Matching and Saving in Continuous Time: Proofs

Transcription

1 Mainz School of Management and Economics Discussion Pape Seies Matching and Saving in Continuous Time: Poofs Chistian Baye and Klaus Wälde Apil 200 Discussion pape numbe 005 Johannes Gutenbeg Univesity Mainz Mainz School of Management and Economics Jakob-Welde-Weg Mainz Gemany wiwi.uni-mainz.de

2 Contact details Chistian Baye TU Belin Institute of Mathematics Staße des 7. Juni Belin Gemany bayemath.tu-belin.de Klaus Wälde Chai in Macoeconomics Johannes Gutenbeg-Univesität Mainz Jakob Welde-Weg Mainz Gemany klaus.waeldeuni-mainz.de All discussion papes can be downloaded fom

3 Matching and Saving in Continuous Time: Poofs Chistian Baye (a) and Klaus Wälde (b) (a) TU Belin and (b) Univesity of Mainz, CESifo and Univesité catholique de Louvain Apil 200 This pape povides the poofs to the analysis of a continuous time matching model with saving in Baye and Wälde (200a). The pape poves the esults on consumption gowth, povides an existence poof fo optimal consumption and a detailed deivation of the Fokke-Planck equations. JEL Codes: Keywods: C62, C65 continuous time uncetainty, Fokke-Planck equations, existence poof Intoduction The pape by Baye and Wälde (200a) BW0 in what follows studies optimal saving in a matching model in continuous time. It states many esults without poving them. All poofs ae pesented hee. This pape stats by ecapitulating the essential equations of BW0 in the next section. The Keynes-Ramsey ule (Eule equation) in BW0 descibes the impact of the inteest ate, the time-pefeence ate and the pecautionay savings tem on consumption gowth. Section 3 povides the poofs behind this desciption. This section also poves concavity of the value function, a esult which is useful fo numeical analysis. The only assumption we base ou poofs on is continuous di eentiability of the consumption function and a nite numbe of sign changes in a nite inteval. These ae vey weak assumptions which ae easily acceptable fo ou applications. Once the Keynes-Ramsey ule is undestood, BW0 continues by using a phase diagam to illustate consumption-wealth dynamics. The existence poof of optimal consumption paths that coesponds to the phase-diagam (in some appoximative sense) is povided in section 4. Such an existence poof is of impotance as it establishes that intuitive easoning is in fact tue. What is moe, as numeical identi cation Lage pats of this pape wee witten while the authos wee woking at the Royal Institute of Technology (KTH) in Stockholm and the Univesity of Glasgow, espectively. We ae gateful to these institutions fo thei stimulating eseach envionment. Chistian Baye: TU Belin, Institute of Mathematics, Staße des 7. Juni 36, 0623 Belin, bayemath.tu-belin.de, Klaus Wälde: Univesity of Mainz, Mainz School of Management and Economics, Jakob-Welde-Weg 4, 5528 Mainz. klauswaelde.com, We ae vey gateful to Walte Schachemaye fo comments and guidance, Michael Gabe, Giuseppe Moscaini, Sevi Rodíguez Moa, Josef Teichmann and Calos Caillo Tudela fo comments and Jeemy Lise fo discussions. The second autho is indebted to Ken Sennewald and Chistian Baue fo ealie collaboation on this topic. 2

4 of an optimal path is not always staightfowad, knowing that the objective one looks fo actually exists is vey eassuing. The analysis of BW0 becomes a geneal equilibium analysis by using Fokke- Planck equations to descibe the evolution of the density of the state vaiables ove time. This density is equied to close the model and obtain an endogenous inteest and wage ate which is linked to the aggegate capital stock. The deivation of these equations is in section 5. This deivation is elatively detailed in ode to show that applications in othe contexts ae possible as well. Having obtained Fokke-Planck equations as a tool to descibe the evolution of distibutions aises the question about existence, uniqueness and stability of the limiting distibution fo the state vaiables. A companion pape (Baye and Wälde, 200b) poves that the low inteest ate case exhibits a unique invaiant distibution and the system is egodic: the distibution always conveges to the invaiant distibution as time appoaches in nity. Ou poofs ae elated to vaious methods and stands in the liteatue. Tools fo the Keynes-Ramsey poofs ae elementay. The existence poof fo optimal consumption builds on the classical theoem of continuous dependence of the solution to an odinay di eential equation on its initial value. The poof continues by showing that hitting times ae continuous in initial values - which is fa fom obvious. We conclude by applying a vesion of the intemediate value theoem, speci cally designed fo ou pupose. We note that the classical theoy of odinay di eential equation, see, fo instance, Mattheij and Molenaa (2002) is only applicable in ou case afte some modi cations, because the ight hand side satis es the standad Lipschitz conditions only afte a vaiable tansfomation. The pinciples behind and the deivation of the Fokke-Planck equation fo Bownian motion ae teated e.g. in Fiedman (975, ch. 6.5) o Øksendal (998, ch. 8.). Fo ou case of a stochastic di eential equation diven by a Makov chain, we use the in nitesimal geneato as pesented e.g. in Potte (995, ex. V.7). Fom geneal mathematical theoy, we know that the density satis es the coesponding Fokke- Planck equation t p(t; x) = A p(t; x), whee p denotes the density of the pocess at time t and A is the adjoint opeato of the in nitesimal geneato A of the pocess (x is the state vaiable). We follow this appoach in ou famewok and obtain the Fokke-Planck equation fo the law of the employment-wealth pocess. Ou appoach constitutes a consideable genealization to the closed-fom solution esults used fo stuctual estimation e.g. by Posch (2009) o Aït-Sahalia (2004). In these models, estimation is possible only if a closed-fom solution fo the density can be found. In many othe stuctual estimation appoaches, at least some stuctue fo the density is known, e.g. the genealized exponential distibution in many duation models (Eckstein and Wolpin, 995, Cahuc et al., 2006 o Launov and Wälde, 200). In ou setup, the density is only implicitly given, i.e. one can deive densities fo estimating models with any dynamic popety which is based on Bownian motions o Poisson pocesses (o Lévy pocesses) and not only with popeties which allow fo a (closed to) closed-fom solution. As we pesent a vey detailed deivation of the di eential equations fo the density, it should be easy to use this tool fo stuctual estimation in othe contexts as well. Lo (988) deives a Fokke-Planck equation as 3

