Difficulty: Intermediate | Prerequisites: Production functions, isoquants, basic calculus (partial derivatives), budget constraints.
Cost minimisation sits at the heart of the theory of the firm. You already know how a firm produces output (production functions) and what combinations of inputs can make a given quantity (isoquants). This topic asks the next question: given that the firm must pay for labour and capital, what is the cheapest way to produce a target level of output? The answer depends entirely on the shape of the production function, so this material ties together everything you have learned about Leontief, Cobb-Douglas, and perfect-substitutes technologies.
To find the total cost function TC(q), you combine two conditions: the production function (how inputs make output) and the cost-minimisation rule (where the isoquant is tangent to the isocost, or hits a corner). Fixed-proportions production uses inputs in a locked ratio, Cobb-Douglas sets the marginal rate of technical substitution equal to the input price ratio, and perfect substitutes compare the "bang for the buck" of each input and go all-in on the cheaper one. All three types in this problem set exhibit constant returns to scale, so TC(q) is linear in q.
Total cost function, TC(q)
The minimum expenditure on inputs needed to produce q units of output, expressed as a function of q. In simple terms, it answers: "What is the least I can spend to make this much stuff?"
Isoquant
A curve showing every combination of labour (L) and capital (K) that produces the same level of output. Think of it as a contour line on a production map.
Isocost line
A line showing every combination of L and K that costs the same total amount. Its slope is -w/r, where w is the wage rate and r is the rental rate of capital. Think of it as a budget line for the firm.
Marginal rate of technical substitution (MRTS)
The rate at which a firm can substitute labour for capital while keeping output constant. Equals the ratio of marginal products, MP_L / MP_K. At the cost-minimising point for a Cobb-Douglas function, MRTS = w/r.
Fixed-proportions (Leontief) production function
A production function of the form q = MIN(aL, bK). Inputs must be used in a fixed ratio; extra units of one input without extra units of the other add nothing. Think of it as a recipe where you need exactly one cup of flour per one egg, no exceptions.
Cobb-Douglas production function
A production function of the form q = L^a K^b. Inputs are substitutable along smooth, convex isoquants. The most common "well-behaved" production function in intermediate micro.
Perfect substitutes production function
A production function of the form q = aL + bK. Inputs are interchangeable at a constant rate, and isoquants are straight lines. The firm goes all-in on whichever input gives more output per pound spent.
Returns to scale
How output responds when all inputs are scaled up by the same factor t. If output scales by exactly t, returns are constant. If output scales by more than t, returns are increasing. If less, decreasing. Formally: f(tL, tK) compared with t * f(L, K).
The setup
Production function: q = MIN(aL, bK)
The firm cannot substitute one input for the other. Both "ingredients" must be present in a fixed ratio.
Isoquants are L-shaped, with the corner along the line where aL = bK.
How to find the optimal input bundle
Set aL = bK. Any point away from the corner wastes one input.
Solve for L and K in terms of q:
From q = aL: L* = q/a
From q = bK: K* = q/b
Substitute into the cost equation: TC(q) = wL* + rK* = w(q/a) + r(q/b) = (w/a + r/b)q
Worked example (Problem 1A)
q = MIN(L, 5K), so a = 1 and b = 5
w = 6, r = 20, q = 12
At the corner: L = 5K, so L* = q = 12 and K* = q/5 = 12/5 = 2.4
TC(q) = 6(q) + 20(q/5) = 6q + 4q = 10q
TC(12) = $120
The graph
The isoquant for q = 12 is an L-shape with its corner at (L = 12, K = 2.4).
The isocost line for TC = $120 passes through that corner, with intercepts at L = 20 (on the horizontal axis) and K = 6 (on the vertical axis).
The cost-minimising point is the corner of the L-shaped isoquant, where it just touches the lowest isocost line.
Returns to scale: Constant. Doubling both L and K doubles MIN(L, 5K), so output exactly doubles.
The setup
Production function: q = L^a K^b
Inputs are substitutable. Isoquants are smooth, convex curves.
The tangency condition applies: MRTS = w/r.
How to find the optimal input bundle
Compute the MRTS. For q = L^(1/2) K^(1/2): MP_L = (1/2)L^(-1/2) K^(1/2), and MP_K = (1/2)L^(1/2) K^(-1/2). So MRTS = K/L.
