Economics, ecologics, and mechanics the dynamics of responding under conditions of varying motivation

1995, 64, 405–431

JO URNAL O F TH E EXPERIMENTAL ANALYSIS O F BEH AVIO R

NUMBER

3 ( NO VEMBER)

ECONOMICS, ECOLOGICS, AND MECH ANICS:
T H E DYNAMICS OF RESPONDING UNDER
CONDIT IONS OF VAR YING MOT IVAT ION
P ET ER R. KILLEEN
ARIZO N A STAT E U N IVERSIT Y

Th e mech an ics of beh avior developed by Killeen ( 1994) is exten ded to deal with deprivation an d
satiation an d with recover y of arousal at th e begin n in g of session s. Th e exten ded th eor y is validated
again st satiation cur ves an d with in -session ch an ges in respon se rates. An omalies, such as ( a) th e
positive correlation between magn itude of an in cen tive an d respon se rates in some con texts an d a
n egative correlation in oth er con texts an d ( b) th e greater promin en ce of in cen tive effects wh en
magn itude is varied with in th e session rath er th an between session s, are explain ed in terms of th e
basic in terplay of drive an d in cen tive motivation . Th e models are applied to data from closed econ omies in wh ich ch an ges of satiation levels play a key role in determin in g th e ch an ges in beh avior.
Relaxation of various assumption s leads to closed-form models for respon se rates an d deman d fun ction s in th ese con texts, on es th at sh ow reason able accord with th e data an d rein force argumen ts for
un it price as a con trollin g variable. Th e cen tral role of deprivation level in th is treatmen t distin guish es it from econ omic models. It is argued th at tradition al experimen ts sh ould be redesign ed to
reveal basic prin ciples, th at ecologic experimen ts sh ould be redesign ed to test th e applicability of
th ose prin ciples in more n atural con texts, an d th at beh avioral econ omics sh ould con sist of th e
application s of th ese prin ciples to econ omic con texts, n ot th e adoption of econ omic models as
altern atives to beh avioral an alysis.
Key words: econ omics, ecologics, mech an ics, deprivation , satiation , motivation , arousal, deman d
fun ction s, drive, in cen tive, models, prin ciples

Th is paper compares th ree approach es to

th e prediction of beh avior th at is un der th e
con trol of in cen tives an d supported by motivation al states of var yin g in ten sity. Behavioral
economics frames beh avior as an exch an ge of
goods, an d motivation as th e optimization of
th e trade-offs required by th e con strain ts of
time an d experimen tal con text in order to
obtain th e best immediate or delay-discoun ted package of goods. Ecologics respects th e
n atural ecology of th e subject an d rejects th e
logic of th e marketplace an d th eoretician for
th at of an organ ism adapted by evolution ar y
fo r ces to co m p lex n atu r al en vir o n m en ts.
Ecologics frames beh avior as n ested sets of
systems or action pattern s, an d motivation as
regulation —th e defen se of setpoin ts with in
th ose system states. Both of th ese approach es
are teleon omic or fun ction al, focusin g on fin al causes, on outcomes: Th e econ omic organ ism beh aves so as to optimize packages of
Th is research was supported by NSF Gran ts IBN9408022 an d BNS 9021562. It ben efited greatly from th e
reviewers’ commen ts, alth ough it is un likely th ey would
en dorse all of th e claims of th is version .
Address correspon den ce to Peter R. Killeen , Departmen t of Psych ology, Box 871104, Arizon a State Un iversity, Tempe, Arizon a 85287-1104 ( E-mail: KILLEEN@
ASU.EDU) .

goods, an d th e ecologic organ ism beh aves to
min imize deviation s from optimal setpoin ts
in its parameter space. Mechanics focuses on
th e efficien t rath er th an th e fin al causes of
beh avior, an d provides a set of formal causes—a set of math ematical models—th at expan ds simple assertion s of causal agen cy in to
more precise fun ction al relation s between
variables. Th e mech an ical organ ism is n ot beh avin g to o p tim ize an yth in g; in citem en t
makes it active, satiation decreases its excitability, an d co-occurren ce of particular respon ses with in cen tives in creases th e probability of th ose respon ses. Th e primar y goal of

th is paper is to develop th e mech an ics to th e
poin t at wh ich it is applicable to th e experimen tal con texts th at are favored by econ omic
an d ecologic th eorists.
MECH ANICS
A recen t mon ograph ( Killeen , 1994) proposed a mech an ics of beh avior based on
th ree prin ciples con cern in g th e n ature of
arousal, temporal con strain t, an d couplin g
between respon din g an d in cen tives. Th e first
prin ciple was th at in cen tives excite respon din g, so th at arousal level ( A) is proportion al

405


406

PETER R. KILLEEN

to rate of in citemen t ( R; a will be defin ed
below) :
A ϭ aR.

( 1)

But th ere are con strain ts. Th ere is on ly so
much time available in wh ich to respon d ( Killeen ’s secon d prin ciple) , an d for a particular
target respon se to be differen tially excited by
an in cen tive, it must be paired with th at in cen tive; th ey must coreside in th e an imal’s
sh ort-term memor y ( th e th ird prin ciple) . It is
on ly wh en effective con tin gen cies couple an
in cen tive with a respon se th at th e in cen tive

becomes a rein forcer. Th ese th ree prin ciples
provided th e bases for models of th e beh avior
gen erated by various sch edules of rein forcemen t. For in stan ce, th e th eor y predicts respon se rates on in ter val sch edules to be
Bϭ

kR
R
Ϫ ,
R ϩ 1/ a
␭

␭, a Ͼ 0,

( 2)

wh ere k is proportion al to th e maximal attain able respon se rate, R is th e rate of rein forcemen t, a is a key parameter wh ose mean in g will be developed below, an d lambda ( ␭)
is th e rate of decay of memor y for a respon se.
Note th at with out th e subtrah en d, th is is essen tially H errn stein ’s h yperbola, wh ich h as
been demon strated to predict respon se rate
over a wide ran ge of con dition s ( see, e.g., de
Villiers & H errn stein , 1976) . Th e subtrah en d
comes in to play on ly at ver y h igh rates of rein forcemen t ( R Ͼ 2 per min ute) , wh ere an
in creasin g fraction of th e in cen tive bears on
th e prior con summator y respon se, stren gth en in g it rath er th an th e in strumen tal respon se. Because th e subtrah en d is importan t
on ly un der ver y h igh rates of rein forcemen t,
it will be set to zero for th e rest of th is paper,
because th is simplifies an alysis an d in curs
on ly a small decrease in goodn ess of fit.
T he Specific Activation of In cen tives
Th e parameter a, wh ich I h ave called th e

specific activation , is of greatest con cern in
th is paper. In H errn stein ’s ( 1974) formulation , RO ϭ 1/ a was treated as th e rate of rein fo r cem en t availab le fr o m so u r ces o th er
th an th ose sch eduled by th e experimen ter.
Th is in terpretation h as n ot been supported
by subsequen t research ( e.g., Bradsh aw, Szabadi, Ruddle, & Pears, 1983; Dougan &
McSween ey, 1985; McSween ey, 1978) . Ac-

cordin gly, some in vestigators ( e.g., Bradsh aw,
Ruddle, & Szabadi, 1981) h ave more agn ostically called th e parameter th e half-life constant, because respon se rate attain s h alf its
maximal value wh en R equals RO .
In earlier work on in cen tive motivation ,
Killeen , H an so n , an d O sb o r n e ( 1978)
sh owed th at each in cen tive delivered un der
con stan t con dition s will gen erate a total of a
secon ds of beh avior. It follows th at R in cen tives will gen erate th e poten tial for aR secon ds of respon din g, an d th ey called aR th e
organ ism’s level of arousal. Th e particular
form of respon din g gen erated by th at arousal
depen ds on th e con tin gen cies th at determin e
just wh at particular respon se will occur before th e deliver y of th e in cen tive. It is th is
couplin g of respon ses to in cen tives th at con stitutes rein forcemen t. Wh en th e couplin g
approach es its maximum ( 1.0) , as it does on
sh ort ratio sch edules, most of th e beh avior of
th e organ ism is con cen trated on th e target
respon se. Wh en th e couplin g is ver y weak, as
in sch edules of beh avior-in depen den t rein forcemen t, beh avior is diffuse an d drifts toward adjun ctive forms. But in all cases, th e
total amoun t of time spen t respon din g is a
fun ction of th e arousal level of th e organ ism,
wh ich is a product of th e specific activation
of th e in cen tives ( a) an d th e rate of th eir deliver y ( R) . It is th ese con sideration s th at gave
rise to Equation 1.

We may simplify Equation 2 by droppin g
its subtrah en d, an d we may multiply its n umerator an d den omin ator by a to reveal
more clearly th e multiplicative in teraction between in cen tive factors summarized by a, an d
rate of in citemen t, R:
Bϭ

kaR
.
aR ϩ 1

( 3)

Equation 3 is h yperbolic in aR because of th e
n on lin earities in troduced by ceilin gs on respon se rate. Wh en we are operatin g well below th ose ceilin gs, it reduces to th e simple
proportion al model, th e first prin ciple of th e
mech an ics. Wh ereas Equation 2 emph asizes
th e relation of th is model to H errn stein ’s h yperbola, Equation 3 remin ds us of th e multiplicative relation between a an d R as th ey
con join tly determin e arousal level an d respon se rate.
Terminology. It is worth an aside to clarify


MECHANICS
th e termin ology used th rough out th is paper.
Th e above equation s were proposed as equilibrium solution s for wh en th e beh avior un der study h as come to a steady state. In ph ysics th e study of systems at equilibrium is
called statics; an alogously, th e above equation s are part of a statics of beh avior. Much
of th e recen t research in beh avior an alysis
con cern s such asymptotic beh avior. It derives
from a tradition of descriptive beh aviorism;
wh en ever a cumulative record is displayed or
a regression is fit th rough a scatter of data,

th e goal is description . Th is is a first step toward a more gen eral scien ce: ‘‘Galileo was
con cern ed n ot with th e causes of motion but
in stead with its description . Th e bran ch of
mech an ics h e reared is kn own as kinematics;
it is a math ematically descriptive accoun t of
m o tio n with o u t co n cer n fo r its cau ses’’
( Frautsch i, O len ick, Apostol, & Goodstein ,
1986, p. 114) . It follows in th e Pyth agorean
tradition th at ‘‘approach ed ph en omen a in
terms of order an d was satisfied to discover
an exact math ematical description ’’ ( Westfall,
1971, p. 1) . Th ere are man y examples of such
a tradition in psych ology today, in cludin g descriptive statistics, th e laws of psych oph ysics,
an d th e origin al match in g law.
Th e study of forces th at cause objects to
move is called dynamics; dyn amics con stitutes
‘‘a th eor y of th e causes of motion ’’ ( Frautsch i
et al., 1986, p. 114) . Beh avior is th e motion
of organ isms, an d th e study of ch an ges in beh avior as a fun ction of motivation , learn in g,
an d oth er causal factors con stitutes a dyn amics of beh avior. Examples in th e beh avioral
literature are provided by H iga, Wyn n e, an d
Staddon ( 1991) , Staddon ( 1988) , an d Myerson an d Miezin ( 1980) ; Marr ( 1992) provides
an over view. A framework th at embraces all
of th e above special cases is called a mechanics.
Th is term does n ot n owadays refer to h ypoth etical in tern al mech an ical lin kages; such
mach in er y is th e vestige of th e Cartesian tradition in wh ich Newton labored wh en h e began to establish th e modern scien ce of mech an ics. Th at mech an ical tradition sough t to
provide causal explan ation s of ph en omen a,
alth ough such causes were often n arrowly
con strued as material causes in volvin g th e
motion s of particles or aggregation s of matter

un derlyin g th e ph en omen a. It was on e of
Newton ’s ch ief disappoin tmen ts th at h e was
n ever able to provide such a ‘‘mech an ical’’

