Download Chapter 2 Models of Economic Systems

Chapter 2
Models of Economic Systems
In this chapter we use correspondences to describe three major parts of an economy: consumers,
producers and markets.
We start our study with the general scheme of making rational decisions by any economic agent
(consumer or producer) in the view of achieving a goal. Then we study a consumer behavior by
supposing knowledge of his preferences over available consumption bundles. We mention three
approaches to model consumer preferences. Excepted the general model of revealed preferences
we study consumer preference relation and also utility approach to consumer behaviour. Under the
consumer’s preferences approach we derive the demand correspondence of an individual consumer.
In the producer theory we describe first the technology of a firm by using the notion of production set. In this general setting we formulate the most often encountered problem of profit
maximization of an individual producer and deduce from this problem the supply correspondence
of a producer.
In the third section of this chapter we begin with study of market system of an economy which
represents a place where consumers (in the role of buyers) contacts producers (in the role of sellers)
in the aim to satisfy their own disparate and often self-interested plans without any coordinating
authorities. We will describe market equilibrium of different market structures. First it will be market of pure exchange economy and then we describe equilibrium of perfectly competitive market
of an economy with production.
2.1 Preferences and choice
Both consumers and producers referred to in the sequel as (economic) agents must make some
rational choices based on their goals and preferences. We will consider an economic agent who
makes the best choice from a collection of mutually exclusive alternatives which are clearly dis23
2.1 Preferences and choice
24
tinguished entities and such collection can be mathematically described (thanks to George Cantor)
as a set of alternatives. Of course, for different economic agents this set can be different but in
this section we consider an arbitrary fixed agent whose set of alternatives will be denoted by X.
Hence the set X includes all potentially (a priori) possible alternatives of the agent. Under various
economic circumstances the agent can be limited to choices from different parts of X, which we
call budget sets and the collection of all economically feasible budget sets B ⊂ X will be denoted
B and so
B ⊂ P0 (X).
Choices of the agent who can choose one best alternative from any budget set B ∈ B can be
described by a choice function
c : B → X, c(B) ∈ B.
When we allow for more than one optimal (i.e. best) choice of the agent we can describe the choice
by a choice correspondence
C : B −−≺ X, B 7→ C(B) ⊂ B.
If we want the choice correspondence to describe a rational behaviour of the agent we should
require some minimal consistency conditions a choice correspondence must satisfy. A natural
assumption about choices of an agent is the following axiom
Weak axiom of revealed preferences (WARP) 2.1. If for some B ∈ B with x, y ∈ B we have
x ∈ C(B), then for any B0 ∈ B with x, y ∈ B0 and y ∈ C(B0 ) we must have also x ∈ C(B0 ).
In words, the weak axiom says that if x is ever chosen when y is available, then there can be no
budget set containing both alternatives for which y is chosen and x is not.
Definition 2.2. A pair (B, C(·)) where C(·) is a choice correspondence on the family of budget sets
of the agent and satisfies the WARP-axiom, is called choice structure (of the agent).
Example 2.3. Let X = {x, y, z}, B := {{x, y}, X} and consider choice correspondences
C1 ({x, y}) := {x} =: C1 (X),
C2 ({x, y}) := {x}, C2 (X) = {x, y},
C3 ({x, y}) := {x}, C3 (X) = {y}.
It is straightforward to verify that C1 satisfies WARP but C2 and C3 do not.
When a choice structure (B, C(.)) is given we can specify binary relation between alternatives
denoted <? and defined by
x <? y ⇔ ∃B ∈ B : x, y ∈ B & x ∈ C(B).
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We call this relation revealed preference relation. We read x <? y as x is revealed as good as y.
We will say that x is revealed preferred to y and write x ? y if the right-hand-side of the following
equivalence holds true
x ? ⇔ ∃B ∈ B : x, y ∈ B & x ∈ C(B) & y < C(B),
i.e. if x is ever chosen over y when both are feasible.
