A note on the ana ys s of the expec ,,d value perfect mformauon w th respect to a cla,, of' R & D projects

27?

A b r a h a m M E H R E Z Department of Industrial Engineering and Manag~,"nen¢ Ben-Gurion University ,,,[ ¢he Negev, Beer ST~eua, Israel

Ak, s~aet. This note is concerned with the question if and when to carry ou~ marketing op:~ra~ions tha~ a:~. aimed at completely reducing marketing uncertainties surrounding the value of a stationary R&. D prr)ject at its technical completion. It is shown that the benKits arising from these operations can be measured via the EVPI (the expected value of perfect L,-fformaticn). In addition, it is observed that the timing of these operations should only be considered at the begimfing of the project's life. Finally, a sensitivity analysis with respect to the statistical properties of the EVPI is performed.

Keywor~: Research & development, marketing

!ntre4uefi~n

A major factor in determining the t~me patterns of sper'.d!ng for an R&D pr%cct is its vah~e a! completion. Given the actions of competitors and changes in consume,,- t;:ste; and pre.fereaccs, this value is, however, often uncertain to the planner during the time that th:~ project is in progress. "rb,'..s, a pianre:" has to establish both a marketing policy directed at reducing the unc¢rtaintie~: surrounding the prq,~.~f~ ,za!ue at compieuon and a criterion for establishing a development sp,'r.ding (~olicy. The empiricai wo4.:~; of Mansfield and Wagner [61 and others have shown that the expenditures on thes: ~ poiicir:s arid their ~[., ~i.~g m i ~ t be critical to the projecfs economic success.

The purpose of this note is to study the characteristics of these policies it,, the framework of the t.uza~;, [3] model. In the next secuon the Lucas model is reforrautated for the stationary case.

The Fae~el

The basic features of t~s model have appeared in Kamien and Schwartz [2], Aidich and .~,,!crb~n [i[, and Mek,'ez [7,8]. These authors have treated the time patterns of p:;encing on tecb-dcai info-matio~, for a class of R&D projects with time independent returns. The factors de~.¢r~%nir,g "d-,e dynamic~ of .,.p~.:ndir:g for this class of projects are the following: The profitabi}ity of a ccro_,pieted project, the probabiiity ',,unction of technical success, the opportunity cost of a proj¢.ct, and ~he function ~etating the ra:e of do?iar .,..~e.~:,c-i~g

- " i "- t . . . . . . . , . ! ' £ 1 e (?£ ,~fla~,!~=:;¢ [11 : 3 : I i , : ! ' [ c , n b~;ov~l~r~[ :~¢~; . . . . , . . . . : - : . . devc .~ i . c~ t} "i(; ~.D.,?, ~ - ; [ o j c c ~ ,

NordM~oliL, nd • o ~ n ~ l of O~era~iona'., P.e.~;earch 19 ( t9~5) 217--22"!

03?7-2..17/8~/$.~.-~0 © 19B5. Elsevie.r S,-ience Publ ' : f~crs B.V, (~ ,or..h-~ .~ ,an ,~

2i8 A. ~i~6rez / The ana~sis of the exp.'ted cal~ of perfec~

W h e n ~he prcf i ta~iI i ty of a mmpie t ed prc~i ::ct is unknown to the p lanner dur ing the t ime ~he pcoject is in Frogress a.~d when m a : k e 6 n g op,:rations wlaei'~ comple te ly reduce these uncer ta int ies :,re available, the p lanner must firs~ decide if a~d when to carry out these operat ions , and secoadly , to establish a de , ,e topment s endin~ pol icy for the per iod before and after these o~erations.

In the frannewcrk of the so.cal~ed Lucas [i.] model, :his p r o b l e m can be formuia ted as fallows:

