Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), familiarity with utility functions and indifference curves from consumer theory.
A production function tells you how much output a firm (or person) gets from a given combination of inputs. This unit teaches you to take partial derivatives to find marginal products, combine them into the marginal rate of technical substitution (MRTS), and test whether scaling all inputs up produces proportionally more, less, or equal output (returns to scale). If you can do those three things with Cobb-Douglas and Leontief functions, you have the toolkit for the cost-minimisation problems that follow.
Production function
A mathematical relationship showing the maximum output obtainable from each possible combination of inputs. In notation: Q = f(L, K) or Q = f(input₁, input₂, ...).
Think of it as: the recipe that converts hours, materials, or machines into finished goods.
Cobb-Douglas production function
A production function of the form Q = A · L^α · K^β, where A is a productivity constant, and α and β are the output elasticities of labour and capital. The exponents govern how responsive output is to each input.
In simple terms, this means: output is a smooth, multiplicative blend of inputs, and the exponents tell you how much each input matters.
Leontief (fixed-proportions) production function
A production function of the form Q = min{aX₁, bX₂}, where inputs must be used in a fixed ratio. Adding more of one input without the other does nothing.
Think of it as: one left shoe plus one right shoe equals one pair. A second right shoe is worthless without another left.
Isoquant
A curve showing every combination of two inputs that produces the same level of output. It is the production-theory analogue of an indifference curve in consumer theory.
In simple terms: draw a line through all the input bundles that give you, say, Q = 4. That line is the isoquant.
Marginal product (MP)
The additional output from one more unit of a single input, holding all other inputs constant. Formally: MPₓ = ∂Q/∂X.
Think of it as: if you hire one more hour of labour and everything else stays the same, how much extra output do you get?
Marginal rate of technical substitution (MRTS)
The rate at which one input can be substituted for another along an isoquant, keeping output constant. MRTS of input 1 for input 2 = MP₁ / MP₂. It equals the (negative of the) slope of the isoquant.
In simple terms: how many units of capital can you give up if you gain one unit of labour, without losing any output?
Returns to scale
What happens to output when all inputs are scaled by the same factor t > 1. If output scales by exactly t, constant returns. By more than t, increasing returns. By less than t, decreasing returns.
Think of it as: if you double every input, does output exactly double (constant), more than double (increasing), or less than double (decreasing)?
The general form is Q = A · L^α · K^β. The exponents α and β are the key to everything: marginal products, the MRTS, and returns to scale all fall out of them.
To find a marginal product, take the partial derivative of Q with respect to that input.
MPL = ∂Q/∂L = A · α · L^(α-1) · K^β
MPK = ∂Q/∂K = A · β · L^α · K^(β-1)
Worked example: Q = 3L^(1/3) · K^(2/3)
MPL = 3 · (1/3) · L^(-2/3) · K^(2/3) = L^(-2/3) · K^(2/3) = (K/L)^(2/3)
MPK = 3 · (2/3) · L^(1/3) · K^(-1/3) = 2 · L^(1/3) · K^(-1/3) = 2(L/K)^(1/3)
MRTS(L,K) = MPL / MPK = (1/2)(K/L)
An isoquant for Q = S^(1/4) · D^(1/2) at Q = 4 is the set of (D, S) pairs satisfying S^(1/4) · D^(1/2) = 4. Rearranging: S = 256 / D². Plot S on the y-axis, D on the x-axis. The curve is convex to the origin, just like an indifference curve.
The MRTS along this isoquant tells you the trade-off rate between D and S. For Q = S^(1/4) · D^(1/2):
MP_D = (1/2) · S^(1/4) · D^(-1/2)
MP_S = (1/4) · S^(-3/4) · D^(1/2)
MRTS(D,S) = MP_D / MP_S = 2S / D
When MRTS = 1 (i.e. MP_D = MP_S), we have 2S/D = 1, so D = 2S. Along the Q = 4 isoquant, substituting S = 256/D² gives D = 2(256/D²), which solves to D³ = 512, so D = 8. Below D = 8 hours of dance practice, MP_D exceeds MP_S.
Not every production function is Cobb-Douglas. The problem set also covers:
Additive (perfect substitutes): Q = 3L + 5K. MPL = 3, MPK = 5, and MRTS = 3/5. The isoquants are straight lines, meaning inputs substitute at a constant rate.
Non-separable: Q = 4L⁴ + K³. MPL = 16L³, MPK = 3K². MRTS = 16L³ / 3K². Here the marginal products depend only on their own input, but the function is not multiplicative.
To test returns to scale, replace every input with t times that input and simplify.
For Q = L^(3/2) · K^(1/2): f(tL, tK) = (tL)^(3/2) · (tK)^(1/2) = t^(3/2) · L^(3/2) · t^(1/2) · K^(1/2) = t² · L^(3/2) · K^(1/2) = t² · Q.
Since t² > t for any t > 1, this production function exhibits increasing returns to scale, not constant. The claim in the problem set is false.
Quick shortcut for Cobb-Douglas: add the exponents. If α + β = 1, constant returns. Greater than 1, increasing. Less than 1, decreasing. Here 3/2 + 1/2 = 2 > 1, confirming increasing returns.
Paula has 16 hours total for school work (S) and dance (D). Her constraint is S + D ≤ 16. She prefers D = 2S, giving 3S = total hours used.
