flowchart LR S1[Stage I<br/>Increasing Returns<br/>MP rises to its peak] --> S2[Stage II<br/>Diminishing Returns<br/>MP falls but is positive] S2 --> S3[Stage III<br/>Negative Returns<br/>MP turns negative; TP falls] RAT[Rational producer<br/>operates only in<br/>Stage II] RAT -.-> S2 style S1 fill:#FFF8E1,stroke:#F9A825 style S2 fill:#E8F5E9,stroke:#2E7D32 style S3 fill:#FFEBEE,stroke:#C62828
23 Law of Variable Proportions
23.1 The Production Function
A production function states the maximum output the firm can produce from any given combination of inputs, given the existing technology (dwivedi2021?):
\[ Q = f(L, K, N, E, T, \dots) \]
where Q is output and L, K, N, E, T are labour, capital, land, entrepreneurship and technology. For the textbook two-input version, \(Q = f(L, K)\).
The time horizon matters. Economists distinguish:
TipShort Run vs Long Run
| Period | Inputs | Operative law |
|---|---|---|
| Short run | At least one input is fixed (typically capital) | Law of Variable Proportions |
| Long run | All inputs are variable | Returns to Scale |
The Law of Variable Proportions describes short-run output behaviour as one input is varied; returns to scale describes long-run output behaviour as all inputs are varied together.
23.2 Total, Average and Marginal Product
TipThree Product Concepts
| Concept | Definition | Formula |
|---|---|---|
| Total Product (TP) | Total output from a given quantity of variable input | \(TP = f(L)\), \(K\) fixed |
| Average Product (AP) | Output per unit of the variable input | \(AP_L = TP / L\) |
| Marginal Product (MP) | Addition to total output from one extra unit of variable input | \(MP_L = \Delta TP / \Delta L\) |
The relations between TP, AP and MP are mechanical consequences of arithmetic:
- When \(MP > AP\), AP is rising.
- When \(MP = AP\), AP is at its maximum.
- When \(MP < AP\), AP is falling.
- When \(MP = 0\), TP is at its maximum.
- When \(MP < 0\), TP is falling.
23.3 The Law of Variable Proportions — Statement
The Law of Variable Proportions (also called the Law of Diminishing Returns or the Law of Non-Proportional Returns) states:
“As more and more units of a variable input are applied to a fixed input, the marginal product of the variable input first rises, reaches a maximum, and then falls; eventually it becomes zero and then negative.”
The classic statement is from F.B. Benham: “as the proportion of one factor in a combination of factors is increased, after a point, the average and marginal product of that factor will diminish” (benham1955?).
23.4 Three Stages of Production
The law generates three identifiable stages (ahuja2020?):
TipThree Stages of the Law of Variable Proportions
| Stage | Returns | Behaviour of TP, AP, MP |
|---|---|---|
| Stage I | Increasing returns | TP rises at increasing rate; MP rises and reaches its peak; AP rises |
| Stage II | Diminishing returns | TP rises at decreasing rate; MP falls but is still positive; AP falls; AP = MP at the boundary |
| Stage III | Negative returns | TP falls; MP becomes negative; AP continues to fall |
The boundary between Stage I and Stage II is at the point where AP is maximised (and AP = MP). The boundary between Stage II and Stage III is where MP = 0 (and TP is maximised).
A rational producer operates only in Stage II — the zone of diminishing positive returns. In Stage I the fixed input is under-utilised; in Stage III the variable input is over-applied to the point of harm.
23.5 A Worked Schedule
Land is fixed at 5 acres; labour is varied. Output is measured in quintals.
TipTP, AP, MP Schedule
| Workers (L) | TP | AP = TP/L | MP = ΔTP | Stage |
|---|---|---|---|---|
| 1 | 8 | 8.0 | 8 | I |
| 2 | 20 | 10.0 | 12 | I |
| 3 | 36 | 12.0 | 16 | I (peak MP) |
| 4 | 48 | 12.0 | 12 | I → II boundary (AP = MP) |
| 5 | 55 | 11.0 | 7 | II |
| 6 | 60 | 10.0 | 5 | II |
| 7 | 60 | 8.6 | 0 | II → III boundary (MP = 0; TP max) |
| 8 | 56 | 7.0 | − 4 | III |
The producer would choose to employ between 4 and 7 workers — somewhere in Stage II — depending on the wage rate.
