CBMR

7626 S1985195 ~

uuu iiii i,NiHi iii iiui uui,iimiiui Nn uiisubfaculteit der econometrie

RESEARCH MEMORANDUM

ILBURG UNIVERSITY

)EPARTMENT OF ECONOMICSbstbus 90153 - 5000 LE Tilburgletherlands

FEW195

OPTIMAL DYNAMIC INVESTMI.:NT POLICYUNDER FINANCIAL RESTRICTIONS ANDADJUSTMENT COSTS

Peter M. Kort

Department of EconometricsTilburg UniversityP.O. Box 901535000 LE TilburgThe Netherlands

Abstract

This paper examines the effects of a convex adjustment cost function on

thi~ optlmal dynHmic~ ínvestment pulicy of a flrm with finnncial restríc-

tions. We assume that the management, which oper~~tes under decreasing

returns to scale, maximizes the shareholder's value of the firm. It

turns out that investments are a continuous function of time, that capi-

tal never keeps a stationary value and that there exists an unique opti-

mal investment decision rule for the firm.

The author would like to thxnk Prof. Dr. P..I.J.M. van Loon, Drs.

G..1.C.'th. van Schijndel and Prof. Dr. P.A. Verheyen for many fruitful

discussions and useful suggestions.

Contents

1. Introductíon

2. The theory of adjustment costs

3. The model

4. The optimal solution

5. Summa ry

References

page

1

2

4

7

l2

13

Appendix 1. Derivation of the optimal solution 15

Appendix 2. The development of capital stock and investments

on the two master trajectories 22

Appendix 3. DerivaCion of ttie investment decision r,ile 27

1

1. Introduction

Surveys by Bensoussan, Kleindorfer á Tapiero (1978) and Feichtinger(1982a, 1982b, 1985) excellently illustrate that many recent papers

using optimal control to solve dynamic models analytically, have

extended the theory of the firm. Those models provide insight into theeconomic behaviour of firms over time.

One of the first dynamic models of the firm is the classical model ofJárgenson (1967). The problem with this model is that the resultingoptimal solution dictates an instantaneous adjutitment of the stock ofcapital goods to the level of maximum revenue.In the literature, two ways ín particular have been proposed to avoidthis unrealistic immediate adjustment. The first way is the introduction

of financing limits ín the dynamlc model of the firm. F.xamples of such

models are those of Leland (L972), Ludwig (1978) and van Loon á Verheyen(1984).

The second way of getting a smoothed adjustment pattern is the introduc-tion of adjustments costs as another aspect governing the dynamice ofthe firm. Research into this subject has been conducted by e.g. Gould(1968), Lucas (1967) and Treadway (1969). The article by Sáderstróm(1976) contains a good survey of the theory of adjustment costs.In this contribution we wíll analyse the tmpact of coupling financingand adjustment costs on the optimal policy within a really dynamic modelof the firm. Section 2 contains a global survey of the theory of adjust-

ment costs; in section 3 we will present our dyn;~mic model of the firm.

In this model we have incorporated both financin}; limits and adjustmentcosts. Section 4 contains a description cind f~~rther analyais of theoptimal solutíon, which is proved in Append[x l. l.n Appendix 2 we derivethe development of investmentK nnd capital durín,; two optlmal trajecto-

ríes and in Appendix 3 we give the derivatlon of an investment decislonrule.

2. The theory of adjustment costs

Adjustment costs arise with investment expenditures of [he firm. In the

lí[erature, a distinction is made between external adjustment costs (in-

vestment expenditures) and internal adjustment costs (seize on available

productive inputs) (Brechling (1975)).External adjustment costa apply to a monopsonistic market of capitalgoods: if the firm wants to increase its rate of growth it will be con-fronted with íncreasing príces on the market because of its increaseddemand of capital gooda. Other examples of external adjustment costsare: architects' fees, expenditures on job advertisements and costs ofmoving, new employees.Internal adjustment coats arise because the acquisition of additionalcapital and~or labour requirea resources which could otherwise be usedfor the production of output. For instance, a firm may have personneland training departments which are adequate for regular replacement ofquits and retirements. Suppose the firm now wiahes to raise its level ofemployment by hiring more people. In consequence more capital and labourhave to be invested in both these departmente. With given total inputsthe level of output muet, therefore, fall.Inatallation cos[s and organisation coats are other examples of internaladjustment costs.

We can consider three different shapes of the adjustment cost functionas given in figure 2.1. it is always assumed that the first derivativeof the cost of adjustment function is positive. The question is whetherthere are constant, increasing or decreasing costs compared to the rateoE adjustment. In accordance with standard terminology, costs of adjust-ment in these three cases will be called linear, convex and concave(S6derstrGm, 1976).The curvature of the adjustment cost function could have an impact onthe optimal investment policy of [he firm. If the cost of adjustmentfunction is convex, marginal adjustment costs are increasing with in-vestment expenditures. Therefore, the total cos[ of raising capitalstock by a gíven amount will be larger the faster the growth of capitalstock, and hence the Eirm will tend to adjuat it slowly.

