Difficulty: Intermediate | Prerequisites: Supply, Demand, and Elasticity study notes; comfort with partial derivatives and basic optimisation.
Consumer theory sits at the heart of intermediate micro. It formalises how individuals choose what to buy given limited income and their own preferences, then derives the demand curves you worked with in the supply-and-demand unit. If you skipped the earlier material on equilibrium and elasticity, go back to those notes first: consumer theory explains where demand curves come from, and the results feed directly into welfare analysis, market structure, and policy evaluation later in the course.
Consumers maximise utility subject to a budget constraint by choosing the bundle where the marginal rate of substitution equals the price ratio. This tangency condition, combined with the budget line, produces individual demand curves. Decomposing a price change into substitution and income effects reveals why demand slopes downward for most goods, and why Giffen goods are a rare exception.
Completeness
The assumption that a consumer can rank any two bundles: for any pair A and B, the consumer 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. This keeps preferences logically consistent.
More is better (non-satiation)
Consumers always prefer more of a good to less. This rules out "bliss points" within the relevant range.
Indifference curve
A curve connecting all bundles that give the consumer the same level of satisfaction. Think of it as a contour line on a utility "map."
Indifference map
The full set of a consumer's indifference curves. Higher curves represent higher utility.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade one good for another while remaining equally satisfied. It equals the (negative) slope of the indifference curve at a given point.
Diminishing MRS
As you consume more of one good relative to another, your willingness to give up the other good for still more of the first declines. This is why indifference curves are convex to the origin.
Utility function
A mathematical function that assigns a number to each bundle, ranking them consistently with the consumer's preferences. The number itself is arbitrary (ordinal, not cardinal).
Marginal utility (MU)
The additional satisfaction from consuming one more unit of a good. It typically diminishes.
Cobb-Douglas utility
A common functional form: U(X, Y) = X^a * Y^b. Always produces interior solutions with well-behaved convex indifference curves.
Quasi-linear utility
U(X, Y) = v(X) + Y. Income effects fall entirely on the linear good, which simplifies the maths and is useful for policy analysis.
Budget constraint
I = P_X * X + P_Y * Y. All affordable bundles lie on or below this line. The slope is the negative price ratio.
Interior solution
The consumer buys positive amounts of all goods, and MRS equals the price ratio at the optimum.
Corner solution
The consumer spends all income on one good. Arises when MRS does not equal the price ratio at any interior point.
Perfect substitutes
Goods with straight-line indifference curves and a constant MRS. The consumer buys whichever is cheaper per unit of utility.
Perfect complements
Goods consumed in fixed proportions, with L-shaped indifference curves. Extra units of one good without matching units of the other add nothing.
Bads
Commodities the consumer wants less of, producing upward-sloping indifference curves.
Revealed preference
The idea that a consumer's actual choices reveal their preferences. If bundle A is chosen when B is affordable, A is revealed preferred to B.
Substitution effect
The change in consumption due to a relative price change, holding utility constant. Always moves in the opposite direction to the price change.
Income effect
The change in consumption due to the change in real purchasing power caused by a price change.
Giffen good
An inferior good where the income effect is so large that it dominates the substitution effect, producing upward-sloping demand. Extremely rare.
Consumer surplus
The difference between what consumers are willing to pay (the area under the demand curve) and what they actually pay. A measure of buyer welfare.
Choke price
The price at which quantity demanded drops to zero.
Preference assumptions
Completeness, transitivity, more is better, and diminishing MRS. These four axioms let us represent preferences with well-behaved utility functions and convex indifference curves.
Indifference curves
Downward sloping (giving up one good requires getting more of the other).
Convex to the origin (diminishing MRS).
Never cross (if they did, a single bundle would lie on two different utility levels, contradicting transitivity).
Marginal rate of substitution
MRS_XY = -dY/dX along the indifference curve = MU_X / MU_Y.
Falls as you move down the curve (you value the now-scarce good more).
Utility functions
Cobb-Douglas: U = X^a * Y^b. Interior solutions, well-behaved. With this form, the consumer always spends a fixed fraction of income on each good.
Quasi-linear: U = v(X) + Y. Demand for X depends only on its own price, not on income (no income effect on X). Useful in welfare and policy analysis.
Budget constraint
I = P_X * X + P_Y * Y.
Intercepts: I/P_X on the X-axis, I/P_Y on the Y-axis.
