Perhaps the question is incomplete. If I assume a Rawlsian welfare function I'll get a different result than I'll get from maximising sum of utilities which would be different from weighted sum of utilities.

The only other thing I can think of is he may want the allocation to be fair i.e no one envies each other and locations are pareto optimal.

If Utility is convex analysis for each kind of welfare function changes but again idk.

A â€˜successâ€™ is defined as picking an â€˜Aâ€™ such that the polynomial x2 âˆ’ Ax + 1 has at least one real root. Mr. X is picking A from uniform distribution over [0, 5]. Mr. Y is picking A âˆˆ [0, 5]
with the probability distribution function fA(a) given by : fA(a) = 2a âˆ€a âˆˆ [0,5]. Which
person has a higher probability of success?

Since we dont have a defined welfare function, we know that if preferences are convex, there exists an allocation which is fair and pareto optimal (Hal Varian). If everyone has the same preferences, the only such allocation is where everyone has the same utility.

Now if transfer from c1 to c2 results in a loss, and we still want to ensure fairness, we still need to give everyone the same bundle since everyone still has the same preference. So both cities will have same bundle, but this will not maximize total utility. Still, it is a fair and a pareto efficient outcome.

If preferences are concave a fair and PE allocation may not exist, it would depend on the nature of the function.

Hey what's the relevance of CRS in this question ?

Suppose a monopolist faces constant returns to scale. Answer the following parts:
(a) Why he never produces on the inelastic side of the demand curve?
(b) Why does total revenue increase on the inelastic side as price increases?