5 well fo a one-dimensional stochastic pocess. We povide a setup with two pocesses which also makes clea how the Fokke-Planck appoach can be genealized to thee o moe pocesses. 2 Backgound In ode to make this pape self-contained, we pesent all equations equied fo subsequent poofs in this section. The economic backgound of these equations is in Baye and Wälde (200a). 2. The model Conside an individual that behaves optimally, accoding to some objective function and given cetain constaints (see () and (2) below), by following a consumption function c (a () ; z ()). Aguments of this function ae the state vaiables wealth a () and the labou maket status z () 2 fw; bg : Wealth of such an individual follows the budget constaint da () = fa () + z () c (a () ; z ())g d () whee is the constant but endogenous inteest ate and z () simultaneously denotes labou maket income. Labou maket status and labou maket income follow a pocess z () diven by two Poisson pocesses with state-dependent aival ates, dz () = dq dq s ; w b: (2) Aival ates ae (b) = > 0 and (w) = 0 fo q and s (w) = s > 0 and s (b) = 0 fo q s : Since q s is only active when z is in the state w, while q is only active when z is in the state b, this equation implies that z () jumps between the states w and b: Labou income w (net wage) and b (unemployment bene ts) ae constant but endogenous as well. This implies that z() is a time-homogeneous Makov chain (in continuous time) with state space fw; bg. The instantaneous utility function is of the CRRA type, whee all poofs will wok with a positive 6= : 2.2 The educed fom u (c ()) = c (), (3) The educed fom in BW0 (sect. 4.2) descibing optimal behaviou is expessed with wealth a as the exogenous vaiable (instead of time t). De ning x(a) c(a; w); y(a) c(a; b); (4) 4

6 the educed fom can be witten as h i x(a) y(a) + s _x(a) = a + w x (a) h _y(a) = a + b y (a) y(a) x(a) i x (a) ; (5a) y (a) : (5b) Hee, _x and _y denote the espective deivatives of x and y with espect to a. Additional paametes appeaing hee ae the time pefeence ate ; the sepaation ate s and the matching ate plus the degee of isk-avesion : We will assume fo all poofs that < : This is a non-autonomous system due to the appeaance of a in the denominatos in both equations. BW0 has shown that thee is a natual boowing limit which implies that any solution to (5) must satisfy y ( b=) = 0: (6) 2.3 Bounday conditions When gaphically illustating the full stochastic system in a phase diagam, BW0 intoduced a point (a w; c (a w; w)) = (a w; x (a w)) in R 2 called tempoay steady state (TSS). At this point, the stochastic system (a(); c(a(); w)) (the stochastic Keynes- Ramsey ules and the budget constaints, see (8) () in BW0) is tempoaily (until the next jump of q s ) at est. We know about the TSS fom BW0 that x (a w) = a w + w; (7) and that the value of consumption afte a jump, i.e. c (a w; b) = y (a w) ; is given by y (a w) = x (a w) whee = < : (8) s In all cases whee is used, is su ciently low such that is a eal numbe. As the TSS must be pat of any optimal consumption path c (a; z) ; we ae inteested in solutions to the educed fom (5) which contain the point (a w; x (a w) ; y (a w)) : Note that this point is fom R 3 in contast to the TSS which is fom R 2 : Note also that this point is not a steady state of this ODE system: _y (a w) is negative and what is wose in a sense _x (a w) is not de ned as _x (a w) = 0=0. Fo the latte eason, BW0 intoduced an auxiliay tempoay steady state (atss) de ned by (a w; x v (a w)) fom R 2 whee x v (a w) = a w + w v (9) and whee v > 0 is an abitaily small positive numbe. This value fo x (a w) eplaces the value in (7). Obviously, (a w; x 0 (a w)) = (a w; x (a w)). The value fo y adjusts as well as it is given by y v (a w) = x v (a w) (0) 5

7 whee is the value in (8). This atss coesponds to a point (a w; x v (a w) ; y v (a w)) fo the ODE (5) fom R 3. This point, witten as v (^a) = (^a; x v (^a) ; y v (^a)) () will late seve as teminal values when solving ou ODE systems. depends on v. Note that 2.4 The famewok fo the poofs The only assumption we make fo ou poofs is an assumption on continuity. Assumption Relative consumption (a) c (a; w) =c (a; b) is continuously di eentiable in wealth a: The numbe of sign changes of 0 (a) in any inteval of nite length is nite. 2 This is obviously a vey weak assumption which does not impose any economically elevant estiction on ou solutions. In addition to this assumption, we have no ambition in undestanding the system (5) in all geneality. We athe estict ouselves to the domain whee equilibium dynamics take place. This egion was identi ed in the g. in BW0 as the one delimited by b= and a w fo wealth and by the zeo-motion lines fo consumption levels, whee we intepet the zeo-motion lines as planes in the (a; x; y)-space. Moeove, we estict ou attention to cases whee x y, i.e. the consumption of the unemployed cannot be lage than the consumption of the employed. 3 Altogethe, we conside the domain Q v = f(a; x; y) 2 R 3 j a b=; x a + w v; y a + b; y 0; x yg: (2) whee v is the small positive constant used in (9) as well. In the poofs, it will be convenient to estict attention to a bounded set. Thus, we conside R v; = Q v \ (a; x; y) 2 R 3 j x < ; a ( w + v)= (3) whee only seves to make R v; R 3 a compact set, which we need to obtain global, unifom Lipschitz constants. We shall see below that has to be chosen lage than. In this case, howeve, does not intefee with the constuction. 0 = w b ( ) 2 The second sentence of this assumption is equied to ule out pathological cases. One can constuct continuously di eentiable functions that change sign in nitely often in a nite neighbohood (think of x sin (=x) in a neighbohood of zeo). None of these functions would be economically plausible in any way. This second sentence is used in the poof of pop We will pove late in lem. 3.0 that x > y: In this sense, the estiction fo ou domain Q v is not binding in any economic sense. 6

8 3 Poof of the pedictions of the Keynes-Ramsey ule We will now pove pop. 3 of BW0. We epeat it hee fo efeence, as ou pop. 3.. Poposition 3. Conside a low inteest ate, i.e. 0 <. De ne a theshold level a w by u 0 (c (a w; b)) u 0 (c (a w; w)) : (4) s Fo ou instantaneous utility function (3), this de nition eads c (a w; b) = c (a w; w) (5) whee is fom (8). (i) Consumption of employed wokes inceases if the woke owns a su ciently low wealth level, a < a w. Employed wokes with a > a w choose falling consumption paths. (ii) Consumption of unemployed wokes always deceases. (iii) Consumption of employed wokes exceeds consumption of unemployed wokes at the theshold a w; i.e. in (5) fo : Let us ecall the ndings in BW0 on consumption gowth. Hee, and fo the emainde of this section, we use the notation as in BW0, i.e. a w and a b satisfy the ODE _a z () = a z () + z c(a z (); z) fo xed z 2 fw; bg. Lemma 3.2 (BW0) Individual consumption ises if and only if cuent consumption elative to consumption in the othe state is su ciently high. Fo the employed woke, consumption ises if and only if c (a w ; w) elative to c (a w ; b) is su ciently high, dc (a w (); w) d 0, u0 (c (a w (); b)) u 0 (c (a w (); w)) s, c (a w(); w) c (a w (); b) = ; (6) whee is fom (8). Fo the unemployed woke, consumption ises if and only if c (a b ; b) elative to c (a b ; w) is su ciently high, dc (a b (); b) d 0, u0 (c (a b (); w)) u 0 (c (a b (); b)) : (7) 3. Poof of pat (i) 3.. A local esult We st show that consumption c (a w ; w) ises in time fo wealth smalle than but close to a w. 7