The revealed preference relation has not the fundamental properties needed in majority of economic theories, namely transitivity and completeness. So it is more common to introduce the
preference relation over alternatives as follows.
Definition 2.4. Preference relation on a set X of alternatives is a binary relation < on X having
two properties for each x, y ∈ X:
1. x , y ⇒ x < y or y < x
2. x < y, y < z ⇒ x < z
(connectedness)
(transitivity)
If moreover < verifies for each x ∈ X the condition
3. x < x
(reflexivity)
we call it weak preference relation on X.
If preference relation verifies the irreflexivity condition
3.’ x % x
(∀x ∈ X)
we call it strict preference relation on X.
We read x < y as x is preferred to y or x is at least as good as y. If < is weak (strict) preference
relation we can read x < y as x is weakly (strictly) preferred to y.
A preference relation < on X need not to be neither week preference nor strict preference relation.
But it can be enlarged to a week preference relation on X by adding to the relation the diagonal on
X, ∆X . It is the smallest weak preference relation containing <.
To every preference relation < we can associate so called indifference relation denoted ∼ and
defined by
x ∼ y ⇔ x < y & y < x.
We read it x is indifferent to y or the agent is indifferent between the alternatives x and y. It is easy
to see that the associated indifference relation is an equivalence relation iff < is a weak preference
relation and it is the maximal equivalence relation contained in the given weak preference relation
<.
Similarly, to any preference relation < we can assign the biggest strict preference relation contained
in < if we subtract from < the diagonal ∆X . However, it is much more common to define the strict
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preference relation associated to the preference relation < as the smallest asymmetric part of the
preference relation denoted and defined by
x y ⇔ x < y & y % x ⇔ x < y & x / y.
Any preference relation is hence the disjoint union of the associated indifference relation and the
associated strict preference relation i.e.
< =
∪∼
and because of this it is quite natural to denote a preference relation by the symbol %.
In the sequel we use this notation exclusively when < is a weak preference relation i.e. when ∼ is
an equivalence relation.
To any preference relation we can associate a choice structure. The interested reader can find
the details in the book [32]. Rather than study the relationship between preferences and choice
structure we introduce the third but the most ancient approach to model the rational behaviour
of an economic agent by mathematical tools. At the dawn of economic theories of consumer
behaviour the economists were persuaded that one can assign to each decision alternative x ∈ X
a real number u(x) which expresses the importance of the alternative to the economic agent. It is
quite natural to assume that the greater the utility value u(x), the more important is the alternative
to the economic agent. We can formalize the notion of utility by the following definition.
Definition 2.5. For any real function u : X → R the binary relation % on X defined by
x % y ⇔ u(x) ≥ u(y)
(?)
is weak preference relation induced by the function u which itself is called utility function of the
induced preference relation.
A natural question arises whether each weak preference relation % on X can be induced by a
utility function u : X → R verifying (?). If such a function exists it is called utility function
representing the preference relation or just utility function of the preference relation.
The next example shows that there exists a weak preference relation which cannot be represented
by a utility function. If a representing utility function u exists it is never unique because for any
function f : R → R which is strictly increasing on u[X] i.e.
u(x) > u(x0 ) ⇒ f (u(x)) > f (u(x0 )),
the function
f ◦u: X →R
is again a utility function representing the same preference relation as u. We leave to an interested
reader to prove in Exercise 2.1 that the reciprocal statement is also true, i.e. when v represents the
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same preference relation as the utility function u does, then there is a strictly increasing function
f : u[X] → R such that v = f ◦ u.
Example 2.6. Let X = R2 and the preference relation on X be the lexicographic ordering defined
for any x = (x1 , x2 ), y = (y1 , y2 ) ∈ R2 by
(x1 , x2 ) < (y1 , y2 ) ⇔ (x1 > y1 ) or (x1 = y1 & x2 ≥ y2 ).
This weak preference relation cannot be represented by any real function u : X → R. Suppose
there is a function u : X → R satisfying
x < y ⇔ u(x) ≥ u(y).
