Pizzas: π or square? psychophysical biases in area estimation

0732-2399/01/2004/0405/$05.001526-548X electronic ISSN

MARKETING SCIENCE � 2001 INFORMSVol. 20, No. 4, Fall 2001, pp. 405–425

Pizzas: � or Square? PsychophysicalBiases in Area Comparisons

Robert E. Krider • Priya Raghubir • Aradhna KrishnaFaculty of Business Administration, Simon Fraser University, Burnaby, BC, Canada V5A 1S6

Haas School of Business, University of California, Berkeley, California 94720-1900University of Michigan Business School, Ann Arbor, MI 48109

[email protected] • [email protected] • [email protected]

AbstractMany product categories, from pizzas to real estate, presentbuyers with purchase decisions involving complex areajudgments. Does a square look larger or smaller than a cir-cle? How much smaller does a circle of 8-inch diameter lookwhen compared to one with a 10-inch diameter? In this pa-per, we propose a psychophysical model of how consumersmake area comparison judgments. The model involves con-sumers making effort-accuracy trade-offs that lead to heu-ristic processing of area judgments and systematic shape-and size-related biases.

The model is based on four propositions: P1. Consumersmake an initial comparison between two figures based ona single dimension; P2. The dimension of initial compari-son—the primary dimension—is the one that is most salientto consumers, where salience is figure and context depen-dent; P3. Consumers insufficiently adjust an initial compar-ison using a secondary dimension, which we assume to beorthogonal to the primary dimension used for the initialcomparison; and P4. The magnitude by which the initialcomparison is adjusted is directly related to the relative sa-lience of the secondary dimension versus the primary di-mension.

The model predicts that a single linear dimension inap-propriately dominates the two-dimensional area comparisontask and that contextual factors affect which linear dimen-sion dominates the task. The relative use of the second di-mension depends on its relative salience, which can be in-fluenced in a variety of ways. The model extends the areaestimation literature in cognitive psychology by exploringnew biases in area estimation and is able to resolve contro-versial effects regarding which shape is perceived to be‘‘bigger,’’ the square or the circle, by incorporating contex-tual factors into model specifications.

A set of six studies—five laboratory experiments andone field experiment—systematically test model predic-tions. Study 1 is a process study that shows that when twodimensions are available to make an area comparison judg-

ment, people choose one of those to be the primary di-mension, with the other being the secondary dimension.Furthermore, it shows that the choice of the primary di-mension is dependent on its relative salience that can becontextually manipulated via manner of visual presenta-tion.

Studies 2 and 3 show how the use of a diagonal versusthe side of a square (contextually determined) can affectwhether a square is perceived to be smaller or larger thana circle of the same area. Study 3 extends the investigationto the domain of the price people are willing to pay for‘‘pizzas’’ of different shapes, presented differently.

Study 4, a field study, demonstrates external validity byshowing that purchase quantities are greater when a circularpackage is expected to contain less than a rectangular pack-age of the same volume in a domain where consumptiongoal is constant (cream cheese with a bagel).

Studies 5 and 6 examine ways in which one can increasethe salience of the secondary dimension, in a size estimationtask, i.e., judging the rate of increase of area. While Study5 does so via contextual visual cues (incorporating lines thatdraw one’s attention to the underused dimension), Study 6does the same using semantic cues that direct attention toa single dimension (e.g., diameter) or the total area andcomparing these with a visual presentation of the figure.

Overall, results suggest that the manner in which infor-mation is presented affects the relative salience of dimen-sions used to judge areas, and can influence the price con-sumers are willing to pay. Underlining the external validityof these findings, container shape can significantly affectquantity purchased and overall sales. The paper highlightsbiases in area comparison judgments as a function of areashape and size. The model is parsimonious, demonstratesgood predictive ability, and explains seemingly contradic-tory results in the cognitive psychology literature. Implica-tions for pricing, product design, packaging, and retailingare suggested.(Consumer Behavior; Experiments; Judgment Biases; Package De-sign; Information Processing)

PIZZAS: � OR SQUARE? PSYCHOPHYSICAL BIASES IN AREA COMPARISONS

MARKETING SCIENCE/Vol. 20, No. 4, Fall 2001406

1. IntroductionConsumers make area comparison judgments aspart of the purchase decision process: which productto buy, which size pizza to order, or how much topay for it. There are a number of unanswered ques-tions regarding consumers’ area comparison judg-ments of potential interest to managers. How accu-rate are these judgments? Does the shape of aproduct or its package affect estimates, e.g., aresome shapes perceived to be systematically ‘‘bigger’’than other shapes? Is perceived amount directly re-lated to actual amount irrespective of actual size, orare larger package sizes systematically underesti-mated? Do contextual factors moderate the directionor intensity of these shape- or size-related biases?These questions have important marketing implica-tions for packaging, pricing, and consumer welfare.For example, do consumers’ size estimates have con-sequences for managerially relevant variables in-cluding purchase likelihood, purchase quantity, andpurchase price? If manufacturers do charge a higherprice for the same amount of the product, can con-sumer education assist in overcoming size and shapeestimation biases? This paper is a start in answeringsome of these questions.

