Consider the standard Ramsey model with endogenous labor supply. Assume that the house- hold has a unitary time endowment that it can split between labor time (Lt) and leisure time (1 Lt). The household's problem is defined as follows: ∞ [Ct(1 - L₁) ¹-0] Vo max Ct,Ct+1,Lt,Lt+1,Kt+1 Y = Σ
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- In a standard consumer optimization problem in microeconomics, a consumer purchases any pair of goods untilU1U2=p1p2(?), at an interior solution, where Ux is the marginal utility of good x for the consumer and px is the price of good x, where x={1,2}. What is the corresponding optimality condition for a worker-consumer in his labour-leisure optimization problem and why is it just a variation on the standard optimality condition (?)?This question will analyze the impact on a person's labour supply from a shock to their partner's job. Assume leisure is a normal good. Let's assume Vanessa has a wage rate of $20 per hour. Recently her partner, Bill, had to take a wage cut at work, with his wage falling from $45 per hour to $30 per hour, but allowed them to continue working 40 hours per week. Analyze the decision of the household over choice consumption and Vanessa's leisure, taking Bill's hours as given (constant).Consider an economy with 100 identical households and 100identical firms. Each household is endowed with one unit of time. Half of the households are endowed with equal shares of firm, while the rest areendowed with no firm shares. A householdís utility is u (x; r) = 2 ln(x*r^2) ,where x is consumption of goods, r = 1- l is leisure time and l is thehouseholdís labour supply. Each firm hires households to produce goodsaccording to technology y = L^1/2, where L denotes the labour input. Goodsprice p, wage rate w, and dividend income are all taken as given. Normalizew = 1. Do the following: (a) Derive the Marshallian demands, x (p, D) andr (p,D), where D is dividend. (b) Derive the firmís input demand functionL (p) and profit function pi(p). (c) List all the market-clearing conditions.(d) Calculate the equilibrium price p
- Consider a one-period model in which the representative consumer splits 100 hours of available time between work hours denoted by NS and leisure hours denoted by L. The consumer receives wage rate w = 9 for every hour worked. In addition, the consumer receives dividend income = 29 and pays lump-sum taxes T = 81. Derive the budget constraint of the consumer. Type in the appropriate numerical values in the boxes provided below to obtain the equation that defines the budget constraint of the consumer. All numerical values should be integers, hence use 0 decimal places. The budget constraint is C = -XL, where the symbol x denotes a multiplication sign.Problem 4 Consider the leisure demand/labor supply model studied in class, and let the consumer have a spccific utility function U(N, Y) = N2/3y!/3. As in lecture, let the price of consumption be normalized to 1, and let w denote the wage. (a) Say that w = 10. What are the optimal N* and Y*? How many hours does the agent work? Draw a sketch to illustrate this situation. (b) Solve for the general demand functions N(w) and Y(w) as a function of the wage, as well as the labor supply function H(w). Calculate the elasticity of labor supply with respect to the wage, w. Do you notice anything special about this particular example? (c) Say the wage rises to some w' > w. What is the change in leisure demand N(w)? Carefully draw a sketch that decomposes this into an income effect and a substitution effect. (d) Sketch the labor supply function. (e) Let H(w) be the labor supply as a function of the (take-home) wage w. Say now that the government imposes an income tax of a. Let T(a) denote the…Consider the standard labor-leisure choice model. Consumer gets utility from consumption (C) and leisure (L). She has H total hours. She works NS hours and receives the hourly wage, w. She has some non-labor income T and pays lump-sum tax T. Further suppose (n-T)>0. The shape of utility function is downward-sloping and bowed-in towards the origin (the standard U-shaped case just like a cobb-douglas function) If this consumer decides to NOT WORK AT ALL, then it must be the case that O A. MUL = MUC O B. MRSL,CI 2 w OC. IMRSL,CIs w O D. MRSL,C = w O E. None of above
- Consider the following model of labour supply. There is a representative worker with the following utility function: U(C,L) = C + 3L The budget constraint and time constraint are: C = wh + V h = T – L where w = 1,V = 500,T = 100. The notation is the same as question 1. a) Calculate the optimal leisure and consumption. b) Explain why we usually do not use this type of utility function to model the labour supply.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.Let the demand and supply function for a commodity be Qa= D(p,Y) Dp 0 Qs = S(p, w) Sp>0, Sw<0 where p is the price, Y in exogenous income, and w is the exogenous wage rate. a. Find dp and dp using the implicit-function theorem. dw dp dy b. Find and dp by totally differentiating the equilibrium condition. dw
- Consider the representative consumer who decides consumption and leisure. Theenvironment is the same as in Lecture 5. Keep the same notation. The preference is givenby U (C,L) = αln C + (1 −α) ln L. Assume h = 1, i.e., the time endowment is one day.(a) Write down the utility maximization problem.(b) Derive the demand for consumption and the supply for labour.(c) Suppose the non-wage income π −T increases while the wage rate w falls at the sametime. The size of the changes can be different. Determine the effects on consumptiondemand and labour supply (i.e., leisure demand). Use the indifference map to explainyour results in terms of income and substitution effects for the following cases:(i) The increase in π −T exactly cancels out the drop in w, i.e., |∆ (π −T)|= |∆w|.(ii) The increase in π −T is greater than the drop in w, i.e., |∆ (π −T)|> |∆w|.(iii) The increase in π −T is smaller than the drop in w, i.e., |∆ (π −T)|< |∆w|.(d) Suppose the utility function is Cobb-Douglas: U…Consider a one period model in which a representative agent maximises the utility function: u(c,l) = lnc + 5lnl subject to the budget constraints: c = (1-t)w(1-l) + v where c is consumption and l is the amount of leisure, they enjoy out of a total of one unit of time available, t is the tax on wage earnings which pays for v in government transfer payments. A. Derive the equation that determines how much revenue the government will receive for a given rate of tax t. What is this relationship called? B. Solve for the maximum amount of revenue the government can raise from this tax. Hint: the tax rate will be a fraction between 0 and 1. C. In this particular example, what are the contributions of the income and substitution effects?A household has an endowment of 1 unit of time .The household maximises its utility u = In(c) + b In(1 – 1), where c denotes consumption and l e (0, 1] denotes time spent working. It finances its consumption from labour income wl, where w is the market wage rate per unit of labour time. If the market wage rate goes up, then equilibrium labour supply of the houschold a) Increases b) Decreases c) Remains constant d) Changes in ambiguous manner