407

substrate for forces such as gravity, an d h e fin ally repudiated kn owledge of such h ypoth etical causes in h is famous ‘‘h ypoth eses n on
fin go,’’ offerin g in stead a precise math ematical description of th e effects of th ose forces.
H is dyn amical th eor y recon ciled ‘‘th e tradition of math ematical description , represen ted by Galileo, with th e tradition of mech an ical ph ilosoph y, represen ted by Descartes’’
( Westfall, 1971, p. 159) .
As is th e case in ph ysics, in beh avior an alysis th e term mechanics is someth in g of an atavism; but in both cases, it may be in terpreted
as an emph asis on th e an alysis of complex
resultan ts in to th eir con stituen t forces, as a
focus on causal rath er th an statistical explan ation s, an d on math ematical rath er th an
mech an ical lin kages between cause an d effect. It is in th ose sen ses, on es common to
th e beh avior-an alytic tradition , th at it is used
h ere. It embraces molecular models such as
melioration , but n ot teleological models such
as th ose predicated upon optimization . It in volves th e th eoretical con structs of value an d
drive. Th eoretical con structs are as n ecessar y
for a scien ce of beh avior as th ey are for an y
oth er scien ce ( Williams, 1986) ; th is was recogn ized by Skin n er th rough out h is career, begin n in g with h is argumen t for th e gen eric n ature of th e con cepts stimulus an d respon se
( Skin n er, 1935) , th rough h is defen se of drive
as a con struct th at can make a th eor y of beh avior more parsimon ious overall ( Skin n er,
1938) , to h is fin al writin gs. Th e issue, as Skin n er an d oth ers ( Feigl, 1950; Meeh l, 1995)
h ave stated, is n ot wh eth er such con structs
are h ypoth etical, but wh eth er th ey pay th eir
way in th e cost-ben efit ratio of con structs to
prediction s. Th is article requires a loan of th e
reader’s patien ce as th ese con structs are developed an d deployed, in th e h ope th at th e

th eor y will in th e en d be judged a worth wh ile
con tribution to th e experimen tal an d th eoretical an alysis of beh avior.
Open Versu s Closed Econ omies
O n e of th e key con dition s th at is assumed
to be con stan t in Killeen ’s ( 1994) mech an ics,
but th at varies substan tially in th e real world,
is th e value of th e in cen tive to th e organ ism.
Th is value depen ds both on th e in trin sic
qualities of th e in cen tive—wh at H ull an d h is
studen ts den oted by K an d called incentive-motivation—an d th e h un ger, th irst, or ‘‘drive’’ of


408

PETER R. KILLEEN

th e organ ism, wh ich th ey den oted by D ( e.g.,
H ull, 1950; Spen ce, 1956) . Much of th e early
research on th ese factors was an essen tially
qualitative an alysis of th e differen tial role
th ey played in motivation . Th e presen t con cern is th e developmen t of a quan titative
an alysis, on e th at proceeds by expan din g th e
sin gle parameter a ( th e specific activation of
an in cen tive) in to compon en ts akin to K an d
D. H ere th ese con structs are developed out
of th e already-establish ed statics ( Equation s 1
th rough 3) an d provide th e motivation al
‘‘causes’’ th at tran sform it in to a dyn amics.
All of th e data an alyzed un der th e origin al
formulation of th e mech an ics were derived

from an imals at h igh levels of deprivation ,
wh ich often requires supplemen tar y feedin g
in th e h ome cages. But beh avioral econ omists
h ave argued th at such con dition s provide a
restricted, perh aps even an omalous, perspective on beh avior, an d th at our an alysis will
h ave more ecological validity to th e exten t
th at we permit our subjects to earn th eir complete daily ration un der th e con strain ts of th e
sch edule we study, in th e process often permittin g th em to approach ad libitum repletion by th e en d of th e ( exten ded) daily session . Th e tradition al procedure h as been
called an open economy because th e subject is
main tain ed by food an d water extrin sic to th e
sch edule con tin gen cies; th e latter arran gemen t h as been called a closed economy. Collier,
Joh n son , H ill, an d Kaufman ( 1986) ch risten ed th e tradition al open -econ omy procedure th e refinement paradigm, ‘‘developed in
classic ph ysics, first en un ciated for an imals by
Th orn dike ( 1911, pp. 25–29) an d per fected
by Skin n er ( 1938) , H ull ( 1943) , th eir studen ts, an d th eir con temporaries’’ ( Collier et
al., p. 113) . Because postsession feedin g is
on e of th e least importan t distin ction s between open an d closed econ omies, because
description of th e procedure as an econ omy
con stitutes a commitmen t to a particular explan ator y framework, an d because th e refin emen t paradigm is th e ideal con text in wh ich
to refin e basic prin ciples, th eir term is utilized th rough out th is paper.
A n umber of research ers h ave adopted th e
econ omic an alysis of sch edule effects, with
th eir design s often in volvin g n ovel sch edules
of rein forcemen t. H ursh ( e.g., H ursh , 1984)
h as sh own th at th e ver y type of fun ction s an alyzed by Killeen ( 1994) look quite differen t

Fig. 1. A revision of th e figure drawn by H ursh
( 1980) , sh owin g th e differen ces in pattern s of respon se
rates of mon keys un der open an d closed econ omies, as
a fun ction of th e in terrein forcemen t in ter val on variablein ter val sch edules. Th e cur ves are drawn by Equation 8Ј.

See H ursh ( 1978) for procedural details an d origin al
data.

u n d er a clo sed eco n o m y. Fo r in stan ce
H ursh ’s ( 1980) Figure 4 sh owed respon se
rate decreasin g sligh tly as th e sch eduled rate
of rein forcemen t decreased in an open econ omy, just as we would expect from Equation s
2 an d 3, but increasing markedly in a closed
econ omy. Figure 1 sh ows th ose data ( derived
from H ursh , 1978) . Th is con stitutes a serious
th reat to beh avioral mech an ics an d to all oth er th eories th at en tail th e H errn stein h yperbola. H ursh argued th at ‘‘It is th e econ omic
system wh ich produced th e differen t results’’
( 1980, p. 223) . But just wh at was it about th e
differen t systems th at made th e differen ce?
H ursh ’s explan ation is in terms of elasticity of
demand. ‘‘In th e closed econ omy with n o substitutable food outside th e session , deman d
was inelastic; in th e open econ omy with con stan t food in take arran ged by th e experimen ter, deman d was elastic’’ ( H ursh , 1980, p.
233) . Elastic goods are th ose such as luxuries
for wh ich in creases in price causes decreases
in willin gn ess to work for th em or in th e
amoun t th at will be paid for th em ( demand) ;
in elastic goods are th ose such as basic n eeds
for wh ich moderate in creases in cost h ave little margin al effect on deman d; customers will
pay wh at th ey h ave to to main tain con sumption ( Kooros, 1965; Lea, 1978) . Elasticity is
measured as th e proportion al ch an ge in de-


MECHANICS
m an d th at r esu lts fr o m a p r o p o r tio n al
ch an ge in price. For th e closed econ omy, as

th e rein forcemen t rate decreases ( movin g to
th e righ t on th e x axis of Figure 1) , price in creases ( an imals get less food per respon se)
an d th ere is a con comitan t in crease in respon se rates. Th e flat fun ction s for th e open
econ omy suggest an elasticity n ear un ity, as
sh ould be th e case: If you can get it after th e
session for free, you sh ouldn ’t work h arder
for it wh en prices go up. ( Th e proper x axis
for th e econ omic an alysis is un it price—respon ses per un it of rein forcer—wh ich is h igh ly correlated with mean time between rein fo r cer s at m o st r esp o n se r ates. At lo w
respon se rates on in ter val sch edules, h owever, price is positively correlated with respon se
rate. Strictly speakin g, th is latter depen den cy
makes econ omic an alyses in appropriate for
in ter val sch edules, because ‘‘In order to deduce th e sh ape of th e deman d for a con sumer good, th e first assumption on e sh ould
make is [ th at] n o in dividual buyer h as an y
appreciable in fluen ce on th e market price;
n amely, th e price is fixed’’ Kooros, 1965, pp.
51–52.)
Beh avioral econ omics provides an in terestin g perspective in a field in wh ich th e data
are rich an d complicated an d th e poten tial
for bridgin g to an oth er disciplin e is so clear.
But is it th e righ t perspective? Does respon din g con stitute a cost—do an imals meter key
pecks th e way h uman s do pen n ies? Do th ey
an ticipate en d-of-session feedin gs? Just wh y
sh ould th e rates un der th e closed econ omy
gen erally be lower th an th ose un der th e open
econ omy, if in th e latter case an imals can
ban k on a postsession feedin g? Wh y sh ould
rates fall to n ear zero for th e variable-in ter val
( VI) 20-s sch edule in th e closed econ omy in
con trast with th e open econ omy? H ow are
th ese effects predicted from econ omic th eor y? Elasticity migh t describe, but can n ot explain , th ese differen ces; n or h ave econ omists

explain ed wh y elasticity itself sh ould var y con tin uously with price, as is usually th e case for
beh avioral data. A simpler h ypoth esis can explain th e differen ces in th e data un der th ese
two experimen tal paradigms: In th e closed
econ omy th e subjects are closer to satiation
more of th e time, especially at small VI values;
subjects from th e open econ omy, bein g h un grier, respon d at a h igh er rate. To formalize
th is treatmen t requires an expan sion of th e

409

mech an ics to h an dle deprivation an d in cen tive motivation .
H U NGER
Wh ere does deprivation level en ter th e basic prin ciples of rein forcemen t? Th e primar y
effect will be on th e specific activation associated with an in cen tive: Th e value of a in
Equation 1 will decrease with satiation . Th e
level of in citemen t th at a small ban an a pellet
will provide to a satiated mon key will be less
th an th at provided to a h un gr y on e.1 Th e
closer an an imal is to its n atural rate of in take
un der ad libitum feedin g, th e smaller a
sh ould be. Similarly, th e in citemen t from a
small ban an a pellet will be less th an th at from
a large ban an a pellet. Th erefore, th e parameter a must be expan ded from a sin gle free
parameter to a product of th e organ ism’s
h un ger an d th e value of th e in cen tive in alleviatin g h un ger. To be con crete, let us th in k
of th e h un ger drive in th e simplest terms:
Con sider th e metabolic system to be a vessel
th at stores a fin ite amoun t of food an d utilizes it at a con stan t metabolic rate M. Th e
con text permits th e organ ism to acquire n ew
food of average magn itude m at th e rate of R

( see th e Appen dix for a review of th e con stan ts an d th eir dimen sion s) . Depen din g on
th e recen t h istor y of depletion an d repletion ,
th ere will be more or less food in store. To
be precise, we would n eed to deal with a cascade of storage devices ( i.e., th e mouth , th e
stomach , th e bloodstream, th e adipose tissue) , each with th eir own release rates; differen t types of food will affect th ese differen tly. Bulky food may fill th e mouth an d
stomach but do little to alleviate deep h un ger, wh ereas sugars may immediately release
stored glucose in to th e bloodstream wh ile
leavin g th e stomach relatively empty. We will
n ot con fron t th ose details h ere: Th in k in
terms of th e stomach ( or crop) an d some
stan dard food such as th ose typically used as
rein forcers. In th is simplest in stan tiation , th e
d eficit is th e em p tin ess o f th e sto m ach .
1 Secon dar y motivation al effects on all th e parameters
are likely. For in stan ce, a weakly motivated organ ism
migh t take lon ger to complete a respon se, lowerin g th e
ceilin gs on respon se rate ( see, e.g., McDowell & Wood,
1984, an d Equation 3Ј below) . But th is paper focuses on
th e primar y motivation al effects, wh ose locus of action is
on th e parameter a.


