Difficulty: Intermediate | Prerequisites: Production functions, marginal products, basic calculus (partial derivatives), isoquant/isocost diagrams from earlier chapters.
Cost minimisation is the producer's analogue to the consumer's expenditure-minimisation problem. Where the consumer picks bundles on a budget to maximise utility, the firm picks inputs (labour L and capital K) to hit a production target at the lowest possible cost. This topic sits right after production theory and leads directly into cost curves, supply, and market structure. You need to be comfortable taking partial derivatives to find marginal products, and you should already know what an isoquant and an isocost line look like on a graph.
To produce a given quantity at minimum cost, set the marginal rate of technical substitution (MRTS) equal to the input price ratio (IPR), then solve for L and K. In the short run, capital is fixed, so you just plug the fixed K into the production function and solve for L. Corner solutions arise when inputs are perfect substitutes and one input is relatively cheaper per unit of output.
Marginal Rate of Technical Substitution (MRTS)
The rate at which a firm can substitute one input for another while keeping output constant. Formally, MRTS = MP_L / MP_K. Think of it as: how many units of K you can give up if you add one more unit of L, without losing any output.
Input Price Ratio (IPR)
The ratio of input prices, w/r (wage divided by rental rate of capital). In simple terms, this tells you the market trade-off between the two inputs, i.e. how much K you could rent for the cost of one unit of L.
Isoquant
A curve showing all combinations of L and K that produce the same level of output. Think of it as the producer's indifference curve.
Isocost line
A line showing all combinations of L and K that cost the same total amount. The slope of the isocost is -w/r. Think of it as the producer's budget line.
Cost minimisation condition (interior solution)
MRTS = IPR, or equivalently MP_L / MP_K = w/r. In simple terms, you keep swapping inputs until the last pound spent on each input yields the same marginal output.
Corner solution
A cost-minimising bundle where the firm uses only one input (the other is zero). This happens when inputs are perfect substitutes and one is relatively cheaper per unit of output.
Long run
The planning horizon in which all inputs are variable. The firm can freely choose both L and K.
Short run
The planning horizon in which at least one input is fixed (typically capital). The firm can only adjust labour to change output.
A Cobb-Douglas production function has the form q = A · L^a · K^b. The isoquants are smooth and convex, so cost minimisation always yields an interior solution where MRTS = IPR.
Method (step by step)
Compute the marginal products: MP_L = a · A · L^(a-1) · K^b, and MP_K = b · A · L^a · K^(b-1).
Form the MRTS = MP_L / MP_K. For q = 2L^(1/2) K^(1/2), this simplifies to K/L.
Set MRTS = IPR. With w = 10, r = 40, you get K/L = 10/40 = 1/4, so L = 4K.
Substitute L = 4K back into the production function and solve for K, then L.
Worked example (Problem 1a)
Production function: q = 2L^(1/2) K^(1/2), with w = 10, r = 40, target q = 100.
MRTS = K/L. IPR = 10/40 = 1/4. Setting K/L = 1/4 gives L = 4K.
Substituting: 100 = 2(4K)^(1/2) · K^(1/2) = 2 · 2K^(1/2) · K^(1/2) = 4K.
So K* = 25, L* = 100.
Total cost = wL + rK = 10(100) + 40(25) = 1,000 + 1,000 = 2,000.
Note that at the optimum, the firm spends equal amounts on each input. This is a general result for Cobb-Douglas with equal exponents: the expenditure shares on L and K match the exponent shares.
A linear production function has the form q = aL + bK. The isoquants are straight lines, and the MRTS is constant everywhere (MRTS = a/b). Because the isoquants are linear, the tangency condition MRTS = IPR will almost never hold exactly, so you get a corner solution.
Decision rule
If MRTS < IPR (i.e. a/b < w/r), then capital is relatively more productive per pound spent. Use only K. Set L = 0 and solve q = bK for K.
If MRTS > IPR, then labour is relatively more productive per pound spent. Use only L. Set K = 0 and solve q = aL for L.
If MRTS = IPR exactly (rare), any combination along the isoquant is equally cheap.
Worked example (Problem 2a)
Production function: q = 2L + 6K, with w = 25, r = 50, target q = 120.
MRTS = 2/6 = 1/3. IPR = 25/50 = 1/2.
Since 1/3 < 1/2, capital gives more output per pound. Set L = 0.
120 = 6K, so K* = 20. L* = 0.
Total cost = 25(0) + 50(20) = 1,000.
The intuition: K is three times as productive as L (coefficients 6 vs. 2) but only twice the price (50 vs. 25). So K is the better deal.
A quasi-linear production function is linear in one input and nonlinear in the other, e.g. q = L + sqrt(K). The isoquants are convex but have a distinctive shape: they are vertical translations of each other (shifting the same curve up or down by changing L).
The standard interior-solution method works here, but notice what happens:
Method
MP_L = 1 (constant). MP_K = 0.5 K^(-0.5) = 1/(2 sqrt(K)).
MRTS = MP_L / MP_K = 1 / (1/(2 sqrt(K))) = 2 sqrt(K).
Set MRTS = IPR. With w = 20, r = 10: 2 sqrt(K) = 20/10 = 2. So sqrt(K) = 1, giving K* = 1.
Substitute into the production function: q = L + sqrt(1) = L + 1. For q = 5, L* = 4.
Key feature
Because MP_L is constant, the optimal K does not depend on the output target. K* = 1 regardless of whether q = 5 or q = 500. All output adjustments come through L. This is a hallmark of quasi-linear production, and it matters for the short run: since K is already at its optimal value for every output level, there is no "wrong" capital stock to be stuck with.
