Law of Variable Proportions: TP, AP, MP, Assumptions and the Three Stages Explained
Updated 17 October 2026 · Fact-checked
The law of variable proportions says that when one input is increased while the others are fixed, total product first rises at an increasing rate, then at a diminishing rate, reaches a maximum and finally falls. These are the three stages: increasing returns (MP rising), diminishing returns (MP falling but positive) and negative returns (MP negative). A rational producer works in stage II.
Understand Law of Variable Proportions: Three Stages of Returns to a Factor
1. Production function (concept only). A production function is the technical relationship between inputs and the maximum output obtainable: Q = f(inputs). In the short run at least one factor (such as land or machinery) is fixed and others vary; in the long run all factors can be varied. *Returns to a factor* belong to the short run; *returns to scale* belong to the long run (meaning only: how output changes when all inputs change in the same proportion).
2. Product concepts.
- Total product (TP): total output from a given quantity of the variable factor.
- Average product (AP): AP = TP / units of the variable factor.
- Marginal product (MP): the addition to TP from one more unit of the variable factor: MPn = TPn - TPn-1.
- Relationships: TP is the sum of the MPs. MP rises, reaches a maximum and then falls. AP rises while MP > AP and falls when MP < AP; MP cuts AP at AP's maximum. TP is maximum when MP = 0, and falls when MP is negative.
3. Statement of the law. As more units of a variable factor are combined with a fixed factor, the marginal product at first rises, then falls, and eventually becomes negative (other things, such as technology, remaining the same).
4. Assumptions. One factor (the variable factor) is varied while others are fixed; technology is unchanged; all units of the variable factor are equal in efficiency (homogeneous); the factors can be combined in varying proportions; the law applies in the short run; output is measured in physical units.
5. The three stages (diagram: TP curve, and AP and MP curves below it).
- Stage I, increasing returns: from the start until AP is maximum (MP = AP). TP rises at an increasing rate, then a diminishing rate; MP rises to its maximum and then begins to fall (the point of inflexion of TP). Reason: the fixed factor is under-utilised, so extra workers allow division of labour and better use of the machinery.
- Stage II, diminishing returns: from the maximum AP to the point where MP = 0. TP rises at a diminishing rate to its maximum; AP and MP both fall, but MP stays positive. Reason: the fixed factor is now crowded, the variable and fixed factors are no longer in the best proportion, and imperfect substitutability sets in.
- Stage III, negative returns: MP is negative and TP falls. Reason: too many units of the variable factor hamper one another.
6. Where does a rational producer operate? In stage II. In stage I the fixed factor is not fully used, so adding more of the variable factor raises output per unit; in stage III extra input reduces output and so is never worth its cost. The exact stage II point depends on input prices and the price of output.
7. Law of variable proportions versus returns to scale. The first changes only one factor, with the others fixed (short run); returns to scale change all factors in the same proportion (long run). A table of the schedule is for understanding and is not tested directly, but numericals on TP, AP and MP are common.
Law of Variable Proportions: Three Stages of Returns to a Factor: key facts
- Average product
- AP = TP / units of the variable factor
- Marginal product
- MPn = TPn - TPn-1
- Total product
- TP = sum of MP
- Key points
- MP = AP at maximum AP; MP = 0 at maximum TP
How to answer Law of Variable Proportions: Three Stages of Returns to a Factor questions
Given TP, build AP and MP columns row by row and read the stages from the pattern.
- 1Compute AP = TP divided by units of the factor for each row.
- 2Compute MP = TP of this row minus TP of the previous row (the first MP equals the first TP).
- 3Stage I ends where AP is maximum (and equals MP).
- 4Stage II runs from there to the unit where MP = 0 or TP is maximum.
- 5Stage III starts when MP becomes negative.
- 6Give a one-line reason for each stage.
Law of Variable Proportions: Three Stages of Returns to a Factor 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 on a small farm
Suppose a family farm of fixed size grows pulses. One worker alone cannot manage ploughing and sowing; a second and third allow the work to be divided, so each adds more than the one before. Later, extra workers crowd the same plot, and a very large crowd tramples the crop.
Which stage of the law is each situation, and where would the farmer stop adding workers?
Try it first, then show the solutionSolution
- Division of labour, extra workers add more output each: stage I (increasing returns).
- Crowding, each extra worker adds less but still positive output: stage II (diminishing returns).
- Trampling crop, output falls: stage III (negative returns).
- The farmer would stop in stage II, before MP turns negative.
Answer: The first workers lie in stage I, the crowding workers in stage II and the destructive crowd in stage III; a sensible farmer stops adding workers within stage II.
Memory hook: Too few waste land, too many waste labour.
Tricks to remember Law of Variable Proportions: Three Stages of Returns to a Factor
Short memory aids. Say each one aloud once, then use it on a question the same day.
