Source: ECON 300 E, University of Washington
Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives, constrained optimisation), introductory microeconomics (budget constraints, indifference curves, utility functions).
Consumer theory sits at the heart of intermediate microeconomics. The core question: given prices and a budget, how does a rational consumer choose quantities of goods to maximise their satisfaction? This topic builds on introductory supply-and-demand by formalising the optimisation problem with calculus, then connecting several functions (Marshallian demand, indirect utility, expenditure) through a framework called duality. If you followed the basic indifference-curve and budget-constraint material, you are ready for this. If not, review those first, because everything here rests on them.
A consumer with a Cobb-Douglas utility function spends fixed shares of income on each good, determined by the exponents. You can derive demand functions using the Lagrangian, then plug them back in to get the indirect utility function, invert that to get the expenditure function, and use the Slutsky equation to decompose any price change into a substitution effect and an income effect.
Utility function
A mathematical representation of a consumer's preferences over bundles of goods. In simple terms, it assigns a number to each bundle so that preferred bundles get higher numbers.
Cobb-Douglas utility function
A utility function of the form U(X, Y) = X^a · Y^b. Think of it as the workhorse functional form in consumer theory: well-behaved, easy to differentiate, and it produces demand functions where each good gets a constant share of income.
Lagrangian method
A technique for solving constrained optimisation problems by combining the objective function and the constraint into a single expression using a multiplier (lambda). In simple terms, it is the standard tool for finding the best affordable bundle.
Marshallian (ordinary) demand function
The quantity of a good a consumer chooses as a function of prices and income: X*(Px, Py, I). Think of it as "what you actually buy given your budget."
Indirect utility function, V(Px, Py, I)
The maximum utility achievable at given prices and income, found by plugging the Marshallian demands back into the utility function. In simple terms, it tells you how happy you can be with a particular budget.
Expenditure function, E(Px, Py, U)
The minimum income needed to reach a target utility level at given prices. It is the mathematical inverse of the indirect utility function. Think of it as "the cheapest way to be this happy."
Slutsky decomposition (Slutsky equation)
A formula that splits the total effect of a price change on demand into a substitution effect (holding utility constant) and an income effect (the change in purchasing power). In simple terms, it separates "I switched to something cheaper" from "I feel poorer."
Substitution effect
The change in quantity demanded due solely to the change in relative prices, holding utility constant. Think of it as the pure price-incentive effect.
Income effect
The change in quantity demanded due to the change in real purchasing power caused by a price change. Think of it as: "the price went up, so my money buys less overall, and that changes how much I want of this good."
The consumer wants to maximise U(X, Y) = X^0.3 · Y^0.7 subject to the budget constraint I = Px·X + Py·Y.
Form the Lagrangian: L = X^0.3 · Y^0.7 + λ·(I - Px·X - Py·Y).
Differentiate with respect to X: 0.3·X^(-0.7)·Y^(0.7) - λ·Px = 0
Differentiate with respect to Y: 0.7·X^(0.3)·Y^(-0.3) - λ·Py = 0
Differentiate with respect to λ: I - Px·X - Py·Y = 0
Divide the first FOC by the second to eliminate λ. This gives the tangency condition: (0.3·Y) / (0.7·X) = Px / Py, which rearranges to Y = (7/3)·(Px/Py)·X.
Substitute this back into the budget constraint and solve for X.
Result: X* = 0.3·I / Px and Y* = 0.7·I / Py.
With Cobb-Douglas, the consumer always spends a fraction of income equal to the exponent on each good. The exponent on X is 0.3, so 30% of income goes to X. The exponent on Y is 0.7, so 70% goes to Y. This constant-shares property is specific to Cobb-Douglas and makes these problems fast to solve once you recognise the form.
Take the Marshallian demands X* = 0.3I/Px and Y* = 0.7I/Py and plug them into the utility function.
V(Px, Py, I) = (0.3I/Px)^0.3 · (0.7I/Py)^0.7 = (0.3/Px)^0.3 · (0.7/Py)^0.7 · I
This gives utility as a function of prices and income only, with no quantities.
The expenditure function is the inverse of the indirect utility function. Solve V = (0.3/Px)^0.3 · (0.7/Py)^0.7 · E for E.
E(Px, Py, U) = U · (Px/0.3)^0.3 · (Py/0.7)^0.7
This tells you the minimum spend needed to reach utility level U at prices Px and Py.
These two functions are mirrors of each other. The indirect utility function answers "how happy can I be with this budget?" The expenditure function answers "how much do I need to be this happy?" Inverting one gives you the other. This relationship is called duality, and it is central to intermediate micro.
From X* = 0.3I/Px, differentiate with respect to Px: dX*/dPx = -0.3I / Px^2.
This is negative, confirming that demand falls when price rises (law of demand holds here).
The Slutsky equation decomposes the total effect into two parts:
dX/dPx = (dX/dPx)|_{U constant} - X · (dX/dI)
Total effect = Substitution effect + Income effect
The income effect is: -X · (dX/dI) = -(0.3I/Px) · (0.3/Px) = -0.09I / Px^2.
