1. Theory of Capital Markets
Security Markets IV
Miloslav S Vosvrda
Theory of Capital Markets

2. A Definition of the Economics
economic goods
consumers
firms
Consumption
Let
be the consumption set for consumer i

3. The quantity consumed of good l by consumer i is
represented by
and
To je spotřební koš spotřebitele i.
Consumer i‘s preferences are represented either by
a complete preordering
or by a utility function

4. Consumer i‘s initial endowment is denoted by

5. Production
Let
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 inner product
where
specifies the price vector.

6. The technology of the firm j is represented either
by the production set
or by the production function
When the firms are privately owned,
indicates the share of firm j owned
by consumer i

7. A private property competitive equilibrium
is characterized by a price vector
and an allocation
such that

8. 1)
maximizes profit
in the production set
that is
for any

9. 2)
maximizes utility
in the budget set given by

10. 3)
supply equals demand on all markets

11. A feasible allocation An allocation
is said to be feasible if and only if 1) 2) 3)

12. A Pareto optimum A Pareto optimum is a feasible allocation
such that there exists no other feasible allocation
that would give at least as much utility to all consumers and more utility to at least one consumer

13. that is
and there exist i‘ such that

14. The fundamental theorems of welfare economics
If is strictly increasing w.r.t. each of its
arguments for , a private
property competitive equilibrium (if it exists) is
Pareto optimum.

15. x y , . ; Theorem 2 If is continuous, quasi concave and strictly
increasing if with
and if is convex, ,
for any given Pareto-optimal allocation ,
there exists a price vector
such that x y I J * 1 , . ;

16. x p i I : , . Î × £ ¢ = R 1 (i) maximizes in the set (ii) maximizes inL : , . Î × = + * R 1

17. If we give each consumer an income of
and if we announce to all economic agents the price
vector , profit maximization by firm
and utility maximization by consumer i subject to
the budget constraint
lead to consumption and production plans that are
compatible and that coincide with the chosen
Pareto-optimal allocation.

18. 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.