EVEN NUMBERED EXERCISES IN CHAPTER 4

Joh Riley 7 July EVEN NUMBERED EXERCISES IN CHAPTER 4 SECTION 4 Exercise 4-: Cost Fuctio of a Cobb-Douglas firm What is the cost fuctio of a firm with a Cobb-Douglas productio fuctio? Rather tha miimie the cost of producig a particular output we examie the dual problem of maximiig output give a fixed budget Max{ = r C} = Takig the logarithm of the obective fuctio, the maximiatio problem becomes Max{l = l r C} = From the Lagragia we obtai the followig first order coditios λr = ρ Hece r = Summig over, = r = C, where ρ = λ λ = = C Usig these two euatios to elimiate λ, = ρr Next substitute for i the productio fuctio C C ρ = ( ) = ( ) ( ) ρr ρ r = = Ivertig we have at last, r ρ C = ρ ( ) = ρ Exercise 4-4: Quasi-liear Productio Fuctio Output is produced accordig to the followig productio fuctio Aswers to Exercises i Chapter 4 page

Joh Riley 7 July = + (a) Show that for sufficietly large Cr (, ) = r 5 r / r (b) Show that Cr (, ) is proportioal to (c) Depict Cr (, ) for i a eat figure for sufficietly small (d) I a secod figure depict margial cost as a fuctio of for two differet values of r (e) If the output is sold by a moopolist, describe the effect o output of a icrease i r Does it make a differece whether the profit maximiig output is large or small? (a) Suppose that both iputs are used Cosider the budget costraied output maximiatio problem, Max{ + r + r C} Substitutig for this problem becomes C r Max + { } r r From the FOC, r r 5 = Hece 5r = ( ) ad r r ( 5 C ) = r r if C 5 r / r Substitutig ito the above expressio, = C r 5 r + r Note that this is the solutio if C 5 r / r, hece Ivertig, the miimied cost is 5 / r r = r, for r / r r C ( ) 5 r 5 Aswers to Exercises i Chapter 4 page

Joh Riley 7 July (b) For smaller outputs =, hece = Ivertig, the iput reuiremet is = ad so total cost is C ( ) = r / / (c) The cost curve is uadratic to the left of the critical output ˆ = 5 r/ r ad liear to r the right Whe ˆ = 5 r/ r, the margial cost is give by MC = ad whe 5 ˆ 5 r / r > =, the margial cost is give by at ˆ MC = r Thus margial cost is cotiuous (e) If the output is sold by a moopolist ad r icreases, the critical poit ˆ decreases ad the effect o the profit maximiig output margial cost icreases or will deped whether > ˆ ad the margial cost does ot chage ˆ ad Exercise 4-6: Properties of firm iput demad ad output supply Let () r be the profit maximiig iput vector for a moopolist producig outputs (,, ) m usig iputs the iput price vector (a) Assume that (,, ) The firm is a price-taker i iput markets Let r be () r is cotiuously differetiable ad show that the matrix i is egative semi-defiite r (b) If the firm is also a price-taker i output markets what ca be said about the m m matrix i / p? Let the vector of feasible productio vectors be all those i the set G (, ) To maximie profit the firm solves the followig problem Max{ Π= p r G(, ) }, (a) The Lagragia of this problem is L = Π λg (, ) = p r λg (, ) Appealig to the Evelope Theorem, Π L = = r r ad Π L = = i r r i i Aswers to Exercises i Chapter 4 page 3

Joh Riley 7 July Differetiatig the first expressio by r i, it follows that Π = ri r ri By Propositio 4-4, the profit fuctio Π( p, r) is a covex fuctio of r The the matrix of secod partial derivatives is positive semi-defiite ad so is egative semidefiite ri (b) Arguig almost exactly as i the aswer to Exercise 4- (a), the profit fuctio is a covex fuctio of p The the matrix of secod partial derivatives is positive semidefiite Followig the steps i the aswer to part (a), is positive semidefiite Π = p i p pi SECTION 43 Exercise 43-: Returs to Scale ad Average cost Prove that if a firm exhibits icreasig/decreasig returs to scale the average cost must decrease/icrease with output We cosider icreasig returs to scale Let Cr (, ) = r Give IRS, for ay λ >, be cost miimiig at output That is λ > λ = λ Thus there is some F( ) F( ) μ < λ such that μ F( ) = λ Hece C r = Mi r F r < r = C r ( λ, ) { ( ) λ } μ λ λ (, ) Thus C( λ, r) C(, r) AC( λ, r) = < = AC(, r) λ Exercise 43-4: Modified Cobb-Douglas Productio Fuctio The productio fuctio of a firm is defied implicitly as follows = K L,, > (a) Give iput prices (, rw, ) show that the cost miimiig iput demads satisfy Aswers to Exercises i Chapter 4 page 4

