DEMAND EQUATIONS FOR QUALITATIVELY DIFFERENT FOODS …

DEMAND EQUATIONS FOR QUALITATIVELY DIFFERENT FOODS UNDER FIXED-RATIOSCHEDULES: A COMPARISON OF THREE DATA CONVERSIONS

T. MARY FOSTER1, CATHERINE E. SUMPTER

1, WILLIAM TEMPLE1, AMANDA FLEVILL

1, AND ALAN POLING2

1UNIVERSITY OF WAIKATO, HAMILTON, NZ2WESTERN MICHIGAN UNIVERSITY

Concurrent schedules were used to establish 6 hens’ preferences for three foods. The resulting biasessuggested wheat was preferred over honey-puffed and puffed wheat, and puffed wheat was the leastpreferred food. The hens then responded under fixed-ratio schedules for each food in 40-min(excluding reinforcer time) sessions, with the response requirement doubling each session until noreinforcers were received. At the smaller ratios, the less preferred the food, the faster the hens’ overallresponse rates (mainly as a result of shorter postreinforcement pauses) and the more reinforcers theyreceived. The relations between the logarithms of the number of reinforcers obtained (consumption)and the response ratio (price) were well fitted by curvilinear demand functions. Wheat produced thesmallest initial consumption (ln L), followed by honey-puffed and puffed wheat, respectively. Theresponse requirement at which the demand functions predicted maximal responding (Pmax) were largerfor wheat than for the other foods. Normalizing consumption and price, as suggested by Hursh andWinger (1995), moved the data for the three foods towards a single demand function; however, the Pmax

values were generally largest for puffed wheat. The results of normalization, as suggested by Hursh andSilberberg (2008), depended on the k value used. The parameter k is related to the range of the data,and the same k value needs to be used for all data sets that are compared. A k value of 8.0 gavesignificantly higher essential values (smaller a values) for puffed wheat as compared to honey-puffedwheat and wheat, and the Pmax values, in normalized standard price units, were largest for puffed wheat.Normalizing demand by converting the puffed and honey-puffed wheat reinforcers to wheat equivalents(by applying the bias parameter from the concurrent-schedules procedure) maintained separatedemand functions for the foods. Those for wheat had the smallest rates of change in elasticity (a) and,in contrast to the other analyses, the largest Pmax values. Normalizing demand in terms of concurrent-schedule preference appears to have some advantages and to merit further investigation.

Key words: fixed-ratio schedules, reinforcer quality, concurrent schedules, behavioral economics,demand functions, normalization, magnitude-of-reinforcer, key peck, domestic hens

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Demand functions (or curves), which plotconsumption of a commodity against its price,are a cornerstone of behavioral economics(Hursh, 1984). If an organism works harder(that is, increases its response rate) acrossprice increases and so maintains a fairlyconstant level of consumption (which produc-es a logarithmic demand function with anegative slope shallower than 21.0), then ithas demonstrated inelastic demand, whichmay suggest that the commodity is a need(Dawkins, 1990). By contrast, elastic demandresults when an animal does not work harder

as price increases, and so consumption fallsrapidly across price increases. Demand func-tions showing elastic demand have a negativeslope steeper than 21.0 and may suggest thatthe commodity is a luxury rather than a need.

In many cases, elasticity changes as priceincreases. The resulting curvilinear logarith-mic demand functions are said to show mixedelasticity (Foltin, 1992, 1994; Hursh, 1984;Hursh, Raslear, Shurtleff, Bauman, & Sim-mons, 1988). Hursh et al. suggested anequation to describe such functions. In naturallogarithmic terms, the equation is:

ln Q ~ ln L z b ln Pð Þ{aP , ð1Þwhere Q refers to total consumption (e.g., thetotal number of obtained reinforcers orreinforcer amount consumed per session), Pdenotes price (e.g., the fixed-ratio [FR]schedule value), and L, b, and a are freeparameters. The parameter L estimates theinitial level of consumption obtained at theminimal price and reflects the height of the

These data formed part of a Master’s thesis completedby the Amada Flevill. Raw data are available in electronicfrom the first author. We thank Jennifer Chandler for hertechnical assistance in conducting this research, and theresearch students who helped run the experimentalconditions.

Correspondence should be addressed to Prof T. MaryFoster, Department of Psychology, University of Waikato,Private Bag 3240, Hamilton, NZ (e-mail: [email protected]).

doi: 10.1901/jeab.2009.92-305

JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 2009, 92, 305–326 NUMBER 3 (NOVEMBER)

305

demand function above the origin. Theparameter b is the initial slope of the demandfunction, and a is the rate of change in theslope of the function across price increases.From these parameters it is possible todetermine the price associated with maximalresponse output, that is, the point at whichdemand changes from inelastic to elastic —Pmax (see Hursh & Winger, 1995) — which iscalculated as follows:

Pmax ~ 1zbð Þ=a: ð2Þ

Demand functions have been used tocompare animals’ needs for different com-modities (e.g., Gunnarsson, Matthews, Foster& Temple, 2000; Hursh & Winger, 1995;Matthews & Ladewig, 1994). When thesefunctions are generated using different com-modities or procedures, however, comparisonmay be problematic. Hursh and Winger(1995) confronted the problem in comparingdemand functions for different drugs (whichdiffered substantially in potency, hence inrelative reinforcing effectiveness at a givendose) and suggested a process of normaliza-tion that allows for direct comparison of thedemand for different drugs. Their normaliza-tion procedure was based on the assumptionthat the total consumption observed at thelowest price (i.e., FR 1) is a reference levelwhich will be defended against price changes.In their analyses, the y-axis is expressed interms of consumption normalized as a per-centage of the consumption at the lowest pricestudied. This analysis involved dividing allconsumption values by the consumption valueat FR 1 and multiplying by a constant (100),thus forcing all demand functions to have aninitial consumption value of 100. The x-axisplots each price per unit of normalizedconsumption.

Hursh and Winger (1995) compared a unitprice analysis and their normalization analysisin a study of the demand curves for four drugs,each tested at three or four different doses.For three drugs (alfentanil, nalbuphine, andmethohexital), but not for cocaine (which mayhave been studied at inappropriate doses),both transformations of the data producedvery similar demand functions. Hursh andWinger suggested that their normalizationapproach improved on the unit price ap-proach, because it used ‘‘the subject’s ownevaluation of the drug in terms of total

consumption to correct for dose differences’’(p. 380).

Hursh and Silberberg (2008) recently pro-posed an alternative strategy for normalizingdemand. They tested a range of equations withexisting behavioral-economic data sets andselected the following exponential function,expressed here as natural logarithms, aspromising:

ln Q ~ k e{aP� �

z minimum ð3Þ

where Q and P are as in Equation 1, k is therange of consumption in logarithmic units, a isthe rate constant and reflects the rate ofdecrease in consumption with increases in P,and minimum is the asymptote of consumptionat an infinitely high price. The maximumvalue of this equation is the logarithm ofconsumption at zero price (termed ln Q0), andso Q0 is equivalent to L in Equation 1. Hurshand Silberberg point out that ln Q0 is equal tok plus minimum and so Equation 3 becomes:

ln Q ~ ln Q0 z k e{aP {1� �

ð4Þ

Hursh and Silberberg (2008) base theiralternative strategy for normalizing demandon this equation. They suggest that price bestandardized as the total cost required todefend consumption at a price of zero (Q0)at each schedule requirement (that is, Q0 3 C,where C represents the varying cost of thereinforcers, e.g., the ratio requirement). Sub-stituting this standardized price for P inEquation 4 gives:

ln Q ~ ln Q0 z k e{aQoC{1� �

ð5Þ

The elasticity (slope) of this function isjointly determined by k and a. Hursh andSilberberg (2008) suggest that, because k issimply a scaling parameter, if it is set to acommon constant across all comparisons,changes in elasticity will be reflected as changesin a. Thus, when k is constant, the larger a thegreater the elasticity of the demand functionand the less the ‘‘value’’ of that commodity tothe organism. The function can also be used todetermine Pmax or the standardized priceassociated with maximal output.

