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1 Inoducoy Mahs ouse Answes.doc Answes o Tuoal Quesons Enjoy wokng hough he examples. If hee ae any moe quesons, please don hesae o conac me. Bes of luck fo he exam and beyond, I hope you won need. Tuoal :. 6x 4x. ( x ). x 4. ( x x e ( 4x ) 5. x ln () 6. f (x) 4x - 6 x /, f (/) 4 > / s mnmum 7. f (x) 48x - x /48, f (/48) 48 > mnmum 8. x x - (x ) (x - ) x -, x 9. x - x - 6 x - x - (x ) (x - 4) x -, x 4. 7x x - 6, x / ±, x, x P Q P ε Q P Q b bp ( ) a bp bp a Un elascy: bp a bp P Q a bp b heck: b a b a ( ). AT T ( y ) AV V ( y ),, y y AF F y. M dt ( y ) dv( y) V'( y) dy dy 4. π '( y) p M( y) π ''( y) M'( y) < (b) p M 5. π ( y) py T( y) π '( y) 6 y 5y 4 y / y y 96 ± 4 84 ± y, y 8 π ''( y) y 5, π ''( ) (max), π ''( 8) (mn)

2 Inoducoy Mahs ouse Answes.doc check: π (4) - 5 < ( ) 6. TR (y) P(y) y π (y) TR (y) - T (y) 7. MR M 8. π (Q) Q -.Q - Q -.5Q Q 8Q 8 π '( Q) Q 8 Q 4 9. π (Q) Q Q Q Q 5Q 45 Q 8Q 8Q 45 π (Q) -Q 6Q - 8 Q -6Q 8 (Q - 4) (Q - ) Q 4, Q, π (4) - < Q 4 s maxmum, π () > Q s mnmum.. π dm π αm εm, π ( α e ) απ m dπ m (b) απ π α

3 Inoducoy Mahs ouse Answes.doc Tuoal :. -8x - c. 4x x f '( x) dx dx x x f ( x) dx f x x ( ( ) ) ln( x ) c. e x c 4. e x 4 ln x c 5. ( x 7) e c 6. 6 x x( x 4 6) dx x 6x dx x x x dx x dx [ x ] 6 9. lef hand sde: α(θx o ( - θ)x ) β gh hand sde: θ(αx β) ( - θ) (αx β) θαx θβ αx β - θαx - θβ α(θx x - θx ) β l.h.s. concave and convex..h.s. l.h.s. concave and convex. [θx ( - θ)x ] - θx ( - θ)x θ x θ( - θ)x x ( - θ) x - θx - ( - θ)x -( - θ)θx - θ( - θ)x θ( - θ) x x -( - θ)θ (x x - x x ) -( - θ)θ (x - x ) < scly convex. As above bu oppose sgn scly concave F f '( y) y. F F F ( y y) < y y y ( y y) < y y y ( ue) 4. F f ''( y) > scly convex y 5. f (x) x, f (x) 6x scly convex fo x>, scly concave fo x < 6. f (x) 9x 4x, f (x) 8x 4 > fo x > 9 (convex), x< 9 (concave). 7. U '( x) x, U ''( x) x < scly concave 4 8. f '( L) L, f ''( L) L > scly convex 4

