Difficulty: Intermediate | Prerequisites: Production functions, marginal products, and MRTS (see companion notes). Basic optimisation (Lagrangian or substitution method).
Once you understand how a firm turns inputs into output, the next question is: what does it cost? This topic takes the production function as given and derives the cheapest way to produce any quantity Q, given input prices r (capital) and w (labour). The result is the cost function C(Q), which is the foundation for everything that follows in the course: supply curves, profit maximisation, and market structure. You need to be comfortable with MRTS and the tangency condition before tackling this.
The cost function tells you the minimum expenditure needed to produce Q units of output. You derive it by solving the cost-minimisation problem (tangency condition MRTS = r/w, or corner/kink analysis for non-smooth production functions). From the cost function, marginal cost and average cost follow directly.
Cost function, C(Q, r, w)
The minimum total expenditure required to produce Q units of output when capital costs r per unit and labour costs w per unit. Think of it as the answer to "what is the cheapest way to make Q?"
Total cost (TC)
The cost function evaluated at a specific output level: TC = rK* + wL*, where K* and L* are the cost-minimising input quantities.
Marginal cost (MC)
The additional cost of producing one more unit of output: MC = dC/dQ. In simple terms, it answers "how much more does it cost to make one extra unit?"
Average cost (AC)
Total cost divided by output: AC = C(Q)/Q. Think of it as the per-unit cost of production.
Isocost line
A line showing all input combinations (K, L) that cost the same total amount. Its slope is -w/r. The firm's problem is to reach the highest isoquant while staying on the lowest isocost.
Conditional factor demands, K(Q, r, w) and L(Q, r, w)**
The cost-minimising quantities of each input as functions of desired output and input prices. "Conditional" because they are conditioned on a specific output target Q.
Fixed cost
A cost that does not vary with the level of output. In the threshold-labour example, the first 10 units of labour are a fixed cost because they are required regardless of how much the firm produces (as long as Q > 0).
The firm uses whichever input is cheaper. No interior solution exists (generically).
If r < w: use only capital. K = Q, L = 0. Cost = rQ.
If w < r: use only labour. K = 0, L = Q. Cost = wQ.
If r = w: any split with K + L = Q works. Cost = rQ = wQ.
Combined: C(Q) = min{r, w} · Q. Cost is linear in output.
Invert: to produce Q, we need K + L = √Q.
Same corner-solution logic as the linear case, applied to √Q instead of Q.
C(Q) = min{r, w} · √Q.
Cost is concave in output (increasing at a decreasing rate), reflecting increasing returns to scale in the production function.
Tangency condition: MRTS = (K/L)² = r/w, so K/L = √(r/w).
Substituting back and solving: the conditional factor demands and cost function involve the ratio √(r/w).
The cost function takes the form C(Q) = (Q/A) · (√r + √w)² after simplification. Cost is linear in Q (constant returns to scale).
Tangency: MRTS = K/L = r/w, so K = L·(r/w). Wait, let us be careful: MRTS = MPL/MPK = (K/L) for this function, so K/L = r/w gives K = L·r/w.
Actually for A√(KL), MPK = (A/2)√(L/K), MPL = (A/2)√(K/L). So MRTS = MPL/MPK = K/L = r/w.
Conditional demands: K* = (Q/A)·√(w/r), L* = (Q/A)·√(r/w).
Cost function: C(Q) = rK* + wL* = (Q/A)(√(rw) + √(rw)) = (2Q/A)·√(rw).
Cost is linear in Q, as expected for constant returns to scale.
No substitution possible. The firm always sets K = L = Q/A.
C(Q) = (Q/A)(r + w).
Cost is linear in Q. Both input prices matter equally.
Key comparison: if r = 1 and w rises from 1 to 2, cost goes from (Q/A)(2) to (Q/A)(3). That is a 50% increase, not a doubling, because only one of the two inputs became more expensive.
To produce Q > 0: L = Q/5 + 10.
C(Q) = w · (Q/5 + 10) = wQ/5 + 10w for Q > 0. C(0) = 0.
