Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), familiarity with two-input production.
This material sits at the core of producer theory in intermediate microeconomics. Before a firm can decide how much to produce or what to charge, it needs to understand the relationship between its inputs (capital K, labour L) and output Q. This topic covers how to measure the productivity of each input, how to compare the trade-off between inputs along an isoquant, and how different functional forms (perfect substitutes, perfect complements, and others) behave. If you are comfortable with partial derivatives, you are ready for this. If not, review basic calculus first.
A production function maps inputs to output. Marginal products tell you the extra output from one more unit of an input. The MRTS tells you the rate at which you can swap one input for the other while keeping output constant, and it equals the ratio of marginal products.
Production function, f(K, L)
A mathematical relationship that maps quantities of inputs (capital K, labour L) to the maximum output Q the firm can produce. Think of it as the firm's recipe: given these ingredients, here is what you get.
Marginal product of capital (MPK)
The additional output produced by one more unit of capital, holding labour constant: MPK = ∂f/∂K. In simple terms, it answers "if I add one more machine, how much more do I produce?"
Marginal product of labour (MPL)
The additional output produced by one more unit of labour, holding capital constant: MPL = ∂f/∂L. In simple terms, it answers "if I hire one more worker, how much more do I produce?"
Average product of capital (APK)
Total output divided by the amount of capital used: APK = f(K, L) / K. Think of it as output per unit of capital.
Average product of labour (APL)
Total output divided by the amount of labour used: APL = f(K, L) / L. Think of it as output per worker.
Marginal rate of technical substitution (MRTS)
The rate at which a firm can replace one input with another while keeping output unchanged. Formally, MRTS = MPL / MPK. In simple terms, it is the slope of the isoquant and tells you "how many units of capital can I give up if I add one more unit of labour?"
Isoquant
A curve showing all input combinations (K, L) that produce the same level of output. Think of it as the production equivalent of an indifference curve.
Perfect substitutes (in production)
Inputs that can replace each other at a constant rate. The production function takes the form f(K, L) = aK + bL. Isoquants are straight lines.
Perfect complements (in production)
Inputs that must be used in fixed proportions. The production function takes the form f(K, L) = A·min{K/a, L/b}. Isoquants are L-shaped.
Both inputs contribute equally and interchangeably to output.
MPK = 1 and MPL = 1, regardless of input levels. The marginal products are constant.
APK = (K + L) / K. This changes with the input mix even though MPK does not.
At K = 1, L = 4: APK = 5, APL = 5/4 = 1.25
At K = 9, L = 4: APK = 13/9, APL = 13/4
APL = (K + L) / L. Same idea, from the labour side.
Notice that average products depend on the ratio of inputs, but marginal products do not. This is a signature of linear production functions.
This function is the harmonic mean of K and L (scaled by A). Output is bounded by the smaller input.
MPK = A · L² / (K + L)²
MPL = A · K² / (K + L)²
Both marginal products are positive and diminishing. As you add more of one input holding the other fixed, the additional output tapers off.
A symmetric Cobb-Douglas function with equal exponents (each ½).
MPK = (A/2)√(L/K), MPL = (A/2)√(K/L)
Diminishing marginal products in each input.
Constant returns to scale: doubling both inputs exactly doubles output.
Inputs must be used in a 1:1 ratio (or more generally, in fixed proportions).
No substitution is possible. Adding more of one input without the other produces nothing extra.
The optimal choice is always K = L (when the ratio is 1:1). Any other combination wastes one input.
Some production functions require a minimum input before any output appears.
Example from the homework: f(L) = 0 for L ≤ 10, and f(L) = 5(L - 10) for L > 10.
The first 10 units of labour produce nothing (a fixed setup requirement). Output is linear beyond that threshold.
To produce Q > 0, you need L = Q/5 + 10.
MRTS = MPL / MPK. It measures how many units of K the firm can give up per extra unit of L, staying on the same isoquant.
For f(K, L) = K + L: MRTS = 1/1 = 1. The firm can always swap one unit of K for one unit of L. The isoquant is a straight line with slope -1.
For f(K, L) = A · KL / (K + L): MRTS = (K/L)². The substitution rate depends on where you are on the isoquant. When K is large relative to L, the firm is willing to give up a lot of K for a little more L.
For f(K, L) = (K + L)²: MRTS = 1. Same as the linear case, because the underlying substitution ratio is still 1:1 (the squaring is a monotonic transformation that does not change the shape of the isoquants).
At the cost-minimising input bundle, MRTS = r/w (the ratio of input prices, where r is the rental rate of capital and w is the wage).
This is the tangency condition: the isoquant is tangent to the isocost line.
For perfect substitutes, the tangency condition never holds in the interior. The firm uses only the cheaper input (a corner solution).
For perfect complements, there is no substitution at all. The firm uses inputs in the fixed ratio regardless of prices.
Linear (K + L): straight lines, slope = -1.
Harmonic mean (AKL/(K+L)): convex curves, MRTS diminishes as you move down the isoquant (standard shape).
Cobb-Douglas (A√(KL)): convex curves, smooth substitutability.
Leontief / perfect complements (A·min{K,L}): right-angle (L-shaped) isoquants at K = L.