Joh Riley 7 July + = = rk wl C( ) (b) Hece or otherwise obtai a expressio for the firm s cost fuctio (c) If + =, show that the Average cost fuctio is U-shaped, with a miimum at = (d) Does a chage i a iput price have ay effect o the cost miimiig output? (a) The cost miimiatio problem is as follows C ( ) = MirK { + wl K L } Euivaletly, KL, C ( ) MirK { wl K = + L } KL, This is a stadard Cobb-Douglas problem The First Order Coditios ca be writte as follows + = = rk wl C( ) (b) From these euatios, K C ( ) = + r ad L C ( ) = + w Substitutig these expressios ito the productio fuctio ad ivertig, the firm s cost fuctio ca be r + + w obtaied as C ( ) = ( + ) + (c) The average cost fuctio is C ( ) r + w + + AC( ) = = ( + ) If + = the cost fuctio ca be writte as C ( ) = K so that AC = K Takig logarithms ad differetiatig, d AC ( ) l AC = = + l = + l d AC( ) Note that this is ero at = Also, d l AC = + > d Aswers to Exercises i Chapter 4 page 5

Joh Riley 7 July d Thus l AC( ) d is egative if < ad positive if > (d) The cost miimiig output is idepedet of iput prices SECTION 44 Exercise 44-: Cosumer surplus with iterdepedet demads A cosumer has utility fuctio Uxx (,, ) = B (, ) + Fx ( ) + x Let p = ( p, p) be the price vector for ad let r be the price vector for x = ( x,, x ) Normalie so that the price of commodity x is You should assume throughout that icome I is large eough for demad for this commodity, x (,, ) pri to be strictly positive (a) Show that demad for all the other commodities is idepedet of icome (b) Show that for all >> the demad price fuctio is p( ) = ( ) Assume, heceforth, that for all p above some upper boud p, ( p ) = (c) Explai why idirect utility is V( pˆ, r, I) = B( ( pˆ)) + f( x( r)) + x ( pˆ, r, I), pˆ < p ad V( p, r, I) = f( x( r)) + x ( p, r, I), p p (d) Hece explai why the gai i utility from a price ˆp < p ca be writte as follows ˆ ˆ ˆ ˆ Δ V = [ (,) pˆ ] d + ( ˆ, ) d = [ p (,) pˆ ] d + p ( ˆ, ) d (a) The cosumer s budget costrait is p + r x+ x = I Substitutig for x i the utility fuctio, Aswers to Exercises i Chapter 4 page 6

Joh Riley 7 July u = B( ) + f( x) p r x+ I = [ B( ) p ] + [ f( x) r x] + I Thus U is maximied by choosig ( p) = arg Max{ B( ) p } ad x( r) = arg Max{ f( x) r x} (b) The FOC for the first maximiatio problem are ( ) p where ( ( ) p) = x Thus for all >> the demad price fuctio is p = (c) Idirect utility is V( p, r, I) = B( ( p)) p ( p) + f( x( r)) r x( r) + I For p p, ( p ) = Therefore V( p, r, I) = f( x( r)) r x( r) + I (d) Hece V( pˆ, r, I) V( p, r, I) = B( ( pˆ)) pˆ ( pˆ) SECTION 45 = [ B( ˆ, ˆ ) B ( ˆ,) p ˆ ˆ ] + [ B ( ˆ,) p ˆ ˆ ] ˆ ˆ = [ ( ˆ, ) pˆ ] d + [ (,) pˆ ] d ˆ ˆ = [ p ( ˆ, ) pˆ ] d + [ p (,) pˆ ] d Exercise 45-: Idirect Price discrimiatio There are two types of buyer Low demaders have a market demad price fuctio p = a while high demaders have a demad price fuctio p = a The umber of type t buyers is t The uit cost of productio is c (a) Explai why, if it is most profitable to sell oly to the high demaders, the profit maximiig two part pricig scheme is ( p, F) ( c, ( a c) ) = Aswers to Exercises i Chapter 4 page 7