3

acl.jtiSU'.~t't1L cusls

CUI1V('K

concave

linear

Yl~~ i investment ex~~enditures

Figure 2.1. The curvature of the adjustment cost func[ion

Concave adjustment cost functions imply declininq mar~inal costs. Thefirm's policy would be to take advantage of decreasing costs of invest-ment and raise its capital stock instantaneously.As linear adjustment costs imply only a rising Ftrice level of the in-

vestments, its impact on the firm's investment polícy is obvious.

In the literature, most dynamic models have incorporated a convex costof adjustment function, but other types of adjust~nent cost functions canalso be considered (Rothschild, 1971).

4

3. The model

We first assume that the firm behaves as if it maximizes the share-holder's value of the firm. This value consists of the sum of thepresent value of th~~ dividend stream over the planning period and thepresent value of fin.~l equlty at the end of the planning period, so:

zmaximize: f D(T)e iTdT f X(z)e-iz

T-0

in whichD(T) ~ dividend

X(T) - equity

T ~ time

i a discount r:~te

z z planning horizon

(1)

Assuming that the firm wtll attract only one kind of money capital:equity and has one production factor: capital goods, we get the balanceequation:

K(T) - X(T) (z)

in whichK(T) a total amount of capital goods

We further assume that c~arnings after deduction of depreciation andadjustment costs are used to issue dividend or to increase the value ofequity through retained earnings.As far as the adjustment c-oste are concerned, we assume that they are aconvex function of investments. Also, we assume that the firm operatesunder decreasing returns to scale and that depreciation is proportionalto capttal goods. 'I'he al,ove results in the next atate equ~tion ofequity:

X:~ áT s S(K) - aK(T) - U(1) - D(T) (3)

5

in which 2S(K) - earninRs, S(K) ~ 0, áK ~ 0, a2~ 0

dK2

U(I) - adjustment costs, U(I) ~ 0, dÍ ~ 0, a2 ~ 0, U(0) ~ 0dI

I(T) - gross investmentsa - depreciation rate

The impact of investments on tlie production str~~cture is described bythe, now generally used, formulation of net investments:

K :- dT - I(T) - aK(T)

max Jz (S(K)-I(T)-U(I))e-iTdT t K(z)e-1zI T-0

As far as its dividend policy is concerned, we assume that the firm is

allowed to pay no dividend, so:

D(T) ~ 0

Investments are irreversible, so:

I(T) ~ 0

A[ last, we assume a positive value of capital go~~d stock at T a 0:

K(U) - K~ ~ 0

s.t.

After some simplifications, we get the followin}~, dynamic model of theFírm:

(4)

(5)

(6)

(7)

(8)

K - I(T) - aK(T) (9)

S(K) - I(T) - U(I) ~ 0 (10)

6

I(T) ~ 0 (11)

K(0) a Kp (12)

This model can be solved analytically by using optimal control theory(Kamien 5 Schwartz (1983)), where the state of the syatem is describedby the amount of capital goods and is controlled by investments. The aimof this control is to reach a maximum value of the objective function.

7

4. The optimal soliition

We obtain necessary and sufficient conditions fnr an optimal solutionusing PontryaRin's standard maximum princ:iple. Next, we apply thegeneral solution procedure of van Loon (1983, p~~. 115-117) to get theoptimal trajectories of the firm (see Appendix 1).Each trajectory consists of one or more feasible paths, which are cha-

racterized by dif.ferent policies concerning investment expenditures anddividend payments. In our problem the set of fea~ible paths amounts tothree:

path I D

1 max 02 ~ 0 ~ Il

3 0 max

Tabel 4.1. Features of the feasibl~ paths

Depending on the values of K~ and z, we get diffc.rent optimal trajecto-ries. Here we will demonstrate two of them, represented in figure 4.1

and 4.2 and derived in Appendix 2. The other patt~~rns are subsections ofthese two solutions.

Concerning figure 4.1, we have to remark that tl,e way I diminishes on

path 2 depends completely on specific features of S(K) and U(I). So,when S(K) and U(i) are not specified, we do not know whether the slope

of ( lncreases, decre~ses ur remalns conr,tant ~~n thís path. Withou[theae features of S(K) and U(I) we do not have insight ínto the way Irises on path 1 ei[her.

First, we concentrate on master trajectory 1, which is repreaented by

figure 4.1. On path 1, the next inequality holds (see Appendix 3):

dU -(i~-a)(z-T) z -(ita)(t-T) dS1 f áI ~ e t J e áK dT (13)

t-T

KD

IO

0 path i t12 path 2

Figure 4.1. Development of I, K, and aK on master trajectory 1

I

~ I !' ~--

0 path 3 t3z path 2

Figure 4.2. Development of I, K and

1- "-~tímeZt23 path 3

t23 path3Z~ time

aK on master trajectory 2

9

The left-hand side of this expression represents an investment expendi-

ture including adjustment costs, at the right-hanci side we find the mar-ginal earnings of investments, conaisting of the present value of the

remaining new equipment at the end of the planning period (the value ofthe new equipment decreases with a rate a during the rest of the plan-

ning period) plus the preaent value of additional ealea over the wholeperiod due to this new equipment (the production ~~apacity of this equip-

ment decreases with a rate a during [he rPat of the planning period).

