Two managerial economics approaches to the R&D decision process

MANAGERIAL AND DECISION ECONOMICS, VOL. 9, 163- 172 (1988)

Two Managerial Economics Approaches to the R&D Decision Process

ABRAHAM MEHREZ College of Business, Kent State University, Kent, Ohio, USA

In this paper we present two different approaches that analyze the effects of rivals on the R&D decision process. The first approach assumes that the introduction time of the new product is uncertain to the manager. Given this assumption, the problem is analyzed in a framework similar to the one suggested by Kamien and Schwartz (1972). The second approach assumes technical certainty to study a set of models that extends and supports different views in the managerial economics literature regarding the properties of the R&D decision process from both private and social points of view.

THE EFFECTS OF RIVALS ON THE R&D DECISION PROCESS

The effects of the market structure and rivals’ actions on a particular potential innovation have been fully recognized and studied by economists such as Scherer (1967), Barzel(1968), Kamien and Schwartz (1972) and others. One tenuous assumption has been made in all of these studies: the introduction date of the new product is entirely under managerial control. Given this assumption, the problem, from the standpoint of society, is to select the introduction date with the property that the marginal social cost of accelerating development just matches the discounted marginal social benefits of earlier introduction. A question of interest is the effect of rivalry among independent potential innovations upon the introduction date.

The authors mentioned above agree that the presence of rivalry gives rise to the conflicting tendencies of premature and belated development. With respect to delayed or reduced effort, there is a common observation that, because the innovator cannot capture all of the benefits of a technical improvement, the R&D manager will invest less time of endeavor than is socially optimal (Arrow, 1962). For the opposite reaction, Barzel has pointed out that rivalry among potential innovators may lead to compression of the development period and premature introduction relative to the socially desirable date. Kamien and Schwartz concluded that the computation of the socially efficient introduction time is a dificult task. In lieu of that computation, they compared the behavior of a firm under rivalry with the timing that would be selected by a firm or cartel operating without any rivals.

In the presence of technical uncertainty, the question of introduction time is irrelevant to some degree, since the developer is unable to control the completion time of the project. Furthermore, it seems that important concepts that can explain a firm’s behavior under technical certainty are not in hand for the uncertainty case. Scherer (19677, as an example, treated the timing question via a traditional oligopoly approach. Using his model, he compared various solutions that the firm will adopt. Among these were the minima and joint maximization solutions. Scherer admitted that his analysis lacked rigor and generality, and the solutions that he found are limited to his model and assumptions. To the extent that a model should consider technical uncertainties and a market structure of N firms, a dynamic equivalent to the Nash solution is unknown for the general control problem with N players and, therefore. for the development problem as well. (See, for example, the work of Reinganum, 1981, 1982, and Justman and Mehrez, 1984, for the analysis of closed loop solutions for R&D dynamic stochastic games.)

However, in the context of the R&D market such a problem, although it has theoretical interest, is not a very interesting practical question, since it requires every firm to have full information about its rivals’ technical and marketing ability. Such information is usually not available. For the analytical and practical reasons mentioned above, we will proceed with the problem of development in the presence of rivals and technical uncertainty.

In the next section, we will analyze and determine the R&D optimal spending policies using the Aldrich and Morton (1975) formulation of the so-called Lucas Model (1971). This formulation enables us to extend the framework suggested by Kamien and Schwartz

0143-6570/88/020 163- 10$05.00 0 1988 by John Wiley & Sons, Ltd.

164 ABRAHAM MEHREZ

(1972) for the case of technical certainty to deal with the typical R&D project with technical risks. Laury (1979), Das Gupta and Stiglitz (1 977), Sethke and Birch (1982), Mehrez (1983) and others have studied this problem in a different framework.

In the third section we restrict our framework to deal with the case of technical certainty and marketing uncertainties due to the patterns of a typical product life cycle. Different results dealing with the optimal introduction time of a new product are reported and related to the R&D economic literature.

In the final section we discuss an extension of the model suggested before to cover situations in which both the introduction time and the quality of a new product are controlled by the manager.

AN OPTIMAL BEHAVIOR IN THE PRESENCE OF RIVALS

Following Kamien and Schwartz (1972) we will make the following assumptions:

( I ) Patent protection is either unavailable o r ineffective.

