ANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 5. The constraint is binding at the maximum therefore we can substitute for y

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1 John Rile Aril 0 ANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 5 Section 5: The Robinson Crusoe Econom Eercise 5-: Equilibrium (a) = ( + ω) = ( + 47, ) Then = 47 Substituting or in the / roduction unction, = = (47 ) Robinsons utilit is thereore / U = = (47 ) Taking the logarithm and dierentiating, lnu = + Setting this equal to ero, L 47 and so * = 4 * = 98, hence ω * * = = 49 (b) Let the rice vector be The roduction set is Y = {(, ) + 0} Robinson then seeks the solution o the ollowing roblem 4 Π= Ma{ + 0} 4 The constraint is binding at the maimum thereore we can substitute or and write the roit as ollows π ( ) = + 4 FOC or roit maimiation: π = + = 0 Hence coconut sul is = Substituting back into the roduction unction, labor demand is L / = /7 ( ) D = For labor demand to be otimal we thereore require that Answers to Chater 5 age

2 John Rile Aril 0 Eercise 5-3: Eistence with a non-conveit (a) The boundar o the roduction set, + γ + = can be rewritten as ollows 8 0 = 8( γ ) The grah o this unction is deicted below, Y Y+ ω γ O γ, Adding the endowment ω = ( γ,0) shits the boundar horiontall b γ (b) Crusoe s consumtion vector is = + ω where Then Crusoe s utilit is FOC U( ) = U( + ω) = + γ + ln = γ + ln 4 + = 0 8 = γ and so 8 = γ 8 (c) Hence * = and so = γ * Answers to Chater 5 age

3 John Rile Aril 0 ( ) = arg Ma{ Y } (d) First ignore the otion o roducing nothing Then Robinson solves the ollowing maimiation roblem Ma{ ( γ ) + } 8 Solving, = 4 / The resulting roit is Π ( ) = / γ = (( / ) γ ) This is ositive i and onl i γ Thereore the sul curve i the irm is ( / ) 0 0, ( / ) ( ) = 4 /, ( / ) γ γ Note that this is a sul corresondence as it is multi-valued at the critical oint where ( / ) = γ (e) I there is a Walrasian equilibrium must be eicient Since the eicient outut o coconuts is, it ollows that 4 / = Thus i a WE eits, the WE rice ration / = () The imlied equilibrium roit is Π ( ) = (( / ) γ ) = ( γ ) (g) Thus a WE eists i and onl i γ 5 General Equilibrium Eercise 5-: Cobb Douglas econom (a) The FOC or cost minimiation roit maimiing b a irm in industr are Answers to Chater 5 age 3

4 John Rile Aril 0 α α = r r Rewriting this equation, α =, =,, F r α r Thereore = and so = θ or some 0 θ > where F = θ = Hence F F α α α α F = = ( θ ) ( θ ) = ( ) ( ) θ = = = q q α = ( ) ( ) α U a q a q a a (b) = ln + ln = ( α ln + ( α) ln ) + ( α ln + ( α) ln Utilit is maimied subect to the constraint + = ω, + = ω Note that the otimiation roblem searates into two simler otimiation roblems The irst is Ma{ a α ln + a α ln + ω}, This is easil solved Similarl, aα ω a α ω (, ) = (, ) * * aα + aα aα+ aα a ( α ) ω a ( α ) ω (, ) = (, ) ( ) ( ) ( ) ( ) * * a α + a α a α + a α r r α α (c) From (a), = hence r = (, ) is a WE α * * α * * The cost unction or industr is r r Cq r q α α (, ) = ( ) ( ) α α Answers to Chater 5 age 4

5 John Rile Aril 0 MC = AC = = r α r α α α ( ) ( ) Eercise 5-3: Robinson meets Frida (a) Since both consumers have the same reerences consider this as a reresentative agent roblem Since = ω and =, the utilit o the reresentative agent is /4 U = ln( ω ) + 4aln( ) = ln( ω ) + aln /4 It is readil conirmed that aω = and hence + a * ω aω = (,( ) ) + a + a * /4 (b) The reresentative consumer has the ollowing budget constraint + ω +Π ( ) He thereore solves FOC Ma{ln + 4a ln + ω +Π ( )} Rearranging 4a = = λ = 4a ( ) ( ) * 4a The WE must be eicient Hence = and so = (, ) * * is a WE rice vector (c) Frida has no roit His endowment o time is hal the total endowment His otimal consumtion bundle is thereore FOC F = arg Ma{ln + 4aln + ( ω )} 4a + 4a + 4a = = = + ω F F F F Answers to Chater 5 age 5