This paper introduces area judgments as a field ofstudy to marketers. We draw on more than a centuryof research in cognitive psychology, apply it to a con-sumer domain, and in the process add to the litera-ture on spatial perceptions. Marketers would be spe-cifically interested in the implications of these biases.For example, do consumers’ size estimates translateinto purchase likelihood with package shapes thatseem larger more likely to be purchased? Do theyaffect purchase quantity? If consumers want an ‘‘ide-al’’ (or fixed) level of consumption, then consumersshould use/purchase a smaller quantity of packageshapes that seem larger (e.g., sugar sachets or creamwith tea or coffee, or cream cheese with bagels). How-ever, on the other hand, holding price constant, ifconsumers derive greater utility from greater con-sumption, then they should purchase a higher quan-tity for package shapes that seem larger. Understand-ing how consumers make size judgments is,therefore, key in identifying and exploring the impli-

cations of package shape on purchase likelihood andpurchase quantity.

Furthermore, marketers are also interested in themaximum price consumers are willing to pay—bothfor an individual item and for a range of productsavailable in different sizes. Pricing a product line ofdifferent sized items is a common managerial taskinvolving an appreciation of the volume discount ac-ceptable to consumers. To the extent consumers un-derestimate larger sizes of a package, the apparentvolume discount required by consumers may be anover statement of what they require (e.g., people maybe willing to pay more for a ‘‘large’’ pizza, albeit nottwice as much, if they appreciate that it is twice thesize of a small pizza compared to when they believeit is less than double).

We propose a psychophysical model of how con-sumers compare areas. The model makes predictionsregarding the extent to which the shape and size ofa product affect consumers’ judgments when com-paring areas. The model postulates over reliance onthe more salient of two dimensions in a two-dimen-sion judgment, which leads to predictable biases inarea comparisons, and predicts how contextual fac-tors affect which linear dimension is overutilized.The model derives from the cognitive process bywhich consumers perceive and integrate two-dimen-sional visual information. Our research extends thearea estimation literature by providing a single mod-el that parsimoniously explains a number of previ-ously unintegrated effects noted by cognitive psy-chologists over the last century. The model is alsoable to resolve a controversy as to which shape isperceived to be ‘‘bigger’’: the square or the circle(Anastasi 1936 versus Warren and Pinneau 1955) byincorporating contextual factors into model specifi-cations.

The paper is divided into four sections. Section 2reviews the literature. Section 3 develops the theoret-ical model. Section 4 describes the empirical resultsof model testing based on five laboratory experimentsand one field experiment, and Section 5 ends by ex-plicating managerial and theoretical contributions ofthis research and offering directions for future re-search.

https://www.researchgate.net/publication/247878198_INFLUENCE_OF_FORM_ON_JUDGMENT_OF_APPARENT_AREA?el=1_x_8&enrichId=rgreq-c0273b9f-d4a2-4717-89e8-43b373a6cd05&enrichSource=Y292ZXJQYWdlOzIyNzQ0MjI2MTtBUzoxNjQ1NTQ3NTU1NTEyMzJAMTQxNjI0NDMxOTAxNw==

KRIDER, RAGHUBIR, AND KRISHNAPizzas: � or Square? Psychophysical Biases in Area Comparisons

MARKETING SCIENCE/Vol. 20, No. 4, Fall 2001 407

2. Shape and Size Biases in AreaPerceptions

More than a century of research in spatial perceptionsdocuments that size perceptions are a function of theshape and the actual size of objects. While there isagreement that people’s estimates of area are system-atically biased, there is no consensus as to why thesebiases occur. Furthermore, there are inconsistent di-rectional findings as to which shapes are perceivedto be bigger. In this section, we summarize biases inarea comparison judgments based on the shape andthe size of a two-dimensional object. We follow thisup with a summary of theories proposed to explainthese biases.

2.1. Shape EffectsThe effect of the shape of an object on estimates ofits size has been studied extensively in cognitive anddevelopmental psychology. Regular shapes (e.g.,squares, rectangles, pyramids, circles, etc.) are of par-ticular relevance to marketers of frequently purchasedconsumer products interested in packaging and pric-ing issues. Findings pertinent to regular shapes in-clude:

2.1.1. Triangle Comparisons. Triangles have beengenerally found to be perceived to be larger than cir-cles and squares (Anastasi 1936, Fisher and Foster1968, Hanes 1950, Pfeiffer 1932, Warren and Pinneau1955 but also see Smets 1970 and Wagner 1931 forexceptions),

2.1.2. Circle-Square Comparisons. There are in-consistent findings regarding the relative size percep-tions of circles vs. squares. While studies in the 1930sdocumented that a square was perceived to be largerthan a circle of the same area (Anastasi 1936, Wagner1931, Pfeiffer 1932), later studies have documentedthe reverse (Fisher and Foster 1968, Hanes 1950,Mansvelt 1928, Smets 1970), and others yet havefound no difference (Croxton and Stein 1932, Warrenand Pinneau 1955).

2.1.3. Elongation Effect. Anderson and Cuneo(1978), Holmberg and Holmberg (1969) and Vergeand Bogartz (1978) document that the more elongated

a figure, the larger it is perceived to be (but see Holm-berg and Wahlin 1969 for null results).