410

PETER R. KILLEEN

Ch an ges in th e deficit will depen d on th e balan ce between th e rates of emptyin g th e stomach ( depletion ) an d of fillin g it ( repletion )
over time. In th e case in wh ich both th e in put
rate ( mR) an d th e output rate ( M) are con stan t over th e in ter val t, th e deficit at time t,
dt, is

dt ϭ d 0 ϩ ( M Ϫ mR) t,

( 4)

wh ere d0 den otes th e in itial deprivation level.
Bou n dary Con dition s
It is worth a con crete discussion h ere of
two of th e variables ( d0 an d M) in Equation
4, because th ey recur th rough out th e paper
an d will often be set to fixed values. In an
open econ omy, th e experimen ter migh t deprive th e organ ism for several days, but n o
matter h ow deprived, an imals can eat on ly
un til th eir stomach s are full. In th ese cases
th e in itial deficit d0 takes th e value of th e
maximum capacity of th e stomach . For rats,
th e typical maximum meal size is about 4 g
( see, e.g., Joh n son & Collier, 1989, 1991) . For
an imals such as pigeon s with a crop or mon keys with ch eek pouch es, a meal can be much
more substan tial. Th is is also th e case for rats
wh en th eir en viron men t permits th em to
h o ar d . T. Reese an d H o gen so n ( 1962)
sh owed th at for deprivation times over 24 h r,
pigeon s will con sume approximately 10% of
th eir free-feedin g weigh ts. Zeigler, Green ,
an d Leh rer ( 1971) foun d th at in th e course
of an h our, 10 Wh ite Carn eaux th at h ad been
deprived to 80% of th eir ad libitum weigh ts
con sumed 40 g of mixed grain on th e average; th is is con sisten t with Reese an d H ogen son ’s estimate of d0.
In closed econ omies in wh ich in itial deprivation times are min imal, d0 will be small an d
may usually be set to zero. Un der th ese con dition s deprivation will grow with time sin ce

th e last meal ( t) accordin g to Equation 4 un til h un ger motivation exceeds th e th resh old,
at wh ich poin t an oth er meal will be in itiated.
Pigeon s of typical size require between 0.5
an d 1 g/ h r to main tain th eir weigh ts between
80% an d 100% of ad libitum, an d th e requiremen ts for rats also fall with in th at ran ge.
Th ese values for M are sufficien tly smaller
th an th e rates of repletion in typical ( open
econ omy) experimen ts th at on e may set M ϭ
0, as is don e in all of th e subsequen t an alyses
in th is paper.

Drive Versu s Deficit
Wh at is th e relation between th e h un ger
drive ht an d deficit dt? Th e simplest model
makes h un ger proportion al to deficit, ht ϭ
␥dt, so th at from Equation 4
ht ϭ ␥[ d 0 ϩ ( M Ϫ mR) t] .

( 5)

Altern ate models of th is basic process are possible. Equation 5 is similar to a regulator y
model proposed by Ettin ger an d Staddon
( 1983) . Town sen d ( 1992) explored a dyn amic motivation al system th at, in place of Equation 5, h ad motivation grow as a fun ction of
th e deviation between th e curren t motivation al level an d th e ideal, with a th resh old
th at motivation must exceed before respon din g will be in itiated. Solution of such a model
leads to motivation th at grows expon en tially
with time, rath er th an lin early:
ht ϭ e␥[ d 0ϩ( MϪmR ) t] Ϫ ␪.

( 6)

With th e th resh old equal to 1.0, motivation
will be zero wh en deprivation level is zero. In
th e case of ␪ Ͼ 1, it requires more th an th e
min imal amoun t of deprivation for th e subject to begin respon din g. In th e case of ␪ Ͻ
1, th e subject will con tin ue respon din g even
wh en satiated ( Morgan , 1974) , eith er because
con dition in g h as created some beh avioral
momen tum or because th e drive is also main tain ed by oth er deprivation s ( e.g., dilute sucrose solution s will assuage both h un ger an d
th irst) . In th e lin ear model, th resh old effects
are absorbed in to th e deficit parameters.
Th e expon en tial model h as some face validity, in th at in trospection suggests th at th e
exigen cy of h un ger seems to grow more
steeply th an lin ear with deprivation time. It is
con sisten t with con trol-systems an alyses of
motivation al systems ( e.g., McFarlan d, 1971;
Toates, 1980) . Serious studen ts of th ese issues
will fin d an excellen t review of th e curren t
state of research on appetite an d its n eural
an d beh avioral bases in Legg an d Booth
( 1994) .
Yet an oth er model of h un ger would h ave it
grow sigmoidally with deprivation , approach in g a ceilin g at th e h igh est levels of deprivation . Such a model is outlin ed in th e Appen dix; its application did n ot improve an y of th e
an alyses, an d so it is n ot pursued h ere.
Equation s 5 an d 6 sh ow th at wh en an an imal becomes satiated ( wh en th e in itial deficit


MECHANICS
is replaced an d depletion is just balan ced by
repletion ) , ht falls below th resh old, drivin g

motivation to zero an d carr yin g respon se rate
alon g with it. Food-motivated beh avior ceases,
preven tin g overin dulgen ce th at would drive
h un ger levels to a n egative value. Con tin gen cies of rein forcemen t th at require con sumption for access to oth er in cen tives, h owever,
could drive ht to a n egative value. In th is case,
respon se rates are depressed below free base
rates ( Allison , 1981, 1993) , requirin g extern al
force or th e passage of time to overcome th at
in h ibition .
Aggregatin g Over a Session
For th e lin ear model, th e average drive level over th e course of a session of durationtsess
is given by Equation 5, with t ϭ tsess/ 2 ( see
th e Appen dix) . Un der th e expon en tial drive
model, th e situation is more complicated. If
session duration is con stan t, th e average drive
level is given by Equation 6, with t ϭ tЈ, some
un defin ed fraction of tsess. In employin g th e
expon en tial model, on e may set tЈ to some
arbitrar y value ( e.g., tsess/ 2) an d let th e remain in g parameters adjust th emselves to th at
con strain t.
Econ omic Tran slation
In econ omic parlan ce, d0 is th e debt, mR is
th e wage, an d M is th e cost of doin g busin ess.
O n ratio sch edules th e rate of rein forcemen t
R is an in verse fun ction of th e ratio size ( n) ,
or price, an d n/ m is th e un it price. M, th e
rate of utilization of food by a free-feedin g
organ ism, is th e coordin ate of th e ideal, or
bliss poin t, alon g th e food con sumption axis.
It could be separated in to fixed cost or overh ead ( basal metabolic rate) , an d production

cost ( respon se effort) . Basal metabolic rate
con stitutes th e major cost of foragin g an d
th us con stitutes a sign ifican t ‘‘sun k cost’’ to
an y en deavor: O n ce stan din g, it doesn ’t require much more en ergy to do an yth in g.
( Th is distin ction implies flat optima for models of foragin g th at maximize calories gain ed
per calories of effort expen ded; more precise
feedback is provided by optimizin g calories
gain ed over time expen ded.)
Th e parameter ␥ represen ts th e cost of deviation s from th e ideal, an d e␥ provides on e
in dex of th e elasticity of deman d. If ␥ is large
( an d th us e␥ Ͼ 1) , th e an imal is ver y sen sitive
to deviation s from th e ideal rate of repletion ,

411

an d deman d is said to be in elastic. If ␥ is
small ( an d th us e␥ ഠ 1) , th en ch an ges in price
elicit on ly min imal beh avioral adjustmen ts;
deman d for th e commodity approach es un it
elasticity. If ␥ is n egative ( an d th us e␥ Ͻ 1) ,
an imals will work less for a commodity as its
price in creases, an d deman d is said to be elastic. Th is occurs in th e presen ce of substitutes,
as wh en food is available for respon din g on
oth er levers ( Joh n son & Collier, 1987) . Th is
in terpretation of elasticity differs from th at of
th e econ omists, because th eirs refers to deman d as a fun ction of price but does n ot take
deprivation levels in to accoun t. Econ omic
models are design ed to map population effects, n ot biological on es. Saturation of th e
market is treated with differen t models th an
elasticity. ‘‘Decr easin g m ar gin al u tility o f

goods’’ captures some of th e idea of satiation ,
but is usually con strued with out referen ce to
th e curren t deficit.
Th e presen t approach predicts th at th e
econ omists’ measure of elasticity will ch an ge
with price, because on ratio sch edules th e
rate of rein forcemen t, R, wh ich appears in
th e righ t sides of Equation s 5 an d 6, equals
m/ n, th e reciprocal of un it price. Motivation
varies with price because th at affects th e rate
of repletion . In deed, H ursh , Raslear, Bauman , an d Black ( 1989) foun d elasticity to
var y as a lin ear fun ction of un it price. But th is
is n ot because ␥ h as ch an ged; our measure
of elasticity, e␥, may stay con stan t over ch an ges
in motivation because we h ave moved th e
con trollin g variables in to our in depen den t
variables ( Equation s 5 an d 6) , an d th erefore
do n ot n eed to let our th eoretical con stan ts
var y with our in depen den t variables.
Ecologic Tran slation
M is th e setpoin t repletion rate th at an imals will defen d. Equation 4 provides a measure of deviation from th at setpoin t. Defen se
of th e setpoin t is equivalen t to an imals’ attemptin g to min imize th at deviation , th at is,
set th e derivative to zero. Th e force of th is
equilibration is given by ␥. In con trol-systems
parlan ce, ␥ represen ts th e regulator y gain , or
restorin g force. Man y differen t arran gemen ts
of con tin gen cies will gen erate man y differen t
con stellation s of beh avior, all of wh ich h ave
on ly on e th in g in common an d predictable;
th e absolute value of Equation 4 will be min imized. Th is approach th erefore is like th e

412

PETER R. KILLEEN

H am ilto n ian ap p r o ach to m ech an ics, in
wh ich all of th e laws of mech an ics may be
derived from min imization of a sin gle differen tial equation called th e action. It is th e core
assumption of regulator y approach es to beh avioral econ omics such as Allison ’s ( 1983) .
Th e curren t approach also recogn izes th e
boun dar y con dition s to th is min imization :
Th e ch an ges in motivation will n ot be revealed in beh avior un til th ey cross a th resh old
for action , an d th ey will n ot con tin ue on ce
th e capacity of th e organ ism is saturated.
An Application of the Basic Model
to Satiation Cu rves
H ow does drive level in teract with magn itude or quality of th e in cen tive? Th e simplest
assumption is multiplicative: Absen t eith er
drive or a viable in cen tive, th e specific activation a must be zero. We may call th e in cen tive variable v. Th en at ϭ vht. Th e value of an
in cen tive will n ot gen erally be proportion al
to its magn itude, alth ough a lin ear relation
may be an adequate approximation if th e
ran ge of variation is small.
In accord with th e above an alysis, for th e
lin ear drive model we expan d th e specific activation to
at ϭ vht ϭ v ␥[ d 0 ϩ ( M Ϫ mR) t] ,

( 7)

wh ere ( M Ϫ mR) is th e balan ce between depletion an d repletion , an d its multiplication
by t gives th e cumulative effects of th at balan ce. Th is equation h as replaced a as a sin gle
free parameter with a th ree-parameter model: value v, th e in itial deficit d0, an d th e depletion rate M. ( For th e lin ear model th e deviation -cost parameter ␥ is redun dan t with
th e value parameter v an d may be absorbed
in to it or simply set to 1.0.) Equation 7 may
th en be in serted in to Equation 3 to predict
r esp o n se r ates o f an im als u n d er in ter val
sch edules wh en deprivation levels var y.
Fisch er an d Fan tin o ( 1968) provided th e
data aroun d wh ich th e lin ear model was developed. Th ey deprived pigeon s to 80% of
th eir ad libitum weigh ts, an d train ed th em to
respon d on ch ain ed VI 45 VI 45 sch edules,
exten d in g th e sessio n s u n til r esp o n d in g
ceased. Th e rein forcer con sisted of access to
a h opper of mixed grain for 2, 6, 10, or 14 s.
Figure 2 sh ows th e resultin g satiation cur ves
in th e termin al lin ks of th e ch ain an d in th e
in itial lin ks. Alth ough th e data th emselves

Fig. 2. Respon se rates un der ch ain ed sch edules for
pigeon s receivin g differen t duration s of access to th e
h opper durin g exten ded session s ( Fisch er & Fan tin o,
1968) . Th e data represen ted by filled symbols come from
th e termin al lin k, an d th ose by open symbols from th e
in itial lin k. Th e cur ves are drawn by Equation s 3 an d 7,
an d represen t per forman ce for 2-s ( in verted trian gles) ,
6-s ( trian gles) , 10-s ( squares) , an d 14-s ( circles) access to
food.

sh ow rath er un excitin g mon oton ic decreases

with n umber of feedin gs, th e model provides
a ration al fit to th em. Th e first step was to
estimate th e amoun t of food obtain ed un der
th e differen t con dition s, because amoun t
con sumed is n ot proportion al to h opper duration . Fortun ately, Epstein ( 1981) publish ed
a useful graph givin g th e amoun t con sumed
from a h opper of th e design used in th is
study. For th ese h opper duration s th e regression gave th e amoun ts as 0.13, 0.28, 0.35, an d
0.36 g of mixed grain .2 I used th ose n umbers
as estimates of m.
Th e pigeon s’ weigh ts were reduced to 80%
of th eir free-feedin g weigh ts. To optimize th e
goodn ess of fit, I set th e parameter k in Equation 3 to 200 respon ses per min ute for th e
termin al lin k an d 64 respon ses per min ute
for th e in itial lin k. Th e in itial deprivation d0
took a value of 57 g. Th e value parameter v
was 1.5 s per rein forcemen t. Th e expon en tial
drive model provides a comparable fit to
th ese data. Given th e n ecessar y approximation s, th e fit of th e model to th e data is per2 For Leh igh Valley feeders th e n umber of grams eaten
approximates a lin ear fun ction of h opper duration , with
a slope of 0.06 g/ s an d an in tercept of 0.2 g ( Epstein ,
1985) . Pigeon s feedin g ad libitum are less efficien t, with
typical eatin g episodes lastin g 7 s, durin g wh ich 0.33 g
are con sumed ( H en derson , Fort, Rash otte, & H en derson , 1992) .


