Consumption rights: a market mechanism
to redistribute wealth∗
Francisco Martı́nez†
Jorge Rivera
‡
January 3, 2006
Abstract
In an exchange economy with only private consumption goods we propose
a competitive mechanism to redistribute wealth among individuals. For that,
we introduce the so called “consumption rights”, which is a real parameter that
modifies the budgetary constraint of individuals but does not participate in their
utility functions. Consumption rights can be traded in the market, which is the
main difference with slack parameters, as fiat money or tax inflation, the last
one widely known as a method to modify the distribution of wealth. The policy
maker control variables are both the amount of rights assigned to each individual
and a pricing rule that defines the rate of exchange between rights and wealth.
The only intervention of the planner will be through the definition of the policy,
because the redistribution of wealth will be the consequence of the competitive
exchange among consumers.
Keywords: redistribution of wealth, competitive equilibrium, consumption rights.
JEL Classification: D31, D51, D63, H21.
1
Introduction
It is well known that in a competitive economy, the Walrasian equilibrium allocation
satisfies a set of desirable properties, including Pareto optimality, belonging to the
core of the economy, and individual rationality among others1 . Thus, a fundamental
conclusion is that under very general hypotheses on the economy, the “invisible hand”
leads to a solution that guarantees all these properties, and is regarded as an efficient
outcome without the need of intervention by a policy maker.
∗
This work was supported by ICM Sistemas Complejos. The authors wish to thank Jean-Marc
Bonnisseau and Esteban Rossi-Hansberg for helpful comments and a productive discussion.
†
Departamento de Ingenierı́a Civil, Universidad de Chile. [email protected]
‡
Departamento de Economı́a, Universidad de Chile. [email protected]
1
See [11] as a general reference on the equilibria properties and for more general economic frameworks, for instance with increasing returns in production and/or infinite many agents, see [8] and
references there in.
1
Nevertheless, despite the merits of the competitive allocation, the view that the
market is capable of assigning resources appropriately is not generally shared by most
economists and policy makers, mainly because this outcome does not necessarily fulfill
a complementary set of consensus on equity criteria.
The economic theory has developed different methods to obtain efficient but also
equitable outcomes. Among these methods, a lump-sum transfer of initial endowments
has the advantage of avoiding undesired distortions in the assignment, thus maintaining efficiency in the economy that justifies why it is theoretically preferred to other tax
systems. However, in practice there are serious and well established difficulties with
the implementation of lump-sum taxes. First, tax collection is far from costless, as
was shown in the poll tax of the UK (see [20], p. 46), which reduces their efficient
performance. Secondly, and more fundamentally, optimal lump-sums depend on all
the relevant variables in the economy, many of them only known by individuals and
not directly observable by the government, which ultimately relies on reported information. Deceptively, Mirrlees (see [14]) presents theorems that prove the impossibility
of designing non-manipulable lump-sum taxes, which makes it ultimately impossible
to define optimal lump-sums.
In this paper we consider an alternative method to redistribute wealth based on
the introduction of a parallel market in the context of an exchange economy, with a
finite number of consumers and goods, without public goods or externalities. For that
purpose we extend the Arrow-Debreu model (see [1]) introducing a parameter that
modifies the budgetary set but does not affect consumer preferences. We decided to
interpret this parameter as tradable “consumption rights”, although another plausible
interpretation is as a parallel currency. In this new economy, called r-economy, the
initial endowment of consumers consists not only of commodities (as usual), but of
a number of rights centrally distributed to individuals at no cost to them; notably,
these rights have no role in the utility function. Every transaction of commodities
implies two payments: a price in wealth (as usual) and a certain amount of rights. An
important feature is that these rights are tradable at a price and are generic, i.e. they
are valid to purchase any commodity.
An existence of equilibrium theorem in the r-economy is proved under usual assumptions on the fundamentals. We also show that the equilibrium allocation in the
r-economy can also be obtained in the standard economy when certain lump sum
transfers is implemented; these lump sum transfers can be identified ex-ante from the
distribution of consumption rights among individuals. Thus, the r-economy provides
the policy maker with a tool to implement social goals, for example to fulfill an initial
subsistence condition for all agents.
With these results, the exchange process with consumption rights has similar merits as the lump-sum method to reassign resources in the economy: both of them avoid
price distortions and decentralize desirable Pareto optima. However, a significant difference between these methods is that by exchanging consumption rights, the wealth
is redistributed without the intervention of a policy maker that collects and assigns
resources in the economy, thus bypassing Mirrlees’ impossibility theorem.
The idea of introducing a parameter that modifies the budgetary set of consumers
2
without participating in their preferences has been previously used in microeconomic
theory to study other problems. For instance, in monetary economics a slack parameter
is defined, usually called fiat money, whose sole role is to facilitate the exchange in the
presence of “frictions” in the economy that make it difficult for agents to execute netexchanges worth exactly zero. See [10] for more details on this relevant aspect of fiat
money in the economy. See also [4] as a complementary reference.
