The aggregate demand curves for each good are equal to:
\(Y^D= \sum\limits_{i=1}^n [\frac{P_X}{P_Y}] + \sum\limits_{j=1}^n [\frac{2}{3}]\)
\(Y^D= \sum\limits_{i=1}^{n} [\frac{2}{3}+\frac{P_X}{P_Y}]\)
Since in equilibrium demand is equal to supply:
\(\sum\limits_{i=1}^{n}\omega_Y= \sum\limits_{i=1}^{n} [\frac{2}{3}+\frac{P_X}{P_Y}]\)
\(2=\frac{2}{3}+\frac{P_X}{P_Y}\)
\(\frac{P_X}{P_Y}=\frac{4}{3}\)
This means that the first group of agents consumes \(X=1\) of X and \(Y=\frac{P_X}{P_Y} = \frac{4}{3}\) of Y.
The second group of agents consumes \(X= \frac{4}{3} * \frac{3}{4}=1\) of X and \(Y= \frac{2}{3}\) of Y.
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 only possible relative price is \(\frac{P_Y}{P_X}=1\) since any other price means that agents will want to consume only x or y. Since each individual have an identical utility function and identical endowments, every agents is consuming one unit of Y and one unit of X.