Varian 17-18 (G & R 10). Walrasian equilibrium The solution to a consumer’s utility maximization problem when taking prices as given is the consumer’s demand function. In equilibrium aggregate demand cannot exceed endowments.
When does a Walrasian equilibrium exist? Normalize prices by dividing each price with the sum of all prices. The sum of the normalizedprices equals 1. All possible prices can now be represented as points on a unit simplex (with a dimension equal to the number of prices minus one). By construction it is a compact set. Is there a price vector p such that excess demand is zero? For any given p the consumers’ choices result in some z(p). If there is excess demand (supply) for a good this would tend to increase (reduce) its price. Thisdescribes a relationship between prices and new adjusted prices. We can use this reasoning to prove the existence of an equilibrium. Specifically, if the price adjustment is a continuous function, say g(p), from the price simplex to itself Brouwer’s fixed point theorem ensures that there exists a p such that g(p) = p. We can construct g so that this can only happen when excess demand is zero. LetThe aggregate excess demand function is,
Walras’ law Since all consumers are on their budget constraints in optimum the value of their endowments must equal the value of the demanded bundles. Aggregation across consumers preserves this property which implies that the value of excess demand pz(p) is zero - Walras’ law. It follows that if we know that all markets but one clears, which we lackinformation about, then that market must also clear. If a good is in excess supply in a Walrasian equilibrium it must be a free good. If not then pjzj(p) < 0 implying that pz(p) < 0 (since the excess demand for each good in a Walrasian equilibrium is non-positive and prices are positive). This contradicts Walras’ law. If we assume that all good are desirable so that pi = 0 implies zi(p) > 0 thenthe excess demand must be equal to zero for each good. When all good are desirable a Walrasian equilibrium can be defined as a (x, p) such that
which is continuous in prices and maps all price vectors back into the price simplex. Consequently, there exists a p such that z(p) = 0. The key requirement for existence of a Walrasian equilibrium is continuity of the aggregate excess demand function.This is the case if consumer preferences are convex or, if there are infinitely many consumers so that each consumer is small compared with the size of the market. The assumption about competitive behavior is more plausible if there are many small consumers. Thus, when competitive behavior seems reasonable so does equilibrium analysis. Pareto efficiency A feasible allocation x is weakly Paretoefficient if there is no other feasible allocation that everybody strictly prefer to x. (When preferences are continuous and monotonic this concept coincides with the concept
of strong Pareto efficiency which requires that there is no feasible allocation that is at least as good for everyone and strictly preferred by some agent. Continuity and monotonicity ensures that we can always take some smallamount away from the strictly better off agent and redistribute to the others to make everybody strictly better off.) We can illustrate the set of feasible allocations and the potential gains from trade in an Edgeworth box.
What are the properties of a Walrasian equilibrium? Agents choose bundles so that MRS(x1,x2)=p1/p2. Since all agents face the same prices they adjust consumption so thattheir MRSs equal. Hence, in a Walrasian equilibrium agents' indifference curves are tangent and separated by the budget line. The bundles preferred by A are thus separated from those preferred by B which suggests that a Walrasian equilibrium is also Pareto efficient. The 1st Theorem of Welfare Economics If (x, p) is a Walrasian equilibrium then x is Pareto efficient. The proof uses the definition...