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13. Alice and Bill live in an exchange economy with two goods,

x_{1}

and $x_{2...

Apr 25, 2024

13. Alice and Bill live in an exchange economy with two goods,

x_{1}

and

x_{2}

. Alice owns 1 unit of good 1 and 9 units of good 2 whereas Bill owns 9 units of good 1 and 1 unit of good 2. Alice's utility function is

u^{A}\left(x_{1}, x_{2}\right)=x_{1}{ }^{\alpha} x_{2}{ }^{1-\alpha}

and Bill's utility function is

u^{B}\left(x_{1}, x_{2}\right)=x_{1}{ }^{\beta} x_{2}{ }^{1-\beta}

where

\alpha \in(0,1)

and

\beta \in(0,1)

. a. Suppose that Alice and Bill enter into a trading agreement. Show all feasible allocations of goods 1 and 2 for Alice and Bill in an Edgeworth box. Show the allocations that are Pareto improvements over the endowment allocation. (10 marks) b. Determine the competitive equilibrium price. (20 marks) c. Assume that

\alpha=\beta

and that Alice and Bill trade at the price ratio that you have found in part

b

. Show that there are gains from trade. (20 marks) d. Assume again that

\alpha=\beta

. Determine how the equilibrium price ratio and the consumption levels change as

\alpha

increases. Give an intuitive explanation for your answer. (20 marks) e. Suppose that initially Alice has 9 units of both goods and Bill has 1 unit of both goods, i.e, the endowments are very unequal. i. Assume that

\alpha \neq \beta

. Determine if there are still gains from trade. What can you infer about the rich and the poor? (15 marks) ii. Now assume that

\alpha=\beta

. The government thinks that the desired level of consumption of both goods is 5 units for both Alice and Bill. Therefore, it levies a proportional tax

t^{*}

on any goods purchased, that is, if the initial price ratio is

P=\frac{P_{1}}{P_{2}}

, then the price ratio after the proportional tax becomes

P^{\prime}=\frac{P_{1}\left(1+t^{*}\right)}{P_{2}\left(1+t^{*}\right)}=P

. The government will use the tax revenue as a lump sum payment to reach the desired allocation. Explain if the government's scheme complies with the Second Welfare Theorem. (15 marks)