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Competitive Equilibrium with Two-Tiered LivingStandards: A Conceptual Framework for Poverty

(Very Preliminary. Please Do Not Circulate)

Dong Chul Won

Ajou University

The Center for Distributive JusticeSeoul National University

April 17, 2017

D. Won (AU) Poverty in Equilibrium 17/12/2010 1 / 28

Introduction

Poverty in Reality

Poverty prevails even in OECD countries.The official poverty rate of U.S. was 14.8 (46.7 million people) in2014.

I The poverty threshold is a minimum income level below which peopleis officially considered to have insufficient access to basic necessities forsubsistence (such as food, shelter, clothing, and utilities).

I The U.S. Census Bureau reports that the poverty threshold for4-member family was $24,230 (annual income) in 2014..

(World Bank) The new global poverty line was set at $1.90 using2011 prices. Just over 900 million people globally lived under this linein 2012.

How to define poverty may be a normative issue. (Where to draw thepoverty line in the wealth map for income redistribution is a policyissue.)

Since the purchasing power and labor supply of the poor affects goodsprice and wage, they cannot be omitted from equilibrium analysis.


Introduction

Poverty and Equilibrium Theory: Static View

The poverty threshold is measured as a minimum expenditure forsubsistence and thus, depends on the prices of consumptions.

The poverty threshold for an individual is the minimum expenditureover the set of consumptions which allow him to maintain at least an‘acceptable’ quality of human life.

It depends on both the shape of the consumption set and theminimum standard of life.

The single-period approach to the poverty threshold can be analyzedin the Arrow-Debereu general equilibrium model.


Introduction

Poverty and Equilibrium Theory : Dynamic View

Poverty must be understood in an intertemporal context.

The primary determinant of poverty will be the size of life-timeincome.

Poor talented youth can generate huge life-time income.

There must be a mechanism by which the talented poor can havepresent access to a certain portion of life-time income to financeinvestment for his income potential.

Financial markets play an important role for smoothing consumptionsover the life-time periods.

Poverty will more prevail in a financially-underdeveloped society wherethe poor youth cannot invest in fulfilling his life-time income potential.


Introduction

How much valid the Arrow-Debreu Model in the RealWorld?

The Arrow-Debreu model is static and thus, assumes implicitlycomplete markets.

For instance, human capital and labor are traded in the Arrow-Debreumodel without any restrictions. (We are used to include them in thestatic budget constraint without any doubt.)

However, moral-hazard-ridden human capital is not traded in the realworld.

Asset markets are incomplete as far as human capital remains anon-traded asset.

Since human capital is a major source of life-time income, its currentmarketability is a major determinant of poverty.


Introduction

Consumption Set in the Arrow-Debreu World

Arrow and Debreu (1954) state that “The set Xi (consumption set foragent i) includes all consumption vectors among which the individualcould conceivably choose if there were no budgetary restraints.Impossible combinations of commodities, such as ... the consumptionof a bundle of commodities insufficient to maintain life, are regardedas excluded from Xi .”

The consumption set is determined by the standard for biologicalsubsistence.

However, the consumption set falls into the domain of the valuejudgement if what human life should be is built into the minimumliving standard.


Introduction

Can People Self-Subsist?

The classical general equilibrium model assumes that agents can liveon the initial endowments to discuss the existence of competitiveequilibrium.

The self-subsistence assumption, however, fails in the modern societywhere people cannot usually live on their own initial endowmentswithout participating in market exchange.

The problem is pointed out in Sen (1981) saying “· · · but it is not thecase that, say, barbers, or shoemakers, · · · , or even doctors orlawyers, can survive without trading.”.

The departure of general equilibrium theory from the classicalself-subsistence assumption is initiated in McKenzie (1959).


Introduction

Cheaper Point and Classical GE

A ‘cheaper point’ in the consumption set is a consumption cheaperthan the initial endowments.

The presence of a cheaper point is necessary for theupper-semicontinuity of demand correspondences in general. (SeeDebreu (1959).)

Thus, the cheaper point (CP, in short) condition is required to provethe existence of equilibrium.

Two types of the CP condition in the literatureI Debreu (1959, 1962) adopts the interiority condition to guarantee a

cheaper consumption point for each agent.I McKenzie (1959) introduces the irreducibility condition. It insures that

if someone has a CP, so does everyone.

The irreducibility condition is more general and useful especially inthe context of infinite-dimensional choice problem.


Introduction

The Pitfall of Demand Function at the Cheaper Point

The continuity of demand function on the price set requires thepresence of cheaper points in the relative interior of the consumptionset.