2.2. Size EffectsThe main findings from prior research that has ex-amined size perceptions of figures of the same shapeacross different sizes are that:

1. Size increases are underestimated—perceivedsize increases at a lower rate than actual size (Bairdet al. 1970, Ekman 1958, Moyer et al. 1978, Stevensand Guirao 1963, Teghtsoonian 1965).

2. The degree of underestimation increases as theobject grows larger. This empirical bias has beenmodeled as Perceived size � Actual sizee, e � 1, byTeghtsoonian (1965) among others.

3. The degree of underestimation has not beenfound to be contingent on the shape of the figure,with the value of e around 0.8 across a number oftwo-dimensional figures (e.g., Circles: Ekman 1958 �0.86; Squares: Stevens and Guirao 1963 � 0.70, Acrossshapes: Baird et al. 1970 � 0.80).

2.3. Theories Proposed to Explain Shape andSize Biases

The theories that have been developed to explain theabove biases are broadly categorized into those thatpropose that the bias is due to (i) Information Selection,people using incorrect information (e.g., ignoring orunderestimating one of the dimensions)—see Demboand Hanfmann (1933), Teghtsoonian (1965), Vergeand Bogartz (1978)—or (ii) Information Integration, anincorrect combinatorial rule to integrate available in-formation (e.g., adding dimensions rather than mul-tiplying them)—see Anastasi (1936), Anderson andCuneo (1978), Smets (1970), Martinez and Dawson(1973), and Warren and Pinneau (1955).

The information selection models, although useful,are unable to explain why in certain situationssquares are perceived to be larger than circles where-as in others the reverse holds. Furthermore, the mod-els appear atheoretic with no behavioral theory be-hind why one source of data is used at the expenseof another. No single information integration or in-formation selection theory appears to satisfactorilyexplain both the shape and size biases reviewed ear-



lier. To be useful such a theory must be based on abehavioral process, be parsimonious but still havehigh explanatory power, and be able to incorporatevariables used in these models but at a deeper ex-planatory construct level. In the next section we for-mulate such a psychophysical model of area judg-ments.

3. Theoretical Framework3.1. Psychophysical Model of Area JudgmentWe propose a parsimonious psychophysical model ofarea comparisons involving an information integra-tion process that incorporates information selection.The model (i) rests on strong behavioral foundationsin terms of the process people go through while mak-ing judgments, (ii) predicts results consistent withmajor previous findings on area judgments, (iii) ex-plains contradictory shape effects, and (iv) predicts anew bias in area comparison judgments.

3.1.1. Area Judgments. The basic behavioral foun-dation that the model rests on is that consumers sim-plify a difficult cognitive task by utilizing simplifyingheuristics, such as using a single piece of informationas a proxy for a more complex analysis (e.g., Einhornand Hogarth 1981). In fact, there is strong evidencefrom the literature on judgment and decision makingthat in many complex judgment situations, peopleuse the simplifying process of anchoring on a singlepiece of information and insufficiently adjusting forremaining information (for a review see Kahneman1992). Analogously, the prominence effect shows thatpeople tend to overweight information that they be-lieve is most important in a task (for a review seeTversky et al. 1988). Recent work on this prominenceeffect has shown that it may be task dependent, withthe prominent attribute receiving more weight whenthe experimental task is to differentiate (versusequate) options (Fischer et al. 1999), as is the case inan area comparison procedure where the output is toidentify which figure is smaller or larger.

We propose that the processing strategy of areacomparisons involves an overreliance on the promi-nent information source—the primary linear dimen-

sion—at the cost of alternate information sources, thesecondary linear dimension. This is because of theperceptual salience of one of the dimensions in a two-dimensional task. Perceptual salience effects havebeen shown in the context of spatial judgments ofdistance where the direct distance between end-points of a map was overutilized as a source of in-formation to make judgments about the distance ofthe route (Raghubir and Krishna 1996) and volume,where the elongation of a container systematically af-fected perceptions of volume of that container (Ra-ghubir and Krishna 1999). Furthermore, research hasshown that the difference in salience of the dimen-sions enhances the use of the more salient dimension,which is then used as the primary dimension, en-couraging the disregard of other secondary dimen-sions (Krishna and Raghubir 1997, Lauer 1929, Sea-shore and Williams 1902).

This suggests a model of area comparisons wherethe primary comparison between figures is made onthe basis of a single comparable dimension chosendue to its perceptual salience (e.g., width), with anadjustment of the results of this initial comparisonbased on a secondary comparison using the remain-ing orthogonal dimension (e.g., length). The use ofthe secondary comparison in the judgment is inade-quate, as would be argued by both traditional anchor-adjustment theories as well as the prominence liter-ature.

