Difficulty: Intermediate | Prerequisites: Basic algebra, intro-level supply and demand concepts.
This is the foundation of consumer theory in intermediate microeconomics. Everything that follows in the course, from welfare analysis to general equilibrium, rests on how consumers choose between goods given their preferences and budget. If you missed the first few weeks, start here: you need to be comfortable with utility functions, indifference curves and budget constraints before moving on to income/substitution effects or compensating and equivalent variation.
Consumer theory boils down to three utility function types: Cobb-Douglas (smooth trade-offs), perfect substitutes (constant swap rate) and perfect complements (fixed proportions). Each type has its own MRS formula, its own shape of indifference curve and its own method for finding the optimum bundle. Get comfortable switching between these three and you can handle most Exam 1 problems.
Utility function
A mathematical expression that assigns a number to each bundle of goods, representing the consumer's preferences. Higher numbers mean the consumer prefers that bundle. Think of it as a scoring system for combinations of goods.
Indifference curve
A curve showing all bundles of two goods that give the consumer the same level of utility. In simple terms, every point on the curve is equally attractive to the consumer.
Marginal Rate of Substitution (MRS)
The rate at which a consumer is willing to give up one good in exchange for more of another, holding utility constant. It is the slope of the indifference curve at a given point. Think of it as the consumer's personal exchange rate between two goods.
Marginal Rate of Transformation (MRT)
The rate at which the market allows you to swap one good for the other, determined by the price ratio Px/Py. It is the slope of the budget constraint. In simple terms, this is the market's exchange rate.
Budget constraint
The equation M = Px(x) + Py(y), representing all bundles a consumer can afford given income M and prices Px, Py. Think of it as the boundary of what you can buy.
Optimum bundle
The bundle of goods that maximises utility subject to the budget constraint. For interior solutions, this is where MRS = MRT (the indifference curve is tangent to the budget line).
Corner solution
An optimum where the consumer spends all income on just one good. This happens with perfect substitutes when MRS and MRT are not equal.
The workhorse of intermediate micro. Cobb-Douglas preferences give smooth, convex indifference curves and always produce interior solutions (you buy some of both goods).
General form: U = x^a × y^b
MRS formula: MRS = -(a/b) × (y/x)
The MRS depends on the ratio y/x, so it changes along the indifference curve. This is what makes the curves convex.
Finding the optimum bundle
Set MRS = MRT, which gives -(a/b)(y/x) = -(Px/Py)
Rearrange to express y in terms of x (or vice versa)
Substitute into the budget constraint M = Px(x) + Py(y)
Solve for x and y
For the standard case U = x^(1/2) × y with equal exponent weights, the demand functions simplify to x = M/(2Px) and y = M/(2Py).
Worked example
U = x^(1/2) × y, bundle (4, 8):
Utility = 4^(1/2) × 8 = 2 × 8 = 16
To find other bundles on the same indifference curve (U = 16): if y = 1, then x^(1/2) = 16, so x = 256; if y = 2, then x^(1/2) = 8, so x = 64; if y = 4, then x^(1/2) = 4, so x = 16
Watch for coefficients. If the utility function has a multiplier out front (e.g. 3x^(1/2) × y^(1/2)), that coefficient carries through into the IUF and expenditure function. Forgetting it is one of the most common errors on exam day.
With perfect substitutes, the consumer views the two goods as interchangeable at a fixed rate. Indifference curves are straight lines, and the optimum is almost always a corner solution.
General form: U = ax + by
MRS formula: MRS = -(a/b), a constant (the slope of the straight-line indifference curve)
The decision rule
Compare MRS to MRT (= Px/Py):
If MRS > MRT (indifference curve is steeper than the budget line), buy all x
If MRT > MRS (budget line is steeper), buy all y
If MRS = MRT, any bundle on the budget line is optimal
This is the single most important thing to remember for perfect substitutes. The consumer always goes to a corner unless the slopes happen to match.