Hursh and Silberberg (2008) used Equation5 to generate demand curves for data frompigeons (Peden & Timberlake, 1984) and

306 T. MARY FOSTER et al.

from humans (Giordano, Bickel, Shahan, &Badger, 2001; Jacobs & Bickel, 1999) andobtained very good fits. They concluded thatan exponential model of demand based onstandard units of price, which the equationrepresents, is the best way to analyze demandand to scale the ‘‘essential value’’ of acommodity when comparing commodities.Christensen, Silberberg, Hursh, Huntsberry,and Riley (2008) tested this model with a newdata set comparing the demand for food andcocaine with rats and found that Equation 5described their data well. Christensen, Kohut,Handler, Silberberg, and Riley (2009) report-ed similar findings in a study that comparedthe demand for food and cocaine in twostrains of rats.

In essence, both the transformation exam-ined by Hursh and Winger (1995) and thetransformation proposed by Hursh and Silber-berg (2008) attempted to assess, then to takeinto account, differences in the relative valueof scaled quantities of different commoditiesas reinforcers. Performance under concurrentschedules provides a well established means ofcomparing reinforcers (for reviews see Davison& McCarthy, 1988; Sumpter, Foster, & Temple,2002), but to our knowledge has not beenused to normalize demand function. In thisprocedure, different (and incompatible) re-sponse alternatives are associated with inter-mittent access to different commodities (rein-forcers). The relative number of responses, therelative time spent responding on each alter-native, or both can be used to measure theindividual’s preference.

Behavior under concurrent schedules ismost commonly analyzed using the General-ized Matching Equation (GME; Baum, 1974,1979), which is:

log B1=B2ð Þ~ a log r1=r2ð Þz log c, ð6Þ

where B1 and B2 represent the numbers ofresponses made or the times spent respondingon two choice alternatives, and r1 and r2

represent the number of reinforcers obtainedfrom the alternatives. The parameter a de-scribes the sensitivity of the animal’s behaviorto changes in the relative reinforcement rate,and log c is a measure of the bias towards oneof the alternatives over and above any rein-forcement-rate differences (such bias is ofteninterpreted as ‘‘preference’’). When the two

alternatives provide access to the same orsimilar reinforcers, any bias is taken to be theresult of uncontrolled subject or apparatusfactors and is termed ‘‘inherent’’ bias. Bias canalso result from experimentally arrangeddifferences, such as the provision of differentfoods on the alternatives. In this case the biasresulting from the differences between theconsequences, over and above any inherentbias, is taken as a measure of the degree towhich the animal prefers one reinforcer overthe other (e.g., Bron, Sumpter, Foster, &Temple, 2003; Hollard & Davison, 1971). Biasin concurrent schedules provides a measure ofthe value of one scheduled commodity relativeto another scheduled commodity and is alwaysrelative (Sumpter et al., 2002). Demandfunctions, in contrast, provide a measure ofthe value of a single scheduled commodity.

Matthews and Temple (1979) studied cows’preferences between two foods and proposed amodification of the GME that allows a quan-titative assessment of the two separate sourcesof bias; inherent bias and, in their case, biasdue to the different food types. In logarithmicform, their modification of the GME is:

log B1=B2ð Þ~ a log r1=r2ð Þz log b

z log q,ð7Þ

where log b is the inherent bias measure, log q isthe bias resulting from the different foodqualities, log b + log q is equal to log c inEquation 6, and the other parameters are aspreviously defined. Food preferences measuredin this way provided valid predictors of grazingtime and effort in a field-choice situation(Matthews, 1983). Similar analyses have beenused successfully to assess the preferences of arange of species between both qualitatively andquantitatively different reinforcers. Examplesinclude pigeons choosing between food andbrain stimulation (Hollard & Davison, 1971),and cows (Matthews & Temple, 1979), pigeons(Miller, 1976), and brushtail possum (Bron etal., 2003) choosing between differing foods.When a sufficient number of pairs of differentreinforcers are compared using the concurrentschedules procedure, then it is possible toconstruct a derived scale of preference (e.g.,Miller, 1976; Sumpter et al., 2002). Thepreference measures obtained from concurrentschedules are relative measures of the subject’sevaluation of the reinforcers and might prove

COMPARING DEMAND EQUATIONS 307

useful in rescaling consumption so as toaccount for differences in the relative reinforc-ing value of scheduled commodities. Thepresent study explored this possibility.

The specific commodities delivered to ex-plore this were of no significance, save that theywere differentially preferred. For purposes ofconvenience, we selected whole wheat (W), asubstance often used as a positive reinforcer forhens’ behavior (e.g., Sumpter et al., 2002), asone commodity. Two other commodities, bothwheat-based generic breakfast cereals (puffedwheat, PW, and honey puffed wheat, HPW),were selected after preliminary (free access)observations had shown that hens readilyconsumed them. Moreover, they differed intexture, weight, appearance and (to humans)taste from one another and from W, weresuitable as a diet for hens, and could bedelivered via our food hoppers. Initially, hens’preference for these foods was assessed underconcurrent-interval schedules.

Demand functions were then generated asthe hens responded under increasing FRschedules for each of the foods separately.Demand functions were normalized using theapproach outlined by Hursh and Winger(1995), which should allow comparison ofthe demand for each of the foods and shouldgenerate a single function if the foods differedon only amount eaten and no other dimen-sion. Normalized demand functions were alsogenerated using the approach recently out-lined by Hursh and Silberberg (2008). Thisanalysis should generate similar values of theparameter a if the foods have the sameessential value but not if they differ in essentialvalue. Finally, demand functions were fittedusing an approach based on the bias param-eters generated in the first part of the study.This approach was based on the assumptionthat the level of consumption of a particularfood is equivalent to consuming more of a lesspreferred food or less of a more preferredfood. Hence, to equate the demand functionsfor less and more preferred foods, consump-tion of the former was multiplied by theinverse of the bias parameter for the latter.

METHOD

Subjects

Six Shaver Starcross hens, numbered 61 to66, served as subjects. During Parts 1 and 2 of

the experiment the hens were maintainedbetween 80 and 85% of their free-feedingbody weight through daily weighing and theprovision of supplementary feed (commerciallaying pellets). They were individually housedin 300 mm by 450 mm by 430 mm wire cageswhere water was freely available. Grit andvitamins were supplied weekly. At the start ofthe experiment, Hens 62, 63, 64, and 66 wereapproximately 4 years old and Hens 61 and 658 years old. All hens had previous experienceresponding under concurrent schedules ofreinforcement.

Apparatus

Part 1: Preference assessment. The experimen-tal particle-board chamber was 620 mm long,580 mm wide, and 540 mm high. The floorconsisted of a wire mesh grid enclosed in agalvanized steel tray 35 mm high. Ventilationto the chamber was provided by a covered fan(100 mm wide and 120 mm long) located onthe left wall 20 mm below the roof and 30 mmfrom the back wall. On the right wall, 380 mmabove the floor and 100 mm apart, were twotranslucent discs (response keys) 30 mm indiameter, which required a minimum force of0.1 N to operate. When operational, the keyswere illuminated from behind by a red 1-Wbulb, and each effective response resulted in abrief audible feedback beep.

Two openings, 100 mm high by 70 mm wideand located 130 mm beneath each responsekey, provided access to the food magazinewhen it was raised. Reinforcers consisted of 3-saccess to W, HPW, or PW. During reinforcerdelivery, the magazine was raised and illumi-nated white, and the key lights were extin-guished. The key and magazine lights were theonly source of light in the experimentalchamber. An infrared movement sensor locat-ed 30 mm above the bottom of the magazineopening was used to record the presence ofthe hen’s head when the magazine was raised.

A 486-series IBM-compatible computer, in-terfaced with a MEDH programmable controlboard and using MED 2.0H software (MEDAssociates, St. Albans, VT), was located in aseparate room. It controlled and recorded allexperimental events. The data were alsologged in a data book at the end of eachsession.

Part 2: Demand assessment. The apparatus wasidentical to that used in Part 1 of the


experiment in all but two respects. The left keyand left magazine-access opening were cov-ered by a thin aluminum sheet so that only theright key and right magazine opening wereavailable. The right key was illuminated frombehind by a green 1-W bulb.

