Long-run production: returns to scale and isoquants – Production Theory (Senior 4 Economics) | Ganzaa

Production Theory

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Senior 4 · Economics

Production Theory

5 lessons

Lesson 4 of 5

Long-run production: returns to scale and isoquants

30 min

Hoes or a bulldozer?

In lesson 1 you saw dozens of people building terraces with hoes. On a road site in Kigali you may see the opposite: one bulldozer and a handful of workers moving the same amount of earth in a day. Both methods "work". In the long run a producer can change every input, so two new questions arise. If a firm doubles all its inputs, does output double? And of all the mixes of labour and capital that can produce a given output, which should it choose?

This lesson answers the first question with returns to scale, and draws the menu of mixes with isoquants. Lesson 5 adds costs to choose from the menu.

Definition

Returns to scale: how output responds when all inputs are increased in the same proportion, in the long run.

Increasing returns to scale (IRS): output rises by a larger proportion than the inputs.
Constant returns to scale (CRS): output rises by the same proportion.
Decreasing returns to scale (DRS): output rises by a smaller proportion.

Example: a bakery doubles its bakers, ovens, flour and floor space. If it now bakes 2.5 times as much bread, it has increasing returns to scale.

1. Returns to scale

Multiply every input by a number $\lambda > 1$ and compare the new output with $\lambda$ times the old:

f(\lambda K, \lambda L) > \lambda Q \ \text{(IRS)}, \qquad f(\lambda K, \lambda L) = \lambda Q \ \text{(CRS)}, \qquad f(\lambda K, \lambda L) < \lambda Q \ \text{(DRS)}

A common form of production function is the Cobb–Douglas function $Q = A K^{\alpha} L^{\beta}$ . Multiplying both inputs by $\lambda$ multiplies output by $\lambda^{\alpha + \beta}$ , so:

$\alpha + \beta$	Returns to scale	Doubling all inputs multiplies output by
greater than 1	Increasing	more than 2
equal to 1	Constant	exactly 2
less than 1	Decreasing	less than 2

Why returns change with scale. As a firm grows from small to medium, it can divide labour more finely, use large machines that a small firm cannot fill (an indivisibility), and buy in bulk: increasing returns. At some size these gains run out (constant returns). Beyond it, the firm becomes hard to manage: decisions are slow, information is lost between layers of managers, and workers are harder to supervise: decreasing returns. Unit 7 turns these ideas into economies and diseconomies of scale, which are about costs.

Worked example 1: testing three production functions

Start from $K = 4$ , $L = 9$ and double both to $K = 8$ , $L = 18$ .

Function	Output at (4, 9)	Output at (8, 18)	Ratio	Returns
$Q = 10K^{0.5}L^{0.5}$	$10\sqrt{36} = 60$	$10\sqrt{144} = 120$	2	Constant ( $0.5 + 0.5 = 1$ )
$Q = K^{0.75}L^{0.75}$	$36^{0.75} \approx 14.70$	$144^{0.75} \approx 41.57$	$2^{1.5} \approx 2.83$	Increasing (1.5)
$Q = 6K^{0.25}L^{0.25}$	$6 \times 36^{0.25} \approx 14.70$	$6 \times 144^{0.25} \approx 20.78$	$2^{0.5} \approx 1.41$	Decreasing (0.5)

Note: the second and third functions give the same output at (4, 9); what separates them is how output responds to scale.

Definition

Isoquant ("equal quantity"): a curve showing all the combinations of two inputs that can produce the same level of output, with given technology.

Isoquant map: a set of isoquants; one further from the origin shows a higher output.

Marginal rate of technical substitution (MRTS) of labour for capital: the amount of capital that can be given up when one more worker is employed, keeping output unchanged:

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

Example: if a firm can replace 3 machine-hours with 1 worker and still produce the same output, its MRTS is 3.

2. Isoquants

A workshop's output follows $Q = 10\sqrt{KL}$ , with K in machine-hours a day and L in workers (illustrative). Every combination with $K L = 36$ gives 60 units:

Combination	A	C	D	E	F	G	B
Labour, L	2	3	4	6	9	12	18
Capital, K	18	12	9	6	4	3	2
$MRTS = -\frac{\Delta K}{\Delta L}$	–	6	3	1.5	0.67	0.33	0.17

Reading the diagram. Each curve is one level of output. Moving down an isoquant, the firm swaps capital for labour. Near A, capital is plentiful and one extra worker replaces 6 machine-hours; near B, labour is plentiful and one extra worker replaces only a fraction of a machine-hour. This diminishing MRTS makes isoquants convex. Along the ray, the isoquants for 60, 120 and 180 units are equally spaced: doubling and tripling both inputs doubles and triples output. With increasing returns they would get closer together along the ray; with decreasing returns, further apart.

Properties of isoquants. (1) They slope downwards: to keep output constant, less of one input needs more of the other. (2) They are convex to the origin, because the MRTS diminishes. (3) A higher isoquant shows a larger output. (4) Isoquants never cross: one combination cannot give two different maximum outputs. (5) Unlike indifference curves, they carry real, measurable output numbers.

Special cases. If two inputs are perfect substitutes (two brands of fertiliser with the same nutrients), the isoquants are straight lines. If they must be used in fixed proportions (one driver per lorry), the isoquants are L-shaped.

