Difficulty: Intermediate to Advanced | Prerequisites: Consumer Theory study notes (budget constraints, preferences, optimal bundles).
This material moves from the individual consumer to the economy as a whole. Income and substitution effects decompose a price change into two distinct forces, which is essential for understanding welfare measures like compensating and equivalent variation. Competitive equilibrium asks what prices make every consumer's optimal choice consistent with the available supply. Pareto efficiency asks whether any reallocation could make someone better off without hurting anyone else. Together, these topics connect individual optimisation to the First Welfare Theorem, which is one of the central results in microeconomics.
When a price changes, the substitution effect captures the consumer switching toward the relatively cheaper good, while the income effect captures the change in purchasing power. Competitive equilibrium is the price vector that clears all markets simultaneously. Pareto efficiency labels allocations where no one can be made better off without making someone else worse off, and the contract curve collects all such allocations in an Edgeworth box.
Substitution effect
The change in the quantity demanded of a good that results purely from the change in relative prices, holding utility constant. In simple terms, it isolates the "switching" behaviour: the good got relatively cheaper, so you buy more of it.
Income effect
The change in the quantity demanded that results from the change in real purchasing power caused by the price change. Think of it as: a lower price makes you effectively richer, and how you spend that extra purchasing power is the income effect.
Compensated budget constraint (Hicksian)
The hypothetical budget line at the new prices but with income adjusted so the consumer can just reach the original utility level. It is tangent to the original indifference curve at the new price ratio.
Equivalent budget constraint
The hypothetical budget line at the old prices but with income adjusted so the consumer can just reach the new utility level.
Indirect utility function V(P_X, P_Y, M)
The maximum utility achievable at given prices and income. Found by substituting the optimal bundle back into the utility function.
Expenditure function E(P_X, P_Y, U)
The minimum income needed to achieve utility level U at given prices. It is the inverse of the indirect utility function with respect to M and U.
Compensating variation (CV)
The amount of money that must be taken away from the consumer after a price decrease (or given after a price increase) to return them to the original utility level. CV = E(new prices, old U) - E(new prices, new U).
Equivalent variation (EV)
The amount of money that, at the old prices, would make the consumer just as well off as the price change does. EV = E(old prices, new U) - E(old prices, old U).
Competitive equilibrium
A set of prices and allocations where every consumer maximises utility given their budget, every firm maximises profit, and every market clears (quantity demanded equals quantity supplied).
Edgeworth box
A diagram that shows all possible allocations of two goods between two consumers. The dimensions of the box are the total endowments of each good. Consumer 1's origin is the bottom-left corner; Consumer 2's is the top-right.
Contract curve
The set of all Pareto efficient allocations in the Edgeworth box. Along this curve, the MRS of Consumer 1 equals the MRS of Consumer 2.
Core
The set of all Pareto efficient allocations where neither consumer is worse off than at their initial endowment. It is the segment of the contract curve that both consumers would voluntarily trade to.
Decomposing a price change (Hicksian method):
Start at the original optimal bundle A.
Draw the compensated budget constraint: same slope as the new prices, but tangent to the original indifference curve. The new tangency point is bundle B.
The move from A to B is the substitution effect. It always goes in the direction of the good that became relatively cheaper.
The move from B to the final optimal bundle C (on the actual new budget line) is the income effect.
Normal vs inferior goods: for a normal good, the income effect reinforces the substitution effect (price falls, buy more). For an inferior good, the income effect works against the substitution effect.
Indirect utility function: plug X*(P_X, P_Y, M) and Y*(P_X, P_Y, M) into U(X, Y). The result, V(P_X, P_Y, M), gives the highest utility achievable.
Expenditure function: solve the dual problem: minimise P_X · X + P_Y · Y subject to U(X, Y) = U-bar. The result, E(P_X, P_Y, U-bar), gives the cheapest way to reach utility U-bar.
Compensating variation: CV = M - E(new prices, old U). This is the most the consumer would pay for the price decrease.
Equivalent variation: EV = E(old prices, new U) - M. This is the lump sum at old prices that would make the consumer just as well off as the price change.
Setup: you are given two consumers with utility functions, initial endowments of two goods, and prices.
Each consumer's income is the value of their endowment: M_i = P_X · e_iX + P_Y · e_iY, where e_iX and e_iY are consumer i's endowments of X and Y.
Each consumer solves their own optimisation problem with this income.
Market clearing: for a competitive equilibrium, total demand for each good must equal total supply (the sum of endowments). Q_D^X = Q_S^X and Q_D^Y = Q_S^Y.
Normalisation trick: set P_Y = 1. Then solve for the value of P_X that clears both markets. By Walras' Law, if one market clears, the other does too.
Excess demand / excess supply: if prices are not at equilibrium, calculate each consumer's optimal bundle, sum demands, and compare to total endowments. If total demand exceeds total supply, there is excess demand. If total supply exceeds total demand, there is excess supply.
Pareto efficient allocation: an allocation where there is no way to make one consumer better off without making the other worse off. Graphically, in the Edgeworth box, this is where the two consumers' indifference curves are tangent (MRS_1 = MRS_2).
Contract curve: connect all tangency points in the Edgeworth box. This curve runs from Consumer 1's origin to Consumer 2's origin and contains every Pareto efficient allocation.
