Difficulty: Intermediate | Prerequisites: Consumer theory basics, constrained optimisation, Lagrangian method
Consumer theory rests on two optimisation problems that turn out to be mirror images of each other. The utility maximisation problem (UMP) asks: given a budget, what bundle makes the consumer happiest? The expenditure minimisation problem (EMP) asks: given a target utility level, what is the cheapest way to reach it? Each problem produces its own demand function, and understanding both, plus how they connect, is the core of intermediate microeconomic theory. You should already be comfortable with indifference curves, budget constraints, and basic Lagrangian optimisation before reading these notes.
Marshallian (ordinary) demand comes from maximising utility subject to a budget constraint; it is a function of prices and income. Hicksian (compensated) demand comes from minimising expenditure subject to a utility constraint; it is a function of prices and a target utility level. The two are linked through duality: at the optimum, plugging the indirect utility into the Hicksian demand recovers the Marshallian demand, and vice versa.
Marshallian (ordinary) demand function
The bundle that maximises utility given prices p and income m. Written x*(p, m). This is the demand function you meet first in principles courses.
In simple terms, it answers: "I have this much money and these are the prices, what should I buy?"
Hicksian (compensated) demand function
The bundle that minimises the cost of reaching a target utility level U at prices p. Written h(p, U). "Compensated" because it holds utility fixed, as though the consumer is compensated for any price change to stay on the same indifference curve.
In simple terms, it answers: "What is the cheapest way to stay exactly this happy?"
Indirect utility function, V(p, m)
The maximum utility achievable at prices p with income m. It is the value function of the UMP. Plug the Marshallian demands back into the utility function and you get V.
Expenditure function, e(p, U)
The minimum cost of achieving utility level U at prices p. It is the value function of the EMP. Plug the Hicksian demands into the budget expression and you get e.
Lagrangian
The function formed by combining the objective and the constraint with a multiplier λ. For the UMP: L = u(x) + λ(m − p·x). For the EMP: L = p·x + λ(U − u(x)).
Think of it as the tool that lets you convert a constrained problem into an unconstrained one.
Normal good
A good whose Marshallian demand increases when income rises (∂x*ᵢ/∂m > 0).
In simple terms, you buy more of it when you get richer.
Inferior good
A good whose Marshallian demand decreases when income rises (∂x*ᵢ/∂m < 0).
Strictly concave utility function
A utility function where the indifference curves are strictly convex to the origin. This guarantees interior, unique solutions to both the UMP and EMP (given positive prices and income). It rules out flat segments on indifference curves.
The setup
The consumer chooses a bundle x = (x₁, x₂) to minimise total spending p₁x₁ + p₂x₂
Subject to the constraint that utility is at least U: u(x₁, x₂) ≥ U
Prices p₁, p₂ > 0 are given, and U is a target utility level
Lagrangian formulation
Form L = p₁x₁ + p₂x₂ + λ(U − u(x₁, x₂))
First-order conditions:
∂L/∂x₁ = p₁ − λ(∂u/∂x₁) = 0
∂L/∂x₂ = p₂ − λ(∂u/∂x₂) = 0
∂L/∂λ = U − u(x₁, x₂) = 0
From the first two conditions, dividing gives the tangency condition:
(∂u/∂x₁) / (∂u/∂x₂) = p₁ / p₂
This says the marginal rate of substitution equals the price ratio, the same tangency condition as in the UMP.
What the solution gives you
The Hicksian demands h₁(p₁, p₂, U) and h₂(p₁, p₂, U), functions of prices and target utility
The expenditure function e(p, U) = p₁h₁ + p₂h₂, the minimum cost of reaching U
The setup
The consumer chooses x = (x₁, x₂) to maximise u(x₁, x₂)
Subject to the budget constraint p₁x₁ + p₂x₂ ≤ m
Prices p₁, p₂ > 0 and income m > 0 are given
Lagrangian formulation
Form L = u(x₁, x₂) + λ(m − p₁x₁ − p₂x₂)
First-order conditions:
∂L/∂x₁ = ∂u/∂x₁ − λp₁ = 0
∂L/∂x₂ = ∂u/∂x₂ − λp₂ = 0
∂L/∂λ = m − p₁x₁ − p₂x₂ = 0
Again, dividing the first two FOCs yields the tangency condition:
(∂u/∂x₁) / (∂u/∂x₂) = p₁ / p₂
The same tangency condition as the EMP. The difference is the constraint: here you exhaust the budget; there you hit the utility target.
What the solution gives you
The Marshallian demands x₁*(p₁, p₂, m) and x₂*(p₁, p₂, m), functions of prices and income
The indirect utility function V(p, m) = u(x₁*, x₂*), the maximum utility attainable
Key relationship between the two problems
At the optimum, the Marshallian and Hicksian demands coincide. If you set m = e(p, U), then x*(p, m) = h(p, U). If you set U = V(p, m), then h(p, U) = x*(p, m). This is the duality that connects the two problems.
