Law of variable proportions

Also called: Returns to a factor, Returns to a variable factor · Topic: Production Function, Returns and Costs · NCERT: Class 12, Ch 3 "Production and Costs"

Meaning

The law of variable proportions (also called returns to a factor) says that in the short run, when one input is fixed and a firm keeps adding more of another input, the marginal product of that variable input first rises and then, after a certain level, falls. It happens because the ratio between the two inputs keeps changing.

• Formulas: MP_L = ΔTP / ΔL (for whole units, MP_L = TP_L − TP_(L−1)). AP_L = TP / L.
• Why it matters: the law explains the shape of a firm's product and cost curves. It also explains why too many workers on small farms add little to output. That links it to disguised unemployment and to India's need to move workers out of agriculture.

Explanation

Basic terms and how the law works

• Production function: the most output a firm can get from given amounts of inputs, with a given technology.
• Short run: a period in which at least one input is fixed, such as land or a machine. The law of variable proportions is a short-run law.
• Fixed factor: an input whose quantity cannot change in the short run. Example: a farmer's 4 hectares.
• Variable factor: an input whose quantity can change. Example: the number of workers.
• Total Product (TP): the whole output from a given amount of the variable input.
• Marginal Product (MP): the extra output from one more unit of the variable input.
• Average Product (AP): output per unit of the variable input.
• Factor proportions: the ratio in which inputs are combined, for example workers per hectare. If one input is fixed and the other rises, this ratio must change. That is where the name "variable proportions" comes from.
• Why MP rises at first:
• The fixed factor is under-used. There is too much land for too few workers.
• Each new worker makes the proportion better.

• So each extra worker adds more output than the one before.

• Why MP falls later:

• The fixed factor gets crowded.
• Each extra worker has less land, or fewer machines, to work with.
• So each extra worker adds less output than the one before.

Worked example: NCERT Table 3.2 (land fixed at 4 hectares)

Labour (L)	TP	MP = ΔTP	AP = TP/L
0	0	–	–
1	10	10	10
2	24	14	12
3	40	16 (highest MP)	13.33 (highest AP)
4	50	10	12.5
5	56	6	11.2
6	57	1	9.5

• Sample calculation: MP of the 3rd worker = 40 − 24 = 16. AP at L = 3 = 40 ÷ 3 = 13.33.
• L = 1 → 3: MP rises (10 → 14 → 16). This is increasing returns to a factor.
• L = 4 onwards: the land gets crowded, and MP falls (16 → 10 → 6 → 1). This is diminishing marginal product.
• TP still rises up to L = 6, because MP is still positive. It only rises more slowly.
• NCERT error: the NCERT text says TP rises "by 10" (L = 1→2) and "by 12" (L = 2→3). The table gives 14 and 16. Go by the table.
• How MP, AP and TP relate:
• When MP > AP, AP rises. At L = 2 and 3, MP (14, 16) is above AP (12, 13.33).
• When MP < AP, AP falls. At L = 4, MP of 10 is below AP of 12.5.
• MP cuts AP at the highest point of AP.
• When MP is positive, TP rises. When MP = 0, TP is at its maximum. When MP < 0, TP falls.
• Both the MP and AP curves are inverted-U shaped, meaning they rise and then fall.

The three stages (beyond NCERT)

Standard convention	Range	Feature
Stage I	Origin → maximum AP	MP > AP; the fixed factor is under-used
Stage II	Maximum AP → MP = 0	AP and MP are both falling and both positive. This is the rational zone
Stage III	MP < 0	TP falls; there is too much of the variable factor

• Why a sensible firm works only in Stage II:
• Stage I: AP is still rising. Each extra worker raises output per worker, so the firm should not stop here.
• Stage III: MP is negative. An extra worker reduces output, so no firm would pay for that worker.

• Stage II: the exact point the firm picks depends on wages and the price of its output.

• In Table 3.2: Stage I ends near L = 3, where AP is highest (13.33). Stage II runs from L = 4 onwards. MP never reaches 0 in the table, so Stage III is not reached.

• Older CBSE convention: it names three phases: increasing, diminishing and negative returns to a factor. Here the first phase ends at maximum MP, not maximum AP.
• In Table 3.2 both peaks fall at L = 3. In general, MP peaks before AP does.
• Say which convention you are using in your answer.

What drives each phase, assumptions and history

• Causes of the rising phase:
• The fixed factor is indivisible, which means it cannot be split into smaller parts (you cannot use half a tractor). It gives its full benefit only when enough workers use it.

• More workers allow specialisation, or division of labour. Each worker does one task and gets better at it.

• Causes of the falling phase:

• Over-crowding of the fixed factor.

• Factors are imperfect substitutes, which means one input cannot fully replace another. Extra labour cannot make up for missing land.

