Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing


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1 MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp issn X eissn hp://dx.doi.org/.287/moor INFORMS Verificaion Theorems for Models of Opimal Consumpion Invesmen wih Reiremen Consrained Borrowing INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Philip H. Dybvig, Hong Liu Olin Business School, Washingon Universiy in S. Louis, S. Louis, Missouri 633 hp://dybfin.wusl.edu/, hp://apps.olin.wusl.edu/faculy/liuh/} Proving verificaion heorems can be ricky for models wih boh opimal sopping sae consrains. We pose solve wo alernaive models of opimal consumpion invesmen wih an opimal reiremen dae (opimal sopping various wealh consrains (sae consrains. The soluions are parameric in closed form up o a mos a consan. We prove he verificaion heorem for he main case wih a nonnegaive wealh consrain by combining he dynamic programming Slaer condiion approaches. One unique feaure of he proof is he applicaion of he comparison principle o he differenial equaion solved by he proposed value funcion. In addiion, we also obain analyical comparaive saics. Key words: volunary reiremen; invesmen; consumpion; free boundary problem; opimal sopping MSC2 subjec classificaion: Primary: 93, 49; secondary: 93E2, 49K, 49K2, 49L2 OR/MS subjec classificaion: Primary: finance/porfolio, finance/invesmen; secondary: finance/asse pricing Hisory: Received April 8, 2; revised November 5, 2, April 7, 2, June 3, 2. Published online in Aricles in Advance Ocober 4, 2.. Inroducion. Reiremen is one of he mos imporan economic evens in a worker s life. This paper conains a rigorous formulaion analysis of several models of life cycle consumpion invesmen wih volunary or maory reiremen wih or wihou a borrowing consrain agains fuure labor income. In hese models, opimal consumpion jumps a reiremen, if reiremen is volunary, he opimal porfolio choice also jumps a reiremen. If reiremen is volunary, he opimal reiremen rule gives human capial a negaive bea if wages are uncorrelaed wih he sock marke because reiremen comes laer when he marke is down. This leads o aggressive invesmen in he marke, a resul ha is dampened when borrowing agains fuure labor income is prohibied may be reversed when wages are posiively correlaed wih marke reurns. In he companion paper, Dybvig Liu [] focus on he economic inuiions for hese resuls. In his paper, we provide rigorous proofs. The main resuls in his paper are explici parameric soluions (up o some consans wih verificaion heorems analyical comparaive saics. In paricular, we combine he dual approach of Pliska [8], He Pagès [3], Karazas Shreve [5] Karazas Wang [6] wih an analysis of he boundary o obain a problem we can solve in a parameric form even if no known explici soluion exiss in he primal problem. Having an explici dual soluion allows us o derive analyically he impac of parameer changes, more imporanly, allows us o prove a verificaion heorem showing ha he firsorder (Bellman equaion soluion is a rue soluion o he choice problem. Compared o he exising lieraure (e.g., Pliska [8], Karazas Wang [6], he noborrowing consrain agains fuure labor income significanly complicaes he derivaions he proof of he verificaion heorem. The proof is suble because of ( he nonconvexiy inroduced by he reiremen decision, (2 he marke incompleeness (from he agen s view caused by he nonnegaive wealh consrain, (3 he echnical problems caused by uiliy unbounded above or below. Two common approaches o proving a verificaion heorem are he dynamic programming (FlemingRichel approach he separaing hyperplane (Slaer condiion approach. Boh approaches encouner difficulies in our seing so we use a hybrid of he wo (a separaing hyperplane afer reiremen dynamic programming before reiremen. The wo are combined wih opional sampling, where he coninuaion afer reiremen is replaced by he known value of he opimal coninuaion. One of he mos challenging asks for proving a verificaion heorem for opimal sopping problems is o show ha he proposed value funcion saisfies cerain inequaliy condiions so ha i is indeed opimal o sop a he proposed boundaries. One unique feaure of our proof is ha his is shown indirecly by applying he comparison principle o he differenial equaions solved by he proposed value funcion. This indirec mehod makes he proof simple elegan. So far as we know, we are he firs o use his approach o prove a verificaion heorem for his ype of conrol problem involving an opimal sopping ime. The res of he paper is organized as follows. Secion 2 presens he formal choice problems used in he paper. Secion 3 presens analyical soluions, comparaive saics, proofs. Secion 4 closes he paper. 62
