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.
Here are some more examples of pure-exchange economies (with 2 consumers and 2 goods) where preferences are continuous and monotone, but set of strongly Pareto efficient allocations is not equal to set of weakly Pareto efficient allocations:
Economy 1: $u_1(x_1,y_1)=\max(x_1,y_1)$, $u_2(x_2,y_2)=\max(x_2,y_2)$, total endowment of X is $\omega_X=2$, total endowment of Y is $\omega_Y=2$. All the allocations on the four boundaries of the Edgeworth Box are weakly Pareto efficient but only two allocations out of those are strongly Pareto efficient $\{((2,0),(0,2)),((0,2),(2,0))\}$
Economy 2: $u_1(x_1,y_1)=x_1+y_1$, $u_2(x_2,y_2)=\max(x_2,y_2)$, total endowment of X is $\omega_X=2$, total endowment of Y is $\omega_Y=2$. The allocation $((x_1,y_1),(x_2,y_2))=((0,0),(2,2))$ is weakly Pareto efficient but not strongly Pareto efficient.
Economy 3: $u_1(x_1,y_1)=x_1+y_1$, $u_2(x_2,y_2)=\min(x_2,y_2)$, total endowment of X is $\omega_X=2$, total endowment of Y is $\omega_Y=3$. The allocation $((x_1,y_1),(x_2,y_2))=((0,0),(2,3))$ is weakly Pareto efficient but not strongly Pareto efficient.
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.
