Topics in Consumer Theory

#economics #micro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Koh, Youngwoo.

Integrability

Theorem (Integrability condition).

is generated by rational , if and only if we have

  1. Homogeneity: .
  2. Walras' law: .
  3. is symmetric and negative semidefinite.

We have already shown the 'if' part throughout the Basic Consumer Theory. Now we want to show the 'only if' part: If satisfies properties 1-3, then are there a rational preference that generates ?

WTS:

  • Recover from .
  • Recover from .
Remark (Recover expenditure from Marshallian).

Let be a single valued and HOD0, and satisfying Walras' law. If is symmetric and n.s.d, then the expenditure function can be driven as

Proof. Let and be fixed.
then by Basic Consumer Theory > Proposition 35 (Shephard's lemma), we have the system of partial differential equations of with initial condition .
By the Introductory Analysis > Theorem 42 (Frobenius' Theorem), the system has a solution if and only if the derivative matrix of the right hand side is symmetric, i.e. is symmetric. □

Example (recover expenditure from Marshallian).

Let and a consumer's demand is summarized by

Proof. note that the demand function satisfies the integrability conditions of Theorem 1 (Integrability condition).
First we differentiate with respect to each , then we have the system of partial differential equations: by taking log for each , we have which implies the same function of .
therefore, by integrating the sum, we have thus we have therefore we have the expenditure function. □

Proposition (Recover preference from expenditure).

Let be strictly increasing in , and differentiablly continuous in , increasing in , HOD1 in , concave in .
Then for each utility , is an upper contour set in the sence that where

Proof. WTS .
WTS#1: .
Remark Basic Consumer Theory > Definition 28 (Expenditure Function): then by Def of : is nonempty, closed, and bounded below.
since , there exists .

WTS#2:
() since for any , we have . i.e. thus we have .

() we use the property that is concave and HOD1 in .
since is HOD1 in , by Introductory Analysis > Theorem 7 (Euler's theorem), we have which is same result from Basic Consumer Theory > Proposition 35 (Shephard's lemma).
next, using concavity in , for any , we have where .
thus we have for all , which means . therefore, we have which completes the proof. □

Pasted image 20240327214000.png

  • for given utility function , yields the smallest convex set containing the upper contour set .
  • however, may differ if is not convex.

Welfare Evaluation

ASM:

  • is rational, continuous, and locally nonsatiated.
  • are differentiable.
Definition (welfare measurment).

for any price change , the consumer welfare is worse off if and better off if .

Since welfare is not quantifiable, we can expressed in the monetary term: for a fixed .

Definition (EV and CV).

Let and the price change . note that we have , and . then Equivalent Variation(EV) and Compensating Variation(CV) are defined as

Tip
  • EV: the income adjustment required to obtain at .
  • CV: the income adjustment required to obtain at .

from the definition, we have , . therefore we have the relationship of Pasted image 20240328142635.png

Remark (EV, CV and CS).

Suppose that while . Then since , and , we have and the change in consumer's surplus is defined as where and holds by the Basic Consumer Theory > Proposition 32 (duality) argument.

Pasted image 20240328144713.png

  • If Good1 is normal, then
  • If Good1 is inferior, then
  • If , then there is no wealth effect. then we have
Example (taxation deadweight loss).

Suppose the government imposes a tax on good to raise the revenue . If the pre-tax price vector is given as , and the post-tax price is .

Proof.Note that ,
DWL at : DWL at : this holds as long as is strictly decreasing in □

Pasted image 20240328145415.png

Revealed Preference

Remark the Def of WARP(Weak Axiom of Revealed Preference)

WARP implies that if the consumer choose even if is also consumable, then can be chosen only if is not available.

만약 을 선택할 때 도 선택 가능했다면 (), 을 선택하는 경우는 가 선택 불가능한 때이다().

Pasted image 20240328165608.png

Theorem (WARP HOD0 & n.s.d. Slutsky).

Let be a differentiable. If satisfied WARP and budget balance(Walras' law), then it satisfies the homogeneity of degree zero and negative semidefiniteness of the Slutsky matrix.

Proof.
WTS#1 HOD0: .
let and . then by the Walras' law, we have therefore we have .
RTA suppose . then by WARP, must hold.
However, from the previous equation, we have which leads to a contradiction.

WTA#2 n.s.d. of Slutsky matrix.
fix , and let . then, for any and , the Walras' law implies that by WARP,
and if , we have . thus we have .

then we have letting , where and , we therefore have which means that for arbitrarily small s.t. , we can define the function as that maximized at , where .
since is maximized at , we have or thus is negative semidefinite. □

However, WARP is not a sufficient condition for the utility maximization.

Example (WARP violates transitive).

Consider following data:

Proof.

  • , and , thus .
  • , and , thus .
  • , and , thus .

while satisfying WARP, the data violates transitivity. □

Definition (SARP).

For any list , if we have then we must have . such system satisfies Strong Axiom of Revealed Preference (SARP).

Definition (GARP).

For any list , if we have then we must have . such system satisfies Generalized Axiom of Revealed Preference (GARP).

  • note that SARP is stronger than WARP while only slightly stronger than GARP.
Example (GARP example).

Let , if then by GARP, we have

Tip

가 에서 가능하고, 가 에서 가능하다면, 을 선택하는 경우는 이 불가능한 경우이다.

Theorem (Afriat's Theorem).

A finite set of observations satisfies GARP if and only if there is a strictly monotonic, continuous, and concave utility function s.t.

Tip

Under GARP, the optimized demand is always realized in the budget boundary.

Proof.
() ASM GARP holds. WTS strict monotone, continuous, and concave s.t. for all .
By GARP, if i.e.

then we have WTS#1: for every , there exists some s.t. where and .
Define then is continuous, strictly monotonic, and concave.

WTS#2: for all .
Let and . then we have RTA if , then for some , we have thus violating the definition of .

Now we show that , then for any s.t. , we have

() the if part is straightforward. □

Aggregate Demand

Definition (Aggregate Demand).

Let consumers with and for . given and wealth , the aggregate demand is

Note that AD must be identical for any distributions if the total wealth is the same.

Definition (p.s.w.).

All consumer's wealth expansion paths are parallel and straight (p.s.w.) if for any distributions , s.t. , we have i.e. for any s.t. , we have which is equivalent to thus the wealth effect must be the same across consumers.

Proposition (Gorman From Indirect Utility Function).

A representative consumer in the above sense exists if and only if every consumer has following form of indirect utility function:

Proof. We only prove the () part. WTS that such representative consumer exists.

ASM the indirect utility function is given as then by the Basic Consumer Theory > Theorem 42 (Roy's Identity), we have where thus, the aggregate demand for good is which can be generated by the indirect utility function of a representative consumer, where . □

Example (gorman form indirect utility function).
  1. Homothetic utility function: , where is strictly increasing function and is HOD1 function, then the indirect utility function is
  2. Quasi-linear utility function: , then the indirect utility function is
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Table Of Contents
Topics in Consumer Theory
Integrability
Welfare Evaluation
Revealed Preference
Aggregate Demand