4 Inoducoy Mahs ouse Answes.doc Tuoal :. (x 4 ) x. a b a c no smplfcaon possble. y - y y - 4. x a x b x c x (abc) 5. 6 x x 8 x 6. log, 4 7. log log log ln e. log e e log () 4 8. ln Ae ln A 4. (hang, p. 465/4a) PV $. 5 e d. ( e e ) $ (hang, p. 465/4b) PV $. 4 e d. ( e e ) $ (hang, p. 465/5b) y y PV $46e d lm e d e y $46 lm $ $46 ( ) $75 y (Hands, p. 7, 8.): (equaon 6) PV 8 ( (. ) ) (9.88) $ (Hands, p. 7, 9.): 9 ( ( 6. ) ) PV 6%. 6 ( 6. ) PV 8% 9 ( ( 8. ) ) ( 8. ) 9. (Hands, p. 9, 5.): In connuous me we have V Ae T n dscee me we have V A() T afe T peods, snce V and A ae he same n boh cases, we have o solve () T e T () e, so.7 mples (Hands, p., 9.): Pof s, as usual, gven as evenue mnus cos. Revenues ae $R n each peod, hus he pesen value of all evenues ove he machnes lfeme ae gven as oss ae smply (T). The funcon fo he pesen value of pofs s hen T R Pof maxmsaon mples d π ( T) Re d ( T) hus be fomulaed as T - (T) [ ] 4 T Re d. e R e T - (T) π ( T ) (T s he only choce vaable). The equesed ule can dt Re -T - (T). The opmal lfe of he machne s whee he pesen value of nceased duably s equal o he magnal cos of duably.

5 Inoducoy Mahs ouse Answes.doc Tuoal 4:. 5x f, f 5x, f f 5 ( x x) ( x x ) ( x x) ( x x). f x x - 4x - y, f y -(x - ), f xy f yx -.. MPP K Q K 4. L 8. 8 K. 7, MPP L Q L K 67. L MU (x )(x ), MU (x ) (x ) (b) MU (, ) (5)(6) dz (6x y) dx (x - 6y ) dy 6. du ( 9x ) dx (9x x ) dx 7. x x dy dx dx ( x x ) ( x x ) 8. dy x x dx ( x x ) ( x x ) dx.. Boh ae funcons of K and L. 9. U x, U x, U U, U U, MRS x x. U x, U x, U x 4. U x, U U, MRS x x. MRTS y K L. yl K. Yes, a b. No 4. Yes, a b. Q 5. K K Q L L Q 6. The Eule heoem n 5 can be expessed as MPP K K MPP L L Q. Gven APP L Q/L, and APP K Q/K, and (b) follow by smple eaangng. 7. We have consan euns o scale, when oupu nceases by he same faco as he npus,.e. λq f(λk, λl). We have nceasng euns o scale when oupu nceases by moe han λ, and deceasng euns o scale when nceases by less han λ. A(λK) α (λl) β λ (αβ) AK α L β λ (αβ) Q, whch mples and (b) (b) (c) The paal elascy s defned as ε Q K Q, K K Q fo capal, wh a coespondng defnon fo labou ε Q, L. alculang hose elasces shows ha ε Q, K α, ε Q, L β. 4 5

6 Inoducoy Mahs ouse Answes.doc Tuoal 5:. -P P α P* 5 - α/; Q* 5 - α/ α 5 α/. P* > : 5 - α/ > α < ; Q* > : 5 α/ > α > - - < α <.. Inseng he equlbum values Y* (D, M S, p) and *(D, M S, p) no he IS and LM cuves, whch become hen I(*(D, M S, p))-s(y* (D, M S, p), *(D, M S, p)) D, L(Y* (D, M S, p), *(D, M S, p)) - M. p Dffeenang hese denes w... p yelds Y I Sy S p p p L Y S Y M L p p p S Thee ae wo unknowns n hs equaon sysem, whch ae p and Y. The soluon can hus p be calculaed and expessed n ems of he paamees I, S Y, S, L Y, L, M S and p. y. Expessng L n ems of y s L. The pof funcon s hen gven as 6 π ( y) py w y vk. d π wy mples p 8p y 6 d y 8 w p hus L f ( y ). (f - sands fo he nvese of f). w T y w y ( ) vk 6 (b) AT T w y vk, V s he pa of T whch s ndependen of y,.e. V w y y 6 y 6. AV V w y dt, M w y y 6 dy 8. π p[ln ( L) ln ( K)] - wl - vk π L pl w L p w π K pk v K p v (b) The SO ae π LL <, π KK <, π LL π KK - π LK >. p p π LL <, π KK < ( L) ( K) π π π LL KK LK p ( L) ( K) > 6

