continuous and discrete time approaches to a maximization problem

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The Review of Economic Studies Ltd.

Continuous and Discrete Time Approaches to a Maximization ProblemAuthor(s): L. G. Telser and R. L. GravesSource: The Review of Economic Studies, Vol. 35, No. 3 (Jul., 1968), pp. 307-325Published by: The Review of Economic Studies Ltd.Stable URL: http://www.jstor.org/stable/2296664

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Continuous a n d Discrete T i m eApproaches t o a MaximizationProblemiI. INTRODUCTION

A common research strategy in economic dynamics is to state the theory of a givenproblem in continuous time and to test it in discrete time. The correspondence betweenthe theory and the real world relies on an implicit belief that the discrete time modelapproximates the continuous time model. This suggests that if the original theory wereposed in discrete instead of continuous time then it would imply observable relations indiscrete time akin to the discrete time approximations of the continuous time model.In general this is not so! Discrete and continuous time models may differ in many subtleand surprising ways especially if the economic agents to which the theories apply areassumed to optimize some function over which they have partial control.We deal with a class of economic problems in which a firm seeks a maximum presentvalue of its return over an infinite horizon at a given interest rate. This problem has twoversions. In the one of our primary concern a monopolist seeks a present-value-maximizingsales (or price) policy and faces a given demand function such that the quantity demandedat time t depends linearly on past and expected future prices. In the second version afirm determines its stock of capital over time so as to maximize the present value of itsnet return. While both problems deal with investment, the first is less often so regardedand so we call it the monopoly problem herein.We show that not only are there surprising differences between the models in con-tinuous and discrete time but also there are surprising differences between the monopolyand investment problems. A solution exists in the monopoly problem only if there is anupperlimit on the interest rate while in the investment problem there is the usual result,namely, a solution exists for all interest rates above some lower limit that may be zero.

1 There are few examplesof applications of the calculus of variationsto economic problems. Anearly one is Hotelling [8]. An extensive discussion of several monopoly problems in an insufficientlyappreciatedbook by G. C. Evans [6] treatsproblemsrelated to the ones we discuss. Evans works with afinite horizon and deals with both fixed and variable end point problems. His book also gives manyreferences o economic applications of the calculus of variations. We are gratefulto Professor L. M.Graves for calling to our attention a little knownwork by Henry Howes Pixley [10]. Pixley's analysisisinspired by the work of Evans and is a very sophisticatedtreatment. He allows solutions q(t) (in ournotation) which have a finitenumberof discontinuities n both q and q'. He assumesa finite horizonandconsiders fixed and variableend-point problems. By allowing these discontinuitiesPixley shows thatthere are no solutions obtainableusing the classical calculus of variations. Part of the explanationstemsfrom the use of a finite time interval which imposes strong demands on the sufficientconditions for amaximum see footnote 3) but the primaryexplanation s that Pixleypermitsdiscontinuities. He himselfsuggests that other mathematical pproachesare likelyto yield meaningfulresults.Recentinterest n problemsrelated o the onestreated n the classicalcalculusof variations s associatedwith the attemptto deriverigorously the relation betweeninvestmentand the interestrate. Among thefirst articles on this topic are Arrow [2] and Eisner and Strotz [5]. Arrow in particularnoted the pheno-menon of an instantaneous ump in the stock of capital if the cost of adjustment s linear in the rate ofinvestment. In a study of the optimal advertisingpolicy for a firmwhich treatsadvertisingas a stock ofgood will, Nerlove and Arrow [9] also noted that a jump of the stock of advertising s requiredwhen thepresent value is linear n the rate of change of the stock of good will.The first named author gratefully acknowledges the financial support of the National ScienceFoundation.

307


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308 REVIEW OF ECONOMIC STUDIESIn the continuous (C) monopoly problem we seek the maximum of the integral

e-rtq(t)p(t)dt, .0

for a given r>0 with respect to q subject top(t) = f(t)-q(t)-alq'(t)-a2q"(t) ... (1-2)

and the initial conditions that q(t), f(t) = 0 for t


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CONTINUOUS AND DISCRETE TIME APPROACHES 309is bounded. In the C problem it is assumed there exists a positive ro such that