Then for every x1 ∈ R we can pick a rational number r(x1 ) such that
u(x1 , 2) > r(x1 ) > u(x1 , 1).
The definition of lexicographic preference assures
x1 > x̂1 ⇒ r(x1 ) > r( x̂1 )
since r(x1 ) > u(x1 , 1) > u( x̂1 , 2) > r( x̂1 ). Hence r : R → Q is injective function of all reals R into
the countable set Q of all rationals. But it would be possible only if the set of all reals is countable
and it is well-known that R is not a countable set.
Under some additional assumptions there exists a utility function representing given preference
relation.
THEOREM 2.7. For any weak preference relation % on a countable set X = {x1 , x2 , . . . } there
exists a representing utility function u : X → R.
Proof. Let us remark that for any utility function we have
u(x) = u(y) ⇔ x ∼ y,
u(x) > u(y) ⇔ x y.
So to determine the values u(x) for any x ∈ X it suffices to define the values of u on indifference
classes
∼ (x) = {y ∈ X| y ∼ x}.
If the number of indifference classes is finite, let us say C1 , C2 , . . . Ck , we can order them linearly
by the strict linear order > defined by
Ci > C j ⇔ ∃x ∈ Ci , y ∈ C j : x y.
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(†)
2.1 Preferences and choice
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We can define the function u : X → R by
u(x) = card{i ≤ k | Ci < ∼ (x)}.
It is straightforward to show that u is utility function for the preference relation %. When the factor
set X/∼ is not finite, it must be countable, i.e.
X/∼ = {C1 , C2 , . . . }.
Let us define first the indicator function



1, if C j > Ci ,
ri j = 

0, otherwise.
Then we can define the real-valued function f : X/∼ → R by the formula
f (C j ) :=
∞
X
2 − j ri j ,
j=1
and it is easy to show that f represents the strict linear order > on X/∼ defined by (†). Now it
suffices to define on X the utility function by
u(x) := f (C j ), when x ∈ C j ,
and it is evident that this function represents the given preference relation.
Remark 2.8. From the proof of Theorem 2.7 one can see that the utility function exists for any set
X and the preference relation % with countable factor set X/∼ .
When X has a topological structure we can prove the existence of continuous utility function
provided the preference relation is continuous in the following sense.
Definition 2.9. Preference relation % on a topological space X is said to be continuous if it is
preserved under limits which is equivalent with closedness of % in the topological product space
X 2 . So for any convergent sequence (xn , yn ) ∈ X 2 , (n = 1, 2, . . . ) with xn % yn for all n, if
x := lim xn ,
n→∞
y := lim yn ,
n→∞
then x % y.
Continuity says that the agent’s preferences cannot exhibit jumps in the sense that the agent
preferring each element of the sequence (xn )∞
n=1 to the corresponding element in the sequence
n ∞
(y )n=1 , cannot suddenly reversing his preferences at the limit points of these sequences x and y.
For example lexicographic preferences are not continuous since (1/n, 0) (0, 1) for every natural
number n but
!
1
lim , 0 = (0, 0) ≺ (0, 1).
n→∞ n
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Equivalent characterization of the continuity of % is that for each x ∈ X both upper and lower
contour sets
% (x) := {y ∈ X | y % x}, (x) %:= {y ∈ X | x % y}
(‡)
are closed in X. Since the indifference relation ∼ associated to a continuous preference relation %
is a closed subset in X 2 , the continuity of % is equivalent to the openedness in X 2 of the following
sets
(x) := {y ∈ X | y x}, (x) := {y ∈ X | x y},
which characterization we will use later.
The proof of the existence of utility function representing a continuous preference relation is
quite technical so we prove the existence of utility function in a special but the most current case
when the set of alternatives is the commodity space X = Rn+ and the preference relation % on Rn+ is
strictly monotone, i.e. for any x, y ∈ Rn+ the following implication holds
x > y ⇒ x y.