2 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 62 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. 2. Choice problems. We consider he opimal consumpion invesmen problem of an invesor who can coninuously rade a riskfree asse n risky asses. The riskfree asse pays a consan ineres rae of r. The risky asse price vecor S evolves as ds = d + dz S where Z is a sard n dimensional Wiener process; is an n consan vecor is an n n inverible consan marix so ha marke is complee wih no redundan asses; he division is elemen by elemen. The invesor also earns labor income y : [( y y exp y y ] y + 2 y Z ( where y is he iniial income from working y y are consans of appropriae dimensions. The invesor can choose o irreversibly reire a any poin in ime. The arrival ime d of he invesor s moraliy follows an independen Poisson process wih consan inensiy. The invesor can purchase insurance coverage of B W agains moraliy, where W is he financial wealh of he invesor a ime so ha, if deah occurs a, he invesor has a beques of W +B W = B. To receive he insurance coverage B W a he ime of moraliy, he invesor pays he insurer a a rae of B W, i.e., insurance is assumed o be fairly priced a he moraliy rae per uni of coverage. The invesor derives uiliy from ineremporal consumpion beques. The invesor has a consan relaive risk aversion (CRRA, ime addiive uiliy funcion (2 wih a subjecive ime discoun rae : [ d ( E e R c + R Kc d + e d kb d ] (2 where > is he relaive risk aversion coefficien, he consan K > indicaes preferences for no working in he sense ha he marginal uiliy of consumpion is greaer afer reiremen han before reiremen, he consan k > measures he inensiy of preference for leaving a large beques, he limi k implemens he special case wih no preference for beques, R is he righconinuous nondecreasing indicaor of he reiremen saus a ime (which is afer reiremen before reiremen. The sae variable R is he reiremen saus a he beginning of he invesmen horizon. Define { if saemen S is rue S = oherwise ( e T + if g (3 T + if = where r + y + y > (4 is he effecive discoun rae for labor income (assumed o be posiive r (5 is he price of risk. Below are he wo choice problems we focus on in his paper. Problem. Given iniial wealh W, iniial income from working y, he deerminisic ime o reiremen T wih associaed reiremen indicaor funcion R = T, choose adaped nonnegaive consumpion c, adaped porfolio, adaped nonnegaive beques B o maximize expeced uiliy of lifeime consumpion beques [ d ( E e R c + R Kc kb d + e d ] d = corresponds o he log uiliy case, which can be examined similarly. Mos of our resuls in he paper apply o he log case by aking.
3 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 622 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS subjec o he budge consrain W = W + rw s ds + s r ds + dz s + W s B s ds c s ds + R s y s ds (6 he labor income process (, he limied borrowing consrain W gy (7 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. where gy is he marke value a of he fuure labor income. Le denoe he ype of borrowing consrains, wih = for he limied borrowing ype = for he noborrowing ype. Problem corresponds o Problem of Dybvig Liu [], Problem 2 wih = corresponds o Problem 2 of Dybvig Liu [], Problem 2 wih = corresponds o Problem 3 of Dybvig Liu []. Problem 2. Given iniial wealh W, iniial income from working y, iniial reiremen saus R, borrowing consrain ype, choose adaped nonnegaive consumpion c, adaped porfolio, adaped nonnegaive beques B, adaped nondecreasing reiremen indicaor R (i.e., a righconinuous nondecreasing process aking values o maximize he expeced uiliy of lifeime consumpion beques (2 subjec o he budge consrain (6, he labor income process before reiremen (, he borrowing consrain W R y (8 where R y / is he marke value a of he subsequen labor income. To summarize he differences across he problems, moving from Problem o Problem 2, he fixed reiremen dae T (R = T is replaced by free choice of reiremen dae (R, a choice variable along wih a echnical change in he calculaion of he marke value of fuure labor income gy o R y / when =. When =, hen he invesor faces a noborrowing consrain W Remark. Because he ime of moraliy d is independen of he Brownian moion, we have ha he objecive funcion [ d ( E e R c + R Kc [ ( = E e + R c d + e d [ ( K R c = E e + + kb kb d ] + R Kc + kb ] d ] d We denoe he value funcions for Problems 2 by vw y V W y R, respecively. Noe ha, because boh problems become he same afer reiremen, we have vw y T = V W y = V W y (9 3. The analyical soluion comparaive saics. The general idea of solving Problems 2 is o perform a change of variables ino a dual variable, he marginal uiliy of consumpion. The general advanage of his dual approach (especially for Problem 2 is ha i linearizes he nonlinear HamilonJacobiBellman (HJB equaion in he primal problem. This is consisen wih he mehod of Pliska [8] of convering a dynamic budge consrain ino a saic budge consrain. Le + r + + /2 ( For our soluions, we will assume >, which is also he condiion for he corresponding Meron problem (Meron [7] o have a soluion because, if <, hen an invesor can achieve infinie uiliy by delaying consumpion. Define he sae price densiy process by e r++ 2 Z (