6 John Rile Aril 0 Thereore F ω = ( + 4 a) Hence Robinson s demand or leisure is R * F = = ( ) + a (+ 4 a) ω Note that this aroaches ω as a 0 Moreover it is readil checked that R is an increasing unction o a or a greater than ero and suicientl small But Robinson s endowment o time is onl ω Thus the reresentative agent aroach must be modiied The onl other ossibilit is that Robinson consumes all this tile in the orm o leisure Then Frida is the onl consumer suling labor We have alread solved or his labor sul The WE rice ratio must be chosen so that the irm s demand or labor = arg Ma{ } = = ω D /4 F F 53 Eistence Eercise 53-: Continuit o the sul unction We irst show that is unique Since Y is comact and is continuous, there is a solution to the maimiation roblem Ma{ Y } To rove that there is a unique solution, we suose instead that there are at least two solutions ever conve combination λ is also a solution, since λ λ 0 and Then or λ = ( ) + = Since Y is strictl conve, ever conve combination δ λ inty Then or suicientl small δ >> 0, = λ + δ Y Then λ 0 = + > But then 0 is not a solution to the otimiation roblem ater all Thus the roit maimiing resonse is a sul unction ( ) B the Theorem o the Maimum (I), this unction is continuous Eercise 53-3: Eistence and Non-eistance with a minimum consumtion threshold Answers to Chater 5 age 6

7 John Rile Aril 0 (a) This is a standard case o two consumers with dierent homothetic reerences From Section 5, the PE allocations are a deicted below and the WE rice ratio / rises along the PE allocations as the utilit o consumer A increases The highest rice ratio is where Ale has the entire endowment ( ω, ω ) Since his marginal rate o A A substitution is α / β along the 45 line, this is the maimum rice ratio Making an B B identical argument or Bev, the minimum rice ratio is α / β B O A O Figure: Edgeworth bo diagram (b) The onl art o the Edgeworth Bo or which neither consumer has a consumtion h α β below γ is the inner deicted below Redeining units, U = ˆ ˆ The budget h h constraint h h becomes ω ˆ = ( ) + ( ) ( ) + ( ) = h h h h h h ˆ γ γ ω γ ω γ ω We can thereore aeal to the answer to art (a) or all the oints in the inner square o the Edgeworth Bo Answers to Chater 5 age 7

8 John Rile Aril 0 B O A O Figure: Edgeworth bo diagram B the irst welare theorem, i there is a WE allocation, it must be PE Thus there are no WE or endowments outside the shaded region Eercise 53-5: Continuum o Equilibria with no consumtion Threshold (b) The Edgeworth-Bo Diagram is deicted below For PE allocations outside the shaded region the rice is determined b the sloe o the high wealth consumer Along the 45 line the sloe o Ale s indierence curve is while the sloe o Bev s is ½ For the PE allocations inside the shaded region we can redeine units so that = Preerences are homothetic in thus we can aeal to our earlier results and conclude that along the PE allocations the WE rice ratio rises rom ½ to i i Answers to Chater 5 age 8

9 John Rile Aril 0 (3,3) (,) Figure:Edgeworth bo diagram Arguing as in the revious eercise, there is a continuum o equilibria or one endowment oint SECTION 54 Eercise 54-: A Public Bad (a) The roblem is to solve L = U(, b) λg(, b) The FOC are thereore Ma{ U (, b) G(, b) 0} The Lagrangian is ( b, ) L G = ( b, ) λ ( b, ) = 0, =, and L U G = ( b, ) λ ( b, ) = 0 b b b Answers to Chater 5 age 9

10 John Rile Aril 0 (b) Introduce rices and or the two commodities and the ollution ta t The rm s otimiation roblem is Ma{ tb) G(, b) 0} Form the Lagrangian ( b, ) L = tb μg(, b) The FOC are thereore L G = μ (, b) = 0, =, L G and = t μ (, b) = 0 b b The consumer equates his marginal utilit er dollar = (c) It is eas to conirm that these conditions are the FOC or the otimum i and μ = ν / λ (d) The analsis generalies directl The new otimiation roblem is = ν ( b, ) Ma{ U (, b) b b, G (, b) 0, =,} = (e) The government intervention creates a new commodit ollution rights and sets the endowment o this commodit at the otimal level To rice taking irms, a ollution right is in eect a third inut Markets clear when demand or this right equals the ied sul so the additional market relaces the need or a direct ta SECTION 55 Eercise 55-: Robinson Cruse Econom (a) Deine u = lnu = ln+ ln Since utilit is strictl increasing = Hence FOC u = ln(08 ) + ln + ln 3 = = Answers to Chater 5 age 0

11 John Rile Aril 0 Hence * = 36 and so * = (7, 7) (b) The roit o the irm is FOC Π ( ) = Ma{ + } = Ma{ + } / / Hence + = / 6 0 ( ) 36( ) = (c) The WE is a PE allocation Let be the equilibrium rice vector Then ( ) = = 36 and so = (,) is WE rice vector Equilibrium roit is Π ( ) = 36 * The budget constraint o the reresentative consumer is Setting 08 +Π ( ) = this can be rewritten as ollows + 44 (d) Alternativel, let r be the real interest rate That is, b orgoing unit o the commodit in eriod, a consumer earns + r units in eriod In the irst eriod the irm borrows and in the second earns is thereore / Π () r = Ma{ + } + r / The maimied resent value o the irm Note that this is eactl the same as in the WE with sot and utures markets i r + = Thus the equilibrium real interest rate is ero Eercise 55-3: Rational eectations equilibrium (a) Let t = ( t, t) be eriod t consumtion Then = ω, = ω, = + ω and = 4 + ω Answers to Chater 5 age