Expression (13) means that on path 1, marginal e:irninga exceed marginal

costs of investment expenditures. Therefore, tlie firm invests at ita

maximiun level, i.e. that leve] which is feasible considering the finan-

cial restrictions, so it dces not pay out any dividend and all earningsare spent for investing.

At t12 this strategy stops, because marginal earnings become too small

(dK decreases when K rises) to finance the risin~; adjustment costs (dIrises when I rises). Therefore on path 2 investmenta are kept on auch alevel that marginal earnings equals marginal costs, so the next equation

must hold:

1} du L è(ifa)(z-T) } Jz é(ifa)(t-T) dS dt (14)dI t~T dK

This implies decreasing investments and increasing capital stock until I

falls below the depreciation level. From this very moment K will alsodrop. Just when investments become zero, path 2 passes into path 3. This

transition is fixed by the moment that the next expressíon becomes

applicable:

1} dU ~ é(ita)(z-T) } r z é(ita)(t-T) dS dt (15)dI t1-T dK

The inequality siiows us that the margínal costti of investments exceed

the marginal earnings on path 3. This is c~.aused by the fact that from

t23 on the remaining time period is too ahort to defray the adjustment

costs of new investments. Therefore, the firm does not inves[ anymore onpath 3.

1 ~l

A noteworthy polnt ie th~~ con[inuity of I. Larger values of I implyrising marginal adj~istment costs because of the convex ad justment costfunction. Therefore, adjustment costs are minimized as much as possibleíf I develops gradually over tíme.

Another interesting feature is the way in which this pattern will changewhen the planning period is extended. If z is fixed upon a higher value,then t12 as well as t23 will be postponed (see Appendix 3). In the caseof an infínite time horiz~~n expreasion (14) continues to hold from t23on, so path 3 disappears c~impletely. This is easy to understand, becausenow there is always enougl~ time to clefray the adjustment costs. On path2, K will approach a ata[t~~nary value~ asymptotícally (fiqure 4.3). Neri~,the influence of [he convex adjustment cost funetion becomes clear; theoptimal value of capital good stock will not be reached within a finitetime period, because it ís always cheaper to split up the final adjust-ment into two parts.

On master trajectory 2(fit;ure 4.2), K~ is so large (this means [hat dKis low) that expression (15) holds at T~ 0 for all possible values ofdU ~ This im lies that investments are zero and ca italdI P p good stock de-creases. At t32, dK has risen enough for expression (14) to becomeapplícable. From this very moment I starts to rise, but tt never reachesthe depreciation level, s~ K still decreases. At t23 the remaining timeperiod ís again too short to defray the adjustment costs of new invest-ments. This means th~t I b~~comes zero again.

In accordance to master trajectory 1 path 2 will be final path on mastertrajectory 2 when the time horizon is infinite. In this case capitalgood s[ock will approach its stationary value asymptotically from above(figure 4.4).

11

LII

0 path 1 t12

~

path 2

Figure 4.3. Master trajectory 1 in the case of an infinite time horizon

KO

aK

~K~~.-----~--------

0 path 3 32 path 2 ---~ t ime

Figure 4.4. Master tra,jec[ory 2 in the case of an infiníte time horizon

12

5. Summary

In a dynamic model of the firm we have incorporated financing and a con-vex cost of adjuatment function. We derived the optimal trajectories ofthe firm by applying Pontryagin's standard maximum principle and thegeneral solution procedure of van Loon. Some striking characteris[ics ofthe optimal solution are the continuity of investments during the plan-ning period and the absence of a stationary value of the capital goodstock. We have also deríved an investment decision rule that explainsthe optimal policy of the Eirm hy comparing marginal earnings and mar-ginal costs in the succeeciing atages of the fírm's evolution. With thehelp of this rule we fix the moments on which the firms policy has to bechanged fundamentally and we díscuss the relation between these momentsand the length of the planning period.

13

References

Bensoussan A., P.R. Kleindorfer S Ch.S. Tapiero, (1978), Applied OptimalControl (North Holland, Amsterdam).

Brechling F., (1975), Investment and employment decisions (Manches[er

Liniversity Press, Manchester).

Feichtinger G., (1982a), Anwendungen des Maximum-Prinzipe ím Operations

Research, Teil 1 und 2(Applications of the maximum principle in Opera-

tions Research, vol. 1 and 2), OR Spektrum 4, pp. 171-190 and 195-212.Feichtinger G., (1982b), Optimal control theory and economic analysis,

(Noth Holland, Amsterdam).Feichtínger G., (1985), Optimal control thenry a~id economíc analysis 2,

(North Holland, Amsterdam).

