Dynamic investment behavior taking into account ageing of the capital good Feichtinger, G.; Hartl, R.F.; Kort, Peter; Veliov, V.

Transcription

1 Tilbug Univesity Dynami investment behavio taking into aount ageing of the apital good Feihtinge, G.; Hatl, R.F.; Kot, Pete; Veliov, V. Published in: Dynamial Systems and Contol Publiation date: 24 Link to publiation Citation fo published vesion (APA): Feihtinge, G., Hatl, R. F., Kot, P. M., & Veliov, V. (24). Dynami investment behavio taking into aount ageing of the apital good. In F. Udwadia, H. I. Webe, & G. Leitmann (Eds.), Dynamial Systems and Contol. (pp ). London: Chapmann & Hall/CRC. Geneal ights Copyight and moal ights fo the publiations made aessible in the publi potal ae etained by the authos and/o othe opyight ownes and it is a ondition of aessing publiations that uses eognise and abide by the legal equiements assoiated with these ights.? Uses may download and pint one opy of any publiation fom the publi potal fo the pupose of pivate study o eseah? You may not futhe distibute the mateial o use it fo any pofit-making ativity o ommeial gain? You may feely distibute the URL identifying the publiation in the publi potal Take down poliy If you believe that this doument beahes opyight, please ontat us poviding details, and we will emove aess to the wok immediately and investigate you laim. Download date: 8. sep. 26

2 Dynami InvestmentBehavio TakingInto Aount Ageing of the Capital Good. Feihtinge, Gustav, Hatl, Rihad F. 2 ; Kot, Pete M. 3 ; Veliov, Vladimi Institute fo Eonometis, OR and Systems Theoy, Univesity of Tehnology, Vienna, Austia. 2 Institute of Management, Univesity of Vienna, Vienna, Austia. 3 Depatment of Eonometis and Opeations Reseah and CentER, Tilbug Univesity, Tilbug, The Nethelands. Januay 3, 2 Abstat In standadapital aumulation models all apital goods ae equally podutive and podue goods of the same quality. Howeve, due to ageing, in eality it holds most of the time that newe apital goods ae moe podutive. Impliations of this featue fo the m s investment poliies ae investigated in an optimal ontol poblem with distibuted paametes. It tuns outthatinvesting in apital goods of di eent age is done suh that the net pesent value of maginal investment equals zeo. Compaing the etunsof investmentinapital goodsof di eent age, the highe podutivity of younge apital goods has to be weighed against the lowe osts of depeiation, disounting and aquisition of olde apital goods. In the steady state it holds that, in the most easonable senaio, the m should invest at the highest ate in new apital goods, and disinvestment an only be optimal when osts of aquisition ae lage and mahines ae old. Keywods: Investment, Vintage Capital, Ageing, Maximum piniple Intodution One of the diving foes in a maket eonomy is the gowth of ms and industies. In the liteatue the analysis of m gowth stated out in the sixties with Eisne and Stotz []. In the famewok they onsideed the m owns a stok of apital goods that is needed to podue goods, whih ae sold on the Coesponding autho, Institute fo Eonometis, OR and Systems Theoy, Vienna Univesity of Tehnology, Agentiniestasse8, A-4Vienna, Austia, Telephone: / 927, Fax: / 999, o@e9ws.tuwien.a.at.