23.6 Causes of the Three Stages
TipCauses of Each Stage
| Stage | Underlying causes |
|---|---|
| Increasing returns (Stage I) | Better utilisation of the fixed input; specialisation and division of labour; teamwork synergies |
| Diminishing returns (Stage II) | Scarcity of the fixed input — each additional variable input gets less of the fixed input to work with; imperfect substitutability between inputs |
| Negative returns (Stage III) | Over-crowding; coordination breakdown; the variable input now interferes with itself |
The textbook intuition: too many cooks in one kitchen first add productivity, then dilute it, then spoil the broth.
23.7 Why the Law Holds Eventually — and Apparent Exceptions
The law is a general tendency, not an absolute rule (dwivedi2021?):
- It assumes one input fixed. If both inputs are variable, the long-run returns to scale concept applies instead.
- It assumes technology unchanged. A technological breakthrough can postpone diminishing returns indefinitely.
- It assumes factors are imperfectly substitutable. With perfect substitutability, the firm could substitute the variable input fully and the law need not hold.
- It assumes the fixed factor is reasonably scarce. If the fixed input is in vast excess (e.g., abundant land in the early stages of agriculture), Stage I can persist longer.
23.8 Returns to Scale (Long Run)
When all inputs are varied in the same proportion, three returns-to-scale outcomes are possible.
TipThree Cases of Returns to Scale
| Case | Working content | Cause |
|---|---|---|
| Increasing Returns to Scale (IRS) | Output grows by more than the proportion of inputs | Internal & external economies of scale |
| Constant Returns to Scale (CRS) | Output grows by the same proportion as inputs | Linearly homogeneous production function |
| Decreasing Returns to Scale (DRS) | Output grows by less than the proportion of inputs | Managerial diseconomies; coordination cost |
Internal economies (specialisation, division of labour, technical economies, financial economies, marketing economies, risk-bearing economies) operate within the firm. External economies (industrial concentration, infrastructure, skilled labour pool, R&D spillovers) operate at the level of the industry or region.
23.9 The Cobb-Douglas Production Function
The most-cited functional form in econometric work and in exams is the Cobb-Douglas production function, named after Charles W. Cobb and Paul H. Douglas (1928) (cobb1928?):
\[ Q = A \cdot L^{\alpha} \cdot K^{\beta} \]
where \(A\) is a technology constant and \(\alpha, \beta\) are the output elasticities of labour and capital.
TipReturns to Scale in the Cobb-Douglas Function
| Sum of exponents | Returns to scale |
|---|---|
| \(\alpha + \beta > 1\) | Increasing |
| \(\alpha + \beta = 1\) | Constant (linear homogeneity) |
| \(\alpha + \beta < 1\) | Decreasing |
The function has three convenient properties: positive marginal products, diminishing marginal products in each input, and constant elasticity of substitution equal to 1.
23.10 Isoquants and Producer Equilibrium (long-run)
In the long run, the production analogue of the indifference curve is the isoquant — the locus of all input combinations that produce the same level of output.
TipProperties of Isoquants
| Property | Justification |
|---|---|
| Slope downward to the right | Substitutability of inputs |
| Convex to origin | Diminishing Marginal Rate of Technical Substitution (MRTS) |
| Higher isoquants represent higher output | More inputs → more output |
| Two isoquants never intersect | Logical consistency |
The slope is the Marginal Rate of Technical Substitution: \(MRTS_{LK} = \frac{MP_L}{MP_K}\).