"-'"-U(,(t))l - - m U i r , . , . ,

s.t. m(:)>_-0, O<r<o , g(0)=0, : ' - ~ (1)

g ' ( O ) < ~ , g ' ( m ) > O , g : ' ( m ) < O f o r , n ~ O ,

= 4 0 ) = 0 ,

F(z) = I - e -~' ,

where m(, ' ) , a contro~ variable, is floe rate of dollar spen0ing on the project at t Vt ~ T to increase the technical knowledge devoted to the project . T is a d ~ i s i o n var iable that de te rmines the time in which the m a r k e t i n g , i teration investments are under taken, C are the t ime- independent market ing costs, r is the d i scount ra~e, z ( t ) is the eut:~uiative effo,:t d e v o t ~ to the projec t by time t, F ( z ) is *.he exponentia! dis t r ibut ion f-ruction that vhe iwoject will be .mccessfully comple ted by the time the cumula*ive effort is z, g ( m ( , ) ) is .i r%nction that rda te s effor{ to cu.~ent spending rate n.,(t), sach that z ' ( . , . ) = g ( m ( t ) ) and u = Max{ E V0r)) ,0} where V ( e ) is ,he expe:tecJ d iscounted value ,of ~.i,e project given a realization of the . ,andom vat able rr, the reward to be collectec upon successfal !echni.cal ,-ompletion to infh,..ity, and E(~r) is

• -~ '- -v,-- . . . . ; ~,,a,..~ ~,: am/. gwen time. ..~-,",,,,,~,,,~.,~-,,,,,, ~Idrich and Mo:van [i], we observe *~at.... u is i n d e p e n d e m or :. The c ,vmponems of P in (1'.

~, .e divided a o ~v:c; p,~rt.,.: , - r - . . . . ~ ~ . ~ . . 1. h,." ex. ec~ra discoum..~d cash flow a : t r ibutabte to the '~ :' '-' , - , ~ protect up to t ime T. where at t ime t ~ T.

the :rqject ,!.eaerates an average income of ,!i'(~') if {:he projec t has been comple ted by that .,:lute, oc of - m i ~ if : ;~.ag. r~o{. Mulup!vmg. tl'.,e~ . . . . . tw:) cash 3.o.a- ,oossibilities at each tit1-~ ~- by "heir r..4¢ ~.o,i~,.,....,..,~. 0roi,abilitie: "--~ ,,,- ~; o bac~ to zero at rat.'. ,', a::xl " -'" '""~ :--- " • ~ c . . n . _ n = ' . mteg, ,m,~o from time z 0 up_ to 7', ..vieids ih.,:: part .

') [ n e se :(rod ~.,a.r~ ' " ' . . . . . "-" "~ .... ~,t_,ve:, into considera t ion with b o b a o ~ h : y of (! - F ( z ( T ) ) the d i s c o u , t e d dilL-rot:co ' " ,m., uncertaint ies are ~ompieteiy reduced, and '" ti:e be tween e~, u e projecFs expected valu. • g 'ven '~"

n-;arketing ,c:~ts. in (1) we in t roduced i~ :he Lucc:~ model tee auestio:a wb.en to carry out mar¢ct ing research oF:r~.fions by def imng th- decismn variable T. Actual ly , we deal with fi~e c ~ e o!" perfect

. . w~m the case of R & D with an informadot : 'v;b~;re market ing o~era t ion: :evev:e unce~ min<~ ~mmpletely and "" . .x~7onentiat effort d i s t {bu t ion function. *'-" . . . . . " " ~' ~.>.~ t5] for emFmC,,, evider~ce f.er using thi:; functmn.)

. - ' k lm .~ys i~ of the sela~i(m

[ 'he structure of ( ! ) is an optima! control p rob iem with a fi,_~dte hoi~o~'. This formuiat/oi~ is si:,-dlar to ~ilc optima! controi pr.)b!em w,th a finite horizon defined by Kamien a ~ ,.~ch~a~ .... [2, p. 64!. Follow'r ig ~'.~.,,:'a~-r and Schw?.rt~: f ~... ,,,:,a~ ?~lehrez [5!. -v,e .,r ...... ~. tha~ for a gh.eu 7".~ ~-~ ue,:cs.~;ary co~diiJon fv~.r .~.~..q.~) ..'o b~ ,,t:mn a] fo~ all (} .-:q ,'~. T is ~bat m:.'~ ) m.~,xi:nizes the Hami l ton ian ~.. ~iven by

- . , . , J .~ i ) ' , ;:J ~: C ",';7.. i L ' - ' ~ : '! - - ~)~ ~2: .... , , . ~ . . ; ~ ~ , ~ , /

a ; ~ d

~,.;/,.'~,1~-;*. = - - e - " e - ' ~ : + : ' ~ : ( m ~ ~ i , ' ' ' • " - s . , \ D /

