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Business Economics · Theory of Production and Cost

Law of Variable Proportions for CA Foundation Business Economics

Updated 1 October 2026 · Fact-checked

The Law of Variable Proportions describes how output changes when you add more units of one variable input to fixed inputs. Output first rises at an increasing rate, then at a decreasing rate, and finally falls. To solve questions, track marginal product (MP) and total product (TP) and match them to the stage.

Understand Law of Variable Proportions

In the short run, at least one input is fixed, such as land, machinery or factory space. A firm can still change other inputs, such as labour. The Law of Variable Proportions studies what happens to output when you keep adding the variable input to the fixed one.

The ratio of variable input to fixed input keeps changing. That is why it is called variable proportions. It is a short-run law. Do not confuse it with returns to scale, where all inputs change together.

The law has three stages. This page uses the common textbook division based on MP. Stage I (increasing returns): MP rises, so TP increases at an increasing rate. The fixed factor is underused, and extra workers allow specialisation and better use of machines. Stage II (diminishing returns): MP falls but stays positive, so TP still rises, but at a decreasing rate. The fixed factor is now crowded. Stage III (negative returns): MP is negative, so TP falls. Too many workers get in each other's way.

The key turning points are easy to remember. TP is at its maximum when MP = 0. AP is at its maximum where MP = AP, because MP cuts AP at AP's highest point. Some texts use this AP peak, instead of the point where MP starts to fall, as the end of Stage I. Check which marker your question uses. A rational firm operates in Stage II. It never stays in Stage III because output falls even though it pays more for inputs.

The law usually assumes that technology is constant, that the variable units are alike, and that the variable input can be changed while at least one input stays fixed. If technology improves, the whole pattern shifts.

Key formulas to remember

Total product
TP = Σ MP (sum of marginal products up to that unit)
If MP is given for each unit, add them up to get TP.
Marginal product
MP = ΔTP ÷ ΔL = TPₙ − TPₙ₋₁
L is the variable input. ΔL is the change in the variable input.
Average product
AP = TP ÷ L
L is the number of units of the variable input.
Stage I
MP rising; TP rising at an increasing rate
In the MP-based textbook division, Stage I ends where MP is highest and starts to fall. Some texts instead end Stage I where AP is highest (MP = AP).
Stage II
MP falling but positive (MP > 0); TP rising at a decreasing rate
Ends where MP = 0 and TP is maximum. A rational firm operates here.
Stage III
MP < 0; TP falling
A firm avoids this stage.

How to solve Law of Variable Proportions questions

Use this method for any question on the three stages, whether it gives a table, a statement or a diagram.

  1. 1Confirm the question is short run: one or more inputs are fixed and only one input varies. If all inputs change, it is returns to scale.
  2. 2If a table is given, find MP for each unit as the change in TP. If MP is given, find TP by adding.
  3. 3Find AP as TP ÷ units of the variable input if the question needs it.
  4. 4Mark where MP is highest and starts to fall. Everything before that point is Stage I (this is the MP-based division). If the question uses the AP-based division, Stage I ends where AP is highest.
  5. 5Mark where MP becomes zero or negative. TP is highest at MP = 0. Stage III is where MP is negative.
  6. 6Everything between the first fall in MP and MP = 0 is Stage II (diminishing returns).
  7. 7Match the answer to the options. Check the wording: increasing, diminishing or negative returns, and whether TP or MP is being asked about.

Quickest way: Read the MP column only

When to use it: Use this for table-based MCQs where you have little time and negative marking applies.

  1. Look at the MP column. If it is not given, subtract consecutive TP values.
  2. Where MP goes up, mark Stage I. Where it first drops, Stage II starts (MP-based division). If the question says Stage I ends at the AP peak, compare MP with AP instead.
  3. Where MP hits 0, TP is at maximum and Stage II ends. Stage III is where MP is negative.
  4. Eliminate any option that says TP falls while MP is positive. This cannot be true.
  5. Eliminate any option saying MP is negative while TP is rising. This cannot be true either.
  6. If the stem asks about returns to scale or all inputs changing, skip the stage logic and choose the returns to scale option.

Common mistakes in Law of Variable Proportions

  • Saying diminishing returns means TP falls.

    The word diminishing sounds like a decline in total output.

    Fix: Diminishing returns means MP falls, not TP. TP keeps rising as long as MP is positive. TP falls only in Stage III.

  • Mixing up the law of variable proportions with returns to scale.

