专业课 尼科尔森微观基本原理与扩展全套资料 尼克尔森微观经济理论-基本原理与扩展 第九版答案 ch04

Chapter 4/Utility Maximization and Choicev 15CHAPTER 4 UTILITY MAXIMIZATION AND CHOICE The problems in this chapter focus mainly on the utility maximization assumption. Relatively simple computational problems (mainly based on CobbDouglas and CES utility functions) are included. Comparative statics exercises are included in a few problems, but for the most part, introduction of this material is delayed until Chapters 5 and 6. Comments on Problems 4.1This is a simple CobbDouglas example. Part (b) asks students to compute income compensation for a price rise and may prove difficult for them. As a hint they might be told to find the correct bundle on the original indifference curve first, then compute its cost. 4.2This uses the Cobb-Douglas utility function to solve for quantity demanded at two different prices. Instructors may wish to introduce the expenditure shares interpretation of the functions exponents (these are covered extensively in the Extensions to Chapter 4 and in a variety of numerical examples in Chapter 5). 4.3This starts as an unconstrained maximization problemthere is no income constraint in part (a) on the assumption that this constraint is not limiting. In part (b) there is a total quantity constraint. Students should be asked to interpret what Lagrangian Multiplier means in this case. 4.4This problem shows that with concave indifference curves first order conditions do not ensure a local maximum. 4.5This is an example of a “fixed proportion” utility function. The problem might be used to illustrate the notion of perfect complements and the absence of relative price effects for them. Students may need some help with the min ( ) functional notation by using illustrative numerical values for v and g and showing what it means to have “excess” v or g. 4.6This problem introduces a third good for which optimal consumption is zero until income reaches a certain level. 4.7This problem provides more practice with the Cobb-Douglas function by asking students to compute the indirect utility function and expenditure function in this case. The manipulations here are often quite difficult for students, primarily because they do not keep an eye on what the final goal is. 4.8This problem repeats the lessons of the lump sum principle for the case of a subsidy. Numerical examples are based on the Cobb-Douglas expenditure function. 4.9This problem looks in detail at the first order conditions for a utility maximum with the CES function. Part c of the problem focuses on how relative expenditure shares are determined with the CES function. 4.10This problem shows utility maximization in the linear expenditure system (see also the Extensions to Chapter 4). Solutions4.1a.Set up Lagrangian Ratio of first two equations impliesHence1.00 = .10t + .25s = .50s.s = 2 t = 5Utility = b.New utility or ts = 10and Substituting into indifference curve:s2 = 16 s = 4 t = 2.5Cost of this bundle is 2.00, so Paul needs another dollar. 4.2Use a simpler notation for this solution: a.Hence,Substitution into budget constraint yields f = 10, c = 25.b.With the new constraint: f = 20, c = 25 Note: This person always spends 2/3 of income on f and 1/3 on c. Consumption of California wine does not change when price of French wine changes. c.In part a, . In part b, . To achieve the part b utility with part a prices, this person will need more income. Indirect utility is . Solving this equation for the required income gives I = 482. With such an income, this person would purchase f = 16.1, c = 40.1, U = 21.5. 4.3a. |So, U = 127.b.Constraint: b + c = 5 c = 3b + 1 so b + 3b + 1 = 5, b = 1, c = 4, U = 79 4.4 Maximizing U2 in will also maximize U.a. First two equations give . Substituting in budget constraint gives x = 6, y = 8 , U = 10. b.This is not a local maximum because the indifference curves do not have a diminishing MRS (they are in fact concentric circles). Hence, we have necessary but not sufficient conditions for a maximum. In fact the calculated allocation is a minimum utility. If Mr. Ball spends all income on x, say, U = 50/3. 4.5a.No matter what the relative price are (i.e., the slope of the budget constraint) the maximum utility intersection will always be at the vertex of an indifference curve where g = 2v. b.Substituting g = 2v into the budget constraint yields: or .Similarly, It is easy to show that these two demand functions are homogeneous of degree zero in PG , PV , and I. c. so, Indirect Utility is d.The expenditure function is found by interchanging I (= E) and V, . 4.6a.If x = 4 y = 1 U (z = 0) = 2.If z = 1 U = 0 since x = y = 0.If z = 0.1 (say) x = .9/.25 = 3.6, y = .9.U = (3.6).5 (.9).5 (1.1).5 = 1.89 which is less than U(z = 0) b.At x = 4 y = 1 z =0 So, even at z = 0, the marginal utility from z is not worth the goods price. Notice here that the “1” in the utility function causes this individual to incur some diminishing marginal utility for z before any is bought. Good z illustrates the principle of “complementary slackness discussed in Chapter 2. c.If I = 10, optimal choices are x = 16, y = 4, z = 1. A higher income makes it possible to consume z as part of a utility maximum. To find the minimal income at which any (fractional) z would be bought, use the fact that with the Cobb-Douglas this person will spend equal amounts on x, y, and (1+z). That is: Substituting this into the budget constraint yields: Hence, for z 0 it must be the case that . 4.7 a. The demand functions in this case are . Substituting these into the utility function gives where . b. Interchanging I and V yields . c. The elasticity of expenditures with respect to is given by the exponent . That is, the more important x is in the utility function the greater the proportion that expenditures must be increased to compensate for a proportional rise in the price of x. 4.8a. b. With . To raise utility to 3 would require E = 12 that is, an income subsidy of 4. c.Now we require . So - that is, each unit must be subsidized by 5/9. at the subsidized price this person chooses to buy x = 9. So total subsidy is 5 one dollar greater than in part c. d. With . Raising U to 3 would require extra expenditures of 4.86. Subsidizing good x alone would require a price of . That is, a subsidy of 0.74 per unit. With this low price, this person would choose x = 11.2, so total subsidy woul