However, the interiority condition is unrealistic in the capitalistsociety where agents cannot subsist under autarky.

Our initial endowments of commodities are located outside theconsumption set. (Do you produce a smartphone for youself?)

The interiority condition may fail intrinsically in theinfinite-dimensional consumption set such as L2+ which has the emptyinterior.

Consequently, the demand function approach to poverty isself-contradictory because the poor endowments do not lie in theinterior of the consumption set.


Introduction

Motivation for the current research

The existing literature is not satisfactory in explaining poverty inequilibrium.

McKenzie (1981) extends the irreducibility condition to the case thatagents cannot subsist on their initial endowments but fails to explainpoverty in equilibrium.

Thus, the poverty threshold cannot be discussed in the currentgeneral equilibrium framework.


Introduction

Contribution of the Paper

This paper introduces the notion of the usefulness of poorendowments by incorporating the poverty threshold into theirreducibility condition.

The poverty threshold issue is addressed from both positive andnormative viewpoints by developing a two-tiered system ofconsumption sets.


Introduction

Literature on Survival Conditions

Interiority condition: Debreu (1959, 1962)

Irreducibility condition:I McKenzie (1959, 1981, 2002)I Arrow and Hahn (1971)I Moore (1975)I Gottardi and Hens (1996) : incomplete marketsI Florig (2001)I Florenzano (2003): infinite-dimensional consumption sets


Introduction

Notation

Exchange economy E = ((Xi ,Yi ),PYi , ei )i∈I where

I I = {1, 2, . . . ,m} is the set of agents.I Yi ⊂ R` is the ‘positive’ consumption set for agent i .I Xi ⊂ R` is the ‘normative’ consumption set for agent i .I PY

i indicates the preference ordering of agent i over Yi .I ei is the initial endowment of agent i .

X =∏

i∈I Xi ; X0 =∑

i∈I Xi .

x = (x1, . . . , xm) ∈ X ; e0 =∑

i∈I ei .

B◦i (p) = {xi ∈ Xi : p · xi < wi (p) = p · ei}.Bi (p) = {xi ∈ Xi : p · xi ≤ p · ei}.

I In the production economy, the wealth w(p) is expressed aswi (p) = ei · p + si maxz∈Z p · z where si is the share of agent is and Zis the aggregate production possibility set.

AY = {x ∈ X :∑

i∈I xi = e0 and ei 6∈ PYi (xi ) ∀i ∈ I}.


Introduction

Three Norms on the Poverty Threshold

A desired level of monetary income mi

Xi = Xmi = {xi ∈ Yi : p∗ · xi ≥ mi ,

where p∗ is an equilibrium price.

A desired level of physical consumptions

Xi = X ci ≡ {xi ∈ Yi : fi (xi ) ≥ 0},

where xi with f (xi ) = 0 is a desired subsistence consumption.

A desired level of welfare measured in utility

Xi = X ui ≡ PY

i (xui ),

where xui is a consumption which meets the desired minimum welfare.


Introduction

Desired Subsistence Consumption and Minimum UtilitySubsistence Consumption makes EIS depend on consumption choices.If it is desired for agent i to have a subsistence consumption c i fromthe welfare perspective, X u

i is expressed as

X ui = PY

i (c i ).

If preferences are represented by an expected utility function vi , hisutility function ui is defined as

ui (x ; c i ) = E [vi (x − c i )].

In this case, X ui = {xi ∈ Yi : xi ≥ c i )}.

I Atkeson and Ogaki (1996), Steger (2000)

ui (xi ) =∑j=1

φj1− αj

((x ij − c ij)

1−αj − 1)

I Stone-Geary utility function: for instance, Varian (1992) and King andRebelo (1993)

vi (xt/nt) = at ln((xt/nt)− c i

).


Introduction

Induced Preferences on Xi

The set PYi (xi ) contains consumptions in Yi preferred to xi .

We define the preference ordering Pi on the normative consumptionset Xi by restricting PY

i to Xi such that for each xi ∈ Xi ,

Pi (xi ) = PYi (xi ) ∩ Xi .

In the case that Xi = X ui , it holds that for all x ′i ∈ Yi \ Xi ,

Pi (xi ) ⊂ PYi (x ′i ). (1)

That is, consumptions which do not meet the normative livingstandards are less preferred to consumptions in Xi .

If the property (1) holds, competitive equilibrium with the normativeconsumption sets is also competitive equilibrium with the positiveconsumption sets.

However, the property (1) may not be kept in the case that Xi = X ci .