Testing Pareto efficiency: given an allocation, compute each consumer's MRS. If MRS_1 = MRS_2, the allocation is Pareto efficient. If they differ, it is not.
The Core: the subset of the contract curve where both consumers are at least as well off as at their initial endowment. To check: compute each consumer's utility at the initial endowment, then verify that the proposed allocation gives each consumer at least that much utility AND lies on the contract curve.
First Welfare Theorem link: any competitive equilibrium allocation is Pareto efficient (it lies on the contract curve).
\text{Compensated budget constraint: } P_X^{\text{new}} \cdot X + P_Y^{\text{new}} \cdot Y = E(P^{\text{new}},\, \bar{U}^{\text{old}})\text{CV} = M - E(P^{\text{new}},\, \bar{U}^{\text{old}})\text{EV} = E(P^{\text{old}},\, \bar{U}^{\text{new}}) - M\text{Market clearing: } \sum_i X_i^* = \sum_i e_{iX} \quad \text{and} \quad \sum_i Y_i^* = \sum_i e_{iY}\text{Pareto efficiency condition: } \text{MRS}_1 = \text{MRS}_2Students often confuse compensating variation and equivalent variation. CV uses the new prices and the old utility. EV uses the old prices and the new utility. A mnemonic: CV compensates you back to where you were; EV asks what is equivalent to the change at the prices you had.
Students sometimes think the substitution effect can go either way. It cannot. The substitution effect always moves the consumer toward the good that became relatively cheaper.
Students sometimes assume any point inside the Edgeworth box is Pareto efficient. Only the points on the contract curve (where the two indifference curves are tangent) are efficient.
The Core is often confused with the contract curve. The Core is narrower: it adds the constraint that neither consumer is worse off than at the initial endowment.
⚠️ 3 questions on income and substitution effects. Expect to compute the compensated bundle, identify the two effects, and possibly calculate CV or EV.
⚠️ 2 questions on competitive equilibrium. The standard problem: given endowments and utility functions, normalise P_Y = 1 and solve for P_X.
⚠️ 2 questions on Pareto efficiency. Likely: check whether a given allocation is efficient, identify the contract curve, or determine if an allocation is in the Core.
⚠️ The indirect utility function and expenditure function are listed as separate exam topics. Know how to derive both from first principles.
True or false: the substitution effect of a price decrease always increases consumption of that good. (True.)
Fill in the blank: the expenditure function gives the minimum ______ needed to reach a given ______ at given prices. (income; utility level.)
True or false: if MRS_1 ≠ MRS_2, the allocation is Pareto efficient. (False: efficiency requires MRS_1 = MRS_2.)
True or false: every allocation on the contract curve is in the Core. (False: it must also leave both consumers at least as well off as at the initial endowment.)
Fill in the blank: by Walras' Law, if one market clears the other ______. (clears as well.)
Q: Consumer 1 has U = X · Y, endowment (10, 0). Consumer 2 has U = X · Y, endowment (0, 10). Find the competitive equilibrium price ratio P_X / P_Y.
A: Total endowments: 10 of X, 10 of Y. With Cobb-Douglas, each consumer spends half their income on each good. Consumer 1's income = 10 · P_X. So X_1* = (1/2)(10 P_X)/P_X = 5, Y_1* = (1/2)(10 P_X)/P_Y = 5 P_X/P_Y. Consumer 2's income = 10 · P_Y. So X_2* = (1/2)(10 P_Y)/P_X = 5 P_Y/P_X, Y_2* = 5. Market clearing for X: 5 + 5 P_Y/P_X = 10, so P_Y/P_X = 1, meaning P_X/P_Y = 1.
Q: Is the allocation where Consumer 1 gets (5, 5) and Consumer 2 gets (5, 5) Pareto efficient, given U_i = X_i · Y_i?
A: MRS_1 = Y_1/X_1 = 5/5 = 1. MRS_2 = Y_2/X_2 = 5/5 = 1. Since MRS_1 = MRS_2, the allocation is Pareto efficient.
Q: A consumer has V(P_X, P_Y, M) = M^2 / (4 P_X P_Y). P_X falls from 4 to 1, P_Y = 1, M = 40. Find CV.
A: Old utility: V = 40^2 / (4 · 4 · 1) = 1600/16 = 100. Expenditure function: invert V to get E = 2 · sqrt(U · P_X · P_Y). At new prices and old utility: E(1, 1, 100) = 2 · sqrt(100) = 20. CV = M - E = 40 - 20 = 20.
Income and substitution effects build directly on the optimal-bundle material from the Consumer Theory notes. The expenditure function and indirect utility function reappear when you study welfare effects of monopoly pricing (Doc 4). Competitive equilibrium and Pareto efficiency are the bridge between consumer theory and the theory of the firm: the same efficiency conditions (MRS = MRT) show up in production as the condition for cost minimisation.
Hicksian decomposition, Slutsky decomposition, real income, relative prices, purchasing power, welfare measures, consumer surplus, general equilibrium, exchange economy, endowment point, initial allocation, Walras' Law, First Welfare Theorem, Second Welfare Theorem, mutually beneficial trade, gains from trade, tangency of indifference curves, individually rational allocation.