Source: Microeconomic Theory, Texas A&M University
Tags: total cost, fixed cost, variable cost, marginal cost, average cost, ATC, AVC, AFC, MC, isocost line, cost minimisation, expansion path, short-run cost, long-run cost
This chapter translates production into costs. In the short run, capital is fixed and only labour varies, so cost functions are derived by inverting the production function. In the long run, both inputs are flexible and the firm picks the cheapest input combination for each output level using isocost lines and isoquants.
Fixed cost (FC)
The cost of the fixed input. Does not change with the output level, so FC(q) is a constant.
Variable cost (VC)
The cost of variable inputs. Changes as output changes: VC(q).
Total cost (TC or C)
TC(q) = FC(q) + VC(q)
Average fixed cost (AFC)
AFC(q) = FC(q) / q
Average variable cost (AVC)
AVC(q) = VC(q) / q
Average total cost (ATC)
ATC(q) = TC(q) / q
Marginal cost (MC)
The rate at which total cost changes when output changes by one unit. MC(q) = TC'(q) = C'(q).
Isocost line
A line showing all combinations of labour and capital that can be purchased for a given total cost.
C = wL + rK, with slope = −w/r
Expansion path
On an isoquant map, the curve that traces all cost-minimising input bundles as the output level changes (given fixed input prices w and r).
In the short run, capital K is fixed and labour L is the variable input. The price of labour is w (wage) and the price of capital is r (rental rate).
Deriving cost functions from a production function (three-step method)
Step 1: set F(K̄, L) = q
Step 2: solve for L as a function of q, giving L(q), the labour needed to produce q
Step 3: plug into cost formulas
The cost formulas then follow:
FC(q) = rK̄ (independent of q)
VC(q) = wL(q)
TC(q) = rK̄ + wL(q)
All the averages and MC are derived from there.
Worked Example 1: w = 2, r = 3, F(K, L) = KL, K fixed at 4.
Step 1: 4L = q
Step 2: L(q) = q/4
Step 3:
FC = 3(4) = 12
VC(q) = 2(q/4) = 0.5q
TC(q) = 12 + 0.5q
AFC(q) = 12/q
AVC(q) = 0.5
ATC(q) = 12/q + 0.5
MC(q) = 0.5
Worked Example 2: w = 2, r = 3, F(K, L) = K^0.5 · L^0.5, K fixed at 4.
Step 1: 2L^0.5 = q
Step 2: L(q) = q²/4
Step 3:
FC = 12
VC(q) = 2(q²/4) = 0.5q²
TC(q) = 12 + 0.5q²
AFC(q) = 12/q
AVC(q) = 0.5q
ATC(q) = 12/q + 0.5q
MC(q) = q
When TC(q) is given directly
If TC(q) = a + bq + cq², then:
FC = a (set q = 0)
VC(q) = bq + cq²
MC(q) = b + 2cq
ATC(q) = a/q + b + cq
AFC(q) = a/q
AVC(q) = b + cq
Observation 1: slopes and connecting lines
MC is the slope (derivative) of the TC or VC curve
ATC, AFC, and AVC are each the slope of the line connecting the origin to a point on the corresponding total curve
Observation 2: MC eventually increases
MC is usually increasing at some point, as a consequence of diminishing marginal returns in the short run.
Observation 3: MC crosses AVC and ATC at their minimums
When MC > AVC, AVC is rising
When MC < AVC, AVC is falling
The same logic applies to ATC
MC intersects both AVC and ATC at their respective minimum points
In the long run, both K (price r) and L (price w) are variable. The firm chooses the input combination that minimises cost for each output level.
Cost minimisation with smooth, convex isoquants (Case 1)
Find the (L, K) at which:
MRTS = w/r (the isoquant is tangent to the isocost line)
F(K, L) = q (the bundle produces the target output)
Then C(q) = wL + rK.
Cost minimisation with straight-line isoquants (Case 2 – perfect substitutes)
The solution is typically a corner: use only the cheaper input. Compare the slopes of the isoquant and isocost line.
Cost minimisation with right-angle isoquants (Case 3 – perfect complements)
The optimal bundle is always at the corner (kink) of the L-shaped isoquant, where no input is wasted.
Worked Example 1: w = 10, r = 20, smooth isoquants.
From the graph, the cost-minimising bundle for q = 300 is L = 150, K = 75
C(300) = 10(150) + 20(75) = 3,000
The expansion path traces these optimal bundles across all output levels.
Worked Example 2: w = 16, r = 20, smooth isoquants.
To produce q = 30, optimal bundle is L = 6, K = 10
C(30) = 16(6) + 20(10) = 96 + 200 = 296
Important long-run properties
In the long run there is no fixed cost, so TC(q) = VC(q)
Long-run costs are weakly lower than short-run costs, because all inputs are flexible
Once you have TC(q), compute AC(q) = TC(q)/q and MC(q) = TC'(q) the same way as in the short run
FC(q) = rK̄ (short run)
VC(q) = wL(q)
TC(q) = FC + VC(q)
AFC = FC/q, AVC = VC/q, ATC = TC/q
MC(q) = TC'(q)
Isocost line: C = wL + rK, slope = −w/r
Cost minimisation condition (interior): MRTS = w/r and F(K, L) = q
⚠️ The three-step method (set F = q, solve for L(q), plug into cost formulas) is the backbone of almost every short-run cost problem. Practise it until it is automatic.
⚠️ MC crosses AVC and ATC at their minimums. This is one of the most frequently tested relationships on the exam.
⚠️ When a TC function is given, find FC by setting q = 0. VC is whatever remains.
⚠️ In the long run, there is no fixed cost. Do not accidentally include an FC term in long-run problems.
⚠️ For the long-run cost-minimisation condition, remember: MRTS = w/r (not r/w). A common mix-up.
Q: A firm has TC(q) = 50 + 4q + 2q². What are FC, VC(q), MC(q), and AVC(q)?
A: FC = 50. VC(q) = 4q + 2q². MC(q) = 4 + 4q. AVC(q) = 4 + 2q.
Q: Why does MC eventually rise in the short run?
A: Because of diminishing marginal returns. As more of the variable input is used alongside a fixed input, each additional unit of output requires increasingly more of the variable input, raising the incremental cost.
Q: In the long run, why are costs weakly lower than in the short run?
A: Because the firm can adjust all inputs. The short-run cost at any output level is at least as high as the long-run cost, since fixing an input removes a degree of freedom from the optimisation.
Q: If w = 5 and r = 10, what is the slope of the isocost line?
A: Slope = −w/r = −5/10 = −0.5.
Q: At the cost-minimising input bundle for smooth isoquants, what condition holds?
A: MRTS = w/r, and the chosen bundle lies on the target isoquant (F(K, L) = q).
cost of production, fixed cost, variable cost, total cost, marginal cost, average total cost, average variable cost, average fixed cost, ATC, AVC, AFC, MC, TC, FC, VC, isocost line, cost minimisation, expansion path, short-run cost curves, long-run cost curves, MRTS equals price ratio, tangency condition, corner solution, perfect substitutes cost, perfect complements cost