Difficulty: Intermediate | Prerequisites: Basic algebra, graphing in two dimensions, familiarity with functions and slopes.
Consumer theory is the foundation of intermediate microeconomics. It answers a deceptively simple question: given limited income, how does a rational consumer decide what to buy? The answer runs through three layers, each building on the last: the budget constraint (what you can afford), preferences (what you want), and optimisation (picking the best affordable bundle). Everything else in the consumer half of the course, from demand curves to welfare analysis, grows out of this framework. If you are coming in cold, make sure you are comfortable with two-variable algebra and reading graphs in the x-y plane before you start.
A budget constraint shows every combination of two goods a consumer can afford. Preferences, captured by utility functions and indifference curves, rank those combinations. The optimal bundle is the affordable combination that gives the highest utility, found where the indifference curve is tangent to the budget line (or at a corner for perfect complements and perfect substitutes).
Budget constraint
The set of all bundles of goods a consumer can purchase given their income (M) and the prices of goods (P_X, P_Y). Graphically, it is the line P_X · X + P_Y · Y = M. In simple terms, it is the boundary of what you can afford.
Marginal rate of transformation (MRT)
The slope of the budget constraint, equal to -P_X / P_Y. It tells you how many units of Y the market requires you to give up to get one more unit of X. Think of it as the "exchange rate" between the two goods set by their prices.
Utility function
A mathematical function U(X, Y) that assigns a number to each bundle, representing how much satisfaction the consumer gets from it. Higher numbers mean more preferred bundles.
Indifference curve
A curve connecting all bundles that give the consumer the same level of utility. In simple terms, every point on the curve feels equally good to the consumer.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade Y for X while staying equally happy. Formally, MRS = MU_X / MU_Y. It is the slope of the indifference curve (in absolute value). Think of it as your personal "exchange rate," as opposed to the market's.
Optimal bundle
The bundle that maximises utility subject to the budget constraint. For interior solutions this is where MRS = MRT (equivalently, MU_X / P_X = MU_Y / P_Y).
Cobb-Douglas utility
A utility function of the form U = X^a · Y^b. Always produces interior (tangency) solutions. The consumer spends a fixed fraction of income on each good.
Perfect complements
Goods consumed in fixed proportions, e.g. U = min(aX, bY). The optimal bundle sits at the kink of the L-shaped indifference curve, where aX = bY.
Perfect substitutes
Goods the consumer views as interchangeable at a constant rate, e.g. U = aX + bY. The solution is a corner: spend all income on whichever good gives more utility per pound spent, unless MRS equals the price ratio exactly.
Price consumption curve (PCC)
The locus of optimal bundles as the price of one good changes, holding income and the other price fixed.
Demand curve
The relationship between the price of a good and the quantity demanded, derived from the PCC by plotting P_X against X*.
Income consumption curve (ICC)
The locus of optimal bundles as income changes, holding prices fixed.
Engel curve
The relationship between income and the quantity demanded of a good, derived from the ICC by plotting M against X*.
The budget constraint equation is P_X · X + P_Y · Y = M, where M is income.
Rearranging for Y gives Y = M/P_Y - (P_X/P_Y) · X. The Y-intercept is M/P_Y, the X-intercept is M/P_X, and the slope is -P_X/P_Y.
Shifts from price changes: if P_X falls, the X-intercept moves outward while the Y-intercept stays put, so the line pivots. The same logic applies symmetrically for P_Y.
Shifts from income changes: if M increases, both intercepts shift outward and the budget line moves out in parallel (the slope does not change because prices have not changed).
The slope of the budget constraint is also called the MRT. It measures the opportunity cost of X in terms of Y at market prices.
Three assumptions about consumer preferences:
Completeness: the consumer can rank any two bundles (prefers A, prefers B, or is indifferent).
Transitivity: if A is preferred to B and B is preferred to C, then A is preferred to C.
More is better (non-satiation): a bundle with more of at least one good and no less of the other is always preferred.
Constructing an indifference curve from a utility function: set U(X, Y) = k for some constant k determined by the given bundle. Solve for Y as a function of X. That equation traces the indifference curve through the bundle.
Calculating the MRS: compute the partial derivatives MU_X = ∂U/∂X and MU_Y = ∂U/∂Y. Then MRS = MU_X / MU_Y. Evaluate at a specific bundle to get a number.
Interpreting the MRS: it tells you how many units of Y the consumer would voluntarily give up for one more unit of X while remaining indifferent. A declining MRS along an indifference curve reflects diminishing willingness to substitute.
General method: maximise U(X, Y) subject to P_X · X + P_Y · Y = M.
Cobb-Douglas (e.g. U = X^a · Y^b):
Set MRS = MRT, i.e. (a/b)(Y/X) = P_X / P_Y. Solve for Y in terms of X.
Substitute into the budget constraint and solve for X*, then Y*.
Shortcut: the consumer spends the fraction a/(a + b) of income on X and b/(a + b) on Y. So X* = [a/(a + b)] · M / P_X.
Perfect complements (e.g. U = min(aX, bY)):
The consumer always chooses the kink point where aX = bY.
Substitute Y = (a/b)X into the budget constraint: P_X · X + P_Y · (a/b)X = M. Solve for X*.
