Production Theory

Lessons

Senior 4 · Economics

Production Theory

5 lessons

Lesson 5 of 5

Isocost lines and producer equilibrium

30 min

Two carpenters or one machine?

A furniture workshop in Kigali wins a contract for school desks. It can meet it by hiring more carpenters, by renting a power saw and planer, or by some mix of the two. Lesson 4 showed that many mixes can produce the same output. Which one should the workshop pick?

The answer depends on what each input costs. NISR's Labour Force Survey found that employees in Rwandan industry earned on average Frw 849 an hour in cash in 2025. If labour is cheap compared with machines, the workshop leans towards labour; if machines become cheaper, it leans towards capital. This lesson draws input prices as an isocost line and finds the cheapest mix.

Definition

Isocost line ("equal cost"): a line showing all the combinations of two inputs that a firm can buy for the same total cost, at given input prices:

C = wL + rK \qquad\Rightarrow\qquad K = \frac{C}{r} - \frac{w}{r}L

where $C$ is total cost, $w$ the wage per unit of labour and $r$ the price per unit of capital (for example, the hire of a machine for an hour). The intercepts are $\frac{C}{w}$ on the labour axis and $\frac{C}{r}$ on the capital axis; the slope is $-\frac{w}{r}$ .

Example: with $w = 800$ , $r = 3,200$ and $C = 19,200$ , the firm can hire 24 worker-hours, or 6 machine-hours, or any mix on the line between them.

1. The isocost line and how it moves

The isocost line does for a producer what the budget line did for a consumer in unit 5. It moves in the same way:

A larger budget (C rises), prices unchanged: a parallel shift outwards.
A change in one input price: a pivot about the other intercept. A wage rise pivots the line inwards along the labour axis; cheaper machine hire pivots it outwards along the capital axis.
Both prices change in the same proportion: a parallel shift, with the slope $-\frac{w}{r}$ unchanged.

2. The least-cost combination

The workshop in lesson 4 produces $Q = 10\sqrt{KL}$ . Suppose labour costs $w =$ Frw 800 per worker-hour, close to NISR's 2025 average for industry, and a machine costs $r =$ Frw 3,200 per hour to hire (illustrative). The order is 60 units. Here is the cost of each combination on the 60-unit isoquant:

Combination (L, K)	(4, 9)	(6, 6)	(9, 4)	(12, 3)	(18, 2)	(24, 1.5)
Labour cost, $800 L$	3,200	4,800	7,200	9,600	14,400	19,200
Capital cost, $3,200 K$	28,800	19,200	12,800	9,600	6,400	4,800
Total cost, C (Frw)	32,000	24,000	20,000	19,200	20,800	24,000

Reading the diagram. The firm wants the lowest isocost line that still reaches the 60-unit isoquant. Frw 14,400 cannot buy enough inputs. At A or B, Frw 24,000 is spent. At E the isocost line of Frw 19,200 just touches the isoquant: 12 worker-hours and 3 machine-hours, the cheapest way to make 60 units, as the table confirms.

Definition

Producer equilibrium (least-cost combination): the combination of inputs at which a firm produces a given output at the lowest possible cost, or, equivalently, the greatest output for a given cost. It is where an isoquant is tangent to an isocost line:

MRTS_{LK} = \frac{MP_L}{MP_K} = \frac{w}{r} \qquad\Leftrightarrow\qquad \frac{MP_L}{w} = \frac{MP_K}{r}

In words: the last franc spent on labour adds as much output as the last franc spent on capital.

3. Conditions for producer equilibrium

Tangency: the slope of the isoquant equals the slope of the isocost, $MRTS_{LK} = \frac{w}{r}$ .
Convexity: the isoquant is convex at that point (MRTS diminishing), so the tangency is a minimum of cost, not a maximum.
The constraint holds: the firm is on the required isoquant (cost-minimising) or on its budget isocost (output-maximising).

Why must the "output per franc" be equal? If $\frac{MP_L}{w} > \frac{MP_K}{r}$ , a franc moved from capital to labour raises output, so the firm should hire more labour and less capital until the ratios are equal. This is the same equi-marginal reasoning a consumer uses, with output in place of utility.

Changes in input prices: factor substitution. If capital becomes cheaper, the isocost line becomes steeper ( $\frac{w}{r}$ rises) and the tangency moves to a point that uses more capital and less labour.

Reading the diagram. When machine hire falls from Frw 3,200 to Frw 1,800, $\frac{w}{r}$ rises from 0.25 to about 0.44. The cheapest way to make 60 units moves from E₁ (12, 3) to E₂ (9, 4), and the cost falls from Frw 19,200 to Frw 14,400. The firm substitutes capital for labour. This is why the relative price of labour and capital helps explain why some economies use hoes and others bulldozers.

The expansion path. If input prices stay fixed and the firm grows, it moves to the least-cost point on each higher isoquant. The line joining these points is the expansion path. For $Q = 10\sqrt{KL}$ with $\frac{w}{r} = 0.25$ , every least-cost point has $K = 0.25 L$ , so the expansion path is a straight ray from the origin: (12, 3) for 60 units, (24, 6) for 120 units.

Worked example 1: finding the least-cost combination with algebra

For $Q = 10\sqrt{KL}$ , lesson 4 showed that $MRTS = \frac{K}{L}$ . With $w = 800$ and $r = 3,200$ , find the least-cost way to make 60 units.

Step 1: tangency. $\frac{K}{L} = \frac{800}{3{,}200} = 0.25 \Rightarrow K = 0.25L$ .

