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1 Full file a hps://frasockeu SOLUTIONS TO CHAPTER 2 Problem 2 (a) The firm's problem is o choose he quaniies of capial, K, and effecive labor, AL, in order o minimize coss, wal + rk, subjec o he producion funcion, Y = ALf(k) Se up he Lagrangian: () L wal rk Y ALf (K AL) The firs-order condiion for K is given by L (2) r ALf (K AL)( AL) 0, K which implies ha (3) r f (k) The firs-order condiion for AL is given by L (4) w f (K AL) ALf 2 (K AL)( K) /(AL) 0, (AL) which implies ha (5) w f (k) kf (k) Dividing equaion (3) by equaion (5) gives us r f ( k) (6) w f ( k) kf ( k) Equaion (6) implicily defines he cos-minimizing choice of k Clearly his choice does no depend upon he level of oupu, Y Noe ha equaion (6) is he sandard cos-minimizing condiion: he raio of he marginal cos of he wo inpus, capial and effecive labor, mus equal he raio of he marginal producs of he wo inpus (b) Since, as shown in par (a), each firm chooses he same value of k and since we are old ha each firm has he same value of A, we can wrie he oal amoun produced by he N cos-minimizing firms as N N N (7) i ALif (k) Af (k) Li ALf (k) i i i where L is he oal amoun of labor employed Y, The single firm also has he same value of A and would choose he same value of k; he choice of k does no depend on Y Thus if i used all of he labor employed by he N cos-minimizing firms, L, he single firm would produce Y = ALf(k) This is exacly he same amoun of oupu produced in oal by he N cos-minimizing firms Problem 22 (a) The individual's problem is o maximize lifeime uiliy given by () U C C 2, subjec o he lifeime budge consrain given by (2) P C + P 2 C 2 = W, where W represens lifeime income Rearrange he budge consrain o solve for C 2 : (3) C 2 = W/P 2 - C P /P 2 Subsiue equaion (3) ino equaion ():

2 2-2 Full file a hps://frasockeu Soluions o Chaper 2 (4) U C W P C P P 2 2 Now we can solve he unconsrained problem of maximizing uiliy, as given by equaion (4), wih respec o firs period consumpion, C The firs-order condiion is given by (5) U C C C2 P P2 0 Solving for C gives us (6) C P P2 C 2, or simply (7) C P2 P C2 In order o solve for C 2, subsiue equaion (7) ino equaion (3): (8) C W P P P C P P2 Simplifying yields ( ) (9) C2[ P2 P ] W P2 or simply (0) C2 W P2 ( ) 2 P P Finally, o ge he opimal choice of C, subsiue equaion (0) ino equaion (7): () C P2 P W P2 ( ) P P 2 (b) From equaion (7), he opimal raio of firs-period o second-period consumpion is (2) C C P P, 2 2 Taking he naural logarihm of boh sides of equaion (2) yields (3) ln C C2 ln ln P2 P The elasiciy of subsiuion beween C and C 2, defined in such a way ha i is posiive, is given by (4) C P C 2 P 2 P C P 2 C 2 ln C ln P C 2 P 2, where we have used equaion (3) o find he derivaive Thus higher values of imply ha he individual is less willing o subsiue consumpion beween periods Problem 23 (a) We can use analysis similar o he inuiive derivaion of he Euler equaion in Secion 22 of he ex Think of he household's consumpion a wo momens of ime Specifically, consider a shor (formally infiniesimal) period of ime from ( 0 - ) o ( 0 + ) Imagine he household reducing consumpion per uni of effecive labor, c, a ( 0 - ) an insan before he confiscaion of wealh by a small (again, infiniesimal) amoun c I hen invess his addiional saving and consumes he proceeds a ( 0 + ) If he household is opimizing, he marginal impac of his change on lifeime uiliy mus be zero

3 Full file a hps://frasockeu Soluions o Chaper This experimen would have a uiliy cos of u '(c before )c Ordinarily, since he insananeous rae of [ r( ) ng] reurn is r(), c a ime ( 0 + ) could be increased by e c Bu here, half of ha increase will be confiscaed Thus he uiliy benefi would be [ ] ( ) [ r ( 2 ) ng u cafer e ] c Thus for he pah of consumpion o be uiliy-maximizing, i mus saisfy r ng () u( cbefore ) c u( cafer ) e [ ( ) ] c 2 Raher informally, we can cancel he c's and allow 0, leaving us wih (2) u( c before ) u( cafer ) 2 Thus here will be a disconinuous jump in consumpion a he ime of he confiscaion of wealh Specifically, consumpion will jump down Inuiively, he household's consumpion will be high before 0 because i will have an incenive no o save so as o avoid he wealh confiscaion (b) In his case, from he viewpoin of an individual household, is acions will no affec he amoun of wealh ha is confiscaed For an individual household, essenially a predeermined amoun of wealh will be confiscaed a ime 0 and hus he household's opimizaion and is choice of consumpion pah would ake his ino accoun The household would sill prefer o smooh consumpion over ime and here will no be a disconinuous jump in consumpion a ime 0 Problem 24 We need o solve he household's problem assuming log uiliy and in per capia erms raher han in unis of effecive labor The household's problem is o maximize lifeime uiliy subjec o he budge consrain Tha is, is problem is o maximize L( ) () U e ln C ( ) H d, 0 subjec o L R( ) ( ) (2) e C H d ( 0) R( ) L( ) ( ) e A( ) w( ) KH H d 0 0 ( 0) R( ) Now le W K H 0 e L( ) A( ) w( ) H d We can use he informal mehod, presened in he ex, for solving his ype of problem Se up he Lagrangian: L() (3) R() L() L e ln C() d W e C() d H H 0 0 The firs-order condiion is given by L L() R() L() (4) e C() e 0 C() H H Canceling he L()/H yields R( ) (5) e C( ) e, which implies (6) C() = e - - e R() Subsiuing equaion (6) ino he budge consrain given by equaion (2) leaves us wih

4 2-4 Full file a hps://frasockeu Soluions o Chaper 2 (7) R R e e e ( ) ( ) L( ) H d W 0 Since L() = e n L(0), his implies L( 0) (8) ( n) e d W H 0 As long as - n > 0 (which i mus be), he inegral is equal o /( - n) and hus - is given by W (9) ( n) L( 0) H Subsiuing equaion (9) ino equaion (6) yields W R( ) (0) C( ) e n L( 0) H Iniial consumpion is herefore W () C( 0) n L( 0) H Noe ha C(0) is consumpion per person, W is wealh per household and L(0)/H is he number of people per household Thus W/[L(0)/H] is wealh per person This equaion says ha iniial consumpion per person is a consan fracion of iniial wealh per person, and ( - n) can be inerpreed as he marginal propensiy o consume ou of wealh Wih logarihmic uiliy, his propensiy o consume is independen of he pah of he real ineres rae Also noe ha he bigger is he household's discoun rae he more he household discouns he fuure he bigger is he fracion of wealh ha i iniially consumes Problem 25 The household's problem is o maximize lifeime uiliy subjec o he budge consrain Tha is, is problem is o maximize 0 () U e C( ) L( ) H d, subjec o r L( ) (2) e C ( ) H d W, 0 where W denoes he household's iniial wealh plus he presen value of is lifeime labor income, ie he righ-hand side of equaion (26) in he ex Noe ha he real ineres rae, r, is assumed o be consan We can use he informal mehod, presened in he ex, for solving his ype of problem Se up he Lagrangian: C() L() r L() (3) L e d W e C() d 0 H 0 H The firs-order condiion is given by L L() r L() (4) e C() e 0 C() H H Canceling he L()/H yields r (5) e C( ) e Differeniae boh sides of equaion (5) wih respec o ime:

