EconomicsCBSE Class 11 · Production and Costs

Production Function, TP, AP and MP, and the Law of Variable Proportions

Q: What is the production function?

Relation between inputs and maximum output.

Q: What is MP?

Extra output from one more unit of input.

Q: What is the law of variable proportions?

Returns to a variable factor first rise then fall.

Q: Why does MP fall?

The fixed factor is overworked.

Updated 17 October 2026 · Fact-checked

The production function gives the maximum output from given inputs. In the short run at least one factor is fixed; with increasing variable input, marginal product first rises, then falls and may turn negative. This is the law of variable proportions (returns to a factor). AP = TP / L and MP = change in TP; MP cuts AP at its maximum.

Understand Law of Variable Proportions: TP, AP and MP

Production function. The relation between inputs and the maximum output that can be produced: Q = f(L, K), for a given technology. Short run: at least one factor (like capital/land) is fixed; output changes by changing the variable factor (labour). Long run: all factors can be varied; returns to scale arise.

Concepts. Total product (TP): total output at a given amount of the variable factor. Average product (AP) = TP ÷ L. Marginal product (MP) = ΔTP ÷ ΔL = TPₙ − TPₙ₋₁ for one more unit of labour. Also TP = ΣMP and TP = AP × L.

Worked schedule (labour with fixed land).
| L | TP | AP | MP |
| 1 | 10 | 10 | 10 |
| 2 | 24 | 12 | 14 |
| 3 | 39 | 13 | 15 |
| 4 | 52 | 13 | 13 |
| 5 | 60 | 12 | 8 |
| 6 | 64 | 10.67 | 4 |
| 7 | 64 | 9.14 | 0 |
| 8 | 60 | 7.5 | −4 |

Law of variable proportions (returns to a factor). As the quantity of the variable factor increases with other factors fixed, MP first rises (increasing returns), then falls (diminishing returns) and finally becomes negative.

Phase I: increasing returns to a factor: MP rises (L = 1 to 3 in the table); TP rises at an increasing rate; fixed factor is underused.
Phase II: diminishing returns: MP falls but is not negative (positive, reaching zero at L = 7); L = 4 to 7; TP rises at a decreasing rate until MP = 0 and TP is maximum (64 at L = 6 and 7).
Phase III: negative returns: MP negative (L = 8); TP falls; overcrowding.

Relations. (1) TP rises while MP > 0; maximum when MP = 0. (2) AP rises when MP > AP, falls when MP < AP, and MP cuts AP at the maximum of AP: AP is 13 at L = 4 where MP = 13. (3) MP can be negative; AP never.

Why diminishing returns? The fixed factor is overworked as more variable factor is used; the proportion is poor; indivisibility initially causes increasing returns.

Producer's choice. A rational producer operates in Phase II, where MP is declining but not negative; Phase I wastes the fixed factor, and Phase III reduces output.

Draw and label: TP curve (rising, then flat, then falling) with MP and AP curves below, MP cutting AP at its peak.

Law of Variable Proportions: TP, AP and MP: key facts

AP and MP: AP = TP / L | MP = TP_n - TP_(n-1); MP = change in TP
Relations: TP = Sum of MP | TP = AP x L; Identities
AP and MP: MP > AP: AP rises
MP < AP: AP falls
MP cuts AP at AP maximum; Curves
Phases: I: MP rising
II: MP falling, non-negative
III: MP negative; Law of variable proportions

How to answer Law of Variable Proportions: TP, AP and MP questions

Questions give TP, AP or MP and ask for the missing columns and the phases.

1Compute MP as the difference in TP and AP as TP over L.
2Mark where MP rises, falls and turns negative.
3Name the phase for each row.
4Check that MP = AP at the maximum AP.
5Relate to the producer's choice.

Law of Variable Proportions: TP, AP and MP in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Workers in a small field

Suppose a farmer with one small field hires more and more workers; the first few raise output sharply but later ones crowd the field.

Which phases appear and where does output peak?

Try it first, then show the solutionSolution

Early workers raise output more each: increasing returns.
Later workers add less: diminishing returns.
Output peaks when the extra worker adds nothing (MP = 0), then more workers reduce output.

Answer: Increasing, diminishing, then negative returns.

Memory hook: Too many cooks.

Tricks to remember Law of Variable Proportions: TP, AP and MP

Short memory aids. Say each one aloud once, then use it on a question the same day.

TP peaks at MP = 0
Total product is highest when MP is zero.
Use it when: TP
MP cuts AP at the top
MP crosses AP at AP max.
Use it when: Relation
I-D-N
Increasing, Diminishing, Negative.
Use it when: Phases

Law of Variable Proportions: TP, AP and MP: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Complete the schedule · 3 marks · Short answerBoard level

Given TP for L = 1 to 4 as 10, 24, 39, 52, find MP and AP.

Try it first, then show the solutionSolution

MP = 10, 14, 15, 13 (1 mark).
AP = 10, 12, 13, 13 (1 mark).
MP rises up to the third unit and then falls (1 mark).

Answer: MP 10,14,15,13; AP 10,12,13,13.

Example 2: Phases · 4 marks · Short answerBoard level

For the schedule TP = 10, 24, 39, 52, 60, 64, 64, 60 (L = 1 to 8), identify the phases with reasons.