3 maket to obtain evenue. The m is able to inease apital stok by investing. This po t maximization poblem thus involves the hoie of investments toexpand the stok of apitalgoods. Afte this st ontibution by Eisne and Stotz [], many othes have followed (e.g., Luas [2], Davidson and Hais [3], Baui [4]), and they mostly di e in the spei ations of evenue and investment ost funtions All these ontibutions have in ommon that apital stok is homogeneous. Hene, its featues do not hange ove the yeas, so that it an be onluded that mattes like ageing and tehnologial pogess ae not taken into aount. The aim of this pape is to analyze a model whee apitalgoods with di eent ages aedistinguished. To do so avintage apital stok model is developed. We use Hauie, Sethi and Hatl [5] as basi depatue point (see also Appendix 5 of Feihtinge and Hatl [6]). In ode to show what in uene ageing has on the age distibution of the apital stok we onside a situation whee thee is no tehnologial pogess and thee is onstant etuns to sale. Podutivity only depends on its age. This means that apital stoks of the same age have the same podutivity independent of the yea in whih they ae opeating. Thus eah apital good of the same age podues a xed amount. The vintage apitalmodelhas beomeineasingly popula among eonomists, espeially beause it povides an appealing famewok fo the analysis of investment volatility. Howeve, Baui and Gozzi [7] state that, apat fom thei pape, in the liteatue the vintage di eentiation of the apital goods has not been analyzed in a omplete dynami optimization famewok; often apital goods ae not duable, they an not be aumulated and theefoe the apital aumulation poblem eithe beomes a simple intetempoal budget alloation poblem (e.g. Gossman and Helpman [8]) o apital is ompletely absent as an expliit input fato (e.g. Chai and Hopenhayn [9]). Xepapadeas and De Zeeuw [] limit thei analysis to the OSSP (Optimal Steady State Poblem). Jovanovi [] agues that full dynamis ae notoiously di ult in suh models. Ou pape o es a omplete dynami optimization famewok, but ontay to Baui and Gozzi [7] who onentate on tehnologial pogess, we fous on the e ets of ageing on the dynami investment ates and on the age distibution of apital goods in the steady state. Like Xepapadeas and De Zeeuw [], ou analysis thus mainly onsides the steady state, but additionally we show that it is in fat optimal fo the m to eah this steady state as soon as possible. The steady state does not exist in Baui and Gozzi s model due to the tehnologial pogess onsideed thee. By analyzing this model we ae able to detemine the m s optimal investment deisions in apital goods of di eent ages. It tuns out that the m always invests in suh a way that the net pesent value of maginal investment equals zeo, sothat the disounted extaevenue steam aused by the addition of a apital good exatly balanes the maginal investment osts. Investments in younge mahines have the advantage that due to ageingthey aemoe podutive than olde ones, but the disadvantage is that olde mahines ae heape and the osts of depeiation and disounting ae less. The pesene of the 2

4 latte e ets may explain why, aoding to Chai and Hopenhayn [9], it is undeniable that new tehnologies ae often adopted on a lage sale only afte a polonged peiod of time (see Mans eld [2] fo empiial evidene). Fo the steady state it tuns out that, povided that the disount ate is su iently low, the m should invest mostly in new apital goods. Disinvestment only ous if aquisition osts ae high and mahines ae su iently old. The pape is oganized as follows. The model is fomulated in Setion 2. In Setion 3 the optimality onditions ae fomulated and expessions fo the investment ate in apital stoks of di eent age ae deived and eonomially analyzed. Moeove, Setion 3.3 onsides the m in steady state in ode to see how the age distibution of apital goods then looks like. 2 The Model In a eent pape Xepapadeas and De Zeeuw [] studied the ideal age omposition of the apitalstok subjet toenvionmentalegulation. Hee weonside a elated vesion of the model of Xepapadeas and De Zeeuw []: whee they onentate on envionmental egulation by speifying pollution output, we leave this out. Instead, we extend thei famewok by adding disounting and depeiation, so that this pape is a natual extension to the apital aumulation liteatue mentioned in the st paagaph of the Intodution. As in thei pape, hee it also holds that the age of the mahine is denoted by 2 [;h], so that the maximum age of mahines is h: v() is the output podued by a mahine of age, with v () : That is, a newe mahine annot podue less output than an olde one. Sine v is independent of time t no tehnologial pogess is inluded. The stok of apital goods of age at time t is denoted by K(t;). Then total output podued in yea t is de ned as Q(t) v()k(t;)d: It is assumed that makets exist fo mahines of any age fom to h. Let b() be the ost of buying a mahine of age, with b () (olde mahines annot be moe expensive than newe mahines) and b(h) (a mahine at the maximum age is not woth anything). Let I(t;) be the numbe of mahines of age bought (if I(t;) ) o sold (if I(t;) < ) in yea t. The total ost o evenue to the m fom tansations This model featue is taken fom Xepapadeas and De Zeeuw [] (see also Baui and Gozzi[3]) who ague that this implies that new mahines ae moe podutive beause they embody supeio tehnology. Howeve, this agument seems to be wong. To see this, note that v() is the same fo di eent t. Now onside two points of time: t and t2 so that t2 t. Then a mahine onstuted at time t2, say m2, has the same podutivity at the same age as a mahine onstuted at time t (m ), i.e. m 2 podues att 2 + :v(), whih is also the amount thatm podues att+: Hene thee is no supeio tehnology embedded in m 2 : Theefoe, in ode to inlude tehnologial pogess, output should be modelled by v(t;) with, at least,vt. 3

