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I have a two consumer economy with utility functions $u_1=x_{12}-x_{21}$ and $u_2=x_{21}x_{22}$. I am asked to find the Pigou tax $t>0$ on agent 2's consumption of good 1, such that the allocation $\{(0.5, 0.5),(0.5, 0.5)\}$ is a Walrasian equilibrium.

But at the efficient allocation $[(0.5, 0.5)(0.5, 0.5)]$, we have that $\frac{x_{22}}{x_{21}}=1=1$ (i.e. MRS of agent 2 equal to MRS of agent 1). And in any Walrasian equilibrium consumers will choose so that $\frac{x_{22}}{x_{21}}=1$ (MRS of consumer 2 equal to MRS of consumer 1). But this holds at the efficient allocation mentioned above (that is, the efficient allocation above will result from optimising behaviour, without the need for any Pigou tax). So is the tax just $0$, then?

Thank you.

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