ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c

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1 John Riley December 00 S O EVEN NUMBERED EXERCISES IN CHAPER 6 SECION 6: LIFE CYCLE CONSUMPION AND WEALH Eercise 6-: Opimal saving wih more han one commodiy A consumer has a period uiliy funcion δ u ( ) where n = = (, ) and u() is sricly concave he spo price vecor is p in each period and he one period ineres rae is r His iniial oal wealh is W Le c be oal spending in period and le v( p, c ) be he indirec uiliy funcion in period (a) Prove ha v( p, c ) is a sricly concave funcion of c (b) Wrie down he opimizaion problem using he indirec uiliy funcion Hence show ha oal spending will rise over ime if and only if ( + r) δ > (c) Wha addiional assumpions (if any) are needed for he opimal consumpion sequence { } o be sricly increasing? (a) Le i i be opimal wih budge, 0, c i = hen for any conve combinaion λ, λ λ λ λ λ λ λ λ p = p (( ) + ) = ( ) p + p ) = ( ) c + c = c hus λ is feasible wih budge c λ and so λ λ λ v( p, c ) = Ma{ U( ) p c } u( ) Since u( ) is sricly concave, λ 0 0 u ( ) > ( λ) u ( ) + λu ( ) = ( λ) vpc (, ) + λvpc (, ) Combining hese inequaliies, λ 0 v( p, c ) > ( λ) v( p, c ) + λv( p, c ) (b) he oal wealh growh equaion is W = ( + r)( W p ) = ( + + r)( W c) he opimal sequence of ependiures is herefore Answers o Chaper 6 page

2 John Riley December 00 δ = = + = + { c } = { c} arg Ma{ v( p, c ) W ( r)( W c )} his problem is idenical o he problem analyzed in eercise 6 he FOC is δ c c = + r, ha is = δ ( + r) δ c + c + herefore > if and only if δ ( + r) > Since v( p, c ) is sricly concave in c, c c + marginal uiliy is decreasing herefore c+ > cif and only if δ ( + r ) > (c) As he budge increases consumpion of every commodiy increases if and only if each commodiy is a normal good Eercise 6-4: Opimal saving wih bounded consumpion A consumer has a period uiliy funcion δ u ( ) where = u ( ) = α ln and j j j= j= α = he spo price vecor in period is p and he one j period ineres rae is r His iniial oal wealh is W Assume ha ( + r) δ > Consumpion of commodiy in each period is bounded from above by ˆ (a) Appeal o your answer o he previous eercise o show ha, for any given, =,, as long as iniial wealh is sufficienly small hus he upper bound is never binding (b) By considering how he unconsrained soluion { ( )} W = changes wih wealh, eplain why, for large enough W, he consrained opimum is = ˆ, > S * < ˆ, S and * (c) Suppose ha he consumer has no iniial financial capial Every period he has an endowmen of ω = (0, ˆ ) where ˆ is his endowmen of ime Inerpre as < ˆ, Answers o Chaper 6 page

3 John Riley December 00 consumpion of leisure in period and use he above analysis o show ha if he ime horizon is sufficienly long, he consumer will choose o reire lae in his life (a) Le c ( W ) be he opimal period ependiure wih iniial wealh W if he upper bound is ignored and le c ( W ) be he opimal period ependiure Since he period * α opimizaion problem is a Cobb-Douglas opimizaion problem ( W) = c( W) As p esablished in eercise 6-3, ependiure grows a a consan rae ( + r) δ so ha if W is sufficienly small, he consrain ˆ is never binding (b) As wealh increases i binds firs in he final period hen consider he opimizaion problem wih ˆ = Ignoring he upper bound, he opimizaion problem for he firs - periods is he same so consumpion of commodiy grows a a consan rae hus eiher he upper bound is saisfied for =,,, or i mus be violaed in period Coninuing his argumen, here mus be some S such ha he upper bound is saisfied if and only if S (c) he oal wealh accumulaion equaion is W = ( + r)( W + c) where c = p o review why his is rue, noe ha he consumer supplies ˆ unis of labor in period so period income is p( ˆ ) he financial capial growh equaion is herefore K = ( + ˆ ˆ r)( K + p( ) p ) = ( + r)( K + + p c) where c = p oal period wealh is he presen value of he labor endowmen plus financial capial ˆ ˆ ˆ p p W = K + p r ( + r) oal period + wealh is W = K + p ˆ p ˆ pˆ r ( + r) herefore + + W K pˆ pˆ = r + r + r ( + r) + + Answers o Chaper 6 page 3

4 John Riley December 00 Subsiuing for K +, W ˆ ˆ + p p = K ˆ + p c = W c + r + r ( + r) pˆ he presen value of lifeime human capial is W = ( + ) r = Since he unconsrained soluion grows a a consan rae, i mus violae he upper bound if is sufficienly large Appealing o he answer o (b), here eis some S consrain is binding if and only if and only if > S ha is he reires from he work-force < such ha he > S hus he consumer consumes all his ime f]if SECION 6: A FAMILY OF DYNAMIC OPIMIZAION PROBLEMS Eercise 6-: Infinie horizon when is a vecor An infiniely lived consumer has a period uiliy is n u ( ) = α ln where j j j= n α j = he sequence of fuure spo process is { p } = j= and he ineres rae is r Iniial wealh is W (a) Solve for he opimal spending sequence { c } where = c = p HIN: Firs show ha ( W+, c+ ) = ( + r) δ ( W, c) saisfies he FOC and growh equaion for all (b) Hence solve for he opimal consumpion sequence { } = (c) Compare he soluion o he soluion of he finie horizon opimizaion problem he finie horizon version of his problem can be found in he eercises for secion Appealing o he same argumens, we focus on he sequence of opimal ependiures c = p he infinie horizon opimizaion problem can hen be wrien as follows Answers o Chaper 6 page 4