MP cuts AP at the top
MP always cuts AP at AP's highest point, like a new marks entry pulling the class average up or down.
Use it when: Relationship questions and diagrams.
I-II-III: AP-max, MP-zero
Stage I ends at maximum AP; Stage II ends at MP = 0; Stage III is MP negative.
Use it when: Locating stages in a schedule.
Rational means stage II
I = waste of land, III = loss of output, so II is the only sensible stage.
Use it when: Why stage II.
Law of Variable Proportions: Three Stages of Returns to a Factor: solved examples
Ordered from board level to HOTS. Try each one before you open the solution.
Example 1: Point of maximum TP · 1 mark · MCQBoard level
When total product is maximum, marginal product is:
(a) maximum
(b) positive
(c) zero
(d) negative
Try it first, then show the solutionSolution
- TP stops rising when the last unit adds nothing: MP = 0.
Answer: (c) zero
Example 2: AP and MP from TP · 3 marks · Short answerBoard level
The total product of 1 to 5 workers on a field is 10, 22, 36, 46 and 50 quintals. Calculate AP and MP for each, and name the point where MP begins to fall. (3 marks)
Try it first, then show the solutionSolution
- MP: 10; 22 - 10 = 12; 36 - 22 = 14; 46 - 36 = 10; 50 - 46 = 4.
- AP: 10/1 = 10; 22/2 = 11; 36/3 = 12; 46/4 = 11.5; 50/5 = 10.
- MP rises up to the 3rd worker (14) and falls from the 4th worker.
Answer: MP = 10, 12, 14, 10, 4; AP = 10, 11, 12, 11.5, 10. MP begins to fall after the 3rd worker.
Example 3: Identify the stages · 4 marks · Short answerHarder board level
Total product with 1 to 8 units of labour on a fixed plot is 8, 20, 36, 48, 55, 60, 60, 56. Find MP and AP and identify the three stages with the reason for each. (4 marks)
Try it first, then show the solutionSolution
- MP: 8, 12, 16, 12, 7, 5, 0, -4. AP: 8, 10, 12, 12, 11, 10, 8.57, 7.
- AP is maximum (12) at the 3rd and 4th units; at the 4th unit MP = AP = 12. So Stage I: 1 to 4 units (MP rises to 16 at the 3rd, then falls; AP rises).
- Stage II: from the 4th unit to MP = 0 at the 7th unit (TP rises to 60). MP positive but falling; AP falls.
- Stage III: from the 8th unit, MP = -4 and TP falls to 56.
- Reasons: stage I under-used fixed factor; stage II crowding of the fixed factor; stage III over-crowding.
Answer: Stage I: units 1-4 (AP rising, ends at maximum AP); Stage II: units 4-7 (MP positive and falling, TP maximum 60 at MP = 0); Stage III: the 8th unit onwards (MP negative).
Example 4: MP and AP · 4 marks · Short answerHarder board level
Five workers produce 100 units with an average product of 20. (a) The 6th worker adds 8 units. Find the new AP. (b) If the 6th worker had added 26 units, find the new AP. Explain the rule illustrated. (4 marks)
Try it first, then show the solutionSolution
- TP with 5 workers = 5 x 20 = 100 units.
- (a) TP6 = 100 + 8 = 108; AP6 = 108/6 = 18, which is below 20: AP falls because MP (8) < AP (20).
- (b) TP6 = 100 + 26 = 126; AP6 = 126/6 = 21, above 20: AP rises because MP (26) > AP (20).
- Rule: AP rises when MP > AP, falls when MP < AP, so MP cuts AP at its maximum.
Answer: AP falls to 18 in (a) and rises to 21 in (b), showing that AP moves in the direction in which MP lies relative to AP.
Example 5: Choosing the stage with prices · 4 marks · Short answerHOTS / competency
Using the schedule of the previous example (TP = 8, 20, 36, 48, 55, 60, 60, 56), a farmer sells output at Rs 1,500 per quintal and pays Rs 4,000 per worker per month. How many workers should he hire if he adds a worker only when the extra revenue covers the wage? Which stage is this? (4 marks)
Try it first, then show the solutionSolution
- Extra revenue = MP x Rs 1,500: 5th worker 7 x 1,500 = Rs 10,500; 6th worker 5 x 1,500 = Rs 7,500; 7th worker 0 x 1,500 = Rs 0.
- Wage is Rs 4,000: the 6th worker covers it (7,500 > 4,000); the 7th does not (0 < 4,000).
- Check the earlier workers: the 1st to 4th have MP 8, 12, 16, 12, all well above the wage level.
- He hires 6 workers, within stage II (units 4 to 7).
Answer: The farmer hires 6 workers (stage II). He never enters stage III, where MP is negative, and has no reason to stop in stage I where additional workers raise output per worker.
Where students lose marks in Law of Variable Proportions: Three Stages of Returns to a Factor
The slips examiners see most often, and what to write instead.