Function | f(K, L) | MPK | MPL | MRTS |
|---|---|---|---|---|
Perfect substitutes | K + L | 1 | 1 | 1 |
Harmonic mean | A·KL/(K+L) | A·L²/(K+L)² | A·K²/(K+L)² | (K/L)² |
Cobb-Douglas (symmetric) | A√(KL) | (A/2)√(L/K) | (A/2)√(K/L) | K/L |
Perfect complements | A·min{K, L} | Undefined at kink | Undefined at kink | 0 or ∞ (flat/vertical segments) |
Squared linear | (K+L)² | 2(K+L) | 2(K+L) | 1 |
Key relationships to internalise:
MRTS = MPL / MPK (always)
At cost minimum: MRTS = r / w
APK = f(K,L) / K, APL = f(K,L) / L
When MPK > APK, average product is rising. When MPK < APK, it is falling.
Perfect substitutes appear when two energy sources (gas vs. electric heating) can be swapped freely. The firm simply picks whichever is cheaper.
Perfect complements describe processes where inputs must match: one truck per driver, one CPU per motherboard. Buying extra of one without the other is waste.
The MRTS matters in practice whenever a firm weighs automation (more K) against hiring (more L). The tangency condition MRTS = r/w is the formal version of "automate when machines are cheap relative to wages."
Students often confuse marginal product with average product. MPK is the derivative (the slope of the production function with respect to K). APK is total output divided by K. They are not the same and they move independently.
Students sometimes think MRTS = MPK / MPL. It is the other way round: MRTS = MPL / MPK. The numerator is the marginal product of the input you are adding (L), not the one you are giving up.
For perfect substitutes, students assume there must be an interior solution where the firm uses both inputs. There is not, unless prices happen to be exactly proportional to marginal products. Generically, the firm goes all-in on the cheaper input.
For (K + L)², students sometimes think the isoquant shape changes because of the squaring. It does not. Squaring is a monotonic transformation: the isoquants of (K + L)² are identical to those of K + L.
⚠️ Be ready to compute MPK, MPL, APK, and APL for any given production function and specific input values. This is bread-and-butter exam material.
⚠️ Know how to derive the MRTS from marginal products and state the cost-minimisation tangency condition MRTS = r/w.
⚠️ You will almost certainly be asked to identify whether a production function has perfect substitutes, perfect complements, or diminishing MRTS, and to sketch or describe the isoquant shape.
⚠️ The distinction between corner solutions (perfect substitutes) and interior solutions (smooth production functions) comes up frequently. Know when each applies.
True or false: If f(K, L) = K + L, then APK is constant regardless of the input mix. (False. APK = (K+L)/K, which changes with L.)
Fill in the blank: MRTS = ___ / ___. (MPL / MPK)
True or false: For a Leontief production function, adding more capital while holding labour fixed always raises output. (False. Output is determined by the binding constraint, min{K, L}. Extra K does nothing if L is the bottleneck.)
Fill in the blank: At the cost-minimising input bundle, the isoquant is ___ to the isocost line. (Tangent)
True or false: The MRTS for f(K, L) = (K + L)² is 1. (True. The isoquants have the same shape as those of K + L.)
Q: Given f(K, L) = K + L, compute MPK, MPL, APK, and APL at K = 9, L = 4.
A: MPK = 1, MPL = 1. APK = (9 + 4)/9 = 13/9. APL = (9 + 4)/4 = 13/4.
Q: For f(K, L) = A · KL/(K + L), derive the MRTS.
A: MPL = A · K²/(K + L)². MPK = A · L²/(K + L)². MRTS = MPL/MPK = K²/L² = (K/L)².
Q: A firm has production function f(K, L) = K + L. The rental rate of capital is r and the wage is w. What inputs does the firm choose to produce Q units of output?
A: If r < w, the firm uses only capital: K = Q, L = 0. If w < r, the firm uses only labour: K = 0, L = Q. If r = w, any combination with K + L = Q is equally cheap.
Q: For f(K, L) = (K + L)², what is the cost function C(Q, r, w)?
A: To produce Q, the firm needs K + L = √Q (invert the production function). The cost function is C(Q) = min{r, w} · √Q.
Q: Explain why the MRTS of f(K, L) = (K + L)² is the same as that of f(K, L) = K + L.
A: Squaring is a monotonic increasing transformation of output. It rescales the output labels on the isoquants but does not change their shape or slope. The trade-off between K and L at any point remains 1-for-1.
This material feeds directly into cost minimisation and cost functions (covered in the companion study notes). Once you know the MRTS and the tangency condition, you can derive the cost function for any production technology.
Returns to scale (constant, increasing, decreasing) are closely related: they describe what happens to output when all inputs are scaled up proportionally, while marginal products describe what happens when one input changes alone.
Consumer theory has a parallel structure: utility functions, MRS, and expenditure minimisation mirror production functions, MRTS, and cost minimisation.
production function, marginal product, average product, MPK, MPL, APK, APL, MRTS, marginal rate of technical substitution, isoquant, isocost, perfect substitutes production, perfect complements production, Leontief production, Cobb-Douglas, harmonic mean production function, cost minimisation tangency condition, corner solution, interior solution, returns to scale, producer theory, intermediate microeconomics, ECON 500