2. Consider a consumer with preferences over current and future consumption given by U(C₁, C₂) = (c₁)¹/2 (c₂)¹/2 where c₁ denotes the amount consumed in period 1 and c₂ the amount consumed in period 2. Suppose that period 1 income expressed in units of good 1 is m₁ = 20000 and period 2 income expressed in units of good 2 is m2 = 30000. Suppose also that p₁ = P2 = 1 and let r denote the interest rate. (a) Find the optimal consumption bundle when r = 0.15. Is the consumer a saver or a borrower?

2. Consider a consumer with preferences over current and future consumption given by U(C₁, C₂) = (c₁)¹/2 (c₂)¹/2 where c₁ denotes the amount consumed in period 1 and c₂ the amount consumed in period 2. Suppose that period 1 income expressed in units of good 1 is m₁ = 20000 and period 2 income expressed in units of good 2 is m2 = 30000. Suppose also that p₁ = P2 = 1 and let r denote the interest rate. (a) Find the optimal consumption bundle when r = 0.15. Is the consumer a saver or a borrower?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.3P

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5. Suppose the representative consumer has the (quasilinear) utility function: U(x, y) = ax + In(v) where x,y are consumption goods. Denominated in US dollars, the prices of x and y Pi and pz, respectively. (a) What does the term quasilinear mean? Is consumption of x independent of y? (b) Assume the consumer has a budget of C dollars at his/her disposal. What is the maximum utility the consumer may achieve with this budget? (c) What is the minimum expenditure if the consumer wants to derive U,units of utility?
On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according tothe functionU(c, b) = 20c− c²+ 18b − 3b²a. How many cigars and glasses of brandy does he consume during an evening? (Cost isno object to J. P.)b. Lately, however, J. P. has been advised by his doctors that he should limit the sum ofglasses of brandy and cigars consumed to 5. How many glasses of brandy and cigarswill he consume under these circumstances?
Exercise 3 An individual lives two periods, 0 and 1. The income is 14,000 in period 0 and 5,000 in period 1. The individual's marginal utility is: K- Co in period 0 and d8(K – C1) in period 1, where C is the consumption in period 0, C1 the consumption in period 1 and d a discount rate. K is a constant (higher than the consumption in each period). Suppose that the person can save or borrow from the bank from period 0 to period 1 with a 25% interest rate. Set the discount rate & to 0.8. a) How much will the individual consume in the two periods? How much will she save? b) A National Insurance scheme (pension system) is established. The individual must pay 4,500 in period 0 and receives the same amount with interests in addition to her income in period 1. What will the consumption and savings be in the two periods if the National Insurance scheme uses the same interest rate as the bank?
5. Suppose that the representative consumer's pref- erences change, in that his or her marginal rate of substitution of leisure for consumption in- creases for any quantities of consumption and leisure. (a) Explain what this change in preferences means in more intuitive language. (b) What effects does this have on the equilib- rium real wage, hours worked, output, and consumption? (c) Do you think that preference shifts like this might explain why economies experience recessions (periods when output is low)? Explain why or why not, with reference to the key business cycle facts in Chapter 3.
. On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according to thefunctionU (c, b) = 20c - c2 +18b- 3b2.How many cigars and glasses of brandy does he consume during an evening? (Cost is no objectto J. P.)b. Lately, however, J. P. has been advised by his doctors that he should limit the sum of glasses ofbrandy and cigars consumed to 5. How many glasses of brandy and cigars will he consumeunder these circumstances?
1. [35 marks] An individual derives utility from consumption spending C and leisurel according to the following utility function: U(C,1)=C"1¹-a where 0>x>1. Leisure time in hours is given by: 1=T-H where T is hours of total time available and H is hours of work. The consumer's real income is given by: C=w (T-1)+N where w is real wage and N is real non-labour income. a) Derive the optimal values of C, I and the Lagrangian multiplier if the individual wishes maximise utility subject to the to constraint: C=w (T-1)+N. [15 marks]
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3. Utility maximization under constraint, substitution and income effect, CV and EV Josh gets utility (satisfaction) from two goods, A and B, according to the utility function U(A, B) = 5A/4B³/4. While Josh would like to consume as much as possible he is limited by his income. a. Maximize Josh's utility subject to the budget constraint using the Lagrangean method. b. Suppose Pa increase. Show graphically the income, substitution effect and total effect and explain. c. Suppose PA increase. Show the graph for CV and EV and explain.
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Suppose a household has the following lifetime utility function: U=c1/2 + ẞc¹/2 12tt+1 A) Find expressions for the partial derivatives of lifetime utility, U, with respect to period t and period t + 1 consumption. Is marginal utility of consumption in both periods always positive? B) Find expressions for the second derivatives of lifetime utility with respect to period t and t+1 consumption, i.e., 2U and a 20_Are these second derivatives always negative for ac²²+1 any positive values of period t and t+1 consumption? C) Derive an expression for the indifference curve associated with lifetime utility level Uo (i.e., derive an expression for C++₁ as a function of U₁ and c). What is the slope of the indifference curve? How does the magnitude of the slope vary with the value of c?
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4. Peter's utility function is equal to: U (L, C) = LVC where L is hours of leisure time per day and C is daily consumption, denoted in €. When he is looking for a job as a consultant he finds the following possibilities: i) Working for "ABC of M", where he may get 100€ per day and 2h of leisure time per day ii) Working for "101 of M", where he may get 25€ per day and 4h of leisure per day iii) Working for "M for Dummies", where he may get 50€ per day and 3h of leisure per day Each day, apart from sleeping and other basic necessities, Peter has 15 hours of available time. He may tutor during as many hours as he desires, receiving 20€ per each hour of work. Assume Peter cannot save and consumes all he receives. a. How many hours does Peter decide to rest? 9h 10h 11h 15h 13.5h