5 Full file a hps://frasockeu Soluions o Chaper (6) e r C( ) C ( ) e C ( ) re 0 This can be rearranged o obain C ( ) (7) e C( ) e C( ) re r 0 C( ) Now subsiue he firs-order condiion, equaion (5), ino equaion (7): C ( ) r r r (8) e e re 0 C( ) Canceling he e r and solving for he growh rae of consumpion, C ( ) C( ), yields C ( ) r (9) C( ) Thus wih a consan real ineres rae, he growh rae of consumpion is a consan If r > ha is, if he rae ha he marke pays o defer consumpion exceeds he household's discoun rae consumpion will be rising over ime The value of deermines he magniude of consumpion growh if r exceeds A lower value of and hus a higher value of he elasiciy of subsiuion, / means ha consumpion growh will be higher for any given difference beween r and We now need o solve for he pah of C() Firs, noe ha equaion (9) can be rewrien as (0) ln C( ) r Inegrae equaion (0) forward from ime = 0 o ime = : ln C( ) ln C( 0) r, () which simplifies o ln C( ) C( 0) r (2) Taking he exponenial funcion of boh sides of equaion (2) yields (3) C C e r ( ) ( 0), and hus (4) C C e r ( ) ( 0) 0 We can now solve for iniial consumpion, C(0), by using he fac ha i mus be chosen o saisfy he household's budge consrain Subsiue equaion (4) ino equaion (2): r r L( ) (5) e C( 0) e H d W 0 Using he fac ha L( ) L( ) e n C( 0) L( 0) r( rn) (6) H 0 yields e d W 0 As long as [ - r + (r - n)]/ > 0, we can solve he inegral: r( rn) (7) e d 0 r ( r n) Subsiue equaion (7) ino equaion (6) and solve for C(0):

6 2-6 Full file a hps://frasockeu Soluions o Chaper 2 W ( r) (8) C( 0) ( r n) L( 0) H Finally, o ge an expression for consumpion a each insan in ime, subsiue equaion (8) ino equaion (4): r W ( r) (9) C( ) e ( r n) L( 0) H Problem 26 (a) The equaion describing he dynamics of he capial sock per uni of effecive labor is k ( ) f k( ) c( ) ( n g) k( ) () For a given k, he level of c ha implies k 0 is given by c = f(k) - (n + g)k Thus a fall in g makes he level of c consisen wih k 0 higher for a given k Tha is, he k 0 curve shifs up Inuiively, a lower g makes break-even invesmen lower a any given k and hus allows for more resources o be devoed o consumpion and sill mainain a given k Since (n + g)k falls proporionaely more a higher levels of k, he k 0 curve shifs up more a higher levels of k See he figure (b) The equaion describing he dynamics of consumpion per uni of effecive labor is given by c( ) f k( ) g (2) c( ) Thus he condiion required for c 0 is given by f ' (k) = + g Afer he fall in g, f ' (k) mus be lower in order for c 0 Since f '' (k) is negaive his means ha he k needed for c 0 herefore rises Thus he c 0 curve shifs o he righ (c) A he ime of he change in g, he value of k, he sock of capial per uni of effecive labor, is given by he hisory of he economy, and i canno change disconinuously I remains equal o he k* on he old balanced growh pah c c* NEW c 0 E NEW In conras, c, he rae a which households are consuming in unis of effecive labor, can jump a he ime of he shock In order for he economy o reach he new balanced growh pah, c mus jump a he insan of he change so ha he economy is on he new saddle pah c* E k 0 k* k* NEW k However, we canno ell wheher he new saddle pah passes above or below he original poin E Thus we canno ell wheher c jumps up or down and in fac, if he new saddle pah passes righ hrough poin E, c migh even remain he same a he insan ha g falls Thereafer, c and k rise gradually o heir new balanced-growh-pah values; hese are higher han heir values on he original balanced growh pah (d) On a balanced growh pah, he fracion of oupu ha is saved and invesed is given by [f(k*) - c*]/f(k*) Since k is consan, or k 0 on a balanced growh pah hen, from equaion (), we can wrie f(k*) - c* = (n + g)k* Using his, we can rewrie he fracion of oupu ha is saved on a balanced growh pah as

7 Full file a hps://frasockeu Soluions o Chaper (3) s = [(n + g)k*]/f(k*) Differeniaing boh sides of equaion (3) wih respec o g yields s f( k*) ( n g)( k * g) k * ( n g) k * f ( k*)( k * g) (4) g f( k*) 2, which simplifies o s ( n g)[ f( k*) k * f ( k*)]( k * g) f( k*) k * (5) g f( k*) 2 Since k* is defined by f '(k*) = + g, differeniaing boh sides of his expression gives us f ''(k*)(k*/g) = Solving for k*/g gives us (6) k*/g = /f ''(k*) < 0 Subsiuing equaion (6) ino equaion (5) yields s ( n g)[ f( k*) k * f ( k*)] f( k*) k * f( k*) (7) g 2 f( k*) f ( k*) The firs erm in he numeraor is posiive whereas he second is negaive Thus he sign of s/g is ambiguous; we canno ell wheher he fall in g raises or lowers he saving rae on he new balanced growh pah (e) When he producion funcion is Cobb-Douglas, f(k) = k, f '(k) = k - and f ''(k) = ( - )k -2 Subsiuing hese facs ino equaion (7) yields (8) 2 s ( n g)[ k * k * k * ] k * k * ( ) k *, g 2 k * k * ( ) k * which simplifies o (9) s ( n g) k * ( ) ( ) k * k *, g [ ( ) k * ( k * )( k * ) / ] which implies (0) s [( n g) ( g )] g ( g) 2 Thus, finally, we have () s ( n ) ( n ) g 2 2 ( g) ( g ) Problem 27 The wo equaions of moion are c () ( ) f ( k( )) g, c( ) and (2) k ( ) f ( k( )) c( ) ( n g) k( )

