Difficulty: Intermediate | Prerequisites: introductory microeconomics (supply and demand, basic algebra, graphing in two dimensions). If you are not comfortable solving two equations in two unknowns or taking a partial derivative, review those first.
This document covers the building blocks of consumer theory: the tools economists use to model how a rational consumer makes choices. You are learning how to describe what a consumer can afford (budget constraints), what they prefer (utility and indifference curves), and how they pick the best option from what is available. These concepts underpin everything else in ECON 302, from demand curves to welfare analysis. If you have not seen constrained optimisation before, start here.
Consumer theory models how people allocate limited income across goods. You learn to write a budget constraint, describe preferences with a utility function, find the bundle that maximises utility subject to the budget, and measure how a price change splits into substitution and income effects. Master the mechanics here and the rest of the exam (demand curves, welfare measures) falls into place.
Budget constraint
The set of all bundles a consumer can just afford given prices and income. With two goods x₁ and x₂, it is written p₁x₁ + p₂x₂ = m, where m is income. In simple terms, it is the line on a graph showing every combination of two goods that exactly exhausts your budget.
Marginal rate of transformation (MRT)
The rate at which the market allows you to swap one good for another, equal to the price ratio p₁/p₂. Think of it as the slope of the budget line (in absolute value). It tells you the opportunity cost of one more unit of good 1, measured in units of good 2 you must give up.
Preferences
A ranking of all possible bundles from best to worst. Economists assume preferences are complete (you can always compare two bundles), transitive (if A ≻ B and B ≻ C then A ≻ C), and that more is better (non-satiation).
Indifference curve
A curve connecting all bundles that give the consumer the same level of utility. Indifference curves are downward-sloping (because of non-satiation), cannot cross, and are typically convex to the origin (reflecting a preference for variety).
Utility function
A mathematical function U(x₁, x₂) that assigns a number to each bundle so that preferred bundles get higher numbers. It is an ordinal ranking: only the order matters, not the size of the numbers.
Marginal utility (MU)
The additional utility gained from consuming one more unit of a good, holding everything else constant. Formally, MU₁ = ∂U/∂x₁. Think of it as how much happier one extra unit of the good makes you.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade good 2 for good 1 while staying on the same indifference curve. MRS = MU₁/MU₂. In simple terms, it measures how many units of good 2 you would voluntarily give up for one more unit of good 1.
Law of diminishing marginal utility
As you consume more of a good (holding everything else fixed), each additional unit adds less to your total utility than the one before it. This is why indifference curves are convex: the more of good 1 you already have, the less good 2 you are willing to sacrifice for another unit.
The budget constraint for two goods is p₁x₁ + p₂x₂ = m. Rearranged for graphing with x₁ on the horizontal axis: x₂ = m/p₂ − (p₁/p₂)x₁.
The vertical intercept is m/p₂ (spend everything on good 2).
The horizontal intercept is m/p₁ (spend everything on good 1).
The slope is −p₁/p₂, which equals the negative of the MRT.
Shifts of the budget constraint:
A change in income m shifts the line in or out, parallel to itself. Higher income moves it outward; lower income moves it inward. The slope does not change because relative prices have not changed.
A change in the price of good 1 pivots the line around the vertical intercept. If p₁ falls, the horizontal intercept moves rightward (you can afford more of good 1); the vertical intercept stays put.
A change in the price of good 2 pivots the line around the horizontal intercept.
A proportional change in all prices and income (multiplying everything by the same constant) leaves the budget set unchanged. Only relative prices and real income matter.
The marginal rate of transformation is the slope of the budget line in absolute value: MRT = p₁/p₂.
It tells you the market's exchange rate between the two goods. If p₁ = 6 and p₂ = 3, the MRT is 2, meaning the market will give you 2 units of good 2 for every 1 unit of good 1 you give up.
At the consumer's optimum, MRS = MRT. The rate at which the consumer is willing to trade must equal the rate at which the market allows them to trade. If MRS > MRT, the consumer values good 1 more than the market does and should buy more of it. If MRS < MRT, the consumer should buy less of it.
Economists model consumer behaviour by assuming the consumer has a well-behaved preference relation over bundles. The standard assumptions are:
Completeness. For any two bundles A and B, the consumer can say A ≻ B, B ≻ A, or A ∼ B. There is no "I don't know."
