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We know that strong Pareto efficient is equivalent to weak Pareto efficient if we have continuous and strongly monotone preferences.

Please give me an example which we don’t have continuous and monotone preferences, so this statement doesn’t hold.

I think that the statement holds for quasi linear utility function.

But I cannot find the counter example.

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There are two goods and two consumers, no production. The aggregate endowment is strictly positive. Consumer 1 cares only about the amount of good 1 they consume, with more being better. Similarly for consumer 2 and good 2. Preferences here are continuous and monotone but not strictly monotone. The allocation in which consumer 1 consumes the entire aggregate endowment is weakly Pareto optimal but not strongly so.

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