24 Law of Variable Proportions: Law of Returns to Scale

24.1 The Production Function

A production function describes the technological relationship between physical inputs and the maximum physical output the firm can produce: $Q = f(L, K, N, E, \ldots)$ where L is labour, K is capital, N is land, E is enterprise. Two horizons matter: the short run in which at least one input is fixed (typically capital), and the long run in which all inputs are variable. The first horizon produces the Law of Variable Proportions; the second produces the Law of Returns to Scale.

24.2 Law of Variable Proportions — Short-Run Production

When one input is varied while others are held fixed, output passes through three phases. This is the Law of Variable Proportions (also called Law of Diminishing Returns). It generalises and replaces the older notion that diminishing returns are universal.

24.2.1 The Three Stages

Three Stages of Variable Proportions

Stage	What happens	TP, AP, MP
I — Increasing returns	More units of variable input increase output more than proportionately	TP rises, MP rises then peaks, AP rises
II — Diminishing returns	TP still rises, but at a declining rate	MP falls (still positive), AP falls after its peak, TP at maximum at MP = 0
III — Negative returns	Variable input becomes a hindrance	TP falls, MP is negative

flowchart LR
  S1[Stage I<br/>Increasing Returns] --> S2[Stage II<br/>Diminishing Returns]
  S2 --> S3[Stage III<br/>Negative Returns]
  S2 -->|rational producer<br/>operates here| EQ[MR = MC<br/>equilibrium]
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

The rational producer operates in Stage II — Stage I leaves productivity gains on the table; Stage III is destructive.

24.2.2 Relationship between TP, AP and MP

Three Production Curves Relationship

Relationship	Working
When AP rises, MP > AP	MP pulls AP up
When AP is maximum, MP = AP	The two curves intersect at AP’s peak
When AP falls, MP < AP	MP pulls AP down
When MP = 0, TP is at maximum	Beyond this, TP falls (Stage III)
When MP is negative, TP falls

24.2.3 Causes of the Three Stages

Why the Stages Occur

Stage I — increasing returns: indivisibility of fixed factor; growing scope for specialisation as variable input rises.
Stage II — diminishing returns: variable input becomes overcrowded relative to fixed input.
Stage III — negative returns: variable input is so plentiful it interferes with production.

24.2.4 Assumptions

Assumptions of the Law

Short run — at least one input is fixed.
Constant state of technology.
Homogeneous inputs — units are identical.
Input proportions can be changed — not fixed in rigid ratio.
Profit-maximising firm.

24.3 Returns to Scale — Long-Run Production

In the long run all inputs are variable. If all inputs are scaled up by the same proportion k:

Three Returns to Scale

Type	Working
Increasing Returns to Scale (IRS)	Output rises by more than k% — efficiency gains from specialisation, indivisibilities, technical economies
Constant Returns to Scale (CRS)	Output rises by exactly k% — the linear homogeneous production function
Decreasing Returns to Scale (DRS)	Output rises by less than k% — managerial diseconomies, communication costs

Mathematically: if $f(kL, kK) = k^n f(L, K)$, then $n > 1$ — IRS; $n = 1$ — CRS; $n < 1$ — DRS.

24.3.1 Cobb-Douglas Production Function

The famous Cobb-Douglas function (Charles Cobb and Paul Douglas, 1928):

\[Q = A L^{\alpha} K^{\beta}\]

where A is total factor productivity. Returns to scale are characterised by $\alpha + \beta$: - $\alpha + \beta > 1$ — IRS. - $\alpha + \beta = 1$ — CRS (the standard form). - $\alpha + \beta < 1$ — DRS.

A property of Cobb-Douglas: the elasticity of output with respect to labour and capital are constants (α and β); factor shares in income are also constants if markets are competitive.

24.4 Iso-Quants and Iso-Cost Lines

In the long run, the firm’s production decision is mapped using iso-quants (curves of equal output) and iso-cost lines (combinations of L and K at equal total cost).

Iso-Quant Properties

Downward-sloping in the relevant range.
Convex to origin — diminishing MRTS (Marginal Rate of Technical Substitution).
Higher iso-quant = higher output.
Two iso-quants do not intersect.

