Source: Intermediate Microeconomics, Problem Set 3 (Questions 2 & 4)
Tags: cost minimisation, least-cost input combination, Cobb-Douglas production function, marginal product, isoquant, isocost, MRTS, long-run equilibrium, zero economic profit, average cost minimisation, number of firms, market demand
A firm minimises cost by choosing inputs so that the ratio of marginal products equals the ratio of input prices (the "bang for your buck" condition). In the long run, free entry and exit drive price down to the minimum of average cost, leaving every firm with zero economic profit. The number of firms in the industry is simply total market demand divided by each firm's output.
Least-cost input combination
The mix of inputs (K, L) that produces a given output at the lowest total cost. Found where the isoquant is tangent to the isocost line.
MRTS (marginal rate of technical substitution)
The rate at which the firm can swap one input for another while keeping output constant. MRTS = MPL / MPK. At the cost minimum, MRTS = w / r.
Isoquant
A curve showing all input combinations that produce the same level of output. Analogous to an indifference curve in consumer theory.
Isocost line
A line showing all input combinations that cost the same total amount. Slope = −w / r.
Cobb-Douglas production function
A production function of the form q = A × K^α × L^β. It has smooth, convex isoquants and well-behaved marginal products.
Long-run equilibrium (perfectly competitive market)
The state where no firm has an incentive to enter or exit the industry. Conditions: P = min AC, economic profit = 0, and each firm produces at the efficient scale.
Zero economic profit
Revenue exactly covers all costs, including the opportunity cost of the owner's capital and time. The firm earns a normal return but no surplus.
Given:
Production function: q = 4K^0.5 × L^0.5
MPL = 2K^0.5 / L^0.5
MPK = 2L^0.5 / K^0.5
w = 4, r = 1
Target output: q = 40
The cost-minimisation condition:
MPL / MPK = w / r
Substituting the marginal products:
(2K^0.5 / L^0.5) / (2L^0.5 / K^0.5) = 4 / 1
Simplify the left side:
(2K^0.5 / L^0.5) × (K^0.5 / 2L^0.5) = K / L
So: K / L = 4, which gives K = 4L.
This is the expansion path, the locus of cost-minimising input combinations for any output level given these prices.
Substitute into the production function:
40 = 4 × (4L)^0.5 × L^0.5
40 = 4 × 2L^0.5 × L^0.5
40 = 8L
L = 5, K = 20
Total cost:
TC = wL + rK = 4(5) + 1(20) = 20 + 20 = 40
Now capital is fixed at K = 16. The firm still needs to produce 40 units.
Substitute K = 16 into the production function:
40 = 4 × (16)^0.5 × L^0.5
40 = 4 × 4 × L^0.5
40 = 16 × L^0.5
L^0.5 = 2.5
L = 6.25, K = 16
Total cost:
TC = wL + rK = 4(6.25) + 1(16) = 25 + 16 = 41
The combination from 2b (K = 16, L = 6.25) represents the short run. Capital is fixed and cannot be adjusted to the cost-minimising level.
Notice the cost difference:
Long-run (optimal) TC = 40
Short-run (constrained) TC = 41
The short-run cost is higher because the firm is forced to use a sub-optimal input mix. With K stuck at 16 instead of the optimal 20, the firm compensates by hiring more labour (6.25 instead of 5). Labour is the more expensive input per unit of marginal product at this mix, so total cost rises.
This illustrates a general principle: the long-run cost of producing any given output is always less than or equal to the short-run cost, because the firm has more flexibility in the long run.