8 2-8 Full file a hps://frasockeu Soluions o Chaper 2 (a) A rise in or a fall in he elasiciy of subsiuion, /, means ha households become less willing o subsiue consumpion beween periods I also means ha he marginal uiliy of consumpion falls off more rapidly as consumpion rises If he economy is growing, his ends o make households value presen consumpion more han fuure consumpion c c 0 The capial-accumulaion equaion is unaffeced The condiion required for c 0 is given by f ' (k)= + g Since f '' (k) < 0, he f ' (k) ha makes c 0 is now higher Thus he value of k ha saisfies c 0 is lower The k*' k* k c 0 locus shifs o he lef The economy moves up o poin A on he new saddle pah; people consume more now Movemen is hen down along he new saddle pah unil he economy reaches poin E ' A ha poin, c* and k* are lower han heir original values (b) We can assume ha a downward shif of he producion funcion means ha for any given k, boh f(k) and f ' (k) are lower han before A E ' E k 0 c c 0 f(k) y = f(k) E y = f(k)' (n + g)k E ' k 0 k*' k* k k The condiion required for k 0 is given by c f ( k) ( n g) k We can see from he figure on he righ ha he k 0 locus will shif down more a higher levels of k Also, since for a given k, f ' (k) is lower now, he golden-rule k will be lower han before Thus he k 0 locus shifs as depiced in he figure The condiion required for c 0 is given by f ( k) g For a given k, f ' (k) is now lower Thus we need a lower k o keep f ' (k) he same and saisfy he c 0 equaion Thus he c 0 locus shifs lef The economy will evenually reach poin E ' wih lower c* and lower k* Wheher c iniially jumps up or down depends upon wheher he new saddle pah passes above or below poin E (c) Wih a posiive rae of depreciaion, > 0, he new capial-accumulaion equaion is

9 Full file a hps://frasockeu Soluions o Chaper (3) k ( ) f ( k( )) c( ) ( n g ) k( ) c c 0 f(k) y = f(k) E (n + g + )k E' k 0 (n + g)k k*' k* k The level of saving and invesmen required jus o keep any given k consan is now higher and hus he amoun of consumpion possible is now lower han in he case wih no depreciaion The level of exra invesmen required is also higher a higher levels of k Thus he k 0 locus shifs down more a higher levels of k In addiion, he real reurn on capial is now f ' (k()) - and so he household's maximizaion will yield c (4) ( ) f ( k( )) g c( ) The condiion required for c 0 is f ( k) g Compared o he case wih no depreciaion, f ' (k) mus be higher and k lower in order for c 0 Thus he c 0 locus shifs o he lef The economy will evenually wind up a poin E ' wih lower levels of c* and k* Again, wheher c jumps up or down iniially depends upon wheher he new saddle pah passes above or below he original equilibrium poin of E Problem 28 Wih a posiive depreciaion rae, > 0, he Euler equaion and he capial-accumulaion equaion are given by c( ) f k( ) g (), and (2) k ( ) f k( ) c( ) n g k( ) c( ) We begin by aking firs-order Taylor approximaions o () and (2) around k = k* and c = c* Tha is, we can wrie (3) c c * * k k k c c c c, and (4) k k * * k k k k c c c, where c /k, c /c, k /k and k /c are all evaluaed a k = k* and c = c* Define ~ c c c * and ~ k k k * Since c* and k* are consans, c and k are equivalen o ~ c and ~ k respecively We can herefore rewrie (3) and (4) as (5) ~ ~ c c ~ k k c c c, and (6) ~ ~ k k ~ k k k c c Using equaions () and (2) o compue hese derivaives yields k

10 2-0 Full file a hps://frasockeu Soluions o Chaper 2 (7) c f ( k*) c *, (8) c f ( k*) g 0, k bgp c bgp (9) k k bgp f ( k*) ( n g ), (0) k c bgp Subsiuing equaions (7) and (8) ino (5) and equaions (9) and (0) ino (6) yields () ~ f ( k*) c * ~ c k, and ~ (2) k f ( k*) ( n g ) ~ k ~ c ( g) ( n g ) ~ k ~ c ~ k ~ c The second line of equaion (2) uses he fac ha () implies ha f ' (k*) = + + g The hird line uses he definiion of - n - ( - )g Dividing equaion () by ~ c and dividing equaion (2) by ~ k yields ~ c f ( k*) c * ~ k (3) ~ ~ k ~ c c ~, and (4) ~ ~ c k k Noe ha hese are exacly he same as equaions (232) and (233) in he ex; adding a posiive depreciaion rae does no aler he expressions for he growh raes of ~ c and ~ k Thus equaion (237), he expression for, he consan growh rae of boh ~ c and ~ k as he economy moves oward he balanced growh pah, is sill valid Thus 2 4 (5) f ( k*) c *, 2 where we have chosen he negaive growh rae so ha c and k are moving oward c* and k*, no away from hem Now consider he Cobb-Douglas producion funcion, f(k) = k Thus (6) 2 f ( k*) k * r *, and (7) f ( k*) ( ) k * Squaring boh sides of equaion (6) gives us (8) r * 2 2 k * 2 2, and so equaion (7) can be rewrien as 2 r (9) f k r 2 ( * ) ( ) ( * ) ( *) k * f ( k*) In addiion, defining s* o be he saving rae on he balanced growh pah, we can wrie he balancedgrowh-pah level of consumpion as (20) c* = ( - s*)f(k*) Subsiuing equaions (9) and (20) ino (5) yields 2 ( r * ) 4 ( s*) f ( k*) f ( k*) (2) 2 Canceling he f(k*) and muliplying hrough by he minus sign yields 2

11 Full file a hps://frasockeu Soluions o Chaper 2 2- (22) 4 ( r * ) ( s*) On he balanced growh pah, he condiion required for c = 0 is given by r* = + g and hus (23) r* + = + g + In addiion, acual saving, s*f(k*), equals break-even invesmen, (n + g + )k*, and hus ( n g ) k * ( n g ) ( n g ) (24) s*, f ( k*) k * ( r * ) where we have used equaion (6), r* + = k* - From equaion (24), we can wrie (r * ) (n g ) (25) ( s*) (r * ) Subsiuing equaions (23) and (25) ino equaion (22) yields 2 4 g g ( n g ) (26) 2 Equaion (26) is analogous o equaion (239) in he ex I expresses he rae of adjusmen in erms of he underlying parameers of he model Keeping he values in he ex = /3, = 4%, n = 2%, g = % and = and using = 3% yields a value for of approximaely - 88% This is faser convergence han he -54% obained wih no depreciaion Problem 29 (a) We are given ha () y () k() From he exbook we know ha when c 0, hen f (k) g Subsiuing () and simplifying resuls in * (2) k g (b) Likewise, from he exbook, we know ha when k 0 hen c * f (k) (n g) k Subsiuing () and simplifying resuls in (3) * c (n g) g g z() k() y( and () c() k () (c) Le ) ge () x Subsiuing () ino he firs equaion and simplifying, we (4) k z k z Subsiuing (4) ino he second equaion above and simplifying resuls in (5) ck xz Now, using equaion (4), ake he ime derivaive of (6) z ( )k k z k y, which resuls in