The 10w term is a fixed cost (the threshold labour that produces nothing but must be hired).
The wQ/5 term is the variable cost.
MC = dC/dQ. Differentiate the cost function with respect to output.
For linear cost functions (C = aQ), MC is constant: MC = a.
For concave cost functions (C = a√Q), MC is decreasing: MC = a/(2√Q). This corresponds to increasing returns to scale.
For the threshold-labour example: C(Q) = wQ/5 + 10w, so MC = w/5. Marginal cost is constant because the underlying production (beyond the threshold) is linear.
AC = C(Q)/Q.
For linear cost functions, AC = MC (constant). There are no economies or diseconomies of scale.
For the threshold-labour example: AC = w/5 + 10w/Q. Average cost falls as Q rises because the fixed cost (10w) is spread over more units. AC approaches w/5 from above as Q becomes large.
The relationship between AC and MC matters: when MC < AC, average cost is falling. When MC > AC, average cost is rising. They cross at the minimum of AC.
For perfect substitutes (only one input used): a change in the price of the unused input has zero effect on cost. A change in the price of the used input scales cost proportionally.
For perfect complements (both inputs always used in fixed proportions): cost rises whenever either price rises, but the increase depends on the share of each input. If r = 1 and w doubles from 1 to 2, cost does not double. It rises from (Q/A)(1+1) = 2Q/A to (Q/A)(1+2) = 3Q/A, a 50% increase.
For smooth production functions (Cobb-Douglas, harmonic mean): the firm partially substitutes away from the input that became more expensive. Cost rises, but by less than it would if the firm could not adjust its input mix. This is the value of substitutability.
When there is only one input x with cost c per unit and f(x) = Q, the cost function is simply C(Q) = c · x(Q), where x(Q) is the minimum input needed for output Q (the inverse of the production function).
AC = c · x(Q)/Q.
MC = c · x'(Q) = c / f'(x(Q)). Marginal cost is the input price divided by the marginal product.
Production function | Cost function C(Q) | MC | AC |
|---|---|---|---|
K + L | min{r,w} · Q | min{r,w} | min{r,w} |
(K + L)² | min{r,w} · √Q | min{r,w}/(2√Q) | min{r,w}/√Q |
A·KL/(K+L) | (Q/A)(√r + √w)² | (1/A)(√r + √w)² | (1/A)(√r + √w)² |
A√(KL) | (2Q/A)√(rw) | (2/A)√(rw) | (2/A)√(rw) |
A·min{K,L} | (Q/A)(r+w) | (r+w)/A | (r+w)/A |
5(L-10), L>10 | wQ/5 + 10w | w/5 | w/5 + 10w/Q |
Key cost relationships:
MC = dC/dQ
AC = C(Q)/Q
When MC < AC, AC is falling. When MC > AC, AC is rising.
MC crosses AC at the minimum of AC (when a minimum exists).
For the single-input case: MC = c / f'(x(Q)).
Cost functions are the bridge between engineering (how things are made) and economics (what things cost). Every supply curve you will ever see rests on a cost function underneath.
The fixed-cost structure in the threshold example mirrors real-world startup costs: a restaurant needs staff, a lease, and equipment before serving a single customer. The cost of that first unit of output is enormous; the cost of the hundredth is small.
The substitution effect when input prices change is why firms offshore production when domestic wages rise, or automate when capital becomes cheap. The cost function captures this adjustment quietly through the conditional factor demands.
Students often assume that if one input price doubles, total cost doubles. This is true only if the firm uses that one input exclusively. With substitutable inputs, the firm adjusts its mix, and the cost increase is less than proportional.
For perfect complements, students sometimes think cost does not change if only one input price rises ("the firm is stuck using both anyway"). Cost does rise, because the firm still pays for that input. It just cannot substitute away from it.
Students confuse the cost function C(Q) with total expenditure rK + wL at an arbitrary input bundle. The cost function is the minimum expenditure, evaluated at the optimal input choice. Any other bundle producing Q costs at least as much.
