Handout on General Equilibrium
To this point, we have learned about consumer theory and producer theory, and then focused on finding an
equilibrium in a model with one good. In the study of
, economists analyze how markets for
different commodities interact. We will describe how and why competitive markets are “good” in an economic
0. The Arrow Debreu Model
In this section I informally describe a general equilibrium model. We will then move to a discussion of a simple
exchange model. The simple analysis we will do later for the exchange model can be done in the more general and
complex setting described in this section. Full general equilibrium models are useful for applied areas of economics
such as Macro Economics and International Economics and Public Finance, where economists try to predict the
effect of a policy or event on
market prices and on welfare. For example study the effect of a proposed
tax or of
opening a market for international trade.
In the model, there are:
N Consumers, n = 1,2,.
I Firms, i = 1,2,.
J Goods, j = 1,2,.
A consumption bundle is a list of quantities of each good: X=(x
A production vector Y=(y
represents the result of a production process. It lists the quantities of the
inputs used as a negative number, the quantity of an output produced as a positive number, and zero for any good
that was neither an input nor an output for the firm. For example, if a firm uses 2 units of good 1 and 3 units of
good 2 to produce 1 unit of good 3, we can write the vector (-2,-3,1,0,0,.
An endowment bundle w = (w
) lists the quantities of each good a consumer had before trade begun.
We will assume that there are many firms and many consumers so that they all are “price takers”. That is we
have a competitive model in which consumers and firms think of the market prices as given and do not take into
account how their own actions might change market prices.
Firms: Each firm has a production function. Its goal is to maximize profit.
Consumers: Each consumer has a utility function that represents his preferences over the consumption bundles.
They have initial endowments of goods, and they are owners of shares of the firms. Every consumer’s income is
calculated as the sum for the value of his endowment (for example he could be endowed with labor) and the
profits he earns from owning shares of the firms. Consumers maximize utility subject to their consumption set
(which is what they can afford to buy with their income at the market prices, with some possible other
constraints such as consuming no more than 24 hours per day).
The model has some technical assumptions we will not discuss in detail, such as assuming that the preferences