Source: Seminar Practice Questions, Texas A&M University
Tags: cost minimisation, isoquant, isocost, MRTS, marginal rate of technical substitution, tangency condition, long-run average cost, short-run average cost, input mix, labour, capital, microeconomics, Chapter 7
The second half of Chapter 7 covers how firms choose the cheapest combination of inputs to produce a given output (cost minimisation) and why long-run average cost can never exceed short-run average cost. The key tools are isoquants, isocost lines, and the tangency condition linking the marginal rate of technical substitution to input price ratios.
Isoquant
A curve showing all combinations of inputs (typically labour and capital) that produce the same level of output. Convex isoquants reflect diminishing marginal rate of technical substitution.
Isocost line
A line showing all combinations of inputs a firm can purchase for a given total expenditure. Its slope equals the negative of the input price ratio (−w/r, where w is the wage and r is the rental rate of capital).
Marginal rate of technical substitution (MRTS)
The rate at which a firm can substitute one input for another while keeping output constant. MRTS = MP_L / MP_K. At the cost-minimising point, MRTS = w / r.
Cost minimisation (tangency condition)
The firm minimises cost where the isoquant is tangent to the lowest attainable isocost line. At tangency: MP_L / w = MP_K / r, meaning the last pound spent on each input yields the same additional output.
Long-run average cost (LRAC)
The lowest possible average cost of producing each output level when all inputs are variable. The LRAC curve is the envelope of short-run average cost curves.
Short-run average cost (SRAC)
Average cost when at least one input (usually capital) is fixed. SRAC ≥ LRAC at every output level because the firm cannot adjust all inputs freely.
All three describe the same optimum. Exam questions often list them as separate options and ask whether all are correct.
Equal marginal product per pound spent: the last dollar on labour yields the same extra output as the last dollar on capital. Formally, MP_L / w = MP_K / r.
Lowest isocost line touching the isoquant: the firm finds the cheapest budget line that still reaches the required output level.
Tangency of isoquant and isocost: at the optimum, the slopes match. The isoquant slope (MRTS) equals the isocost slope (w/r).
All three are correct, so "all of the above" is the right answer when they appear together.
At the cost-minimising point, MRTS = w / r.
If MRTS = 5 and w = $10:
5 = 10 / r
r = 10 / 5 = $2
The rental rate of capital is $2. This is a direct plug-in once you know the tangency condition.
If labour costs 5 times as much as capital (w = 5r), and isoquants are convex, and the firm does not change its input mix, then the tangency condition holds:
MRTS = w / r = 5
This means MP_L / MP_K = 5, so: 5 × MP_K = MP_L
The firm equates the marginal product ratio to the price ratio. The statement 5 × MP_K = MP_L captures this. It does not tell us the firm hires 3 times as much of either input, nor that it avoids labour entirely.
In the short run, at least one input is fixed. The firm may be stuck with too much or too little capital for a given output level, so it cannot reach the cheapest input combination.
In the long run, all inputs are variable. The firm can adjust every input to achieve the tangency condition at every output level. Because the long-run firm has strictly more flexibility, it can always do at least as well as the short-run firm.
The correct explanation: in the long run, tangency of the isocost and isoquant is always attainable. In the short run, a fixed input may prevent the firm from reaching that tangency.
This is not about the shape of the cost curve (U-shaped or otherwise), nor about paying down debts, nor about whether diseconomies of scale occur.
Condition | Formula |
|---|---|
Tangency (cost min) | MRTS = w / r |
Equal bang per buck | MP_L / w = MP_K / r |
MRTS definition | MRTS = MP_L / MP_K |
Solving for r | r = w / MRTS |
LRAC envelope | LRAC ≤ SRAC at every q |
⚠️ The three statements of cost minimisation (equal marginal product per dollar, lowest isocost touching isoquant, tangency) are all equivalent. "All of the above" is correct when they appear as separate options.
⚠️ MRTS = w / r is the tangency condition. To find an unknown input price, rearrange: r = w / MRTS. This is a very common numerical question.
⚠️ When told the input price ratio and that the firm does not change its mix, jump to 5 × MP_K = MP_L (or whatever the ratio is). Do not assume a specific quantity ratio between inputs without more information.
⚠️ LRAC ≤ SRAC because long-run flexibility allows full input adjustment (tangency is attainable). The other explanations offered in multiple-choice (cost curve shape, debt repayment, diseconomies) are distractors.
Q: List three equivalent conditions for cost minimisation.
A: (1) MP_L / w = MP_K / r, (2) the lowest isocost line touching the isoquant, (3) the isoquant is tangent to the isocost line.
Q: MRTS = 5, wage = $10. What is the rental rate of capital?
A: $2. From MRTS = w / r: r = 10 / 5 = 2.
Q: Labour costs 5 times as much as capital, isoquants are convex, and the firm keeps its input mix unchanged. What can we conclude?
A: 5 × MP_K = MP_L. The marginal product ratio equals the input price ratio at the cost-minimising tangency.
Q: Why is long-run average cost never greater than short-run average cost?
A: In the long run all inputs are variable, so the firm can always achieve the cost-minimising tangency of isoquant and isocost. In the short run a fixed input may prevent reaching that tangency, so costs are at least as high.
Q: A firm uses two inputs. In the short run, capital is fixed. Could the firm's short-run cost be equal to its long-run cost at some output level?
A: Yes. At the output level where the fixed capital happens to be exactly the cost-minimising amount, SRAC = LRAC. At all other output levels, SRAC > LRAC.
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