CHAPTER 2 THE MATHEMATICS OF OPTIMIZATION. Solutions. dπ =

Transcription

1 CHAPTER THE MATHEMATICS OF OPTIMIZATION The problems in this chapter are primaril mathematical. The are intended to give students some practice with taking derivatives and using the Lagrangian techniques, but the problems in themselves offer few economic insights. Consequentl, no commentar is provided. All of the problems are relativel simple and instructors might choose from among them on the basis of how the wish to approach the teaching of the optimization methods in class. Solutions.1 U(, ) = a. U U = 8, = 6 b. 8, 1 c. U U du = d + d = 8 d + 6 d d. d for du = 0 d 8 d + 6 d = 0 d 8 4 = = d 6 3 e. = 1, = U = = 16 f. d 4(1) = = /3 d 3() g. U = 16 contour line is an ellipse centered at the origin. With equation = 16, slope of the line at (, ) is d 4 = d 3.. a. Profits are given b π = R C = q + 40q 100 dπ = * 4q + 40 q = 10 dq * π = (10) + 40(10) 100 = 100 b. d π 4 = so profits are maimized dq c. dr MR = = 70 q dq 1

2 Solutions Manual dc MC = = q + 30 dq so q* = 10 obes MR = MC = Substitution: = 1 so f = = f = 1 = 0 = 0.5, = 0.5, f = 0.5 Note: f = < 0. This is a local and global maimum. Lagrangian Method: = + λ(1 ) λ = = 0 λ = = 0 so, =. using the constraint gives = = 0.5, = Setting up the Lagrangian: = + + λ(0.5 ). = 1 λ = 1 λ So, =. Using the constraint gives = = 0.5, = = a. f t gt () = t df dt * 40 = g t + 40 = 0, t = g. b. Substituting for t*, * f ( t ) = 0.5 g(40 g) + 40(40 g) = 800 g. * f( t ) = 800 g. g c. f 1 = ( * t ) depends on g because t * depends on g. g f * so = 0.5( t) = 0.5( ) =. g g g

3 Chapter /The Mathematics of Optimization 3 d = 5, = 4.9, a reduction of.08. Notice that 800 g = so a 0.1 increase in g could be predicted to reduce height b 0.08 from the envelope theorem..6 a. This is the volume of a rectangular solid made from a piece of metal which is b 3 with the defined corner squares removed. V b. = 3 16t+ 1t = 0. Appling the quadratic formula to this epression ields t 16 ± ± 10.6 t = = = 0.5, To determine true 4 4 V maimum must look at second derivative -- = t which is negative onl t for the first solution c. If t = 0.5, V so V increases without limit. d. This would require a solution using the Lagrangian method. The optimal solution requires solving three non-linear simultaneous equations a task not undertaken here. But it seems clear that the solution would involve a different relationship between t and than in parts a-c..7 a. Set up Lagrangian = + 1 5ln + λ ( k ) 1 ields the first order conditions: = 1 λ = = λ = 0 = k 1 = 0 λ Hence, λ = 1 = 5 or = 5. With k = 10, optimal solution is 1 = = 5. b. With k = 4, solving the first order conditions ields = 5, 1 = 1. c. Optimal solution is 1 = 0, = 4, = 5ln 4. An positive value for 1 reduces. d. If k = 0, optimal solution is 1 = 15, = 5. Because provides a diminishing marginal increment to whereas 1 does not, all optimal solutions require that, once reaches 5, an etra amounts be devoted entirel to 1..8 The proof is most easil accomplished through the use of the matri algebra of quadratic forms. See, for eample, Mas Colell et al., pp Intuitivel, because concave functions lie below an tangent plane, their level curves must also be conve. But the converse is not true. Quasi-concave functions ma ehibit increasing returns to scale ; even though their level curves are conve, the ma rise above the tangent plane when all variables are increased together.

4 4 Solutions Manual α 1.9 a. f 1 = α 1 > 0. α 1 f = >. 1 0 α f 11 = α ( α 1) 1 1 < 0. α f = ( 1) 1 <0. f f α = 1 = α 1 > 0. Clearl, all the terms in Equation.114 are negative. α b. If = c = 1 1/ / α = c 1 since α, > 0, is a conve function of 1. c. Using equation.98, α α f11 f f 1 = α ( α 1)( )( 1) 1 α 1 α = α(1 α) 1 which is negative for α + > a. Since > 0, < 0, the function is concave. b. Because f11, f < 0, and f1 = f 1 = 0, Equation.98 is satisfied and the function is concave. c. γ γ is quasi-concave as is. But is not concave for γ > 1. All of these results can be shown b appling the various definitions to the partial derivatives of.

