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1
(a) alpha = beta
(b) Rawls would choose an allocation from the set of feasible allocations that maximizes min(u_1, u_2).
2 I am answering this question assuming that good cars are worth 1200 to the buyers. It is mentioned in the question but incorrectly referred as "some cars" instead of "good cars".
(a) At Pareto efficient allocation, all cars will be sold.
(b) If the buyers can't observe the quality, they will be willing to pay upto (600+1200)/2 = 900 for a randomly chosen car.
(c) Since sellers of the good cars value their cars at 1000 (> 900), they will not participate. Buyers also understand this and they will also know that in the market only bad cars will be available for sale, so they will revise their value for the cars that are available on sale to 600. In other words, for a randomly chosen available car the consumer will be willing to pay at most 600. Therefore, only bad cars will be sold in this market at some price between 500 and 600.
(d) The values of g for which all cars will be sold in equilibrium can be obtained by the following inequality:
1200g + 600(1-g) >= 1000
Simplifying it, we get
g >= 2/3.
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