Socio-Econ. P/am. Sci. Vol. 20, No. 5, pp. 271-274, 1986 Printed in Great Britain

003%0121/86 $3.00 +O.OO Pergamon Journals Ltd

OPTIMAL FUNDING POLICY FOR R AND D PROJECTS AIMED AT COST REDUCTION: THE CASE OF A

NEW DRUG IN PSYCHIATRY

ABRAHAM MEHREZ’ and AMIRAM GAFNI’ ‘Department of Industrial Engineering and Management, Ben-Gurion University of the Negev,

Beer Sheva, Israel and 2Faculty of Management, The Leon Recanati Graduate School of Business Administration, Tel Aviv University, Tel Aviv, Israel

(Received 22 February 1986)

Abstract-Governments’ role in supporting research developments is well established. In this paper we describe a model for finding the optimal funding path for R and D projects aimed at cost reduction. A numerical illustration for a case of pharmaceutical research for a new drug to reduce the cost of treating patients in an irreversible chronic condition (schizophrenia) is presented. This is done by using the mathematical results obtained for the special case--the stationary exponential case. The sensitivity analysis method is utilized to study the optimal spending path and the project’s discounted expected value. We also present a possible alternative subjective consistent estimation procedure for finding the optimal funding path for such projects.

INTRODUCTION

In general, there are 2 types of R and D projects: those which develop entirely new goods or services, and others whose aim is to reduce the cost of producing existing goods or services. A framework for analyzing the optimal funding path of R and D projects of the first type was developed by Lucas [l] and extended by Kamien and Schwartz [2], Aldrich and Morton [3] and more recently by Mehrez [4]. However, R and D projects which are directed at cost reduction have not been examined extensively in the literature. A discussion of such projects from a welfare point of view within a deterministic frame- work can be found in Nordhause [5]. This type of project is more common in the health care field where the aim is to find a less costly way of providing at least the same level of treatment. For example, pharmaceutical research for a new drug which will reduce the cost of treating patients suffering from an irreversible chronic condition may solve a problem which is major both in terms of population size and the amount spent on treatment.

In this paper we examine R and D projects which are directed at cost reduction. We describe a frame- work for analyzing the optimal funding path for such projects by using a model where one minimizes the discounted total average costs needed to develop the new procedure. This model is, in fact, equivalent to the maximization model of optimal control under finite horizon suggested by Kamien and Schwartz [2]. We further present an application for the stationary exponential case based on the case of a new drug in psychiatry. We present the optimal funding path of

*U.S. Department of Health, Education and Welfare. Basic Data Relating to the National Institute of Health (Washington D.C. HEW, 1981).

R and D project whose objective is development of a new drug that will lower the cost of treating schizophrenic patients. This is done under different assumptions regarding the value of the parameters in the model. We also present, in an Appendix, a possible alternative subjective estimation procedure for finding the optimal funding path of such projects.

The model and its application as presented in this paper can be of great use for policy and decision makers in different governmental agencies. This is due to the important role that government plays in financing R and D projects. For an example, with respect to pharmaceutical R and D (the application described in this paper) data from U.S.* show that out of the $7.7 billion total investment in biomedical R and D in 1980, $5.1 billion came from the federal government ($3.2 billion from the National Institute of Health). From these figures it is easy to see the importance of a relatively easy applicable model to help decision makers decide whether to approve an investment in a given R and D project, and if so, what is the optimal funding path for such a project. As forecasts predict that more and more public money will be allocated for pharmaceutical R and D (Bezold [6]) the need for such framework of analysis as presented in this paper is further strengthened.

THE MODEL

The typical framework for analyzing the optimal funding path of R and D projects used in the literature is the following: an R and D project can be seen as a dynamic procedure where technical infor- mation is produced. This information is accumulated over time till the developer’s goals are achieved. The basic assumption is that the level of information which is needed to achieve these goals is unknown in

272 ABRAHAM MEHREZ and AWRAM GAFNI

advance. This framework was used in the literature for R and D projects arrived at developing entirely new goods or services. The factors determining the dynamics of spending for this class of projects are the following: the probability of a completed project, the probability function of technical success, the oppor- tunity cost of a project and a function relating the rate of dollar spending on the project to the rate of change in the effort spent to acquire knowledge (the accumulated information). The effort invested in the project is related to the probability of technical success via a distribution function which is called the “cumulative effort distribution function”. In this paper we use a similar framework to model the optimal funding path of R and D projects aimed at cost reduction.