(2) The benefit stream available to innovators decays exogenously. Since delay in introduction means foregoing part of that stream, the capitalized value of available rewards declines as the technical completion is postponed.

We further note that complicating the analysis is the assumption that rival firms may be developing a similar product or process. If the rival introduces its product first. the gain to our firm from its own project will be diminished. Even if our firm is the original innovator, imitators can reduce the reward collected by our firm.

To describe ou r firm's benefit in the presence of rivals we define P(t) to be the value of the quasi-rents from the innovation (or class of innovations that are close substitutes) at time t , atailable to all firms collectively which have introduced the innoLation by that time. P(r) is assumed to decay exponentially at rate 9. reflecting exogamous technological obsole- scence, meaning P(r) = e4" Po.

This assumption corresponds exactly to that made by Scherer (1967) and in special case studied by Marglin (1963). Let R( t ) denote the present value at time t of the total reward stemming from the introduction of the innovation at time t . Then, by definition.

and if the product has not yet appeared by time t . then no one receives the potential benefits.

The returns to our firm from introducing the product at time f 3 0 depends not only on the firm's introduction time but also on the rivals' actual introduction date, t,. Let the quasi-rent flow (exclusive of the development cost) to our firm at time s from

the innovation introduced at time t, t < s, be

eqsp0 t < s < t , egsP1 if t < t , d s (2) eq.'P, t , < t d s

where

O < P , d P o 0 6 P , d P o and g < O (3)

According to Eqns (2) and (3), if our firm is first to introduce then i t reaps a monopoly position from time t until rivals enter the market a t time t,, and a fraction of P,/P, of the reward stream after the monopoly position is lost at time t , . If the firm is not first, it receives a fraction P,/Po of the total reward available after its entry. The shares PJP, or P,/P, of total quasi-rents available to our firm in the presence of rivals are the best the firm subjectively thinks it can obtain given the rivals' predicted behavior.

The 'best' may be arrived at via tacit colusion, leader-follower relations or other means. The following schemes have been discussed by Kamien and Schwartz

P, + P, =Po. In this case, the share usurped by the imitator is constant regardless of whether our firm is the imitator or its rivals are the imitators. P , > Po - P , . If our firm is smaller or less powerful than its combined rivals, the share it could get as imitator would be smaller than that it would lose to its followers should the firm be innovator. Pz > Po - P,. Similar to case (2) , but the relative strengths are reversed. P, = P o and P,=O. The 'winner take all' case. This case may occur when a patent system is available and effective.

i f n o firm can capture all the benefits. 0 < P , P , < Po.

The limitations of the quasi-rent subjective description is two fold. It depends on the firm's beliefs, which may be based mostly on past experience. This case is most appropriate for those situations with modified products rather than entirely new ones. If the market includes just one potential rival, then the meaning of the rival's introduction time is independent of the rival's power. That is not the case where the market includes more than one rival. Under this case, the quasi-rent analysis can be extended to derive a multi- dimensional quasi-rent analysis which describes all possible situations which the firm might face.

To complete the description of the firm's belief about its rivals we will denote by F(r,) the firm's assessment at time r = 0 (when the firm makes its development plans) of the probability that the composite rival will have introduced its improved product or process by time t,. Thus, F(t , ) is our firms subjective comulative distribution over the rival's introduction time t,. F(t,)

TWO MANAGERIAL ECONOMICS APPROACHES TO THE R&D DECISION PROCESS 165

is given by flow at time t is equal to

where k and h are positively and non-negative parameters, respectively.

The density function associated with t , is

Note that the hazard functions are given by

The parameters, k # h, allow the possibility that rivals may accelerate ( k > h) or decelerate (k < h) their own effort as a consequence of our firm's introduction of its product. If the rival's development rate is unaffected by our firm's introduction time then k = h.

Under the market structure described by Eqns (2)-(6) we derive the optimal spending policies using the Aldrich and Morton (1975) formulation of the so-called Lucas Model (1971).

To start, we define V the discounted expected value of the project to be

V(s) ift, 2 t u(s) ift, < t

V(t) = (7)

where u will be defined later. Further, we will consider for analytical reasons the case where the effort distribution function is exponential with parameter 6.