5 CHAPTER 3 PREFERENCES AND UTILITY These problems provide some practice in eamining utilit functions b looking at indifference curve maps. The primar focus is on illustrating the notion of a diminishing MRS in various contets. The concepts of the budget constraint and utilit maimization are not used until the net chapter. Comments on Problems 3.1 This problem requires students to graph indifference curves for a variet of functions, some of which do not ehibit a diminishing MRS. 3. Introduces the formal definition of quasi-concavit (from Chapter ) to be applied to the functions in Problem This problem shows that diminishing marginal utilit is not required to obtain a diminishing MRS. All of the functions are monotonic transformations of one another, so this problem illustrates that diminishing MRS is preserved b monotonic transformations, but diminishing marginal utilit is not. 3.4 This problem focuses on whether some simple utilit functions ehibit conve indifference curves. 3.5 This problem is an eploration of the fied-proportions utilit function. The problem also shows how such problems can be treated as a composite commodit. 3.6 In this problem students are asked to provide a formal, utilit-based eplanation for a variet of advertising slogans. The purpose is to get students to think mathematicall about everda epressions. 3.7 This problem shows how initial endowments can be incorporated into utilit theor. 3.8 This problem offers a further eploration of the Cobb-Douglas function. Part c provides an introduction to the linear ependiture sstem. This application is treated in more detail in the Etensions to Chapter This problem shows that independent marginal utilities illustrate one situation in which diminishing marginal utilit ensures a diminishing MRS This problem eplores various features of the CES function with weighting on the two goods. 5

6 6 Solutions Manual Solutions 3.1 Here we calculate the MRS for each of these functions: a. MRS = f f = 31 MRS is constant. b ( ) MRS = f f = = ( ) MRS is diminishing. c. d. 0.5 MRS = f = 0.5 f 1 MRS is diminishing MRS f f MRS is increasing ( ) = = 0.5( ) = e. ( + ) ( + ) MRS = f f = = ( + ) ( + ) MRS is diminishing. 3. Because all of the first order partials are positive, we must onl check the second order partials. a. f = f = f = Not strictl quasiconcave b. f, f < 0, f > 0 Strictl quasiconcave 11 1 c. f < 0, f = 0, f = 0 Strictl quasiconcave 11 1 d. Even if we onl consider cases where, both of the own second order partials are ambiguous and therefore the function is not necessaril strictl quasiconcave. e. f, f < 0 f > 0 Strictl quasiconcave a. U =, U = 0, U =, U = 0, MRS =. b. c. U U U U MRS. =, =, =, =, = U U U U MRS = 1, = 1, = 1, = 1, = This shows that monotonic transformations ma affect diminishing marginal utilit, but not the MRS. 3.4 a. The case where the same good is limiting is uninteresting because U(, ) = = k = U(, ) = = U[( + ),( + )] = ( + ). If the limiting goods differ, then > = k = <. 1 1 Hence, ( ) / 1+ ) / > k and ( 1+ > kso the indifference curve is conve.

7 Chapter 3/Preference and Utilit 7 b. Again, the case where the same good is maimum is uninteresting. If the goods differ, 1 < 1 = k = >.( 1+ )/ < k,( 1+ )/< k so the indifference curve is concave, not conve. c. Here ( 1+ 1) = k = ( + ) = [( 1+ )/,( 1+ )/ ] so indifference curve is neither conve or concave it is linear. 3.5 a. Uhbmr (,,, ) = Minh (, bm,,0.5 r. ) b. A full condimented hot dog. c. $1.60 d. $.10 an increase of 31 percent. e. Price would increase onl to $1.75 an increase of 7.8 percent. f. Raise prices so that a full condimented hot dog rises in price to $.60. This would be equivalent to a lump-sum reduction in purchasing power. 3.6 a. U( p, b) = p+ b U b. > 0. coke c. U( p, ) > U(1, ) for p > 1 and all. d. Uk (, ) > Ud (, ) for k = d. e. See the etensions to Chapter 3.