Set MRTS = w/r: K/L = w/r, which gives K = (w/r)L.
Substitute back into the production function to solve for L* and K* in terms of q.
Plug L* and K* into TC = wL + rK.
Worked example (Problem 1B)
q = L^(1/2) K^(1/2), w = 36, r = 4, q = 6
Tangency: K/L = 36/4 = 9, so K = 9L
Substitute into the production function: q = L^(1/2)(9L)^(1/2) = 3L
For q = 6: L* = 2, K* = 9(2) = 18
TC(q) = 36(q/3) + 4(3q) = 12q + 12q = 24q
TC(6) = $144
The graph
The isoquant for q = 6 is a smooth, downward-sloping convex curve passing through (2, 18).
The isocost line for TC = $144 is tangent to the isoquant at (2, 18), with intercepts at L = 4 (horizontal) and K = 36 (vertical).
The slope of the isocost is -w/r = -36/4 = -9.
Returns to scale: Constant. The exponents sum to 1 (1/2 + 1/2 = 1). When you double both inputs, output exactly doubles.
The setup
Production function: q = aL + bK
Inputs are perfectly interchangeable at a constant rate. Isoquants are straight lines.
There is no interior tangency. The solution is always a corner: use only L, only K, or (in the knife-edge case where costs per unit of output are equal) any mix along the isoquant.
How to find the optimal input bundle
Compare the cost of producing one unit of output using only labour vs only capital:
Cost using only L: w/a (spend w per unit of labour, get a units of output per unit of labour)
Cost using only K: r/b
If w/a < r/b, use only labour: L* = q/a, K* = 0, TC(q) = (w/a)q
If r/b < w/a, use only capital: L* = 0, K* = q/b, TC(q) = (r/b)q
If w/a = r/b, any combination on the isoquant is equally cheap.
Worked example (Problem 1C)
q = 3L + 5K, w = 2, r = 3, q = 20
Cost per unit of output from L: w/a = 2/3
Cost per unit of output from K: r/b = 3/5
Since 3/5 < 2/3, capital is cheaper per unit of output. Use only K.
K* = q/b = 20/5 = 4, L* = 0
TC(q) = (3/5)q = r(q/b) = 3(4) = $12
The graph
The isoquant for q = 20 is a straight line from (L = 20/3 = 6.67, K = 0) to (L = 0, K = 4).
The isocost line for TC = $12 touches the isoquant at the vertical-axis corner (0, 4).
The isocost is steeper than the isoquant (slope of isocost = -w/r = -2/3; slope of isoquant = -a/b = -3/5), so the firm slides to the K-intercept.
Returns to scale: Constant. Doubling both inputs doubles aL + bK, so output exactly doubles.
Leontief: q = MIN(aL, bK)
Optimal ratio: aL = bK, so L/K = b/a
L* = q/a, K* = q/b
TC(q) = (w/a + r/b)q
Cobb-Douglas: q = L^a K^b
Tangency condition: (a/b)(K/L) = w/r
Solve for K in terms of L (or vice versa), substitute into the production function, then into TC = wL + rK.
When a + b = 1 (constant returns to scale), TC(q) is linear in q.
Perfect substitutes: q = aL + bK
If w/a < r/b: use only L. TC(q) = (w/a)q
If r/b < w/a: use only K. TC(q) = (r/b)q
If w/a = r/b: any point on the isoquant. TC(q) = (w/a)q = (r/b)q
Returns to scale test
Replace L with tL and K with tK in f(L, K). If f(tL, tK) = t^n * f(L, K):
n = 1: constant returns to scale
n > 1: increasing returns to scale
n < 1: decreasing returns to scale
Leontief production functions describe processes where inputs must be combined in exact proportions: one truck per driver in logistics, one CPU per motherboard in computer assembly. Cobb-Douglas functions capture most manufacturing and service industries where firms can trade off workers for machines (more automation, fewer workers, and vice versa). Perfect substitutes arise when two energy sources (gas vs electric) can power the same process, and the firm simply picks whichever is cheaper per kilowatt-hour.
