Efficiency and Equilibrium

The First Welfare Theorem says that the equilibrium in a set of competitive markets is Pareto efficient. What about the other way around? Given a Pareto efficient allocation, can we find prices such that it is a market equilibrium? It turns out that the answer is yes, under certain conditions. The argument is illustrated in Figure 31.7.

Let us pick a Pareto efficient allocation. Then we know that the set of allocations that A prefers to her current assignment is disjoint from the set that B prefers. This implies of course that the two indifference curves are



tangent at the Pareto efficient allocation. So let us draw in the straight line that is their common tangent, as in Figure 31.7.

Suppose that the straight line represents the agents' budget sets. Then if each agent chooses the best bundle on his or her budget set, the resulting equilibrium will be the original Pareto efficient allocation.

Thus the fact that the original allocation is efficient automatically determines the equilibrium prices. The endowments can be any bundles that give rise to the appropriate budget set—that is, bundles that lie somewhere on the constructed budget line.

Can the construction of such a budget line always be carried out? Unfortunately, the answer is no. Figure 31.8 gives an example. Here the illustrated point X is Pareto efficient, but there are no prices at which A and B will want to consume at point X. The most obvious candidate is drawn in the diagram, but the optimal demands of agents A and B don't coincide for that budget. Agent A wants to demand the bundle Y, but agent B wants the bundle X— demand does not equal supply at these prices.

The difference between Figure 31.7 and Figure 31.8 is that the preferences in Figure 31.7 are convex while the ones in Figure 31.8 are not. If the preferences of both agents are convex, then the common tangent will not intersect either indifference curve more than once, and everything will work out fine. This observation gives us the Second Theorem of Welfare

Economics: if all agents have convex preferences, then there will always be a set of prices such that each Pareto efficient allocation is a market equilibrium for an appropriate assignment of endowments.

The proof is essentially the geometric argument we gave above. At a Pareto efficient allocation, the bundles preferred by agent A and by agent B must be disjoint. Thus if both agents have convex preferences we can draw a straight line between the two sets of preferred bundles that separates one from the other. The slope of this line gives us the relative prices, and any endowment that puts the two agents on this line will lead to the final market equilibrium being the original Pareto efficient allocation.

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