8 8 Solutions Manual 3.7 a. b. An trading opportunities that differ from the MRS at, will provide the opportunit to raise utilit (see figure). c. A preference for the initial endowment will require that trading opportunities raise utilit substantiall. This will be more likel if the trading opportunities and significantl different from the initial MRS (see figure). 3.8 a. α 1 U / α MRS = = = α 1 U / α ( / ) This result does not depend on the sum α + which, contrar to production theor, has no significance in consumer theor because utilit is unique onl up to a monotonic transformation. b. Mathematics follows directl from part a. If α > the individual values relativel more highl; hence, d d > 1 for =. c. The function is homothetic in ( 0) and ( 0), but not in and. 3.9 From problem 3., f1 = 0 implies diminishing MRS providing f11, f < 0. Conversel, the Cobb-Douglas has f1 >0, f11, f < 0, but also has a diminishing MRS (see problem 3.8a) a. δ 1 U / α α MRS = = = ( / ) δ 1 U / 1 δ so this function is homothetic. b. If δ = 1, MRS = α/, a constant. If δ = 0, MRS = α/ (/), which agrees with Problem 3.8. c. For δ < 1 1 δ > 0, so MRS diminishes. d. Follows from part a, if = MRS = α/.

9 Chapter 3/Preference and Utilit 9 α 0.5 α e. With δ =.5, MRS (.9) = (.9) =.949 MRS α 0.5 (1.1) = (1.1) = 1.05 α α α With δ = 1, MRS (.9) = (.9) =.81 MRS α (1.1) = (1.1) = 1.1 α Hence, the MRS changes more dramaticall when δ = 1 than when δ =.5; the lower δ is, the more sharpl curved are the indifference curves. When δ =, the indifference curves are L-shaped impling fied proportions.

10 CHAPTER 3 PREFERENCES AND UTILITY These problems provide some practice in eamining utilit functions b looking at indifference curve maps. The primar focus is on illustrating the notion of a diminishing MRS in various contets. The concepts of the budget constraint and utilit maimization are not used until the net chapter. Comments on Problems 3.1 This problem requires students to graph indifference curves for a variet of functions, some of which do not ehibit a diminishing MRS. 3. Introduces the formal definition of quasi-concavit (from Chapter ) to be applied to the functions in Problem This problem shows that diminishing marginal utilit is not required to obtain a diminishing MRS. All of the functions are monotonic transformations of one another, so this problem illustrates that diminishing MRS is preserved b monotonic transformations, but diminishing marginal utilit is not. 3.4 This problem focuses on whether some simple utilit functions ehibit conve indifference curves. 3.5 This problem is an eploration of the fied-proportions utilit function. The problem also shows how such problems can be treated as a composite commodit. 3.6 In this problem students are asked to provide a formal, utilit-based eplanation for a variet of advertising slogans. The purpose is to get students to think mathematicall about everda epressions. 3.7 This problem shows how initial endowments can be incorporated into utilit theor. 3.8 This problem offers a further eploration of the Cobb-Douglas function. Part c provides an introduction to the linear ependiture sstem. This application is treated in more detail in the Etensions to Chapter This problem shows that independent marginal utilities illustrate one situation in which diminishing marginal utilit ensures a diminishing MRS This problem eplores various features of the CES function with weighting on the two goods. 5

11 6 Solutions Manual Solutions 3.1 Here we calculate the MRS for each of these functions: a. MRS = f f = 31 MRS is constant. b ( ) MRS = f f = = ( ) MRS is diminishing. c. d. 0.5 MRS = f = 0.5 f 1 MRS is diminishing MRS f f MRS is increasing ( ) = = 0.5( ) = e. ( + ) ( + ) MRS = f f = = ( + ) ( + ) MRS is diminishing. 3. Because all of the first order partials are positive, we must onl check the second order partials. a. f = f = f = Not strictl quasiconcave b. f, f < 0, f > 0 Strictl quasiconcave 11 1 c. f < 0, f = 0, f = 0 Strictl quasiconcave 11 1 d. Even if we onl consider cases where, both of the own second order partials are ambiguous and therefore the function is not necessaril strictl quasiconcave. e. f, f < 0 f > 0 Strictl quasiconcave a. U =, U = 0, U =, U = 0, MRS =. b. c. U U U U MRS. =, =, =, =, = U U U U MRS = 1, = 1, = 1, = 1, = This shows that monotonic transformations ma affect diminishing marginal utilit, but not the MRS. 3.4 a. The case where the same good is limiting is uninteresting because U(, ) = = k = U(, ) = = U[( + ),( + )] = ( + ). If the limiting goods differ, then > = k = <. 1 1 Hence, ( ) / 1+ ) / > k and ( 1+ > kso the indifference curve is conve.

