Consider an economy with two goods, consumption c and leisure 1, and a representative consumer. The consumer is endowed with 24 hours of time in a day. A consumer's daily leisure hours are equal to 1 = 24-h where h is the number of hours a day the consumer chooses to work. The price of consumption p is equal to 1 and the consumer's hourly wage is w. The consumer faces an ad valorem tax on their earnings of 7 percent. The con- sumer also receives some exogenous income Y that does not depend on how many hours she works (e.g. an inheritance). The consumer's preferences over consumption and hours of work can be represented by the utility function U(c, h) = c-3h¹+, where ß>0 and p>0 are parameters. 1+p
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- 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?Q2: Let a consumer’s daily hours of work is denoted by H, and hours of leisure by L. Consumer has no other source of income except wages for hours worked. She consumes what she earns each day. Her utility function is U(C, N) = ln(C) + 3 ln(N) Where C stands for the dollar amount of her consumption. Now answer following questions (a) Suppose the wage rate is 50Rs. per hour. Write down the consumer’s utility function and budget constraint with C and H as the choice variables. (b) How many hours will she choose to work, and what will be the resulting utility?21. Let U=x 2 +y 2 is the utility function of a worker who has 10 hours that to be allocatedbetween labour supply (L) and leisure (x). Let y is a consumption good whose price is 1.Wage rate (w) is Rs 1 and non-wage income is 20. Find out L.a) 10 b) 0 c) 5 d) 8 e) none 22. On the basis of the above question, hen w=0 and non-wage income is 40, find out L.a) 10 b) 0 c) 5 d) 8 e) none
- 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?a. Based on only the first-order condition with respect to labor computed in part a (Based on the given Lagrangian, compute the representative consumer's first-order conditions with respect to consumption and with respect to labor). Qualitatively sketch two things in a diagram with the real wage on the vertical axis and labor on the horizontal axis. First, the general shape of the relation ship between w and n (perfectly vertical, perfectly horizontal, upward-sloping, downward-sloping, or impossible to tell). Second, how changes in / affect the relationship (shift it outward, shift it in inward, or impossible to determine). Briefly describe the economics of how you obtained your conclusions. (Note: In this question you are not to use the first-order condition with respect to consump tion nor any other conditions.) b. Now based on both of the two first-order conditions computed in part a, construct the consumption-leisure optimality condition. Clearly present the important steps and…
- Need help on part (d) and (e) 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. c) Use indifference curves and the feasible set to show why, given the properties of the optimal choice in part b), it is not optimal to work, say, 10, or 6 hours per day. d) Now assume that you receive an income of £140 per week from an unknown benefactor. Show the impact on your feasible set, and show a new optimal choice in which consumption increases but labour supply decreases.…d. Based on both the consumption-leisure optimality condition obtained in previous part (Based on both of the two first-order conditions, construct the consumption-leisure optimality condition) and on the budget constraint, qualitatively sketch two things in a diagram with the real wage on the vertical axis and labor on the horizontal axis. First, the general shape of the relationship between w and n (perfectly vertical, perfectly horizontal, upward-sloping, downward-sloping, or impossible to tell). Second, how changes. in / affect the relationship (shift it outward, shift it inward, or impossible to deter mine). Briefly describe the economics of how you obtained your conclusions.(50 points) Consider the two-period economy with investment discussed in class. Forthe sake of simplicity, assume that the utility is time separable and that the per-periodutility is equal to:u(C) = log CAssume also that the technology to produce the consumption goods has constant returns to scale. Moreover, capital is the only input to production. Hence the currentperiod technology can be represented as:Y = zK1where Y , K, and z stand for output, capital and total factor productivity in the currentperiod. The future period technology is akin to the current period technology. There isno government expenditure in the current and future period so that G = 0 and G0 = 0.Finally there is full capital depreciation, namely δ = 1.(a) (10 points) Write down the social planner’s problem. State the variables withrespect to which the social planner maximises.(b) (10 points) Derive the first order conditions to the planner problem. Rearrangethere condition to find the conditions characterising the…
- 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…Shelly’s preferences for consumption and leisure can be expressed as U(C, L) = (C - 200) * (L - 80) This utility function implies that Shelly’s marginal utility of leisure is C - 200 and her marginal utility of consumption is L - 80. There are 168 hours in the week available to split between work and leisure. Shelly earns $5 per hour after taxes. She also receives $320 worth of welfare benefits each week regardless of how much she works. a. Graph Shelly’s budget line. b. What is Shelly’s marginal rate of substitution when L = 100 and she is on her budget line? c. What is Shelly’s reservation wage? d. Find Shelly’s optimal amount of consumption and leisure.If TP = 8L2-0.2L3 based on this function; A. Find the value of L that maximizes output B. How many workers can maximize an extra production and the value of labor that maximize APL D. Find the maximum value of APL and MPL and maximum production