A, Mehrez / The analysis of the expected ralue of perfect 219

where the multiplier k obeys the differential equa t ion

= - /az ( : i )

and the argument ~" has been suppressed for notationa:,, ease• Fur the rmore , for a non-nu]! spend%g func.'.iop.. po l icy m > 0, (2)- (5) imply that

- g ' ( . , ) . , - , , ' / g ' ( . , ) = ,. + m )

where

= + m)g'(,,,,). (6) (6) implies that m* the opt imal policy is i ndependen t of z and thus by the posit ivi ty c,f m*, this ft:nction }s

,¢ cons tan t over time. Inser t ing the cons tan t p rope r ty of r e ( t ) into ~1) for a aon,-nu!l pc!icy, r e sub , m

Max - e - " E ( ~ r ) ( ! - e - -- ra* e: , .s, , , ] d t =, e e ..... ~-'. ,.." ;' ," ' O ~ T ~ m v 0

where m* is the opt imal pol icy cor responding to E~ ~ ). By taking the der ivat ive of P with respe~;t to T, ,,w. can observe that if the solut ion of P is a finite T, ~her~

this solut ion is at T = 0. T'hus, at T = 0. one cf the :;wo decisions are to be ,..,n..:ae,~ ' a" " ,ed.'" 1. First, observe :r, and then follow a spending Folicy wi~ich is opdma i wi:h respect t c a ; ¢ . , , e n

realization. 2. D o not purchase _perfect information, o r folio,,, a policy w}fich is opt imal with re.~pe,-~~ . . . . . ~,') £ ( ~ ) . The first decision leads to the 'wai t anti see.' p~ob!em def ined by Mada~-,si,'y [4]. The second decisi,on

_ d,.,,n.~a by Watkup and Wets [ t 0 f The dif ference between these ~wo leads to the Recourse ,~rob!era , ,~; o " p rob lems is given by EVPI, where

= e ( v ( - "/(, ) . . ' . . , , ;

' ~ is easy to observe that th}t .r,~,-,. ~.,. i s , ,-, . . . . . t ~ , . - . . . . " ~ , , ~;,~; omc;tu-~ ,~ l._~.atl.,.,_ L O t . , , e ~ d . d . . i:urthe:::m,::.:., i~. h, at, upF.?r ~?oucd e ... . , } 1 . . . . . . " : 1 ° m o n e y a ~ :.annet ,e,:ill be w.~;ung io pay t<) p:;:c!ha5.:: .,,,:J;r; . . . . . , ,r,,..o ..... _,i'~c"'('~"i;('(.,..,, ..:.,..,,,,

A~ ,:!ius~'agve examlg.e

In Table 1, the resu!ts of an ar, at).sia measwi<'.g the EVPi were ~;u,.m~:~.::zr::.!..or ~_~f : ; ; . ; = r ' : " . < "" ~. -

,[7," • ~. . .~,~¢r)=0 a~,d thus E V P [ = E ( , / ( v ) ) . r}~,e re'subs in th,.s " "' ar~ ;~ -' ~ .... ; ' ' - S [ ~ . ! [ t . y . ~ . : b i 3 ~ r ~ . , . a : ' 0 f i s h r c : : , ? - ) : ~ c ; . : " ' : J : ~ . t / l~ t ; foy . , ! "* l

and normal dis~ributioa functions, h i.s ,:iemonsi.ratcd that the EVPf [.: :v:m.q:,'.iv-:. to -.5~,~g,:~ ;;~ ,,. c. e .?.<..d ~ e is the s tandard deviat ion of lhe d is t : ibut ion of ',r... b is th : coeffi.,.:h"~:t o:," b .and .::. ~.t i,; :Jc:nop.'~,:c.t{.'<i ::[-~a! the gain by purchasing perD, ct in format icn is F~rger whey th : di;tribufi,:m is j.:;i..<~<, th:.~z: , v , ' - e : n <v.: d is t r ibut ion is -r~ormat. for m. U set of parameters , This diffcr:~r-ce mc,.',:a.'.e~: as : incre>:.e:: ~;P u, ?(.,~ k:mi!ar results were repor ted by Mehrez and S: 'dm.an [9] for a stat;c project :~elec,.:.):,: ;><,l?~,u~ ,:.,.;:.::i-.z :ha:: d is t r ibut ions v;ith largax" s taadard measure of ku:io.~ds (f.c. longer tail<; ha:,'~:' a :;::w!icr }:'~' [~!.

