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Fundamentals of Business Economics and Management · Theory of Production

Law of Variable Proportions: Three Stages Explained

Updated 10 October 2026 · Fact-checked

The law of variable proportions says that when you add more units of one variable input to fixed inputs, total product first rises at an increasing rate, then at a decreasing rate, and finally falls. You solve questions by computing MP = change in TP, AP = TP ÷ units, and spotting the stage.

Understand Law of Variable Proportions

In the short run, some inputs are fixed, such as a factory building or a machine. Others, such as labour, can be changed. The law of variable proportions studies what happens to output when you keep adding units of the variable input to the fixed inputs.

Three measures describe this. Total product (TP) is the total output from all units of the variable input. Average product (AP) is output per unit of the variable input. Marginal product (MP) is the extra output from employing one more unit of the variable input.

Output does not rise in a straight line. At first the fixed inputs are under-used, so each new worker adds more than the last one. MP rises. After a point the fixed inputs become crowded, so each new worker adds less. MP falls. If you keep adding workers, MP can become zero and then negative.

This gives three stages. Stage I (increasing returns): TP rises at an increasing rate at first, and AP keeps rising until it reaches its single maximum. Stage I ends at that point, where MP = AP. MP reaches its own peak earlier, inside Stage I, and starts falling while AP is still rising. Stage II (diminishing returns): TP rises at a decreasing rate, AP and MP both fall, and Stage II ends where MP = 0 and TP is maximum. Stage III (negative returns): TP falls and MP is negative.

A rational producer operates in Stage II. In Stage I, fixed inputs are still under-used. In Stage III, an extra unit reduces output, so no one would hire it. Note that the law applies when technology is unchanged and the variable units are of equal quality.

Key formulas to remember

Average product
AP = TP ÷ L
L is the number of units of the variable input.
Marginal product
MP = TPn − TP(n−1) = ΔTP ÷ ΔL
Use the change in TP when one more unit is added.
Total product from MP
TP = Σ MP
TP at any level is the sum of all MPs up to that level.
MP and AP link
MP > AP: AP rises; MP = AP: AP is maximum; MP < AP: AP falls
MP cuts AP at its highest point.
Stage boundaries
Stage I ends at maximum AP; Stage II ends at MP = 0 (TP maximum); Stage III has MP < 0
Stage II is where a rational producer operates.

How to solve Law of Variable Proportions questions

Use this method for schedule, stage-identification and concept questions.

  1. 1Write down the units of variable input and the TP, AP or MP values given.
  2. 2Find the missing columns: MP = change in TP, AP = TP ÷ units, TP = running sum of MP.
  3. 3Find where AP is highest. Stage I ends there, where MP = AP.
  4. 4Find where MP becomes zero. TP is maximum there and Stage II ends.
  5. 5Mark any units with negative MP as Stage III.
  6. 6Match the stage to the question: increasing returns is Stage I, diminishing is Stage II, negative is Stage III.
  7. 7For concept questions, check the assumptions: short run, fixed inputs, same technology.

Quickest way: Scan MP and AP only

When to use it: Use this when a schedule is given and you need the stage or the maximum point.

  1. Compute MP for each row by subtracting the previous TP.
  2. Note where MP first starts falling. This is the peak of MP and it lies inside Stage I. Diminishing marginal returns begin here, but Stage II does not start yet.
  3. Look for the row where MP drops below AP. AP peaked just before that row, and that peak ends Stage I. Stage II starts after it.
  4. Look for MP = 0 or negative. TP peaks at MP = 0, which ends Stage II.
  5. Eliminate options that say TP is maximum when MP is positive, or that Stage II starts where MP is highest.

Common mistakes in Law of Variable Proportions

  • Saying Stage I ends where MP is maximum.

    Students link the end of increasing returns to the highest MP.

    Fix: MP peaks inside Stage I. Stage I ends where AP is maximum and MP = AP.