Strict monotonicity says that the consumer prefers more to less of each commodity.
THEOREM 2.10. Suppose that a preference relation % on Rn+ is continuous and strictly monotone.
Then there is a continuous utility function u that represents %.
Proof. Let us consider the diagonal ray D in Rn+ , i.e. the locus of vectors with all n components
equal
D := {x ∈ Rn+ : x1 = x2 = · · · = xn }.
If we denote 1 the n-vector whose components are all 1, then α1 ∈ D for any positive scalar α.
We prove that for any x ∈ Rn+ the indifference set ∼ (x) hits the diagonal D. Note first that the
monotonicity and continuity of the preference relation implies that for any x ∈ Rn+ we have x % 0.
Also note that for any ᾱ with ᾱ1 ≥ x we have ᾱ1 % x. Consequently the sets
A− := {α− ∈ R | x % α− 1}, A+ := {α+ ∈ R | α+ 1 % x}
are nonempty and closed because of the continuity of %. Also A− ∪ A+ = R by the completeness
of %. Obviously, for any α− ∈ A− , α+ ∈ A+ we have α− ≤ α+ and so A− and A+ are bounded from
above and below, respectively, which together with their closedness gives the inequality
max A− ≤ min A+ .
If this inequality were strict, then we would have for some α > 0, x ≺ α1 ≺ x, a contradiction.
Therefore there is a unique value
α(x) = max A− = min A+ ,
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2.1 Preferences and choice
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for which α(x) % x % α(x)1, so α(x)1 ∼ x.
We will show that the function
u : x 7→ α(x)
is continuous utility function representing %. That u represents % follows from its construction,
since if we suppose u(x) ≥ u(y), then by monotonicity of % we have u(x)1 % u(y)1, and since
x ∼ u(x)1 % u(y)1 ∼ y we have x % y. On the other hand if x % y then
u(x)1 ∼ x % y ∼ u(y)1,
and by monotonicity we can conclude that u(x) ≥ u(y). Hence we have proved that
u(x) ≥ u(y) ⇔ x % y.
To prove the continuity of u : Rn+ → R it suffices to show that pre-image of any open interval (a, b)
under u is an open set. We have
u−1 [(a, b)] = {x ∈ Rn+ | a1 < u(x) < b} = {x ∈ Rn+ | a1 < u(x)1 < b1},
and since u(x)1 ∼ x we have
u−1 [(a, b)] = {x ∈ Rn+ | a1 ≺ x ≺ b1} = {x ∈ Rn+ | a1 ≺ x} ∩ {x ∈ Rn+ | x ≺ b1}.
Both sets in the intersection are open because of the continuity of preference relation. Hence also
u−1 [(a, b)] is open in the definition domain Rn+ which means the continuity of the utility function
u.
Consumer demand
To describe a consumer behaviour we will specify his set of alternatives X to be a set of commodities (i.e. goods and services) which are physically (potentially) available to a consumer, i.e. we
will suppose that
X ⊂ R+N
and call it consumption set. The elements of X are called consumption bundles. Excepted this
physical constraints on consumer’s consumption set there are economic constraints which consumer must consider in making his optimal decisions. The most important economic constraints
are prices of commodities available for consumption. So we will suppose that in the economy to
each commodity i = 1, 2, . . . , N we assign a real number pi ∈ R called price of the commodity. By
convention, for commodities which are scarce goods, i.e. they are demanded by the consumer, we
suppose pi > 0, for the commodities which are noxious commodities (called also bads), i.e. they
are consumed by the consumer because of different environmental circumstances which obligate
the consumer to consume them, we set negative price pi < 0. We set the zero price pi = 0 for
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2.1 Preferences and choice
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that commodities which are inevitable for the existence of the consumer or that are available in
any demanded quantity and call them free commodities. We will consider the price system of all
commodities in the economy as the vector
p := (p1 , . . . , pN ) ∈ RN ,
and call it price vector. If we consider only commodities which are scarce goods then we can
suppose p ∈ R+N . We will suppose that prices form a prevailing price system. It means that if a
consumer want to obtain one unit of a good j he must either pay p j monetary units or he must
exchange p j /pi units of the good i. The second property of prevailing price system is its linearity,
i.e. consumer can obtain a consumption bundle x = (x1 , . . . , xN ) ∈ X only if he pays the value of
the bundle which equals
N
X
p1 x1 + · · · + pN xN =
pi xi =: p · x,
0
or he can exchange a consumption bundle x =
i=1
0
(x1 , . . . , x0N )
with equal value
p · x = p · x0 .
