two-stage optimal control problems with an explicit switch point dependence: optimality criteria and...

Journal of Economic Dynamics and Control 13 (1989) 319-337. North-HoUand

T W O - S T A G E O P T I M A L C O N T R O L P R O B L E M S W I T H AN E X P L I C I T S W I T C H P O I N T D E P E N D E N C E

Optimality Criteria and an Example of Delivery Lags and Investment*

Ken T O M I Y A M A

Aoyarna Gakuin University. Tokyo 157, Japan

Rober t J. R O S S A N A

North Carolina State University, Raleigh, NC 27695-8110, USA

Received March 1988, final version received November 1988

This paper provides necessary conditions for the solution of two-stage optimal control problems where the switch point is a choice variable and where the switch point appears as an argument of the integrands in each integral which comprise the criterion index to be maximized. Problems of this variety have arisen in the exhaustible resource and neoclassical investment literature. Using the conditions necessary for the solution of these problems, we analyze an example of optimal investment in the presence of delivery lags and show how the optimal switch point (delivery lag) responds to shifts in exogenous parameters.

1. Introduction

In a n u m b e r of recent studies, economists have examined dynamic opt imiza- t ion p r o b l e m s in which the cr i ter ion index can be wri t ten as

t 1 t/ t )d t , (1) J = fto L : ( x ( t ) , u ( t ) , t : , t ) d , + ftl L 2 ( x ( t ) ' u ( t ) ' tl'

where the objec t ive functions, L : and L2, are uti l i ty or cash flow expressions. Here , x( t ) is a vector of state variables, u(t) is a vector of ins t ruments , and t refers to ca l enda r time. A set of differential equat ions will general ly cons t ra in the max imiza t i on of J , and the final t ime, t/, may be taken to be finite or infinite. The in te rmedia te time, t t, will be called the switch po in t of a class of p r o b l e m s descr ibed here as two-stage op t imal control problems.

A t least two areas of research have examined models of this sort. The a d j u s t m e n t cos t l i terature, concerned with del ivery lags a t tached to the acqui-

*The authors are indebted to Dennis Richardson for his expert assistance with the derivation of the results presented in this paper. The referee provided helpful comments on an earlier draft. Errors and omissions are the responsibility of the authors.

0165-1889/89/$3.50©1989, Elsevier Science Publishers B.V. (North-Holland)

320 K. Tomiyama and R.J. Rossana, Two-stage optimal control problems

sition of new capital goods, gives rise to problems of this variety [see Maccini (1973a), Nickell (1977), and Rossana (1985)]. In these applications, the intermediate time is the capital goods delivery lag and the delivery lag may appear in the objective functions if the purchase price of capital depends upon the delivery lag. In the exhaustible resource literature [e.g., Dasgupta, Gilbert, and Stiglitz (1982)], the intermediate time is the time at which alternative energy resources become available.

All of these studies are limited in that their intermediate (switch point) times are fixed and exogenous. This is clearly restrictive in the cited applications. For example, firms may be expected to exercise some control over factor input delivery lags, since inputs can be purchased from alternative suppliers who generally would offer different delivery lags. Social planners can exercise some control over the time at which energy substitutes become available. To extend these analyses to allow for an endogenous switch point requires development of optimality criteria to solve such problems. We provide these conditions in this paper. These results are an extension of optimality criteria developed in Tomiyama (1985). This earlier paper provides results for the case in which the integrands of the criterion functional do not depend upon the intermediate time. Incorporating an explicit dependence upon this switch point is nontrivial and would seem to be of more use in economic applications.

The next two sections of the paper formally pose the two-stage problem we are considering and establish notation to be employed. Section 4 of the paper provides our main results, as it contains necessary conditions for these problems. Following this section we provide a Calculus of Variations treat- ment of the problem. We will use this form of the results in an example drawn from the adjustment cost literature to illustrate these results. This example is of some interest, since it provides some foundation for the endogenous determination of delivery lags observed in economies. This discussion is contained in section 6 and is followed by a concluding section that summarizes results. Readers unconcerned with mathematical details may proceed directly to the illustrative example, which should be sufficient to convey some understanding of the use of these methods.