Our model of area comparison is based on the fol-lowing monadic area judgment:

Perceived Area

�� Primary dimension∗Secondary dimension . (1)

We do not claim that monadic judgments of areawill be made per Equation (1) but that relative areajudgments (area comparisons) will be consistent withthis monadic judgment. We now develop a model forarea comparisons. A two-dimensional figure has twodefining dimensions, which are typically orthogonal.These are often labeled ‘‘width’’ and ‘‘height’’ andmay be analogous to the sides of a square, to the di-agonals of a square, to the base and altitude of a tri-angle, or to two orthogonal diameters of a circle. Con-sider two two-dimensional figures, F1 and F2, which



Figure 1 Example of the Most Salient Dimension (Used for the InitialComparision)

Figure 2 Example of a Contextual Cue (the Arrangement of Shapes toBe Compared) for the Most Salient Dimension

Figure 3 Examples of Graphical Highlighting Used to Draw Attentionto Specific Dimensions

are to be compared, each with a most salient dimen-sion of length Li (i � 1, 2) and a secondary dimension,less salient and orthogonal, of width wi. That is, Li

and wi are defined by [salience of Li] � [salience ofwi]. An initial judgment as to which figure is largeris made by comparing the length of the most salientdimensions. Define the initial comparison based onthe most salient dimensions as I1,2, � L1/L2. Here I1,2

anchors the comparison of F1 and F2, with F1 initiallyjudged as larger than F2 if I1,2 � 1.

This comparison is then adjusted by a secondarycomparison of the less salient dimensions. Define thesecondary comparison as A1,2 � w1/w2.

Let the final criterion ratio, R1,2 be defined as1

�R � I ∗(A )1,2 1,2 1,2

�� (L /L )∗(w /w ) , 0 � � � 1, (2)1 2 1 2

where � � 0 implies that the initial comparison isadjusted, and � � 1 implies that the adjustment isinsufficient. The area of F1 is finally judged largerthan the area of F2 if R1,2 is greater than one.

The model is consistent with the underlying be-havioral process, which can be summarized as:

PROPOSITION P1. Consumers make an initial comparisonbetween two figures based on a single dimension.

PROPOSITION P2. The dimension of initial comparison—the primary dimension—is the one that is most salient toconsumers, where salience is figure and context dependent.

‘‘Salience’’ refers to the properties of a dimensionthat cause a consumer to pay more or less attentionto it. These may be properties of the figure itself. Ina comparison between a square and highly elongatedrectangle of the same area, for example, the longestdimension of the rectangle is more ‘‘attention-get-ting’’ than the shorter dimension and will, therefore,be the dimension chosen for the initial comparison

1The criterion ratio could have been operationalized as either RCS �

ICS·�·(ACS) or RCS � ICS·(ACS � �). We chose the power functionformulation to be consistent with prior literature. Note also that thecriterion ratio, R1,2 � (L1/L2)∗(w1/w2)�, can be derived through acomparison based on two monadic tasks with A1 � L W and A2

�*1 1

� L W .�*2 2

with a side of the square. Hence, the ‘‘elongation ef-fect’’ (see §2.1)—because the long dimension of therectangle is longer than the side of the square of thesame area, the rectangle is judged larger (Figure 1).Contextual cues, for example, the direction in whichtwo or more figures to be compared are lined up,may also determine relative salience of two dimen-sions. This would be particularly likely in the casewhere the figures are symmetric and there is no‘‘long dimension’’ that stands out. In the example(Figure 2), the dimension perpendicular to the row offigures is the most obvious one on which to makeinitial comparisons. This particular contextual ma-nipulation is used in several of the studies that follow.Graphical highlighting can also draw attention tospecific dimensions and encourage their use. Exam-ples include, but are not limited to, individual lines,arrows, or sets of lines, either inside or outside of thefigures to be compared (Figure 3).

The third and fourth propositions relate to a sec-



ondary comparison based on the remaining infor-mation—the secondary dimension.

PROPOSITION P3. Consumers insufficiently adjust an ini-tial comparison based on a second comparison using thesecondary dimensions, which we assume to be orthogonalto the primary dimension used for the initial comparison.

Proposition P3 states that consumers know thatcomparing areas strictly on the basis of one dimen-sion is not correct, and they therefore adjust this judg-ment. The adjustment, however, is not sufficient toreach the normatively correct comparison because ofthe overutilization of the prominent primary dimen-sion. The model captures the insufficient adjustmentwith the exponent, � � 1.2 The first three proposi-tions provide a parsimonious unifying basis for awide range of area comparison biases reported in theliterature. These are discussed below.

We offer a fourth proposition that states that the de-gree of over- and underutilization of the primary andsecondary dimensions, respectively, will be endoge-nously determined by their relative salience. This fol-lows from the argument that a dimension is chosen tobe the primary or secondary dimension by virtue ofits perceptual salience. In this sense, any factor thatwill increase the salience of the second dimension (vi-sual cues, semantic cues, priming cues, etc.) will in-crease the likelihood of a complete adjustment. Spe-cifically:

PROPOSITION P4. The magnitude by which the initial com-parison is adjusted is directly related to the relative salienceof the secondary dimension to the primary dimension.

In terms of our model, this implies that � is amonotonically increasing function of the relative sa-lience of the secondary dimension compared to theprimary dimension.

In summary, the choice of the primary dimensionis a categorical outcome of the determination of the‘‘most salient’’ dimension. The degree of adjustmentbased on the secondary dimension is a continuousoutcome, based on the relative salience of the two de-

2The power function formulation is a mere operationalization ofProposition 3, rather than a principle in itself. We thank a reviewerfor pointing this out.

fining (nominally orthogonal) dimensions of a two-dimensional object. The salience of a dimension de-pends on characteristics of the figure and can beaffected by contextual factors.