Often a price system is normalized by a price of a fixed good called numéraire and then we
can suppose that
N
X
pi = 1.
i=1
To describe the economically feasible consumption bundles we will suppose the consumer is endowed by an initial amount of money denoted w (wealth) and considered positive, i.e. w > 0. Then
for a given price system p the consumer budget set is defined by
B(p, w) := {x ∈ X | p · x ≤ w}.
When X = R+N and p > 0 this set is called Walrasian budget set. So for each price - wealth situation
we can consider the consumer budget-correspondence as
B : R+N+1 −−≺ R+N , (p, w) 7→ B(p, w),
which represents physically and economically feasible consumption bundles of the consumer. The
consumer makes his rational decisions among the bundles of his budget set B(p, w) in the way
to maximize his satisfaction. If we suppose the satisfaction of consumer being represented by a
preference relation % then the optimal consumption bundles form the set
C ? (p, w) := {x? ∈ B(p, w) | x? % x, ∀x ∈ B(p, w)}
of greatest elements of the budget set B(p, w).
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2.2 Technologies and production
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If we denote E := {(p, w) ∈ R+N+1 | C ? (p, w) , ∅} then the correspondence
D : E −−≺ RN , (p, w) 7→ C ? (p, w)
is called the demand correspondence (of the consumer).
If the consumer satisfaction is measured by a utility function u, then the demand correspondence
is given by
D(p, w) = Argmax{u(x) | x ∈ B(p, w)}.
2.2 Technologies and production
Another important part of an economic system is its supply side. The description of the supply side
is the subject of (economic) production theory or theory of a firm. This economic theory studies
the process by which goods and services are produced in the aim to be delivered to a consumer.
The supply side of an economy can be viewed as composed of a number of productive units we
shall call firms. Firms can be corporations or other legally recognized businesses which represent
the productive possibilities of individuals or households. Firms can have many aspects but we will
confine our attention only to the aspect which specifies what are the production possibilities of a
firm. Then we are going to view a firm as a "black box" able to transform inputs into outputs.
One way how to describe the production possibilities of a given firm is by introducing the firm’s
production possibility set which represents the production activities or production plans that are
technologically feasible for the firm.
Consider an economy with L commodities which can serve as inputs and /or outputs of a
production process provides by a firm. If the firm uses y(i)
j units of the commodity j as an input and
(o)
(o)
produces y j units of the same commodity as an output (while y(i)
j , y j ≥ 0) then the net output of
the commodity is given by
(i)
y j := y(o)
j − yj .
If the net output of a commodity is positive, then the firm is producing more of the commodity
than it uses as an input; if the net output is negative, then the firm is using more of the commodity
than it produces; if the net output is zero, then the production process has no net output of the
commodity.
Considering the net outputs of all L commodities we can form the production plan or production
vector
y := (y1 , . . . , yL ) ∈ RL ,
that describes the net outputs from a production process.
The set of all technologically feasible production plans (of a given firm) is called the firm’s production (possibility) set and will be denoted by Y, a subset of RL . The set Y is supposed to describe
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2.2 Technologies and production
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all patterns of inputs and outputs that are technologically feasible. It gives a complete description
of the technological possibilities facing the firm. Any y ∈ Y is possible and any y < Y is not.
