By Michael Pohst
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Extra info for Algorithmic Methods in Algebra and Number Theory
47 MARKET EXCHANGE AND OPTIMALITY Let's turn to the formal statements of these observations. Observation Ui(yj} > 1. Let (x,p) be a competitive Ui(Xj} for some bundleYi, then equilibrium. If P • Yi>P· Wi Proof. If this were not the case, then Xi would not be the bundle that maximizes the utility function Ui subject to i's budget constraint, which would contradict the assumption that (x,p) is a competitive equilibrium. D. Observation 2. Let (x,p) be a competitive equilibrium. Assume i has a monotonic utility function.
It is typically assumed that people's utilities rise as the quantities of goods they possess increase, so indifference curves further from person l's origin (the lower lefthand corner) represent higher levels of Ul ; and 2's indifference curves further from his origin (the upper righthand corner) represent higher levels of U2. The point w represents an initial allocation. FIGURE 2-2 28 WELFARE ECONOMICS AND SOCIAL CHOICE THEORY PARETO OPTIMAL ALLOCATIONS AND THE CORE Some allocations are unambiguously inferior to others.
We are ready at this point to illustrate our little exchange economy in an Edgeworth box diagram, in Figure 2-5. The box is one unit on each side. We have sketched in four (hyperbolic) indifference curves for person 1, and four (straight line) indifference curves (with Islopel = \12) for person 2. The two individual's indifference curves are tangent where their slopes are equal, or where MRS person 1 = MRS person 2. PERSON 2 I z PERSON I FIGURE 2-5 33 BARTER EXCHANGE This gives Graphically, this is the straight line from person l's origin to the point z.