Slope: -P_X / P_Y.
A change in income shifts the line in or out (parallel). A change in one price rotates the line around the other-good intercept.
Utility maximisation
The consumer picks the bundle on the highest attainable indifference curve that is still on or below the budget line.
At the optimum (interior solution): MRS = P_X / P_Y, which means MU_X / P_X = MU_Y / P_Y (equal marginal utility per pound spent on each good).
Corner and interior solutions
Interior: positive amounts of both goods, MRS = price ratio.
Corner: all income on one good. Happens when MRS is always above or below the price ratio across the budget line (common with perfect substitutes).
Special preferences
Perfect substitutes: straight-line indifference curves, constant MRS. The consumer buys only the cheaper good unless MRS exactly equals the price ratio.
Perfect complements: L-shaped curves. The consumer always buys goods in a fixed ratio. Extra units of one without the other are worthless.
Bads and useless goods
A "bad" is something the consumer wants less of (pollution, noise). Indifference curves slope upward: you need more of the good to compensate for more of the bad.
A useless good contributes nothing to utility. Indifference curves are vertical or horizontal lines.
Revealed preference
If a consumer chose bundle A when B was affordable, A is revealed preferred to B.
Consistency and transitivity of actual choices let you reconstruct indifference curves from observed purchasing data, even without knowing the utility function.
Deriving individual demand curves
Hold income and the price of the other good constant, vary P_X, and trace the optimal X at each price. The resulting relationship is the individual demand curve for X.
Income and substitution effects
When the price of X falls, two things happen simultaneously:
Substitution effect: X is now relatively cheaper, so the consumer substitutes toward X (always increases X).
Income effect: the consumer's real purchasing power rises. For a normal good, this also increases X. For an inferior good, this decreases X.
For most goods, both effects push in the same direction, so demand slopes downward.
For a Giffen good (an inferior good with a very large income effect), the income effect dominates, and demand slopes upward. This is theoretically possible but extremely rare in practice.
Normal, inferior, luxury, and necessity goods
Normal: demand rises with income (positive income elasticity).
Inferior: demand falls with income (negative income elasticity).
Luxury: income elasticity > 1 (demand grows faster than income).
Necessity: income elasticity between 0 and 1 (demand grows, but less than proportionally).
Using the Lagrangian
Set up: L = U(X, Y) + lambda * (I - P_X * X - P_Y * Y).
First-order conditions: dL/dX = 0, dL/dY = 0, dL/dlambda = 0.
Solving these gives the demand functions X*(P_X, P_Y, I) and Y*(P_X, P_Y, I).
Lambda (the Lagrange multiplier) represents the marginal utility of income: how much additional utility the consumer would gain from one more unit of income.
Consumer surplus
Consumer surplus = area under the demand curve above the market price.
It measures the net benefit consumers receive from participating in the market.
Used to assess welfare changes from price changes, taxes, or subsidies.
Market demand aggregation
Market demand is the horizontal sum of all individual demand curves: at each price, add up the quantities each consumer demands.
The choke price is the price at which market demand falls to zero.
Food stamps vs. cash transfers
Food stamps restrict spending to food; cash transfers let recipients spend freely.
If the food stamp amount is less than what the recipient would have spent on food anyway, the two are equivalent in their effect on consumption.
If the stamp pushes food consumption above what the recipient would freely choose, it creates a distortion: the recipient reaches a lower indifference curve than with an equal-value cash transfer.
Evidence suggests that cash transfers do not substantially increase spending on alcohol, tobacco, or other "temptation goods," which weakens the paternalist case for in-kind transfers.
MRS
MRS_XY = MU_X / MU_Y = -dY/dX (along the indifference curve)
Budget constraint
I = P_X * X + P_Y * Y
Slope = -P_X / P_Y
Tangency condition (interior optimum)
MRS_XY = P_X / P_Y, equivalently MU_X / P_X = MU_Y / P_Y
Lagrangian
L = U(X, Y) + lambda * (I - P_X * X - P_Y * Y)
FOCs: MU_X = lambda * P_X, MU_Y = lambda * P_Y, I = P_X * X + P_Y * Y
Consumer surplus
CS = (area under demand curve from 0 to Q*) - (P* x Q*)
The income and substitution effects framework explains why a petrol tax hurts lower-income drivers disproportionately: for them, fuel is closer to a necessity (low income elasticity), and the income effect of a price increase is large relative to their budget. The Lagrangian approach is the standard tool behind demand estimation in applied economics, used by firms setting prices and governments evaluating tax policy.
