Marbella has 101 residents.All wear the same fancy clothes and each has the same utility function, u(m, b, B)  m  24b  b²  B/50, where m is the amount of macaroni (in kilograms) that he or she eats per day, b is the number of hours that he or she spends on the beach per day, and B is the total number of person-hours spent per day on the beach by other residents of Marbella.Each has an income of $10 per day and macaroni costs $1 per kilogram.The city council is considering a law that would limit the amount of time that any person can spend on the beach.How many hours per day should they allow in order to maximize the utility of a typical Marbellite?

A) 12
B) 14
C) 11
D) 15
E) They could not possibly be made better off by legislation that limits their freedom to choose.

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Devyn Whitten · Accepted Answer

Final Answer : C, Explanation :  To maximize the utility of a typical Marbellite, we need to maximize their total utility function u(m, b, B). Since each person's utility function depends on the number of person-hours spent on the beach by other residents of Marbella, we can assume that there is a negative externality associated with spending too much time on the beach. This means that each person's utility decreases as the total hours spent on the beach by all residents increase.Thus, the city council should limit the amount of time that any person can spend on the beach in order to reduce the negative externality and increase the total utility of all residents. We need to find the maximum value of b that also satisfies the budget constraint:10 = 1m + 24b + (B/50)Rearranging the budget constraint, we get:B = 500 - 50m - 1200bSubstituting this into the utility function, we get:u(m, b) = m + 24b - b(B/50)= m + 24b - b(500 - 50m - 1200b)/50= m + 24b - 10b(50 - 5m - 120b)Taking the partial derivative of u with respect to b, we get:∂u/∂b = 24 - 10(50 - 5m - 120b) - 10b(-120)= 24 + 500 - 50m - 1200b + 1200b= 524 - 50mSetting this equal to zero, we get:524 - 50m = 0m = 10.48Since m must be a whole number, we round up to m = 11. Substituting this back into the budget constraint, we get:B = 500 - 50(11) - 1200bB = 330 - 1200bWe need to find the maximum value of b that satisfies this constraint. Since b cannot be negative, we set B = 0 and solve for b:0 = 330 - 1200bb = 11/3Thus, the city council should limit the amount of time that any person can spend on the beach to 11/3 hours per day, or approximately 3.67 hours per day, in order to maximize the utility of a typical Marbellite.

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