ERIC EJ859313: Consumption and Response Output as a Function of Unit Price: Manipulation of Cost and Benefit Components

2009·0.34 MB·English

by  ERIC

JOURNALOFAPPLIEDBEHAVIORANALYSIS 2009, 42, 609–625 NUMBER3 (FALL2009)
CONSUMPTION ANDRESPONSEOUTPUT ASA FUNCTIONOF UNIT
PRICE: MANIPULATION OF COST AND BENEFIT COMPONENTS
XERES DELMENDO, JOHN C. BORRERO, KENNETH L. BEAUCHAMP, AND
MONICA T. FRANCISCO
UNIVERSITYOFTHEPACIFIC
Weconductedpreferenceassessmentswith4typicallydevelopingchildrentoidentifypotential
reinforcersandassessedthereinforcingefficacyofthosestimuli.Next,wetestedtwopredictions
ofeconomictheory:thatoverallconsumption(reinforcersobtained)woulddecreaseastheunit
price(responserequirementperreinforcer)increasedandthatthecostandbenefitcomponents
that defined unit price would not influence overall consumption considerably when unit price
valueswereequal.Wetestedthesepredictionsbyarrangingunitpricesuchthatthedenominator
wasone(e.g.,tworesponsesproducedonereinforcer)ortwo(e.g.,fourresponsesproducedtwo
reinforcers). Results showed that consumption decreased as unit price increased and that unit
pricevalueswith differentcomponents producedsimilar consumption.
DESCRIPTORS: behavioral economics, costs and benefits, preference assessment,
reinforcerdemand, unit price
_______________________________________________________________________________
Methods of assessing preferred stimuli that schedule requirements (e.g., fixed-ratio [FR] 1),
will increase appropriate engagement and involving relatively low-effort responses (e.g.,
reduce levels of problem behavior have consti- touchingtheexperimenter’shand)thatmaynot
tuted an important area of research in behavior approximate more effortful responses required
analysis (Ivancic, 2000). Several methods of in appliedsettings (e.g., Fisher & Mazur,1997;
determining stimulus preference and subse- Roscoe, Iwata, & Kahng, 1999).
quent reinforcer efficacy have been reported The conditions under which responding is
(e.g., Fisher et al., 1992; Pace, Ivancic, predicted to persist under increasing response
Edwards, Iwata, & Page, 1985; Roane, Voll- requirements may be informed by methods
mer, Ringdahl, & Marcus, 1998). In these used in behavioral economics. The primary
studies, evaluations of reinforcer efficacy have dependent variable in behavioral economic
generally been arranged under relatively low research is consumption, or the number of
reinforcers obtained (Hursh, 2000; Johnson &
Thisexperimentwascompletedinpartialfulfillmentof
Bickel, 2006). A second dependent variable in
therequirementoftheMAdegreebythefirstauthor.We
thank Lucas Bradley for his comments on an earlier behavioral economic research measures spend-
version of this manuscript and Adam Brewer, Patrick ing,aseitherlocaloroverallresponseoutput.In
Johnson, and Greg Madden for their conceptual contri-
a traditional economic system, money serves as
butionsto the demand analyses. We also thank Stephany
Crisolo,NoelRoss, andSandeep Sranfortheirassistance the currency or means by which one obtains
with data collection and various other aspects of this necessities (e.g., water) and luxury items (e.g.,
project. Portions of this research were presented at the
large flat-screen televisions). In behavioral
25th meeting of the California Association for Behavior
economic research with nonhumans and con-
Analysis in Burlingame and the 33rd meeting of the
Association for Behavior Analysis in San Diego. Xeres temporary behavior-analytic reinforcer assess-
Delmendo is now with Nyansa Learning Corporation,
mentresearch,thecurrencyisbehavior,andthe
Santa Ana, California. Monica Francisco is now at the
commoditiesavailableforpurchasearegenerally
Universityof Kansas.
