Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), budget constraints, indifference curves, introductory consumer theory.
This topic sits at the core of intermediate microeconomics: how a rational consumer allocates a fixed budget across goods to get the most satisfaction possible. You are formalising the intuition from introductory micro ("buy more of what you like until the money runs out") into a constrained optimisation problem solved with calculus. Everything here feeds forward into demand theory, welfare analysis, and general equilibrium, so if this material is shaky the rest of the course will feel harder than it needs to. You should already be comfortable with indifference curves, the budget line, and taking partial derivatives.
A consumer with Cobb-Douglas preferences maximises utility by setting up a Lagrangian and solving the first-order conditions. For U(X, Y) = X^0.5 Y^0.5, the optimal (Marshallian) demands turn out to be X* = I / (2Px) and Y* = I / (2Py), meaning the consumer spends exactly half of income on each good. From these demands you can derive the Engel curve, confirm whether a good is normal or inferior, build the indirect utility function, and invert it to get the expenditure function and Hicksian demands.
Utility function
A mathematical representation of a consumer's preferences over bundles of goods. It assigns a number to each bundle such that preferred bundles get higher numbers.
In simple terms, it is a scoring system: the bundle you like more gets a higher score.
Cobb-Douglas utility
A specific functional form U(X, Y) = X^a Y^b, where a and b are positive constants. It produces smooth, convex indifference curves and guarantees an interior solution.
Think of it as the "standard workhorse" utility function in intermediate micro, because its algebra is clean and its properties are well behaved.
Lagrangian
A function that combines the objective (utility) and the constraint (budget) into one expression: L = U(X, Y) + lambda (I - Px X - Py Y). Solving its first-order conditions yields the optimal bundle.
In simple terms, it is a trick that converts a constrained problem into an unconstrained one you can solve with ordinary calculus.
Marshallian (ordinary) demand
The quantity of each good the consumer chooses as a function of prices and income: X*(Px, Py, I). These are what you get directly from solving the Lagrangian.
Think of it as the answer to "how much does the consumer actually buy?"
Engel curve
A graph (or equation) showing how the quantity demanded of a good changes as income changes, holding prices fixed.
In simple terms, it answers the question "if you got a pay rise, would you buy more or less of this?"
Normal good
A good whose quantity demanded rises when income rises (dX*/dI > 0). Equivalently, the Engel curve slopes upward.
Inferior good
A good whose quantity demanded falls when income rises (dX*/dI < 0). The Engel curve slopes downward.
Indirect utility function, V(Px, Py, I)
The maximum utility the consumer can achieve at given prices and income. Found by substituting the Marshallian demands back into the utility function.
Think of it as "the best score you can get, given your budget."
Expenditure function, E(Px, Py, U)
The minimum income needed to reach a given utility level at given prices. It is the inverse of the indirect utility function (solve V = U for I).
In simple terms, it answers "how much money would you need to be exactly this happy?"
Hicksian (compensated) demand
The quantity of each good that minimises expenditure for a given utility level. Found by differentiating the expenditure function with respect to price (Shephard's lemma) or by solving the dual problem directly.
Think of it as the demand you would observe if someone adjusted your income to keep your satisfaction constant every time a price changed.
The consumer's problem: maximise U(X, Y) = X^0.5 Y^0.5 subject to the budget constraint Px X + Py Y = I.
Form the Lagrangian:
L = X^0.5 Y^0.5 + lambda (I - Px X - Py Y)
Take partial derivatives and set each equal to zero:
dL/dX = 0.5 X^(-0.5) Y^(0.5) - lambda Px = 0
dL/dY = 0.5 X^(0.5) Y^(-0.5) - lambda Py = 0
dL/d(lambda) = I - Px X - Py Y = 0
Divide the first FOC by the second to eliminate lambda:
(0.5 X^(-0.5) Y^(0.5)) / (0.5 X^(0.5) Y^(-0.5)) = Px / Py
This simplifies to Y / X = Px / Py
This tells you the tangency condition: the marginal rate of substitution equals the price ratio.
Rearrange to get: Y = (Px / Py) X, or equivalently Px X = Py Y. The consumer spends equal amounts on each good (a general property of symmetric Cobb-Douglas).
Substitute Y = (Px / Py) X into the budget constraint:
Px X + Py ((Px / Py) X) = I
Px X + Px X = I
2 Px X = I
X = I / (2 Px)*
By the same logic (or by symmetry):
Y = I / (2 Py)*
Each good receives exactly half of income. This equal-share result holds whenever the exponents in the Cobb-Douglas function are equal.
The Engel curve for X is X* = I / (2 Px), which is linear in income with slope dX*/dI = 1 / (2 Px).
Since dX*/dI > 0 for any positive price, X is a normal good: more income means the consumer buys more of it.
Numerical example: if Px = 2 and I = 100, then X* = 100 / 4 = 25. If income doubles to I = 200, then X* = 200 / 4 = 50.
Substitute the Marshallian demands back into U:
V(Px, Py, I) = (I / (2 Px))^0.5 (I / (2 Py))^0.5
V = I / (2 Px^0.5 Py^0.5)
This gives the maximum achievable utility as a function of prices and income.
