1.  Consider a standard exchange setting with two consumers A and B and two goods x and y, as introduced in slide 7 of Lecture 4. Assume that  = 1 and  = 1, and that

uA (xA ,yA ) = (xA )  (yA )  ,  and

uB (xB ,yB ) = ln(xB + 1) + yB .

(i) Find mrsA (xA ,yA ) and mrsB (xB ,yB ), and characterise points (xA ,yA ) on the Pareto efficient (PE) curve as a relation that expresses yA  as a function of xA . Find the first- and second-order derivatives of the resulting function yA (xA ).

(ii) Express all allocations (xA ,yA ,xB ,yB ) that lie on the PE curve, in terms of only the associated value of xA , for xA   ∈  [0, 1].   (Remember that  =  =  1.)   Derive the PE allocations that correspond to xA  = 0, xA  = , and xA  = 1, and sketch the PE curve in the Edgeworth box. (Don’t forget your answers from part (i).)

(iii) For every allocation  (xA ,yA ,xB ,yB ) on the PE curve,  express the associated utility values uA (xA ,yA ) and uB (xB ,yB ) in terms of the corresponding value of xA  only.   Derive numeri- cal approximations for the coordinates (uA ,uB ) on the utility possibilities frontier (UPF) that correspond to xA  = 0, xA  =  and xA  = 1. Try to sketch the UPF using Wolfram Alpha or any other computer algebra system.

(iv) Now assume that the Impartial Spectator has a social welfare function (SWF) W defined by W(uA ,uB ) = λuA + (1 − λ)uB , where λ ∈ (0, 1).

Find the marginal rate of substitution mrsW (uA ,uB ) associated with W, and use your answer to characterise the shape of the Impartial Observer’s indifference curves. Discuss, based on the graph for the UPF that you derived in part (iii), how the solution to the Impartial Observer’s constrained optimisation problem depends on the value of λ (you don’t need to solve for the actual solutions!).

2.  Consider again the exchange setting from Problem 1, but assume in addition that the endowment allocation is given by

(xz(A),yz(A),xz(B),yz(B)) = (1, 0, 0, 1),

so A has all of the available amount of good x, and B all the available amount of good y .  Assume furthermore that A has TIOLI power for any potential exchange between the two consumers.

(i) Compute B’s fallback utility corresponding to his initial endowment (xz(B),yz(B)), and express all points  (xA ,yA ) that lie on B’s participation constraint as a relation that describes yA   as a function of xA .

(ii) Derive the post-exchange allocation/outcome that results from the backward induction equilib-

rium of the sequential exchange game where A has TIOLI power. (Finding a numerical approx- imation of the allocation is sufficient.)

(iii) Compute the optimal amount * that will be traded in the backward induction equilibrium from

A to B, and the associated per-unit price pˆ*  that B pays for good x in terms of good y .