P3 2. Labor-leisure in the RCK model. Consider the following economy that is very similar tothe economy in the RCK model except that we assume the labor supply is not fixed. We includeleisure in the utility function such that the lifetime utility of a representative household ist=0where u(c) is twice differentiable and strictly concave in each argument. In this setting, &is the number of hours worked by the representative household at time t. The rest of time1-4 is consumed as leisure. The production function is therefore y, f(k,l), which is alsotwice differentiable and strictly concave in each argument. Suppose that the initial conditionof capital per worker is given, ko > 0 and the transversality condition is satisfied.(a) Set up the maximization problem faced by the planner and derive the set of equations thatcharacterize an interior solution to the planner's problem.(b) Derive the set of equations that characterize a steady state.(c) Suppose that y = f(k,l4) = kil-a. Solve for the steady state capital-labor ratio. Go asfar as you can to describe how to solve for the steady state (k*, l*, c*).

The lifetime utility of the household model is given as l represents the work hour provided by each…

2. Labor-leisure in the RCK model. Consider the following economy that is very similar to the economy in the RCK model except that we assume the labor supply is not fixed. We include leisure in the utility function such that the lifetime utility of a representative household is ∞ Σβ'u(c,1 - 4) t=0 where u(c) is twice differentiable and strictly concave in each argument. In this setting, is the number of hours worked by the representative household at time t. The rest of time 1- is consumed as leisure. The production function is therefore y = f(kt, l), which is also twice differentiable and strictly concave in each argument. Suppose that the initial condition of capital per worker is given, ko> 0 and the transversality condition is satisfied. (a) Set up the maximization problem faced by the planner and derive the set of equations that characterize an interior solution to the planner's problem. (b) Derive the set of equations that characterize a steady state. (c) Suppose that y₁ = f (kr, lt) = kalla. Solve for the steady state capital-labor ratio. Go as far as you can to describe how to solve for the steady state (k*, l*, c*).

2. Labor-leisure in the RCK model. Consider the following economy that is very similar to the economy in the RCK model except that we assume the labor supply is not fixed. We include leisure in the utility function such that the lifetime utility of a representative household is ∞ Σβ'u(c,1 - 4) t=0 where u(c) is twice differentiable and strictly concave in each argument. In this setting, is the number of hours worked by the representative household at time t. The rest of time 1- is consumed as leisure. The production function is therefore y = f(kt, l), which is also twice differentiable and strictly concave in each argument. Suppose that the initial condition of capital per worker is given, ko> 0 and the transversality condition is satisfied. (a) Set up the maximization problem faced by the planner and derive the set of equations that characterize an interior solution to the planner's problem. (b) Derive the set of equations that characterize a steady state. (c) Suppose that y₁ = f (kr, lt) = kalla. Solve for the steady state capital-labor ratio. Go as far as you can to describe how to solve for the steady state (k, l, c*).

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter16: Labor Markets

Section: Chapter Questions

Problem 16.10P

See similar textbooks

Similar questions

5 Supposing that a single consumer works for a firm, the quantity of labor input for the firm, N, is identical to the quantity of hours worked by the consumer, h - l. Graph the relationship between output produced, Y on the vertical axis and leisure hours of the consumer, l, on the horizontal axis, which is implied by the production function of the firm. (In Chapter 5, we refer to this relationship as the production possibilities frontier.) What is the slope of the curve you have graphed? PLEASE DO NOT COPY OTHER PEOPLE'S WORK AND GRAPH
Consider the following labor-leisure choice model. Utility function over consumption (C) and leisure (L) U(C.L) = (1/3L1/3 Total hours: H = 40 Labor hours: NS = H - L Non-labor income: π = 30 Lumpsum tax: T = 10 Hourly wage: w = 4 Suppose the hourly wage changes to w = 3. What is the substitution effect of this change on labor supply? A. +2.65 B. -2.65 C. +3.48 D.-3.48 E. None of the above
Consider a 2-period economy in which Yt = AtKt. The households start with aquantity a1 > 0 of initial assets and maximize lifetime utility given a per-periodutility function u(c) = ln (c).1. Do households earn any labor income? Solve the household’s Euler Equa-tion. 2. Solve the firms’ problem. Is the production function constant returns toscale? Do firms make any profits?3. Define an equilibrium for this economy.4. Solve for the equilibrium of this economy. Solving for the equilibriummeans finding the value of all equilibrium objects as a function of theparameters of the model: A1, A2, α, β, δ, a15. Assume that A1 = Ab1 = 1 and A2 = 1. What is rb1 equal to?6. Assume that A1 = As1 = 1 −∆ and A2 = 1. What is rs1 equal to?7. What is the percentage difference between As1 and Ab1 ? That is, find thevalue of x such that As1 = (1 + x) Ab1. What is the percentage differencebetween rb1 and rs1 ? How do these two differences relate to each other andwhat does this connection remind you of?
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
2. Let an individual’s utility function be given by where C and L are consumption and leisure respectively, and g, a and b are positive constants with a + b = 1. Derive the individual’s Marshallian labor supply function and comment on the magnitudes of the income and substitution effects of wage change. Derive the general form of Slutsky equation of labor supply.
6. Suppose that the representative consumer's div- idend income increases, and his or her wage rate falls at the same time. Determine the effects on consumption and labor supply, and explain your results in terms of income and substitution effects.
Refer to the labor–leisure budget constraint shown to answer the questions. This curve shows trade-offs between income and leisure that must be made over the course of one day. How much does this person earn per hour? $ At point A, how many hours of labor are selected? hourshours At point A, how many hours of leisure are selected? hours
The lines on the graph are budget constraints, showing the tradeoff between labor and leisure. Suppose that when the wage changes, an individual chooses to move from point A to another point on the graph. For each of the other points, where would it belong on the backward bending labor supply curve? Backward‑bendingportionVerticalportionUpward‑slopingportion Answer Bank B D F C E
Given a production function of this nature; AX1aX2b determine the Marginal Rate of Technical Substitution.
According to the Wall Street Journal, Mitsubishi Motors recently announced a major restructuringplan in an attempt to reverse declining global sales. Suppose that as part of the restructuring planMitsubishi conducts an analysis of how labour and capital are used in its production process. Priorto restructuring Mitsubishi’s marginal rate of technical substitution is 0.15 ( in absolute value). Tohire workers. Suppose that Mitsubishi must pay the competitive hourly wage of US$ 15. In thestudy of production process and markets where capital is procured, suppose that Mitsubishidetermine that its marginal productivity of capital is 0.5 small cars per hour at its new targetedlevel of output and that capital is procured in a highly competitive market. The same studyindicates that the average selling price of Mitsubishi’s smallest car is US$ 9500. Determine therate at which Mitsubishi can rent capital and marginal productivity of labour at its new targetedlevel of output. To minimize costs…
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?