4 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 623 This is he usual sae price densiy adjused o be condiional on living, given he moraliy rae fair pricing of long shor posiions in erm life insurance. Also, define b / INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. The soluion o Problem can be saed as follows. Theorem 3.. he iniial values ( f ˆ exp + k b T k b ˆ K b + k b Suppose > ha he limied borrowing consrain is saisfied wih sric inequaliy a W > gy (2 The soluion o an invesor s Problem can be wrien in erms of a dual variable ˆx (a normalized marginal uiliy of consumpion, where ( W + gy ˆx e + f y (3 Then, he opimal wealh process is he opimal consumpion policy is he opimal rading sraegy is = y he opimal beques policy is W = f y ˆx / gy (4 [ r Furhermore, he value funcion for he problem is c = K br y ˆx / (5 f ˆx / ] y g B = k b y ˆx / (6 W + gy vw y = f Proof. Because Problem can be ransformed ino a sard dual problem wih minor modificaions (e.g., Pliska [8], we omi he proof here. See Dybvig Liu [] for a skech of he proof using a separaing hyperplane o separae preferred consumpions from he feasible consumpions. Unlike Problem, Problem 2 also requires one o solve for he opimal reiremen decision. We conjecure ha i is opimal o reire when he wealhoincome raio is high enough, which corresponds o when a new dual variable x his a lower bound x. Then, as in a ypical opimal sopping problem, one imposes C condiion for he dual value funcion across x. In he presence of he noborrowing consrain (i.e., =, one also imposes he noriskyinvesmen condiion (i.e., he second derivaive of he dual value funcion is when wealh W his (or, equivalenly, when x his an upper bound x. Afer obaining he soluions, we verify ha all of our conjecures are indeed correc. Recall he definiions of in ( in (4. Define y y y 3 y y (7 3
5 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 624 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/ (8 3 (9 ( b A + + xb + x ( A + x + b + b + xb + x (2 + ( ˆ/b b / x + b (2 / + ( ˆb x + x (22 b where is he unique soluion o q = 2 hus where ( K b q b + k b b ( + + ( K b b + k b b + ( + Then, he soluion o Problem 2 can be saed as follows. + b b + (23 Theorem 3.2. Suppose >, >, 2 >, 3 > ha he borrowing consrain holds wih sric inequaliy a he iniial condiion: 3 W > R y (24 The soluion o an invesor s Problem 2 can be wrien in erms of a dual variable x, where ( x x e + y x + e + y /y y max sup s min x e +s s y s /y / x where x solves y x x R = W (25 { ( = R inf x e + y x} y (26 ˆ xb b x R = A + x + A x + xb b + x Then, he opimal consumpion policy is he opimal rading sraegy is c = K br y x / if R = or x x oherwise (27 = y rx xx x R y x xx x R + xx R he opimal beques policy is B = k b y x / 2 The exisence uniqueness of he soluion o q = is shown in Lemma > is o ensure he finieness of he labor income; 2 > is o ensure he finieness of he expeced uiliy from working consuming labor income forever; 3 > is o avoid he degeneraion of he dual process x o a deerminisic funcion of ime. The razor edge case where 3 = is less ineresing needs a separae reamen.
6 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 625 he opimal reiremen policy is R = he corresponding reiremen wealh hreshold is W = y x x INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. he opimal wealh is Furhermore, he value funcion is where x solves W = y x x R (28 V W y R = y x R x x x R (29 y x x R = W (3 The invesor s problem can be associaed wih he dual opimal sopping problem, where he invesor solves [ ( ( ] x y max E e +s y s + k b xb s b + x s ds + e + y ˆ xb b subjec o (, he borrowing consrain where dx x = x d + x dz y xx y y x r 2 y y + y y (3 x y (32 The ransformed dual value funcion x y x/y y hen saisfies a variaional inequaliy 4 wih boundary condiions where L = x < x < x L < < x < x x > ˆ xb b x < x < x x = ˆ xb b < x x x x < < x < x xx x > < x < x x x x x = xx x = L 2 3x 2 xx 2 x x 2 + k b xb b + x 4 We do no prove ha he dualiy gap is zero or even ha he firsorder soluion of he dual problem is an acual soluion. However, we do no need hese resuls because our verificaion heorem works wih he primal objecive funcion.