. . . . . . . . . . . . , '~ ~;-~i~; ~t;~. 4,, ~ '~ 'D: , , , l : : .c ,gJccv: , ~ q . i t m ' , 4 , 'c!~:c.:-~cq f ! ~ o u ! d <..,,c~.~,~.: t.:? ;i~ !;.~{:..i-- "J;:t:~._-, ")~, ~-.: . , ;-~ '~'"~_< . . . . . . . . . . . . . . . ;. - ., . . . . . ; "

."}.10\'~,'II ~TV *L~L: . I3cZ ~. t~, "U.TO '/" " / I t ' " ~12.: & [i..~K.::t.:::l: .W~(:;~]i d iF , ~ . . . . . . . . . . . . . . I~'~::~[ '<: Cq' 2 ..'[i~t.,;u,.r,:- :n; :~, :c.jL~;. ;=iS o , ~ ' , : ! i ' ( d~ . ; :q :

C,!5 ' ,2,L}.ai ! t~-: V L : i J O C'..' "~ " " '~ ' - '<"* ",v I~r; i t," : 2 i~ ' i b2 l - , ! t . . i : :CE~: . - ;~" , - ~ : ; ' , ,S Ltd':.~I-:L: "..(;' [ } ! : , . : -2,. .?h::

t lh " .~',-rp~.d..@,~.nL_. 0]" ,J. ' ' ' f . . . . . . . . . . . ",,.*v~,.,',.t ' ' ~ . . . . . . . . . ~.S . :)~V.. ' ! HJ : . :~ ,,,,.'-'~;.,a ,:~ ('V!;,f.'4! ,7.';t [;'~.;,~.. TIt]CT,:T~'tI"S.L'~iqw~, i~: -b . - ; ' ( ' ;U ' ; .b. : : [ t; ' , , ~-" \ ' y ] i ' , ; , ,,

220 A. Mehrez / The analysis ,%; ~he expected ~.'a:,',e ~] perfect

Table t Resal',s of ,2~e EVPI

r b ~x o EVPI: uMform EVPI: non~,ai Ratio of dis,.ribution d;a,ribution rue E v P I

0.C6 0.333 0.250 0.500 1.07063t 0.8939604 1.139~50 C.06 0.250 0.250 0,500 0.98i47~ 0.861639 1 129072 ft.06 0.200 0.250 0.500 0.902596 0.792654 1.138702

O.f,~5 8.333 0.56"0 0.500 0.999868 "3.87845 ~ 1.! 38218 ,7. 136 0.250 0.500 0.5f,~ 0.~0~559 0.7g2233 1.1372n5 f )6 0.2~3 0.5130 &500 0.792396 0.6974,~ 6 i . i 36 t89

~.v ~"¢&" } 0.8922,2"5 1.137!72 L:.'>5 0.333 0.750 G.5@0 ~ ' v " "~ 6.06 0.250 0.750 0 5-'.~ 0.°9-4263 0.78761 t 1.i354 t2 ~1.06 0.200 (1.750 o.:~(~ 0382¢31 3.,i9¢}3,¢4 ~.~ 43393

6.10 2.222 C.250 0.500 0.541643 0.475147 ! . ! 3 8 6 8 7

O. 10 O. 250 0.250 0.500 0.475201 0.417526 1.138136

9.10 0.2~0 0.250 0.5-~ 0.421190 0.370234 1.137631

0.10 0.333 0.50.) O.N rO 0,475266 0.418299 1.', 36187 O. 10 0.25'~ 0.500 0.5(1'0 0,393821 0.34713i 1. i 34501 0.10 0.20') 0.500 0.51~ 0.328321 0,289817 0.132856

0.10 0.333 0.750 0.503 0.469115 0.413908 1.13338!