Finding the critical price
Set MRS = MRT to find the price at which the consumer switches goods. For U = 2x + 3y: MRS = 2/3, so MRT = Px/Py = 2/3. If Py = 1, the switching price is Px = 2/3.
Worked example
U = 2x + 3y, bundle (3, 4):
Utility = 2(3) + 3(4) = 18
x-intercept: 2x = 18, so x = 9
y-intercept: 3y = 18, so y = 6
Another example
U = 6x + y, M = 60, Py = 1, Px = 1 (then Px rises to 10):
MRS = 6/1 = 6
At Px = 1: MRT = 1/1 = 1, so MRS > MRT, buy all x. Consumer gets x = 60, utility = 360
The switching price is Px = 6 (where MRS = MRT)
At Px = 10: MRT = 10 > 6 = MRS, so buy all y. Consumer gets y = 60, utility = 60
Perfect complements are goods that must be consumed in a fixed ratio. Indifference curves are L-shaped, and the optimum always sits at the kink of the L.
General form: U = Min(ax, by)
MRS: Undefined at the kink. On the horizontal segment, MRS = 0 (extra x adds nothing). On the vertical segment, MRS is infinite (extra y adds nothing).
Finding the optimum bundle
Set the two arguments equal: ax = by, which gives the fixed-proportion relationship (e.g. for Min(4x, y), set 4x = y)
Substitute into the budget constraint M = Px(x) + Py(y)
Solve for x and y
Worked example
U = Min(4x, y), bundle (2, 4):
4(2) = 8 and y = 4, so minimum is 4. Utility = 4
For the optimum, set 4x = y and substitute: M = Px(x) + Py(4x) = x(Px + 4Py)
So x = M/(Px + 4Py) and y = 4M/(Px + 4Py)
Worked example: Min(2x, y)
U = Min(2x, y), M = 12, Px = 1, Py = 1:
Set 2x = y. Budget: 12 = (1)(x) + (1)(2x) = 3x, so x = 4, y = 8
Utility = Min(8, 8) = 8
Price Consumption Curve (PCC)
Shows how the optimum bundle changes as the price of one good varies, holding income and the other price fixed. To build a demand curve from this: make a table of Px values and the corresponding optimum quantity of x, then plot price on the vertical axis and quantity on the horizontal axis. For Cobb-Douglas preferences, the demand curve is convex.
Income Consumption Curve (ICC)
Shows how the optimum bundle changes as income varies, holding prices fixed. It traces out the path of optimal bundles across different income levels.
Engel Curve
Plots income (M) on the y-axis against the quantity of one good on the x-axis. For Cobb-Douglas with equal exponents, the Engel curve is a straight line through the origin, because demand for each good is a fixed fraction of income.
Necessary and Sufficient Conditions
These come up in logic-based exam questions:
Sufficient condition: if A then B. Example: if you are the president, then you are over 35. Being president is sufficient for being over 35
Necessary condition: if not A then not B. Example: if you are not over 35, then you are not the president. Being over 35 is necessary for being president
The distinction matters when exam questions ask whether a condition is necessary, sufficient, or both.
Type | Utility Function | MRS | Demand for x | Demand for y |
|---|---|---|---|---|
Cobb-Douglas (a = b = 1/2) | U = x^(1/2) y^(1/2) | -(a/b)(y/x) | x = M/(2Px) | y = M/(2Py) |
Perfect Substitutes | U = ax + by | -(a/b), constant | Corner: all x if MRS > MRT | Corner: all y if MRT > MRS |
Perfect Complements | U = Min(ax, by) | Undefined at kink | x = M/(Px + (b/a)Py) | y = (b/a)M/(Px + (b/a)Py) |
Budget constraint: M = Px(x) + Py(y)
MRT (slope of budget line): -(Px/Py)
Optimality condition (interior solution): MRS = MRT, i.e. (a/b)(y/x) = Px/Py
Students often think the MRS is the same thing as the MRT. It is not. MRS is the consumer's willingness to trade (slope of the indifference curve). MRT is the market's trade-off (slope of the budget line, Px/Py). At the optimum they are equal, but they measure different things.
