Consider an exchange economy. Is it possible to have a Pareto optimal allocation where one consumer...

Consider an exchange economy. Is it possible to have a Pareto optimal allocation where one consumer is worse oﬀ than she is at a diﬀerent allocation that is not Pareto optimal? Explain.

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Pareto Optimality & Allocation
Pareto optimality refers to the situation where a re-allocation of resources or goods cannot make a consumer better-off without making anyone worse-off. Any other re-allocation cannot be considered as Pareto Optimal if it makes anyone worse-off while making other better-off. In a situation where one consumer is worse-off than other where the previous was even not Pareto optimal, the chance for Pareto optimality exists. If a situation is not Pareto optimal, it is said that an efficient re-allocation of exchange of goods can attain Pareto optimality. Also, an inefficient re-allocation can make the situation worse than that of before.
Pareto optimal conditions demand for efficient re-allocation. Inefficiency can always make the situation worse than before. Two consumers who have been opted their preferences on goods by the utility they derive, may be inefficiently allocate to their preferences. That can leads to a situation where it is not Pareto optimal. To attain the situation, the marginal rate of substitution of two goods must be the same for every individual who consumes both. The ratio of substitution should be equal to their prices. Consumer equates the price ratios of both with MRS of the goods. If the re-allocations fails in satisfying the MRS, that is inefficient. This can only leads to worsen the situation as suggests. But if the condition is satisfied, it leads to an efficient re-allocation attaining Pareto optimality.

Rahul Sunny answered 1 month ago

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