Procedure

Part 1: Preference assessment. Throughout eachcondition of Part 1, all hens responded underconcurrent random-interval (RI) 90-s RI 90-sschedules of reinforcement with a 2-s change-over delay (COD). The schedules were pro-grammed to be dependent in that once areinforcer was available for responding on oneschedule the other schedule stopped timinguntil the scheduled reinforcer was delivered.The COD meant that a response to oneschedule could not result in a reinforcer until2 s had elapsed since the response initiating achangeover to that schedule. The random-interval contingencies were arranged using anelectronic probability gate and a recyclingtimer in a way similar to that described byMillenson (1963). Each daily experimentalsession started with both keys illuminated redand ended after 30 reinforcers had beenobtained or 40 min had elapsed, whicheveroccurred first. At least six sessions wereconducted for each hen every week.

In all conditions the left magazine con-tained W. In Conditions 1, 2 and 3, the rightmagazine contained W, PW and HPW, respec-tively. Experimental conditions were changedwhen the behavior of all 6 hens was deemed tobe both statistically and visually stable. Statis-tical stability was reached when the median ofthe proportion of behavior (responses andtimes spent responding) on the left key overthe most recent five sessions did not differ bymore than 0.05 from the median of theprevious five sessions. This had to occur fivetimes but not consecutively. Thus, a minimumof 14 sessions was required for statisticalstability. Visual stability was reached when theproportions of both the responses and timesspent responding on the left key plotted acrosssessions did not reveal a trend in eitherdirection, as judged by two or more labmembers. Conditions 1, 2 and 3 were judgedstable after 28, 28 and 39 sessions, respectively.

In all sessions, the following were recorded:the total number of responses, the totalnumber of reinforcers obtained, the individual

postreinforcement pause (PRP) durations,and the times spent responding on each key.The total number of changeovers betweenkeys, the total number of pecks made on theleft and right keys during a changeover, andthe total session time were also recorded.

Part 2: Demand assessment. In all sessions ofPart 2, the hens responded for 3-s access tofood delivered according to FR schedules ofreinforcement. The only procedural differ-ence across conditions was the food availablein the magazine. In Condition 1, the magazinecontained W, in Condition 2 it contained PW,and in Condition 3 it contained HPW.

Experimental sessions started with the rightkey illuminated green and ended after 40 minof key time (i.e., the accumulated time duringwhich the key was lit). When ratios were low,much more time was spent in eating than inkey pecking, leading to long overall sessiontimes. Although experimental sessions wereconducted at least 6 days per week, only 3 henswere studied each day. Experimental sessionswere arranged on alternate days for Hens 61 to63 and Hens 64 to 66 throughout this part ofthe experiment.

Each experimental condition began withthe hen responding on an FR 10 schedule for aminimum of three sessions. A series of session-to-session increases in FR requirements wasthen introduced. The first FR requirement ineach series was 2, with the requirementdoubling (i.e., FR 4, 8, 16, etc.) in eachsubsequent session, provided the hen hadobtained at least one reinforcer in the previoussession. If the hen did not obtain a reinforcerin a particular session, she was reexposed tothe same FR value during the next experimen-tal session. If at least one reinforcer wasobtained in that session, the session-to-sessionincrements in FR value continued for that hen.Following two consecutive sessions in whichthe hen received no reinforcers, she wasexposed to an FR 1 schedule for one session.Following this, the hen remained in her homecage, where she was given supplementary fooduntil all hens had completed that series, atwhich point all hens received three sessionswith FR 10. Another series then began. Oncompletion of two series of FR-value changeswith one food, the sequences were repeatedwith a new food in a new condition. Table 1summarizes the sequence of experimentalconditions, the foods used, and the largest


FR value each hen completed (i.e., the highestFR schedule value at which she received atleast one reinforcer) during each of the twoseries of FR schedules per food condition.

In each session of each condition, all of theexperimental events and the times at whichthey occurred were recorded. The followingdata were displayed on the computer screenand logged into a data book: the FR schedulein effect, the time to the first response, thetotal numbers of responses made and rein-forcers obtained, the PRP durations, the totalruntime (i.e., the accumulated times from thefirst to the last response for each FR require-ment), the total PRP time, the total key time,and the total session time.

RESULTS

Part 1: Concurrent-schedule Data

The data from the final five sessions for eachhen were combined for analysis for each of thethree concurrent-schedule conditions. Re-sponse and time measures were comparable;therefore, only the former are presented.Response and reinforcer ratios were expressedas left over right, and logarithms to the base 10were used.

As r1 and r2 were not varied, and, as a resultof the dependency between the schedules, r1

5 r2, and so log (r1/r2) 5 1.0. This means thelogarithms of the response ratios give the log cvalues or total bias measures (see Equation 6).In Conditions 2 and 3 the different foodsprovided a source of bias over and above anyinherent bias, that is, log c is composed of log band log q (as shown in Equation 7). Therefore,subtracting the individual log c values obtainedin Conditions 2 and 3 (W vs. PW and W vs.HPW, respectively) from those found in

Condition 1 (W vs. W) gives the values of logq. Table 2 presents both the log c and the log qresponse biases for each hen and condition.

As shown in Table 2, the inherent responsebiases (log c values) obtained during the W vs.W condition were all towards the right key(range 5 20.21 to 20.03; mean 5 20.15). Bycontrast, all of the log q biases (i.e., those withinherent bias removed and thus due to thefood type alone) were towards the left key,indicating biases towards W during both the Wvs. PW and W vs. HPW conditions. The log qbiases obtained from the W vs. PW conditionranged from 0.34 to 0.92 (mean 5 0.64). Thelog q biases from the W vs. HPW conditionranged from 0.23 to 0.69 (mean 5 0.48). Thelog q values for W vs. PW were larger thanthose obtained for W vs. HPW for all hens.When the individual log q values are used torank each hen’s food preference, W is thehighest-ranked food and PW is the lowest-ranked food.

Part 2: FR Data

Untransformed demand functions. To generatedemand functions, the natural logarithms ofconsumption, measured as the total number ofreinforcers obtained during both series of FR-schedule changes in each food condition, wereplotted as functions of the natural logarithmsof the FR schedule requirements. (Obtainedreinforcers were also used as the measure ofconsumption when demand was normalized.)Natural logarithms form the basis of thenonlinear analysis proposed by Hursh et al.(1988). Functions were fitted to the data fromeach FR series using nonlinear regression andHursh et al.’s equation (Equation 1). Theparameters of the equations describing thedata from each series for each hen are

Table 1

The order of experimental conditions conducted in Part 2 (Demand Assessment) of theexperiment, together with the highest FR schedule value completed by each hen in each series ofFR schedules.

Condition Food Series

Hen

61 62 63 64 65 66

1 W 1 256 512 512 512 256 1282 128 512 256 512 128 64

2 PW 1 256 256 128 512 128 1282 256 256 128 256 256 64

3 HPW 1 512 512 128 512 256 642 512 512 512 512 256 128


presented in the Appendix, together with thevariances accounted for by the regression lines(% VAC), the standard errors of the estimatesof the fits (se), and the predicted FR schedulevalue corresponding to the maximal responseoutput (Pmax; see Equation 2). The demandfunctions described the data well, accountingfor over 80% of the variance in all but 3 out of36 cases, and did not differ systematicallyacross the two series of FR-schedule changesin each food condition. Hence, the followinganalyses were based on the averaged data fromthe two FR series with each food. Where onlyone value existed at larger FR values, it wastaken as the estimate.

Figure 1 presents the natural logarithms ofthe averaged consumption data plotted againstthe natural logarithms of the FR value for theW (left panel), HPW (middle panel), and PW(right panel) conditions. The solid linesshown were fitted, as described above, to theaveraged data. The values of the parameters,the se, the % VAC, and the Pmax values arepresented in Table 3. Similar-shaped demandfunctions were produced irrespective of foodtype. All initial slopes (b values) were largerthan 21.0, consistent with inelastic initialdemand, and all a values were positive,indicating that the demand functions becameincreasingly more elastic as FR value increased.The ln L values ranged from 4.19 to 6.57.

The effects of food type on the shapes of thedemand functions can be seen by comparingthe parameters and the Pmax values acrossfoods (see Table 3). Systematic patterns can beobserved for ln L and Pmax but not for a and b.Table 3 shows that values of ln L for all 6 henswere lowest for W (the most preferred food),greatest for PW (the least preferred food), and

intermediate for HPW (the middle-rankedfood). In other words, ln L monotonicallydecreased with increases in the size of thepreference measures. The Pmax values weremarkedly higher for W than for PW for all butHen 65. However, the Pmax values for HPWwere smaller than those for both of the otherfoods for 4 hens and larger for the other 2,though only marginally so for Hen 66. Unlikeln L, the Pmax values did not decreasemonotonically with changes in the preferencemeasure.

