Difficulty: Intermediate
Prerequisites: Part 1 of these notes (Sections 6.1 and 6.2). Comfort with derivatives and partial derivatives is essential for the calculus-based portions.
Sections 6.3 and 6.4 are the analytical heart of the chapter. Section 6.3 holds capital fixed and examines what happens as the firm adds more labour: you meet total product, marginal product and average product curves, and the law of diminishing marginal returns. Section 6.4 frees both inputs and introduces isoquants, the production-side equivalent of indifference curves. Together, these sections give you the tools to describe any production technology, which Chapter 7 then uses to derive cost curves.
In the short run, adding workers to a fixed amount of capital first raises then eventually lowers the marginal product of labour (diminishing marginal returns). In the long run, the firm can substitute capital for labour along downward-sloping isoquants, and the marginal rate of technical substitution (MRTS) measures that trade-off. Different production functions (Cobb-Douglas, linear, fixed-proportions, CES) produce isoquants of different shapes.
Total product of labour
The total amount of output a given quantity of labour can produce, holding other inputs fixed. It is the short-run production function itself: q = f(L, K-bar).
Marginal product of labour (MPL)
The change in total output from using one additional unit of labour, holding all other inputs constant. Formally, MPL = dq/dL. Think of it as the extra output the next worker adds.
Average product of labour (APL)
Output per worker: APL = q/L. Think of it as the average contribution of each worker.
Law of diminishing marginal returns (diminishing marginal product)
If a firm keeps increasing one input while holding all other inputs and technology constant, the marginal product of that input will eventually decrease. The key qualifier is "eventually": MPL may rise at first (at low levels of labour) before it begins to fall.
Isoquant
A curve showing all efficient combinations of two inputs (typically labour and capital) that produce the same level of output. Think of it as the production-side equivalent of an indifference curve.
Marginal rate of technical substitution (MRTS)
The slope of an isoquant: the rate at which a firm can replace capital with labour while keeping output constant. MRTS = dK/dL = -MPL/MPK. Because isoquants slope downward, the MRTS is negative.
Diminishing marginal rate of technical substitution
As a firm moves along a curved isoquant, substituting labour for capital, each additional unit of labour replaces less and less capital. The isoquant gets flatter. This is why most isoquants are convex.
Cobb-Douglas production function
q = A x L^a x K^b, where A, a and b are constants. The most commonly used production function in economics. The MRTS equals -(a/b)(K/L), and the elasticity of substitution is always 1.
Constant elasticity of substitution (CES) production function
q = (aL^rho + bK^rho)^(d/rho), where rho < 1. The elasticity of substitution sigma = 1/(1 - rho) is the same at every point on an isoquant. Linear, fixed-proportions and Cobb-Douglas are all special cases.
Elasticity of substitution (sigma)
The percentage change in the capital-labour ratio (K/L) divided by the percentage change in the MRTS. Measures how easily a firm can substitute one input for another. A high sigma means the isoquant is relatively flat (easy substitution); a low sigma means it is sharply curved (hard to substitute).
Perfect substitutes (linear production function)
A production function where q = x + y, and the isoquants are straight lines with a constant slope. One input can always be swapped for the other at a fixed rate. Sigma is infinite.
Fixed-proportions production function (Leontief)
q = min(g, b), where inputs must be used in fixed ratios. The isoquants are right angles (or, more precisely, single points). No substitution is possible. Sigma approaches zero.
In the short run, capital is fixed at K-bar. The firm's only decision is how much labour to use: q = f(L, K-bar).
Total product (TP): the entire output at each level of L. Rises and then eventually flattens or falls.
Marginal product of labour (MPL): the slope of the total product curve at any point. MPL = dq/dL.
Average product of labour (APL): total output divided by labour. APL = q/L. Geometrically, it equals the slope of a line drawn from the origin to the relevant point on the TP curve.
When MPL > APL, the average product is rising. Adding a worker whose output exceeds the current average pulls the average up.
When MPL < APL, the average product is falling. A below-average worker drags the average down.
MPL = APL at the peak of the APL curve. This is the point where the line from the origin is tangent to the TP curve.
MPL = 0 at the peak of the TP curve. Beyond that point, extra workers would reduce total output.
Production function: q = 0.1LK + 3L^2K - 0.1L^3K. With K = 10: q = L + 30L^2 - L^3.
MPL = 1 + 60L - 3L^2 (differentiate q with respect to L).