MECHANICS
h aps acceptable, alth ough respon din g in th e
in itial lin ks decreased at a faster rate th an
predicted, especially for th e 14-s h opper con d itio n . ( Len d en m an n , Myer s, & Fan tin o ,

1982, foun d a similar h ypersen sitivity in th e
in itial lin ks in respon se to variation s in duration of rein forcemen t, as did Nevin , Man dell, & Yaren sky, 1981, in respon se to satiation .) It may be th at in all cases decreased
motivation h as its primar y effects on pausin g,
an d on ce an an imal h as begun to respon d, it
con tin ues un til rein forcemen t. If th is is th e
case, th en pausin g will occur primarily in th e
initial links, with animals responding throughout the terminal links. Segmenting responding
will thus put the greatest leverage of motivation
on the earliest segments. (See Williams, Ploog,
& Bell, 1995, for further analyses of these
chain-schedule effects.)
We can write th e above models in a more
con den sed form. Set th e metabolic rate M to
0, th e magn itude of th e in cen tive m to 1, an d
let th e gain parameter ␥ be absorbed in to v;
th en write Equation 3 as
Bϭ

kR
.
R ϩ 1/ [ v( d 0 Ϫ Rt) ]

( 8)

Th is equation reiterates th e above description s, but also provides quan titative prediction s: Because of satiation effects, respon se
rate is a quadratic fun ction of rein forcemen t
rate. Un der con dition s of large in itial deficit
( d0) relative to repletion ( Rt) , th e paren th etical expression is essen tially con stan t an d can
be absorbed by v, wh ich return s to us our simple Equation 3 ( or 3Ј, below) . Th e H errn stein
h yperbola is th us valid primarily for session s

of sh ort duration or low rate of rein forcemen t, wh ere th e in itial deficit outweigh s th e
cumulative repletion . But satiation effects
grow with t, an d become domin an t later in a
session .
If on e is in terested in estimatin g th e parameters in H errn stein ’s h yperbola, th en it is
better to use data from early in a session in
wh ich repletion ( Rt) is low relative to in itial
deficit ( d0) , or from sh ort session s, so th at th e
den omin ator is relatively con stan t. Better yet,
use Equation 8 at th e cost of on e addition al
parameter ( d0) an d predict th e complete
fun ction .
Note th at th e adden d 1/ [ v( d0 Ϫ Rt) ] in th e
den omin ator was in terpreted by H errn stein

413

Fig. 3. With in -session satiation effects sh own for gen eral activity as measured by a stabilimeter, an d for lever
pressin g. Th e data are averaged over two session s in
wh ich 4 rats were given two 45-mg pellets for th e first
respon se 30 s after th e previous rein forcemen t ( FI 30) .
Th e cur ves are drawn by Equation 8Ј.

as RO , th e value of rein forcemen t for oth er
( n on target) respon ses. H e an d Lovelan d predicted th at wh en an imals were n ot deprived
of th e primar y rein forcer, th ese oth er implicit
rein forcers sh ould seem to grow in relative
value, th us in creasin g th e value of RO ( H errn stein & Lovelan d, 1972) . Th eir data sh owed
th is to be th e case; h owever, our in terpretation is more straigh tfor ward: Wh en an imals
are n ot greatly deprived, d0 will by defin ition

be small, an d th us 1/ [v( d0 Ϫ Rt) ] ( th eir RO )
will be correspon din gly large.
Th e expon en tial-drive model is n ecessar y
for some of th e data on satiation . In th at case,
Equation 8 may be rewritten as
Bϭ

kR
,
R ϩ 1/ ( vht)

( 8Ј)

with drive level ht an expon en tial fun ction of
deficit ( Equation 6) rath er th an a lin ear fun ction ( Equation 5) . In an un publish ed experimen t, Lewis Bizo an d I delivered two 45-mg
pellets to rats immediately after a lever press
on a fixed-in ter val ( FI) 30-s sch edule. Gen eral activity was con curren tly measured with
a stabilimeter. Figure 3 sh ows th e declin e in
gen eral activity an d lever pressin g as a fun ction of th e n umber of trials. Equation 8Ј drew
both cur ves. Th e motivation al parameters ( ␥
ϭ 0.3 gϪ1 an d d0 ϭ 4 g) were th e same for
both respon ses, wh ereas th e remain in g parameters were un dercon strain ed by th e data.


414

PETER R. KILLEEN

Th e lever-press data are flatter because ceilin gs on respon se rate compress th e top en d
of th e fun ction . Th e key poin t is th at Equation 8, wh ich predicts a lin ear or con cavedown decrease in respon din g, could n ot h ave

fit th e con cave-up time course of satiation as
measured by gen eral activity.
Equation 8Ј also drew th e cur ves th rough
th e data in Figure 1. In both econ omies d0
took th e value of 140 rein forcers an d k was
5,500 respon ses per h our; for th e open econ omy, ␥ ϭ 0.10, an d for th e closed econ omy ␥
ϭ 0.07. Th e key differen ce between th e
cur ves is th e degree of repletion permitted
with in th e session . For th e closed econ omy
th e session duration was 6,000 s, so th at tsess/
2 is 3,000 s, an d th e average session deficit
( th e coefficien t of ␥ in Equation 6) is 140 Ϫ
R ϫ 3,000. Th e fixed duration of th e closed
econ omy permitted differen tial satiation as a
fun ction of rate of rein forcemen t ( R) . For
th e open econ omy th e session en ded after
180 rein forcemen ts, so tsess/ 2 is 90/ R s, an d
th e average session deficit is 140 Ϫ R ϫ 90/
R; th at is, a con stan t 50 g. Termin atin g session s after a fixed n umber of rein forcers, or
in gen eral keepin g session duration proportion al to in terrein forcemen t in ter val ( 1/ R) ,
con fers a con stan t average level of motivation . Th is is th e key differen ce between th e
experimen tal paradigms; it is ‘‘th e econ omic
system wh ich produced th e differen t results’’
sh own in Figure 1. It did so by lettin g th e
an imals differen tially satiate in on e case but
n ot in th e oth er.
Th e amoun t of food con sumed in th ese
an d th e Fisch er an d Fan tin o ( 1968) session s
was two to five times th e amoun t con sumed
in a typical session . Is th ere eviden ce for th e

decrease in respon din g durin g operan t session s of more typical duration ? Th an ks to
McSween ey an d h er colleagues, th ere is n ow
ample eviden ce of with in -session satiation effects ( see McSween ey & Roll, 1993, for a review) . But h er data also sh ow with in -session
warm-up effects, so we must digress to a
model of th ose.
WARM-U P
Some of th e first eviden ce for with in -session effects from McSween ey’s laborator y
came from a study con ducted to test th e effects of postsession feedin g on rats th at were

Fig. 4. Data from McSween ey et al. ( 1990) , sh owin g
with in -session warm-up an d satiation effects in rats. Th e
cur ve is drawn by Equation 3, with Equation 7 represen tin g th e satiation effects an d Equation 9 th e warm-up
effects.

required to press a lever for Noyes pellets or,
in a differen t con dition , to press a key for
sweeten ed con den sed milk ( McSween ey, H atfield, & Allen , 1990) . Alth ough n o effects of
postsession feedin g were foun d, a remarkable
pattern of rate ch an ges with in th e session was
discovered ( see Figure 4) . Respon se rates in creased th rough th e first 20 min of th e session an d decreased th ereafter, an d th e patter n was vir tu ally id en tical fo r th e two
respon ses an d rein forcers.
Th e decrease in rates may be attributed to
satiation of th e kin d seen in th e previous figures. To wh at do we attribute th e in crease in
rates? Killeen an d h is colleagues ( Killeen , in
press; Killeen et al., 1978) h ave described similar in creases in rates wh en an imals are first
in troduced to a sch edule of periodic rein forcemen t, an d attributed th em to th e cumulation of arousal. Such warm-up plays a
large role in beh avior main tain ed by aversive
stimuli an d a lesser but still measurable role
in beh avior main tain ed by relief from h un ger. In troduction to th e ch amber itself becomes a con dition ed rein forcer an d th erefore a con dition ed exciter. If th ere were n o
loss of th is arousal between session s, even tually each session would begin with rates at

th eir asymptotic level. But th e an imals calm
down between session s. For th e presen t purposes, assume th is between -session s loss is
complete ( see Killeen , in press, for a more


MECHANICS

415

gen eral treatmen t) ; th en arousal sh ould accrue as
A ϭ aR( 1 Ϫ eϪ␣t ) ,

( 9)

wh ere ␣ is th e rate of th e decay of arousal,
usually takin g a value aroun d 6 min Ϫ1 ( Killeen , in press; Killeen et al., 1978) , an d t is
th e time in to th e session . As t grows large, th is
reduces to A ϭ aR. Th is sh ould look familiar:
It is Equation 1 ( th e first prin ciple of th e mech an ics) an d a key compon en t of Equation 3.
Note th at Equation 9 predicts th e time course
of warm-up to be in depen den t of th e rate of
rein forcemen t; R merely sets th e asymptote.
To accoun t for th e data of McSween ey et
al. ( 1990) , we replace a in Equation 9 ( a model of warm-up) with its expan sion by Equation
7 ( a model of satiation effects) an d in sert th is
in place of aR in Equation 3 ( a model of ceilin gs on respon se rates) . We may fix d0 an d M
at th eir stan dard values of 4 g an d 0 g/ s.
Th en solvin g for scale parameter k ϭ 7 respon ses per secon d, value parameter v ϭ 11.5
s per rein forcemen t, an d decay rate ␣ ϭ 1/ 9
min utes, min imizes th e sum of squares deviation from th e data. Figure 4 sh ows th e prediction s with th e lin ear h un ger model ( Equation 7) with th ese parameter values; th e

expon en tial h un ger model provides an equivalen t fit to th e data, as it does to th ose from
th e n ext study.
A recen t experimen t of McSween ey an d
Joh n son ( 1994) rein forces th is in terpretation
of th e biton icity bein g due to warm-up an d
satiation . In th is study th e auth ors rein forced
pigeon s’ peckin g on a VI 60-s sch edule with
5 s access to mixed grain . After 50 min th ey
were removed from th e ch amber an d th en
return ed after 3, 10, or 30 min . O ur in terpretation of th e ascen din g limb as bein g due
to warm-up en tails th at th ere sh ould also be
a warm-up wh en th e subjects are rein troduced to th e ch amber. If pigeon s are detain ed with in ch amber, we expect a similar
but less pron oun ced warm-up effect. For lon ger duration s of in terruption , th ere sh ould
also be a sligh t in crease in h un ger motivation . Figure 5 sh ows th e data from McSween ey an d Joh n son ’s first experimen t, with
th e cur ves sh owin g respon se rates in 5-min
bin s before an d after th e in termission s, averaged over subjects an d duration s of in termission . I set k ϭ 240 respon ses per min ute
an d d0 ϭ 22 g; th e latter less th an typical, but

Fig. 5. Data from McSween ey an d Joh n son ( 1994) ,
sh owin g with in -session warm-up an d satiation effects. Th e
pigeon s were removed from th e ch amber for periods of
3 to 30 min an d th en were rein troduced to it. Th e data
are averaged over subjects an d duration s of in termission ;
th e cur ves are drawn by Equation s 3, 7, an d 9.