9 Given that, by ass., the numbe of sign changes of 0 (a) in any inteval fo a of nite length is nite, fo any a 0 we can nd an " > 0 such that (a) c (a; w) =c (a; b) is monotonic in [a 0 "; a 0 ]. Exploiting this fo a w; whateve the popeties of elative consumption, we can always nd an " such that one of the following thee cases must holds fo " [a w "; a w[ 9 8 = < (i) (ii) (iii) ; 0 (a)j a2" : < > = 9 = ; 0: Note that we do not make any statement about the deivative in a w: In fact, in case (i) 0 (a)j a2a w can be negative o zeo, in case (ii), it can be positive o zeo. Lemma 3.3 (a) Consumption of employed wokes ises ove time fo a wealth level a 2 " if and only if case (i) holds, dc (a w (); w) d > 0 fo a w () 2 ", case (i) holds. (b) Consumption c (a w (); w) falls ove time fo a w () 2 " if and only if (ii) holds. dc(a Poof. (a) By (6), w();w) > 0, c (a d w (); w) =c (a w (); b) > =. As c (a w; w) =c (a w; b) = = at a w, as w and b ae paametes and using ass., this is a condition on the deivative of elative consumption with espect to wealth a in " : dc (a w (); w) =d is positive fo a w () 2 " if and only if case (i) holds. (b) By (6), consumption falls ove time if elative consumption lies below = : This can be the case in " only if case (ii) holds. Lemma 3.4 Relative consumption falls in wealth fo a 2 ", 0 (a)j a2" case (i) holds. < 0; i.e. Poof. a) Assume that case (ii) holds, i.e. 0 (a)j a2" > 0. Then, by lem. 3.3, dc(a w();w) < 0 fo a d w () < a w: Consumption of unemployed wokes would still decease in time fo all wealth levels. In ou set Q v, daw() > 0 and theefoe dc(a;w) < 0: d da As dc(a b();b) < 0 and da b() < 0 in Q d d v ; we know that dc(a;b) > 0: As a consequence, da 0 (a) < 0: This contadicts the assumption that case (ii) holds and case (ii) can be excluded. b) Now assume that case (iii) holds, i.e. elative consumption is at, 0 (a)j = a2"[a w 0. As c (a w; w) =c (a w; b) = =, dc (a w (); w) =d = 0 fo a w () 2 " : As dc (a b (); b) =d < 0; elative consumption is not constant which contadicts the assumption that elative consumption is at in wealth. As case (iii) is theeby excluded as well, the poof is complete. 8

10 3..2 A global esult We now complete the poof by a global esult on consumption gowth. Lemma 3.5 Consumption c (a w ; w) (a) ises in time fo all a < a w and (b) deceases in time fo all a > a w. Poof. (a) Imagine to the contay of c (a w ; w) ises in time fo all a < a w that thee is an inteval ] ; 2[ with 2 < a w such that this is is the last inteval befoe a w whee c (a w ; w) falls in time, dc (a w (); w) =d < 0; 8 < a w () < 2 < a w: (8) We now poceed as in the poof of lem As daw() > 0 in Q d v ; this would imply that dc(a;w) < 0 fo da < a < 2 : We know that dc(a;b) > 0 in Q da v : Hence, we would conclude that 0 (a) < 0; 8 < a < 2 : (9) By (6), the assumption in (8) would hold if and only if elative consumption c(a w;w) dc(a c(a w;b) is below = fo < a < 2 : w();w) < 0, c(aw();w) c(a;w) d c(a w();b) < = : As is c(a;b) continuous in wealth by ass. and as case (i) holds by lem. 3.4, c(a;w) can be smalle c(a;b) than = only if thee is some ange ] 3; 2[ in which 0 (a) > 0: (An example of such a path is shown in g..) This is a contadiction to the conclusion in (9). Hence, consumption must ise in time fo all a < a w: (b) This poof is in analogy to the poof of (a). Figue An example fo elative consumption (a) c(a;w) c(a;b) 9

11 3.2 Intemediay steps Befoe we pove the est of pop. 3., we need some futhe intemediay esults which, howeve, ae of some inteest in thei own ight. Given that maginal utility fom (3) is positive and deceasing, u 0 (c) > 0 and u 00 (c) < 0; we can establish that x (a) > y (a) ; i.e. consumption in the state of employment is lage than in the state of unemployment, keeping wealth constant. We pove in passing that the value functions V (a; z) ae stictly concave in wealth a: Lemma 3.6 Consumption ises in wealth, c a (a; z) > 0: Poof. Pop. 3. (i) shows that dc (a w (); w) =d > 0 in Q v : As da w ()=d > 0 as well, the deivative _x (a) in (5) is positive in Q v : Lemma 3.7 As maginal utility fom consumption is positive, the value function V (a; z) ises in wealth, V a (a; z) > 0: Poof. The st-ode condition fo optimal consumption is given by (47) in the appendix and eads u 0 (c (a; z)) = V a (a; z) : (20) As maginal utility is positive by (3), the value function ises in wealth. Lemma 3.8 As u 00 (c) < 0 and as consumption ises in a by lemma 3.6, the value function is stictly concave in a. Poof. The patial deivative of the st-ode condition with espect to wealth implies u 00 (c (a; z)) c a (a; z) = V aa (a; z) : (2) As u 00 (c (a; z)) < 0 fom the concavity of (3) and c a (a; z) is positive by lem. 3.6, V aa (a; z) must be negative. With lem. 3.7, the value function is stictly concave. Lemma 3.9 The shadow pice fo wealth is highe in the state of unemployment, V a (a; b) > V a (a; w) : Poof. The deivation of the Keynes-Ramsey ule gives us (see app. A.) ( ) V a (a; z) s (z) [V a (a; b) V a (a; w)] (z) [V a (a; w) V a (a; b)] = [a + z c (a; z)] V aa (a; z) : In state z = w; this means ( ) V a (a; w) s (z) [V a (a; b) V a (a; w)] = [a + w x (a)] V aa (a; w) : (22) Given the egion we ae inteested in (whee a + w x (a) > 0) and given lemma 3.8, the ight-hand side is negative. Hence, the left-hand side must be negative as well. As ( ) V a (a; w) is positive due to <, the second tem must be negative. This is the case only fo V a (a; b) > V a (a; w) : 0