    Both use the words increasing and diminishing returns.

    Fix: Variable proportions is short run with one variable input and others fixed. Returns to scale is long run with all inputs changed in the same proportion.

  • Mixing up the two ways of marking the end of Stage I.

    Textbooks differ, and students blend the two definitions.

    Fix: The usual textbook division uses MP: Stage I is where MP rises, and Stage II starts when MP begins to fall. Some texts end Stage I at the AP peak (MP = AP). Read the question's wording and options to see which marker is used.

  • Thinking a firm may operate in Stage III.

    Students assume more input is always worth using.

    Fix: In Stage III, extra input reduces output and raises cost. A rational firm stops before it. It operates in Stage II.

  • Computing MP by dividing TP by units.

    AP and MP get confused.

    Fix: AP = TP ÷ units. MP = change in TP when one more unit is added.

  • Forgetting the assumption of constant technology.

    The assumptions are skipped as theory.

    Fix: State that technology and the quality of the variable input are unchanged. Better technology shifts the whole product curve and is not a case of this law.

Worked examples

Example 1

A firm adds labour to a fixed plant. TP for 1 to 6 workers is 10, 30, 60, 80, 90, 90. MP becomes zero at which worker?
(a) 4th (b) 5th (c) 6th (d) 3rd

Show the solution
  1. MP of 1st worker = 10.
  2. MP of 2nd = 30 − 10 = 20.
  3. MP of 3rd = 60 − 30 = 30.
  4. MP of 4th = 80 − 60 = 20.
  5. MP of 5th = 90 − 80 = 10.
  6. MP of 6th = 90 − 90 = 0.
  7. TP reaches its maximum of 90 at the 5th worker. The 6th worker adds nothing, so MP = 0 there.
  8. MP rises up to the 3rd worker (Stage I) and falls but stays positive for the 4th and 5th workers (Stage II).

Answer: (c) 6th worker. MP is zero at the 6th worker, while TP already reached its maximum of 90 at the 5th worker. Stage II ends here. Stage III would begin only if MP turned negative.

Example 2

In a table, MP is 8, 12, 10, 6, 2, −3 for units 1 to 6 of labour. In which unit does Stage III begin?
(a) 3rd (b) 4th (c) 5th (d) 6th

Show the solution
  1. Stage III is where MP is negative.
  2. Check each unit: MP is positive for units 1 to 5.
  3. MP of the 6th unit is −3, which is negative.
  4. So TP falls from the 6th unit onwards.

Answer: (d) 6th unit.

Example 3

When MP is positive but falling, which statement is correct?
(a) TP is falling (b) TP is rising at a decreasing rate (c) TP is rising at an increasing rate (d) TP is at its maximum

Show the solution
  1. Positive MP means each added unit adds to TP, so TP is rising.
  2. Falling MP means each addition is smaller than the last.
  3. So TP rises, but by smaller amounts each time.
  4. This is Stage II, diminishing returns.

Answer: (b) TP is rising at a decreasing rate.

Exam tips

  • Questions usually give a TP or MP table. Compute MP first. It takes seconds and settles the stage.
  • Remember three markers: MP rising is Stage I, MP falling but positive is Stage II, MP negative is Stage III.
  • Watch the stem for the words short run and fixed factor. If you see all inputs increased in the same proportion, the question is on returns to scale.
  • Do not guess when two options look close. With 0.25 negative marking, use elimination: TP cannot fall while MP is positive.

Practice questions from Theory of Production and Cost

Law of Variable Proportions: frequently asked questions

What are the three stages of the Law of Variable Proportions?

Stage I is increasing returns, where MP rises. Stage II is diminishing returns, where MP falls but stays positive. Stage III is negative returns, where MP is below zero and TP falls.

Why does marginal product fall?

The fixed factor has to be shared among more and more units of the variable factor. Each new worker gets less of the fixed factor to work with, such as less machine time or space. So each addition raises output by less than the previous one.

What is the difference between the Law of Variable Proportions and returns to scale?

The Law of Variable Proportions is a short-run law: one input is varied and the others stay fixed. Returns to scale is a long-run concept: all inputs are changed in the same proportion. The first is about factor proportions and the second is about the scale of production.

Where does a rational firm operate?

A rational firm operates in Stage II. In Stage III, extra inputs reduce output, so no firm wants to be there. In Stage I, the fixed factor is still underused, so adding more of the variable input is still worthwhile.