Students often think utility numbers have real-world meaning. They do not: utility is ordinal. Saying U = 10 is "twice as good" as U = 5 is meaningless.
A frequent error is to assume MRS is constant along a standard indifference curve. It is constant only for perfect substitutes.
Students sometimes set up the Lagrangian with the wrong sign on lambda (using a minus instead of plus, or vice versa). Be consistent: the standard form adds lambda times (I minus expenditure).
Many students believe Giffen goods are common. They are a theoretical curiosity. On an exam, if a question asks "is this a Giffen good," the answer is almost always no unless the problem explicitly states an inferior good with a dominating income effect.
⚠️ Be able to derive demand functions from Cobb-Douglas and quasi-linear utility using the Lagrangian. This is the single most common computation question in the consumer theory unit.
⚠️ Know how to decompose a price change into substitution and income effects graphically (the Slutsky or Hicks decomposition). Exams ask you to draw the compensated budget line and identify each effect on a diagram.
⚠️ Understand when a corner solution arises. If a question uses perfect-substitutes preferences, the tangency condition does not apply; you compare MRS to the price ratio to determine which good is bought.
⚠️ Consumer surplus questions appear frequently. Know the formula and be able to calculate the change in CS when a price rises or a tax is imposed.
True or false: Indifference curves can cross if the consumer's preferences change over time. (False. Within a single preference ordering, crossing is ruled out by transitivity.)
Fill in the blank: At an interior optimum, MU_X / P_X = _______ . (MU_Y / P_Y)
True or false: The substitution effect of a price decrease always increases consumption of the good that became cheaper. (True.)
Fill in the blank: A Giffen good must be a(n) _______ good. (inferior)
True or false: With quasi-linear utility, a change in income affects demand for the nonlinear good. (False. It affects only the linear good.)
Q: A consumer has utility U(X, Y) = X^0.5 * Y^0.5, income I = 100, P_X = 5, P_Y = 10. Find the optimal bundle.
A: With Cobb-Douglas U = X^a * Y^b, the consumer spends fraction a/(a+b) of income on X. Here a = b = 0.5, so half on each. X* = (0.5 * 100) / 5 = 10. Y* = (0.5 * 100) / 10 = 5.
Q: Explain graphically how to decompose a fall in P_X into substitution and income effects for a normal good.
A: Start at the original optimum on budget line B1. After the price fall, the new budget line is B2 (rotated outward at the X-intercept). Draw a compensated budget line B' parallel to B2 but tangent to the original indifference curve. The move from the original bundle to the tangency on B' is the substitution effect (X increases). The move from B' to the new optimum on B2 is the income effect (X increases further, since the good is normal).
Q: Why does a food stamp programme that gives $50 in food vouchers sometimes produce a different consumption outcome from a $50 cash transfer?
A: If the consumer would have spent less than $50 on food with cash (say $30), the food stamp forces them to consume more food than they would choose freely. They reach a lower indifference curve than with cash, because the stamp constrains their choice set. If they would have spent $50 or more on food anyway, the two programmes yield identical outcomes.
Q: A consumer's MRS between goods X and Y is 3, but the price ratio P_X / P_Y is 2. What should the consumer do?
A: The consumer values X more highly (in terms of Y) than the market does. They should buy more X and less Y. This will decrease MRS (diminishing MRS) until it equals the price ratio of 2.
Consumer theory provides the micro-foundations for demand curves used in market equilibrium analysis. The welfare concepts here (consumer surplus, indifference-curve ranking of policies) reappear in the study of market power, price discrimination, and government intervention. The Lagrangian method is the same technique used later to solve the firm's cost-minimisation problem.
consumer theory, utility maximisation, indifference curve, marginal rate of substitution, MRS, budget constraint, budget line, Cobb-Douglas, quasi-linear utility, Lagrangian, Lagrange multiplier, marginal utility, interior solution, corner solution, perfect substitutes, perfect complements, bads, useless goods, revealed preference, substitution effect, income effect, Slutsky decomposition, Hicks decomposition, Giffen good, inferior good, normal good, luxury, necessity, consumer surplus, choke price, market demand, horizontal summation, food stamps, cash transfers, temptation goods, in-kind transfers