APL = 1 + 30L - L^2 (divide q by L).
MPL peaks at L = 10, APL peaks at L = 15, and TP peaks at L = 20.
Beyond L = 20, the production function equation would give falling output, so a rational firm would never hire past that point.
States that if a firm keeps increasing one input (holding all others and technology constant), the marginal product of that input will eventually fall.
This is an empirical regularity, not a theoretical necessity.
Diminishing marginal returns means the MPL curve is falling. It does not mean output is falling. Output can still be rising (positive MPL), just at a decreasing rate.
"Diminishing returns" (without the word "marginal") is a much stronger claim: it means total output is falling (MPL is negative). A well-run firm would never operate in this range.
The law requires that technology and other inputs are held constant. If technology improves or other inputs increase alongside labour, MPL may keep rising indefinitely (as in the Green Revolution).
An isoquant maps all efficient (L, K) combinations that produce a given output level q-bar.
Four key properties:
Farther from the origin = higher output.
Isoquants never cross (crossing would imply the same input bundle is both efficient and inefficient).
Isoquants slope downward (an upward slope would mean wasting inputs).
Isoquants are thin (same logic as the downward slope).
Perfect substitutes (linear production function, e.g. q = x + y): isoquants are straight lines. Slope is constant. One input always replaces the other at the same rate.
Fixed proportions (Leontief, e.g. q = min(g, b)): isoquants are single efficient points along the 45-degree line. No substitution is possible.
Imperfect substitutes (most real production functions): isoquants are smooth, convex curves that lie between the extremes of straight lines and right angles. The ability to substitute varies along the curve.
MRTS = dK/dL = the slope of the isoquant at a point.
Derived by totally differentiating the isoquant q-bar = f(L, K) and setting dq = 0: MRTS = -MPL / MPK.
The MRTS tells you how many units of capital you can give up if you add one more unit of labour, while keeping output constant.
For a Cobb-Douglas function q = AL^a K^b, the MRTS = -(a/b)(K/L). It depends only on the exponents and the capital-labour ratio.
As you move down and to the right along a convex isoquant (more L, less K), the MRTS falls in absolute value: the isoquant gets flatter.
Intuition: the more labour you already have relative to capital, the harder it is to replace the remaining capital with yet more labour.
Perfect substitutes do not exhibit diminishing MRTS (constant slope). Fixed proportions have no substitution at all.
sigma = (percentage change in K/L) / (percentage change in |MRTS|). Equivalently, sigma = d ln(K/L) / d ln|MRTS|.
A large sigma means the isoquant is relatively flat and inputs are easy to swap. A small sigma means the isoquant is sharply curved.
For a CES production function q = (L^rho + K^rho)^(1/rho), sigma = 1/(1 - rho).
rho = 1: linear, sigma = infinity (perfect substitutes).
rho approaches 0: Cobb-Douglas, sigma = 1.
rho approaches negative infinity: fixed proportions, sigma approaches 0.
The Cobb-Douglas elasticity of substitution is always exactly 1, regardless of the values of a and b.
Empirical CES estimates vary by industry: 0.15 in paper, 0.29 in agriculture, 0.56 in primary metals, 1.16 in finance/insurance/real estate.
Short-run production function
q = f(L, \bar{K})Marginal product of labour
MP_L = \frac{\partial q}{\partial L} = \frac{\partial f(L, K)}{\partial L}Average product of labour
AP_L = \frac{q}{L}Cobb-Douglas production function
q = A L^a K^bMarginal products: MPL = aq/L, MPK = bq/K.
MRTS (slope of isoquant)
MRTS = \frac{dK}{dL} = -\frac{MP_L}{MP_K}For Cobb-Douglas: MRTS = -(a/b)(K/L).
CES production function
q = (L^{\rho} + K^{\rho})^{1/\rho}with MRTS = -(L/K)^(rho - 1) and sigma = 1/(1 - rho).
Elasticity of substitution
\sigma = \frac{d \ln(K/L)}{d \ln |MRTS|}Malthus and the Green Revolution: Malthus predicted mass starvation because land is fixed and diminishing marginal returns to labour would slow food production. Technology (hybrid seeds, fertilisers, pesticides, irrigation) shifted the production function outward, preventing his prediction from materialising. Norman Borlaug's Green Revolution is estimated to have prevented a billion deaths.
