Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), budget constraints, indifference curves.
This topic sits at the core of consumer theory in intermediate microeconomics. You are solving the consumer's problem: given a budget and prices, what combination of goods maximises utility? The Cobb-Douglas utility function is the workhorse example because it produces interior solutions with clean algebra. If you are comfortable with setting up a Lagrangian or using the tangency condition MRS = price ratio, you are ready for this material.
A consumer with Cobb-Douglas preferences (U = X^(1/2) Y^(1/2)) splits income across goods using simple demand formulas derived from the tangency condition. When a price changes, the total change in quantity demanded breaks into a substitution effect (moving along the indifference curve) and an income effect (shift to a new curve). Understanding these two pieces is what the exam tests.
Cobb-Douglas utility function
A utility function of the form U(X, Y) = X^a Y^b, where a and b are positive constants. With a = b = 1/2, the consumer spends a fixed fraction of income on each good regardless of prices. In simple terms, it describes a consumer who always splits their budget in a predictable way.
Budget constraint
The set of all bundles a consumer can afford: Px·X + Py·Y = M. Think of it as the boundary of what your wallet allows.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade Y for X while staying equally happy. For U = X^(1/2) Y^(1/2), MRS = Y/X. In simple terms, it is the slope of the indifference curve at a given point.
Tangency condition
At the optimum, MRS = Px/Py. The consumer's willingness to trade matches the market's rate of exchange. This is the key equation you solve alongside the budget constraint.
Substitution effect (SE)
The change in quantity demanded due solely to the change in relative prices, holding utility constant. You pivot along the original indifference curve to the new price ratio.
Income effect (IE)
The change in quantity demanded due to the change in purchasing power after a price change, moving from the compensated bundle to the final bundle.
Compensated budget constraint (Hicksian)
A hypothetical budget line at the new prices but with enough income to reach the original utility level. It isolates the substitution effect from the income effect.
Composite good (numeraire)
A stand-in for "all other goods," priced at $1 per unit. The Y-axis then represents dollars of remaining spending power.
Setting up the problem. Given U(X, Y) = X^(1/2) Y^(1/2), prices Px and Py, and income M, the consumer maximises utility subject to the budget constraint Px·X + Py·Y = M.
Tangency condition. MRS = (Y/X) = Px/Py. Rearranging: Y = (Px/Py)·X. Substitute into the budget constraint to solve for X, then back-substitute for Y.
Demand formulas for this utility function:
X* = M / (2·Px)
Y* = M / (2·Py)
Worked example (from the homework). M = 36, Px = 4, Py = 1.
X* = 36 / (2·4) = 4.5
Y* = 36 / (2·1) = 18
U(4.5, 18) = (4.5)^(1/2) · (18)^(1/2) = 9
Why the consumer spends exactly half of income on each good. With equal exponents (a = b = 1/2), the expenditure share on X is always a/(a+b) = 1/2, and likewise for Y. This property is specific to symmetric Cobb-Douglas and breaks down when the exponents differ.
Setup. Start at the original optimum (X* = 4.5, Y* = 18) with Px = 4. Now Px rises to 9, while Py = 1 and M = 36 stay the same.
New optimum (X, Y**).** Apply the same demand formulas at the new price:
X** = 36 / (2·9) = 2
Y** = 36 / (2·1) = 18
The total change in X is 2 - 4.5 = -2.5
Finding the compensated bundle (X^c, Y^c). The compensated budget gives the consumer enough income at the new prices to reach the original utility level (U = 9).
The expenditure function for this utility: M^c = 2·U·(Px·Py)^(1/2) = 2·9·(9·1)^(1/2) = 54
X^c = M^c / (2·Px_new) = 54 / 18 = 3
Y^c = M^c / (2·Py) = 54 / 2 = 27
Decomposition:
Substitution effect = X^c - X* = 3 - 4.5 = -1.5
Income effect = X** - X^c = 2 - 3 = -1
Total effect = SE + IE = -1.5 + (-1) = -2.5 (checks out)
Interpretation. Both effects are negative for a normal good when its price rises. The substitution effect says "X got relatively more expensive, so buy less of it." The income effect says "the price rise made you poorer in real terms, so you buy less of everything."
