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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?

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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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