Unvesy of Calfona, Davs Dae: June xx, 009 Depamen of Economcs Tme: 5 hous Mcoeconomcs Readng Tme: 0 mnues PRELIMIARY EXAMIATIO FOR THE Ph.D. DEGREE Pa I ASWER KEY Ia) Thee ae goods. Good s lesue, measued n hous, s consumpon denoed x, and s dolla pce pe hou) p > 0. Goods o ae consumpon goods, wh dolla pces p,, p, all posve. The consume s endowed wh ω > 0 uns of lesue and zeo uns of any ohe good ω = 0 = = ω = 0). The consume s also endowed wh m dollas of pce-ndependen wealh. We defne he vaable L ω - x and call he supply of labo. He pefeences ae epesened by a dffeenable uly funcon u: X R, nceasng n all s agumens n he neo of X, whee X R+ denoes he consumpon se. Denoe he Jevonsan supply-of-labo funcon by Lp ˆ, p,..., p ) we ake he paamees m and ω as gven, so ha hey do no appea as agumens). Ia).. We he Slusky equaon fo he decomposon of he oal effec change n p on he supply of labo, expessng he subsuon and he wealh effecs n ems of he paal devaves of he Hcksan and Walasan demand-fo-lesue funcons. whee: The sandad fom of he Slusky equaon fo he supply of labo s L H = ˆ H L L L% = + L, ˆL h s he Hcksan demand fo lesue) s he SUBSTITUTIO EFFECT, nonnegave gven ha s always nonposve, by he negave semdefneness of he Slusky max. of a
L% x% = x% s he Walasan demand fo lesue) s he WEALTH EFFECT. Hence, we can we he Slusky equaon n ems of he devaves of he Hcksan and Walasan demand fo lesue as Lˆ x% = L. Ia).. Assume ha lesue s a nomal good. Wha can you say abou he sgn of he subsuon and wealh effecs of Ia).? Wha can you say abou he sgn of he oal effec Inepe n wods. Lˆ x% We have ha = L, whee ˆL? SUBSTITUTIO EFFECT = 0 x% x% WEALTH EFFECT = L < 0, because > 0 by he assumpon of nomaly. Hence, he subsuon and he wealh effec ae of weakly) oppose sgns, and he sgn of he oal effec may n pncple be posve, negave o zeo. In wods, an ncease n he wage p ses he oppouny cos of lesue, weakly) educng he demand fo, whch s equvalen o weakly) nceasng he supply of labo SUBSTITUTIO EFFECT). On he ohe hand, an ncease n he wage p nceases he eal wealh of he consume, nceasng he demand fo all nomal goods, ncludng lesue, whch s equvalen o educng he labo supply WEALTH EFFECT). Hence, he sgn of he oal effec can n pncple be anyhng. Ib). We now consde a new model. We sll have pces p, p,, p ) >> 0, an endowmen veco ω,0, 0), and he consume s pce-ndependen wealh m. Bu we noduce wo dffeences. Fs, lesue no longe enes he uly funcon, so ha pefeences ae defned on an open subse Z R, wh ypcal elemen x,, x ), and epesened by a dffeenable, scly + quasconcave and scly nceasng uly funcon β : Z R.
3 Second, he consumpon of any commody {,, } akes me, so ha he avalable amoun of lesue ω mus be allocaed beween he supply of labo n he make, denoed L, whch s pad a ae p pe hou, and unpad me devoed o consumpon acves. Moe specfcally, he daa of he economy nclude, fo =,,, a nonnegave coeffcen expessng he pe-un amoun of me equed by he consumpon of good of, so ha he consumpon of x uns of good eques spendng x uns of me, n addon o spendng p x dollas. Ib).. We he consume opmzaon poblem ha yelds he demand fo goods x p, p,..., p ), =,..., and he supply of labo L p, p,..., p ) agan, we ake he paamees m and ω as gven). Assume ha he soluon exss.) max β x,..., x ) subec o x,..., x, L he budge consan: px +... + px pl+ m and he me consan x +... + x + L ω. ) Ib).. Ague ha he me endowmen ω s oally spen. I s he usual agumen fo Walas Law, gven he assumed local nonsaaon. If x +... + x + L<ω, hen he consume could ncease he lef hand sde of he budge consan by sellng moe labo. Ib).3. Combne he budge consan and he me consan no a sngle equaly consan nvolvng x,, x ), and nepe. Mulplyng he me consan by p, we ge p x+... + p x pω = pl whch added o he budge equaly yelds p + p ) x +... + p + p ) x = pω + m. Inepeaon. I s a budge consan fo goods,,, whee he compehensve pce of good has wo componens: he dec dolla pce p and he ndec cos p due o he make value of he me spen consumng good. Ib).4. Denoe a pce veco by π,..., π ), and a wealh magnude by w, and defne he Walasan demand funcon x% π,..., π, w),..., x% π,..., π, w)), as usual, by he soluon o he poblem max β x,..., x ) subec o π x = = w. Smlaly, defne he Hcksan demand
