The aggregate demand curves for each good are equal to:
\(X^D= \sum\limits_{i=1}^n [0.5+0.5\frac{P_Y}{P_X}] + \sum\limits_{j=1}^n [\frac{2}{3}+\frac{2}{3}\frac{P_Y}{P_X}]\)
\(X^D= \sum\limits_{i=1}^{n} [\frac{7}{6}+\frac{7}{6}\frac{P_Y}{P_X}]\)
Since in equilibrium demand is equal to supply:
\(\sum\limits_{i=1}^{2n}\omega_X= \sum\limits_{i=1}^{n} [\frac{7}{6}+\frac{7}{6}\frac{P_Y}{P_X}]\)
\(2=\frac{7}{6}+\frac{7}{6}\frac{P_Y}{P_X}\)
\(\frac{P_Y}{P_X}=\frac{5}{7}\)
This means that the first group of agents consumes \(X=0.5+0.5*\frac{5}{7} = \frac{6}{7} \approx 0.857\) of X and \(X=0.5+0.5*\frac{7}{5} = \frac{6}{5} = 1.2\) of Y.
The second group of agents consumes \(X= \frac{2}{3}+\frac{2}{3}\frac{5}{7}=\frac{24}{21} \approx 1.143\) of X and \(Y= \frac{1}{3}+\frac{1}{3}*\frac{7}{5}= \frac{12}{15} \approx 0.8\) of Y.
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.