Difficulty: Intermediate | Prerequisites: Consumer choice theory (MRS, tangency), partial derivatives.
Cost minimisation is the producer-side mirror of utility maximisation. The firm sets MRTS equal to the input price ratio (w/r) and substitutes into the production constraint, exactly as a consumer sets MRS equal to the price ratio and substitutes into the budget constraint. With perfect substitutes in production, the firm uses only the cheaper input (per unit of output), which can create a mismatch between long-run and short-run costs.
Production theory asks: given that a firm wants to produce a target quantity, what combination of Labour and Capital minimises cost? The structure mirrors consumer theory so closely that if you have mastered the consumer problem, you already know the producer problem. The twist comes in the short run, where one input is fixed and the firm loses the flexibility to optimise. Understanding why short-run costs are weakly higher than long-run costs is one of the most-tested conceptual points in this part of the course.
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. Think of it as the firm's internal exchange rate between Labour and Capital.
Isoquant
A curve showing all combinations of inputs that produce the same level of output. It is the producer equivalent of an indifference curve. An isoquant does not represent equal cost (that is the isocost line).
Isocost line
A line showing all combinations of inputs that cost the same total amount: wL + rK = C. Slope = -w/r.
Cost minimisation condition
At the optimal input mix, MRTS = w/r. This is the producer-side tangency condition.
Long run vs. short run
In the long run, both inputs are variable and the firm can fully optimise. In the short run, Capital is fixed, and the firm can only adjust Labour. Short-run costs are always weakly higher than long-run costs because the firm has less flexibility.
Perfect substitutes in production
A production function of the form q = aL + bK, where inputs are interchangeable at a constant rate. The firm uses only the input with the lower cost per unit of output.
For q = 2L^(1/2) K^(1/2), w = 10, r = 40, target q = 100:
Step 1: Derive MRTS. MP_L = L^(-1/2) K^(1/2) and MP_K = L^(1/2) K^(-1/2), so MRTS = MP_L / MP_K = K/L.
Step 2: Set MRTS = w/r. K/L = 10/40 = 1/4, giving L = 4K.
Step 3: Substitute into the production function. 100 = 2(4K)^(1/2) K^(1/2) = 2 · 2K^(1/2) · K^(1/2) = 4K, so K = 25 and L = 100.
The optimality condition is K/L = w/r. If the wage doubles (w goes from 10 to 20), the new condition is K/L = 20/40 = 1/2. The ratio L/K was 4 and is now 2, so it halves. Intuitively, Labour has become relatively more expensive, so the firm substitutes toward Capital.
For q = 2L + 6K, w = 25, r = 50, target q = 120:
Step 1: Compute MRTS. MP_L / MP_K = 2/6 = 1/3.
Step 2: Compare to w/r. w/r = 25/50 = 1/2.
Since MRTS < w/r (1/3 < 1/2), Capital delivers more output per dollar than Labour. The firm uses only Capital: K = 120/6 = 20, L = 0.
The logic is identical to perfect substitutes in consumption: compare the marginal rate to the price ratio and go to a corner.
Suppose Capital is fixed at K = 20 (the long-run optimal value for q = 120). Now the firm needs q = 240.
With K fixed at 20: 240 = 2L + 6(20) = 2L + 120, so 2L = 120 and L = 60.
In the long run, the firm would use only Capital: K = 240/6 = 40, L = 0, at a cost of 40 × 50 = 2,000. In the short run, the cost is 25(60) + 50(20) = 1,500 + 1,000 = 2,500. The short-run cost is higher because the firm cannot adjust its Capital.
This illustrates the general principle: short-run costs are always greater than or equal to long-run costs. Equality holds only at the output level where the fixed input happens to be at its long-run optimal value.
If the production function were q = min(2L, 6K), inputs would be used in a fixed ratio. Set 2L = 6K, giving L = 3K. Regardless of input prices, the firm always uses three units of Labour per unit of Capital.
Concept | Formula |
|---|---|
MRTS (Cobb-Douglas q = AL^a K^b) | (a/b) · (K/L) |
Cost minimisation condition | MRTS = w/r |
MRTS (linear q = aL + bK) | a/b (constant) |
Fixed proportions kink | aL = bK for q = min(aL, bK) |
Short-run cost (K fixed) | C_SR = wL + rK_bar |
Cost minimisation is how firms decide between automation and hiring. A rise in wages (higher w) shifts the optimal mix toward more Capital, which is precisely the trade-off manufacturers face when evaluating robotic assembly lines against human workers.
Students often confuse isoquants with isocost lines. An isoquant shows equal output. An isocost line shows equal cost. The exam tests this distinction directly.
When MRTS > w/r, students sometimes conclude the firm should use more Capital. The opposite is true: MRTS > w/r means Labour is relatively more productive per dollar, so the firm should use more Labour.
A persistent error is claiming that short-run costs can be lower than long-run costs. They cannot. The long run is strictly more flexible, so costs are weakly lower.
With perfect substitutes in production, students sometimes try to use the tangency condition. There is no tangency: compare MRTS to w/r and go to a corner.
⚠️ Deriving MRTS and setting it equal to w/r is tested in short-answer questions. The algebra parallels consumer MRS problems.
⚠️ The effect of a wage change on the L/K ratio is a common multiple-choice question. Remember: L/K moves inversely with w/r for Cobb-Douglas.
⚠️ Short-run vs. long-run cost comparison is a 10-mark question. Show the numerical cost difference and explain why.
⚠️ True/False: "Short-run costs are always lower than or equal to long-run costs" is False. This trips up students who confuse the direction.
True or False: An isoquant represents all input combinations with the same total cost. (False, that is an isocost line)
Fill in the blank: If MRTS > w/r, the firm should use more ____ and less ____. (Labour, Capital)
True or False: Short-run costs are always greater than or equal to long-run costs. (True)
Fill in the blank: For q = min(2L, 6K), the fixed input ratio is L = ____K. (3)
True or False: With perfect substitutes in production, the firm may use both inputs. (False, except the knife-edge case)
Q: For q = 2L^(1/2) K^(1/2), w = 10, r = 40, find the optimal L and K to produce q = 100.
A: MRTS = K/L. Set K/L = 10/40 = 1/4, so L = 4K. Substitute: 100 = 2(4K)^(1/2) K^(1/2) = 4K, so K = 25, L = 100.
Q: If w doubles to 20 while r stays at 40, what happens to L/K?
A: New condition: K/L = 20/40 = 1/2, so L = 2K. The ratio L/K falls from 4 to 2 (halved).
Q: For q = 2L + 6K, w = 25, r = 50, what is the long-run choice to produce q = 120?
A: MRTS = 2/6 = 1/3, w/r = 1/2. Since MRTS < w/r, use only Capital: K = 20, L = 0.
Q: With K fixed at 20, how much Labour is needed for q = 240?
A: 240 = 2L + 6(20) = 2L + 120, so L = 60.
Cost minimisation is the dual of profit maximisation. A firm that minimises cost for a given output can then choose the profit-maximising output level, connecting this material to monopoly and perfect competition.
The MRTS = w/r condition is structurally identical to MRS = P_X/P_Y in consumer theory. Practising one reinforces the other.
The short-run constraint ties into the theory of short-run supply curves in perfect competition, where the firm's fixed costs determine the shutdown decision.
MRTS, marginal rate of technical substitution, isoquant, isocost, cost minimisation, input choice, Cobb-Douglas production, perfect substitutes in production, fixed proportions, Leontief production, short run, long run, fixed capital, variable labour, ECN 312, intermediate microeconomics, ASU, producer theory