2. Formulation of two-stage problem

The two-stage optimal control problem is formally posed in this section. First, we consider a dynamic system given by

2 ( t ) _ d x ( t ) = [ f l ( x ( t ) , u ( t ) , t , tx) , to<-t<tl , d t ~ f 2 ( x ( t ) , u ( t ) , g, t l ) , t 1 < t <_~ t f , (2)

where x(t), u(t), t 1, to, and t/ are the n-dimensional state vector, the m-dimensional instrument vector, the switching time, the initial time, and the final time, respectively. The functions fl and f2 are assumed to be at least

K. Tomiyama and R.J. Rossana, Two-stage optimal control problems 321

continuously differentiable in x, t, and q, and continuous in u. The boundary conditions for the state vector at the end points are given as follows:

X(to) ~ X o and x(ty) ~ X f , (3)

where X 0 and X! are nonempty closed subsets of the n-dimensional Euclidean space E".

The instrument function, u(.), is measurable on [to, tf], and its value, u(t), is constrained to dosed subsets of an m-dimensional Euclidean space E m as

[2 t C E m almost everywhere on t 0 < t < q ,

u ( t ) ~ ~'~2 C E " almost everywhere on t 1 < t < tf. (4)

The two-stage performance index J has been defined in eq. (1), with the integrands L~ and L 2 at least continuously differentiable in x, t, t 1 and continuous in u. It is noted again that the unspecified switching time t 1 is a choice variable to be chosen optimally. We also assume that an optimal control pair (u*,t{') exists, where to< q* < tf. The cases where q*---t o or q* = t /wi l l be discussed in section 5.

The two-stage optimal control problem can now be stated.

Two-Stage Optimal Control Problem

Find a pair ( u*, t~') that maximizes the performance index (1) subject to the state equation (2), the boundary condition (3), and the instrument constraint (4).

The maximizing pair (u*, q*) is called the optimal pair, and u*(t) and q* are called the optimal instrument function and the optimal switching time, respectively.

As noted before, a significant feature of this problem is that the functions ft , f2, L1, and L 2 are all explicitly dependent on the switching time q. This makes the problem unique and nontrivial. Previous results in Tomiyama (1985) are no longer applicable because of this q-dependence so that a new set of optimality conditions must be developed. However, the solution to this nontrivial problem turns out to be expressible in the form of the classical Maximum Principle except for the addition of a new matching condition at the switching point. It should be noted that, although the result is similar to classical optimality conditions, the derivation of this additional matching condition turns out to be technically demanding. The significance of the results of this paper is the fact that a potentially very difficult two-stage problem can be reduced to a tractable one.

First we derive a set of optimality conditions using the Maximum Principle. Then the conditions will be interpreted further, under a set of stricter smooth- ness conditions. This yields an optimality condition in the style of the Calculus of Variations.


3. Definitions and notation

To avoid unnecessary repetition, we summarize various definitions and notation used in this paper. First, the admissibility of an instrument function u(.) defined on a time interval (t , , tb) with a constraint set f2~ is defined as follows:

Definition 1. An instrument function u is said to be admissible if it is measurable on (t a, tb), the vector u(t) belongs to the constraint set ~2~ almost everywhere on (t a, tb), and it transfers the system from a given initial set at an initial time t a to a final set at a final time t b.

We frequently use Hamiltonians in optimality conditions. Instead of defin- ing the Hamiltonian for each individual case, we define a generic Hamiltonian in terms of generic variables as follows.

Definition 2. The Hamiltonians H 1 and H 2 for the first and second stages are defined by

H i ( x ( t ) , u( t ) , p ( t ) , t, t l ) = - L i ( x ( t ) , u ( t ) , t, t l )

+ p ( t ) r f i ( x ( t ) , u( t ) , t, tl) , (5)

where x(t) , u(t), p(t) , t, and t 1 are generic variables representing state, instrument, costate, time, and switching time, respectively, and the functions Li and f~ are a generic integrand of the performance index and a generic right-hand side of the state equation.

As usual, we use asterisks (*) to denote optimal quantities. The following shorthand notation also proves to be very useful:

Hi* = Hi(x*( t ) , u*(t), p*( t ) , t,/1), (6a)

Hil r = H i ( x ( r ), u(r) , p ( r ) , r, tl), (68)

l I - -C ' Hits_ lim H 1 (6c)

c~0, c>0

H2t~ += c-~o,limc>oH2 ,1+c" (6d)

4. Optimality condition I

We follow the same approach as that used in Tomiyama (1985), namely, we attempt to decompose the original two-stage problem into a sequence of two almost conventional problems, and then the Maximum Principle will be used


to obtain a set of optimality conditions. We start with an auxiliary problem corresponding to the second stage of the two-stage problem.