4 funcon h π,..., π, u),..., h π,..., π, u)) by he soluon o he poblem mn x subec π = o β x,..., x ) = u. Usng I.b)3, expess he funcons x p, p,..., p), =,..., and L p, p,..., p ) n ems of he Walasan demand funcons x% π,..., π, w), =,...,. We can we he agumen π n he Walasan demand funcon as he compehensve pce of good as specfed n Ib.3), =,,. Smlaly, we can we wealh as w = p ω + m. Theefoe: x p, p,..., p ) = x% p + p,..., p + p, pω + m), =,...,. Usng he me consan ), we can hen we Lp, p,..., p) =ω x% p+ p,..., p + p, pω + m). = b).5. Compue L x% π,..., π, w),..., x% π,..., π, w)). L = n ems of he devaves of he Walasan demand funcons x% x% k k + ω = = πk b).6. By usng he Slusky decomposon of he oal effecs L as he sum of a subsuon em nvolvng he devaves π wealh em nvolvng he devaves Slusky decomposon: % x % % x h x = x k k k w π π. ) of Walasan demand. k x% π k, k =,, ), we of Hcksan demand, and a, whch subsued no ) yelds
5 L x% x% = x k k = = k + ω πk x% x% = + ω π k x k k kk = = = = = k w x% = + ω + k k k kx = = = = k πk = k k = = = πk whee ) has been used. We may defne: x% L, SUBSTITUTIO EFFECT: WEALTH EFFECT: and hence = = k= πk x% L = L TOTAL EFFECT = SUBSTITUTIO EFFECT + WEALTH EFFECT k b).7. Assume ha goods,, ae nomal. Wha can you say abou he sgn of he subsuon and wealh effecs obaned n b).6? Wha can you say abou he sgn of he oal L effec? Compae wh a) above. Because π k = e πk π, he, k) eny n he negave semdefne max D e e s he expendue funcon assocaed wh he Hcksan demand funcons), he subsuon effec 3) can be wen -,..., ) D e,..., ), nonnegave. L x% Because goods,, ae nomal, > 0, and heefoe he wealh effec = x% s posve. Hence, he subsuon and wealh effecs have weakly) oppose sgns, and he sum can n pncple be posve, negave o zeo.
6 The suaon dsplays a song paallelsm wh ha n he model of secon Ia) above. The modelng of he labo supply s que dffeen, ye he Slusky equaons ae que smla: n boh cases, he subsuon and wealh effecs move n oppose decons. The man dffeence s ha, n he model of Ia), lesue.e., me no sold n he make) has dec uly, wheeas n ha of secon Ib) s uly s ndec, n wha enables he consume o consume goods,..,. Pa II ASWER KEY Two goods, good and good, whch s he numeae good. Thee ae I consumes. Fo =,, I, consume s uly funcon s u : R + R R : u x, x) = b x) + x, whee b s dffeenable, nceasng and scly concave. All consumes ae pce akes. We assume ha he pce of he numeae good s equal o, and we denoe by p he pce of good. Denoe by x% ) p consume s Walasan demand fo good. Thee s a sngle fm whch poduces good by usng he numeae as an npu, wh cos funcon Cy), whee y s he amoun of good poduced. We assume n wha follows ha fs ode equales chaaceze he soluon o evey opmzaon poblem. Gven a pce p, we defne he makup as a he amoun of oupu equal o aggegae demand. p C ' p, whee he magnal cos C s evaluaed IIa). We he fs ode condon of he consume opmzaon poblem ha yelds he Walasan demand fo good. The poblem max b x) + x + w px whee w s he wealh of he consume) yelds he fs-ode condon db p dx =. IIb). Suppose ha pces ae egulaed n ode o maxmze he sum of consume suplus and pofs. Wha can you say abou he esulng makup? I We X p) x % p). By defnon, s consume suplus s b x% p)) px% p), and he = fm s pofs ae px p) C X p)). Hence, he sum of he consume supluses and pofs s
