User In any given week, Anne receives satisfaction only from the consumption of goods and from the consumption of leisure L, which is measured in hours, obviously, Anne consumes many different types of goods, but for simplicity was aggregate the dollar value of all their goods that she consumes during the week and call it C. Assume that if Anne chooses to work, the hourly wage rate is w=6 and she has to sleep 8 hours per day, Potentially ,if she wants, she can work up to 5 days a week, Call L Anne’s weekly leisure time, where L is all the hours during the 5-day workweek during which Anne is neither working nor sleeping. Anne's utility is given by u(C,L) =𝐶^a𝐿^(1−𝑎), where a=0.6. (a)(1 point) What is Anne's utility maximizing choice of C? (b)(1.5) By how many hours does her optimal choice of L increase if the government gives her a weekly unconditional subsidy of 20?
Question
User In any given week, Anne receives satisfaction only from the consumption of goods and from the consumption of leisure L, which is measured in hours, obviously, Anne consumes many different types of goods, but for simplicity was aggregate the dollar value of all their goods that she consumes during the week and call it C. Assume that if Anne chooses to work, the hourly wage rate is w=6 and she has to sleep 8 hours per day, Potentially ,if she wants, she can work up to 5 days a week, Call L Anne’s weekly leisure time, where L is all the hours during the 5-day workweek during which Anne is neither working nor sleeping. Anne's utility is given by u(C,L) =𝐶^a𝐿^(1−𝑎), where a=0.6. (a)(1 point) What is Anne's utility maximizing choice of C? (b)(1.5) By how many hours does her optimal choice of L increase if the government gives her a weekly unconditional subsidy of 20?
Solution
(a) To find Anne's utility maximizing choice of C, we need to set up the Lagrangian for this utility maximization problem. The Lagrangian is a function that summarizes the problem and allows us to use calculus to find the optimal choices.
Anne's budget constraint is C = w(112 - L), where 112 is the total hours in a 5-day week (24 hours/day * 5 days/week - 8 hours/day * 5 days/week for sleep).
The Lagrangian is then L = C^a * L^(1-a) + λ[w(112 - L) - C], where λ is the Lagrange multiplier.
Taking the derivative of the Lagrangian with respect to C and setting it equal to zero gives us the following first order condition:
a * C^(a-1) * L^(1-a) - λ = 0.
Solving this for λ gives us λ = a * C^(a-1) * L^(1-a).
Substituting this into the budget constraint gives us C = w(112 - L)/λ = w(112 - L)/(a * C^(a-1) * L^(1-a)).
Solving this equation for C gives us the utility maximizing choice of C.
(b) If the government gives Anne a weekly unconditional subsidy of 20, her budget constraint changes to C = w(112 - L) + 20.
Substituting this into the equation for λ gives us λ = a * (w(112 - L) + 20)^(a-1) * L^(1-a).
Solving this equation for L gives us the new optimal choice of L. The difference between this and the original optimal choice of L gives us by how many hours her optimal choice of L increases.
Note: The above steps provide the general approach to solve this problem. However, the actual calculations can be quite complex and may require numerical methods.
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