Respuesta :
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.
