.ASPECTS OF RISI PHOGH.AMMING .,

Mac Eason Rein \\I

Thais submitted to tbe Graduate Faculty of the

Virginia Polytechnic Institute

in candidacy tor the degree of

MASTER OF SCIENCE

IN

STATISTICS

=; - > < - -v

Director of Graduate Studies u u z- s=-·w· -..r(j<'M

Head of Depa.r\llent

Science and Bu.aineas .A.dm1 ni.stration

May 19513

Blacks bllr8• Virginia

,,

TABLE OF CONTENTS

Page

I Introduction 2 l.l Linear Progra••lng 2

1.1.1 Buie Concepts and .Asa~tiona 2 l.1.2 Mathematical 1'ol"lllll.atione 3 l.l.3 The Solution of the Maximization Probl.. 6 1.1.4 The niaJ. Concept and Price Imputation 8

1.2 Non L1near Progra-dng 9 1.2 .1 Oeneral 1'heor,r 9 1.2.2 Sol'Ying the Qnadratic Programing Probl• 10

1.) R1ak Prognmd ng lJ l.J.) Description of a R1ak Situation 13 l.J.2 The Choice aaong ilternailn R1a1q Ootcomee 14 l.J.) Method of Solution l.6

n An Exaat>la 2.1 TM Deacription of the F1ra and the No-RiH Program 18

2.1.l The Deacription Of the Firm 18 2.1.2 The Technology Matrix 26 2.1 • .) The Optinmm Ho-Risk Program 30

2.2 '!'ho Risk Program 31 2.2.l The Ducription of a Riak Situationt The Conpi-

tation of Var.iancee of Unit Level Net ReYetmea .31 2.2.2 .l Computation of the Risk Prograa .3S 2 .2 .) The Opt.ilmm Risk Program 38

Ill Fo.rther .lapectc of Riek Programming 40 J.l The •Risk-Aversion" ~ 40 J.2 The Price Map 46

3.2.1 Price Maps for a No-Riak Prograa 46 3.2.2 Price Map for a Risk Program 48

3.3 Resource Mapa 48 J.4 Variance Mapa Sl

IV Summary S2

v Jppendix S4 VI .Acknowledgement.a S7

VII Ribliograptv SB

VIII Vita 60

LIST OF TABLES

Page

2-1 Buie Bu.dget for One Acre of Fall Oats 19 2-2 Buio Budget for One Acre of Wheat 20 2 .. 3 Buie BLldget for One Acre of BarlAly 21 2-Ji Buie Budget for One Acre 0£ Corn 22 2-5 Buie Bndget. for one Acre of Iriah Potatoea 23 2-6 Buie &dget tor One Acre of Tomato. 24 2-7 Buie Budget .for One .A.ere of Alialta 25 2-8 5cal'ce Reeourc.. Jleeded and lfet Revam.e for One

Acre el Wall O&ta 26 2-9 Scarce Reeource Yector, ,!, for Proceaa l - 1all Oau 27 2-10 The 'J:eclmoloQ llatr.1.z .for the Prograw1ng Probl.a 29 2-ll The Optimll lo-Rillk Prograa JO 2-12 Eetimati.on of aro.a Bneme frc:a One Acre of oat. 33 2-13 The 18timat.ed Per Acre Adjusted Rnemiee 34 2-14 Per Acre Variances 34 2-lS Unit Lsftl Var.lancu 34 2-16 c~ the R1H and the No-Riek Progr• 39 .3-1 Optima Prognme tor Var:l.ows R1U: J.Teraion Constant.a 1a4 3-2 Optima Riak Program for Yariooa TOll&to Price• b9

LIST 01' FIGURm

).1 Indifference Map ).2 Indifference Kap for a • ~ 3 .3 Opportuni t)r CurYe 3.4 Price Hap, Jfo-Riak Program 3.5 Price Jfap, Rislt Progra

.. Page

2

CHAPTER I

1.1 IJllUR noaumJIG

l.J..1 JUIC CO&llfil .AID .tmaPTIGm

•Pncr 1'111 91' pftCJW pl•mdnc1 ..,. 'be dlf1md. U the eoD-

8\l'UUan ot tJt.e .-.i. et ..U.. 1'7-- ot Dlcll 1111 eal&llij',

~ • 9'Mr a 5iJa et ae\:l:dU.. _,.,.. Ira w d9t1md

atate to wtber, • tfta a clllftm4 nate ts iD'd ... ~

d9t1Md. ob.jeftiftll (JS). 11 .. w 11111 -- ... --.zmcl 111\h \be 11..,.

png:r '• ot a f1:na _.... ...,.UU.. 1' 11111 lte et ~ to ftnt.

M._ ._ _..,-. ml W1lllpUem ,....,,..,. to Umar pngrawhag

~.

The \llNe buie oeaaepla (6) '9 be w14eN4 ae ~,

pr1illn., ad~ pl"OIH-.......... an all phJ'wioal ad

in'm«'hle w,. .... -, \lie nna. Pnmotia, or _.., ..an ot .:11

ot ~ i.Uta ot tM pncllc\1.w ett.n ot t.11e n.-. IMll ......_ ud

.__. ___ , dS:Y.S8'h'l.e, llld ~ bT

• ~ a:S.i td ••ma . .&. ~ prn1•1 1a a php1aa1 ...t

or Hl'ie9 fill...- ldd.oh "9 ftn oaada.9'11 in .-dlrr \o ~-- n-.-zw imo prodlafta. Jn a pftda.oUa Pftl Ill \be GDq YU'iaUOD

all ... 1a a ftZ'1a'UAa ., ...-an ~. _,, .. t.a.. 1den81'7 ot the

pl'MSll1 111d.oJl ia _..ill -- ef \he 11111\ l9ft1 ef \)le pndu\1m

pna11111bioll1a ill \'an Mfhwd \o be - imtllMe of ta. pre•- (one

am-. et a cnp, .-.) • ~ u oamidtlr ,. pnpud:1m of a ••dbe&l for

3

corm •Buh add9d Ul"8 pl.owed • • • nqa1ree ma equ.l iDplt and oaa be

cGIDaS.d8nd. \e add an equal ~ to outpi\. Con. plmt.1Dc. OGl'll

cal'1.U.. ad m ~ 111ftl:n ,,.,,._. Nl&U..ldpa be.._ :lnpa' ........ , 9ftl" all tieclmSoaJ 11111\s ••• • (lJ). Tlma. pN-

dla.ua pnn•• 1a a .,r1aHMd md ~ tom ot a uDMI' lw-

1sa ... ~ fw\!m..

A .t1na - '- tlllftm4 "8 \Ila ~ et pn1•a11 m4 br \laa .. ~ ., ~ na11abJ.e to 1\. !'be ...,.u. of \b9 tinl

m lie ..,tJml • \119 ldml.WMtllllll .,.a\iea el - li,•R O"h'•ticm

et 11111\ J....i. .t ~ ~ pztodlloU.m pi 11•121. !1w ~

.tamt.ica l• \u ft.JS m be~ Dr 1111D1H:bic Gl. pauihle

llnev elllbt.DaUw ot Ull\ laft1a ol pne•••·

Tb9I' .....

... _, .., .t.ate tM 1-d.e ......,U... or Um- procnm1Dg. ~

l.. !119 ~- 9jili*''6id.U. of - MWI J _. MODMlO

md.\ _.. 9tJ.Md Dr "1lie NS.a'- D4 ~ predillaU."le

pl'HHI• ....,lahJ.a '8 1\. 'fha ..-U.tiaa ~f at J.wt

.- ot ti. wn. ae ftld.M ad M S. t.be ""._.

ot ~.,. pna1 • anil•hle.

2. _, pre..IHtr..1.w PN•••• _, i. _.. a ..,. paa.u.w i...i ~ ld.\ll u. nppq flt ~ aft'llal>l9. The

·- •'19 et ----- md ~ aa.tpd of pndact9 1a

pnperU-.1 tie '119 i..l .. 1lld.ell • pr•.... 18 --·

l· an.al ~- Pfti••••• .., i. ..... slml.Mzasuq.

it t.119 ..,,q of l9ClaJW 18 ...... -. It 'hia 1a ....

\ba ......,U. of each ~ u \be ma ot the

4 conaumptiom ot the indirld.ual process• used, and the out-

put ot products 1a the 8Ull of the outputs ot the J.ndirldual

prGCMS••·

"Within this tnmllvork, the productiTe problea became8 the

probla ot ehooaing 1lb1ch productive processes to WM and the leTe1 at

which to me each ot tbea. It will be ue.tul to .tcmmlate thi.8 problea

algebraicallT' (6). 1.1.2 MlTHJlfATICAL FOHMULlTIOB

We 'MT now ~ebraically tormulate the liwr programing pro.

bl.a ot a ccapetitiw tin in which proouaes ~Ol"ll ruovcu into

tinal produck.

Let u consider a firm with n prooea•s where each process 1a

\Ud.queq detined b7' the nsoarcu needed and the product.a produced b7' the operation at the llld:t; leftl. Ve 1lill di Tide the ruourcu into two

groupst scarce resouroee, thoee that are in l.iJlited euppl;r, and non-

scarce re•ourca, thaee that can be bought without lbd.t in the open

market. Denote the aoante ot scarce resource• neceeaar;r tar the opera-

tion of the h-th proce88 at the unit leTel 'bJ' the TeCtor

t• - (t t ••• t ), ~ lh 2h 1lh

the uaotmts ot non-scarce ruolll'CM needed by the ftetor

<.:g>• • Ct.ta t~h· •• t:<>ii>' and the mnount ot the products produced by the TeCtor

!h • (~ 112h···1\ii>•

The subscripts a, mO, and le denote respecti....q the nllllher ot •carce

involTed in the whole production program of t.he tint. Thus some Teet.ors

corresponding to ~ proceaaes will haft sero elelleD.ta 1.f 'the7 do not

ilm>l'ft all resourcu and product.a available to t.he .nr.. In thia wanner

arrr procua can be eoapletelJ' deecribed by the Teetor

Ct;.t:'!h) The whole firll ia deacribed by the collection ot the Teeton

corrupand1 ng te all aftilable proceaea. The collection ot the TeCton

can be COllb1ned into a •trix which d•cribea the entire tin. Thia

matrix, which 1a called the technology matrix,, COiiea in t.hree part9 r

T • (~~··•.;>

which indicates the scarce resources needed for all purpoees,,

T° - (!~~ ..• ;),

which indicates the non-ecarce reaources needed for all. procenea, and

U • (u u ••• u ), -1-2 '"'h

vhich indicates the products produced by all procuaea. The complete tech-

nology u.t.r:lx is 1-hen (T ) (!°) (U ).