12 2-2 Full file a hps://frasockeu Soluions o Chaper 2 Equaion (225) in he exbook ells us ha k k c (n g) k ; herefore (6) becomes (7) z ( )k (k c (n g)k) Simplifying and subsiuing equaions (4) and (5), equaion (7) becomes (8) z ( )( xz (n g)z) Now look a x c k Taking logs and hen he ime derivaive resuls in (9) x x c c k k Using equaion (224) and (225) from he exbook, equaion (9) becomes x k g k c (n g)k (0) x k Again, subsiuing equaions (4) and (5) ino (0), and using he assumpion ha () x x x n (d)(i) From he conjecure ha x is consan, equaion (8) becomes * gives us (2) z ( )( (n g x )z) To find he pah of z, consider equaion (2) and observe ha i is a linear non-homogeneous ordinary differenial equaion Therefore, our soluion will consis of he complemenary soluion ( z c ) and he paricular soluion ( z p ) For simplificaion, le ( )(n g x ) To solve for he complemenary soluion, we consider he homogeneous case in which z z 0 Isolaing for z z yields a complemenary soluion of (3) zc Ae, where A is a consan of inegraion To solve for he paricular soluion, we consider he non-homogeneous case, where z z ( ) Using an inegraing facor, we find he soluion o be (4) (5) z p 2e ( ) A, where A 2 is a consan of inegraion Therefore, p c 2 z z z z ( ) (A A ) e Using he iniial condiion z(0), we can subsiue for A A 2 and (5) becomes (6) z ( ) e (z(0) ( ) ) Lasly, we can simplify he ( ) erm by subsiuing equaions (2) and (3) for x* and using equaion (4), resuling in * (7) z z e (z(0) z ) * (ii) We wan o find he pah of y, so consider equaion () and subsiue in equaion (4) We know he pah value of z, so subsiue in equaion (7) This resuls in * * () (8) y (z e (z(0) z )) Again, use equaion (4) o pu everyhing in erms of k and (8) becomes * * ( ) (9) y ((k ) e (k(0) (k ) )) Now we would like o see wheher he speed of convergence o he balanced growh pah is consan Using equaion (9) and subracing he balanced growh pah y*, where * * y (k ) () g from equaion (2), and aking logs resuls in *

13 Full file a hps://frasockeu Soluions o Chaper ( ) * (20) ( ) ln( y y ) ln z g Taking he ime derivaive, we ge: * z z ln(y y ) (2) z g Clearly his is no consan, so he speed of convergence is no consan (e) We wan o see wheher equaions (224) and (225) from he exbook c c ( k g) and k k c (n g) are saisfied Because hese equaions were used o find x x, hen x x holds if and only if equaions (224) and (225) also hold However, we know ha equaion (225) holds because i was used previously o find z Thus, we can say ha x x 0 if and only if c c 0 Assuming ha x x 0 and using equaion () resuls in (22) n x * Using pars (a) and (b), he balanced growh pah of x* is ( g) g (n g) so (22) becomes (23) n Equaion (23) is he same resul as (), so equaions (224) and (225) are saisfied x * Problem 20 (a) The real afer-ax rae of reurn on capial is now given by ( - )f ' (k()) Thus he household's maximizaion would now yield he following expression describing he dynamics of consumpion per uni of effecive labor: c( ) ( ) f ( k( )) g () c( ) The condiion required for c 0 is given by ( - )f ' (k) = + g The afer-ax rae of reurn mus equal + g Compared o he case wihou a ax on capial, f ' (k), he pre-ax rae of reurn on capial, mus be higher and hus k mus be lower in order for c 0 Thus he c 0 locus shifs o he lef The equaion describing he dynamics of he capial sock per uni of effecive labor is sill given by (2) k ( ) f ( k( )) c( ) ( n g) k( ) For a given k, he level of c ha implies k 0 is given by c() = f(k) - (n + g)k Since he ax is rebaed o households in he form of lump-sum ransfers, his k 0 locus is unaffeced

14 2-4 Full file a hps://frasockeu Soluions o Chaper 2 (b) A ime 0, when he ax is pu in place, he value of k, he sock of capial per uni of effecive labor, is given by he hisory of he economy, and i canno change disconinuously I remains equal o he k* on he old balanced growh pah c c 0 A In conras, c, he rae a which households are consuming in unis of effecive labor, can jump a he ime ha he ax is inroduced This jump in c is no inconsisen wih he consumpionsmoohing behavior implied by he household's opimizaion problem since he ax was unexpeced and could no be prepared for In order for he economy o reach he new balanced growh pah, i should be clear wha mus occur A ime 0, c jumps up so ha he economy is on he new saddle pah In he figure, he economy jumps from poin E o a poin such as A Since he reurn o saving and accumulaing capial is now lower han before, people swich away from saving and ino consumpion Afer ime 0, he economy will gradually move down he new saddle pah unil i evenually reaches he new balanced growh pah a E NEW (c) On he new balanced growh pah a E NEW, he disorionary ax on invesmen income has caused he economy o have a lower level of capial per uni of effecive labor as well as a lower level of consumpion per uni of effecive labor (d) (i) From he analysis above, we know ha he higher is he ax rae on invesmen income,, he lower will be he balanced-growh-pah level of k*, all else equal In erms of he above sory, he higher is he more ha he c 0 locus shifs o he lef and hence he more ha k* falls Thus k*/ < 0 On a balanced growh pah, he fracion of oupu ha is saved and invesed, he saving rae, is given by [f(k*) - c*]/f(k*) Since k is consan, or k 0, on a balanced growh pah hen from k ( ) f ( k( )) c( ) ( n g) k( ) we can wrie f(k*) - c* = (n + g)k* Using his we can rewrie he fracion of oupu ha is saved on a balanced growh pah as (3) s = [(n + g)k*]/f(k*) Use equaion (3) o ake he derivaive of he saving rae wih respec o he ax rae, : (4) s ( n g) ( k * ) f ( k*) ( n g) k * f ( k*) ( k * ) f ( k*) 2 Simplifying yields (5) s ( n g) k * ( n g) k * f ( k*) k * ( n g) k * k * f ( k*) f ( k*) f ( k*) f ( k*) f ( k*) f ( k*) Recall ha k*f ' (k*)/f(k*) K (k*) is capial's (pre-ax) share in income, which mus be less han one Since k*/ < 0 we can wrie E E NEW k 0 k* NEW k* k

