Competitive Equilibrium with Two-Tiered Living
Standards: A Conceptual Framework for Poverty
(Very Preliminary. Please Do Not Circulate)
Dong Chul Won
Ajou University
The Center for Distributive Justice
Seoul National University
April 17, 2017
D. Won (AU)
Poverty in Equilibrium
17/12/2010
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Introduction
Poverty in Reality
Poverty prevails even in OECD countries.
The official poverty rate of U.S. was 14.8 (46.7 million people) in
2014.
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The poverty threshold is a minimum income level below which people
is officially considered to have insufficient access to basic necessities for
subsistence (such as food, shelter, clothing, and utilities).
The U.S. Census Bureau reports that the poverty threshold for
4-member family was $24,230 (annual income) in 2014..
(World Bank) The new global poverty line was set at $1.90 using
2011 prices. Just over 900 million people globally lived under this line
in 2012.
How to define poverty may be a normative issue. (Where to draw the
poverty line in the wealth map for income redistribution is a policy
issue.)
Since the purchasing power and labor supply of the poor affects goods
price and wage, they cannot be omitted from equilibrium analysis.
D. Won (AU)
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Introduction
Poverty and Equilibrium Theory: Static View
The poverty threshold is measured as a minimum expenditure for
subsistence and thus, depends on the prices of consumptions.
The poverty threshold for an individual is the minimum expenditure
over 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 the
minimum standard of life.
The single-period approach to the poverty threshold can be analyzed
in the Arrow-Debereu general equilibrium model.
D. Won (AU)
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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-time
income.
Poor talented youth can generate huge life-time income.
There must be a mechanism by which the talented poor can have
present access to a certain portion of life-time income to finance
investment for his income potential.
Financial markets play an important role for smoothing consumptions
over the life-time periods.
Poverty will more prevail in a financially-underdeveloped society where
the poor youth cannot invest in fulfilling his life-time income potential.
D. Won (AU)
Poverty in Equilibrium
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Introduction
How much valid the Arrow-Debreu Model in the Real
World?
The Arrow-Debreu model is static and thus, assumes implicitly
complete markets.
For instance, human capital and labor are traded in the Arrow-Debreu
model without any restrictions. (We are used to include them in the
static budget constraint without any doubt.)
However, moral-hazard-ridden human capital is not traded in the real
world.
Asset markets are incomplete as far as human capital remains a
non-traded asset.
Since human capital is a major source of life-time income, its current
marketability is a major determinant of poverty.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Consumption Set in the Arrow-Debreu World
Arrow and Debreu (1954) state that “The set Xi (consumption set for
agent i) includes all consumption vectors among which the individual
could conceivably choose if there were no budgetary restraints.
Impossible combinations of commodities, such as ... the consumption
of a bundle of commodities insufficient to maintain life, are regarded
as excluded from Xi .”
The consumption set is determined by the standard for biological
subsistence.
However, the consumption set falls into the domain of the value
judgement if what human life should be is built into the minimum
living standard.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Can People Self-Subsist?
The classical general equilibrium model assumes that agents can live
on the initial endowments to discuss the existence of competitive
equilibrium.
The self-subsistence assumption, however, fails in the modern society
where people cannot usually live on their own initial endowments
without participating in market exchange.
The problem is pointed out in Sen (1981) saying “· · · but it is not the
case that, say, barbers, or shoemakers, · · · , or even doctors or
lawyers, can survive without trading.”.
The departure of general equilibrium theory from the classical
self-subsistence assumption is initiated in McKenzie (1959).
D. Won (AU)
Poverty in Equilibrium
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Introduction
Cheaper Point and Classical GE
A ‘cheaper point’ in the consumption set is a consumption cheaper
than the initial endowments.
The presence of a cheaper point is necessary for the
upper-semicontinuity of demand correspondences in general. (See
Debreu (1959).)
Thus, the cheaper point (CP, in short) condition is required to prove
the existence of equilibrium.
Two types of the CP condition in the literature
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Debreu (1959, 1962) adopts the interiority condition to guarantee a
cheaper consumption point for each agent.
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 in
the context of infinite-dimensional choice problem.
D. Won (AU)
Poverty in Equilibrium
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Introduction
The Pitfall of Demand Function at the Cheaper Point
The continuity of demand function on the price set requires the
presence of cheaper points in the relative interior of the consumption
set.