Auxiliary Problem (I)

Given a system

d x ( t ) / d l = f2( x( t ), u( t ), t, tx) ,

a set of boundary conditions

x ( t : ) = x : and x ( t : ) s X : ,

the instrument function constraint

u( t ) ~ [2 2 almost everywhere on

and a performance index

t, <_ t <_ 9 , (7)

(8)

[t: , t /] , (9)

Jz-- L 2 ( x ( t ) , u ( t ) , t , t : ) d t , (10)

f ind an admissible instrument function u that maximizes the performance index.

Note that the starting time t 1 and the starting point x 1 for this auxiliary problem are considered to be exogenous. This implies that t~ is merely a given constant and the appearance of t a in the integrand of the performance index does not cause any conflict when Pontryagin's Maximum Principle is applied. Referring to a standard result [e.g., Athans and Falb (1966) and Lee and Markus (1967)], the optimality conditions for the problem at hand can be summarized as follows:

Lemma 1. Let u* be an optimal instrument function for Auxiliary Problem ( I ). Then it is necessary that there exists an n-dimensional costate p* such that

(i) x* and p* satisfy the set of canonical equations

d x * ( t ) / d t = O H ~ ' / O p , x * ( t : ) = x , , x * ( t / ) ~ X / ,

dp*( t ) /d t = - aH~'/ax, p*(t:) ± H:,

where H / i s a hyperplane tangent to X/ at x*(t/), and

(ii) u* minimizes the Hamiltonian,

man H2( x*( t ), u, p*( t ) , t) = H~' almost everywhere on U E ~ 2

(11)

t : , t f].

(12)


Remark. The transversality conditions on the costate p*(ty) in eq. (11) (also appearing repeatedly in the sequel) need to be modified for the infinite horizon problem where t /= o¢. The characteristics of such cases can be summarized as follows. Suppose first that the problem is time-invariant. Since the costate equation needs to be solved backwards from infinity, the solution p(t) for any finite time t is already at the steady state, regardless of the final condition for p(t). Therefore, the final transversality condition on p(t) does not yield any contribution. Instead, the necessary condition on the costate should be modified to that of the steady state condition, namely, dp*(t) /dt = 0 for t < ~ . Combining this with the necessary condition, we have d H ~ / d x = 0. For a more general case where the problem may be time-varying, the following condition is known to be sufficient:

lim p ( t ) r x ( t ) = 0. (11 ' ) l --'* O0

The necessity of this condition under a set of modest conditions was proven by Benveniste and Scheinkman (1982).

Let -/2* = J2*(X(tl), tl) be the optimal performance index for this problem, i .e. ,

f r i l L " J2*(X(tl),tl) = 2 (x*( t ) ,u*( t ) , t , q )d t . (13)

As noted before, the applicability of the Maximum Principle depends critically on the fact that tl is not a choice variable for this auxiliary problem. However, when we let t a be a choice variable in a later step, the explicit dependence of the optimal performance index J2* on t 1 must be properly considered.

Using J2*(X(tl), tl) as defined above, we reduce the two-stage problem to a more tractable single-stage problem.

Auxiliary Problem (11)

Giuen a system

d x ( t ) / d t =fx(x( t ) , u(t), t, t l) ,

a set of boundary conditions

X(to) ~ X o and x (q ) free,

t o < t < t 1, (14)

(15)


the instrument function constraint

u( t ) ~ 12: almost everywhere on [t 0, tl], (16)

the f inal time constraint t o < t 1 < t f, and a performance index

f/~l • /1 = L l ( X ( t ) , u ( t ) , t , t l ) d t + J 2 * ( X ( t l ) , t l ) , (17)

f ind an admissible instrument function u that maximizes the performance index.

As discussed before, we assume that tl* is an interior point of [t 0, ty]. For this case, the constraint t o < t: < tf is inactive and the problem reduces to one with free end point and free end time.

Despite this reduction, the problem is still nonstandard because t: appears both in the integrand and at the upper limit of the integration. We alleviate this difficulty by employing the approach used by Long (1965) and others. In applying a numerical algorithm to a two-point boundary value problem with an unknown end time t/, Long used a simple linear transformation of the time variable t as

t = ty . s , (18)

where s is the new scaled time variable. Obviously, s varies from 0 to 1 as t varies from 0 to tf. Long solved for the unknown coefficient t! for this transformation by augmenting it to the state vector as an auxiliary state variable.

