extra1w03 - Ohio State University Department of Economics...

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Ohio State University Department of Economics Econ. 805 Prof. James Peck Winter 2003 Extra Problems in General Equilibrium Theory (Do not hand in) (1) An allocation is called envy free if every consumer prefers his/her allocation to the allocation of any other consumer. That is, x is envy free if for all h and i, we have u i (x i ) ! u i (x h ) . (a) Will every envy free allocation be Pareto optimal? Explain why or why not. (b) Explain why this economy has at least one envy free allocation that is Pareto optimal. Specify any theorems that your argument relies upon. (Hint: First, redistribute resources so that everyone has the same endowment. Now can you find an allocation that is both envy free and Pareto optimal?) (2) Give an example of a pure exchange economy with two consumers and two commodities, where: (i) each consumer has a utility function that is continuous and strictly increasing, and (ii) the conclusion of the second fundamental theorem of welfare economics is false (that is, there is a Pareto optimal allocation, strictly positive in all components, that cannot be achieved as a competitive equilibrium). A carefully drawn Edgeworth box diagram that is clearly explained is sufficient to answer this problem. (3) Consider an exchange economy with I different "types" of agents, and M agents of each type. In other words, the total number of agents is IM. Any two agents of the same type have the same utility function and the same strictly positive endowments. Assume that all utility functions are strictly monotonic, continuous, and strictly quasiconcave. For each of the following two statements, either show why it is true or find a counterexample.
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