Another example is one where the non-satiation assumption does not hold in the
economy. If for any given price, some consumers wish to consume a commodity bundle
in the interior of their budget set, the Walras equilibrium may fail to exist. In this case
one may establish the existence of an equilibrium by allowing for the possibility that
some agents spend more than the value of their initial endowment. This generalization
of the Walras equilibrium is called dividend equilibrium or equilibrium with slack (see
[3], [6], [9], [16] and [13] among others).
The r-economy also has some similarities with the system of pollution rights to
control emissions, where pollution rights are also tradable, however, here too, there are
some fundamental differences. In the case of pollution rights, the price to trade rights is
exogenously given by technical relationships, it affects only one good (more precisely,
one bad) and directly affects the production of goods. Conversely, the r-economy
endogenously defines consumption right prices, affecting all goods and is defined as an
exchange economy without production being affected. See [18], [19] and [21] for more
details on this type of model.
Moreover, our notion of consumption rights should not be confused with the concept
used by Hammond in [12], defined as a right to “choose net trade vectors”. It is also
different to Aumann and Kurz’s notion in [2], which represents the right to choose
lump-sum transfers according to the individuals’s or group’s political power in the
society. On the other hand, these concepts bear in common the fact that they are all
defined from equity considerations, by means of political institutions.
The main difference of our approach compared with the above mentioned models is
that in our case, in addition to the good prices that define the usual wealth constraint,
there is a new set of prices that introduces a second constraint based on consumption
rights; altogether these constrains define an extended budgetary set for any individual.
Moreover, the possibility of trading rights links the two constraints, in a way that at the
equilibrium both constrains are binding, inducing an efficient consumption outcome in
the economy which will be useful for the study of the equilibrium in the r-economy.
In our mind, the most innovative element of the r-economy is that the new set of
constraints and the condition of tradable consumption rights introduce a new market
that yields an equilibrium allocation intended to comply with some exogenous equity
criteria. In this way, a double objective -efficiency and equity- can be achieved by combining complementary market mechanisms -on goods and rights-, each one specifically
designed for each criteria but interacting without reducing their capability of reaching
their underpinning objectives.
This paper is organized as follows. In Section 2 we introduce some mathematical
concepts and definitions used in this work. In Section 3 we describe the model of
consumption rights, whereas in Section 4 we study the demand in the r-economy. The
3
pricing rule and the existence of equilibrium in the r-economy is discussed in Section
5. Finally, in Section 6 we present some application issues.
2
Mathematical notation and preliminaries
Given n ∈ IN , we set IRn+ = {x = (xk ) ∈ IRn | xk ≥ 0} and IRn++ = {x = (xk ) ∈
IRn | xk > 0}. For a couple of vectors x, y ∈ IRn , x · y denotes the Euclidean inner
product of them and the orthogonal complement of the vector x ∈ IRn is denoted
by x⊥ . Finally, given 1 ≤ m ≤ n we denote x(m) = (x1 , x2 , . . . , xm ) ∈ IRm and
x(−m) = (xm+1 , xm+2 , . . . , xn ) ∈ IRn−m , with n > m.
For a matrix A ∈ IRm×n , its transpose is denoted as At and, when it exists, its
inverse by A−1 . The inverse transpose of A is denoted as A−t . The matrix Diag[σi ] ∈
IRn×n denotes the diagonal matrix of σi ∈ IR, i ∈ {1, 2, . . . , n}.
A matrix A ∈ IRm×n is said to be positive (resp. strictly positive), denoted A ≥ 0
(resp. A > 0), if all of its elements are non negative (resp. strictly positive). The
spectral radius of A ∈ IRn×n is denoted by ρ(A). The subset of M -matrices of dimension
n × n is denoted by M [n]. Thus,
n
o
M [n] = µIn − A, A ≥ 0, A ∈ IRn×n , µ ≥ ρ(A) ,
where In ∈ IRn×n is the identity matrix. Finally M ∗ [n] ⊆ M [n] denotes the subset
of non-singular elements of M [n]. We refers to [7] for more details and properties of
M -matrices.
Finally, the Simplex in IRn is denoted by ∆n , that is,
(
n
∆n = σ = (σk ) ∈ IR | σk ≥ 0,
n
X
)
σk = 1 .
k=1
3
The model of consumption rights
Following the standard Arrow-Debreu model (see [1]), we assume that in the economy
there are ` ∈ IN consumption goods and m ∈ IN consumers, indexed by j ∈ L =
{1, 2, . . . , `} and i ∈ I = {1, 2, . . . , m} respectively. For an individual i ∈ I, his utility
function and initial endowments are given by ui : IR`+ → IR and ωi ∈ IR`+ respectively.