th ese were small birds main tain ed at 85% an d
given 5-s feeder access per meal. Th e time
con stan t for warm-up was 1/ ␣ ϭ 6.5 min , an d
value of v was 0.15 s per rein forcemen t. In
th eir secon d experimen t th e birds were n ot

removed from th e ch amber, an d th e postin termission warm-up was reduced.
In an oth er study, McSween ey ( 1992) varied
rates of rein forcemen t for lever pressin g an d
measured rats’ respon se rates th rough out th e
session . As expected, th e decreases in rates
durin g th e last h alf of th e session were greatest un der th e h igh est rates of rein forcemen t,
wh ere satiation occurs most quickly. Th e
fun ction s look similar to th ose sh own in Figures 4 an d 5, an d th e above model provides
an excellen t fit to th em. McSween ey also plotted th e data usin g rate of rein forcemen t as
th e x axis for data from differen t portion s of
th e session —first 5 min , th e th ird 5 min , th e
9th min , an d th e 12th min . Alth ough th e
small database en tails irregularity in th e data,
Figure 6 makes an importan t poin t: Th e
sh ape of th e H errn stein h yperbola depen ds
on wh ich portion of th e session th e data are
collected from. In particular, th e decrease in
respon se rates at th e h igh est rate of rein forcemen t in th e latter part of th e session is n ot
predicted by H errn stein ’s model. A decrease
in respon din g at ver y h igh rates is predicted
by Equation 2, but for reason s oth er th an satiation , an d th at equation can n ot predict th e


416

PETER R. KILLEEN

Fig. 6. Data from McSween ey ( 1992) , sh owin g respon se rate as a fun ction of rate of rein forcemen t, with 5-min
segmen ts of th e session as th e parameter. Th e cur ves are drawn by Equation s 3, 7, an d 9.

obser ved with in -session ch an ges th at are due
to satiation ( th e use of Equation 2 in con cert
with th e satiation an d warm-up models does
in gen eral provide a sligh tly better fit to th e
data, as on e would expect) . In fittin g th e
presen t model, I give th e in itial deficit an d
metabolic rate th eir stan dard assign men ts: d0
ϭ 4 g, M ϭ 0. Th e remain in g parameters
were assign ed values th at min imized th e sum
of squares deviation between th eor y an d data,
an d th e cur ves were drawn th rough th e data
sh own in Figure 6: 1/ ␣ ϭ 10 min , v ϭ 11.5 s
per rein forcemen t, an d k ϭ 72 respon ses per
min ute. Equation 8 provides an almost-equivalen t fit, but overpredicts th e rate in th e first
pan el because it does n ot allow for warm-up.
Figure 6 provides a strikin g picture of th e
impact th at such with in -session satiation can
h ave on our overall models of beh avior. Th e
top left pan el sh ows respon se rates from th e

first 5 min of each sch edule, displayin g th e
form th at Catan ia an d Reyn olds ( 1968) made
famous an d th at H errn stein made epon ymic.
Bu t as th e sessio n p r o gr esses, th e fo r m
ch an ges: At all rates of rein forcemen t greater
th an or equal to on e per min ute, respon se
rates dropped, an d un der a VI 15-s sch edule
th ey dropped precipitously.
Effects of session duration an d satiation
similar to th ose sh own above were foun d by

Dougan , Kuh , an d Vin k ( 1993) an d O sborn e
( 1977) . Satiation an d warm-up effects can be
substan tial, an d th e mech an ics of beh avior
provides a framework with in wh ich to derive
models of th em. Both th e presen tation of in cen tives an d th eir removal often affect an imals on ly after a lag; th us, we h ave warm-up
effects wh en session s start, cool-down or extin ction effects wh en in cen tives cease, an d respon din g th rough satiation in well-practiced


MECHANICS
subjects. All may be assimilated in mech an istic models of beh avior. Th e fin al issue th at
must be addressed before mech an ics can begin to stan d as an altern ative to econ omic
an d ecologic an alyses is th e relation between
th e amoun t of an in cen tive an d its value.
MAGNIT U DE O F INCENT IVES
Wh ereas an imals typically ch oose larger
amoun ts of food over smaller amoun ts ( see,
e.g., Bon em & Crossman , 1988; Collier, Joh n son , & Morgan , 1992; Killeen , Cate, & Trun g,
1993) , respon se rates often ch an ge little or
n ot at all as a fun ction of th e magn itude of
th e in cen tive. Wh y sh ould th is be? In part,
th e an swer depen ds on th e fact th at th e rein forcin g value of an in cen tive is n ot proportion al to its size. In th e case in wh ich magn itude is man ipulated by var yin g duration of
th e in cen tive, th e reason s for th is are obvious:
Th e secon d, th ird, an d nth in stan ts of con sumption are n ot con tiguous with th e respon se th at brough t th em about; th ey are
separated from it by n Ϫ 1 prior in stan ts of
con sumption ( Killeen , 1985) th at block th eir
effectiven ess. Th e last in stan ts of a lon g-duration reward con stitute a delayed reward.
Th ose later in stan ts of con sumption in creasin gly rein force n ot th e prior operan t respon ses but rath er th e immediately prior con summator y respon ses. Assume th at each of
th e in stan ts of con summator y activity in terpolated between a respon se an d th e last in stan t of con summator y activity will block th at
latter’s effectiven ess by a con stan t proportion ,
␯. Th en it follows th at th e effectiven ess of an

in cen tive sh ould in crease as an expon en tial
in tegral fun ction of its duration :
v m ϭ v ϱ( 1 Ϫ eϪ␯m) ,

( 10)

wh ere v m expan ds th e value of an in cen tive
from a con stan t v to a fun ction of its duration
or magn itude ( m) ; v ϱ is th e value of an arbitrarily lon g duration of th at in cen tive, an d ␯
is th e rate of discoun tin g th e in cen tive as a
fun ction of its duration . Value ( v m) refers to
th e psych ological/ beh avioral magn itude of
an in cen tive wh ose ph ysical magn itude ( m)
may be measured in grams, secon ds, or milligrams per kilogram. Incentive motivation refers to th e evaluative or in stigatin g effectiven ess of th e in cen tive th at depen ds on its value

417

in th e con text, as represen ted by equation s
such as Equation 8.
Equation 10 embodies th e maxim of ‘‘margin ally decreasin g utility’’ of in cen tives ( as a
fun ction of th eir duration , n ot, as often used
in econ omic parlan ce, as a fun ction of n umber of rein forcers) . If ␯ is small, th e relation
is approximately proportion al; if ␯ is large,
in creasin g duration adds ver y little value. Killeen ( 1985) foun d th at Equation 10 with ␯
between 0.25 an d 0.75 sϪ1 fit man y of th e
ch oice data h e reviewed. For th e represen tative value of ␯ ϭ 1/ 2, th e value of 3 s of h opper access h as attain ed 78% of th e maximum
possible ( v ϱ) . Studies th at man ipulate lon ger
duration s are operatin g with in a ver y restricted ran ge.
Th is model of th e ch an ge in value with
ch an ges in th e duration of an in cen tive may

be combin ed with Equation s 7 an d 8 to predict per forman ce wh en th e duration of an in cen tive is varied. Wh en th e value of an in cen tive is man ipulated by ch an gin g its quality
rath er th an by ch an gin g its duration , some
utility fun ction oth er th an Equation 10 ( e.g.,
a power fun ction or a logarith mic fun ction )
may be more appropriate. Wh en , for in stan ce, a drug level or sucrose con cen tration
is man ipulated, a plausible model is v m ϭ m␯,
an d th en
at ϭ m ␯␥[ d 0ϩ ( M Ϫ mR) t] .

( 11)

Wh ereas larger in cen tives are margin ally
stron ger rein forcers, th ey also decrease th e
motivation to work by satiatin g an imals more
quickly. Th ese effects will ten d to can cel, depen din g on th e ran ge of duration s studied
an d th e value of th e deficit th e an imal is attemptin g to satisfy. If in itial deficit d0 is large
or repletion time t is sh ort or th e rate of repletion mR is small, th e satiation effects will
be buffered by d0 an d n et in cen tive effects
( in creasin g respon se rates with in creasin g
magn itude) will be foun d. Con versely, if d0 is
small an d repletion is moderate or large, as
is typical of closed econ omies, th e satiation
effect will domin ate, an d respon se rates will
decrease as a fun ction of magn itude. Th e depen den ce of th e sign of th e correlation between magn itude an d respon se rate—positive
in th e realm of small in cen tives, n egative in
th e realm in wh ich satiation effects domin ate—is sh own in a study by Collier an d Myers ( 1961) , wh o foun d positive covariation of


418

PETER R. KILLEEN

respon se rates with volume for dilute an d in frequen t sucrose con cen tration s an d n egative
covariation for frequen t h igh con cen tration s.
Th e auth ors spoke in terms of momen tar y satiation , wh ich is exactly h ow we h ave been
speakin g about repletion h ere. More particularly, we can take th e derivative of Equation
11 with respect to m an d set it to zero to fin d
th e magn itude of m at wh ich th e correlation
will go from positive to n egative. Th e turn over poin t is
m* ϭ

΂1 ϩ ␯΃
␯

d 0/ t ϩ M
.
R

( 12)

O f th e variables un der experimen tal con trol, in creases in d0 will exten d th e ran ge of
m over wh ich a positive correlation —an in cen tive effect—is foun d; in creases in session
duration an d rate of rein forcemen t ( t an d R)
will move th e turn over poin t to th e left, leavin g more of th e ran ge to sh ow a n egative correlation —a satiation effect. O f course, large
values for d0 an d relatively small values for
session duration are typical of tradition al experimen tal design s, in wh ich in cen tive effects
sh ould th us be th e rule; small values for d0
an d relatively large values for session duration are typical of closed econ omies, in wh ich
satiation effects sh ould th us be th e rule.
Within -Session Effects Versu s

Between -Session Effects
Ch oice beh avior sh ows greater con trol by
magn itude of rein forcemen t th an does sin gle-operan t respon din g. Th e presen t framework explain s th is result th e followin g way:
Th e satiation effects are sh ared by both operan ts in a ch oice situation , leavin g th e in cen tive effects to act differen tially, un buffered by satiation . Th e same is true for respon se
rates in multiple sch edules, in wh ich satiation
effects sh ould gen eralize wh en compon en t
duration s are n ot too lon g, leavin g in cen tive
effects th e opportun ity for differen tial effectiven ess—an effect kn own as contrast ( Nevin ,
1994) . It remain s to be seen just h ow much
of th e complex literature on beh avioral con trast can be un derstood in th ese terms. To
th e exten t th at th is mech an ics applies, con trast sh ould be greatest wh en th ere is least
bufferin g by d0; th at is, toward th e en d of session s, in lon ger session s, an d in closed econ omies. It sh ould be greater for an imals th at

take lon ger to satiate because th ey h ave crops
or oth er cach es ( e.g., pigeon s) , compared to
th ose th at don ’t ( e.g., rats) . Con trast sh ould
be greater for in cen tives for wh ich th ere is
little satiation ( e.g., electrical stimulation of
th e brain , n on n utritive sweeten ers) an d lower
for bulky but low-valued in cen tives.
An alogous prediction s h old for postrein forcemen t pausin g ( see Peron e & Courtn ey,
1992) . ( a) Un sign aled with in -session man ipulation s sh ould reflect primarily satiation effects ( lon ger pauses after larger rein forcers) ,
because th e differen tial magn itudes provide
differen tial momen tar y satiation effects immediately after th eir deliver y, wh ereas th e
forth comin g in cen tive value is averaged over
all duration s of in cen tives. ( b) For between session s ch an ges, th e two compon en t effects
will ten d to can cel. ( c) Sign aled with in -session ch an ges sh ould reflect primarily in cen tive effects, because th e forth comin g in cen tive is particular to per forman ce un der its
stimulus con trol, wh ereas th e satiation effects
will ten d to be averaged across magn itudes.
Un like respon se rates, th ere is n o ceilin g

effect on pause len gth s, wh ich may make
th em more sen sitive to ch an gin g motivation al
levels th an rates; most of th e effects predicted
by th e presen t th eor y may reflect differen ces
in th e amoun t of time spen t pausin g or en gagin g in oth er respon ses, rath er th an con tin uous ch an ges in respon se rates over a substan tial ran ge. In an y case, th e presen t th eor y
predicts th at all of th ese effects sh ould be
stron gly affected by deprivation level, explain s wh y, an d stipulates th e con texts in
wh ich satiation versus in cen tive effects will be
foun d.
ECO NO MICS
Th e cen tral con cern of tradition al econ omics is th e exch an ge of goods for oth er goods,
in cludin g labor, an d th at is also th e con cern
of beh avioral econ omics. Experimen tal subjects exch an ge beh avior for goods, or strike
balan ces between several goods in return for
th eir beh avior. With out th is requiremen t for
exch an ge of tan gible items, th ere would be
un dercon strain t in th eories an d ch aos in th e
marketplace: If all th at mattered to h un gr y
subjects were maximization of rein forcemen t,
all an imals would always respon d at th eir
maximum rates un der most con tin gen cies.


