Difficulty: Intermediate | Prerequisites: Profit maximisation basics (Lecture 11), production functions, Cobb-Douglas form, partial derivatives.
This lecture bridges two major ideas in the theory of the firm: profit maximisation and supply. Up to this point, the course analysed how a firm chooses inputs to maximise profit given a price. The problem is that, under perfect competition, only decreasing-returns-to-scale production functions yield a well-defined profit maximum. Cost minimisation sidesteps that limitation by reframing the question: for a given level of output, what is the cheapest combination of inputs? Once you have a cost function expressed in terms of output, you can derive the firm's supply curve, which is the payoff of the whole exercise.
Cost minimisation asks: given that a firm must produce y units, what mix of capital and labour costs the least? Solving that problem turns a production function into a cost function, c(w, r, y). From the cost function you recover the familiar result that profit is maximised where price equals marginal cost, and that equation is the firm's supply curve.
Returns to scale
How output responds when every input is scaled by the same factor t. Decreasing: f(tk, tl) < t f(k, l). Constant: f(tk, tl) = t f(k, l). Increasing: f(tk, tl) > t f(k, l). In simple terms, this means: if you double all your inputs, does output less than double, exactly double, or more than double?
Cost minimisation problem
The problem of choosing inputs k and l to minimise total cost rk + wl, subject to the constraint that output equals a target level y. Think of it as the mirror image of profit maximisation: instead of asking "how much should I produce?", you ask "given that I must produce this much, what is the cheapest way to do it?"
Cost function, c(w, r, y)
The minimum cost of producing y units of output when the wage is w and the rental rate of capital is r. It collapses the two-input cost problem into a single expression that depends on output and input prices. In simple terms, this is the answer to the cost minimisation problem, packaged as a formula.
Conditional factor demands
The cost-minimising quantities of each input, expressed as functions of output y and input prices w and r. They are "conditional" because they tell you how much capital and labour to use conditional on producing a specific output level.
Marginal cost
The derivative of the cost function with respect to output: dc/dy. It tells you the additional cost of producing one more unit. In a competitive market, a profit-maximising firm sets price equal to marginal cost.
Zero profit condition
The result that arises under constant returns to scale: when the cost function is linear in y (i.e. c = g(w, r) * y), the marginal cost equals average cost, and profit is exactly zero at the competitive price. Think of it as the long-run equilibrium benchmark for perfectly competitive firms.
For a Cobb-Douglas production function f(k, l) = k^a * l^b:
Decreasing returns to scale when a + b < 1
Constant returns to scale when a + b = 1
Increasing returns to scale when a + b > 1
A firm does not have to exhibit one type permanently. A firm may see increasing returns at low output and decreasing returns at high output.
Under perfect competition (price-taking), only decreasing returns to scale yield a finite profit-maximising solution. Constant returns give zero profits; increasing returns imply infinite profits (or no maximum).
Profit maximisation alone tells you how a firm responds to a given price, but not how firms collectively determine the market price.
Cost minimisation reframes the problem: express costs as a function of output rather than inputs. That opens the door to deriving supply curves.
The firm chooses k and l to minimise rk + wl, subject to f(k, l) = y.
r is the rental rate of capital; w is the wage rate of labour; y is the target output level.
The solution gives you the conditional factor demands (how much k and l to use for a given y) and the cost function c(w, r, y).
Worked example using f(k, l) = k^(1/3) * l^(2/3):
Step 1: Isolate one input from the constraint. Solve k^(1/3) * l^(2/3) = y for l, giving l = y^(3/2) / k^(1/2).
Step 2: Substitute into the cost expression. The problem becomes: minimise rk + w * y^(3/2) / k^(1/2) with respect to k alone.
Step 3: Take the first-order condition. Differentiate with respect to k, set equal to zero, and solve for k. This gives the conditional demand for capital: k = y * (w / (2r))^(2/3).
Step 4: Back-substitute for the other input. Plug the optimal k back into the expression for l to get: l = y * (2r / w)^(1/3).
Step 5: Derive the cost function. Substitute both optimal inputs into c = rk + wl. The result simplifies to c = g(w, r) * y, where g(w, r) collects all the terms involving input prices.
Rewrite profit as: profit = py - c(w, r, y).
The first-order condition gives: p = dc/dy, i.e. price equals marginal cost.
For the Cobb-Douglas example above, marginal cost is the constant g(w, r), because the cost function is linear in y. That means:
If p > g(w, r): profits are positive at every output level.
If p = g(w, r): profits are exactly zero (zero profit condition).
If p < g(w, r): profits are negative; the firm should shut down.