24.4.1 MRTS and Producer Equilibrium

MRTS_{LK} = the rate at which capital can be substituted by labour, holding output constant:

\[MRTS_{LK} = -\frac{\Delta K}{\Delta L} = \frac{MP_L}{MP_K}\]

The producer is in equilibrium where the iso-cost line is tangent to the highest attainable iso-quant:

\[MRTS_{LK} = \frac{w}{r}\]

(w = wage rate of labour; r = rental rate of capital).

24.5 Economies and Diseconomies of Scale

The cause of IRS and DRS is the presence of economies and diseconomies of scale.

Economies vs Diseconomies of Scale

Economies (cause of IRS)	Diseconomies (cause of DRS)
Technical — indivisible plant; specialisation	Managerial — communication/coordination costs
Managerial — division of labour at top	Excessive bureaucracy
Marketing — bulk advertising	Labour relations strain
Financial — cheaper credit	Resource scarcity raising input prices
Risk-bearing — diversified product mix	Diseconomies of large scale logistics

A further distinction: - Internal economies — accrue to the individual firm as it grows. - External economies — accrue to all firms as the industry grows (skilled labour pool, ancillary suppliers, infrastructure).

The famous U-shaped long-run average cost (LRAC) curve is generated by IRS at low output (LRAC falling), CRS at the minimum efficient scale, and DRS at high output (LRAC rising).

24.6 Differences — Variable Proportions vs Returns to Scale

Variable Proportions vs Returns to Scale

Dimension	Law of Variable Proportions	Law of Returns to Scale
Horizon	Short run	Long run
Inputs changed	Only variable input(s); some fixed	All inputs scaled together
Three phases	Increasing → Diminishing → Negative returns	IRS → CRS → DRS
Curve illustrating	MP curve	LRAC curve
Cause	Factor proportions become sub-optimal	Scale economies and diseconomies

24.7 Practice Questions

Q 01 LVP Easy

The Law of Variable Proportions applies to:

ALong run only
BShort run with at least one fixed input
CAll inputs scaled together
DPerfect competition only

View solution

Correct Option: B

LVP is a *short-run* law — at least one input fixed.

Q 02 Stages Medium

A rational producer operates in:

AStage I — increasing returns
BStage II — diminishing returns
CStage III — negative returns
DAll three stages

View solution

Correct Option: B

Stage I leaves productivity gains; Stage III is wasteful; **Stage II** is rational.

Q 03 AP-MP Medium

When AP is at its maximum:

AMP = 0
BMP > AP
CMP < AP
DMP = AP

View solution

Correct Option: D

AP is at maximum when **MP = AP** — MP intersects AP at AP's peak.

Q 04 TP-MP Medium

Total Product is at its maximum when:

AMP is rising
BMP is at maximum
CMP = 0
DMP is negative

View solution

Correct Option: C

TP peaks when **MP = 0**; thereafter MP turns negative and TP falls.

Q 05 RTS Medium

Returns to scale studies what happens when:

AOne input varies, others fixed
BAll inputs are scaled by the same proportion
COnly labour is changed
DDemand changes

View solution

Correct Option: B

RTS — *long-run* concept; all inputs scaled together.

Q 06 Cobb-Douglas Medium

In the Cobb-Douglas function $Q = A L^{\alpha} K^{\beta}$, CRS holds when:

Aα + β > 1
Bα + β = 1
Cα + β < 1
Dα = β = 0

View solution

Correct Option: B

α + β = 1 → **CRS / linearly homogeneous**.

Q 07 Cobb-Douglas Medium

The Cobb-Douglas production function was introduced in:

A1890
B1928
C1948
D1962

View solution

Correct Option: B

**1928** — Cobb and Douglas, *American Economic Review*.

Q 08 MRTS Hard

At producer equilibrium on the iso-quant:

AMRTS_LK = MP_L − MP_K
BMRTS_LK = MP_L / MP_K = w / r
CMRTS_LK = r × w
DMRTS_LK = 1

View solution

Correct Option: B

Tangency: MRTS = ratio of MPs = ratio of input prices.

Q 09 Match Medium

Match each concept with its horizon:

	Concept		Horizon
(i)	Law of Variable Proportions	(a)	Long run
(ii)	Returns to Scale	(b)	Short run

A(i)-(b), (ii)-(a)
B(i)-(a), (ii)-(b)
CBoth short
DBoth long

View solution

Correct Option: A

LVP — short; RTS — long.