12 Lemma 3.0 Consumption of the employed woke is highe than consumption of the unemployed woke, x (a) > y (a) : Poof. As V a (a; b) > V a (a; w) ; the st-ode condition implies u 0 (c (a; b)) > u 0 (c (a; w)) : As the maginal utility is deceasing, c (a; w) > c (a; b), x (a) > y(a): 3.3 Poof of pats (ii) and (iii) We now complete the nal bits of the poof. Poof of pop. 3. continuation. (ii) By (7), dc (a b (); b) =d < 0, u 0 (c (a b (); w)) < {u 0 (c (a b (); b)) whee { as : As u0 (c (a b (); w)) < u 0 (c (a b (); b)) with c (a b (); w) > c (a b (); b) fom lem. 3.0, this condition always holds. (iii) This follows fom solving (4) fo elative consumption. 4 The existence of an optimal consumption path De nition 4. (Optimal consumption path) (i) A consumption path is a solution 4 (a; x(a); y(a)) of the ODE-system (5) fo the ange b= a ^a in R v; with teminal condition fom (). 5 The teminal condition satis es (9), (0) fo an abitay ^a > b=: (ii) An optimal consumption path is a consumption path which additionally satis- es y ( b=) = 0: If such a path exists, we denote the wealth level ^a of the teminal condition by a w. We emak that the notion of an optimal consumption path depends on v (via ). The objective of this section consists in poving the following Theoem 4.2 Thee is an optimal consumption path. 4. Peliminaies In what follows, we will use classical theoems fo initial value poblems fo ODEs. Cuently, we have fomulated ou system (5) as a teminal value poblem, since we have set a value, which the system should attain at ^a = a w, i.e. at the end of the inteval [ b=; a w] unde consideation. Fo ease of notation and to help intuition, we shall now ecast the poblem into a classical initial value poblem, i.e. we will equie the value to be attained at the xed beginning = 0 of an inteval [0; ], on which we study the poblem. 4 By a solution to (5), we hee undestand continuous maps x : [ b=; ^a]! R, y : [ b=; ^a]! R, which ae continuously di eentiable in ] b=; ^a[ and solve (5a) and (5b), espectively, in the open inteval. In paticula, we do not equie the deivatives of x and y to convege fo a! b=. 5 In the sense that an optimal consumption path satis es x(^a) = x v (^a), y(^a) = y v (^a).

13 To this end, it is moe useful to wok with an autonomous system. Hence, we ewite (5) by including m (a) = a as thid vaiable which eplaces wealth a, which now puely seves as a paamete, i.e. as the independent vaiable. This gives the system _m(a) = ; h i + s x(a) y(a) _x (a) = m (a) + w h _y (a) = m (a) + b x (a) i y(a) x(a) y (a) x (a) ; y (a) : Now de ne ^a a, x () m (^a ), x 2 () x (^a ), x 3 () y (^a ) : d Then, x d () _x () = d d m (^a ) = m (a) = d m (a) = _m (a) : Doing d d[^a a] da the same fo x and y; the inveted autonomous system theefoe eads _x () = ; (23a) h i + s x2 x 3 () x 2 () _x 2 () = x () + w x 2 () ; (23b) h i x2 () x 3 () x 3 (a) _x 3 () = x 2 () + b x 3 () ; (23c) whee now _x i denotes the deivative of x i () with espect to, i = ; 2; 3. De nition 4.3 Given (23) and fo 0; let X(; ) = (x (); x 2 (); x 3 ()) denote the solution of (23) stated at X(0; ) = 2 R v; fom () whee b= ^a +v w. Fo late use, we also intoduce the notation x i () = x i (; ), i = ; 2; 3. By passing fom (5) to (23) we have eveted the time-diection moe pecisely, in ou setting, the wealth-diection and tuned a non-autonomous system into an autonomous one by including the independent vaiable as an additional component of the solution. Thus, the cuve a 7! (a; x(a); y(a)) with teminal value x(^a) = x v (^a), y(^a) = y v (^a) is equal to the cuve 7! X(; ) with = (^a), which is the solution of an initial value poblem in the classical sense. Howeve, the paametization is eveted in the sense that in the fome case we stat at the left endpoint ( left in the sense of the smallest value of the a-component) and end in the ight endpoint, wheeas in the latte case we stat at the ight endpoint and end in the left one. In paticula, the absolute value of the speed long the cuve is equal, but the diection is evesed. 4.2 Continuity of the solution in initial values In ode to be able to apply classical theoems, we need nite deivatives on the ighthand side of an ODE system. The ight-hand side of the ODE (5), howeve, exhibits 2

14 singulaities at the bounday y = a+b of Q v : This is of paticula impotance as the de nition of the optimal consumption path in De nition 4. uses y ( b=) = 0 which lies on this bounday. We obtain nite deivatives by (i) a coodinate tansfomation and by (ii) (tempoaily) educing the set on which we ae inteested in a solution by demanding that y ". We will late show how this eduction can then be emoved again by passing "! 0. Lemma 4.4 (Coodinate tansfomation) Let x(a) and y(a) be solutions of (5). The mapping a 7! y(a) is bijective. Change vaiables a = a(y) and conside x and a as functions of y. Then i x 0 (y) dx(y) dy a 0 (y) da(y) dy = = + s h x(y) y h y x(y) a(y) + b y h y x(y) i x(y) y i y : a(y) + b y a(y) + w x(y) ; (24a) (24b) Poof. Since _y(a) > 0, y is a bijective function of a: As a 0 (y) =, we obtain _y(a) the second equation by inseting (5b). The st equation follows fom dividing (5a) by (5b). We ae going to avoid the singulaity at y ( b=) = 0 by tempoaily equiing these popeties only to hold up to an abitaily small numbe ". We do this by consideing the domain R ";v; as given in the following De nition 4.5 Fix a numbes " > 0 and de ne R ";v; = R v; \ (a; x; y) 2 R 3 j y " : (25) This de nition implies that we tempoaily eplace the equiement that y ( b=) = 0 by y (a) = " fo some b= a b= + "=. Lemma 4.6 The ight-hand side given in (24) is unifomly Lipschitz on R ";v;. Poof. Conside the ight-hand side of (24a). The only possible points, whee the Lipschitz constant can explode, ae when the denominatos in the ight-hand side become 0 o when a tem unde a factional powe (i.e. with exponent ) becomes 0. In R = R ";v;, y is unifomly bounded away fom 0 and x is unifomly bounded away fom a+w. Moeove, note that y x = 0 if and only if y x =. Now > by the assumption that <. On the othe hand, y < x, implying y that x <. Consequently, all the denominatos ae unifomly bounded away fom 0. Fo the factional powes, note that x=y > is tivially unifomly bounded away fom 0. As x, y x > 3