3.2. Consistency of Model with Prior EmpiricalFindings

This model has many desirable properties. Using ex-amples, we illustrate the intuition behind the model,and how it can explain the earlier reviewed (i) Sizeand (ii) Shape effects.

3.2.1. Size Effects. When comparing many figuresof the same shape, e.g., squares, rectangles, circles, etc.,setting � � 1 (i.e., adequate adjustment based on thesecondary dimension) gives a normatively correct per-ceived ratio, R1,2. Thus, when comparing squares ofsides 2 cm and 4 cm, the computed R1,2 � (4/2)(4/2)� 4 equals the true area ratio. On the other hand, if �

� 1, Rlarger-smaller � (4/2)∗(4/2)� � 21�� 22; the largersquare is underestimated relative to the smallersquare.

This is the well-known ‘‘size effect’’ (Teghtsoonian1965), where the increase in size of similar objects isunderestimated. Our model suggests that the relativesalience of the primary and secondary dimensionsdetermines perceived area ratio and that these ratiosdo not depend on shape. This implies that perceptionof increased area does not depend on whethersquares or circles are being compared and is consis-tent with prior empirical findings. Equation (2) alsopredicts that the degree of underestimation of larger sizesof the same shape is a function of the relative difference insize of the object. This is a consequence of the powerlaw formulation with � � 1.

The extent of underestimation, UCvs.S � (LC/LS)∗(wC/wS) � (LC/LS)∗(wC/wS)�, which is �0, iff �

� 1. Thus, for a given �, a small square of side n,and a larger square of side n � m, the underestima-tion of area increases with m. That is, as (n � m)2 �

(n � m)1�� increases more than proportionately withm. The larger the square, the more it will be under-estimated. A simple numerical illustration amplifiesthis for squares with � � 0.8 and n � 2, 4, 8:



AlphaActualArea

PerceivedArea U

Underestimationas % Normative

Area (%)

Side � 2Side � 4Side � 8

0.80.80.8

41664

3.4812.1342.22

0.523.87

21.78

12.944924.214234.0246

Thus, the formulation presented predicts (i) thecorrect formula when there is complete adjustment;(ii) that larger sizes of the same shape are underes-timated; (iii) that size effects are not contingent onshape (provided there is no explicit manipulation ofthe relative salience of primary and secondary di-mensions); and (iv) that underestimation increases asthe size of the object increases.

3.2.2. Shape Effects. Consider first the elongation ef-fect. Compare a square of side 4 cm with a rectangleof 2 � 8 cm, both of area 16 cm2. In our formulation,taking the most salient dimension as the longest di-mension, the above formula gives: RRect-Sq � (8/4)∗(2/4)�. If � � 1 (no bias), RRect-Sq � 1. However, if � � 1,RRect-Sq � 1, or, the rectangle will be judged largerthan the square: the ‘‘elongation effect.’’

Consider next across-shape judgments of triangles,circles and squares of the same true area, where priorliterature has shown that there is no clear agreementon whether squares appear smaller or bigger thancircles, although triangles have been found to appearbigger than both. The model formulation is able todemonstrate how each of these findings is explain-able.

Equilateral Triangle vs. Circle or Square: The side andthe height of an equilateral triangle are longer thanthe side or diagonal of a square or the diameter of acircle of the same area. Thus, our model predicts thattriangles will be judged larger than circles andsquares, as has been documented (Anastasi 1936,Fisher and Foster 1968, Hanes 1950, Pfeiffer 1932,Warren and Pinneau 1955).

Square vs. Circle: An interesting area comparison isbetween circles and squares. This is because while thecircle has an unambiguous linear dimension (its di-ameter) the square can be defined either in terms ofits sides, or in terms of its diagonals. If consumers’ es-timates of area are based on the length of a single

dimension, then the outcome of the initial compari-son (IC-S, Equation (1)) between a circle and a squareof the same true area depends on whether the diagonalor side of the square is used as the primary dimen-sion. For a square and circle of the same area, theratio of the side-of-square to the circle diameter is0.88, while the ratio of the diagonal of square to thecircle diameter is 1.25. Thus, if the diagonal is mostsalient, RSq-Cir � (1.25)∗(1.25)�, which is �1 for all �� 0, implying squares will be judged as larger thancircles in this case. On the other hand, if the side ofthe square is most salient, RSq-Cir � (0.88)∗(0.88)�,which is �1 for all � � 0, implying that circles willbe judged as larger than squares in this case. Notethat RSq-Cir is greater when diagonals are salient, im-plying different effect sizes for circle-square compar-isons. (In other words, it will be easier to find circle-square differences when diagonals are salient and thesquare is judged larger than the reverse, as 1.25 is25% greater than 1, whereas 0.88 is only 12% lessthan 1. We test this proposed resolution of reportedcircle-square inconsistencies below.

In conclusion, the model meets our first criterion ofunifying a number of previously unintegrated andcontradictory biases reported in the literature.

