Katherine and Gareth consume bananas and potatoes, where Bi and Pi denote their consumption of these goods respectively. Katherine$$s utility function is UK$($BK$,$ PK$)=$ BKPK and Gareth$$s is UG$($BG$,$PG$)=($BG$)1/2($PG$)1/2.$ Their initial endowments are $25$ bananas and $10$ potatoes for Katherine and $15$ bananas and $10$ potatoes for Gareth.

a$)$ Draw an Edgeworth box with bananas in the horizontal axis and Katherine in the lower left corner. Illustrate the endowment point and the indifference curves for each agent which pass through the endowment point.

b$)$ What is Katherine$$s initial level of utility? What is Gareth$$s$?$

c$)$ Take the price of potatoes as a numeraire, i$.$e$.,$ pP $=1.$ Find Gareth$$s and Katherine$$s demands for bananas and potatoes as a function of the prices of the price of bananas pB$.$

d$)$ What is the equilibrium price of bananas? What is the allocation of bananas and potatoes which will result?

e$)$ What are their levels of utility after trading?