Security Markets
IV
Miloslav S Vosvrda
Theory of Capital Markets
A Definition of the Economics
• economic goods l  1,...,
i  1,..., I
• consumers
• firms
j  1,..., J
L
Consumption
Let X i be the consumption set for consumer i
L
X R

i
The quantity consumed of good l by consumer i is
i
x
represented by l
and
x  ( x ,, x )  X
i
i
1
i
L
i
To je spotřební koš spotřebitele i.
Consumer i‘s preferences are represented either by
i
a complete preordering R
i
U
or by a utility function
i1
i2
 U (x )  U (x )
i1
i2
 U (x )  U (x )
x Rx
x Px
i
i
i1
i1
i
i
i2
i2
Consumer i‘s initial endowment is denoted by
L
w R

i
Production
Let
y  ( y ,, y )
j
j
1
j
L
be the production vector for producer j.
Outputs(products) have a positive sign
Inputs(factors) have a negative sign.
Then, the profit of firm j can be written as the
L
inner product
p y
where
p R L

j

py
l
j
l
l 1
specifies the price vector.
The technology of the firm j is represented either
by the production set
Y R
j
L
or by the production function
f (y )  0
j
j
When the firms are privately owned,

ij
indicates the share of firm j owned
by consumer i
j  1, ... , J
i  1,..., I
A private property competitive
equilibrium
L
p

R
is characterized by a price vector


and an allocation
i
I
1
J
( x ,, x ; y ,, y )
such that
1)
y
j

p y
maximizes profit
in the production set
Y
j
j
that is

p y
for any
j
 p y
y Y
j

j
j
j  1, ... , J
2)
x
i
i
i
U (x )
maximizes utility
in the budget set given
by
J
B  {x : x  X and p  x  p  w    p y }
i
i
i
i

i

i
j 1
i  1,..., I
ij

j
3)
supply equals demand on all markets
I
x
i 1
i
J
I
  y w
j 1
j
i 1
i
A feasible allocation
An allocation
i
I
1
J
( x ,, x ; y ,, y )
is said to be feasible if and only if
1)
2)
3)
x X
j
j
y Y
i
I
i
i  1,..., I
j  1,..., J
J
I
x   y w
i 1
i
j 1
j
i 1
i
A Pareto optimum
A Pareto optimum is a feasible allocation
i
I
1
J
( x ,, x ; y ,, y )
such that there exists no other feasible allocation
i
I ~1
J
~
~
~
( x ,, x ; y ,, y )
that would give at least as much utility to all
consumers and more utility to at least one consumer
that is
i
i
i
~
U (x )  U (x )
i
i  1,..., I
and there exist i‘ such that
i'
i'
i '
~
U (x )  U (x )
i'
The fundamental theorems of
welfare economics
Theorem 1
i
If U (.) is strictly increasing w.r.t. each of its
arguments for i  1,..., I , a private
property competitive equilibrium (if it exists) is
Pareto optimum.
Theorem 2
If U i  is continuous, quasi concave and strictly
i
L
increasing if X  R  with
w  0, l  1,..., L, i  1,... I ,
i
l
j
Y
and if
is convex, j   1,..., J
for any given Pareto-optimal allocation
1
I
1
x ,..., x ; y ,..., y
there exists a price vector

p R
such that
L

J
,
,
x
(i)
i
maximizes

i
i
U ( x ) in the set

i
x : x R , p  x  p  x , i  1,..., I,
i
(ii)
i
y
L

j
i
maximizes
in

p y
j
Y , j  1,..., J .
j
If we give each consumer an income of
i

R  p x
i
and if we announce to all economic agents the price
*
p
vector
, profit maximization by firm
j , j  1, J
and utility maximization by consumer i subject to

i
i
the budget constraint p  x  R , i  1,, I
lead to consumption and production plans that are
compatible and that coincide with the chosen
Pareto-optimal allocation.
Pareto optimality of
the private property competitive equilibrium
is satisfactory with respect to the efficiency criterion,
but it may lead to undesirable income distributions.
Whichever Pareto optimum we wish to decentralize
(therefore which ever Pareto optimum corresponds
to the equity criterion taken), it is possible to
decentralize this allocation as a competitive
equilibrium so long as the income of the agents is
chosen appropriately.