\(\frac{p_x}{p_y}=\frac{(50)(.5)(1)+(50)(\frac{2}{3})(1)}{(50)(.5)(\overline{x}) +(50)(\frac{1}{3})(2-\overline{x})}=\frac{7}{\overline{x} +4}\) This means that the initial type of agents each have an income of \((\frac{7}{\overline{x} +4})(\overline{x}) +1\) where they spend half of their income on good x and good y. This means that they consume \(\frac{(\frac{7}{\overline{x} +4})(\overline{x}) +1}{(2) \frac{7}{\overline {x} +4}}=\frac{4\overline{x} +2}{7}\) units of x and \(\frac{\frac{7}{\overline{x} +4}(\overline{x}) +1}{2}=\frac{4\overline{x} +2}{\overline{x} +4}\) units of y. Plugging this into their utility function gives: \(U=\ln {\frac{4 \overline{x} +2}{7}} +\ln{\frac{4\overline{x} +2}{\overline{x} +4}}\) .

Setting this to 0.5 and solving gives: \(0.5=\ln {\frac{4 \overline{x} +2}{7}} +\ln{\frac{4\overline{x} +2}{\overline{x} +4}}\) \((\frac{4\overline{x}+2}{7} )(\frac{4\overline{x} +2}{\overline{x}+4})=e^{0.5}\) \(16\overline{x^{2}} +4.46\overline{x}-42.16=0\)

\(\frac{p_x}{p_y}=\frac{7}{1.49+4}=1.275\) The income of the initial type of agents is equal to 2.9 which means that they consume \(\frac{1.45}{1.275}=1.137\) units of x and \(\frac{1.45}{1}=1.45\) units of y. In essence, after the redistrubtion of .49 units of x to the initial agents, they sell \(1.49-1.137=.353\) units of x to buy .45 additional units of y.