I e- rotl2f(t)lis bounded. For technical reasons in the C problem, f must be further restricted to theclass of continuous functions of t except at a finite number of points. The requirementsimply that

ert I (t)2dt andE t fO Q

are both finite for r> ro and /


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310 REVIEW OF ECONOMIC STUDIESwhich are straightforward. Our main interest in this investment problem is to demonstratethe differentrole of the interest rate r in (1.6) and in (1.1).In the monopoly problem the solutions of the C and D versions differ in severalimportant respects. First, as we shall show, there is an exponential approach to theequilibriumquantity path in the C problem if and only if a2 # 0. If a2 = 0, so that theconstraint (1.2) reduces to a first order linear differential equation, then the optimal pathrequires instantaneous adjustment from the initial to the equilibrium path. In the Dproblem there is never instantaneous adjustment to the equilibrium path if a =#0. Aslong as a =A , qt approaches the equilibrium path at a geometric rate.Second, the boundary conditions for the C problem raise delicate mathematical issues.It turns out that in the C problem there is in general an initial finite jump of the first deriva-tive at t = 0. This jump is analogous to the jump in q itself in that C problem for whicha2 = 0.Third, if the sufficientconditions for a maximum of the C problem are satisfied thenthe demand equation (1.2) is unstable. The sufficient conditions for a maximum in theD problem are satisfied for all values of the discount factor ,Bsuch that 0


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CONTINUOUS AND DISCRETE TIME APPROACHES 311Hence (2.1) becomes

PV = a2q(0)q'(0) + e- r[fq - q2-(a1+a2r)qq'+ a2q 2]dt. ...(2.4)Both q(0) and q'(0) are given, making the first member of the right hand side a constant.Therefore, maximizing PV is equivalent to maximizing the integral on the right side.Define the function

F(t, q, q') = e-rt[fq - q2-(a, +a2r)qq + a2q 2]. .. (2.5)In order to maximize

F(t, q, q')dt,it is necessary that the function q(t) satisfy the Euler equation

Fq-d Fq,= 0.Other conditions for a maximum are more complicated in our problem because theterminal value of q is only restricted by the condition that PV be bounded. This implies,inter alia, that q may grow at an exponential rate not greater than r/2. Hence it is notpossible to appeal to the " simpler" sufficient conditions in the calculus of variationsbecause these apply to problems in which boundary conditions are given in the form ofequalities. One of the classical conditions does apply in our problem. Thus for a maximumit is necessary that

Fqq,


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312 REVIEW OF ECONOMIC STUDIESwithout necessarily violating an economic restriction we have not yet mentioned, namelynon-negativity of prices. For q given by (2.10), the PV in (4) becomes

00PV = a2q(0)q'(0)+ e-rtqfdt00

- M2 e(2 -r)t[1 + b(a+ a2r) 2a2]dt0

= a2q(o)q'(0) e-rtqfdt0

+M2(25-r) -'[1 + (a1 + a2r)5-a2 2] ...(2.11)As 5 approaches r/2 from below (26 - r)-1 approaches minus infinity. Hence the presentvalue can be made an arbitrarily arge positive number if

lim (1+(al +a2r)b a2 62) < 0.2n-r-Therefore, to rule out this possibility it is necessary that

1 +aj(r/2) + a2(r/2)2 = 0. ...(2.12)Since the discriminantof the roots in (2.8) can be written as follows:

- 4a2(1+ a1(r/2) + a2(r/2)2),this proves that the characteristicroots of the demandequationmust be real if the maximumpresent value of revenue is to be bounded from above.'This argument depends on the fact that q given in (2.10) is admissible, but nothinghas been said so far about the non-negativity of prices. Thus perhaps a quantity paththat satisfies (2.10) might require steadily falling prices. Hence if non-negativityis imposedon prices, the q given in (2.10) might be inadmissible. However, admissibility of q forprices non-negative depends on how rapidly the market grows. Since the growth of themarket equals the growth rate off, it is sufficientfor our purposes to display anf that growsenough to maintain non-negative prices, and that is admissible. An example of such anf is given by

f(t) = Kert/2/(l1+t). ...(2.13)It is readily verified that forf given by (2.13) the integral

00e-rty2dt

is finite. Moreover, for this f, and q given by (2.10), since prices must satisfy the demandequation (1.2), it follows thatp(t) = e't[Ke(r/2 - )t/(1 + t)-M(1 + alb + a2 2)]. . ..(2.14)

