Difficulty: Intermediate to Advanced | Prerequisites: Section 7.1 and 7.2 (cost concepts, short-run cost curves), Chapter 6 (isoquants, MRTS, marginal products, returns to scale), basic calculus (Lagrangian optimisation is used in several solved problems).
In the long run, every input is variable: the firm can choose its plant size, equipment and workforce from scratch. This is where the firm solves its fundamental cost problem, picking the cheapest combination of inputs for any given output level. The tools here (isocost lines, tangency conditions, expansion paths) parallel the consumer theory you saw in Chapter 3 (budget lines, indifference curves, utility maximisation). Sections 7.4 and 7.5 explain why long-run costs are always at least as low as short-run costs, and how learning by doing and producing multiple goods can lower costs further.
A firm minimises long-run cost by choosing inputs where the isoquant is tangent to the lowest isocost line, which means the marginal rate of technical substitution equals the input price ratio. The long-run average cost curve is the envelope of all possible short-run average cost curves. Costs fall over time through flexibility, technological progress and learning by doing. Firms producing multiple goods may benefit from economies of scope.
Isocost line
A line showing all combinations of inputs (labour and capital) that cost the same total amount. Its equation is C = wL + rK, and its slope is -w/r. Think of it as the firm's "budget line" for inputs.
Tangency rule (cost minimisation)
The firm minimises cost where the isoquant is tangent to the isocost line, i.e. where the MRTS equals the input price ratio: MRTS = -w/r.
Last-dollar rule
Cost is minimised when the last dollar spent on labour produces the same extra output as the last dollar spent on capital: MP_L/w = MP_K/r. In simple terms, every pound or dollar of spending is equally productive at the margin.
Expansion path
The curve connecting the cost-minimising input combinations for every output level, holding input prices constant. For a Cobb-Douglas production function, this is a straight line through the origin.
Economies of scale
A property of the cost function where average cost falls as output expands. Sufficient condition: increasing returns to scale in the production function. But economies of scale can also arise from switching to more capital-intensive methods.
Diseconomies of scale
A property where average cost rises as output increases.
Learning curve
The relationship between average cost and cumulative output. As workers and managers gain experience, the average cost of production falls. In simple terms, practice makes cheaper.
Economies of scope
Cost savings that arise when a firm produces two or more goods together rather than separately. For example, cattle provide both beef and hides from a single animal.
The isocost line equation: C = wL + rK, rearranged as K = C/r - (w/r)L.
Three properties: (1) intercepts at C/r on the K-axis and C/w on the L-axis; (2) higher isocost lines (farther from origin) represent higher costs; (3) slope = -w/r, the same for all parallel isocost lines at given input prices.
Analogous to the consumer's budget line, but the firm faces many isocost lines (one for each spending level), not just one.
Lowest-isocost rule: Pick the input bundle on the isoquant that lies on the lowest isocost line.
Tangency rule: At the cost-minimising point, the isoquant is tangent to the isocost line. This means MRTS = -w/r, or equivalently MP_L/MP_K = w/r (Equation 7.10).
Last-dollar rule: MP_L/w = MP_K/r (Equation 7.11). The last dollar spent on each input yields the same marginal output.
If the isoquant is not smooth (e.g. fixed proportions), only the lowest-isocost rule works.
Minimise L = wL + rK + λ[q_bar - f(L,K)].
First-order conditions yield w/r = MP_L/MP_K (same as the tangency rule) and the constraint q_bar = f(L,K).
The Lagrangian multiplier λ = w/MP_L = r/MP_K = MC (the marginal cost of production).
The dual problem (maximise output for a given cost) produces the same tangency condition.
If the wage falls relative to the rental rate of capital, the isocost lines become flatter. The firm substitutes towards more labour and less capital. Both the input mix and total cost change.
Beer manufacturer example: when w falls from $24 to $8 (r stays at $8), the firm moves from (L=50, K=100) at $2,000 to (L=77, K=52) at $1,032.
The expansion path connects the cost-minimising input bundles across all output levels. For the Cobb-Douglas q = AL^a K^(1-a), the expansion path is K = (1-a)/aL, a straight line through the origin.
The long-run cost function is derived from the expansion path. For the beer manufacturer: C(q) = 20q (constant returns to scale, so cost is proportional to output).
For a general homogeneous production function of degree g: C = θq^(1/g), where θ depends on input prices and the production function parameters.
Increasing returns to scale (g > 1) produce economies of scale: AC falls as output rises.