0.!0 0,250 0.750 0.5~0 0.370009 0.327624 1.129371 0.10 0.200 0.75C, 0.500 0.286106 0.254457 1.124378

0.06 0.333 0.250 1.000 2.588521 2.195200 !.179173 0.06 0.253 0.250 1.000 2.372361 2.01759i 1.175838 0.06 0.200 0.250 1.000 2.181143 1.860679 1.172607

,3.06 0.33 ', 6.500 1 .ff,)O 2.421686 2 0%503 1. i 6.~ 757 0.06 0.250 9.56vd 1.0130 2.154257 i .870488 i .i 51709 0.(}6 0.26(. 0.5 PA,' 1.0SO 1.918699 1.6803% 1. ~ 41 g 19

0.06 6333 0.750 1.0"00 2.459143 2.139399 i. ', 49 ~55 0.06 0.250 0.750 t .000 2.167235 ] .914:42 I..~ 3)0~0 0:36 0.200 0.750 !.00(; 1.8961 ! ! 1.704 i 66 1.112633

0.10 0.333 0.250 l.gO 1307439 l. I 15125 ! 172459 0.10 0.250 0.250 i.Ogo 1 147813 6.982~35 !.167859 O. I r,~ 0.200 0 2.50 1.000 1,016870 £.8737q9 1.16381 -~.

0.10 0.333 0 506 t.C4XI I ~ a g n ~ !.@3,,~8- i .1 ~.1801 0.10 0.150 0,500.. 1.00t3 0.953350 0.84,5833 i,125782 0.10 0.200 0 50f i .000 0794555 L.715,~05 ~.110636

0.10 0 333 0.75(:, 1.000 t.!36835 1.02t857 ' 112519 O, 10 9.250 0.750 1.003 0,896565 0.833457 12575719 0.10 o.200 0.750 t .00/2 0.693153 C.671' (16 1.032~52

0.0. ~, 0 333 0250 1.500 3.608366 3.255210 1.108489 :-' % B.25£ 0.250 ! " " " ' " ~ 9~8,695 5~A 3,a~,,974 _. 1,105138

. . . . . . . . ~ " 7 -''~ ...... I.{019!5 - , ,, 0 2{2ii P, 2q" i ~..& o.0,.a,., ; }

9.06 0.?; 3 6 .~Ot ! 500 3.~t67.45 3,12-~.549 !.09! 574 0.06 0 2; 5¢ 0.5(~ :..5~0 3.05q219 2.~ 2T;. L ~ ~. i .g; ~X;{} {2.2.- 0.20q 0.509 ! .5~20 2.74274 ~ 2.557660 ] .97236"-;

O.¢;i f! 23 U. 750 ? .500 i:,,'~98.L2.,". 3,2373<':; 1.0{,;0579 {) '36 B.'.<~ Q 759 ! 5C0 3. ! 19599 2.93& ~ ~4;~ t £644')'9 .'.".C 6 {;.2CO 0.v�.) t .500 2.7~Ob~b5 2.6 ~'20t0 ~ A~41;872

.4 . M e k r e z / T h e amd)u ' i . , - o f &c e x r ~ e c t e . d ~;cdue o f e.erfect ~';"

Table l (:;zi:td.,,..~ed)

r b , : : o EVPI: unifor,-t EVPI: notre.a| R a t i o of

dis~p:ba~ib~ d';-s:nbution the EVPt

9, 10 0.333 ~L250 i .5,30 1.g.~3261 1.661192 1. I 0t 776 0 1[;- 0.25~ 0.250 !.5{)0 1 . 6 1 t 4 ~ L459729 1,~,~7166 0.10 0.200 0,250 i .5(~ t .43/_2 ~2 1.3092<7' 1.093157

0 i 0 0.333 0.500 1.500 i .645070 1.534072 i .072355 O.lO ~.250 0.500 1.500 i.377647 1.303247 1.05708g 0.10 0.2,9,0 0.500 !. 560 t .160183 i. t ! 261 ~ i .042758

0 . i 0 0.333 0.750 t.5(~'1 t .658426 !.58432g l X..-V 769 0 1 0 0.250 0.750 i .Sf~) ~ ."~42535 1 •323603 1.0143o3 O.10 0,200 0.750 t. "2L.~) ! . ig2631 }. '~ 3622.Z i .{;':)262~

0.06 0.333 0 250 2.0f~:.; 4.822596 4 ! 77537 I ~, 5 &~: 0.06 0.250 0.250 2.f~JO 4.44333g ].835356 ~., :, 25.t :'.