Gould J.P., (1968), Adjustment costs in the theor~y of investment of thefirm, Review uf F.conomic Studies, pp. 47-55.Jorgenson D.W. (1967), The theory of investment hehaviour, in: R. Ferbes

(ed.): Determinants of investment behaviour, National Bureau of Economic

Research, (Columbia University Press, New York).

Kamien M.I. á~ N. Schwartz, (1983), Dynamic optimization, the calculus of

variation and optimal control in economics and management, 2nd reprint

(North Holland, New York).

Leland H.F.., (1972), The dynamics of a revenue maximizing firm, Inter-

national F.conomic Review 13, pp. 376-385.T.oon P..1..1.M. van (1983), A dynamic theory of the firm: production,

finance and investment, Lecture notes in economics and mathematical sys-

tems 218, (Springer, Berlin).Loon P..l.J.M. van b P.A. Vertieyen (1984), Gr~ieifasen bij bedrijven

(Growthstages of the firm), Maandblad voor Accou~itancy en Bedrijfahuis-houdkunde 58, pp. 386-394.

Lucas R.E., (1967), Optimal investment policy a~id the flexible accele-rator, International F.conomic Review, VIII (February 1967), pp. 78-85.Ludwig Th., (1978), Optimale Expansionspfade der Unternehmung (OptimalGrowthstages of the firm), (Gabler Verlag, Wiesbaden).Rothschild M., (1971), On the cost of adjustment, Quarterly Journal ofF.conomics, pp. 605-622.

14

Sáderstrám H.T., (1976), Production and investment under costs of ad-justment, a survey, 7.eitschrift fur National8konomie 36, pp. 369-388.Treadway A.B., (1969), On rational entrepreneurial behaviour and thedemand for inveatment, Review of F.conomic Studies, pp. 227-239.

15

Appendix 1. Derivation of the optimal solution

We apply Pontryagin's standard maximum princtple to obtain the necessaryand sufEicient conditions.

Let the Hamiltonian be:

H - (S(K)-I-U(I))e-iT t ~,(I-aK)

and the Lagrangian:

L~ H t at(S(K)-I-ll(I)) -~ aZI

in which:

(16)

(17)

y~ .- adjoint variable or co-atate variable which denotes themarginal contribution of capital good stock to the per-formance level

1`j :- dynamic Lagrange multiplier representing the dynamic'shadow price' or 'opportunity costs' of the j-th res-

triction

then the necessary conditions are:

6I - -(lf áÍ)(e-iT}zl) } ~ } ~2 ~ U

-y - áK (e-iT~~l) - a~

al(S(K)-I-U(I)) ~ 0

(18)

(19)

(20)} complementary slackness conditions

a2I - 0 (21)

-iz~y(z) - e (transversality condition) (22)

~y s continiious with piecewise co~~tinuo~is derivativea (23)

~r,a : continuous on intervals of continuity of I (24)

16

K . conti~~uous

I : piece~~ise c~~ntinuous

(25)

(26)

As xoptimal i s conca~e in (K,I) and the functions S(K) - I- U(I) and Iare quasi-concave in (K,I), these conditions are also sufficient (VanLoon, 1983 (pp. 105)l.

Next, we can apply Van Lnon's general solution procedure in order totransform these conc~itions into the optimal trajectories of the firm.These trajectories consiet of different paths, which are each of themcharacterized by the set oE active constraints. The properties of thesepaths are presented hy tab~~l A.1.

path ~1 ~2

1 f 0

2 0 0

3 0 t

4 f t

Table A.1. The different paths

We will prove that pnth 4 is infeasible:From table A.1, (20) and (?1) we conclude that on path 4:

S(K) - I- U(I) s 0 (27)

I~ 0 ~ U(I) ~ 0 (28)

From (27) and (28) w~~ obtatn:

S(K) - 0 i K s 0

But, ( 4), (6) and (7) imply that:

(29)

17

K ~ 0 (30)

conclusion: path 4 is infeasible

To find the optimal trajectories of the firm, we ~;tart at the time hori-zon z and go backwards in time. According to this strategy we firstselect all final paths. In order to find these paths, we substitute thetransversality condition (22) in (18) for T z z:

-(lf ai)(éiz}~}~ } é iz }~2 ~ 0 (31)

From t}iis, we can conclude tliat on a final ~iath it muat hold that:

a2 ~ 0 (32)

From (32), we conclude tliat only path 3 is a feasible final path.

Next, we start a coupling procedure to complete the optimal trajec[o-

ries. The essence of coupling two paths is to tc~st whether such a cou-

pling will or will not violate [he continuity ~~roperties of the state

variables and the co-state variables. In our model, this means that K

and ~y have to be continuous.As an example of a feasible coupling we wiLi prove that path 1 can pre-

cede path 2.

