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7/31/2019 Killeen 95 Economics http://slidepdf.com/reader/full/killeen-95-economics 1/27 405 JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 1995, 64, 405–431 NUMBER 3 ( NOVEMBER) ECON OM ICS, ECOL OGICS, A N D M ECH A N ICS: T H E DY N A M ICS OF R ESPON DIN G UN DER CON DIT ION S OF VA R Y IN G M OT IVAT ION P ETER R. KILLEEN ARIZONA STATE UNIVERSITY The mechanics of behavior developed by Killeen (1994) is extended to deal with deprivation and satiation and with recovery of arousal at the beginning of sessions. The extended theory is validated against satiation curves and within-session changes in response rates. Anomalies, such as (a) the positive correlation between magnitude of an incentive and response rates in some contexts and a negative correlation in other contexts and (b) the greater prominence of incentive effects when magnitude is varied within the session rather than between sessions, are explained in terms of the basic interplay of drive and incentive motivation. The models are applied to data from closed econ- omies in which changes of satiation levels play a key role in determining the changes in behavior. Relaxation of various assumptions leads to closed-form models for response rates and demand func- tions in these contexts, ones that show reasonable accord with the data and reinforce arguments for unit price as a controlling variable. The central role of deprivation level in this treatment distin- guishes it from economic models. It is argued that traditional experiments should be redesigned to reveal basic principles, that ecologic experiments should be redesigned to test the applicability of those principles in more natural contexts, and that behavioral economics should consist of the applications of these principles to economic contexts, not the adoption of economic models as alternatives to behavioral analysis. Key words: economics, ecologics, mechanics, deprivation, satiation, motivation, arousal, demand functions, drive, incentive, models, principles This paper compares three approaches to the prediction of behavior that is under the control of incentives and supported by moti- vational states of varying intensity. Behavioral economics frames behavior as an exchange of goods, and motivation as the optimization of the trade-offs required by the constraints of time and experimental context in order to obtain the best immediate or delay-discount- ed package of goods. Ecologics respects the natural ecology of the subject and rejects the logic of the marketplace and theoretician for that of an organism adapted by evolutionary forces to complex natural environments. Ecologics frames behavior as nested sets of systems or action patterns, and motivation as regulation—the defense of setpoints within those system states. Both of these approaches are teleonomic or functional, focusing on fi- nal causes, on outcomes: The economic or- ganism behaves so as to optimize packages of This research was supported by NSF Grants IBN- 9408022 and BNS 9021562. It benefited greatly from the reviewers’ comments, although it is unlikely they would endorse all of the claims of this version. Address correspondence to Peter R. Killeen, Depart- ment of Psychology, Box 871104, Arizona State Univer- sity, Tempe, Arizona 85287-1104 (E-mail: KILLEEN@ ASU.EDU). goods, and the ecologic organism behaves to minimize deviations from optimal setpoints in its parameter space. M echanics focuses on the efficient rather than the final causes of behavior, and provides a set of formal caus- es—a set of mathematical models—that ex- pands simple assertions of causal agency into more precise functional relations between variables. The mechanical organism is not be- having to optimize anything; incitement makes it active, satiation decreases its excit- ability, and co-occurrence of particular re- sponses with incentives increases the proba- bility of those responses. The primarygoal of this paper is to develop the mechanics to the point at which it is applicable to the experi- mental contexts that are favored byeconomic and ecologic theorists. MECHANICS A recent monograph (Killeen, 1994) pro- posed a mechanics of behavior based on three principles concerning the nature of arousal, temporal constraint, and coupling between responding and incentives. The first principle was that incentives excite respond- ing, so that arousal level (  A ) is proportional
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405

JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 1995, 64, 405–431 NUMBER 3 ( NOVEMBER)

E C O N O M IC S , E C O L O G I C S , A N D M E C H A N IC S :T H E DY N A M IC S O F R ES PO N DI N G UN DERC O N D IT IO N S O F V A R Y IN G M O T IV A T IO N  

PE T E R R. KILLEEN

ARIZONA STATE UNIVERSITY

The mechanics of behavior developed by Killeen (1994) is extended to deal with deprivation andsatiation and with r ecover y of arousal at the beginning o f sessions. The extended theor y is validatedagainst satiation cur ves and within-session changes in response rates. Anomalies, such as ( a) thepositive correlation between magnitude of an incentive and response rates in some contexts and anegative correlation in other contexts and (b) the greater prominence of incentive effects whenmagnitude is varied within the session rather than between sessions, are explained in terms of thebasic interplay of drive and incentive motivation. The models are applied to data from closed econ-omies in which changes of satiation levels play a key role in determining the changes in behavior.Relaxation of various assumptions leads to closed-form models for r esponse rates and deman d func-tions in these contexts, ones that show reasonable accord with the data and reinforce arguments forunit price as a controlling variable. Th e central role of deprivation level in this treatment distin-guishes it from economic models. It is argued that traditional experiments should be redesigned to

reveal basic principles, that ecologic experiments should be redesigned to test the applicability of those principles in more natural contexts, and that behavioral economics should consist of th eapplications of these principles to economic contexts, n ot the adoption of economic models asaltern atives to beh avioral analysis.

Key words: economics, ecologics, mechanics, deprivation, satiation, motivation, arousal, demandfunctions, drive, incentive, models, principles

This paper compares three approaches tothe prediction of behavior that is under thecontrol of incentives and supported by moti-vational states of var ying inten sity. Behavioral

econ omi cs frames behavior as an exchange of goods, and motivation as the optimization of the trade-offs required by the constraints of time and experimental context in order toobtain the best immediate or delay-discount-ed package of goods. Ecologics respects thenatural ecology of the subject and rejects thelogic of the marketplace and theoretician forthat of an organism adapted by evolutionar yf o r c e s t o c o m p l e x n a t u r a l e n v i r o n m e n t s .Ecologics frames beh avior as n ested sets o f systems or action patterns, and motivation as

regulation—the defense of setpoints withinthose system states. Both of these approachesare teleonomic or functional, focusing on fi-nal causes, on outcomes: The economic or-ganism behaves so as to optimize packages of 

This research was supported by NSF Grants IBN-9408022 an d BNS 9021562. It benefited greatly from thereviewers’ commen ts, although it is un likely they wouldendorse all of the claims of this version.

Address correspondence to Peter R. Killeen, Depart-men t of Psychology, Box 871104, Arizon a State Univer-sity, Tempe, Arizona 85287-1104 ( E-mail: KILLEEN@

ASU.EDU).

goods, and the ecologic organism behaves tominimize deviations from optimal setpointsin its parameter space. M echan ics focuses onthe efficient rather than the final causes of 

behavior, and provides a set of formal caus-es—a set of mathematical models—that ex-pands simple assertions of causal agency intomore precise functional relations betweenvariables. The mechanical organism is not be-h a vi n g t o o p t i m ize a n yt h i n g ; i n c i t e m e n tmakes it active, satiation decreases its excit-ability, and co-occurren ce of particular re-spon ses with incentives increases the prob a-bility of those respon ses. The primar y goal of this paper is to develop the mechanics to thepoint at which it is applicable to the experi-men tal contexts that are favored by economicand ecologic theorists.

MECHANICS

A recent monograph (Killeen, 1994) pro-posed a mechanics of behavior based onthree principles concerning the nature of arousal, temporal constraint, and couplingbetween responding and incentives. The firstprinciple was that incentives excite respond-

ing, so that arousal level ( A ) is proportional

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406 P ET E R R . K IL L E EN  

to rate of incitement ( R ; a will be definedbelow):

 A ϭ a R . ( 1)

But there are constraints. There is only so

much time available in which to respond (Kil-leen’s second principle), and for a particulartarget response to b e d ifferentially excited byan incentive, it must be paired with that in-centive; they mu st coreside in the animal’sshort-term m emor y (the third principle). It isonly when effective contingencies couple anincentive with a response th at th e incentivebecomes a reinforcer. These three principlesprovided the bases for models of the behaviorgenerated by various schedules of reinforce-ment. For instance, the theory predicts re-

sponse rates on interval schedules to be

kR R B ϭ Ϫ , , a Ͼ 0, ( 2)

 R ϩ 1/  a

where k  is p roportional to the maximal at-tainable response rate, R is the rate of rein-forcement, a is a key parameter whose mean-ing will be developed below, an d lambda ( )is the rate of decay of memor y for a response.Note that without the subtrahend, this is es-sentially Her rn stein’s hyperbo la, which has

been demonstrated to predict response rateover a wide range of conditions (see, e.g., deVilliers & H err nstein, 1976). The subtrah endcomes into play only at very high rates of re-inforcement ( R Ͼ 2 per minute), where anincreasing fraction of the incentive bears onthe prior consummatory response, strength-e n in g it r at h er t h an t h e in st r um en t al r e-sponse. Because the subtrahend is importantonly under ver y high rates of reinforcement,it will be set to zero for the rest of this paper,because this simplifies analysis and incurs

only a small decrease in goodness of fit.

T h e Sp ecifi c A ctiv a ti on of In cen ti v es

The parameter a, which I have called thespecific activation, is of greatest concern inthis paper. In Herr nstein’s ( 1974) formula-tion, R O ϭ 1/  a was treated as the rate of re-inforcement available from sources otherthan those scheduled by the experimenter.This interpretation has n ot been supportedby sub sequ en t research ( e.g., Brad shaw, Sza-badi, Ruddle, & Pears, 1983; Dougan &

McSween ey, 1985; McSween ey, 1978) . Ac-

cord ingly, some investigator s (e .g., Bradsha w,Ruddle, & Szabadi, 1981) have more agnos-tically called the parameter the half-life con-stant, because response rate attains half itsmaximal value when R equals R O.

In earlier work on incentive motivation,Kille e n , H a n so n , a n d O sb o r n e ( 1 97 8)showed that each incentive d elivered un derconstant conditions will generate a total of  aseconds of behavior. It follows that R incen-tives will generate the poten tial for aR sec-onds of responding, and they called a R th eorganism’s level of arousal. The particularform of respond ing generated by that arousaldepend s on th e contingencies that determine

 just wh at par ticu lar re spo n se will occur be -fore the delivery of the incentive. It is this

coupling of responses to incentives that con-stitutes reinforcement. When the couplingapproaches its maximum (1.0), as it does onshor t r atio schedu les, most of th e behavior of the organism is concentrated on the targetresponse. When the coupling is ver y weak, asin schedules of behavior-independen t rein-forcemen t, b ehavior is diffuse an d drifts to-ward adjunctive forms. But in all cases, thetotal amount of time spent responding is afunction of the arousal level of th e organism,which is a prod uct of th e specific activation

of th e incentives ( a) and the rate of their de-liver y ( R ). It is these considerations th at gaverise to Equation 1.

We may simplify Equation 2 by drop pingits subtrahen d, and we may multiply its n u-m e rat or a n d d e n om in a to r b y a to revealmor e clearly the m ultiplicative in teraction be-tween incentive factors summarized by a , an drate of incitement, R :

kaR B ϭ . ( 3)

a R ϩ 1

Equation 3 is hyperbolic in aR because of thenon linearities introduced by ceilings on re-sponse rate. When we are operating well be-low those ceilings, it reduces to the simpleproportional model, the first principle of th emechan ics. Whereas Equation 2 emph asizesthe relation of this model to Herrnstein’s hy-perbola, Equation 3 reminds us of the mul-tiplicative relation between a an d R as theyconjointly determine arousal level and re-sponse rate.