We set
ω=
X
ωi ∈ IR`++ .
i∈I
as the total initial resources of the economy2 . Thus an exchange economy is defined by
E = ((ui ), (ωi ))i∈I.
Hereafter, we assume the following condition.
In the following, ωi,k ∈ IR denotes the k ∈ {1, 2, . . . , `} component of vector ωi ∈ IR` , i ∈ I,
whereas ωk ∈ IR is the k component of vector ω.
2
4
`
Assumption U. For any i ∈ I, ui is a C 2 mapping such that for each x ∈ R++
,
`
2
⊥
∇ui (x) ∈ R++ ; the Hessian matrix D ui (x) is negative definite on ∇ui (x) and
the utility function ui satisfies a sufficient condition that ensures the uniqueness
of the Walras equilibrium provided it existence3 .
Any distribution of goods {xi }i∈I is said to be a feasible allocation if
X
xi =
i∈I
X
ωi .
i∈I
In our model we will assume that, besides an initial endowment of resources, each
consumer is initially assigned with a strictly positive amount of what we call consumption right, which for any individual i ∈ I is denoted by ri ∈ IR+ . The exchange
economy with consumption rights, called r-economy, is therefore defined by
Er = ((ui ), (ωi ), (ri ))i∈I .
We assume that consumption rights can be freely traded in the market, but they do
not participate in the utility function; we denote by q ∈ IR+ the price of a consumption
right in the market.
Consumption rights modify the consumer’s budget constraint in the following way.
Assume that goods prices are given by p ∈ IR`+ and that in the economy there are
transformation rates from goods to consumption rights, which generically will be represented by a vector s ∈ IR`++ . The solely role of vector s in the economy is to restrain
the set of consumption possibilities of each agent, in such a way that with the initial
consumption rights ri ∈ IR+ available for the individual i ∈ I, he can only consume
those bundles xi ∈ IR`+ that comply with
s · xi ≤ ri .
(1)
Inequality (1) represents a restriction for consumption expressed in terms of consumptions rights, which equivalently can be presented in terms of wealth using price q as
follows
q (s · xi ) ≤ qri .
(2)
To finally determine the feasible consumption bundles for individual i ∈ I, condition
(2) should be also considered along with the usual wealth constraint, that is,
p · xi ≤ p · ω i .
Now, and this is a relevant point in our model, the individual i ∈ I may decide to
trade δ ∈ IR consumption rights in the market, obtaining ri +δ consumption rights, then
his wealth will be necessarily modified in −qδ. Given this transaction of consumption
rights, which ex post will be determined endogenously as part of the equilibrium, the
budgetary set of an individual i ∈ I corresponds to
3
See [5], [15] and [17] for more details on this type of conditions.
5
n
o
B(p, s, q, ωi , ri , δ) = x ∈ IR`+ | p · x ≤ p · ωi − qδ, (qs) · x ≤ qri + qδ .
4
(3)
The demand in the r-economy
Given a set of prices (p, s, q) ∈ IR`+ × IR`+ × IR+ , in the r-economy each individual will
choose both an “optimal amount of consumption rights to trade in the market” and
“an optimal quantity of consumption goods” in order to maximize his utility function
subject to the budget constraint given by (3). In consequence, the r-consumer’s problem
for an individual i ∈ I in the r-economy is defined by
max ui (x)
(4)
x,δ
s.t
x ∈ B(p, s, q, ωi , ri , δ).
The solution set of (4) in terms of consumption goods (resp. consumption rights)
will be called demand for goods (resp. demand for consumption rights) at prices p, s, q
for individual i ∈ I, and denoted by xi (p, s, q, ωi , ri ) (resp. δi (p, s, q, ωi , ri )). Note that
the demand sets could be empty-sets.
Note now that for every δ ∈ IR, B(p, s, q, ωi , ri , δ) is a convex and compact set,
possibly empty. However, given p, s ∈ IR`++ and q > 0 it is easy to check that
"
#
p · ωi
B(p, s, q, ωi , ri , δ) 6= ∅ ⇔ δ ∈ Γi = −ri ,
.
q
In consequence, problem (4) can be presented equivalently as
max ui (x)
x,δ
s.t
x ∈ hB(p, s, q,iωi , ri , δ)
i
δ ∈ −ri , p·ω
,
q
(5)
and then, from the continuity of ui and the fact that the correspondence
"
#
p · ωi
δ ∈ −ri ,
7→ B(p, s, q, ωi , ri , δ)
q
is upper-hemicontinuous and compact valued, we can apply the well known Berge’s
Maximum Theorem to conclude that optimization problem (5) has a solution, that is,
the demand for goods and consumption rights are nonempty sets (see [23] for details).
With this, we had proved the following statement.