15 Full file a hps://frasockeu Soluions o Chaper s ( n g) k * K 0 f ( k*) Thus he saving rae on he balanced growh pah is decreasing in (6) ( k*) (d) (ii) Ciizens in low-, high-k*, high-saving counries do no have he incenive o inves in lowsaving counries From par (a), he condiion required for c 0 is ( -)f ' (k) = + g Tha is, he afer-ax rae of reurn mus equal + g Assuming preferences and echnology are he same across counries so ha, and g are he same across counries, he afer-ax rae of reurn will be he same across counries Since he afer-ax rae of reurn is hus he same in low-saving counries as i is in high-saving counries, here is no incenive o shif saving from a high-saving o a low-saving counry (e) Should he governmen subsidize invesmen insead and fund his wih a lump-sum ax? This would lead o he opposie resul from above and he economy would have higher c and k on he new balanced growh pah c 0 c E E NEW k 0 A k* k* NEW k appropriaely discouned) The answer is no The original marke oucome is already he one ha would be chosen by a cenral planner aemping o maximize he lifeime uiliy of a represenaive household subjec o he capialaccumulaion equaion I herefore gives he household he highes possible lifeime uiliy Saring a poin E, he implemenaion of he subsidy would lead o a shor-erm drop in consumpion a poin A, bu would evenually resul in permanenly higher consumpion a poin E NEW I would urn ou ha he uiliy los from he shorerm sacrifice would ouweigh he uiliy gained in he long-erm (all in presen value erms, This is he same ype of argumen used o explain he reason ha households do no choose o consume a he golden-rule level See Secion 24 for a more complee descripion of he welfare implicaions of his model (f) Suppose he governmen does no rebae he ax revenue o households bu insead uses i o make governmen purchases Le G() represen governmen purchases per uni of effecive labor The equaion describing he dynamics of he capial sock per uni of effecive labor is now given by (2 ' ) k ( ) f ( k( )) c( ) G( ) ( n g) k( ) The fac ha he governmen is making purchases ha do no add o he capial sock i is assumed o be governmen consumpion, no governmen invesmen shifs down he k 0 locus

16 2-6 Full file a hps://frasockeu Soluions o Chaper 2 c c 0 E k 0 Afer he imposiion of he ax, he c 0 locus shifs o he lef, jus as i did in he case in which he governmen rebaed he ax o households In he end, k* falls o k* NEW jus as in he case where he governmen rebaed he ax Consumpion per uni of effecive labor on he new balanced growh pah a E NEW is lower han in he case where he ax is rebaed by he amoun of he governmen purchases, which is f ' (k)k E NEW k* NEW k* k Finally, wheher he level of c jumps up or down iniially depends upon wheher he new saddle pah passes above or below he original balanced-growh-pah poin of E Problem 2 (a) - (c) Before he ax is pu in place, ie unil ime, he equaions governing he dynamics of he economy are c () ( ) f ( k( )) g, c( ) and (2) k ( ) f( k( )) c( ) ( n g) k( ) The condiion required for c 0 is given by f ' (k) = + g The capial-accumulaion equaion is no affeced when he ax is pu in place a ime since we are assuming ha he governmen is rebaing he ax, no spending i Since he real afer-ax rae of reurn on capial is now ( - )f ' (k()), he household's maximizaion yields he following growh rae of consumpion: c (3) ( ) ( ) f ( k( )) g c( ) The condiion required for c 0 is now given by ( - )f ' (k) = + g The afer-ax rae of reurn on capial mus equal + g Thus he pre-ax rae of reurn, f ' (k), mus be higher and hus k mus be lower in order for c 0 Thus a ime, he c 0 locus shifs o he lef

17 Full file a hps://frasockeu Soluions o Chaper c c 0 [before ime ] c 0 [afer ime ] The imporan poin is ha he dynamics of he economy are sill governed by he original equaions of moion unil he ax is acually pu in place Beween he ime of he announcemen and he ime he ax is acually pu in place, i is he original c 0 locus ha is relevan When he ax is pu in place a ime, c canno jump disconinuously because households know ahead of ime ha he k 0 ax will be implemened hen A disconinuous jump in c would be inconsisen wih he consumpion smoohing implied by he household s ineremporal opimizaion The k household would no wan c o be low, and hus marginal uiliy o be high, a momen before knowing ha c will jump up and be high, and hus marginal uiliy will be low, a momen afer The household would like o smooh consumpion beween he wo insans in ime (d) We know ha c canno jump a ime We also know ha if he economy is o reach he new balanced growh pah a poin E NEW, i mus be righ on he new saddle pah a he ime ha he ax is pu in place Thus when he ax is announced a ime 0, c mus jump up o a poin such as A Poin A lies beween he original balanced growh pah a E and he new saddle pah c c 0 [unil ] c 0[afer ] B A E E NEW k 0 A A, c is oo high o mainain he capial sock a k* and so k begins falling Beween 0 and, he dynamics of he sysem are sill governed by he original c 0 locus The economy is hus o he lef of he c 0 locus and so consumpion begins rising The economy moves off o he norhwes unil a, i is righ a poin B on he new saddle pah The ax is hen pu in place and he sysem is governed by he new c 0 locus Thus c begins falling The economy moves down he new saddle pah, evenually reaching poin E NEW k* NEW k* k (e) The sory in par (d) implies he following ime pahs for consumpion per uni of effecive labor and capial per uni of effecive labor

18 2-8 Full file a hps://frasockeu Soluions o Chaper 2 c k k* c* c* NEW k* NEW 0 ime 0 Problem 22 (a) The firs poin is ha consumpion canno jump a ime Households know ahead of ime ha he ax will end hen and so a disconinuous jump in c would be inconsisen wih he consumpionsmoohing behavior implied by he household's ineremporal opimizaion Thus, for he economy o reurn o a balanced growh pah, we mus be somewhere on he original saddle pah righ a ime Before he ax is pu in place unil ime 0 and afer he ax is removed afer ime he equaions governing he dynamics of he economy are c( ) f k( ) g (), and (2) k ( ) f k( ) c( ) ( n g) k( ) c( ) The condiion required for c 0 is given by f ' (k) = + g The capial-accumulaion equaion, and hus he k 0 locus, is no affeced by he ax The c 0 locus is affeced, however Beween ime 0 and ime, he condiion required for c 0 is ha he afer-ax rae of reurn on capial equal + g so ha ( - )f ' (k) = + g Thus beween 0 and, f ' (k) mus be higher and so k mus be lower in order for c 0 Tha is, beween ime 0 and ime, he c 0 locus lies o he lef of is original posiion A ime 0, he ax is pu in place A poin E, he economy is sill on he k 0 locus bu is now o he righ of he new c 0 locus Thus if c did no jump up o a poin like A, c would begin falling The economy would hen be below he k 0 locus and so k would sar rising The economy would drif away from poin E in he direcion of he souheas and could no be on he original saddle pah righ a ime c c 0 [beween ime 0 and ] B A E c 0 [before ime 0 and afer ime ] Thus a ime 0, c mus jump up so ha k 0 he economy is a a poin like A Thus, k and c begin falling Evenually he economy crosses he k k 0 locus and so k begins rising This can be inerpreed as households anicipaing he removal of he ax on capial and hus being willing o accumulae capial again Poin A mus be such ha given he dynamics of he sysem, he economy is righ a a poin like B, on he original saddle pah, a ime when he ax is removed Afer, he original c 0 locus governs he dynamics of he sysem again Thus he economy moves up he original saddle pah, evenually reurning o he original balanced growh pah a poin E