Under Eqn. (7) it is easy to show that I/ and u are related by the following recursive equation:

V(t) zz Max { - mAt + R'(t)bg(m)At + Athu(t) + [I - rAt - (bg(m) + h)At] V(t + At)} (8)

subject to m >, 0 and

u(t) z Max { - mAt + bg(m)P2eqt + (1 - rAt - bg(m)At)u(s + As)) (9)

subject to m 3 0 By combining Eqns ( 3 ) and (4), R' is given by

~ ' ( ~ 1 = ,rt 1; e - ( r - 4 ) V CPd1 - F ( V ) )

+ PI j F( V) = F(t)] dv (10)

Further, by substituting from Eqns ( 5 ) and (6) the information with the rewards, we get finally

The expression in Eqn. ( 1 1) may be further interpreted. If the firm is first, it will collect a quasi-rent stream which may be thought of as two expected discounted flows, one temporary (received only during the period of monopoly) and the other permanent. The transition

The former stream effectively decays at rate r -g + k. (The discounted rate, r, adjusted for the quasi-rent growth or decay, g, and the 'hazard rate', k , while the permanent flow decays at rate r - g only, and equal to

PI - (h - sfr __ r - g

The developer's incremental optimal policy depends on three events:

(1) The firm will be the first to introduce the new product with discounted expected reward at time t , R'(t).

(2) The rivals will complete development of the new product, and the discounted expected reward will be u(t).

( 3 ) Our firm will carry on effort where the rivals have not appeared in the market and the discounted expected reward will be V(s + As),

The optimal spending policy is derived in two steps: (a) u(t) can be computed from Eqn.(9). The man-

ager computes u(t) conditional on our firm not being the first in the market, where u(t) is given implicitly by

du(d dt - _- = - mx(t) + hg'(mx(t))[P2eY' - ru ( t ) ] (12)

where

y'(m") = I/[b(eY' - u(t))] (13) (b) Then he solves Eqn. (8) by introducing the

solution of u from Eqns (12) and (13). The solution is given implicitly by

mx + hg(m"(t))

(14) (R'(t) - V ( t ) - rV(t))

and g'(mx) = l/[b(R'(t) - V(t)] (15)

Equations (I), (12) and (13) establish a complete theoretical structure to derive an optimal spending strategy under the threat of rivals in the structure of Alrich and Morton.

To complete the analysis, a few remarks are in order. The solution of u and V can be derived, since a natural bound on mx exists for the case where g < 0. For u. as an example, mx = 0 if g'(0) be

In this way, a bound on mx can be established for V . The analysis of cases (1)-(5) regarding Po, P, and

P , can be derived. Under Eqns (12)-( 15) we will discuss very simple cases that can be identified as private cases

166 ABRAHAM MEHREZ

(1)-(5) if more assumptions about F and g will be introduced.

Po = P , = P , is the case in which there is not effective rivalry. This case implies that V ( t ) = u(t). P,=P, and h = O is the case where our firm establishes monopolistic power and V is derived as if k = 0, or there is no threat of imitators. That is also the case where a complete patent system is available. The case where P o - P , = P z = 0 is the case where 'winner takes all', and clearly u ( t ) = O and consequently the project will be terminated if our firm realizes that someone appears in the market before it. The case where g = 0 is the time-independent return for u. That is not correct for V .

Note that, in general, it is not true that u(r) d Vtr), since if P , is large enough, P , is small enough and k is large enough to make imitation more profitable than innovation.

A MANAGERIAL ECONOMIC ANALYSIS UNDER CONDITIONS OF TECHNICAL CERTAINTY

We assume that the manager's objective is to maximiLe the expected present value of the R&D project and that there is just one decision variable--the introduction time of the new product.

Let c ( T ) be the development costs and $(t , T ) the net benefit receipts at time t. where

and r is the rate of interest.

time which maximizes The optimal introduction time is determined as that

Vp( T ) = u( t)e ~ d t - c( T)e - " (17) i: where VJT) is the present value of the project.

A necessary condition for T' to be an optimal interior solution is obtained by differentiating Eqn. (1 7 ) with respect to T and equating it to zero: i.e. by

or

Equation ( 1 9) states that one ought to introduce a project at the point TI where the sum of the net benefit received the market ZI( T I ) at time T' and the saving of costs due to incremental delay in introducing the product, dc( T')ldT, are equal to the interest on the project's cost of development.

