Suppose that the consumer side of the market for good x can be modeled using a representative consumer with (initially - until part (d)) non-quasilinear preferences, and suppose the industry that produces x is perfectly competitive with identical firms.
a. Illustrate a demand and supply graph with the market clearing price and quantity.
b. Would a social planner who takes the distribution of income as given and seeks to maximize social surplus choose the same output quantity as the market clearing quantity in (a)?
c. Does your answer to (b) change if the social planner initially redistributes income in a lump-sum way and then maximizes social surplus?
d. Next, suppose tastes are quasilinear in the good x and identical for all individuals. But there are two different types of consumers (represented in equal proportion in the population) - rich type 1 consumers and poor type 2 consumers. At their current income levels, type 2 consumers consume only the goodx (at quantity x? ) and no "other goods". For what type of lump-sum redistribution will we no longer be able to represent the consumer side as if it arose from choices by a representative consumer?
e. Continuing with (d), suppose further that utility for an individual is given by the utility level associated with her consumption of xplus the dollar value of all other goods she consumes. Let type 2's utility level at her current consumption x? be given by u?. If she were to then also consume $10 worth of other goods, her utility would be (u? + 10). If a social planner could redistribute "$'s of other goods" from type 1 to type 2 in a lump-sum way, what shape would the utility possibility frontier have in the range where individuals get at least u?.
f. Continuing with (e), draw the entire (first-best) utility possibility frontier (all the way to the horizontal and vertical axes) assuming that the utility possibility set has the feature you discovered in (e) and is convex. Indicate where on that utility possibility frontier we currently are in the absence of any redistribution. Which utility combination would be preferred to this by a Rawlsian social planner? Would a social planner who only cares about total utility object to the Rawlsian choice?
g. Now suppose that every dollar that is redistributed entails a penny of deadweight loss. How would your answer to (f) change? What would have to be true for the Rawlsian social planner to let go of his desire for full equality?
What will be an ideal response?
b. Yes, a social planner who takes the income distribution as given will maximize social gain by producing X*. This is because the aggregate marginal willingness to pay curve crosses the market marginal cost curve at the intersection of demand and supply.
c. Since we can represent demand by a representative consumer, redistributing income will not alter the demand curve. Thus, demand and supply continue to intersect at X* -- and a social planner would continue to produce the same quantity.
d. If income were redistributed from type 2 to type 1, type 1 consumers would not change their demand for x but type 2 individuals would have to given that they start at a corner solution prior to the redistribution. Thus, market demand would change and we could no long represent the demand side with a representative consumer.
e In this case, every dollar that is redistributed would redistribute 1 "util" from type 1 to type 2. The utility possibility frontier would therefore be linear with slope -1 for the range of utilities graphed here.
f. The current utility allocation (prior to redistribution) is at point A. A Rawlsian social indifference map would choose the utility allocation B on the 45 degree line.
If the social objective were instead to maximize the sum of utilities, any utility allocation along the flat portion of the utility possibility frontier could be chosen. Thus, a social planner who only cares about total utility would not object to the solution preferred by the Rawlsian social planner.
g. a. The picture looks almost identical to the one in part (f) - but the slope of the linear portion is now slightly shallower. A Rawlsian social planner would still choose complete equality at B?(which is slightly lower down on the 45 degree line than B in the previous graph of part (f).)
As a result, a social planner that aims to maximize total utility will strictly prefer A to B (and to any other point on the utility possibility frontier).
In order for the Ralwsian social planner to choose something other than full equality on the 45 degree line, the marginal cost of redistribution would have rise sufficiently as redistribution gets large for the second-best utility possibility frontier to slope down prior to reaching the 45 degree line (as in the graph below.) 
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