Lecture 6 · Week 8 · Besanko & Braeutigam ch. 6 · about 45 min

Inputs and Production Functions

The firm's side of the market: how output relates to labour and capital, marginal and average product, the law of diminishing marginal returns, isoquants and the marginal rate of technical substitution, and returns to scale.

By the end you can

Compute marginal product and average product of labour from a production function
Explain the law of diminishing marginal returns and locate where it sets in
Compute the marginal rate of technical substitution and explain diminishing MRTS
Classify a production function's returns to scale

The production function

A production function describes the maximum output a firm can produce from given quantities of inputs, typically labour $L$ and capital $K$ .

Production functionFormula sheet →

Q = F(L, K)

$Q$: output
$L$: labour
$K$: capital

Use it when you need the general relationship between inputs and maximum feasible output, before specifying a functional form.

The short run is a period in which at least one input (usually capital) is fixed; the long run is long enough that every input can be varied. This lesson’s marginal/average product analysis is a short-run story; returns to scale, at the end, is inherently a long-run concept, since it requires every input to change together.

Marginal and average product

Marginal and average product of labourKey formulasFormula sheet →

MP_L = \frac{\partial Q}{\partial L} \qquad AP_L = \frac{Q}{L}

$MP_L$: extra output from one more unit of labour, holding capital fixed
$AP_L$: output per worker

Use it when you need to measure labour productivity either at the margin (the next worker) or on average (across all current workers).

The law of diminishing marginal returns

Law of diminishing marginal returnsKey lawFormula sheet →

\text{As } L \text{ rises with } K \text{ fixed, } MP_L \text{ eventually falls}

Use it when a question asks why marginal product does not keep rising forever, or asks you to identify where diminishing returns set in on a production function.

Diminishing marginal returns is not a claim that $MP_L$ is always falling — only that it eventually must, once capital is fixed and workers keep being added. Early workers may make each other more productive (specialisation); eventually, workers start competing for the same fixed machines and floor space, and $MP_L$ turns down.

Isoquants and the marginal rate of technical substitution

An isoquant shows every combination of $L$ and $K$ that produces the same output — the production-side twin of an indifference curve.

Marginal rate of technical substitutionKey formulaFormula sheet →

MRTS_{LK} = -\frac{dK}{dL}\bigg|_{Q = \bar{Q}} = \frac{MP_L}{MP_K}

$MRTS_{LK}$: the rate at which a firm can substitute labour for capital while holding output constant

Use it when you need the slope of an isoquant, or the rate at which a firm could give up capital for labour without changing output.

For a Cobb-Douglas production function $Q = AL^aK^b$ , exactly the same algebra as the Cobb-Douglas utility case gives $MRTS_{LK} = (aK)/(bL)$ .

Just as diminishing MRS shapes indifference curves, diminishing MRTS — MRTS falling as you move along an isoquant toward more labour and less capital — is what gives isoquants their typical convex (bowed-in) shape.

Returns to scale

Returns to scale asks what happens to output when every input is scaled up by the same factor — inherently a long-run question, since fixed inputs are, by definition, not being scaled.

Returns to scaleKey rulesFormula sheet →

F(tL, tK) = t^k F(L,K):\quad k=1 \text{ constant} \qquad k \gt 1 \text{ increasing} \qquad k \lt 1 \text{ decreasing}

$t$: the common scaling factor applied to every input
$k$: the returns-to-scale exponent

Use it when you double (or scale by any factor) every input at once and need to classify how output responds — a much broader question than diminishing marginal returns, which holds only one input fixed.

For Cobb-Douglas $Q = AL^aK^b$ , the returns-to-scale exponent is simply $k = a+b$ .

Exam practice

Summary and review

$MP_L = \partial Q/\partial L$ (next worker’s contribution); $AP_L = Q/L$ (output per worker).
Law of diminishing marginal returns: with capital fixed, $MP_L$ eventually falls as L rises.
MP crosses AP exactly at AP’s maximum.
$MRTS_{LK} = MP_L/MP_K$ , the slope of an isoquant; diminishing MRTS gives isoquants their convex shape.
Returns to scale: scale every input by $t$ ; $k=1$ constant, $k \gt 1$ increasing, $k \lt 1$ decreasing. For Cobb-Douglas, $k = a+b$ .
Diminishing marginal returns (short run, one input) and returns to scale (long run, all inputs) are different concepts.