The function $ and the conditions we derived in Eqn. (18) are, in fact, very general. Under these

conditions, and under our assumptions, the manager decides to introduce the new product according to the functional relationship between the expected discounted benefit from the project upon introducing the project at time T and the costs of development which follow from this decision. As we noted previously in the work, due to the complexity of the R&D economic market there is no way to derive the expected potential benefit from the project at completion in a unique closed form. The most that we are able to do is to find the 'best' decisions that the manager can take under very specific conditions that we will define in the following illustrative example. This example is based on an empirical observation of the market for new products done by Pessemier (1966), and further assumptions regarding the 'subjective' beliefs of the manager about the behavior of rivals and consumers.

Illustrative Example

Empirically, it has been found that many products, especially those that are characterized by minor technical difficulties, reveal a typical pattern of market development which is called the product life cycle. Pessemier (1966) has used five different stages to describe a typical product life cycle: (1) introduction, ( 2 ) growth, ( 3 ) maturity, (4) saturation and (5) decline. Empirical studies have shown that in general the life-cycle model is a good one, especially when different product forms compete for essentially the same market within a general class of products.

Pessemier, in attempts to capture the empirical patterns of the life-cycle model, found that the Weibull distribution yields a close fit to sales data over time. To proceed, we note that the Weibull distribution has the density function

f ( t , cx, p) = pro- exp { - at@) (20)

A decrease in r gives a tighter density function and an increase in jl makes the peakedness in the density more marked.

To interpret the evidence of Pessemier in our context, we assume that the manager observed the stochastic process, ( x , , t ~ [ T ] } ( [ T I is an index time set), where x, (the accumulated percentage of sales up to r ) is a random variable that satisfies the following properties based on Eqn. (20):

E(x,) = 1 - exp { - ato) (21) Thus, by Eqn. (21) we observe that dE(xt)/dt, the rate of change in the expected sales in time t , is given by

To derive from Eqns (21) and (22) a knowledge about the expected gross benefit (not including the development costs), a few further assumptions have to be made:

(1) The production function of the new product exhibits constant returns to scale.


(2) The manager regards the time that a new similar product will be introduced into the market as a random variable, s, that obeys the exponential distribution, i.e.

f,(s) = he-hs h > 0, s > 0 (23)

where h may be interpreted as a measure of the market concentration and the Min {s, T'} is the initiation time point of the product life cycle.

The fact that the exponential distribution is memories and stationary reflects on the supposition that the manager's beliefs are stationary and, further, that the probability that a new, similar product will be introduced up to the period depends just on the length of the period.

(3) Based on experience, the manager observes three prices for the new product:

6 , - i f T ' < s and T ' < t d s 6 , - i fT '<s and t 3 s d , - i f T ' > s and t B T '

Based on the theory of the demand function, 6, 3 S,, where 6, is extracted from the monopoly power of our firm that equates the elasticity of price of the new product to the marginal rate of return from producing an additional unit.

(4) For analytical ease, we restrict f(t, a, 8) =f(t, a, 1). This supposition exhibits the properties of the exponential distribution for the distribution of percentage of sales, where a is interpreted as the expected product life.

(5) The percentage of sales and s are independent random variables. Using the above assumptions, we can compute

Vp(T) = ESC~T(IC/(t, T))1 = EsCETCIC/(4 T)ls 3 TI + E T C W , T)ls < TI1 - e-rTc(T) (24)

where the equalities in Eqn. (24) were derived from the expectation properties.

Further, from Eqns(21) and (22) and assumptions (I), (2) and (3,

where L is a proportional constant that converts the percentage of sales to sales.

Finally by assumption (2), and with some computations, we get

V,( T ) = L ::: e - ( r + n + h ) T

- (r + a + h ) T has, ( r + a + h)(r + a)

+

e-rtc(T) (26)

by Eqn. (26) we observe that the expected gross benefits is a summation of three components, each one of which corresponds to a time period where the price is 6 , i = 1,. . . , 3 .

Further, Vp for a given T' depends on its parameters (h , a, r, 6, , a,, 6,). To analyze the effects of an increase in one of these parameters on the gross benefits we define, from Eqn. (26),

e - ( r + a + h ) 7

r + a + h

had, - ( r+ a + h)T' + (r + a + h)(r + a)

Clearly,

This says that the manager is better off if at least one of the prices increases for any given introduction time and thus one can claim, by extension, these assertions hold with regard to the optimal introducton times, i.e. for the original and the adjusted introduction times. These assertions do not hold, in general, for the effect of the interest rate on the manager's position, since r is also a parameter of the cost function.