15 is unifomly bounded away fom 0 on R ";v;. This shows that (24a) is unifomly Lipschitz. The same aguments show that the ight-hand side of (24b) is unifomly Lipschitz, too. Remak 4.7 Since the ight hand side of (24) is unifomly Lipschitz, we can now apply the classical theoy of ODEs. Fo instance, we have existence and uniqueness of the solution by the Picad-Lindelöf theoem, see Mattheij and Molenaa (2002, th. II.2.3, th. II.3.). Moeove, the solution will be continuous as a function of the initial value, see, again, Mattheij and Molenaa (2002, th. II.4.7). In the lemma below, we will see how this even implies the coesponding popeties fo the nontansfomed system (23). Lemma 4.8 (Continuity in initial values) Conside the set R = R ";v; fom (25) and the solution X(; ) fom De nition 4.3 with initial condition given in (). The solution X(; ) depends continuously on its initial values. Moe pecisely, thee is a constant L > 0 and an inceasing map : [0; [! [0; [ (a modulus of continuity) with lim t&0 (t) = (0) = 0 such that kx( ; ) X( 2 ; 2 )k Lk 2 k + (j 2 j); povided that ; 2 2 R and X(; i ) 2 R fo all 0 max( ; 2 ), i = ; 2. Hee, kk denotes the Euclidean nom on R 3. Poof. By classical esults fom the theoy of odinay di eential equations, see fo instance Mattheij and Molenaa (2002, th. II.4.7), the solution of an ODE-system depends continuously on the initial data as long as the ight-hand side is unifomly Lipschitz. Moe pecisely, let Y (; ) denote the solution of an ODE with unifomly Lipschitz ight-hand side (with Lipschitz constant C), stated at Y ( 0 ; ) =, then ky (; ) Y (; 2 )k exp (C( 0 )) k 2 k: Now conside the tansfomed system (a(y); x(y)) fom (24). By Lemma 4.6, the ight-hand side is unifomly Lipschitz. The solution of (24) theefoe depends continuously on its initial data (a 0 ; x 0 ). It is then obvious that the tajectoy (a(y); x(y); y) depends continuously on (a 0 ; x 0 ; y 0 ): As system (24) is a epaameteized vesion of (5), the solution (a; x (a) ; y (a)) to (5) fom def. 4. is also continuous in its bounday conditions even though the ight hand side of (5) is not unifomly Lipschitz. Similaly, as (23) is just a epaameteization of (5), the solution X (; ) to (23) fom def. 4.3 is also continuous in its initial condition. In ode to get the estimate, we now conside the ODE (23) and note that we only conside it on the compact set R ";v;. In the paametization by y given in (24), y is the independent vaiable, i.e. plays the ole of in the above estimate. By compactness of R ";v;, y only uns though a bounded set, theefoe we can ewite the constant in the above inequality as exp(c(y y 0 )) L fo some suitable L > 0. 4

16 Given 2 R ";v;. Then a w w+v, which implies that the solution X(; w) can only stay inside R ";v; until time = w+v+b, at most. Conside D = f(; ) 2 [0; [R ";v; j X(; ) 2 R ";v; g: Then D is a closed subset of 0; w+v+b R";v;, implying that D is compact. Consequently, X : D! R ";v; is unifomly continuous, which implies the existence of a modulus of continuity with kx( ; ) X( 2 ; 2 )k (j 2 j + k 2 k): The inequality in the lemma then follows by the tiangle inequality. 4.3 Continuity of the st hitting-wealth in initial values While we have shown in the pevious section that the solutions to all systems (5), (23) and (24) ae continuous in initial values, this does not automatically imply that the solutions will be continuous on the bounday of the domain we ae inteested in, in the sense that the place whee the solution leaves the domain R might not depend continuously on the initial data. This will now be poved in this section. In the poofs and also in a late step, we will use the following De nition 4.9 (Fist hitting-wealth) Conside the set R ";v; fom (25) and the solution X (; ) to the system (23). Conside the path y (a) that coesponds to x 2 () of this solution. Then we de ne ^a st = f (^a) as the st hitting-wealth (in analogy to st hitting-time), i.e. the wealth level whee the path y (a) hits any bounday of R ";v; fo the st time. Similaly denote () inff 0 j X(; ) 2 R ";v; g and F () X((); ). We know that ^a st exists because in the set R ";v; the deivatives in (23) ae wellde ned and a solution theefoe exists. Notice that ^a st equals the st component of F ((^a)). We also need De nition 4.0 Let N R ";v; with N = (^a) b ^a 2 ; [w v] b [ ] be the set of all potential initial conditions fom () fo a solution in the sense of def. 4.. Hee we implicitly assume that is lage enough that indeed N R ";v;. 6 De ne M as M = M [ M 2 [ M 3 R ";v; (26) 6 This is the only necessay condition on fo the constuction to wok. In the sequel, we shall assume this condition without futhe notice. 5

17 with M = f(a; x; y) 2 R ";v; j y = a + bg; M 2 = f(a; x; y) 2 R ";v; j a = b=g; M 3 = f(a; x; y) 2 R ";v; j y = "g: This set will tun out to be the set of all potential st hitting-wealths. Since we know that x > y, the tajectoy will not hit the bounday of R at the pat fx = yg. Theefoe, we have the Coollay 4. F : N! M is a well-de ned map fom R 3 to R 3, i.e. fo evey 2 N, the coesponding solution path X(; ) exists and stays in R ";v; until it nally hits M (and no othe bounday of R ";v; ). Befoe fomulating the main lemma of this section, let us st deive a simple bound on the deivative _y(a) of the consumption of the unemployed. Lemma 4.2 Fo (a; x; y) in the inteio of Q v fom (2), we have Poof. By (5b) we have h _y (a) = y(a) x(a) _y(a) i a + b y (a) 0 = a + b y (a) y (a) a + b h a + b y(a) y(a) x(a) y(a) : i y (a) A y (a) > a + b y (a) y (a) : The last inequality follows fom the fact that [ ( x(a)) y(a) ] is negative (and theefoe a+b y(a) [ ( x(a)) y(a) ] is positive) as a + b y (a) is negative in the inteio of Q a+b y(a) v. The key esult in this section is pesented in Lemma 4.3 The map F : N! M is continuous. Poof. We need to pove that fo evey 2 N and evey > 0 thee is an > 0 such that k 0 k < =) kf ( 0 ) F ()k < : (27) We stat the poof by xing 0 ; 2 N such that k 0 k < fo some > 0. Let us st assume that ( 0 ) (). By the tiangle inequality and Lemma 4.8, we have kx(( 0 ); 0 ) X((); )k kx(( 0 ); 0 ) X(( 0 ); )k + + kx(( 0 ); ) X((); )k L k 0 k + (j( 0 ) ()j); (28) 6