Joh Riley 7 July (b) Alteratively, the moopolist offers two plas ad serves both types (so p < a) Show that the gai to a type buyer purchasig pla is U = ( a p ) F (c) Hece show that if the moopoly extracts all of the surplus from type buyers, the profit from a type customer is ( p c)( a p ) ( a p ) gai of U = ( a a )( a + a p ) if they choose pla + ad type buyers have a (d) Explai why the maximum profit that the moopoly ca extract from type is ( a c) ( a a )( a + a p ) (e) Appeal to your aswers to (c) ad (d) to obtai a expressio for the total profit The solve for the profit maximiig price p (f) Cofirm that as the umber of type icreases, the pla use fee rises There are two types of buyer Low demaders have a market demad price fuctio p = a while high demaders have a demad price fuctio p = a The umber of type t buyers is t The uit cost of productio is c (a) Explai why, if it is most profitable to sell oly to the high demaders, the profit maximiig two part pricig scheme is ( p, F) ( c, ( a c) ) = (b) Alteratively, the moopolist offers two plas ad serves both types (so p < a) Show that the gai to a type buyer purchasig pla is U = ( a p ) F Hece show that if the moopoly extracts all of the surplus from type buyers, the profit from a type customer is Π = ( p c)( a p) + ( a p) (c) Show that buyers have a gai of U = CS( p ) CS( p ) + U if they choose pla Explai why the payoff to a type buyer choosig pla is U = CS() c F Hece show that if all surplus is extracted from type buyers, choosig pla is best for type if ad oly if F F = CS () c U = CS () c [ CS ( p ) CS ( p )] (d) Appeal to your aswers to (c) ad (d) to obtai a expressio for the total profit The solve for the profit maximiig price p Aswers to Exercises i Chapter 4 page 8

Joh Riley 7 July (3) Cofirm that as the umber of type icreases, the pla use fee rises (a) Social surplus of type t if payig p per uit is s maximied by settig p chargig a access fee F eual to social surplus t( p) at p t t t t CS ( p) = ( p ( x) p) dx = ( a x p) d = ( a p) = c ad Hece F CS c a c = () = ( ) (b) A type cosumer has a cosumer surplus of CS p a p ( ) = ( ) ad a payoff of U = CS ( p ) F The moopoly sueees the cosumer s payoff to ero by chargig a access fee eual to this cosumer surplus The profit from the use fee is ( p c ) = ( p c)( a p) Addig the access fee, F CS p a p = ( ) = ( ) the total profit o each type customer is Π ( p) = ( p c)( a p) + ( a p) Sice it will be useful later we ote that Π = p ( p c) (b) We ow compare the payoff to a type customer from the two plas If he chooses pla his payoff is U = CS ( p ) F = CS ( p ) CS ( p ) + [ CS ( p ) F] = CS ( p ) CS ( p ) + U Note that lowerig U to ero both raises profit o pla ad lowers the icetive for a type buyer to switch Thus settig U = is profit-maximiig (c) Pla offers a price per uit eual to margial cost Thus type s payoff if he chooses pla is U = CS () c F Thus type will choose this pla as log as U U Thus the highest possible access fee is Aswers to Exercises i Chapter 4 page 9

Joh Riley 7 July F = CS () c U = CS () c [ CS ( p ) CS ( p )] The profit per buyer for pla is Note that Π = F Π F = = ( CS CS) = a a p p p (c) Sice the moopoly sets p = c, the profit o a type pla is Π = F Total profit is Π= Π + Π The Π = ( p c) + ( a a ) = [ c + ( a a ) p ] p Thus profit is maximied by settig p = c+ ( a a) Aswers to Exercises i Chapter 4 page