<p>It looks right to me. There's an alternative (but equivalent) way to solve part D: we know that the consumer's choice will satisfy</p>

<p>$$ x^* =  \frac{ a(m-t)}{ p_x - s} ,\  y^*= \frac{(1-a)(m-t)}{p_y},  $$
or, with a balanced budget,
$$ x^* =  \frac{ a(m-sx^*)}{ p_x - s},\ y^* = \frac{(1-a)(m-sx^*)}{p_y}.$$</p>

<p>The government wants to maximize $\min\{x,y\}$, which is achieved when $x=y$. Solving
$$ x^*=y^*\iff \frac{ a(m-sx^*)}{ p_x - s}= \frac{(1-a)(m-sx^*)}{p_y}$$</p>

<p>for $s$ yields</p>

<p>$$ s^* = p_x - \frac{a p_y}{1-a}. $$</p>

CC BY-SA 3.0
2017-09-09T19:38:50.240Z
2017-09-09T19:38:50.240Z
Ubiquitous
<p>A consumer with income $m$ has preferences $U(x , y) = x^a y^{1-a} $ where a is between 0 and 1. A paternalistic Government wants to regulate the choices of the consumer to maximize it's own welfare function: $w(x,y) = \min(x,y)$ . The market prices are $p_x$ and $p_y$.</p>

<p>A. Suppose the government can impose a ceiling on the consumption of either good. Under what parameter condition is it necessary to place a ceiling on good $x$?
B. Suppose, instead, government provides a per unit subsidy on good $x$ and charge a lump sum tax $t$. Derive Marshallian demand for any $(s,t)$.
C. A policy $(s,t)$ is budget balanced if total subsidy paid equal tax. Derive relationship between $s$ and $t$ that must be satisfied by any budget balanced policy.
D. From the government's perspective, find optimum s and t values subject to a balanced budget.</p>

<p>Here's how I tried it:
Part A:
Using optimisation methods, from consumer perspective:
$$ X^*= (x^*,y^*) = \left( \frac{ am}{ p_x} , \frac{(1-a)m}{p_y} \right) $$</p>

<p>However, Government's welfare is maximised at :
$$ X^g= (x^g,y^g) = \left( \frac{ m}{ p_x + p_y} , \frac{m}{p_x + p_y} \right) $$</p>

<p>Then, I considered two cases:
$$ 1. X^* &gt; X^g $$
$$  2. X^* &lt; X^g $$</p>

<p>In the first case, ceiling on good 2 would be effective whereas in second case, ceiling on 1 would be effective, the ceiling being the optimal quantity from government's perspective. The parameter condition for second case,by comparing the quantities of good , was:</p>

<p>$$ \frac{p_y}{p_x} &gt; \frac{1}{a} -1$$</p>

<p>Part B:
Simple optimisation problem:</p>

<p>$$ X^*= (x^*,y^*) = \left( \frac{ a(m-t)}{ p_x - s} , \frac{(1-a)(m-t)}{p_y} \right) $$</p>

<p>Part C:
Putting $sx^* = t$ I get
          $$s = \frac{p_x t}{a(m-t)+ t}$$
Part D:
I am not sure if the method I have used is correct or not.</p>

<p>Government's welfare is maximised at :
$$ X^g= (x^g,y^g) = \left( \frac{ m-t}{ p_x + p_y-s} , \frac{m-t}{p_x + p_y-s} \right) $$</p>

<p>Also,since,budget is balanced, therefore,
  $$ s*\frac{ m-t}{ p_x + p_y-s}= t $$</p>

<p>Now putting the s and t relationship from c part in the above balanced budget identity, I got</p>

<p>$$ t^* = \frac {m(p_x(1-a) -ap_y)}{(p_x +p_y)(1-a)} $$</p>

<p>$$ s^* = p_x - \frac{a p_y}{1-a} $$</p>

<p>Can someone please have a look at my solution and see if I have done it correctly?</p>

<microeconomics><optimal-taxation>
Paternalistic Government policies
SDM
# Paternalistic Government policies

(Asked by: username_0 on 2017-09-09)

A consumer with income $m$ has preferences $U(x , y) = x^a y^{1-a} $ where a is between 0 and 1. A paternalistic Government wants to regulate the choices of the consumer to maximize it's own welfare function: $w(x,y) = \min(x,y)$ . The market prices are $p\_x$ and $p\_y$.