The iso-cost line shows all combinations of L and K that cost the same amount: \(C = wL + rK\). The producer’s equilibrium (least-cost combination for a given output, or maximum output for a given cost) is at the tangency of the isoquant and the iso-cost line:
\[ \frac{MP_L}{MP_K} = \frac{w}{r} \]
23.11 Variable Proportions vs Returns to Scale — A Sharp Distinction
TipLaw of Variable Proportions vs Returns to Scale
| Dimension | Law of Variable Proportions | Returns to Scale |
|---|---|---|
| Time horizon | Short run | Long run |
| Inputs varied | One; the other(s) fixed | All inputs varied in same proportion |
| Concept centred on | MP of the variable input | Output response to scale |
| Stages | Three (Increasing, Diminishing, Negative) | Three (Increasing, Constant, Decreasing) |
| Driving cause | Imperfect substitutability between fixed and variable inputs | Internal & external economies vs managerial diseconomies |
23.12 Exam-Pattern MCQs
Q 01
Which of the following is not a feature of the Law of Variable Proportions?
View solution
Correct Option: C
All inputs varied in the same proportion describes returns to scale, not the law of variable proportions.
Q 02
Match each stage of the law with the behaviour of its products:
| Stage | Behaviour | ||
| (i) | Stage I | (a) | TP falls; MP < 0 |
| (ii) | Stage II | (b) | TP rises at increasing rate; MP rises |
| (iii) | Stage III | (c) | TP rises at decreasing rate; MP positive but falling |
View solution
Correct Option: A
Q 03
A rational producer operates in:
View solution
Correct Option: B
In Stage I the fixed factor is under-utilised; in Stage III output is falling. The producer settles in Stage II — the zone of diminishing positive returns.
Q 04
When the Average Product of labour is at its maximum:
View solution
Correct Option: C
AP is at its maximum where MP cuts AP from above — i.e., when MP = AP.
Q 05
A Cobb-Douglas production function $Q = AL^{0.6} K^{0.5}$ exhibits:
View solution
Correct Option: C
Sum of exponents = 0.6 + 0.5 = 1.1 > 1 → increasing returns to scale.
Q 06
Match each concept with its definition:
| Concept | Definition | ||
| (i) | Isoquant | (a) | Locus of input combinations costing the same amount |
| (ii) | MRTS | (b) | Locus of input combinations producing the same output |
| (iii) | Iso-cost line | (c) | Slope of the isoquant; rate at which one input substitutes for another at constant output |
| (iv) | Producer equilibrium | (d) | MRTS equals the ratio of input prices |
View solution
Correct Option: A
Q 07
Arrange the following events in the order in which they appear as the variable input is increased from zero: (i) MP reaches its maximum (ii) AP reaches its maximum (= MP) (iii) MP becomes zero (TP at maximum) (iv) MP becomes negative
View solution
Correct Option: A
MP rises to a peak (i), then equals AP at AP's maximum (ii), then falls to zero where TP is maximum (iii), then turns negative (iv).
Q 08
Match each long-run case with its primary cause:
| Case | Cause | ||
| (i) | Increasing Returns to Scale | (a) | Linearly homogeneous production function |
| (ii) | Constant Returns to Scale | (b) | Specialisation, division of labour, technical economies |
| (iii) | Decreasing Returns to Scale | (c) | Managerial diseconomies; coordination problems |
View solution
Correct Option: A
ImportantQuick recall
- Production function: maximum output for given inputs with given technology.
- Short-run law = Variable Proportions; Long-run law = Returns to Scale.
- TP-AP-MP relations: MP > AP → AP rising; MP = AP → AP max; MP = 0 → TP max; MP < 0 → TP falling.
- Three stages: I — Increasing returns; II — Diminishing returns; III — Negative returns. Rational producer operates in Stage II.
- Boundaries: I→II at AP max (AP = MP); II→III at MP = 0 (TP max).
- Causes: Stage I — better use of fixed factor; Stage II — scarcity of fixed factor; Stage III — over-application of variable factor.
- Long-run Returns to Scale: IRS (sum of exponents > 1), CRS (= 1), DRS (< 1) in Cobb-Douglas \(Q = AL^\alpha K^\beta\).
- Isoquant ↔︎ indifference curve; MRTS = \(MP_L / MP_K\) = slope of isoquant.
- Producer equilibrium: \(MRTS = w/r\), i.e., \(MP_L/MP_K = w/r\).
- Internal economies: technical, managerial, marketing, financial, risk-bearing. External economies: industry concentration, skilled labour pool, infrastructure.