Security Markets III MSV Teorie finančních trhů

Definice ekonomiky economic goods consumers firms Let be the consumption set for consumer i Consumption

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

Consumer i‘s initial endowment is denoted by

Production 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 Let where specifies the price vector.

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

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

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

maximizes utility in the budget set given by 2)

3) supply equals demand on all markets

Předpoklad a private property competitive equilibrium vyjadřuje tu skutečnost, že každý agent bere ceny jako dané. To znamená, že při modelování ekonomického chování je každý jednotlivý agent zanedbatelný.

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

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

that is and there exist i‘ such that

The fundamental theorems of welfare economics Theorem 1 If is strictly increasing w.r.t. each of its arguments for, a private property competitive equilibrium (if it exists) is Pareto optimum.

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 xxyy IJ  11,...,;,...,

(i) maximizes in the set (ii) maximizes in xx pxpx iI iiL ii :,,...,,      R 1,

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.

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.