Consider a worker who consumes one good and has a preference for leisure. She maximizes the utility function u(x, L) = xL, where à represents consumption of the good and L represents leisure. Suppose that this worker can choose any L = [0, 1], and receives income w(1-L); w represents the wage rate. Let p denote the price of the consumption good. In addition to her wage income, the worker also has a fixed income of y ≥ 0. (a) Write down the utility maximization problem for this consumer. (b) Find the Marshallian demands for the consumption good and leisure. (c) Find the indirect utility as a function of p, w, and y.
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- Consider the problem of a consumer who chooses between consuming goods and enjoying leisure in the current and future periods. Denote the consumption and leisure in the current period as C and l, and the consumption and leisure in the future period as C′ and l′, respectively. The preference is summarized by the following utility function: U(C,C′,l,l′)=lnC+ψlnl+β(lnC′ +ψlnl′). This individual is endowed with h units of time in each period. Wage rate per unit of labour time is w and w′ in the current and future period. In addition, the consumer receives profit transfer π and π′ and pays lump-sum taxes T and T′ in the current and future periods. Denote the saving in the current period as Sp. Answer the following questions. Derive the life-time budget constraint of this consumer. Set up the consumer’s problem. Solve for consumption (C and C′), leisure (l and l′), and saving (Sp). How does an increase in wage rate w affect C, Sp, and l?A worker receives, when unemployed, an offer to work forever at wage w, where w is drawn from the distribution F(w). Wage offers are identically and independently distributed over time. The worker maximizes Where ct is consumption and lt is leisure. Assume Rt is i.i.d. with distribution H(R). The budget constraint is given by And lt + nt ≤ 1 if the worker has a job that pays wt. If the worker is unemployed, the budget constraint is at+1 ≤ Rt(at + z − ct) and lt = 1. Here z is unemployment compensation. It is assumed that u(·) is bounded and that at , the worker’s asset position, cannot be negative. This assumption corresponds to a no-borrowing assumption. Write the Bellman equation for this problem.Assume that consumption and leisure are perfect complements, that is, the consumer always desires a consumption bundle where the quantities of consumption and leisure are equal, that is, C=L 1) (Denote the total hours of time available by h, the real wage by w, the real dividend income from firms by pi (π), and the lump-sum tax by T. Write down the consumer’s budget constraint. 2) Determine the consumer’s optimal choice of consumption and leisure. 3) Assume that there is an increase in w . Show how the consumer’s optimal consumption bundle changes. Explain with reference to income and substitution effects
- The consumer's utility function for Consumption (C) and Leisure (L) is given as U(C,L) = √CLHis hourly wage is $10, non-labor income is $20; and he has a total of 16 hours to allocate between labor and leisureBased on this information, the consumer's total utility at the optimal level (or optimal C,L combination) is:a. 57.0 utilsb. 28.5 utilsc. 99.75 utilsd. 114.5 utilse. Cannot be determined with the information given I prefer typed answers.Suppose a person can work up to 80 hours per week at a pre-tax wage of $20 per hour but faces a constant 20% payroll tax. Assume that under these conditions the person maximizes utility by choosing to work 50 hours each week. The government proposes a negative income tax so that everyone receives $300 per week regardless of how much they work. To pay for the negative income tax, the payroll tax would be increased to 50%. Using the labor-leisure model, graphically show whether a person would be better off if the negative income tax is adopted and indicate whether hours worked increases or decreases due to the policy.Consider worker 1 with non-labour income Y facing a wage offer w and a utility function defined over consumption and leisure. U(c,l) = lnC + 4lnl Derive worker’s income elasticity. Is leisure a normal or inferior good for this worker?
- Consider worker 1 with non-labour income Y facing a wage offer w and a utility function defined over consumption and leisure. U(c,l) = lnC + 4lnl a) Derive worker’s income elasticity. Is leisure a normal or inferior good for this worker? b) Provide the functional form of the income effect from a marginal decrease in income. c) Provide the functional form of the substitution and total income effects of a marginal increase in wage.Consider a representative agent with the utility function U = ln(Ct)+ Nt The budget constraint is Ct = wtNt +Dt where wt is the wage and Dt is non-wage income (i.e. a dividend from ownership in the firm). The agent lives for only one period (period t), and hence its problem is static. (a) Derive an optimality condition characterizing optimal household behav- ior. (b) Solve for the optimal quantities of consumption and labor. Plz do fast asap, urgent.Given the consumption function C = a + bY (with a>0, 0 < b<1): a. find its marginal function and its average function. b. find the income elsaticity of consumption ECY, and determine its sign, assuming Y > 0. c. Show the income that this consumption function is inelastic at all positive income levels
- The weekly preferences over consumption (C) and leisure(L) are defined by u(C, L) = √C + 3√L. The person receives a weekly allowance of m from The hourly wage is $18 per hour, and the person can work up to50 hours each week (T = z + L = 50), where z is the number of hours spent working). a)How many hours will the person work if her allowance is m= $450 per week b) What is the smallest allowance m for which the person will stopworking altogether (z∗ = 0) for a wage of w = 18?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) noneHow would you demonstrate part c) 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. 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.