Emulating this approach, we utilize the following change of variable, which is appropriate for our problem:

t = ( t I -- to)S "[- t O. (19)

This change of variable transforms the interval t o < t < t 1 into 0 < s < 1. We define

X n + I ( S ) = t 1 -- t o (20)

to be the auxiliary state variable, where we use the tilde ( ' ) to denote quantities in the new time variable s. The corresponding state equation is given by

d g n + l ( S ) / d s = 0, (21)

because both t 1 and t o are constants.


The set of equations that defines Auxiliary Problem (II) can be rewritten in the new time variable s using the chain rule as follows:

d Y ( s ) / d s = (t 1 - t o )~ ( Y ( s ) , f i (s) , ( t 1 - to)S + to, tl) (22)

= Y.+l(s)f~(Y(s) , fi(s), Y . + l ( s ) s + to, Y.+l(S) + to) ,

J1 = fol(tl - to)Ll(X(S), fi(s), (t 1 - to)S + to, t l )dS + ~* (~(1), tl)

= folY.+l(S)Ll(Y(s) , fi(s), y . + i ( s ) s + to, y . + l ( s ) + to) ds

+a~*(Y(1),Y.+I(1 ) + to). (23)

Let ~(s) be the augmented state vector defined by

[~(s) ~(s ) = [Y.+l(s) ]" (24)

Using the notation ^ as an indicator of augmented quantities, we rewrite the above set of equations as follows:

d . ~ ( s ) / d s = ) ~ ( ~ ( s ) , fi(s), s), (25)

~ = f o fi(s) s) ds + ~ * ( 2 ( 1 ) ) xL~(2(s), , , (26)

where

and

f~(~(s), a(s), s)

= [ Y"+'(s)'~(Y(s)'fi(s)'Y~"+'(s)s + t°'Y"+'(s) + t°) ] (27)

L~(~(s),~,(~),s)

= ~°+1(s)£ , (~(~) , ~ ( , ) , ~°+, (s ) , + to, ~°+a(S) + to), (28)

a~* (2(1)) = a~*(2(1), ~n+l(1) + to). (29)


Because the value of t 1 is to be chosen optimally, the boundary conditions of the corresponding auxiliary state £,+1(s) are also to be chosen optimally. This corresponds to a case of free boundary conditions for g, + x(s). Therefore, the appropriate set of boundary conditions for the augmented state :~(s) are

where x(O) E 20 C E n+l and £(1) is free, (30)

{ [2 ] : 2 ~ X o a n d 2 n + free}. (31) ) ( ° = "x = "Xn+l 1

In summary, Auxiliary Problem (II) has been reduced to the standard problem stated below.

Revised Auxifiary Problem (11)

Given a system (25), a set of boundary conditions (30), and an instrument function constraint

~(s ) ~ I21 almost everywhere on [0,1], (32)

find an admissible instrument function ~* that maximizes the performance index, eq. (26).

We are now able to supply the Maximum Principle. The set of standard optimality conditions for this problem is listed in the following lemma:

Lemma 2. Let ~* and ~* be an optimal instrument function and the corresponding optimal state trajectory. Then it is necessary that there exists an (n + 1)- dimensional costate /3* such that

(i) 2" and/3* satisfy the set of canonical equations,

d ~ * ( s ) / d s = a&*/a/3, ~*(0) ~ 2 o,

d/3*(s ) /ds = - 3tt1"/0~, /3*(0) _1_ _0o, /3"(1) = - a~*/g2[s=l,

where EI o is a hyperplane tangent to )(o at ~*(0),

(ii) ~* minimizes the Hamiltonian,

min/-Ix(~*(s), ~,/3*(s), s) = Hi* almost everywhere on [0,1],

(33)

(34)


where the augmented Hamiltonian 1211 is defined as

#~ = - / , l (~(s) , fi(s), s) + ~(s)~f](~(s), ~(s), s). (35)

Now we are ready to obtain a set of necessary conditions for the original two-stage problem. The following theorem is derived by interpreting the above Lemma 2:

Theorem 1. Let ( u*, t ~ ) be an optimal pair for the original two-stage problem defined in section 1. Then it is necessary that there exists a costate p* such that