We deal with an organization which provides a service at a costant level of spending, C, $/yr. The organization has to decide how much money to allocate for an R and D project aimed at the devel- opment of a cheaper procedure to supply at least the same level of service. C, will denote the new expected annual level of spending. Obviously, it is assumed that C, < C,. More explicitly, the organization’s problem is to find a planned expenditure policy, m(t), 0 < t 6 T to minimize the discounted expected cash flow from the project. Alternatively, the exact R and D problem is formulated as follows:

MinP= ‘e-“{C,F[Z(t)] + [m(t) + C,] m(r) s 0

x [ 1 - F(Z(t)]))dt + F[Z( T)]e-” T

+ { 1 - P[Z(?7)]}e-‘r?

s.t. m(t) > 0

Z‘(t) = g[m(t)] Z(0) = 0

g(O)=0 lim g(m)=B m--m

g’(O)< co, g’(m)>0 g”(m)< form 30

F(0) = 0; F’(0) = 0;

F’(Z) 2 0; Jim F(Z) = I (1) where:

P-is the discounted expected cash flow; m(l)--is the rate of dollar spending on the project

at time t; r-is the discount rate; Z(t+is the cumulative effort devoted to the

project by time I; F(Z)-is the probability that the project will be

successfully completed by the time the cumulative effort is Z. It should be noted that without loss of generality the mode1 can be extended to a structure where k different, or identical, technologies are devel- oped independently by k teams.

&z(t)]-is a function that relates development efforts to the current spending rate m/(t) such that Z’(t) = g[m(t)] and h(Z) = F’(Z)/[(l -P(Z)] is the conditional completion density function.

A problem equivalent to the one described in

problem (1) is

s T

YFP ebrr[(C, - C#lZtt)l 0

- m(t){ 1 - F[Z(t)])dt

(Cl - CJ + F[Z(T)]e-“A

r

s.t. the set of constraints given in problem (1).

The probiem formulated in this way is quivalent to the one described by Kamien and Schwartz [2] where the profit rate is defined as the alternative loss due to the use of the old procedure rather than the newly developed procedure. Thus we can claim that all the results proved by Kamien and Schwartz also hold for our problem [eqn (2)].

To compute the infinite horizon optimal spending path for problem (l), we relax the assumption that F’(0) = 0 and take T = co and F to be the exponential distribution with mean l/b. Using the results ob- tained by Kamien and Schwartz [4], it can be shown that a non-null optimal spending function policy satisfies the following relations

where

-g”(m).rn’/g’(m) = r + b G(m) (3)

G(m)=g(m)-(D+m)g’(m)andII=C,--C,.

(4)

Equation (4) implies that m*, the optimal policy, is independent of Z and thus by the positivity of m, this function is constant over time.

To summarize the characteristics of m *, the follow- ing proposition is in order.

Proposition (1). A necessary and sufficient condi- tion for the existence of an optimal non-null policy m* is that IIbg(m*) >, rm *. Furthermore, if such a policy exists it is constant and unique.

Proof. The constancy of m * has been derived from eqn (4). The uniqueness follows by the strictly mono- tone increasing property of G with respect to m*. Finally, by solving problem (1) for a constant policy, we receive the necessary and sufficient condition.

Two remarks follow from proposition (1):

(i) Kamien and Schwartz [4] and Aldrich and Morton [3] stated that if bg’(O)II > r, a project will be undertaken. This condition, which is comparing the expected marginal benefits with marginal costs of investing additional dollars in the project, serves as a substitute for the necessary and sufficient condition for a project to be undertaken given by proposition (1).

(ii) Equation (4) implies that for exact, optimal, control planning, the solution of m* is uniquely determined by the non-linear differential eqn r + bG(m*) = 0. The comparative static. analysis of m* in terms of the parameters of the model (b, II, r), provides that

am* - > 0; g > 0; fg < 0. i3b

It is also easy to show that for the optimal annual

Optimal funding policy for R and D projects 213

spending on the project m*, the value of P (the discounted expected cash flow) is given by

P= c,+m*--C, c2

r + bg(m *) +tT

and the expected time for the completion of the project (E) is given by

1

E = b[g(m*)]2

NUMERICAL ILLUSTRATION

In this section, using the results accepted in the previous section, we present a numerical illustration describing the optimal funding path of an R and D project whose aim is to develop a new drug (or improve the efficiency of an existing one) that will lower the cost of treating schizophrenic patients.