Transitivity. If A ≻ B and B ≻ C, then A ≻ C. Preferences are internally consistent.
Non-satiation ("more is better"). A bundle with more of at least one good and no less of any other is strictly preferred.
Convexity. Averages are preferred to extremes. A mixture of two equally-liked bundles is at least as good as either one alone. This is what makes indifference curves bow inward.
These assumptions let us represent preferences with a continuous, well-behaved utility function and ensure that the consumer's optimum is unique (for strictly convex preferences).
An indifference curve is a level set of the utility function: all bundles (x₁, x₂) such that U(x₁, x₂) = k for some constant k.
Properties:
Downward-sloping. If you gain some of good 1, you must lose some of good 2 to stay at the same utility (non-satiation).
Higher curves are better. Curves further from the origin represent higher utility.
Cannot cross. If two indifference curves crossed at a bundle, that bundle would have two different utility levels, which is a contradiction.
Convex to the origin (for standard preferences). The MRS falls as you move along the curve from left to right, reflecting diminishing willingness to substitute.
Special shapes to know:
Perfect substitutes: straight-line indifference curves with constant slope.
Perfect complements: L-shaped indifference curves with a kink along a ray from the origin.
Cobb-Douglas: smoothly bowed curves, the workhorse case for exam problems.
A utility function U(x₁, x₂) translates preferences into numbers. Any monotonic transformation of a utility function represents the same preferences (ordinal, not cardinal).
Common forms:
Cobb-Douglas: U = x₁^a · x₂^b. Well-behaved, smooth indifference curves. The consumer always spends a fixed fraction of income on each good.
Perfect substitutes: U = ax₁ + bx₂. Indifference curves are straight lines with slope −a/b.
Perfect complements: U = min(ax₁, bx₂). The consumer always chooses the kink point where ax₁ = bx₂.
Marginal utility of good 1 is MU₁ = ∂U/∂x₁. It measures the rate at which utility increases as you add a small amount of good 1.
The MRS is derived from marginal utilities:
MRS = MU₁ / MU₂
This is the (negative of the) slope of the indifference curve at a point. At the optimum for an interior solution, MRS = p₁/p₂.
As consumption of a good increases (holding other goods constant), each successive unit adds less utility. The second slice of pizza is less satisfying than the first; the tenth is less satisfying still.
Formally, ∂²U/∂x₁² < 0 (marginal utility is decreasing in x₁).
This drives the convexity of indifference curves. When you already have a lot of good 1, you are willing to give up only a small amount of good 2 for one more unit of good 1, so the MRS is small. When you have very little of good 1, the MRS is large.
Budget constraint: p₁x₁ + p₂x₂ = m, rearranged as x₂ = m/p₂ − (p₁/p₂)x₁
MRT: p₁/p₂ (absolute value of the budget line's slope)
MRS: MU₁/MU₂ = (∂U/∂x₁) / (∂U/∂x₂)
Cobb-Douglas optimality condition (interior solution):
Set MRS = MRT: (a·x₂) / (b·x₁) = p₁/p₂ for U = x₁^a · x₂^b
Combine with budget constraint to solve for x₁* and x₂*
Demand functions: x₁* = (a/(a+b)) · m/p₁ and x₂* = (b/(a+b)) · m/p₂
Perfect complements optimality: ax₁ = bx₂ and p₁x₁ + p₂x₂ = m. Solve simultaneously.
Perfect substitutes: Corner solution. Compare MRS = a/b to MRT = p₁/p₂. If MRS > MRT, buy only good 1. If MRS < MRT, buy only good 2. If equal, any bundle on the budget line is optimal.
Diagram checklist for exam problems: label both axes with the good names, mark intercepts with their numerical values, draw the budget line, draw the indifference curve tangent to the budget line at the optimal bundle, and label the optimal bundle coordinates.
Budget constraints show up whenever you have a fixed amount to allocate: a household deciding between groceries and rent, a student splitting study hours between two courses, or a firm dividing a fixed budget between advertising and R&D.
