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Lecture 4: Consumer Problem

Advanced Microeconomics I

Yosuke YASUDA

National Graduate Institute for Policy Studies

October 22, 2013

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What is Consumer Problem?

Assume % is a consumer’s preference relation on the consumption set X = Rn+ where

Rn+:= {(x1, ..., xn)|xi ≥ 0, i = 1, ..., n} ⊂ Rn.

For any x, y ∈ X, x % y means x is at least as preferred as y.

Consumption set contains all conceivable alternatives.

A budget set is a set of feasible consumption bundles, represented as B(p, ω) = {x ∈ X|px ≤ ω}, where p is an n-dimensional positive vector interpreted as prices, and ω is a positive number interpreted as the consumer’s wealth.

We assume that the consumer is motivated to choose the most preferred feasible alternative according to her preference relation. That is, she seeks

x ∈ B(p, ω) such that x % x for all x ∈ B(p, ω).

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Utility Maximization

Def We refer to the problem of finding the % best bundle in B(p, ω) as the consumer problem.

Function U : X → R represents the preference % if for all x and y ∈ X, x % y if and only if U (x) ≥ U (y).

If U represents a preference relation %, we call it a utility function, and we say that % has a utility representation.

Utility or utility functions are useful since it is often more convenient to talk about the maximization of a numerical function than of a preference relation.

Given utility representation, consumer problem becomes:

x∈B(p,ω)max U(x) or maxx∈X U(x) s.t. x ∈ B(p, ω).

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Remarks on Utility Function

Rm Utility function has NO meaning other than that of representing a preference relation %.

Thm If U represents %, then for any strictly increasing function f : R → R, the function V (x) = f(U(x)) represents % as well.

proof Note that increasing function implies the second step. a% b⇔ U (a) ≥ U (b) ⇔ f (U (a)) ≥ f (U (b)) ⇔ V (a) ≥ V (b).

Q Under what conditions do utility representations exist?

→ Please wait until Lecture 8.

Thm If % is represented by continuous utility function U , then any consumer problem has a solution.

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Monotonicity

Monotonicity says that “more is better (than less).”

Def A preference relation % satisfies monotonicity at the bundle yif for all x ∈ X,

If xk≥ yk for all k, then x % y, and If xk> yk for all k, then x ≻ y.

Monotonicity can be expressed by a increasing utility function U , which is assumed in most of consumer problems.

Thm U (x) is strictly increasing if and only if % is monotonic. Monotonicity says that if one bundle contains at least as much of every commodity as another bundle, then the one is at least as good as the other.

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Inequality Constraints

Consider a two-variable optimization problem in which the only constraint is given by the inequality g(x1, x2) ≥ 0. Formally, our problem is

maxx1,x2

f(x1, x2) s.t. g(x1, x2) ≥ 0.

Let us define Lagrangian function as if the constraint holds with equality.

L = f (x1, x2) + λg(x1, x2).

The optimal solutions must satisfy the following Kuhn-Tucker conditions:

∂L

∂x1 =

∂f

∂x1 + λ

∂g

∂x1 = 0,

∂L

∂x2 =

∂f

∂x2 + λ

∂g

∂x2 = 0. λg(x1, x2) = 0, λ ≥ 0, g(x1, x2) ≥ 0.

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Solving Consumer Problem (1)

Ex Practice: A consumer problem with n goods.

x∈Rmaxn+

u(x) s.t. px ≤ ω.

The corresponding Lagrangian function is:

L = u(x) + λ(ω − px) + λ1x1+ · · · + λnxn.

The Kuhn-Tucker conditions are as follows:

∂L

∂x1 =

∂u

∂x1 − λp1+ λ1= 0. ...

∂L

∂xn

= ∂u

∂xn

− λpn+ λn= 0.

λ(ω − px) = 0, λ ≥ 0, ω − px ≥ 0,

λ1x1 = 0, ..., λnxn= 0, λi≥ 0, xi≥ 0 for all i.

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Solving Consumer Problem (2)

Suppose that solution xi and ∂u(x∂x)

i is strictly positive for all i. Then, the corresponding Lagrangian multipliers λi must be 0 for all i, which implies λ, pi >0 for all i. Therefore, for any two goods j and k, we can combine the conditions to conclude that

∂u(x)

∂xj

∂u(x)

∂xk

= λpj λpk =

pj

pk.

This says that at the optimum, the marginal rate of substitution (MRS) between any two goods must be equal to the ratio of the goods’ prices.

Note that, for two goods case (along the indifference curve)

0 = du = ∂u(x

)

∂x1 dx1+

∂u(x)

∂x2 dx2 ⇐⇒

∂u(x)

∂x1

∂u(x)

∂x2

= −dx2 dx1.

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Indirect Utility Function

To construct the indirect utility function, we fix market prices and “initial wealth,” and seek the maximum level of utility the consumer could achieve.

Def The indirect utility function is the maximum-value function corresponding to the consumer’s utility maximization problem (UMP), and it is denoted by v(p, ω). That is,

v(p, ω) = max

x∈Rn+

u(x) s.t. px ≤ ω, or, equivalently

v(p, ω) = u(x(p, ω))

where x(p, ω) is the solution of the UMP, known as Marshallian demand functions.

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Example of UMP

Ex Cobb-douglas utility function with two goods

max

x∈R2+

xα1x1−α2 s.t. px (= p1x1+ p2x2) ≤ ω. where α ∈ (0, 1).

Rm u(x) = x α1xβ2(α, β > 0) can express the identical preference.

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Expenditure Function

To construct the expenditure function, we fix prices and a “level of utility,” and seek the minimum level of money expenditure the consumer must make to achieve this particular level of utility.

Def The expenditure function is the minimum-value function corresponding to the consumer’s expenditure minimization problem (EMP), and it is denoted by e(p, u). That is,

e(p, u) = min

x∈Rn+

px s.t. u(x) ≥ u, or, equivalently

e(p, u) = pxh(p, u)

where xh(p, u) is the solution of the EMP, known as Hicksian (Compensated) demand functions.

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Example of EMP

Ex Cobb-douglas utility function with two goods

min

x∈R2+

p1x1+ p2x2 s.t. xα1x1−α2 ≥ u where α ∈ (0, 1).

Rm EMP is the mirror image of UMP. This property is formally established as duality (in Lecture 5).

参照

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