5 John Riley December 00 { ( ) = δ ln + = ( + )( )} = = Ma u c W r W c he FOC for his opimizaion problem esablish ha opimal ependiure growh is c + = ( + r) δ c If W = ( + r) δw, hen, from he wealh accumulaion equaion, + ( + r) W = W = ( + r)( W c ) δ + Solving his equaion, c = ( δ ) W hus { c, W} = {( δ ) W, W} is an infinie sequence saisfying he FOC for all his is an opimal sequence if he ransversaliy Condiion holds, ie u δ ( c) W 0 Bu u W ( c) = Hence u δ ( c) W = δ = hus he c c δ ransversaliy Condiion is indeed saisfied (b) he analysis of he opimal period consumpion vecor is eacly he same as in he finie horizon model (c) he finie horizon soluion approaches he infinie horizon soluion as SECION 63: HE RAMSAY PROBLEM Eercise 63-: Adverising and Repuaion Demand for a firm's produc in period is a funcion of he price ha he firm charges and of he repuaion of he produc K p = p( q, K ) = q α K β, 0 < β < α < he cos of producion is c per uni (a) Show ha maimized profi a ime, Π( K ) = Ma{ pq cq } is an increasing q concave funcion of K Repuaion depreciaes over ime and is increased via adverising ependiures according o: K K = f a + ( ) ν K Answers o Chaper 6 page 5

6 John Riley December 00 where a is adverising in period and f () is concave (b) Wrie down he opimizaion problem for he firm, assuming an ineres rae of r and an infinie ime horizon (c) Show ha { a, K } mus saisfy + r ν = Π ( K ) + f ( a ) f ( a ) + Hence show ha a + r +ν > a if and only if Π ( K ) < f ( a ) (d) Draw he phase diagram for his problem Hence show ha opimal rae of adverising declines over ime as he firm builds up is repuaion (e) Suppose he firm is a is seady sae repuaion level If he ineres rae suddenly falls and i is believed ha i will remain forever a his new lower level, wha will be he firm's adverising policy? (a) Revenue in period is π q K p c q q K cq q K c q α β α β (, ) = ( ) = = ( ) FOC π α β ( qk, ) = ( α) q K c= 0 q Subsiuing back ino he epression for profi, αc Π ( K) = ( ) q( K) α α ( α) β Also, from he FOC, q = K Hence c β α α α qk ( ) = ( ) K c hus he profi funcion is sricly concave if and only α > β (b) he presen value of profi is = ( ) ( Π( K) a) + r he Lagrangian for he opimizaion problem is Answers o Chaper 6 page 6

7 John Riley December 00 FOC L = ( ) ( Π( K) a) + λ( f( a) + ( ν) K K+ ) + r = L = λ = a + r + ( ) 0 L = ( ) Π ( K+ ) λ + ( ν) λ+ = 0 K + r Eliminaing he shadow prices yields he following adverising growh equaion Subracing hus + r ν Π ( K + ) = f ( a ) f ( a ) + r +ν from boh sides of his equaion, f ( a ) r + ν Π ( K + ) = ( ν )( ) f ( a ) f ( a ) f ( a ) + r + ν a+ > a Π ( K+ ) < f ( a ) (c) he phase diagram is almos eacly he same as he phase diagram for he previous invesmen model hus he analysis also essenially he same SECION 64: DYNAMIC PROGRAMMING REFORMULAION Eercise 64-: A-K model wih known shocks Consider he following problem V( K ) = Ma { u ( c ) K A( K + u ) c, c 0} + { c, K+ } = = where { u } = is a sequence of known (ie fully anicipaed) shocks Answers o Chaper 6 page 7

8 John Riley December 00 (a) Define W rewrien as follows u = + Show ha he dynamic programming problem can be s K s s= A VW ( ) = Mau { ( c) + δvw ( ) W AW c, c 0} + + c, W+ (b) Hence draw a conclusion as o how he size of he shocks affecs he opimal consumpion sequence SECION 65: SAVING AND BORROWING Eercise 65-: A debor who saves for reiremen Suppose an individual earns an income a he rae w over [0, w ] and mus hen reire He lives off his savings over he inerval [ w, ] He discouns he fuure a he rae δ and he ineres rae for borrowing and lending is r where r describe his life-ime pah of consumpion and asse accumulaion Hin: Assume ha he wishes o begin his reiremen wih asse level < δ Use wo phase diagrams o k ( w) = kˆ Analyze he inerval [0, w ] wih his erminal condiion hen appeal o he discussion above o characerize he pah for > (a) he growh equaion is dk rk() w() () d = + We wrie down he Hamilonian δ H = e u( ) + λ( rk+ w( ) ) w hen d λ H H = = rλ and e δ = u ( ) λ = 0 d k r Inegraing he differenial equaion for λ, λ( ) = λ(0) e hen δ u ( ) = λ( ) e = λ(0) e ( δ r) I follows ha marginal uiliy grows coninuously a he rae δ r > 0 Hence ( ) is a coninuous decreasing funcion Answers o Chaper 6 page 8

9 John Riley December 00 Le k ( w) = kˆ be he opimal erminal sock when he individual reires his mus be posiive For large enough, he phase diagram is herefore almos eacly as in Eercise 65- he only difference is ha he erminal sock is posiive his depiced below dk For > w he capial growh equaion is = rk() () he new phase diagram is d also depiced below We have argued ha ( ) mus be coninuous Consumpion and capial hus boh decrease as depiced dk rk w 0 d = + = I k ( w) k Fig : Phase diagram w II dk rk w 0 d = + = I k ( w) k Fig : Phase diagram, > w Answers o Chaper 6 page 9