Graphing the three budget constraints:
Original: 4X + Y = 36 (X-intercept 9, Y-intercept 36)
New: 9X + Y = 36 (X-intercept 4, Y-intercept 36)
Compensated: 9X + Y = 54 (X-intercept 6, Y-intercept 54)
The compensated line is parallel to the new line (same slope, -9) but shifted outward to be tangent to the original indifference curve.
For U(X, Y) = X^(1/2) Y^(1/2) with prices Px, Py and income M:
X^* = \frac{M}{2P_X}, \quad Y^* = \frac{M}{2P_Y}MRS = \frac{Y}{X} = \frac{P_X}{P_Y}U^* = \frac{M}{2\sqrt{P_X P_Y}}Compensated (Hicksian) income to maintain utility U at new prices:
M^c = 2U\sqrt{P_X^{\text{new}} \cdot P_Y}Substitution effect = X^c - X*. Income effect = X** - X^c.
The substitution and income effect decomposition is how economists analyse policy changes such as a tax on petrol. The substitution effect captures drivers switching to public transport (petrol is relatively more expensive), while the income effect captures the fact that higher fuel costs leave households with less to spend on everything else. Governments sometimes return tax revenue as lump-sum rebates to offset the income effect while preserving the substitution incentive.
Students often think the substitution effect can be positive when a price rises. It cannot: the substitution effect for the good whose price increased is always negative (you always substitute away from the more expensive good).
Confusing the compensated bundle with the new optimum. The compensated bundle (X^c, Y^c) is on the original indifference curve at the new price ratio. The new optimum (X**, Y**) is on a lower indifference curve at the new budget.
Forgetting that the compensated budget line is parallel to the new budget line, not the original. It uses the new prices, just with a different income level.
Mixing up the signs. When a price increases, both SE and IE for a normal good are negative. Students sometimes report positive values when the question asks for a negative answer.
⚠️ You will almost certainly be asked to compute X*, Y*, then X**, Y** after a price change, and decompose the total change into SE and IE. Practise the full sequence until it is automatic.
⚠️ Graphing is heavily tested. You must draw three budget lines (original, new, compensated) and label all three bundles with their coordinates. The compensated line is parallel to the new line.
⚠️ Know how to compute the compensated income M^c. The expenditure function for Cobb-Douglas with equal exponents is M^c = 2U·sqrt(Px·Py).
⚠️ The question may note that answers "might not be integers." Do not round intermediate steps.
True or false: For U = X^(1/2) Y^(1/2), the consumer always spends exactly half of income on each good. (True)
If Px doubles and everything else stays the same, X* is ______. (Halved)
True or false: The compensated budget line has the same slope as the original budget line. (False, it has the slope of the new budget line)
The substitution effect for a normal good whose price rises is always ______ (positive/negative). (Negative)
True or false: SE + IE = total change in quantity demanded. (True)
Q: A consumer has U = X^(1/2) Y^(1/2), M = 50, Px = 5, Py = 2. What is the optimal bundle?
A: X* = 50/(2·5) = 5, Y* = 50/(2·2) = 12.5.
Q: Using the bundle above, what utility does the consumer achieve?
A: U = 5^(1/2) · 12.5^(1/2) = sqrt(62.5) ≈ 7.91.
Q: Now Px rises to 10. What is the new optimal bundle?
A: X** = 50/(2·10) = 2.5, Y** = 50/(2·2) = 12.5.
Q: What is the compensated income needed to maintain the original utility at the new prices?
A: M^c = 2·7.91·sqrt(10·2) = 2·7.91·sqrt(20) ≈ 70.71.
Q: Decompose the change in X into SE and IE.
A: X^c = 70.71/(2·10) ≈ 3.54. SE = 3.54 - 5 = -1.46. IE = 2.5 - 3.54 = -1.04. Total = -2.5.
This material connects directly to the Slutsky equation, which formalises the SE/IE decomposition for any utility function (not just Cobb-Douglas). It also sets up the derivation of individual and market demand curves: the demand function X* = M/(2Px) is the individual demand curve for X.
The compensated demand curve (Hicksian demand) holds utility constant and traces out only the substitution effect as price changes, which connects to welfare measures like compensating and equivalent variation (covered in the next problem and the Leontief notes).
Tags: Cobb-Douglas, consumer optimisation, utility maximisation, budget constraint, MRS, marginal rate of substitution, tangency condition, substitution effect, income effect, Hicksian decomposition, compensated demand, normal good, composite good, numeraire, ECON 302, intermediate microeconomics