12 Chapter 3/Preference and Utilit 7 b. Again, the case where the same good is maimum is uninteresting. If the goods differ, 1 < 1 = k = >.( 1+ )/ < k,( 1+ )/< k so the indifference curve is concave, not conve. c. Here ( 1+ 1) = k = ( + ) = [( 1+ )/,( 1+ )/ ] so indifference curve is neither conve or concave it is linear. 3.5 a. Uhbmr (,,, ) = Minh (, bm,,0.5 r. ) b. A full condimented hot dog. c. $1.60 d. $.10 an increase of 31 percent. e. Price would increase onl to $1.75 an increase of 7.8 percent. f. Raise prices so that a full condimented hot dog rises in price to $.60. This would be equivalent to a lump-sum reduction in purchasing power. 3.6 a. U( p, b)= p+ b U b. > 0. coke c. U( p, ) > U(1, ) for p > 1 and all. d. Uk (, ) > Ud (, ) for k = d. e. See the etensions to Chapter 3.

13 8 Solutions Manual 3.7 a. b. An trading opportunities that differ from the MRS at, will provide the opportunit to raise utilit (see figure). c. A preference for the initial endowment will require that trading opportunities raise utilit substantiall. This will be more likel if the trading opportunities and significantl different from the initial MRS (see figure). 3.8 a. α 1 U / α MRS = = = α 1 U / α ( / ) This result does not depend on the sum α + which, contrar to production theor, has no significance in consumer theor because utilit is unique onl up to a monotonic transformation. b. Mathematics follows directl from part a. If α > the individual values relativel more highl; hence, d d > 1 for =. c. The function is homothetic in ( 0) and ( 0), but not in and. 3.9 From problem 3., f1 = 0 implies diminishing MRS providing f11, f < 0. Conversel, the Cobb-Douglas has f1 >0, f11, f < 0, but also has a diminishing MRS (see problem 3.8a) a. δ 1 U / α α MRS = = = ( / ) δ 1 U / 1 δ so this function is homothetic. b. If δ = 1, MRS = α/, a constant. If δ = 0, MRS = α/ (/), which agrees with Problem 3.8. c. For δ < 1 1 δ > 0, so MRS diminishes. d. Follows from part a, if = MRS = α/.

14 Chapter 3/Preference and Utilit 9 α 0.5 α e. With δ =.5, MRS (.9) = (.9) =.949 MRS α 0.5 (1.1) = (1.1) = 1.05 α α α With δ = 1, MRS (.9) = (.9) =.81 α MRS = = (1.1) (1.1) 1.1 α Hence, the MRS changes more dramaticall when δ = 1 than when δ =.5; the lower δ is, the more sharpl curved are the indifference curves. When δ =, the indifference curves are L-shaped impling fied proportions.

15 CHAPTER 4 UTILITY MAXIMIZATION AND CHOICE The problems in this chapter focus mainl on the utilit maimization assumption. Relativel simple computational problems (mainl based on Cobb Douglas and CES utilit functions) are included. Comparative statics eercises are included in a few problems, but for the most part, introduction of this material is delaed until Chapters 5 and 6. Comments on Problems 4.1 This is a simple Cobb Douglas eample. Part (b) asks students to compute income compensation for a price rise and ma prove difficult for them. As a hint the might be told to find the correct bundle on the original indifference curve first, then compute its cost. 4. This uses the Cobb-Douglas utilit function to solve for quantit demanded at two different prices. Instructors ma wish to introduce the ependiture shares interpretation of the function's eponents (these are covered etensivel in the Etensions to Chapter 4 and in a variet of numerical eamples in Chapter 5). 4.3 This starts as an unconstrained maimization problem there is no income constraint in part (a) on the assumption that this constraint is not limiting. In part (b) there is a total quantit constraint. Students should be asked to interpret what Lagrangian Multiplier means in this case. 4.4 This problem shows that with concave indifference curves first order conditions do not ensure a local maimum. 4.5 This is an eample of a fied proportion utilit function. The problem might be used to illustrate the notion of perfect complements and the absence of relative price effects for them. Students ma need some help with the min ( ) functional notation b using illustrative numerical values for v and g and showing what it means to have ecess v or g. 4.6 This problem introduces a third good for which optimal consumption is zero until income reaches a certain level. 4.7 This problem provides more practice with the Cobb-Douglas function b asking students to compute the indirect utilit function and ependiture function in this case. The manipulations here are often quite difficult for students, primaril because the do not keep an ee on what the final goal is. 4.8 This problem repeats the lessons of the lump sum principle for the case of a subsid. Numerical eamples are based on the Cobb-Douglas ependiture function. 10

16 Chapter 4/Utilit Maimization and Choice This problem looks in detail at the first order conditions for a utilit maimum with the CES function. Part c of the problem focuses on how relative ependiture shares are determined with the CES function This problem shows utilit maimization in the linear ependiture sstem (see also the Etensions to Chapter 4). Solutions 4.1 a. Set up Lagrangian = ts + λ ( t.5 s ). = t s 0.5 ( s/ t) (/ t s).5 λ λ = t.5s = 0 λ Ratio of first two equations implies t.5 t.5s s = = Hence 1.00 =.10t +.5s =.50s. s = t = 5 Utilit = 10 b. New utilit 10 or ts = 10 t.5 5 and s =.40 = 8 5s t = 8 Substituting into indifference curve: 5s = 10 8 s = 16 s = 4 t =.5 Cost of this bundle is.00, so Paul needs another dollar.