Mistake: Taking MP as TP divided by units
Mixing AP and MP.
Wrong working
MP of the 3rd unit = 36/3 = 12.
Right working
MP of the 3rd unit = 36 - 22 = 14.
Do this: MP is a difference, AP is a division.
Usually costs: 2 marks.
Mistake: Starting stage II at MP maximum
Confusing the points.
Wrong working
Stage I ends where MP is maximum.
Right working
Stage I ends where AP is maximum (MP = AP). MP maximum is the point of inflexion inside stage I.
Do this: Link stage boundaries to AP maximum and MP = 0.
Usually costs: 2 marks.
Mistake: Confusing the law with returns to scale
Both talk about rising and falling returns.
Wrong working
Law of variable proportions: all inputs increased in the same proportion.
Right working
One input varies, the rest are fixed (short run).
Do this: State 'variable factor with fixed factors' in the first line.
Usually costs: 2-3 marks.
Mistake: Wrong shape of curves
Not showing MP crossing AP at AP's maximum.
Wrong working
MP drawn cutting AP while AP is still rising.
Right working
MP cuts AP exactly at AP's highest point; MP cuts the axis exactly below the TP peak.
Do this: Draw TP above and AP, MP below on a common horizontal axis.
Usually costs: 1-2 marks.
Exam tips
- Present the law as: statement, assumptions, schedule or example, diagram, explanation of the three stages.
- Draw TP in the upper panel and AP and MP in the lower panel, with the three stages marked by dotted vertical lines.
- Give one reason for each stage and say why a rational producer chooses stage II.
- Students often confuse returns to a factor with returns to scale; state clearly that the law varies one factor with the others fixed.
- In schedules, show MP and AP columns before naming the stages.
Practice questions on Law of Variable Proportions: Three Stages of Returns to a Factor
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
In the stage of negative returns to a factor, marginal product is:
(a) maximum
(b) positive but falling
(c) zero
(d) negativeHint
Total product is falling.
Answer
(d) negative
Q2 · 3 marks · Short answerBoard level
Total product with 1 to 5 workers is 12, 28, 45, 56 and 60 units. Compute MP and AP and say at which worker MP is maximum. (3 marks)
Hint
MP is the difference between consecutive TPs.
Answer
MP = 12, 16, 17, 11, 4; AP = 12, 14, 15, 14, 12. MP is maximum (17) at the 3rd worker.
Q3 · 4 marks · Short answerHarder board level
The marginal products of successive units of labour are 10, 14, 12, 8, 4, 0 and -2. Find the total product at each unit, say where TP is maximum, and identify the unit at which stage III begins. (4 marks)
Hint
TP is the running sum of MP.
Answer
TP = 10, 24, 36, 44, 48, 48, 46. TP is maximum (48) at the 5th and 6th units; MP = 0 at the 6th. Stage III begins at the 7th unit, where MP = -2 and TP falls to 46.
Q4 · 4 marks · Short answerHarder board level
(a) A firm keeps land at 2 acres and raises labour from 4 to 8, so output goes from 80 to 120 units. (b) Another firm doubles every input (land, labour, capital) and output goes from 80 to 176 units. Identify the law operating in each case and state what the second case shows. (4 marks)
Hint
One factor varied, or all factors varied?
Answer
(a) Only labour varies with land fixed: law of variable proportions (returns to a factor); output rises by 50% for a 100% rise in labour, so diminishing returns. (b) All inputs double: returns to scale; output rises 176/80 = 2.2 times, more than double, so increasing returns to scale.
Q5 · 3 marks · Short answerHOTS / competency
Explain why a producer will not produce in stage III even if the variable factor were free, and why he will not choose stage I either, using the idea of MP and the fixed factor. (3 marks)
Hint
Think about output when MP < 0, and about unused capacity.
Answer
In stage III MP is negative, so another unit reduces total output; even free labour should not be hired. In stage I the fixed factor is not fully used and AP is still rising, so adding the variable factor raises output per unit and total profit; the producer therefore expands until at least maximum AP and settles in stage II.
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 when TP is maximum?Show answer
Zero.
2.Where does MP cut AP?Show answer
At the maximum point of AP.
3.Stage I ends where?Show answer
At maximum AP.
4.Which stage does a rational producer choose?Show answer
Stage II.
5.Is this law a short-run or long-run law?Show answer
Short run, since at least one factor is fixed.
Law of Variable Proportions: Three Stages of Returns to a Factor: frequently asked questions
What is the law of variable proportions?
With a fixed factor, successive units of a variable factor first add more, then less, and finally negative output.
Why does MP fall?
The fixed factor is shared among more units of the variable factor.
What are the three stages of the law?
Increasing returns, diminishing returns and negative returns.
How is it different from returns to scale?
The law varies one factor with others fixed (short run); returns to scale vary all factors together (long run).