(i) x* and p* satisfy the canonical equations

OHt*/Op, d x * ( t ) / d t = OH~'/Op,

d p * ( t ) / d t =

t o < t < t ~, x*(to) E X o, (36)

t ~ < t < t f , x * ( t / ) ~ X f ,

- OHI*/Ox, t 0 <_ t < t~ , p * ( t o ) ± !-/0, (37)

- O H ~ / O x , t~ < t < tf , p * ( t f ) ± FIr,

(ii) u* minimizes the Hamiltonians

rnin Hl ( X* ( t ), u, p * ( t ) , t) = Hi* u ~ $21

rain/ 2(x*(t), u, p*( t ), t ) = u~22

almost everywhere on [ t o, t~ ),

(38)

almost everywhere on ( t ~', t f],

(39)

(iii) x*, p*, and Hi* satisfy the following matching conditions at t~:

x*( t~ - ) = x*(t~ +), (40)

p*( t~ - ) = - aJ2*/ax, (41)

t* HI*IC_ + f ~ (OHl*/Otl) dt = OJ2*(x*(t ~ ),t{" ) lOt 1, (42)

to

where J2* was defined in eq. (13).

Proof Since the set of canonical equations, (36) and (37), together with the matching conditions (40) and (41), and the minimization of the Hamiltonians,


(38) and (39), are direct consequences of Lemmas 1 and 2, we concentrate on the condition (42). We first decompose the costate ~b in Lemma 2 into two components, an n-dimensional costate vector /3(s) and an auxiliary costate ,b~+l(S), corresponding to the state :~(s) and the auxiliary state Y~÷l(s), respectively. Note that we drop the asterisks (*) for simplicity, except for J2*. Thus, we have

[/3(s) ] (43) #(~) = [~.+l(s) "

Then the last row of eq. (33) can be written as

dP.+l(s) Off1 [ 0£1 r af~ ] ds - O-~n+ 1 a3~n + ~ "~-/~ (S) a.Xn---~ 1

c9L1 all ] = z.~ +.~,,+l(~)--~-s +.~,,+1(.~)-~

af] s + - af~ 1 T x.+l(s) j

(44)

ra~l a~] •

From the two end point conditions for /3(s) in eq. (33), we can deduce the following end point conditions for/~,+l(S):

/~,,+x(O) = 0 and /3,,+1(1 ) = - O.~(Y(1), .~,,+,(1))/0Y,,+1(1 ).

(45)

Noting that the right-hand side of eq. (44) does not depend on t~n+l(S), we can solve eq. (44) using the first of the above end conditions as

~o+l(1)= - fol[ttt + £~+l(S)(( altl/at)s+ aitt/gtl)]ds. (46)


Therefore, the second end condition in eq. (45) can be expressed as follows after the change of time variable back to t:

Pn+x(tl) =/3n+ 1(1)

= - e ~ ( ~ ( 1 ) , ~ . + , ( 1 ) ) / 8 ~ . + 1 ( 1 )

= -OJ2*(x*(t~) , t~) /Ot 1

= - i t l [ n l q- (/1 - t0)((cgnl/Ot) [ ( t - t o ) / ( t l - / 0 ) ] ' o

(47)

+ aH1//atl}] [d/ / ( t l - / 0 ) ]

( it: 1 = - [ ( t - t o ) / ( t l - t o ) l H l l ~ ; + (OH, /Oq)d t

tl = - H i l t , - f (aI-I1/Oq)dt. Q.E.D.

-to

Benveniste and Scheinkman (1979) discuss differentiability of the value function associated with optimal solutions (which for our case is J2*). A discussion of this in the present context is nontrivial and is beyond the scope of our analysis. In the sequel, the partial derivative of J2* appearing in eq. (42) will be explicitly derived under more stringent conditions.

5. Optimality condition II

By enforcing more strict twice continuous differentiability for functions, we will show that the necessary conditions in Theorem 1 may be expressed in the form of the first-order optimality conditions of the Calculus of Variations. The key quantity here is the partial derivative of the optimal second-stage performance index with respect to the switching time, aJ2*/atl, the right-hand side of eq. (42). This quantity would be identical to the second-stage Hamiltonian if the second-stage problem had been a standard one. However, the switching time dependence of the lower limit of integration for the second-stage performance index is nonstandard and has to be handled explicitly. This point will be discussed in the proof of the following theorem, which summarizes the optimality conditions in the Calculus of Variations style.