AddresscorrespondencetoJohnC.Borrero,whoisnow foods or preferred activities. Consumption and
at the University of Maryland, Baltimore County, spending may thus be characterized as compo-
Department of Psychology, 1000 Hilltop Circle, Balti-
nents of a cost–benefit arrangement, in which
more,Maryland21250(e-mail: [email protected]).
doi:10.1901/jaba.2009.42-609 the ratio of cost (response requirement) over
609
610 XERES DELMENDO et al.
benefit (reinforcer magnitude) defines a simple which yields a demand function with a slope
formulation of unit price, which is represented between 0 and 21. If a 1% change in price
by the equation producesalessthan1%changeinconsumption,
FR
P~ , ð1Þ demand is characterized as inelastic. Elastic
A
demand describes a greater than proportional
where price (P) is determined by, in this
change inconsumption asa function of changes
example, the FR requirement divided by the
in price, which yields a demand function with a
amount (A) of the reinforcer or its magnitude.
slope more negative than 21 (Hursh, Raslear,
Unit price may be altered by manipulating
Shurtleff,Bauman,&Simmons,1988).Ifa1%
reinforcer magnitude or response requirements
change in price produces a greater than 1%
separately or in combination (Foster & Hack-
changeinconsumption,demandischaracterized
enberg, 2004). Thus, if completion of an FR 2
aselastic.Inotherwords,acommodityforwhich
requirement is followed by one reinforcer, this
demandisinelasticisonewhoseconsumptionis
is equivalent in price to completion of an FR 4
not greatly affected by changes in price (e.g.,
requirement that is followed by two reinforcers.
consumptionmaydecreaseaspriceincreases,but
Behavioral economic research characterizes
decreasesarelessthanproportional).Bycontrast,
performance in terms of the exchange of
a commodity with elastic demand is one whose
responses and reinforcers. For example, if
consumptionismoreaffectedbychangesinprice
‘‘spending’’ eight lever presses (response) pro-
(e.g., consumption decreases in a greater than
duces one food pellet (reinforcement), a unit
proportional fashion as price increases). For
price analysis predicts that eight lever presses
example, if the price of kiwis increased from
that produce one food pellet would be
$1.00 per kilgram to $2.00 per kilogram, this
functionally equal to 16 lever presses that
would represent a 50% increase in price. To
produce two food pellets. Thus, if both the
characterize demand for kiwis as elastic, con-
response requirement and the reinforcer
sumption would necessarily decrease by more
amount increase by the same constant, eco-
than 50%. However, if consumption of kiwis
nomic theory suggests that consumption will
decreased by only 10%, given a change in price
not be affected because the unit price of the
from$1.00perkilogramto$2.00perkilogram,
reinforcer will be constant.
then consumption for kiwis wouldbe character-
Given a range of unit price values, reinforcer
ized as inelastic.
demand can be empirically assessed. Demand
Changes in the slope of the demand function
refers to the amount of the commodity
(obtained by plotting reinforcer consumption
consumed relative to the price for each
against unit price) can be used to assess the
commodity (Hursh, 1980). The effects of
relative reinforcing efficacy of stimuli. Given
changes in price (on consumption) are predict-
ed to follow the law of demand, which states increases in price, stimuli that produce slopes
that, all else being equal, consumption of a more shallow than 21 are more valuable than
commodity (reinforcer) decreases as its price thosewithslopessteeperthan21(Allison,1983).
increases (Freed & Green, 1991). Demand Whenplottedondoublelogarithmiccoordinates,
elasticity is the extent to which consumption of thefunctionrelatingconsumptionandincreasing
a commodity is influenced by alterations in unit price is positively decelerating, and the
price for that commodity (Green & Freed, function relating response output to increases in
1993),and two types of demandarecommonly unit price is typically an inverted U-shaped
discussed in the literature. Inelastic demand function (Bickel, Marsch, & Carroll, 2000).
describes a less than proportional change in Several recent studies have evaluated rein-
consumption as a function of changes in price, forcerefficacyunderprogressive-ratioschedules,
COST AND BENEFIT COMPONENTS 611
in which response requirements increase within academic performance has been shown to
(e.g., Francisco, Borrero, & Sy, 2008; Glover, persist when one reinforcer is delivered for
Roane, Kadey, & Grow, 2008; Roane, Call, & every 10 responses. Assume further that a
Falcomata,2005;Roane,Lerman,&Vorndran, teacher’s attention is diverted from the student
2001) and across (e.g., DeLeon, Iwata, Goh, & for a period of time such that delivery of the
Worsdell, 1997; Johnson & Bickel, 2006; reinforcer following Response 10 is omitted.
