Difficulty: Intermediate | Prerequisites: Read the companion theory notes on cost minimisation first. You should be comfortable with MRTS, IPR, and the three production function types before attempting these.
Three fully worked cost-minimisation problems, each with a long-run part, a short-run part, and graphing. Covers Cobb-Douglas (interior tangency), perfect substitutes (corner solution), and quasi-linear (interior tangency with constant optimal K). Work through each before checking the solution.
Setup: Caoimhe faces w = 10, r = 40.
Q: What is the cost-minimising choice of L and K to produce q = 100 in the long run?
A: First, find the marginal products. MP_L = L^(-1/2) K^(1/2) and MP_K = L^(1/2) K^(-1/2). The MRTS = MP_L / MP_K = K/L.
Set MRTS = IPR: K/L = w/r = 10/40 = 1/4. This gives L = 4K.
Substitute into the production function: 100 = 2(4K)^(1/2) K^(1/2) = 2 x 2K^(1/2) x K^(1/2) = 4K.
So K* = 25, L* = 4(25) = 100.
Q: What is the cost-minimising choice of L and K to produce q = 200 in the short run, with K fixed at 25?
A: K is fixed at 25 (from the long-run answer). Plug into the production function:
200 = 2 L^(1/2) (25)^(1/2) = 2 L^(1/2) x 5 = 10 L^(1/2).
So L^(1/2) = 20, and L_SR = 400.
The short-run bundle is (L, K) = (400, 25). Notice how much labour the firm needs when it cannot adjust capital.
Q: What is the total cost in the long run (q = 100) vs. the short run (q = 200)?
A: Long-run cost for q = 100: C = 10(100) + 40(25) = 2,000.
Short-run cost for q = 200: C = 10(400) + 40(25) = 4,000 + 1,000 = 5,000.
If the firm could re-optimise both inputs for q = 200 in the long run, it would set L = 4K again: 200 = 4K, K = 50, L = 200, and C = 10(200) + 40(50) = 4,000. The short-run cost (5,000) exceeds the long-run cost (4,000) by 1,000.
Setup: Scott faces w = 25, r = 50.
Q: What is the cost-minimising choice of L and K to produce q = 120 in the long run?
A: With a linear production function, MRTS = MP_L / MP_K = 2/6 = 1/3. The IPR = w/r = 25/50 = 1/2.
Since MRTS (1/3) < IPR (1/2), capital is relatively more productive per pound spent. The firm uses only K.
The reasoning: K produces 6 units of output per unit, at a price of 50, so 6/50 = 0.12 units per pound. L produces 2 units per unit, at a price of 25, so 2/25 = 0.08 units per pound. K wins.
Set L = 0: 120 = 6K, so K* = 20. L* = 0.
Q: What is the cost-minimising choice of L and K to produce q = 240 in the short run, with K fixed at 20?
A: K is fixed at 20. Plug into the production function:
240 = 2L + 6(20) = 2L + 120.
120 = 2L, so L_SR = 60.
The short-run bundle is (L, K) = (60, 20). The firm is forced to use labour, which is the more expensive input per unit of output, because it cannot increase K beyond 20.
Q: What is the total cost in each case?
A: Long-run cost for q = 120: C = 25(0) + 50(20) = 1,000.
Short-run cost for q = 240: C = 25(60) + 50(20) = 1,500 + 1,000 = 2,500.
Long-run cost for q = 240 (if K were free to adjust): L = 0, K = 240/6 = 40, C = 50(40) = 2,000. The short-run premium is 500.
Setup: Lottie faces w = 20, r = 10.
Q: What is the cost-minimising choice of L and K to produce q = 5 in the long run?
A: MP_L = 1. MP_K = 0.5 K^(-0.5) = 1/(2 sqrt(K)). So MRTS = MP_L / MP_K = 2 sqrt(K).
Set MRTS = IPR: 2 sqrt(K) = 20/10 = 2. So sqrt(K) = 1, K* = 1.
Substitute: 5 = L + sqrt(1) = L + 1. So L* = 4.
Notice that K* = 1 does not depend on the output target at all. This is the defining feature of quasi-linear production: the optimal level of the nonlinear input is pinned by prices alone.