Terminology. It is worth an aside to clarify

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407 M EC H A N ICS 

the terminology used throughout this paper.The above equations were proposed as equi-librium solutions for when the behavior un-der study has come to a steady state. In phys-ics the study of systems at equilibrium is

called statics; analogously, the above equa-tions are part of a statics of behavior. Muchof the recent research in behavior analysisconcerns such asymptotic behavior. It derivesfrom a tradition of descriptive beh aviorism;whenever a cumulative record is displayed ora regression is fit through a scatter of data,the goal is description. This is a first step to-ward a more general science: ‘‘Galileo wasconcerned not with the causes of motion butinstead with its description. The branch of mechanics he reared is known as kinematics;

it is a mathematically descriptive account of m o t i o n wi th o u t c o n c e r n fo r i ts c au s es’’( Frautschi, Olenick, Apostol, & Goodstein,1986, p . 114). It follows in the Pythagoreantradition that ‘‘approached phenomena interm s of or der and was satisfied to discoveran exact m athe matical description’’ ( Westfall,1971, p. 1). Th ere are many examples of sucha tr adition in psychology tod ay, includin g de-scrip tive statistics, th e laws o f psycho ph ysics,and the original m atching law.

The study of forces that cause objects to

move is called dynamics; dynamics constitutes‘‘a theor y of the causes of motion’’ (Frautschiet al., 1986, p. 114). Behavior is the motionof organisms, and the study of changes in be-havior as a function of motivation, learning,and other causal factors constitutes a dynam-ics of behavior. Examples in the beh avioralliterature are provided by Higa, Wynne, andStaddon (1991), Staddon (1988), and Myer-son and Miezin (1980); Marr (1992) providesan over view. A framework that embraces allof the above special cases is called a mechanics.

This term does not nowadays refer to hypo-thetical internal mechanical linkages; suchmachinery is the vestige of the Cartesian tra-dition in which Newton labored when he be-gan to establish the modern science of me-chanics. That mechanical tradition sought toprovide causal explanations of phen omena,although such causes were often narrowlyconstrued as material causes involving themotions of particles or aggregations of matterunderlying the phenomena. It was one of Newton’s chief disappointments that he was

never able to provide such a ‘‘mechan ical’’

substrate for forces such as gravity, an d he fi-nally repudiated knowledge of such hypo-th etical causes in h is famous ‘‘hypo the ses n onfingo,’’ offering instead a precise mathemat-ical description of the effects of those forces.

His d ynamical theor y reconciled ‘‘the tradi-tion of mathematical description, represent-ed by Galileo, with the tradition of mechani-cal philosophy, represented by Descartes’’(Westfall, 1971, p. 159).

As is the case in physics, in behavior anal-ysis the term mechanics is something of an at-avism; but in bo th cases, it may be inte rp retedas an emphasis on the analysis of complexresultants into their constituent forces, as afocus o n causal rath er than statistical expla-nations, and on mathematical rather than

mechanical linkages between cause and ef-fect. It is in those senses, ones common tothe behavior-analytic tradition, that it is usedhere. It embraces molecular models such asmelioration, but not teleological models suchas those predicated upon optimization. It in-volves the theoretical constructs of  v a lu e an ddrive. Theoretical constructs are as necessaryfor a science of behavior as they are for anyother science (Williams, 1986); this was rec-ognized by Skinn er th rough out h is career, be-ginn ing with h is argumen t for the generic na-

ture of the concepts stimulus and response(Skinn er, 1935), th rough his defense of driveas a construct that can make a theory of be-havior more parsimon ious overall ( Skinn er,1938) , to h is final writings. Th e issue, as Skin-ner and others ( Feigl, 1950; Meehl, 1995)have stated, is not whether such constructsare hypothetical, bu t whether they pay theirway in the cost-ben efit ratio o f constructs topred ictions. This article requ ires a loan o f thereader’s patience as these constructs are de-veloped and deployed, in the hope that the

theor y will in the e nd be judged a worth whilecontribution to the experimental and theo-retical analysis of behavior.

O pen Versu s Cl osed Econ om ies

On e of the key conditions that is assumedto be constant in Killeen’s (1994) mechanics,but that varies substantially in the real world,is the value of the incentive to the organism.This value depends both on the intrinsicqualities of the incentive—what Hull and hisstudents denoted by K  and called incentive-mo-

tivation —and the hunger, thirst, or ‘‘drive’’ of 

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408 P ET E R R . K IL L E EN  

Fig. 1. A revision of the figure drawn by H ursh

(1980), showing the differences in patterns of responserates of monkeys under open and closed economies, asa function of th e inter reinforcement inter val on variable-inter val sched ules. Th e cur ves are drawn by Equation 8Ј.See Hursh (1978) for procedural details and originaldata.

the organism, which they denoted by D (e.g.,Hu ll, 1950; Spen ce, 1956). Much of th e earlyresearch on these factors was an essentiallyqualitative analysis of the differential rolethey played in motivation. The present con-

cern is the development of a quantitativeanalysis, one that proceeds by expanding thesingle parameter a (th e specific activation of an incentive) into components akin to K  an d

 D. Here these constructs are developed outof th e already-establish ed statics (Equation s 1through 3) and provide the motivational‘‘causes’’ that transform it into a dynamics.

All of the data analyzed under the originalformulation of the mechanics were derivedfrom animals at high levels of deprivation,which often requires supplementary feeding

in the home cages. But behavioral economistshave argued that such conditions p rovide arestricted, perhaps even anomalous, perspec-tive on behavior, and that our analysis willhave more ecological validity to the extentthat we pe rm it our subjects to earn their com-plete daily ration un der th e constraints of theschedu le we study, in the process often per-mitting them to approach ad libitum reple-tion by the end of the (extended) daily ses-sion. The traditional procedure has beencalled an open econ omy because the subject is

maintained by food and water extrinsic to theschedule contingencies; the latter arrange-ment has been called a closed econ om y. Collier,Johnson, Hill, and Kaufman (1986) chris-tened the traditional open-economy proce-d u r e t h e refinement paradigm, ‘‘develop ed inclassic ph ysics, fir st en un ciated for animals byThorn dike (1911, pp. 25–29) and perfectedby Skinner (1938), Hull (1943), their stu-dents, and their contemporaries’’ (Collier etal., p. 113). Because postsession feeding isone of the least important distinctions be-

tween open and closed economies, becausedescription of the procedure as an economyconstitutes a commitment to a particular ex-planator y framework, and because the refin e-ment paradigm is the ideal context in whichto refine basic principles, th eir term is uti-lized throughout this paper.

A number of researchers h ave adopted theeconom ic analysis o f schedu le effects, withtheir designs often involving novel schedulesof reinforcement. Hursh (e.g., Hursh, 1984)has shown that the very type of functions an-

alyzed by Killeen (1994) look quite different

u n d e r a clo se d e co n o m y. Fo r in st an c eHursh’s (1980) Figure 4 showed responserate decreasing slightly as the schedu led rateof reinforcement decreased in an open econ-omy, just as we would expect from Equation s

2 a n d 3, b u t increasing markedly in a closedecon omy. Figure 1 shows those d ata ( der ivedfrom Hu rsh, 1978). This constitutes a seriousthreat to behavioral mechanics and to all oth-er theories that entail the Herrnstein hyper-bola. Hursh argued that ‘‘It is the economicsystem which produced the different results’’(1980, p. 223). But just what was it about thedifferent systems th at made the difference?Hursh’s explanation is in terms of  elasticity of demand. ‘‘In the closed economy with no sub-stitutable food outside the session, demand

was inelastic; in the open economy with con-stant food intake arranged by the experi-menter, demand was elastic’’ (H ursh, 1980, p.233). Elastic goods are those such as luxuriesfor which increases in price causes decreasesin willingness to work for them or in theamount that will be paid for them ( demand ) ;inelastic goods are those such as basic needsfor which moderate increases in cost have lit-tle marginal effect on demand; customers willpay what they have to to maintain consump-tion (Kooros, 1965; Lea, 1978). Elasticity is

measured as the proportional change in de-

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409 M EC H A N ICS 

m an d t h at r e su lt s fr o m a p r op o r tio n alchange in p rice. For th e closed econom y, asthe reinforcement rate decreases (m oving tothe right on the x axis of Figure 1), price in-creases (animals get less food per response)

and there is a concomitant increase in re-sponse r ates. The flat functions for the openeconom y suggest an elasticity near un ity, asshould be the case: If you can get it after thesession for free, you shou ldn’t work hard erfor it when prices go up. (The proper x axisfor the economic an alysis is un it price—re-sponses per unit of reinforcer—which is high-ly correlated with mean time between rein-fo r ce r s a t m o st r e sp o n se r a te s. At lo wresponse rates on interval schedules, howev-er, price is positively corr elated with r espon se

rate. Strictly speaking, th is latter depe nd encymakes economic analyses inappropriate forinterval schedules, because ‘‘In order to de-duce the shape of the demand for a consum-er good, the first assumption one shouldmake is [ that] no individual buyer has anyappreciable influence on the market price;n amely, th e pr ice is fixed’’ Koor os, 1965, p p.51–52.)

Behavioral economics provides an interest-ing perspective in a field in which the dataare rich and complicated and the potential

for bridging to another discipline is so clear.But is it the right perspective? Does respond -ing constitute a cost—do animals meter keypecks the way humans do pennies? Do theyant icipate en d-of-session feedings? Just whyshould the rates under the closed economygenerallybe lower th an th ose un der th e openeconomy, if in the latter case animals canbank on a postsession feeding? Why shouldrates fall to near zero for the variable-interval(VI) 20-s schedule in the closed economy incontrast with the open economy? How are

these effects pred icted from economic theo-ry? Elasticity might describe, but cannot ex-plain, these differences; nor have economistsexplained why elasticity itself sho uld var y con-tinuously with price, as is usually the case forbehavioral data. A simpler hypothesis can ex-plain the differences in the data under thesetwo experimental paradigms: In the closedeconomy the subjects are closer to satiationmor e of th e time, e specially at small VI values;subjects from the open econom y, being hu n-grier, respond at a higher rate. To formalize

this treatment requires an expansion of the

mechanics to handle deprivation and incen-tive motivation.

HUNGER

Where does deprivation level enter the ba-sic principles of reinforcement? The primaryeffect will be on the specific activation asso-ciated with an incentive: The value of  a inEquation 1 will decrease with satiation. Thelevel of incitement that a small banana pelletwill provide to a satiated monkey will be lessthan that provided to a hungry one.1 Th ecloser an animal is to its natural rate of intakeunder ad libitum feeding, the smaller ashould be. Similarly, the incitement from asmall banana pellet will be less than that from

a large banana pellet. Therefore, the param-eter a must be expanded from a single freeparameter to a product of the organism’shunger and the value of the incentive in al-leviating hunger. To be concrete, let us think of the hunger drive in the simplest terms:Consider the metabolic system to be a vesselthat stores a finite amount of food and util-izes it at a constant metabolic rate M . Th econtext permits the organism to acquire n ewfood o f average m agnitude m at the rate of R(see the Appendix for a review of the con-

stants and their dimensions). Depending onthe recent history of depletion and repletion,there will be more or less food in store. Tobe precise, we would need to deal with a cas-cade of storage devices (i.e., the mouth, thestomach, the bloodstream, th e adipose tis-sue) , e ach with their own release rates; dif-ferent types of food will affect these differ-ently. Bulky food may fill the mouth andstomach but do little to alleviate d eep hun -ger, whereas sugars may immediately releasestored glucose into the bloodstream while

leaving th e stomach re latively em pt y. We willnot confront those details here: Think interms of the stomach (or crop) and somestandard food such as those typically used asreinforcers. In this simplest instantiation, th ed e fi cit i s t h e e m p t in e ss o f t h e st o m a ch .

