Competitive Equilibrium and Pareto Efficiency

[Rain Man]

Hi

If none of the agents can be made better-off without making the other worse-off the outcome is said to be pareto-efficient .

How do we find out if a competitive equilibrium for a 2good , 2 agent economy exists or not .

please help.

[Chocolate Frog]

We make use of Walras' law, mostly.

[Rain Man]

thanks for the reply .

walrasian eqm :

the income generated from the endowments ( at the given prices ) should be sufficient to purchase the equilibrium amount of goods for both the agents.


if there exists a situation in which the prices are such that the total demand for goods is more that the supply (in a 2 agent , 2goods economy ) we say that a competitive equilibrium does not exist.

is that correct..?


can you provide an example when the distribution is efficient but it is not a competitive equilibrium.
please elaborate..

[Chocolate Frog]

My grasp over the theory of general equilibrium is tenuous, at best. Nevertheless, I'll share with you what I know.

Walras' law states that the value of aggregate excess demand for all goods is always zero, since agents must always satisfy their budget constraints. If it's zero at all price vectors, it must be zero, in particular, at equilibrium price vector P*.

If at P*, the aggregate excess demand for a good is negative, that is, if a good is in excess supply, its price must be zero. This is a direct implication of Walras' law. Otherwise in Walrasian equilibrium, demand must equal supply for all goods. Note that if there are k markets, you need only find a price vector at which k-1 markets clear, for according to Walras' law, the kth market should then necessarily clear.

As for your second question, try this example. U1 = min{x1,y1} and U2 = max{x2,y2}. E1 = (0,5) and E2 = (5,0). Every point inside the Edgeworth box is Pareto efficient yet a competitive equilibrium doesn't exist. This is as crude an explanation as there can be. It would be wonderful if one of the smarter people on this forum (or Amit sir himself!) could shed more light on this, for everyone's benefit.