First, we derive the necessary conditions for th~~ continuity of ~. From

(18) and table A.1 we Ket:

On pat}i 1 lt holds that:

~y - (1-~ ái)(e-iT}~l)

On path 2 tt holds tha[:

~ - (1} ddÍ)eiT

(33)

(34)

From (10), (20) and table A.1 we derive:

18

On path 1 it holds t~iat:

S(K) ~ I f U(I)

Qf, path 2 it holds that:

S(K) ~ I i- il(I)

Because K has to be continuous and S(K) ís a continuous function of K,we conclude from (35), (36) and the convexity of U(I):

i t-i ffor t12 : I~ I (lt áI) ~(lt áI)

(tlZ is the coupling moment of path 1 and path 2. An arrow to the right(left) indicates the left (right) side limít of the relevant variable atthe relevant point or time.)

From (33), (34), (3') and table A.1 we derive the following necessaryconditions for the c~,ntinutty of ~:

I must be c„ntinuous at t12

al must be continuoua at t12

(35)

(36)

(37)

(38)

(39)

Next, we check if K~.an be continuous when (38) and (39) must hold. Whenwe differentiate (33~ to tlme it holds that on path 1:

2 .dI2

I(e--iT}al) - ie-iT(lt dÍ) f al(1~- áÍ) (40)

From (19) and table n.l we obtain that on path 1 it holds tha[:

~ - a~ - ~ (e-iT}~1) (41)

After substituting ("t3) and (40) in (41) we get:

19

2 .((ifa)(lf áÍ) - áK - a2 I)e it -

dI

2 .(-a(lt áÍ) } áK t a 2 I)~1 } Z 1(1} áÍ)dI

Analogous to the above, we can derive that on pat~i 2, it holds that:

(ita)(If dI) - dK -d2U ~dI2

I 3 0

From (39) and table A.1 we obtain:

-i

~1(t12) ~ U

~

~1(t12) t 0

(42)

(43)

(44)

(45)

Due to (42), (44) and (45) we get that at the end of path 1, it holde

that:

2 .(ifa)(lt dÍ) - dK - d 2 I~ UdI

(46)

2From ( 38), (43) and ( 46), the continuity in I of {Í and `~2 and the con-

dItinuity i n K of áK we conclude that a necessary condition of the conti-nuity of K is:

Eor t12 : I ~ I (47)

So, the coupling path 1-path 2 is feasible, íf ttie necessary conditions(38), (39) and (47) hold.

Next, we show an example of an ínfeasibl~~ cout~ling. We try to couplepath 1 to path 3. From (18), (21) and table A.1 we derive that on path 3it holds that:

20

~ - (lt ddi)~-iT - ~2

I - 0

(48)

(49)

Due to (33), (35), ( 48), (49), table A.1 and the convexity of U(I) weconclude tha[ ~y is continuous if:

I, al and a~ are continuous on t13 (50)

Due to (35), (49) and (50) we derive that at the end of path 1 1[ holdsthat:

S(K) - I f U(I) ~ 0 i K- 0 (51)

This is in conflict with (30), so the coupling path 1-path 3 is infea-sible.

Table A.2 gives us ,i survey of the feasible and infeasible couplings.From this table we c~n derive the following possible trajectories:

path 1- path 2- path 3(master trajectory 1)

path 3- path 2- path 3(master trajectory 2)

path 2 - path 3

pa[h 3

Depending on the value of ICD and the length of the planning period oneof these trajectories ia optimal.

21

path 1path 2 `` ~-path 3

in which

path 1 t

path 3

,'`path 2 ~

f : coupling is feasible

- : coupling is infeasible

Table A.2. The feasible and infeasible coupltngs of the paths

z2

Appendix 2. The development of capital stock and investments on the twomaster trajectories

First, we concentrate on master trajectory 1.At T 3 0 we make the following ass~ption:

I((l) ~ aK~

Combining this with (4) we get:

K(0) ~ 0

From (35) we obtain that on path 1 it holds:

dK K - (lt áÍ) I

From (53) and (54) we conclude that on path 1 ít holds that:

K ~ 0

Further, from appendix 1 wc~ know that on path 3 it holds that:

I s 0

The above is represented in figure A.1.

(52)

(53)

(54)

(55)

(56)

(57)

23

IO

aK0aK,

-~-'

~ aK~-

I

0 path 1 t12 path 2 t23 patls? time

Fi.gure A.1. Development of K, I and aK on path 1 and path 3

Due to (6), (34), (48), (49), table A.1, tlie convexity of U(I) and the

continuity of ~ we can conclude that I must be continuous at t23. So at

the end of path 2(57) becomes applicable.Let us assume that at the beginning of path 2 it holds that:

From (43) and (58) we derive:

(ita)(lt dI) ~ dK

(58)

(59)

Due to (58), the convexity of U(I) and the concavity of S(K), we canconcludecreases).us to the

that (1-~a)(lt áÍ)As (59) i s still

increases and áK decreasea (becauae K in-satisfied, I contínues to rise. This brings

conclusion that (58) holds onwhile restriction (10) will be violated

I being zero at the end oE path 2, so

hold at the begin of path 2.

the entire path 2. Then, after aand it i5 also in conflic[ withwe have proved that (58) cannot

From the above, we conclude that at the beginning of path 2, it holds:

(ita)(lt dÍ) t áK (60)

24

Now there are four possibilities:

(ita)(lf dÍ) ~ dK on the entire path 2

2. (ifa)(lf áÍ) becomes equal to dK when I~ aK

3. (ita)(lf ái) becomes equal to áK when I- aK

4. (ifa)(lf dÍ) becomes equal to áK when I~ aK

ad 1.Due to ( 43) we conclude that I~ 0 on the entire pat}i 2.

ad 2.

ad 3.