Schizophrenic reactions are a group of diseases that cause massive disruption in thinking, mood, sensorimotor functioning and behavior. This group of diseases is very common in our society; over the last 25 yr the annual number of new cases among the American population has ranged from 0.043 to 0.069%. Its prevalence-the fraction of the popu- lation defined as schizophrenic during a year-is between 0.23 and 0.47%. Therefore, between 494,500 to 1,010,500 people need treatment annually. Life time prevalence-the % of those now living who have had or are likely to have schizophrenia at some point in their lives-is 1%. Most cases of schizophrenia are chronic, that is, the patient cannot be fully cured.

The many treatments used are very expensive. As schizophrenia is not a fatal disease and those affected tend to live a normal life span, treatment costs for this disease are exorbitant. Chemotherapy (the use of drugs) is a widely used technique. Increased knowl- edge of the biochemistry of the brain, the physiology and pharmacology of the synapse and their re- lationship with behavior have stimulated research to discover more efficient drugs that will reduce the high costs of treatment. (More about schizophrenia can be found in Nicholi [7].)

The question which we seek to answer through the illustrative example is: in a public health care system, how much should the government spend, under different assumptions, to encourage R and D projects leading to cost reduction? Our empirical data is drawn from the Israeli health care system, however, the methodology presented is a general one and can be used in many other policy issues which are not

specifically considered here. For example, the govern- ment can consider the optimal structure of R and D research by allocating resources to k independent teams. (In the model this question can be handled by substituting bk for b.) In addition the amount of subsidy to R and D projects can be evaluated using this model in different settings.

TO calculate the optimal annual spending (m *) on such R and D projects we use the stationary ex- ponential case, described in the previous section. The assumption of the exponential distribution function of the knowledge accumulated was not rejected by the industry experts with whom we talked, that is the accumulation of knowledge in the R and D process is memoryless. This seems reasonable due to the trial and error nature of most experiments. More about the nature of pharmaceutical R and D can be found in Cox and Loveday [8], and Wiggins [9]. Another assumption we used for reaching optimality is that g(m) = Am” which is common in the literature (e.g. Kamien and Schwartz [4]). Data on r (discount rate) and an estimation of C,, the current annual spending on treatment, including hospitalization as well as ambulatory treatment costs were obtained from the Ministry of Health. C, does not include costs incurred by the patients, such as lost work-time and travelling expenses. The values of C2 (b . A) and o! were arbi- trarily decided upon, and sensitivity analysis was performed. Although arbitrary, these values reflect our interpretation of potential scenarios as described to us by experts. The results of our sensitivity analysis are presented in Table 1.

It is easy to see from Table 1 that m*, the optimal annual spending on R and D projects, is very sensi- tive to changes in a and Cz, but not sensitive to changes in bA. P, the discounted expected cash flow from the project, is sensitive only to C,. If one recalls that bA can be interpreted as the parameter de- scribing R and D projects with different number of independent research teams, one can easily conclude that both m* and P are not sensitive to the number of research teams (rows 1, 3, and 4 in Table 1). This implies a clear situation of dis-economies to scale. Thus, if minimizing P is our only objective, then it seems reasonable to work with one team only and thus avoid administrative complications (more on such complications can be found in a special issue devoted to management of research groups [lo]).

Although in our numerical example, the parame- ters b, A and a were arbitrarily assumed, a procedure which enables the user of this model to obtain consistent estimates of CI, A and b for the stationary

Table I. Optimal levels of annual spending on R and D projects directed to reduce costs of treating schizophrenic patients

Cl G/C, r b.A 61 m*1.2 PI.3

I 67,353,OOO 0.8 0.06 I 0.25 4,482,400 898,430,OOO 2 67,353,OOO 0.5 0.06 0.25 I1,210,000 562,050,OOO 3 67,353,OOO 0.8 0.06

: 0.25 4,486,300 898,235,OOO

4 67,353,OOO 0.8 0.06 3 0.25 4,487,600 898,170,OOO 5 67,353,OOO 0.8 0.06 I 0.50 13,470,160 898,047,OOO 6 67,353,OOO 0.8 0.06 I 0.75 40.411.780 89X.040.000

‘In 1982 dollars. 2Was calculated by solving r+bG(m*)=O where: G(m*) =g(m*) -

(C, - C, + m*)g’(m*) and g(m) = Am” ‘Was calculated using eqn (5).