Marshallian demand (UMP)
Maximise x₁⁰·³ x₂⁰·⁷ subject to p₁x₁ + p₂x₂ = m.
Lagrangian: L = x₁⁰·³ x₂⁰·⁷ + λ(m − p₁x₁ − p₂x₂)
FOCs:
0.3 x₁⁻⁰·⁷ x₂⁰·⁷ = λp₁
0.7 x₁⁰·³ x₂⁻⁰·³ = λp₂
Divide the first by the second:
(0.3 x₂) / (0.7 x₁) = p₁ / p₂
Solve for x₂: x₂ = (7p₁x₁) / (3p₂)
Substitute into the budget constraint:
p₁x₁ + p₂ · (7p₁x₁)/(3p₂) = m
p₁x₁(1 + 7/3) = m
p₁x₁(10/3) = m
So x₁* = 0.3m / p₁ and x₂* = 0.7m / p₂.
This is the standard Cobb-Douglas result: the consumer spends a fixed fraction of income on each good equal to that good's exponent.
Hicksian demand (EMP)
Minimise p₁x₁ + p₂x₂ subject to x₁⁰·³ x₂⁰·⁷ = U.
The tangency condition is the same: (0.3 x₂)/(0.7 x₁) = p₁/p₂, giving x₂ = (7p₁x₁)/(3p₂).
Substitute into the utility constraint:
x₁⁰·³ · [(7p₁x₁)/(3p₂)]⁰·⁷ = U
x₁⁰·³ · x₁⁰·⁷ · (7p₁/(3p₂))⁰·⁷ = U
x₁ · (7p₁/(3p₂))⁰·⁷ = U
So h₁ = U · (3p₂/(7p₁))⁰·⁷ and h₂ = U · (7p₁/(3p₂))⁰·³.
The expenditure function is e(p, U) = p₁h₁ + p₂h₂. After simplification, for the general Cobb-Douglas u = x₁ᵅ x₂¹⁻ᵅ, the expenditure function takes the form e = U · (p₁/α)ᵅ · (p₂/(1−α))¹⁻ᵅ.
Marshallian demand (UMP)
Maximise √x₁ + x₂ subject to p₁x₁ + p₂x₂ = m.
Lagrangian: L = √x₁ + x₂ + λ(m − p₁x₁ − p₂x₂)
FOCs:
(1/2)x₁⁻¹/² = λp₁
1 = λp₂
From the second: λ = 1/p₂. Substitute into the first:
(1/2)x₁⁻¹/² = p₁/p₂
x₁⁻¹/² = 2p₁/p₂
x₁* = (p₂/(2p₁))² = p₂²/(4p₁²)
From the budget constraint: x₂* = (m − p₁x₁*)/p₂ = m/p₂ − p₂/(4p₁)
Notice that x₁* does not depend on income m. This is the hallmark of quasilinear preferences: the good that enters non-linearly (here x₁) has demand that depends only on the price ratio, not on income. All additional income goes to x₂, the "linear" good.
This holds provided income is large enough that x₂* > 0. If m is very small, you hit a corner solution.
Hicksian demand (EMP)
Minimise p₁x₁ + p₂x₂ subject to √x₁ + x₂ = U.
The tangency condition gives the same x₁: h₁ = p₂²/(4p₁²).
From the utility constraint: √h₁ + h₂ = U, so h₂ = U − p₂/(2p₁).
The expenditure function: e(p, U) = p₁ · p₂²/(4p₁²) + p₂(U − p₂/(2p₁)) = p₂U − p₂²/(4p₁).
Cobb-Douglas case: u = x₁⁰·³ x₂⁰·⁷
Both goods are normal. The Marshallian demands are x₁* = 0.3m/p₁ and x₂* = 0.7m/p₂. Taking the derivative with respect to income:
∂x₁*/∂m = 0.3/p₁ > 0
∂x₂*/∂m = 0.7/p₂ > 0
Both derivatives are strictly positive (since prices are positive), so both goods are normal. This is a general feature of Cobb-Douglas preferences: with u = x₁ᵅ x₂¹⁻ᵅ, the consumer always spends fraction α of income on good 1 and (1 − α) on good 2, so demand for both rises linearly with income.
Quasilinear case: u = √x₁ + x₂
Good 1 (x₁) has demand x₁* = p₂²/(4p₁²), which does not depend on m at all. So ∂x₁*/∂m = 0. Strictly speaking, good 1 is neither normal nor inferior; its income elasticity is zero. Some textbooks classify this as "borderline normal."
Good 2 (x₂) has demand x₂* = m/p₂ − p₂/(4p₁), so ∂x₂*/∂m = 1/p₂ > 0. Good 2 is normal, and in fact absorbs all marginal income.
The intuition: with quasilinear utility, the non-linear good has a fixed "ideal" quantity determined purely by the price ratio. Once that quantity is purchased, every remaining unit of income goes to the linear good.
