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Marbella has 101 residents. All wear the same fancy clothes and each has the same utility function, uſm, b, B) = m + 166 – 62 – B/50, where m is the amount of maccaroni (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 maccaroni costs $1 per kilogram. 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. 7 hoursb. 8 hoursc. 10 hours

Sagot :

Answer:

a = 7 hours is the time must be allowed to maximize the utility of a typical Marbellite.        

Explanation:

Note: Some corrections in the questions

Wrong Utility function is = uſm, b, B) = m + 166 – 62 – B/50,

Correct Utility function = u ( m, b, B) = m + 16[tex]b^{}[/tex] - [tex]b^{2}[/tex] - [tex]\frac{b}{50}[/tex]

Data Given:

Marbella residents = 101

Utility Function = u ( m, b, B) = m + 16[tex]b^{}[/tex] - [tex]b^{2}[/tex] - [tex]\frac{b}{50}[/tex]

where m = amount of maccaroni (Kg)

b = hours spend on beach

B = total person-hours spent on beach

Each residents has an income of 10$ per day

maccaroni costs = 1$ per kg

Required = How many hours per day should they allow in order to maximize the utility of a typical Marbellite ?

Solution:

In order to find the required statement, we need to find the value of [tex]b^{}[/tex] from the utility function.

And it can be done by applying the partial differentiation on the utility function.

u ( m, b, B) = m + 16[tex]b^{}[/tex] - [tex]b^{2}[/tex] - [tex]\frac{B}{50}[/tex]

[tex]\frac{du}{dm}(m,b,B)[/tex] = [tex]\frac{dm}{dm}[/tex] +16[tex]b^{}[/tex] - [tex]b^{2}[/tex] - [tex]\frac{B}{50}[/tex]

[tex]\frac{du}{dm}(m,b,B)[/tex] = 1

[tex]\frac{du}{db}(m,b,B)[/tex] = 1m +16[tex]\frac{db}{db}[/tex] - [tex]\frac{db}{db}[/tex][tex]b^{2}[/tex] -[tex]\frac{B}{50}[/tex]

[tex]\frac{du}{db}(m,b,B)[/tex] = 16- 2[tex]b^{}[/tex]

[tex]\frac{du}{dB}(m,b,B)[/tex] = 1m +16b- 2[tex]b^{}[/tex] - [tex]\frac{dB}{dB}[/tex][tex]\frac{B}{50}[/tex]

[tex]\frac{du}{dB}(m,b,B)[/tex] = - [tex]\frac{1}{50}[/tex]

Solving the above equations, we will get b.

b = 7 hours

Hence, a = 7 hours is the time must be allowed to maximize the utility of a typical Marbellite.        

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