7 Inoducoy Mahs ouse Answes.doc (c) L*: λ p p L λ,.e. L* s homogeneous of degee. The same s easly shown λ w w fo K*. (d) ompaave sac analyss: L p L L K K p K,,,,,. w w v p w w v v p v π TR(y, R) - T(y, R) π R y 4 y y, R 4 π y 6 R R (b) π yy - <, π RR - <, π yy π RR - π yr - 6 >,.e. he SO hold. The fs sep s o fnd he pof funcons fo he wo fms, whch ae gven by evenue mnus coss. Revenues ae pce mes quany (oupu), hus we have P - (q q ) a π Pq - (q ) -q - q q aq - bq - c -q - q q aq - bq - c π Pof maxmsaon mples seng he devave of π w... he especve npu equal o zeo, assumng q (ouno conjecue). q π -q - q a - b q q -q a - b (*) π -q - q a - b q 8. q -q a - b usng (*) fo q q -(-q a - b) a - b a b q Dong he same fo q yelds a b q (b) Q q q ( a b), P -Q a a (c) Ths eques o dffeenae he ouno equlbum oupus q and q w... he paamee b: q, q. b b 7

8 Inoducoy Mahs ouse Answes.doc (b) (c) (d) The fs sep s o fnd he pof funcon fo each fm. Revenues ae gven as Pq j. Wh P -Q a, we need o deemne Q befoe, whch s q. The pof funcon fo fm j s hen q q a bq j c. n π j j π max s found by seng π j q. Ths devave s ease found by exacng q j fom he expesson n backes o ge n π j q j q a q j bqj c, and hus, j j π q qj a b. Gven ha he equlbum s symmec, he expesson q j j ( a b) smplfes o (n ) q j (a - b), o q j ( n ). oss and Pces ae deemned n he same way,.e. he paamees a and b have he same meanng. Wh n he answe n s conssen wh he pevous queson s answe. The cos funcon fo he monopols s he same. The monopols faces he whole make demand hough, so he pof funcon s π M (-Q a)q - bq - c. π M ( a b) Q a b QM. Ths s a specal case of he geneal esul, whee Q n. n( a b) Toal ouno Oupu s nq j. Fo n appoachng nfny, he lm s gven as ( n ) n lm ( a b ) n ( a b) lm a b. n ( n ) n ( n ) In pefec compeon p mc, whch s n hs case -Q a b,.e. Q a - b, whch s he same oupu as deved above. n 8

9 Inoducoy Mahs ouse Answes.doc Tuoal 6:. The Lagangan s gven as L(x, x, λ) ln x ln x λ (M - p x - p x ) The FO ae L λ p x x. L x λ p x L M px px λ p Dvdng he fs by he second FO and eaangng yelds x p x, whch can be subsued M M fo x n he hd FO o ge x and coespondngly x. p p L(x, x, x, λ) a ln x a ln x a ln x λ (M - p x - p x - p x ). The FO ae gven as. (b) (b) L x a λ p,,, x L M px p x px λ Ma whch yelds x,,,. p x p Ma x x a, ( j),. p p M p j The Lagange funcon fo hs poblem s 4 L( e,, s, λ) e s λ( e s) The FO ae L L e s λ, e s λ, e L L e s λ, 4 e s. s λ Dvdng he fs by he second and eaangng o e Dvdng he second by he hd and eaangng o s.5 Subsung e and s n he fouh by he above esuls n 5 5h mnues. e and s can be calculaed fom above o ge e h 4 mnues, and s 8h. The Lagange mulple measues appoxmaely he effec of a change n he consan. The consan was hheo 4h, now s effecvely h. We can deve λ fom ehe of he fs hee FO, e.g. n he fs one λ e s 9