MECHANICS
Econ omic beh avioral th eor y was in troduced
in part because its framework of sacrificin g
on e th in g to get an oth er provides a ‘‘ration al’’ basis for th e modulation of respon se rates
we see on man y sch edules of rein forcemen t.
Wh en return rates are ver y low, an imals
sh ould respon d with little en th usiasm because doin g so is n ot worth th eir wh ile compared to oth er th in gs th ey could purch ase

with th eir labor; wh en th e return rates are
ver y h igh , th ey sh ould respon d with little en th usiasm because th ey are close to satiation .
Th e greatest stren gth of econ omic an alyses
lies in th e developmen t of models th at frame
th e trade-offs between differen t rein forcers,
clarify wh at con stitutes a ‘‘bun dle’’ of goods,
an d explain th e in teraction s between similar
rein forcers th at permit on e to be substituted
for an oth er. Th e application of econ omic
models to beh avior con trolled by a sin gle
source of rein forcemen t is more problematic,
because th ese models are forced to in troduce
oth er h ypoth etical goods in volved in th e
trade-offs, in a way n ot dissimilar to H errn stein ’s in troduction of RO as a source of competin g rein forcemen t. Rach lin an d associates
( Rach lin , 1989; Rach lin , Battalio, Kagel, &
Green , 1981; Rach lin & Burkh ard, 1978;
Rach lin , Kagel, & Battalio, 1980) treat leisure
as a good, so th at depen din g on th e experimen ter’s con strain ts, th e an imals must make
trade-offs between th e leisure given up by respon din g an d th e material rein forcers th at respon din g provides. Th ose trade-offs are motivated by th e subject’s preferen ce for an
optimal package of goods un der con strain ts
of time an d sch edule. Staddon ( 1979) assumes th at optimal rates exist for all activities,
an d th at an imals are motivated to approach
th at locus in beh avioral space th at min imizes
a weigh ted sum of squares of th e deviation s
of each from its optimal rate ( or th at min imizes some oth er cost fun ction ) given th e
con strain ts of time an d sch edule. Experimen tal con tin gen cies usually require operan t respon din g at a h igh er-th an -optimal rate, so
th at such respon din g fun ction s as a cost,
much as it does for Rach lin an d associates. In
Staddon ’s multidimen sion al beh avior space,
th e coordin ates of th e ideals of all relevan t

dimen sion s defin e a bliss poin t, an d because
ever y oth er poin t is in some way in ferior, variation s in an organ ism’s beh avior th at carr y it

419

Fig. 7. Data from Kelsey an d Allison ( 1976) plotted
by H an son an d Timberlake ( 1983) , alon g with th e cur ves
resultin g from th eir model an d from Staddon ’s ( 1979) .
Reprin ted with permission . Superimposed is th e parabola
drawn by Equation 16.

away from th is global min imum are selected
again st.
H an son an d Timberlake ( 1983) focus on
regulation , provide a math ematical model of
th e equilibrium approach of Timberlake an d
Allison ( 1974) , an d derive as special cases
Staddon ’s ( 1979) an d Allison ’s ( 1976, 1981,
1993) optimality accoun ts. At th e h eart of th e
model are th e coupled differen tial equation s
kn own as th e Lotka-Volterra system. As an exam p le o f its ap p licatio n , th e asym m etr ic
cur ve is drawn th rough th e data from Kelsey
an d Allison ( 1976) , sh own in Figure 7. Th e
dash ed lin e is given by Staddon ’s ( 1979) min imum distan ce model. In fittin g th eir five-parameter model, H an son an d Timberlake n oted th at th ese fun ction s ‘‘quickly exh aust th e
degrees of freedom in h eren t in , for example,
six or seven data poin ts’’ ( p. 272) . Th us, th e
most we can h ope for in comparin g th eor y
to data is a con sisten cy ch eck, a h urdle th at
is n ecessar y for th e th eories to clear, but
wh ose clearan ce is n ot sufficien t groun ds for

us to accept th em. Wh eth er or n ot we accept
th ese th eories seems to depen d on wh eth er
we fin d th eir assumption s con gen ial to our
in tuition s about beh avior, an d wh eth er th ey
make n ovel prediction s. Th ere h ave been few
n ovel prediction s th at I am aware of. H owever, th ey do provide n ew con structs an d in -


420

PETER R. KILLEEN

Fig. 8. Deman d fun ction s collected an d graph ed by Lea ( 1978) . Reprin ted with permission . Superimposed is th e
model deman d fun ction drawn by Equation 17.

dices, such as elasticity of deman d, th at provide altern ative perspectives on beh avior.
Elasticity is an in dex, ‘‘a n umber derived
from a formula, used to ch aracterize a set of
data’’ ( American Heritage Dictionary, 1992) . In dices are useful because a sin gle n umber can
often ch aracterize some crucial aspect of a
ph en omen on ( e.g., th e in dex of refraction of
optical materials, th e con sumer price in dex,
etc.) . Lea ( 1978) drew deman d cur ves as th e
amoun t of an item purch ased as a fun ction
of th e price of th e item. Wh en th e axes are
logarith mic, th e slope of th ese cur ves equals
th eir coefficien ts of elasticity ( see, e.g., Koo-

ros, 1965) . In h is Figures 3 an d 4, Lea drew
idealized deman d fun ction s as straigh t lin es

of differen t slopes, with items such as coffee
an d bread sh owin g th e least decrease in con sumption as price is in creased ( deman d for
th em is in elastic, as we would expect) , an d
items such as h errin g an d cakes sh owin g th e
greatest decrease. H ere a sin gle n umber—th e
coefficien t of elasticity—effectively ch aracterizes a set of data. H owever, in h is Figures 1
an d 5, as is th e gen eral case, real data from
closed econ omies are con cave: Elasticity in creases con tin uously with th e price of th e
commodity ( see Figure 8) . Th is result is


MECHANICS
about as satisfyin g as would be th e discover y
of an ‘‘in verse square law’’ for force as a fun ction of distan ce, but in a world in wh ich th e
expon en t varies con tin uously with distan ce
an d takes th e value of Ϫ2 on ly at on e particular distan ce. Elasticity sh ould n ot itself be so
elastic!
Th e deman d cur ve was design ed for an alysis of decision s by population s, wh ere in creasin g proportion s of th e population may
be in fluen ced to purch ase a commodity, perh aps just on ce, as its price decreases. It was
n ot design ed to an alyze th e repeated purch ases by in dividuals, because such data will
be greatly affected by decreasin g margin al
utility as magn itude in creases, an d by satiation as rate of con sumption in creases. As n oted by Staddon ( 1982) , rein forcemen t rate appears on both axes ( R vs. n/ R) of th e deman d
cur ve, so th at in depen den t an d depen den t
variables are in trin sically correlated. Such
fun ction s provide good stimulus con trol of visual an alysis on ly wh en th ey are lin ear an d
differen ces in slope may be directly compared. Lookin g for secon d-order effects such
as differen ces in degree of cur vature is made
un n ecessarily difficult by th e tactical ch oice
of th ose coordin ates.
Beh avioral econ omics h as useful th in gs to

tell us about substitutability an d complemen tarity ( see, e.g., Green & Freed, 1993; Lea &
Roper, 1977) , issues n ot addressed in th is article. But wh en applied to sin gle respon se–
rein forcer paradigms, th at approach is less
useful ( see, e.g., th e commen taries on Rach lin et al., 1981) . Th ere are too man y free variables to be tied down ; motivation al ch an ges
affect th e parameters wh ile th ey are bein g
collected, an d th e core n otion th at an imals
prefer n ot to respon d above a relatively low
bliss-poin t rate is false, as sh own by Staddon
an d Simmelh ag ( 1971) for pigeon s an d by
n umerous oth er in vestigators for n umerous
oth er organ isms wh ose un econ omical adjun ctive beh avior often over wh elms th eir con tin gen t beh avior. Th e paired baselin e distributio n s o f r esp o n d in g u sed in r egu latio n
models h ave been sh own n ot to predict bliss
poin ts, an d th e ratio of in strumen tal to con tin gen t respon din g is n ot th e con trollin g variable it h as been purported to be ( Tiern ey,
Smith , & Gan n on , 1987) .
Th e econ omic approach does n ot respect
m olecu lar con tin gen cies of rein forcem en t

421

( Allison , Buxton , & Moore, 1987) , an d th erefore is prima facie un able to predict th e h uge
differen ces in respon din g th at can be obtain ed with brief delays of rein forcemen t, an d
is un able even to predict th e profoun d differen ces th at depen d on th e order of exch an ge of goods—th at is, th e differen ces in
for ward versus backward con dition in g. Beh avioral econ omics th erefore does n ot con stitute a gen eral th eor y of beh avior. It offers
some tools for th e comparison of differen t
in cen tives an d th eir effects on beh avior wh en
satiation an d rein forcemen t con tin gen cies
are con trolled. It open s th e door to a beh avio r al an alysis o f co n su m er ch o ice, ab o u t
wh ich a mature beh avioral econ omics will
h ave much to say.
ECO LO GICS

Collier an d Joh n son an d associates ( Collier
et al., 1986, 1992; Joh n son & Collier, 1989,
1991) h ave required rats to work for food un der a variety of con dition s, usually on es th at
respect th e an imal’s n ormal feedin g routin e,
lettin g th e an imals complete meals un in terrupted, an d often exten din g th e session s to
permit an imals to acquire most of th eir food
with in th e experimen tal con text ( i.e., closed
econ omies) . Th is exten ds th e an alysis of beh avior to a larger time scale. But, alth ough
perh aps more n atural, it makes it more difficult for th e th eorist to an alyze th e beh avior
th at is obtain ed from th ese con texts. Th e reason for th is is th at un der th ese con dition s,
rates of rein forcemen t are closely tied to th e
pattern s an d rates of th e an imal’s beh avior—
rate of rein forcemen t, a key con trollin g variable, is n o lon ger an in depen den t variable.
To un derstan d th is, we must digress to examin e h ow an an imal’s beh avior affects its
rate of rein forcemen t.
Schedu le Feedback Fu n ction s
Killeen ( 1994) derived a sch edule feedback
fun ction ( SFF) th at predicts th e rate of rein forcemen t on con stan t probability VI sch edules, given a con stan t rate of respon din g of B
respon ses per min ute, as
R ϭ B( 1 Ϫ eϪR Ј/ B ) ,

B Ͼ 0,

wh ere RЈ is th e programmed rate of rein forcemen t. O ver most of its ran ge, th is may
be approximated by its Taylor expan sion :


422

PETER R. KILLEEN

Rϭ

BRЈ
.
B ϩ RЈ

( 13)

Th is is also th e form of th e SFF suggested by
Staddon ( 1977) an d Staddon an d Moth eral
( 1978) . It is also th e equation derived if on e
assumes th at rein forcers are set up an d respon ses are emitted ran domly an d in sequen ce with rate con stan ts of RЈ an d B ( i.e.,
it is th e mean of series-laten cy devices such as
two-step gen eralized gamma distribution s) .
Wh en respon se rates are h igh , rein forcemen t
rate approximately equals th e sch eduled rate
RЈ ( divide n umerator an d den omin ator by B
an d th en let B go to in fin ity) ; wh en th ey are
ver y low, rein forcemen t rate approximately
equals th e respon se rate B. Equation 13 is accurate on ly in th e ideal case of con tin uous
en gagemen t of organ ism an d sch edule. If an
organ ism takes exten ded timeouts from respon din g, obtain ed rates of rein forcemen t
are lower ( Baum, 1992; Nevin & Baum,
1980) . Th e SFF for ratio sch edules is simply
R ϭ B/ n, wh ere n is th e ratio requiremen t.
Such SFFs are n ot of in terest because we
believe th at an imals are sen sitive to h ow th e
margin al rates of rein forcemen t are affected
by respon din g un der differen t SFFs. ( Th is
fun damen tal assumption of all molar optimality models h as been effectively discredited

by Ettin ger, Reid, & Staddon , 1987.) Rath er,
SFFs are importan t because th ey determin e
th e rate of rein forcemen t ( a key con trollin g
variable in Equation s 1 th rough 3) in th e con text of an in teractive organ ism. Closed systems such as th ose employed by Collier an d
associates are closed-loop systems, with th e
feedback from respon se rates on rein forcemen t rates closin g th e loop th rough th e SFF.
To predict beh avior un der such con dition s,
we in sert th e appropriate feedback fun ction
in to th e motivation equation s, an d in sert
th ese in to Equation 3. For ratio sch edules,
th e solution gen erates th e basic equation of
prediction ( Killeen , 1994, Equation 8) . For
in ter val sch edules, it yields equation s proportion al to Equation 3, but with a sligh tly lower
asymptote:
Bϭ

( k Ϫ 1/ a) aR Ј
,
aR Ј ϩ 1

a Ն 1/ k.