Worked example (Problem 3a)
Production function: q = L + sqrt(K), with w = 20, r = 10, target q = 5.
K* = 1, L* = 4.
Total cost = 20(4) + 10(1) = 80 + 10 = 90.
In the long run, you choose both L and K freely, so you can always reach the tangency of the isoquant and the isocost. In the short run, K is fixed at its long-run optimal value (from a previous output target), and you can only adjust L.
Short-run procedure
Take the fixed K (from the long-run solution at the original output level).
Plug that K into the production function with the new output target.
Solve for L. That is your short-run labour demand.
Why short-run cost is higher (or equal)
With K fixed, the firm cannot re-optimise its input mix. It is stuck at the old K and must compensate entirely through L. Graphically, the short-run bundle lies on the new isoquant at the height of the old K, which is generally not the point where the new isoquant is tangent to an isocost. The firm is on a higher isocost than it would choose in the long run.
The one exception is quasi-linear production where optimal K does not depend on output. In that case, the short-run K is already optimal for the new output too, so short-run and long-run costs coincide.
Summary of the three problems
Problem 1 (Cobb-Douglas): LR bundle (100, 25) for q = 100. SR for q = 200 with K = 25: L_SR = 400. The firm overuses labour relative to the LR optimum for q = 200.
Problem 2 (Linear): LR bundle (0, 20) for q = 120. SR for q = 240 with K = 20: L_SR = 60. The firm is forced to use labour it would never use in the long run.
Problem 3 (Quasi-linear): LR bundle (4, 1) for q = 5. SR for q = 7 with K = 1: L_SR = 6. Here K* = 1 for any output level, so the short-run bundle is actually long-run optimal too.
MRTS = \frac{MP_L}{MP_K}IPR = \frac{w}{r}\text{Cost minimisation condition (interior):} \quad \frac{MP_L}{MP_K} = \frac{w}{r}\text{Total cost:} \quad C = wL + rKFor Cobb-Douglas q = A L^a K^b:
MP_L = aAL^{a-1}K^b \qquad MP_K = bAL^aK^{b-1}MRTS = \frac{a}{b} \cdot \frac{K}{L}For linear q = aL + bK:
MRTS = \frac{a}{b} \quad (\text{constant})For quasi-linear q = L + sqrt(K):
MP_L = 1 \qquad MP_K = \frac{1}{2\sqrt{K}}MRTS = 2\sqrt{K}Cost minimisation is the foundation of how firms choose their technology and staffing levels. A restaurant deciding between hiring more servers or investing in self-service kiosks is solving exactly this problem: which mix of labour and capital produces the desired output (meals served per hour) at the lowest cost? When wages rise, the IPR shifts, and the optimal mix tilts toward more capital, which is why you see more automation in high-wage economies.
Students often think MRTS = w/r means the firm equates marginal products. It does not. It equates the ratio of marginal products to the ratio of prices. MP_L = MP_K would only be correct if the inputs happened to cost the same.
Students often confuse the MRTS direction. MRTS = MP_L / MP_K, not MP_K / MP_L. Getting it backwards flips the tangency condition and produces the wrong answer.
With perfect substitutes, students sometimes try to set MRTS = IPR and solve for an interior bundle. This does not work because the MRTS is constant. You need to compare MRTS with IPR and pick a corner.
In short-run problems, students sometimes re-optimise both inputs. The whole point of the short run is that K is fixed. You only solve for L.
Cost minimisation is the building block for deriving cost curves (TC, AC, MC), which in turn drive supply and market equilibrium. If you cannot solve the input choice problem, the rest of the course breaks down.
Expect to be tested on all three production function types. The exam will likely include at least one Cobb-Douglas and one perfect-substitutes problem. Quasi-linear is less common but still fair game.
Graphing matters. Be prepared to draw isoquants, isocosts, and mark both the long-run and short-run bundles on the same diagram. Label axes (L horizontal, K vertical) and key values.
The short-run question almost always follows a long-run question. Read both parts before you start, so you carry the correct K forward.
True or false: at the cost-minimising bundle, MP_L always equals MP_K. (False. MP_L/MP_K = w/r, so they are equal only if w = r.)
Fill in the blank: when inputs are perfect substitutes and MRTS < IPR, the firm uses only ______. (Capital, K.)
True or false: short-run total cost is always strictly greater than long-run total cost for the same output. (False. It is greater or equal. It is equal when the fixed K happens to be optimal for the new output, as with quasi-linear production.)
Fill in the blank: the slope of an isocost line is ______. (-w/r.)
True or false: for a Cobb-Douglas production function with equal exponents, the firm spends equal shares of its budget on each input at the optimum. (True.)
This connects to consumer theory because cost minimisation is mathematically identical to expenditure minimisation: swap utility for output, goods for inputs, and prices for wages and rental rates. If you understood the consumer's dual problem, you already know the structure here.
Cost minimisation also feeds directly into cost curves. The minimum cost you find here, as a function of q, is the total cost function C(q). From that you derive average cost and marginal cost, which drive the firm's supply decisions in competitive and imperfectly competitive markets.
Cost minimisation, cost minimization, MRTS, marginal rate of technical substitution, input price ratio, isoquant, isocost, Cobb-Douglas, perfect substitutes, quasi-linear production, corner solution, interior solution, long run vs short run, conditional factor demands, expenditure minimisation, duality, producer theory, ECN 312, intermediate microeconomics, Arizona State University