$(50$ points$)$ The Edgeworth Box below $($next page$)$ shows the preferences $($indifference curves$)$ and initial endowments for two individuals, Anne $($A$)$ and Bart $($B$).$ Anne's initial endowment bundle of goods $x$ and $Y$ is $({}_x^A,{}_Y^A)=(50x,10Y)$; Bart's initial endowment bundle is $({}_x^B,{}_Y^B)=(10x,30Y).$

a$)$ In the Edgeworth Box below, is the bundle S Pareto superior to the initial endowment bundle $?$ Explain. Is the bundle S Pareto superior to the bundle R$?$ Explain.

b$)$ Suppose that Anne's marginal rate of substitution between goods $x$ and $Y$ at Anne's initial endowment point is $\frac{1}{4},$ i$.$e$.,$ MRS $S_{yY}^A({}_x^A,{}_y^A)=\frac{1}{4}.$ Suppose that Bart's marginal rate of substitution at Bart's initial endowment point is $6,$ i$.$e$.,MR{S}_{xy}^B({}_x^B,{}_y^B)=6.$ Is the initial endowment bundle $$ Pareto efficient $($Pareto optimal$)?$ Explain. If not, determine and explain an explicit exchange rate or price ratio that would clearly result in a Pareto superior consumption bundle.

c$)$ Illustrate and explain the contract curve in the Edgeworth Box below. What marginal condition is satisfied along the contract curve? Illustrate and explain the core given the initial endowment point $.$

d$)$ If the initial price ratio is $P\frac{x^0}{P}{y}^0=1,$ illustrate and explain each individual's utility maximizing consumption bundle, $(x_A^0,y_A^0)$ and $(x_B^0,y_B^0),$ and the trade offers each individual would make. Would Anne and Bart complete a trade or exchange at $\frac{P_x^0}{P_Y^0}=1?$ Explain.

e$)$ Given the initial endowment point $=(({}_x^A,{}_y^A),({}_x^B,{}_y^B)),$ illustrate and explain the competitive market equilibrium price and consumption bundles that Anne and Bart would achieve through bargaining or a competitive market. That is$,$ illustrate and explain the trade each individual would be willing to make, the price ratio $P\frac{x^{**}}{P}{y}^{**},$ and the resulting consumption bundles, $(x_A^{**},y_A^{**})$ and $(x_B^{**},y_B^{**}).$ State the conditions that $(x_A^{**},y_A^{**})$ and $(x_B^{**},y_B^{**})$ satisfy. Approximate graphically the numerical value of the equilibrium price ration $P\frac{x^{**}}{P}{y}^{**}.$

f$)$ State or briefly explain the First Fundamental Theorem of Welfare Economics.