Difficulty: Intermediate-Advanced | Prerequisites: You need the short-run cost derivation notes (Part 2) and comfort with partial derivatives. Familiarity with isoquants from Ch. 6 is essential.
In the short run, the firm is stuck with a fixed level of capital and must make do. In the long run, both labour and capital are adjustable. This opens up an optimisation problem: for any target output, what mix of L and K produces it at the lowest possible cost?
The answer comes from the tangency condition between the isoquant (all input combinations that produce a given q) and the isocost line (all input combinations that cost a given amount). This is the producer's analogue of consumer theory's tangency between the indifference curve and the budget line. Once you have the optimal input mix, you can derive the long-run total cost and average cost functions, and connect them back to returns to scale.
In the long run, the firm picks the cheapest combination of L and K for each output level by setting MRTS = w/r (the ratio of input prices). For a Cobb-Douglas function with constant returns to scale, this produces a linear long-run average cost curve. The LRAC is the lower envelope of all possible short-run average cost curves.
Isoquant
A curve showing all combinations of L and K that produce the same level of output. It is the producer's equivalent of an indifference curve.
Think of it as a menu of recipes that all yield the same number of widgets.
Isocost line
A line showing all combinations of L and K that cost the same total amount. Its equation is C = wL + rK, and its slope is -w/r.
In simple terms, this is your budget line for inputs.
Marginal rate of technical substitution (MRTS)
The rate at which the firm can trade one input for another while keeping output constant. Formally, MRTS = -dK/dL along the isoquant = MPL/MPK.
Think of it as: how many units of capital can you give up if you hire one more worker, without losing any output?
Tangency condition (isoquant-isocost)
Cost is minimised where the isoquant is tangent to the isocost line. At that point, MRTS = w/r. This ensures the firm gets the most output per pound spent on each input.
Long-run total cost (LRTC)
The minimum cost of producing each quantity when all inputs are variable. It is derived by substituting the cost-minimising input demands into the cost equation.
Long-run average cost (LRAC)
LRTC divided by quantity. Its shape reveals returns to scale: flat means constant returns, downward-sloping means increasing returns, upward-sloping means decreasing returns.
Returns to scale
How output responds when all inputs are scaled up proportionally. For q = L^a × K^b: if a + b = 1, constant returns; if a + b > 1, increasing returns; if a + b < 1, decreasing returns.
Envelope theorem (LRAC as envelope of SRACs)
The LRAC curve is the lower boundary of all possible SRAC curves. Each SRAC (one per level of fixed capital) touches the LRAC at exactly one point.
In the long run, there are infinitely many (L, K) pairs that produce any given q.
The firm picks the one that minimises cost, given input prices w and r.
The guiding principle: spend each additional pound where it buys the most output.
With q = L^0.75 × K^0.25, producing q = 4 requires 256 = L³K (after raising both sides to the 4th power).
Two valid bundles: (L = 4, K = 4) and (L = 2, K = 32).
Cost at (4, 4): 16(4) + 4(4) = 64 + 16 = $80.
Cost at (2, 32): 16(2) + 4(32) = 32 + 128 = $160.
The firm chooses (4, 4) because it is cheaper. Labour is expensive relative to capital, so using more capital-intensive methods is not always better; the firm balances the two.
The isoquant's slope is -MRTS. The isocost line's slope is -w/r.
At the cost-minimising point, the two slopes are equal: MRTS = w/r.
Intuitively: the rate at which the firm can technically substitute one input for another equals the rate at which the market prices them.
With w = 16 and r = 4, w/r = 4. So the firm always picks the point where MRTS = 4.
For q = L^0.75 × K^0.25:
MPL = 0.75 × L^(-0.25) × K^0.25
MPK = 0.25 × L^0.75 × K^(-0.75)
MRTS = MPL / MPK = (0.75/0.25) × (K/L) = 3K/L.
Set MRTS = w/r: 3K/L = 16/4 = 4.
Solving: 3K = 4L, so K = 4L/3.
This is the expansion path: as the firm scales up, it always uses K and L in this ratio.
Substitute K = 4L/3 into the production function:
q = L^0.75 × (4L/3)^0.25 = L^0.75 × (4/3)^0.25 × L^0.25 = (4/3)^0.25 × L.
So q = (4/3)^0.25 × L.
Invert to get L as a function of q:
L = q / (4/3)^0.25 = q × (3/4)^0.25.
Substitute K = 4L/3 into the total cost function C = wL + rK:
C = 16L + 4(4L/3) = 16L + 16L/3 = (48L + 16L)/3 = 64L/3.
Substitute L = q × (3/4)^0.25:
LRTC = (64/3) × q × (3/4)^0.25.
This simplifies to a linear function of q: LRTC = c × q, where c is a constant.
LRAC = LRTC / q = (64/3) × (3/4)^0.25. This is a constant, independent of q.
A flat LRAC means constant returns to scale.
Confirmed by the production function: the exponents sum to 0.75 + 0.25 = 1. Doubling both inputs exactly doubles output.
Each level of fixed capital K produces a different SRAC curve.
The SRAC from Part 2 (K = 16) touches the LRAC at exactly one point.
At that point, the SRAC is at its minimum, and it equals the LRAC.
For all other output levels, the SRAC lies above the LRAC, because the firm is stuck with the wrong amount of capital.
The LRAC is the lower boundary (envelope) of all such SRACs.
