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Unformatted text preview: ECON475 HW #0 Answers (P) A feasible allocation is Pareto Optimal if there is no other feasible allocation such that ; and for some . (WP) A feasible allocation allocation such that is Weak Pareto Optimal if there is no other feasible . 1. Given the definitions above, show whether i) ii) holds holds. 2. Give an example of a WPO allocation that is not SPO. 1. i) Assume not, i.e. assume that is not a Weak Pareto Optimal allocation. Then, there is at least one other feasible allocation such that . This means that is (weak) Pareto superior to . In other words, moving from to constitutes a (weak) Pareto improvement (i.e. we have an allocation at which all the consumers are strictly better off). Note that this implies the case where such that and for some . Thus we have proved that . Therefore holds. 2nd way: Assume that there are n consumers. Define feasible allocations , ,..., and , ,..., with . implies that implies that and (i.e. for some , (let , ..., ) , , ..., ) If does not occur, then this means that is strong Pareto Optimal (SPO) allocation, i.e. strong Pareto improvement is impossible. If does not occur, then this means that is weak Pareto Optimal (WPO) allocation, i.e. weak Pareto improvement is impossible. Then, if does not occur does not occur. ii) Following from the previous argument: does not occur does not occur. (i.e. Weak Pareto improvement being impossible does not imply that strong Pareto improvement is impossible.) 1 2. Consider a case in an Edgeworth Box economy where consumer A's preferences are represented by thick indifference curves: In the figure, point B is a WPO allocation, while it is not a SPO allocation. Why? Consider moving from point B to A. Note that this reallocation does not constitute a weak Pareto improvement as we are not making all consumers better off. Therefore, B is a WPO allocation. Movement from B to A constitutes a strong Pareto improvement: while keeping consumer A at the same utility level, we are making consumer B better off. This makes point A a SPO allocation (and B not a SPO allocation!). 2 ...
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- Spring '10