General Cobb-Douglas u = x₁ᵅ x₂¹⁻ᵅ
Marshallian: x₁* = αm/p₁, x₂* = (1−α)m/p₂
Hicksian: h₁ = U(p₂(1−α)/(p₁α))¹⁻ᵅ, h₂ = U(p₁α/(p₂(1−α)))ᵅ
Indirect utility: V(p,m) = (α/p₁)ᵅ ((1−α)/p₂)¹⁻ᵅ · m
Expenditure: e(p,U) = U · (p₁/α)ᵅ (p₂/(1−α))¹⁻ᵅ
Quasilinear u = √x₁ + x₂
Marshallian: x₁* = p₂²/(4p₁²), x₂* = m/p₂ − p₂/(4p₁)
Hicksian: h₁ = p₂²/(4p₁²), h₂ = U − p₂/(2p₁)
Indirect utility: V(p,m) = m/p₂ + p₂/(4p₁)
Expenditure: e(p,U) = p₂U − p₂²/(4p₁)
Tangency condition (both problems)
MRS₁₂ = (∂u/∂x₁)/(∂u/∂x₂) = p₁/p₂
The Hicksian demand is what economists use to measure welfare changes from price shifts (e.g., a tax on petrol). Because it holds utility constant, it isolates the pure price effect, which is what policy analysis needs. The Marshallian demand is what you observe in actual market data, since consumers respond to prices and their actual income, not to a hypothetical utility target.
Students often think the Hicksian and Marshallian demands are two completely separate things. They are not. At the optimum they give the same bundle; they differ only in what you hold fixed (income vs. utility) when you vary prices.
Students sometimes set up the Lagrangian for the EMP the same way as the UMP. The objective and constraint swap roles: in the UMP you maximise utility subject to the budget; in the EMP you minimise expenditure subject to a utility floor.
A common error is forgetting that the Cobb-Douglas expenditure shares are constant. If the exponent on x₁ is 0.3, the consumer always spends exactly 30% of income on good 1, regardless of prices.
With quasilinear utility, students often forget to check the corner solution. The interior solution for x₂ can go negative if income is too low, which is economically meaningless. Always verify x₂* > 0.
⚠️ You will be asked to set up and solve both the UMP and EMP using the Lagrangian. Be able to do this mechanically for Cobb-Douglas and quasilinear forms.
⚠️ Know the duality identities cold: x*(p, e(p,U)) = h(p,U) and h(p, V(p,m)) = x*(p,m).
⚠️ Be ready to determine whether goods are normal or inferior by differentiating Marshallian demand with respect to m.
⚠️ The quasilinear case is a favourite exam question because x₁ does not depend on income, which tests whether you understand the economics rather than just grinding through algebra.
True or False: The Hicksian demand function depends on income. (False, it depends on prices and a target utility level.)
Fill in the blank: For Cobb-Douglas u = x₁⁰·⁴ x₂⁰·⁶, the consumer spends ____% of income on good 1. (40%)
True or False: In the quasilinear case u = √x₁ + x₂, good 1 is an inferior good. (False, its income derivative is zero, not negative.)
Fill in the blank: The tangency condition sets the ________ equal to the price ratio. (Marginal rate of substitution, MRS)
True or False: The Marshallian and Hicksian demands always give different bundles. (False, at the optimum they coincide.)
Q: What is the economic interpretation of the Lagrange multiplier λ in the UMP?
A: It is the marginal utility of income, the rate at which the maximum achievable utility increases per additional unit of income. Formally, λ = ∂V/∂m.
Q: For u = x₁ᵅ x₂¹⁻ᵅ, derive the Marshallian demand for good 1.
A: Set up the Lagrangian, take FOCs, divide to get the tangency condition (αx₂)/((1−α)x₁) = p₁/p₂, solve for x₂ in terms of x₁, substitute into the budget constraint, and obtain x₁* = αm/p₁.
Q: Why does x₁ not depend on m in the quasilinear case?*
A: Because the marginal utility of x₂ is constant (equal to 1), the consumer's valuation of x₁ relative to x₂ depends only on the price ratio. Once the consumer has purchased the "right" amount of x₁, every extra unit of income is spent entirely on x₂.
Q: Suppose u = x₁⁰·³ x₂⁰·⁷ and p₁ = 3, p₂ = 7, m = 100. Find the optimal bundle.
A: x₁* = 0.3(100)/3 = 10. x₂* = 0.7(100)/7 = 10. The consumer buys 10 units of each good.
Q: Write down the expenditure function for the quasilinear case.
A: e(p, U) = p₂U − p₂²/(4p₁).
This material connects directly to the envelope theorem and duality (covered in the next set of notes). The Hicksian demand is the foundation for compensating variation and equivalent variation, which are welfare measures used in public economics. The Marshallian demand connects to consumer surplus, which you will use when analysing market interventions like taxes and subsidies.
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