Source: Strickland, Chapter 7, Texas A&M University
Tags: isoquant, isocost line, expansion path, long-run total cost, LRATC, long-run average total cost, short-run average total cost, SRATC, economies of scale, diseconomies of scale, constant returns to scale, cost minimisation, production function, ECON 323, microeconomic theory
In the long run, firms choose input combinations by finding the cheapest way to reach each output level, which means finding the tangency of isoquants and isocost lines. The expansion path connects these optimal points and maps directly to the long-run total cost curve. The long-run average total cost curve is the envelope of all short-run ATC curves, and its shape tells you whether the firm enjoys economies of scale, constant returns, or diseconomies of scale.
Isoquant
A curve showing all combinations of inputs (typically labour and capital) that produce the same level of output. Higher isoquants represent higher output levels.
Isocost line
A line showing all combinations of inputs that cost the same total amount. Its slope equals the negative of the input price ratio (−w/r, where w is the wage and r is the rental rate of capital).
Expansion path
The line connecting all cost-minimising input combinations as output changes. Each point on the expansion path is a tangency between an isoquant and an isocost line.
Long-run total cost curve
Derived from the expansion path. Each point gives the minimum total cost of producing a given output level when all inputs are variable.
Short-run average total cost (SRATC)
Average total cost when at least one input (usually capital) is fixed. Each level of fixed capital generates its own SRATC curve.
Long-run average total cost (LRATC)
The envelope of all possible SRATC curves. At each output level, LRATC equals the lowest achievable average cost when the firm is free to choose any plant size.
Economies of scale
When long-run average total cost falls as output increases. Doubling output less than doubles total cost.
Diseconomies of scale
When long-run average total cost rises as output increases. Doubling output more than doubles total cost.
Constant returns to scale
When long-run average total cost stays the same as output increases. Doubling output exactly doubles total cost.
The expansion path on an isoquant map connects the cost-minimising input bundles at different output levels. Each tangency point tells you two things: the output level (from the isoquant label) and the total cost (from the isocost line label).
For example, if the expansion path passes through:
Q = 100 at the C = 600 isocost line
Q = 220 at the C = 1,200 isocost line
Q = 300 at the C = 1,800 isocost line
Then the long-run total cost curve passes through (100, 600), (220, 1200), and (300, 1800).
On the exam graph (Figure 7.3), curve B passes through these three points. Answer: B.
When capital is fixed in the short run, the firm can only adjust labour. On an isoquant map, this means moving horizontally (more labour, same capital).
If the firm is at point A on isoquant Q₁ and wants to reach Q₂:
In the short run, capital is fixed, so the firm moves horizontally to a point on Q₂ that has the same capital level. That is point W (same capital as A, but more labour).
In the long run, the firm can adjust both inputs and move to the cost-minimising tangency on Q₂, which is point Z.
So the firm moves to W in the short run and Z in the long run. Answer: D on the exam (but the circled answer appears to be D).
The short-run point (W) is not on the expansion path. It uses more labour than the cost-minimising combination because the firm cannot reduce capital and rebalance. This is why short-run costs exceed long-run costs at most output levels.
If the total cost curve shifts downward from TC₁ to TC₂ (same intercept on the vertical axis, but TC₂ lies below TC₁ at every output level beyond zero):
The same fixed cost at Q = 0 rules out a change in fixed costs
A lower TC at every positive output level means lower variable costs
This is consistent with a technological change that increases the productivity of inputs (you need fewer inputs for the same output)
It is not economies of scale (that is a movement along a single cost curve, not a shift of the curve). Answer: C.
The LRATC curve is built by tracing the lowest point reachable across all possible SRATC curves:
Each SRATC corresponds to a different fixed plant size (level of capital)
The LRATC is tangent to each SRATC, but only at one point per SRATC
The LRATC touches each SRATC at its minimum only if the LRATC is flat (constant returns) at that output. Otherwise, the tangency occurs to the left of the SRATC minimum (during economies of scale) or to the right (during diseconomies)
On Figure 7.12:
The short-run ATC curve is given by CZW (it traces the higher, U-shaped path)
The long-run ATC curve is given by AZY... but wait, re-reading the exam: the SRATC is AZY and the LRATC is CZW? Let me reconsider.
Actually, the SRATC passes through points that correspond to a single plant size, and the LRATC is the lower envelope. From the answer key: SRATC = AZY, LRATC = CZW. Answer: C.
Short-run average total cost curves intersect the long-run average total cost curve at its minimum point. FALSE. SRATCs are tangent to the LRATC, and they can be tangent at points other than the LRATC minimum.
It is not possible for two short-run average total cost curves to cross. FALSE. Two SRATCs (for different plant sizes) can and do cross.
The long-run average total cost curve is derived by tracing out all of the firm's short-run average total cost curves. TRUE. This is the envelope definition.
Answer: C (statement III only).
Economies of scale: LRATC is falling. The downward-sloping portion of the LRATC curve.
Diseconomies of scale: LRATC is rising. The upward-sloping portion of the LRATC curve.
Constant returns to scale: LRATC is flat. The bottom of the LRATC curve (if it has a flat region).
For the LRATC to slope downward, total cost must increase less than proportionally with output. For it to slope upward, total cost increases more than proportionally.
A firm may have economies of scale even with a production function that has constant returns to scale. This can happen if, for instance, bulk purchasing discounts or specialisation benefits exist that the production function alone does not capture.