7 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 626 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. When =, Problem 2 can be ransformed ino one ha has been sudied by Karazas Wang (2 wih minor modificaions. 5 Therefore, we will only presen he proof for he case wih he noborrowing consrain, i.e., =. For noaion simpliciy, we use V W y R x R o represen V W y R x R, respecively. Le Then, by (27, (33, (34, we have x A + x + A x + xb b + x (33 x R = ˆx ˆ xb b (34 { ˆx if R = or x x x oherwise The following lemmas are useful for he proof of Theorem 3.2. Lemma 3.. Suppose =, >, >, 2 >, 3 >. Suppose here exiss a soluion o Equaion (23 (o be shown in Lemma 3.4. Then, (i ˆx is sricly convex sricly decreasing for x (ii x x, we have x ˆx; x x x, we have x x ˆ x x ( K b x < (35 b (iii ( + k b A < A + > x > b (iv x is sricly convex sricly decreasing for x < x. (v Given W >, here exiss a unique soluion x > o (25. In addiion, W defined in (28 saisfies he borrowing consrain (8. Proof of Lemma 3.. (i > implies ha b = / <. Then, because >, direc differeniaion shows ha ˆx is sricly convex sricly decreasing for x. (ii Le hx x ˆx I can be easily verified ha wih 2 3x 2 ˆ xx x 2 x ˆ x x 2 ˆx K b + k b xb b = (36 2 3x 2 xx x 2 x x x 2 x + k b xb + x = (37 b x = ˆx (38 x x = ˆ x x (39 x x = (4 xx x = 5 One such ransformaion is W W y y W y y We hank an anonymous referee for poining his ou. B B y c c y Ṽ W y R y V y W y R
8 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 627 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Then, by (36 (37, hx mus saisfy 2 3x 2 h 2 xh 2 h = K b x b x (4 b By (38 (4 he fac ha ˆx is monoonically decreasing for x >, we have Differeniaing (4 once, we obain hx = h x = h x > (42 2 3x 2 h xh h = K b x b (43 We consider wo possible cases. Case. K b x b <. In his case, he righh side of Equaion (43 is negaive. Because >, h x canno have any inerior nonposiive minimum. To see his, suppose ˆx x x achieves an inerior minimum wih h ˆx. Then, we would have h ˆx h ˆx =, which implies ha he lefh side is posiive. This is a conradicion. Because h x =, h x >, we mus have h x > for any x x x; oherwise, here would be an inerior nonposiive minimum. Then, he fac ha hx = implies ha hx > for any x x x. Because h x > for any x x x h x =, we mus have h x. In addiion, if h x were equal o, hen we would have h x < by (42 (43 because K b x b <. However, his would conradic he fac ha h x > for any x x x h x =. Therefore, we mus have h x >. Then, (4, (42, h x > imply ha ( K b x < b Case 2. K b x b. In his case, we mus have < b < because K >. Therefore, x K b < K b /b. This implies ha h x > by (4 (42. There exiss > such ha h x > for any x x x + because h x =. The righh side of Equaion (43 is monoonically decreasing in x. Le x be such ha he righh side of (43 is. Then, for any x x, he righh side is nonnegaive hus h x canno have any inerior nonnegaive (local maximum in x x for similar reasons o hose in Case. There canno exis any ˆx x + x such ha h ˆx. If x < x, hen, for any x x x, he righh side is nonposiive hus h x canno have any inerior nonposiive (local minimum in x x. There canno exis any ˆx x x such ha h ˆx. Therefore, here canno exis any ˆx x x such ha h ˆx hus we have h x > hx > for any x x x. Now, we show, for boh cases, ha hx > for any x < x. Equaion (35 implies ha he righh side of (4 is posiive for x < x h canno achieve an inerior posiive maximum for x < x. On he oher h, h x >, h x is coninuous a x h x =, which imply ha here exiss an > such ha x x x h x < Thus, x x x, hx >, herefore, x < x, hx > ; oherwise, h would achieve an inerior posiive maximum in x. (iii Recall ha =. I can be shown ha A + = ˆ + b x b + x b + + = + + k b + bb Equaion (35 hen implies ha A + >. Because we also have (2, x mus saisfy x > ( + b + Because + b + > b we have ( b x > (44 which (by he definiion (9 implies ha A <.