Clearly, if K> 0 and large enough so that p is never negative, the market actually grows,and not only are prices positive, but also they become arbitrarily large. In this case thepreceding argument applies full force and inequality (2.12) is necessary for the existenceof a bounded positive maximum present value.The inequality given in (2.12) together with the firstnecessarycondition for a maximum,a2


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CONTINUOUS AND DISCRETE TIME APPROACHES 313problem described before. In the investment problem the present value of the firm's netrevenue satisfies

PV = ert{s({0 - als)- s(0 + YlS+ y2s')}dt. ...(2.15)0

Compare this expression with the present value given in (2.4) that applies to the monopolyproblem. The same combination of q and derivatives appears in (2.4) as in (2.15) with thetrivial exception that (2.4) lacks a term in q' alone. The significant difference is that thecoefficient of qq' is (a, + a2r), which depends on r, while the coefficient of ss' in (2.15) is Yi,a constant independent of r. In both problems for a bounded maximum it is necessarythat the coefficient of the squared derivative term must be negative. However, in themonopoly problem this fact together with the fact that r appears in the coefficient ofqq' " explains " why there has to be an upper bound on r. For if r gets too big, then sincea2


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314 REVIEW OF ECONOMIC STUDIESand condition (3) must be satisfied for the present value to be finite. These two require-ments imply, respectively, that

q(O) = (Cl + C2)(Al + A2)C2 = -(2a2) 1 e 2sf(s)ds.

The latter condition ensures that the second term on the right side of (2.16) approacheszero as required by condition (2.3). Substituting these values of the constants into (2.16)gives the following expression for the quantity path:q(t)= e 't{q(O)+(2a2)"1GA1_Z2)V1 Astf(s)ds- -e 2sf(s)ds]}

+ (2a2f '-1 -_2) -1 eA2t e-A2sf(s)ds. .. .(2.17)JtRecalling that Al


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CONTINUOUS AND DISCRETE TIME APPROACHES 315it follows that r/2 is also a characteristic root of the demand equation. Hence the char-acteristic equations of the Euler equation and the demand equation have a commonroot, r/2, allowing us to represent the demand equation (1.2) as follows:

p(t)= f(t) -a2 d - r/2) d _ ar q(t).2~~~ dt dWe now show that in this case there is a maximum growth rate of q(t) and that theproblem has a finite maximum present value so that an edge solution exists providedf(t) ? Ce6t,where 6 is less than r/2 and r is given. The example,

f = e ,12/(I + t),which is admissible, illustrates the necessity of imposing some conditions on f for thereto be edge solutions since for this choice of f and q = ertI2, he value of the integral

e7rtq(t)f(t)dt0is infinite. Hence without loss of generality let C = 1 and set

f(t) = eat.For thisf, the solution of the Euler equation becomes

q(t) = clertl2+c2tertl2 + [2a2(6- r/2)2] le5t.It is readily verified that setting c2 = 0 and

c, = q(O)-[2a2(6- r/2)2]-yields a finite PJV n (2.1) although lim e tqq'o 0

t-4cGoMoreover, in this case f < 0 since a2< 0.Next we observe that the rate of growth of q(t) in general is the largerof the followingtwo numbers: the smaller characteristic root of the Euler equation, and the growth rateof f. The maximum value of the characteristic root of the Euler equation occurs for thatr which makes the discriminantof the roots of the Euler equation equal zero. This followsfrom (2.12) and a2


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316 REVIEW OF ECONOMIC STUDIESWe seek a sequence {qt} to maximize the present value. Straightforward methods ofdifferential calculus apply and give the necessary conditions for a maximum. Thus for amaximum PV it is necessary that

OPV =f0-2q0-c43q1 =0aqoapv = fl[f1-aqo-2q1-cf4q2] = 0aq,dPV = flt[f-xq,1-2qtf-afq, 1] = 0.aqt

It is sufficient for a maximum that the infinite band matrix (known as a Toeplitz matrix),2 p*5 o 0 0 0

fl,5o 2 fl.50 0 ?0 4l5a 2 -50C Q

be positive definite. This can be shown to be equivalent to requiring that the differenceequation as follows:ocqt_- +2qt+aflqt+j =ft ...(3.1)

must not have a characteristicroot of modulus p35. This condition is analogous to therequirement for an interior solution of the C problem that the characteristicroots of theEuler equation must be real and not equal to r/2. The characteristicequation of (3.1) isaz-' +2+ xflz = 0 ...(3.2)

whose roots are(1B)'1[- 1+?(1-ot2fl)5].

