Robinson has preferences described by the utility function u(c, h) = log c – yh, where c denotes consumption and h hours of work. Assume that the total number of available hours is equal to one so that l+h = 1. The production function is y = Ahª, with 0 < a < 1. (a) What is the optimal choice of hours worked, h*? (b) What happens to h* if A rises? (c) What happens to h* if we increase the value of y?
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- You have £20 per week to spend, and two possible uses for this money: telephoning friends back home, and drinking coffee. Each hour of phoning costs £2, and each cup of coffee costs £1. Your utility function is U(X,Y) = XY, where X is the hours of phoning you do, and Y the number of cups of coffee you drink. What are your optimal choices? What is the resulting utility level? You can use the standard result on the constrained maximization of such a function, but must state it clearly. Now suppose the price of telephone calls drops to £1 per hour. What are your optimal choices? What is the resulting utility level? How much income per week will enable you to achieve the same quantities at the new prices as the ones you chose before? What income will enable you to attain the same utility as you did before? Comment on your answer in the context of equivalent variation and compensating variation.What is the optimal number of work hours for the student whose utility function for other goods (X) and leisure (L) is U (C,L) = CL, and who has $50 of nonlabor income per week and the possibility to work at $5 per hour. Assume that after studying for class & other activities, the student has only 50 hours per week remaining to choose between work and leisure.Consider a utility function l(X_{A}, X_{B}) = X_{A}*X_{B} Let P_{A} =\$3 and P_{B} =\$2. and income is set at M =\$40. Suppose P_{B} falls to P_{B}' = 1 1. Before the price change, what was x_{A} ^ * and x B^ * the optimal consumption bundles? Sketch the original budget line and label the point ( x_{A} ^ * ,x B ^ * ) as A. Let x_{A} be on the horizontal axis. 2. If, after the price change, income changed so that the original optimal bundle is just as affordable. What is the new income, m' ? At (p_{A}, p_{B}', m') what is the new optimal bundle (x_{A}', x_{B}')' Sketch the budget line associated with p_{A}, p_{B}', m' ) . Label the point (x_{A}', x_{B}') as B. 3. Does the substitution effect result in more x_{B} ? How many more or fewer? 4. After the price change, how much x_{A} and x_{B} are actually bought. ( x_{A} ^ prime prime ,x B ^ prime prime )? Sketch the budget line associated with (p_{A}, p_{B}', m) Label the point x_{4} ^ prime prime , x_{R} ^ prime prime ) as C. 5.…
- Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F Suppose as Case 1, Total income is $100 and per unit prices of Food (F) and Cloth (C) are $2 and $15, respectively, then: a) What is the value of MRS at the optimal point and what does this value mean? b) What is the optimal consumption bundle i.e (F*,C*)? c) Also plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) spaceConsider a utility function: U (F,C) = FC so MU_F = C and MU_C = F Suppose as Case 1, Total income is $100 and per unit prices of Food (F) and Cloth (C) are $2 and $10, respectively, then: a) What is the value of MRS at the optimal point and what does this value mean? b) What is the optimal consumption bundle i.e (F*,C*)? c) Also plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space Now assume a new Case 3, where instead of one time income change, Pc' = $15, holding all else the same as in Case 1, do the same analysis (parts a-c) and contrast your answers to Case 1. For part c, you should draw old (Case 1) and new (Case 3) budget lines/point of optimality.Please answer and explain case 3 and draw draw old (Case 1) and new (Case 3) budget lines/point of optimality.Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F Suppose as Case 1, Total income is $100 and per unit prices of Food (F) and Cloth (C) are $2 and $10, respectively, then: a) What is the value of MRS at the optimal point and what does this value mean? b) What is the optimal consumption bundle i.e (F*,C*)? c) Also plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space Case # 2 assuming if income increases to $120, holding all else the same, do the same analysis (parts a-c) and contrast your answers to Case 1. For part c, you should draw old (Case 1) and new (Case 2) budget lines/point of optimality.
- Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F Suppose as Case 1, Total income is $100 and per unit prices of Food (F) and Cloth (C) are $2 and $10, respectively, then: a) What is the value of MRS at the optimal point and what does this value mean? b) What is the optimal consumption bundle i.e (F*,C*)? c) Also plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space Case # 2 assuming if income increases to $120, holding all else the same, do the same analysis (parts a-c) and contrast your answers to Case 1. For part c, you should draw old (Case 1) and new (Case 2) budget lines/point of optimality.Please answer and explain case 2 and compare their budget lines/point of optimality.Dick, Jane, and their dog, Spot, share an apartment. Their utilities depend upon the cleanliness of the apartment and the hours spent doing housework. Dick and Jane are both averse to housework, though Dick finds it especially troublesome. Their utility functions are where c is an index of cleanliness and h represents hours of housework. The degree of cleanliness is determined by the “production function” a) Define the marginal rate of substitution MRSi as the increase in cleanliness that exactly compensates person i for one more hour of housework: and define the marginal rate of transformation as the number of hours of leisure that must must be given up to generate a one unit increase in cleanliness. Find the Samuelson condition for this economy. b) An allocation is a triplet (h D, hJ, c). What two conditions must a Pareto optimal allocation satisfy? Show that there are Pareto optimal allocations in which Dick, despite his aversion to housework, does more…Q11. Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F. Suppose as Case A, Total income is $120 and per unit prices of Food (F) and Cloth (C) are $2 and $10, respectively. a. What is the value of MRS at the optimal point and what does this value mean? b. What is the optimal consumption bundle i.e (F*,C*)? c. Plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space.
- Assume an individual's utility from consuming good #1 and good #2 is given by the following function: U (q1 , q2) = min (q1 , 2q2) Suppose the price of good #1 is $1 (p1=1) and the individual's income is $10 (y=10). If this individual's utility maximizing decision is to purchase 2 units of good #2 (q2=2), what must be the price of good #2?A student spends 165 HUF on books (x) and renting bikes (y). The price of an average book is 12 HUF and renting a bike for 1 hour costs 15 HUF. The preferences of the student are given by her utility function U(x,y) = x^(0.9)y^(0.1). Assuming that the student is rational, for how many hours is she renting a bike if she can only rent the bike for round hours?Aisha is considering how to allocate the next 6 hours of her free time. She could choose between leisure (L) and helping her neighbour with the house chores. If she decides to help her neighbour, she is going to get paid at £25 per hour, which she can then spend on her favourite pizza (P). Suppose the price of pizza is £12.50. Aisha’s preferences for leisure and pizza are given by the following utility function: . U(L,P)= 3L + P MU(L)= 3 MU(P)=1 Write down Aisha’s budget equation and draw the corresponding budget line. Clearly label the axes and calculate the coordinates of the points of intersection of the budget line with each axis. Calculate Aisha’s marginal rate of substitution between leisure and pizza. Explain the concept of MRS and interpret the figure obtained. Find Aisha’s optimal consumption bundle, both algebraically and graphically. Explain your reasoning. Would Aisha’s optimal choice change if she could get a discount on her pizza purchases so that each pizza would cost…
