Economics 201b Spring 2010 Solutions to Problem Set 3 John Zhu

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1 Economics 20b Spring 200 Solutions to Problm St 3 John Zhu. Not in th 200 vrsion of Profssor Andrson s ctur 4 Nots, th charactrization of th firm in a Robinson Cruso conomy is that it maximizs profit ovr fasibl sts. This is incorrct, and th appropriat corrction has bn mad. Th corrct dscription should coincid ith our undrstanding of firms and agnts from prvious xchang conomis and mor gnral Arro-Dbru conomis - that is, thy do not tak into account th siz of th social ndomnt hn making thir dcisions. For this problm st ill mak an xcption for thos studnts ho rot thir solutions for this xrcis ith th incorrct assumption in mind. Hovr, in th futur ill b oprating undr th corrct assumption. t and p b th prics of th first and scond good. Th Agnt s Utility Maximizing Dcision p < β us all alth for good consumption (x, x 2 ) = {( + Π, 0)} p = β p > β indiffrnt btn any consumption bundl on budgt frontir (x, x 2 ) = {(x, ( x ) + Π ) x [0, + Π p ]} us all alth for good 2 consumption Th Firm s Profit Maximizing Dcision p (x, x 2 ) = {(0, + Π )} p < α profit can b arbitrarily larg, ill ant arbitrarily larg amount good (i.. dmand is mpty). Altrnativly, undr th incorrct assumption, th firm can maximiz profit dmanding th social ndomnt of good (z, q) = (, α) and Π = pα = α firm maks zro profits, indiffrnt btn any point on production possibility p frontir (z, q) = {(z, αz) z 0} and Π = 0

2 > α good (th input good) is too xpnsiv, th firm chooss not to produc anything, zro profits p (z, q) = (0, 0) and Π = 0 Thus this xrcis rducs to mixing and matching th quantitis,, α so that hat th agnt p β ants to do fits ith hat th firm ants to do. cost function / budgt frontir production function cost function / budgt frontir + production function quilibrium(a) indiffrnc curvs Figur : Cass - 3 Cas : > α β W can rul out > > α and > α >. Hovr, for any pric ratio such that p β β p p α thr is a uniqu quilibrium hr th firm producs nothing, and th agnt β p consums his ndomnt: {(x (p, ), x 2 (p, )) = (, 0), (z(p, ), q(p, )) = (0, 0), Π(p, ) = 0} Cas 2: = α β 2

3 On can sho that a ncssary condition for quilibrium is = = α, in hich cas th β p st of quilibria is {(x (p, ), x 2 (p, )) = (x, ( x ) p ), (z(p, ), q(p, )) = ( x, α( x )), Cas 3: β < α Π(p, ) = 0 x [0, ]} On can sho it must b th cas p = α, in hich cas th uniqu quilibria is {(x (p, ), x 2 (p, )) = (0, p ), (z(p, ), q(p, )) = (, α), Π(p, ) = 0} Undr th incorrct assumptions, can lt th rlativ pric bcom lor. So long as α, still hav quilibrium bcaus th firm ill nvr dmand mor than β p {(x (p, ), x 2 (p, )) = (0, + p α ), (z(p, ), q(p, )) = (, α), p Π(p, ) = p α } 2. Strong monotonicity implis p 0 and budgt constraints ar binding: p x i = p ω i + θ ij p y j + T i for all i j= p x i = p ω i + θ ij p y j + j= ( ) p l x li = p l ω li + θij p l y lj l= l= j= l= ( ) pl x li = ( ) ( pl ω li + pl l= l= l= j= ( ) pl x li = For l, th markt clars, so p l ω l + l= l= l= j= T i ( ) pl y lj ) θ ij y lj () x li = ω l + y lj for l =, 2,... j= 3

4 or quivalntly p l x li = p l ω l + p l Summing ovr l =,..., ( p l l= y lj for l =, 2,... j= ) ( x li = p l ω l + p l l= l= ) y lj j= (2) subtracting (2) from () gt p x i = p ω + p j= y j Sinc p > 0, may divid by p and dduc th markt claring condition for good : x i = ω + 3(a). Fix an arbitrary pric vctor p = (p, p 2 ). Th firm s profit maximization problm is j= y j argmax y p 2 log( y ) + p y Taking th drivativ, hav th first ordr condition p 2 y + p = 0 y = min{ p 2 p, 0} W nd th min bcaus hn p 2 p, hav a boundary solution of y = 0. Hovr ill s that this is a non-binding condition bcaus if p 2 p, thn th markts ill not clar. No th first agnt s utility maximization problm is log(x ) argmax + x 2 x,x 2 s.t. p x + p 2 x 2 p + θ Π log(x ) argmax + p + θ Π p x x p 2 Taking th drivativ, hav th first ordr condition Similarly, can find agnt 2 s x 2 : = p x = p 2 x p 2 p x 2 = p 2 p 4

5 Thus markt claring condition for good dictats x + x 2 = p 2 p ( + ) = p 2 p = ω + y p 2 ( ) = p 2 = p p Indd, if p 2 p, ould hav x + x 2 + < = ω + y. From hr, can thn find y 2, Π, x 2 and x 22 : y = 2 y 2 = 2 and Π = p p ( 2 ) = p 2 (2 2 ) = p 2 ( + ) x 2 = p + θ Π p x p 2 = + θ ( + ) x 22 = p 2 + θ 2 Π p x 2 p 2 = + θ 2 ( + ) No bcaus of xrcis 2 kno that th scond good must automatically clar as ll, hich mans hav found th quilibrium: ( { p 2 (, =, x = + θ ( + )), (, + θ 2 ( + } ) )), y = ( 2, 2) p O 2 cost function / budgt frontir 2 O production function (b). Th phras - no accss to firm s tchnology - litrally mans that thy cannot us th tchnology of th firm, hich mans thr is no scond good, and th agnts thn simply consum 5

