Consider a utility function defined over hours of leisure and consumption expenditure: U(c, h) ch. The after-tax wage rate is given by w and the amount of nonwage income, by N. Assume the number of hours of leisurein a year is 8000. Potential annual income, I, is given by1-8000wN.It is allocated over hours of leisure, which cost wh, and consumption expenditure, c. The budget constraint isthus8000wN-cwh 0.1. Find the demand functions for leisure and consumption.a. Write the Lagrangean function for this problem.b. Find the first order conditions.c. Solve the first order conditions to obtain the condition that the MRS wage rate. Show your steps inthe solution.d. Use the results from c to obtain the expansion path

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Answered: Consider a utility function defined…

ENGR.ECONOMIC ANALYSIS

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Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

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just subparts a, b and c please
Consider an individual whose utility function is U(C, R) = 2C - (4 - R) 2 where R is the amount of leisure the consumer experiences per day. Suppose that the individual normally sleeps T hours a day, and they can spend the remaining hours between work and leisure. The individual receives an hourly wage of w > 0 and has also an income of Y > 0 a day from non - labour sources. The price of consumption goods is p per unit.
A. Consider a consumer whose preferences can be represented by Cobb-Douglas utility function u(x₁, x₂) = xx where ₁ and 2 are the quantities of good 1 and good 2 she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote her income. 1. Derive the consumer's Marshallian demand functions. 2. Derive the consumer's Hicksian demand functions. 3. Derive the consumer's expenditure function. 4. Let m = 20, P₁ = 2, and p2 = 1. Suppose that the price of good 1 drops to p₁ = 1. Find the following (a) Compensating variation (CV) (b) Equivalent variation (EV) (c) Change in consumer surplus (ACS) (d) Compare CV, ACS, and EV. 5. Let m = 120, P₁ = 1, and p2 = 1. Suppose that the price of good 1 increases to P₁ = 2. Find the following (a) Compensating variation (CV) (b) Equivalent variation (EV) (c) Change in consumer surplus (ACS) (d) Compare CV, ACS, and EV.
Refer to the following diagram. Point A is 24 hours, C is 14 hours, D is 8 hours, and E is 6 hours. Income B₂ 0 D -Leisure If the current wage rate results in a budget constraint of AB2, how many hours will the individual work?
23. A worker has utility over consumption c and leisure I given by U(c,1) = a 6-6 + 18 where 0 << 1 She has T hours to allocate between leisure and work. For each hour she works, she earns a wage of w to spend on consumption c. The price of c is 1. She also receives an additional 'non-labor income' m regardless of how much she works. She maximizes utility subject to the following constraints: contact@cloure.m Assume interior solution, then cw(T-1) +m C≥O 0≤1≤T A. c is normal and I is inferior good. B. c is inferior and I is inferior good. C. c is inferior and I is normal good. D. c is normal and I is normal good. Page 8
11. A worker derives utility from consumption, C and leisure, L U = U(C, L) and initially faces the constraints p.C = N+w.H H = 24 - L where p is the nominal price of consumption, N> 0 is unearned income, w is the nominal wage and H is hours worked. (a) Derive the budget constraint in terms of C and L and show on a diagram an initial optimal choice of C and L, in which both are positive. (b) State this equilibrium algebraically, and briefly explain how it can be derived. (c) Show that, assuming normality of leisure, for some value N*, if N> N*, the individual will choose not to work. Is the initial level of N in your answers to parts a) and b) greater or less than N*?

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