10 Inoducoy Mahs ouse Answes.doc 4. (c) (d) alculang λ fo he values above yelds λ Ths, mes (-), s he appoxmae change n he value of he objecve funcon, whch s he oal uly. The eason fo λ only expessng an appoxmae change s ha λ self changes when T changes. T has changed fom 4 o,.e. calculang λ fo T 4 esuls n a small eo. Seng up he Lagange funcon agan and usng T nsead of T4 n he consan, follows ha he new opmal allocaons ae e h 5h 6 4 s 7h 4 Toal uly s obaned by nseng he opmal values fo e, and s no he objecve funcon. Whou he uoal (4h) oal uly s.94586, wh he uoal (h) s.46494,.e. he change s The appoxmaon n (b) was heefoe faly close. The Lagange funcon s Z K wl λ( Q K L ) Z K λ K Z λ, w L, L Z Q K L. λ Ths solves o K Q w L Q, w w (b) solvng he fs of he FO: λ K. Inseng K* gves he desed esul. λ* epesens he magnal change n cos whch esuls fom a small elaxaon n he poducon funcon consan. Theefoe λ* M. (c) Pof maxmsaon mples P M. Fom (b) we know ha M λ* and hus p λ*. Subsung p fo λ* n (b) and solvng fo Q gves he expesson n he queson.

11 Inoducoy Mahs ouse Answes.doc Tuoal 7: z z (b) z (c) f(x) f(x) f(x). x* x x* x x* (b) x* f (x*) > (b) (c) > omplemenay slackness condon: If x* > (slack) hen f (x*) (bndng); If f (x*) > (slack) hen x* (bndng). (c) f (x*) and x* f (x*), x. x. The pocedue s o seach fs fo an unesced mnmum. If hs mnmum s whn he doman ( ), hen we can accep, and we have f (x*) and x*> o x*, dependen on wha he soluon was. If, howeve, we fnd he unesced x* o be negave, we canno accep. We have hen a bounday soluon wh x* and f (x*) >. The funcon n (d) has mnma (-, ). We have o be caeful when accepng he soluon whn he doman. I could be he case ha we sll have a bounday soluon. In hs case, howeve, we have a maxmum a x, so he mnmum a x s he soluon. An mpoan pon whch was made n he uoal s ha he mnmum we fnd wh hs appoach s no necessaly a global one. In case (b), fo example, mgh well be he case ha f(x) < f(x*) fo hghe levels of x. Ths, howeve, apples smlaly o unconsaned opmsaon poblems. x* f (x*) (b) > (Noe: lm f ( x) x s no a global mnmum. (c) > (d) > x Smulaneously slack s no possble (complemenay slackness condon); Boh zeo s possble, as n.. 4. M : Tangen soluon on I 4 ; M : Bounday soluon (I ), x. (b) x (c) p x p x M Gven n Hands, p. 48, 9.4 (b) Hands, p. 48, 9.6a - 9.6g

12 Inoducoy Mahs ouse Answes.doc (c) Takng condons 9.6cd, x * > mples U p λ*. Snce U and p ae boh scly posve, λ* mus be scly posve as well. Ths, n un, mples fom 9.6f ha he budge consan holds as an equaly,.e. he soluon s on he budge lne. We can fuhe say ha U λ λ p and U, whch aken ogehe mples p U U p U. p p p U Ths means ha he MRS mus be smalle o equal he pce ao. 5. U U > mples λ* >, so he budge consan holds as an equaly (s bndng). Ths means ha x and x can no boh be zeo. Wh λ* > and p > p we have p λ* > p λ*. Snce U U we can subac U on he lef hand sde and U on he gh hand sde o ge p λ* - U > p λ* - U. Mulplyng by (-), whch uns he > no < yelds U - p λ*- < U - p λ* (**) x * > mples (9.6d) ha he lef hand sde of (**) s equal zeo and hence he gh hand sde mus be geae han zeo, mplyng (9.6b) x * o be zeo. Ths conadcs p < p, because he uly s he same fo boh. x * can hus no be geae han zeo. If, on he ohe hand, x * >, hs mples (9.6b) ha he gh hand sde of (**) s equal o zeo, and hence he lef hand sde mus be smalle han zeo, mplyng (9.6d) x *. Ths s he only possble soluon.

