Difficulty: Intermediate | Prerequisites: basic calculus (partial derivatives, logarithm rules), budget constraint algebra.
Consumer theory sits at the heart of intermediate microeconomics. The central question: given a budget and a set of prices, how does a rational consumer choose quantities of goods to maximise their satisfaction? This unit builds the mathematical machinery to answer that, starting from a utility function, moving through marginal utility and the marginal rate of substitution, and arriving at demand functions that express optimal quantities in terms of prices and income. If you are comfortable with partial derivatives and basic algebra, you have everything you need.
The marginal rate of substitution (MRS) measures the rate at which a consumer trades one good for another while staying equally happy. At the optimum, MRS equals the price ratio, and the consumer spends their entire budget. Solving these two conditions together yields the demand function for each good, which tells you how much the consumer buys at any combination of prices and income.
Utility function
A mathematical representation of a consumer's preferences over bundles of goods. Higher utility means a more preferred bundle. Think of it as a scoring system: the consumer picks whichever bundle scores highest within their budget.
Marginal utility (MU)
The additional utility gained from consuming one more unit of a good, holding everything else constant. Formally, MU_x = dU/dx. In simple terms, this means how much happier one extra unit of good x makes you.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to give up good B to get one more unit of good A, while remaining on the same indifference curve. MRS = -MU_A / MU_B. In simple terms, this means the personal "exchange rate" between two goods in the consumer's own preferences.
Budget constraint
The set of all affordable bundles given prices and income: P_A * A + P_B * B = M. Everything on or below this line is within reach. Think of it as a wall: the consumer can shop anywhere inside it, but not beyond.
Tangency condition
At the optimal interior solution, MRS equals the price ratio: MU_A / P_A = MU_B / P_B. The consumer's personal trade-off matches the market's trade-off. In simple terms, this means the last pound spent on each good delivers the same marginal utility.
Marshallian (ordinary) demand function
The optimal quantity of a good expressed as a function of prices and income: x*(P_x, P_y, M). Derived by solving the tangency condition together with the budget constraint. Think of it as the consumer's shopping list, written in terms of prices and income.
Normal good
A good whose demand increases when income rises, i.e. dA*/dM > 0. In simple terms, this means you buy more of it when you get richer.
Inferior good
A good whose demand decreases when income rises, i.e. dA*/dM < 0. In simple terms, this means you switch away from it when you can afford better alternatives.
Lagrangian method
A technique for solving constrained optimisation problems. Set up L = U(x,y) + lambda * (M - P_x * x - P_y * y), take partial derivatives, set them to zero, and solve. Think of it as a systematic machine for finding the best affordable bundle, especially when the tangency shortcut gets messy.
Consider U(A, B) = beta * ln(A) + alpha * ln(B).
Marginal utility of A: MU_A = beta / A
Marginal utility of B: MU_B = alpha / B
MRS = -MU_A / MU_B = -(beta / A) / (alpha / B) = -(B * beta) / (A * alpha)
Equivalently: dA/dB = -A * alpha / (B * beta)
The MRS is negative, confirming a decreasing function (the indifference curve slopes downward).
To find the consumer's best affordable bundle, solve two equations simultaneously:
Tangency condition: MU_A / P_A = MU_B / P_B
The marginal utility per pound spent must be equal across both goods.
Budget constraint: A * P_A + B * P_B = M
The consumer spends their entire income.
Starting from U(A, B) = beta * ln(A) + alpha * ln(B):
From the tangency condition: beta / (A * P_A) = alpha / (B * P_B)
Rearranging: B * P_B * beta = A * P_A * alpha
Substitute into the budget constraint to isolate each good:
Demand for B: B* = (alpha * M) / (P_B * (alpha + beta)) , with a correction term
Demand for A: A* = (M * (beta - alpha)) / (beta * P_A) , or more precisely A* = (beta * M - alpha * M + alpha * beta^2) / (beta * P_A)
The exact algebra depends on the parameter values. The key pattern: each demand function is a fraction of income divided by price, scaled by the utility parameters.
Once you have a demand function like A = (M - B * P_B) / P_A:
Take the partial derivative with respect to income: dA/dM = 1 / P_A
Since 1 / P_A > 0, demand for A rises with income. A is a normal good.
A good is inferior only if dA/dM < 0. This does not happen here.
For U(m, n) = a * m^r * n^s with budget constraint P_m * m + P_n * n = W:
Set up the Lagrangian: L = a * m^r * n^s + lambda * (W - P_m * m - P_n * n)
First-order conditions:
dL/dm = r * a * m^(r-1) * n^s - lambda * P_m = 0
dL/dn = s * a * m^r * n^(s-1) - lambda * P_n = 0
dL/d(lambda) = W - P_m * m - P_n * n = 0
From the first two FOCs, set lambda = lambda:
r * a * m^(r-1) * n^s / P_m = s * a * m^r * n^(s-1) / P_n
Simplifies to: r / (m * P_m) = s / (n * P_n)
So: n = m * P_m * s / (P_n * r)
Substituting n = m into the budget constraint gives m = r * W / (P_m * (r + s))* and n = s * W / (P_n * (r + s))*.
These are the Marshallian demands for a Cobb-Douglas utility function. Each good receives a fixed share of income, determined by its exponent's share of the total exponent sum.
For U(m, n) = a * m^r * n^s:
MRS = -MU_m / MU_n = -(r * a * m^(r-1) * n^s) / (s * a * m^r * n^(s-1)) = -r * n / (s * m)
With m = 24, n = 16, r = 3, s = 2, a = 1: MRS = -3(16) / (2(24)) = -48/48 = -1
An MRS of -1 means the consumer trades the two goods one-for-one at the margin. At this particular bundle, the goods behave like perfect substitutes.