For comparison with subsequent analysesthe left-hand panel in Figure 2 presents thenatural logarithms of the consumption datafor each of the three foods averaged over thetwo FR series plotted against ln FR values (notethat these are the same data as in Figure 1 butnow in a single graph for each hen). The datafrom the W, HPW and PW conditions areshown as plus marks, crosses, and unfilledcircles, respectively. The solid lines were fittedto these unmodified data using Equation 1,and the parameter values of these fits are givenin Table 4.

Normalized demand. The central panel ofFigure 2 presents the data plotted as normal-ized consumption against normalized price, asproposed by Hursh and Winger (1995). Eachconsumption measure was normalized (to aninitial value of 100) by multiplying by 100 anddividing by the consumption value obtained atFR 1. Normalizing each price analogue (i.e.,FR value) was achieved by dividing theresponses required by 100 and multiplying bythe consumption at FR 1. This normalizationof consumption constrains the L values (initiallevels at FR 1) for these data to 100 and aconsumption level of 100 is indicated by the

Table 2

The logarithms of the response ratios (i.e., log c; see Equation 6) for each hen for the W vs. W,PW vs. W, and HPW vs. W conditions. Biases resulting from the different foods, log q (seeEquation 7), found by subtracting log c from W vs W condition from log c for each of the PW vs.W and HPW vs. W conditions are also presented.

Hen

W/W W/PW W/HPW

log c log c log q log c log q

61 20.14 0.39 0.53 0.34 0.4862 20.21 0.71 0.92 0.35 0.5663 20.21 0.38 0.59 0.36 0.5764 20.11 0.65 0.76 0.24 0.3565 20.03 0.31 0.34 0.19 0.2366 20.19 0.52 0.71 0.50 0.69Mean 20.15 0.49 0.64 0.33 0.48


Fig. 1. The natural logarithms of the numbers of reinforcers per session plotted as functions of the naturallogarithms of the FR value for each food and for each hen. The data are the averages across the two series of increasingFR values with each food. The solid lines were fitted using Equation 1, and their parameter values appear in Table 3. Thedotted lines and dashed lines were fitted using Equation 5 with k values of 3.5 and 8.0, respectively, and their parametervalues appear in Table 6.


dashed horizontal line (ln 100 5 4.61). Theparameter values and the % VAC and semeasures are presented in Table 4. It shouldbe noted that, for ease of comparison, con-sumption (y-axes) and price (FR values; x-axes)for each panel in Figure 2 are set so that anequal rate of change for both consumption andthe FR value would fall parallel to the diagonaland would have a slope of 21.0, allowing a visualcomparison of trends across panels. Table 4indicates that the normalized data provide acloser fit to Equation 1 than do the unmodifieddata (left panel) as can be seen in the smaller sevalues higher % VAC values (Hen 65 was thelone exception for both measures).

Equation 1 was also fitted to the normalizeddata for each food separately. Table 5 showsthe results of these fits. Because normalizationchanges the scales on the x and y axes but notthe relative positions of the data in relation tothe fitted function, the parameter b, the se and% VAC are the same as in Table 3 and so arenot shown in Table 5. The values of a, ln L,and Pmax were changed by the normalizationmanipulation. The values of (a) for the W,HPW, and PW data were not systematicallydifferent across hens. PW produced the smallestand HPW produced the largest rates of changein elasticity for 5 hens in each case, the

exceptions being Hen 65 and Hen 66. Thevalues of L are sometimes much bigger than 100(particularly for PW) as a result of the initialslope (b in Table 3) being steep and the priceadjustment having established a new pricegreater than 1. For three W data sets (Hens63, 64, and 66) L was smaller than 100 when thenew price was less than 1. The initial consump-tion of W remained lower than that of PW for allhens and was the lowest over all three foods for 5of 6 hens. Pmax was largest for PW and smallestfor HPW for 5 hens in each case. Again theexceptions were for Hen 65 and Hen 66.

Preference-adjusted demand. The right-handpanel of Figure 2 shows consumption adjustedfor preference based on the log bias measuresobtained in Part 1 of the experiment. In thiscase, all consumption measures obtained inthe HPW and PW conditions were divided bythe preference ratio (q) for W relative to thatparticular food, effectively converting thoseconsumption measures into W equivalents. Toillustrate, if 3-s access to W was valued 1.5 timesmore than 3-s access to PW, then a consump-tion of six 3-s PW reinforcers would convert tofour W equivalents. This analysis means that,for all hens, consumption at all prices of HPWand PW were lower than the unmodifiedvalues in the left-hand panel of Figure 2. The

Table 3

The parameters a, b and ln L for Hursh et al.’s (1988) nonlinear equation (Equation 1) fitted tothe averages from Series 1 and 2 of the natural logarithms of the numbers of reinforcers persession (consumption) and the averages of the natural logarithms of FR schedule value for eachhen. The FR value at which the equation predicts maximal response rate (Pmax; see Equation 2),the standard error of the estimates of the fits (se), the percentages of variance accounted for bythe functions (%VAC), and ln Lpa, the value of this parameter for the preference-adjusted dataset, are also given.

Hen Food a b ln L Pmax se %VAC ln Lpa

61 W 0.0033 20.36 4.66 191.4 0.28 94.6 —HPW 0.0157 20.33 5.66 43.1 0.14 98.4 4.56PW 0.0062 20.60 6.22 64.5 0.17 98.7 5.02

62 W 0.0041 20.34 4.85 159.6 0.12 99.1 —HPW 0.0083 20.56 5.55 52.9 0.26 95.5 4.26PW 0.0045 20.53 5.96 101.8 0.21 98.5 3.87

63 W 0.0049 20.26 4.19 148.7 0.24 96.4 —HPW 0.0240 20.12 4.47 36.6 0.39 89.9 3.52PW 0.0075 20.51 6.20 64.0 0.17 98.7 5.26

64 W 0.0023 20.38 4.86 269.5 0.38 88.9 —HPW 0.0106 20.16 5.09 79.7 0.14 95.6 4.28PW 0.0028 20.63 6.57 129.8 0.24 97.9 4.80

65 W 0.0135 20.22 5.26 58.0 0.15 98.9 —HPW 0.0121 20.29 5.36 58.5 0.07 99.4 4.85PW 0.0127 20.46 6.36 42.5 0.26 97.9 5.58

66 W 0.0011 20.35 4.41 572.9 0.15 93.6 —HPW 0.0264 20.14 4.70 32.5 0.29 95.0 3.11PW 0.0291 20.11 5.34 30.6 0.12 99.3 3.70


Fig. 2. The left panel shows the natural logarithm of the numbers of reinforcers per session plotted as functions ofthe natural logarithms of the FR value for each food and for each hen. The data are the averages across the two series ofincreasing ratios with each food. W data are indicated by plus marks, HPW data by crosses, and PW data by circles. Thecentral panel shows the data after the normalization suggested by Hursh and Winger (1995), where consumption wasnormalized to a value of 100. The natural logarithm of 100 is indicated by the dashed horizontal line. Price was alsomodified (see text for details). The right panel shows the data after they were adjusted by the preference values. Here


data for the W condition (plus marks) areunchanged between the left and right panels.The solid lines in each graph were fitted to thepooled data (that is, the data from for all threefood conditions) for each hen using Equation1. The parameter values and the % VAC and semeasures are presented in Table 4. Becausepreferences were towards W (Table 2), thepreference-adjusted consumption data forboth PW and HPW are lower than theunmodified data in the left-hand panel. Singlefunctions fitted to these data have generallylarger se values and smaller % VAC values thanthose for the normalized data (see Table 4)and generally do not describe the data as wellas the unmodified and normalized functionsdo.

Only the consumption measure is changedby the preference adjustment. Thus whenseparate functions were fitted to the prefer-ence-adjusted data for each food they differfrom those of the unmodified data (Table 3)only in that they have different values of ln L.

