# Competitive prices, tax and lump sum cash transfer in case of externalities

An economy is made up of two people. The utility functions are $$u_1(x_{11},x_{12}) = x_{11}x_{12}$$ $$u_2(x_{21},x_{22}) = 2x_{21} + 2x_{22} −x_{11}$$

The initial endowments are $ω_1 = (1,0)$ and $ω_2 = (0,1)$.

(a) Calculate a competitive equilibrium. Draw an Edgeworth box diagram to illustrate your answer.

(b) Find the locus of interior Pareto optimal points.

(c) Calculate prices $p_1$ and $p_2$, a per unit subsidy $s$ or tax $t$, and lump sum cash transfers $T_1$ and $T_2$ to bring the economy to the allocation $x_1 = (\frac{1}{3} , \frac{1}{2} )$, $x_2 = (\frac{2}{3} , \frac{1}{2})$

I am having trouble with the 3rd part. Here is how I tried doing it: It can be seen from utility that increased consumption of $x_1$ by 1st consumer results in less utility of 2nd consumer. So, I charged a tax $t$ on consumption of good 1 by 1st consumer and gave a lump sum payment $T$ to compensate for it. So the budget constraint became $(P_1+t)x_{11} + x_{12} = p_1+ T_1$ (normalising $p_2=1$)

After maximizing the utility I got two equations with three variables $p_1$, $T_1$ and $t$:

$$\frac{1}{3}P+T_1-\frac{2t}{3} =0$$

$$P_1 + T_1 = 1$$

Now, since it's given that for consumer 2 the two goods are demanded in positive quantities so it must mean that $p_1=1$. ( $p_2 =1$) Incorporating it into the first two equations I got $T_1=0$ and $t=\frac{1}{2}$ and then using 2nd's BL I got $T_2= -\frac{1}{6}$.

Now, a negative transfer for cons. 2 seems a bit absurd. So, could someone please tell me where I could be going wrong.

• The sum of the transfers should be 0. Did you put that into your calculations? – Kitsune Cavalry Feb 10 '17 at 19:08
• No, I didn't. Could you please explain why they should be equal to 0? – SDM Feb 10 '17 at 19:16
• The government doesn't lose or gain revenue from taxation in this case. Balanced budget tax/subsidy. – Kitsune Cavalry Feb 10 '17 at 19:42
• Shouldn't the transfers, in that case, be equal to the total tax collected? Since I am getting a positive tax rate, how will the sum of transfers be zero? – SDM Feb 10 '17 at 19:54
• Oh, is the tax not part of the transfer somehow? – Kitsune Cavalry Feb 10 '17 at 20:55