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

<p>The initial endowments are $ω_1 = (1,0)$ and $ω_2 = (0,1)$. </p>

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

<p>(b) Find the locus of interior Pareto optimal points. </p>

<p>(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 $x_1 = (\frac{1}{3} , \frac{1}{2} )$, $x_2 = (\frac{2}{3} , \frac{1}{2})$</p>

<p>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 $p_2=1$)</p>

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

<p>$$\frac{1}{3}P+T_1-\frac{2t}{3} =0 $$</p>

<p>$$P_1 + T_1 = 1$$</p>

<p>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 $T_2= -\frac{1}{6}$.</p>

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

CC BY-SA 3.0
2017-02-10T19:08:18.543Z
The sum of the transfers should be 0. Did you put that into your calculations?
Kitsune Cavalry
CC BY-SA 3.0
2017-02-10T19:16:47.770Z
No, I didn't. Could you please explain why they should be equal to 0?
SDM
CC BY-SA 3.0
2017-02-10T19:42:42.363Z
The government doesn't lose or gain revenue from taxation in this case. Balanced budget tax/subsidy.
Kitsune Cavalry
CC BY-SA 3.0
2017-02-10T19:54:06.420Z
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
CC BY-SA 3.0
2017-02-10T20:55:13.710Z
Oh, is the tax not part of the transfer somehow?
Kitsune Cavalry
CC BY-SA 3.0
2017-02-11T07:30:27.123Z
No, the tax here is $$ tx_{11}$$ and transfers are $$T_1$$ and $$T_2$$ for consumers 1 and 2 respectively.
SDM
CC BY-SA 3.0
2017-02-11T07:33:19.503Z
Anyways, I think I did some silly calculation error in the last part as I am getting the total tax to be equal to the negative of total transfers. I will redo the question. Thanks for the help. :)
SDM
<microeconomics><competitive-equilibrium><externalities>
Competitive prices, tax and lump sum cash transfer in case of externalities
SDM
# Competitive prices, tax and lump sum cash transfer in case of externalities

(Asked by: username_0 on 2017-01-27)

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.

**Tags:** microeconomics, competitive-equilibrium, externalities
> **Comments:**
> **username_1**: The sum of the transfers should be 0. Did you put that into your calculations?  
> **username_0**: No, I didn't. Could you please explain why they should be equal to 0?  
> **username_1**: The government doesn't lose or gain revenue from taxation in this case. Balanced budget tax/subsidy.  
> **username_0**: 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?  
> **username_1**: Oh, is the tax not part of the transfer somehow?  
> **username_0**: No, the tax here is $$ tx_{11}$$ and transfers are $$T_1$$ and $$T_2$$ for consumers 1 and 2 respectively.  
> **username_0**: Anyways, I think I did some silly calculation error in the last part as I am getting the total tax to be equal to the negative of total transfers. I will redo the question. Thanks for the help. :)  

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## 0 answers