Marginal cost is not the same as the price of an input. MC = dC/dQ, the cost of one more unit of output, which depends on input prices and the production technology together.
⚠️ Deriving the cost function from a given production function is a core exam task. Practise the full workflow: set up MRTS = r/w, solve for the input ratio, substitute back into the production function, then compute total cost.
⚠️ Know the corner-solution logic for perfect substitutes cold. It is quick marks if you state the three cases (r < w, r > w, r = w) cleanly.
⚠️ Be ready to compute MC and AC from a cost function and describe their behaviour (constant, rising, falling). The threshold-labour example is a favourite for testing whether students can handle fixed costs and variable costs together.
⚠️ Comparing how cost changes when an input price rises, across different production function types, is a common exam question. Know that substitutable technologies soften the blow relative to Leontief.
True or false: For f(K, L) = K + L with r = 3 and w = 5, the firm uses both inputs. (False. It uses only capital, because r < w.)
Fill in the blank: AC = C(Q) / ___. (Q)
True or false: If MC is constant, then AC = MC. (Not necessarily. If there is a fixed cost, AC > MC and falls toward MC. If there is no fixed cost, then yes, AC = MC.)
Fill in the blank: For perfect complements, the firm sets K = L = ___ (when the ratio is 1:1). (Q/A)
True or false: When one input price rises, cost always rises by the same proportion. (False. With substitutable inputs, the firm reoptimises and the cost increase is less than proportional.)
Q: Derive the cost function for f(K, L) = A · min{K, L}.
A: The firm sets K = L = Q/A (using more of one input is wasteful). Total cost is C(Q) = r(Q/A) + w(Q/A) = (Q/A)(r + w).
Q: For f(K, L) = K + L with input prices r and w, what is the cost function? What are MC and AC?
A: C(Q) = min{r, w} · Q. MC = min{r, w}. AC = min{r, w}. Both are constant and equal.
Q: A firm has f(L) = 0 for L ≤ 10 and f(L) = 5(L - 10) for L > 10. The wage is w. Derive C(Q), MC, and AC for Q > 0.
A: L = Q/5 + 10. So C(Q) = w(Q/5 + 10) = wQ/5 + 10w. MC = w/5 (constant). AC = w/5 + 10w/Q (falling in Q because the fixed cost 10w is spread over more units).
Q: For f(K, L) = A · min{K, L}, suppose r = 1 and w rises from 1 to 2. Does the cost double?
A: No. Original cost: (Q/A)(1 + 1) = 2Q/A. New cost: (Q/A)(1 + 2) = 3Q/A. Cost rises by 50%, not 100%. The firm cannot substitute away from the more expensive input, but only half its cost base is affected.
Q: For f(K, L) = A√(KL), suppose r = 1 and w rises from 1 to 2. How does the cost change compared to the Leontief case?
A: Original cost: (2Q/A)√(1·1) = 2Q/A. New cost: (2Q/A)√(1·2) = (2Q/A)√2 ≈ 2.83Q/A. That is a 41% increase, less than the 50% in the Leontief case. The firm substitutes toward the cheaper input (capital), cushioning the impact.
The cost function feeds directly into profit maximisation and supply: a price-taking firm produces where P = MC, and the supply curve is the MC curve above minimum AC.
Long-run vs. short-run costs come next. In the short run, one input is fixed, so the firm cannot fully optimise. The long-run cost function (what we derived here) is always weakly below the short-run cost.
Economies of scale link back to returns to scale in the production function. Constant returns to scale give a linear cost function (constant AC). Increasing returns give falling AC. Decreasing returns give rising AC.
cost function, total cost, marginal cost, average cost, cost minimisation, isocost line, conditional factor demands, perfect substitutes cost, perfect complements cost, Leontief cost, Cobb-Douglas cost, fixed cost, variable cost, threshold production, returns to scale, economies of scale, input price changes, factor substitution, MRTS cost minimisation, intermediate microeconomics, ECON 500