Try it first, then show the solutionSolution

MP = 10, 14, 15, 13, 8, 4, 0, -4 (1 mark).
Phase I: L = 1 to 3 with MP rising (1 mark).
Phase II: L = 4 to 7 with MP falling but not negative (1 mark).
Phase III: L = 8 with MP negative and TP falling (1 mark).

Answer: I: 1-3; II: 4-7; III: 8.

Example 3: AP and MP link · 4 marks · Short answerHarder board level

Show from the schedule that AP is maximum where MP equals AP, and explain why MP must cut AP from above.

Try it first, then show the solutionSolution

AP = 13 at L = 4 and MP(4) = 13 (1 mark).
Before L = 4, MP (15 at L = 3) is above AP, so AP rises (1 mark).
After L = 4, MP (8 at L = 5) is below AP, so AP falls (1 mark).
Therefore MP cuts AP from above at AP maximum (1 mark).

Answer: MP = AP = 13 at L = 4.

Example 4: Missing values · 3 marks · Short answerHarder board level

In a table AP of 3 workers is 12 and TP of 4 workers is 52. Find TP of 3 workers, MP of the 4th worker and AP of 4 workers.

Try it first, then show the solutionSolution

TP of 3 workers = 12 x 3 = 36 (1 mark).
MP of 4th worker = 52 - 36 = 16 (1 mark).
AP of 4 workers = 52/4 = 13 (1 mark).

Answer: 36; 16; 13.

Example 5: Choice of phase · 5 marks · Long answerHOTS / competency

A producer has TP maximum at L = 6 and 7. Explain why a rational producer will not hire the 8th worker, and why he may stop before L = 7 when wages are positive. Which phase will he choose?

Try it first, then show the solutionSolution

MP(8) = -4: the 8th worker reduces output (1 mark).
Hiring costs wages and lowers output, so it is irrational (1 mark).
The 7th worker adds nothing (MP = 0) but costs a wage (1 mark).
So he stops where the extra output still justifies the wage, in Phase II (1 mark).
Phase I leaves the fixed factor underused, so he moves on to Phase II (1 mark).

Answer: Phase II.

Where students lose marks in Law of Variable Proportions: TP, AP and MP

The slips examiners see most often, and what to write instead.

Mistake: TP maximum at AP maximum
Curves confused.
Wrong working
TP is maximum when AP is maximum.
Right working
TP is maximum when MP = 0.
Do this: MP zero.
Usually costs: 1 mark.
Mistake: MP = TP/L
Formula mix.
Wrong working
MP is total divided by labour.
Right working
AP = TP/L; MP is the change in TP.
Do this: Change.
Usually costs: 1 mark.
Mistake: Negative AP
Sign error.
Wrong working
AP becomes negative in Phase III.
Right working
AP stays positive; MP can be negative.
Do this: AP positive.
Usually costs: 1 mark.

Exam tips

Prepare MP and AP columns.
Mark phases with row numbers.
State the MP-AP link.
Name the rational phase.

Practice questions on Law of Variable Proportions: TP, AP and MP

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Total product is maximum when
(a) MP is maximum (b) AP is maximum (c) MP is zero (d) AP is zero
Hint
Peak of TP.
Answer
(c).
Q2 · 1 mark · Assertion-reasonHarder board level
Assertion (A): MP can be negative but AP cannot.
Reason (R): AP is TP divided by the number of workers, which is positive while TP is positive.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Definition.
Answer
(a).
Q3 · 2 marks · Very short answerBoard level
Define returns to a factor and state its three phases.
Hint
Phases.
Answer
Returns to a factor describe how output changes as one variable factor is increased with others fixed. Phases: increasing returns, diminishing returns, and negative returns.
Q4 · 4 marks · Case-basedHarder board level
Read and answer.
Suppose TP of labour 1 to 5 is 8, 20, 36, 48, 54.
(a) Find MP. (2)
(b) Find AP. (1)
(c) Find the first L where diminishing returns appear. (1)
Hint
Differences.
Answer
(a) MP = 8, 12, 16, 12, 6. (b) AP = 8, 10, 12, 12, 10.8. (c) At L = 4, where MP falls from 16 to 12.
Q5 · 3 marks · Short answerHOTS / competency
Suppose a factory doubles workers but not machines and finds output rises by less than double. Explain with the law of variable proportions, and why this differs from returns to scale.
Hint
Fixed capital.
Answer
Machines are fixed, so extra workers share the same capital: MP of labour falls (diminishing returns to a factor, short run). Returns to scale concern changing all inputs in the same proportion in the long run.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.MP formula?Show answer
Change in TP.
2.AP formula?Show answer
TP / L.
3.When is TP maximum?Show answer
MP = 0.
4.Rational phase?Show answer
Phase II.
5.MP cuts AP at?Show answer
Maximum of AP.
6.Short run means?Show answer
At least one factor fixed.

Law of Variable Proportions: TP, AP and MP: frequently asked questions

What is the production function?

Relation between inputs and maximum output.

What is MP?

Extra output from one more unit of input.

What is the law of variable proportions?

Returns to a variable factor first rise then fall.

Why does MP fall?

The fixed factor is overworked.

What is the long run?

All factors variable.