Denote the roots by ,u1 nd 12 where I91 I< 1 2 l. For p, not to have modulus /3., it isnecessary and sufficient that the discriminant satisfy1-x23> 0. ...(3.3)

Hence both roots must be real. Condition (3.3) also shows the role of the discount factorin the D problem. If this condition must be satisfied for all ,Bsuch that 0


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CONTINUOUS AND DISCRETE TIME APPROACHES 31700

qt-l-lqt-l= E2S .. (.4s = t f4s=twhere I , I is not necessarily less than one.'In the D problem only interior solutions can yield a finite value of the maximum PV.This follows from the fact that the characteristicequations of (1.4) and (3.1) cannot havea common root so that if one root of (3.2) had a modulus f- 5 then so would the other,and the maximum PV would be unbounded.

4. INTERPRETATIONThe C and D problems differ in several important respects. In the C problem, ifa2 = 0, then (1.2) reducesto a first order linear differential equation and the Euler equation(2.6) reduces to q = f/(2 + a1r), ... (4.1)

implying instantaneous adjustment to the optimal path. There is no cost in allowingq(t) to be discontinuous at t = 0 in this case because the integrand does not contain theterm q'(t)2. If a2


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318 REVIEW OF ECONOMIC STUDIESprice because his individual purchases are too small a part of the total demand to exertany effect on the current price. Moreover, competition among the sellers prevents anyone of them from exploiting consumer lack of foresight and maintains their return at acompetitive level. This argument does not, of course, deny the benefit to producers offorecasting prices. However, in a competitive market, forecasting helps producers byimprovingtheir inventory, output and employment policies and not by exploiting customers.Now consider an unstable demand relation. This means that one of the roots of(4.2) say a,, has a negative real part, while the other, a2, has a positive real part. Therefore,the quantity demanded at time t depends on future as well as contemporary and pastprices. Instead of (4.3) there isq(t) = (a! -a2)'- {eealt[1 +a- e- is (f-p)ds] + a' f- e(t-S)af2(f-p)ds}. ...(4.4)In contrast with (4.3), only the constant c1 depends on the initial conditions. Nevertheless,although the consumer demand depends on future prices, it is still possible for the mono-polist to secure an unlimited, positive present value. This can happen if a2


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CONTINUOUS AND DISCRETE TIME APPROACHES 319Now write a new demand relation in terms of a perfectfl-symmetricdifference equationas follows:

2qt+ a(f-5qt+I - 5- qt 1)= ft - pt .. (4.5)We assert that (4.5) is the correct discrete analogue of the C problem for which a2 = 0.To verify this assertion, maximize the PV given by (1.3) subject to (4.5) instead of (1.4).The necessary maximum conditions require the optimal sequence {qt} to satisfy

2qt+ (x-5qt+1 -f* 5qt- )+2qt +a(f 5qt- I-_ 5qt+1) =ft, ...(4.6)which simply reduces to

4qt= ft ...(4.7)Equation (4.6) is derived in precisely the same way as (3.1), that is, by substituting thevalue of pt given by (4.5) into (1.3) and then calculating the partial derivatives of PV withrespect o qo,ql, q2 .. Observe hat (4.7)exhibitsprecisely he samemode of behaviouras the C problem in which a2 = 0, namely, instantaneous adjustment to the optimal path!Moreover, for this problem, a finite maximum present value always exists for all valuesof a and without restrictions on f,; clearly the restrictions on ft must be retained.Finally we come to the role of the discount factor. In the original D problem, or,for that matter in the general class of D problems containing lagged q's, it is always possibleto guarantee the existence of a finite maximum present value by choosing a sufficientlysmall positive value of ,. Thus the D problem has only interior solutions. This is nottrue in the C problem in which for special values of r there are edge solutions. We seefrom (2.12) that if r is too large then there would be an unbounded positive present value.Normally, one would think that higher discount rates would help rather than hinderthe attainment of a finite maximum present value. What explains this puzzling aspectof the C problem?The explanation is that, paradoxically, the C problem as stated is not symmetricenough. This is suggested by consideration of the analogous discrete problem that isimperfectly symmetric. We now show that in the imperfectly symmetric discrete problemthe same phenomenon occurs as in the C problem. Thus write a demand equation asfollows:

