3rd worker, MP (18) is still above the earlier AP of 15, so AP rises again to 16. This is the single highest AP in the table. At the 4th worker MP (8) is below AP (16), so AP falls to 14. MP cuts AP between the 3rd and 4th worker, so AP peaks at the 3rd worker. Stage I ends there.
MP is zero at the 5th worker, so TP is at its maximum of 56. Stage II runs from the 3rd worker (maximum AP) to the 5th worker (MP = 0).
MP is never negative in this table, so Stage III does not appear. It would begin only if a 6th worker reduced TP.

Answer: MP is 10, 20, 18, 8, 0 and AP is 10, 15, 16, 14, 11.2. AP has its single maximum of 16 at the 3rd worker, so Stage I ends at the 3rd worker. MP is zero at the 5th worker, where TP is maximum at 56. Stage II covers 3 to 5 workers, so a rational farm would hire within that range. Stage III (MP negative) would begin only beyond 5 workers.

Example 2

A factory in Pune doubles its labour, machines and raw material. Output rises from 1,000 units to 1,800 units. Which type of returns is this?

Show the solution

All inputs are doubled, so this is a returns to scale question.
Inputs rise by 100%.
Output rises from 1,000 to 1,800, an increase of 800 units, which is 80%.
Output rises by a smaller proportion (80%) than inputs (100%).

Answer: Decreasing returns to scale.

Exam tips

First check whether the question says 'fixed factor' or 'all inputs'. This alone separates the two laws.
Remember: MP cuts AP at AP's maximum, and TP peaks when MP is zero. Many MCQs test just these two points.
Learn the three stages with their boundaries, not just their names.
For returns to scale questions, compare percentages, not absolute numbers.
Know the assumptions of the law of variable proportions: technology is constant, and the variable units are identical.

Practice questions from Utility, Wealth, Production

Production Function and Laws of Returns in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Production Function and Laws of Returns: frequently asked questions

What is the production function in simple words?

It is a rule that links the quantity of inputs to the maximum output you can produce, given the technology. It is usually written Q = f(L, K). A better technology changes the function itself.

What is the difference between returns to scale and returns to a factor?

Returns to a factor (law of variable proportions) is a short-run idea where only one input changes and others stay fixed. Returns to scale is a long-run idea where all inputs change in the same proportion.

What are the three stages of the production function?

In Stage I, AP rises up to its maximum. In Stage II, AP and MP fall but MP stays positive, and TP keeps rising to its maximum. In Stage III, MP is negative and TP falls. Producers operate in Stage II.

Why does diminishing return set in?

The fixed factor gets used too heavily. Each extra unit of the variable factor has less of the fixed factor to work with. So the extra output per unit starts to fall.