Berkeley ECON 101A - Lecture Notes on Convexity and Competitive Equilibrium

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Economics 101A Lecture Notes on Convexity and CompetitiveEquilibriumSteve GoldmanAbstract. Convexity is closely connected to continuity of the excessdemand function - a condition used in establishing the existence of competitiveequilibrium.The proof of the existence of competitive equilibrium required that the excessdemand function be continuous in prices. The proof did not require a description ofwhere the excess demand function came from, but for a competitive economy, thiscondition for continuity is closely tied to convexity.If the individuals' indi®erence curves are not quasi-concave (i.e. exhibit diminish-ing marginal rates of substitution) then a small change in prices in the neighborhoodof the nonconvexity could result in a discontinuous change in demand. In the examplebelow, a change of the budget line from I to II, brought about by a fall in the priceof good 1, cause demand to shift from x1to x2.If the ¯rms' production sets are not convex, then a small change in prices, againnear the nonconvexity, could result in a discontinuous change in supply. A fall in theratio of the price of factor x to output y from I to II results in a change of the netoutput vector from (¡x1; y1) to (¡x2; y2).1Economics 101A Lecture Notes on Convexity and Competitive Equilibrium 2In the Robinson Crusoe model below, a competitive equilibrium does not exist.Convexity insures that such discontinuities will not arise.The "competitive equilibrium" at the point of tangency of Robinson's indi®erencecurve and production possibility set would entail negative pro¯ts and the ¯rm wouldsimply shut


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Berkeley ECON 101A - Lecture Notes on Convexity and Competitive Equilibrium

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