Source: Lecture Notes of Prof. Guoqiang Tian, Texas A&M University
Tags: budget line, indifference curve, marginal rate of substitution, MRS, utility function, ordinal utility, consumer optimisation, tangency condition, corner solution, composite good, perfect substitutes, perfect complements
Consumer choice theory explains how rational consumers allocate their income across goods to maximise satisfaction. The framework rests on three elements: a budget constraint (what you can afford), a preference ordering represented by indifference curves (what you want), and an optimisation rule (choose the bundle on the highest reachable indifference curve). The key result is the tangency condition: at the optimal bundle, the marginal rate of substitution equals the price ratio.
Budget line
A straight line representing all possible combinations of two goods a consumer can purchase by spending their entire income at given prices. Equation: px x + py y = I.
Slope of the budget line
Equal to -px/py. Represents the rate at which the market allows you to trade one good for the other.
Indifference curve
A curve on which the consumer is indifferent between any two bundles. All bundles on the same indifference curve yield the same level of satisfaction.
Indifference map
A family of indifference curves, where curves further from the origin represent higher levels of satisfaction.
Marginal rate of substitution (MRS)
The number of units of good y that must be given up for one extra unit of good x while keeping the consumer indifferent. Equal to the absolute value of the slope of the indifference curve, and equal to MUx/MUy.
Diminishing MRS
As a consumer moves down along an indifference curve (gaining x, losing y), the amount of y they are willing to sacrifice for one more unit of x decreases. This is why standard indifference curves are convex to the origin.
Utility function U(x, y)
Assigns a numerical value to each commodity bundle such that higher-valued bundles are preferred. Used to rank bundles consistently with preferences.
Ordinal utility
A utility function where only the ranking of bundles matters, not the absolute numbers. If U(A) > U(B), bundle A is preferred to B, but the size of the gap is meaningless.
Marginal utility (MU)
The additional utility from consuming one more unit of a good. MUx = change in U / change in x.
Perfect substitutes
Goods with linear (straight-line) indifference curves, giving a constant MRS.
Perfect complements
Goods consumed in fixed proportions, producing L-shaped indifference curves. E.g. left shoes and right shoes.
Composite good convention
When there are many goods, treat all goods other than x as a single "composite good" measured by total expenditure on everything except x. Its price equals one.
For two goods x and y, with prices px and py and income I:
px x + py y = I
Rearranging: y = (I/py) - (px/py) * x
The y-intercept is I/py (spend all income on y).
The x-intercept is I/px (spend all income on x).
The slope is -px/py.
If both prices and income change in the same proportion (e.g. all double), the budget line is unchanged. If only one price changes, the intercept on that good's axis shifts and the slope changes. If only income changes, the budget line shifts parallel (same slope, different intercepts).
Three fundamental assumptions:
Completeness: between any two bundles A and B, the consumer can state A is preferred to B, B is preferred to A, or they are indifferent.
Transitivity: if A is preferred to B and B is preferred to C, then A is preferred to C.
Non-satiation (more is preferred to less): if bundle B has more of at least one good than bundle A (and no less of any other), then B is preferred to A.
They slope downward (by non-satiation).
They do not intersect. If they did, transitivity would be violated.
They are convex to the origin, reflecting diminishing MRS.
A utility function U(x, y) assigns numbers to bundles that are consistent with the indifference map.
MRS = MUx / MUy
Example 1: U(x, y) = x + 2y. Then MUx = 1, MUy = 2, so MRS = 1/2 (constant, meaning these goods are perfect substitutes).
Example 2: U(x, y) = xy. Then MUx = y, MUy = x, so MRS = y/x (diminishing as x increases).
To trace an indifference curve, set U(x, y) = constant and solve for y in terms of x.
The consumer maximises utility subject to the budget constraint.