Log utility MRS
U(A, B) = beta * ln(A) + alpha * ln(B) gives MRS = -(A * alpha) / (B * beta)
Cobb-Douglas MRS
U(m, n) = a * m^r * n^s gives MRS = -(r * n) / (s * m)
Tangency condition (general)
MU_x / P_x = MU_y / P_y , or equivalently |MRS| = P_x / P_y
Cobb-Douglas Marshallian demands
m* = r * W / [P_m * (r + s)] and n* = s * W / [P_n * (r + s)]
Each good receives a share of income proportional to its exponent divided by the sum of exponents.
Lagrangian setup
L = U(x, y) + lambda * (M - P_x * x - P_y * y)
Take partial derivatives with respect to x, y, and lambda, set each to zero, then solve the system.
Normal good test
Take dA*/dM. If positive, the good is normal. If negative, it is inferior.
Students often confuse the sign of the MRS. The MRS is negative because the indifference curve slopes downward, but many exam questions ask for the absolute value. Read the question carefully.
Students sometimes think the tangency condition alone is enough to find the optimum. It is not. You always need the budget constraint as well, giving you two equations in two unknowns.
Students often assume every good must be normal. Inferior goods exist in theory and in practice. Always check the sign of dA*/dM rather than guessing.
When using the Lagrangian, students sometimes forget to set up the third FOC (the derivative with respect to lambda). This is simply the budget constraint, and skipping it leaves you with an underdetermined system.
Deriving the MRS from a utility function is one of the most frequently tested skills in intermediate micro. Expect it on every problem set and exam.
The Cobb-Douglas demand result (fixed income shares) is a workhorse. Many exam questions rely on you knowing or being able to derive m* = rW / [P_m(r+s)].
Classifying a good as normal or inferior by differentiating the demand function with respect to income is a standard exam question. The answer is always: take the derivative, check the sign.
The Lagrangian method and the tangency-plus-budget-constraint approach give the same answer. You should be comfortable with both. Some lecturers favour one; the Lagrangian scales better to three or more goods.
Lambda (the Lagrange multiplier) has an economic interpretation: it is the marginal utility of income, the extra utility from one more unit of budget. Some exams ask for this interpretation explicitly.
True or false: The MRS is always positive for a well-behaved utility function. (False: it is negative, reflecting the downward slope of indifference curves.)
Fill in the blank: At the consumer's optimum, MU_A / P_A = ______ . (MU_B / P_B)
True or false: If demand for a good rises when income rises, the good is inferior. (False: it is normal.)
Fill in the blank: For Cobb-Douglas utility U = m^r * n^s, the optimal demand for m is m* = ______ . (rW / [P_m(r + s)])
True or false: The Lagrange multiplier lambda represents the marginal utility of income. (True.)
Q: Given U(A, B) = beta * ln(A) + alpha * ln(B), derive the MRS.
A: MU_A = beta / A and MU_B = alpha / B. MRS = -MU_A / MU_B = -(beta * B) / (alpha * A).
Q: For the same utility function, state the two conditions needed to find the optimal bundle.
A: (1) Tangency: beta / (A * P_A) = alpha / (B * P_B). (2) Budget constraint: A * P_A + B * P_B = M.
Q: Given demand A = (M - B * P_B) / P_A, is A a normal or inferior good? Show your reasoning.
A: Take dA/dM = 1 / P_A. Since P_A > 0, this derivative is positive, so A is a normal good.
Q: For U(m, n) = m^3 * n^2, set up the Lagrangian and derive the first-order conditions.
A: L = m^3 * n^2 + lambda * (W - P_m * m - P_n * n). FOCs: 3m^2 * n^2 - lambda * P_m = 0; 2m^3 * n - lambda * P_n = 0; W - P_m * m - P_n * n = 0.
Q: Using those FOCs, show that m = 3W / [P_m * 5] and n = 2W / [P_n * 5].**
A: From the first two FOCs, set lambda = lambda: 3m^2 * n^2 / P_m = 2m^3 * n / P_n. Cancel common terms to get 3n * P_n = 2m * P_m, so n = 2m * P_m / (3 * P_n). Substitute into the budget constraint: P_m * m + P_n * [2m * P_m / (3P_n)] = W, giving m * P_m * (1 + 2/3) = W, so m * P_m * 5/3 = W, hence m* = 3W / (5 * P_m). By symmetry, n* = 2W / (5 * P_n).
Q: Compute the MRS of U(m, n) = m^3 * n^2 at m = 24, n = 16. What does the value tell you?
A: MRS = -(r * n) / (s * m) = -(3 * 16) / (2 * 24) = -48/48 = -1. At this bundle, the consumer is willing to trade one unit of n for exactly one unit of m: the goods are locally perfect substitutes at the margin.
This material connects directly to consumer demand theory. Once you have Marshallian demands, you can derive price-consumption curves, income-consumption curves (Engel curves), and the Slutsky decomposition of price effects into substitution and income effects.
The Lagrangian method reappears in producer theory (cost minimisation, profit maximisation) and later in general equilibrium. Mastering it here pays dividends throughout the course.
The classification of goods as normal or inferior feeds into the analysis of Giffen goods: a Giffen good is an inferior good whose income effect is so large that it overwhelms the substitution effect, making the demand curve slope upward.
Utility maximisation, consumer optimisation, marginal rate of substitution, MRS, indifference curve, budget line, tangency condition, equal marginal principle, Marshallian demand, ordinary demand, Cobb-Douglas utility, log utility, Lagrange multiplier, shadow price of income, normal good, inferior good, constrained optimisation, first-order conditions, FOC, intermediate microeconomics, ECON, University of Washington.