Consumer Theory: Utility Functions and Demand

Consumer theory is the mathematical foundation of microeconomics. It explains how a rational agent chooses a bundle of goods, maximizing utility under a budget constraint. This theory provides tools for analyzing changes in demand, evaluating the effects of government policy, and welfare.

Consumer's Problem

The consumer chooses a vector of goods x = (x₁,...,xₗ) ∈ ℝᴸ₊, maximizing utility u(x) under a budget constraint. Formally:

max_{x∈ℝᴸ₊} u(x) subject to p·x ≤ m

Here p = (p₁,...,pₗ) is the price vector, m is monetary income. The constraint p·x ≤ m is a "budget hyperplane" in ℝᴸ.

Assumptions about preferences:

• Completeness: for any x, y: x ≿ y or y ≿ x
• Transitivity: x ≿ y and y ≿ z → x ≿ z
• Continuity: ensures the existence of a utility function
• Strict monotonicity: "more is better"— ∂u/∂xₗ > 0
• Strict convexity of preferences: mixtures are preferred to extremes

Marshallian demand: xˡ(p, m) — solution to the consumer's problem. Properties: (1) homogeneous of degree zero: x(tp, tm) = x(p, m) — no "money illusion"; (2) Walras' identity: p·x(p,m) = m (everything is spent).

Specific Utility Functions

Cobb–Douglas: u(x₁, x₂) = x₁ᵅ x₂^β. Demand: x₁* = αm/((α+β)p₁), x₂* = βm/((α+β)p₂). Expense shares are constant: p₁x₁/m = α/(α+β)—independent of prices. Substitution elasticity = 1.

Linear (perfect substitutes): u = ax₁ + bx₂. Corner solution: if a/p₁ > b/p₂ → x₁* = m/p₁, x₂* = 0. The consumer buys only the more "profitable" good.

Leontief (perfect complements): u = min(x₁/a, x₂/b). Optimum: x₁*/a = x₂*/b (consumed in fixed proportion). Demand: x₁* = am/(ap₁ + bp₂), x₂* = bm/(ap₁ + bp₂).

CES (constant elasticity of substitution): u = [αx₁^{-ρ} + (1-α)x₂^{-ρ}]^{-1/ρ}. Substitution elasticity σ = 1/(1+ρ). ρ → 0: Cobb–Douglas (σ = 1). ρ → ∞: Leontief (σ = 0). ρ → −1: linear (σ → ∞).

Duality and Expenditure Function

Hicksian demand: the dual problem—minimize expenditure at a fixed utility level:

min_{x} p·x subject to u(x) ≥ ū → h(p, ū) (compensated demand)

Expenditure function: e(p, ū) = p·h(p, ū)—minimum expenditure required to reach ū.

Shephard's Lemma: ∂e(p, ū)/∂pₗ = hₗ(p, ū)—the derivative of expenditure with respect to price equals Hicksian demand. Follows from the envelope theorem.

Slutsky equation: connects Marshallian and Hicksian demand:

∂xₗ/∂pₖ = ∂hₗ/∂pₖ − xₖ · ∂xₗ/∂m

Decoding: the change in Marshallian demand with respect to price = substitution effect (∂hₗ/∂pₖ, always ≤ 0 if l = k) + income effect (−xₖ · ∂xₗ/∂m). Giffen good: ∂xₗ/∂pₗ > 0—is possible only with a strong positive income effect (a good below Giffen, a normal good cannot be Giffen).

Numerical Example

u(x₁, x₂) = x₁^{0.5} x₂^{0.5}, m = 100, p₁ = 2, p₂ = 5.

Marshallian demand: x₁* = 0.5·100/2 = 25, x₂* = 0.5·100/5 = 10.

Expenditures: 2·25 + 5·10 = 50 + 50 = 100 ✓.

Now suppose p₁ increases to 4. New demand: x₁** = 0.5·100/4 = 12.5. A decrease of 12.5 units. Substitution effect (Hicksian): need to find e(p', ū*), where ū* = 25^{0.5}·10^{0.5} = √250 ≈ 15.81. At p₁=4: h₁ = ū*·(p₂/p₁)^{0.5}/(2^{0.5}) = 15.81·(5/4)^{0.5}/√2 ≈ 13.98. Substitution effect: 13.98 − 25 = −11.02. Income effect: 12.5 − 13.98 = −1.48.

Applications of the Theory

Assessment of consumer surplus under tax policy. Analysis of benefit systems (subsidies vs cash payments). Econometric estimation of demand systems (AIDS model). Welfare analysis: CV and EV of price changes.

Assignment: A consumer with u(x₁,x₂) = x₁^{0.4}x₂^{0.6}, m=200, p₁=4, p₂=5. (1) Find the Marshallian demand. (2) Compute the indirect utility function V(p,m). (3) Compute e(p,ū) via Shephard's Lemma. (4) When p₁ is reduced to 2: decompose the change in demand for x₁ into substitution and income effects (Slutsky equation). Which effect is greater?