Let ua denote by Xti the mmber or unit 1eftla at which the h-th

proceu ia operated. Thia ia the intenaity ot the proce•. It tolloa that

the nctor

!' • (~~···~>

detines the production program of the tirll. ill of the elements of tm ! vector must be non-negat.in.

Since the firm is aaaumed to be operating under c<mpetition, all

6

price• ot reaourou and products are gi:nm. We em thu obtain the net

reftllUe due to the operation ot UT proceas at the unit le'ftl. U we

denote b,y p the prioe ot the 1-th acaroe resource, b,y po the price ot 1 1

the 1-th non-ecarce l'98ource and b,y qj the price ot the j-th product,

the net Nftllll8 ot the h-th procas at the unit leTel is thm

The net 1'9ftlllln ot all proceuea at the unit leTe1 can be .-r1sed b,y

the Teet.or

which ill called the um:t leftl profit Teeter. Further, the net reTenue

due to .. production progra dennect b,y a particular ftet.or ! ill

n r • % 8.X • a'X

h•l n n - - (1.1)

The acnmt ot the i-th scarce resource ued b,y this production

progrm 1a

t • X.. t + Lt +. •• • X t 1. i U z 12 n in i • 11 2, ••• m,

and t.he amount ot the j-th product produced 1a

Thaae quantitiu can be swrisad tar all resourcea b,y the Teetors

7

!·' • <ti.~ •.•• t .. > • ~

!:' • <tf.~ •..• t~ ) • ~

_!. • • ( "l. "2 ••• • 1111:.> • "!. Wbieh l"epl M.U \be 'tioUJ. WUD."8 et HU'08 N~CU• DOB-HU'Ce n-

WW 0991 llDcl ~' ~.q. imolftd ill a prodaeUm progna

daf1md. 3-

n. p!'Ob3-~ \be f1ra 1a to t1n4 - ec-b'•Uon ot -1\

d- no\ w _.. \ban t.he &TailabJe .appq ot cel'ti&1n aw ~.

Denote bJ' "'1• 1•1.2, ... .-, the 'k»tal aw11Mle momdJ ot t.ba i-th aearae

th8D Gp.118818 t.he total aoua\ at all 8carce NHUNH• The l.1Jlitatioa

'JIPCl•ecl b7 \119 pix._ at t.a. 1-th SOU"C8 reaovce caa then "9 apJ'98Rd 1

1 • i,2, ••••• ,

or ecmlliderillg all HU"Oe l'UCRU'CM

~. !& ~ "'· - - -

and z ~ •• -

(1.2)

(l.2a)

(1.3)

The &ban wytxlutioa problaa ill a llJ)ICd•l aae ot wba\ -r b9

_.,, .. •\be• 11MV pl'Ogl'a•Sng problaa 1dd.eh 1a •hl'll' naW1

t.o -r'1"• a 11-ar tunction ot non-mp.tin T&P.l.ablee nbjeo\ to -

8

linear rutrictiona, eqv.alitiea or inequalitiu. We haft eouidered tbe

special. probl.ea of ..,,., m sing the net Nft!IU8 of a firll 81noe w intend

to .. that model.

l.~3 THI SOWI'IOI or THI MAIIMIZ.1Tl<ll PROBLBM

Although the Un9 -nmsation probl.ea om be eolftd by Mftl'lll

methoda, it is unalJ¥ aolTed by an itrerat.ift prooed111"8 vh1ch u.t1Hsea the

•sillplax crlter.t.on• de'ftloped b,y Dautsig and otben. Bw:planet.ian8 ot t.he

simplex criterion are an1J•ble (Cbarnes, Henderson, me! Cooper, 19.5),

Part ll, pp. 41-62 ad Dortun• l9Sl, pp. 2)-$2), and. need not be diacUAecl

hen. It v1ll ntt1.ce to gift a brief out.line ot the prooeclun used in aolT-

1.ng t.he Unear prOCJ••'na problea.

It can be eholm that the optimising eolution tor a progra•Sag

problea 18 one ot a limited Ht ot eolutiom (program) which haft the

~ t.hat a number ot tb9 procenee (el1 1nta ot the TeCtor x) are -carried on at sero bmWI. .Ueociated. 111 th Vl1' aolution ot t.lda aet 1a a

•a1wpla" 'ft9Ctor with elmM"te eorruponding to the sero-leftl procea ....

Bach elew1nt ot tbU ftetor indicatee how the net l"WYenue 1a affected b,r

adding to the prograa cme 111lit le't"el ot one ot the8e sero-leTel prooe1•u

without rlolating 8lJ1' ot the rut.riAiom. If the simplex indica\u that

the addition ot a lUlit ot UV" sero-l.eTel. prooeu dean•• net n.._, an

optbma solution bu been. obtained. Thia eiJlplu ftct.or 1a ued 1D m

iteratiTe prooeclure with proceeds frca solution to eolution until the optimua

i8 obvi1 nect.

l.l.h THI DUAL COJEEPT AID PRICK D!PUTATIOI

The li"MJ' prograMing eolution hu an additicmal r .. ture1 it

9

price or the 1-th tact.or indicate• the amoant by' 1fhich net 1Whlllle can be

increand b,r adc11ng to the suppq ot that tae\or one unit. It toll.ova

that 1"98ourcn that are not tull1' ued will haft sero imputed prioe•.

Another tea'11N ot the• illputecl pricu 18 that the .-

(1.4)

~ r. !bu, the t.oUl. iJlplted co8' ~ tm l:bd:ted naoareu ot aa

opt.im proctaotlcm JiNi1W 18 metq -.aaJ. to ita net ~ (7).

progr tng problaJI U •t&iied 1n IMt.iOD l~l.2 • haYe a:lml~

aolftd amrt.her probla 11hich oan he 8\atedt

to m:lnhd M !'~

nbj9d to b netrietiona p ~· o (l.S) and. r•p ~.!

Thia probha, lmolln u the dual ot the or.I.gt ml problaJI,, 1a to

_, n1..S.se ti. blpl1ied coat ot l.ild.ted. naovaea 1'1\ll ta. rut.rict.iona \bat

all impatecl pri.oee mut be non-negatift,, and \bat the total imputed coat

ot & unit leTe1 Of eaoh pJ"Ooeu 8st be greater 'than• or equal to, the m't

nrrenue ot t.bat pr00888.

l.2 mtl-LID&Jl PROOIWlmlO

1.2.1 OllKRAL tHllllll!

Thu tar we haft eomidencl the eolutiona or problem which an

atated 1n ....._of tba eaz1.Ssat1on ot a 11newr tunetion ot non-negatiw

Ta:l'1ablea nbjec\ to u _.,. reetriettom. lon-J:ln•J"i. t.iu -r ariH ill

tin diatinot but not, ~ aclaaiw 1AQ'81

10

1. The function to be vn w:I zed, the .rn •nd,, ia non-linear,, and

2. The rest.rictions are non-linear.

In general, very little wrk baa been done on cu• including

non-linear restrictions. We 1lil1 bent diaC\188 ~a special cue of a non-

linear •r'llllDd, Tis. a quadratic !Ul1d•nd. The pl"'Ogftmling probl• for

to ...,.,..... Y" • !'~'Pb

aubJeot to '1'! • !I (l.6) and ! .. o.

1.2.2 SOLVIllG THI QIWIW'IC PROCIRlMKIIO PROBllM.

The qudrat.ic pncram.1 ng probl.• 11111' be aolTed b,y tba ue of a

theorem clne1oped b7 luhn md TuU.Z. (16) and an itv&Un aolution pro-

oedltre deYel.oped b,y HUdret.h (J.4).

The lllhn-Tuckar theorem at.tee that the probl ...

to MxiJdse 7 • S'! - !'l'b nbJeot to ~ ft !I

(l.. 7)

vheN F 1a poai:U.w de.fini:te, ia equ.inl.ent to the w1n1wz probl.au

•1 n1wi se vi th reapect to ~ m&rlmse vi~h respect to:!) ;cb~) • ~'l'.!-2'~'(!-~),, (1.8)

subject to ! > o, where! ia a Tector eomda'ting of u JIBIJ1' el•anta u T bas roa.

Thia problea ditfen in one 1"98p8ct fl"GI the progrewing problea

stated in Sec\i.on 1.2.1, "Vis., it doee not include the rutriction x a o. -(1.9)

yll' • (v) - @),

ll

When -I ia a negative identity matrix,, and 2 is a vector of zeros. Then

(l.10)

ia equivalent to

(J..lDa)

and :z: a o. -The above problea JD.q now be made into a prograw1 ng problea by the sub-

ati tution of T* and ..,,.. for T and v. -The tirat step in the solution procedure ia to accomplish the

unreatricted minimization of ' (!,:!) with respect to!• Thia can be

accmc>llahed by the uu.al procedure of setting the derivative equal to

seroa

• • 2~•u-o, (1.11)

whence ~ • t~T'!)•

Thia is subetituted into the original. function, which becomee

re~ • -t<!!·r~ + f(d•rlr•!>-t~·rrl.r'!>~·!· (l.12)

It ia desired to JIA'dmze fa- ~) llith respect to !I subject to the

reatriction that ~ > O. Define

C • TrJ.ir•, !? • !""i'I'r~

and

We can nov wr1 te

(l.lJ)

12

The procedure of estilllating the vector ~ which accomplishes

thiB miniJllization i• an iterative one1 • .liter choosing an initial. val.ue

of ~ •.:Y !.o, ve hold. all except the tint of the components constant at.

the level given by ! 0 , and fiDd. that. non-negative Talue of u1 which mini-

mises e(!). Call thi.9 nlue ui· Hut, holding constant the elements

ui· u;, ... , u;, find the non-negative value ot ~ wbich m:hdm:lzea ew. Cont;jmiing in tbia manner, a mw TeCtor, !l, 1a constructed. The same

procedure i• uaed to conatru.ct in turn '!.2, !3, ••• , until the desired de-

gree ot ataMHty is obtained.