1 Secondary motivational effects on all the parametersare likely. For instance, a weakly motivated organismmight take longer to complete a response, lowering theceilings on response rate (see, e.g., McDowell & Wood,1984, and Equation 3Ј below). But this paper focuses onthe primary motivational effects, whose locus of action is

on the parameter a.

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410 P ET E R R . K IL L E EN  

Changes in the deficit will depend on the bal-ance between the rates of emptying the stom-ach (depletion) and of filling it (repletion)over time. In the case in which both the inputrate ( m R ) and the output rate ( M ) are con-

stant over the interval t, the deficit at time t,d t , is

d  ϭ d  ϩ ( M  Ϫ m R ) t , ( 4)t  0

where d 0 denotes the initial deprivation level.

 B ou n d a r y C on d i t i on s

It is worth a concrete discussion here of two of the variables ( d 0 an d M ) in Equation4, because they recur throughout the paperand will often be set to fixed values. In anopen economy, the experimenter might de-

prive the organism for several d ays, but nomatter how deprived, animals can eat onlyuntil their stomachs are full. In these casesthe initial deficit d 0 takes the value of themaximum capacity of the stomach. For rats,the typical maximum meal size is about 4 g( see, e.g., Joh n son & Collier, 1989, 1991) . Foranimals such as pigeons with a crop or mon-keys with cheek pouches, a meal can be muchmore substantial. This is also the case for ratswhen their environment permits them toh o a r d . T. Re e se a n d H o ge n so n ( 1 962 )

showed that for deprivation times over 24 hr,pigeons will consume approximately 10% of their free-feeding weights. Zeigler, Green,and Lehrer (1971) found that in the courseof an hour, 10 White Carneaux that had beendeprived to 80% of their ad libitum weightsconsumed 40 g of mixed grain on the aver-age; this is consistent with Reese and Hogen -son’s estimate of  d 0.

In closed economies in which initial depri-vation times are minimal, d 0 will be small andmay usually be set to zero. Un der these con-

ditions d epr ivation will gro w with time sincethe last meal ( t ) according to Equation 4 un-til hunger motivation exceeds the threshold,at which point anoth er meal will be initiated.

Pigeons of typical size require between 0.5and 1 g/ h r to m aintain their weights between80% and 100% of ad libitum, and the re-quirements for rats also fall within that range.These values for M  are sufficiently smallerthan the rates of repletion in typical (openeconomy) experiments that one may set M  ϭ0, as is don e in all of th e subsequent analyses

in this paper.

 D ri v e Versu s D efi ci t 

What is the relation between the hungerdrive h t  and deficit d t ? The simplest modelmakes h unger proportional to deficit, h t  ϭ␥d t , so that from Equation 4

h ϭ ␥[ d  ϩ ( M  Ϫ m R ) t ] . ( 5)t  0

Altern ate m ode ls of this basic process are pos-sible. Equation 5 is similar to a regulatorymodel proposed by Ettinger and Staddon(1983). Townsend (1992) explored a dynam-ic motivational system that, in place of Equa-tion 5, had motivation grow as a function of the deviation between the current motiva-tional level and the ideal, with a thresholdthat motivation must exceed before respond-ing will be initiated. Solution of such a modelleads to motivation that grows exponentiallywith time, rather than linearly:

␥[ d  ϩ( M Ϫm R ) t ]0h ϭ e Ϫ . ( 6)t 

With the threshold equal to 1.0, m otivationwill be zero whe n dep rivation level is zero. Inthe case of  Ͼ 1, it requires more than theminimal amount of deprivation for the sub-

 ject to be gin re spo n din g. In th e case of  Ͻ1, the subject will continue responding evenwhen satiated (Morgan, 1974), either because

conditioning has created some behavioralmomentum or because the drive is also main-tained by other deprivations (e.g., dilute su-crose solutions will assuage both hu nger andthirst). In the linear model, threshold effectsare absorbed into the deficit parameters.

The exponential model has some face va-lidity, in that introspection suggests th at theexigency of hunger seems to grow moresteeply than linear with deprivation time. It isconsistent with control-systems analyses of motivational systems (e.g., McFarland, 1971;

Toates, 1980). Serious stude n ts of th ese issueswill find an excellent review of th e currentstate of research on appetite and its neuraland behavioral bases in Legg and Booth(1994).

Yet an other mod el of h un ger would have itgrow sigmoidally with deprivation, approach-ing a ceiling at the highest levels of depriva-tion. Such a m odel is outlined in the Appen -dix; its application did n ot impro ve an y of theanalyses, and so it is not pursued here.

Equations 5 and 6 show that when an ani-

mal become s satiated (when the initial deficit

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411 M EC H A N ICS 

is replaced and depletion is just balanced byrepletion), h t  falls below th reshold, drivingmotivation to zero and carrying response ratealong with it. Food-motivated behavior ceases,preventing overindu lgence that would drive

hu nger levels to a negative value. Contingen-cies of reinforcement that require consump-tion for access to other incentives, however,could drive h t  to a negative value. In this case,response rates are depressed below free baserates (Allison, 1981, 1993), requiring externalforce or the passage of time to overcome thatinhibition.

 A ggreg a t i n g O v er a S es si on

For the linear model, the average drive lev-el over the course of a session of duration t sess

is given by Equation 5, with t  ϭ t sess / 2 ( seethe Appendix). Under the exponential drivemodel, the situation is more complicated. If session du ration is con stant, t he average dr ivelevel is given by Equation 6, with t  ϭ t Ј, someundefined fraction of  t sess. In employing theexponential model, one may set t Ј to somearbitrary value ( e.g., t sess / 2) an d let th e re-maining parameters adjust them selves to thatconstraint.

Econ om ic T ran sla ti on

In economic parlance, d 0 is the debt, m R isthe wage, and M  is the cost of doing business.On ratio schedules the rate of reinforcement

 R is an inverse function of the ratio size ( n ) ,or price, and n /  m is the unit price. M , th erate of utilization of food by a free-feedingorganism, is the coordinate of the ideal, orbliss point, along the food consumption axis.It could be separated into fixed cost or over-head (basal metabolic rate), and productioncost (r espon se effort) . Basal m etabolic rateconstitutes the major cost of foraging and

thus constitutes a significant ‘‘sunk cost’’ toany endeavor: O nce standing, it doesn’t re-quire much more energy to do anything.(Th is distinction implies flat optima for mod-els of foraging that maximize calories gainedper calories of effort expen ded ; more precisefeedback is pr ovided by optimizing caloriesgained over time expended.)

The parameter ␥ represents the cost of de-viations from the ideal, and e␥ provides oneindex of the elasticity of demand . If ␥ is large(and thus e␥ Ͼ 1), the animal is very sensitive

to deviations from the ideal rate of repletion,

and demand is said to be inelastic. If  ␥ issmall (and thus e␥ ഠ 1), then changes in priceelicit only m inimal behavioral adjustmen ts;demand for the commodity approaches unitelasticity. If  ␥ is negative (and thus e␥ Ͻ 1) ,

animals will work less for a commodity as itsprice increases, and demand is said to be elas-tic. This occurs in the presence of substitutes,as when food is available for respon ding onother levers (Johnson & Collier, 1987). Thisinterpr etation of elasticity differs from that of the economists, because theirs refers to de-mand as a function of price but does not takedeprivation levels into account. Economicmodels are designed to map population ef-fects, not biological ones. Saturation of themarket is treated with different models than

elasticity. ‘‘Decreasing mar ginal utility of goods’’ captures some of the idea of satiation,but is usually construed without reference tothe current deficit.

The present approach predicts that theeconomists’ measure of elasticity will changewith price, because on ratio schedules therate of reinforcement, R , which appears inthe right sides of Equations 5 and 6, equalsm /  n , the reciprocal of unit price. Motivationvaries with price because that affects the rateof repletion. Indeed, Hursh, Raslear, Bau-

man, and Black ( 1989) found elasticity tovar y as a linear function of un it price. But thisis not because ␥ has changed; our measureof elasticity, e␥, may stay constant over chan gesin motivation because we have moved thecontrolling variables into our independen tvariables (Equations 5 and 6), and thereforedo not need to let our theoretical constantsvar y with our indepe nd ent variables.

Ecologic T ran s lati on

 M  is the setpoint repletion rate that ani-

mals will d efend . Equation 4 provides a mea-sure of deviation from that setpoint. Defenseof the setpoint is equivalent to animals’ at-tempting to minimize that deviation, that is,set the derivative to zero. The force of th isequilibration is given by ␥. In control-systemsparlance, ␥ represents the regulatory gain, orrestoring force. Many differen t ar rangemen tsof contingencies will generate many differentconstellations of beh avior, all o f which haveonly one thing in common and predictable;the absolute value of Equation 4 will be min-

imized. This approach therefore is like the

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412 P ET E R R . K IL L E EN  

Fig. 2. Response rates under chained schedules forpigeons receiving different durations of access to thehopper during extended sessions (Fischer & Fantino,1968). The data rep resented by filled symbols come from

the terminal link, and those by open symbols from theinitial link. The curves are drawn by Equations 3 and 7,and represent performance for 2-s (inverted triangles),6-s (triangles), 10-s (squares), and 14-s (circles) access tofood.

H a m il to n i an a p p r o a ch t o m e c h a n ic s, i nwhich all of the laws of mechanics may bederived from minimization of a single differ-ential equation called the action. It is the coreassumption of regulatory approaches to be-

havioral e cono mics such as Allison ’s ( 1983) .The current approach also recognizes theboundary conditions to this minimization:The changes in motivation will not be re-vealed in behavior until theycross a thresholdfor action, and they will not continue oncethe capacity of the organism is saturated.

 A n A p p li ca t i on of t h e B a si c M o d elto S a t i a t i o n C u r v es

How d oes d rive level interact with magni-tude or quality of the incentive? The simplest

assump tion is multiplicative: Absent eitherdrive or a viable incentive, the specific acti-vation a must be zero. We may call the incen-tive variable v. Then a t  ϭ v h t . The value of anincentive will n ot generally be prop ortionalto its magnitude, although a linear relationmay be an adequate approximation if therange of variation is small.

In accord with the above analysis, for thelinear drive model we expand the specific ac-tivation to

a ϭ v h ϭ v ␥[ d  ϩ ( M  Ϫ m R ) t ] , ( 7)t t  0

where ( M  Ϫ m R ) is the balance between de-pletion and repletion, and its multiplicationby t  gives the cumulative effects of that bal-ance. This equation h as replaced a as a singlefree parameter with a three-parameter mod-el: value v, the initial deficit d 0, a n d t h e d e -pletion rate M . (For the linear model the de-viation -cost param eter ␥ is redundant withthe value parameter v and may be absorbedinto it or simply set to 1.0.) Equation 7 maythen be inserted into Equation 3 to predict

r e s p o n s e r a t e s o f a n i m a l s u n d e r i n t e r v a lschedules when deprivation levels vary.