9 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 628 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS (iv Differeniaing (33 wice, we have, for x < x, xx x = ( A + + x + b + A + x b b x b 2 > xx xx/ x b 2 = (45 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. where he inequaliy follows from he fac ha d dx A + + x + b + A + x b < This is implied by A <, A + >, + > > b >, <, he las equaliy in (45 follows from xx x =. Thus, x is sricly convex x < x. Because x x = x < x, xx x >, we mus also have x < x, x x <. (v By par (i, par (iv, x x = ˆ x x, x x R is coninuous sricly increasing in x x. By inspecion of (33 (34, x x R akes on all nonposiive values. Because y >, here exiss a unique soluion x > o (25 for each W >. Also, because x x R, (28 implies ha W Though he dual approach yields almos explici soluions, i is simpler o show he opimaliy of he cidae policies in he primal for his combined opimal sopping opimal conrol problem. Define ( ] c M = e [ +s s R s + kb s ds + V W s y s dr s + R e + V W y (46 The following lemma is a generalized dominaed convergence heorem ha is required for he proof of Lemma 3.3. Lemma 3.2. Suppose ha a.s. convergen sequences of rom variables X n X Y n Y saisfy X n Y n EY n EY <. Then, EX n EX. Proof of Lemma 3.2. Because X n Y n, by Faou s lemma, liminf EX n EX liminf EY n X n EY X. These inequaliies imply ha boh limsupex n EX liminf EX n EX because EY n EY <. Therefore, we mus have EX n EX. Lemma 3.3. Suppose =. Given he definiions for Theorem 3.2, (i M as defined in (46 is a supermaringale for any feasible policy a maringale for he claimed opimal policy. (ii For any feasible policy, lim E R e + V W y (47 wih equaliy for he claimed opimal policy. Proof of Lemma Define W = y x x. Then, for any W, V W y V W y (48 wih equaliy for W W. This can be shown as follows. Le x x R be such ha y x x=w y x x R =W. Then, we have x x R x x R x x R x x R =x x x x R x x R where he firs inequaliy follows from x x by Lemma 3. he second inequaliy follows from he convexiy of x. Afer rearranging, we obain (48. Applying he generalized Iô s lemma o M (see, e.g., Harrison [2, 4.7], we have M = M ( c R s {e +s e +s V W s y s V W s y s dr s s +kb s +LV W sy s } ds R s e +s V W W s y s s +y s V y W s y s y dz s (49
10 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 629 where LV = 2 V W W + y y V Wy + 2 y yy 2 V yy +rw + r+w B c + R y V W + y V y +V INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. By he definiions of V,, c B R W, he fac ha x saisfies (36 (39, we obain ha he firs inegral is always nonposiive for any feasible policy cbr is equal o zero for he claimed opimal policy c B R. By (48, he hird erm in (49 is always nonposiive for every feasible reiremen policy R equal o zero for he claimed opimal policy R. In addiion, using he expressions for he claimed opimal, V, B, W, we have ha, under he claimed opimal policy, he sochasic inegral is a maringale because y is a geomeric Brownian moion; wih =, x is bounded beween x x before reiremen; x is also a geomeric Brownian moion afer reiremen. This shows ha M is a local supermaringale for all feasible policies a maringale for he claimed opimal policy. Nex, we show ha M is acually a supermaringale for all feasible policies. Firs, we show ha V W y for every feasible policy. By (29 (3, V W W y=y x > hus V W y increases in W. If <, hen V W y because V W y V W y. If >, hen V W y< because V W y V W y=v W y< Therefore, V W y for every feasible policy. If <, hen V W y > he local supermaringale M is hen always nonnegaive hus a supermaringale. Suppose >. By (49, here exiss an increasing sequence of sopping imes n such ha M EM n, i.e., n ( ] c V W y E e [ R +s s s s +kb s ds +V W s y s dr s +E R n e + n V W n y n (5 Because he inegr in he inegral of (5 is always negaive, his inegral is monoonically decreasing in ime. In addiion, where R e + V W y e + V y = V e + y V N (5 N e y + yz (52 is a maringale wih EN =, he second inequaliy follows from V being negaive increasing in W W >, he equaliy follows from he form of V as defined by (29 (3, he las inequaliy follows from V < 2 >. In addiion, V >. Therefore, aking n in (5, by he monoone convergence heorem for he firs erm Lemma 3.2 for he second erm, we have ( ] c V W y E e [ R +s s s +kb s ds +V W s y s dr s +E R e + V W y Tha is, M EM for any. Because his argumen applies o any ime s, we have M s E s M for any s. Thus, M is a supermaringale for all feasible policies. 2. Because V W y for every feasible policy, we have lim E R e + V W y = lim E R e + y x x x x lim EL e + y + ˆe +/ b = (53