18 fo a constant L > 0 and the modulus of continuity. In ode to get an estimate fo j( 0 ) ()j, we have to distinguish between thee di eent cases. Case (i): F ( 0 ) 2 M. By Lemma 4.2, thee ae constants L 2 ; `2 > 0 such that _y L 2 fo jy (a+b)j `2. Moe pecisely, we can choose `2 > 0 feely and obtain the bound fo L 2 = `2 ( )". If L `2, we can bound the absolute value of the deivative of x 3 (; ) fom below by L 2 (fo t ( 0 )). This implies that the path X(; ) hits M befoe time ( 0 ) + fo (L 2 ) = `2 () = `2 L 2 ; unless it hits anothe bounday of R ";v; befoe that. Inseting into (28), this gives the estimate `2 kf ( 0 ) F ()k L + : Choosing `2 = L, the bound is smalle than povided that! L C L L 2 + L < ; (29) whee C ( )". Note that the left hand side in (29) conveges to zeo fo! 0, theefoe we can nd an 0 () > 0 (only depending on the constants C, L and and the modulus of continuity, but not on 0 o ) such that the desied inequality (27) holds fo < 0. We have tacitly assumed that L 2 = C=`2 = C L >, which can be ealized by choosing small enough. Case (ii): F ( 0 ) 2 M 2. Let ^a denote the st component of, and ^a 0 the st component of 0. Note that x (; ) = ^a, fo evey 0. Since X(( 0 ); 0 ) 2 M 2, we have b= = x (( 0 ); 0 ) = ^a 0 ( 0 ), implying that ( 0 ) = ^a 0 + b=. On the othe hand, x ((); ) b=, implying that () ^a + b=. Combining these two esults, we obtain j( 0 ) ()j = () ( 0 ) ^a ^a 0 k 0 k : Consequently, the inequality (28) implies kf ( 0 ) F ()k L k 0 k + (k 0 k) L + (); and (27) holds fo small enough such that L + () < : (30) Case (iii): F ( 0 ) 2 M 3. Since x 3 (( 0 ); 0 ) = ", we have 0 x 3 (( 0 ); ) " L. By Lemma 4.2, we can nd a constant L 3 > 0 such that _y L 3 on R ";v; note that L 3 depends on ". Thus, X(s; ) will hit the bounday M 3 befoe time ( 0 ) + with = L =L 3, unless it hits anothe bounday of R ";v; befoe. In any case, j( 0 ) ()j L =L 3, and we obtain L kf ( 0 ) F ()k L + ; L 3 7

19 and (27) is satis ed fo L L + < : (3) L 3 Choosing small enough that both (29) and (30) and (3) ae satis ed, settles the poof fo ( 0 ) (). Notice that none of the conditions (29), (30) and (3) depends on 0. Theefoe, in the othe case ( 0 ) (), we can just evet the ôles of and 0 and obtain the same esults in cases (i), (ii) and (iii). 4.4 Existence of a solution This section poves ou main esult fomulated in Theoem 4.2. Poof of Theoem 4.2. Fix some " > 0 and conside R ";v;. By an intemediate value theoem applied to F : N! M, we will obtain a point o points 2 N such that F () 2 M 3 as used in (26), i.e. x 3 ((); ) = " povided that we can show the existence of points (that could be called uppe and lowe bounds) min v ; max v 2 N with F ( min v ) 2 M 2 and F ( max v ) 2 M. (Note that F = F " and all the M i = M i ("), i = ; 2; 3, depend on " and v, but not on, povided that is lage enough.) Choose min v = ( b=) = ( b=; w b v; [w b v]); max v = (w v) b ( ) By constuction, both min v and max v ae contained in N. Moeove, we tivially have F " ( min v ) 2 M 2 ("), F " ( max v ) 2 M (") fo evey " > 0 small enough. Note, in paticula, that Lemma 4.3 also implies continuity of F in the bounday points min v and max v of N. Theefoe, the image set F " (N) is a connected set, with non-empty intesection with both M and M 2. Since the distance dist(m ; M 2 ) = inf fk 2 k j 2 M ; 2 2 M 2 g = " > 0; we may conclude that F " (N) \ M 3 (") 6= ;. This establishes that thee must be a such that F " () 2 M 3 : In wods, thee is an initial condition (^a) such that the path (a; x(a); y (a)) hits the bounday at y = ": Now de ne N 3 (") F " (M 3 (")) = f 2 N j F " () 2 M 3 (")g : By continuity of F " : N! M("), the set N 3 (") is compact. Moeove, the family (N 3 (")) ">0 is diected in the sense that 0 < " 2 < " =) N 3 (" 2 ) N 3 (" ): By standad esults fom topology, the intesection of a diected family of non-empty, compact sets is non-empty, i.e. N 3 (0) \ ">0 N 3 (") 6= ;: : 8

20 Indeed, take a deceasing sequence (" n ) n of positive numbes conveging to zeo. Fo evey n choose some n 2 N 3 (" n ). By compactness of the lagest set N 3 (" ), we can nd a subsequence n k such that ( nk ) k conveges to some. Note that 2 N 3 (" nk ) fo evey k, since = lim l!; lk nl and each such nl lies in the closed set N 3 (" nk ). Now choose any " > 0 and pick a k such that " nk < ". Then 2 N 3 (" nk ) N 3 ("), implying that 2 T ">0 N 3("). We claim that evey element 2 N 3 (0) coesponds to an atss. Indeed, the path (a; x(a); y(a)) with teminal value (^a; ^x; ^y) = (coesponding to the path X(; )) satis es the ODE (5) on ] b=; ^a]. Moeove, it stats at N by constuction, and fo evey " > 0, it takes on the value " somewhee on the inteval ] b=; b= + "[. Thus, using monotonicity of y, we may conclude that lim y(a) = 0: a& b= This establishes that thee is an initial condition (^a) such that the path y (a) hits the bounday at y = 0 in the sense that y( b=) = 0. Remak 4.4 It is essential fo the poof of Theoem 4.2 that the tajectoy X(; ) o, equivalently, (a; x(a); y(a)) does not depend on ", which only detemines how long we obseve the tajectoy. This means that we obseve the tajectoy X(; ) fo 0 (), with the hitting time () obviously depending on ". Theefoe, we can, fo xed 2 N 3 (0), easily take the limit "! 0, which means that we take the limit in (), but do not change the tajectoy itself. As a consequence, the ODE is automatically satis ed fo the limit, at least fo 0 < lim "!0 (). Let us illustate why we had to use the speci c popeties of the dynamic system (23) in the poof of lem Continuity in initial conditions does not imply continuity of st hitting values in geneal. Indeed, the st hitting times ae inheently non-continuous functionals, even if both the paths and the set, which detemines the hitting times, ae smooth. This is sketched in ex Figue 2 Non-continuity of the st hitting time in ex

