4
UTILITY CHAPTER 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 4.1 This 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. This 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 function's exponents (these are covered extensively in the Extensions to Chapter 4 and in a variety of numerical examples in Chapter 5). This 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. This problem shows that with concave indifference curves first order conditions do not ensure a local maximum. This 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. This problem introduces a third good for which optimal consumption is zero until income reaches a certain level. This 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. This 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.
4.2
4.3
4.4 4.5
4.6 4.7
4.8
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11 y Solutions Manual
4.9
This 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. This problem shows utility maximization in the linear expenditure system (see also the Extensions to Chapter 4).
4.10
Solutions
4.1 a. Set up Lagrangian = ts + (1.00 .10t .25s ) . = ( s / t )0.5 .10 t = (t / s )0.5 .25 s = 1.00 .10t .25s = 0 Ratio of first two equations implies t = 2.5 s Hence 1.00 = .10t + .25s = .50s. s=2 t=5 Utility = 10 b. New utility 10 or ts = 10 and t= t .25 5 = = s .40 8 t = 2.5s
5s 8
Substituting into indifference curve: 5s 2 = 10 8 s2 = 16 s = 4 t = 2.5 Cost of this bundle is 2.00, so Paul needs another dollar.
Chapter 4/Utility Maximization and Choicey 12
4.2
Use a simpler notation for this solution: U ( f , c) = f 2 / 3 c1/ 3 a. = f 2 / 3 c1/ 3 + (300 20 f 4c ) = 2 / 3(c / f )1/ 3 20 f = 1/ 3( f / c) 2 / 3 4 c Hence, 5=2 c , 2c = 5 f f
I = 300
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, U ( f , c) = f 2 3c1 3 = 102 3 251 3 = 13.5 . In part b, U ( f , c) = 202 3251 3 = 21.5 . To achieve the part b utility with part a prices, this person will need more income. 23 13 2 3 1 3 23 2 3 1 3 Indirect utility is 21.5 = (2 3) (1 3) Ip f pc = (2 3) I 20 4 . 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. 4.3 U (c, b) = 20c c 2 + 18b 3b 2 a. U = 20 2c = 0, c U = 18 6b = 0, b So, U = 127. b. Constraint: b + c = 5 = 20c c 2 + 18b 3b 2 + (5 c b) = 20 2c = 0 c = 18 6b = 0 b = 5 c b= 0 c = 3b + 1 so b + 3b + 1 = 5, b = 1, c = 4, U = 79 c = 10 | b= 3
13 y Solutions Manual
4.4
U ( x, y ) = ( x 2 + y 2 )0.5 Maximizing U2 in will also maximize U. a. = x 2 + y 2 + (50 3x 4 y ) = 2x 3 = 0 x = 2y 4 = 0 y
= 2x 3 = y 2
= 50 3x 4y = 0 two First equations give y = 4 x 3 . 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. 4.5 U (m) = U ( g , v) = Min[ g 2, v] a. 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: 2 pg v + pv v = I or v = Similarly, g = I . 2pg + pv
2I 2pg + pv
It is easy to show that these two demand functions are homogeneous of degree zero in PG , PV , and I. c. U = g 2 = v so, Indirect Utility is V ( pg , pv , I ) = I 2pg + pv
d. The expenditure function is found by interchanging I (= E) and V, E ( pg , pv ,V ) = (2 pg + pv )V .
Chapter 4/Utility Maximization and Choicey 14
4.6
a. 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 MU x / px = MU y / p y = 1 M U z / p z = 1/2 So, even at z = 0, the marginal utility from z is "not worth" the good's 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: px x = p y y = pz (1 + z ) Substituting this into the budget constraint yields: 2 pz (1 + z ) + pz z = I or 3 pz z = I 2 pz Hence, for z > 0 it must be the case that I > 2 pz or I > 4 .
4.7
U ( x, y ) = x y1 a. The demand functions in this case are x = I px , y = (1 ) I p y . Substituting these
(1 ) into the utility function gives V ( px , p y , I ) = [ I px ] [(1 ) I p y ] = BIpx p y
where B = (1 )(1 ) .
1 (1 ) b. Interchanging I and V yields E ( px , p y ,V ) = B px p y V .
c. The elasticity of expenditures with respect to px 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.
15 y Solutions Manual
4.8
a.
0.5 0.5 b. E ( px , p y , U ) = 2 px p y U . With px = 1, p y = 4, U = 2, E = 8 . To raise utility to 3 would require E = 12 that is, an income subsidy of 4. 0.5 0.5 0.5 c. Now we require E = 8 = 2 px 4 3 or px = 8 12 = 2 3 . So px = 4 9 -- 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. 0.3 0.7 d. E ( px , p y , U ) = 1.84 px p y U . With px = 1, p y = 4, U = 2, E = 9.71 . Raising U to 3 would require extra expenditures of 4.86. Subsidizing good x alone would require a price of px = 0.26 . That is, a subsidy of 0.74 per unit. With this low price, this person would choose x = 11.2, so total subsidy would be 8.29.
4.9
a. MRS =
U/ x 1 = ( x y ) = px /p y for utility maximization. U/ y where = 1 (1 ) .
1 ( 1) = ( px p y ) Hence, x/y = ( px p y )
b. If = 0, x y = p y px so px x = p y y .
1 c. Part a shows px x p y y = ( px p y )
Hence, for < 1 the relative share of income devoted to good x is positively correlated with its relative price. This is a sign of low substitutability. For > 1 the relative share of income devoted to good x is negatively correlated with its relative price a sign of high substitutability. d. The algebra here is very messy. For a solution see the Sydsaeter, Strom, and Berck reference at the end of Chapter 5. 4.10 a. For x < x0 utility is negative so will spend px x0 first. With I- px x0 extra income, this is a standard Cobb-Douglas problem: px ( x x0 ) = ( I px x0 ), p y y = ( I px x0 )
Chapter 4/Utility Maximization and Choicey 16
b. Calculating budget shares from part a yields p xx (1 ) px x0 = + , I I lim( I ) py y I =
px x0 I
py p xx = , lim( I ) y = . I I
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