Constant returns to scale (g = 1) mean AC is flat.
Decreasing returns to scale (g < 1) produce diseconomies of scale: AC rises.
If returns to scale vary with output (increasing at low output, constant at medium, decreasing at high), the long-run AC curve is U-shaped.
The LRAC is the envelope of all possible SRAC curves. It touches each SRAC at one point, but not necessarily at that SRAC's minimum.
Long-run cost ≤ short-run cost, always. Three reasons: (1) greater flexibility, (2) technological progress over time, (3) learning by doing.
The long-run expansion path slopes upward (the firm increases both L and K). The short-run expansion path is horizontal (K is fixed).
Beer manufacturer: doubling output from 100 to 200 costs $4,000 in the long run (double) but $4,616 in the short run (more than double, because capital is stuck at 100).
As cumulative output rises, workers and managers become more efficient, and the AC curve shifts downward over time.
Intel CPU learning curve: average cost fell sharply with the first few million units, then declined more slowly.
If a firm has both economies of scale and learning by doing, expanding output today lowers cost in two ways: lower AC today (scale) and a lower AC curve tomorrow (learning).
Solar installation learning curves vary by country (e.g. Sweden 74%, meaning each doubling of cumulative installations reduces cost to 74% of the previous level; China 96%).
When a firm produces two or more goods using shared inputs, the cost of one good can depend on how much of the other it produces.
Economies of scope exist when producing goods together is cheaper than producing them separately (e.g. beef and hides from the same animal, heating fuel and petrol from the same barrel of oil).
C = wL + rKK = \frac{\bar{C}}{r} - \frac{w}{r}L \quad \text{(isocost line rearranged)}\text{Tangency condition: } \frac{MP_L}{MP_K} = \frac{w}{r}\text{Last-dollar rule: } \frac{MP_L}{w} = \frac{MP_K}{r}\text{Expansion path (Cobb-Douglas): } K = \frac{(1-a)}{a} \cdot \frac{w}{r} \cdot L\text{Long-run cost (homogeneous degree } g\text{): } C = \theta q^{1/g}Figure 7.3 (Cost Minimisation): The q = 100 isoquant is tangent to the $2,000 isocost line at point x (L = 50, K = 100). Higher isocost lines ($3,000) cross the isoquant at two points (y and z), confirming x is the cheapest.
Figure 7.5 (Factor Price Change): When w falls from $24 to $8, the isocost lines flatten. The tangency moves from x to v (more labour, less capital), and cost falls from $2,000 to $1,032.
Figure 7.6 (Expansion Path): The straight line K = 2L through points x, y, z. Panel b shows the corresponding linear long-run cost curve C = 20q.
Figure 7.7 (LRAC as Envelope): The smooth LRAC curve touches each SRAC at one point. The LRAC point on a given SRAC is not necessarily that SRAC's minimum.
Figure 7.9 (Learning by Doing): Panel a: Intel CPU learning curve (average cost vs cumulative output). Panel b: economies of scale move along one AC curve; learning by doing shifts the entire AC curve downward.
Semiconductor manufacturing (chapter opening challenge): U.S. firms use capital-intensive production at home because w/r is high. Abroad, where labour is cheaper, the cost-minimising input mix shifts towards more labour and less capital. The tangency condition (MRTS = -w/r) explains why the same firm uses different technologies in different countries.
3D printing and economies of scale: Traditional manufacturing relies on large factories to exploit economies of scale. 3D printing can make it as cheap to produce one unit as a thousand, potentially flattening the long-run AC curve and eliminating the scale advantage of large factories.
Solar installation learning curves: In Sweden, each doubling of cumulative installations drops costs to 74% of the prior level. In China the figure is 96%. The difference reflects varying labour costs, regulatory environments and cumulative experience.
Water-heater manufacturing: What looked like a fixed-proportions process (one cylinder, one liner, one control unit per heater) turned out to allow substitution between labour and metal. When metal is expensive, workers cut carefully and waste less. When labour is expensive, they cut quickly and waste more. Cost minimisation at work.
"The LRAC touches each SRAC at its minimum." It does not, except at the output where the LRAC itself is at its minimum. At other outputs, the LRAC touches the relevant SRAC to the left or right of that SRAC's minimum.
"Long-run cost can be higher than short-run cost." Never. The firm has strictly more flexibility in the long run, so long-run cost is always ≤ short-run cost.