13 Inoducoy Mahs ouse Answes.doc Tuoal 8:. A geneal pon woh emembeng s ha a consume maxmsng uly facng only a budge consan (hee s a consan on he avalably of he good) wll always spend all he ncome f he magnal uly fom a leas one good s posve - he budge consan wll bnd. The only way he whole budge wll no be spen s f he magnal uly deved fom consumpon eaches zeo befoe all goods ae consumed. So one possbly s o smply wok wh he uly funcon and see a wha pon magnal uly s equal o zeo. The uly funcon s symmec fo he wo consumes,.e. Y and Y have he same wegh. An ncease n Y would be allocaed half and half o each consume. Wh ha knowledge, we can epesen consume s uly funcon as U Y - ky. The magnal uly s hen gven as MU - ky Y k. Takng boh consumes ogehe, wh Y beng equally dsbued amongs he wo, Y k. Tha means ha an ncease n Y o levels above would no ncease he consumes uly, he k magnal uly s zeo. Pung hs n he conex of he Lagange mulple, whch expesses he magnal uly of he consan, he consan would no be bndng fo Y >, and ha would k mean ha he Lagange mulple s equal zeo. An alenave appoach s o wok wh he Kuhn-Tucke condons: Noe ha hee s no nonnegavy consan on Y and Y,.e. he equaly FOs apply - as long as we ge sensble esuls. To be on he save sde, howeve, he queson can be solved wh he full Kuhn-Tucke condons as well. () L U U λ(y - Y - Y ) Y Y - k(y Y ) λ(y - Y - Y ) () L ky λ Y () L ky λ Y (4) L λ λ L Y Y Y ; ; and. λ λ () and () λ Y Y Y Y λ k k If λ * > we fnd a conadcon fo Y,.e. hs doesn help us o answe he queson. k If λ, subsue n above equaon: Y Y k. Y Y k, f Y > k hen consan s slack.

14 Inoducoy Mahs ouse Answes.doc As menoned above, magnal uly mus be equal o zeo a he opmum f he consan s slack. So he consumes ae so envous of each ohe ha f each s gven an exa un of he good, he dsuly fom he ohes addonal consumpon ouweghs he uly fom he own consumpon. In hs case s opmal o how away some of he good.. (b) The Lagangan fo he poblem s gven as L(x, x, λ) (x ) (x ) λ(m - p x - p x ) (x x x x ) λ(m - p x - p x ). The Kuhn-Tucke condons ae () L x * - λ*p () x *L x *( x * - λ*p ) () L x * - λ*p (v) x *L x *( x * - λ*p ) (v) L λ M - p x - p x (v) λ*l λ λ*( M - p x - p x ) (v) λ*, x *, x *. ase : x *>, x * Fom (), () and x *: λ p ase : x *, x *> M >. Thus fom (v) and (v) x. p Fom () and (v): λ M. Thus fom (v) and (v) x. p p ase : x *>, x *> () and () ae bndng, λ* >, (v) bndng, and hus we have hee bndng equaons fo he hee unknowns x *, x * and λ*, esulng n ( M p p ) ( M p p ) x, x p p ase : (), (), (v), (v) and (v) hold, () depends on he paamees. Ths mples he escon ha M p < p. ase : (), (), (v), (v) and (v) hold, () depends on he paamees. Ths mples he escon ha M p < p.. L (x x x x ) λ ( - x - x ) λ ( - x - x ) ( ) L x λ λ ; ( ) x ( x λ λ ) ; ( ) L x λ λ ; ( v) x ( x λ λ ) ; ( v) L x x ; λ ( v) λ ( x x ) ; ( v) L x x ; λ ( v) λ ( x x ) ; ( x) x ; x ; λ ; λ. 4

