03Pareto

03Pareto - PARETO EFFICIENCY of GENERAL EQUILIBRIUM with...

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PARETO EFFICIENCY of GENERAL EQUILIBRIUM with COMPLETE MARKETS Our original hypothesis that utility is money measurable, and independent of the amount of money, is highly suspect. Who believes he can measure another man’s happiness, much less equate it with a quantity of money? But if we give up the money measurable utility hypothesis, what is the argument for laissez faire? The answer was provided by Wilfredo Pareto, an Italian economist from the early 1900s, who discovered the concept of Pareto e ciency. One of the earliest proofs of the Pareto e ciency of competitve equilibrium was given by the Polish socialist Oskar Lange. (Lange was Polish ambassador to the United States in the 1940s). A pictorial proof in two dimensions due to the Oxford economist Edgeworth is also very instructive. The best, shortest, and most general proof was given independently by Kenneth Arrow and Gerard Debreu in 1951. Another important property of equilibrium for f nance is arbitrage e ciency. Two goods that are perfect substitutes will trade at the same price. Arbitrage e ciency is much weaker than Pareto e ciency. Arrow emphasized that if some goods are not traded, then equilibrium will prob- ably not be Pareto e cient. But it will still be arbitrage e cient. Geanakoplos and Polemarchakis proved that if some goods are not traded in a dynamic economy, then typically the goods that are traded will not be traded Pareto e ciently, even taking into account the constraint on reallocating the untraded goods. 1 Preferences and Monotonic Transformations Edgeworth’s family of indi f erence curves completely describes the preferences of agent i , namely it provides enough information to determine which of any two bun- dles gives agent i more welfare (or whether the two bundles are indi f erent). Some years later economists saw the importance of an obvious mathematical fact: there can be many di f erent welfare functions that give rise to the same family of indi f erence curves. Indeed, consider the welfare function U ( x, y )=2 W ( x, y ) obtained by doubling the welfare function W .I ti sev iden ttha t U and W describe the same preferences between commodity bundles. In fact if f is any increasing function (called a monotonic function), then the welfare function U ( x, y )= f (
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also represents the same preferences as W does. In particular, letting f ( w )= e w , the behavior of W ( x, y )= a log x + b log y U ( x, y )= f ( W ( x, y )) = x a y b are identical. Incidentally, it will be useful later to observe that the welfare functions W ( x, y )= u ( x )+ dv ( y ) U ( x, y )= qu ( x )+(1 q ) v ( y ) represent the same preferences if q is chosen so that q =1 / (1 + d ) .Inpa r t icu la r ,i f W ( x, y )= a log x + b log y we will get the same behavior by replacing a with α = a/ ( a + b ) and b with β = b/ ( a + b
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This note was uploaded on 12/08/2009 for the course ECON 251 at Yale.

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03Pareto - PARETO EFFICIENCY of GENERAL EQUILIBRIUM with...

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