Proposition 4.1 For each individual i ∈ I, the demand for goods and consumption
rights are nonempty sets provided that the utility function ui is continuous, p, s ∈ IR`++ ,
q > 0, ri > 0 and ωi ∈ IR`+ \ {0IR` }.
In the following, assume the hypotheses of Proposition 4.1 and take δi ∈ δi (p, s, q, ωi , ri )
and xi ∈ xi (p, s, q, ωi , ri )4 .
4
We recall that the demand for goods and consumption rights could be an empty set.
6
Claim 4.1 δi 6= −ri and δi 6=
p·ωi
.
q
In the contrary case, it is easy to check that B(p, s, q, ωi , ri , δ) = {0IR` }, which
implies xi = 0IR` . Thus, taking
#
"
p · ωi
δ ∈ −ri ,
,
q
if we define
(
1
p · ωi − qδ ri + δ
,
λ = min
2
p · ωi
s · ωi
)
>0
then we can readily show that λωi ∈ B(p, s, q, ωi , ri , δ), and therefore, from the monotonicity of ui we have that ui (0IR` ) < ui (λωi ), which contradicts the demand definition.
Claim 4.2 p · xi = p · ωi − qδi and s · xi = ri + δi .
In the contrary case, one of the following is satisfied:
(a) p · xi = p · ωi − qδi ,
s · xi < ri + δi ,
(b) p · xi < p · ωi − qδi ,
s · xi = ri + δi ,
(c) p · xi < p · ωi − qδi ,
s · xi < ri + δi .
Given v = (1, 1, . . . , 1)t ∈ IR` and ² > 0, define xi (²) = xi + ²v. Thus, under case
(a)
"
Ã
p·v
p · xi (²) = p · x + ²p · v = p · ωi − q δi − ²
q
!#
,
and for ²0 > 0 small enough
Ã
s · xi + ²0 s · v < ri + δi − ²0
n
p·v
q
!
Ã
⇔ s · xi (²0 ) < ri + δi − ²0
o
³
!
p·v
.
q
´
i
Defining ²1 = min ²0 , δi +r
, δi0 = δi − ²1 p·v
, x0i = xi (²1 ), follows that δi0 ∈ Γi ,
2
q
x0i ∈ B(p, s, q, ωi , ri , δi0 ) and, from the monotonicity of ui , ui (xi ) < ui (xi (²1 )), which
contradicts the demand definition. On the other hand, under case (b), for ²0 > 0 small
enough we have that
p · xi < p · ωi − qδi − ²0 [qs · v + p · v] ⇔ p · xi (²0 ) < p · ωi − q[δi + ²0 s · v],
and therefore, defining
(
1
²1 = min ²0 ,
2
Ã
p · ωi
− δi
q
7
!)
>0
follows directly that δi0 = δi + ²1 s · v ∈ Γi , xi (²1 ) ∈ B(p, s, q, ωi , ri , δi0 ) and, from the
monotonicity of ui , ui (xi ) < ui (xi (²1 )), which also contradicts the demand definition.
Finally, under case (c), consider
(
p · ωi − qδi − p · xi ri + δi − s · xi
²1 = min
,
2p · v
2s · v
)
>0
to conclude that xi (²1 ) ∈ B(p, s, q, ωi , ri , δi ) and ui (xi ) < ui (xi (²1 )), which ends the
proof of Claim 4.2.
Consequently, from the fact that p · xi = p · ωi − qδi and s · xi = ri + δi , we eliminate
δi from these relations concluding that the following is satisfied at the optimum
(p + qs) · xi = p · ωi + qri ,
(6)
which finally allow us to deduce that r-consumer’s problem for an individual i ∈ I can
be equivalently rewritten as
max ui (x)
x
s.t (p + qs) · x = p · ωi + qri .
x ∈ IR`+ .
(7)
Thus, under Assumption U, the solution of (7) (demand for goods) is unique,
and so is the demand for rights. With all the foregoing we had proved the following
proposition.
Proposition 4.2 For each individual i ∈ I, if Assumption U holds true, p, s ∈ IR`++ ,
q > 0, ri > 0 and ωi ∈ IR`+ \ {0IR` }, then the r-consumer’s problem has a unique
solution.
Remark 4.1 Note that the existence of a parallel market for consumption rights implies that the consumer’s problem in the r-economy can be viewed equivalently as
an standard consumer’s problem with both an income effect -given by term qri - and
a substitution effect5 . These effects working simultaneously modify the consumer’s
budgetary set and constitute the theoretical justification for the employment of consumption rights to redistribute wealth in the economy, as we will detail later on this
paper.
5
Equilibrium
The equilibrium notion in the r-economy is simply a natural extension of the Walrasian
equilibrium concept for the standard economy without consumption rights. Thus, we
are concerning on prices p, s and q such that the respective demands (consumption
rights and goods) at these prices are feasible allocations according to the notions given
immediately.