Fischer and Fantino (1968) provided thedata around which the linear model was de-veloped. They deprived pigeons to 80% of their ad libitum weights, and trained them torespond on chained VI 45 VI 45 schedu les,e xt e n d i n g t h e se ssio n s u n t i l r e sp o n d i n gceased. The reinforcer consisted of access toa hopper of mixed grain for 2, 6, 10, or 14 s.Figure 2 shows the resulting satiation curvesin the terminal links of the chain and in the

initial links. Although the data themselves

show rather unexciting monotonic decreaseswith number of feedings, the model providesa rational fit to them. The first step was toestimate the amount of food obtained underthe different conditions, because amountconsumed is not proportional to h opper du-

ration . Fortu nate ly, Epstein ( 1981) pu blish eda useful graph giving the amount consumedfr om a h o p pe r o f t h e d e sign u se d in t h isstudy. For the se h opp er d urations the regres-sion gave the amou nts as 0.13, 0.28, 0.35, and0.36 g of mixed grain.2 I used those numbersas estimates of  m .

The pigeons’ weights were reduced to 80%of their free-feeding weights. To optimize thegoodness of fit, I set the parameter k  in Equa-tion 3 to 200 responses per minute for theterminal link and 64 responses per minute

for the initial link. The initial deprivation d 0took a value of 57 g. The value parameter vwas 1.5 s per reinforcement. Th e e xponen tialdrive model provides a comparable fit tothese data. Given the necessary approxima-tions, the fit of the model to the data is per-

2 For Lehigh Valley feeders the n umber of grams eatenapproximates a linear function of hopper duration, witha slope of 0.06 g/ s and an intercept of 0.2 g ( Epstein,1985). Pigeons feeding ad libitum are less efficient, withtypical eating episodes lasting 7 s, durin g which 0.33 gare consumed (Hen derson, Fort, Rashotte, & Hen der-

son, 1992).

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413 M EC H A N ICS 

Fig. 3. Within-session satiation effects sho wn for gen -eral activity as m easured by a stabilimeter, and for leverpressing. The data are averaged over two sessions in

which 4 rats were given two 45-mg pellets for the firstresponse 30 s after the previous reinforcement (FI 30).The curves are drawn by Equation 8Ј.

haps acceptable, although respond ing in theinitial links d ecreased at a faster rate thanpred icted, especially for the 14-s ho pper con-d i t i o n . ( L e n d e n m a n n , M y e r s , & F a n t i n o ,1982, found a similar hypersensitivity in the

initial links in response to variations in du-ration of reinforcement, as did Nevin, Man-dell, & Yaren sky, 1981, in respon se to satia-tion.) It may be that in all cases decreasedmotivation has its primar y effects on pausing,and once an animal has begun to respond, itcontinues until reinforcement. If this is thecase, then pausing will occur primarily in theinitial links, with animals responding through-out the terminal links. Segmen ting respon dingwill thus put the greatest leverage of motivationon the earliest segments. (See Williams, Ploog,

& Bell, 1995, for further analyses of thesechain-schedule effects.)We can write the above models in a more

condensed form. Set the metabolic rate M  to0, the magnitude of the incentive m to 1, andlet the gain parameter ␥ be absorbed into v ;then write Equation 3 as

kR B ϭ . ( 8)

 R ϩ 1/ [ v ( d  Ϫ R t ) ]0

This equation reiterates the above descrip-

tions, but also provides qu antitative pred ic-tions: Because of satiation effects, responserate is a quadratic function of reinforcementrate. Under conditions of large initial deficit( d 0) relative to repletion ( R t ), the parenthet-ical expr ession is essen tially constant and canbe absorbed by v, which r etur ns to u s our sim-ple Equation 3 (or 3Ј, below). Th e H err nsteinhyperbola is thus valid primarily for sessionsof short duration or low rate of reinforce-ment, where the initial deficit outweighs thecumulative repletion. But satiation effects

grow with t, and become dominant later in asession.

If one is interested in estimating the pa-rameters in Herrnstein’s hyperbola, then it isbetter to use data from early in a session inwhich repletion ( R t ) is low relative to initialdeficit ( d 0) , or from shor t sessions, so th at theden ominator is relatively constant. Better yet,use Equation 8 at the cost of one additionalparameter ( d 0) a n d p r ed ict t h e co m ple tefunction.

Note that the adden d 1/ [ v ( d 0 Ϫ R t )] in the

denom inator was interpreted by Herr nstein

as R O, the value of reinforcement for other(nontarget) responses. He and Loveland pre-dicted that when animals were not deprivedof the p rimar y reinforcer, these oth er implicitreinforcers should seem to grow in relativevalue, thus increasing the value of R O ( H e r r n -stein & Loveland, 1972). Their data showed

this to be the case; however, our interpreta-tion is more straightfor ward: When animalsare not greatly deprived, d 0 will by d efin itionbe small, and thus 1/ [v ( d 0 Ϫ R t )] (their R O)will be correspondingly large.

The expone ntial-drive mode l is n ecessar yfor some of the data on satiation. In that case,Equation 8 may be rewritten as

kR B ϭ , ( 8Ј)

 R ϩ 1/ ( v h )t 

with drive level h t  an exponential function of 

deficit (Equation 6) rather than a linear func-tion (Equation 5). In an unpublished exper-iment, Lewis Bizo and I delivered two 45-mgpellets to rats immediately after a lever presson a fixed-interval ( FI) 30-s schedule. Gen-eral activity was concurrently measured witha stabilimeter. Figure 3 shows the decline ingeneral activity and lever pressing as a func-tion of the nu mber of trials. Equation 8Ј drewboth curves. The motivational parameters ( ␥ϭ 0.3 gϪ1 an d d 0 ϭ 4 g) were the same forboth responses, whereas the remaining pa-

rameters were und erconstrained by the data.

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414 P ET E R R . K IL L E EN  

Fig. 4. Data from McSweeney et al. ( 1990), showingwithin-session warm-up and satiation effects in rats. The

curve is drawn by Equation 3, with Equation 7 repre-senting the satiation effects and Equation 9 the warm-upeffects.

The lever-press data are flatter because ceil-ings on response rate compress the top endof the function. The key point is that Equa-tion 8, which predicts a linear or concave-down decrease in responding, could n ot h ave

fit the concave-up time course of satiation asmeasured by gen eral activity.Equation 8Ј also drew the curves through

the data in Figure 1. In both economies d 0took the value of 140 reinforcers and k  was5,500 responses per h our; for the open econ-omy, ␥ ϭ 0.10, and for the closed economy ␥ϭ 0.07. The key difference between thecurves is the degree of repletion permittedwithin the session. For the closed economythe session duration was 6,000 s, so that t sess / 2 is 3,000 s, and the average session deficit

(the coefficient of  ␥ in Equation 6) is 140 Ϫ R ϫ 3,000. The fixed duration of the closedeconomy per mitted differential satiation as afunction of rate of reinforcement ( R ). Forthe open economy the session ended after180 reinforcements, so t sess / 2 is 90/  R s, andthe average session deficit is 140 Ϫ R ϫ 90/ 

 R ; that is, a constant 50 g. Terminating ses-sions after a fixed number of reinforcers, orin general keeping session duration propor-tional to interreinforcement interval ( 1/ R ) ,confers a constant average level of motiva-

tion. This is the key difference between theexperimental paradigms; it is ‘‘the economicsystem which produced the different results’’shown in Figure 1. It did so by letting theanimals differentially satiate in one case butnot in the other.

The amount of food consumed in theseand the Fischer and Fantino (1968) sessionswas two to five times the amount consumedin a typical session. Is there evidence for thedecrease in respond ing du ring operan t ses-sions of more typical duration? Thanks to

McSweeney and her colleagues, there is nowample evidence of within-session satiation ef-fects (see McSweeney & Roll, 1993, for a re-view) . But he r data also sho w within-sessionwarm-up effects, so we must digress to amodel of those.

WARM-UP

Some of the first evidence for within-ses-sion effects from McSweeney’s laboratorycame from a study conducted to test the ef-

fects of postsession feeding on rats that were

requ ired to press a lever for Noyes pellets or,in a different condition, to press a key forsweeten ed cond en sed milk ( McSween ey, H at-field, & Allen, 1990). Although no effects of postsession feeding were found, a remarkablepattern of rate changes within the session wasdiscovered (see Figure 4). Response rates in-

creased through the first 20 min of the ses-sion and decreased thereafter, and the pat-t e r n wa s vi r tu a lly id e n t i ca l fo r t h e t woresponses and reinforcers.

The decrease in rates may be attributed tosatiation of the kind seen in the previous fig-ures. To what do we attribute the increase inrates? Killeen and his colleagues (Killeen, inpress; Killeen et al., 1978) have described sim-ilar increases in rates when animals are firstintroduced to a schedule of periodic rein-forcement, and attributed them to the cu-

mulation of arousal. Such warm-up plays alarge role in behavior maintained by aversivestimuli and a lesser but still measurable rolein behavior maintained by relief from hun -ger. Introduction to the chamber itself be-comes a conditioned reinforcer and there-fore a conditioned exciter. If there were noloss of this arousal between sessions, eventu-ally each session would begin with rates attheir asymptotic level. But the animals calmdown between sessions. For the present pur-poses, assume this between-sessions loss is

complete (see Killeen, in press, for a more

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417 M EC H A N ICS 

subjects. All may be assimilated in mechanis-tic models of behavior. The final issue thatmust be addressed before mechanics can be-gin to stand as an alternative to economicand ecologic analyses is the relation between

the amount of an incentive and its value.

MAGNITUDE O F INCENTIVES

Wher eas animals typically ch oose largeramounts of food over smaller amounts (see,e.g., Bonem & Crossman , 1988; Collier, John -son , & Morgan, 1992; Killeen, Cate, & Tru n g,1993), response rates often change little ornot at all as a function of the magnitude of the incentive. Why should this be? In part,the answer depends on the fact that the re-

inforcing value of an incentive is not propor-tional to its size. In the case in which mag-nitude is manipulated by varying duration of the incentive, the reasons for this are obvious:The second, third, and n th instants of con-sumption are not contiguous with the re-sponse that brought them about; they areseparated from it by n Ϫ 1 prior instants of consumption (Killeen, 1985) that block theireffectiveness. The last instants of a long-du-ration reward constitute a delayed reward.Those later instants of consumption increas-in gly r ein fo rce n o t t h e p r io r o p er an t r e-sponses but rather the immediatelyprior con-summator y responses. Assume that each of the instants of consummator y activity inter-polated between a response and the last in-stan t of consumm ator y activity will b lock th atlatter’s effectiveness by a constant prop ortion,. Then it follows that the effectiveness of anincentive should increase as an exponentialintegral function of its duration:

Ϫmv ϭ v ( 1 Ϫ e ) , ( 10)m ϱ

where v m expands the value of an incentivefrom a constant v to a fun ction of its dur ationor magnitude ( m ) ; vϱ is the value of an arbi-trarily long duration of that incentive, and is the rate of discounting the incentive as afunction of its duration. Value ( v m ) refers tothe psychological/ beh avioral magnitude of an incentive whose ph ysical m agnitude ( m )may be measured in grams, seconds, or mil-ligrams pe r kilogram. In cen tive motivation re -fers to the evaluative or instigating effective-

ness of the incentive th at d epen ds on its value

in the context, as represented by equationssuch as Equation 8.