Tustin, 2000) sessions. Results of these studies Presumably,performance shouldbe maintained
have been consistent with the law of demand in at similar levels if two reinforcers were arranged
that decreases in consumption were generally for completion of 20 responses, and thus, a
associated with increases in response require- potentially detrimental instance of integrity
ment. failure (Peterson, Homer, & Wonderlich,
Although not framed as such, the work of 1982) might be avoided by a modest manipu-
DeLeon et al. (1997) may also be assessed in lation of reinforcer amount, thereby keeping
terms of reinforcer demand. DeLeon et al. unit price constant.
evaluated consumption of two concurrently In addition to offsetting potential treatment
available edible reinforcers (e.g., a cookie and integrity problems, evaluations of reinforcer
a cracker) under progressively increasing sched- demand may suggest the maximum schedule
ule requirements. Responding for the two requirementunderwhichstimulimaintainhigh
reinforcers was similar under low response value (i.e., P [the price at which peak
max
requirements, but a preference emerged for responding is reached], and corresponding
one reinforcer (cookie) under increasing re- valuesalongtheinelasticportionofthedemand
sponse requirements. These data illustrate that function). Inotherwords,inpractice,clinicians
the value of one reinforcer (cookie) varied attempt to identify the largest amount of work
depending on the other concurrently available that can be maintained with the smallest
reinforcer. Results reported by DeLeon et al. reinforcer amount. If a client will complete 20
raise important issues that should be evaluated. academictasksforonereinforcer,itwouldbeill
First, performance under a single schedule may advised to arrange one reinforcer contingent on
not correspond to performance under concur- 10academicresponses.Similarly,itwouldbeill
rent schedules (e.g., Francisco et al., 2008; advised to arrange one reinforcer contingent on
Roscoe et al., 1999). That is, consumption of 50 academic responses. By setting a price that
one reinforcer might differ, depending on the exceeds P , spending (or work) will decrease.
max
presence or absence of alternative reinforcers. By setting a price that approaches P , the
max
Second, in terms of behavioral programming, value of the reinforcer is maintained, and the
predicted performance should be similar when maximum amount of work is produced.
unit price remains the same across changes in Evaluations of demand elasticity across
the cost–benefit components. In other words, if various price arrangements may be useful in
maximal performance (as evidenced by the determining the conditions under which highly
largest value of the inelastic portion of a preferred stimuli, identified via preference
demand function) occurs when x responses assessment, will support clinically acceptable
produce one reinforcer, similar performance levels of responding under increasing response
should occur when 2x responses produce two requirements. For example, more valuable
reinforcers (i.e., unit price is unchanged). From stimuli (i.e., those that produce relatively
a practical standpoint, such a finding might shallow slopes) may be made contingent on
have an impact in classroom or institutional more effortful behavior, and relatively less
settings. For example, assume that a student’s valuable stimuli (i.e., those that produce
612 XERES DELMENDO et al.
relatively steeper slopes) might be arranged conducted in a small room at their respective
contingent on less effortful behavior. By schools. The room contained a desk, a table,
incorporating greater variety in the pro- several chairs, and various teaching materials.
grammed consequences for behavior, satiation, Sessions for Keelan were conducted in the
at least with respect to a particular stimulus, dining room of his home. The room contained
may be minimized. Although such findings a dining table, four chairs, and a shelf. Sessions
would be of importance to applied researchers, wereconductedonetothreetimesperday,4to
a preliminary assessment of reinforcer demand, 5 days per week for each child. As part of the
withunitpriceheldconstant,isaprerequisiteto intakeandconsentprocedure,parentsindicated
more complex analyses like those involving which foods were prohibited due to food
qualitatively different stimuli. allergies or parental preference. Prior to the
Previous operant research with humans (e.g., initiation of data collection, the university
Bickel & Madden, 1999; Madden, Bickel, & institutional review board approved this study.