5
The substitution effect comes from the fact that the price used to value the demand (p + qs) is
different from the price used to value the endowments (p).
8
To begin we introduce the following feasibility condition, used to preserve the total
amount of consumption rights in the market.
Definition 5.1 A consumption rights exchange distribution {δi }i∈I is said to be feasible if
X
δi = 0.
i∈I
Considering that if6 (p∗ , s∗ , q ∗ , (x∗i )i∈I , (δi∗ )i∈i ) is an equilibrium of Er , then for each
i ∈ I, s∗ · x∗i = ri + δi∗ and then,
s∗ ·
X
i∈I
x∗i =
X
i∈I
ri +
X
δi∗ =
i∈I
X
ri ⇒ s∗ · ω = R,
i∈I
P
with R = i∈I ri the total amount of consumption rights in the economy. Consequently,
condition s∗ · ω = R should always be verified at the r-equilibrium.
The next definition is a natural extension of the Walrasian equilibrium notion in
standard economies to our setting.
Definition 5.2 Prices (p∗ , s∗ , q ∗ ) ∈ IR2`
+ × IR+ are said to be an r-equilibrium price for
∗ ∗ ∗
the economy Er if {δi (p , s , q , ωi , ri )}i∈I is feasible and
X
xi (p∗ , s∗ , q ∗ , ωi , ri ) = ω.
i∈I
The tupla
(p∗ , s∗ , q ∗ , (xi (p∗ , s∗ , q ∗ , ωi , ri ))i∈I , (δi (p∗ , s∗ , q ∗ , ωi , ri ))i∈I ) ∈ IR`+ ×IR`+ ×IR+ ×IRm·` ×IRm
constitutes what we call a competitive r-equilibrium for the economy Er .
Observe that the number of unknowns involved in the r-equilibrium definition is
2` + 1 (components of prices p, s and q) and, from the feasibility condition in goods
and consumption rights exchange, the number of equations is ` + 1. Hence, there
is a fundamental indeterminacy in our equilibrium concept. This indeterminacy is
intriguing because one would expect that the marginal rates of substitution could
define simultaneously vectors p and s or a linear relation between them, which would
naturally eliminate this indeterminacy. However, the Kuhn-Tucker conditions for the
consumers’ problem do not imply a linear relation between these vectors.
Thus, in order to characterize an equilibrium in the r-economy we define what
we call the pricing rule, which, along with the assignment of consumption rights to
consumers, define the policy maker’s instruments that are sufficient to reach any social
objective in the economy as we will see later on this work.
A simple way to define a pricing rule, and probably the one with practical interest,
is by means of a linear relation among prices as we define in the following.
6
For simplicity, we denote x∗i = xi (p∗ , s∗ , q ∗ , ωi , ri ) and δi∗ = δi (p∗ , s∗ , q ∗ , ωi , ri ).
9
Definition 5.3 A linear pricing rule in the r-economy will be a relation
1
s = Ap
q
with A ∈ IR`×` a given matrix7 .
Given a linear pricing rule we can reformulate the r-equilibrium notion above introduced. From (7) it follows that the consumer’s budgetary restriction of individual
i ∈ I can be rewritten as
(p + Ap) · x = p · ωi + qri ⇔ [(I + A)p] · x = p · ωi + qri ,
(8)
and then the r-consumer’s problem (4) can be re-write equivalently as
max
ui (x)
x
s.t [(I + A)p] · x = p · ωi + qri
x ∈ IR`+ .
(9)
If we assume that (I + A) is non-singular and defining π = (I + A)p, condition (8)
can be re-written as
h
i
π · x = π · (I + A)−t ωi + qri ,
and then, due to s · ω = R should be satisfied at the r-equilibrium, it follows that
π · [(I + A)−t At ω]
.
(10)
R
Thus, if we define ρi = rRi ∈ [0, 1], i ∈ I, and ρ = (ρi ) ∈ ∆m , the r-consumer’s problem
for individual i ∈ I can be finally expressed as
q=
max
ui (x)
x
s.t π · x = π · ωi (ρ, A).
x ∈ IR`+ .
(11)
with
h
i
ωi (ρ, A) = [I + A]−t ωi + ρi At ω ∈ IR` .
(12)
Note now that
X
i∈I
"
ωi (ρ, A) = [I + A]
−t
X
i∈I
ωi +
Ã
X
!
#
t
ρi A ω = [I + A]−t [ω + At ω] = ω,
i∈I
and then, in order to guarantee that {ωi (ρ, A)}i∈I constitute a redistribution of initial
endowments in the economy we must verify that for each i ∈ I
7
Properties of matrix A will be discussed later on this Section.
10
ωi (ρ, A) ∈ IR`+ .