"Economies of scale and increasing returns to scale are the same thing." Increasing returns to scale is a sufficient condition for economies of scale, but not necessary. A firm can have economies of scale (falling AC) even with constant returns to scale if it switches to a more efficient production method at higher output.
"The last-dollar rule and the tangency rule are different conditions." They are mathematically equivalent rearrangements: MP_L/MP_K = w/r is the same as MP_L/w = MP_K/r.
⚠️ The tangency condition (MP_L/MP_K = w/r) and last-dollar rule (MP_L/w = MP_K/r) are the most commonly tested conditions in this chapter. Be ready to apply them to Cobb-Douglas and CES production functions.
⚠️ Deriving the long-run cost function from a production function (Solved Problems 7.5 and 7.6) is a standard exam exercise. Know the steps: tangency condition → expansion path → substitute into cost identity.
⚠️ The LRAC-as-envelope concept comes up frequently. Know why the LRAC does not touch each SRAC at the SRAC's minimum.
⚠️ Distinguish economies of scale (a cost-function property: AC falls with output) from returns to scale (a production-function property: how output responds to proportional input increases). They are related but not identical.
⚠️ Learning by doing vs economies of scale: scale is movement along one AC curve; learning shifts the entire curve down over time. Exams test whether you can tell them apart.
Fill in the blank: The slope of an isocost line is ____.
-w/r (the negative of the ratio of input prices).
True or False: At the cost-minimising input bundle, the isoquant crosses the isocost line.
False. It is tangent to (just touches) the isocost line.
Fill in the blank: For a Cobb-Douglas production function q = AL^a K^(1-a), the expansion path is K = ____.
[(1-a)/a] × (w/r) × L.
True or False: Long-run average cost can exceed short-run average cost.
False. The firm has more flexibility in the long run, so LRAC ≤ SRAC always.
True or False: Learning by doing reduces cost by moving along the AC curve to higher output.
False. Learning by doing shifts the entire AC curve downward. Economies of scale are the movement along a given AC curve.
Q: A beer manufacturer has q = 1.52L^0.6 K^0.4, w = $24, r = $8. Using the last-dollar rule, verify that (L = 50, K = 100) is cost-minimising for q = 100.
A: MP_L = 0.6 × 100/50 = 1.2. MP_K = 0.4 × 100/100 = 0.4. MP_L/w = 1.2/24 = 0.05. MP_K/r = 0.4/8 = 0.05. They are equal, so the last-dollar rule holds and the firm is minimising cost.
Q: What is the expansion path for q = AL^a K^(1-a) with wage w and rental rate r?
A: From the tangency condition, w/r = (a/(1-a)) × (K/L). Rearranging: K = [(1-a)/a] × (w/r) × L. For the beer firm (a = 0.6, w = 24, r = 8): K = (0.4/0.6)(24/8)L = 2L.
Q: The wage falls from $24 to $8 while r stays at $8. How does the cost-minimising input bundle change for q = 100?
A: The isocost line becomes flatter (slope changes from -3 to -1). The firm substitutes towards more labour and less capital, moving from (50, 100) to (77, 52). Cost falls from $2,000 to $1,032.
Q: Explain the difference between economies of scale and learning by doing.
A: Economies of scale mean that average cost falls as the firm produces more output at a given point in time (movement along one AC curve). Learning by doing means that cumulative production experience shifts the entire AC curve downward over time. A firm expanding output can benefit from both simultaneously.
The cost-minimisation framework here parallels utility maximisation from Chapter 3: isoquants correspond to indifference curves, isocost lines to budget constraints, and the tangency condition to MRS = price ratio. The long-run cost curves derived in this chapter are the foundation for supply decisions in perfect competition (Chapter 8), monopoly (Chapter 11) and oligopoly (Chapter 13). Returns to scale (Chapter 6) directly determine the shape of the LRAC curve.
Long-run costs, isocost line, isoquant, cost minimisation, tangency condition, MRTS equals price ratio, last-dollar rule, expansion path, long-run average cost, LRAC, economies of scale, diseconomies of scale, constant returns to scale, increasing returns to scale, decreasing returns to scale, learning by doing, learning curve, economies of scope, Cobb-Douglas cost function, CES cost function, Lagrangian cost minimisation, envelope theorem, short-run vs long-run, factor price changes, Perloff Chapter 7.3 7.4 7.5, intermediate microeconomics long-run costs, semiconductor technology choice, 3D printing economies of scale