21 Example 4.5 Conside the di eential equation _z (t) = ( z (t)) z (t) whose solution is z (t) = + z0 e t : This solution is continuous in the initial level z 0 (fo z 0 > 0 which we assume) and the solution is plotted fo z 0 2 f0:; 0:2g in g. 2. Now conside the st-hitting time on the staight line 0:05 + t=5 as dawn. Obviously, this time is not continuous in the initial values z 0. 5 Deiving the Fokke-Planck equations This section deives the Fokke-Planck equations of the wealth-employment pocess (a(t); z(t)) as descibed in Section 2, i.e. the patial di eential equations which descibe the dynamics in time of the density of these andom vaiables. The deivation is in geat detail as this facilitates applications fo othe puposes. Befoe we go though individual steps, hee is the geneal idea. Step : We stat with some function f having as aguments the vaiables whose density we would like to undestand. We compute the di eential of this function in the usual way. Step 2: The stating point hee is Dynkin s fomula. This fomula, intuitively speaking, gives the expected value of some function f; whose aguments ae the andom vaiables we ae inteested in, as the sum of the cuent value of f plus the integal ove expected futue changes of f. The expected change of f is expessed by using the density of ou andom vaiables. The Dynkin fomula is di eentiated with espect to time. Step 3: By using integation by pats o the adjoint opeato, we get an expession fo the change of the expected value of f: Step 4: A di eent expession fo this change of the expected value can be obtained by stating fom the expected value and di eentiating it. Step 5: Equating the two gives the di eential equations fo the density. 5. The expected change of some function f Assume thee is a function f having as aguments the state vaiables a and z. This function has a bounded suppot S, i.e. f(a; z) = 0 outside this suppot. 7 Heuistically, the di eential of this function, using a change of vaiable fomula, 8 gives df (a () ; z ()) = f a (:) fa () + z () c (a () ; z ())g d + ff (a () ; z () + ) f (a () ; z ())g dq + ff (a () ; z () ) f (a () ; z ())g dq s : Due to the state-dependent aival ates (see table ), only one Poisson pocess is active at a time. When we ae inteested in the expected change, we need to fom expectations. We view a () and z () with t as two stochastic pocesses which stat in t and whee initial conditions a (t) and z (t) can be andom vaiables. We theefoe fom expectations about df by using the unconditional expectations opeato 7 We can make this assumption without any estiction. As we will see below, this function will not play any ole in the detemination of the actual density. 8 Thee ae fomal deivations of this equation in mathematical textbooks like Potte (995). Fo a moe elementay pesentation, see Wälde (2009, pat IV). 20

22 E as the andomness of initial values ae then also taken into account. This is useful fo its geneality and also when it comes to applications (see the discussion in BW0 on initial conditions and especially distibutions). Applying the conditional expectations opeato E and dividing by d yields the heuistic equation E df (:) d = f a (:) fa () + z () c (a () ; z ())g + [f (a () ; z () + ) f (a () ; z ())] fbg (z()) + s [f (a () ; z () ) f (a () ; z ())] fwg (z()) (32) In what follows, we denote this expession by Af (a () ; z ()) E df (a () ; z ()) d which is, moe pecisely, the in nitesimal geneato A de ned by (33) E (f(z( + ); a( + ))jz() = z; a() = a) f(a; z) Af(a; z) = lim : &0 Notice that Af(a; z) does not depend on, because the Makov-pocess (a(); z()) is time-homogeneous. We undestand A as an opeato mapping functions (in a and z) to othe such functions. Moeove, note that all test-functions, i.e. C functions of bounded suppot, ae in the domain of the opeato A, i.e. the domain of all functions f such that the above limit exists (fo all a and z). 5.2 Dynkin s fomula and its manipulation To abbeviate notation, we now de ne x () (a () ; z ()) : The expected value of ou function f (x ()) is by Dynkin s fomula (e.g. Yuan and Mao, 2003) given by Ef (x ()) = Ef (x (t)) + Z t E (Af (x (s))) ds: (34) To undestand this equation, use the de nition in (33) and fomally wite it as Ef (x ()) = Ef (x (t)) + Z t Edf (x (s)) ds = Ef (x (t)) + ds Z t Edf (x (s)) : Intuitively speaking, Dynkin s fomula says that the expected value of f (x ()) is the expectation fo the cuent value, Ef (x (t)) (given that we allow fo a andom initial condition x (t)), plus the sum of expected futue changes, R Edf (x (s)) : t Let us now di eentiate (34) with espect to time and nd Ef (x ()) = Z t E (Af(x (s))) ds = E (Af (x ())) ; (35) 2

23 whee the st equality used that Ef(x (t)) is a constant and pulled the expectations opeato into the integal. This equation says the following: We fom expectations in t about f (x ()) : We now ask how this expectation changes when moves futhe into the futue, i.e. we look at E [f (x ())]. We see that this change is given by the expected change of f(x ()); whee the change is Af (x ()) : Now de ne p (a; z; ) as the joint density of z () and a (). Notice that z; a and ae independent vaiables now: as soon as we integate with espect to the density p(a; z; ), the whole time-dependence has been absobed into. The expectation opeato E integates ove all possible states of x () : When we expess this joint density as p (a; z; ) p (a; jz) p z (), we can wite equation (35) as Ef (x ()) = E (Af (x ())) = p w () Z Af (a; w) p (a; jw) da + p b () Z Af (a; b) p (a; jb) da: Now pull p w () and p b () back into the integal and use p (a; z; ) p (a; jz) p z () again fo z = w and z = b. Then Ef (x ()) = Z Af (a; w) p (a; w; ) da + Z Af (a; b) p (a; b; ) da w + b : (36) Remak 5. The law of x() will not have a density, unless we aleady stat with a density at the initial time t. In geneal, p will theefoe be a measue o a distibution (in the sense of a genealized function, see (Yosida 995), not a tue function. Even in that case, the following deivation emains valid, though, using the usual calculus fo genealized functions. 5.3 The adjoint opeato and integation by pats This is now the cucial step in obtaining a di eential equation fo the density. It consists in applying an integation by pats fomula which allows to move the deivatives in Af(x ()) into the density p (x; ) : Let us bie y eview this method, without getting into technical details. Given two functions f; g : R! R and two xed eal numbes c < d, the facto ule of di eentiation d(f(x) g(x)) = df(x) g(x) + f(x) dg(x) (37) implies that f(d)g(d) f(c)g(c) = R d f 0 (x)g(x)dx+ R d c c f(x)g0 (x)dx; a fomula efeed to as patial integation ule. In paticula, it also holds fo c = and d = +, if the function evaluations ae undestood as limits fo c! and d! +, espectively. If f has bounded suppot, i.e. is equal to zeo outside a xed bounded set, then the function evaluations at vanish and we get Z + f 0 (x)g(x)dx = Z + f(x)g 0 (x)dx: (38) 22

