Problem 4: Time Allocation with Log-Lin Preferences 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 = Ahº, 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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- How does a consumer’s optimal choice of goods change if all prices and the consumer’s income double? (Hint: focus on the budget constraint. You don’t have to, but you can use an example to support your answer).5. Output is produced according to a production process given by: Q = 4LK, where L is the quantity of labor input and K is the quantity of capital input. If the price of K is $10 and the price of L is $5, then what is the cost-minimizing combination of K and L capable of producing 32 units of output?Jane receives utility from days spent traveling on vacation domestically (D) and days spent traveling on vacation in a foreign country (F), as given by the utility function U(D,F) = 10DF. In addition, the price of a day spent traveling domestically is $100, the price of a day spent traveling in a foreign country is $400, and Jane’s annual travel budget is $4000. Suppose F is on the horizontal axis and D is on the vertical axis. Jane's marginal rate of substitution between F and D is equal to 10 1 F/D D/FConsider an economy with one consumption good, 100 identical consumers and 100identical firms. Each consumer is endowed with one unit of time and one unit ofcapital. The agent can spend time either working or enjoying leisure. A represen-tative consumerís utility function is u (x; l) = ln x + 2 ln l, where x is consumptionof goods and l is leisure. Each firm hires consumers to work and rents capital fromconsumers to produce goods: y = f (K; L) = K^1/2L^1/2, where L is the amount oflabor hired and K is the amount of capital rented. Both consumers and firms takegoods price p, wage rate w and rental rate r as given. Normalize p = 1. 1. Set up a representative consumerís utility-maximization problem.Derive the Marshallian demands for consumption and leisure as functions ofwage w and rental rate r 2. Set up a representative firm's profit-maximization problem. Writedown the first-order conditions regarding the choices of K and L 3. Set up all the market-clearing conditions. 4. Use the…
- 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 an individual who receives utility from consumption, c, and leisure, l. The individual has L time to allocate to work, n, and leisure. The individual’s consumption is a function of how much he works. In particular, c = root n. The individual’s maximization problem is max U =ln(c)+θl subject to c = √n n+l=L where θ > 0. Solve the maximization problem. Hint: Substitute both constraints into the objective function.Ivan faces a labor supply decision. His well-behaved preferences over the two goods "hours of leisure' L and 'consumption' c can be represented by u= 4(L)1/2 +c. He has no non-labor income and can choose how many hours to work at the wage rate u per hour. The price per unit of consumption is p, and his available free time is T hours .a)Sketch Ivan's budget set, with axes, intercepts, and slope labeled (these will depend on the parameters w, p, and T) .b)Use the tangency method to find Ivan's demand functions for leisure and consumption (as functions of u, p. and T) .c) Let's think about Ivan's "time expansion path" (that is, the analog of the income expansion path a.k.a. income-consumption loci but for changes in T). Sketch it and explain why it has this shape. with reference to Ivan's demand functions.d) (3 points) In terms of parameters from the model, what is the most that Ivan would be willing to pay to have an extra hour of free time (that is, to increase T by 1)? Why?
- Ivan faces a labor supply decision. His well-behaved preferences over the two goods "hours of leisure' L and 'consumption' c can be represented by u= 4(L)1/2 +c. He has no non-labor income and can choose how many hours to work at the wage rate u per hour. The price per unit of consumption is p, and his available free time is T hours .a) Sketch Ivan's budget set, with axes, intercepts, and slope labeled (these will depend on the parameters w, p, and T) .b) Use the tangency method to find Ivan's demand functions for leisure and consumption (as functions of u, p. and T) .c) Let's think about Ivan's "time expansion path" (that is, the analog of the income expansion path a.k.a. income-consumption loci but for changes in T). Sketch it and explain why it has this shape. with reference to Ivan's demand functions.d) In terms of parameters from the model, what is the most that Ivan would be willing to pay to have an extra hour of free time (that is, to increase T by 1)? Why?Q16 Pareto optimality refers to the maximum efficient allocation of economic resources such that there is no way that one's wealth another person worse off: this statement is choices - irrelevant -conditionally true -always true -always falseIvan faces a labor supply decision. His well-behaved preferences over the two goods "hours of leisure' L and 'consumption' c can be represented by u= 4(L)1/2 +c. He has no non-labor income and can choose how many hours to work at the wage rate u per hour. The price per unit of consumption is p, and his available free time is T hours Use the tangency method to find Ivan's demand functions for leisure and consumption (as functions of u, p. and T)
- The consumer choice is not restricted to the choice of consumptiongoods. In fact, it can apply to all our decisions that involve trade-offs. Suppose Mary has awage per hour of 10 euros. With her earned income she consumes. That isC=wH per day.She also decides how many hours to work of take leisure time each day.H=24-N, whereHis work and N is leisure. Her utility is given by （picture） Solve for the optimal decision of labor/leisure. Plot the budget constraint and the indif-ferent curve. What is the labor supply function?How would you demonstrate part b) diagramatically 6. Assume you can work as many hours you wish at £12 per hour (net of tax). If you do not work, you have no income. You have no ability to borrow or lend, so your consumption, c, is simply equal to your income. a) Derive and plot the feasible set, between daily values of consumption c, and “leisure”, l. Label the values at the intercepts (the points where the feasible frontier cuts the two axes). b) Assume that your optimal choice of consumption and leisure is to work 8 hours per day. Illustrate this choice diagrammatically using the feasible set and indifference curves.why is there no inclusion of the parameter z in the solutions and the budget constraint?