( 3Ј)

No problem: Still th e same old h yperbola!
Equation 3Ј sh ows on e of th e reason s th at a
h yperbolic model is so robust: Wh en specific

activation ( a) is large, Equation 3Ј is equivalen t to Equation 3. But even at low activation
wh en obtain ed rein forcemen t rate falls substan tially below its sch eduled value, per form an ce r em ain s a h yp er b o lic fu n ctio n o f

sch eduled rein forcemen t rates, merely fin din g a lower asymptote ( k Ϫ 1/ a) .
Un fortun ately, th e complete equation s of
motion for organ isms con tain a double feedback loop. Not on ly does rate of respon din g
affect rate of rein forcemen t ( th at Equation 3Ј
compen sates for) , but rate of rein forcemen t
determin es th e satiation of th e organ ism,
wh ich affects th e value of specific activation
a. Th e obtain ed rate of rein forcemen t appears in Equation 7, wh ich is an expan sion of
a. If we in sert Equation 13 in to th at an d attempt to solve it, we get stuck. Th e result is a
quadratic equation with n o simple solution s.
( Equation 8 is quadratic in th e rate of rein forcemen t, but because th at is an in depen den t variable, it caused n o misch ief. H ere th e
equation s are quadratic in th e depen den t
variable, respon se rate.) Quadratic equation s
are, of course, n on lin ear; th e n on lin earity is
in troduced by h avin g beh avior be a fun ction
of a variable ( motivation ) th at itself is a fun ction of beh avior ( wh ich reduces motivation
by repletin g th e an imals) . Now it becomes
impossible to write equation s with all th e
kn own s on on e side an d th e un kn own s on
th e oth er. Th ere is n o simple, complete solution to th is impasse.
Copin g with Non lin earity
Wh en con fron ted with a difficult n on lin earity such as th is, we h ave several option s:
Experimentally opening the loop. We may reduce th e n on lin earity by makin g th e con stan t
terms large relative to th e var yin g terms. Th is
mean s large in itial deficits ( d0) relative to repletion rates ( mR) ; Equation s 8 an d 8Ј sh ow
th at th is is ach ieved with some combin ation
of h igh ly deprived organ isms, small an d in frequen t meals, an d sh ort session s: All of th e
beˆ tes n oires th at Collier an d oth er econ omic
th eorists h ave repeatedly excoriated.
It is h ard to dispute th eir poin t th at th ese

co n d itio n s o f th e r efin em en t exp er im en t
( i.e., th e stan dard procedures) are n on represen tative extrema un der wh ich th e an imals
can display little of th e ran ge of th e n atural
repertoire of th eir n ormal in strumen tal an d
con summator y pattern s. O bjects fallin g in a


MECHANICS
vacuum display little of th e ran ge of th e n atural repertoire of leaves fallin g in an autumn
win d. It is th rough refin emen t experimen ts
th at p h ysicists, ch em ists, an d b eh avio r ists
h ave come to un derstan d th e variables of
wh ich th eir subject is a fun ction . We can h ave
simple laws, such as Equation 3, or we can
h ave more precise but complicated on es,
such as th ose obtain ed by in sertin g Equation
11 in to it; to th e degree th at we wan t precision , we must forgo its complemen t, simplicity ( Killeen , 1993) .
By opening the loop between controlled and
controlling variables, the refinement experiment permits us to explore alternate ways of
formulating models to cover the phenomena
of interest, to estimate the values of the models’
basic parameters, and to evaluate the adequacy
of one model against alternate models (e.g.,
the linear vs. exponential drive models).
Surgically opening the loop. An oth er way of
con trollin g th e feedback loop is to open th e
esoph agus so th at th e con sumed food does
n ot fill th e gut. Th is is sh am feedin g, a kin d
of con tin uous bin ge an d purge. It provided
Pavlov ( 1955) an d Miller ( 1971) with an experimen tal preparation th at effectively addressed certain question s about th e locus of

satiety sign als. But, because it in sults th e in tegrity of th e organ ism–en viron men t match
in a differen t way, it is less useful in addressin g th e question s we pursue con cern in g th e
beh avior of a wh ole organ ism.
Postdictions. Wh en basic refin emen t experimen ts are completed, we would like a way of
th en applyin g th e results to more complex
experimen tal arran gemen ts th at are n ot so
th eoretically felicitous. A mean s to accomplish th is is to give up sch eduled rein forcemen t rate as an in depen den t variable, an d in use the measured rates of reinforcement in our
equations of prediction. The measured rates of
instrumental and contingent behavior are the
variables compared by economic theorists such
as Staddon (1979) and Rachlin et al. (1981).
This is a useful tactic in that it demonstrates
consistency of the models with data, and in
many cases is the best that can be achieved. But
settling for correlations between dependent
variables is less than an optimal solution to the
problem; in giving the prime instrument of experimental analysis—control—to the subject by
making the paradigm more ‘‘ecologically valid,’’ we are consequently forced to abandon the

423

prime goal of experimental analysis, giving up
prediction to settle for postdiction.
Numerical solutions. An oth er option is to fall
back on iterative n umerical solution s of th e
equation s, wh ich is possible even with th e un kn own on both sides. Th is option will be useful in some situation s, but is n ot furth er explored h ere.
Simplifications. Th ere are differen t aspects
of th e complete equation s th at we can ign ore
for th e sake of a closed-form solution to th e
laws of beh avior. For in stan ce, in movin g

from Equation 2 to Equation 3, we sacrificed
th e correction for blockin g of rein forcemen t
by previous rein forcemen ts, in currin g some
in accuracy at rein forcemen t rates above two
per min ute. Let us n ext table Killeen ’s ( 1994)
secon d prin ciple of rein forcemen t by ign orin g th e temporal con strain ts on respon din g,
an d fall back on h is simplest first prin ciple of
arousal, Equation 1. Th en Equation 3 simplifies to an expan sion of th at first an d most
basic prin ciple:
B ϭ aR ϭ v ␥[ d 0 ϩ ( M Ϫ mR) t] R.

( 14)

Th is equation is a parabola. It describes respon din g at time t in a session as a fun ction
of rate of rein forcemen t. It also describes th e
average respon din g in a session wh en t is set
equal to h alf th e session duration ( tsess/ 2; see
th e Appen dix) . Because we h ave ign ored ceilin gs on respon se rate, we expect th e actual
data to be sligh tly less peaked th an a parabola, bein g squash ed in to more of an ellipsoid
form. Equation 14 provides a good fit to th e
data an alyzed by Staddon ( 1979) usin g h is
min imum distan ce model. H owever, some of
th ose data were collected in open econ omies,
an d th eir down turn at low ratio values is
probably due more to th e impoverish ed couplin g of rein forcers to respon ses, wh ich I
h ave an alyzed at len gth ( Killeen , 1994) .
O n ratio sch edules requirin g n respon ses
per rein forcemen t, we may substitute th e ratio sch edule feedback fun ction B/ n for R. At
last, we may write an equation th at can be
solved for B! Its solution is

Bϭ

΂

΃

n
n
MϪ
,
m
vЈ

m, v Ј, Ͼ 0,

( 15)

wh ere M is th e average depletion , M ϭ d0/ t ϩ
M, m is th e magn itude of th e in cen tive, an d
vЈ is proportion al to th e in cen tive value of th e


424

PETER R. KILLEEN

rein forcer, v ( see Equation A5 in th e Appen dix) .
Equation 15 is a parabola th at in creases to
a maximum at n ϭ vЈM / 2 an d decreases toward zero both as n approach es zero ( satiation effects) an d as n becomes ver y large
( strain in g th e ratio, wh ich occurs as n →

vЈM , exactly twice th e poin t at wh ich th e maximum occurs) . Equation 15 provides a good
fit to data such as th ose sh own in Figure 10
of Collier et al. ( 1986) . It may be preferable
to Equation 14, because it predicts respon din g in terms of an in depen den t variable, th e
size of th e ratio sch edule n, rath er th an in
terms of a depen den t variable, rate of rein forcemen t.
To calculate th e total n umber of respon ses
( b) in a session of duration tsess, multiply
th rough by tsess:
bϭ

΂

΃

n
n
MϪ
t
m
vЈ sess

m, vЈ Ͼ 0.

( 16)

Equation 16 provides a reason able fit to th e
data in Figure 7 with m an d tsess fixed at 1, vЈ
set to 1.2 ϫ 10Ϫ3, an d M ϭ 5,450 licks per
session . For th e expon en tial drive model

( Equation A6 in th e Appen dix) , th e parabola
is skewed to th e righ t an d looks ver y much
like H an son an d Timberlake’s ( 1993) cur ve.
It is a sh ort step to write th e equation for
th e deman d fun ction , th e n umber of rein forcers earn ed ( r) as a fun ction of ratio requiremen t, by dividin g Equation 16 by th e
n umber of respon ses required per rein forcemen t ( n) . If we take th e session as th e un it
of time, so th at we can set tsess equal to 1, th en
rϭ

΂΃

M
1 n
Ϫ
m
vЈ m

m, v Ј Ͼ 0.

( 17)

Th is is a model deman d fun ction : Con sumption r is a lin ear fun ction of un it price n/ m,
with a slope of Ϫ1/ vЈ an d an in tercept of
M / m. It is drawn as th e bold lin e in th e logarith mic coordin ates of Figure 8 with m ϭ 1,
M ϭ 200, an d vЈ ϭ 3. It h as approximately th e
same sh ape as man y of th ose empirical deman d cur ves; it is simple, an d does n ot make
th e obviously erron eous econ omic assertion
th at th ere is a th in g such as elasticity th at can
be assign ed to a good an d th at is in depen den t of its price ( i.e., it does n ot assert th at
th e data fall on straigh t lin es in double-log

coordin ates) . Th e expon en tial drive model

provides more flexible deman d cur ves, wh ich
are n ecessar y to fit some of th ese data.
DeGran dpre, Bickel, H ugh es, Layn g, an d
Badger ( 1993) h ave systematically reviewed
data such as th ose sh own in Figures 7 an d 8,
man y in volvin g drug rein forcers. Th ey argued for th e use of un it price ( n/ m) as th e
proper metric of th e x axis ( as did Timberlake & Peden , 1987, an d H ursh , 1980) . Un it
price plays a key role in Equation s 15 th rough
17 as well. Th e slope of th e deman d cur ve
predicted by Equation 17 depen ds n ot on th e
variables n an d m, but on ly on th eir ratio.3
Th ere is an importan t differen ce between
th e an alysis of DeGran dpre et al. ( 1993) an d
th e presen t on e. DeGran dpre et al. plotted
th eir data on logarith mic coordin ates. A parabola in logarith mic coordin ates is n ot parabolic in lin ear coordin ates, but is skewed to
th e righ t. Con versely, Equation s 15 an d 16
are skewed to th e left wh en plotted on a logarith mic x axis. Th e expon en tial drive model
is less skewed th an th e lin ear drive model.
Wh eth er th e presen t models can provide as
good a fit to th e ran ge of available data as
h ave th ose of H ursh et al. ( 1989) an d DeGran dpre et al. ( 1993) remain s to be seen .
CO NCLU SIO N
Mech an istic explan ation s h ave fallen in to
disrepute, in part because good on es are h ard
to come by, an d in part because th ey elicit
images of gears an d pulleys—poor models for
th e processes th at beh aviorists seek to un derstan d. Goal seekin g, regulation , optimization ,
or, in gen eral, teleological ( Rach lin , 1992)

an d teleon omic ( H . Reese, 1994) approach es
seem more modern . Econ omics, th e scien ce
of fin al causes ( Rach lin , 1994) , studies th e
goals aroun d wh ich beh avior is organ ized. As
Rach lin h as n oted in h is sch olarly an d in sigh tful an alyses, we must h ave some sen se of th e
purposes of beh avior before we can un derstan d wh at an act is about. All four of Aristotle’s causes are n ecessar y for a complete accoun t of beh avior: th e fun ction al goals an d
rein forcers ( fin al causes) , effective stimuli
( efficien t causes) , un derlyin g ph ysiology ( ma3 For ver y small values of m, v will covar y with m; for
simplicity in th ese an alyses I h ave assumed th at v h as
topped out, or at least th at m is n ot experimen tally varied
over th e lower en d of its ran ge.