24 We now apply (38) to equation (36). We can do this as the expessions in (36) lost all stochastic featues. To this end, inset the de nition of A given in (33) togethe with (32) into (36). To avoid getting lost in long expessions, we look at the both integals in (36) in tun. Fo the second, obseve that Af (a; b) = f a (:) fa + b c (a; b)g + [f (a; w) f (a; b)] ; i.e. the tem with s in (32) is missing given that we ae in state b. Hence, b = = + Z Z Z [f a (a; b) fa + b c (a; b)g + [f (a; w) f (a; b)]] p (a; b; ) da f a (a; b) fa + b [f (a; w) c (a; b)g p (a; b; ) da f (a; b)] p (a; b; ) da: Now integate by pats. As this integal shows, we only need to integate by pats fo the f a tem. The est emains untouched. This gives with (38), whee g (x) stands fo fa + b c (a; b)g p (a; b; ) and x fo a; b = + Z Z f (a; b) c (a; b) p (a; b; ) + fa + b c (a; b)g a a p (a; b; ) da [f (a; w) f (a; b)] p (a; b; ) da: (39) Now look at the st integal of (36). Afte simila steps (as the pinciple is the same, we eplace b by w and the aival ate by s in the last equation), this eads Z w = f (a; w) c (a; w) p (a; w; ) + fa + w c (a; w)g a a p (a; w; ) da + Z s [f (a; b) f (a; w)] p (a; w; ) da: (40) Summaizing, we nd = Z Z Z Z f (a; w) s [f (a; b) f (a; b) Ef (x ()) = w + b c (a; w) p (a; w; ) fa + w c (a; w)g a a p (a; w; ) da f (a; w)] p (a; w; ) da c (a; b) p (a; b; ) fa + b c (a; b)g a a p (a; b; ) da [f (a; w) f (a; b)] p (a; b; ) da: (4) 23

25 5.4 The expected value again and nish The expected value again Let us now deive the second expession fo the change in the expected value. By de nition, and as an altenative to the Dynkin fomula (34), we have Ef (x ()) = Z f (a; b) p (a; b; ) da + Z When we di eentiate this expession with espect to time, we get Note that we can use Z Ef (x ()) = + Z Z f (a; z) p (a; z; ) da = f (a; b) p (a; b; ) da f (a; w) p (a; w; ) da: (42) f (a; w) p (a; w; ) da: (43) Z f (a; z) p (a; z; ) da as z and a inside this integal ae no longe functions of time. Step 5 - Equating the two expessions We now equate (4) with (43). Collecting tems belonging to f (a; w) and f (a; b) gives Z Z f (a; w) ' w da + f (a; b) ' b da = 0; (44) whee ' w c (a; w) + s p (a; w; ) a fa + w c (a; w)g p (a; w; ) a + p (a; b; ) p (a; w; ) and ' b + sp (a; w; ) c (a; b) + p (a; b; ) a fa + b c (a; b)g p (a; b; ) a p (a; b; ) : Obviously, the above equation is satis ed if These ae the Fokke-Planck equations used in BW0. ' b = ' w = 0: (45) 24

26 It is easy to see that the integal equation can only be satis ed fo all functions f, if these Fokke-Planck equations ae satis ed. Indeed, assume that ' b > 0 on an inteval I = [d ; d + ]. One can nd a non-negative function f smooth in a such that f(a; w) = 0 fo all a and ( ; a 2 [d =2; d + =2]; f(a; b) = 0; a 2] ; d ] [ [d + ; [: Inseting this test function into the integal equation gives Z f (a; w) ' w da + Z f (a; b) ' b da = 0 + Z d+ d f (a; b) ' b da > 0 by constuction. Theefoe, ' b = 0 has to hold fo all a 2 R, and similaly fo ' w. 6 Conclusion This pape poves the esults stated in Baye and Wälde (200a). In section 3, we pove concavity of the value function and the link between the inteest ate and consumption gowth. This poof is of inteest as it is based only on vey weak assumptions about popeties of optimal consumption. In section 4, we give an existence poof fo an optimal consumption path given an auxiliay tempoay steady state (atss). The atss is chaacteized by the fact that we only end v-close to the zeo-motion line of the employed, fo any v > 0, howeve small. It has poven to be vey simple to wok with this atss numeically. Due to the non-continuity of hitting times of dynamic systems, see g. 2, the poof is designed fo ou puposes. We use the speci c fom of the dynamic system (5) at hand in ode to establish continuity in this special case. If this poof is to be used fo othe systems, one always would have to make sue that lem. 4.3 o a coesponding vaiant of it will hold. In section 5, we give an easily accessible exposition into the mathematical methods used to establish Fokke-Planck patial di eential equations fo the dynamics of densities of stochastic pocesses given by stochastic di eential equations. While the deivation is based on the paticula model intoduced in section 2, we have tied to keep it athe geneal. Fo systems diven by di eent stochastic di eential equations, the in nitesimal geneato A will have a di eent fom, thus equiing di eent integations by pats. In some abstact sense, these integations by pats seve to compute the adjoint opeato to A, which in tun detemines the Fokke- Planck equation. Theefoe, the deivation used in section 5 can be applied fo a wide class of stochastic models. Concening futue wok, it would be desiable fom a theoetical pespective to extend the poof of the auxiliay tempoay steady state to an existence poof fo a tempoay steady state of the wealth dynamics. In the notation used in section 4, this means that we would like to take the limit v! 0, like we took the limit "! 0. Thee ae, howeve, cetain complications as compaed with the latte situation. Fo 25

27 once, note that the tajectoy X(; ), = v, does not depend on ". Only the time (), when X(; ) hits the bounday R ";v;k does. Theefoe, we only had to establish the existence of an initial value v, such that the coesponding tajectoy satis es x 3 ( b=) = 0. In contast, X(; v ) obviously does depend on v. Thus, taking a limit v! 0 involves taking a limit of solutions to the ODE, and we would have to additionally show that the limiting tajectoy again solves the ODE. We could avoid this poblem, if we knew how to choose v = v (^a), such that two tajectoies X(; v (^a)) and X(; 0 (^a 0 )) coincide (on R ";v;k ). This aleady indicates the dilemma, because we do not know how to solve the ODE fo v = 0 because of the singulaity in (5a) at the zeo-motion line of the employed. If we could ovecome this poblem, we would still face a simila kind of shooting poblem as in theo This time, howeve, we would have to hit a line f(a; a + w; (a + w))j b= a a max g in (a; x; y)-space instead of the line f( b=; x; 0)jx 0g. 7 Appendix Thee is a shot appendix which is available upon equest. Refeences Aït-Sahalia, Y. (2004): Disentangling di usion fom jumps, Jounal of Financial Economics, 74, Baye, C., and K. Wälde (200a): Matching and Saving in Continuous Time : Theoy, CESifo Woking Pape (200b): Matching and Saving in Continuous Time: Stability, mimeo - available at Cahuc, P., F. Postel-Vinay, and J.-M. Robin (2006): Wage Bagaining with On-thejob Seach: Theoy and Evidence, Econometica, 74, Eckstein, Z., and K. I. Wolpin (995): Duation to Fist Job and the Retun to Schooling: Estimates fom a Seach-Matching Model, Review of Economic Studies, 62, Fiedman, A. (975): Stochastic di eential equations and applications. Vol.. Academic Pess [Hacout Bace Jovanovich Publishes], New Yok, Pobability and Mathematical Statistics, Vol. 28. Launov, A., and K. Wälde (200): Estimating Incentive and Welfae E ects of Non- Stationay Unemployment Bene ts, mimeo - available at Lo, A. W. (988): Maximum Likelihood Estimation of Genealised Ito pocesses with discetely sampled data, Econometic Theoy, 4,