2qt + x(qt + I1-qt - 1) = ft - pt. ... (4.8)The quantity path maximizing (1.3) subject to (4.8) necessarily satisfies the following secondorder linear difference equation:

2qt +a(qt + -qt- 1) +2qt +a( l-qt - 1-fqt+ 1) = ft. ..-(4.9)This can be verified by calculating the partial derivatives of PV with respect to the q'sseriatum. If f, = 1, corresponding to r = 0 in the C problem, then a finite maximumPV exists although ft has to approach zero. In addition there is a finite maximum PVif f, is close to one. However, there is no bounded maximum if f, gets too small! 1Intuitively, the reason is that iff, is too small it makes past receipts accumulate at toohigh a rate at the same time that it makes the present value of future receipts small (a lowdiscount factor means a high discount rate) so that eventually even small sums becomearbitrarily large. This cannot occur in a perfectly symmetric C problem. A perfectlysymmetricC problem requires the coefficients to satisfy the equation as follows:al+a2r = 0. ...(4.10)

1 The mperfectly-symmetricdemandoperatorcan have an edge solution f the characteristic quationsof (4.8) and (4.9) havea common root. This commonroot is p-5 and occurs for a Pthat satisfies2+OC(P5-P-'5) = 0.Thusqtgrowsat themaximum ate -5 and the analysis s exactly he same as for the edgesolutiondiscussedat the end of section2.


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320 REVIEW OF ECONOMIC STUDIESIf the coefficients satisfy (4.10) then a finite maximum PV exists for the C problem for allpositive values of the discount rate r. Thus (4.10) is the sought for condition on thecoefficients of the demand relation that makes the C problem perfectly symmetric. Interms of the roots of the characteristic equation of the demand relation (1.2),

alla2'-(- a1+a2) and a2 = (a1a2) '.Therefore, (4.10) implies thatl+Uf2 = r. ...(4.11)

In other words, the arithmetic mean of the roots is r/2. In the perfect fl-symmetricdemandrelation given by (4.5), denote the characteristic roots by s1 and s2. For (4.5) we haveS1 S2 =-2(o4f5f)-' and S1S2 = -- '.

Therefore, the geometric mean of the roots is ,B5. Recalling that ,Be-r, this is preciselythe same relation as in the perfectly symmetric C problem.The symmetry implied by (4.11) is directly visible if the demand relation is convertedinto an expression relating the quantity demanded at time t to past and future prices.Since2= r-u1>O, a,


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CONTINUOUS AND DISCRETE TIME APPROACHES 321on a is necessary. The explanation is that if a2 = 0, the corresponding C problem is notsymmetric enough! Condition (4.10) is pertinent and suggests that for perfect symmetryin a C problem for which a2 = 0, it is also necessary that a, equal zero. Therefore, ifthe differential equation (1.2) lacks the term q" then it also cannot include the term q'.In other words there is no perfectly symmetric first order linear differential equation.This interpretation explains the paradoxical role of the interest rate in the continuousproblem by showing that in its original form the C problem was insufficiently symmetric.By introducing condition (4.10) the dynamic demand relation becomes symmetric in thesame sense as the classical static demand schedule. The most important moral to bedrawn from this is why symmetry of a demand relation over time is unreasonable in thereal world. This is because such symmetry would destroy the distinction between thepast and future. Symmetry in the static case arises essentially because the order in whichthe goods are chosen does not matter. But in choices over time the order in which goodsare chosen does matter for two reasons. First, it is impossible to change the decisionsof the past. Second, the impossibility of changing past decisions would not matter onlyif there were perfect foresight because then there would never be need of changing pastdecisions. Perfect foresight implies the absence of mistakes. In the real world there isuncertainty and perfect foresight is impossible. Hence current choices depend on thefact that one must adapt to new unforeseen circumstancesas well as possible. In a discretemodel the mathematics allows us to take into account the irreversibility of past decisionsand the effects of uncertainty.'