The preference-adjusted ln L values for PWand HPW, ln Lpa in Table 3, were reduced bythe preference adjustment. Because the othertwo parameters and the fits of the separatefunctions remained the same as those for theunmodified data, the separate functions (Ta-ble 3) had better fits than the single function(see Table 4) in terms of both larger % VACand smaller se. W produced the smallest avalues for all but Hen 65 and the largest Pmax

values for all hens (see Table 3).Exponential demand and essential value. The

size of the scaling parameter, k, reflects therange of the data and the shape of thefunction defined by Equation 5. Because thevalues of ln Q0 and a both change withchanges in k, Hursh and Silberberg (2008)suggest that the same k value should be usedfor all data sets when comparing theseparameters. Therefore, it was necessary toselect a value of k to be used here. For thepresent data the values of k found for the best-fitting functions when all three parameters, k,

r

consumption was converted to W equivalents by dividing by the W preferences found in Part 1 of the experiment. Thesolid lines were fitted to the pooled data in each panel using Equation 1. Their parameter values appear in Table 4. Theparameters of functions fitted to the unmodified data and preference adjusted data from each food separately appear inTable 3 and those for the normalized data in Table 5.

Table 4

The parameters a, b and ln L for Hursh et al.’s (1988) nonlinear equation (Equation 1) fitted tothe unmodified, normalized, and preference-adjusted (pref-adjusted) data pooled across allfoods for each hen. The text describes the derivation of the latter two measures. The FR value atwhich the equation predicts maximal response rate (Pmax; see Equation 2), the standard error ofthe estimates of the fits (se), and the percentages of variance accounted for by the functions(%VAC) are also given.

Hen Analysis a b ln L Pmax se %VAC

61 Unmodified 0.0036 20.53 5.62 129.8 0.49 87.8Normalized 0.0025 20.46 5.14 220.4 0.32 94.2Pref-adjusted 0.0019 20.57 4.89 223.0 0.46 87.7







a and Q0, were left free to vary, ranged from1.63 to 10.15 in natural-log units and variedacross foods and hens. To examine thechanges in a, Q0 and the degree of fit withchanges in k, Equation 5 was fitted to thepresent data sets with k fixed at a range ofvalues. A k value of zero was not included, asthis gives a flat line through the averageconsumption.

The resulting fits showed that, as k increasedfrom 0.5, ln Q0 initially increased rapidly to amaximum value at the value of k equal to thatfor the best-fitting function (i.e., when thefunction was fitted with all three parametersfree to vary). Then ln Q0 decreased graduallyand reached an asymptote at a value slightlylower than its maximum.

Of importance for the present analysis werethe changes in a. This parameter decreasedrapidly as k increased from 0.5 for all data setsand reached asymptotes at values close to zero.A problem for the present analysis is that therelative magnitudes of a for the three foodschanged as k increased. For example, for Hen61, the values of k (in ln units) for the best-fitting functions were 3.59, 3.72 and 4.24 forW, HPW and PW, respectively. For this hen’sdata, with k 5 0.5, the value of a was largest forW (0.00053), then HPW (0.00040), andsmallest for PW (0.00034). With k 5 8.0, a

was largest for HPW (2.13E-05), then W(2.17E-05), and smallest for PW (1.54E-05).This order held for further increases in k, butthe differences between the three a valuesdecreased.

Another example is provided by the data ofHen 66, where the best-fitting k values were themost disparate over the three foods (1.63, 6.01,and 10.15 for W, HPW and PW, respectively).For this hen the ordinal relation between the avalues for the three foods changed at a value ofk close to 2.0, from being W.HPW.PW tobeing HPW.W.PW. Also, while the a valuesfor the HPW data remained higher than thosefor the other two foods for all further increasesin k, the difference between the a values for Wand PW decreased rapidly as k increased from2.0 to 3.0, and then decreased very graduallywith further increases in k. Thus, the k valuesused for these data sets altered both themagnitude and direction of the differencesbetween the a values for the three foods.

The % VAC for the various fits also changedin an orderly fashion with changes in k.Constraining k at small values (,1.0) gavepoor fits (% VAC < 20% for k 5 0.5), but thedegree of fit improved rapidly as k approachedthe best-fitting k value for the particular dataset. The best-fitting functions accounted forover 89% of the variance in all cases except

Table 5

The parameters a, ln L and L for Hursh et al.’s (1988) nonlinear equation (Equation 1) fitted tothe normalized data for each food and for each hen. The FR value at which the equation predictsmaximal response rate (Pmax; see Equation 2) is also given.

Hen Food a ln L L Pmax

61 W 0.0039 4.75 115.8 167.3HPW 0.0068 5.10 164.2 99.3PW 0.0017 5.70 297.8 238.0

62 W 0.0037 4.78 119.0 177.1HPW 0.0044 5.26 193.4 100.0PW 0.0018 5.53 251.5 258.6

63 W 0.0071 4.44 85.2 104.8HPW 0.0441 5.00 148.3 19.9PW 0.0019 5.54 255.0 248.7

64 W 0.0014 4.57 96.9 428.6HPW 0.0084 4.90 133.8 100.4PW 0.0005 5.96 388.4 680.1

65 W 0.0075 4.80 121.7 104.7HPW 0.0054 4.79 120.6 131.1PW 0.0025 5.50 243.8 211.3

66 W 0.0013 4.48 88.5 510.2HPW 0.0320 4.86 129.4 26.8PW 0.0176 4.89 133.4 50.4

Note. The value of b, the standard error of the estimates of the fits (se), and the percentages of variance accounted for bythese functions (%VAC) are as in Table 3. The text describes the derivation procedures.


one (80% for Hen 64 with W). For 12 of the 18fits the values of % VAC were over 95. Furtherincreases in k beyond the best-fitting valuesdecreased % VAC gradually. For instance,when k was 112, the % VAC was over 80% for15 of 18 fits and over 90% for 8 fits. Thus, anyvalue of k above the best-fitting value resultedin a reasonable description of that data set.For this reason % VAC did not provide anunequivocal basis for the selection of anappropriate k value.

In their description of the method for fittingEquation 5, Hursh and Silberberg (2008)suggest two strategies for selecting the valueof k. One is to use a value of k based on themaximum range of consumption over the datasets to be compared (i.e., the largest range inany of the data sets) and then to use that kvalue for fitting functions to all those data sets.Another is to use a value of k based on themean range of consumption over all the datasets to be compared. Neither strategy seemedappropriate here, as both the size and relativeorder of the a values were affected by changesin k. As a result of these considerations it wasdecided to present the analysis using two kvalues, 3.5 and 8.0 (equivalent to 1.5 and 3.5 innormal log units); both are within the rangefound for the best-fitting functions across allhens.

Figure 1 shows the functions resulting fromfitting Equation 5 to the average data fromeach food for each hen with both k values; thedotted line shows the function found with k 53.5 and the dashed line shows the functionwith k 5 8.0. The parameters of these fittedfunctions, se, % VAC, and Pmax [in units of theoriginal price (C or the FR value) and in unitsof the normalized standard price] are provid-ed in Table 6. Figure 1 shows that while somefunctions appear to fit the data well, someunderestimate the data at small FR values andoverestimate them at moderate values, espe-cially for W and PW. The functions with k 53.5 tend to be higher at small FR values andlower at moderate FR values than those with k5 8.0. The functions with k 5 3.5 display anasymptote at consumptions higher than thosereached at the larger FR values for some datasets. None of the asymptotes for the functionswith k 5 8.0 are within the range shown onthese graphs. Figure 1 and Table 6 show thatEquation 5 fitted the data reasonably well with% VAC exceeding 80 in 16 of the 18 data sets

for both k values. Comparison with the % VACand se for the fits of Equation 1 (solid lines inFigure 1; see also Table 3) indicate thatEquation 1 provided the better fit (larger %VAC and smaller se) for 12 of 18 data sets for k5 3.5 and 16 of 18 data sets for k 5 8.0.

Figure 3 shows the functions obtained byfitting Equation 5. The y-axis is the naturallogarithm of the normalized consumption,obtained by dividing the original consumptionmeasures by the parameter Q0 and multiplyingby 100. These are plotted against the loga-rithms of the normalized standard pricesuggested by Hursh and Silberberg (2008),i.e., (C 3 Q0) 4 100.