Let ~ k • 1, 2, • .. , be a component of u. The m1.nimua of 8(,!)

with respect to uk 1lill be atta1ned either where uk • o, or where~ • o • .)uk

It the latter equation yields a non-negative value for uk, then this is the

min1ndzing Yalue, otberwin uk • 0 is the minimizing Tal.ue.

The deriYatift

a 8 : eu + 2b. cl! - - (1.14)

Define -: u the value ot tha k-th coordinate of the q-th iteration that

ia obtained by setting cli/.>uk • o. Thus

(l.15)

where °k:i are el.anent. of C and 11_ are the element. of !?• The val.l1e of the

k-th coordinate of .! at the qth 1 teration ili then obtained by tak1 ng

u~ • maxJ•>• <W:• o),

L.l'he following two paragraphs follow closely the wording in Hildreth•s paper ( (J..4) P• 6o5) •

13

or in other words u~ • wf if the latter 1• non-negative, zero if it is

negati"Te.

The iteration atop• when two BUCceeding iterations are identical.

The cleaired value of the factor .! can then be found fl'Oll the f ornula

~ • 11.-1:!, - i~ '!.1 (l.l~

which wu developed abaft.

The abon iteratin procednre has been developed for the case

vhere the restriction matrix• T, is of fl1l.l row rank. However. the re-

striction aatrll, T-, of the program:l.ng probl.ea 111W1t contain more rows

than colWIDll becaue of the added negative identit;r u.trix and is, there-

fore, o£ lua than full row rank. There 1a no reason to believe that the

Hildreth procedure vill not work for tlbe cue where the restriction

matrix ia not of full rov rank. Thia baa not yet been Tigorousl.y proved,

but Freund (8) bu shown that a solution for a programing problem

obtained by H11clreth 1a •thod vu optimal.

l.J RISI PROOIUMMIHG

l.J.l ~'ION OF A RISI SITUATION

Under a no-riak 8it.uation the entrepreneur ia faced with a

given, kncnm set o£ pricea and prodllct.ion conditions that v111 prevail

during a prodnction period. Theae given conditions, in turn, specii'y

uniquely the result that v1ll be obtained by deciding to follow a certain

program du.ring the production period. Thua t,he entrepreneur can baa• his

decisions on t.haae k:nOllll consequences. Under a rialt condition, however,

thia does not hold; the entrepreneur is faced with a DU.Jlber of poasible

outcomea aa a result of a decision to conduct a cert.ain program. In order

to make such a decision, then, it ia of importance to be able to characterize

the set of pouibl.e outcones that are aasociated with a certain decision.

Probabilit7 statements ~ be used to characterize uncertain

eventa. It can be said t.hat those who propoee to uae probability for th1a

purpose define it u some measure or the degree of belld that a parti.-

cular outcOIJle Cll occur. rather than some objectift meanre ot relative

frequency of the occurrence of a particular outco.. The pouibilit7 tor

inferring degrees of belief troa obsened actions of indi:ri.dual s bu bean

developed and discussed by Sa"nlge (17). Some objections to t.he ue of

probability are discussed by Sange (17) and ArrOfl (2).

In deacribing a ri8k •ituation for programing purposes. all

the components or the program; that is, prices Pi• P1• and Clj• techni.cal 0

coet.ticienta 1'th and 'tth• and quantities produced ~h IS8l' be conaidered

random n.riables. Tint.Der (20a) gifts a procedure for handling this

general uae. but his results have• u yet, lilllited applications. We will

consider here that ~ pi• p:, ~· tih, t~, and uih are random nriables.

titles do not ~ enter into any other part of the lineu' progrumd:llg

probl.8la, ve will si.llp]J' state that the eh are randc8 ftl"iables.

If the sh are no~ distributed, then the Yector .! can be

considered a multivariate normal. with mean ! and ftriance-conriance

matrixf. The net revenue due to a progrus b is then, a m1:nriate normal

with mean Pr. e'! and nriance 4 2 • !~

1.).2 THE CHOICE AMONG ALnRHA.TIVE RISK! OUTCOMES

The choice uong risky outeomea is equivalent to detendning the

"best• probability distribution. We mq enluate a probability diatrl.bution

by means of the expected lltility. The expected utility of the set ot out-

comea of a particular deciaion is, by definition, a .uure of the relative

value 0£ the particular decision. Tlm.8, if the expected utility dne to all

possible deciaions is evaluated,, the best decision can be picked by maxi·

mi.sing the expected utility.

'1'be expected utility of a probabilit7 diatribu.tion of incomes ie

defined

E(y) • ~ y(r)f(r)dr (l.17) r

vhera t(r) denotes the probabHJty diatribu.tion of iDCOM and 1(r) deno~•

t.he relationship beWMD. ou~ and utility. This relatiouh:1p, called

t.he utility tunct.ion ot income, giTita a nbJective value, a utili'ty', for

arq given UIOWl\ of iLCCJM.

The aiDpl.est function relating utili 'ty' and 1nc:om or money is the

liDMr t'wlction 7 • r. It can be ahcnm that tl".J.a utill'ty' tunction u.>llea

that aaillizing expected utility 18 identical 111.th uzimizing apected in-

come without. regard to the vaz1.abilit, of inc0ll91 since

K(y) • { rf(r)dr • l(r). r (1.18)

However, experience at.on that tb18 ia not an accurate picture of actual

behavior.

A second utility fULction ia om. vhcise first derivatiTe is every-

where poaitive and aecond derivative 1a eveeywhere negative. Thia function

exhibit. the usual asaunrption of positive bu.t d.ecl"U81ng marginal utilit7

of mone;y. Such a function a,q be reguWld u the one that best describes

the behavior of an entrepreneur with l.ind.ted capital who mat practice

relative~ COll8erYative policiea. This utility function _,. be represented

by y • 1-e-ar where a ia an arbitrarr constant which indicates the entre-

preneur•s aversion to risk. Thia can be shown by the fact that the max1m:f-

sation ot the expected utllit7 llllder this utility function and the

unmption that r 1a nOl'Jll&llT c:liatributecl can be accomplished it n

chooee that probabili:t)r d19tribution with the uxl-. nlue ot the

tanetion

(l.19) late t.hat a large Talue tor a iDdicatea that the ftl"ian• pl.qs a

relati~ larger role in the decision making procea•. Another 1'.tili't7 tunction 1a one repre•entecl by ebr -1. where

b 1a anoth• arbitralT constant. Th:1s 1J0\1ld. be the util1t7 hnction tor

a natval gambler• one who pre.ten thoae Olltccmas with wide d.iaperaiom

and greater poa8ibW.tiell o:t gnat gdu and losses. H w ue the aboYe

utility !unction• t.he corruponding function to be wart 11'1 HCl 1a

)I+; c:s-2 (l.20)

The abon function indicates that the gambler is v11J:lng to reduae incoae

to increue YU'imce.

u vu etated. in Section 1.3.l,. net NT8JIUe aaociated with a

program. .! ia uallMCl to be no~ distributed with~ ·~'.! and

cs2 - x• f x. where~ is the matrix o:t the nriancn or net 1"9Ye:l'IJ88. - .... TJ.ma. uing the wx111'1sation or expected utilit,- according to (1.19) aboft•

the utJ.lit.7 function to be waxlalsed will be

,. • •<·~ -~· i' x. -- ~ .-The ~ probl.m reslll.ting from this martwisation of

expected utility is

marhwise 7 • B(!).! - ~' ! .p subjed to ~ (l.22)

17

The procednre to be used for solving this problem 1lill be that

derived by Hildreth for the use of t.he lubn and Tucker m:1n1mn probl.em

(See Section 1.2.2) with alterations as ll&de by Freund (8). It vu noted

that to ue t.bia Mt.hod the tvo rest.rictiona to tbe above probl• 1111at be

comhinecb

( (T) " cr) (-1)~ ~

which WU denoted

'I-tt!: ' "* - - (1.10)

31noe we viU onJJ- be concel'D9d with the vector'!* and. the matrix

T•, we will omit the asterisk. JJ.ao, we will denote the utrix

t. The m::lnim equiYalent foz- the programndng problElll can tim. be

writ.tem

18

CHAPTER ll

.Al IUMPLB

2.1 THI DISCRIPTIOJf OF THI FIRM A!ID THI 10 RISI PROOR.&M

2.1.l THI DESCRIPTIOJf OF THI 1IRM

acre tara in the Blacksburg area of Southwea~ Y'irginia.

The f81'1l J1Q' be identUied by the proceaaesl available to its

1. rortedeer ran 0ata, 2. Yahart. Wheat, ). Vong Barley, 4. u. s. 13 corn, 5. Ir1ah Cobbler Pot&.toea, 6. Rutgers Tomatoes, 7. lansaa CCllROll ilt&Ua,

and by t.he scarce ruourcu 1

1. Land, 2. Production capital., period 1 (September 10 - January 9), .). Product.ion capital., period 2 (Januar.y 10 - J1U11J 9}, 4. Production capital, period 3 (June 10 - September 9), $. Managerial. labor, period 1, 6. Managerial labor. period 2, 7. Managerial. labor. period 3.

Land rel'en to t.hat portion ~ cleared croplancl that h suitable tor

cultintion. Production capital is the aftil.ability of cuh tcr net.

a:penses. Managerial. labor retera to the fal."ll operator•• awn labor.