Equation 10 embo dies the maxim of ‘‘mar-ginally decreasing utility’’ of incentives (as afunction of their duration, not, as often used

in economic parlance, as a function of num-ber of reinforcers). If  is small, the relationis approximately proportional; if  is large,increasing d ur ation add s ver y little value. Kil-leen (1985) found that Equation 10 with between 0.25 and 0.75 sϪ1 fi t m a n y o f t h echoice data he reviewed. For the representa-tive value of  ϭ 1/ 2, the value of 3 s of hop-per access has attained 78% of the maximumpossible ( vϱ). Studies that manipulate longerdur ations are operating within a ver y restrict-ed range.

This model of the change in value withchanges in the duration of an incentive maybe combined with Equations 7 and 8 to pre-dict performance when the duration of an in-centive is varied. When the value of an incen-tive is man ipulated by changing its qualityrather than by changing its duration, someutility function other than Equation 10 (e.g.,a power function or a logarithmic function)m ay b e m o re ap p r op r ia te . Wh e n , fo r in -stance, a drug level or sucrose concentrationis manipulated, a plausible model is v m ϭ m ,

and thena ϭ m ␥[ d  ϩ ( M  Ϫ m R ) t ] . ( 11)t  0

Whereas larger incentives are marginallystronger reinforcers, they also decrease themotivation to work by satiating animals morequickly. Th ese effects will ten d to cancel, d e-pending on the range of durations studiedand the value of the deficit the animal is at-temp ting to satisfy. If initial d eficit d 0 is largeor repletion time t  is short or the rate of re-pletion m R is small, the satiation effects will

be buffered by d 0 and net incentive effects(increasing response rates with increasingmagn itude) will be foun d. Conversely, if d 0 issmall and repletion is moderate or large, asis typical of closed economies, the satiationeffect will dominate, and response rates willdecrease as a function of magnitude. The de-pendence of the sign of the correlation be-tween magnitude and response rate—positivein the realm of small incentives, n egative inthe realm in which satiation effects d omi-nate—is shown in a study by Collier and My-

ers (1961), who found positive covariation of 

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418 P ET E R R . K IL L E EN  

response rates with volume for dilute and in-freque nt sucrose concentr ations an d negativecovariation for freque nt high concentrations.The authors spoke in ter ms of momentary sa-tiation, which is exactly how we have been

speaking about repletion here. More partic-ularly, we can take th e der ivative of Equation11 with respect to m and set it to zero to fin dthe magnitude of  m at which the correlationwill go from positive to negative. The tur n-over point is

d  /  t  ϩ M 0m * ϭ . ( 12) 1 ϩ R

Of the variables under experimental con-trol, increases in d 0 will extend the range of 

m over which a positive correlation—an in-centive effect—is foun d; increases in sessionduration and rate of reinforcement ( t  an d R )will m ove the turn over point to the left, leav-ing more of the range to show a n egative cor-relation—a satiation effect. O f course, largevalues for d 0 and relatively small values forsession duration are typical of traditional ex-perimental designs, in which incentive effectsshould thus be the rule; small values for d 0and relatively large values for session dur a-tion are typical of closed economies, in which

satiation effects should thus be the rule.

W it h in -Session Ef fects Versu s B et w een -S essi on E f f ect s

Choice behavior shows greater control bymagnitude of reinforcement than does sin-gle-operant respond ing. The present frame-work explains this result the following way:The satiation effects are shared by both op-erants in a choice situation, leaving th e in-centive effects to act differe ntially, u n buffer-ed by satiation. Th e same is true for response

rates in multiple schedules, in which satiationeffects should generalize when componentdurations are not too long, leaving incentiveeffects the opportunity for differential effec-tiven ess—an effect kno wn as contrast  (Nevin,1994). It remains to be seen just how muchof the complex literature on behavioral con-trast can be understood in these terms. Tothe extent that this mechanics applies, con-trast should be greatest when there is leastbuffering by d 0; that is, toward the end of ses-sions, in longer sessions, and in closed econ-

omies. It should be greater for animals that

take longer to satiate because th ey have cropsor other caches (e.g., pigeons), compared tothose that don’t (e.g., rats). Contrast shouldbe greater for incentives for which there islittle satiation (e .g., electrical stimulation of 

the brain, nonnutritive sweeteners) and lowerfor bulky but low-valued incentives.Analogous predictions hold for postrein-

forcement pausing (see Perone & Courtney,1992). (a) Unsignaled within-session manip-ulations should reflect primarily satiation ef-fects (longer pauses after larger reinforcers),because the differential magnitudes providedifferen tial mom entar y satiation effects im-mediately after their deliver y, whereas theforthcoming incentive value is averaged overall durations of incentives. (b) For between-

sessions changes, the two component effectswill tend to cancel. (c) Signaled within-ses-sion changes should reflect primarily incen-tive effects, be cause the forth coming incen -tive is particular to performance under itsstimulus control, whereas the satiation effectswill tend to be averaged across magnitudes.

Unlike response rates, there is no ceilingeffect on pause lengths, which may makethem more sensitive to changing motivationallevels than rates; most of the effects predictedby the present theor y may reflect differences

in the amount of time spent pausing or en-gaging in other responses, rather than con-tinuous changes in response rates over a sub-stantial range. In any case, the present theor ypredicts that all of these effects should bestrongly affected by deprivation level, ex-plains why, and stipulates the contexts inwhich satiation versus incentive effects will befound.

ECONOMICS

The central concern of traditional econom-ics is the exchange of goods for other goods,including labor, and that is also the concernof behavioral economics. Experimental sub-

 jects exch an ge be h avior for good s, or strikebalances between several goods in return fortheir behavior. Without this requirement forexchange of tangible items, there would beunderconstraint in theories and chaos in themarketplace: If all that mattered to hungrysubjects were maximization of reinforcement,all animals would always respond at their

maximum rates under most contingencies.

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419 M EC H A N ICS 

Fig. 7. Data from Kelsey and Allison (1976) p lottedby Hanson an d Timberlake (1983), along with the cur vesresulting from their model and from Staddon’s (1979).Reprinted with permission. Superimposed is the paraboladrawn by Equation 16.

Economic behavioral theory was introducedin part because its framework of sacrificingone thing to get another provides a ‘‘ration-al’’ basis for the modulation of response rateswe see on many schedules of reinforcement.

When return rates are very low, animalsshould respond with little enthusiasm be-cause doing so is not worth their while com-pared to other things they could purchasewith their labor; when the return rates arevery high, they should respond with little en-thu siasm because they are close to satiation.

The greatest stren gth of econom ic analyseslies in the development of models that framethe trade-offs between different reinforcers,clarify what constitutes a ‘‘bundle’’ of goods,and explain the interactions between similar

reinforcers that perm it one to be substitutedfor another. The application of economicmodels to behavior controlled by a singlesource of reinforcement is more prob lematic,because these models are forced to introduceother hypothetical goods involved in thetrade-offs, in a way not dissimilar to Her rn -stein’s introduction of R O as a source of com-peting reinforcement. Rachlin and associates(Rachlin, 1989; Rachlin, Battalio, Kagel, &Green, 1981; Rachlin & Burkhard, 1978;Rach lin , Kagel, & Battalio, 1980) treat leisur eas a good, so that depending on the experi-menter’s constraints, the animals must maketrade-offs between the leisure given up by re-spond ing and the material reinforcers that re-sponding provides. Those trade-offs are mo-tivated by the subject’s preference for anoptimal package of goods under constraintsof time and schedule. Staddon (1979) as-sum es th at o ptimal rates exist for all activities,and that animals are motivated to approachthat locus in beh avioral space that minimizes

a weighted sum of squares of the deviationsof each from its optimal rate (or that mini-mizes some other cost function) given theconstraints of time and schedu le. Experimen-tal contingencies u sually require operant re-spond ing at a higher-than-optimal rate, sothat such responding functions as a cost,much as it does for Rachlin and associates. InStaddon ’s multidimen sional behavior space,the coordinates of the ideals of all relevantdimensions define a bliss point, and becauseever y other point is in some way inferior, vari-

ations in an organism’s behavior th at carr y it

away from this global minimum are selectedagainst.

Hanson and Timberlake ( 1983) focus onregulation, provide a mathematical model of the equilibrium approach of Timberlake and

Allison (1974), and derive as special casesStadd on ’s ( 1979) and Allison ’s ( 1976, 1981,1993) op timality accoun ts. At th e h eart of th emodel are the coupled differential equationskn own as th e Lotka-Volter ra system. As an ex-a m p l e o f i ts a p p l ic at io n , t h e a sym m e t r i ccurve is drawn through the data from Kelseyand Allison (1976) , shown in Figure 7. Th edashed line is given by Staddon ’s (1979) min-imum distance model. In fitting their five-pa-rameter mod el, Han son and Timberlake not-ed that these functions ‘‘quickly exhaust the

degrees of freedom inherent in, for example,six or seven data points’’ (p. 272). Thus, themost we can hope for in comparing theoryto data is a consistency check, a hurdle thatis necessary for the theories to clear, butwhose clearance is not sufficient grounds forus to accept them. Whether or not we acceptthese theories seems to depend on whetherwe find their assumptions congenial to ourintuitions about behavior, and whether theymake n ovel pred ictions. Ther e h ave bee n fewnovel predictions th at I am aware of. How-

ever, they do provide new constructs and in-

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420 P ET E R R . K IL L E EN  

Fig. 8. Demand functions collected and graphed by Lea (1978). Reprinted with per mission. Superimp osed is themodel demand function drawn by Equation 17.

dices, such as elasticity of demand, that pro-

vide alternative perspectives on behavior.Elasticity is an index, ‘‘a nu mber derived

from a formula, used to characterize a set of data’’ ( A m erica n H erita ge Dict ion a r y, 1992). In-dices are useful because a single nu mber canoften characterize some crucial aspect of aphen omenon (e.g., the index of refraction of optical materials, the consumer price index,etc.). Lea (1978) drew demand curves as theamount of an item purchased as a functionof the price of the item. When the axes arelogarithmic, the slope of these curves equals

their coefficients of elasticity (see, e.g., Koo-

ros, 1965). In his Figures 3 and 4, Lea drew

idealized demand functions as straight linesof different slopes, with items such as coffeeand bread showing the least decrease in con-sumption as price is increased (demand forthem is inelastic, as we would expect), anditems such as herring and cakes showing thegreatest de crease. He re a single n umbe r—thecoefficient of elasticity—effectively character-izes a set of data. However, in his Figures 1and 5, as is the general case, real data fromclosed economies are concave: Elasticity in-creases continuously with the price of the

commodity ( see Figure 8). This result is

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421 M EC H A N ICS 

about as satisfying as would be the discoveryof an ‘‘inverse square law’’ for force as a func-tion of distance, but in a world in which theexpon ent varies continu ously with distanceand takes the value of  Ϫ2 on ly at on e partic-

ular distan ce. Elasticity sho uld n ot itself be soelastic!The demand curve was designed for anal-

ysis of decisions by populations, where in-creasing proportions of the population maybe infl uen ced to p urchase a commodity, per-haps just once, as its price decreases. It wasnot designed to analyze the repeated pur-chases by ind ividuals, b ecause such data willbe greatly affected by d ecreasing marginalutility as m agnitude increases, and by satia-tion as rate of consumption increases. As not-

ed by Staddon (1982), reinforcement rate ap-pears on both axes ( R vs. n /  R ) of the demandcurve, so that independent and dependentvariables are intrinsically correlated. Suchfunctions provide good stimulus control of vi-sual analysis only when they are linear anddifferences in slope may be directly com-pared. Looking for second-order effects suchas differen ces in degree of cur vature is madeun necessarily difficult by th e tactical choiceof those coordinates.