Jacobs,2000)suggeststhatconsumptionwillbe
Response Measurement and
similar under equal unit price values, regardless
Interobserver Agreement
of the cost–benefit components. However, such
findings have yet to be demonstrated in the During the preference assessment, data were
context of reinforcer assessment research in- collected on consumption (defined as an edible
volving academic task completion. Thus, the item passing the plane of the participant’s lips).
purpose of the current investigation was to During the reinforcer assessment and reinforcer
determine whether similar results would be demand evaluations, target academic responses
obtained in the context of repeated reinforcer were selected according to the participant’s
assessments in which unit price was held academic level, based on consultation with the
constant while cost–benefit components were participant’s teachers. The target response was
varied. If consumption and response output are tracing (letters and numbers) for Elijah, sorting
largely uninfluenced by cost–benefit compo- beads by color for Anna, completion of writing
nents, behavioral programming may be en- worksheets for Keelan, and color matching for
hanced. In other words, the unit price itself Elizabeth. For the tracing task, a response was
(e.g., six) may be more relevant in determining recorded when the participant completely
reinforcer demand than the methods used to traced either one letter or one number. For
arrangeaunitpriceofsix(e.g.,12responsesfor the bead-sorting task, a response was recorded
two reinforcers or 36 responses for six reinforc- when the participant placed one like-colored
ers; Roane, Falcomata, & Fisher, 2007). bead among an array of beads arranged in a
sorting receptacle. For the color-matching task,
the participant was required to place an object
METHOD
of one color in a receptacle containing like-
Participants and Settings colored stimuli.
Four typically developing children partici- Trained observers were seated unobtrusively
pated, and parental consent, as well as child in the room and collected data using paper and
assent, was obtained prior to data collection. pencil (preference assessment) or handheld
Elijah was 4 years old, attended preschool 4 personal digital assistants. Interobserver agree-
days per week, and attended an afterschool care ment was assessed by having a second observer
program. Annaand Elizabethwere both 3 years simultaneously but independently record con-
old and attended preschool 5 days per week. sumption and academic responses. Agreement
Keelan was 6 years old and attended first grade. was calculated using the method of partial
Sessions for Elijah, Anna, and Elizabeth were agreement within intervals (Vollmer, Borrero,
COST AND BENEFIT COMPONENTS 613
Wright, Van Camp, & Lalli, 2001). Data were food was ever unavailable. Consumption was
assessed by dividing the entire session into recorded using a 10-s partial-interval recording
consecutive 10-s intervals, and the smaller system. It was possible for participants to
number in each 10-s interval was divided by consume multiple foods during a given 10-s
thelargernumber.Interobserveragreementdata interval. Preference was determined by record-
during the free-operant preference assessment ing the percentage of 10-s intervals during
were collected during 50%, 33%, 67%, and which consumption of each food occurred. In
67% of sessions for Elijah, Anna, Keelan, and other words, the number of 10-s intervals with
Elizabeth, respectively (multiple preference consumption of each food was divided by the
assessments were conducted for each partici- total number of intervals in the session, and
pant, described below). Agreement for con- multiplied by 100%. Results of each preference
sumptionduringthepreferenceassessmentwere assessment were averaged, and the food item
90% for Elijah, 100% for Anna, 72% for consumed during the greatest mean percentage
Keelan, and 96% for Elizabeth. During the of intervals was subsequently evaluated via
reinforcer assessment, agreement data were reinforcer assessment.