For perfect substitutes, students frequently try to set MRS = MRT and solve for an interior bundle. With perfect substitutes the MRS is constant, so if it does not equal MRT, the solution is a corner, not an interior point. Check whether MRS > MRT or MRT > MRS first.
With perfect complements, students sometimes forget to set the two arguments of the Min function equal. The optimum is always at the kink, so ax = by is your starting equation, not MRS = MRT.
Forgetting coefficients on utility functions is extremely common. If U = 3x^(1/2)y^(1/2), that 3 must carry through when you derive the indirect utility function and expenditure function. The summary sheet flags this in bold for a reason.
⚠️ You will almost certainly be asked to identify a utility function type and find the optimum bundle. Know the method for each type cold.
⚠️ Expect a question where a price changes and you need to trace the effect through the PCC to build a demand curve.
⚠️ Perfect substitutes corner-solution logic (MRS vs MRT comparison) appears regularly. Practise deciding which good the consumer buys before solving.
⚠️ Watch for utility functions with coefficients or non-standard exponents. The exam often tests whether you can adapt the standard formulas rather than just memorise them.
True or false: For Cobb-Douglas utility, the MRS is constant along an indifference curve. (False. MRS varies with the ratio y/x.)
Fill in the blank: For perfect substitutes U = ax + by, if MRS > MRT the consumer buys all ________. (x)
True or false: The optimum for perfect complements is found by setting MRS = MRT. (False. Set the two arguments of the Min function equal.)
Fill in the blank: The Engel curve for Cobb-Douglas plots ________ on the y-axis and ________ on the x-axis. (Income M; quantity of x)
True or false: A corner solution means the consumer buys positive quantities of both goods. (False. A corner solution means the consumer spends everything on one good.)
Q: Given U = x^(1/2)y^(1/2), Px = 4, Py = 1, M = 3600, find the optimum bundle.
A: MRS = MRT gives y/x = 4/1, so y = 4x. Substitute into budget: 3600 = 4x + 1(4x) = 8x, so x = 450, y = 1800.
Q: For U = 6x + y with Px = 1 and Py = 1, what does the consumer buy? What if Px rises to 10?
A: MRS = 6, MRT = Px/Py = 1. Since MRS > MRT, buy all x: x = 60, y = 0, utility = 360. At Px = 10, MRT = 10 > 6 = MRS, so buy all y: x = 0, y = 60, utility = 60. The switching price is Px = 6.
Q: For U = Min(2x, y), M = 12, Px = 1, Py = 1, find the optimum bundle and utility.
A: Set 2x = y. Budget: 12 = x + 2x = 3x, so x = 4, y = 8. Utility = Min(8, 8) = 8.
Q: Explain the difference between a necessary condition and a sufficient condition.
A: A sufficient condition guarantees the outcome (if A then B). A necessary condition must hold for the outcome to be possible (if not A then not B). Being over 35 is necessary to be president; being president is sufficient to be over 35.
This material feeds directly into the income and substitution effects (decomposing what happens when a price changes), as well as compensating and equivalent variation (measuring welfare changes in dollar terms). You cannot do CV/EV problems without first being able to find optimum bundles and utility levels for each function type.
The indirect utility function and expenditure function, covered in the next set of notes, are derived from the demand functions here. If your demand functions are wrong, everything downstream is wrong too.
Utility function, indifference curve, marginal rate of substitution, MRS, marginal rate of transformation, MRT, budget constraint, budget line, Cobb-Douglas, perfect substitutes, perfect complements, Leontief preferences, corner solution, interior solution, optimum bundle, consumer choice, price consumption curve, PCC, income consumption curve, ICC, Engel curve, demand curve derivation, ECON 201, intermediate microeconomics, Penn State, consumer theory, utility maximisation