Q: What is the cost-minimising choice of L and K to produce q = 7 in the short run, with K fixed at 1?
A: K is fixed at 1. Plug in: 7 = L + sqrt(1) = L + 1. So L_SR = 6.
The short-run bundle is (L, K) = (6, 1). Here is the interesting part: K* = 1 is also the long-run optimum for q = 7 (since K* does not depend on q). So the short-run bundle is identical to the long-run bundle, and there is no short-run cost premium.
Q: What is the total cost in each case?
A: Long-run cost for q = 5: C = 20(4) + 10(1) = 90.
Short-run cost for q = 7: C = 20(6) + 10(1) = 130.
Long-run cost for q = 7 (if K were free to adjust): K* = 1, L* = 6, C = 20(6) + 10(1) = 130. Same as the short-run cost, confirming there is no penalty from having K fixed.
Q: How do you set up the graph for these problems?
A: L goes on the horizontal axis, K on the vertical. For each problem, you need to draw:
The isoquant for the long-run output target (e.g. q = 100).
The isoquant for the short-run output target (e.g. q = 200).
The isocost line tangent to the long-run isoquant at the optimal bundle.
The isocost line passing through the short-run bundle.
A horizontal dashed line at K = K* (the fixed capital), showing the constraint in the short run.
Two dots: one at the long-run bundle (a), one at the short-run bundle (b), both labelled.
Q: What do the isoquants look like for each production function type?
A: For Cobb-Douglas, the isoquants are smooth, convex curves (the standard textbook shape). For perfect substitutes, the isoquants are straight lines with slope -a/b (here, -2/6 = -1/3). For quasi-linear q = L + sqrt(K), the isoquants are convex curves that are vertical translations of each other: they all have the same shape, just shifted left or right by the change in L.
Q: How do you read the graph to see the short-run cost penalty?
A: The short-run bundle sits on a higher isocost than the long-run optimum for the same output (unless the production function is quasi-linear). The gap between the two isocost lines represents the extra cost the firm pays for being stuck with the wrong K.
Q: A firm has production function q = 3L^(1/3) K^(2/3), with w = 12 and r = 8. Find the cost-minimising bundle for q = 90.
A: MP_L = L^(-2/3) K^(2/3). MP_K = 2 L^(1/3) K^(-1/3). MRTS = K / (2L). Set MRTS = IPR: K/(2L) = 12/8 = 3/2. So K = 3L.
Substitute: 90 = 3 L^(1/3) (3L)^(2/3) = 3 L^(1/3) x 3^(2/3) L^(2/3) = 3^(5/3) L. So L = 90 / 3^(5/3).
3^(5/3) is approximately 6.24, so L is approximately 14.4 and K is approximately 43.3. (On an exam you may leave the answer in exact form.)
Q: A firm has production function q = 4L + 2K, with w = 6 and r = 4. Find the cost-minimising bundle for q = 80.
A: MRTS = 4/2 = 2. IPR = 6/4 = 1.5. Since MRTS (2) > IPR (1.5), labour is relatively more productive per pound. Use only L. Set K = 0: 80 = 4L, L* = 20. Cost = 6(20) = 120.
Q: Explain why short-run cost can never be lower than long-run cost for the same output level.
A: In the long run, the firm chooses both inputs freely, so it can always reach the cost-minimising bundle. In the short run, one input is fixed, which restricts the firm to a subset of input combinations. Since the long-run optimum is always available as one possible choice in the long run but may not be available in the short run, the short-run cost is at least as high. Formally, the long-run problem is the short-run problem with the constraint on K removed, and removing a constraint can only reduce (or maintain) the objective value.
Q: Under what conditions does short-run cost exactly equal long-run cost?
A: When the fixed K happens to be the long-run optimal K for the new output target. This occurs whenever the optimal K does not depend on q (as with quasi-linear production), or when the new output target is the same as the original one.
Cost minimisation practice problems, worked solutions, ECN 312 problem set 3, MRTS equals IPR, Cobb-Douglas cost minimisation, perfect substitutes corner solution, quasi-linear production function, short-run vs long-run input choice, isoquant isocost diagram, conditional factor demands, intermediate microeconomics, Arizona State University