11 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 63 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS for some consan L, where he second inequaliy follows from he fac ha when =,x, x, x x are all bounded R = for >. The las equaliy in (53 follows from he condiions ha > 2 >. Therefore, for he claimed opimal policy, we obain lim E R e + V W y = INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. For any feasible policy, if <, hen V W y> herefore (47 holds. If >, because 2 >, we have lim Ee + y =. Therefore, aking he limi as in (5, we have ha (47 also holds. This complees he proof. Lemma 3.4. Suppose >, >, 2 >, 3 >. Then, here exiss a unique soluion o Equaion (23 (( K b < =min b+k b Proof of Lemma 3.4. Because >, >, 2 >, 3 >, + >>b > < Nex, because b + dominaes + as, we have If =, i is easy o verify ha limq= lim If = K b /b+k b <, hen we have I follows ha K b b+k b b + b + =+ q = + + K b +k b < + +k b K b b+k b b + + = + + q = K b b+k b b = + >> + ( + ( + + < Then, by coninuiy of q, here exiss a soluion such ha q =. Suppose here exiss anoher soluion ˆ such ha q ˆ=. Le V W y W be he value funcion boundary, respecively, corresponding o le ˆV W y ˆ W be he value funcion boundary, respecively, corresponding o ˆ. Wihou loss of generaliy, suppose W > ˆ W. Because ˆ W is he reiremen boundary, he value funcion corresponding o ˆ for W >W > ˆ W is equal o V W y. However, Lemma 3. implies ha V W y> V W y for any W < W. This implies ha ˆ W canno be he opimal reiremen boundary, which conradics Theorem 3.2. Therefore, he soluion o Equaion (23 is unique. We are now ready o prove Theorem 3.2. Proof of Theorem 3.2. If R =, hen Problem 2 is idenical o Problem. Therefore, he opimaliy of he claimed opimal sraegy follows from Theorem 3.. In addiion, as noed in (9, we have V W y=vw yt where vw yt (independen of T is he value funcion afer reiremen for Problem. From now on, we assume w.l.o.g. ha R =. I is edious bu sraighforward o use he generalized Iô s lemma Equaions (7 (23 (27 (28 o verify ha he claimed opimal sraegy W, c,, R in Theorems saisfy he budge consrain (6. In addiion, by Lemma 3., x exiss is unique W saisfies he borrowing consrain in each problem. Furhermore, by Lemma 3.4, here is a unique soluion o (23.
12 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 63 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. By Doob s opional sampling heorem, we can resric aenion w.l.o.g. o he se of feasible policies ha implemen he opimal policy saed in Theorem afer reiremen. The uiliy funcion for such a sraegy can be wrien as ( ] c E e [ R +s s s +kb s ds +V W s y s dr s By Lemma 3.3, M is a supermaringale for any feasible policy cbr a maringale for he claimed opimal policy c B R, which implies ha M EM, i.e., ( ] c V W y E e [ R +s s s +kb s ds +V W s y s dr s +E R e + V W y (54 wih equaliy for he claimed opimal policy. In addiion, by Lemma 3.3, we also have ha lim E R e + V W y wih equaliy for he claimed opimal policy. Therefore, aking he limi as in (54, we have ( ] c V W y E e [ R +s s s +kb s ds +V W s y s dr s wih equaliy for he claimed opimal policy c B R This complees he proof. Theorem 2 provides essenially complee soluions because he soluion for x, given W, requires only a onedimensional monoone search o solve Equaion (25, for, one only needs o solve Equaion (23. We nex provide resuls on compuing he marke value of human capial a any poin in ime, which is useful for undersing much of he economics in he paper. Proposiion 3.. Consider he opimal policies saed in Theorems Afer reiremen, he marke value of he human capial (i.e., he capialized labor income is zero. Before reiremen, in Theorem 3., he marke value of he human capial is Hy =gy where y g are given in ( (3. In Theorem 3.2, if =, hen he marke value of he human capial is Hx y = y x x + if =, hen he marke value of he human capial is where Proof. Hx y = y Ax +Bx + + A= + x x + + x + x + B = x + x + x + For Problem, by Iô s lemma, (, (3, (4, (, simple algebra, Furhermore, d gy = R y d + gy y dz E s gsy s 2 y y ds < because s y s is a sard lognormal diffusion he oher facors are bounded for any zero for >T. Therefore, he local maringale gy + s R s y s ds =gy + s gsy s y dz s (55 s=