Due to (43) we conclude that I becomes equal to zero when T~ aK.Now the level of (lfa)(lt áÍ) does not change and K í ncreasesbecause I~ aK. Thi:; implies tha[ áK diminishes and according to

2(43) we derive that (ita)(lt dÍ) - a2 I has to diminish too. So

dIthe level of I must change.I has to diminísh because when it starts to rise, it continues torise at the rest of path 2, which is in conflict with the factthat (57) holds at the end of path 2. Moreover, restric[ion (10)will be violated after a while.

Due to (43) we conclude that I- 0 when I~ aK. This implies thatthe levels of I and K do not change, so a stationary situationarises which is in conflict wíth the fact that (57) holds at theend of path 2.

ad 4.This ímplies that ( ]ta)(1-} dI) ~ dK when I-diminished so áÍ has diminished, and K hasd

aK. Since then I hasdiminished so áK has

the conclusioncan't become equal to áK when i~ aK.

increased. This brings us to that (ifa)(lt áÍ)

25

From the above we can conclude that I will di-minísh on path 2, but an

exception is possible: i can remain at the same level during a very

short period when I~ aK.

Now we have proved Eigure 4.1.

In the case of an infinite time horizon path 2 will be final path. Then

I cannot become equal to zero, so the only satisfactory possibility is

the third one (see figure 4.3).

Let us now concentrate on master trajectory 2.So far, the following is known about the development of I, K and aK on

master trajectury 2 (fiRure A.2).

K, 1, aK

KQ

~aK~`

I

I~ ~ ~~~,, r ,, . ~

~ ~I aK ~

1 --1 ~L

z ~ t imepath 3 t32 path 2 t2j ,path 3

Figure A.2. Development of K, I and aK on path 3

Due to (6), (34), (48), (49), table A.1, the continuity of U(I) and the

continuity oE ~ we can cnnclude that i muRt be e-nntinunus at t32. This

means that the following must hold:

f

for t32 : I ~ 0 (61)

From figure A.2, we derive:

26;

for t23 : I ~ 0

Due to (25), (26) and (43) we can concludetion of time on path 2.Now, there are three possibilities:

1. I z 0 when I~ aK

2. I- 0 when I s aK

3. I s 0 when I~ aK

ad l.

This means that depreciattonspath 2.

(62)

that I is a continuous func-

are larger than investments during

ad 2.In this case a stationary situation arises, which is in conflictwith I being zero at the end of path 2.

ad 3.This means that I~ ~ when I- aK, so ( due to (43)) (ifa)(lf dU) ~dS dU dS dIdK when I ~ aK. Since then aI has risen and aK has diminished, sothís possibility cannot arise.

From the above, we can conclude that possibility 1 will occ~ir, so wehave proved figure 4.2.

In the case of an infinite time horizon path 2 will be final path. Thenpossibility 2 will arise, because I cannot become equal to zero (figure4.4).

27

Appendix 3. Derivation of the investment decision rule

During the whole planning period, it holds that:

z .V~(T) - V~(z) - f V~(r)dt

T-t(63)

After substítuting ( l8), (19) and (22) in (63) Eor al ~ a2 ~ 0 i[ holdsthat on path 2:

(1} áII)e-iT a éiz } r- (áK éir-a~(T))~IT (64)r 1aT

After solving the dif[erentia] equation (19) on prrth 2 and path 3(ala0)and substituting in this solution the transversality condition (22) weget:

zJ~(t) - eat( J e-(ita)t dY dt t e-(ifa)z)

t-r(65)

When we substitute (65) in (64) and dífferentiate this to time we get:

2-i(lt áÍ)é iT } d ~ ÍéiT a

di

aT -(ifa)z dS -iT aT z -(i;-a)t dS- ae e - dK e f ae J e áK dt (66)tvT

After substituting (43) in ( 66) and deviding t}iis by ae-iT we obtainthat on path 2 it holds that:

1 t dU - é(ifa)(z-T) } z é(ifa)(t-T) dS dtdi t~T dK

In appendix 2 we obtained that at the end of path 2 it holds that:

(67)

I ~ 0 (68)

28

From (43), (68) and the continuity in t23 of K and I we get:

for t23 : (ifa)(lt dI) ~ dK

From (57) we derive that on path 3 it holds that:

I ~ 0

(69)

(70)

Due to (69), (70) and the fact that áK risea on path 3 we can concludethat on path 3 it holds tliat:

(i-ta)(lf dI) ~ dK (71)