Perfect substitutes (e.g. U = aX + bY):
Compare MRS = a/b to MRT = P_X / P_Y.
If a/b > P_X / P_Y, spend all income on X (corner solution: X* = M / P_X, Y* = 0).
If a/b < P_X / P_Y, spend all income on Y.
If a/b = P_X / P_Y, any bundle on the budget line is optimal.
Price consumption curve (PCC): trace the optimal bundle as P_X varies (holding P_Y and M constant). Connecting these bundles in (X, Y) space gives the PCC.
Demand curve: derived from the PCC. For each P_X, read off X* and plot the (P_X, X*) pair. The result is the consumer's demand curve for good X.
Income consumption curve (ICC): trace the optimal bundle as M varies (holding P_X and P_Y constant). For Cobb-Douglas, the ICC is a ray from the origin because expenditure shares are constant.
Engel curve: derived from the ICC. For each M, read off X* and plot (M, X*). An upward-sloping Engel curve means X is a normal good. A downward-sloping one means X is an inferior good.
\text{Budget constraint: } P_X \cdot X + P_Y \cdot Y = M\text{MRT (slope of budget line)} = -\frac{P_X}{P_Y}\text{MRS} = \frac{MU_X}{MU_Y} = \frac{\partial U / \partial X}{\partial U / \partial Y}\text{Optimality (interior): MRS} = \text{MRT} \quad \Longleftrightarrow \quad \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}\text{Cobb-Douglas shortcut: } X^* = \frac{a}{a+b} \cdot \frac{M}{P_X}, \quad Y^* = \frac{b}{a+b} \cdot \frac{M}{P_Y}Students often think the MRS and MRT are the same thing. They are not. The MRS is the consumer's willingness to trade (slope of the indifference curve). The MRT is the market's exchange rate (slope of the budget line). At the optimum they happen to be equal, but they measure different things.
Students often set MRS = price ratio and forget to also use the budget constraint. Two equations, two unknowns: you need both to solve for the optimal bundle.
With perfect substitutes, students sometimes try to find an interior tangency. There usually is none. The solution is almost always a corner.
Students sometimes confuse the ICC with the Engel curve. The ICC lives in (X, Y) space. The Engel curve lives in (M, X) space. They contain the same information plotted differently.
⚠️ The exam has 2 questions on budget constraints, 2 on preferences, 2 on finding optimal bundles, and 1 on how price or income changes shift the optimum. That is 7 of 27 questions from this material alone.
⚠️ Any of the three utility function types (Cobb-Douglas, perfect complements, perfect substitutes) could appear. Know all three solution methods cold.
⚠️ You will need to graph budget constraints from given M, P_X, P_Y. Practice computing intercepts and slope quickly.
⚠️ "Calculate and interpret the MRS" is listed separately, meaning the exam will likely ask for both the number and a one-sentence explanation of what it means.
True or false: if P_X doubles and M doubles, the budget constraint does not change. (True: both intercepts stay the same.)
Fill in the blank: the MRS equals ______ divided by ______. (MU_X divided by MU_Y.)
True or false: for perfect complements, the optimal bundle is always at the kink of the indifference curve. (True.)
Fill in the blank: the Cobb-Douglas shortcut says X* = [a/(a + b)] · ______ / ______. (M / P_X.)
True or false: an upward-sloping Engel curve means the good is inferior. (False: it means the good is normal.)
Q: A consumer has utility U = X^(1/3) · Y^(2/3), income M = 120, P_X = 4, P_Y = 2. What is the optimal bundle?
A: Using the Cobb-Douglas shortcut, X* = (1/3)(120/4) = 10 and Y* = (2/3)(120/2) = 40. The optimal bundle is (10, 40).
Q: Given U = min(2X, Y), M = 60, P_X = 3, P_Y = 6. Find the optimal bundle.
A: At the kink, 2X = Y. Substitute into the budget constraint: 3X + 6(2X) = 60, so 15X = 60, X* = 4, Y* = 8.
Q: A consumer has U = 3X + 2Y, P_X = 6, P_Y = 5, M = 30. What does the consumer choose?
A: MRS = 3/2 = 1.5. MRT = P_X/P_Y = 6/5 = 1.2. Since MRS > MRT, the consumer values X more than the market does, so spend all income on X: X* = 30/6 = 5, Y* = 0.
Q: If P_X falls from 4 to 2 (all else equal), what happens to the budget constraint?
A: The Y-intercept stays the same (M/P_Y is unchanged). The X-intercept doubles from M/4 to M/2. The line pivots outward around the Y-intercept, becoming flatter.
This material connects directly to income and substitution effects (Doc 2): when a price changes, the shift in the optimal bundle decomposes into a substitution effect along the indifference curve and an income effect from the change in purchasing power. It also feeds into competitive equilibrium and Pareto efficiency, where individual optimal bundles must collectively clear markets.
Budget line, budget set, feasible set, affordable bundles, price ratio, opportunity cost, utility maximisation, consumer optimisation, indifference map, marginal utility, diminishing marginal rate of substitution, tangency condition, corner solution, interior solution, Cobb-Douglas preferences, Leontief preferences, linear utility, normal good, inferior good, Giffen good, demand function, individual demand, comparative statics.