17 1 Solutions Manual /3 1/3 4. Use a simpler notation for this solution: U ( f, c ) = f c I = 300 a. /3 1/3 = + (300 0 f 4 c) f c λ = f 1/3 /3( c/ f) 0 λ 1/3( / ) /3 4 = f c c λ Hence, 5= c,c = 5f f Substitution into budget constraint ields f = 10, c = 5. b. With the new constraint: f = 0, c = 5 Note: This person alwas spends /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 c = 10 5 = In part b, U( f, c ) = 0 5 = 1.5. To achieve the part b utilit with part a prices, this person will need more income Indirect utilit is 1.5 = ( 3) (1 3) Ip f pc = ( 3) I 0 4. Solving this equation for the required income gives I = 48. With such an income, this person would purchase f = 16.1, c = 40.1, U = Ucb (, ) = 0c c+ 18b 3b a. U = 0 c = 0, c c = 10 U = 18 6b = 0, b b = 3 So, U = 17. b. Constraint: b + c = 5 = 0c c + 18b 3 b + λ(5 c b) = 0 c λ = 0 c = 18 6b λ = 0 b = 5 c b = 0 λ c = 3b + 1 so b + 3b + 1 = 5, b = 1, c = 4, U = 79

18 Chapter 4/Utilit Maimization and Choice U(, ) = ( + ) 0.5 Maimizing U in will also maimize U. a. = + + λ( ) = 3 λ= 0 λ= 3 = 4 λ= 0 λ= = = 0 λ First two equations give = 4 3. Substituting in budget constraint gives = 6, = 8, U = 10. b. This is not a local maimum because the indifference curves do not have a diminishing MRS (the are in fact concentric circles). Hence, we have necessar but not sufficient conditions for a maimum. In fact the calculated allocation is a minimum utilit. If Mr. Ball spends all income on, sa, U = 50/ Um ( ) = Ugv (, ) = Ming [, v] a. No matter what the relative price are (i.e., the slope of the budget constraint) the maimum utilit intersection will alwas be at the verte of an indifference curve where g = v. b. Substituting g = v into the budget constraint ields: p g v+ p v v= I or Similarl, I g = p + p g I v = p + p. v g v It is eas to show that these two demand functions are homogeneous of degree zero in P G, P V, and I. c. U = g = v so, I Indirect Utilit is V( pg, pv, I) = p + p d. The ependiture function is found b interchanging I (= E) and V, E( p, p, V) = ( p + p ) V. g v g v g v

19 14 Solutions Manual 4.6 a. If = 4 = 1 U (z = 0) =. If z = 1 U = 0 since = = 0. If z = 0.1 (sa) =.9/.5 = 3.6, =.9. U = (3.6).5 (.9).5 (1.1).5 = 1.89 which is less than U(z = 0) b. At = 4 = 1 z =0 M / p = M / p = 1 U U M U z / p = 1/ z So, even at z = 0, the marginal utilit from z is "not worth" the good's price. Notice here that the 1 in the utilit function causes this individual to incur some diminishing marginal utilit for z before an is bought. Good z illustrates the principle of complementar slackness discussed in Chapter. c. If I = 10, optimal choices are = 16, = 4, z = 1. A higher income makes it possible to consume z as part of a utilit maimum. To find the minimal income at which an (fractional) z would be bought, use the fact that with the Cobb-Douglas this person will spend equal amounts on,, and (1+z). That is: p = p = p (1 + z) z Substituting this into the budget constraint ields: p (1 + z) + p z = I or 3p z = I p z z z z Hence, for z > 0 it must be the case that I > p or I > 4. z 4.7 U(, ) α = 1 α a. The demand functions in this case are = α I p, = (1 α) I p. Substituting these (1 ) into the utilit function gives V( p, p, I) [ I p] α [(1 ) I p] BIp α = α α = p α α (1 α ) where B = α (1 α). 1 α (1 α) b. Interchanging I and V ields E( p, p, V) = B p p V. c. The elasticit of ependitures with respect to p is given b the eponent α. That is, the more important is in the utilit function the greater the proportion that ependitures must be increased to compensate for a proportional rise in the price of.