4. Empirical Tests of the ModelA set of six studies—five laboratory experiments andone field experiment—systematically tests model pre-dictions.

Subjects for all laboratory experiments were drawnfrom undergraduate subject pools at three differentuniversities in three different countries. No subjectparticipated in more than one study. The pools weredrawn from students of introductory marketing clas-ses who participated in the experiments for partialcourse credit.

In Section 4.1, we present a process study that sug-gests that a primary and a secondary dimension areused in comparing area, and that relative salience canbe manipulated to influence the choice of a primarydimension. In Section 4.2 we present two laboratorystudies to challenge Propositions P1 and P2, and atthe same time demonstrate the validity of our pro-



Figure 4 Example of Stimuli Used for Study 1

Note. The circle and square have equal areas.

posed resolution of the circle-square controversy (see§2.1). A field study (Study 4) demonstrates externalvalidity. The last two studies in this section also ex-amine economic implications of the bias. Section 4.3then focuses on the use of the secondary mechanism,as outlined in Propositions P3 and P4, and on asso-ciated economic implications.

4.1. Study 1: Process-Based Evidence forPropositions P1–P4

This study explicitly examines the underlying processused to make area comparisons, using squares andcircles as stimuli. Because squares and circles are 90rotationally symmetric, there are no elongation cues(i.e., long dimensions) to base the initial comparisonon. Comparisons are, therefore, likely to be based oncontextual cues such as arrangement. We used con-textual graphics by drawing double-headed arrowseither horizontally or diagonally (relative to the sidesof the paper) in both the square and the circle in anattempt to increase the salience of the associated di-mension. Figure 4 shows an example for a square andcircle of equal area.

Method. Each of 33 subjects saw two pairs of cir-cles and squares of equal area (42.25 cm2) on lettersize (8½ � 11) paper, with the circle beside the squareand one circle-square pair above the other. The fourfigures are thus arranged in a rectangular patterncongruent with the sides of the paper. For half thesubjects the squares were on their base, and for theother half on their corner (with left-right circle-squareorder counterbalanced between subjects). We refer tothe corner orientation as a ‘‘kite.’’ One circle-square(or circle-kite) pair had a horizontal double-headedarrow through the center of the pair of figures. Theother pair had a diagonal double-headed arrow

through the figures.3 The pair presented on top wascounterbalanced between subjects. Subjects (n � 33)were told to compare the circles to the squares. Sub-jects were then asked to allocate 100 points betweenthe two dimensions (side or corner-to-corner diago-nal) to reflect the extent to which each dimensionplayed a role in their size comparison. They couldalso indicate if they had used any other method forthe area comparison.

Subjects were then asked how they thought theymade their size comparison. They were given eightchoices.4 Four of these were based on the comparisonof a single dimension with adjustments (e.g., ‘‘I ba-sically compared the side of the square with the di-ameter of the circle but made some adjustments forthe difference in shape’’). Four were based on com-paring squared dimensions (e.g., ‘‘I compared the(side of the square)2 with the (radius of the circle)2

but made some adjustments for the difference inshape’’). Among the eight given choices, four werebased on diagonals of the square and four on the sideof the square. They also had a ninth choice for anyother method they may have used. The order of pre-sentation of the eight choices was also counterbal-anced between subjects.

Results. Vertical order of presentation of the choicefigures had no effect on the allocation of points be-tween the use of the horizontal and diagonal dimen-sion and is not discussed further (p � 0.80). Whenthe square was presented on its side, the mean pointallocation was 61.76 (versus 38.24) for the side (versuscorner-to-corner diagonal) dimension, whereas it was

3To test whether consumers use the side or diagonal dimension ofthe square in making area comparisons, we need to use the hori-zontal (or vertical) and diagonal dimensions. Taking all three hor-izontal, vertical and diagonal dimensions would bias the results infavor of the side (vertical/horizontal) dimension merely by allowingtwo options for side versus one option for diagonal.4The eight choices were constructed through a pre-test (n � 32),where four choices, plus an ‘‘other’’ choice were provided. The eightchoices were constructed from the answers subjects gave in thispretest, taking into account balancing of the number of linear/square options and side/diameter options to minimize bias result-ing from inferences as to what the appropriate response was basedon the number of options available on any given type of process.



37.50 when the square was presented on its vertexlike a kite (F(1,31) � 5.38, p � 0.05, 2 � 0.148).

Furthermore, 21 of 33 subjects believed they hadused one of the four single-dimension-with-adjust-ment strategies. Eight of 33 subjects felt they hadused squared-dimensions-with-adjustment strategies(binomial p � 0.01). The remaining four subjects useda visual strategy of trying the place the circle (square)inside the square (circle).

Results strongly indicate that people commonlyuse one of the dimensions of a two-dimensional fig-ure as a primary comparison point, and then subse-quent adjust this to incorporate the second dimen-sion. Further, results show that the choice of theprimary dimension can be influenced by contextuallymanipulating dimension salience.5 In the remainingstudies we investigate detailed predictions of Propo-sitions 1–4.