collected during 41%, 30%, 29%, and 76%
of sessions for Elijah, Anna, Keelan, and Reinforcer Assessment: Procedure and Design
Elizabeth, respectively. These values were then The purpose of the reinforcer assessment was
averaged for the entire session. Agreement to determine whether items consumed most
values for academic responding during the during the preference assessment would func-
reinforcer assessment were 92% for Elijah tionasreinforcerswhendeliveredcontingenton
(range, 88% to 93%), 100% for Anna, 98% academic responding. The academic response
forKeelan (range,95%to 100%),and 95%for foreachchildwasdefinedpreviouslyandserved
Elizabeth (range, 86% to 100%). During the as the dependent variable during the reinforcer
reinforcer demand evaluation, agreement data assessment. During baseline (A), the therapist
were collected during 38%, 36%, 44%, and presented the participant with the instructional
37% of sessions for Elijah, Anna, Keelan, and materials (e.g., a tracing worksheet) and
Elizabeth,respectively,andwere88%forElijah permitted the participant to engage in the
(range, 74% to 100%), 95% for Anna (range, activity. However, the therapist did not deliver
81%to100%),89%forKeelan(range,60%to reinforcers for academic responding, and edible
100%) and 97% for Elizabeth (range, 86% to items were not present or in view. During the
100%). reinforcement condition (B), the therapist
presented the participant with the same task
Preference Assessment: Procedure used in baseline; however, the therapist deliv-
Thepurposeofthepreferenceassessmentwas ered reinforcers for academic responding on an
toidentifyonefooditemthatwouldfunctionas FR 1 schedule. Edible items were both present
a potential reinforcer for the subsequent and in view during this condition. Reinforcer
analyses. Each participant’s behavior was ex- efficacy was evaluated using an A-B-A design,
posed to two (Elijah) or three (Anna, Elizabeth, and a minimum of three sessions per condition
and Keelan) preference assessments using the were conducted. After demonstration of a
procedures described by Roane et al. (1998). reinforcement effect, participants proceeded to
During the preference assessment, only food the reinforcer demand evaluation.
items were evaluated. A total of 10 foods (in
bite-sizeportions)werearrangedinacircleona Reinforcer Demand Evaluation: Procedure
table, and participants could consume the food The purpose of the reinforcer demand
freely. Foods were replenished such that no evaluation was to assess consumption and
614 XERES DELMENDO et al.
Table 1
Sequenceof Conditions andExposure
Series1 Series2
Participant Edibleitem Unitprice Sessions Fixedratio Edibleitem Sessions Fixedratio Edibleitem
Elijah Candy 2 7 2 1 7 4 2
4 5 4 1 5 8 2
6 5 6 1 5 12 2
8 6 8 1 5 16 2
15 5 15 1 5 30 2
5 45 1
Anna Chips 2 5 2 1 7 4 2
4 6 4 1 8 8 2
6 5 6 1 5 12 2
8 5 8 1 5 16 2
16 5 16 1 6 32 2
Keelan Chips 2 5 2 1 5 4 2
4 7 4 1 7 8 2
6 9 6 1 9 12 2
8 5 8 1 5 16 2
15 5 15 1 5 30 2
Elizabeth Cookies 2 5 2 1 5 4 2
4 6 4 1 5 8 2
6 5 6 1 5 12 2
8 5 8 1 5 16 2
performance given increasing response require- target academic response or would do so at low
ments and to assess consumption and perfor- levels.
mance under equal unit price values composed Prior to the first reinforcer demand session,
of different cost–benefit components. Each the participant was told that he or she could
participant completed two reinforcer demand earn a bite of their favorite food for completing
evaluations (hereafter referred to as ‘‘series’’). In a specific number of responses, and that he or
Series 1, completion of the predetermined FR she could consume the food. A minimum of
schedulerequirementresultedinonereinforcer. five sessions was conducted for each unit price
For example, given an FR 15 response in each series, and sessions were terminated
requirement, completion of 15 academic re- when one of the following criteria was met: (a)
sponses produced one bite of the participant’s 3 min elapsed without a response, or (b) the
most preferred food. In Series 2, completion of participant emitted the vocal response, ‘‘I’m
the predetermined FR schedule requirement done.’’ The latter termination criterion was
resulted in two reinforcers. Thus, to ensure an included to reduce the likelihood of disruptive
equal unit price (in this example), an FR 30 behavior that might have occurred by requiring
schedule was arranged, which resulted in participants to stay seated for 3 min.
delivery of two reinforcers. The sequence of
exposures, specific unit price values, cost and
RESULTS
benefit components, response requirements,
and reinforcers are summarized in Table 1. Figure 1 depicts the mean results of the
Prior to Series 1, all participants had just preference assessments for all participants.
completed the baseline condition of the prior Results suggested that candies, chips, chips,
reinforcer assessment phase, and thus we and cookies might serve as potential reinforcers
concluded that in the absence of programmed for Elijah, Anna, Keelan, and Elizabeth,
contingencies, participants would not emit the respectively.