Step 2: output. $10\sqrt{L(0.25L)} = 60 \Rightarrow \sqrt{0.25L^2} = 6 \Rightarrow 0.5L = 6 \Rightarrow L = 12$ , so $K = 3$ .

Step 3: cost. $C = 800 (12) + 3,200 (3) = 9,600 + 9,600 = 19,200$ francs.

Check with output per franc: at (12, 3), $MP_L = 5\sqrt{\frac{3}{12}} = 2.5$ and $MP_K = 5\sqrt{\frac{12}{3}} = 10$ . Then $\frac{MP_L}{w} = \frac{2.5}{800} = 0.003125$ and $\frac{MP_K}{r} = \frac{10}{3{,}200} = 0.003125$ . Equal ✔.

Worked example 2: is the firm using the right mix?

A maize mill hires labour at Frw 500 an hour and machine time at Frw 1,500 an hour. At its present mix, the marginal product of labour is 20 kg of flour an hour and that of machine time 40 kg.

Output per franc: labour $\frac{20}{500} = 0.04$ kg; machine time $\frac{40}{1{,}500} \approx 0.027$ kg.

Decision: labour gives more flour per franc, so the mill should use more labour and less machine time. As it does, $M P_{L}$ falls and $M P_{K}$ rises (diminishing returns) until $\frac{MP_L}{w} = \frac{MP_K}{r}$ .

Warning

The cheapest input is not always the best buy. A firm compares output per franc, $\frac{MP}{\text{price}}$ , not prices alone and not marginal products alone. And the slope of the isocost is $-\frac{w}{r}$ only when labour is on the horizontal axis.

Tip

Same picture, new names. Consumer: indifference curve + budget line, $MRS = \frac{P_x}{P_y}$ . Producer: isoquant + isocost line, $MRTS = \frac{w}{r}$ . Learn one diagram and you have both.

Important

Discuss. Suppose that, as in many low-income countries, wages are low while machines are imported and the loans to buy them are expensive. Use the isocost line to explain why firms would choose labour-intensive methods. Is this choice always good for the country in the long run?

Guy Pascale, "Isoquant Isocost Cost Minimization". Watch for the comparison of points on a dearer isocost line with the single point on a cheaper one, and for the condition that, at the least-cost point, the ratio of marginal products equals the ratio of input prices.

✦

Practice

Define an isocost line and write its equation.
A firm pays Frw 1,000 per worker-hour and Frw 2,500 per machine-hour, and has Frw 50,000 to spend. Find the intercepts and the slope of its isocost line.
For the firm in question 2, describe the effect on the isocost line of (a) a budget of Frw 75,000; (b) a wage of Frw 2,000.
State the conditions for producer equilibrium.
At a firm's present mix, $M P_{L} = 30$ and $M P_{K} = 60$ ; $w = 600$ and $r = 1,500$ . Is the firm using the least-cost mix? If not, how should it change?
For $Q = 10\sqrt{KL}$ with $w = 400$ and $r = 1,600$ , find the least-cost combination for 80 units and its total cost.
What is an expansion path? Sketch the expansion path of the firm in question 6.
Explain, using isocost lines, how a rise in wages affects the mix of inputs a firm chooses.

Show answersHide answers

An isocost line shows all combinations of two inputs costing the same total at given prices: $C = w L + r K$ .
Labour intercept $\frac{50{,}000}{1{,}000} = 50$ ; capital intercept $\frac{50{,}000}{2{,}500} = 20$ ; slope $-\frac{1{,}000}{2{,}500} = -0.4$ .
(a) Parallel outward shift to intercepts 75 and 30. (b) Pivot inwards about the capital intercept: labour intercept falls to 25; slope becomes −0.8.
The isoquant is tangent to the isocost ( $MRTS = \frac{w}{r}$ , so $\frac{MP_L}{w} = \frac{MP_K}{r}$ ); the isoquant is convex at that point; the firm is on its required isoquant or its budget isocost.
$\frac{30}{600} = 0.05$ and $\frac{60}{1{,}500} = 0.04$ . Not least-cost: labour gives more output per franc, so it should use more labour and less capital.
$\frac{K}{L} = \frac{400}{1{,}600} = 0.25 \Rightarrow K = 0.25L$ ; $10\sqrt{0.25L^2} = 80 \Rightarrow 5L = 80 \Rightarrow L = 16,\ K = 4$ . Cost $= 400 (16) + 1,600 (4) = 6,400 + 6,400 = 12,800$ francs.
The line joining least-cost points on successively higher isoquants, with input prices fixed. Here it is the ray $K = 0.25 L$ through (16, 4) and (32, 8).
A wage rise raises $\frac{w}{r}$ , so the isocost line becomes steeper and pivots inwards along the labour axis. The tangency with the isoquant moves to a point with less labour and more capital: the firm substitutes capital for labour, and the cost of producing the same output rises.

Quick check

Key points

An isocost line $C = w L + r K$ shows input combinations of equal cost; its slope is $-\frac{w}{r}$ .
Producer equilibrium is where an isoquant is tangent to the lowest attainable isocost: $MRTS = \frac{w}{r}$ , or $\frac{MP_L}{w} = \frac{MP_K}{r}$ .
A change in relative input prices makes the firm substitute the cheaper input for the dearer one.
The expansion path joins least-cost points as output grows at fixed input prices.

1Two carpenters or one machine?

21. The isocost line and how it moves

32. The least-cost combination

43. Conditions for producer equilibrium

Two carpenters or one machine?

1. The isocost line and how it moves

2. The least-cost combination

3. Conditions for producer equilibrium