19 Full file a hps://frasockeu Soluions o Chaper (b) The firs poin is ha consumpion canno jump a eiher ime or ime 2 Households know ahead of ime ha he ax will be implemened a and removed a 2 Thus a disconinuous jump in c a eiher dae would be inconsisen wih he consumpion-smoohing behavior implied by he household's ineremporal opimizaion In order for he economy o reurn o a balanced growh pah, he economy mus be somewhere on he original saddle pah righ a ime 2 Before he ax is pu in place unil ime and afer he ax is removed afer ime 2 equaions () and (2) govern he dynamics of he sysem An imporan poin is ha even during he ime beween he announcemen and he implemenaion of he ax ha is, beween ime 0 and ime he original c 0 locus governs he dynamics of he sysem c B C c 0 [beween ime and 2 ] A E c 0 [before ime and afer ime 2 ] A ime 0, he ax is announced Consumpion mus jump up so ha he economy is a a poin like A k 0 A A, he economy is sill on he c 0 k locus bu is above he k 0 locus and so k sars falling The economy is hen o he lef of he c 0 locus and so c sars rising The economy drifs off o he norhwes A ime, he ax is implemened, he c 0 locus shifs o he lef and he economy is a a poin like B The economy is sill above he k 0 locus bu is now o he righ of he relevan c 0 locus; k coninues o fall and c sops rising and begins o fall Evenually he economy crosses he k 0 locus and k begins rising Households begin accumulaing capial again before he acual removal of he ax on capial income Poin A mus be chosen so ha given he dynamics of he sysem, he economy is righ a a poin like C, on he original saddle pah, a ime 2 when he ax is removed Afer 2, he original c 0 locus governs he dynamics of he sysem again Thus he economy moves up he original saddle pah, evenually reurning o he original balanced growh pah a poin E Problem 23 Wih governmen purchases in he model, he capial-accumulaion equaion is given by () k ( ) f k( ) c( ) G( ) ( n g) k( ), where G() represens governmen purchases in unis of effecive labor a ime

20 2-20 Full file a hps://frasockeu Soluions o Chaper 2 Inuiively, since governmen purchases are assumed o be a perfec subsiue for privae consumpion, changes in G will simply be offse one-forone wih changes in c Suppose ha G() is iniially consan a some level G L The household's maximizaion yields c c 0 c( ) f k( ) g (2) c( ) G L (G H - G L ) Thus he condiion for consan E consumpion is sill given by f ' (k) = + g Changes in he level of G L will affec he level of c, bu will no shif he c 0 locus E NEW k 0 Suppose he economy sars on a balanced growh pah a poin E A some ime 0, k G unexpecedly increases o G H and households know his is emporary; households know ha a some fuure ime, governmen purchases will reurn o G L A ime 0, he k 0 locus shifs down; a each level of k, he governmen is using more resources leaving less available for consumpion In paricular, he k 0 locus shifs down by he amoun of he increase in purchases, which is (G H - G L ) The difference beween his case, in which c and G are perfec subsiues, and he case in which G does no affec privae uiliy, is ha c can jump a ime when G reurns o is original value In fac, a, when G jumps down by he amoun (G H - G L ), c mus jump up by ha exac same amoun If i did no, here would be a disconinuous jump in marginal uiliy ha could no be opimal for households Thus a, c mus jump up by (G H - G L ) and his mus pu he economy somewhere on he original saddle pah If i did no, he economy would no reurn o a balanced growh pah Wha mus happen is ha a ime 0, c falls by he amoun (G H - G L ) and he economy jumps o poin E NEW I hen says here unil ime A, c jumps back up by he amoun (G H - G L ) and so he economy jumps back o poin E and says here Why can' c jump down by less han (G H - G L ) a 0? If i did, he economy would be above he new k 0 locus, k would sar falling puing he economy o he lef of he c 0 locus Thus c would sar rising and so he economy would drif off o he norhwes There would be no way for c o jump up by (G H - G L ) a and sill pu he economy on he original saddle pah Why can' c jump down by more ha (G H - G L ) a? If i did, he economy would be below he new k 0 locus, k would sar rising puing he economy o he righ of he c 0 locus Thus c would sar falling and so he economy would drif off o he souheas Again, here would be no way for c o jump up by (G H - G L ) a and sill pu he economy on he original saddle pah In summary, he capial sock and he real ineres rae are unaffeced by he emporary increase in G A he insan ha G rises, consumpion falls by an equal amoun I remains consan a ha level while G remains high A he insan ha G falls o is iniial value, consumpion jumps back up o is original value and says here

21 Full file a hps://frasockeu Soluions o Chaper Problem 24 Equaion (260) in he ex describes he relaionship beween k + and k in he special case of logarihmic uiliy and Cobb-Douglas producion: (260) k ( ) k ( n)( g) 2 (a) A rise in n shifs he k + funcion down From equaion (260), a higher n means a smaller k + for a given k Since he fracion of heir labor income ha he young save does no depend on n, a given amoun of capial per uni of effecive labor and hus oupu per uni of effecive labor in ime yields he same amoun of saving in period Thus i yields he same amoun of capial in period + However, he number of individuals increases more from period o period + han i used o So ha capial k is spread ou among more individuals han i would have been in he absence of he increase in populaion growh and hus k NEW * k* k capial per uni of effecive labor in period + is lower for a given k (b) Wih he parameer B added o he Cobb-Douglas producion funcion, f(k) = Bk, equaion (260) becomes () k ( ) Bk ( n)( g) 2 This fall in B causes he k + funcion o shif down See he figure from par (a) A lower B means ha a given amoun of capial per uni of effecive labor in period now produces less oupu per uni of effecive labor in period Since he fracion of heir labor income ha he young save does no depend on B, his leads o less oal saving and a lower capial sock per uni of effecive labor in period + for a given k (c) We need o deermine he effec on k + for a given k, of a change in From equaion (260): (2) k k k ( ) ( n)( g) 2 We need o deermine k / Define f() k and noe ha lnf() = lnk Thus (3) lnf()/ = lnk Now noe ha we can wrie f( ) f( ) ln f( ) ln f( ) (4), ln f( ) ln f( ) f( ) and hus finally (5) f()/ = f()lnk Therefore, we have k / = k lnk Subsiuing his fac ino equaion (2) yields k (6) k k ( ) ln k, ( n)( g) 2 or simply