(13)
If condition (13) were true, we point out that under the linear pricing rule, the rconsumer’s problem can be viewed as a consumer problem in the original economy
-without consumption rights- with a lump-sump transfers defined by (12) that modifies
the initial resources of any individual. Precisely this fact will be the central justification for the employment of consumption rights and a pricing rule as the tools set to
redistribute wealth in the economy.
From standard results in general equilibrium theory (see [1]), if condition (13) holds
true we know that there exists an equilibrium price π(ρ, A) ∈ IR`++ for the economy
where the initial endowments are given by (12). Denoting xi (π(ρ, A), ωi (ρ, A)) the
respective equilibrium allocations, it is easy to check that (p∗ , s∗ , q ∗ , (x∗i )i∈I , (δi∗ )i∈I )
defined by
p∗ = (I + A)−1 π(ρ, A)
q∗ =
x∗i = x∗i (π(ρ, A), ωi (ρ, A)),
π(ρ,A)·[(I+A)−t At ω ]
R
s∗ =
1
Ap∗
q∗
(14)
δi∗ = s∗ · x∗i − ri , i ∈ I,
constitutes an r-equilibrium for Er provided that
q ∗ > 0.
(15)
Consequently, conditions (13) and (15) appears as the key properties to be satisfied
in order to demonstrate the existence of a competitive equilibrium in the r-economy,
under the framework previously described. As we show in the next proposition, they
will depend non only on the non-singularity of I + A but also on the positiveness of
matrix (I + A)−1 .
Proposition 5.1 If the linear pricing rule is defined by a matrix A such that A ∈ M ∗ [`]
and A > 0, then for each i ∈ I, ωi (ρ, A) ∈ IR`++ and q ∗ > 0.
Proof. We recall that A ∈ M ∗ [`] if A is non-singular and there exist B ≥ 0 and
µ > ρ(B) ≥ 0 such that
A = µI − B.
Since ρ(B) < µ follows that I + A is non-singular (and so (I + A)t ) and therefore,
considering that
Ã
−t
(I + A)
t −1
= (I + A )
t −1
= ((1 + µ)I − B )
=
1
1+µ
!
lim
n→+∞
n
X
k=0
"
Bt
1+µ
that is,
(I + A)
−t
1
=
I+
1+µ
"
lim
n→+∞
n
X
k=1
11
Ã
[B t ]k
(1 + µ)k+1
!#
≡
1
I + Γ,
1+µ
#k
it follows immediately that Γ ≥ 0, which implies that (I + A)−t > 0. Since A > 0 and
ω ∈ IR`++ , we have that ρi At ω ∈ IR`++ , and therefore
h
i
h
i
ωi (ρ, A) = (I + A)−1 ωi + ρi At ω = (I + A)−1 ωi + (I + A)−1 ρi At ω ∈ IR`++
|
{z
}
∈IR`+
|
{z
}
∈IR`++
Since ωi (ρ, A) ∈ IR`++ , i ∈ I, is a redistribution of initial endowments for economy
E, under Assumption U we already know ([1]) that there exists an equilibrium price
π ∗ ∈ IR`++ when the consumer’s problem is given by (11). Thus, π ∗ ·(I +A)−1 [ρi At ω] >
0, which, from (14), finally implies that q ∗ > 0 as required.
2
The following corollary comes directly from Proposition 5.1.
Corollary 5.1 If one of the following is satisfied:
(a) A = Diag[σi ], with σi > 0, i ∈ I,
`−k
(b) for a given 1 ≤ k ≤ `, suppose that ωi (−k) = (ωi,k+1 , · · · , ωi,` ) ∈ IR++
and
"
A=
Ak 0
0
0
#
with Ak ∈ M ∗ [k] such that Ak ω(k) ∈ IR`++ ,
then for each i ∈ I, ωi (ρ, A) ∈ IR`++ and q ∗ > 0.
P
Proof. Under case (a), define µ = i∈I σi and B = Diag[µ − σi ] to conclude that
A = µI − B satisfies the hypotheses of Proposition 5.1, which ends this part of the
proof. On the other hand, under case (b), a direct calculus yields
³
³
´
´
ωi (ρ, A) = [I + Ak ]−t ωi (k) + ρi Atk ω(k) , ωi (−k) ,
which, from hypotheses, belongs to IR`++ . Thus, provided that ωi (ρ, A) ∈ IR`++ , i ∈ I,
the remaining part of the demonstration is direct from Proposition 5.1.
2
The case (a) in Corollary 5.1 can be considered the simplest pricing rule to be
implemented by the policy maker and will be used in next section to describe some
applications of the model. The case (b) can be interpreted as a situation where only
the first 1 ≤ k ≤ ` markets of goods are “restricted” by consumption rights, this
in the sense that the transaction of consumption rights generates redistributions of
endowments only in these markets. This case corresponds to assume that vector s ∈ IR`
has the form s = (s1 , s2 , · · · , sk , 0, 0, · · · , 0) ∈ IR` .
