Consider a simple exchange economy with two consumers $(1$ and $2)$ and two goods $($x and y$).$The economy has the following endowments: e$1$x$=8,$ e$1$y$=8,$ e$2$x$=2,$ e$2$y$=18.$ Also suppose that the consumers$$ utility functions are as follows:

U$1=$xy$,$ where MUx$=$y$,$ and MUy$=$x

U$2=$xy$0\mathrm{.}5,$ where MUx$=$y$0\mathrm{.}5$ and MUy$=0\mathrm{.}5$xy$-0\mathrm{.}5$

a$)$ What is the competitive market allocation and price ratio?

b$)$ Suppose the social welfare function is SW$=$ U$1*$U$2.$ Does this competitive bundle

maximize social welfare? Find one allocation that would improve the overall social

welfare based on this function.