Behavioral economics has useful things to

tell us about substitutability and complemen -tarity (see, e.g., Green & Freed, 1993; Lea &Roper, 1977), issues not addressed in this ar-ticle. But when applied to single response–reinforcer paradigms, that approach is lessuseful (see, e.g., the commentaries on Rach-lin et al., 1981). Th ere are too man y free vari-ables to be tied down; motivational changesaffect the parameters while they are beingcollected, and the core notion that animalsprefer not to respond above a relatively lowbliss-poin t rate is false, as shown by Stadd on

and Simmelhag (1971) for pigeons and bynum erous other investigators for num erousother organisms whose u necon omical adjunc-tive behavior often overwhelms their contin-gent behavior. The paired baseline distribu-t io n s o f r e sp o n d in g u se d in r e gu la tio nmodels have been shown not to predict blisspoints, and the ratio of instrumental to con-tingent r espon ding is not the controlling vari-able it has been purported to be (Tierney,Smith, & Gannon, 1987).

The economic approach does not respect

molecular contingencies of reinforcement

(Allison, Buxton, & Moore, 1987), and ther e-fore is prima facie unable to predict the hugedifferences in responding that can be ob-tained with brief delays of reinforcement, andis un able even to predict the profound dif-

fe re n ce s t h at d e pe n d o n t h e o rd e r o f e x-change of goods—that is, the differences infor ward versus b ackward conditioning. Be-havioral economics therefore does not con-stitute a general theory of behavior. It offerssome tools for the comparison of differentincentives and their effects on behavior whensatiation and reinforcement contingenciesare controlled. It opens the door to a behav-i o r a l a n a l ysi s o f c o n su m e r c h o i ce , a b o u twhich a mature behavioral economics willhave mu ch to say.

ECOLOGICS

Collier and Johnson and associates (Collieret al., 1986, 1992; John son & Collier, 1989,1991) h ave r equired rats to work for food u n-der a variety of conditions, usually ones thatrespect the animal’s normal feeding routine,letting the animals complete meals uninter-rupted, and often extending the sessions topermit animals to acquire most of their foodwithin the experimental context (i.e., closed

economies). This extends the analysis of be-havior to a larger time scale. But, althoughperhaps more natural, it makes it more dif-ficult for the theorist to analyze the behaviorthat is obtained from these contexts. The rea-son for this is that under these conditions,rates of reinforcement are closely tied to thepattern s and rates of the animal’s behavior—rate of reinforcement, a key controlling vari-able, is no longer an independen t variable.To und erstand this, we must d igress to ex-amine how an animal’s behavior affects its

rate of reinforcement.

Sch edu le Feed ba ck Fu n cti on s

Killeen (1994) derived a schedule feedback function (SFF) that predicts the rate of rein-forcement on constant probability VI sched-ules, given a con stant rate o f respondin g of Bresponses per minute, as

Ϫ R Ј / B R ϭ B ( 1 Ϫ e ) , B Ͼ 0,

where R Ј is the programmed rate of rein-forcement. Over most of its range, this may

be appr oximated by its Taylor expansion:

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422 P ET E R R . K IL L E EN  

 B R Ј R ϭ . ( 13)

 B ϩ R Ј

This is also the form of th e SFF suggested byStaddon (1977) and Staddon and Motheral(1978). It is also the equation derived if oneassumes that reinforcers are set up and re-sponses are emitted randomly and in se-quence with rate constants of  R Ј an d B (i.e.,it is the mean of series-latency devices such astwo-step generalized gamma distributions) .When response rates are high, reinforcementrate approximately equals the scheduled rate

 R Ј (divide numerator and denominator by Band then let B go to infinity); when they arever y low, reinforcement rate approximatelyequals the response rate B . Equation 13 is ac-curate only in the ideal case of continuousengagement of organism and schedule. If anorganism takes extended timeouts from re-sponding, obtained rates of reinforcementare lower (Baum, 1992; Nevin & Baum,1980). The SFF for ratio schedules is simply

 R ϭ B /  n , where n is the ratio requirement.Such SFFs are not of interest because we

believe that animals are sensitive to how themarginal rates of reinforcement are affectedby responding under different SFFs. (Thisfundamental assumption of all molar opti-

mality mod els has been effectively discred itedby Ettinger, Reid, & Staddon, 1987.) Rather,SFFs are important because they determinethe rate of reinforcement (a key controllingvariable in Equations 1 through 3) in the con-text of an interactive organism. Closed sys-tems such as those employed by Collier andassociates ar e closed-loop system s, with thefeedback from response rates on reinforce-ment rates closing the loop through the SFF.To predict behavior und er such conditions,we insert the appropriate feedback function

into the motivation equations, and insertthese into Equation 3. For ratio schedules,the solution generates the basic equation of prediction (Killeen, 1994, Equation 8). Forinterval schedules, it yields equations propor-tional to Equation 3, but with a slightly lowerasymptote:

( k  Ϫ 1/ a ) aR Ј B ϭ , a Ն 1/  k . ( 3 Ј)

a R Ј ϩ 1

No problem: Still the same old hyperbola!Equation 3Ј shows one of the reasons that a

hyperbo lic m odel is so robu st: When specific

activation ( a ) is large, Equation 3Ј is equiva-lent to Equation 3. But even at low activationwhen obtained reinforcement rate falls sub-stantially below its schedu led value, per for-m a n c e r e m a in s a h yp e r b o lic fu n c t io n o f  

scheduled reinforcement rates, merely find-ing a lower asymptote ( k  Ϫ 1/  a) .Unfortunately, the complete equations of 

motion for organisms contain a double feed-back loop. Not only does rate of respondingaffect rate of reinforcement ( that Equation 3Јcompensates for), but rate of reinforcementdetermines the satiation of the organism,which affects th e value of specific activationa . The obtained rate of reinforcement ap-pears in Equation 7, which is an e xpansion of a . If we insert Equation 13 into that and at-

tempt to solve it, we get stuck. The result is aquadratic equation with no simple solutions.(Equation 8 is quadratic in the rate of rein-forcement, but because that is an indepen-den t variable, it caused no mischief. He re theequations are quadratic in the dependentvariable, response rate.) Quadratic equationsare, of course, non linear; th e non linearity isintroduced by having behavior be a functionof a variable (motivation) that itself is a func-tion of beh avior (which redu ces motivationby repleting the animals). Now it becomes

impossible to write equations with all theknowns on one side and the unknowns onthe other. There is no simple, complete so-lution to this impasse.

Copi n g wi th N on l in earity

When confronted with a difficult n onlin-earity such as this, we have several options:

Experimentally opening the loop. We may re-duce the n onlinearity by making the constantterms large relative to the varying terms. Thismean s large initial deficits ( d 0) relative to re-

pletion rates ( m R ); Equations 8 and 8Ј showthat this is achieved with some combinationof h ighly deprived organisms, small and in-frequent meals, an d shor t sessions: All of th ebetes n oires that Collier and other economictheor ists h ave repeated ly excoriated.

It is hard to dispute their point that thesec o n d it io n s o f t h e r e fi n e m e n t e x p e r im e n t(i.e., the standard procedures) are nonrep-resentative extrema under which the animalscan display little of the range of the naturalrepertoire of their normal instrumental and

consummator y patterns. Objects falling in a

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423 M EC H A N ICS 

vacuum display little of the range of the nat-ural r eper toire of leaves falling in an autumnwind. It is through refinement experimentsthat physicists, chemists, and behavioristshave come to understand the variables of 

which their subject is a function. We can havesimple laws, such as Equation 3, or we canhave more precise but complicated ones,such as those obtained by inserting Equation11 into it; to the degree that we want preci-sion, we must forgo its complement, simplic-ity (Killeen , 1993) .

By open ing the loop between controlled andcontrolling variables, the refinement experi-ment permits us to explore alternate ways of formulating models to cover the phenom enaof interest, to estimate the values of the m ode ls’

basic parameters, and to evaluate the adequacyof on e model against alternate models ( e.g.,the linear vs. exponential drive models).

Surgically opening the loop. Anoth er way of controlling the feedback loop is to open theesophagus so that the consumed food doesnot fill the gut. This is sham feeding, a kindof continuous binge and purge. It providedPavlov ( 1955) and Miller (1971) with an ex-perimental preparation that effectively ad-dressed certain questions about the locus of satiety signals. But, because it insults the in-

tegrity of the organism–environment matchin a differen t way, it is less useful in add ress-ing the questions we pursue concerning thebehavior of a whole organism.

Postdictions. When basic refinement exper-iments are completed, we would like a way of then applying the results to more complexexperimental arrangements that are not sotheoretically felicitous. A means to accom-plish this is to give up scheduled reinforce-ment rate as an independent variable, and in-use the measured r ates of reinforcement in ou r

equations of prediction. The measured rates of instrumental and contingent behavior are thevariables comp ared by econom ic the orists suchas Staddon (1979) and Rachlin et al. (1981).This is a useful tactic in that it demonstratesconsistency of the models with data, and inman y cases is the b est that can be achieved. Butsettling for correlations between dependentvariables is less than an optimal solution to theproblem; in giving the prime instrument of ex-per imental analysis—con trol—to th e subject b ymaking the parad igm mor e ‘‘ecologically val-

id,’’ we are consequen tly forced to abandon the

prime goal of experimental analysis, giving upprediction to settle for postdiction.

 N u m erica l solu ti on s. Another option is to fallback on iterative num erical solutions of theequations, which is possible even with the un-

known on both sides. This option will be use-ful in some situations, but is not further ex-plored here.