The next theorem is a direct consequence -and a summary- of all previous developments.
Theorem 5.1 If Assumption U holds true, for each i ∈ I ωi ∈ IR`+ , ri > 0, and the
linear pricing rule is defined by a matrix A that satisfies the conditions of Proposition
5.1, then there exists a unique r-equilibrium in Er , which can be identified as one obtained in the economy E where a lump-sum transfers defined by (12) are implemented.
12
6
Applications and particular cases
6.1
A particular case for a pricing rule
The simplest pricing rule one can consider is A = σI, for some σ > 0. In such case,
given ρ ∈ ∆m and
λ=
1
∈ [0, 1],
1+σ
we can readily deduce that
ωi (ρ, A) = λωi + (1 − λ)ρi ω,
(16)
which implies that the respective equilibrium allocation in the r-economy can be viewed
as an equilibrium allocation in the economy E where a lump-sum transfer defined by
(16) has been implemented. The tax device consist therefore on collecting (1 − λ) of
the initial resources of the economy and give then back to the individuals according to
the percentages of consumption rights assigned to them.
Another approach consists of interpreting the r-equilibrium allocation as the resulting equilibrium in economy E where a transfer of money has been carried out. To see
this, let us take the equilibrium price in the economy E without consumption rights,
pc , and the equilibrium price in the r-economy, pr , to define
Mi = [λpr − pc ] · ωi + (1 − λ)ρi q ∗ σR ∈ IR
as the “money” assigned to individual i ∈ I in the economy E 8 . Consider now the
consumer’s problem defined by
max
ui (x)
x
s.t
p · x = p · ωi + Mi .
x ∈ IR`+ .
(17)
It is easy to check that the equilibrium outcome in economy E with the consumer’s
problem given by (17) coincides with the r-equilibrium previously described.
The simplest pricing rule given here - particularly the case with σ = 1 where p = qs is probably the only one that may have a realistic option to be implemented in practical
terms. The main reason for this is inherent simplicity to compute the corresponding
transaction of consumption rights and goods by the population.
6.2
The policy maker problem regarding wealth distribution:
a simple example
Some intuition about the effects on the final wealth distribution in the r-economy can
be obtained from the following simplified case.
8
Price q ∗ comes the equilibrium in the r-economy. Note that Mi could be positive (transfer) or
negative (tax).
13
Consider an economy with m individual indexed by i ∈ I = {1, 2, · · · , m} and two
goods (i.e. ` = 2). Assume that their utility functions are given by ui (x, y) = xαi y 1−αi
and the respective initial endowments are ωi = (ωi1 , ωi2 ). Define
X
ω = (ω1 , ω2 ) =
ωi ∈ IR2 .
i∈I
In this case, considering good one as the numerary, the competitive equilibrium
price is pc = (1, pc ) with
P
pc =
i∈I
(1 − αi )ωi1
P
i∈I
Assume now that A = σI and λ =
σ
1+σ
.
αi ωi2
∈]0, 1[. Given ρ ∈ ∆m and
e i1 , ω
e i2 ) = λωi + (1 − λ)ρi ω
ωi (ρ, A) = (ω
the equilibrium price in this r-economy is given by pr = (1, pr ) with
P
pr =
i∈I
e i1
(1 − αi )ω
P
i∈I
.
e i2
αi ω
In the economy E, the wealth of an individual i ∈ I is given by
Wic = pc · ωi = ωi1 + pc ωi2 ,
whereas in the r-economy Er this value is
e i1 + pr ω
e i2
Wir = pr · ωi (ρ, A) = ω
If we define the total wealth of the r-economy (resp. for the economy E) as T W r = pr ·ω
(resp. T W c = pc ·ω), the relative wealth of an individual i ∈ I in the r-economy satisfies
that
·
¸
pr · ω i
Wir
=
=λ
+ (1 − λ)ρi .
(18)
r
TW
TWr
When all individuals have the same utility function, let say αi = αj (= α), ∀i, j,
we can check that
RWir
pr = pc ,
(19)
which implies that9 T W r = T W c and therefore
·
RWir
¸
Wic
+ (1 − λ)ρi = λRWic + (1 − λ) ρi ,
=λ
TWc
9
(20)
Condition (19) holds true if, for instance, there are ` goods in the economy and the identical
preferences of all consumers is given by a Cobb-Douglas homogenous of degree one or by a CES utility
function.
14
being RWic the relative wealth of the individual i ∈ I in the economy E. Thus, any
desirable relative wealth distribution w = (wi ) ∈ ∆m can be attained in the following
way: define λ∗ ∈]0, 1[ such that
λ∗ max {RWic } < min{wi }.
i∈I
i∈I
∗
This value of λ corresponds to assume A = σ ∗ I with
σ∗ =
λ∗
1 − λ∗
the linear pricing rule. Thus, if we define
wi − λ∗ (RWic )
∈]0, 1[
1 − λ∗
and assign consumption rights according to these percentages, then the desired relative
wealth distribution is attained.