These kinds of clear-cut assertions cannot be stated with regard to an increase in CI or h, without further considering the first- and second-order conditions. At most, the following assertion holds: For T': if

then

In other words, if the original and adjusted optimal introduction times are sufficiently postponed then the manager is worse off if the expected 'waiting' time and/or the expected life of the product are shorter.

Another interesting question that has an impact on social welfare and has been extensively treated in the economic literature of R&D is related to the introduction time of a new product. As noted by many economists, the computation of the socially optimal introduction time is difficult, if, indeed, it is possible at all. A t most, one should attempt to study the effects of market concentration (interpreted in our model as

168 ABRAHAM MEHREZ

u), information and concentration (interpreted in our model as h) and prices and interest rate on the introduction time.

To answer these questions we will make first a few

where . denotes any of the parameters we have mentioned in operation (1).

(3) A sufficient condition for T‘ to be a local maximum is that

observations to characterize b,( T).

(dT‘)’ < O ’ Theorem 1

Thus the sign of iT‘l6 is determined by the sign of (?P,(T’))/?T’, and we will consider the former. Before doing this we note that by assuming c’( T ) < 0 and c”(T) > 0, in the presence of operation (3) the sign of (i?2V,(T))/i?T2 is not determined uniquely, V T 3 0.

Based on operations (1)-(3) we investigate a T / a . via ((?/3,( T’))/?T. These observations can be verified from Eqn. (28) by some analytical arguments that will be omitted.

c‘B,(T) < If 6 , 3 6 , then C T

(28) Thus, the effect of price increases (for the case where our firm is the innovator) will premature the introduction time.

(5) This is not the case where our firm is the price taker, and ?TJ(?d3 changes sign. One can argue that there exists a T‘ for which

Theorem 2

at most for one value of T .

Proof’ After some manipulations of Eqn. (28) we get and, further, this holds V T‘ 3 T,, i.e. for some range

of T‘ an increase in 6, will result in postponing the introduction time. In this range the firm prefers to wait, not compress the development period and cost and to capture the benefits conditioned on being the imitator.

( 6 ) To identify the sign of ?T’/i?h we suppose that 6 , 3 6,. Then we have to consider the following cases. There exists T; such that VTI, > T‘ (?T‘/i.?h) < 0 and V T’ < T3, (dT‘/dh) < 0. T,E ([0, XI). Thus, we conclude that above some time point the decision to introduce the new product will be postponed if the expected ‘waiting’ time decreases, and below this time point, the opposite conclusion holds.

( 7 ) The analysis of ZT’/?a is more complicated. We note that generality, as in operation (6), it cannot be deduced, i.e. there is more than one possible case to consider. We only claim that under some restrictions on the parameters of Bp, one can show that the case described in operation (4) holds for ?T/?h. The verification of the many feasible cases is not, however, what we want to focus on.

The main conclusion from these results are that they give rise to both views that have been described in the economic literature, the one articulated by Barzel (1968), that rivalry among potential innovators may lead to compression of the development period, and the second shared by Arrow (1962) and Kamien and Schwartz, who pointed out that rivalry may lead to the postponement of the introduction time.

If L = ehr, where L is some positive number. Thus by the monotonic property of the exponential function

at most. once.

Theorem 3 Suppose that 6 , > 6,, then by Theorem 1, P,(O) = Lz > 0, and if

lim p,( T ) = 0

we claim that P,(T) is either convex on [0, x] or a concave function on [0, T;] and convex on [T;. x]. where O < T‘< x. The proof is omitted. Thus Theorem 1-3 establish the properties of B,(T).

T - 7

Finally. the following operations are in order:

(2) From operation ( I ) ,


AN EXTENDED ANALYSIS OF THE INTRODUCTION TIME AND THE QUALITY OF A NEW PRODUCT

In the previous structure we assumed that the manager can control only the introduction time. More common is the situation where the subset of decision variables includes both the introduction time and the quality of the new product.

In this case we define a 'new' variable, q, the quality of the new product. Furthermore, we extend II/ to be a function of t , T, and 4:

The objective function is naturally extended to maximize

V,(T,q)= U(t,q)e-"dt -c(T,q)e-" (30)

where c(T, 4) is the development costs incurred if the development period is of length T and quality 4.