15 Inoducoy Mahs ouse Answes.doc (b) ase : Nehe consan bnds, mplyng boh λ ; ase : Only me consan bnds, mplyng λ >, λ ; ase : Only budge consan bnds, mplyng λ >, λ ; ase 4: Boh bnd, mplyng λ >, λ >. (c) ase : () x *, whch s a conadcon. ase : (v) bndng - x * - x * x * - x *. Usng hs expesson fo x * n (v): - x * - ( - x *) x * 8,.e. () s bndng: x * λ * -. Fom () we have λ * 9, and heefoe x * 8, whch conadcs (v). ase : (v) s bndng, x * - x *, o x * 6 -.5x *. (v) - (-x *) - x * x *,.e. () s bndng. If x *, hen x * 6 and λ *.5, whch conadcs (),.e. x * > and () bnds. Wh (), () and (v) bndng we can solve fo he hee unknowns. The soluon s hen gven as x * 6.5, x *.75, λ *, λ *.75, U* 8.5. ase 4: (v) and (v) ae bndng: x * - x *, - ( - x *) - x *, x *, x * 8,.e. () and () ae bndng, whch solves fo a negave value of λ *, so case 4 s a conadcon as well. (d) λ epesens he magnal uly of me, λ epesens he magnal uly of ncome. The budge consan bnds, s assocaed Lagange mulple s scly posve, so he magnal uly of ncome s posve and addonal ncome wll make he consume bee off. The me consan s slack, s assocaed Lagange mulple s zeo. Addonal lesue me wll no make he consume any bee off - hee s aleady suffcen me o spend all ncome. Fuhe lesue me s only of use f combned wh an ncease n ncome. (e) Followng fom he logc n (d), f he govenmen decdes o levy a new ax on ncome bu uses he evenue o mpove he oad sysem, he consume wll be wose off. The ax wll make he budge consan bnd soone, educng aanable consumpon and hence uly. The mpoved oad sysem wll cu he jouney me o and fom wok and wll heefoe esul n a gan n lesue me. Ths addonal lesue me s howeve wohless as explaned above and canno compensae fo he loss of ncome. 4. U > mples λ*, whch mples λ* >, so M p x * p x *. Bu λ* > and U < mples (U - λ*p ) < and hus x *. Fom he budge consan we have x * > as would be expeced gven he assumpons. 5

16 Inoducoy Mahs ouse Answes.doc Tuoal 9: ln. U ln ( ) The Lagangan s gven as ln L(,, λ) ln ( ) λ Y Y ( ) ( ) (b) (c) (d) () L λ, λ () L, ( ) ( ) Y () L λ Y ( ) ( ). Dvdng () by () esuls n. Reaangng ha and elmnae fom he consan: Y Y Y Y We can calculae he pesen value of * as Y Y. ( ) Fo half of he p.d.v. of ncome s spen n each peod. The expesson fo savngs s S Y - *, and hus Y S Y Y. The compaave sac analyss fo a change n nees s S Y > ( ) We can oally dffeenae he expesson fo * o ge d Y dy Y dy, whee dy dy dy. We can hen fnd he soluon as d dy.. (b) The answe s nuvely * *.5, because pces n boh peods ae he same. Ths s confmed hough he usual Lagange se-up: L λ ( - - ). (c) The new neempoal budge consan s gven as ncome () plus nees on savngs: n( - ). L λ ( n( - ) - - ) The new soluon s hen **.5, **.5 ( n). (d) Indvduals ae bee off afe he noducon of he socal secuy sysem: U( *, *).5, U( **, **) ( n).5. 6

17 Inoducoy Mahs ouse Answes.doc 7 (e) In any gven peod, oal pad n s SN Y, oal pad ou s ( n)sn O. Snce N Y ( n)n O, he scheme s self-fnancng and wll no need o be subsdsed fom any ohe souce of evenue.. u The Lagange funcon fo hs poblem s L Y λ ( ) ( ) The FOs ae: () λ L () λ L () λ L ( ) ( ) (v) λ L Y ( ). Dvdng () by (): Dvdng () by () esuls n: Subsung and n (v) by he expessons above: Y ( ) ( ) The lef hand sde s he pesen value of ncome, PV(Y), he expesson can be eaanged o ( ) PV Y ( ) ( ) And solvng fo :

18 Inoducoy Mahs ouse Answes.doc 8 PV Y ( ) ( ) o smplfed: PV Y β β ( ), ( ) PV Y β ( ), PV Y β ( ). (b) MP β (c) Y (Y - )( ) Y.

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