COST AND BENEFIT COMPONENTS 615
contingent food was withdrawn.Based on these
results, we concluded that these edible items
served as reinforcers for each participant’s
academic responding.
Figure 3 depicts results of the reinforcer
demand evaluations for all participants. Data
from the last four sessions are reported because
this is the method of data aggregation used in
the majority of behavioral economic research
with nonhumans and also eliminates data that
might be anomalous due to transitions from
one unit price value to the next.
The objective of Series 1 (Figure 3) was to
determine the level of academic responding,
and by extension, the number of reinforcers
obtained, when the price of the reinforcer was
systematicallyincreased.Inbrief,asthenumber
of academic responses required was increased,
the number of reinforcers obtained decreased,
consistent with the law of demand. More
specifically, during Series 1 for Elijah (top left)
and Anna (second left), demand was inelastic
(there was a less than proportional change in
consumption as a function of the change in
price)whenpricewasincreasedfromUnitPrice
2 to Unit Price 4. Specifically, the slope of the
function from Unit Price 2 to Unit Price 4 was
20.97 for Elijah and 20.15 for Anna. Recall
that slopes between 0 and 21 define inelastic
demand, which means that although academic
responding did decrease for Elijah, the propor-
tional decrease in reinforcers obtained was less
than the proportional increase in price. Or, the
Figure1. Preferenceassessmentresultsforallparticipants. value of the edible item was maintained despite
Notethatthescaleoftheyaxisissettoamaximumof40. requiring more academic work to produce it.
Subsequent increases in price during Series 1
Figure 2 depicts the results of the reinforcer (e.g., from Unit Price 4 to Unit Price 6)
assessment for all participants. In the initial resulted in largely elastic demand (greater than
baseline condition, in which edible items were proportional change in consumption as a
not available contingent on academic respond- function of changes in price), which means
ing, mean response rates were generally low for that at these higher prices, proportional de-
all participants. When reinforcers were deliv- creasesinreinforcers obtainedweregreaterthan
ered on an FR 1 schedule, mean response rates the proportional increases in price, with the
increased relative to baseline levels for all exceptionofthetransitionbetweenUnitPrice6
participants, and subsequently decreased when and Unit Price 8 for Anna (slope 5 0.28). In
616 XERES DELMENDO et al.
Figure2. Resultsofthereinforcerassessmentforallparticipants.Seetextforadescriptionofthecontingenciesin
placeduring baseline andFR1.
other words, as the price of the edible item work when the price increased but earned less
increased, the number of academic responses for every instance of academic responding.
required was too great and resulted in less Results for Keelan and Elizabeth illustrate the
responding than would have been predicted opposite change in response output. Response
given the proportional increase in price. By output decreased slightly for Keelan and
contrast, slopes for Keelan (third left) and Elizabeth (slopes 5 20.72, and 20.83,
Elizabeth(bottomleft)wereindicativeofelastic respectively), which means that they responded
demand during transitions from even the less when the price of the reinforcer was
smallest unit price values (Unit Price 2 to Unit increased, but the decrease was within the
Price 4) during Series 1 (slope 5 21.70 for expected(proportional)range.Thisunderscores
Keelan and 22.82 for Elizabeth) suggesting the fact that increases in academic responding
that even these very modest increases in price are only contextually related to evaluations of
were too much to maintain academic respond- demand (i.e., a child may emit more absolute
ing at sufficiently high levels. In addition, responding under increasing price require-
response output increased from Unit Price 2 ments, but this does not in and of itself suggest
to Unit Price 4 for Elijah and Anna (slopes 5 greater stimulus value). If the number of
0.14, and 0.86, respectively). These data reinforcers obtained under these higher price
illustrate that even though the number of requirements decreases in a greater than
academic tasks completed might increase, the proportional fashion, the stimulus has still lost
total amount of reinforcers obtained can some of its value.
decrease(becausemore workisrequired to earn The objective of Series 2 (Figure 3) was to
reinforcers when the price of a reinforcer determine the extent to which changes in
increases). From a practical standpoint, this consumption would be observed when unit
means that Elijah and Anna completed more pricewasheldconstantandthecostandbenefit
COST AND BENEFIT COMPONENTS 617
components were varied. In other words, for at which decreases in the number of reinforcers
each unit price assessed, the objective was to obtained (and by extension, tasks completed)
determine the extent to which reinforcers were proportionally greater than the change in
obtained and task completed would be similar price from Unit Price 4 to Unit Price 6. Thus,
at the same unit price values. For 3 of 4 despite the differences in Anna’s overall con-
participants (Elijah, Keelan, and Elizabeth), sumption (Figure 3),demandwas characterized
there was considerable similarity in terms of aselasticinbothSeries1andSeries2,following
reinforcersobtainedacrossSeries1andSeries2. thetransitionfromUnitPrice4toUnitPrice6.