Self-driving trucks: a trucking isoquant can be built from two technologies (traditional and self-driving), each using labour and capital in fixed proportions. Mixing both creates a kinked isoquant that curves away from the origin, just like a smooth isoquant.
Students often confuse "diminishing marginal returns" with "diminishing returns." Diminishing marginal returns means MPL is falling but may still be positive (output is still rising). Diminishing returns means output itself is falling (MPL is negative). A well-run firm never operates in the diminishing-returns region.
Students often think the law of diminishing marginal returns means MPL must always fall as you add more labour. It says MPL will eventually fall. At low levels of labour, MPL may be rising.
Students sometimes think isoquants can cross. They cannot, because crossing would mean the same input combination produces two different output levels, violating the efficiency requirement.
Students often confuse the MRTS with the marginal product. The MRTS is the ratio of marginal products (-MPL/MPK), not a marginal product itself.
⚠️ Be able to derive MPL and APL from a given production function and find where each peaks.
⚠️ Know that MPL = APL at the peak of APL. This is a classic exam question.
⚠️ Be precise about diminishing marginal returns vs. diminishing returns.
⚠️ Be able to derive the MRTS from a Cobb-Douglas production function: MRTS = -(a/b)(K/L).
⚠️ Know the four properties of isoquants and be ready to explain the intuition behind each one.
⚠️ Know that the Cobb-Douglas elasticity of substitution is always 1, and be able to explain what that means.
⚠️ Understand how the CES nests linear (rho = 1), Cobb-Douglas (rho approaches 0) and fixed proportions (rho approaches negative infinity).
True or False: Diminishing marginal returns means total output is falling. (False. It means MPL is falling, but output may still be rising.)
Fill in the blank: MPL = APL at the ______ of the APL curve. (peak)
True or False: Isoquants can cross if the firm uses different technologies. (False. Isoquants never cross, regardless of the technology.)
Fill in the blank: For a Cobb-Douglas production function, the elasticity of substitution is always ______. (1)
True or False: The MRTS is the slope of the total product curve. (False. The MRTS is the slope of the isoquant. The slope of the total product curve is the MPL.)
Q: Given the production function q = L + 30L^2 - L^3 (with K fixed at 10), derive the MPL and find the value of L where MPL = 0.
A: MPL = dq/dL = 1 + 60L - 3L^2. Setting MPL = 0: 3L^2 - 60L - 1 = 0. Using the quadratic formula, L is approximately 20. Beyond 20 workers, additional labour would reduce output.
Q: Explain why the APL curve peaks where it intersects the MPL curve.
A: When the marginal worker's output exceeds the current average (MPL > APL), the average is pulled up. When it falls below (MPL < APL), the average is pulled down. At the crossing point, the marginal worker's output exactly equals the average, so the average is neither rising nor falling: it is at its peak.
Q: For the Cobb-Douglas function q = L^0.5 K^0.5, calculate the MRTS at the point (L = 6, K = 6) and at (L = 12, K = 3).
A: MRTS = -(a/b)(K/L) = -(0.5/0.5)(K/L) = -K/L. At (6, 6): MRTS = -1. At (12, 3): MRTS = -0.25. The isoquant is flatter at the second point because the firm has relatively more labour and less capital.
Q: A CES production function has rho = 0.5. What is the elasticity of substitution? Is substitution between inputs easier or harder than under a Cobb-Douglas?
A: sigma = 1/(1 - 0.5) = 2. Since sigma > 1 (compared to sigma = 1 for Cobb-Douglas), substitution is easier: the isoquant is flatter and inputs can be swapped more readily.
Isoquants are the production-theory parallel of indifference curves from consumer theory (Chapters 3 to 5). The MRTS is the analogue of the MRS. If you understood consumer theory, this section should feel familiar.
The MRTS and input substitution feed directly into cost minimisation in Chapter 7, where the firm chooses the input mix where the isoquant is tangent to the isocost line.
Diminishing marginal returns underpin the shape of short-run cost curves (Chapter 7) and the supply curve (Chapter 8).
total product of labour, marginal product of labour, MPL, average product of labour, APL, diminishing marginal returns, diminishing marginal product, law of diminishing returns, isoquant, marginal rate of technical substitution, MRTS, Cobb-Douglas production function, CES production function, constant elasticity of substitution, elasticity of substitution, perfect substitutes, fixed proportions, Leontief, linear production function, convex isoquant, input substitution, Perloff Chapter 6, intermediate microeconomics, ECN 302