For 15 of the 18 data sets Q0 was larger whenk 5 3.5 than when k 5 8.0. With one exception(Hen 62 and k 5 3.5), Q0 was largest for PW,next largest for HPW, and smallest for W forboth k values. Pmax values based on the originalprice (i.e., C or the FR value) were largest forW for all hens and smallest for HPW for 4 ofthe 6 hens for both k values. Pmax values basedon the normalized standard price were largestfor PW for all hens for both k values and weresmallest for HWP for 3 of 6 hens with k 5 3.5and for all 6 hens with k 5 8.0.

In all cases, a was larger for k 5 3.5 than fork 5 8.0. Paired t-tests showed that the a valuesfor the same food differed significantly across kvalues: W (3.5 vs. 8.0), t(5) 5 10.72; HPW (3.5vs. 8.0), t(5) 5 4.56, and PW (3.5 vs. 8.0), t(5)5 10.57, p , .05 in all cases. The a values weresmallest for PW for all hens at both k values,while the a values for HWP were largest for 3out of 6 and 6 out of 6 hens with k 5 3.5 and8.0, respectively. Repeated measures ANOVAsrevealed that the a values for the three foodsdiffered significantly at both k values: F (2, 10)5 5.1, partial g2 5 .51 for k of 3.5 and F (2, 10)5 9.3, partial g2 5 .65 for k 5 8.0 both with p ,.05. Paired t-tests over the a values across thepairs of foods gave no significant results with k5 3.5: W vs. PW, t(5) 5 2.49; HPW vs. PW, t(5)5 2.47, and W vs. HPW, t(5) 5 21.87, all withp . .05. With k 5 8.0 the values for PW weresignificantly different from those of both Wand HPW, but those from HPW and W did notdiffer significantly from each other: W vs. PW,t(5) 5 4.55; HPW vs. PW, t(5) 5 3.24, bothwith p , .05, and W vs. HPW, t(5) 5 22.32, p. .05. As noted previously, for these data thevalues of a continued to decrease as kincreased, reaching asymptotes at values near


zero. With k constrained to 100, the mean avalues were 1.58E-06 for W, 2.32E-06 for HPWand 9.83E-07 for PW. Paired t-tests showed thea values from PW were still significantlydifferent from those of W and HPW, W vs.PW, t(5) 5 5.64; HPW vs. PW, t(5) 5 3.49,both with p , .05, and those from W and HPWwere still not significantly different from eachother, W vs. HPW, t(5) 5 22.47, p ..05.

FR response patterns. The negative relationsbetween both ln L (Table 3) and Q0 (Table 6)

and food preference (Table 2) evident in theunmodified-data demand functions (Figure 1)suggest that FR response rates were higher forless preferred foods than for more preferredfoods, at least at the smaller FR values. Toexamine this possibility, mean PRP durationsand average running response rates (i.e.,response rates calculated with magazine-oper-ation time and PRP duration excluded) arepresented in Figure 4 (running responserates) and Figure 5 (PRP durations). The data

Table 6

The parameters of ln Q0, Q0 and a for Hursh and Silberberg’s (1988) exponential equation(Equation 5) fitted to the natural logarithms of the unmodified data for each food and for eachhen and with k constrained to be 3.5 and 8.0. The standard error of the estimates of the fits (se),and the percentages of variance accounted for by the functions (%VAC) also appear. The value atwhich the equation predicts maximal response rate (Pmax) in units of the original price, C or theFR value, and in units of the normalized standard price [(C3Q0)/100] are also included.

Hen Food ln Q0 Q0 a se %VAC

Pmax

CC|Q0

100

k = 3.561 W 4.20 66.9 8.24E-05 0.40 89.0 80 54

HPW 5.53 251.4 7.73E-05 0.07 99.7 23 57PW 5.65 283.8 7.23E-05 0.35 94.8 22 61

62 W 4.46 86.7 8.01E-05 0.18 98.0 64 55HPW 5.39 219.3 0.000146 0.11 99.1 14 30PW 5.16 174.4 6.27E-05 0.52 90.5 41 71

63 W 3.85 47.2 0.000117 0.33 93.1 80 38HPW 4.51 90.7 0.000197 0.36 91.6 25 22PW 5.70 300.1 5.93E-05 0.30 95.8 25 75

64 W 4.32 75.5 5.94E-05 0.51 79.9 99 75HPW 4.97 143.8 4.91E-05 0.13 96.3 63 90PW 5.73 307.3 0.000042 0.46 92.1 34 106

65 W 5.00 148.7 7.47E-05 0.40 92.1 40 59HPW 5.13 169.7 7.09E-05 0.18 96.2 37 63PW 5.71 300.4 5.76E-05 0.60 88.6 26 77

66 W 4.01 54.9 0.000096 0.29 76.8 84 46HPW 4.66 106.0 0.000183 0.38 91.7 23 24PW 5.27 194.7 9.59E-05 0.39 91.6 24 46

k = 8.061 W 4.02 55.5 2.13E-05 0.46 85.2 126 70

HPW 5.32 203.9 2.17E-05 0.23 95.9 34 69PW 5.40 220.8 1.54E-05 0.50 89.0 44 97

62 W 4.25 70.3 1.93E-05 0.34 92.9 110 78HPW 4.92 137.0 3.41E-05 0.45 86.4 32 44PW 5.11 165.8 1.31E-05 0.50 91.3 69 114

63 W 3.74 42.0 3.13E-05 0.34 92.7 114 48HPW 4.41 82.1 5.79E-05 0.35 91.9 31 26PW 5.50 244.8 1.35E-05 0.42 91.8 45 111

64 W 4.14 62.6 1.61E-05 0.56 75.3 148 93HPW 4.90 134.7 1.71E-05 0.17 93.6 65 87PW 5.52 250.7 8.00E-06 0.64 85.0 75 187

65 W 5.02 152.0 2.08E-05 0.17 98.6 47 72HPW 5.01 150.2 2.25E-05 0.23 94.2 44 67PW 5.85 347.5 1.34E-05 0.32 96.8 32 112

66 W 3.95 52.0 3.44E-05 0.32 73.2 84 44HPW 4.62 101.3 5.30E-05 0.27 95.7 28 28PW 5.30 199.9 2.83E-05 0.12 99.2 26 53


presented are from W and PW (the most andleast preferred foods), and are the meansacross the two series of FR-schedule changes(recall that the two yielded comparable data).Running response rate cannot be calculatedfor the FR 1 condition. Figure 4 shows thatrunning response rates tended to decreasewith increases in FR value and were similar forW (open circles) and PW (filled diamonds) for4 hens. For Hens 63 and 64 response rates forPW were higher than for W over the smallerFR values. As FR value increased, the ratedifferences decreased until there were virtuallyno differences at the largest FR value.

Given the variability in the differences inrunning response rates for the two foodsacross hens at the small FR values, theconsistent differences in the overall responserates must have resulted from longer PRPsduring the W condition. The mean PRPdurations in Figure 5 confirm this. The PRPsfrom the W condition (open circles) wereconsistently longer than those from the PWcondition (filled diamonds). The difference inPRP durations for W and PW also was evidentin the cumulative records. Response rates forPW exceeded those for W from the start of thesession, as is evident in the representativeexamples from Hen 61 provided in Figure 6.

DISCUSSION

The purpose of the present study was tocompare unmodified demand functions todemand functions normalized in three differ-ent ways. A striking feature of the unmodifieddemand functions was that initial level ofdemand (ln L) was higher for the leastpreferred food (PW; see Figure 1 and Table 3)than for the food that was most preferredunder the concurrent-schedule arrangement.On the other hand, with the unmodified data,and consistent with the concurrent-scheduledata, the FR schedule at which peak respond-ing occurred (Pmax; see Equation 2) was largestfor the most preferred food (W). This appar-ent discrepancy in the relative ‘‘importance’’Fig. 3. Both panels show the natural logarithm of the

consumption normalized as suggested by Hursh andSilberberg (2008) and plotted as functions of the naturallogarithms of the normalized standard price (see text fordetails) for each food and for each hen. The data are theaverages across the two series of increasing ratios with eachfood. W data are indicated by plus marks, HPW data bycrosses, and PW data by circles. The left and right panelsshow the functions found by fitting Equation 5 with a k

r

value of 3.5 and 8.0, respectively. The solid lines show thefunctions for the W data, the dotted lines the functions forthe HPW data, and the dashed lines the functions for PWdata. The parameter values are in Table 6.


of the foods, depending on the aspect of thefunction that is emphasized, illustrates theproblem of direct comparisons between de-mand functions for different commodities.