12 John Rile Aril 0 Substituting into the utilit unction FOC U = ln(0 ) + ln(0 ) + ln( + 0) + ln(4 + 0) = + 0, with equalit i > = + = 0, with equalit i > It is readil checked that < 0 i = 45 Thus the otimal consumtion vector is or all 0 and that the second condition is satisied (, ) =(0,0,75,300) * * (b) First consider an econom with sot and utures rices The FOC or utilit maimiation is = (,,, ) = (,,, ) = λ * * * * 4 I =, then λ = Hence = (,, 5, 8) 60 (c) I the interest rate is ero, the sot rice vector is = (, ) and the uture sot rice s 4 vector is = ( + r) = = (, ) Note that the sot rices are no guide to uture sot 5 8 rices Thus the reresentative consumer has no eas wa o redicting the uture sot rices in this econom Eercise 55-5: Walrasian equilibrium in a two-eriod model U (a) ( ) = ln + ln + δ ln + δ ln I there is no storage, the gradient vector must be roortional to the WE rice vector at the endowment Thereore δ δ ( ω) = (,,, ) = (,, δ, δ) = λ ω ω ω ω Thereore = (,, δ, δ ) is a WE rice vector Answers to Chater 5 age

13 John Rile Aril 0 (b) The sot rice vector is = (,) and the utures rice vector is = ( δ, δ ) The s uture sot rice vector is = ( + r) = ( + r)( δ, δ) = ( + r) δ(,) Thus the sot rices and uture sot rices are identical i ( + r)δ =, that is + r = /δ (c) Storing units o commodit and units o commodit ields a roit o ( ) + ( ) The otimum is a WE Thus i the otimum involves storage, roit must be maimied at >> 0 Then = and = (d) I there is no storage the WE rices are = (,, δ, δ ) Storing units o commodit and units o commodit ields a roit o ( ) + ( ) = ( δ ) + (δ ) Thus it is roit-maimiing to do no storage i and onl i δ SECTION 56: Equilibrium with Constant Returns to Scale Eercise 56-: Market Adustment Let inut be labor so that commodit A is more labor intensive Let r be the Walrasian Equilibrium inut rice vector I the sul o inut increases there is ecess sul resulting in a downward shit in r This lowers the average and marginal cost o roduction in both industries so both industries have an incentive to eand The additional labor inut raises the marginal roduct o inut in both industries so there is ecess demand or inut The rice o inut must thereore rise until it is roitable or one o the industries to contract Since industr is more inut intensive, its unit cost is lowered more thus it is industr that contracts The contraction o industr and eansion o industr continues This laces uward ressure on the rice o inut and downward ressure on the rice o inut B Proosition 563, i both goods are roduced, then, in the long-run, both inut rices must return to their original levels I the increase in sul o labor is large, industr B contracts comletel so the econom is secialied in the roduction o commodit A The equilibrium cost line is Answers to Chater 5 age 3

14 John Rile Aril 0 tangent to the isoquant at With constant returns to scale the sloe is constant along a ra thus, as labor increases, the wage-rental rate must all Figure: Inut rice change with secialiation Eercise 56-3: Corollar o the Rbcnski Theorem From the derivation o the Rbcnski Theorem qv A A+ qv B B = Hence dq v + dq v = d A A B B From the Rbcnski Theorem dq B < 0 Thereore dqava > d It ollows that dqa qv A A+ qv B B > = > q d q v q v A A A A A Eercise 56-5: Which Inut Prices? (a) Since F ( v ) = F (8,8) = this is the unit isoquant Note that r v = = so this is A A A the dollar isoquant For commodit B, it is a standard result that or a smmetric Cobb- Douglas roduction unction, the cost shares are equal Since the total cost o each inut are ½ when the inut vector is v B = (5,0) this is indeed eicient Since outut is and A A Answers to Chater 5 age 4

15 John Rile Aril 0 the rice o commodit B is one, this is the dollar isoquant or commodit B The dollar isoquants or both commodities are deicted below along with the dollar cost line (b) Figure: Dollar isoquants and iso cost line (b) The second equilibrium cost line is deicted below Since the isoquants are smmetric the equilibrium inut rice vector must be r = (, ) Figure: Second equilibrium iso-cost line (c) Given the unit inut requirements o art (a), total demand or inuts is Answers to Chater 5 age 5

16 John Rile Aril qa 8 5 qa qb 8 0 q = + B 8 0 Thus the inut demands are a ositivel weighted average o vectors with sloes o and 4 With the unit inut requirements o art (c) total demand or inuts is 8 0 qa 8 0 qa qb 8 5 q = + B 8 5 Thus the inut demands are a ositivel weighted average o vectors with sloes o and ¼ Since the sul ratio is ½, onl the second case is consistent with equilibrium (e) Now the sul ratio is so onl the irst case is consistent with equilibrium Thus r = (, ) 0 40 Answers to Chater 5 age 6