Theorem 2. Suppose that the functions f~ and L,, i --- 1, 2, are twice continuously differentiable in x, t, and t 1, and continuously differentiable in u, and that the admissible instrument function u(.) is continuous except possibly at the switching point. I 't (u*, t~) be an optimal pair and x* be the corresponding optimal trajectory. Then it is necessary that there exists a costate p* such that

(i) x* and p* satisfy the canonical equations in eqs. (36) and (37),

(ii) partial derivatives of the Hamiltonians vanish at u*,

OHl*/Oul"*=O' t°<-t<t~" (48) 0 H ~ ' / 0 ulu. = O, t{' < t <_ tf,

(iii) x*, p*, and Hi* , i= 1,2, satisfy the following matching conditions at t~':

x*( t~' - ) = x*( t{' + ), (49)

p*(t~ - )=p*( t l* +), (50)

t: OH* H~'lt~+-Hl*lt~_= fi~(OHl*/Otl)dt+ ft ( 2 /Ot l )dt . (51)

Proof. The condition, eq. (48), is a direct consequence of the minimization condition, eq. (36), of Theorem 1, and the matching condition, eq. (49), is the same as that in Theorem 1. The fight-hand side of eq. (50) can be derived using the dynamic programming technique [see, e.g., Athans and Falb (1966)]. We will show the new matching condition, eq. (51), from eq. (42) of Theorem 1 by computing the partial derivative OJ2*/O q using Leibnitz' rule. We drop the asterisk (*) from t~* for notational simplicity,

dJ2*( X( tl), t l ) /dt l

t: = d / d t l f L2(x*(t ) ,u*( t ) , t , t l)dt

tx

= d / d t lft:/[ L~[ t + p*(t)v(:c*(t ) - f2*l, }] dt (52)

t/ = -L~lt l++ ft~ [OL~/Otl -p*( t )r(of2*/Oq)]dt

= - L~lt~ + - ft:: ( O H ~ ' / O t l ) d t .


On the other hand,

d J2* (x (t l) , t l ) / d t I = [ 0J2" (x ( t l ) , t l ) /Ox (t l) ] T [dx( t l ) / d t ]

+ 0J2" (x ( t l ) , t l ) /Ot l (53)

= - p * ( t I + ) rfz*lt,++ OJ2*(X(tl), t l ) /Otl .

By combining the above two equations, we obtain

t/, OH, 0J2* (X(tl) , tl)//Otl = n~ l t l+ - f [ 2 / 0 t l ) dt. "tl

(54)

This and eq. (42) of Theorem 1 given eq. (51). Q.E.D.

Remark 1. The condition, eq. (51), can also be obtained through the technique of Calculus of Variations. The derivation is straightforward but rather lengthy and therefore will be omitted.

Remark 2. Due to the explicit q-dependence of the functions, the Hamiltoni- ans H 1 and H 2 do not match at the switching time. Eq. (51) specifies the amount of 'miss-match' due to this dependence. As we can easily see, the difference in the Hamiltonians will vanish if the q-dependence is eliminated, and the condition reduces to the standard case.

Remark 3. For cases in which the optimum switching time is at one of the terminal times t o or t/, the optimality conditions should be modified as follows. Suppose that q* = t o is optimal, then it is necessary that

n2*l,o+-/-/1" It0- ~ f~i y(0n~'/0tl) dt. (55)

On the other hand, if q* = t/, then

H * - r q ' - * 2 It/+ Hi*It/-> )d t . Jto ~at1 (56)

The above conditions can easily be derived from the observation that the first-order variation of J with respect to the allowable perturbation in q* must be nonnegative from the optimality of q*.


Having established the optimality conditions for the general two-stage control problem, we turn our attention in the next section to an economic example of the results.

6. An illustration

In this section, we apply the optimality criteria developed above to a problem originally formulated in Maccini (1973a). We do this not only to illustrate the methods developed in this paper but also because previous research has examined optimal delivery lags only from the point of view of suppliers providing goods subject to such lags [see Maccini (1973b) and Rossana (1985)]. If these delays are of importance in a macroeconomic context, delays actually observed would reflect the interaction of optimizing behavior on both sides of the market. A complete understanding of these delays therefore requires some understanding of optimal behavior on the buying side, so the example should be of some interest in its own right. The model described below was the first example of the imposition of delivery lags in an adjustment cost framework and is simple enough so we can obtain some qualitative results about the optimal choice of delivery lag. To accomplish this, we impose some additional structure on the problem that was not imposed in Maccini (1973a).