The production possibility set is limited first and foremost by technological constraints. However,
in any particular model, legal restrictions or prior contractual commitments may also contribute to
the determination of the production set. Often we want to distinguish between production plans
that are "immediately" feasible and those that are "eventually" feasible. For example, if in a short
run, some inputs or outputs are fixed by a contractual commitments to take the prescribed values
ȳ j , j ∈ S ⊂ {1, 2, . . . , L}, the immediately feasible production plans are those vectors y ∈ Y for
which
y j = ȳ j , ( j ∈ S ).
So if we denote
ȳS := (ȳ j | j ∈ S ),
we can consider the subset of the production possibility set
YS := {y ∈ Y | ∀ j ∈ S : y j = ȳ j } =: Y(ȳS ),
and this restricted set is called short-run production set and the set Y itself is then called long-run
production set.
It is sometimes convenient to describe the production set Y ⊂ RL using a function F : RL → R
such that
Y = {y ∈ RL | F(y) ≤ 0}
and
F(y) = 0 ⇔ y ∈ ∂Y − boundary of Y.
Such function is called transformation function and the set of boundary points
∂Y = {y ∈ Y | F(y) = 0}
is called the transformation frontier.
Technologies with distinct inputs and outputs
In many real production processes the set of goods that can be inputs
S i := { j ≤ L | ∃y ∈ Y : y j ≤ 0}
is distinct from the set of goods that can be outputs
S o := { j ≤ L | ∃y ∈ Y : y j > 0}
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2.2 Technologies and production
34
in the strong sense of the word "distinct" i.e.
S i ∩ S o = ∅.
If we denote cardS i =: m, cardS o =: n, then m + n = L and we can split any production vector
y ∈ Y into two sub-vectors:
(y j : j ∈ S i ) =: yi ∈ Rm− ,
(y j : j ∈ S o ) =: yo ∈ Rn+ ,
which concatenation gives y, i.e.
yi ∨ yo = y.
In the case of clear distinction of inputs from outputs it is common to measure inputs by their
magnitude and we denote
x = −yi ∈ Rm+
the vector of nonnegative input levels from which the outputs yo can be produced, i.e. (−x, yo ) ∈ Y.
If we denote
A := {x ∈ Rm+ | ∃q ∈ Rn+ : (−x, q) ∈ Y}
we can identify the production set Y ⊂ Rm− × Rn+ with the correspondence
Q : A −−≺ Rn+ , Q(x) := {q ∈ Rn+ | (−x, q) ∈ Y}
which we call production correspondence. For any q ∈ Rn+ the set Q−1 (q) is called input requirement set for the outputs levels q. One of the most frequently encountered production models is
that in which there is a single output q ∈ R+ . A single-output production technology Y ⊂ Rm− × R+
is commonly described by means of production function f (·) which gives the maximum amount
q of output that can be produced using input amounts x = (x1 , . . . , xN−1 ) ∈ A. Hence
f : A 7→ R+ , f (x) := max Q(x).
For single-output production technology the set f −1 (q) is called isoquant. It can be characterized
by means of input requirement set
f −1 (q) = {x ∈ R+N−1 | x ∈ Q−1 (q) and x < Q−1 (q0 ) for q0 > q}.
There is an N-dimensional analogy of a production function that is useful in general equilibrium
theory. We say that a production plan y ∈ Y is technologically efficient (or Pareto efficient) if there
is no way to produce more outputs with the same inputs or to produce the same output with less
inputs. We often assume that we can describe the set of technologically efficient production plans
by a function T : RL → R such that
T (y) = 0 iff y is Pareto efficient.
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2.2 Technologies and production
35
Just as production function picks out the maximum scalar output as a function of inputs, the Pareto
efficiency function picks out the maximal vectors of net outputs.
More properties of production set are discussed in the monograph [32] but we do not need them
in the sequel and leave the details for the interested reader. Instead of detailed analysis of properties of production sets we describe how an individual producer behaves in perfectly competitive
environment.