However, the interiority condition is unrealistic in the capitalist
society where agents cannot subsist under autarky.
Our initial endowments of commodities are located outside the
consumption set. (Do you produce a smartphone for youself?)
The interiority condition may fail intrinsically in the
infinite-dimensional consumption set such as L2+ which has the empty
interior.
Consequently, the demand function approach to poverty is
self-contradictory because the poor endowments do not lie in the
interior of the consumption set.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Motivation for the current research
The existing literature is not satisfactory in explaining poverty in
equilibrium.
McKenzie (1981) extends the irreducibility condition to the case that
agents cannot subsist on their initial endowments but fails to explain
poverty in equilibrium.
Thus, the poverty threshold cannot be discussed in the current
general equilibrium framework.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Contribution of the Paper
This paper introduces the notion of the usefulness of poor
endowments by incorporating the poverty threshold into the
irreducibility condition.
The poverty threshold issue is addressed from both positive and
normative viewpoints by developing a two-tiered system of
consumption sets.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Literature on Survival Conditions
Interiority condition: Debreu (1959, 1962)
Irreducibility condition:
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McKenzie (1959, 1981, 2002)
Arrow and Hahn (1971)
Moore (1975)
Gottardi and Hens (1996) : incomplete markets
Florig (2001)
Florenzano (2003): infinite-dimensional consumption sets
D. Won (AU)
Poverty in Equilibrium
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Introduction
Notation
Exchange economy E = ((Xi , Yi ), PiY , ei )i∈I where
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X =
I = {1, 2, . . . , m} is the set of agents.
Yi ⊂ R` is the ‘positive’ consumption set for agent i.
Xi ⊂ R` is the ‘normative’ consumption set for agent i.
PiY indicates the preference ordering of agent i over Yi .
ei is the initial endowment of agent i.
Q
Xi ; X0 =
P
Xi .
P
x = (x1 , . . . , xm ) ∈ X ; e0 = i∈I ei .
i∈I
i∈I
Bi◦ (p) = {xi ∈ Xi : p · xi < wi (p) = p · ei }.
Bi (p) = {xi ∈ Xi : p · xi ≤ p · ei }.
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In the production economy, the wealth w (p) is expressed as
wi (p) = ei · p + si maxz∈Z p · z where si is the share of agent is and Z
is the aggregate production possibility set.
AY = {x ∈ X :
D. Won (AU)
P
i∈I
xi = e0 and ei 6∈ PiY (xi ) ∀i ∈ I }.
Poverty in Equilibrium
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Introduction
Three Norms on the Poverty Threshold
A desired level of monetary income m̄i
Xi = Xim = {xi ∈ Yi : p ∗ · xi ≥ m̄i ,
where p ∗ is an equilibrium price.
A desired level of physical consumptions
Xi = Xic ≡ {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 = Xiu ≡ PiY (xiu ),
where xiu is a consumption which meets the desired minimum welfare.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Desired Subsistence Consumption and Minimum Utility
Subsistence Consumption makes EIS depend on consumption choices.
If it is desired for agent i to have a subsistence consumption c i from
the welfare perspective, Xiu is expressed as
Xiu = PiY (c i ).
If preferences are represented by an expected utility function vi , his
utility function ui is defined as
ui (x; c i ) = E [vi (x − c i )].
In this case,
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Xiu
= {xi ∈ Yi : xi ≥ c i )}.
Atkeson and Ogaki (1996), Steger (2000)
ui (xi ) =
`
X
j=1
I
φj
(xji − c ij )1−αj − 1
1 − αj
Stone-Geary utility function: for instance, Varian (1992) and King and
Rebelo (1993)
vi (xt /nt ) = at ln (xt /nt ) − c i .
D. Won (AU)
Poverty in Equilibrium
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Introduction
Induced Preferences on Xi
The set PiY (xi ) contains consumptions in Yi preferred to xi .
We define the preference ordering Pi on the normative consumption
set Xi by restricting PiY to Xi such that for each xi ∈ Xi ,
Pi (xi ) = PiY (xi ) ∩ Xi .
In the case that Xi = Xiu , it holds that for all xi0 ∈ Yi \ Xi ,
Pi (xi ) ⊂ PiY (xi0 ).