5 in the mahine maket is de ned as b()i(t;)+ 2 [I(t;)]2, with the seond tem e eting the adjustment osts in buying o selling mahines. These osts ae, fo example, adaptation osts o seah osts. The quadati fom of this ost tem leads to a simple expession fo optimal puhases. It is futhe imposed that mahines of age depeiate with ate ± (), whih is the same fo evey vintage. The m hooses to buy o sell mahines of di eent ages in ode to maximize po ts, with p the pie of output. That is, the m hooses at eah point in time an age distibution of mahines to maximize po ts. In addition to Xepapadeas and De Zeeuw [], ou modelalso inludes disounting, whee is the disount ate. The dynami model of the m is now given by max I(t;);I (t) Z Z e t e t h b I (t) + 2 [I (t)] 2i dt h pv()k(t;) b()i(t;) 2 [I(t;)]2i ddt() I(t;) ± ()K(t;); (2) K (t;) I (t);k (;) K (): (3) This is an in nite hoizon optimal ontol poblem with tansition dynamis desibed by a linea patial di eential equation (Calson, Hauie and Leizaowitz [4]). The tansition equation indiates that the ate of hange in the numbe of mahines of a given age,, at a given time, t, is detemined by two fatos. These ae the edution o inease in the numbe of mahines bought about by the sale o aquisition of mahines of the given age (the st tem of the tansition equation), and the edution due to depeiation at ate ± (). The initial ondition on the numbe of mahines implies that the m stats with given amount K () of mahines of age. At eah time t it is possible to buy new mahines. This puhase ate of new mahines is denoted by the bounday ontol I : 3 Analysis of the Model Fist, by using the maximum piniple analytial expessions ae obtained fo investment and apital stok in Setion 3.. It is shown that afte h yeas the steady state will be eahed. In Setion 3.2 the expessions fo investment and apital stok ae eonomially analyzed. Setion 3.3fouses entiely at the steady state to see how the optimal age distibution in the steady state looks like. 4

6 3. Maximum Piniple The uent value Hamiltonian H fo this poblem is given by (see, e.g., Feihtinge and Hatl [6]): H pv()k(t;) b()i(t;) 2 [I(t;)]2 + (t;) [I(t;) ± ()K(t;)]; (4) while the bounday Hamiltonian is H b I (t) 2 [I (t)] 2 + (t;) I (t): Consequently, the st-ode onditions fo ; o I(t;) (t;) ; o I (t) (t;) b ; (t;) ( + ± ()) (t;) pv(); (7) (t;h) : (8) Solving the patial di eential equation (7), while taking into aount the bounday ondition (8) yields: (t;) (+±(½))d½ pv(s)ds: (9) Fom (5) and (9) the optimal investment ate is obtained: 2 I(t;) 4 Zh (+±(½))d½ pv(s)ds b() 5: () By (6) and (9) it an be onluded that a simila expession holds fo the investment in new apital goods: I (t) 2 4 Zh (+±(½))d½ pv(s)ds b 3 3 5: () 5