Producer supply
The most common goal of any producer is the profit maximization based on his production activities represented by its production set Y ⊂ RN . To evaluate the profit of a producer we need to know
prices of inputs and outputs. So we will suppose each commodity k ∈ {1, 2, . . . , N} has fixed price
pk ∈ R. Usually, we will suppose a given price system to be non-negative, i.e.
p := (p1 , . . . , pN ) ∈ R+N .
Under the given (exogenous) price system an individual producer with the production set Y ⊂ RN
solves the profit maximization problem
max{p · y | y ∈ Y} =: π(p).
This problem can be reformulated in terms of transformation function F:
π(p) := max{p · y | F(y) ≤ 0}.
Let us denote
Y ? (p) := {y? ∈ Y | ∀y ∈ Y : p · y? ≥ p · y},
i.e.
Y ? (p) = Argmax{p · y | F(y) ≤ 0}.
Hence the elements of Y ? (p) are the optimal production plans of the producer called also producer
equilibrium. Let us remark that it can happen that Y ? (p) = ∅, i.e. there is no equilibrium in
production which means that
Sup{p · y | y ∈ Y} = +∞.
If we denote
P := {p ∈ RN | Y ? (p) , ∅}
then we can define the profit function
π : P → R, p 7→ π(p),
an the supply correspondence
S : P −−≺ Y, p 7→ S (p) := {y ∈ Y | p · y = π(p)}.
Let us remark that any equilibrium production plan y? ∈ S (p) is rather the firm net supply to the
market, since the negative entries of y? should be interpreted as demand for inputs.
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2.3 Pure exchange economy
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2.3 Pure exchange economy
We have described in the previous sections rational behaviour of an individual consumer and an
individual producer. An economy is composed of finite set of economic agents who can play
the role of consumer or the role of producer or both. In a decentralized economy agents acts
in their own interests however an agent decision must take into consideration the actions of the
other agents. Market structure serves to consolidate the individual actions of economic agents.
In this section we describe the simplest market structure which allows to consolidate interactions
among economic agents who are considered to be consumers only. Hence there is no production
in such economy. We denote I the finite number of consumers in the economy and any individual
consumer will be labelled by an index
i ∈ {1, 2, . . . , I} =: I.
Each consumer i ∈ I has a consumption set Xi ⊂ RL of physically available consumption bundles.
The agent i has preferences over his consumption bundles represented by a preference relation
%i on Xi , i.e. complete and transitive binary relation. Since there is no production in this simple
economy we will suppose that each consumer i owns certain quantity ωik ∈ R of the kth commodity
for k = 1, 2, . . . , L which he can voluntarily exchange with other consumer (if they agree on the
exchange). Hence we are going to suppose that every consumer i ∈ I is equipped with an initial
consumption bundle
ωi = (ωi1 , . . . , ωiL ) ∈ Xi ,
which we call initial endowment of the consumer i.
All the given data of this economy are resembled in the following definition.
Definition 2.11. Pure exchange economy is the system
E0 = (I, X, %, ω),
where
I = {1, 2, . . . , I} − set of agents
X : I −−≺ RL , i 7→ Xi ⊂ RL consumption set correspondence
%: I −−≺ RL × RL , i 7→%i ⊂ Xi × Xi − preference relation correspondence
ω : I −−≺ RL , i 7→ ωi ∈ Xi − initial endowment choice function for X
Equilibrium in pure exchange economy
In pure exchange economy barter exchange can lead to several reallocations of the initial endowment system (ω1 , . . . , ωI ). An I-tuple of consumption bundles
x := (x1 , . . . , xI ) ∈ X1 × · · · × XI
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2.4 Economy with production
37
is called feasible allocation in the economy if it satisfies the equation
X
X
xi =
ωi .
i∈I
i∈I
The set of all feasible allocations will be denoted F(ω). It is non-empty set because the initial
endowment ω is obviously a feasible allocation called also initial allocation. The feasible choices
of each consumer i ∈ I are hence the elements of the budget set, i.e.