• Assumptions of the law:

• Technology is fixed. A better seed or a new machine would shift the whole TP curve up.
• At least one input is fixed.

• Units of the variable input are identical (homogeneous), which means every worker is equally skilled. So MP falls because of crowding, not because later workers are weaker.

• History:

• Turgot first stated the idea of diminishing returns on land.
• Ricardo's theory of rent rests on it. As farming spreads to poorer land, or more work goes into the same land, the extra output falls. The better land then earns rent.
• Malthus's population argument also rests on it. Food output meets diminishing returns while population keeps growing.

In India

• Where the law shows up: disguised unemployment in farming.
• Disguised unemployment means people who look employed but add nothing to output. Their MP is zero.
• When too many family members work a fixed plot, the MP of labour is zero or close to zero.
• Example: 5 family members work 2 acres and grow 20 quintals of wheat. One member leaves for a city job, and the other 4 still grow 20 quintals. The 5th worker's MP = 20 − 20 = 0.

• In stage terms, this is the Stage II/III boundary (MP = 0), where TP is at its maximum.

• Who measures it: the Periodic Labour Force Survey (PLFS) gives the workforce data. The Economic Survey uses that data to analyse the labour market.

• Latest figures:
• According to PLFS, more than 45% of India's workforce is in agriculture. The rest is 11.4% in manufacturing, 28.9% in services and 13.0% in construction. The total workforce is about 56.5 crore (2022-23) [2].
• Agriculture's share of the workforce was 45.8% (2023). The Economic Survey assumes it will slowly fall to one-fourth by 2047 [2].
• Agriculture produced 17.7% of GVA at current prices in FY24, against 27.6% for industry and 54.7% for services [3].

• Agriculture and allied sectors grew at an average of about 5% a year (FY17–FY23) [4].

• What the numbers mean:

• About 46% of workers produce less than one-fifth of output.

• So output per farm worker is low, which points to disguised unemployment.

• Policy response: move workers out of farming. India needs to create about 78.5 lakh non-farm jobs every year until 2030 [2].

Don't confuse with

• Returns to scale: a long-run idea in which all inputs change together, in the same proportion. The law of variable proportions is a short-run idea in which at least one input stays fixed.
• Law of diminishing marginal product (LDMP): this covers only the falling phase of MP. The law of variable proportions covers the whole path, where MP first rises and then falls. NCERT uses the two names as synonyms, but LDMP is only one part of LVP.
• Falling TP: "diminishing returns" means MP falls, not TP. In Table 3.2, TP keeps rising from L = 4 to 6 while MP falls. TP falls only when MP turns negative (Stage III).
• Maximum MP vs maximum AP: MP peaks before AP in general. MP then cuts AP at AP's highest point. In the standard convention Stage I ends at maximum AP. In the older CBSE convention it ends at maximum MP.

Prelims Hooks

• LVP is a short-run law that assumes fixed technology, at least one fixed input and homogeneous units of the variable input.
• MP = ΔTP/ΔL; AP = TP/L. MP cuts AP at the highest point of AP. TP is at its maximum when MP = 0.
• Trap: "Diminishing returns means total output falls." This is wrong. TP still rises, only more slowly, as long as MP is positive.
• Rational zone = Stage II, which runs from maximum AP to MP = 0. There, AP and MP are both falling but still positive.
• Disguised unemployment = MP of labour equal to zero. It is linked to the Lewis (1954) surplus-labour model. Turgot first stated diminishing returns, and Ricardo (rent) and Malthus (population) built on it.
• Agriculture had 17.7% of GVA at current prices in FY24 [3], against more than 45% of the workforce under PLFS [2].

Mains Points

• Structural transformation and the Lewis model:
• About 46% of workers are on farms but produce only 17.7% of GVA (FY24) [2][3]. So many farm workers have near-zero MP.
• In the Lewis (1954) model, moving these workers into industry does not reduce farm output. Industry gets cheap labour, and the economy grows.

• This is why the Economic Survey calls for about 78.5 lakh non-farm jobs every year until 2030 [2].

• Small plots, crowded land and the policy fix:

• Small landholdings with large families push farms towards the Stage II/III boundary.

• Ways to raise the MP of labour: consolidate landholdings, move workers into agro-processing and rural non-farm jobs, and raise capital per worker through irrigation and farm machines.

• Technology can offset diminishing returns, but add a caveat:

• The law assumes fixed technology. HYV seeds, irrigation and the Green Revolution shifted the whole TP curve up. This is why Malthus's forecast of food shortage failed, and why farm R&D and extension services are GS-III priorities.
• Caveat: zero MP is an extreme case. Farm labour is often scarce at sowing and harvest time. So it is more accurate to describe the problem as seasonal underemployment with low MP, not zero MP all year.

Related concepts

• Law of diminishing marginal product
• Factor proportions

Read more

• Law of variable proportions and diminishing marginal product