7 An expession fo the stok of apital goods an be deived fom (2), assuming fo the moment that t: µz K(t;) e R ¾ ±(½)d½ I (t + ¾ ;¾)d¾ + A 2 e R ±(½)d½ : (2) Note that the initial stok is A 2 K (t ;) I (t ) (see (3)). Combining the last thee expessions, we obtain Z e R ¾ ±(½)d½ 2 4 ¾ ¾ (+±(½))d½ pv(s)ds b(¾) 5d¾A e R 3 ±(½)d½ ÃR R h s! + e (+±(½))d½ pv(s)ds b e R ±(½)d½ : (3) Note that this fomula is only valid fo t: In ase t; i.e. the vintage aleady exists at the initial time, it is easily obtained via the seond bounday ondition in (3) that K(t;) K ( t)e R ±(½)d½ t + (4) 2 3 Z t e R ¾ ±(½)d½ 4 ¾ ¾ (+±(½))d½ pv(s)ds b(¾) 5d¾: An impotant obsevation is that (9), () and () ae time invaiant. Moeove, K (t;) depends on t only in ase t < : This means that afte h yeas eveything beomes time invaiant, that is, the steady state with espet to alenda time is eahed. 3.2 Eonomi Analysis Let us analyze by what haateistis the investment ate in mahines of diffeent yeas is in uened. The amount of investment is given by 2 I(t;) 4 fo olde mahines, and I (t) Zh 2 4 (+±(½))d½ pv(s)ds b() 5 (+±(½))d½ pv(s)ds b 3 5 (5) in new mahines. It follows that the net pesent value of maginal investment equals zeo: the tem with the integal equals the evenue steam (oeted 3 6

8 fo disounting and depeiation) geneated by an exta unit of apital stok of age (o ) bought at time t; and this exta evenue equals total maginal investment osts b + I: It is lea that no investment will take plae in a mahine of age h, so that I(t;h) : At a given point of time t the investment ate is in uened by its age as (t;) ( + ± ())@ pv() b (): (+±(½))d½ pv(s)ds A (6) Expession (6) shows how investment is a eted when the m ompaes investing in a mahine of age with investing in a mahine of a maginally olde age. Aoding to the RHS of (6), thee e ets aise. The st e et is positive and onsists of a disounting and a depeiation e et. The depeiation e et esults fom the fat that by buying a mahine of olde age the mahine is depeiated less at the moment that its age is s; thus when its podutivity equals v(s). The disounting e et is also positive, beause the evenue obtained at the moment that the mahine is of age s is obtained ealie so that the disounted evenue is highe. The seond e et is negative and aises fom the fat that when buying the mahine of a maginally olde age than ; it will not ollet the evenue when the mahine opeates at age : The last e et is positive whih is due to the fat that the aquisition osts of olde mahines ae heape. These e ets may help to explain why ms often invest in olde tehnologies even when appaently supeio tehnologies may be available (Chai and Hopenhayn [9]. Aoding to (6) easons may be that (i) e ets of disounting and depeiation ae substantial, and (ii) an olde mahine has a lowe aquisition pie. Expession (6) also helps to explain the obsevation that new tehnologies ae often adopted so slowly, as eognized by, e.g., Chai and Hopenhayn [9]. Reasons fo suh behavio an thus be that e ets of disounting and depeiation (espeially duing the st yeas that a new apital good opeates) ae lage and/o that the edution of the aquisition pie when the apital good gets olde is substantial. In ase v and ± it an be easily shown that the st e et is always dominated by the seond e et, i.e. the disountingand depeiation e ets ae moe than outweighed by the e et that evenues ae eaned duing a shote time. We illustate this by taking± and v onstant, afte whih expession (6) (t;) h³ ³ (h )(+±) i pv e pv b () h i pve (h )(+±) b () : 7

9 Now thee ae only two ontay e ets of age on the investment ate. The advantage of investingin a mahineof olde age is that investments aeheape as e eted by the tem b () : Howeve, the disadvantage is that the planningpeiod duing whih the m enjoys evenue fom this investment beomes shote, whih is pesented by the st tem. Conside now the evolution of the apital stok, whee we onentate on those apital goods fo whih < t; thus at the initial point of time this stok was not pesent yet. Fom (2) and A 2 I (t ); it an be obtained (t;) I (t;) ±()K (t;) : (7) Hene, to nd out how apital stoks of di eent age elate to eah othe at a given point of time, would equie substitution of (3) and () into (7), and this beomes too messy fo dawing lea eonomi onlusions. 3.3 The SteadyState As emaked at the end of Setion 3., fom time h onwads the m is in steady state with espet to alenda time. Fist we onside the optimal age distibution in geneal, afte whih we onside a spei example The optimal age distibution Fom (9) it follows that () is given by () (+±(½))d½ pv(s)ds (8) The value of as given by (8) e ets the bene ts fom installingone mahine of age and keeping it until it beomes of maximum age. Fom (5) the optimal sales o aquisitions of mahines of age is given by I() () b() (+±(½))d½ pv(s)ds b(): (9) Note that < A ; as < A b(); whih is intuitively lea beause denotes the bene ts and b denotes the pie of new mahines. The stokofmahines of age is patly detemined by sales and aquisitions of mahines of that age and patly inheited fom sales and aquisitions in the 8