The MRS = MRT condition is the logic behind any "bang for your buck" calculation. If the last pound spent on coffee gives you more satisfaction than the last pound spent on tea, you should shift spending toward coffee until the marginal values equalise.
"Utility numbers have real meaning." They do not. Utility is ordinal. Saying U(A) = 10 and U(B) = 5 does not mean A is "twice as good." It only means A is preferred to B. Any monotonic transformation of U represents the same preferences.
"Indifference curves can cross." They cannot. If they did, transitivity would be violated. This is a favourite exam trap.
"The MRS is constant along an indifference curve." For standard convex preferences (Cobb-Douglas), the MRS changes as you move along the curve. It is constant only for perfect substitutes.
"A higher price always shifts the budget line inward." A higher price of good 1 pivots the line inward around the vertical intercept, it does not shift it in parallel. A parallel shift happens only when income changes.
⚠️ You will be given prices, income, and a utility function and asked to draw the budget constraint, find the optimal bundle, and draw the indifference curve through it. Practise the full sequence: write the budget equation, compute MRS, set MRS = p₁/p₂, solve simultaneously with the budget constraint, plot everything.
⚠️ Know how to handle all three functional forms: Cobb-Douglas (tangency condition), perfect complements (kink condition ax₁ = bx₂), perfect substitutes (corner solution by comparing MRS to MRT).
⚠️ Be able to explain in words what the MRS and MRT represent and why they must be equal at the optimum.
⚠️ Exam questions often ask you to show what happens when one price changes. You need to re-solve the optimum at the new price and show the pivot of the budget line on the graph.
True or false: if U(A) = 20 and U(B) = 10, then A gives exactly twice as much satisfaction as B.
Fill in the blank: the slope of the budget line in absolute value equals ____.
True or false: at the consumer's optimal bundle, the indifference curve crosses the budget line.
Fill in the blank: for perfect complements U = min(3x₁, 2x₂), the consumer always chooses bundles where 3x₁ = ____.
True or false: if MRS > p₁/p₂, the consumer should buy more of good 1.
Answers: 1. False (utility is ordinal). 2. p₁/p₂. 3. False (it is tangent, not crossing). 4. 2x₂. 5. True.
Q: A consumer has utility U = x₁^(1/2) · x₂^(1/2), income m = 100, p₁ = 5, p₂ = 10. Find the optimal bundle.
A: With Cobb-Douglas (a = b = 1/2), the consumer spends half income on each good. x₁* = (1/2)(100/5) = 10. x₂* = (1/2)(100/10) = 5. The optimal bundle is (10, 5).
Q: Suppose p₁ falls to 2. What is the new optimal bundle? Describe how the budget line changes on a graph.
A: x₁* = (1/2)(100/2) = 25. x₂* = (1/2)(100/10) = 5. The budget line pivots outward around the vertical intercept (which stays at 10). The horizontal intercept moves from 20 to 50.
Q: For U = min(x₁, 2x₂), p₁ = 3, p₂ = 6, m = 36, find the optimal bundle.
A: At the kink, x₁ = 2x₂. Substitute into the budget: 3(2x₂) + 6x₂ = 36, so 12x₂ = 36, x₂ = 3, x₁ = 6. Optimal bundle is (6, 3).
Q: A consumer has U = 2x₁ + 3x₂ (perfect substitutes), p₁ = 4, p₂ = 5. Which good does the consumer buy?
A: MRS = 2/3. MRT = p₁/p₂ = 4/5. Since MRS (0.67) < MRT (0.8), the consumer values good 1 relatively less than the market does, so they spend all income on good 2 (corner solution at the vertical intercept).
The optimal bundle you find here is the starting point for deriving demand curves (Doc 2): vary one price and trace how the optimal quantity changes. The same optimality conditions reappear when you decompose price changes into income and substitution effects (Doc 3). Budget constraints also connect to producer theory later in the course, where the analogous concept is the isocost line.
budget line, budget set, feasible set, price ratio, opportunity cost, marginal rate of substitution, marginal rate of transformation, indifference map, utility maximisation, constrained optimisation, Cobb-Douglas utility, perfect substitutes, perfect complements, Leontief preferences, tangency condition, corner solution, interior solution, ordinal utility, cardinal utility, convex preferences, monotonic transformation