Walrasian general equilibrium model

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The Walrasian general equilibrium model is a form of the general equilibrium model named after Leon Walras in economic theory . The model is based on the idea that the quantities of all goods offered and demanded by households and companies in an economy depend on the prices of all these goods. Whether there can be an equilibrium in all markets can therefore only be decided by looking at all markets simultaneously.

Excess demand

If one assumes that the optimization problems of all households and companies have a clear solution for every price constellation, there is a function for every household h and every company u, which assigns the associated optimal consumption or production plan to each price constellation. However, since it can depend on prices whether someone offers or asks for a certain good, and households can appear not only as consumers but also as providers of the resources belonging to them, the term excess demand is used instead of supply and demand functions. If this is positive for a certain good and a certain economic subject, the corresponding amount is requested, if it is negative, it is offered.

If there is a total of n different, consecutively numbered goods, a price constellation is created by a price vector

and for each good g the excess demand aggregated across all households and companies is a function

of the entire price vector p.

If one writes the vector of the aggregated excess demand as

,

Thus one has defined a function z which assigns to each price vector a vector of the aggregated excess demand quantities, and a general market equilibrium can be defined as a price vector p * for which z (p *) = 0 holds - or, in other words, as one Price constellation in which aggregated supply and aggregated demand are the same for every good. (For formal reasons, it is sometimes permitted for a free good whose price is zero that there could be an oversupply in equilibrium.)

existence

The question of the existence of a (general market) equilibrium - mathematically the question of whether z (p) = 0 has a solution - can be answered in the affirmative under the following conditions:

Continuity of z: The (aggregated) excess demand function z is a continuous function.

Homogeneity of z: If all prices are multiplied by a positive number k, excess demand does not change. Formally: z (kp) = z (p).

Walras' law: The value of all aggregated excess demand, added up over all goods, must always be zero. Formal: pz (p) = 0.

Uniqueness

Due to the homogeneity of the excess demand, the equilibrium prices cannot be unambiguous - taken literally - if z (p *) = 0, then z (kp *) = 0 also applies. However, the relative prices, i.e. the quotients from the respective equilibrium goods prices, could be clear. A sufficient condition for this is the so-called gross substitutability of the goods:

If and for all , then applies: for all (and then based on Walras' law ).

stability

If, in the sense of a tâtonnement process, it is assumed that out of equilibrium the price of a good always increases when there is a (positive) surplus demand for this good, while it falls when there is a surplus supply, it can be shown that gross substitutability of all goods is also stability of equilibrium. Deviations from equilibrium then lead back to equilibrium.

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