10 past. The set of stoks of all ages is the optimal age distibution of mahines and fom (3) it is obtained that R h K() er ¾ ±(½)d½ R R i h ¾ e s (+±(½))d½ ¾ pv(s)ds b(¾) d¾e R ±(½)d½ + h R h e R s (+±(½))d½ pv(s)ds b ie R ±(½)d½ : Example In ase thee is no depeiation ± () and no initial investment I ; as in Xepapadeas - De Zeeuw the solution simpli es to: K() () e (s ) pv(s)ds: (2) R h () b() I() e (s ) pv(s)ds b() : (2) Z I (¾)d¾ Z " Z # h e (s ¾) pv(s)ds b(¾) d¾: (22) To see what (2) and (22) look like, onside the following example: v() a + a (h ) ; (23) ¾ b() b(h ); (24) whee all paametes ae nonnegative and at least a is stitly positive. This implies that aquisition ost b deline linealy with age of the mahines and output v is linealy deeasing with age : Substitution of these funtions into (2) gives I() e (½ ) p(a + a (h ½))d% b(h ) (25) p(a + a h)e ½ e d½ pa ½e ½ e d½ b(h ) h p (a + a h)e (½ )i µ h pa + e (½ ) ½ + h b(h ) p h e (h ) a + a i + p h a a i + a (h ) b(h ) ; h a a i p h e (h )i h pa i + b (h ) : fom whih it is obtained ³ p a + a e (h ) a p + b; (26) 9

11 so that This yields the following I() 2 ( a + a )pe (h ) : (27) Poposition Unde the spei ations given by (23) and (24) it holds that I (h) : Futhemoe, fo di eent ases the following esults ae obtained:. Low disount ate: < a a 2 I 2 ;.. Low aquisition ost: b < pa < 8 ; I () fo 2 [;h) ; High aquisition ost: b pa < A < A h ln µ a p a p a p b 2. High disount ate: a a @ 2 I 2 < ; 2.. High aquisition ost: b pa 8 ; I () < fo 2 [;h) ; 2.2. Low aquisition ost: b pa : < A < A h ln µ a p a p b a p :

12 We note that the most easonable ases ae pobably. and.2. In ase 2. the solution makes no sense, sine I being equal to zeo and investments being negative fo eah age imply that K will beome negative too. Next, let us onentate on the apital stok athe than investment. To do so, we ombine (22) and (25) to obtain: K() p ³ a + a Z e (h z) dz + Z h + pa i + b zdz p ³ 2 a + a µ pa Z h p ³e (h ) e h + p h + b fom whih it an be deived that (f. p ³ a + a ; e (h ) + p h a a i + a h bh + ³ a a i + a h bh dz a a i + a h bh µ pa + b I(): Due to the last two equations and Poposition we an onlude the following poposition: Poposition 2 Conside the poblem with the spei ations pesented in (23) and (24). Then it holds that apital stok is age dependent in the following j h : Futhemoe, fo di eent ases the following esults ae obtained: µ.. Low disount ate < a and low aquisition ost (b < pa ) : 2 K() 2 < 8 fo 2 [;h); µ.2. Low disount ate < a and high aquisition ost (b pa ) : 2 2 < A < Ah ln µ a p a p a p b ;