Simplifications. There are different aspectsof the complete equations that we can ignorefor the sake of a closed-form solution to thelaws of behavior. For instance, in movingfrom Equation 2 to Equation 3, we sacrificedthe correction for blocking of reinforcementby previous reinforcements, incurring someinaccuracy at reinforcement rates above twoper minute. Let us next table Killeen’s (1994)

second principle of reinforcement by ignor-ing the temporal constraints on responding,and fall back on his simplest first principle of arousal, Equation 1. Then Equation 3 simpli-fies to an expansion of that first and mostbasic principle:

 B ϭ aR ϭ v ␥[ d  ϩ ( M  Ϫ m R ) t ] R . (14)0

This equation is a parabola. It describes re-sponding at time t  in a session as a functionof rate of reinforcement. It also describes theaverage responding in a session when t  is set

equal to half the session duration ( t sess / 2; seethe Appendix). Because we have ignored ceil-ings on response rate, we expect the actualdata to be slightly less peaked than a parab-ola, being squashed into m ore of an ellipsoidform. Equation 14 provides a good fit to thedata analyzed by Staddon (1979) using hisminimum distance model. However, some of those data were collected in open econom ies,and their downturn at low ratio values isprobably due more to the impoverished cou-pling of reinforcers to responses, which I

have analyzed at length (Killeen, 1994).On ratio schedules requiring n responses

per reinforcement, we may substitute the ra-tio sched ule feedback function B /  n for R . Atlast, we may write an equation that can besolved for B ! Its solution is

n n B ϭ M  Ϫ , m , v Ј, Ͼ 0, ( 15) m v Ј

where is the average depletion, ϭ d 0 /  t  ϩ M M  M , m is the magnitude of the incentive, and

v Ј is proportional to the incentive value of the

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424 P ET E R R . K IL L E EN  

reinforcer, v (see Equation A5 in the Appen-dix).

Equation 15 is a parabola that increases toa maximum at n ϭ v Ј / 2 and decreases to- M ward zero both as n approaches zero (satia-

tion effects) and as n becomes ver y large(straining the ratio, which occurs as n →v Ј , exactly twice th e p oint at which th e m ax- M imum occurs). Equation 15 provides a goodfit to data such as those shown in Figure 10of Collier et al. (1986). It may be preferableto Equation 14, because it predicts respond-ing in terms of an independent variable, thesize of th e ratio schedule n , rather than interms of a dependent variable, rate of rein-forcement.

To calculate the total number of responses

( b) in a session of duration t sess, multiplythrough by t sess:

n nb ϭ M  Ϫ t m , v Ј Ͼ 0. ( 16)sess m v Ј

Equation 16 provides a reasonable fit to thedata in Figure 7 with m an d t sess fixed at 1, v Јset to 1.2 ϫ 10Ϫ3, a n d ϭ 5,450 licks per M session. For the exponential drive model(Equation A6 in the Appendix), the parabolais skewed to the right and looks very muchlike Hanson and Timberlake’s (1993) curve.

It is a short step to write the equation forthe demand function, the number of rein-forcers earned ( r ) as a function of ratio re-quirement, by dividing Equation 16 by thenumber of responses required per reinforce-m e n t ( n ). If we take the session as the unitof time, so that we can set t sess equal to 1, then

 M  1 nr  ϭ Ϫ m , v Ј Ͼ 0. ( 17) m v Ј m

This is a model d emand function: Consump-

tion r  is a linear function of unit price n /  m ,with a slope of  Ϫ1/  v Ј and an intercept of 

 /  m . It is drawn as the bold line in the log- M arithmic coordinates o f Figure 8 with m ϭ 1,

ϭ 200, and v Ј ϭ 3. It has approximatelythe M same shape as many of those empirical de-mand curves; it is simple, and does not makethe obviously erron eous econom ic assertionthat th ere is a th ing such as elasticity that canbe assigned to a good and that is indepen-dent of its price (i.e., it does not assert thatthe data fall on straight lines in double-log

coordinates). The exponential d rive model

provides more flexible demand curves, whichare necessary to fit some of these data.

DeGrand pre, Bickel, Hu ghes, Layng, andBadger ( 1993) have systematically re vieweddata such as those shown in Figures 7 and 8,

many involving dru g reinforcers. They ar-gued for the use of unit price ( n /  m ) as theproper metric of the x axis (as did Timber-lake & Peden, 1987, and Hursh, 1980). Unitprice plays a key role in Equations 15 through17 as well. The slope of the demand curvepredicted by Equation 17 depend s not on thevariables n an d m , but only on their ratio.3

There is an important difference betweenthe analysis of DeGrandpre et al. (1993) andthe present one. DeGrandpre et al. plottedtheir data on logarithmic coordinates. A pa-

rabola in logarithmic coordinates is not par-abolic in linear coordinates, but is skewed tothe right. Conversely, Equations 15 and 16are skewed to the left when plotted on a log-arithmic x axis. The exponential drive modelis less skewed than the linear drive model.Whether the present models can provide asgood a fit to the range of available data ashave those of Hursh et al. ( 1989) and De-Grandpre et al. (1993) remains to be seen.

CONCLUSIONMechanistic explanations have fallen into

disrepu te, in part because good ones are h ardto come by, an d in part because they elicitimages of gears and pu lleys—poor mod els forthe processes that beh aviorists seek to un der-stand. Goal seeking, regulation, optimization,or, in general, teleological (Rachlin, 1992)and teleonomic (H. Reese, 1994) approachesseem more modern. Economics, the scienceof final causes (Rachlin, 1994), studies thegoals around which behavior is organized. As

Rachlin has no ted in h is scholarly and insight-ful analyses, we must have some sense of thepurp oses of beh avior before we can und er-stand what an act is about. All four of Aris-totle’s causes are necessar y for a complete ac-count of behavior: the functional goals andreinforcers (final causes), effective stimuli(efficient causes), underlying physiology(ma-

3 For very small values of  m , v will covary with m ; forsimplicity in these analyses I have assumed that v h astopped out, or at least that m is not experimentallyvaried

over the lower end of its range.

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425 M EC H A N ICS 

terial causes), and precise metaphors andmodels ( formal causes). Insofar as we con-ceive of operant behavior as being under thecontrol of its consequences, und erstandingthe final causes of that behavior—both the

more proximate causes (on togenetic, histo-ries of reinforcement) and the ultimate caus-es (phylogenetic, selection pressures)—takesfirst priority. But that doesn’t mean that itmust take all our efforts; identification of fi-nal causes is largely a qualitative end eavor,and may proceed quickly ( we may discoverthat one of the causes of birds’ singing is de-fense of their territory) but working out themachinery that permits the attainment of such goals remains a substantial p roject of analysis. There is much to be said for a me-

chan ics, a scien ce of for mal causes, as the sec-ond and most detailed part of the scientificendeavor, to guide us in that analysis.

The development of simple models basedon naturalistic obser vations and laboratoryexperiments leads us to a clearer understand-ing of the variables of which behavior is afunction; that is, to a clearer understandingof its causes. The ‘‘essential feature of theNewtonian style is to start out with a set of assumed ph ysical entities and ph ysical con-ditions that are simpler th an those of nature,

and which can be transferred from the worldof physical nature to the domain of mathe-matics. . . . The rules or proportions derivedmathematically may be . . . compared andcontrasted with the data of experiment andobservation’’ (Cohen, 1990, pp. 37–38); thatis, refinement experiments. This leads tomodifications of the model system and, inturn , of the experimental design, and aroundagain, with these cycles ‘‘leading to systems of greater and greater complexity and to an in-creased vraisemblance of nature’’ (Cohen,

1990, p. 38); th at is, ecological validity. Math-ematics was Newton’s tool for the discover yof  veræ causæ, tru e causes: ‘‘Specification of those causes was not a precondition for theconstruction of model systems, but rather aproduct of it’’ (Cohen, 1990, p. 29). Andmathematics, even the relatively trivial math-ematics in this paper, provides an invaluableformal structure for our metaphorical mod-els: ‘‘It was th e exten sion of Ne wton ’s intel-lectual po wers by mathematics and not mere-ly some kind of physical or philosophical

insight that enabled him to find the meaning

of each of Kepler’s laws’’ ( Cohen , 1990, p .31). Mathematics puts a fine point on thedull pen cil of metaphor.

The present mechan ics provides a relative-ly parsimon ious quan titative accoun t of m any

of the data. It also introduces the constructof satiation, a concept that is in accord withour understanding of nature and is overduefor formal recognition in our analyses. Me-chanics generates a bridge to ecologic andeconomic analyses through the explicit utili-zation of the concepts of ideal rate of reple-tion or reinforcement ( M , which providesone coordinate of the multidimensional ide-al, th e bliss point) , the cost of d eviations fromit ( ␥), the decreasing marginal utility of re-inforcers (Equations 10 through 12), and a

role for unit price as an independent variable(Equations 15 through 17). It is also consis-tent with the changes in response rate thatare foun d within a single session (Equation8; see, e.g., Killeen , 1991; McSween ey, 1992) .Futhermore, it leads to a biologically basedtreatment of hu nger that p rovides a dynamicapproach to the steady state assumed by eco-nomic models. Unlike the ecologic and reg-ulatory approaches, mechanics does not in-voke defense of a setpoint as a fundamentalforce, but introduces that defense implicitly

in equations that make deprivation a key fac-tor in motivation (Equations 5 through 7). Itis not so much that animals defend a set-point, as that deviation from a setpoint in-creases the reinforcing value of events that,as n ature usually has it, reduces th at devia-tion. Finally, in Figures 7 and 8 it providesaltern atives to econ omic analyses th at are par-simonious of parameters, d erive from simpleversions of the basic principles of reinforce-ment, and provide interpretable parametersa n d t e st a bl e p r e d i ct io n s ( E q u a tio n s 1 5

through 17 and A5 through A7).Ecologics calls our attention to the rich in-

teractive e nviron men ts in wh ich animals haveevolved and that have shaped their responsesto metabolic challenge. Its experimental re-sults may be charted with accuracy, but be-cause it is a dynamic, path-dependent, non-linear enter prise, th ose results can seldom bepredicted from principles. Like the meandersof a river that are consistent with simple andprecise models, the paths of unchanneled be-havior may come to be seen as being consis-

tent with models such as those presented

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426 P ET E R R . K IL L E EN  

here, even while the particular courses of riv-er and beast may never be predictable fromtheir principles. Prediction and control areengineering ideals, not scientific on es. It isthe purpose of refinemen t experiments to es-

tablish pr inciples; in mor e ecologically validexperiments our goal is to understand, andunderstanding is n othing other than recog-nition of consistency with established princi-ples.

Like ecologics, economics provides inspi-ration to search for the ends around whichbehavior is organized—its final causes—andthis is wise. It provides an approach to un-derstanding the trade-offs animals make be-tween altern ate p ackages of goods, an impor-tant and underrepresented area of research.

But it also seduces us into using the analyticframework of economists, and this is folly.Economics is not only the science of finalcauses; it is also ‘‘th e d ismal science .’’ Its com-plexities and routine failures to predict be-h a vio r fr o m e co n o mic p r in cip le s a relegendary. An economic behaviorism thatborrows its constructs, rather than its goals,takes the worst of it. Let us first identify theproximate an d ultimate causes of beh avior inthe ecological context in which those finalcauses have provenance. But then let us seek 

the true causes of behavior through the de-velopmen t of a mechanistic theor y—a sci-ence of formal causes—based on principledexperimentation, that may guide us in the d e-velopm en t of an ‘‘en lighten ed science’’ of be-havioral economics.