13 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 632 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS is a maringale ha is consan for >T. Picking any >maxt, he definiion of a maringale implies ha ] gy + s R s y s ds =E [ gy + s R s y s ds INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Now, >maxt implies ha g=, he inegral on he lefh side is known a. Therefore, we can subrac he inegral from boh sides divide boh sides by o conclude gy = [ ] E s R s y s ds = E [ ] s R s y s ds where he second equaliy follows from he fac ha R s for s >. For he cases wih volunary reiremen, because here is no more labor income afer reiremen, he marke value of human capial afer reiremen is zero. We nex prove he claims for afer reiremen. Using he expressions of H he dynamics of x y, i can be verified ha, for x >x when = for x <x < x when =, we have ha he change in he marke value of human capial plus he flow of labor income will be given by d ( Hx y + y d = ( 2 3x 2 H xx 2 3 x H x H +y d + ( x H x x +H y dz (56 The drif erm in (56 is equal o zero afer plugging in he expressions for H (if =, he addiional local ime erm a x from applying he generalized Iô s lemma is also equal o zero because i can be verified ha H x xy =. This implies ha M Hx y + s y s ds is a local maringale. In addiion, here exiss a consan <L< such ha x H x x +H y <L y because < x >x if = x <x x if =. Because boh y are geomeric Brownian moions, we have ha M is acually a maringale. Recall he definiion (26 of he opimal reiremen ime. For all we have ] Hx y + s y s ds =E [ Hxy + s y s ds which implies ha Hx y = E [ ] s y s ds because i can be easily verified ha Hxy= Therefore, H as specified in Proposiion 3. is indeed he marke value of he fuure labor income. The following resul shows ha because of reiremen flexibiliy, human capial may have a negaive bea even when labor income correlaes posiively wih marke risk. Proposiion 3.2. As an invesor s financial wealh W increases, he invesor s human capial H decreases in Problem 2. Furhermore, if y </, hen human capial has a negaive bea measured relaive o any locally meanvariance efficien risky porfolio. The following lemma is useful for he proof of Proposiions Lemma 3.5. Suppose =, >, >, 2 >, 3 >. Then, (i ˆx is sricly decreasing sricly convex. (ii x is sricly convex x x /. (iii x, we have x ˆx x x, we have x x ˆ x x (iv Given (24, here exiss a unique soluion x > o (25. In addiion, W borrowing consrain (8. defined in (28 saisfies he
14 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 633 Proof of Lemma 3.5. (i This follows from direc differeniaion because ˆ > b <. (ii Firs, because >, >, 2 >, 3 >, i is sraighforward o use he definiions of + o show ha + >>b > < A + > INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Then, he claimed resuls also follow from direc differeniaion. (iii This follows from a similar argumen o ha of par (ii of Lemma 3.. (iv By par (i, par (ii, x x= ˆ x x, xr is coninuous sricly increasing in x. By an inspecion of (33 (34, x xr akes on all values ha are less han or equal o /. Because y >, here exiss a unique soluion x > o (25 for each W y /. Also, because x xr /, (28 implies ha W > R y / Proof of Proposiion 3.2. Firs, as shown in Lemmas , he dual value funcion defined in Theorem 3.2 is convex hus he wealh level W defined in Theorem 2 decreases wih he dual variable x. By Proposiion 3., differeniaing he expression for human capial wih respec o x for he case = yields ha human capial is increasing in x because <. Therefore, human capial decreases wih financial wealh W for = in Problem 2. For = in Problem 2, differeniaing human capial H wih respec o x, we have ha, before reiremen, Hxy x = y A x +B + x + = y + x x x + x + (( x + > (57 x where he second equaliy follows from he expressions of A B in Proposiion 3. he inequaliy follows from he fac ha + >> x < x. Thus, human capial decreases wih financial wealh W also for =. Furhermore, because x = y, if y </, hen, as he marke risk Z increases, x decreases herefore human capial decreases by (57, i.e., human capial has a negaive bea. The following resul shows ha reiremen flexibiliy ends o increase sock invesmen. Proposiion 3.3. Suppose y = >r. Then, he fracion of oal wealh W +H invesed in he risky asse in Problem 2 when = is greaer han ha in Problem. Proof. By Theorem 3., he fracion of oal wealh W +H invesed in he risky asse in Problem is consanly equal o r/. Wih =, by Theorem 3.2 Proposiion 3., we have W +H = r A + x +x b A + x +x b / x x Plugging in he expressions for A + x using he fac ha <b, we have A + x +x b A + x +x b / x x > A reiremen decision is criical for an invesor s consumpion invesmen policies. The following proposiion shows ha he presence of he noborrowing consrain ends o make an invesor reire earlier. Proposiion 3.4. In Problem 2, he reiremen wealh hreshold for = is higher han for =. Proof. Le A, A +, x be defined as in Theorem 3.2. To make he exposiion clear, we now make heir dependence on explici, i.e., using A, A +, x insead: hx=a + x +A x + xb b + x+ ˆ xb b We prove by conradicion. Suppose x x. By Lemmas , we have hx=h x x =, hx=h x x=. From he proof of Lemma 3., we have h x x> for all x x x. By (22 (44, we have x< x. Therefore, hx =hx h x x =h x x (58