13 µ 2.. High disount ate a a and high aquisition ost (b pa ) 2 K() 2 8 < fo 2 [;h); 2.2. High disount ate µ a a and low aquisition ost (b pa ) 2 K() < A < A h ln µ a p a p b a p Eonomi Intepetation To undestand the agedependent investment level, let us ewite (26) as b pa e (h ) pa e (h z) dz: (28) The st tem of the.h.s. of (28) e ets that investing in an olde mahine is advantageous fom the point of view that less investment osts ae inued. The seond tem indiates that investing in an olde mahine implies that the lifetime of this mahine is shote whih edues the evenue steam. The thid tem of the.h.s. of (28) esembles the fat that podution with an olde mahine leads to a lowe evenue ow pe time unit. Explaining Poposition is now an easy job. (28) (f. (27)) implies that, in ase of a low disount I() ineases with (aoding to the thid tem of the.h.s. of (28) the evenue ow edution takes plae duing ashote time inteval when ineases), implying eahes its maximum fo If pa b; is negative fo h, whih implies that it will be negative fo all possible ages. Sine I(h), this in tun implies that the investment ate is positive fo all ages of the apital stok, exept o ouse fo h: In ase aquisition osts ae high (b pa ), it holds I fo su iently lage, whih togethe with I(h) implies that the m sells mahines (only su iently young mahines may be bought, beause fo these mahines a lage lifetime with positive evenues may ountebalane the high aquisition osts). The fat that mahines ae sold in the ase of lage aquisition osts also holds when the disount ate is lage. When aquisition osts ae low, the m again makes use of this by keeping the investment ate positive fo all : 2

14 ages (exept the maximal age). Fo high disount ate it futhe holds deeases with. This is due to the fat that futue evenues ae heavily disounted, so that the e et of the shote lifetime of the mahine (given by the seond tem on the.h.s. of (28)) is less. The esults onening the levels of the apital stoks pesented in Poposition 2 follow dietly fom the investment levels, but additionally it must be taken into aount that olde mahines have a longe investment histoy. It holds that apital stok ineases in a onave way with age if investment is positive but deeasing, apital stok deeases in a onave way with age if investment is negative (mahines ae sold) and deeasing, while apital stok deeases in a onvex way if investment is negative but ineasing. Aknowledgement The authos like tothank Chistian Almede who povided us with valuable omments. 4 Refeenes [] Eisne, R., Stotz, R., 963, The deteminants of business investment, in Impats of Monetay Poliy, Pentie Hall, Englewood Cli s, NJ. [2] Luas, R.E., 967, Adjustment osts and the theoy of supply, Jounal of Politial Eonomy, 75, [3] Davidson, R., Hais, R., 98, Non-onvexities in ontinuous time investment theoy, Review of Eonomi Studies, 48, [4] Baui, 998, Optimal investments with ineasing etuns to sale, Intenational Eonomi Review, 39, [5] Hauie, A., Sethi, S., Hatl, R.F., 984, Optimal ontol of an agestutued population model with appliations to soial sevies planning, Lage Sale Systems, 6, [6] Feihtinge, G., Hatl, R.F., 986, Optimale Kontolle Oekonomishe Pozesse: Anwendungen des Maximumpinzips in den Witshaftswissenshaften, de Guyte, Belin. [7] Baui, E., Gozzi, F., 2, Tehnology adoption and aumulation in a vintage apital model, fothoming in: Jounal of Eonomis. [8] Gossman, G., Helpman, E., 99, Quality laddes in the theoy of gowth, Review of Eonomi Studies, 58, [9] Chai, V.V., Hopenhayn, H., 99, Vintage human apital, gowth, and the di usion of new tehnology, Jounal of Politial Eonomy, 99, [] Xepapadeas, A., De Zeeuw, A., 999, Envionmental poliy and ompetitiveness: the Pote hypothesis and the omposition of apital, Jounal of Envionmental Eonomis and Management, 37, [] Jovanovi, B., 998, Vintageapitaland inequality, Review ofeonomi Dynamis,, [2] Mans eld, E., 968, Industial Reseah and Tehnologial Innovation: An Eonometi Analysis, Longmans, London. [3] Baui, E., Gozzi, F., 998, Investment in a vintage apital model, Reseah in Eonomis, 52,

15 [4] Calson, D., Hauie, A., Leizaowitz, A., 99, In nite Hoizon Optimal Contol, Spinge-Velag, Belin. 4