Chapter 2: Preference Theory -...

Preference Relations Utility Maximization Expenditure Minimization Utility Max. and Expenditure Min. Integrability Welfare

Chapter 2: Preference TheoryAdvanced Microeconomics I

Andras Niedermayer1

1Department of Economics, University of Mannheim

Fall 2009

Chapter 2: Preference Theory Fall 2009 1 / 22


2.2 Preference Relations

Definition: A preference relation on the set X is a binaryrelationship, denoted �, where x � y reads “x is at least asgood as y”.



Standard assumptions on preferences in economics:

Completeness: For all x , y ∈ X , either x � y or y � x , orboth.

Transitivity: For all x , y , z ∈ X , if x � y and y � z, thenx � z.

Local non-satiation: Suppose the X has a metric. For allx ∈ X and all ε > 0, there is a y ∈ X such that ‖y − x‖ < ε

and y ≻ x .

Convexity: For all x ∈ X , the upper contour set{x ′ ∈ X ; x ′ � x} is convex.

Continuity: For all x ∈ X , the contour sets {x ′ ∈ X ; x ′ � x}and {x ′ ∈ X ; x ′ � x} are closed sets.



Two special classes of preferences:

Homotheticity: A preference relation is homothetic if x ∼ yimplies λx ∼ λy for all λ > 0.

Quasi-linearity: A preference relation on X ⊂ RH is

quasi-linear with respect to commodity 1 if x ∼ y impliesx + λe1 ∼ y + λe1 for all x , y , x + λe1, y + λe1 ∈ X andλ ∈ R.



Proposition 2.1

If � is complete, transitive, and continuous, then � can berepresented by a continuous utility function, i.e., there exists acontinuous function u : X → R such that for anyx , y ∈ X , x � y ⇐⇒ u(x) ≥ u(y).



2.3 Utility Maximization

Given a budget b and price p ≫ 0, the consumer solves

maxx

u(x)

s.t. x ∈ X (UM) (1)

p · x ≤ b

Definition: The set of all solutions of (UM) for a given (p, b) isdenoted ϕ(p, b), and ϕ is called the Marshallian demandcorrespondence. For x ∈ ϕ(p, b), v(p, b) = u(x) is called theindirect utility function.



Proposition 2.2

If � is locally non-satiated and u continuous on X = RH+, then

(Hϕ) ϕ(λp, λb) = ϕ(p, b) for all p, b and λ > 0,

(BI) p · x = b for all x ∈ ϕ(p, b),

(UHC) ϕ is upper hemi-continuous at all p, b ≫ (0, 0),

(Hv) v(λp, λb) = v(p, b) for all p, b and λ > 0,

(Mv) v is strictly increasing in b and non-increasing inph for all h,

(Qv) The set {(p, b) : v(p, b) ≤ v} is convex for all v ,

(Cv) v is continuous.



2.4 Expenditure Minimization

The dual problem to that of utility maximization is

minx

p · x

s.t. x ∈ X (EM) (2)

u(x) ≥ u

Definition: The set of all solutions of (EM) for a given (p, u) isdenoted h(p, u), and h is called the Hicksian demandcorrespondence. For x ∈ h(p, u), e(p, u) = p · x is called theexpenditure function.



Proposition 2.3


(Hh) h(λp, u) = h(p, u) for all p, u and λ > 0,

(UHCh) h is upper hemi-continuous at all (p, u) ≫ (0, 0),

(He) e(λp, u) = λe(p, u) for all p, u and λ > 0,

(Me) e is strictly increasing in u, and non-decreasing inph for all h,

(Conc e) e is concave in p,

(Ce) e is continuous.



Proposition 2.4

(“Shephard’s Lemma”): If � is locally non-satiated and strictlyconvex, and u continuous on X = R

H+, then e is differentiable

almost everywhere, and

∂pe(p, u) = h(p, u)T for almost all (p, u) ≫ (0, u(0)).

Proof.

Fix (p0, u0). For every p define Z (p) = p · h(p0, u0) − e(p, u0).Total expenditure under (p, u0) is minimized by h(p, u0), henceZ (p) ≥ 0 for all p. Furthermore, Z (p0) = 0.Hence, if Z is differentiable w.r.t. p at p0, then

∂pZ (p0) = h(p0, u0)T − ∂pe(p0, u0) = 0.

Since e is concave, it is differentiable at almost every p0, andso is Z .



Remarks:

Price change has two impacts on expenditure: a directeffect of the order of magnitude h(p, u), and an indirecteffect through the induced change in demand. At theoptimum of the (EM) problem, this indirect effect is ofsecond order.

Standard, less general, proof: assume that e iscontinuously differentiable and use the envelope theorem:At (p, u) with h(p, u) ≫ 0

∂pe(p, u) = ∂p(p · h(p, u)) = h(p, u)T + (p · ∂ph(p, u))T .