Certain non-Mnagerial tasks can be performed by h1recl labor, which,

it la U8UMCl, can be hired. wit.bout limit at. a giveD vage rate.

lror t.ha sake of aillplicity ~ harvutable crape vere uncl.

TABLE 2-1

BASIC BUDGl'1' FOR OD ACRE OF FALL QlTS

I\tm Price Unit Quan- jmou,ni; Period Doll.an tit7 DDllan

Incaaa

Ilel.d .11 bu. 6o.o 46.20 3

E!p811H8

Seed i.ss bu. 2.2.s 4.16 l Fertiliser S-10-10 2.25 cwt. s.o u.2s l Machine operation

Land preparing and pl..anting .56 bra • 1.96 1.10 l Harres ting • 56 hrs. 1.04 .58 3

Total all apanau 17.09 Ne'\ .a.venue 29.ll

~tor•• labor

Land preparation and plant.ing bra. 2.37 1 Harvesting hrs. 2.08 3

20

TABLE 2-2

BASIC BUDGET FOR OB ACRE OF WBBlT

I tea Price Unit Quan- Mount Period Doll.an t1t7 Doll.an

lnCOM

Iield 1.90 bu.. 30.0 s1.oo 3

E!pem!!

See4 3.00 bu.. i.s 4.50 1 Ferti11Mr s-10-10 2.2s cwt. 5.0 11.2.s l Macbine operati.OD

Lad preparatiell and planting .S6 bra • 1.96 1.10 l Hane.ting • 56 hrll. 1.04 .ss 3

Total all !!J>!D!U 17.bJ

Met Bevenue 39.57 cprator•e t.bor

Land preparation and planting bn. 2.37 1 Haneating bra. 2.08 3

21

TABLE 2-J

BASIC BUDGET FOR ONE !CRE OJ' BARI.EI

!tea Price Unit ~ .b.ount Period Doll.an tit;y Doll.an

Income

Yield .9S bu.. 40.0 38.00 3

Expenses

Seed 2.25 bn. 2.0 4.50 l I'ertiliser .$-10-10 2.25 cwt. 5.0 u.25 l Machine operation

Land preparation and planting • .)6 hrs. 1.96 1.10 1 Harv eating .S6 hrs. i.04 .sa 3

Total all apenau 17.43 Net Revenue 20.57

()p!rat.or•a Labor

Land preparation and p1anting hrs. 2.37 1 Harvesting hrs. 2.08 3

22

TABLE 2-4

BASIC BUOOB.'T FOR ONE .ACRE OF CORN

I tea Price Unit Quan- Allount Period Dal.la:ns tit;y Doll.an

inoome

lield l.6o bu.. 60.0 :us.oo l

.!s?,enses

Seed, hybrid n.oo bu. 0.2 ~.20 2 Fertiliser l.o-10-10 ).00 cwt. B.o ~4.00 2 Tractor operation

Land preparation and planting .48 bra. 2.83 1.36 2 Gultivat.ing .48 hrs. 2.22 l.o6 3 Harveating .48 bra. 1.78 .BS l

'.total all m:p!n8!! 29.47

Net ReYenue 96.53

gperator•s Labor

Land preparation and planting hrs. ).03 2 Cultivating bra. 4.17 3 Harvesting bra. 2.63 l

23

TABLE 2-5 BASIC .BUDG.1'..'T Ji'OR ONE ACRE OF I1:.ISH PfJU.TOES

Item ~rice Unit Quan- Mount Period .Loll.an tity Doll.an

Inoom

Yield 1.os bu. • 250.0 262 • .so 3

.l!lxpen.ses

Seed, Cobbler 4.16 cwt. 8.1 )6.0$ 2 Fertilizer 6-8-6 2.15 cwt. ~J.o 49.16 2 IAtst, 5% D.D.T. .17 lb. 60.o 10.20 2 Ba.:,<78 I new .22 baf: J.40.0 30.80 3 Washing and Grading .35 bag 140.0 49.00 3 Digging, eontraeted 9.00 acre 1.0 9.00 3 Picking up, contracted .13 70 lb. 215.0 27.95 3

bag HanH ng1 contract.eel .07 70 lb. 215.0 15.05 3

bag Tract.or operation

Land preparation and planting .48 hrs. 4.0 1.92 2 Poisoning .48 hrs. 1.0 .48 3

Total. all Uioenaee 229.90

Net Bevenn.e 32.60

C>perator•a Labor

Land prepara:t.ion and planting bra. 4.0 2 Tt\gging, supervising hl'll. .8 3 Poisoning bra. 1.0 3

24

TABLE 2-6

BASIC BUDGET FOR ONE .lCRE OF TOMA.TOES

Item Price Unit Qnan- Amowlt Period Dollars tity Doll.arl!I

lncOM

Ileld • 90 bu • 533.0 480.oo 3

Expemea

Plan~ 12.00 M 3.0 J6.oo 2 Fertiliser 12• lba. B-20-16 6.oo cwt. 12.0 72.00 2 Machine operation

3rolling .49 hrs. 7.9 3.87 2 Orov1ng .49 bra. 7.9 J.87 3

Spr¢ng and doting 17.85 acre 1.0 17.8.S 3 T1-uc.ld.ng .20 mi. 63.0 12.6o 3 Picldng, contracted 6.oo ton i6.o 96.00 3 Conta.1.Dera & othera 8.12 acre l.O 8.12 3

Tot.al all ap!D88!, 250 • .31

Net !ievenue 229.69 -Qperator•s Labor

(h•awing hrs. 18.2 2 Growing hrB. 18.2 3 HarYeeting hrs. 20.0 3

25

TABLE 2-7

BASIC BUDGl.'r FOR OB ACRE 01· ALFALFA

I tea Price Unit ~an- .A.mount Period Dollars tity Doll.an

Income

Yield 42.as Ton 2.67 ll.4.27 2 Yield 42.as Ton 1.)3 57.13 3

E!eEID888

Seed (Pro-rated seeding coat over 4 -yrs. ) 1).00 acre 1.0 13.00 1

Top Drusing 0-10-20 2.20 cwt. 8.o 17.6o 1 Machine apenaea

i'or utablisbing .b8 hrs • 1.18 .51 1 For maintenance and • 56 bra • 3.14 2.35 2

harvesting • )6 hrs. 3.14 i.17 3 Hired labor .Bo hrs. 1.41 1.13 2 Hired l.a.1'or .ao hrs. .n .57 )

'f ot.al. all. ~· 36.39

Net Renmue 135.0l

.9i_.er&~' 8 i.bor

.Eatabl.18bing hrs. 1.41 1 Maintenance and Barnsting hra. 7.33 2 Maint.enance and Harvesting hrs. 3.67 3

26

The budgets for the model were constructed by Dr. R. G. Dine

of the jgricultural EcoDOJllica Department at the Virg:l.nia Polytechnic

Institute and the author with the asaiatance of Mr. l. E. Loope and

Dr. c. w. lien, also of the Agricultural F.conomiCll Department.

The basic budgets for proc .. ee 1-7 are pruented in t.abl•

2-11 2-21 2-31 2-41 2-.$1 2-6, and 2-7, i-espective}¥.

2.1.2 THE TIDHNOLOOI MATRIX

The deri.-tion o! the (11carce reaourcea) technology matrix, T,

from the budgets 1d.ll be illustrated. for proceaa 11 1aJ.l Oat.. We will

present the da\a for the other tudgeta without further explanation. Froa

Tabl.e 2-1 w obtain the various amounts of the acarce :r.ourcea needed

for one acre of oata. The amount ot acarce reaourc• needed and the net

revenue are given in Table 2-8.

FOR OHE ACRE OF 1.lLL OlTS

Resourcea

Land, acres Production capital, dollars

Per100. 1 Period 2 Period 3

Managerial labor, hours Period l Period 2 Period 3

Net reverme, dollara

J.mounts

i.o 16.Sl

.oo -!&S.62

2.37 .o 2.08

29.n

Note that Table 2-8 gives the net requirement11 of acarce resour-

ces. In the ca11e of production capital for the third period, production

27

capital ia created by the aale of oata. Thua1 the net requiremnts ot

capital for the third period becomes to • .58 - $46.20 " -th$.62, the nega.-

ti'f'e amount denoting the creation rather than the uae of the resource.

For computational simpliciV it 1a advantageoua that unit

levels of the various procesau prodllce equal amounts of net revenue.

i'or '\bia reason, all proceuu are redefined such that a unit leTel pro-

du.cu tlOO net revenue. In the oaae of oats this is acCC9'>11•hed by

rai.81.ng 100/29.ll • 3.435 acrea of oats. This redefinition cauaea all

ruourcea requiralllnta to be mltiplied by J.43.5. Finally all prodnction

capital requi.remanta and the unit level. profits are upreased in hundreds

of dollars. Tbi.8 scaling device makes the nwnben in the tecboology' matrix

roughly of the .._ ugnitude, which tends to 1IBke compu~tiona leas sub-

ject to rounding errors. The resulting vector, _h, and. the unit level

profit, s1, are presented in table 2-9.

TABLE 2-9

SC.ARCE RESOURCE VIOOTOR, !• FOR PROCE38 l

FALL OATS

Reaourcea

Land.1 acres Production capital, $100

Period 1 Period 2 Period 3

Managerial. labor, hours Period l. Period 2 Period 3

Unit level profit, $100

Amounts

3.435 .567 .ooo

-1.567 8.J.h.O .ooo

1·!40 1.000

28

The resulting technology matrix, T, of the entire fira, the unit

level. profit vector, !I and the resources 11.rni tationa Teetor1 !i are pre-

serdied in 11iile 2-10.