4.2. Propositions P1 and P2: Initial ComparisonBased on the Most Salient Dimension

In this section we present three studies. The first two(Studies 2 and 3) are laboratory studies that providesupport for Propositions P1 and P2 and simulta-neously demonstrate the validity of our proposed res-olution of the circle-square controversy. A field studyprovides external validity. The second and third stud-ies also examine economic implications of the bias.

4.2.1. Study 2. The Primary Use of the Salient Di-mension for the Initial Comparison. FollowingStudy 1, we manipulate the salience of alternate lin-ear dimensions of the square—the diagonal or theside—presented alongside a circle of equal area. Therelative salience of the side versus the diagonal of asquare should influence which dimension is used for

5We also conducted a second study, not reported in detail here,which indirectly supports the hypothesized single-dimension an-choring mechanism. When asked to draw squares and circles of thesame given area, subjects produced squares and circles that hadsides and diameters closer in length than the true sides and di-ameters would be for figures of equal area (the drawn circles hadsmaller area than the drawn square). In a control condition, subjectswere also given the true side and diameter dimensions associatedwith the equal-area figures and were then able to reproduce thefigures much more accurately.

the initial comparison and, hence, whether the circleor square is judged larger. Thus, it follows:

HYPOTHESIS 1. A square where the diagonal (versus theside) is salient is more likely to be judged as larger than acircle of equivalent area.

Method.(a) Design. The design manipulated the salience of

the diagonal versus the side of the square similar toStudy 1. In the diagonal salient condition, the squareswere presented on their corners (like a kite). In theside salient condition, the squares were presented ontheir sides.

(b) Procedure and Measures. In each condition theexperimenter gave the subject a white cutout circleas a standard and explained that the task was toplace the circle beside one of six white squares thatwas closest in area to the standard circle. The sixsquares (or kites) of different sizes were alignedvertically on a gray background. The six squareswere generated by enlarging or reducing a squareof area equal to the standard circle, by factors of0.90, 0.94, 0.98, 1.02, 1.06, and 1.10. The subject wasallowed to move the circle around but could notplace the circle on top of the squares. Matching acircle with a larger square indicates that circles areperceived to be larger than squares of the samearea, whereas matching it with a smaller square in-dicates that squares are perceived to be larger thancircles of the same area. Note that this is a strongtest of the prominence effect because more impor-tant attributes have been shown to have a greaterinfluence in choices than in matching judgments(Hawkins 1994).

Subjects (n � 38) were told that the experiment wasdesigned to study judgments under time pressureand that the task was to be completed in 10 seconds.There was a trial session prior to the experimentaltask to familiarize subjects with the unfamiliar task.After completing the task, the subjects filled out aquestionnaire that included gender and handedness,knowledge of the formula for the area of a circle, andan open-ended question on what they believed thepurpose of the study was.



Results. We expected that when the diagonal wassalient (kite orientation) the circle would be (i) judgedsmaller than the kite of the same area and (ii)matched with a smaller square and that the reversewould happen when the side of the square was sa-lient.

(a) Choice. As predicted, in the side-salient condi-tion, 74% of the subjects matched the circle with alarger square, indicating that the area of the circle wasperceived to be larger than the area of the square. Inthe diagonal-salient condition, 63% matched the circlewith a smaller square, indicating that in this condi-tion the square was perceived to be larger than thecircle (Overall � � 5.2, p � 0.05).2

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(b) Mean Subjective Equivalence (MSE). As a mea-sure of the magnitude of the bias, for each conditionwe calculated the mean (across subjects) size of thesquares that were chosen as equivalent to the circle—i.e., its MSE (cf. Fisher and Foster 1968). In the side-salient condition the MSE was 1.057 (i.e., linear mag-nification � 1.028 � and area � 1.057 � standardcircle area), which is significantly higher than theMSE of 0.964 in the diagonal-salient condition (t36 �

2.25, p � 0.05, 2 � 0.123). Area comparisons werebiased by approximately 5% in either direction, de-pending on whether the side or diagonal of thesquare was salient.

Discussion. We found that relative judgments of acircle versus a square were contingent on the context-influenced salience of the different linear dimensionsof the figures—in this case, the side or the diagonalof the square. Given a square and a circle of equalarea, the square was judged larger when its diagonalwas salient and smaller when its side was salient. Hy-pothesis 1 is supported. These results are consistentwith the first two parts of the proposition—that peo-ple base area comparison judgments on a single di-mension, and the choice of the dimension is a func-tion of its salience.

In the next study we examine the robustness of theeffect using a different experimental method with asimpler judgment and examine whether the biastranslates into the price consumers are willing to pay.

4.2.2. Study 3: Salience-Contingent Choice of Pri-mary Dimensions and Reservation Price. Firenze’s,6

an Italian restaurant, caters to the ‘‘eat-in’’ pizza seg-ment. It needed to decide whether to make roundpizzas or square pizzas. Preliminary cost calculationshad shown that the two shapes did not significantlyvary on overall cost. Firenze’s wondered whether cus-tomers would care.

Consumers’ reservation price should be directly re-lated to perceived size, or:

HYPOTHESIS 2. Consumers will be willing to pay less(more) for a circular package versus a square one when thesquare package is presented on its corner (side).

