Mathematical Foundations -1- Choice over Time. Choice over time. A. The model 2. B. Analysis of period 1 and period 2 3

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1 Mahemaial Foundaions -- Choie over Time Choie over ime A. The model B. Analysis of period and period 3 C. Analysis of period and period + 6 D. The wealh equaion 0 E. The soluion for large T 5 F. Fuure markes and finanial inermediaries 6 Firs draf John Riley Sepember 7, 06

2 Mahemaial Foundaions -- Choie over Time T period model Iniial finanial apial K Consumpion sequene { } T Inome sequene { y } T Finanial apial sequene { K } T Life ime uiliy: T U ( ) u( ) u( )... u( T ) where u ( ) is srily inreasing and srily onave. Ineres rae r John Riley Sepember 7, 06

3 Mahemaial Foundaions -3- Choie over Time B. Analysis of he firs wo periods Assumpion : The fuure is weighed less heavily (i.e. disouned ) Le K 3 be he opimal period 3 finanial apial. Fix his and onsider he firs wo periods of his model. Period budge onsrains K ( r)( K y ) () K ( r)( K y ) () 3 3 In presen values, K K y r K 3 K y ( r) r r r Add and rearrange y K y K r r ( r) 3 John Riley Sepember 7, 06

4 Mahemaial Foundaions -4- Choie over Time Eonomiss refer o he presen value of he fuure wage inome sream as he onsumer s human apial. The firs period wealh of he onsumer is he sum of he iniial finanial apial and he human apial. W K PV ( y, y ) Then he wo period budge onsrain is K PV (, ) W ( r ) 3 John Riley Sepember 7, 06

5 Mahemaial Foundaions -5- Choie over Time A neessary ondiion for lifse-ime uiliy maximizaion is ha (, ) mus solve he following maximizaion problem K3 Max{ u( ) u( ) W )} ( r) r The Lagrangian: K3 L U( ) ( W PV ( ) ( r) K3 u( ) u( ) ( W ) ( r) r We will hoose a CES funion so ha. Then he FOC are as follows: 0 L u( ) 0,,..., T L u( ) 0 r,,..., T Therefore u( ) u( ) where ( r) John Riley Sepember 7, 06

6 Mahemaial Foundaions -6- Choie over Time C. Analysis of period and period + Nex we noe ha we an also fix { } { } and hene s s neessary ondiion for life-ime uiliy maximizaion is ha maximizaion problem K Max{ u( ) u( ) W )}, ( r) r W (, ) and also se K K. A mus solve he following The FOC are he same as for he firs wo periods. Hene he following is he growh equaion for all periods. Consumpion growh equaion u u( ) ( ) This equaion impliily defines a onsumpion growh equaion f ( ) Case (i) hen from he growh equaion, Given he onaviy assumpion i follows ha u( ) u( ). John Riley Sepember 7, 06

7 Mahemaial Foundaions -7- Choie over Time { } T is a srily inreasing sequene. Case (ii) hen { } T is a srily dereasing sequene. John Riley Sepember 7, 06

8 Mahemaial Foundaions -8- Choie over Time Complee soluion for he CES family Consumpion growh equaion u u( ) ( ) CES uiliy u( ) / U () ( / ) T /, T ( ) ln( ) U Then u( ) and so / u( ) u( ) / ( ). Growh equaion FOC u( ) u( ) / ( ) ( r) John Riley Sepember 7, 06

9 Mahemaial Foundaions -9- Choie over Time Therefore (( r) ). Assumpion : r Finanial apial growh equaion Hene K ( r)( K y ) K K y r John Riley Sepember 7, 06

10 Mahemaial Foundaions -0- Choie over Time D. Wealh growh equaion y y y W K y r ( r) ( r) K Sine K y r Therefore y y 3 y W 4 ( K y )... 3 r ( r) ( r) K y y y... 3 r r ( r) ( r) ( y y r)( W ) K y... r ( r) W The same argumen holds for all.then he wealh growh equaion is W ( r)( W ) Sine he onsrain funions are all linear hey are onave. Sine he uiliy funion is onave he Lagrangian is onave and so he neessary ondiions are also suffiien for a maximum. John Riley Sepember 7, 06

11 Mahemaial Foundaions -- Choie over Time Analysis of possible wealh onsumpion pahs For he finie horizon problem he onsumpion growh equaion (derived from he FOC) is. Therefore W ( r)( W ) W ( r) ( r) r W ( ) ( r) Given Assumpion, he slope of his line is Greaer han. (i) Choose and so, so ha W W John Riley Sepember 7, 06

12 Mahemaial Foundaions -- Choie over Time W Then for all, Sine onsumpion grows a he rae so does oal wealh. (ii) Choose so ha W W ) ( r) From he Figure i should be lear ha, W { } is a srily inreasing funion. The oal wealh grows faser han onsumpion. Bu final wealh mus be zero. Thus hese pah do no saisfy he erminal onsrain W. T 0 Thus no suh pah is opimal. John Riley Sepember 7, 06

13 Mahemaial Foundaions -3- Choie over Time (ii) Choose so ha W W ) ( r) From he Figure i should be lear ha W { } is a srily dereasing funion. John Riley Sepember 7, 06

14 Mahemaial Foundaions -4- Choie over Time Soluion when T 4 Wih no beques moive no apial will be lef WT W5 Afer period T so 0 SIne his poin lies on he purple line, T 4 W4 we an solve for 3. This is also shown on he verial axis. (See he lef hand marker on he 45 line.) We an hen repea hese seps solving bakwards as depied. Noe ha he wealh onsumpion raio for he las four periods mus be exaly he same. Thus wih W T 6 here is simply one more sep needed o alulae John Riley Sepember 7, 06

15 Mahemaial Foundaions -5- Choie over Time I follows ha as he number of periods inreases he wealh onsumpion raio in he early periods mus be on he doed 45 line very lose o he poin (, ) Indeed for suffiienly large T he las few periods have almos no impa he early onsumpion o wealh raio. Thus boh rise a he same onsan rae John Riley Sepember 7, 06

16 Mahemaial Foundaions -6- Choie over Time Exerise: Exhange eonomy U( x ) 5ln x 5ln x 4ln x 4ln x h h h h h 3 4 The aggregae endowmen is (5,50,6,6). 5 5 Normalizing so ha p, show ha p (,,, ) is a WE prie veor. 4 4 Class Exerise: Exhange eonomy wih wo periods U u( ()) u( ()) where u( ( )) ln ( ) ln ( )). 4 5 The aggregae firs period endowmen is (5,50). The aggregae seond period endowmen is (6,6). The period prie of ommodiy is. Suppose ha in period here are boh spo markes (markes for delivery on he spo ) and fuures markes (markes for fuure delivery of eah ommodiy.) 5 5 (a) Explain why he spo prie veor is p() (, ) and he fuures prie veor is p() (, ) (b) Suppose ha here are no fuures markes. If he ineres rae is zero wha are he equilibrium spo pries and fuure spo pries? 4 4 John Riley Sepember 7, 06

17 Mahemaial Foundaions -7- Choie over Time John Riley Sepember 7, 06