MECHANICS
terial causes) , an d precise metaph ors an d
models ( formal causes) . In sofar as we con ceive of operan t beh avior as bein g un der th e
con trol of its con sequen ces, un derstan din g
th e fin al causes of th at beh avior—both th e
more proximate causes ( on togen etic, h istories of rein forcemen t) an d th e ultimate causes ( ph ylogen etic, selection pressures) —takes
first priority. But th at doesn ’t mean th at it
must take all our efforts; iden tification of fin al causes is largely a qualitative en deavor,
an d may proceed quickly ( we may discover
th at on e of th e causes of birds’ sin gin g is defen se of th eir territor y) but workin g out th e
mach in er y th at permits th e attain men t of
such goals remain s a substan tial project of
an alysis. Th ere is much to be said for a mech an ics, a scien ce of formal causes, as th e secon d an d most detailed part of th e scien tific
en deavor, to guide us in th at an alysis.
Th e developmen t of simple models based
on n aturalistic obser vation s an d laborator y
experimen ts leads us to a clearer un derstan din g of th e variables of wh ich beh avior is a

fun ction ; th at is, to a clearer un derstan din g
of its causes. Th e ‘‘essen tial feature of th e
Newton ian style is to start out with a set of
assumed ph ysical en tities an d ph ysical con dition s th at are simpler th an th ose of n ature,
an d wh ich can be tran sferred from th e world
of ph ysical n ature to th e domain of math ematics. . . . Th e rules or proportion s derived
math ematically may be . . . compared an d
con trasted with th e data of experimen t an d
obser vation ’’ ( Coh en , 1990, pp. 37–38) ; th at
is, refin emen t experimen ts. Th is leads to
modification s of th e model system an d, in
turn , of th e experimen tal design , an d aroun d
again , with th ese cycles ‘‘leadin g to systems of
greater an d greater complexity an d to an in creased vraisemblan ce of n ature’’ ( Coh en ,
1990, p. 38) ; th at is, ecological validity. Math ematics was Newton ’s tool for th e discover y
of veræ causæ, true causes: ‘‘Specification of
th ose causes was n ot a precon dition for th e
con struction of model systems, but rath er a
product of it’’ ( Coh en , 1990, p. 29) . An d
math ematics, even th e relatively trivial math ematics in th is paper, provides an in valuable
formal structure for our metaph orical models: ‘‘It was th e exten sion of Newton ’s in tellectual powers by math ematics an d n ot merely some kin d of ph ysical or ph ilosoph ical
in sigh t th at en abled h im to fin d th e mean in g

425

of each of Kepler’s laws’’ ( Coh en , 1990, p.
31) . Math ematics puts a fin e poin t on th e
dull pen cil of metaph or.
Th e presen t mech an ics provides a relatively parsimon ious quan titative accoun t of man y
of th e data. It also in troduces th e con struct

of satiation , a con cept th at is in accord with
our un derstan din g of n ature an d is overdue
for formal recogn ition in our an alyses. Mech an ics gen erates a bridge to ecologic an d
econ omic an alyses th rough th e explicit utilization of th e con cepts of ideal rate of repletion or rein forcemen t ( M, wh ich provides
on e coordin ate of th e multidimen sion al ideal, th e bliss poin t) , th e cost of deviation s from
it ( ␥) , th e decreasin g margin al utility of rein forcers ( Equation s 10 th rough 12) , an d a
role for un it price as an in depen den t variable
( Equation s 15 th rough 17) . It is also con sisten t with th e ch an ges in respon se rate th at
are foun d with in a sin gle session ( Equation
8; see, e.g., Killeen , 1991; McSween ey, 1992) .
Futh ermore, it leads to a biologically based
treatmen t of h un ger th at provides a dyn amic
approach to th e steady state assumed by econ omic models. Un like th e ecologic an d regulator y approach es, mech an ics does n ot in voke defen se of a setpoin t as a fun damen tal
force, but in troduces th at defen se implicitly
in equation s th at make deprivation a key factor in motivation ( Equation s 5 th rough 7) . It
is n ot so much th at an imals defen d a setpoin t, as th at deviation from a setpoin t in creases th e rein forcin g value of even ts th at,
as n ature usually h as it, reduces th at deviation . Fin ally, in Figures 7 an d 8 it provides
altern atives to econ omic an alyses th at are parsimon ious of parameters, derive from simple
version s of th e basic prin ciples of rein forcemen t, an d provide in terpretable parameters
an d testab le p r ed ictio n s ( Eq u atio n s 15
th rough 17 an d A5 th rough A7) .
Ecologics calls our atten tion to th e rich in teractive en viron men ts in wh ich an imals h ave
evolved an d th at h ave sh aped th eir respon ses
to metabolic ch allen ge. Its experimen tal results may be ch arted with accuracy, but because it is a dyn amic, path -depen den t, n on lin ear en terprise, th ose results can seldom be
predicted from prin ciples. Like th e mean ders
of a river th at are con sisten t with simple an d
precise models, th e path s of un ch an n eled beh avior may come to be seen as bein g con sisten t with models such as th ose presen ted


426

PETER R. KILLEEN

h ere, even wh ile th e particular courses of river an d beast may n ever be predictable from
th eir prin ciples. Prediction an d con trol are
en gin eerin g ideals, n ot scien tific on es. It is
th e purpose of refin emen t experimen ts to establish prin ciples; in more ecologically valid
experimen ts our goal is to un derstan d, an d
un derstan din g is n oth in g oth er th an recogn ition of con sisten cy with establish ed prin ciples.
Like ecologics, econ omics provides in spiration to search for th e en ds aroun d wh ich
beh avior is organ ized—its fin al causes—an d
th is is wise. It provides an approach to un derstan din g th e trade-offs an imals make between altern ate packages of goods, an importan t an d un derrepresen ted area of research .
But it also seduces us in to usin g th e an alytic
framework of econ omists, an d th is is folly.
Econ omics is n ot on ly th e scien ce of fin al
causes; it is also ‘‘th e dismal scien ce.’’ Its complexities an d routin e failures to predict beh avio r fr o m eco n o m ic p r in cip les ar e
legen dar y. An econ omic beh aviorism th at
borrows its con structs, rath er th an its goals,
takes th e worst of it. Let us first iden tify th e
proximate an d ultimate causes of beh avior in
th e ecological con text in wh ich th ose fin al
causes h ave proven an ce. But th en let us seek
th e true causes of beh avior th rough th e developmen t of a mech an istic th eor y—a scien ce of formal causes—based on prin cipled
experimen tation , th at may guide us in th e developmen t of an ‘‘en ligh ten ed scien ce’’ of beh avioral econ omics.
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Received December 31, 1994
Final acceptance July 14, 1995


MECHANICS

429

APPENDIX
Con stan ts an d Dimen sion s
Table 1 lists th e symbols an d th eir in terpretation s. Lower-case letters are used for all variables except rate variables, wh ich are written
in capitals. Greek letters are used for con stan ts an d parameters. Th e secon d column
lists th e co n stitu en t d im en sio n s, n o t th e
un its. For example, A is th e n umber of secon ds of respon din g per secon d; th ese can cel
to make it a ‘‘dimen sion less’’ variable. b an d
r refer to n umber of respon ses an d rein forcers; because coun tin g in volves an absolute
scale, th e un its for both are ‘‘coun ts’’; but because th ey are coun tin g differen t th in gs, th ey
h ave differen t dimen sion s.
Session Averages
To calculate average respon se rates durin g
a session , on e sh ould write th e complete
model predictin g respon se rates an d in tegrate it over th e session duration . Th is is because with fin ite ceilin gs on respon se rate,
even th e most extreme deprivation can on ly
elevate respon se rates sligh tly closer to th eir
ceilin g. It is for th is reason th at th e lin ear an d
expon en tial drive models provide equally
good fits to man y of th e operan t con dition in g
data: Th e differen ces between th e drive level
predicted by th ose models are greatest at
h igh deprivation s, but th at is wh ere respon se
rates are n ear th eir ceilin gs an d th us least respon sive to ch an ges in drive levels. ( It is also
for th is reason th at sigmoidal fun ction s between deprivation an d drive do n ot provide a
measurable improvemen t in fit to th e data.)
Un fortun ately, in tegration of th e complete
models yields un gain ly or in soluble forms. It
is th erefore a worth wh ile simplification to

compute th e average drive an d arousal levels
over th e course of a session , an d use th ese to
predict average respon se rates.
The linear model. For th e lin ear model h un ger level is given by Equation 5 in th e text.
Th e average h un ger over a session of durationtsess is th e in tegral of th at fun ction with
respect to time divided by tsess:
h¯ sess ϭ ␥[ d 0 ϩ ( M Ϫ mR) t sess / 2] .

( A1)

For sh ort session s ( tsess small) , h un ger is determin ed by th e in itial deficit d0, but as sessio n d u r atio n in cr eases, h u n ger ch an ges
lin early with it. For exten ded session s in

Table 1
Sym- Dimen bol
sion s
A

1

R
B

r/ s
b/ s

M

g/ s

RЈ
a

r/ s
s/ r

k

b/ s

d

g

h

1

m

g/ r

t
v

s
s/ r

n

b/ r

b
r
␭

b
r
r/ b

␥

1/ g

␪

1

␣
␯

1/ s
r/ g

␦

s/ b

Mean in g
Arousal level; th e amoun t of respon din g elicited by a sch edule of in cen tives

in th e absen ce of competition from
oth er respon ses
Rate of rein forcemen t ( obtain ed)
Rate of respon din g; arousal level corrected for respon se duration an d ceilin gs on respon se rate
Metabolic rate; assumed con stan t an d
often set to zero
Rate of rein forcemen t ( sch eduled)
Specific activation : th e n umber of secon ds of respon din g th at are elicited by
a sin gle in cen tive, wh ich depen ds on
drive an d in cen tive factors
Asymptotic respon se rate on in ter val
sch edules
Deficit resultin g from a depletion / repletion imbalan ce over time
H un ger, a lin ear or expon en tial fun ction of deficit
Magn itude of an in cen tive, h ere measured in grams per rein forcer
Time
Value of an in cen tive, wh ich depen ds
on its n ature an d magn itude
Number of respon ses required to complete a ratio sch edule
Number of respon ses
Number of rein forcers
Lambda, th e rate of decay of sh ortterm memor y; does n ot play an importan t role in th e presen t developmen t
Gamma, th e gain or restorin g force
th at tran slates deficit in to drive
Th eta, th e th resh old level of motivation for respon din g
Alph a, th e rate of warm-up
Nu, th e rate of discoun tin g an in cen tive as a fun ction of its magn itude; its
dimen sion s depen d on th e in depen den t variable an d th e particular discoun t model ( Equation s 10 or 11)
Delta, th e m in im u m in ter resp on se
time

wh ich tsess is large, h un ger is determin ed primarily by th e balan ce between on goin g metabolic depletion an d repletion , MϪmR.
The exponential model. Calculatin g th e average h un ger durin g a session of durationtsess