General cost minimisation problem
\min_{k, l} \; rk + wl \quad \text{s.t.} \quad f(k, l) = yReturns to scale (Cobb-Douglas f(k,l) = k^a l^b)
\alpha + \beta < 1 \implies \text{Decreasing returns to scale}\alpha + \beta = 1 \implies \text{Constant returns to scale}\alpha + \beta > 1 \implies \text{Increasing returns to scale}Conditional factor demands (for k^(1/3) l^(2/3))
k^* = y \left( \frac{w}{2r} \right)^{2/3}l^* = y \left( \frac{2r}{w} \right)^{1/3}Cost function (same example)
c(w, r, y) = \left[ \left( \frac{1}{2} w r^{1/2} \right)^{2/3} + \left( 2 r w^2 \right)^{1/3} \right] yProfit maximisation from the cost function
\pi = py - c(w, r, y)\frac{\partial \pi}{\partial y} = p - \frac{\partial c}{\partial y} = 0 \implies p = MCCost minimisation is how firms decide their mix of machinery and workers. A factory choosing between hiring more staff or leasing another machine is solving exactly this problem: for a given production target, which combination keeps the bill lowest? When wages rise, the cost-minimising firm shifts towards more capital, and vice versa.
Students often confuse cost minimisation with profit maximisation. Cost minimisation is a necessary condition for profit maximisation, not the same thing. You minimise cost for a given output; you maximise profit by also choosing the right output level.
Students sometimes think constant returns to scale means the firm earns positive profits. Under perfect competition with constant returns, the zero profit condition holds: p = MC = AC, so profit is exactly zero.
The substitution method is sometimes confused with the Lagrangian method. Both work, but the substitution approach isolates one input from the production constraint and plugs it into the cost equation directly. The lecture uses substitution throughout.
Students frequently forget that the cost function c(w, r, y) depends on input prices, not just output. If wages or the rental rate change, the cost function shifts even though the production technology has not changed.
⚠️ You must be comfortable solving the substitution method step by step. Exam questions will give you a Cobb-Douglas function and ask you to derive conditional factor demands and the cost function.
⚠️ Know the link between the cost function and the supply curve. The first-order condition p = MC is the supply equation.
⚠️ Understand the zero profit condition for constant returns to scale: when c = g(w, r) * y, marginal cost is constant and equals average cost.
⚠️ Be able to explain why only decreasing returns to scale give a well-defined profit maximum under perfect competition.
True or false: A Cobb-Douglas production function with a + b = 1.3 exhibits decreasing returns to scale. (False, it exhibits increasing returns.)
Fill in the blank: Cost minimisation allows us to express costs as a function of ______ rather than inputs. (Output.)
True or false: Under perfect competition with constant returns to scale, the firm earns zero economic profit. (True.)
Fill in the blank: The first-order condition of the profit equation using the cost function gives p = ______. (Marginal cost, or dc/dy.)
True or false: If a firm has increasing returns to scale and is a price taker, we can find a finite profit-maximising output. (False, profits are unbounded.)
Q: Given the production function f(k, l) = k^(1/3) l^(2/3) and input prices r and w, derive the conditional demand for capital.
A: Solve the production constraint for l: l = y^(3/2) / k^(1/2). Substitute into the cost expression rk + w * y^(3/2) / k^(1/2). Take the derivative with respect to k and set it equal to zero: r - (w/2) * y^(3/2) / k^(3/2) = 0. Solve for k to get k* = y * (w / (2r))^(2/3).
Q: Explain why cost minimisation is a necessary condition for profit maximisation.
A: If a firm is maximising profit, it must be producing its chosen output level at the lowest possible cost. If it were not, it could reduce costs while keeping revenue the same, which would increase profit. So any profit-maximising output level must also be cost-minimising.
Q: For a Cobb-Douglas production function with constant returns to scale, what happens to profit under perfect competition?
A: The cost function takes the form c = g(w, r) * y, meaning marginal cost equals g(w, r), a constant. Under perfect competition, p = MC = g(w, r). Revenue is p * y = g(w, r) * y = c, so profit = revenue - cost = 0.
Q: Why does cost minimisation help us analyse a wider variety of production functions than profit maximisation alone?
A: Under perfect competition, profit maximisation with respect to inputs only gives a finite solution for decreasing returns to scale. Cost minimisation works for any returns to scale because it takes output as given and finds the cheapest input mix. You then use the resulting cost function to find the supply curve, which handles all cases.
This material connects directly to supply curve derivation, which is covered in subsequent lectures. The cost function you derive here is the foundation for short-run and long-run supply analysis. It also links back to consumer theory: just as consumers minimise expenditure subject to a utility constraint (the expenditure function), firms minimise cost subject to a production constraint. The zero profit condition reappears in general equilibrium and welfare economics.
Cost minimisation, cost minimization, cost function, conditional factor demand, returns to scale, decreasing returns, constant returns, increasing returns, Cobb-Douglas, substitution method, marginal cost, zero profit condition, price equals marginal cost, p = MC, supply curve derivation, input prices, rental rate of capital, wage rate, ECN 312, intermediate microeconomics, intermediate macroeconomics, Arizona State University, Lecture 12