Bi := pri F(ω) ⊂ Xi ,
where
pri :
Y
Xi → Xi , (x1 , . . . , xI ) 7→ xi
i∈I
is the projection function on ith component set. Rational consumer chooses the most preferred
consumption bundle x̄i ∈ Bi , i.e.
∀xi ∈ Bi : x̄i % xi .
It is not obvious if such optimal consumption bundle exists and it is even less obvious whether
there is an "optimal" feasible allocation
x̂ = ( x̂1 , . . . , x̂I ) ∈ F(ω)
called general equilibrium in pure exchange economy. This difficult question was answered positively by Walras introducing a prevailing price system in the economy and we will show it later
after having established necessary mathematical properties of correspondences.
2.4 Economy with production
Now we enlarge the pure exchange economy E0 by the production sector which will be composed
by J firms that are labeled by an index j ∈ {1, 2, . . . , J} =: J. Each firm j ∈ J has its own production
set Y j ⊂ RL . We will suppose that in the production sector individual firms can create certain
coalitions modelled by non-empty subsets of the set J and the family of all available coalitions
will be denoted by C ⊂ P0 (J). Production possibilities of any coalition K ∈ C are given by a set
Y(K) ⊂ RL and the production possibilities of all coalitions in the production sector of the economy
can be described by a production set correspondence
Y : C −−≺ RL
that assigns to each coalition K ∈ C its production (possibility) set Y(K). The coalition production
set Y(K) need not to be simple sum of the individual firm’s production sets Y j , ( j ∈ K) because of
the economies of scale or other economic impacts of forming coalitions.
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2.4 Economy with production
38
Like in the case of individual firms, we will suppose that the economic interest of any coalition of
firms K ∈ C is to maximize its profit
max{p · y | y ∈ Y(K)} =: πK (p),
where p is prevailing price system of commodities in the economy which is supposed to be independent of any activity of the production sector. The production plans which realizes this maximal
profit form the supply side of the economy and can be denoted
Y ? (p) := {y? ∈ Y(K) | p · y? = πK (p)},
i.e.
Y p? (K) = Argmax{p · y | y ∈ Y(K)}.
If we denote
P := {p ∈ RL | ∃K ∈ C : Y ? (K) , ∅},
then we can consider the supply correspondence defined by
[
S : P −−≺ RL , p 7→ S (p) :=
Y p? (K).
K∈C
To relate consumers and producers we will suppose that each consumer is a shareholder of any
producers coalition and the share of ith consumer in the coalition K ∈ C is given by a ratio siK ∈
[0, 1] on the profit πK . We will suppose that the profit of each coalition is distributed entirely
among consumers, i.e.
I
X
siK = 1.
i=1
If we denote
σ := (siK | (i, K) ∈ I × C)
the share system in the economy, then the concatenation of the pure exchange economy E0 =
(I, X, %, ω) and the production sector (J, C, Y, σ), i.e.
E = E0 ∨ (J, C, Y, σ)
is called production economy.
Equilibrium in production economy
To describe the equilibrium notion in production economy E we define the wealth of ith consumer
(i ∈ I) as the sum of the value of his initial endowment ωi and the total value of his shares in the
production sector, i.e.
X
i
wi = p · ω +
siK πK .
K∈C
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2.4 Economy with production
39
The wealth is obviously a function of the prevailing price system p ∈ P.
The demand in the production economy of the ith consumer is
Di (p) = max Bi (p, wi )
where
Bi (p, wi ) = {x ∈ Xi | p · x ≤ wi }
is the ith consumer budget set. The total demand in the economy is
D(p) =
I
[
Di (p).
i=1
We call an equilibrium price system of the production economy E such prevailing price system
p? ∈ P that equalizes total demand and supply in the economy, i.e.
D(p? ) = S (p? ).
The existence of an equilibrium of any production economy will be shown later using Brower fixed
point theorem.
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