Introduction

Definitions of Equilibrium

A quasiequilibrium for E is a pair (p, x) ∈ (R` \ {0})× Y such that

(i) xi ∈ Bi,Y (p) for all i ∈ I ,(ii) PY

i (xi ) ∩ B◦i,Y (p) = ∅ for all i ∈ I , and(iii)

∑i∈I (xi − ei ) = 0

The pair (p, x) is a normative quasi-equilibrium if it is aquasi-equilibrium with x ∈ X .

A competitive equilibrium for E is a pair (p, x) ∈ (R` \ {0})× Y suchthat

(i) xi ∈ Bi,Y (p) for all i ∈ I ,(ii) PY

i (xi ) ∩ Bi,Y (p) = ∅ for all i ∈ I , and(iii)

∑i∈I (xi − ei ) = 0

The pair (p, x) is a normative competitive equilibrium if it is acompetitive equilibrium with x ∈ X .

As mentioned earlier, normative competitive equilibrium need not bea competitive equilibrium.


Introduction

Poverty Threshold

For any nonzero price p in R`+, we define the minimum income

functions

mi (p) = infxi∈Xi

p · xi and mYi (p) = inf

xi∈Yi

p · xi .

The income mi (p) is the poverty threshold to normative consumptionswhile mY

i (p) is the poverty threshold to positive consumptions.


Introduction

Assumptions

We make the following assumptions

A1. Each Xi is closed and convex.

A2. For every i ∈ I , PYi (xi ) 6= ∅ and xi 6∈ PY

i (xi ) for all xi ∈ Yi , andPi (xi ) 6= ∅ for all xi ∈ Xi .

A3. Let xi be a point in Yi . Then for each yi ∈ PYi (xi ) and zi ∈ Yi , there

exists α ∈ (0, 1) such that αyi + (1− α)xi ∈ PYi (xi ).

A4. For every i ∈ I and every nonzero p ∈ R`+, there exists x i (p) ∈ Xi

such that mi (p) = p · x i (p).

A5. The aggregate endowment e0 is in the interior of∑

i∈I Xi .


Cheaper Point Conditions

Existing Cheaper Point Conditions

Interiority Condition (Debreu): For each i ∈ I , ei ∈ int Xi .

(McKenzie) The economy E is said to be irreducible if, for everyx ∈ A and for every nontrivial partition {I1, I2} of I , there exists somek ∈ I1 and α > 0 such that e0 ∈ int X0 and

e0 ∈∑i∈I1

Pi (xi ) +∑i∈I2

[α (Xi − {ei}) + {xi}] .

(The economy is irreducible if for each partition of all the agents into twogroups, one group of agents can improve their welfare by taking not only theexisting net trade but also some new net trade of the other group.)



Continued

(Florig (2001)) The economy E is said to be weakly irreducible if, forevery x ∈ A and for every nontrivial partition {I1, I2} of I , there existsome k ∈ I1 and αi > 0 for all i ∈ I such that e0 ∈ int X0 and

0 ∈ αk [Pk(xk)−{ek}]+∑

i∈I1\{k}

αi [cl Pi (xi )−{ei}]+∑i∈I2

αi (Xi − {ei}) .

The economy is weakly irreducible if it is irreducible.



Irreducibility and Usefulness

Let’s look at irreducibility from a different angle. By irreducibility,there exists zi ∈ Xi for each i ∈ I2 such that

e0 =∑i∈I

xi ∈∑i∈I1

Pi (xi ) +∑i∈I2

[α{zi − ei}+ {xi}] .

It is rearranged as∑i∈I1

xi + α∑i∈I2

{ei − zi} ∈∑i∈I1

Pi (xi ).

Then the consumption α∑

i∈I2(ei − zi ) is useful for agents in I1 inthat it improves their welfare.

By irreducibility, agents have useful trades to each other. Moreover,every agent has cheaper points in equilibrium. Thus, no poverty existsin equilibrium.



Usefulness of Poor Endowments

Notation: For an allocation x ∈ A and a nonzero p in R`+, we define

sets of agents

I0(p, x) = {i ∈ I : p · ei = mi (p) and x ′i �i xi implies p · x ′i ≥ p · xi},I (p, x) = I \ I0(p, x),

I0(p, x) = {i ∈ I0(p, x) : x ′i �i xi implies p · x ′i > p · xi}, and

I (p, x) = I0(p, x) \ I0(p, x).

Agent i in I0(p, x) would live on the poverty threshold with theexpenditure-minimizing choice xi if (p, x) were a competitiveequilibrium.