(1)
That is, consumptions which do not meet the normative living
standards are less preferred to consumptions in Xi .
If the property (1) holds, competitive equilibrium with the normative
consumption sets is also competitive equilibrium with the positive
consumption sets.
However, the property (1) may not be kept in the case that Xi = Xic .
D. Won (AU)
Poverty in Equilibrium
17/12/2010
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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) P
PiY (xi ) ∩ Bi,Y
(p) = ∅ for all i ∈ I , and
(iii)
(x
−
e
)
=0
i
i
i∈I
The pair (p, x) is a normative quasi-equilibrium if it is a
quasi-equilibrium with x ∈ X .
A competitive equilibrium for E is a pair (p, x) ∈ (R` \ {0}) × Y such
that
(i) xi ∈ Bi,Y (p) for all i ∈ I ,
(ii) P
PiY (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 a
competitive equilibrium with x ∈ X .
As mentioned earlier, normative competitive equilibrium need not be
a competitive equilibrium.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Poverty Threshold
For any nonzero price p in R`+ , we define the minimum income
functions
mi (p) = inf p · xi
xi ∈Xi
and miY (p) = inf p · xi .
xi ∈Yi
The income mi (p) is the poverty threshold to normative consumptions
while miY (p) is the poverty threshold to positive consumptions.
D. Won (AU)
Poverty in Equilibrium
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Introduction
Assumptions
We make the following assumptions
A1. Each Xi is closed and convex.
A2. For every i ∈ I , PiY (xi ) 6= ∅ and xi 6∈ PiY (xi ) for all xi ∈ Yi , and
Pi (xi ) 6= ∅ for all xi ∈ Xi .
A3. Let xi be a point in Yi . Then for each yi ∈ PiY (xi ) and zi ∈ Yi , there
exists α ∈ (0, 1) such that αyi + (1 − α)xi ∈ PiY (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).
P
A5. The aggregate endowment e0 is in the interior of i∈I Xi .
D. Won (AU)
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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 every
x ∈ A and for every nontrivial partition {I1 , I2 } of I , there exists some
k ∈ I1 and α > 0 such that e0 ∈ int X0 and
X
X
e0 ∈
Pi (xi ) +
[α (Xi − {ei }) + {xi }] .
i∈I1
i∈I2
(The economy is irreducible if for each partition of all the agents into two
groups, one group of agents can improve their welfare by taking not only the
existing net trade but also some new net trade of the other group.)
D. Won (AU)
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Cheaper Point Conditions
Continued
(Florig (2001)) The economy E is said to be weakly irreducible if, for
every x ∈ A and for every nontrivial partition {I1 , I2 } of I , there exist
some k ∈ I1 and αi > 0 for all i ∈ I such that e0 ∈ int X0 and
X
X
0 ∈ αk [Pk (xk )−{ek }]+
αi [cl Pi (xi )−{ei }]+
αi (Xi − {ei }) .
i∈I1 \{k}
i∈I2
The economy is weakly irreducible if it is irreducible.
D. Won (AU)
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Cheaper Point Conditions
Irreducibility and Usefulness
Let’s look at irreducibility from a different angle. By irreducibility,
there exists zi ∈ Xi for each i ∈ I2 such that
X
X
X
e0 =
xi ∈
Pi (xi ) +
[α{zi − ei } + {xi }] .
i∈I
It is rearranged as
X
i∈I1
i∈I1
xi + α
i∈I2
X
X
{ei − zi } ∈
Pi (xi ).
i∈I2
i∈I1
P
Then the consumption α i∈I2 (ei − zi ) is useful for agents in I1 in
that it improves their welfare.
By irreducibility, agents have useful trades to each other. Moreover,
every agent has cheaper points in equilibrium. Thus, no poverty exists
in equilibrium.
D. Won (AU)
Poverty in Equilibrium
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Cheaper Point Conditions
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 xi0 i xi implies p · xi0 ≥ p · xi },
I (p, x) = I \ I0 (p, x),
Iˆ0 (p, x) = {i ∈ I0 (p, x) : xi0 i xi implies p · xi0 > p · xi }, and
I˜(p, x) = I0 (p, x) \ Iˆ0 (p, x).