Production function:
q = L^{0.75} K^{0.25}Marginal products:
MPL = 0.75 \, L^{-0.25} K^{0.25}MPK = 0.25 \, L^{0.75} K^{-0.75}MRTS:
MRTS = \frac{MPL}{MPK} = \frac{3K}{L}Tangency condition:
MRTS = \frac{w}{r} \implies \frac{3K}{L} = 4 \implies K = \frac{4L}{3}Long-run total cost:
LRTC = \frac{64}{3} \left(\frac{3}{4}\right)^{0.25} qLong-run average cost:
LRAC = \frac{64}{3} \left(\frac{3}{4}\right)^{0.25} \quad (\text{constant})Returns to scale check: exponents sum to 0.75 + 0.25 = 1 (constant returns to scale).
The tangency condition explains why firms in different countries with different wage levels use different production methods. In a country where labour is cheap relative to capital, MRTS = w/r means the firm substitutes towards more labour-intensive methods. This is why factories in low-wage economies often use more manual assembly, while factories in high-wage economies automate more heavily.
The envelope result has a practical implication for business planning: a firm that expects to produce at a certain scale should invest in the capital level that makes that scale the cheapest. Getting the capital decision wrong means sitting on a short-run cost curve that is above the long-run optimum for most output levels.
Students often think the MRTS should equal 1 at the optimum ("use equal amounts of each input"). The MRTS equals w/r, not 1, because the inputs have different prices. Equal amounts is optimal only when w = r.
A common error is confusing the MRTS with the MRS from consumer theory. Both are tangency conditions, but MRTS is about inputs in production; MRS is about goods in consumption.
Students sometimes think constant returns to scale means cost is constant. Constant returns means LRAC is constant (cost per unit does not change with scale). Total cost still rises linearly with output.
When inverting the production function and substituting into the cost equation, students frequently lose track of which variable they are eliminating. Write out each step and label what you are substituting for.
⚠️ The full long-run cost derivation (MRTS, tangency condition, substitute into production function, invert, substitute into cost function) is a multi-step exam problem. Practice it end to end.
⚠️ Be ready to explain in words why MRTS = w/r minimises cost. The intuition about equalising the "bang for the buck" across inputs is frequently tested.
⚠️ Know how to determine returns to scale from the production function exponents and confirm with the shape of LRAC.
⚠️ The envelope relationship (LRAC as lower boundary of SRACs) is a conceptual favourite. Be able to explain why each SRAC touches the LRAC at only one point and why that point is the SRAC's minimum under constant returns to scale.
True or false: In the long run, at least one input is fixed. (False. All inputs are variable.)
Fill in the blank: The cost-minimising condition is MRTS = ______. (w/r)
For q = L^0.75 × K^0.25, what are the returns to scale? (Constant, because 0.75 + 0.25 = 1.)
True or false: A flat LRAC means the firm has increasing returns to scale. (False. Flat LRAC means constant returns to scale.)
True or false: Every SRAC curve lies on or above the LRAC curve. (True.)
Q: In the long run, what guides the firm's choice of input combination?
A: Cost minimisation. The firm chooses the combination of L and K that produces its target output at the lowest total cost, which occurs where the isoquant is tangent to the isocost line (MRTS = w/r).
Q: Two input bundles produce q = 4: (L = 4, K = 4) and (L = 2, K = 32). With w = $16 and r = $4, which does the firm prefer and why?
A: (L = 4, K = 4) costs $80; (L = 2, K = 32) costs $160. The firm chooses (4, 4) because it is cheaper.
Q: For q = L^0.75 × K^0.25, derive the MRTS.
A: MPL = 0.75L^(-0.25)K^(0.25); MPK = 0.25L^(0.75)K^(-0.75). MRTS = MPL/MPK = 3K/L.
Q: Use the tangency condition to find the cost-minimising relationship between L and K when w = 16 and r = 4.
A: Set MRTS = w/r: 3K/L = 4, so K = 4L/3.
Q: Derive the long-run total cost function for this production function.
A: Substitute K = 4L/3 into q = L^0.75 × K^0.25 to get q = (4/3)^0.25 × L, so L = q(3/4)^0.25. Substitute into C = 16L + 4(4L/3) = 64L/3. LRTC = (64/3)(3/4)^0.25 × q.
Q: What is the LRAC, and what does its shape tell you about returns to scale?
A: LRAC = LRTC/q = (64/3)(3/4)^0.25, a constant. The flat LRAC confirms constant returns to scale (exponents 0.75 + 0.25 = 1).
Q: Explain in one sentence why the SRAC from Part 2 (K = 16) touches the LRAC at exactly one point.
A: K = 16 is the optimal capital level for only one output level; at every other output, the firm would prefer a different K, so SRAC exceeds LRAC.
The tangency condition here mirrors the consumer's utility-maximising condition (MRS = price ratio). If you understand one, you understand the structure of the other. Both are constrained optimisation problems solved by equating the slope of an indifference/isoquant curve with the slope of a budget/isocost line.
Returns to scale connect to market structure: industries with increasing returns to scale tend toward natural monopoly, while those with constant or decreasing returns to scale tend toward competitive markets. This link will become central in the chapters on market structure.
The envelope result also prepares you for the distinction between short-run and long-run supply curves in competitive markets.
Long-run costs, long-run total cost, LRTC, long-run average cost, LRAC, MRTS, marginal rate of technical substitution, isoquant, isocost, tangency condition, cost minimisation, expansion path, returns to scale, constant returns to scale, Cobb-Douglas, envelope theorem, SRAC envelope, short-run vs long-run, input substitution, ECON 300, intermediate microeconomics, Chapter 7, University of Washington