20 Chapter 4/Utilit Maimization and Choice a. b. E( p, p, U) = p p U. With p = 1, p = 4, U =, E = 8. To raise utilit to would require E = 1 that is, an income subsid of c. Now we require E = 8= p 4 3 or p = 8 1= 3. So p = 49-- that is, each unit must be subsidized b 5/9. at the subsidized price this person chooses to bu = 9. So total subsid is 5 one dollar greater than in part c. d. E( p, p, U) = 1.84 p p U. With p = 1, p = 4, U =, E = Raising U to would require etra ependitures of Subsidizing good alone would require a price of p = 0.6. That is, a subsid of 0.74 per unit. With this low price, this person would choose = 11., so total subsid would be a. U/ δ 1 MRS = = ( ) = p /p U/ for utilit maimization. Hence, / = p p = p p =. 1( δ 1) σ ( ) ( ) where σ 1 (1 δ ) b. If δ = 0, = p p so p = p. c. Part a shows p p = ( p p ) 1 σ Hence, for σ < 1 the relative share of income devoted to good is positivel correlated with its relative price. This is a sign of low substitutabilit. For σ > 1 the relative share of income devoted to good is negativel correlated with its relative price a sign of high substitutabilit. d. The algebra here is ver mess. For a solution see the Sdsaeter, Strom, and Berck reference at the end of Chapter a. For < 0 utilit is negative so will spend p 0 first. With I- p 0 etra income, this is a standard Cobb-Douglas problem: p = I p p I p ( 0) α ( 0), = ( ) 0

21 16 Solutions Manual b. Calculating budget shares from part a ields ( α) p 0 0 p 1 p p = α+, = I I I I p p lim( I ) = α, lim( I ) =. I I

22 17

23 CHAPTER 5 INCOME AND SUBSTITUTION EFFECTS Problems in this chapter focus on comparative statics analses of income and own-price changes. Man of the problems are fairl eas so that students can approach the ideas involved in shifting budget constraints in simplified settings. Theoretical material is confined mainl to the Etensions where Shephard's Lemma and Ro s Identit are illustrated for the Cobb-Douglas case. 5.1 An eample of perfect substitutes. Comments on Problems 5. A fied-proportions eample. Illustrates how the goods used in fied proportions (peanut butter and jell) can be treated as a single good in the comparative statics of utilit maimization. 5.3 An eploration of the notion of homothetic functions. This problem shows that Giffen's Parado cannot occur with homothetic functions. 5.4 This problem asks students to pursue the analsis of Eample 5.1 to obtain compensated demand functions. The analsis essentiall duplicates Eamples 5.3 and Another utilit maimization eample. In this case, utilit is not separable and cross-price effects are important. 5.6 This is a problem focusing on share elasticities. It shows that more customar elasticities can often be calculated from share elasticities this is important in empirical work where share elasticities are often used. 5.7 This is a problem with no substitution effects. It shows how price elasticities are determined onl b income effects which in turn depend on income shares. 5.8 This problem illustrates a few simple cases where elasticities are directl related to parameters of the utilit function. 5.9 This problem shows how the aggregation relationships described in Chapter 5 for the case of two goods can be generalized to man goods A revealed preference eample of inconsistent preferences. 17

24 18 Solutions Manual Solutions 5.1 a. Utilit = Quantit of water = b. If p< p = I p, = I If p > p = 0, =. 8 p c. d. Increases in I shifts demand for outward. Reductions in p do not affect demand for until 8 p p <. Then demand for falls to zero. 3 e. The income-compensated demand curve for good is the single, p point that characterizes current consumption. An change in p would change utilit from this point (assuming > 0). 5. a. Utilit maimization requires pb = j and the budget constraint is.05pb +.1j = 3. Substitution gives pb = 30, j = 15 b. If p j = $.15 substitution now ields j = 1, pb = 4. c. To continue buing j = 15, pb = 30, David would need to bu 3 more ounces of jell and 6 more ounces of peanut butter. This would require an increase in income of: 3(.15) + 6(.05) =.75.

