1. Suppose 50% of all agents in an economy have U=ln x + ln y, and the other 50% have U=2 ln x + ln y. All agents start with one unit of x and one unit of y. Find the general equilibrium relative prices and allocations.
On the second page of the notes, we see that \(\frac{p_x}{p_y}=\frac{\Sigma a_i \overline{y_i}}{{\Sigma \overline{x_i}} -\Sigma a_i \overline{x_i}}\) .  Additionally, to normalize, \(a + b =1\) and there are 100 agents in total.
Using the formula, \(\frac{p_x}{p_y}=\frac{(50)(.5)(1)+(50)(\frac{2}{3})(1)}{(50)(.5)(1)+(50)(\frac{1}{3})(1)}=1.4\)   This means that each agent has \((1.4)(1)+(1)(1)=2.4\) units of income. The constant income fractions rule then demonstrates that the initial type of agent spends 50% of his or her income on each good while the latter type spends 2/3 on x and 1/3 on y.
The initial agent spends half of his or her income on good x and good y, therefore consuming  \(\frac{(.5)(2.4)}{1.4}=.857\) units of good x and \(\frac{(.5)(2.4)}{1}=1.2\) units of good y. In essence, each initial agent sells .143 units of good x to attain .2 units of good y (in addition to the y he or she began with).
The latter agent type spends 2/3 of his or her income on good x which is \((2.4)(\frac{2}{3})=1.6\) units of good x and 1/3 of his or her income on good y which is \((2.4)(\frac{1}{3})=0.8\) units of good y. He or she consumes \(\frac{1.6}{1.4}=1.143\) units of good x  and \(\frac{0.8}{1}=0.8\) units of good y.  He or she sells 0.2 units of good y and buys 0.143 units of good x.
2. Re-do problem #1, assuming that the first type of agent starts with 2 units of x and 0 of y and the second type of agent starts with 2 units of y and 0 of x.
\(\frac{p_x}{p_y}=\frac{(50)(.5)(0)+(50)(\frac{2}{3})(2)}{(50)(.5)(2)+(50)(\frac{1}{3})(0)}=\frac{4}{3}\)   Now the initial agent has \(\frac{8}{3}\) units of income and the latter agent has 2 units of income.
The initial agent consumes \(\frac{(\frac{8}{3})(.5)}{(\frac{4}{3})}=1\) unit of good x and \(\frac{(\frac{8}{3})(.5)}{1}=\frac{4}{3}\) units of good y. He sells 1 unit of good x to attain \(\frac{4}{3}\) units of good y.
The latter agent consumes \(\frac{(2)(\frac{2}{3})}{\frac{4}{3}}=1\) unit of good x and \(\frac{\frac{2}{3}}{1}=\frac{2}{3}\) units of good y. She sells \(\frac{4}{3}\) units of good y to attain 1 unit of good x.
3. Re-do problem #1, assuming that all agents have U=x+y. (Hint: At disequilibrium prices, agents want to consume only x or only y).
The price ration in this case would be \(\frac{p_x}{p_y}=1\).  This infers that if a good is cheaper than the other, all agents would desire to consume only the cheaper good and none of the other. Assuming that agents start with one unit of good x and one unit of good y, each agent would only consume one unit of good x and one unit of good y.
4. Suppose you can redistribute x, but not y. Returning to problem #1, what exactly must you do to: (a) make the equilibrium utility of the first type of agents equal to .5  (b) give all agents of the second type the same utility,  (c) and make type-2 agents' utility as high as possible conditional on (a)?