Universityof Chicago L. G. TELSERR. L. GRAVES

APPENDIXSection 1In the classical calculus of variationsproblem with fixed initial and terminal conditions,two conditions would be sufficient for a weak relative maximum:(i) Fq,q, 0

known as Legendre's condition, and the negative definiteness of the second variation,which is equivalent to requiring negative definiteness of the matrix:(ii) [Fqq Fqq,1

Fqq, Fq,q,The latter reduces to a matrix of constants in our case as follows:[-2 -(al+a2r)]L-(al +a2r) 2a2and (ii) implies that 4a2+(al+a2r)2


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322 REVIEW OF ECONOMIC STUDIESPixley [10] shows that if the admissible paths permit a finite number of discontinuitiesof both q and q', then extremalsmust satisfy the following analogue of the Euler equation:

Fq= Fq,= 0.In the present application this requires a2 = 0, a result which agreeswith ours as we shallsee. See also Arrow [2] and Nerlove and Arrow [9].Section 2

The following brief proof of a necessarycondition for a maximum PV easily generalizesto nth order linear differential equations. LetA(x) = a2x2 + alx +.

This permits us to express the demand equation in powers of the differential operator ddtas follows:

=fA (d q)PV Xe-rt qf-qA (d dt

For a finite PV, the choice of q = Meat, r - 26>0 requires that-M2 e(26-r)tA(b)dt =-M2(r-26)-1A(6) < 0

for all 3 such that r-2 > 0. In particular,A(r/2) > 0,which is (2.12) in the text.A similar argument proves directly the necessity of the condition a20, choose 6 0 for 0 < x ? r/2and (ii) a,( -1)


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CONTINUOUS AND DISCRETE TIME APPROACHES 323meaning that q jumps from q(O-) to q(O+) which is on the Euler path. On the given linesegment, q'(t) = m. Therefore,

rh e-rtm2dt = -(1Ir)(e- rh - e-rk)m2, OO, t


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324 REVIEW OF ECONOMIC STUDIESSection 5There s an equationanalogous o (3.4) in the C problem. The differentialquation(2.7) canbe writtenas follows:

-2a2 (d 21 d 2 q-fdit dA.ct qwhichreduces o (- -21) q = (2a2)' (d -A2)fiThe latterexpressiondenotesan integral. To prove this let

g = (d _22)f,so thatg is the solutionof the following irstorder ineardifferentialquation:

I'd _-22 0Its solution,as is readilyverifiedby differentiation,seA2tC + X e - A 2 f s r )The solution s bounded f 00c= - e2S f(s)dsosince 2 >0. Therefore g _ _eA2t e-2Sf(s)ds.Hence finally

q'-1 q = (2a2)-'1 e-2(S-t)f(s)ds.tSimilarly,we can write the solution of (3.4) in a form like (2.17).

REFERENCES[1] Allen, R. G. D. MathematicalEconomics (MacMillan, London, 1956).[2] Arrow, K. J. " Optimal Capital Adjustment ", Studies in Probabilityand Manage-ment Science, edited by K. J. Arrow, S. Karlin and H. Scarf (Stanford UniversityPress, Stanford, 1962).[3] Birkhoff, Garrett and Saunders MacLane. A Survey of Modern Algebra, rev. ed.(MacMillan Co., New York, 1953).[4] Bliss, Gilbert A. Lectures on the Calculus of Variations (University of Chicago

Press, Chicago, 1946).[5] Eisner, Robert and Strotz, Robert H. "Determinants of Business Investment,Impacts of Monetary Policy, Commission on Money and Credit (Prentice-Hall,Incorporated, Englewood Cliffs, New Jersey, 1963, esp. pp. 71-72, 79-81).[6] Evans, Griffith C. Mathematical Introduction to Economics (McGraw-Hill, NewYork, 1930, Chapter XIV).


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CONTINUOUS AND DISCRETE TIME APPROACHES 325[7] Graves, Robert L., and Telser, Lester G. "An Infinite-Horizon Discrete-TimeQuadratic Program as Applied to a Monopoly Problem", Econometrica, 35(April 1967).[8] Hotelling, H. "The Economics of Exhaustible Resources ", Journal of PoliticalEconomy, 39 (1931), 137-175.[9] Nerlove, Marc, and Arrow, K. J. "Optimal Advertising Policy Under DynamicConditions ", EconomicaN.S., 29 (1962), 129-142.

[10] Pixley, Henry Howes. "A Problem in the Calculus of Variations Suggested by aProblem in Economics ", Contributions to the Calculus of Variations 1931-1932(University of Chicago Press, Chicago, pp. 131-189).

continuous and discrete time approaches to a maximization problem

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