15 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 634 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. The firs equaion of (58 implies ha which, in urn, implies A + x +A x + A + x A + >A + (59 because A < as shown in Lemma 3.. On he oher h, he second equaion of (58 implies ha which, in urn, implies A + x +A + x + A + x A + <A + (6 because A < + >. Resul (6 conradics (59. This shows ha we mus have x>x. Because, a reiremen, he financial wealh is equal o y x x for Problem 2 because y x x= ˆx /, we mus have ha he financial wealh level W a reiremen for = is higher han for =. One measure ha is useful for examining he life cycle invesmen policy is he expeced ime o reiremen. The following proposiion shows how o compue his measure. Proposiion 3.5. In Problem 2, suppose ha an invesor is no reired ha x 2 2 x <. Then, he expeced ime o reiremen for he opimal policy is for = for =, where Proof of Proposiion 3.5. which implies ha E x =x= logx/x 2 x 2 x >x x E x m x m x =x= 2 x 2 xm x + logx/x m 2 x 2 x x x (6 x m= 2 x 2 x Firs, we prove he resul for = in Theorem 3.2. Recall ha dx s =x s x ds + x dzs x s =x exp [ x 2 2 x s + xz s Z ] Because x >x x 2 2 x <, we have < almos surely (see, for example, Karazas Shreve [4, p. 349]. Le f x logx/x 2 x x x Then, by Iô s lemma, for any sopping ime, we have ( f x + ds =f x + 2 x xx 2 s f x xx + x x s f x + ds + 2 x dz s (62 x x which implies ha f x + ds is a maringale because i can be easily verified ha he drif erm is zero, given he definiion of f x, he sochasic inegral is a scaled Brownian moion hus a maringale. Thus, aking = + aking he expecaion in (62, we ge f x=e x =x because x + =x f x=. A similar argumen applies o he case =; noe ha whenevaluaed a x = x, he firs derivaive of he righh side of (6 wih respec o x is zero x is bounded. This complees he proof. The following proposiion shows how he expeced ime o reiremen is relaed o human capial financial wealh, which can help explain he graphical soluions presened in he previous secion. Proposiion 3.6. Suppose ha an invesor is no reired ha 2 2 x x >. Then, as he expeced ime o reiremen increases, financial wealh decreases human capial increases. Proof. By Proposiion 3.5, i can be easily verified ha he expeced ime o reiremen is increasing in x. By (57, we have ha human capial is increasing in x Proposiion 3.2 hen implies ha financial wealh is decreasing in x. Therefore, he claim holds.
16 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS Conclusion. We examine he impac of reiremen flexibiliy borrowing consrains agains fuure labor income on opimal consumpion invesmen policies. We solve wo alernaive models almos explicily (a leas paramerically up o a mos a consan provide verificaion heorems ha are proved using a combinaion of he dual approach an analysis of he boundary. In addiion, we also obain prove some ineresing comparaive saics. INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Acknowledgmens. The auhors hank Ron Kaniel, Masakiyo Miyazawa (an area edior of Mahemaics of Operaions Research, Anna Pavlova, Cosis Skiadas, wo anonymous referees an associae edior a Mahemaics of Operaions Research, he seminar paricipans a he 25 American Finance Associaion (AFA conference, Duke Universiy, MIT, Washingon Universiy in S. Louis for helpful suggesions. References [] Dybvig, P. H., H. Liu. 2. Lifeime consumpion invesmen: Reiremen consrained borrowing. J. Econom. Theory 45( [2] Harrison, J. M Brownian Moion Sochasic Flow Sysems. Wiley, New York. [3] He, H., H. F. Pagès Labor income, borrowing consrains, equilibrium asse prices: A dualiy approach. Econom. Theory 3( [4] Karazas, I., S. E. Shreve. 99. Brownian Moion Sochasic Calculus. SpringerVerlag, New York. [5] Karazas, I., S. E. Shreve Mehods of Mahemaical Finance. SpringerVerlag, New York. [6] Karazas, I., H. Wang. 2. Uiliy maximizaion wih discreionary sopping. SIAM J. Conrol Opim. 39( [7] Meron, R. C Lifeime porfolio selecion under uncerainy: The coninuous ime case. Rev. Econom. Sais. 5( [8] Pliska, S. R A sochasic calculus model of coninuous rading: Opimal porfolio. Mah. Oper. Res. (
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