$($Based on exercise $28\mathrm{.}3$ of Lima$)$ Consider a two good, two individual endowment economy. Suppose that individual i$\text{'}$s endowment is given by: ${}^i=({}_x^i,{}_y^i)$ and individual j$\text{'}$s$,$ endowment is given by: ${}^j=({}_x^j,{}_y^j).$ where we assume that ${}_x^i>{}_x^j$ and ${}_y^j>{}_y^i.$ Also assume individual i$\text{'}$s preferences are given by: $U(x_i,y_i)=ln(x_i)+(1-)ln(y_i)$ and individual j$\text{'}$s preferences are given by: $U(x_j,y_j)=ln(x_j)+(1-)ln(y_j).$ Finally, let the absolute prices of good x and good y be given by $p_x$ and $p_y.$

$($a$)$ Write down each individual's maximization problem.

$($b$)$ Solve each individual's maximization problem. You should obtain $x_i^{**},x_j^{**},y_i^{**}$ and $y_j^{**}$

$($c$)$ Find $x^{**}$ and $y^{**},$ aggregate demands for the two goods.

$($d$)$ What are the equilibrium conditions for this economy?

$($e$)$ Use the the market clearing condition in the x $-$ market to find the equilibrium price. Hint: You can't find absolute prices. You need to find the relative price, $p=\frac{p_x}{p_y}.$

$($f$)$ Use the market clearing condition in the y $-$ market to find the equilibrium price. You should find that your answer is the same as in part $($e$).$