Although unit price analysis can be useful incomparing demand curves for different com-modities when there are measurement scalesfor the dimensions along which the commod-ities differ (e.g., weight, concentration, caloricvalue), a potential problem remains when thecommodities differ qualitatively in a mannerthat cannot be measured directly. As notedpreviously, both Hursh and Winger (1995) andHursh and Silberberg (2008) developed nor-malization procedures allowing for the com-parison of demand regardless of how thereinforcers in question differ. The presentstudy compared their approaches to a prefer-ence-adjusted normalization procedure.Hursh and Winger’s normalization procedureclearly eliminated differences in the initiallevels of demand for W, PF, and HPW. Their

procedure also tended to merge the demandfunctions for the three foods (see Figure 2),making it appear that the demands for thethree commodities were essentially identical,as was the case for different doses of the samedrug in the Hursh and Winger study. Itappears reasonable that demand for differentdoses of the same drug would be comparable ifdifferences in the quantity of the reinforcerwere taken into account. If, however, it isassumed that less preferred foods in thepresent study were of lesser quality than morepreferred foods, then the former should nothave maintained responding at as high a priceas a higher quality food, and the normalizeddemand curves for the three commoditiesshould not be equivalent. In fact, the normal-ized demand functions for each food separate-ly (Table 5) show that PW produced the lowestrates of change of elasticity (a) and highestPmax values for 5 hens, suggesting that this lesspreferred food was characterized by moreinelastic demand, maintaining behavior athigher normalized prices than the other twofoods did. This finding would not be expectedif it is assumed that the higher quality foodpossesses the more inelastic demand.

However, it may be argued conversely thatthe higher response rates for the less preferredfoods at lower FR values were the result of ahen ‘‘defending’’ its consumption, that is,producing greater access to (or consumptionof) the less preferred food in order to gain thesame overall value per session that it does withthe most preferred food. From this perspec-tive, equating the start points of the demandfunctions for different foods, in the mannersuggested by Hursh and Winger (1995), is

Fig. 4. Means of the running response rates (responses per second) over the two series with W and PW for each henplotted against the normal logarithm of the FR value. Open circles mark data from the W condition and filled diamondsfrom the PW condition.

Fig. 5. Mean PRP durations (in seconds) over the twoseries using W and PW for each hen plotted against thenormal logarithm of the FR values. Open circles mark datafrom the W condition and filled diamonds from thePW condition.


appropriate. In addition, if the subject contin-ues to pursue an equated consumption asprice rises, a single function for all three foodsmight well emerge. That is, the normalizedfunctions may reflect a common demand forthe three foods. Whether such an analysis isvalid remains to be tested. It is, however, thecase that demand curves normalized in themanner suggested by Hursh and Wingerimplied that the three foods were eitherroughly identical in their ‘‘importance’’ tothe hens or that the less preferred food was themore important. In any event, the concurrent-schedule preference data suggest otherwise.

Hursh and Silberberg’s (2008) procedureproduces a sigmoidal function with the asymp-tote affected by the scaling parameter k. Whenthis analysis was applied, the demand func-tions for the three foods nearly merged (seeFigure 3), although this appears to be mainlythe result of setting the initial consumption to100. The extent to which demand functionsfor the three foods separate with increases in

FR value depends on the value of k used forthe curve-fitting. In the present analysis, k wasset equal to 3.5 and 8.0. With k 5 8.0, demandfor PW separated from that for W, as FR valueincreased (see Table 6 and Figure 3). When k5 3.5, the resulting functions reached anasymptote at values higher than the lowest datapoints.

Using the larger k value suggests separatefunctions for the three foods, in that the avalues for the functions for PW were signifi-cantly smaller than those for W and HPW.Although Hursh and Silberberg (2008) sug-gest that a common value of k is required forcomparing commodities, they do not discussthe selection of the k value. They do, however,provide a link to a spreadsheet that can beused to fit Equation 5. According to thespreadsheet, the value of k used should eitherbe based on the largest range of the data sets’consumption rates, or be based on the meanrange of consumption rates across all the datasets to be compared. Neither strategy is

Fig. 6. Examples of cumulative records from Hen 61 responding under FR 1, 2, 4, 8, and 64 schedules for W (leftpanels) and PW (right panels). The vertical dotted line indicates the end of the session.


obviously appropriate when the ranges ofconsumption and the resulting k values forthe best-fitting functions from the differentdata sets vary widely, as they did in the presentstudy. One k value used in the present analysis,3.5 (1.5 in normal logarithmic units), waswithin the lower range for the best-fittingfunctions across all data sets. The other value,8.0 (3.5 in normal logarithmic units), waswithin the higher range and is the same as thatused by Christensen et al. (2008). The value ofk used affects not only the value of a but alsothe values of other parameters and the degreeof fit. The value of % VAC increases as k movesfrom a small value to the best-fitting value andthen decreases as k moves beyond the bestfitting value, but it does so gradually. Asmentioned previously, for the present data, k5 8.0 was greater than the best-fitting k valuesfor the data sets with the smaller ranges andclose to the best-fitting k values for the datasets with the larger ranges, and so mostfunctions fitted the data reasonably well (seeTable 6). However, Figure 1 shows that, for alldata sets, the asymptotes of the fitted functionswith k 5 8.0 were beyond the range of thedata, unlike those for the k 5 3.5.

According to Hursh and Silberberg (2008),lower values of a reflect higher essential values.The a values in the present study were lowerfor PW than for the other foods when k 5 8.0,suggesting that PW, the less preferred food,has the highest essential value of the threefoods that were used. The lower a values alsoproduced the high Pmax values for PW in thisanalysis (see Table 6). These findings seemcontrary to what might be expected if thehigher response rates for PW were simply theresult of the hen ‘‘defending’’ her consump-tion, as discussed previously.

In contrast to the other transformations,and not surprisingly, demand curves normal-ized on the basis of preference data supportconclusions comparable to those supported bythe preference data themselves. Specifically,the Pmax values were highest and the values forrate of change in elasticity (a) were smallest forthe most preferred food (W) (Table 3). In thisanalysis consumption measures were convert-ed to most-preferred-food equivalents. Thiswas done to account for possible qualitative, aswell as quantitative, differences in the threereinforcers. The procedure yielded orderlydemand functions, but its general utility

remains to be determined. It is noteworthythat bias in the present analysis was calculatedbased on a comparison of a single pair ofschedule values, and several values are oftencompared to obtain this value (e.g., Davison &McCarthy, 1988). Bias measures based on acomparison of only two schedules has yield-ed apparently meaningful information inprior studies (e.g., McAdie, Foster, & Temple,1993; McAdie, Foster, Temple, & Matthews,1996) and was used in the present study tominimize the time required to normalizedemand. Further research examining whetherdifferent procedures for determining bias yieldvalues dissimilar enough to affect normalizationsignificantly, if at all, is nonetheless warranted.

As noted previously, when demand fordifferent quantities of the same reinforcer iscompared, it is reasonable to assume that avalid normalization procedure will yield com-parable demand functions (i.e., Pmax valuesand a values) across a range of quantities. Ifthe preference-based normalization procedureused here was applied to data generated withdifferent amounts of the same food (oranother reinforcer) and produced comparabledemand functions, this finding would suggestthat the present data reflect differences indemand for the three foods. Finding separatefunctions would, however, suggest that thenormalization procedure did not provide agood index of demand. Although it appears tohave some merit, further testing of thepreference-based normalization procedure,including studies examining different quanti-ties of the same reinforcer, is needed.

An important positive aspect of rescalingconsumption onto a common scale based onindependently obtained preference measuresis that doing so allows for meaningful com-parisons of the resulting demand functionseven when the commodities differ alongunknown or unmeasurable dimensions. Anadditional advantage of the preference-basedanalysis over both the Hursh and Winger(1995) and Hursh and Silberberg (2008)normalization procedures is that the ‘‘rescal-ing’’ variable—the bias measure—is obtainedseparately from the demand assessment, andso it may be possible to predict the shape ofthe demand function for a commodity from itsindependently-assessed preference value.