The onq item in Table 2-10 that bu not been diacwssed 1a the

conatruction of the ruources llm1tat1on ftetor, !• Thie vector 1a some-

what arbitrary, particul.arq when one visbu to cbaracterise a repruen-

tative fara. In our case we have 8UWl8d a sixty acre ta:ra v.ith homogeneoua

fertility of the soil. The availability of production capital is based on

an initial 81.ock of $1600 011 hand as of the beginning of Period 1, of

which $1500 met be reaerved for living a:pensee in each of the three

perioda. The 811CRlllt of manageri&l labor available is limited to the DWllber

ot good wather daylight hour8 in each period and ia bued on put weather

record9.

TABLE 2·10 THE TECHNOLOGY MATRll FOR THE PROGRAMMING PROBLEM

• INPUT VJOOTORS FOR PROCISSJ:ia AT UNIT LEVELS Sea.roe s Availability Reaouroea I tl t2 t) ' ts t6 t7 Vector

i Oat.a Barley Wheat Corn Potatoes Tomato.. Alfalfa .,. -! t

J..nd, acre1 • 3.43) 2.527 4.861 l.Ol.5 3.o67 .43s .740 6o Production I Capital, $100 I

.567 .426 Period l l .819 -1.291 .o .o .231 )0 Period 2 I .o .o .o -1.0ll 2.994 .487 -1.243 15 Period .3 1-1.)67 -1.426 -1.619 -1.000 .3.994 -1.486 -1.330 0

°' Managerial N I Labor, houra I Period l I 8.J.40 5.969 11.$20 2.670 .o .o 1.040 199 Period 2 I .o .o .o 3.01s 12.268 10.1370 5.424 867 Period 3 I 7.140 5.2.$6 10.llO 4.230 s.s21 1).660 2.116 783

I

30

Under no circwmtancee v1ll the restriction on production

capital in Period l or Period 3, and the restriction on labor in Period

1 becme effectiTe. This u true Id.nee th•• restrictions, for arrr pro-

cess, are l•• stringent t.ban the other reatrictione. We can, t.beratore,

regard the correepondiDg resourcee u being 1n tml1w1W auppq and can

Ollit these reatrict.ions !roll the prognmiDg probl•.

2.1.3 THE OPlIMUK NO RISI PROOP.AM

Solution b,- the aillpl.a: teclnd.que ( ... Cb.apter I) ruulted. 1n

the optima prograa pr1J1Jented in Table 2-11. Thia 1a the lo R1ak Prograa

since variabili't7 vu not considered. In thia prograa, note the beav,y

reliance on tomato.., which an a highly UI18t.able, ar riaq., crop. ilao

mnall grain crops, the uau&l hacti>one of a general plllpOH fara, are not

r&ised. The above feat.urea, vbich have been noted in otb8r l :t near pro-

granning probl.m8, cauae ua to consider the e.tfect of risk into a

program1 ng mod.el.

TABIB 2-11

THI OPTIMUK NO RISI PROGIWf

Process Unit Lewla Acrea

i. oat.a .oo .oo 2. Barlq .oo .oo J. Wheat .oo .oo 4. Corn .oo .oo S. Potatou .oo .oo 6. Tomatoes 46.6S 20.29 1. il!all'a !)3.66 39.72

Met Rneme ($100) 100.)l

2.2 THE RISI PROORAM

2.2.1 THK DESCRIPTION OF .l BISI Sll'U.lTIOH1 THI COMPU'r.A.TIOH OF V~.l!S OF UNl'T LKVEL Bl' REVENUIS

The nriab1liti• ot t.be unit level. net NV'enna 1lill. be ulllm8d

to depend al.-e1i entirel3' on t.be ou.tpnt pric• and quantitiee (aee Section

l.J.l), vbile .,.t, ot the input. are find. H01MTer, certain input., Reh

u picld.ng coats tor tomatoes and potatoes, do ft17 v:l.t.h the quantity

produced. We will thua deal. v:l.th adjusted groaa 1'9T8DU•1 which in thia

case v1ll be the produ.ct ot pric• and 71elds ot the varioaa cropa, ad-

juated for nriable input., since the variability ot adjusted gross reYe-

nuea ia equal to the variance of net 1'999llli.88·

We an intereated in obtaining the ent.npremur•s estimate of

the Tar1ability or pricu and yi.elda ot 1Dd1Tidual crops u an indication

or a probalrl 11 ty dietribntion. Thia utiute ia usu.aJ.lT based on past

beba'l"i.or. Tberetore variability utimatea wre obtained froa data tor

1948-l9S6. Thia vu eatillated in two Btageaa (1) the aat.imation of past

prices in ccmatant dolJan, and (2) t.be utimation or put. yi.elda under

comtant cultintion practicu.

Price •timates,, v:l.th the nception of t.omatoee,, were obtained

by using &Terage Virginia pricu (2l) for each crop,, deflating th•• pricea

by tbe United States Wholesale Price Indu {l) and tin•JJy nall.tip:cying th••

deflated.price• by a constant sou to obtain an &Terage price equal. to the

price specified in the budget.l. Recorda of put tomato pricee wre 1Umished

by J. M. Jobnaon of the Department of jgricul.tural Economica at Virginia

Po~cbn:lc Inati\ute.

1'rhi8 adjustment procedure aasumea that T&r.iation in yield is proportioned to the mean.

The yielda of each of the crops uaed were obtained 1'roa recorda

of aperimnta conducted by the Virginia Agricu1tural. Elperiment St.tion

at Blackabnrg. Data OD mll grainll were obtained froa T. M. St.arl.ing

(18 & 19), data on corn fr. c. r. Oenter (101 11 & 12)1 data on alfalfa

.trm T. S. Siil.th md P. T. Giah• and the data tor potato. and tomato..

trca P. B. Muaq1 Jr., and othen. W. data are obt.ained trca aper.i.-

canwd b7 ditfennt locaUona w eliwlnatecl • .l linear t1- trend aa an

iDdicator of bproT .. t 1n crop growing practic. vu COllpUted for each

crop by J.eut 8qU&1'9ll ngreuion. In no cue did the linear trend proYe

to be significant, and consequen~ no adjulltaent for U. trend wu

made. The yielda were then adjuated bJ the ratio of the yield spec11'1ed

in the budget. to the average yield obtained above so u to obtain the

yield epeo:i.fied in the budgetl.

The product of uti:Mted per acre yield and eatiuted price for

each year wu then uecl aa the esUllate of gron reTeme for each crop.

The groee revenue wu then &djwrt.ed for variable 1.npute ae vu mntioned

prni~. VU'iancu wre COllpUted .troa tbia a.ta and mlt:iplied by the

appnpriate COD8t.anta in order to redu.ce th- to var1ancee of unit levels.

The COllpltationa for the eatimated grou revenue of proceaa 1, oats, are

nprodllcecl in Table 2-12 to illustrate the procednra.

l.rbia adju•tllent pl'OCedllre aaeumea that. variation in y:lel.d 1a proportioned to the man.

33

TABLE 2-12

~Tli1ATION OF GROSS REVENUE FROM ONE ACRE Oii' OATS

I : I Adjustedl : Price Defiated Adj. : Estimated :

Year 1 Price to : Yield Yield I Gross I $ • $.77 I : Revenue t $ : Bu. Bu. : $ I I : I I :

1948 I .87 .84 .87 • 56.4 49.86 I 43.38 . 1949 I .10 .n .74 I 83.J 73.Qi 54.49 1950 I .81 .79 .82 I 49.5 43.76 I 35.88 1951 I .79 .69 .12 I 47.3 41.81 I 30.10 1952 : .91 .Bl .84 I so.a 44.91 I 37.72 1953 : .82 .74 .11 I 94.5 83.~ I 64.33 1954 I .80 .73 .76 I 90.4 79.91 I 60.73 1955 I .71 .64 .66 I 67.0 59.23 I 39.09 1956 I .76 .68 .n I n.6 63.29 I W+.94

I I I

1rn the case o:r oats, wheat• barl.q, com and al.falf'a, gross revenue was not adjusted tor variable inputs.

The estimated per acre adjusted gross revenues for the seven

cropa in t.be example are presented in Tabla 2-13· Thue adjusted gross

rnerme• 1191'9 used to estimate variance8 of per acre gross revenue

(Table 2-14). Covariance• were as8Wl8d to be zero. The per acre variances

were reduced to uirl.t level. T&riancea (Table 2-15).

Year Oata

1948 43.38 1949 54.49 l9SO 35.88 19Sl. 30.10 19$2 37.72 1953 64.33 1954 6o.73 1955 39.09 1956 ~94

34

TABLE 2-13

THE ESTIMATED PER ACRE .ADJUS'l'ED REVENUES

Wheat Barley Corn Potatoes

66.09 56.87 137.94 167.59 SB.62 .)6.10 129.27 281.43 52.41 2.$.84 l2J.8S iw...53 46.44 53.Ja 113.89 272.61 )2.76 Ja.o6 118.25 338.60 56.10 40.93 91.so 139.50 70.31 28.87 146.08 no.6.S sa.66 2S.46 124-47 160.85 70.52 32.39 12).9) 164.58

TABLE 2-J.4

PER ACRE VJ.RliNCES

Crop

Oats Wheat Barley Corn Potatoee TOll&toea Alf al.fa

Crop

Oats Wheat Barley Corn Potatoes Tomatoes ilf alfa

Variance

137.92 146.20 46.85

2)6.97 12,,8)6.92 20,493.02 1, 700.56

TABLE 2-15 UNIT LEVEL V.lRli.NCES

Variance

1,,627.32 933.63

1,101.02 244.08

120,744.o6 J,873.18

931.91

Tomatoes

253.30 396.53 207.31 26o.l.6 542. 78 3$6.02 297·9'l 503.93 612.98

.lltal.ta

165.)6 116.So 128.73 186.52 207.6o 177.04 J.Je.r-120.22 240.20

35

2.2.2 .l COMPU'l'.ATIOU OF THE RISK PIWORAM

We aball derive an optillull risk prograa reaulting frca the maxi-

mization of the expected utility of the form

y ·"'- .!O" 2 2

aa discuaaed in Section l.J.2. The data used vill be ti. aame u in the

l1118U' program together with the var.lances as derived 1n the previous

section. The comt.ant a will be giTen an arbitrary value l/1500. Further

d:Lscuaaion of tbi.s constant can be found in Section 1.).2 and Section 3.1.1.