274 ABRAHAM MEHREZ and AMIRAM GAFNI

exponential case is available and described in the Appendix. Although this procedure may be difficult to implement it (a) enables the user to test whether the normative model describes real human behavior and (b) can be used as a supplement to the sensitivity analysis and an interpretation of the optimal values obtained from the model. The authors admits that they are not aware of similar estimation procedure for the more general structure of the problem. (Namely, when we relax the stationary exponential assumption.) Under this genera1 condition simple equations (as presented in the Appendix) that can be estimated seems not to be available. Thus, sensitivity analysis seems to be the only method that can be used to evaluate the model. This specific issue character- ized the phenomenon of a R and D project with knowledge that accumulates in the course of the project without the ability of being measured and/or observed.

Finally, the numerical example, described in this section, illustrates the usefulness of the model for practical decision making regarding allocation of resources for different R and D projects. From this example we learn that even with relatively little info~ation available (a common situation), using a sensitivity analysis technique can reveal a lot about the optimal annual spending on an R and D project aimed at cost reduction. Such information can im- prove the decision of where and how to invest the money provided by the government for R and D projects.

REFERENCES

1. R. E. Lucas. Optimal management of a research and development project. Mgmt Sci., 1, 679-697 (1971).

2. M. I. Kamien and N. L. Schwartz Expenditure patterns for risky R and D projects. f. appl. Probability 8(l), 60-72 (1971).

3. C. Aldrich and T. E. Morton. Optimal funding paths for a class of R and D projects. Mgmt Sci. 21(S), 491-500 (1975).

4. A. Mehrez. Development and marketing strategies for a class of R and D projects, with time independent stochastic returns. R.A.Z.R.O. 17(l), l--13 (1983).

5. W. D. Nordhause. Invention, Growth and Welfare: A Theoretical Treatment of Teehnolagicai Change. MIT Press, Cambridge, Mass. (1969).

6. C. Bezold. Pharmaceutical in the Year 2000: The Chang- ing Context fir Drug R & D. institute for Alternative Future, Alexandria, Va. (1983).

7. A. M. Nicholi. The Harvard Guide to Modern P.ychiafry. The Belknap Press of Harvard University Press, Cambridge, Mass. (1978).

8.

9.

10.

J. S. G. Cox and D. E. E. Loveday. Creativity and organization in pharmaceutical R. & D. Res. Dev. Mgmt. 8(3), 165-175 (1978). S. N. Wiggins. The Pharmaceutical Research and De- velopment Decision Process. In Drugs and Health. (Edited by R. Helm), American Enterprise Institute, Washington D.C. (I98 1). Special Issue on The Management of Research Groups. Res. Dev. 9 (1979).

APPENDIX

In this appendix we describe a 2 stage procedure for consistent estimators, based on researcher’s believes as a source of data, for the stationary exponential case.

Estimation procedure, ~1, A and b

Stage I: estimation of a and bA. The estimation of a and bA is based on the assumption that the probability to complete the project within a small enough interval of time At, given that it was not completed at the beginning of the interval is approximately bg(m)At. This assumption was employed by Aldrich and Morton [3] to build a recursive model from which our problem (1) was derived for the case of infinite horizon. Under our assumption that g(m) = Am” the probability is b. Am&At. We can now use the relation

Z,(m,) = bAm;Atu,V i = I,. , n, (I)

where n,-the number of observations. Alt-a constant, small enough, period of time. Z-the probability (a subjective estimation of the evalu-

ator) to complete the project in the interval of At for a given level of expenditures m,.

u;--independent measurement errors. The evaluator is presented with different hypothetical values of m and is asked to estimate Z(m). The log transformation of eqn (I) generates consistent (possible least squares) estimators of bA and a.

Stage 2: esfimating b. We use the estimates for_ a (2) and ~~ (a) to estimate the folIowing function embicmpr. The estimation of b is obtained by using the relation

s

a, K(m,) = b te-hA”tl dl + a,, i = 1, _, nz (11)

0 where

a,--the number of observations. E,-as defined by eqn (6). m*--as defined in stage I. a,-independent measurement errors. Using the n2 observations we can estimate b from the

linear relation described in eqn (II). If the null hypothesis that the intersect is equal to zero is not rejected, then we obtain an estimate of b (b). Having an estimate of b we can calculate, from stage (1) and (2), an estimate of A.