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Consider the Real Business Cycle (RBC) model studied in class. Recall that this model features a representative household that lives for two periods. In the first period, the household derives utility for consumption and leisure C and l, respectively, and in the second period the household derives utility from consumption C'. Assume that the household’s preferences are represented by the utility function:
U(C, l, C') = u(C) + θv(l) + βu(C'),
where u and v are strictly increasing and strictly concave, θ > 0 is a parameter which captures how much the household values leisure, and β ∈ (0, 1) is the household’s discount factor.
In the first period, the consumer supplies labour in a competitive market at the hourly wage w, pays lump-sum taxes T, receives dividends π, and decides how much to borrow or lend at the real interest rate r. In the second period, the household receives dividends π', pays debts or collects savings returns, and pays lump-sum taxes T' .
(1) Write down the problem that the representative household solves when choosing (C, l, C'). Use the lifetime budget constraint in your formulation.
(2) Write down the Lagrangian associated with the household’s problem.
(3) Using first order conditions, show that the following holds at the optimum: [θv' (l) /u'(C) ]= w. Interpret this optimality condition.
(4) Suppose that θ suddenly increases, e.g., because the household becomes sick. Based on the optimality condition from part (3), explain the impact of the increase in θ on the labour supply decision of the household.
(5) Consider the effects of an increase in θ on the labour market in the first period. How would the shock to θ impact employment and the wage in equilibrium? Explain by using the equilibrium diagram for the current labour market in the RBC model.
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- Q7: An individual lives for only two periods and has preferences given by the follow- ing intertemporal utility function: U = lng + aln(1-h₁) + B[lno₂+ a ln(1 — h₂)] (1) where, c₁, c₂ denotes consumption in period 1 and period 2 respectively h₁, h₂ denote the labour supply in period 1 and 2 respectively. Therefore 1-h is the amount of leisure time in period 1. The term 3 is the discount factor. The problem of the individual at period 1 is to choose consumption in both periods and labour supply in both periods subject to the following budget constraints: +8= why and 0₂= w₂h₂ + (1+r)s where s denote the saving, w denotes the wates and r denotes the real interest rate. (a) Provide an economic interpretation of the two budget constraints written above. (b) Combine the two budget constraints written above and prove the economic interpretation of of life time budget constraint. (c) Set the Lagrangean function and find the first order conditions with respect to C₁ C₂, hi and h₂. (d) Find the…arrow_forwardConsider a worker who consumes one good and has a preference for leisure. She maximizes the utility function u(x, L) = xL, where a 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. Solution: The problem is max x L s.t. px ≤w(1 L) + y. x>0, LE [0,1] The budget constraint may also be written with equality since preferences are monotone. (b) Find the Marshallian demands for the consumption good and leisure. Solution: Using FOCs will find the maximum since preferences are Cobb-Douglas (and therefore convex). Dividing the FOCs L= Xp and x = Xw gives wL = px. Substituting into the budget constraint and checking the restriction L = [0, 1], we get and x(p, w, y) L(p, w, y) = = = 1 w+y…arrow_forwardConsider the following one-period model. Consumer Utility function over consumption (C) and leisure (L) U(C,L)= C^(1/2)L^(1/2) = Total hours: H = 40 Labour hours: = H – L Non-labour income: π Lump-sum tax: T Hourly wage: w Firm Production function: Y = zF() = z Total factor productivitiy: z = 2 Government Government spending (exogenous): G = 20 Suppose that the total factor productivity, z, increases to 5. What is the substitution effect of this wage change on labour supply()? A. +8.51 B. -5.51 C. -8.51 D. +5.51 E. None of the abovearrow_forward
- 1. Consider a two-period model in which you work and save in the initial period (period 0) and live off savings and the interest from savings in retirement (period 1). Suppose that income in period 0 is $250,000, income in period 1 is $0, and the interest rate is 50%. a. Draw the budget line. b. Draw an indifference curve consistent with a person whose preferences causes them to equalize their consumption in period 0 and period 1. c. Savings are $_ d. Draw the after-tax budget line if a personal income tax is introduced with a marginal tax rate of t= 40%. e. Suppose that preferences are such that after tax consumption is equalized in period 0 and period 1. On the diagram show after-tax savings. Сі ($x 1,000) 450 400 350 300 250 200 150 100 50 25 50 75 100 125 150 175 200 225 250 275 300 325 350 Со ($ x 1,000)arrow_forwardConsider the two-period household-maximization model discussed in class. The model is modified in order to look at applications including credit constraints, interest-rate markups, and taxation. A representative household lives for two periods and maximizes utility of consumption in period 1 and in period 2. The utility is represented by log(c) where c denotes consumption. Assuming no discounting between period 1 and period 2. The maximization problem for the representative household can be written as a) Explain what is meant by a representative household. Briefly explain the budget constraints of the representative households and of the government. Explain the role played by the assumption that the representative households lives for only two periods and the assumption of “no discounting”.arrow_forwardThe one-period model with quasi-linear utility predicts that a decrease in marginal income tax rates could increase tax collection if:Group of answer choices Substitution effects dominate income effects so that the percent change in taxes is greater than the percent change in GDP Substitution effects dominate income effects so that the percent change in taxes is less than the percent change in GDP Income effect dominate substitution effects so that the percent change in taxes is less than the percent change in GDP Income effects dominate substitution effects so that the percent change in taxes is greater than the percent change in GDParrow_forward
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