22 2-22 Full file a hps://frasockeu Soluions o Chaper 2 k ( n)( g) 2 (7) k k ( ) ln Thus, for ( - )lnk - > 0, or lnk > /( - ), an increase in means a higher k + for a given k and hus he k + funcion shifs up over his range of k 's However, for lnk < /( - ), an increase in means a lower k + for a given k Thus he k + funcion shifs down over his range of k 's Finally, righ a lnk = /( - ), he old and new k + funcions inersec Problem 25 (a) We need o find an expression for k + as a funcion of k Nex period's capial sock is equal o his period's capial sock, plus any invesmen done his period, less any depreciaion ha occurs Thus () K + = K + sy - K To conver his ino unis of effecive labor, divide boh sides of equaion () by A + L + : K K ( ) sy K ( ) sy k ( ) sf ( k ) (2), A L A L ( n)( g) A L ( n)( g) which simplifies o s (3) k k f( k ) ( n)( g) ( n)( g) (b) We need o skech k + as a funcion of k Noe ha (4) k s f ( k ) 0, and (5) 2 k sf ( k ) 0 k ( n)( g) ( n)( g) k 2 ( n)( g) The Inada condiions are given by k (6) k lim, and (7) lim k0 k k k ( n)( g) Thus he funcion evenually has a slope of less han one and will herefore cross he 45 degree line a some poin Also, he funcion is well-behaved and will cross he 45 degree line only once k k + As long as k sars ou a some value oher han 0, he economy will converge o k* For example, if k sars ou below k*, we see ha k + will be greaer han k and he economy will move oward k* Similarly, if k sars ou above k*, we see ha k + will be below k and again he economy will move oward k* A k*, y* = f(k*) is also consan and we have a balanced growh pah k* k (c) On a balanced growh pah, k + = k k* and hus from equaion (3) we have s (8) k* k * f( k*), ( n)( g) ( n)( g) which simplifies o

23 Full file a hps://frasockeu Soluions o Chaper n g ng s (9) k * f( k*) ( n)( g) ( n)( g) Thus on a balanced growh pah: (0) k*(n + g + ng + ) = sf(k*) Rearranging equaion (0) o ge an expression for s on he balanced growh pah yields () s = (n + g + ng + )k*/f(k*) Consumpion per uni of effecive labor on he balanced growh pah is given by (2) c* = ( - s)f(k*) Subsiue equaion () ino equaion (2): ( n g ng ) k * f ( k*) k * ( n g ng ) (3) c* f ( k*) f ( k*) f ( k*) f ( k*) Canceling he f(k*) yields (4) c* = f(k*) - (n + g + ng + )k* To ge an expression for he f ' (k*) ha maximizes consumpion per uni of effecive labor on he balanced growh pah, we need o maximize c* wih respec o k* The firs-order condiion is given by (5) c * k * f ( k*) ( n g ng ) 0 Thus he golden-rule capial sock is defined implicily by (6) f ' (k GR ) = (n + g + ng + ) (d) (i) Subsiue a Cobb-Douglas producion funcion, f(k ) = k, ino equaion (3): s (7) k k k ( n)( g) ( n)( g) (d) (ii) On a balanced growh pah, k + = k k* Thus from equaion (7): s (8) k* k * k * ( n)( g) ( n)( g) Simplifying yields ( n)( g) ( ) s (9) k* k *, ( n)( g) ( n)( g) or (20) k* - = s/(n + g + ng + ) Thus, finally we have (2) k* = [s/(n + g + ng + )] /(-) (d) (iii) Using equaion (7): (22) dk s dk n g n g k * ( )( ) ( )( ) k k* Subsiuing he balanced-growh-pah value of k* equaion (2) ino equaion (22) yields (23) dk s ( n g ng ) dk ( n)( g) ( n)( g) s k k* I will be useful o wrie (n + g + ng + ) as ( + n)( + g) - ( - ):

24 2-24 Full file a hps://frasockeu Soluions o Chaper 2 (24) dk dk ( ) ( n)( g) ( ) ( n)( g) k k* Simplifying furher yields (25) dk ( )( ) dk ( n)( g) k k* Replacing equaion (7) by is firs-order Taylor approximaion around k = k* herefore gives us (26) k k * ( )( ) ( n)( g) k k * Since we can wrie his simply as (27) k k* ( )( ) ( n)( g) k k *, equaion (26) implies (28) k k* ( )( ) ( n)( g) k 0 k * Thus he economy moves fracion ( )( ) ( )( ) n g of he way o he balanced growh pah each period Some simple algebra simplifies he expression for his rae of convergence o ( )( n g ng ) ( n)( g ) Wih = /3, n = %, g = 2% and = 3%, his yields a rae of convergence of abou 39% This is slower han he rae of convergence found in he coninuous-ime Solow model Problem 26 (a) The individual's opimizaion problem is no affeced by he depreciaion which means ha r = f '(k ) - The household's problem is sill o maximize uiliy as given by C C2,, () U, subjec o he budge consrain (2) C, C2, A w r As in he ex, wih no depreciaion, he fracion of income saved, s(r + ) (-C, )A w, is given by (3) s( r) ( ) ( r ) ( ) Thus he way in which he fracion of income saved depends on he real ineres rae, r +, is unchanged The only difference is ha he real ineres rae iself is now f ' (k + ) -, raher han jus f ' (k + ) The capial sock in period + equals he amoun saved by young individuals in period Thus (4) K + = S L, where S is he amoun of saving done by a young person in period Noe ha S s(r + )A w ; he amoun of saving done is equal o he fracion of income saved imes he amoun of income Thus equaion (4) can be rewrien as (5) K + = L s(r + )A w To ge his ino unis of ime + effecive labor, divide boh sides of equaion (5) by A + L + : K A L (6) s( r) w A L A L Since A + = ( + g)a, we have A /A + = /( + g) Similarly, L /L + = /( + n) In addiion, K + /A + L + k + Thus (7) k n g s ( r w ( )( ) )

25 Full file a hps://frasockeu Soluions o Chaper Finally, subsiue for r + = f ' (k + ) - and w = f(k ) - k f ' (k ): (8) k n g s f ( k ) f ( k ) k f ( k ) ( )( ) This should be compared wih equaion (259) in he ex, he analogous expression wih no depreciaion, which is (259) k n g s f ( k ) f ( k ) k f ( k ) ( )( ) Thus adding depreciaion does aler he relaionship beween k + and k Wheher k + will be higher or lower for a given k depends on he way in which saving varies wih r + (b) Wih logarihmic uiliy, he fracion of income saved does no depend upon he rae of reurn on saving and in fac (9) s(r + ) = /(2 + ) In addiion, wih Cobb-Douglas producion, y = k, he real wage is w = k - k k - = ( - )k Thus equaion (8) becomes (0) k ( ) k ( n)( g) 2 We need o compare his wih equaion (3) in he soluion o Problem 25, he analogous expression in he discree-ime Solow model, wih he addiional assumpion of 00% depreciaion (ie = ) The saving rae in his economy is oal saving divided by oal oupu Noe ha his is no he same as s(r + ), which is simply he fracion of heir labor income ha he young save Denoe he economy's oal saving rae as s Then s will equal he saving of he young plus he dissaving of he old, all divided by oal oupu and in addiion, all variables are measured in unis of effecive labor The saving of he young is ( 2 ) ( ) k Since here is 00% depreciaion, he old do no ge o dissave by he amoun of he capial sock; here is no dissaving by he old Thus ( 2 ) ( ) k () s ( ) k 2 Thus equaion (0) can be rewrien as (2) k n g sk s n g f k ( )( ) ( ) ( )( ) Noe ha his is exacly he same as he expression for k + as a funcion of k in he discree-ime Solow model wih = Tha is, i is equivalen o equaion (3) in he soluion o Problem 25 wih se o one Thus ha version of he Solow model does have some microeconomic foundaions, alhough he assumpion of 00% depreciaion is quie unrealisic Problem 27 (a) (i) The uiliy funcion is given by () ln C, ln C2, Wih he social securiy ax of T per person, he individual faces he following consrains (wih g, he growh rae of echnology, equal o 0, A is simply a consan hroughou): (2) C, S Aw T, and (3) C2, ( r) S ( n) T,