  • Saying TP is maximum when MP is negative.

    Students confuse falling MP with falling TP.

    Fix: TP is maximum when MP = 0. TP rises while MP is positive, even if MP is falling.

  • Calculating MP as TP ÷ units.

    AP and MP formulas get mixed up.

    Fix: AP = TP ÷ units. MP = change in TP for one extra unit.

  • Treating diminishing returns as falling TP.

    The word diminishing sounds like a decline in output.

    Fix: In Stage II, TP still rises but at a decreasing rate. TP falls only in Stage III.

  • Forgetting the assumptions.

    Students memorise stages without conditions.

    Fix: Remember: short run, at least one fixed input, unchanged technology, and equal quality of variable units.

  • Choosing Stage I or III as the rational stage.

    Students do not link stages to producer behaviour.

    Fix: A producer chooses Stage II, where MP is positive but falling and fixed inputs are fully used.

Worked examples

Example 1

A bakery uses one oven and varies workers. TP for 1 to 5 workers is 10, 26, 36, 42, 42 kg. Find MP for the 3rd worker and the stage at which the 5th worker is added.

Show the solution
  1. MP of the 3rd worker = TP3 − TP2 = 36 − 26 = 10 kg.
  2. MP of the 4th worker = 42 − 36 = 6 kg.
  3. MP of the 5th worker = 42 − 42 = 0.
  4. MP = 0 means TP is maximum, so the 5th worker is at the end of Stage II.

Answer: MP of the 3rd worker is 10 kg. At the 5th worker MP = 0, so TP is maximum and Stage II ends.

Example 2

A farm has fixed land. The MP of 1 to 5 workers is 6, 14, 13, 5, 2 quintals. Find the AP at 3 workers and say whether adding the 4th worker is in Stage I or II.

Show the solution
  1. TP at 1 to 5 workers = 6, 20, 33, 38, 40 quintals.
  2. AP at 3 workers = 33 ÷ 3 = 11 quintals.
  3. AP at 1, 2 and 3 workers = 6, 10, 11. AP at 4 workers = 38 ÷ 4 = 9.5, and at 5 workers = 40 ÷ 5 = 8.
  4. MP of the 3rd worker is 13, above AP 10 at 2 workers, so AP rises up to 3 workers and is maximum there.
  5. The 4th worker has MP = 5, below AP 11, so AP falls. Stage I has ended, and MP is still positive, so the 4th worker is in Stage II.

Answer: AP at 3 workers is 11 quintals. The 4th worker is in Stage II, since MP is positive and below AP.

Exam tips

  • Practise building MP and AP columns from a TP schedule quickly; most numerical MCQs need only one or two values.
  • Memorise the three boundary markers: maximum AP, MP = 0, and MP negative.
  • Watch for options that mix up the stages. Match each stage with TP, AP and MP behaviour before choosing.
  • Remember that MP cuts AP at AP's maximum. This is a frequent conceptual MCQ.
  • Do not treat the peak of MP as the start of Stage II. MP peaks inside Stage I, and Stage II starts only after AP is maximum.
  • With no negative marking, attempt every question and eliminate wrong options first.

Practice questions from Theory of Production

Law of Variable Proportions in other exams

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

Law of Variable Proportions: frequently asked questions

Is the law of variable proportions the same as the law of diminishing returns?

They are closely related. The law of diminishing returns describes Stage II, where MP falls. The law of variable proportions covers all three stages, including increasing and negative returns.

Why does a producer not operate in Stage I?

In Stage I, fixed inputs are not fully used and AP is still rising. Adding more variable input raises average output, so the producer gains by expanding into Stage II.

What is the relation between TP and MP?

MP is the change in TP from one extra unit. TP rises while MP is positive, is maximum when MP = 0, and falls when MP is negative.

Does this law apply in the long run?

No. It applies in the short run, when at least one input is fixed. In the long run all inputs can vary, and returns to scale apply instead.