Question

In class, for the definition competitive equilibrium, I stated that the prices and allocations must satisfy the N market clearing conditions if there were N goods. Show that it is sufficient to require market clearing for N − 1 goods. In other words, the Nth market clearing condition is redundtant.

Answer 1

In general equilibrium model , market clearing constraint is redundant ie if N-1 markets are satisfied automatically N market would be cleared as well. This characteristic is for the general equilibrium model which is known was the Walras law. To see this we look at I consumers and see if it satisfies N-1 consumers.

It is seen in the following

So hence the N th market is satisfied if N-1 markets clear.

In the fourth step we assume that the difference between the total cost and total endowment is 0 then the markets clear for N-1 markets .

Eg : suppose there are markets of two goods A and B

If the market of good A is satisfied then it means the total expenditure from it is equal to total income from its marker. Since the total expenditure and income of the market as a whole is same, we find that good B would also be in a market clearing area since the difference between total expenditure and expenditure of good A would be equal to total income from market less the income from market A

So B market clears automatically.