The necessary conditions for T' and q1 to be optimal are given by

s:

and

ac V(T ' ,4 ' )+rc(T ' ,4 ' ) - - (T ' ,q f )=0 (32)

2 V -- p - - 2T irT

Condition (32) is essentially the same as Eqn. (18). To clarify Eqn. (31) we note that

is the present value of the sum of all future alterations in U due to the increases in q. This should be equal to the marginal increase in cost due to an increase in quality.

The sufficient conditions for T' and q' to have obtained a local maximum are that

and

q, the decision variable that has been introduced, is related, in general, to two economic factors. First, it is connected with the gross benefits from the project in that the higher the quality, the higher are the gross benefits. Second, it is also related to the effort of the firm to complete a project which is technically satisfactory, in that the bigger the effort, i.e. costs of

developing the product for a given time period, the better is the quality.

Empirical evidence for the two relationships we have defined above has been reported in the literature. The first type of relationship is manifested in the model by the functional relations between U and q. To derive the second type of relationship we refer the reader to the definition of z-the level of knowledge accumulated in the course of the project-and we further suppose that 4 is a function of z, i.e. that i is an indication of the quality of the new product. We will establish via an illustrative example the relations between c and T' and z. The example extends a cost function which has been studied by Kamien and Schwartz ( I 972).

Illustrative Cost Function

We suppose that the rate of dollar spending, rn(t), and the cumulative effect effort, z, are related via a concave monotonic function of the following form:

z ' ( t ) = ma(,) - yz(t) (33)

where 0 < p < 1 and y > 0. In their analysis of this function, Kamien and

Schwartz considered 7 to be zero, where 7 is a constant which determines the rate at which the accumulated efforts of the project decreases if no incremental efforts are done. Several authors, such as Griliches (1973), have postulated that a significant part of R&D expenditures goes toward the maintenance of existing knowledge. This point of view is based on the hypothesis that no spending in the project will end in knowledge depreciation, mostly due to a cut in manpower. The parameter, B, indicates decreasing returns from increased spending rates. The more rapidly a given sum is spent, the less it contributes to the total effort.

We further define boundary conditions on z by

and z(0) = 0

Z( T' ) = A

(34)

(35)

whereby z(0) = 0. We suppose, without loss of generality, that there is no partial development by time 0.

Equations (33)-(34) produce enough information to study the project's development costs, for a given T' and A.

The mathematical problem posed here to minimize costs can be solved via optimal control theory with m(t) and z(t) playing the roles of the control and the state variables, respectively.

To solve the problem we suggest here the use of Theorem 1 (Luenberger, 1969).

Theorem 1 Let xo, uo minimize

J = r' L(x, u) dt f0

170 ABRAHAM MEHREZ

subiect to

x(t) = f / ( x , u), x( tO) fixed

G(x( t , ) ) = constant

and assume that the following regularity are satisfied:

(37) (38)

conditions

( I ) f has a continuous partial derivative with respect to x and u.

(2) U is taken to be cm[to. t ] , the space of continuous m dimensional functions, where V is the space of admissible control functions.

(3) For any U E U the space of admissible control functions and initial condition x(t,) = constant, we assume that Eqn. (37) defines a unique continuous solutions x ( t )V to 6 t < t

Then, the necessary conditions for (so, uo) to be optimal to the problem defined by Eqns (36)-(38) satisfying conditions (1)-(3), are the following:

- j.'(o = C. fXx0V) , uo( r ) )14 t ) + L(.x,(~), u,(t)) (39)

j . ( f , ) = G:(.x,(t 1)) (40)

(41) j . ' ( r ) . fubo(f) , uo( f ) ) + L(-x,,(r), u 0 W ) = 0.

for all t E [ t o , t l ] where i ( r ) is an m-dimensional.

redefining first: We use the theorem to solve our problem by

u( t ) = m(t) (42)

x ( t ) = z(r) (43)

,f = - ;': + 172' (44)

(45)

t o = o (46)

r , = T (47)

.x( to) = 0 (48)

(49)

.u(t , ) = A (50)

To proceed toward the solution of our problem, we note that one can easily check that conditions (1)-(3) are satisfied for our problem. Now, using Eqns (42)-- (50) and having translated Eqns (39)-(41) into our problem's context, we observe that