A. Suppose the government can impose a ceiling on the consumption of either good. Under what parameter condition is it necessary to place a ceiling on good $x$?
B. Suppose, instead, government provides a per unit subsidy on good $x$ and charge a lump sum tax $t$. Derive Marshallian demand for any $(s,t)$.
C. A policy $(s,t)$ is budget balanced if total subsidy paid equal tax. Derive relationship between $s$ and $t$ that must be satisfied by any budget balanced policy.
D. From the government's perspective, find optimum s and t values subject to a balanced budget.

Here's how I tried it:
Part A:
Using optimisation methods, from consumer perspective:
$$ X^\*= (x^\*,y^\*) = \left( \frac{ am}{ p\_x} , \frac{(1-a)m}{p\_y} \right) $$

However, Government's welfare is maximised at :
$$ X^g= (x^g,y^g) = \left( \frac{ m}{ p\_x + p\_y} , \frac{m}{p\_x + p\_y} \right) $$

Then, I considered two cases:
$$ 1. X^\* > X^g $$
$$ 2. X^\* < X^g $$

In the first case, ceiling on good 2 would be effective whereas in second case, ceiling on 1 would be effective, the ceiling being the optimal quantity from government's perspective. The parameter condition for second case,by comparing the quantities of good , was:

$$ \frac{p\_y}{p\_x} > \frac{1}{a} -1$$

Part B:
Simple optimisation problem:

$$ X^\*= (x^\*,y^\*) = \left( \frac{ a(m-t)}{ p\_x - s} , \frac{(1-a)(m-t)}{p\_y} \right) $$

Part C:
Putting $sx^\* = t$ I get
$$s = \frac{p\_x t}{a(m-t)+ t}$$
Part D:
I am not sure if the method I have used is correct or not.

Government's welfare is maximised at :
$$ X^g= (x^g,y^g) = \left( \frac{ m-t}{ p\_x + p\_y-s} , \frac{m-t}{p\_x + p\_y-s} \right) $$

Also,since,budget is balanced, therefore,
$$ s\*\frac{ m-t}{ p\_x + p\_y-s}= t $$

Now putting the s and t relationship from c part in the above balanced budget identity, I got

$$ t^\* = \frac {m(p\_x(1-a) -ap\_y)}{(p\_x +p\_y)(1-a)} $$

$$ s^\* = p\_x - \frac{a p\_y}{1-a} $$

Can someone please have a look at my solution and see if I have done it correctly?

**Tags:** microeconomics, optimal-taxation

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## 1 answers

### Answer by username_1 on 2017-09-09. Score: 4

It looks right to me. There's an alternative (but equivalent) way to solve part D: we know that the consumer's choice will satisfy

$$ x^\* = \frac{ a(m-t)}{ p\_x - s} ,\ y^\*= \frac{(1-a)(m-t)}{p\_y}, $$
or, with a balanced budget,
$$ x^\* = \frac{ a(m-sx^\*)}{ p\_x - s},\ y^\* = \frac{(1-a)(m-sx^\*)}{p\_y}.$$

The government wants to maximize $\min\{x,y\}$, which is achieved when $x=y$. Solving
$$ x^\*=y^\*\iff \frac{ a(m-sx^\*)}{ p\_x - s}= \frac{(1-a)(m-sx^\*)}{p\_y}$$

for $s$ yields

$$ s^\* = p\_x - \frac{a p\_y}{1-a}. $$


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