Students often think the tangency condition (MRTS = w/r) applies to all production functions. It does not. It only applies to Cobb-Douglas and other smooth, interior-solution cases. Leontief has no tangency (corner solution), and perfect substitutes have no interior tangency (also corner).
Students sometimes set L = K for a Leontief function regardless of the coefficients. The correct condition is aL = bK, which only gives L = K when a = b.
For perfect substitutes, students often split output between the two inputs. Unless the cost per unit of output is identical for both inputs, the firm uses only one.
When checking returns to scale, students sometimes scale only one input. You must scale both L and K by the same factor t simultaneously.
⚠️ You will almost certainly be asked to derive TC(q) for at least one of these three production function types. Know the method cold for each.
⚠️ Graphing the isoquant and isocost together, with the optimal bundle labelled, is a standard exam question. Practise drawing all three cases by hand.
⚠️ Returns to scale is a quick two-point question. The test is mechanical: scale inputs by t and see what happens to output. For Cobb-Douglas, it comes down to whether the exponents sum to more than, less than, or exactly 1.
⚠️ Corner solutions (Leontief and perfect substitutes) are where students lose marks. If you default to tangency for every problem, you will get these wrong.
True or false: For a Leontief production function q = MIN(2L, 4K), the cost-minimising ratio is L = 2K. True.
Fill in the blank: For a Cobb-Douglas production function, the cost-minimising condition is MRTS = ______. w/r.
True or false: With perfect substitutes q = 3L + 5K and input prices w = 10, r = 20, the firm should use only capital. **False.** Cost per unit from L = 10/3 ≈ 3.33; cost per unit from K = 20/5 = 4. Labour is cheaper per unit of output, so the firm uses only labour.
True or false: A Cobb-Douglas function q = L^(0.3) K^(0.8) has constant returns to scale. False. Exponents sum to 1.1, so it has increasing returns to scale.
Fill in the blank: For perfect substitutes, the isoquant is a ______. Straight line.
Q: A firm has the production function q = MIN(4L, 2K), with w = 8 and r = 10. Derive TC(q) and find L and K when q = 16.**
A: The optimal ratio is 4L = 2K, so K = 2L. Then L* = q/4 = 4 and K* = q/2 = 8. TC(q) = 8(q/4) + 10(q/2) = 2q + 5q = 7q. TC(16) = $112.
Q: A firm has q = L^(1/3) K^(2/3), with w = 12, r = 6. What is the cost-minimising ratio of K to L?
A: MRTS = (1/3)L^(-2/3)K^(2/3) / (2/3)L^(1/3)K^(-1/3) = K/(2L). Set K/(2L) = w/r = 12/6 = 2, so K = 4L. The firm uses four times as much capital as labour.
Q: A firm has q = 2L + 3K, with w = 9, r = 12. Should it use labour, capital, or both? What is TC(20)?
A: Cost per unit of output from L = w/a = 9/2 = 4.50. Cost per unit from K = r/b = 12/3 = 4.00. Capital is cheaper, so use only K. K* = 20/3 ≈ 6.67, L* = 0. TC(20) = (12/3)(20) = $80.
Q: Does q = L^(0.6) K^(0.6) exhibit increasing, constant, or decreasing returns to scale?
A: The exponents sum to 1.2 > 1, so increasing returns to scale. Doubling both inputs more than doubles output.
This material connects directly to profit maximisation (the next major topic): once you have TC(q), you can derive marginal cost and find the profit-maximising output level. It also links back to consumer theory, because the firm's cost-minimisation problem is mathematically identical to the consumer's expenditure-minimisation problem, with isoquants playing the role of indifference curves and isocosts playing the role of budget lines. Understanding returns to scale here will matter when you study long-run average cost curves and market structure.
Cost minimisation, total cost function, TC(q), isoquant, isocost, MRTS, marginal rate of technical substitution, Leontief production function, fixed proportions, Cobb-Douglas, perfect substitutes, linear production function, returns to scale, constant returns to scale, increasing returns to scale, decreasing returns to scale, optimal input bundle, cost-minimising input combination, corner solution, interior solution, tangency condition, input prices, wage rate, rental rate of capital, intermediate microeconomics, ECON 302, Penn State, McLeod