5. Consider a representative household in the static consumption-leisure model with prefer- ences given by u(c, l) = Ac¹/² + 1¹/2. The household has a unit time endowment given by 1 = l +n, faces a price of P on consumption, and earns nominal wages at rate W on their labor supply, n. In addition, the household faces a proportional tax on wage income of T.

5. Consider a representative household in the static consumption-leisure model with prefer- ences given by u(c, l) = Ac¹/² + 1¹/2. The household has a unit time endowment given by 1 = l +n, faces a price of P on consumption, and earns nominal wages at rate W on their labor supply, n. In addition, the household faces a proportional tax on wage income of T.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter12: The Partial Equilibrium Competitive Model

Section: Chapter Questions

Problem 12.12P

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Consider a forward-looking individual who aims at maximizing her lifetime utility from her lifetime resources. Assume the initial endowment of the individual isand her expected labour income is in the sequence Her utility function takes the form where is consumption in period and . Assume the real interest rate, is constant but not equal to the discount rate . Suppose that this individual lives for two periods, write down her intertemporal budget constraint and carefully interpret it. Explain why the lifetime budget constraint must be satisfied with a strict equality. From the intertemporal budget constraint, derive the permanent income hypothesis (PIH) and explain the drivers of consumption growth in this model.
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…
Suppose that a consumer/investor has an initial endowment only for the current period, which is Eo =450. She may consume today or in the next period only (two-period model). The interest rate for borrowing and lending in the capital market is 5% (a)Depict the budget constraint for the investor in an inter-temporal consumption diagram! What is the maximum amount the consumer is able to consume in the next period? (b)The consumption preferences of the consumer/investor are best described by a square root function, defined over current and future consumption. What is his optimal consumption plan? Show your calculations! Depict the results in appropriate diagram. Which amount is invested in the capital market?
Consider an individual who lives for two periods and has utility of lifetime consumption U = log(C1) + 1/1+δ log(C2), where C1 and C2 are the consumption levels in the first and second period respectively, and δ, 0 1 > 0 in the first period and no income in the second period, so Y2 = 0. He can transfer some income to the second period at a before-tax rate of return of r, so saving $S in the first period gives $[1 + r]S in the second period. The government levies a capital tax at rate τ on capital income received in the second period. The tax proceeds are paid as a lump-sum transfer to the following generation. The present generation does not care about the next one. a. What is the lifetime consumption profile of this individual? What is his lifetime indirect utility function expressed as a function of Y1 and b. Evaluate the change in initial income Y1 that is required to compensate the individual for the welfare loss due to the capital income tax τ. c. What is…
ADVANCED MACROECONOMICS: MODERN MACROECONOMICS MODEL It is known that the maximization of the household utility function is as follows: U₁ = fety InC₁ + (1 − y) ln(L – 1,)]dt Withconstraint : K₁ = (R₂ - S)K + Welt - Ct (2) The firm produces output through the production function (Yt = F(Kt. Lt. At)) Cobb Douglas constant return to scale with the aim of maximizing profit. It is assumed that the economy is a closed economy. A. Derive the labor supply equation. Explain how the relationship between increased household preferences for (i) consumption and (ii) leisure time on labor supply. B. Derive the labor demand equation. Explain how the relationship between (i) wages and (ii) capital to labor demand. C. Determine the amount of labor (level of labor) from the above economy in steady
Consider the two-period endowment economy we discussed in class. In this economy, a household lives for two periods and can save s. The return on saving is the real interest rate,r. Income is exogenous and given as y and y'. Households have utility overconsumption,c, in each period by U(c)=In(c)+B In(c') where future utility is discounted with rate B. Households choose consumption in both periods,c and c'. 1-Write down the current period and future period budget constraints. Derive the lifetime budget constraints. 2-What is the slope of the lifetime budget constraint? What are the intercepts? What is the significance of the endowment point? 3-Write down the expression for the consumer's optimization problem. What is the tangency condition? Provide an intuitive explanation for this condition. 4-Solve for c,c', and s. Given that B=1 and r=0, how would you modify the answers for c,c; and s? Explain.
3. Suppose we are in a society where the social rate of time preference is 5% per year. The discount rate of utility is 3.5% per year, and the elasticity of marginal utility of consumption is 1.25. A. What is the assumed growth rate of future consumption under this scenario? B. Now assume the social rate of time preference changes to 10% while all else stays the same. What is the assumed new growth rate of future consumption? C. What does a SRTP of 5% mean compared to a SRTP of 10%? D. Why does the growth rate of future consumption change from one scenario to the other? Answer C & D
3. Suppose we are in a society where the social rate of time preference is 5% per year. The discount rate of utility is 3.5% per year, and the elasticity of marginal utility of consumption is 1.25. A. What is the assumed growth rate of future consumption under this scenario? B. Now assume the social rate of time preference changes to 10% while all else stays the same. What is the assumed new growth rate of future consumption? C. What does a SRTP of 5% mean compared to a SRTP of 10%? D. Why does the growth rate of future consumption change from one scenario to the other? ( If you Answer allow the above I will upvot definitely . ) Thank you
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?
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…
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
Suppose that firms produce according to the production function Y = AK1/2L1/2, where A = 5 andL = 400. Assume that the prices of capital and output are equal and that the real interest rate, r, isequal to 0.25 and the depreciation rate, δ, is equal to 0.1.1. If firms operate according to the neoclassical theory of investment, what is the optimal levelof capital stock, K (YP) to rich (YR).22. Suppose that the government offers an investment tax credit which changes the relative priceof capital. This results in Pk = 3 and P = 6. What is the new optimal level of capital stock,K??3. Does the investment tax credit have an expansionary impact on the economy? Explain whyor why not.4. Based on the optimal capital stock computed in part (2), what is the level of investmentneeded to sustain this level of capital stock?