I. If a firm has economies of scale, LRATC rises with output. FALSE (it falls).
II. Diseconomies of scale are associated with the upward-sloping portion of the LRATC. TRUE.
III. For the LRATC to slope downward, total cost must increase less than proportionally with output. TRUE.
IV. A firm may have economies of scale despite a constant-returns-to-scale production function. TRUE.
Statements II, III, and IV are true. Answer: B.
A firm produces 10,000 units at TC = $5,000. It increases output to 15,000 units and TC rises to $7,000.
Original ATC = $5,000 / 10,000 = $0.50 per unit
New ATC = $7,000 / 15,000 = $0.467 per unit
ATC fell as output rose, so the firm has economies of scale. Answer: A.
Alternatively: output rose by 50%, but total cost rose by only 40%. Cost increased less than proportionally, which is the definition of economies of scale.
Given Q = KL, where MP_L = K and MP_K = L. Wage (w) = $50, rental rate (r) = $12.50, K is fixed at 10 in the short run.
Short-run ATC for Q = 100:
Q = KL = 10L, so L = 10 for Q = 100
Total cost = wL + rK = 50(10) + 12.50(10) = 500 + 125 = 625
Short-run ATC = 625 / 100 = $6.25
Long-run ATC for Q = 100:
In the long run, cost minimisation requires MP_L / w = MP_K / r:
MP_L / w = K / 50
MP_K / r = L / 12.50
Setting equal: K / 50 = L / 12.50, so K = 4L
Substitute into Q = KL = 100:
4L × L = 100
4L² = 100
L² = 25
L = 5, K = 20
Total cost = 50(5) + 12.50(20) = 250 + 250 = 500
Long-run ATC = 500 / 100 = $5.00
Answer: B ($6.25; $5). The long-run cost is lower because the firm rebalances to the optimal input mix.
Charlie's Umbrellas: Q = 10K^0.5 × L^0.5, with K = 9 fixed, w = $80, r = $5.
For Q = 60:
60 = 10(9^0.5)(L^0.5) = 10(3)(L^0.5) = 30L^0.5
L^0.5 = 2, so L = 4
Total cost = wL + rK = 80(4) + 5(9) = 320 + 45 = $365
Answer: B.
Cost-minimisation condition (long run):
MP_L / w = MP_K / r
This can be rearranged to: MP_L / MP_K = w / r (the marginal rate of technical substitution equals the input price ratio).
Isocost line equation:
C = wL + rK, or equivalently K = C/r − (w/r)L
Economies of scale test:
If increasing output by x% raises total cost by less than x%, the firm has economies of scale (LRATC is falling).
From a Cobb-Douglas production function Q = AK^a × L^b:
If a + b > 1: increasing returns to scale
If a + b = 1: constant returns to scale
If a + b < 1: decreasing returns to scale
Note: returns to scale (production concept) and economies of scale (cost concept) are related but not identical.
⚠️ The expansion path connects cost-minimising points. In the short run, the firm cannot be on the expansion path (except at the one output level where the fixed capital happens to be optimal). This is why short-run costs exceed long-run costs.
⚠️ When a TC curve shifts down but keeps the same vertical intercept, fixed costs have not changed. The cause is a productivity improvement, not a change in fixed costs and not economies of scale.
⚠️ The LRATC is the envelope of SRATCs. It is tangent to each SRATC but only passes through the minimum of an SRATC at the output level of minimum efficient scale (the bottom of the LRATC).
⚠️ Economies of scale means LRATC is falling. A common exam trap is confusing this with "LRATC is rising" (that is diseconomies of scale).
⚠️ For short-run cost calculations with a production function, plug the fixed input into the production function first, solve for the variable input, then compute total cost as wL + rK.
⚠️ For long-run cost calculations, use the cost-minimisation condition (MP_L/w = MP_K/r) along with the production function to solve for both inputs simultaneously.
Q: What is the expansion path?
A: The line connecting all cost-minimising input combinations as output varies. Each point is a tangency between an isoquant and an isocost line.
Q: A firm has Q = KL, w = $50, r = $12.50, and K is fixed at 10. What is the short-run total cost of producing 100 units?
A: L = Q/K = 100/10 = 10. TC = 50(10) + 12.50(10) = $625.
Q: Same firm, long run. What is the long-run total cost of producing 100 units?
A: Cost-minimisation gives K = 4L. Substituting into Q = KL = 100: 4L² = 100, L = 5, K = 20. TC = 50(5) + 12.50(20) = $500.
Q: A firm produces 10,000 units at $5,000 total cost, then 15,000 units at $7,000. Economies or diseconomies of scale?
A: Economies of scale. ATC fell from $0.50 to $0.467 as output rose.
Q: The LRATC curve is derived from what?
A: It is the envelope of all short-run average total cost curves, tracing the lowest achievable ATC at each output level.
Q: What causes a downward shift of the TC curve (same intercept, lower costs at every positive Q)?
A: A technological change that increases the productivity of inputs. Not a change in fixed costs (the intercept did not move) and not economies of scale (that is movement along a curve, not a shift).
Q: Charlie's production function is Q = 10K^0.5 L^0.5, K = 9, w = $80, r = $5. What is TC for Q = 60?
A: L = 4 (from 60 = 30L^0.5). TC = 80(4) + 5(9) = $365.
isoquant, isocost, expansion path, cost minimisation, input optimisation, marginal rate of technical substitution, MRTS, long-run total cost, long-run average total cost, LRATC, short-run average total cost, SRATC, economies of scale, diseconomies of scale, constant returns to scale, increasing returns to scale, decreasing returns to scale, envelope curve, plant size, Cobb-Douglas, production function, ECON 323, Chapter 7, Strickland, microeconomic theory, Texas A&M