Modeling an Economy of Exchanges
Imagine a situation where people traded goods in a closed economy. In ancient times, this may have been an open-air market, where the community would gather together and trade items. The most important thing here is that there is no production, in that no new goods are being created. Although we can certainly model this, we will focus here on an exchange economy without production to begin.
To fix ideas, assume the economy consists of only everyone in this room, and the only goods in our utility functions are apples and bananas. Say someone comes in and arbitrarily gives us a bunch of different bundles, so maybe I have 0 bananas and 3 apples, maybe you have 5 bananas and 0 apples, etc. If our utility functions are dependent on both goods (for instance, u(x) = x1x2), immediately you can see we can be better off trading. If we didn’t, we would both have utility 0. The motivation behind general equilibrium thus follows. We need to find the allocations xθi in the end after trade, and the corresponding prices that dictate the trade. Generally, in computation, the prices are all that we want, and then we just do some algebra to get the allocations (because they were solved from first order maximization conditions).
We begin with n goods and Θ consumers. They are each endowed with goods, ωiθ, and we need to determine the equilibrium allocation and prices.
We begin with market clearing, and talk about the properties of an exchange economy. We will state propositions without their proofs for brevity.