Due to the continuity of 1 and K on t12, we can conclude that (60) holdsat the end of path l. Also, we know that (56) holds on path 1. There-fore, we can rewrite (46) as follows:

2 .-(ifa)(1-F ~) f d2 I ~- dKdI

When we divide (67) by e(ifa)T we get:

(72)

(lt ái)e (ita)T - é( ífa)z } rz é(ifa)t áK dt (73)t13T

After differentiating the left-hand and right-hand side of (73) to timewe get:

dU -(íta)Ta((lt áI)e)~(-(ifa)(lf d11) } dZU Í)e(ita)T (74)8T dI dI2

za(e-(ifa)z} J é(ifa)t dS dt)t-T dK as -(ifa)T

3-dKe (75)8T

29

Concerníng master trajectory 1, from (70) througl~ (72), (74) and (75) we

can conclude that:

on path 1 it holds that:

1} dU ~ é(ifa)(z-T) } z é(ita)(t-T) dS dtdI tJT dK

on path 3 it holds that:

(76)

1} dU ~ é(i-~a)(z-T) } fz é(it:i)(t-T) dS dt (77)dI t~T dK

Next, we want to know what happens to t12 and t23 on master trajectory 1when the planning period is extended. Therefore we differentiate theleft-hand and right-hand side of (67) to z:

8(lt dI)aZ - ~

a(e (ita)(z-T)} Jz é(ifa)(t-T) dS dt)áKt-Taz

(78)

- (-(ita)t áK) e-(ita)(z-T) (79)

From (43), the fact tha[ K reaches Lts maximum level on path 2 and thefact that I t 0 on path 2, we can conclude thal during the whole plan-

ning period it holds:

dKa (ii-a)(If áÍ)

(80)

Due to (79) and (80) we conclude that the derivative of the right-hand

side of (67) to z has a positive value. So, when the planning period is

extended (z becomes bigger), the right-hand sidc of (67) becomes bigger,

so I must be on a higher level when path 1 pas::es into path 2. So, the

coupling moment of path 1 and path 2 has to be ~~oatponed.

When the coupling of path 2 and path 3 takes place, I must be equal to

zero, so the left-hand side oE (67) always ha; the eame value at this

coupling moment. When z becomes bigger, the right-hand side of (67)

30

becomes bigger. Due to the fact that the right hand side of (67) dimi-nishes during the ttme, t23 has to be postponed.

Finally, we concentrate ~n master trajectory 2.Due to (43), the fact that I~ 0 at the begin of path 2 and the conti-

nuity of K and I, it must hold that:

for t32 : (ifa)(lf dI) ~ dK (AI)

Due to the fact that áK rises on path 3, we can derive that on path 3 itholds that:

(ifa)(lf di) ~ ~1K (82)

From (70), (74), (75) and (82) we can conclude [hat (77) holds on path3.

1

IN 1984 REEDS VERSCHENEN

138 G.J. Cuypers, .J.P.C. Kleijnen en J.W.M. van KooyenTesting the Mean oE an Asymetríc Population:t''our Yrocedures Evaluated

l39 T. Wansbeek en A. KapteynEstíma[ton ln a linear model with serially ~~orrelated errors whenobsa~rvatLuns are missiní;

14U A. 14ipteyn, S. van de (;eer, H. van de Stadt, T. WansbeekInterdependent preferences: an econometric analysis

141 W.J.H. van GroenendaalDiscrete and continuous univariate modelling

142 J.P.C. Kleijnen, P. Cremers, F. van BelleThe power of weighted and ordinary least squares with estimatedunc:yual variances in experimental desii;n

l43 J.P.C. KleijnenSuperefEicientexperiments

~sttmation of power fun~tions Ln símulation

l44 P.A. Bekker, D.S.G. PollockIclentlílr:itton ot linenr r;tochastic mo~lels with covariancerestrictLuns.

145 Max ll. Merbis, Aart J. de 'LeeuwFrom structural form to state-space form

146 T.M. Doup and A.J.J. TalmanA new variable dimenston simplicial algorith~n to find equilibria onthe product space of unit simplices.

147 G. van der Laan, A..J.J. Talman and L. Van der HeydenVariable dimension algorithms for unproper labellings.

148 G.J.C.Th, van SchijndelDynamic firm behaviour and Eínancíal leverap,e cllenteles

149 M. Plattel, J. Peil'The ethico-political and theoretical reconstruction of contemporaryeconomic doctrínes

15U F.J.A.M. Hoes, C.W. VroumJapanese Business Policy: The Cash Flow Tri.~nglean exercise in sociological demystification

151 T.M. Do~pl, G, van der Laan and A.J.J. Talma~iThe (2 -2)-ray algorithm: a new simpllci:il algorithm to computeeconomic equilibria

il

IN 1984 REEDS VERSCHENEN (vervolg)

152 A.L. Hempenius, I'.G.H. MulderTotal Mortality Analysis of the Rotterdam Sample of the Kaunas-Rotterdam Intervention Study (KRIS)