7 b x% p)) px p) + px p) C X p)), I = I db dx% wh fs ode equaly C' X' = 0, whch usng IIa), can be wen = dx dp px ' C ' X ' =0. Dvdng hough by dx dp, yelds E p = C '. e., he maxmzaon of he sum of consume supluses and pofs mples he equaly of pces and magnal coss, and, hence, zeo makup. As we know, hs s a condon fo economc effcency. IIc). Le consume own a shae θ 0 n he pofs of he fm =,, I, ). I θ = = As a consume, she buys he good n he make, whee she s a pce ake, bu she can voe a he shaeholdes meeng on he pce ha he fm wll chage. Wha s he bes pce fo consumeshaeholde? Consume-shaeholde solves he poblem max b x% p)) px% p) + w +θ[ px p) C X p))], wh fs-ode equaly p o, usng IIa), db dx % dx p % x% +θ [ px' + X C' X'] = 0, dp dp dp x% =θ [ X + p C') X'. ) ] IId). Suppose ha, a he shaeholdes meeng, all shaeholdes unanmously agee on a pce. Wha can you say abou he esulng makup? Wha can you say abou he shae θ of consume? Inepe. Hn. Add up he FOC. Addng up ) fo =,,I, we oban x X = X + p C') X',. e., p C, and zeo makup. Theefoe, ) becomes x% =θx, o θ = %,. X e., he pof shaes mus be equal o he shaes n consumpon. Inuvely, he pce has wo effecs on consume s welfae. As a consume, she wll pefe a vey low pce zeo), bu as a shaeholde she would pefe he monopoly pce. Hence, a
8 bg consume who s a small shaeholde pefes low pces, wheeas a bg shaeholde who s a small consume pefes hgh pces. When he shaes n pofs equal he shaes n consumpon, hs wo effecs balance ou n a way ha all shaeholdes pefe he suplus maxmzng pce. IIe). We now specalze he model o a vey smple case, whee b x ax x ) = ), =,, I, and Cy) = cy, whee a > c. Bu we assume ha only a facon σ of he populaon of I consumes ae shaeholdes n he fm, each ownng a shae IIe).. Compue he monopoly pof-maxmzng pce p M. θ= n he fm s pofs. σ I Indvdual demand s gven by p= a x,. e., x% ) p = a p, and aggegae demand by X p) = I a p). The monopoly pof maxmzng poblem s max p c p ) I a p ) wh FOC: a p) p c) =0,. e., a p+ c = 0, o: M a+ c p =. IIe).. Show ha all shaeholdes agee on a pce pσ), and compue. How does pσ) vay wh σ? Wha ae he lms of pσ) as σ 0? As σ? Commen. Fom ), we can we he FOC fo a shaeholde as a p = [ I a p) + p c) I)]. σi Aggegang ove he σi shaeholdes, we oban σi[ a p] = I a p) + p c) I)],. e., σa σ p= a p p+c, o: p σ ) = σ ) a+ c,. e., σ ) a+ c p σ ) =. σ dp σ) a σ ) + σ ) a+ c a+ aσ+ a σ a+ c a+ c We compue: = = = 0 dσ [ σ] [ σ] [ σ] <, a+ c M lm σ 0 p σ ) = = p,
a) Le F be he c.d.f.. Then Answe key fo Q5 Mco Pelm June 09 0 f p > f p and p > D p p f p p, ) = = F p p) + [ F p p ] f p p ) < [ F p p ] f p p ) < 0 f p > f p and p > D p p f p p, ) = = F p p ) + [ F p p ] f p p ) < [ F p p ] f p p ) < b) Le p = 0. Then π p,0) = p D p ). Thus π 0,0) = 0 = 60. Consde a lowe pce, e.g. 00. Then π 00,0) = 00 F0) + F0) ) = 000.7 ) = 70 > 60. Thus 0,0) s no a ash equlbum. c) Fx a fm. If he ohe fm chages, p = yelds a pof of. Fo hs o be a ash equlbum s necessay and suffcen ha fm canno ncease s pofs by choosng a pce p <. If he fm chages p < hen s pofs wll be: p F p ) + [ F p )]p Thus we need p F p ) + [ F p )]p fo all p ha s, p [ + F p )] fo all p..) Le us dop he subscp and defne he RHS of.) as gp). Thus g p) = p + pf p). Then g0) = 0 and g) =. Fuhemoe, g p) = + F p) pf p). Thus g 0) = + F ) > 0. If gp) s convex hen g p) s non-deceasng and gp) looks lke gp) and heefoe.) s sasfed and,) s a ash equlbum. p 0
d) If he funcon gp) s concave hen hee ae wo possbles. CASE : g ) 0. Then gp) looks lke gp) and heefoe.) s sasfed, hence,) s a ash equlbum. 0 CASE : g ) < 0 hen gp) looks lke gp) p In hs case hee s a p 0,) such ha gp) > and heefoe.) s volaed and,) s no a ash equlbum. p 0 Thus a necessay and suffcen condon fo,) o be a ash equlbum s g ) 0. Snce g ) = f 0) he condon can also be wen as f 0) e) If f s consan, hen mus be fx) = fo all x. Then Fx) = x so ha he funcon gp) of pa ) becomes g p) p p = +. Thus g p) =,.e. gp) s concave. Hence, by he esuls of pa ),,) s a ash equlbum f and only f f 0) whch s of couse ue. So,) s a ash equlbum n hs case. f) Assume ha Fx) < fo suffcenly small x. Then f fm chages 0, fm ges zeo pofs f also chages 0, bu posve pofs f chages a lle b moe han zeo s demand s posve snce Fx) < fo x small). Thus p = p = 0 s no a ash equlbum. g) Inuvely, he Beand paadox coesponds o he case whee all he mass s concenaed a 0. One mgh be able o show he Beand paadox as a lm esul: consde a famly f of densy funcons such ha, as, he smalles x a whch F x) = ends o zeo. Then he ash equlbum mgh end o zeo.