It should be noted that the preferencemeasures used here were taken with the two


foods available equally often under equal RIschedules. It has been argued that preferencebetween two commodities may change withchanges in price for both, even when theprices are kept equal. The studies which claimto show this have generally assessed demandfor the commodities when both are concur-rently available in the same condition [e.g.,Hursh and Bauman’s (1987) reanalysis ofHursh and Natelson (1981)] and have thenused differences in the demand functions toinfer the degree of any preference that mayexist. However, there are problems in inter-preting the data from this procedure. One, asSørensen, Ladewig, Ersbøll, and Matthews(2004) point out, is that demand functionsobtained by varying price with both commod-ities concurrently available (termed ‘‘cross-price demand’’) may be confounded by thedegree to which one commodity will substitutefor the other. That is, comparisons of cross-price demand functions may yield informationabout the substitutability of the two commod-ities rather than preference.

Sørensen et al. (2004) also point out thatcomparisons of demand functions from suchexperiments can be made only when theconsumption of the commodities can bemeasured on a common scale—the veryproblem the present research was attemptingto address. As part of their argument that pricechanges can result in changes in preference,Hursh and Bauman (1987) present demandfunctions for both electrical stimulation of thebrain (EBS) and food obtained from ratsresponding for these two commodities whenthey were concurrently available (derived fromHursh & Natelson, 1981). For both commod-ities, consumption (scaled on the y-axis) wasmeasured as the number of times access toeach was obtained per hour. In this way theplots are similar to the unmodified demandfunctions in Figure 1. To compare the EBSand food functions directly requires theassumption that the consumption measuresboth occupy the same scale. Hursh and Bau-man’s conclusion of more elastic demand forEBS is valid only if a change of one EBS perhour is equivalent to a change of one foodpresentation an hour. The present data(where lower prices resulted in less pausingfor the less preference food; see Figure 5),suggest that the interpretation of the datamight not be as simple as Hursh and Bauman

assumed. Further research is required toclarify the relation between cross-price de-mand and preference as measured by anindependent procedure.

An interesting aspect of the present data isthe finding that overall response rates atsmaller FR values (and therefore initial de-mand levels) were lower for the most preferredfood. Similar findings were reported previous-ly by Foltin (1992, Experiment 1), who usedbaboons and found lower initial levels ofdemand and greater Pmax values for 5-pelletrather than 1-pellet reinforcers. The 10-pelletdata were equivocal, however.

In the present study different mean PRPlengths with W and PW contributed to thedifferences in initial demand levels. However,as Schlinger, Derenne, and Baron (2008)point out, changes in the average PPRs underFR schedules may not directly reflect thechanges in the underlying distribution ofpauses. Thus the differences between the Wand PW data shown in Figure 5 could haveresulted from several different underlyingpatterns of responding. For example, theycould have been the result of the hens ceasingto respond earlier in a session with W(increasing pause time), or of hens paus-ing longer a few times with W, or of hensgenerally pausing longer after a W reinforcerthan after a PW reinforcer. An analysis ofthe within-session data showed that therewere consistent differences in the PRPs withW and PW at the small ratios throughout asession, with generally longer PRP durationswith W.

The finding of differences in PRP lengthparallels the results from prior studies regard-ing the effect of magnitude of reinforcementon performance under FR schedules (see, e.g.,Schlinger et al., 2008). These studies foundthat larger reinforcers were accompanied byincreased PRP lengths (and hence loweroverall response rates), particularly at smallto moderate FR values (see, e.g., Lowe, Davey& Harzem, 1974; Perone & Courtney, 1992).Larger reinforcers are preferred over smallerreinforcers under concurrent schedules ofreinforcement (e.g., Schneider, 1973), and itmay be the case that more preferred reinforc-ers, regardless of whether they are morepreferred by virtue of quantity, quality, orboth, generate longer pauses under relativelyshort FR schedules. Unfortunately, there are


no comprehensive parametric studies of theeffects of reinforcer quantity, reinforcer qual-ity, or reinforcer preference and FR value onperformance under FR schedules. Some stud-ies of reinforcer quantity and FR responserates have yielded results similar to those fromthe present study, but others have yieldedequivocal results or results opposite to ours.For instance, Meunier and Starratt (1979)found that rats emitted shorter pauses underFR 7 and FR 9 schedules with higher concen-trations of sucrose solution (i.e., larger rein-forcers). Similarly, Powell (1969) found short-er PRPs accompanying larger reinforcersunder schedules between FR 40 and FR 70,although this difference virtually disappearedwhen the FR schedules were reduced tobetween FR 10 and FR 50, depending on theindividual pigeon. Clearly, further research isneeded to clarify how quantitative (and other)dimensions of reinforcers interact with sched-ule value to determine response rate (andhence apparent demand).

In this regard, it is important to note thatthe three foods in the present study differedalong more than one dimension, and any orall of these differences may have influencedpreference and FR response rates. To humans,they differed in appearance, texture, and taste,and it is probable that different amounts ofthe three foods could be consumed in the 3-saccess time. Thus, the preferences we foundare likely to be a product of both reinforcermagnitude (in terms of measures such ascaloric value or weight) and other, qualitativedifferences. Measuring the amount of each ofthe three foods consumed (as well as correct-ing for caloric value) would provide a potentialindex of the potential influence of ‘‘magni-tude of reinforcement’’ on the differentialresponse rates observed in the present study,but unfortunately consumption was not mea-sured. This should be done in future investi-gations, which appear justified in view of thepotential value of the preference-adjustednormalization procedure reported here.

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Received: September 5, 2008Final Acceptance: August 3, 2009


APPENDIX

The parameters a, b and ln L for Hursh et al.’s (1988) nonlinear equation (Equation 1) fitted to thenatural logarithms of consumption and FR value for both series of FR-schedule changes and foreach food for each hen. The FR value at which the equation predicts maximal response rate (Pmax;see Equation 2), the standard error of the estimates of the fits (se), and the percentages of varianceaccounted for by the functions (%VAC) are also given.

Food Hen Series a b ln L Pmax se %VAC

W 61 1 0.0032 20.38 23.00 189.2 0.20 97.42 0.0032 20.34 23.27 203.5 0.48 84.7

62 1 0.0032 20.40 22.63 186.1 0.18 98.02 0.0057 20.23 23.43 132.9 0.22 97.3

63 1 0.0263 20.03 23.42 37.0 0.40 88.62 0.0058 0.05 25.01 180.6 1.02 40.1

64 1 0.0031 20.32 23.07 216.4 0.64 73.82 0.0012 20.42 22.99 449.6 0.48 81.4

65 1 0.0167 20.16 22.54 50.0 0.31 96.32 0.0111 20.26 22.56 66.6 0.29 95.3

66 1 0.0040 20.38 23.23 153.5 0.16 93.02 0.0030 20.25 23.60 245.0 0.18 89.1

PW 61 1 0.0071 20.57 21.47 60.6 0.18 98.72 0.0098 20.56 21.74 45.3 0.16 98.3

62 1 0.0043 20.51 21.81 113.2 0.40 94.22 0.0061 20.38 22.45 100.9 0.27 97.4

63 1 0.0048 20.64 21.28 73.5 0.17 99.32 0.0112 20.34 22.01 58.6 0.21 98.0

64 1 0.0019 20.61 21.38 203.3 0.39 93.62 0.0037 20.68 21.10 85.4 0.51 93.0

65 1 0.0136 20.37 21.58 46.3 0.31 96.72 0.0277 20.29 21.50 25.5 0.14 99.2

66 1 0.0332 0.03 22.55 31.0 0.14 98.92 0.0350 20.17 22.40 23.8 0.13 98.1

HPW 61 1 0.0112 20.42 22.03 51.8 0.18 98.82 0.0092 20.43 22.04 61.1 0.20 98.1

62 1 20.0032 -0.83 22.75 252.4 0.69 77.42 0.0089 20.57 21.81 48.0 0.23 98.2

63 1 0.0204 20.14 23.24 42.3 0.43 85.52 0.0326 20.07 23.45 28.6 0.41 92.5

64 1 0.0068 20.29 22.54 104.2 0.14 99.22 0.0103 20.11 22.78 86.1 0.18 96.9

65 1 0.0166 20.28 22.45 43.4 0.30 93.22 0.0158 20.17 22.59 52.5 0.32 95.8

66 1 0.0318 0.04 23.22 32.9 0.35 92.62 0.0543 20.04 23.12 17.6 0.18 97.8


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