Worked example 2: the MRTS from the production function

For $Q = 10\sqrt{KL} = 10K^{0.5}L^{0.5}$ , using calculus:

MP_L = 5K^{0.5}L^{-0.5} = 5\sqrt{\frac{K}{L}}, \qquad MP_K = 5\sqrt{\frac{L}{K}}, \qquad MRTS = \frac{MP_L}{MP_K} = \frac{K}{L}

(a) At D (4, 9): $MRTS = \frac{9}{4} = 2.25$ : one more worker can replace about 2.25 machine-hours.

(b) At F (9, 4): $MRTS = \frac{4}{9} \approx 0.44$ . The MRTS has fallen as labour replaced capital, which is why the isoquant is convex.

(c) How much capital is needed to produce 80 units with 16 workers? $10\sqrt{16K} = 80 \Rightarrow \sqrt{16K} = 8 \Rightarrow 16K = 64 \Rightarrow K = 4$ machine-hours.

Warning

Returns to scale are not returns to a factor. Diminishing marginal returns (lesson 3) happen in the short run when ONE input grows and another is fixed. Returns to scale describe the long run, when ALL inputs grow together. A firm can have diminishing returns to labour and constant returns to scale at the same time, as $Q = 10\sqrt{KL}$ does.

Tip

"Along the ray, count the gaps." Draw a straight line from the origin through an isoquant map whose curves are numbered at equal steps (60, 120, 180). Shrinking gaps mean increasing returns; equal gaps, constant; widening gaps, decreasing.

Important

Discuss. The terraces in lesson 1 were built by hand under a public works programme that aims to give poor households paid work. A private contractor might use machines instead. Both methods lie on the same isoquant. What, besides output, might a government consider when choosing between them?

Policonomics, "B.2 Isoquants | Production - Microeconomics". Watch how an isoquant is drawn from combinations of capital and labour giving the same output, and how its slope, the marginal rate of technical substitution, falls along the curve.

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Practice

Distinguish between returns to scale and returns to a variable factor.
A firm's inputs (K, L) and outputs are: (1, 2) → 100; (2, 4) → 250; (4, 8) → 500; (8, 16) → 900. State the returns to scale at each doubling.
State the returns to scale of (a) $Q = 5K^{0.4}L^{0.4}$ ; (b) $Q = 2K^{0.5}L^{0.5}$ ; (c) $Q = K^{0.7}L^{0.6}$ .
Define an isoquant and the MRTS, and explain why isoquants are convex to the origin.
These combinations of (L, K) give the same output: (1, 20), (2, 12), (3, 8), (4, 6), (5, 5). Calculate the MRTS between each pair.
With $Q = 10\sqrt{KL}$ , how many workers are needed to produce 120 units with 9 machine-hours?
State four properties of isoquants.
Explain two causes of increasing returns to scale and one cause of decreasing returns to scale.

Show answersHide answers

Returns to scale: the response of output when all inputs change in the same proportion (long run). Returns to a variable factor: the response when one input changes and others are fixed (short run).
100 → 250 (×2.5): increasing; 250 → 500 (×2): constant; 500 → 900 (×1.8): decreasing.
(a) $0.4 + 0.4 = 0.8 < 1$ : decreasing. (b) $0.5 + 0.5 = 1$ : constant. (c) $0.7 + 0.6 = 1.3 > 1$ : increasing.
An isoquant shows all combinations of two inputs giving the same output. The MRTS is the capital given up for one more worker with output unchanged. As labour replaces capital, each extra worker replaces less capital (diminishing MRTS), so the curve flattens.
8, 4, 2, 1: diminishing, so the isoquant is convex.
$10\sqrt{9L} = 120 \Rightarrow \sqrt{9L} = 12 \Rightarrow 9L = 144 \Rightarrow L = 16$ workers.
Downward sloping; convex to the origin; higher isoquants show more output; they never cross; they show measurable output.
IRS: finer division of labour; indivisible large machines used fully; bulk buying. DRS: management problems, slow decisions and poor coordination in a very large firm.

Quick check

A firm doubles all its inputs and its output rises from 500 to 1,300 units. This shows…

Key points

Returns to scale describe the long run: when all inputs rise by a proportion, output rises by more (IRS), the same (CRS) or less (DRS).
For $Q = AK^{\alpha}L^{\beta}$ , returns to scale depend on whether $\alpha + \beta$ is greater than, equal to or less than 1.
An isoquant joins input combinations that give the same output; its slope is the MRTS, $\frac{MP_L}{MP_K}$ .
Isoquants slope down, are convex (diminishing MRTS), never cross, and higher ones show more output.

Sources and credits

Photo: Road construction process in Rwanda, Alex shema, Wikimedia Commons, CC BY-SA 4.0.
Video: B.2 Isoquants | Production - Microeconomics, Policonomics, embedded from YouTube.

Written by Ganzaa: the text, the isoquant map (drawn to scale from $Q = 10\sqrt{KL}$ ), the tables, the worked examples and the practice items. All production figures are illustrative.

This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Hoes or a bulldozer?

21. Returns to scale

32. Isoquants

Hoes or a bulldozer?

1. Returns to scale

2. Isoquants