Question

Assume a two-person, two-good economy. Holding the price of one commodity as numéraire, describe analytically how the price mechanism coordinates the economy. Derive Walras’ law for the two-good economy and discuss its policy implications.

Answer 1

In general, the incomes of individuals depend on the prices of goods and services that they have to sell. Therefore in the study of general equilibrium theory, we need to make incomes depend on the prices of commodities.

This is nicely illustrated in the example of a pure exchange economy where there is no production, but agents have initial endowments of goods which can they bring to market and trade with each other.

Each consumer initially has some vector of endowments of goods.

These goods are traded at competitive prices and in equilibrium the total demand for each good is equal to the supply of that good.

A Pure Exchange Economy

There are m consumers and n goods. Consumer i has a utility function u_i (x_i ) where x_i is the bundle of goods consumed by consumer i. In a competitive market, Consumer i has an initial endowment of goods which is given by the vector ?_i ? 0.

Where p is the vector of prices for the n goods,

consumer i’s budget constraint is px_i ? p?_i which simply says that the value at prices p of what he consumes cannot exceed the value of his endowment.

Consumer i chooses the consumption vector D_i (p) that solves this maximization problem.

Where x_i (p, m_i ) is i’s Marshallian demand curve, we 1 have

D_i (p) = x_i (p, p?_i ).

Let us denote i’s demand for good j by D_ij (p),

which is the jth component of the vector D_i (p).

A pure exchange equilibrium occurs at a price ¯p such that total demand for each good equals total supply. This means that

for all j = 1, . . . n.

This vector equation can be thought of as n simultaneous equations, one for each good.

Finding a competitive equilibrium price amounts to solving these n equations in n unknowns. There are two important facts that simplify this task if the number of commodities is small.

Homogeneity and a numeraire

The first is that the functions D_i (p) are all homogeneous of degree zero in prices and hence, so is P_i D_i (p). To see this, note that if you multiply all prices by the same amount, you do not change the budget constraint (since if px_i = p?_i , then it must also be that kpx_i = kp?_i for all k > 0. Therefore we can set one of our prices equal to 1 and solve for the remaining prices.

Since any multiple of this price vector would also be a competitive equilibrium, we lose no generality in setting this price to 1.

Walras Law and one Equality for Free

The second fact is a little more subtle. It turns out that if demand equals supply for all n ? 1 goods other than the numeraire, then demand equals supply for the numeraire good as well. This means that to find equilibrium where there are n goods, we really only need to solve n?1 equations in n?1 unknowns. Thus if n = 2, we only need to solve a single equation. If n = 3, we still only need to solve 2 equations in 2 unknowns.

To see why this happens, we prove an equality that is known as Walras’ Law.

If all consumers are locally nonsatiated, we know that pD_i (p) = p?_i and so

or equivalently,

(1)

This equality is preserved if we reverse the order of summation, in which case we have

(2)

Let us define aggregate excess demand for good j as

(3)

Then Equation 2 can be written as

(4)

This is the equation commonly known as Walras’ Law. Equation 4 implies that

  (5)

Let good k be the numeraire. Suppose that at price vector ¯p, demand equals supply for all commodities j /= k. Then E_j (¯p) = 0 for all j /= k. Therefore

(6)

It follows from Equation 5 that

p_kE_k(¯p) = 0.

But p_k = 1. Therefore E_k(¯p) = 0.