Q1 2 Points A representative household receives utility from consumption c and supplying labor 1: u(c, l)=√c+y√1-1 As usual, is the share of time that the household spends working. The household owns an exogenously determined amount of capital k, earns the nominal rental rate R on each unit of its capital holdings, earns the nominal wage W on each unit of labor it supplies, and pays the price P per unit of consumption.. The household's budget constraint is: Pc W1+ Rk Solve the household's utility maximization problem and select which of the following represents the household's labor supply equation: 1¼-6² (WP) = 2+ 1-2-()k ()+2 R/P O11-4 (WP) l= 1+6 -² (WP) Save Answer
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- Suppose that a person has 2000 hours to allocate each year between leisure and work. a. Derive the equation of his budget constraint given an hourly wage of $(15)/hour. b) Graph his budget constraint line based on the equation you derived in part a. (Consumption (C) on the vertical axis and leisure (L) on the horizontal axis). Please make sure to include the value for the vertical and horizontal intercepts. c) Now suppose that the local government introduces an income guarantee program for single parents in which the income transfer is $10,000 per year if an individual does not work during that year (this dollar amount represents the benefit guarantee). If the individual decides to work, this transfer program imposes a 100% benefit reduction rate (e. g.. each additional hourly wage earned is reduced by 100%). Derive the new budget constraint equation that corresponds to this scenario. d) Draw the budget line that corresponds to the new scenario on a new graph. (Consumption (C) on the…(Short Answers) 1. Jie works in a university. He can work as many hours as he wishes at a wage rate of w. Let C be the number of dollars he has to spend on consumption and let R be the number of hours of leisure that he chooses. Assume that Jie has the utility function U(C, R) = In(C) + In (R). He carms $4 per hour and has 80 hours per week to devote to labor or leisure, and has no income from sources other than labor. a) How many consumptions does he choose? How many hours of leisure does he choose? b) Suppose that Jie's wage rate will rise to $6 per hour from next year. How many hours of leisure per week will he choose next year? You are required to decompose his change in demand into the substitution effect, ordinary income effect and endowment income effect. c) Suppose that Jie will get $4 per hour for the first 35 hours that he works and $6 per hour for every hour beyond 35 hours a week from next year. How many hours of leisure per week will he choose next year?7. An individual's utility function is given by U =1000x, +450x, +5 x,x, -2x - x where x, is the amount of leisure measured in hours per week and x, is income earned measured in cedis per week. Determine the value of the marginal utilities, when x, = 138 and x, = 500. Hence estimate the change in utility if the individual works for an extra hour, which increases earned income by GH¢15 per week. Does the law of diminishing utility hold for this function?
- 5. Cindy gains utility from consumption C and Leisure L. Then most leisure she can consume in any given week is 110 hours. Her utility function is U (C, L) = C × L. Cindy receives $660 each week from her great-grandmother. (a) Find her marginal rate of substitution |M RS|. (b) Find her reservation wage. (c) Interpret the reservation wage.A) Ana decides every day how many hours to work and how much beef to consume. She spends all income earned from work each day on consumption of beef. Her utility function for free time (t, 24 hours minus hours worked) and consumption (y) is: U(t, y) = 2t + y The price per unit of beef consumed is 4 and her hourly wage is 1. • Using a diagram and appropriate algebra, explain what Ana's optimal number of hours work and units of beef consumed are each day. Ana reads a book about the impact of beef consumption on climate change and realises that there is a cost to society of her beef consumption. She takes account of this impact on society in her own decisions, incorporating a disutility of -0.03y per unit of beef consumed. She still only consumes beef and spends all income earned each day on this consumption. • Explain, using a diagram and appropriate algebra, how recognition of the impact of her consumption choice on climate change impacts on Ana's daily free time and beef consumption…1) Sharon spends her time (20h) between leisure (L) and work and he consume Y product from his working income (Py=1). Assume that she gets W$ per hour of working and has the following utility function: U (L, Y)=LY+2L a. b. C. Calculate the demand function for L and show it on a graph (L vs W). Calculate the labor supply (H) and show it on a graph (H vs W). What will happen to L, Y and H if the wage per hour (W) will decrease? =
- 1 The consumption-leisure framework Suppose that the representative consumer has the following utility function over consumption (c) and labour (n): u(c, l) = ln c − A 1 + � n 1+� (1) where, as usual, c denotes consumption and n denotes the number of hours of labour the consumer chooses to work, The constants A and � are outside the control of the individual, but each is strictly positive. Suppose the budget constraint (in real terms) faced by the individual is given by: c = (1 − t) · w · n (2) where t is the labour tax rate, w is the real hourly wage rate, and n is the number of hours the individual works. Remember, as seen in class, n + l = 1 is always true. Using the static consumption-leisure framework, answer the questions that follow. 1. State the utility maximization problem, and state carefully the choice variables in the problem. [2] 2. Write the Lagrangian function for this problem. [2] 3. Using the Lagrangian function from above, derive the first order condition with respect…Consider 5 workers who care about their consumption and continuous job satisfaction J.Their preferences are described by the utility function U(C,J) = 2C + J. There are 5 firms thatare producing the output using the production function Q(J,L) = L√20 − J1. What are the marginal rate of substitution between consumption and job satisfaction andthe marginal rate of transformation between wages and job satisfaction?2. What are the equilibrium levels of wage and job satisfaction?3. What is the slope of the wage-job satisfaction locus?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?
- - Find the marginal revenue function for the total revenue function given by 1 TR(Q) = 500Q Q3 - - 3 Find the marginal cost function for the total cost function given by TC(Q) = Q3 – 90Q² + 7,500QIvan 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?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 we aggregate the dollar value of all the goods that she consumes during the week and call it C. Assume that if Anne chooses to work, the hourly wage rate is w=9 and she has to sleep 8 hours per day. (Sleeping time is not considered leisure.) Potentially, if she wants, she can work up to 7 days a week.