John Riley 9 Otober 6 Eon 4A: Miroeonomi Theory Homework Answers Constnt returns to sle prodution funtion () If (,, q) S then 6 q () 4 We need to show tht (,, q) S 6( ) ( ) ( q) q [ q ] 4 4 4 4 4 4 Appeling to () it follows tht (,, q) S (b) The prodution funtion of the firm is q /4 /4 Mimum output is q( B) M{ B} /4 /4 We insted mimie the logrithm M{ln ln ln r r B} 4 4 FOC 4 4 r r r r B B B nd nd so B B q ( ) ( ) ( ) ( ) ( ) ( ) B /4 / /4 /4 /4 /4 () With budget of ˆB the mimum output is
John Riley 9 Otober 6 /4 /4 ˆ qˆ ( ) ( ) B () Thus the minimum totl ost of produing ˆq is less thn or equl to Bˆ ( ) ( ) qˆ /4 /4 But for ny budget B Bˆ it follows from () tht it is impossible to produe ˆq So the minimum totl ost of produing ˆq 4 /4 4 /4 is ˆ ˆ r r TC( q) B ( ) ( ) qˆ (d) AC( q) MC( q) ( ) ( ) /4 /4 (e) For prie tking firm, if p /4 /4 MC( q) ( ) ( ) there is no profit mimiing output s it is lwys more profitble to inreses output sine MR p MC If p MC( q) ( ) ( ) /4 /4 the profit mimiing output is ero if the firm s profit-mimiing output q, it must be the se tht p MC( q) ( ) ( ) /4 /4 Remrk: It is importnt to understnd why this result holds for ny onstnt returns to sle prodution funtion Observtion : A CRS prodution funtion is homotheti To see this, suppose F ( ) F( ) Appeling to CRS F( ) ( ) nd F( ) ( ) F F F F ( ) ( ) ( ) ( ) Observtion : Given homothetiity, we showed tht if solves the budget problem q M F p B { ( ) }
John Riley 9 Otober 6 Then solves the budget problem q M F p B { ( ) } Then q F( ) F( ) q Thus mimied output rises linerly with the budget This is depited below Appeling to observtion it follows tht minimum totl ost C( q, r ) is proportionl to output Hene MC nd AC re onstnt Three ommodity eonomy 64 (),, q /4 /4 Simplify by mimiing u lnu ln ln 4ln 64 /4 /4 ln( ) ln( ) 4[ln ] 64 ln( ) ln( ) 4 ln ln ln FOC: 4 64 64, (Note tht I hve use the rtio rule twie)
John Riley 9 Otober 6 nd so 6 /4 /4 q (6) (6) (b) From line in (), 6 (,, q ) (,6,) () This is the sme prodution funtion s in question so the verge ost is 4p 4p AC( r) MC( q) ( ) ( ) /4 /4 As disussed bove, for positive prodution p AC( p, p) Totl Revenue p q qac( q) Totl ost (d) p q p p p p p /4 /4 FOC ( ) p ( ) p p p /4 /4 4 () nd ( ) p ( ) p p p () /4 /4 4 These must both be ero t (6,6) 6 ( ) p p 4 nd p ( ) p The supporting prie vetor is therefore ( p, p, p ) (,,) (e) Note tht 6 (,6,) nd the vlue of the endowment is p 48 4
John Riley 9 Otober 6 In this eerise mimum profit is p q p p () (6) (6) However the following remrk is ompletely generl Generl Remrk: The level set for mimum profit is p q p p p q p p The sublevel set for mimum profit is p q p p p q p p Note tht q nd the sublevel set for mimum profit n be, rewritten s follows: p p ( ) p ( ) Equivlently, p p p p p Using vetor nottion p p The right hnd side is the dividend from the firms nd the vlue of the ggregte endowment Thus the sublevel set for mimied profit is lso the budget set of the representtive onsumer The remining step is to hek tht the FOC for the following onsumer problem re stisfied t M{ U( ) p p } FOC: Mrginl utility per dollr must be equl MU ( ) MU ( ) MU ( ) p p p It is redily onfirmed tht these onditions hold 5
John Riley 9 Otober 6 Elstiity of substitution in two ommodity eonomy M ( p, u) Min{ p U( ) u} () Minimiing p is the sme s mimiing p So we onsider the following problem M ( p, u) M{ p U( ) u} Lgrngin L p ( U( ) u) To mke it ler tht we re onsidering fied utility problem (so inome is ompensted) we write the solution s With the FOC re L U p ( ) L U p ( ) From these two equtions U ( ) p MRS( ) U ( ) p (b) U(, ) Then / / p MRS( ) ( ) / / / p p p ( ) ( ) 6
John Riley 9 Otober 6 () p (, ) ln ln( ) ( ) p p p p p p p p ln( ) ( ) p p p [ln( ) ln p ln p ] p (d) (i) p MRS( ) ( ) / / / p p p / / ( ) ( ) Arguing s in () (, p) (d) (ii) I prefer to simplify by finding more onvenient representtion of the onsumer s preferenes The inverse of U( ) is Thus hs the sme level sets but is deresing funtion Then I hoose the following representtion u() p MRS( ) ( ) p p p / / ( ) ( ) Arguing s in () (, p) 7
John Riley 9 Otober 6 (e) The three utility funtions re ll sums of onve funtions so re onve Then the superlevel sets re onve sets (i) Level set U( ) U() U() / / Consider Then nd so / ( ) Similrly if then ( ) (ii) Level set U( ) U() U() / / Consider Then nd so / ( ) (iii) Level set u( ) u() u() Note tht ny in whih either or is very smll nnot be on the level set sine one of the terms in the utility funtions hs limit of Suppose Then nd so u() Then nd so (f) The three level sets re depited below 8
John Riley 9 Otober 6 9
John Riley 9 Otober 6
John Riley 9 Otober 6 Tehnil Remrk (not eminble nd only for those who like lulus) For ny define ( ) (, ) The derivtive of F( ( )) with respet to is ( ) ( ) Under CRS, F ( ( )) ( ) the derivtive of F( ( )) is F( ) F( ) ( ) ( ) for ll In prtiulr, setting, F( ) ( ) ( ) for ll p F( ) p ( ) p ( ) () From (d) the FOC re ( ) p ( ) p () nd ( ) p ( ) p () Substituting () nd () into (), p q p p So revenue is equl to ost nd so profit is ero