The procedure which will be tollowed in the computation is ident-

ical. to the procedll.rea outlined in Section l.J.3. It should be recalled

that .. need to !ind

(2.1)

and (2.2)

where (2.J)

The data for this problem have been diacwsaed before but will be

presented again at tbia point to facilitate the presentation.

s' - - (100 100 100 100 100 100 100)

.542 .Jll

2 .369 • .081

40.248 1.291

.Jll v• • (6o 15 867 '(83 0 0 0 0 0 0 0)

3.435 2.527 4.861 l.OlS 3.067 .435 .740 0 0 0 -1.0ll 2.994 .l,87 -1.243 0 0 0 3.07$ 12.268 10.870 5.b24 1.lh.o s.256 10.ll 4.2}) s.s21 lJ.66o 2.7].6

-1 T • -1

-1 -1

-1 -1

-1

Fram this we DUJt DQV coaput.e1:,-l, c, Tf•l, 8l1d 2]? • 2.! - 'Jf:-1.!s

1.845 3.215

-1 -2.710

~ 12.)46 .025

• 775. 3.215

-4,169.370

1,627.500

-4, 66!i.l00 -n,329.200

184.JSo 2b - 321.300 -

270.990

i.229.100

2.liBS

77.454 321.900

121.107 -15.160 55.884 2BS.Jl$ -6.332 -8.119 -13.17) -12.474 - .076 - .337 -2.382

...JS.180 17.943 -54.906 -57.867 0 0 0 12.426 - .074 - .377 4.002

55.884 -54.906 306.180 323.970 0 0 0 .37.800 - • 30S -8 .418 17 .460

285.315 .57.864 323.940 846.610 -13.161 -16.887 -21.396 -51.990 - .137 -10.578 -8.742

-6.332 0 0 -13.161 1.644 0 0 0 0 0 0

~C: I ..a.119 0 0 -16.887 0 ).215 0 0 0 0 0

-13.173 0 0 -27.396 0 0 2.710 0 0 0 0

-12.474 12.426 .37.800 -s1.990 0 0 0 12.346 0 0 0

-.076 -.074 •• 305 •• 137 0 0 0 0 .025 0 0

•• 337 •• 377 -8.4J,8 -10.575 0 0 0 0 0 .115 0

-2.)82 4.002 17.46o -8. 742 0 0 0 0 0 0 3.2l5

The iterationa tor obtaining the vector u were performed. using -the IBM t)-:pe 650 magnetic drum cmputer located at lorth Carolina State

College in laleigh1 Borth Carolina. The prograa for UH vi.th the 650 WU

deviaed b)" Jt. J. Fn1lnd ot the l>epa.rtaent or statiatice at Virginia PolJr-

technic Imtitute. '.l'hU progr8JI computes the iterations at a speed of

approxbmtelJ' 100 tiaea .taster than an a:perienced operator on an auto-

matic desk cal.cul.ator. The i teraUona tor tbia probla beC81M stable

after about )So iterationa. Without the use or the 650, the work: .. out-

lined in the resul. ta or t.hia paper would hne been practi~ 1...,...1ble

to obtain.

2.2.3 THI OPTIMUM BISI PROOIWf

The risk program and t.he no-risk prograa, together with other

pertinent intonation are presented in Table 2-16. The figures in the

table conform n17 well u to what was expected. The consideration or risk

incNUed. the importance of corn, the crop with the lonat unit lenl

T&riance, and decreased the 1llportance or tc:aatoea, a comparatinl,y high

ri8k crop.

The total :reTenue vu, ot course, decreasM, but the expected

11t41.1.:t7 ot the form ,U - ; s2 vu increased and the at.andard deTiation ot

the net revenue vu substantiall,y decreased. The rlJl.k p.rograa also require•

leas capital. and labor than the no-rhk program.

COMPilING THE RISI A.ND THE NO RISI PROGRAM

I tea Riak Ho Riak Program Prograa

Proceaa Intemiti .. Unit Leve.ls

oats .oo .oo Wheat .oo .oo Barley .oo .co Corn 15.61 .oo Pot.atoea .oo .oo Tomato.. 22.55 46.6S .Alf al.ta 46.58 53.66

.&.crea OGe .oo .oo Wheat .oo .oo Barley .oo .oo Corn 15.80 .oo Pot.a\oea .oo .oo Tcma'8M 9.80 20.29 ilfalta 34.40 39.70

Expected. Net. Revem•, dollan s,475.00 10,031.00 Expected Utility, arbitrary unita 71126 61.326

Standard Deviation of Net Revenue, dollara 2,012 3,334

U•e of Beeaurcea Land 60.oo 59.99 Production capital

Period 1 -9.3925 12.3997 Period 2 -62.6980 -43.9923 Period 3 -106.UOO -JJ5.3.36

Managerial. Labor Period 1 90.]Ji 55.81 Period 2 516.77 798.19 Period 3 500.51 78).00

40 CHl.PTER Ill

F'Uk'i' HER ASPECTS 01' F..ISK Pll0Glw1l UNG

The example of the p:.>eviows Gb.iaptttr will be uaed to investigate

additional phases of prot;ra."11l'Ging for the risk case. We ahall diacuas the

"Bisk Aversion" map in some detail and hritdly touch on price, resource

an.cl variance 118.pS as aspects of riak programing. Moat of theae have been

perfected and used in the no-risk cue. and exttm.sion of these aspects to

risk programrdng should lean tQ u.se.ful results.

).l THE "RISI .lVEUSIOtP' MAP

In Section 2.2, -w did not justify the use of a particular YB.l.ue

for the risk aversion constant. We shall present in this Section a method

o! risk programming which avoids thill difficulty.

To a cfll"ta.1n extt:Jnt each cOllbination of economic good8 is a ~

stitute for every other combination. Among the possible combinations. there

&.re some which are preferable to othen; but there are some which are

equally attractive to ua, so that we feel it ia a 11&tter of indifference

which one we choose (J). Thia micy- be represented by an indifference cu.rn•

which ia the loci of all points, combinations of goods, or of indifference.

Another way of atating this condition is that an indifference curve is the

loci of all points of equal utility.

There are other combination:: of economic goods lying outside a

given indif'ierence curve which give rise to other indi!'ference curvea • .l

set of indi..ff erence curves combine to produce an indifference map abcndng

iuciiiference cu.rvea ranked from those of loveat utility to those of highest

utilit;r. Thde axists a unique relationship between a utility function and

an indifference map. An ioealizeci innif'ference map is illustrated by the

set of curves il', BB', CC', DD' and EE' in Figure 3.1.

figure 3.1.

E

Bconomic Goods B

The problem lacing an entrepreneur, or consumer, 1a to Mximse

utUitJ' subject to income llnd.tat.iou. An idealised incc.e llld.tation 1a

giTen by the line aa• in Figure J.lJ that ia, linear combination ot good.a

can be purchaaed by a giTeD. 8110unt ot income. It 1a intuitiftl,y obviou

that the point ot MX1111111 ut.ilit7 oecurs at the point where the inccae or

opportunitT line is tangent to an inditterenoe curve. More Y'!gorous argu-

mente ~ be found in econOllli.c reference vorta, e. g., (.3).

In riak eonditiona, where r1ak,r outccau aq be repruented by

the expected return (g) and the at.andard deviation of tb1a net return

(CS), w Ila)" conatnct an inditterence map tor n.rioua ccmhinat.iona otp

and.CS. The inditference map o! these •econotd..ca goods• is somewhat the

l'e'YWae ~ the illustration in Figure J.l. since stanct•rd derlation, being

the reverse ~ reliabilitJ', is an untawrable good. The 1.nd1.fference map

resulting troa the utility function or income (r) diacused in Section

1.3.21 u • J.1- Ic{l' tor a • l/1500, is given in Figure ).2. The eunea

represent loci of equal Tilluea ot u at 4000, Sooo, 6ooo, 10001 8000 and.

9000.

La ,u in thousands

C» 'O b

\ \

\ \ \ \ \

\ \ \ \ \ \ •

\ \ \ \ \ H

i:!j

\ \ \ I ~ ~ N \ \ • -0 \ IS" ~ ~ a \ \ \ ~ ~

::C N r Ill

\ \ \ II .... tr \ \ 8

\ \ \

\ \ \

\ \ \ \ \ \ \ \

\ \

\ \

In Sect.ion 2.2.) the optimum risk program aasu:mi.ng the utility

function u • /"' - l tJl was ob~d. ruul. tj_ng in _,If • 184 75 • tr • 2(1500)

'2012, and u • 7126. It can be seen that this combination o.f ~ and '5

lies wry close to th6 indit!erence curve for u ,_ 7000 in Figure ).2.

Thia point 1a marked by P. In fact. according to the abcmt theory, tbe

point P ahou1d lie on the opportunity line of the entrepreneur which 18

tangent to the indifference curve for u • 7126.

In.diff'erence ups aimilar to Figure 3.2. cou1d be constructed

for various Talues of the r:l.ak aversion constant. By reoonputing optilmlm

riak programs for eacn of these val.uee, ww can obtain several points on

the opportunity line which are tangent to the inditferunce curves eorrea-

ponding to utility functions llith different Tal.uea of •a•. The different

points of tangency mrq be joined together to form. an opportunity curve.

Thia opportmdty curve represents combinations of net revenue

and the variance o1.' net revenue which are available to the entrepreneur.

An entrepnmeur could choose a point on the curve which to hlll represents

the best combination of net revenue and variance. In doing so, he will

eff ectivel.J' be choosing his own risk aversion constant and corresponding

optinua program.