Q 10 Iso-quant Medium

An iso-quant shows:

ACombinations of two outputs at equal cost
BCombinations of two inputs that yield the same output
CCombinations of price and quantity
DCombinations of profit and cost

View solution

Correct Option: B

**Iso-quant** = equal output curve in input space.

Q 11 U-shape Medium

The long-run average cost (LRAC) curve is U-shaped because of:

ALaw of variable proportions
BEconomies followed by diseconomies of scale
CCobb-Douglas function
DMarginal product of labour

View solution

Correct Option: B

IRS → CRS → DRS gives U-shape in LRAC.

Q 12 Stage III Hard

In Stage III of variable proportions:

AMP is positive and rising
BMP is negative; TP falls
CAP = MP
DTP is at maximum

View solution

Correct Option: B

Stage III — variable input is excessive; MP turns negative.

Q 13 External Medium

Economies of scale that accrue to **all firms** as the industry grows are called:

AInternal economies
BExternal economies
CTechnical economies
DFinancial economies

View solution

Correct Option: B

**External** — industry-wide economies (skilled labour, infrastructure, ancillaries).

Q 14 Cobb-Douglas IRS Hard

A Cobb-Douglas function with α = 0.6 and β = 0.5 exhibits:

ACRS
BIRS
CDRS
DIndeterminate

View solution

Correct Option: B

α + β = 1.1 > 1 → **IRS**.

Q 15 Cause Medium

Diminishing returns set in because:

ADemand falls
BVariable input becomes excessive relative to the fixed input
CTechnology improves
DOutput price falls

View solution

Correct Option: B

Overcrowding of variable input against fixed input → diminishing returns.

Q 16 Isocost Medium

The slope of the iso-cost line is:

A−w/r
B−r/w
Cw × r
Dw + r

View solution

Correct Option: A

From wL + rK = C, with K on Y-axis, slope = **−w/r**.

Q 17 Linear homogeneous Hard

A "linearly homogeneous" production function satisfies:

A$f(kL, kK) = k f(L, K)$
B$f(kL, kK) = k^2 f(L, K)$
C$f(kL, kK) = \sqrt{k} f(L, K)$
D$f(kL, kK) = 1$

View solution

Correct Option: A

Degree 1 homogeneity = CRS.

Q 18 Match Medium

Match each output change with the kind of returns:

	All inputs doubled, output:		Returns
(i)	Trebles	(a)	CRS
(ii)	Doubles	(b)	DRS
(iii)	1.5 ×	(c)	IRS

A(i)-(c), (ii)-(a), (iii)-(b)
B(i)-(b), (ii)-(c), (iii)-(a)
C(i)-(a), (ii)-(b), (iii)-(c)
D(i)-(c), (ii)-(b), (iii)-(a)

View solution

Correct Option: A

Trebling > 2× → IRS; doubling = CRS; 1.5× < 2× → DRS.

Q 19 MRTS Medium

MRTS diminishes because:

AIso-quants are linear
BInputs are imperfect substitutes; the MP of the substituted input rises and the MP of the other falls
CReturns to scale fall
DPrices change

View solution

Correct Option: B

As K is given up for more L, MP_L falls and MP_K rises → MRTS_LK = MP_L/MP_K declines.

Q 20 Compare Medium

Which is **not** a difference between the Law of Variable Proportions and Returns to Scale?

AHorizon
BInputs changed
CBoth assume profit maximisation
DUnderlying cause of the stages

View solution

Correct Option: C

Both assume profit-maximising firm; that is a *similarity*, not a difference.

24.8 Quick Recall

Quick recall

Law of Variable Proportions (short run; one input fixed): three stages — Increasing → Diminishing → Negative returns.
Relations: AP rises if MP > AP; AP peaks when MP = AP; TP peaks when MP = 0; MP negative ⇒ TP falls.
Rational producer operates in Stage II.
Returns to Scale (long run; all inputs scaled): IRS, CRS, DRS.
Cobb-Douglas (1928): $Q = AL^\alpha K^\beta$; α+β > 1 IRS; α+β = 1 CRS (linearly homogeneous); α+β < 1 DRS.
MRTS_LK = MP_L/MP_K = w/r at producer equilibrium.
Iso-quants: convex, downward-sloping, do not intersect; iso-cost slope = −w/r.
U-shaped LRAC = IRS → CRS → DRS.
Economies of scale: technical, managerial, marketing, financial, risk-bearing; internal vs external; diseconomies — communication, bureaucracy, resource scarcity.