26 2-26 Full file a hps://frasockeu Soluions o Chaper 2 where S represens he individual's saving in he firs period As far as he individual is concerned, he rae of reurn on social securiy is ( + n); in general his will no be equal o he reurn on privae saving which is ( + r + ) From equaion (3), ( r) S C2, ( n) T Solving for S yields C2, ( n) (4) S T r ( r ) Now subsiue equaion (4) ino equaion (2): C2, ( n) (5) C, Aw T T r ( r ) Rearranging, we ge he ineremporal budge consrain: C2, ( r n) (6) C, Aw T r ( r ) We know ha wih logarihmic uiliy, he individual will consume fracion ( + )/(2 + ) of her lifeime wealh in he firs period Thus r n (7) C, Aw T 2 r To solve for saving per person, subsiue equaion (7) ino equaion (2): r n (8) S Aw Aw T T 2 r Simplifying gives us r n (9) S Aw T, 2 2 r or ( 2 )( r) ( )( r n) (0) S 2 Aw T ( 2 )( r) Noe ha if r + = n, saving is reduced one-for-one by he social securiy ax If r + > n, saving falls less han one-for-one Finally, if r + < n, saving falls more han one-for-one Denoe () S 2 Aw Z T Z ( 2 )( r ) ( )( r n) ( 2 )( r ) and hus equaion (0) becomes I is sill rue ha he capial sock in period + will be equal o he oal saving of he young in period, hence (2) K + = S L Convering his ino unis of effecive labor by dividing boh sides of (2) by AL + and using equaion () yields (3) k n ( 2 ) w Z T A Wih a Cobb-Douglas producion funcion, he real wage is given by (4) w = ( - )k Subsiuing (4) ino (3) gives he new relaionship beween capial in period + and capial in period, all in unis of effecive labor: (5) k n ( 2 ) ( ) k Z T A

27 Full file a hps://frasockeu Soluions o Chaper (a) (ii) To see wha effec he inroducion of he social securiy sysem has on he balanced-growh-pah value of k, we mus deermine he sign of Z If i is posiive, he inroducion of he ax, T, shifs down he k + curve and reduces he balanced-growh-pah value of k We have ( 2 )( r) ( )( r n) ( )( r) ( )( r n) (6) Z, ( 2 )( r) ( 2 )( r) and simplifying furher allows us o deermine he sign of Z : ( r) ( ) ( r) ( r n) ( r) ( )( n) (7) Z 0 ( 2 )( r) ( 2 )( r) Thus, he k + curve shifs down, relaive o he case wihou he social securiy, and k* is reduced (a) (iii) If he economy was iniially dynamically efficien, a marginal increase in T would resul in a gain o he old generaion ha would receive he exra benefis However, i would reduce k* furher below k GR and hus leave fuure generaions worse off, wih lower consumpion possibiliies If he economy was iniially dynamically inefficien, so ha k* > k GR, he old generaion would again gain due o he exra benefis In his case, he reducion in k* would acually allow for higher consumpion for fuure generaions and would be welfare-improving The inroducion of he ax in his case would reduce or possibly eliminae he dynamic inefficiency caused by he over-accumulaion of capial (b) (i) Equaion (3) becomes (8) C2, ( r) S ( r) T As far as he individual is concerned, he rae of reurn on social securiy is he same as ha on privae saving We can now derive he ineremporal budge consrain From equaion (4), S C r T (9) 2, Subsiuing equaion (9) ino equaion (2) yields C2, (20) C, Aw T T, r or simply C2, (2) C, Aw r This is jus he usual ineremporal budge consrain in he Diamond model Solving he individual's maximizaion problem yields he usual Euler equaion: C ( r ) C (22) 2,, Subsiuing his ino he budge consrain, equaion (2), yields (23) C, 2 Aw To ge saving per person, subsiue equaion (23) ino equaion (2): S Aw 2 Aw T, (24) or simply S 2 Aw T (25) The social securiy ax causes a one-for-one reducion in privae saving The capial sock in period + will be equal o he sum of oal privae saving of he young plus he oal amoun invesed by he governmen Hence (26) K + = S L + TL Dividing boh sides of (26) by AL + o conver his ino unis of effecive labor, and using equaion (25) yields

28 2-28 Full file a hps://frasockeu Soluions o Chaper 2 T T (27) k w, n 2 A n A which simplifies o (28) k n 2 w Using equaion (4) o subsiue for he wage yields (29) k n 2 ( ) k Thus he fully-funded social securiy sysem has no effec on he relaionship beween he capial sock in successive periods (b) (ii) Since here is no effec on he relaionship beween k + and k, he balanced-growh-pah value of k is he same as i was before he inroducion of he fully-funded social securiy sysem (Noe ha we have been assuming ha he amoun of he ax is no greaer han he amoun of saving each individual would have done in he absence of he ax) The basic idea is ha oal invesmen and saving is sill he same each period; he governmen is simply doing some of he saving for he young Since social securiy pays he same rae of reurn as privae saving, individuals are indifferen as o who does he saving Thus individuals offse one-for-one any saving ha he governmen does for hem Problem 28 (a) In he decenralized equilibrium, here will be no inergeneraional rade Even if he young would like o rade goods his period for goods nex period, he only people around o rade wih are he old Unforunaely, he old will be dead and hus in no posiion o complee he rade nex period The individual s uiliy funcion is given by () ln C ln C, 2, The consrains are (2) C, F A, and (3) C2, xf where F is he amoun sored by he individual Subsiuing equaion (3) ino (2) yields he ineremporal budge consrain: (4) C C x A, 2, The individual s problem is o maximize lifeime uiliy, as given by equaion (), subjec o he ineremporal budge consrain, as given by equaion (4) Se up he Lagrangian: (5) L ln C, ln C2, A C, C2, x The firs-order condiions are given by L C C 0, and (6),, C, C2, C2, x 0 C2, L x (7) Subsiue equaion (6) ino equaion (7) and rearrange o obain (8) C2, xc, Subsiue equaion (8) ino he ineremporal budge consrain, equaion (4), o obain (9) C xc x A,,, or simply (0) C, A 2 To obain an expression for second-period consumpion, subsiue equaion (0) ino equaion (8): () C2, xa 2