In other words, from the perspective of 3 Reinforcer demand in Series 2 changed some-
participants, a situation in which, for example, whatforElijahcomparedtoSeries1(Figure 3).
two academic responses were required for one BecausemeanreinforcersobtainedinSeries2at
reinforcerwasviewedassimilartoasituationin Unit Price 15 exceeded consumption from
which four academic responses were required Series 1 under Unit Price 15 (the largest unit
for two reinforcers. Data for Anna, during the price value assessed in Series 1), an increase in
transition from Unit Price 2 to Unit Price 4, unit price was introduced. Specifically, it
appeared that the value of the commodity
weretheexception(althoughtheoverallpattern
changed from Series 1 to Series 2. Thus, Unit
remained unchanged). A second objective of
Price 45 was introduced. The transition from
Series 2 was to determine the point at which
Unit Price 15 to Unit Price 45 produced a
demand transitioned from inelastic to elastic;
decrease in reinforcers obtained that was
that is, when the change in the number of
considered proportional to the rather large
reinforcers obtained moved from decreases that
increase in price and resulted in a transition
would be expected proportionally to decreases
slope of 20.89 (or degree of change).
that were larger proportionally than expected.
Figure 4depictsdataforthelastfoursessions
For all participants, these transition values (i.e.,
conducted at each unit price (instead of the
specific price increases such as Unit Price 4 to
means of those sessions depicted in Figure 3).
Unit Price 6 for Elijah in Series 1 and Series 2)
For all participants, reinforcers obtained (left)
were identical. This can be illustrated with
were fitted using the logarithmic version of the
Anna’s data. For the other 3 participants, the
equation proposed by Hursh et al. (1988):
data paths are either fairly close to one another
orontopofoneanother,atthesameunitprice. lnC~lnLzbðlnPÞ{aP, ð2Þ
For Anna, reinforcers obtained for Unit Price 2 to evaluate the functional relation between total
in Series 1 are greater than the number of reinforcers consumed per session (C) and unit
reinforcers obtained in Series 2. The same is price(P).Lisanestimateofreinforcersobtained
true for Unit Price 4 (note that open circles are thatwouldoccuratUnitPrice1,andbandaare
consistently higher than closed triangles for relatedtotheinitialslopeandaccelerationofthe
Anna, across all unit price values). Despite the demandfunction,respectively.Dataonacadem-
differences in the number of reinforcers ob- ic responding (right) were fitted to the same
tainedfromSeries1toSeries2acrossthesetwo equation,exceptthatb+1wassubstitutedforb
price values, the point at which her academic (Hursh, Raslear, Bauman, & Black, 1989).
responding was proportionally greater than the When the number of reinforcers obtained was
increaseinpricewasstillthesamefromSeries1 zero(i.e.,AnnaatUnitPrice16duringSeries2,
to Series 2. In other words, when either six Keelan at Unit Price 15 during Series 2, and
responses were required for one reinforcer ElizabethatUnitPrice8duringSeries2),avalue
(Series 1) or 12 responses were required for of0.1wasplottedsothatdataforallfoursessions
two reinforcers, this is the price (Unit Price 6) would be visible for inspection.
618 XERES DELMENDO et al.
Figure 3. Results of the demand analysis for all participants. Each participant’s data occupies one row. Each data
pointrepresentsmeannumberofreinforcersearned(leftcolumn)ormeannumberoftaskscompleted(rightcolumn),
across the last four sessions of the condition, for all participants (session-by-session data are available from the second
author).Dataareexpressedondoublelogarithmicaxes.OpencirclesdepictdatafromSeries1,andfilledtrianglesdepict