153 A. Kapteyn, P. KooremanA disaggregated analysis of the allocation of time withín thehousehold.

154 T. Wansbeek, A. Kapteyntitatistically and Cumputattonally Efficient Estímation uf [heGrav[ty Model.

155 P.F.Y.M. NederstigtOver de kosten per ziekenhuisopname en leveneduurmodellen

156 A.R. MeijboomAn input-output Like corporate model including multipletechnologies and make-or-buy decisions

157 Y. Kooreman, A. KapteynEstimation of Rationed and IJnrationed Household Labor SupplyFunc[ions Using Flexible Functional Furms

158 R. Heuts, J. van LieshoutAn implementation of an inventory model with stochastic lead time

159 Y.A. BekkerComment on: Identificatíon in the Linear Errors in Variables Model

16U P. MeysFuncties en vormen van de burgerlijke staatUver parlementarisme, corporatisme en autoritair etatisme

161 .I.P.C. Kleijnen, H.M.M.T. Denis, R.M.G. KerckhoffsEffictent estimation uf power functions

162 H.L. TheunsThe emergence of research on third world tourísm: 1945 to 197U;An introductory essay cum bibliography

163 F. Boekema, L. VerhoefDe "Grijze" sectur zwart op wítWerklozenprojecten en ondersteunende i natanties ín Nederland inkaart gebracht

164 G. van der Laan, A.J..I. Talman, L. Van der HeydenShortest paths fur simplicial algorithms

165 J.H.F. SchilderinckInterregional structure of the European Communi[yPart II:Interre~;tonal input-output tables of the European Com-

muníty 1959, 1965, 197U and 1975.

iti

iN (1984) REEDS VEKSCHENEN (vervolg)

166 P.J.F.G. MeulendijksAn exercise in welfare economics (I)

167 L. Elsner, M.H.C. PaardekooperOn measures of nonnormality of matrices.

ív

LN 1985 REEDS VERSCI{H:NI;N

168 T.M. Doup, A.J..I. TalmanA continuous deFormation algorithm on the product space of uni[simplices

169 P.A. BekkerA note on the identification of restricted factur loadiny; matrices

170 J.H.M. Donders, A.M. van NunenEconomische politiek ín een twee-sectoren-model

171 L.H.M. Bosch, W.A.M, de LangeShift work in health care

172 B.B. van der GenugtenAsymptotíc Normality of Least Squares Estimators in AutoregressiveLinear Regression Models

173 R.J. de GroofGe3soleerde versus gecoórdineerde economische politiek í n een twee-regiomodel

174 G. van der Laan, A.J..1. TalmanAdjustment processes for fínding economic equilibria

175 B.R. MeijboomHorízontal mixed deco~nposition

176 F. van der Ploeg, A.J. de 7,eeuwNon-cooperative strategies for dynamic policy games and the problemof time inconsistency: a comment

177 B.R. MeijboomA two-level planning procedure with respect to make-or-buy deci-sions, including cost allocations

178 N.J. de BeerVoorspelprestaties van het Centraal Planbureau in de periode 1953t~m 1980

178a N.J. ~le BeerKIJLAGI:N bij Voorspelprestxtfes van het Centraal Planbureau in deperiode 1953 t~m l98O

179 R.J.M. Alessie, A. Kapteyn, W.H.J. de FreytasDe ínvloed van demografische factoren en inkomen op consumptieveuitgaven

180 P. Kooreman, A. KapteynEstimation of a game theoretic model of household labor supply

L81 A..1, de 'l,eeuw, A.C. M~~1 jdamOn Expectations, Information and Dynamic Game Equillbria

V

182 Cristina PennavajaPeriodization approaches of capitalist development.A critical survey

183 J.P.C. Kleijnen, G.L.J. Kloppenburg and F.L. MeeuwsenTesting the mean of an asymmetric population: Johnson's modiEied Ttest revisited

184 M.O. Nijkamp, A.M. van NunenFreía versus Vintaf, een analyse

185 A.H.M. GerardsHomomorphisms of graphs to odd cycles

186 P. Bekker, A. Kapteyn, T. WansbeekConsistent sets of estimates for regressions with correlated oruncorrelated measurement errors in arbitrary subseta of allvariables

187 P. Bekker, .J. de LeeuwThe rank of reduced dispersion matrices

l88 A.J. de 'l,eeuw, F. van der YlueKCunslstency of conjecf.ures and reaction~;: a crltique

189 E.N. KertzmanBelastingstructuur en privatisering

190 J.P.C. KleijnenSimulation with too many factors: review of random and group-screening designs

191 J.P.C. KleijnenA Scenario for Sequential Experimentation

192 A. DortmansUe loonverbelijkingAfwenteling van collectieve lasten door loontrekkers?

193 R. Heuts, J. van Lieshout, K. BakenThe quality of some approximation formulas ín a continuous reviewinventory model

194 J.P.C. KleijnenAnalyzíng simulation experiments with common random numbers

I II M~nÍ IÍÍIÍ~ÍI IIIIÍII N II MÍÍIIÍn ~ llll'vd~ 7 000 01 059746 7