25 Chapter 5/Income and Substitution Effects 19 d. e. Since David N. uses onl pb + j to make sandwiches (in fied proportions), and because bread is free, it is just as though he bus sandwiches where p sandwich = p pb + p j. In part a, p s =.0, q s = 15; In part b, p s =.5, q s = 1; 3 In general, q = s p s so the demand curve for sandwiches is a hperbola. f. There is no substitution effect due to the fied proportion. A change in price results in onl an income effect. 5.3 a. As income increases, the ratio p p stas constant, and the utilit-maimization conditions therefore require that MRS sta constant. Thus, if MRS depends on the ratio, this ratio must sta constant as income increases. Therefore, since income is spent onl on these two goods, both and are proportional to income. b. Because of part (a), > 0 so Giffen's parado cannot arise. I 5.4 a. Since = 0.3 I p, = 0.7I p, U.. Ip p BIp p = 3 7 = The ependiture function is then E = B Up p. b. The compensated demand function is c = E / p =.3B p p. c. It is easiest to show Slutsk Equation in elasticities b just reading eponents from the various demand functions: e p, = 1, e, 1, c.7, 0.3 I= e = s, p = Hence e p, = e c s e,, or p I =

26 0 Solutions Manual 5.6 a. e pi I Ip I p s,, 1 I= = = e I I p I I p If, for eample e = 1.5, e = 0.5. e I, s, I ( pi) p p p + I = = = e b. s, p, p p p I I If, for eample, e = 0.75, e = p, s, p c. Because I ma be cancelled out of the derivation in part b, it is also the case that e = e + p, p, 1. p ( pi) I. + 1 p p p p I p d. es, p = = = = e p, p p I I p p. e. Use part b: kp p e p p p kp p k k 1 k k k k s, (1 ) p = k k + = k k (1 + pp ) 1+ pp. To simplif algebra, let d = p p k k kd kd 1 d Hence ep, = e, 1 1 s p = = 1+ d 1+ d remembering that e =. I, 1. Now use the Slutsk equation, kd d 1 1 d( k 1) e c = e, p, + s (1 )( ) p = + = = s σ. 1+ d 1+ d 1+ d 5.7 a. Because of the fied proportions between h and c, know that the demand for ham is h= I p + p ). Hence e ( h c h p I p ( p + p ) ph = = =. p h ( p + p ) I ( p + p ) h h h c hp, h h h c h Similar algebra shows that e, hpc c pc =. So, if ph= pc, ehp, = e, 0 h hp =.5. c ( p + p ) h b. With fied proportions there are no substitution effects. Here the compensated price elasticities are zero, so the Slutsk equation shows that ep, = 0 s= c. With ph= pc part a shows that ehp, =, e h hp, =. c 3 3 d. If this person consumes onl ham and cheese sandwiches, the price elasticit of demand for those must be -1. Price elasticit for the components reflects the proportional effect of a change in the price of the component on the price the whole c

27 Chapter 5/Income and Substitution Effects 1 sandwich. In part a, for eample, a ten percent increase in the price of ham will increase the price of a sandwich b 5 percent and that will cause quantit demanded to fall b 5 percent. 5.8 a. ep, = (1 s) σ s ep, = sσ s Hence ep, + ep, = σ 1. The sum equals - (triviall) in the Cobb-Douglas case. b. Result follows directl from part a. Intuitivel, price elasticities are large when σ is large and small when σ is small. c. A generalization from the multivariable CES function is possible, but the constraints placed on behavior b this function are probabl not tenable. 5.9 a. Because the demand for an good is homogeneous of degree zero, Euler s theorem states n i i pj + I = 0 p I. j= 1 j Multiplication b 1 i ields the desired result. b. Part b and c are based on the budget constraint pii = I. Differentiation with respect to I ields: pi i I = 1. Multiplication of each term b I i I i ields se i i, I = 1. c. Differentiation of the budget constraint with respect to p j : p j i pi i pj + j = 0. Multiplication b ields I i i se = s i i, j j. i i i i 5.10 Year 's bundle is revealed preferred to Year 1's since both cost the same in Year 's prices. Year 's bundle is also revealed preferred to Year 3's for the same reason. But in Year 3, Year 's bundle costs less than Year 3's but is not chosen. Hence, these violate the aiom.

28 CHAPTER 6 DEMAND RELATIONSHIPS AMONG GOODS Two tpes of demand relationships are stressed in the problems to Chapter 6: cross-price effects and composite commodit results. The general goal of these problems is to illustrate how the demand for one particular good is affected b economic changes that directl affect some other portion of the budget constraint. Several eamples are introduced to show situations in which the analsis of such cross-effects is manageable. Comments on Problems 6.1 Another use of the Cobb-Douglas utilit function that shows that cross-price effects are zero. Eplaining wh the are zero helps to illustrate the substitution and income effects that arise in such situations. 6. Shows how some information about cross-price effects can be derived from studing budget constraints alone. In this case, Giffen s Parado implies that spending on all other goods must decline when the price of a Giffen good rises. 6.3 A simple case of how goods consumed in fied proportion can be treated as a single commodit (buttered toast). 6.4 An illustration of the composite commodit theorem. Use of the Cobb-Douglas utilit produces quite simple results. 6.5 An eamination of how the composite commodit theorem can be used to stud the effects of transportation or other transactions charges. The analsis here is fairl intuitive for more detail consult the Borcherding-Silverberg reference. 6.6 Illustrations of some of the applications of the results of Problem This problem demonstrates a special case in which uncompensated cross-price effects are smmetric. 6.8 This problem provides a brief analsis of welfare effects of multiple price changes. 6.9 This is an illustration of the constraints on behavior that are imposed b assuming separabilit of utilit This problem looks at cross-substitution effects in a three good CES function.

