# Second welfare theorem problem

I would like to get a better understanding of part c of the following problem. It appears to me that Walrasian allocations do not change but I want to confirm whether I am right or not. Thanks.

Effect of distortionary taxes Consider an economy with two individuals, $$i = {A, B}$$, each with identical Cobb–Douglas utility function $$u(x_{1i},x_{2i}) = x_{1i}x_{2i}$$, and initial endowments $$e_{A} = (200, 100)$$ and $$e_{B} = (100, 200)$$.

a. Find the Pareto optimal allocation (PEA). b. Find the WEA. (For simplicity, you can assume that $$p_{1} = p_{2} = 1$$) c. Assume that the government sets a tax $$t$$ on purchases of good 1, which is refunded to the consumers as a lump-sum payment, $$T_{i} = tx_{1i}$$. Find the post-tax WEA, and compare it with your results in part b. d. Show that the WEA when taxes are absent in part b is efficient, whereas the WEA when taxes are present found in part c is not necessarily efficient for all values of t.

What I did is that I set up the agent's problem as follows:

Max $$U(x_{1i},x_{2i}) = ln(x_{1i}) + ln(x_{2i})$$

with the constraint being $$(1+t)(x_{1i}) + (x_{2i}) = 300 + tx_{1i}$$

Obviously the $$tx_{1i}$$'s cancel out which turns out as if no tax has been implemented. To me, it is an example of the second welfare theorem but again, my approach might be wrong.

• could you include your answers if you are trying to confirm them? More generally within the site, everyone is asked to show their efforts (it only makes sense if you are trying to get someone to check this for you) – user20105 Dec 7 '19 at 22:58
• Sure, I maximized the utility function lnx1+ lnx2 s/c (1+t)x1 + x2 = 300 + tx1, I find that the tx1's cancel out which leaves me in the initial situation as if no tax has been implemented. To me, it is an example of the second welfare theorem but again, I might be wrong. – Lea Dec 7 '19 at 23:07
• I don't mean in a comment, I mean in your question. When asking for answers from someone the least you can do is add is your answer so they can go through them! (also check out the formatting of your equations) – user20105 Dec 7 '19 at 23:19
• Just did. Hopefully, it makes more sense now. – Lea Dec 7 '19 at 23:37
• This problem seems ill-defined in several ways. Since there are endowments, not money, what does the government take the tax in? This would have en effect on supply, and hence also equilibria. Also, I think they mean that $T_i$ is a constant, but it is set at such a level that it happens to be equal $tx_{1i}$ when $x_{1i}$ is chosen optimaly, that is when $$x_{1i} = \frac{1}{2}\frac{\text{income}_i + T_i}{p_1+t}.$$ (Cobb-Douglas property) – Giskard Dec 8 '19 at 8:39