The optimization problem may be stated as follows:

maximize J = fo°{ PF(Ko, L ( t ) ) - WL(t)}e-r'dt

+ fo°°( P [ F ( K ( t ) , L ( t ) ) - R ( I ( t ) ) e ~ - B ( I ( t ) ) ]

-WL( t ) -Gv I ( t ) } e - r td t , (57)

subject to

l~ ( t )=l ( t ) , K( t )=Ko>O , O<t<v, Gv=Goa(V), (58)

where G o = nominal purchase price of capital goods, I = deliveries of capital goods, K = stock of capital goods, L = labor, P = output price, r = discount rate, t = calendar time, v = delivery lag, W = money wage rate.

The firm is assumed to produce nonstorable output. It faces competitive input and output markets and discounts its cash flow using the rate r. It pays


for capital goods at the time that deliveries are taken. The exogenous parameters (G 0, P, r, W) are held with static expectations. The firm uses a neoclassical production function F(K( t ) , L( t ) ) which displays positive, marginal products and is concave. The firm will also incur planning costs, R(I( t ) ) , and installation costs, B(I( t ) ) , associated with placing orders for and taking delivery of capital goods. These are measured in units of output and thus capture the resource costs attached to these activities. These functions have properties that are standard in the adjustment cost literature: R ( 0 ) = 0, R' ( I ( t ) ) > 0 as l ( t ) > O, R"( I ( t ) ) > 0 and similarly for B(.). The purchase price of capital goods, G o, is comprised of a spot price, G o, associated with immediate delivery of capital and an additional term, a(v), which captures any discounts or premia imposed by suppliers if the firm takes forward delivery of new capital. If a ' (v )< 0, a discount applies, and if a ' (v )> O, a premium is imposed. Arguments can be advanced for either assumption and these may be found in Maccini (1973a) and the references cited there. More will be said about this below. 1

In what follows, we assume that a"(v) > 0 so that cash flow for v < t < oo is concave in v. The interval 0 < t < v corresponds to the Marshallian short run, a finite time where the firm's capital stock is fixed and the firm can only engage in more limited decision-making, namely, vary output by its choice of labor input. By choosing the delivery lag, the firm chooses the length of the short run. The second interval is the period during which deliveries may be taken. Note that this analysis describes the behavior of the firm in response to unanticipated movements in the exogenous parameters. Orders for capital goods may be placed appropriately in advance in response to anticipated movements in these magnitudes.

Optimality criteria may be obtained for this problem by forming the Hamiltonians

n 1 = Pe-rt{ F(Ko, L(t)) - (W/P)L( t ) + O(t)I(t)} ,

//2 = e e - " { f (K ( t ) , L(t)) - R(l ( t ) )e - B(I(t))

-(W/P)L(t)- (GJP)I(t) + O ( t ) I ( t ) } ,

tin the analysis below, we proceed as though an interior solution prevails where v > 0 and is finite without imposing any additional constraints to guarantee that this is so. Imposing such a constraint seems to offer little in the way of additional insight, while making the problem much less tractable analytically. A simple way to guarantee an interior solution would be to revise the price function to be G=G(v) and to impose the Inada conditions limv.oG'(v)= oo and lim v _ ~G'(v) = O. We might also argue that it is reasonable to suppose that heterogeneous goods cannot be supplied instantaneously so that v is exogenously bounded away from zero.

K. Tomiyama and R.J. Rossana, Two-stage optimal control problems 3 3 5

where O(t) is an adjoint variable that has the usual shadow price interpreta- tion. For 0 < t < v, we have

FL( K o, L ) = W / P , (59)

and f o r v < t < o o ,

FL ( K, L ) = W / P , (60a)

0 = R ' ( I ) e~°+ B ' ( I ) + G J P , (60b)

= - F K ( K , L ) + rO, (60c)

/{ = I , (60d)

lim O e - " K = O, (60e) t-"* O0

where the time notation is suppressed where convenient. These conditions are easily interpreted, but as this is provided in Maccini (1973a), we will not do so here. However, our interest is in the switch point condition that describes the optimal choice of v. Applying earlier results, we have