29 Chapter 6/Demand Relationships Among Goods 3 Solutions 6.1 a. As for all Cobb-Douglas applications, first-order conditions show that p m = p s = 0.5I. Hence s = 0.5 I p and s p = 0. m s b. Because indifference curves are rectangular hperboles (ms = constant), own substitution and cross-substitution effects are of the same proportional size, but in opposite directions. Because indifference curves are homothetic, income elasticities are 1.0 for both goods, so income effects are also of same proportionate size. Hence, substitution and income effects of changes in p m on s are precisel balanced. c. s s s = 0 = m and U p p I m m m m = 0 = s U p p I s s m s m s But = so m = s m. U U p p I I m s s m d. From part a: s m m = m = m = s = s. I ps pmm s pm I 6. Since r/ p r > 0, a rise in p r implies that p r r definitel rises. Hence, p jj = I prr must fall, so j must fall. Hence, j p r < a. Yes, p = p + p. bt b t b. Since pc c = 0.5I, c / pbt= 0. c. Since changes in p b or p t affect onl p bt, these derivatives are also zero. 6.4 a. Amount spent on ground transportation t = pbb + ptt = pb b + t pb p pt = pb g where g = b + t. p b. Maimize U(b, t, p) subject to p p p + p b b + p t t = I. This is equivalent to Ma U(g, p) = g p Subject to ppp+ pbg = I. b

30 4 Solutions Manual c. Solution is I I p = g = 3p 3p p b d. Given p b g, choose p b b = p b g/ p t t = p b g/. 6.5 a. Composite commodit = p + p ( ) 3 p 3= 3 k+ 3 p + t kp + t 3 b. Relative price = = p + t p + t 3 3 Relative price < 1 for t = 0. Approaches 1 as t. Hence, increases in t raise relative price of. c. Might think increases in t would reduce ependitures on the composite commodit although theorem does not appl directl because, as part (b) shows, changes in t also change relative prices. d. Rise in t should reduce relative spending on more than on 1 since this raises its relative price (but see Borcherding and Silberberg analsis). 6.6 a. Transport charges make low-qualit produce relativel more epensive at distant locations. Hence buers will have a preference for high qualit. b. Increase in bab-sitting epenses raise the relative price of cheap meals. c. High-wage individuals have higher value of time and hence a lower relative price of Concorde flights. d. Increasing search costs lower the relative price of epensive items. 6.7 Assume i = a i I j = a j I i Hence: = a a I = I j i j i I j so income effects (in addition to substitution effects) are smmetric. 6.8 a. CV = E( p, p, p, Kp, U) E( p, p, p, K p, U). ' ' 1 3 n 1 3 n

31 Chapter 6/Demand Relationships Among Goods 5 b. Notice that the rise p 1 shifts the compensated demand curve for. c. Smmetr of compensated cross-price effects implies that order of calculation is irrelevant. d. The figure in part a suggests that compensation should be smaller for net complements than for net substitutes. 6.9 a. This functional form assumes U = 0. That is, the marginal utilit of does not depend on the amount of consumed. Though unlikel in a strict sense, this independence might hold for large consumption aggregates such as food and housing. b. Because utilit maimization requires MU p = MU p, an increase in income with no change in p or p must cause both and to increase to maintain this equalit (assuming U i > 0 and U ii < 0). c. Again, using MU p = MU p, a rise in p will cause to fall, MU to rise. So the direction of change in MU p is indeterminate. Hence, the change in is also indeterminate. d. If U = α α 1 MU = α But ln U = α ln + ln MU = α /. Hence, the first case is not separable; the second is a. Eample 6.3 gives = are gross complements. I p + p p + p p z clearl / p, / p < 0 so these b. Slutsk Equation shows / p= / p U= U so / p U = U could be I positive or negative. Because of smmetr of and z here, Hick s second law suggests / p and / p > 0. U =U z U =U z

32 6