Since the utility constraint in the EM problem binds by thecontinuity of u and by local non-satiation, one has byLagrange p − λ∂xu(x) = 0 for a λ ≥ 0. Hence,differentiating the identity u(h(p, u)) = u shows thatp · ∂ph(p, u) = 0.



Proposition 2.5

f � is locally non-satiated and strictly convex, u continuous onX = R

H+, and h continuously differentiable, then for all

(p, u) ≫ (0, u(0))(i) ∂ph(p, u) is symmetric,(ii) ∂ph(p, u) is negative semi-definite,(iii) ∂ph(p, u)p = 0.

Proof.

Proposition (2.4) ⇒ ∂ph(p, u) = ∂2ppe(p, u). h is continuously

differentiable ⇒ e is twice continuously differentiable.Hence(“Schwarz’s Theorem” in calculus), ∂2

ppe(p, u) is symmetric.Since e is concave in p (Proposition 2.3), ∂2

ppe(p, u) isnegative-semidefinite. Differentiating the identitiyh(λp, u) = h(p, u) with respect to λ and setting λ = 1 yields(iii).



Interpretation: (∂ph(p, u))ij : marginal change of consumptionof good i with change of price of good j and constant utility u.⇒ ∂ph(p, u) similar to Slutsky matrix



2.5 The Relationship Between Utility Maximization andExpenditure Minimization

Proposition 2.6:


for all u > u(0), b > 0, p ∈ RH+,

(i) ϕ(p, b) = h(p, v(p, b))(ii) h(p, u) = ϕ(p, e(p, u))(iii) e(p, v(p, b)) = b(iv) v(p, e(p, u)) = u.

Proof.

For all p ≫ 0, staightforward use of the definitions (includingcontinuity of u and local non-satiation). For p on the boundary,continuity. (Exercise)



Proposition 2.7

If � is locally non-satiated and strictly convex, u continuous onR

H+, and ϕ is differentiable on R

H++, then h is differentiable at

almost every (p, u) ≫ (0, u(0)) and

∂ph(p, v(p, b)) = ∂pϕ(p, b) + ∂bϕ(p, b)ϕ(p, b)T .

Proof.

Differentiating (ii) in Proposition 2.6 with respect to p yields, for(p, u) ≫ (0, u(0))

∂ph(p, u) = ∂pϕ(p, e(p, u)) + ∂bϕ(p, e(p, u))∂pe(p, u)

= ∂pϕ(p, e(p, u)) + ∂bϕ(p, e(p, u))h(p, u)T ,

by Proposition 2.4.Since the budget b = e(p, u) allows toachieve u = v(p, b) by Proposition 2.6 (iii), the proof iscomplete.



Result : ∂ph, at u = v(p, b), is just the Slutsky matrix ofProposition 1.3. This is somewhat intuitive from what has beensaid before, but not trivial.



Proposition 2.5 ⇒ Slutsky matrix symmetric

⇒A Marshallian demand system with asymmetric Slutskymatrix cannot be derived from preference-basedoptimization.



2.6 Integrability

utility maximization →Marshallian demand

Question : Marhallian demand →utility maximization

Immediate : Marshallian demand has to satisfyPropositions 2.2-2.7

Goal : recover utility function from Marshallian demand

Best way to proceed: through the expenditure function



From Propositions 2.4 and 2.6:

∂pe(p, u0)T = ϕ(p, e(p, u0)) (3)

This is a system of differential equations in p for theunknown e with parameter u0.Choose p0, b0 rather than u0, take x0 = ϕ(p0, b0) andu0 = u(x0).Use

e(p0, u0) = p0 · ϕ(p0, b0) (4)

as an initial condition for (3).The theory of partial differential equations now implies that(3) has a solution, locally around p0, if and only if theSlutsky matrix, ∂pϕ + ∂bϕϕT , is symmetrice concave iff Slutsky matrix negative semi-definite(⇔(WARP))



(BI), (H), (WARP), and symmetry of the Slutsky matrix of ϕ areequivalent to preference optimization



Recovering preferences from observed date is useful forempirical work. Examples

When is two-stage budgeting (Chapter 1.5) valid?

Demand system ϕ separable if utility function separable(u(x1, ..., xH) = F (u1(q1), ..., uL(qL)), where the qi arecommodity vectors)In order for each commodity group to have one single groupprice index (determination of group budgets bi in step 1 ofthe process), it is sufficient (and almost necessary) thatintra-group preferences are homothetic

Preferences of a consumer with linear Engel curves(important for aggregation)? →“Gorman polar form":

e(p, u) = α(p) + uβ(p)

where α and β concave and homogenous of degree 1.



2.7 Welfare

Does a given price/budget manipulation make the (hopefullyrepresentative) household “better” or “worse” off?see Mas-Colell, Whinston, Green, Ch. 3.I and the exercises


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