Using the computational. shortcut discussed in the appendix,

optinma programs were developed for a • ...!...,.. 1 , ••• , 1 • The result.a 150 ~ SOD

of the computations are 8WllD&l'ized in Table 3-1. and the corresponding

opportunity curve in figure 3.3 •

.As was expected, the larger the Talue o:f the ruk aversion con-

stant •a•, the more the optimum. program depended. on the comparatively low

risk crop, com. The smaller the valne of •a", the closer the optiam

TABLE 3-1

OPl'IMUM PROGRAMS FOR VARIOUS RISI .A.VERSION CONSTil!S

Ita VU'iOUI value• of •a• Opt.illml Progr ..

Proo••• In\enaiti ... -Unit levw l/7SO 1/1000 l/l2SO l/lSOO l/l'TSO 1/2000 l/22SO l/2SOO Linear

Corn 33.48 21.s2 21.s1 15.61 9.67 3.69 0 0 0

Tomatoea 11.97 15.49 19.02 22.ss 26.08 29.61 32.76 3s.21 46.6S iltalta 28.20 34.32 40.46 46.S8 s2.12 S8~8S 62.JS 60.88 S3.66

J.crea :i

Corn 33.98 27.93 21.ao l).60 9.61 3.70 0 0 0

Tou.toe1 s.20 6.74 8.27 9.80 11.24 12.87 24.os is.01 20.29 AUalta 20.86 2.s.40 29.93 34.40 39.01 43.50 46.03 Jt.S.OS 39.70

Expec1'ed let Reveme, 736S Dol.lan 773.S 810) 847.S 8847 9215 9511 9609 10031

S'Undard Dft'iation of Net Rnlmue,

Doll.an l2S2 1487 174.l 2013 2290 2S71' 2789 287) 3334

*O&t1, Wheat, Barlq and Potato crop8 never entered the prograa.

46 program approached t.he lineu' prograa, which ..,- be considered a risk

progra with a • .!.. It ill belitrYed, holrevar, that a finite nl.ue of 00

•a• will produce an opt.iJmll aactl1' equal to the linear program. Aa

the nlue of •a• increases, indicating greater &Tenion to riak, the

net l"'9ftll'19 decreases and. tha standard deviation or net reTeDM de-

cn••• • It ia interesting t.o note that tha lOINt1" 2% 11a1\ (p ."t:, ... cs)

ot the net NTeDUe in the 11 near progra ia lea• than the lovei" 2% lillit of 1J1Z1¥ or U. r1ak program.

3.2 PRIC& MAPS

In practical appllcationa ot 11,,..,. progrwing, it ia of

interest. t.o inftstigate changes in the optiJl1m program d11e t.o Yariationa

in pricu or Tari.OWi product. produced °b7 the f ira. Such iJrrestigationa

are unaJ.q grapbi~ preaented in the fora or price mape which show

the nriows opthnm progr... Price -.pa can conceptualq be derived

tor n.rioua CGlllbinations or all pricea, wt tor the AD or aimplicit)r

ve will concern oureelTU onq vi.th price changes in one product.

3.2.1. PRICB MAPS 1'0ll .l 10-RISI PROOIWI

The net ettect of price chang• on t.he 11 near programing llOdel

conaiste or cbangee in the elrranta or the ftCtor ~· Computational short-

cuts pend.t the computation or optbnm programs for price changes without

the compl.ete :reccmpatation for each price (8).

A price map, •hawing optimua prograu for a range of tomato

prices, is pl'ellentecl in Figure 3.4. Tau.to prices ot leas than $.70 per

bushel indicate progras vit.h no tomatoeBJ bet.ween t. 70 and $. 80 tou.toea

replaee altalta. aont $.Bo no additional toaat.oea can be grown due to

PRICE MAP, NO-RISI PROORlM

56

S2 Toutoee

48

32 • .iltaU'a

! 28

24

20

16

12

8

.85 1.00

Price of Tcmatoes, Doll.an

48 the third-period labor restriction.

3.2.2 PRICE MAP FOR THE lil.Sl PROGRAM

Etf ect. of cbaDgea in the price of tomatoes were ulMNI. to deter-

mine the e.ff ect on the optima program. Theme changea are eui.J3 incor-

porated in the riak program.

Conaidflr equation (2.2) where

b • T - fr%.-1s. - - -Price cbangea for prodllct• produced by a process can be incorporated into

nw values for the vector !• It is then necessary to recompute ,!? and change

the last column of the Dllltiplier matrix z., (see appendix equation 5.2).,

and proceed with the Hildreth solution procedure.

Programs for a • l/lSOO for varioua prices of tomatoes are shown

in Table .3-2. Using thue data wa may construct, by interpolation, a price

map which shove the production programs as continuous .functions of tomato

prices. This price map ia given in Figure 3.5. The risk price map shows some aimilarities to the no-risk map.

The primary difference consists of a •flattening" of the price responser

a given price change baa more effect on the no-risk prograa. Extrapola-

t.ion indicates that tomatoes will enter the no-risk program at about 1.70

per bushel and the risk program at about $.65 per bu.Bhel.J at $.80 per

bushel the maxinmm tomato proruetion ia a.lreaq attained with the no-risk

prograaJ in the case of the risk program, production of tomatoes is still

increasing at. a price of $1.00 per b.tshel.

3 .3 RESOURCE MAPS

'l'he risk prograa is,, again• easily adaptable to the comparison

li9

T.lBLE 3-2

OPTIMUM RISI~ FOR fARIOUS TOMATO PRICES (a• 1/1500)

Price of To11atoee, Dollara Per Bushel

Item 0.10 o.ao o.as 0.90 1.00

Proceaa Intemiti•

Un:it L8Yela

Corn 22.)2 18.98 17.30 15.61 12.28

TOU'tooea 4.71 lJ.60 18.0S 22.55 .31.J.ao

.&lfalla 47.87 47.23 46.91 46.S9 45.95 Acrea

Corn 22.60 19.20 17.50 lS.84 12.46

Tomatoes 2.os 5.91 7.85 9.6J. 13.60

Altal.ta 35.40 )4.9S 34.71 34.40 34.00

Net. Revenue,

Doll.an 7490 7981 8226 847S 8963

12

8

-

so Figure 3.s

PRICE MAP, RISI PROGRAM

Corn

•

Tomatou

- - -<- - • __________ ..,_ __ _. ______ _

.il.f .Ua

.10 .80 .as .90 1.06 Price of T011at.oea,, Dollan

51 programs when the availability o! one or more scarce resources are changed.

Procedurea for doing thia have already been developed for the no-riak

case (8). In the caae of riak prognwn:Sng 1.f ve are interested in changing

the k-th resource {Tlc), it can be seen that only one element, 2bic, of the ckk

matrix% {equation s.2) 1a affected by auch a change. FrOlll equation {5.6),

~ .!,. and c. If we are interested in changing the availability of more

than one resource, a direct extension u;y be applied.

The inadequacy of computational. facilities pl'8'9'ented a con1plet.e

investigation of resource maps.

The Tar.Lance, of course, does not enter into no-riak progrumdng

and, as such, variance maps h.al'e no parallel in no-risk progranaing. Com-

putational shortcut• were investigated in connection vi.th cmparing pro-

grm119 when one or more var.1.ancu ware changed. No aiJllplif'ying transfor-

mation were :found.

CHAPI'llt IV

SOMMAR!

Linear, or no-risk, progra11111ing has been uaed for finding the

optimm coml>ination of enterprises for a firm. This method of programrd ng

baa a definite limitation in that prov.l.aiona have not been made for con-

aidering the relative riak involved in various alternative prograu. It

we incorporate the riak feature into a progranmrl ng problem by considering

the theory of cboice under risk a.a equivalent to the choice from among

several alternative probability distributions, the expected utility

hypothesis rJlt1:3" be used. Under thia bypotheaia, the entrepreneur acta aa

if he maximizes expected u till. ty which is defineds

B(y) • f y(r)f(r)dr, r

where y is the upected utility, :r(r) is the utility of a particular amount

of income, r, and f(r) is the probability density function. The resulting

expected utility dapendlS, tben, on the form of the utility of income and

on the distribution of income. AB shown by Freund (8) 1 if the utility

function of income ia assumed to be -ar y(r) • l - e

and incOllitl ia assumed to be normally distributed, the maximization of

expected utility becomes the equivalent of maximizing the linear combina-

tion of e:zpected income (M) and variance of' income ( trl)

B(y) •fl-~ 1 where •a11 represents the ri8k aversion

constant. The maximization of expected utility can be uaed for a programm1ng

analysu where it results in a quadratic programming problem.

In the example presented in this theaia, experimental. data hued

on Tarietal teat.a were used to obtain estimates of the variance of net

53 revenue of the various erope considered. The results coni'ormed very well

as to what was expected. The consideration of risk increased the import-

ance of com, the crop with the lowest unit level variance, and decreased

the illportance of tomat.oea, a c~arati vely high risk crop. In the risk

program the total revenue vu, of coune, decreued, but the expected

utility vu inereaaed and tha atandard de'Yiation of the net reTerme was

substantially decreased. The riak program also requirea leaa capital and

labor than the no-risk program.

The r:l.ak aversion constant, a, pla;ys an 1.ap1>l"tant part in the

programnd ng problea, since the wrong choice for the risk aversion con-

stant would invalic:late any results obtained. We ~ overcome thia diffi-

culty by constructing indifference ups for various values of the risk

aversion conatant, recomputing optimml riak program for each of these

values, and obtaining several point. on the opportunity line which are

tangent to the indifference curves corresponding to utility functions

with di.f'ferent values of •a•. The different point.a of tangency 11181' then

be joined together to fom. an opportunity cune. Thia opportunity curve

repreaent.a combinations of net revenue and the variance of net revenue

which ara available to the entrepreneur. An entrepreneur could choose a

point on the curve which to him represents the beat combination of net

revenue and variance. In doing so,, he will effectively be choosing his

own risk aversion constant and corresponding opti.mwn program.

Computational shortcuts for arriving at optiDllm progr8118 for

varioua risk aversion conatanta were developed. A method of dc)nstructing

resource maps using simplifying transformations was indicated. Price map•

for the risk program were constructed incorporating computational. ahort-

cuta which W8l"9 developed. Variance aapa were also investigated.