Given:
Cost function: C = 10,000q − 100q² + 1.25q³
Marginal cost: MC = 10,000 − 200q + 3.75q²
Market demand: Q = 1,000 − 0.005p
In long-run equilibrium, two conditions hold simultaneously:
P = min AC (zero economic profit, no incentive to enter or exit)
MC = AC (MC always crosses AC at its minimum)
Step 1: Derive AC
AC = C / q = 10,000 − 100q + 1.25q²
Step 2: Find the minimum of AC
Take the derivative and set it to zero:
dAC/dq = −100 + 2.5q = 0
q = 40
Step 3: Calculate AC at q = 40
AC = 10,000 − 100(40) + 1.25(40²)
AC = 10,000 − 4,000 + 1.25(1,600)
AC = 10,000 − 4,000 + 2,000
AC = 8,000
Step 4: Verify MC = AC at q = 40
MC = 10,000 − 200(40) + 3.75(40²)
MC = 10,000 − 8,000 + 3.75(1,600)
MC = 10,000 − 8,000 + 6,000
MC = 8,000 ✓
So in long-run equilibrium: P = 8,000 and each firm produces q = 40.
Step 1: Find total market quantity at P = 8,000
Q = 1,000 − 0.005(8,000)
Q = 1,000 − 40
Q = 960
Step 2: Divide by output per firm
Number of firms = Q / q = 960 / 40 = 24 firms
Formula | Meaning |
|---|---|
MPL / MPK = w / r | Cost-minimisation condition (tangency) |
MRTS = MPL / MPK | Marginal rate of technical substitution |
TC = wL + rK | Total cost with two inputs |
AC = C / q | Average cost |
dAC/dq = 0 | Condition for minimum AC |
P = min AC | Long-run equilibrium price |
MC = AC at min AC | MC crosses AC at its lowest point |
N = Q_market / q_firm | Number of firms in long-run equilibrium |
⚠️ The cost-minimisation condition MPL/MPK = w/r is the producer's equivalent of MUx/MUy = Px/Py in consumer theory. Same logic: equalise the marginal product per pound spent across all inputs.
⚠️ When capital is fixed, the firm generally cannot hit the cost-minimising input ratio. This is exactly what makes short-run costs higher than long-run costs for the same output level.
⚠️ In long-run equilibrium, profit is zero. This does not mean the firm is struggling. It means it earns a normal return, exactly covering all opportunity costs. Positive economic profit would attract entrants; negative economic profit would cause exits.
⚠️ To find the long-run equilibrium, you need to minimise AC (not MC). Set dAC/dq = 0, solve for q, then evaluate AC at that q to get the equilibrium price.
⚠️ The market demand function gives you total industry output Q. Divide Q by each firm's output q to get the number of firms. This only works when all firms are identical.
⚠️ Common error: confusing the firm's demand with market demand. The firm is a price taker facing a horizontal demand curve at P. The market demand curve is downward-sloping.
Q: What does the cost-minimisation condition MPL/MPK = w/r mean in plain language?
A: The last pound spent on labour should yield the same additional output as the last pound spent on capital. If it does not, the firm can cut costs by shifting spending toward the input with the higher marginal product per pound.
Q: Why is the total cost higher in Question 2b (K = 16) than in 2a (K = 20)?
A: Because K = 16 is not the cost-minimising level of capital for producing 40 units. The firm overuses labour to compensate, and since the input ratio K/L = 4 is not satisfied, the firm sits on a higher isocost line. The constraint on capital forces a costlier input mix.
Q: In long-run equilibrium, why does P = min AC?
A: If P > min AC, firms earn positive profit, attracting entry. New firms increase market supply, pushing P down. If P < min AC, firms make losses and exit, reducing supply and pushing P up. The process stops when P = min AC and profit is zero.
Q: How do you find the number of firms in a competitive industry in the long run?
A: First find the long-run equilibrium price (P = min AC) and each firm's output (q where AC is minimised). Then plug P into the market demand function to get total quantity Q. Number of firms = Q / q.
Q: A Cobb-Douglas production function is q = 10K^0.3 L^0.7, with w = 14 and r = 3. What is the cost-minimising ratio of K to L?
A: MPL/MPK = (0.7 × q/L) / (0.3 × q/K) = 0.7K / 0.3L = 7K / 3L. Set equal to w/r = 14/3. So 7K / 3L = 14/3, giving K/L = 2, or K = 2L.
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