Invert V = U for I to find the minimum spending needed for utility level U:
E(Px, Py, U) = 2 U Px^0.5 Py^0.5
Apply Shephard's lemma (differentiate E with respect to each price):
Xh = dE/dPx = U Px^(-0.5) Py^(0.5)
Yh = dE/dPy = U Px^(0.5) Py^(-0.5)
Alternatively, substitute the expenditure function into the Marshallian demands and simplify. The Hicksian demands depend on utility and prices, not on income.
Utility function: U(X, Y) = X^0.5 Y^0.5
Budget constraint: Px X + Py Y = I
Lagrangian: L = X^0.5 Y^0.5 + lambda (I - Px X - Py Y)
Tangency condition (MRS = price ratio): Y / X = Px / Py
Marshallian demands: X* = I / (2 Px), Y* = I / (2 Py)
Indirect utility: V = I / (2 Px^0.5 Py^0.5)
Expenditure function: E = 2 U Px^0.5 Py^0.5
Hicksian demands: Xh = U Px^(-0.5) Py^(0.5), Yh = U Px^(0.5) Py^(-0.5)
The Cobb-Douglas form is used far beyond textbook exercises. Economists use it to model household spending patterns, and the equal-expenditure-share result (when exponents are equal) roughly matches how some households split spending between broad categories like food and housing. The Lagrangian method itself is the same technique engineers and operations researchers use whenever they need to optimise something subject to a constraint, from portfolio allocation to production planning.
Students often think lambda (the Lagrange multiplier) is just a mathematical placeholder. It has an economic interpretation: it is the marginal utility of income, i.e. how much extra utility the consumer would gain from one more unit of income.
Students sometimes confuse Marshallian and Hicksian demands. Marshallian demands are functions of prices and income. Hicksian demands are functions of prices and a target utility level. They ask different questions.
A common error is forgetting that the tangency condition (MRS = price ratio) only works for interior solutions. If the consumer might spend zero on one good, you need to check corner solutions separately.
Students often assume all Cobb-Douglas functions produce equal spending shares. That only happens when the exponents are equal. For U = X^a Y^b, the share spent on X is a / (a + b).
⚠️ You will almost certainly be asked to set up and solve a Lagrangian for a Cobb-Douglas utility function. Practise this until it is automatic.
⚠️ Be ready to identify whether a good is normal or inferior from the sign of dX*/dI. For Cobb-Douglas, every good is always normal.
⚠️ Know the relationship chain: Marshallian demands → indirect utility → expenditure function → Hicksian demands. Exam questions often ask you to move through two or three of these steps.
⚠️ The numerical example (Px = 2, I = 100 → X = 25; I = 200 → X = 50) is a classic plug-and-verify question. Be comfortable substituting values quickly.
True or false: For U(X, Y) = X^0.5 Y^0.5, the consumer always spends more on X than on Y.
False. The consumer spends exactly half of income on each good.
Fill in the blank: The Marshallian demand for X is X* = ____.
I / (2 Px)
True or false: If dX*/dI is negative, the good is normal.
False. A negative dX*/dI means the good is inferior.
True or false: The indirect utility function tells you the minimum cost of achieving a given utility.
False. That is the expenditure function. The indirect utility function tells you the maximum utility achievable at given prices and income.
Q: Given U(X, Y) = X^0.5 Y^0.5 with Px = 4, Py = 1, and I = 80, find the optimal bundle.
A: X* = 80 / (2 × 4) = 10. Y* = 80 / (2 × 1) = 40. The consumer buys 10 units of X and 40 units of Y.
Q: Using the same utility function, derive the indirect utility function.
A: V(Px, Py, I) = (I / (2Px))^0.5 × (I / (2Py))^0.5 = I / (2 Px^0.5 Py^0.5). With Px = 4, Py = 1, I = 80: V = 80 / (2 × 2 × 1) = 20.
Q: What is the expenditure function for this consumer, and how much income is needed to reach U = 10 when Px = 4 and Py = 1?
A: E(Px, Py, U) = 2U Px^0.5 Py^0.5. With U = 10, Px = 4, Py = 1: E = 2 × 10 × 2 × 1 = 40.
Q: Is X a normal or inferior good here? How do you know from the Marshallian demand?
A: X is a normal good. The Marshallian demand X* = I / (2Px) increases as income I increases, since dX*/dI = 1/(2Px) > 0.
Q: Find the Hicksian demand for X when U = 10, Px = 4, Py = 1.
A: Xh = U Px^(-0.5) Py^(0.5) = 10 × (1/2) × 1 = 5.
This material connects directly to consumer surplus and welfare analysis: the expenditure function is the tool you use to measure compensating and equivalent variation, which are precise welfare measures of a price change.
It also links to Slutsky decomposition: once you have both Marshallian and Hicksian demands, you can decompose the effect of a price change into substitution and income effects.
More broadly, the Lagrangian technique reappears in producer theory when firms minimise costs subject to an output target, so mastering it here pays off twice.
Utility maximisation, consumer optimisation, Lagrange multiplier method, Cobb-Douglas preferences, Marshallian demand, ordinary demand, Hicksian demand, compensated demand, Engel curve, normal good, inferior good, indirect utility function, expenditure function, Shephard's lemma, budget constraint, MRS, marginal rate of substitution, tangency condition, constrained optimisation, duality in consumer theory