;.'(() = yj.( t ) (51) and

i ( 0 ) = c1 (52)

or by solving Eqn. (51) and using Eqn. (52h

i ( t ) = cle.;' (53) with

j . (T) = c1e7(7' (54)

L(x, u ) = - e-"m

G = I the identity function

From Eqns(53) and (54) and the equivalent of Eqn. (41), we see that the optimal spending m'(t) is given by

= ~ ~ ~ . , , p ~ ( ~ + r ) ' ] l 11 - @ I ( 5 5 )

To study the properties of m'(r) and c( T ' , A ) we provide

some further notation:

N = y + r (56)

(57)

rl = PA1 - B) (58 )

Then, to derive c explicitly we impose Eqn. (55) on the differential equation defined by Eqn. (44) and the boundary conditions (48)-(50) to get, using the new notation,

z' = z + eNq'.K

K = (c ?P)P/( 1 - 8)

(59)

z(0) = 0 (60)

(61)

subject to

z (T) = A

Equation (59) is a linear ordinary differential equation, and the general solution of it is given by

(62) Z(j)eP = K e"I++Y'"ds + c' Sb and by Eqn. (60) we see that

c' = 0 (63)

Now solving Eqn. (62) with c' = 0 obtain

Using Eqn. (61) and rearranging, we obtain

or

or

Finally, by introducing c1 in Eqn. (55 ) we get an explicit form for rn'(t):

where, for 7 = 0, we are able to verify to solution of Kamien and Schwartz:

Furthermore, c( T', A), the total development cost function, is given by discounting and integrating m'(t) on [O, T'] . Doing so, we get the original notation:

c( T' , A )

(70) To establish the properties of m' and c we first note

that both are functions of y, /?, r , T' and A . Their other main properties are summarized as follows:


(1) m’(t), the spending function, is monotonically increasing in time. This assertation verifies an observation in the literature that the spending on R&D projects tends to increase as the project nears completion.

(2) By (1) we observed the time patterns of spending within a given program defined by T’ and A . To compare the time patterns of spending between two programs with common T and A, if we denote any parameter of m‘ as P, then the elasticity of m‘ with respect to P(E,.,) is a linear function of time. Further,

By the last observation we are able to compare two programs that have the same targets and differ with respect to their parameters. First, we consider y and r. For these parameters, K is positive and thus the magnitude of their elasticities grow linearly in time. In the case of p it is impossible to make such a general statement.

The conclusion we derive from (2) are that, at least with respect to r and y , the percentage change in spending by altering the parameter of the cost function increase in absolute value by time.

(3) For fixed T’ and A one can show that

..

Proof Rewrite from Eqn. (68):

where

(72) A question discussed frequently in the literature concerns the signs of [dc(T’, A)]/JT‘ and [a2c(T’, A)]/(Dt2). The general assertion made

(73) in the literature is that [ac(T’,A)]/dT’ < 0 and [dc(T’, A)] / (aT)2 > 0.

In the case where y = 0, the above assertion holds, and it probably also holds where y is large enough, postponing the project’s completion beyond some time might bear additional costs. The property of &(T) that (apkT’)/dT < 0 implies that a bound on the interval within which the optimal introduction time lies may be imposed.

(4) We introduced the reader to the problem ofmaking a decision on both the introduction time and the level of effort (quality). Up to now we essentially assumed that z was given. In fact, the project’s goals can be established by varying z. If dp,/aq > 0, dq/dz > 0 and &/az < 0, then the decision problem can be solved in two steps: (a) For the fixed levels of z and T the cost function

can be found by employing an optimal control Em., = PC(Qo/P)Q(t) + Qo(Q(t)/P)I (74)

Qo Q(t) and by algebraic operations we get from Eqn. method.

(b) Inserting these cost functions a solution for the optimal T’ and z can be found by employing an

(5) A comparative static analysis of the parameters (y, p), that will be omitted, can be handled in a way similar to the one we established for ( T , 6,, 6,, 6,, h, a) in the first model under certainty.

(6) A more complex analysis, that also will be omitted, regards the parametric effects of r . This complexity is due to the fact that r is a parameter of both the cost and the gross benefit functions.

(74)

Em,, = K(P) + N(P)t (75) optimization technique. where

K(P) = P%/Qo ap

and

W Q ( t ) / W = N(P)t Q(t)

REFERENCES

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Documents

Two managerial economics approaches to the R&D decision process