Agent i in I0 (p, x) would live on the poverty threshold with the
expenditure-minimizing choice xi if (p, x) were a competitive
equilibrium.
Agent i in Iˆ0 (p, x) would live on the poverty threshold with the
optimal consumption choice xi if (p, x) were a competitive
equilibrium.
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Cheaper Point Conditions
Usefulness of Poor Endowments
A6. Usefulness of poor endowments
For a pair (p, x) in (R`+ \ {0}) × AIR with I˜0 (p, x) 6= ∅, there exist
xi0 ∈ Xi and αi > 0 for each i ∈ I0 (p, x) such that
X
X
X
xi +
αi (ei − xi0 ) ∈
Pi (xi ).
(2)
i∈I (p,x)
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I
i∈I0 (p,x)
i∈I (p,x)
Assumption A6 states that if I˜0 (p, x) 6= ∅, the poor in I0 (p, x) has a
useful trade at p for the relatively rich in I (p, x).
It will play a critical role for the existence of normative competitive
equilibrium with possible poverty.
It allows us to check that competitive markets are viable under
normative poverty standards.
A6s. Strong Usefulness of poor endowments
This is a strong version of A6 obtained by replacing “with
I˜0 (p, x) 6= ∅” by “with I0 (p, x) 6= ∅” in the statement of (A6).
D. Won (AU)
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Main Results
The Existence of Normative Competitive Equilibrium
Theorem 1. Normative competitive equilibrium exists if the economy
satisfies 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 ) = PiY (xi ) for each i ∈ I , it is a competitive equilibrium for
the economy E.
Theorem 1 states that normative competitive equilibrium exists in the
economy where the poor endowments can provide useful trades for
the relatively rich.
Existential failure may arise in the economy where poor endowments
are not useful to the class of relatively rich agents.
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Examples
Example 1: Normative Equilibrium with Poverty
The economy has two agents with homothetic preferences.
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Y1 = Y2 = R2+ .
ui (ai , bi ) : a homothetic utility function for each i = 1, 2 which is
strictly increasing, strictly convex and differentiable in R2+ .
C 1 = C 2 = (1, 1). X1 = X2 = {(a, b) ∈ R2+ : a ≥ 1, b ≥ 1}.
e1 = (3, 0), e2 = (0, 3) and thus, ei 6∈ Xi .
Assume that for each a > 0, there exists λi > 0 such that for each
i = 1, 2
D 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).
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Agent 2 subsists on the poverty threshold with m2 (p ∗ ) = 1.
Agent 2 is poor because he is endowed with cheaper good (good 2).
Assumption A6 holds here but all the existing survival conditions for
normative competitive equilibrium fail here because agent 2 has no
cheaper consumption that e2 .
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Examples
Example 1: In-Kind Transfers
Suppose that the society calls fro a public policy to rescue agent 2
from poverty.
One conceivable policy option is to conduct an an in-kind transfer
(a , b ) from agent 1 to agent 2.
Let E t denote the post-transfer economy where the initial
endowments are given by e1t = (3 − a , −b ) and e2t = (a , 3 + b ).
The post-transfer economy satisfies A6s, i.e., it has pover-free
normative equilibrium if the transfer (a , b ) lies in the set
F = (a , b ) ∈ R2 : 0 < 2a + b < 3
∩ (a , b ) ∈ ([0, 1) × (−1, 0]) ∪ ((2, 3] × [−3, −2))
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Conclusions
Concluding Remarks
The paper provides a general equilibrium model for the poverty
threshold from both positive and normative viewpoints.
The normative approach to poverty is built on the notion of normative
competitive equilibrium in which agents are allowed to make optimal
consumption choices which meet desired minimum living standards.
It presents the usefulness of poor endowments as a new cheaper-point
condition to show the existence of normative competitive equilibrium
with possible poverty.
Promising research topics
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Specialization of the general results in a testable or computable GE
model with concrete utility functions developed in the literature on
poverty.
Search for testable implications of the usefulness of poor endowments.
Unified approach to the poverty threshold.
When can the imposition of normative living standards improve the
welfare of agents who are poor in competitive equilibrium?
Extension of Assumption A6 to incomplete-market economies.
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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
Normative consumption set for agent 1
3
Normative eq’m
Competitive eq’m
2
1
0
1
2
3