At D = 2S, her score is Q = S^(1/4) · (2S)^(1/2) = √2 · S^(3/4). For Q ≥ 4: S^(3/4) ≥ 4/√2 = 2√2, so S ≥ (2√2)^(4/3) ≈ 3.17 hours. Then D ≈ 6.35 hours, total ≈ 9.52 hours. Since 9.52 < 16, yes, Paula can hit Q = 4 with time to spare for socialising.
Cobb-Douglas marginal products
For Q = A · L^α · K^β:
MPL = Aα · L^(α-1) · K^β
MPK = Aβ · L^α · K^(β-1)
MRTS (of labour for capital)
MRTS(L,K) = MPL / MPK
For Cobb-Douglas this simplifies to (α/β) · (K/L).
Returns to scale test
Replace L with tL and K with tK. If f(tL, tK) = t^(α+β) · Q:
α + β = 1: constant returns to scale
α + β > 1: increasing returns to scale
α + β < 1: decreasing returns to scale
Isoquant rearrangement
For Q = S^(1/4) · D^(1/2) at Q = 4: solve for S to get S = 256 / D².
Cobb-Douglas functions are used to model national economies (GDP as a function of labour and capital), agricultural output, and manufacturing. The returns-to-scale property tells firms whether merging two small factories into one large one would produce more than the sum of their parts. Leontief functions model assembly lines and chemical processes where inputs must combine in fixed ratios: you cannot bake a cake with twice the flour and no eggs.
Students often assume that if MP_L > MP_K, the firm should use more labour. That comparison alone says nothing without knowing the prices of L and K. The cost-minimising condition compares MP_L/w to MP_K/r, not MP_L to MP_K.
A common error is to check returns to scale by doubling only one input. Returns to scale requires scaling all inputs simultaneously by the same factor.
Students sometimes confuse the MRTS with the ratio of input quantities. The MRTS is the ratio of marginal products (MP_L / MP_K), not K/L. For Cobb-Douglas, the MRTS simplifies to a function of K/L, which may be the source of the confusion.
With Leontief functions, students try to take partial derivatives to find marginal products. The min function is not differentiable at the kink, so standard calculus does not apply there.
⚠️ You will almost certainly be asked to compute MP_L, MP_K, and MRTS for a Cobb-Douglas function. Have the general formula memorised so you can write it down in seconds.
⚠️ Returns to scale is a favourite exam question. The exponent-sum shortcut (α + β vs. 1) only works for Cobb-Douglas. For other functional forms you must go through the full f(tL, tK) calculation.
⚠️ Drawing isoquants: know the shape. Cobb-Douglas isoquants are smooth and convex. Leontief isoquants are L-shaped. Perfect-substitute isoquants are straight lines.
⚠️ The MRTS along an isoquant changes as you move along it (for Cobb-Douglas). Expect to be asked at what point MP_L = MP_K, or equivalently where MRTS = 1.
True or false: For Q = L^(0.3) · K^(0.7), the production function has constant returns to scale. ___
Fill in the blank: The MRTS for a Cobb-Douglas function Q = A · L^α · K^β simplifies to (___) · (K/L).
True or false: Isoquants for a Leontief production function are smooth, downward-sloping curves. ___
Fill in the blank: If f(tL, tK) = t^1.5 · f(L, K), the function exhibits ___ returns to scale.
True or false: Along an isoquant, the MRTS is constant for a Cobb-Douglas production function. ___
Answers: 1. True (0.3 + 0.7 = 1). 2. α/β. 3. False (they are L-shaped). 4. Increasing. 5. False (it changes as the input mix changes).
Q: Given Q = S^(1/4) · D^(1/2), find MP_S and MP_D.
A: MP_S = (1/4) · S^(-3/4) · D^(1/2). MP_D = (1/2) · S^(1/4) · D^(-1/2).
Q: For Q = 3L + 5K, what is the MRTS(L,K)? What shape are the isoquants?
A: MRTS = MPL/MPK = 3/5. The isoquants are straight lines because the inputs are perfect substitutes.
Q: Does Q = L^(3/2) · K^(1/2) exhibit constant returns to scale? Explain.
A: No. Scaling inputs by t gives f(tL, tK) = t² · Q. Since t² > t for t > 1, the function has increasing returns to scale. The exponent sum is 3/2 + 1/2 = 2 > 1.
Q: Paula's score function is Q = S^(1/4) · D^(1/2) and she has 16 hours. If D = 2S, can she achieve Q = 4?
A: Yes. At D = 2S, Q = √2 · S^(3/4). Setting Q = 4 gives S ≈ 3.17, D ≈ 6.35, total ≈ 9.52 hours, well within 16.
Q: Along the Q = 4 isoquant for Q = S^(1/4) · D^(1/2), below what level of D does MP_D exceed MP_S?
A: MP_D > MP_S when MRTS > 1, i.e. 2S/D > 1. On the isoquant S = 256/D², this gives D < 8 hours.
Production functions mirror utility functions from consumer theory. Isoquants behave like indifference curves, and the MRTS is the production analogue of the marginal rate of substitution (MRS). If you understood consumer optimisation, cost minimisation uses the same tangency logic, except the budget constraint becomes an isocost line (wL + rK = C). Returns to scale connects directly to long-run average cost: increasing returns imply declining average costs, which matters for market structure (natural monopoly, economies of scale).
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