This is negative for a normal good: when Px rises, real income falls, and the consumer buys less X.
Substitution effect = Total effect - Income effect = (-0.3I/Px^2) - (-0.09I/Px^2) = -0.21I / Px^2.
Also negative: even holding utility constant, the consumer substitutes away from the good that became relatively more expensive.
Both effects work in the same direction here (both negative), which is typical for a normal good. The substitution effect is always negative (for a price increase). The income effect is negative for normal goods but positive for inferior goods. If the income effect were positive and large enough to outweigh the substitution effect, you would have a Giffen good.
Cobb-Douglas utility: U(X, Y) = X^a · Y^b
Lagrangian: L = X^a · Y^b + λ·(I - Px·X - Py·Y)
Marshallian demand (Cobb-Douglas): X* = a·I / ((a+b)·Px), Y* = b·I / ((a+b)·Py). When a + b = 1, this simplifies to X* = a·I/Px, Y* = b·I/Py.
Indirect utility (for U = X^0.3 · Y^0.7): V = (0.3/Px)^0.3 · (0.7/Py)^0.7 · I
Expenditure function (same case): E = U · (Px/0.3)^0.3 · (Py/0.7)^0.7
Slutsky equation: dX/dPx = (dX/dPx)|_{U=const} - X·(dX/dI)
The expenditure function is the basis for constructing cost-of-living indices. When statisticians measure inflation, they are essentially asking: "How much more does it cost to reach the same utility as before?" The Slutsky decomposition matters for tax policy: if the government taxes a good, the substitution effect captures the distortion (people shifting to untaxed goods), while the income effect captures the welfare loss from reduced purchasing power.
Students often confuse the indirect utility function with the utility function itself. V is a function of prices and income, not quantities. If you still see X and Y in your answer, you have not finished the substitution.
The income effect in the Slutsky equation refers to the change in real purchasing power caused by a price change, not a change in actual nominal income. The consumer's paycheque has not changed; the price shift just made them effectively richer or poorer.
When inverting V to get E, students sometimes forget to swap I for E (or U for V). Keep the logic clear: V maps (prices, income) to utility; E maps (prices, utility) to income.
The substitution effect is always negative for a price increase (the consumer moves away from the more expensive good). Students sometimes get a positive sign and do not question it.
⚠️ The Lagrangian setup and FOCs for Cobb-Douglas is one of the most commonly tested problems. Know the steps cold.
⚠️ Be able to move fluently between Marshallian demand, indirect utility, and expenditure function. Exam questions often ask you to derive one from another.
⚠️ The Slutsky decomposition is a favourite for asking you to compute and interpret each effect separately. Make sure you can identify the income effect term (-X · dX/dI) and compute it correctly.
⚠️ For Cobb-Douglas, the constant income-share property (spend fraction a/(a+b) on good X) is a quick sanity check. If your demand function does not have this property, something went wrong.
True or false: For U(X,Y) = X^0.4 · Y^0.6, the consumer spends 40% of income on X.
Fill in the blank: The Lagrangian multiplier λ represents the marginal utility of ______.
True or false: The expenditure function is increasing in utility and increasing in prices.
True or false: The substitution effect of a price increase can be positive.
Fill in the blank: Inverting the indirect utility function gives the ______ function.
Q: Given U(X,Y) = X^0.3 · Y^0.7 with prices Px and Py and income I, what are the Marshallian demand functions?
A: X* = 0.3I / Px and Y* = 0.7I / Py. Each good receives a share of income equal to its exponent.
Q: How do you derive the indirect utility function from the Marshallian demands?
A: Substitute X* and Y* back into the utility function: V(Px, Py, I) = (0.3I/Px)^0.3 · (0.7I/Py)^0.7.
Q: What is the expenditure function for this consumer, and how do you obtain it?
A: Invert the indirect utility function by replacing V with U and I with E, then solve for E. Result: E = U · (Px/0.3)^0.3 · (Py/0.7)^0.7.
Q: Using the Slutsky equation, what is the income effect of a small increase in Px on the demand for X?
A: The income effect is -X · (dX/dI) = -(0.3I/Px) · (0.3/Px) = -0.09I / Px^2.
Q: What is the substitution effect of the same price change?
A: Substitution effect = Total effect - Income effect = (-0.3I/Px^2) - (-0.09I/Px^2) = -0.21I / Px^2.
This material connects directly to compensated (Hicksian) demand, which solves the mirror problem: minimise expenditure for a given utility level rather than maximise utility for a given budget. It also connects to revealed preference theory, which tests whether observed choices are consistent with utility maximisation without assuming a specific functional form. The Slutsky decomposition reappears in labour economics when analysing how wage changes affect hours worked (substitution between leisure and consumption).
Consumer optimisation, utility maximisation, Lagrange multiplier, Lagrangian method, constrained optimisation, Marshallian demand, ordinary demand, Cobb-Douglas, indirect utility function, expenditure function, duality in consumer theory, Slutsky decomposition, Slutsky equation, income effect, substitution effect, normal good, inferior good, Giffen good, budget constraint, tangency condition, first-order conditions, FOC, ECON 300, intermediate microeconomics.