ρ∗i =
6.3
Decentralization
A central question dealing with public choice arises when we try to characterize which
Pareto allocations can be decentralized through the competitive process. We discuss
this very relevant issue for the r-economy whenever we are interested in implementing
a social policy defined ex ante.
In the following, just to illustrate the method, we will consider the simplest pricing
rule
p = qs.
(21)
The redistribution mechanism inherent to this pricing rule can not reach any point
on the contract curve, because, as we already know, the corresponding lump-sum transfer consists on collecting only half of the total resources in the economy and assign them
to individuals according to their percentage of rights10 . Thus, in the extreme case an
individual can be “taxed” with only 50% of his resources, which is not enough to reach
any Pareto allocation.
The following picture of an Edgeworth box, with individual’s initial endowments
given by ω1 and ω2 , illustrates that there are some points on the contract curve (passing
trough CEDF ) that can not be decentralized with any assignment of positive consumption rights according to the rule previously defined. Point A corresponds to ω1 /2, that
is, when r1 = 0 (and therefore r2 = R), whereas point D is ω1 /2 + [ω1 /2 + ω2 /2],
that is, when r1 = R. Just to illustrate the idea, suppose that all supporting lines of
any Pareto belonging to the portion CED of the contract curve are included in the
shadowed region in Figure 1 (which is delimited by the supporting lines passing trough
A and D). In such case, only Pareto points belonging to the CED portion of the curve
10
From (16) with σ = 1, we deduce that ωi (ρ, I) = ωi /2 + ρi /2 ω, which can be interpreted as the
lump-sum indicated.
15
can be decentralized by an adequate distribution of consumption rights. This is the
case, for instance, with allocation E. Conversely, the Pareto allocation F can not be
decentralized because the intersection between its supporting line defined by price pF
does not intersect the shadowed region.
(2)
ω
F
D
pF
B
E
ω/2
p
ω1
C
B
p
A
E
p
A
(1)
Figure 1
Thus, the question of which are the Pareto allocations that can be decentralized
using consumption rights to implement the lump-sum transfers, can be now answered.
From the previous figure, and restricted to the case of positive rights and a pricing rule
as (21), this procedure can not decentralize Pareto optimum allocations that correspond
to a “radical redistribution” of wealth, that is, all those that make a “poor individual”
to become “rich” and a rich to become a poor. This is the case, for instance, with
allocation F in the above figure.
Therefore, the procedure just described reduces the wealth inequality but prohibits
extreme progressive outcomes. This result is in the line with Roemer’s notions of
equalitarian societies (see [22]), since they also correspond to allocations located in
portion CED of the contract curve. Incidentally, it is worth commenting that an
extreme redistribution of wealth, like point F , may be considered irrelevant in practice,
because it is unlikely to become a social choice since it yields the economy without
improving the equity across individual.
6.4
Equalitarian assignment of consumption rights
The equalitarian assignment of consumption rights across the population corresponds
to assume that
ri =
R
,
m
16
where R is the total amount of rights to be distributed and m is the number of agents
in the economy. In this scenario, and assuming again the simplest pricing rule p = qs,
we have that
·
¸
1
ω
ωi (ρ, I) =
ωi +
.
2
m
Let pc the equilibrium price in economy E (the original economy without consumption rights) and let Wic = pc · ωi the corresponding wealth of individual i ∈ I. Since it
is easy to check that
pc · ω
≤ pc · ωi ,
m
that is, ex ante the assignment of consumption rights and the implementation of the
price policy, the consumption bundle ωi (ρ, I) is feasible for an individual i ∈ I if and
only if his wealth is over the average of the consumer’s wealth in economy E. Consequently, once implemented the parallel market, at the new equilibrium will happen
that necessarily the transfers of wealth will be from the rich to the poor consumers,
more precisely, from those above the average per capita wealth to those below this level.
Thus, an equalitarian assignment of consumption rights and the implementation of the
simplest pricing will imply a ”somehow fear” transfer in the economy, independently of
the individual’s preferences. Moreover, if the equilibrium price does not change due to
the implementation of the consumption rights market11 , then it is easy to check that
the relative wealth (see Section 6.2) in the r-economy comply with
pc · ωi (ρ, I) ≤ pc · ωi ⇔
·
¸
1
1
=
RWic +
,
2
m
and therefore the respective variance comply with
RWir
1
var (RWir ) = var (RWic ) .
(22)
4
Considering that the average of the relative wealth is equal ex ante and ex post the
assignment of consumption rights, then identity (22) leads us to conclude that wealth
differences will be significatively reduced -to one forth- by the market mechanism here
described.
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19