Agent i in I0(p, x) would live on the poverty threshold with theoptimal consumption choice xi if (p, x) were a competitiveequilibrium.



Usefulness of Poor Endowments

A6. Usefulness of poor endowmentsFor a pair (p, x) in (R`

+ \ {0})× AIR with I0(p, x) 6= ∅, there existx ′i ∈ Xi and αi > 0 for each i ∈ I0(p, x) such that∑

i∈I (p,x)

xi +∑

i∈I0(p,x)

αi (ei − x ′i ) ∈∑

i∈I (p,x)

Pi (xi ). (2)

I Assumption A6 states that if I0(p, x) 6= ∅, the poor in I0(p, x) has auseful trade at p for the relatively rich in I (p, x).

I It will play a critical role for the existence of normative competitiveequilibrium with possible poverty.

I It allows us to check that competitive markets are viable undernormative poverty standards.

A6s. Strong Usefulness of poor endowmentsThis is a strong version of A6 obtained by replacing “withI0(p, x) 6= ∅” by “with I0(p, x) 6= ∅” in the statement of (A6).


Main Results

The Existence of Normative Competitive Equilibrium

Theorem 1. Normative competitive equilibrium exists if the economysatisfies Assumptions 1 – 6. If E is weakly irreducible, then it is viable.

Corollary 1. Let (p, x) denote a normative competitive equilibrium.If Pi (xi ) = PY

i (xi ) for each i ∈ I , it is a competitive equilibrium forthe economy E .

Theorem 1 states that normative competitive equilibrium exists in theeconomy where the poor endowments can provide useful trades forthe relatively rich.

Existential failure may arise in the economy where poor endowmentsare not useful to the class of relatively rich agents.


Examples

Example 1: Normative Equilibrium with Poverty

The economy has two agents with homothetic preferences.I Y1 = Y2 = R2

+.I ui (ai , bi ) : a homothetic utility function for each i = 1, 2 which is

strictly increasing, strictly convex and differentiable in R2+.

I C 1 = C 2 = (1, 1). X1 = X2 = {(a, b) ∈ R2+ : a ≥ 1, b ≥ 1}.

I e1 = (3, 0), e2 = (0, 3) and thus, ei 6∈ Xi .I Assume that for each a > 0, there exists λi > 0 such that for each

i = 1, 2D ui (a, a) = λi (2, 1).

The economy has a unique equilibrium with p∗ = (2/3, 1/3),x1 = (2, 2) and x2 = (1, 1).

I Agent 2 subsists on the poverty threshold with m2(p∗) = 1.I Agent 2 is poor because he is endowed with cheaper good (good 2).I Assumption A6 holds here but all the existing survival conditions for

normative competitive equilibrium fail here because agent 2 has nocheaper consumption that e2.


Examples

Example 1: In-Kind Transfers

Suppose that the society calls fro a public policy to rescue agent 2from poverty.

One conceivable policy option is to conduct an an in-kind transfer(εa, εb) from agent 1 to agent 2.

Let E t denote the post-transfer economy where the initialendowments are given by et1 = (3− εa,−εb) and et2 = (εa, 3 + εb).

The post-transfer economy satisfies A6s, i.e., it has pover-freenormative equilibrium if the transfer (εa, εb) lies in the set

F ={

(εa, εb) ∈ R2 : 0 < 2εa + εb < 3}

∩{

(εa, εb) ∈ ([0, 1)× (−1, 0]) ∪ ((2, 3]× [−3,−2))}


Conclusions

Concluding Remarks

The paper provides a general equilibrium model for the povertythreshold from both positive and normative viewpoints.

The normative approach to poverty is built on the notion of normativecompetitive equilibrium in which agents are allowed to make optimalconsumption choices which meet desired minimum living standards.

It presents the usefulness of poor endowments as a new cheaper-pointcondition to show the existence of normative competitive equilibriumwith possible poverty.Promising research topics

I Specialization of the general results in a testable or computable GEmodel with concrete utility functions developed in the literature onpoverty.

I Search for testable implications of the usefulness of poor endowments.I Unified approach to the poverty threshold.I When can the imposition of normative living standards improve the

welfare of agents who are poor in competitive equilibrium?I Extension of Assumption A6 to incomplete-market economies.


Figures for presentation

Cheaper point and endowment

Consumption set

Endowment

Example 1: Existence of Normative Equilibrium

3

2

1

0 1 2 3

Welfare-worsening norm for the poor

3

2

1

0 1 2 3

Competitive eq’m

Normative eq’m

Normative consumption set for agent 1

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