Case 1: Interior solution (strictly convex indifference curves)
The optimal bundle occurs where the indifference curve is tangent to the budget line. Two conditions must hold simultaneously:
Tangency: MRS = px/py (slope of indifference curve equals slope of budget line)
Budget: px x + py y = I
Example: U(x, y) = xy, px = 2, py = 1, I = 100.
MRS = y/x, so y/x = 2/1, meaning y = 2x. Substituting into the budget: 2x + 2x = 100, so x = 25, y = 50.
Case 2: Corner solution (linear indifference curves)
When MRS is constant (perfect substitutes), the optimal bundle depends on whether MRS is greater than, less than, or equal to px/py.
If MRS > px/py: spend all income on x (x = I/px, y = 0).
If MRS < px/py: spend all income on y (x = 0, y = I/py).
If MRS = px/py: any bundle on the budget line is optimal.
Example: U = x + 2y, so MRS = 1/2. If px = 1, py = 1, then px/py = 1. Since MRS (1/2) < px/py (1), the consumer buys only y: y* = I/py = 20.
When there are many goods, we can reduce the analysis to two dimensions by grouping all goods other than x into a single composite good measured by total expenditure on those goods. The price of the composite good is 1, so the slope of the budget line is -px/1 = -px. At the optimum, MRS = px.
Budget line: px x + py y = I
Slope of budget line: -px/py
MRS = MUx / MUy = absolute value of slope of indifference curve
Tangency condition (interior optimum): MRS = px/py
For U = xy: MRS = y/x
For U = x + 2y: MRS = 1/2 (constant)
Composite good: MRS = px (since price of composite = 1)
⚠️ The tangency condition MRS = px/py is the single most important result in consumer theory. Be able to use it alongside the budget constraint to solve for optimal bundles.
⚠️ Distinguish between interior solutions (convex indifference curves, tangency) and corner solutions (linear indifference curves, compare MRS with price ratio).
⚠️ Indifference curves never intersect. Be prepared to prove this by contradiction using transitivity and non-satiation.
⚠️ Know the special cases: perfect substitutes produce linear indifference curves with constant MRS; perfect complements produce L-shaped curves.
⚠️ The utility function is ordinal. Only the ranking matters, not the numerical value. Any monotonic transformation of a utility function represents the same preferences.
Q: A consumer has utility U(x, y) = xy, faces prices px = 5, py = 10, and has income I = 200. What is the optimal bundle?
A: MRS = y/x = px/py = 5/10 = 1/2, so x = 2y. Budget: 5(2y) + 10y = 200, giving 20y = 200, y = 10, x = 20.
Q: Why do indifference curves slope downward?
A: Because of the assumption that more is preferred to less. If x increases, y must decrease to keep the consumer on the same indifference curve (same level of satisfaction).
Q: What happens to the budget line if income doubles but prices stay the same?
A: The budget line shifts outward in a parallel fashion. The slope remains -px/py, but both intercepts double.
Q: A consumer has utility U = x + 3y, px = 2, py = 3, and I = 60. What is the optimal bundle?
A: MRS = 1/3. Price ratio = px/py = 2/3. Since MRS (1/3) < price ratio (2/3), the consumer spends all income on y: y* = 60/3 = 20, x* = 0.
Q: Prove by contradiction that two indifference curves cannot intersect.
A: Suppose curves U0 and U1 intersect at point B, with A on U1 and C on U0. Since A and B are on U1, the consumer is indifferent between A and B. Since B and C are on U0, the consumer is indifferent between B and C. By transitivity, the consumer is indifferent between A and C. But if U1 represents a higher utility level, then A should be preferred to C, which is a contradiction.
budget constraint, budget set, indifference curve, indifference map, marginal rate of substitution, MRS, diminishing MRS, utility function, ordinal utility, cardinal utility, marginal utility, tangency condition, consumer equilibrium, consumer optimisation, corner solution, interior solution, perfect substitutes, perfect complements, composite good, consumer theory, ECON 323, intermediate microeconomics