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APPENDIX

Table 1

Sym-bo l

Dimen-sions Meaning

 A 1 Ar ou sa l le ve l; t h e a mo u nt o f r esp on d -

ing elicited by a schedu le of incentives

in the absence of competition from

other responses

 R r  /  s Rate of reinforcement (obtained)

 B b /  s Rate of r esponding; arou sal level cor-

rected for response duration and ceil-

ings on response rate

 M g /  s Metabolic rate; assumed constant and

often set to zero

 R Ј r  /  s Rate of reinforcement (scheduled)

a s /  r  Specific activation: the number of sec-

onds of responding th at are elicited by

a single incentive, which depends on

drive and incentive factors

k b /  s Asymptotic response rate on interval

schedules

d g Deficit resulting from a depletion/ re-

pletion imbalance over time

h 1 H u n ge r, a lin e ar o r e xp o ne n tia l fu n c-

tion of deficit

m g /  r  Magnitude of an incentive, h ere mea-

sured in grams per reinforcer

t s Time

v s /  r  Value of an incentive, which depen ds

on its nature and magnitude

n b /  r  Number of responses required to com-

plete a ratio schedule

b b Number of responses

r r  Number of reinforcers

r  /  b Lambda, the rate of decay of short-

term memor y; does not play an impor-

tant role in the present development

␥ 1/  g Gamma, the gain or restoring force

that translates deficit into drive

1 T h et a, t h e t h re sh o ld le ve l o f m ot iva -

tion for responding

␣ 1/  s Alpha, the rate of warm-up

r  /  g Nu, the rate of discounting an incen-

tive as a function of its magnitude; its

dim ensions depend on the indepen-

dent variable and the particular dis-

count m odel (Equations 10 or 11)

␦ s /  b D elta, the m inim um interresponse

time

Con stan ts an d Di m en sion s

Table 1 lists the symbols and their interpre-tations. Lower-case letters are used for all vari-

ables except rate variables, which are writtenin capitals. Greek letters are used for con-stants and parameters. Th e second columnl i s t s t h e c o n s t i t u e n t d i m e n s i o n s , n o t t h eunits. For example, A is the number of sec-onds of respond ing per second; these cancelto make it a ‘‘dimensionless’’ variable. b an dr  refer to number of responses and reinforc-ers; because counting involves an absolutescale, the units for both are ‘‘counts’’; but be-cause they are coun ting differen t th ings, th eyhave differen t dimen sions.

Session A v erages

To calculate average response r ates dur inga session, one should write the completemodel predicting response rates and inte-grate it over the session duration. This is be-cause with finite ceilings on response rate,even the most extreme deprivation can onlyelevate response rates slightly closer to theirceiling. It is for this reason that the linear andexponential drive models provide equallygood fits to m any of the operant conditioning

data: Th e differen ces between the drive levelpredicted by those models are greatest athigh deprivations, but that is where responserates are near their ceilings and thus least re-sponsive to changes in drive levels. (It is alsofor this reason that sigmoidal functions be-tween depr ivation and drive d o n ot p rovide ameasurable improvement in fit to the data.)Unfortunately, integration of the completemod els yields u ngainly or insoluble form s. Itis therefore a worthwhile simplification tocompute the average d rive and arousal levels

over the course of a session, and use these topred ict average response r ates.

T he lin ear m odel. For the linear model hun-ger level is given by Equation 5 in the text.The average hun ger over a session of dur a-tion t sess is the integral of that function withrespect to time divided by t sess:

¯ h ϭ ␥[ d  ϩ ( M  Ϫ m R ) t  / 2] . ( A1)sess 0 sess

For short sessions ( t sess small), hunger is de-termined by the initial deficit d 0, but as ses-si o n d u r a t i o n i n c r e a se s, h u n g e r c h a n g e s

linearly with it. For extended sessions in

which t sess is large, hunger is determined pri-marily by the balance between ongoing met-abolic depletion and repletion, M Ϫm R .

T he exponential model. Calculating the aver-

age hunger during a session of du rationt sess

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430 P ET E R R . K IL L E EN  

fo r t h e e x p o n e n t ia l m o d e l ( E q u a tio n 6 )yields a more complicated expression than isthe case for the linear model:

␥d 0Ϫe␥( M Ϫm R ) t ¯  sessh ϭ ( 1 Ϫ e ) Ϫ .sess

␥( M  Ϫ m R ) t sess

But the integral may be simplified using apower-series expansion. If we retain only thefirst two terms of that expansion, it yields aprediction of hu nger level that depend s onlyon the initial conditions and the constant of integration:

␥d ¯  0h ഠ e Ϫ .sess

Because session du ration t sess has disappeared,hun ger depend s on ly on initial dep rivationlevel. This is the implicit assumption of most

traditional open -econom y research, which isunconcerned about changes in hunger dur-ing the course of a session.

If we include the first three terms of theexpansion, we get:

␥d ¯  0h ഠ [ 1 ϩ ␥( M  Ϫ m R ) t  / 2] e Ϫ .sess sess

Because ␥ an d d 0 may be treated as free pa-rameters, this is equivalent to the linear mod -el, Equation A1. Therefore, the linear modelis a special case of this exponential model.This approximation is best when ␥ is verysmall; that is, in the case of a unit elastic de-mand. Adding a fourth term reintroduces thenon linearity as [ ␥( M  Ϫ m R ) t sess)

2 / 3!. It is o n lyat this point that the models become substan-tively differen t; un fortun ately, it is also at thispoint that the approximation becomes ascumbersome as the exact form.

As an alternate tactic to achieve a simpleraverage we may invoke the mean value theo-rem: When we integrate a function betweentwo points on the x axis, there is some un-

specified value of  x between those points atwhich the function will equal the averageover that r ange. In the present case, for somet Ј between 0 and t sess,

␥[ d  ϩ( M Ϫm R ) t Ј]¯  0h ϭ e Ϫ . ( A2)sess

This can finally be simplified to:

␥Ј( M Ϫm R )¯ h ϭ e Ϫ . ( A3)sess

where is a measure of the average deple- M tion over the course of a session of durationt sess, ϭ d 0 /  t Ј ϩ M , and ␥Ј is proportional to M 

the cost of deviations ( ␥Ј ϭ ␥t Ј). This is the

simplest statement of the basic expon entialmodel for average drive level during a ses-sion. Equation A3 may be directly evaluatedas long as session duration (which would af-fect the implicit t Ј) is not varied.

 A v erag e arou sa l lev el. We may calculate theaverage arousal level throughout a session of duration t sess. It is the integral of Equation 9divided by t sess:

Ϫ␣t sess( 1 Ϫ e )¯  A ϭ aR 1 Ϫ , ␣, t  Ͼ 0.sess sess[ ]␣( t  / 2)sess

If session duration is constant, the parenthet-ical factor can be ignored because it is con-stant and can be absorbed into a. In likemanner, if there is little loss of arousal be-tween sessions or session du rations are long,as in closed economies, then (1/ ␣t sess) is smalland the correction is negligible. O nly in thecase o f ver y brief session s (t sess Ͻ 3/  ␣; typically,that is, less th an 20 m in) will warm-up affectsession-average data. In oth er words, in mostcases little is usually lost by ignoring the par-enthetical factor and setting B ϭ A ϭ aR .

T he Com plete M odel for  C losed Econ om ies

T he lin ear m odel. In contexts in which ceil-

ing effects on response rate can be ignored,we may solve the general model for ratioschedules. From the first principle (Equation1) :

 B ϭ a R ϭ v hR /  ␦ ( ␦ Ͼ 0) ,

where v a measure of the quality of the in-centive, h is the drive level, and R is the rateof reinforcement. ␦ is the minimum interre-sponse time; it appears here to convert themeasure of re spon se stren gth (r espon se-sec-onds per second, as given by A in Equation

1) to a measure of discrete responding ( B ,responses per second). This is a level of ex-plicitness not necessary for the body of thistext, but is presented here for completeness.

On ratio schedules the rate of reinforce-ment is perfectly correlated with the rate of responding. The schedule feedback functionfor ratio schedu les is simply R ϭ B /  n , wheren is the ratio requirement. Substituting andrearranging, this becomes:

v h ϭ ␦n . ( A4)

This is a fundamental equation of motion for

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431 M EC H A N ICS 

behavior. On the left is the force of an incen-tive—its value times the drive level of the or-ganism—and on the right is the number of response-seconds it is r equired to sustain.(Th e complete equation is v h ϭ ␦n , where

is a measure of the coupling between incen-tives and beh avior, as deter mined by the con -t i n g e n c ie s o f r e i n fo r c e m e n t ; se e Ki ll e e n ,1994. In the present treatment, is assum edto be constant at 1.0.)

Under the linear drive assumption (Equa-tions 5 or A1),

v ␥[ d  ϩ ( M  Ϫ m R ) t ] ϭ ␦n .0

We again use the ratio SFF ( R ϭ B /  n ) to elim-inate R , and rearrange to get

n n B ϭ M  Ϫ , ( A5)

m v Ј

where ϭd 0 /  t  ϩ M , and v Ј ϭ v ␥t  /  ␦, with m , M t , ␦ Ͼ 0. This is Equation 15 in the text. Wemay derive the session-average rates by replac-ing t  with t sess / 2 in th e above eq uation s. Forlong sessions, d 0 /  t  becomes negligible andmay be omitted, especially in the case of closed econom ies; conversely, for sho rt ses-sions and open economies, may be omitted. M In general, M  may be treated as a free param-eter representing average d epletion over thecourse of a session (part or all of which maybe offset by the average repletion during thesession, m R ) .

In experiments that terminate after a fixednumber of reinforcers, the value of  t  ϭ t sess

will tend to covary with n so that the paren-thetical term will not change greatly withchanges in the schedule requirement ( n ) orunit price ( n /  m ). This is especially true inclosed economies in which the initial deficitd 0 /  t  is small. In that case, response r ate will

be a monotonic function of  n /  m . In experi-ments that terminate after a fixed amount of time, response rate will be a quadratic func-tion of  n , as shown by Equation A5. If themagnitude of the incentive, m , is manipulat-

ed , v will change with it, over at least part of its range.

T he expon en tial m odel. In the case of an ex-ponen tial r elation between deprivation andhunger, Equations A3 and A4 develop into

␥Ј( M  Ϫ m R )v [ e Ϫ ] ϭ ␦n ;

again substitute the ratio sched ule feedback function and rearrange to get

n 1 ␦n B ϭ M  Ϫ log ϩ , ( A6) [ ]m ␥Ј v

with the average d epletion: ϭ d 0 /  t Ј ϩ M , M an d m , ␥Ј, v , t Ј Ͼ 0.

This is the basic equation of pre diction forsession averages und er the exponential as-sumption. The parameter ␥’ is the product of the restoring force an d t Ј. The curves it gen-erates are skewed parabolas, which fit manyof the data better than the linear model. Theconsiderations of the previous section on ses-sion duration and magnitude manipulationsapply here also.

T he gen eral drive m odel. Under extreme de-privation, drive no longer increases exponen -tially with further deprivation, but approach-es some maximum (i.e., is sigmoidal) andmay even decrease due to inanition (or, inthe case of drugs, due to withdrawal). Forsuch extreme deprivation conditions, otherfunctions (e.g., the Weibull distributions)might be a more appropriate model of therelation between drive and depr ivation. Letus write the appropriate function of depriva-tion as h ϭ f [ d ], and its inverse as d  ϭ f Ϫ1[ h ] ;then Equation A4 becomes:

v f [ d  ϩ ( M  Ϫ m R ) t Ј] ϭ ␦n ,0

whose solution is

n 1 ␦nϪ1 B ϭ M  Ϫ f  , ( A7)

[ ]m t Ј v

with, as before, ϭd 0 /  t Ј ϩ M , a n d m , v , t Ј M Ͼ 0.


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