R ( I ( v + ) ) e rv + B ( I ( v + ) ) + ( G o l P ) I ( v +) - O(v+)I (o +)

= - f o ~ [ R ( I ( t ) ) r e ' ° + ( G o / P ) a ' ( o ) l ( t ) ] e - r ' t - ° + ) d t . (61)

This expression may be simplified and interpreted more easily if we impose quadratic form assumptions that are commonplace in macroeconomic analysis. Specifically, assume that

F( K, L ) = coK + ( c l / 2 ) K 2 + doL + ( d l / 2 ) L 2 + elKL ,

c o, d o, e 1 > O, c l, d 1 < O, e~ - cld 1 < O,

R ( I ) = ( s x / 2 ) I 2, B ( I ) = ( q x / 2 ) I 2, sl, ql > O.

Using these assumptions and eliminating 0 from (61) using (60b), we obtain

[ (s l /2)e r° + (qx/2)] 1(o+) 2

= - fo~+l(sa/2)I(t):re"+ ( G o / P ) a ' ( v ) I ( t ) l e - " t - ° + ) d t . (62)

336 K. Tom(vama and R.J. Rossana, Two-stage optimal control problems

This condition instructs the firm to choose its delivery lag such that the planning and installation costs at the instant that capital goods arrive be just equal to the discounted benefits and costs attached to resources used in placing new orders for capital goods and any discounts or premia associated with the purchase price of capital goods for future delivery. This condition also shows that if I ( t )> 0, then it must be that a ' ( v ) < 0. Otherwise, this condition cannot hold. The reason is that if a' (v)> 0, the lowest purchase price of capital wold be for immediate delivery (v = 0). If the firm plans to expand its capacity, it will choose to do so immediately. A similar discussion may be provided for the case I ( t ) < 0. Thus, for the delivery lag to be nonzero, if I ( t ) is positive (negative), a '(v) must be negative (positive). In what follows, we will focus only on the case I ( t ) > 0, since this is the one of most interest for macroeconomic analysis. The I( t ) < 0 case is omitted only for the sake of brevity.

Intuition suggests that variations in the exogenous parameters (G 0, P, r, W) should affect the optimal delivery lag chosen by the firm. To obtain qualitative results about these effects, we can use the switch point condition (62) in conjunction with the expression for optimal investment, which arises here under the quadratic form assumptions. The latter can be written as I ( t ) = f fa (K0- K*)exp[~l ( t - v)], where K* is the steady state level of the capital stock. We have K* = [elW + P( cod 1 - eldo) - ra( v)Godl][ P[( el) 2 - cldll1-1 The stable characteristic root is denoted by/~1; the unstable root was eliminated by a suitably chosen constant so as to satisfy (60e). These roots are real and symmetric about r/2. As is customary in the adjustment cost literature, gl will be assumed to be invariant to shifts in exogenous parameters. Using the investment path given above, the integral in (62) may be explicitly evaluated. The resulting expression may be differentiated directly to yield the following results.

8v/SGo>O, 8 v / S W > O , 8v /SP <O, 8 v / S r ~ O .

The firm will choose a longer delivery lag when the spot price of capital or the wage rate rises. If the firm plans to expand its capacity, higher wages or purchase prices of capital will lower cash flow and may be offset by a longer delivery lag, raising cash flow. A higher output price will shorten the delivery lag. A firm planning to add capacity will want more rapid delivery to take advantage of higher output prices. A higher discount rate has an ambiguous impact upon the delivery lag, reflecting two conflicting forces. When the discount rate rises, the firm will wish to take more rapid delivery to conserve on implicit short-run planning costs. It will want a longer delivery lag, however, to lower its capital costs so as to partly offset the rise in capital costs associated with a higher discount rate. Consequently, the end result is uncer- tain.


7. Conclusion

This paper has provided optimality criteria for two-stage, continuous time opt imal control problems where the integrands of these problems depend upon the switch point and where the latter is subject to optimal choice. Problems of this variety have arisen in the investment literature in the presence of delivery lags as well as in the exhaustible resource literature. It is shown that familiar criteria may be employed with the addition of an integral matching condit ion describing the optimal choice of the switch point. The methods developed here are illustrated in a problem from the adjustment cost literature in which we derive some qualitative results concerning the optimal choice of the capital goods delivery lag. The methods displayed here should be of some use in extending economic research in a variety of areas into more realistic settings where delays encountered may be chosen in an optimal fashion.

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