S4

CHAP'.rER V

.lPPENDil

Using the Hildreth solution procedure, n nm.at find

C • T 1.-1.r•

.2 • ! - tr ~-1.!,

!: -1. £ 'L-1 a a

The iterations of the Hildreth procedure (Section 1.2.2)

k-1 c P cld. q-1 b v~ • - !1 e~ ti{ - i~l °kt ui - 2 __!, where cij

.n.a. ckk

and b1 are elementa of C and ,21 can be simplified by defining a matrix of

lllll.tipliers, z, where

Then

Then

I•

~ • - °ti, i ; k, 1 ; p+l °kt

.kk • 0

~ ··'!ii " 0 I • • - -ell en ~l 0

c21 -- • • • -c22 c22

• • • • • • • • • • • • • • •

'id - ~ cld -c;; ~ • • • - lfkk

(5.1)

• • • 2b1 --;., 217

• • • --~2 (5.2) • • • • • • 2bic • • • --ckk

55

It can be shown that all zki' 1 r p+l and so• zit, p+l are invariant

through changes of a. Let WI define

c-. '1' %_-1.1 , where 2.-1 • aL-1. 8 , 8 2

Comider the general. element for Z

" -cki z ·-ld. ckk

2 • -i ctt.. ~. -

2 C* ki & <tic kk

where ~ are el.8119nta of C and ~ are elwnts of Cit.

Nov

b • T - frl,-ls - - -• v - f I T f. -1a - ·-

Consider the elaaent, zk,p+l

21\: 2(vk - iar f.;1ak -- . --------~ 2

ac*kk avk -T~-lsk . - . .

c*kk

(S.3)

(5.4)

(5.5)

(5.6)

Note that zk, p+l • s*k, p+l if vk • o which happens in the non-negativity

restrictions.

In order to change the riak avenion constant., a, as can be aeen

from (5.4) and (5.6), it ia onlJr neceuary to nmltipq the rrailab1llty

Tector, !• by the desired value of the risk aversion comrtant, insert. th18

in npreasion (5.6), and proceed with the Hildreth procedure as outlined

in Section i.3.3.

$6

The iterationa for the various programs become more numerous aa

the value of the risk avendon constant decreased. There were about 680

iterations to find the optimwa program for a • l/2500 aa compared to about

100 iterations tor a • l/750.

57 CH.APTER. VI

ACKNOWJ..EDGEMENTS

The author wishes to express hie appreciation to Dr. R. J.

Freund• hie thesis advisor,, whose guidance and assistance have been

inatrumental. in the completion Gf thi• thesi•· Gratitude ia also

expreased to Dr. Freund !or deTI.sing the program for the ue of the

IBM 650 computer and for arranging for. and assisting with• the use

of the IBM 650.

The author vi.shes to thank also the Institute of Statistics,

North Carolina State College, Raleigh, North Carolina,, for allowing

the author to use the facilities of the IBM COD1pUting Laboratory at

North Carolina State College.

Appreciation is a1so expressed for the encouragement and

criticisms froa Dr. Boyd Harshbarger and the other members of the

staff of the Department of Statistics,, Virginia Polyteclmic Institute.

Tho author is also indebted to Mr. Norman E. Simpson for

preparing the final copies of this theai•J to Mr. Norman P • .Miller

£or hi8 aaaiat.ance in preparing the charts and graphsJ and to my

father, Dr. William c. Rein, for his helpful criticism and suggestiona.

Finally, my wi.fe,, Sld.tsi, deserves my thanks for her hard

work, patience and encouragement.

LITERATURE Cl'l'ED1

58 CHAPTER VII

B1.BUOOP..APHI

(1) Agricultural Sta.t.ietics, prepared bf the United.ates Department of Agriculture, WaaW.ngton, D.C., J.nnual Issue, 1953.

(2) .Arrow, lennet.h J. 1951. Alternative approaches to the tbeor.r of choice in ri•k-taking situatiou. Econo•trica 1,21404-437.

(3) Bye, Rqllond f. 1956. Principlea of Economies, j,pp1eton-century-Crott., Inc., N• York.

(4) Cbarnea, .&.. 19$2. Opt.imali ty and degeneracy in linev prograww1ng. :&collOll8tri.ca 2Qt 160-170.

(5) Charnea, .l•J Henderson, A., and Cooper, w. v. 1953. An introduction to l.1nea.r progr&llJling. John Vile,y and Som, Inc., lew York.

(6) Dorfman, Robert, 19'1. J.pplication of linear prognnmhJg to the theory of the fira. Univerait7 o:f California Presa, Berkeley, Cal.if ornia.

(7) Freund, Rudolf J. 195$. Toe intro~ction of risk into a liMar pro .. grma:ing problem. Unpubliahed Ph.D. thuia, North Carolina State College, Raleigh, North Carullna.

(o) Ji'rellnd., Rudolf J. 1956. The introduction of riak into a progrundng model.. Econo•trica ~1 253-26).

(9) Freund, R.J • ., and ling, R. A. )963. The selection of optiJlal fara enterprisea1 a case etucy in linear progrun:ing. Journal Paper No. 5221 mimeographed. North Carolina Agricultural Experiment Station, Raleigh, ~!orth Carolina.

(10) Genter, c. F. Annual reports for 19491 19501 19$1, 19S2, 19531 19'4. The davelopaent and •election or adap1;ed corn lJTbrims. Virginia Agricultural Experiment Stat.ion, Blackaburg1 Virginia.

(11) Gellter, c. F ., a.nd Shull.:cwa, ~. 1?56 .. Virginia corn performance teata. Research report No. 31 March, 1956, Virginia jgricultural Experiment Station, Blackaburt;1 Virginia.

la~/ Genter, C. F., and Shu1Jrcwa1 lid. 1957. Virginia corn performance teate. Research rel)Ort No. 4, March., 1957. Virginia'Agricultu.ral. Experiment Station, BlacJcaburg, Virginia.

(13) Heady, Earl o. l9S2 Economics of Agricultural production and reaoura uae. Prentice-Hall, Inc., New Ynrk

(14) Hildreth, c. G. 1954. Point estimates vf 01d1natee 0£ concave fui"l.Ctiona. J ou.rnal of the .American Stat.i•tical .&.a•ociation 42,t 598...619.

(15) Koopmans, TJ&lli~ (.. 19$1. ActiYit7 analysia of production and allocation. John Wiley and Sona, Inc., Plew York:.

(16) luhn, H. w., and Tucker, A. w. 19$0. Non linear progr-.dng. Second Berkeley Syatposium on Mathematical Stati.Jltica and Probab111ty. U?livenity of Cal.i1'ornia Pren, Berkeley, California.

(17) Sa-Y'age.; Leona.rd J. 1954. ~.:h.r foundations of stat.iatica. John Wiley and Son11, Inc., Nev York

(18) Starling, T. M. 19$6. Small grain varietal teat.a. luearch report ?,o. $,, V~_rginia ~rlcultural Experillent Station, Bl.acbbnrg, Virginia.

(19) Starling, T. M., Traml, J. L.,, and Shulkcwa,, Id. 1956. Ii9ld performance of sr.uall gr~d.ns tested in Virginia (1945-1955). Reaearch report Mo. 4,, Anguat, 1956,, Agronomy Department., Virginia AgricuJ.tui .. al Ex_f>6r1.'llent, Station, Blacla!rbnrg, Virginia.

(20) Tintner, Gerhard, 1952. ~onometrics. John Wiley and Sona, Inc., New York.

(20a) Tintner, Gerhard. 1955. Stochastic Linear Programdng, Second Sympoai.ua on Linear Programring, ?lational Jltrean of Stand.arda, Washington, D. r.., pp 19'i'-227.

(21) Virginia Crops and Livestock, prepared by the Virginia Department of Jgr:leulture. ltichllond• Virginia• Annual Issue 1948-19$6.

The vita has been removed from the scanned document

ASPECTS OJ' RISI PROOIWIMilfG

by

Mae Buon Rein

Jl>atract submitted to the Graduate Faculty of the

Virginia Pely't.eclmie Institute

in candidacy for the degree of

MASTER or SCIBNCE

STATISTICS

.lPPROfm1 .A.PPROVED1

Director of Graduaw Studiu Head of Department

Dean of the School. of Applied Major Profeasor

Stience and Bllaineu Adm1nist.ration

Mq 19$8

Black8burg, Virginia

.ABSTRACT

The purpoae of th:la tb•ie vu to investigate certain aspect•

of riak programing.

In the computational er.apl•• aperimntal data baaed on

T&r.ietal teata were used to obtain •t:blatea of the Tal'ianc• of net

rennue of t.be T&rious cropa comsidered. Aa vu upected, the consider-

ation of riak increased the 1llportanoe of corn, the crop with the l.anat.

unit level variance• and decre•ed the ilrportance of tou\oea, a com-

paratiT~ high riak crop. In tba r1ak program, tba total renrme was,

of COIJ.l'llfl1 d8crund, bnt th& expected utilit7 wu increued and the

standard devi.Ation of the net 1"8Tenne vu substantially decreased. The

riak prograa alao requ1rea l_. capital and labor than the no-riak

prograa.

Al\ opportunity CUrYe vu formed by joining several point• of

tangetlCT between the opportunity line and ind11'ference curve correspond-

ing to ut1lit7 functions with different Taluea of the riak aversion con-

stant •a•. Thia opportunit7 curve represents combinations of net reTenue

and the variance of net revemie which are available to the entrepreneur.

An entrepreneur could choose a point on the curve which to hia represents

the best combination of net reveme and variance. In doing ao, he will

e!fectivel;r be choosing his cnm risk aversion constant and corresponding

opt:imll prograa. By this procedure, the difficulty of hypotheaising an

incorrect risk a'V9raion constant can be avoided.

Computational shortcuta for arri:rlng at optimwa progr.. for

various risk aversion constants were developed u were •thoda for YU71.ng

the price of a procea• and the availability of a •carce reaource.