6 thir ndomnt. In particular no accss to firm s tchnology dos not hav anything to do ith hthr th agnts gt a shar of th firm s profits. Som of you askd for clarification at offic hours. In gnral this is a good ida if you ar vr unsur. Hovr, if you misundrstood th maning to b zro shars but did th analysis corrctly othris, I ill giv you full crdit. On can asily sho that th rsrvations utilitis U i ar both 0. So it rducs to maximizing th folloing xprssion ith rspct to θ : argmax U (, + θ ( + )U ) 2 (, + ( θ )( + ) ) θ argmax θ ( 2 + θ ( + ) )( + + ( θ )( + ) 2 ) Taking th drivativ, gt th first ordr condition for θ ( + ) ( 2 + ( θ )( + ) ) ( + ) ( 2 + θ ( + ) ) = = θ 2( + ) = θ 2( 2 + ) θ = ( 2 + ) and θ 2 = 3 2( 2 + ) If you intrprtd th problm to man zro shars, thn a similar analysis can b prformd and th ansr is (θ, θ 2 ) = ( 2, 2 ). 4(a). Rgardlss of γ, th xact allocations {(, 0), (0, γ)} and {(0, γ), (, 0)} ar alays Parto Optimal - hovr gts (, 0) achivs maximal utility givn th social ndomnt, so if thr is a Parto improvmnt, it must improv th utility of th agnt ith (0, γ). But this can only b don if ithr h rcivs mor of good 2, hich is impossibl, or rcivs strictly mor than γ of good, hich is ithr also impossibl or ould lav th othr agnt strictly ors off. Cas : γ < 2 Considr an arbitrary xact allocation {(a, b), (c, d)}. a γ: thn considr th folloing squnc of changs: {(a, b), (c, d)} {(a, b + d), (c, 0)} {(0, b + d), (c + a, 0)} = {(0, γ), (, 0)} sinc c γ > γ d th first chang is akly Parto improving sinc b + d = γ a th scond chang is akly Parto improving if d > 0 thn th first chang is strictly Parto improving if a > 0 thn th scond chang is strictly Parto improving Thus th only xact Parto Optimal allocations hn a γ is {(0, γ), (, 0)}. c γ (quivalntly, a γ): by symmtry th only xact Parto Optimal allocation hn c γ is {(, 0), (0, γ)}. 6

7 a (γ, γ) (quivalntly, c (γ, γ)): Any xact allocation is Parto Optimal bcaus both agnts driv thir utility from thir allocation of th first good. Evn if a particular agnt had all of good 2 transfrrd to himslf, it is lss than hat h s consuming in good, so it dos not constitut a strict Parto improvmnt. So th st of all xact Parto Optimal allocations is {(, 0), (0, γ)} {(0, γ), (, 0)} {(x, x 2 ), ( x, γ x 2 ) x (γ, γ) x 2 [0, γ]} Cas 2: γ Thr ar only to possibilitis: ithr a γ or c γ. W can 2 us th xact sam argumnt to gt to th only to xact Parto Optimal allocations {(, 0), (0, γ)} and {(0, γ), (, 0)}. O O γ γ xact Parto Optimal allocations γ γ O 2 O 2 γ O O 2 γ No I claim that by symmtry hav alrady basically solvd th cass hr γ as ll. First convinc yourslf that th analysis of cass hr th social ndomnts only diffr by a constant factor ar ssntially th sam. Furthrmor, bcaus of th symmtry of th utility function, convinc yourslf that th analysis of cass hr th social ndomnt for ach good is rvrsd is also ssntially th sam. Thus for γ, th (, γ) cas is basically th sam as th (, ) cas hich is basically γ th sam as th (, ) cas, hich hav alrady considrd sinc. γ γ 7

8 So for γ 2, th only xact Parto Optimal allocations ar {(, 0), (0, γ)} and {(0, γ), (, 0)} just lik Cas 2. And for γ > 2, th st of xact Parto Optimal allocations is {(, 0), (0, γ)} {(0, γ), (, 0)} {(x, x 2 ), ( x, γ x 2 ) x [0, ] x 2 [, γ ]} hich is similar to Cas. 4(b). Th Parto Optimal allocations {(, 0), (0, γ)} and {(0, γ), (, 0)} can both b supportd by th pric vctor (p, p 2 ) hr p = p 2 = q. W ill no sho that no othr Parto Optimal allocation can b supportd as an quilibrium ith transfrs. So lt s considr on of th non cornr Parto Optimal allocations in, say, Cas : {(x, x 2 ), ( x, γ x 2 )} hr x is som numbr in (γ, γ) and x 2 is som numbr in [0, γ]. Th total alth of th of xchang conomy is p + p 2 γ. If thr is a good that is strictly chapr (say good i), both agnts ould us all thir alth to purchas that good. So th total dmand for good i is p +p 2 γ p i > + γ > max{, γ} social ndomnt of good i. Thus p = p 2 = q. In this cas, ach agnt ould still choos on good and us thir ntir alth to purchas this good. Sinc x > γ, for th consumption bundl (x, x 2 ) to b vn fasibl for agnt, it must b th cas that his alth qx > qγ, hich mans for thr to b an quilibrium h must choos to only buy good (if h chos instad to buy only good 2, thr ouldn t b nough). Similarly, agnt 2 is too althy to only buy good 2, hich mans h must also choos to only buy good but thn th total dmand for good is too grat and th total dmand for good 2 is to small. Thus th Scond Wlfar Thorm fails for γ (0, ) (2, ). 2 8

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