0.06 0.200 0.250 2.000 4.106529 2 568498 ! .~ 5~802

0.06 0.333 0.500 2.C~30 4,6472g8 &053415 i. i 465 ~ 2 0.06 0.250 0.500 2.0C0 4.202664 3.681454. 1.1415"/7 0.06 0.2~} 0.50,0 2.(t-90 3.804262 3.3467;67 ,'..136629

0.06 0.333 0.750 2.000 4.817278 4.2! 5590 i. 142729 0.06 C.250 0.750 2.0t09 4.368801 3.847880 t. 135379 0.06 0,2t30 0.750 2.0,"',0 3.946491 3.5 f3(.~3945 i. t 27297

0.10 0.333 0.250 2.0(30 2.462166 2,139633 ] .150742 0.10 0.250 0.250 2.094.1 2.177425 !.896490 1.140t -40 0 . !0 0.200 0.250 2.'.R;~O ~..941392 i .6942" O i. ~ 4. ~ 898

0.I0 0.333 0.:" O0 2/X;O 2.281798 2 007498 ). l 3( 63e 0 1 0 0.250 0ZC.0 2.000 1.94059C ! 719594 i. 129520 O.lO 0.200 t .500 2 .~C.',) 1.65g.076 1 ..:7F~5 ~ 9 } ! 2<A3.~

0 . t 0 0.333 0.750 2.000 2.366610 2.09':)z! 3 :. ~ 2725!. ~ 0 10 6.250 0.750 2,~30 !.98365~ 1.7~357>, 1 ; !2J ?5 0 10 0.2(10 t3.';50 2.000 } .647306 i .5045a4 ~ ( , . ; ~ ~: t :~

a genera l m e a s u r e ~o evah.~a;e ~be ,~ai~,- f r o m p u r c h a s i n g : ~ ," ",'~ " " ~ - : . . . . - " ~, .... m , . o . m ~ a o n . A!.:,:;rrn, t~e.P< a sL,~ ~ ii>,-,c,7~f)r~ v:- t lm t o f ( ! ) c an i)e u s e d to ana!y:,.¢ the zirn.ing a n d ~he ga in f r o m p u r c h a s i n g i~~f:~rmati,_m

[!] At.drich. C.. and Mo:',,;-n. T.E.. " O p t i m a i fuw3ir, g - ~ " ° :.~f ~ ' . "' ~ , . . . . p.a**L., ~.o~- a class n~i~:,, ~-t&j_.) I] ,7Oig, . '{ ", • ,~:[/.2n/l~le~-~,~t!~ .~cQ.-}P.{ ) 2 { 5 ; : (11;.} '75:

49! --<aO. • ~1 K a m i m , M.i. , and Schwer~'2, N.L.. ~'E~menditut't p,.r.te, n~, L,,P ri:.!cv R,3.'Z..* ()ro),:c.:,_.". , ; ov : rm> r~, .~Av.". ied .Prch._q.bihtt B f~ '7 ] ~

60-72.

[2, I Lu'.:'ae, R . E , ,)p,n:ual managemen t o.~' :-. re-'~:a.~,.:.q a~,,~ ............. . ...... : pr' .:jev<. (~4a,v..':g,-~-e,lt V , , e n c ~ . 7(}}. t~>,,r":'~'~: 679-Ur, ' . ~4] M.:'..da,lskv, &, " inequa l i t i e s fc, r stochastic ~ : ,-.",r p rv}5-~a~ i ,g m'ob~em" v ' , v .., ....... , r;."s"v, :" (~ ;,:9(,q" i~o~; - -:,r:~ [5] Mar..~iMd, E . "&chnoh:~:icc;!. . ,..,.a&."-" . . . . . . , Norto,-.. New ¥c:'~,:.. 1970

~6I Marxsfie!d, E.. aad V{agn~:r. S., Orgamzatic,>:3 a~u s:.'aae:~:;2 g~cw,':s ; :s:x:} u.ed w}i? ~?,r,.")a>'-_h~:-=::. ,2:5u::-'.e;5 ~i ~, ; , , ~L.. :a, . . . . . . . YT,,:" ./r~,..:.,:-a! v / }~ i . ' ; :e~a" . ; ";cv e, Gr>-.b,<-- '"oh:> .t ,"f gv. ~ c% of ~:.~,e i J niv,>-,,,y ,,~ CI,~c:.../.:-; a~,{ ..:, i:. 9"L5' '; 7 :~ :U' .

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