Dean Foster Z. Interface and Professor J. Fetid Nightsoil exchange platitudes and bromides. When Dean Interface consumes

Question:

Dean Foster Z. Interface and Professor J. Fetid Nightsoil exchange platitudes and bromides. When Dean Interface consumes TI platitudes and BI bromides, his utility is given by
UI (BI, TI) = BI + 2ˆšTI.
When Professor Nightsoil consumes TN platitudes and BN bromide, his utility is given by
UN(BN, TN) = BN + 4ˆšTN.
Dean Interface€™s initial endowment is 12 platitudes and 8 bromides. Professor Nightsoil€™s initial endowment is 4 platitudes and 8 bromides.

(a) If Dean Interface consumes TI platitudes and BI bromides, his marginal rate of substitution will be __________ If Professor Nightsoil consumes TN platitudes and BN bromides, his marginal rate of substitution will be ___________
(b) On the contract curve, Dean Interface€™s marginal rate of substitution equals Professor Nightsoil€™s. Write an equation that states this condition. _______________ This equation is especially simple because each person€™s marginal rate of substitution depends only on his consumption of platitudes and not on his consumption of bromides.
(c) From this equation we see that TI/TN = _________ at all points on the contract curve. This gives us one equation in the two unknowns TI and TN.
(d) But we also know that along the contract curve it must be that TI + TN = __________ since the total consumption of platitudes must equal the total endowment of platitudes.
(e) Solving these two equations in two unknowns, we find that everywhere on the contract curve, TI and TN are constant and equal to ____________
(f) In the Edgeworth box, label the initial endowment with the letter E. Dean Interface has thick gray penciled indifference curves. Professor Nightsoil has red indifference curves. Draw a few of these in the Edgeworth box you made. Use blue ink to show the locus of Pareto optimal points. The contract curve is a (vertical, horizontal, diagonal) ________ line in the Edgeworth box.
(g) Find the competitive equilibrium prices and quantities. You know what the prices have to be at competitive equilibrium because you know what the marginal rates of substitution have to be at every Pareto optimum.