Method. The product chosen was pizza, a commontwo-dimensional consumer product. The design wasa 2 (shape: circle/square) � 2 (salience of diagonal:low/high) mixed design, with the first factor admin-istered within subjects and the second between sub-jects. The salience of the diagonal of the square wasmanipulated as in section 4.2.1. Each subject saw botha circular and a square ‘‘pizza’’ presented side byside. Both figures were of the same area (256 cm2,side of square � 16 cm � circle diameter � 18.1 cm� diagonal of the square � 22.6 cm). Note the meth-odological difference between Studies 2 and 3—in-stead of matching a circle to a series of squares ofdifferent areas, in this study subjects are shown fig-ures of the same area.

Subjects (n � 90) were told that there was a newpizzeria opening on campus and asked to (i) judgewhich figure was smaller and (ii) estimate the maxi-mum price they were willing to pay, i.e., their reser-vation price (RP) for the two ‘‘pizzas.’’

Results.(a) Choice. As in Study 2, when the diagonal of the

square was salient, the majority (71.8%) judged thesquare larger than the circle, whereas when the sideof the square was salient, a larger proportion of sub-jects (56%) estimated the circle as larger (� � 6.88,2

(1)

p � 0.01). Thus, Hypothesis 1 is again supported.(b) Reservation Price. The format and presentation

6Examples are based on actual situations. Names may have beendisguised to maintain anonymity.



of the square pizza also affected the maximum pricea subject was willing to pay (Reservation Price, RP).A within-subjects analysis of variance (ANOVA) onRP for the circular and square pizza across the twosalience conditions revealed a significant shape by sa-lience interaction (F(1,88) � 4.20, p � 0.05; 2 � 0.05):When the diagonal was salient, the maximum pricesubjects were willing to pay was significantly lowerfor circles vs. squares (Means � HK$43.39 vs. HK$46.56; Ratio of Means � 1.07; contrast F(1,88) � 6.13,p � 0.05; 2 � 0.065). However, shape exerted no ef-fect when the diagonal was not salient (Means �$40.25 vs. $39.86 for circle and square, respectively,ratio of means � 0.99; contrast F � 1). This is notsurprising because of the small effect size in thisstudy (half of that in Study 1). In sum, Hypothesis 2as partially supported—size estimates translate intothe reservation price of consumers, but the effect isstronger when the diagonal is salient.

Discussion. This study provides further supportfor Propositions P1 and P2, that while making anarea comparison judgment, people overly reply onthe more prominent dimension, with the prominenceof the dimension determined contextually in part.Additionally, the study shows that this can affect theprice they are willing to pay for a product. Thus, ourstudy suggests that Firenze’s should make square piz-zas and display them in a diamond pattern.

While the shape biases reported to this point areconsistent and robust, all have been found in the con-text of controlled laboratory experiments. We con-ducted a field study to address whether the biaseshave an economic impact in a real-world setting.

4.2.3. Study 4: External Validity for PropositionsP1 and P2. This study examines whether containershape affects perceived size and, in turn, affects pur-chase quantity.

Stimuli. A popular cafeteria of an undergraduatecollege on the East Coast carries only one type of(plain) cream cheese every day. There are two typesof cream cheese tubs—rectangular and round. Bothcontain ¾ oz. of plain cream cheese. The rectangulartub has a smaller surface area but is taller comparedto the round tub. Both tubs are priced at 25 cents, but

prices are not marked on the tubs. Consumers see onlythe total price of ‘‘bagel and cream cheese’’ on the cashregister. In the first part of the study, the pretest, weexamine whether one container is perceived to besmaller or larger than the other. In the follow-up mainstudy, we examine whether purchase quantity per ba-gel purchased is affected by container shape.

Pretest. Two cream cheese containers were coveredwith white paper. Cafeteria patrons were used assubjects (n � 19). They were asked which of the twocontainers was bigger. Two subjects correctly esti-mated that the two containers were the same size.Thirteen judged the rectangular container as biggerthan the round one, significantly greater than the foursubjects who judged the reverse (76.47% versus23.53%, p � 0.05). This is consistent with our expec-tations that the longest dimension of a rectangle isthe most salient and is used as the primary compar-ison with the circle’s diameter.

According to the cafeteria manager, most consum-ers buy either one or two containers of cream cheese.If a round container is perceived to be smaller thana rectangular container of cream cheese, then morecontainers of the round container should be pur-chased if a fixed amount of cream cheese is desired.That is, the purchase likelihood of buying multipleunits should be greater for the round versus the rect-angular container. The main field study examineswhether container shape affects purchase quantity.

Field Study. In the field study, the cafeteria carrieda ¾-oz. round tub of cream cheese on two days anda ¾-oz. rectangular tub on two other days. Cashierswere instructed to note the number of cream cheesetubs picked up by each person purchasing creamcheese on these 4 days from 8:00 a.m. (opening time)to 12 noon.

The results were in the expected direction. Whereas18 of 41 (43.90%) consumers purchased 2 tubs of theround cream cheese (others bought 1 tub), only 7 of34 (20.59%) bought 2 tubs of the rectangular creamcheese (� � 4.55, p � 0.05). This pattern is consistent2

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with the results of the previous study that showed thatthe round cream cheese container is perceived to besmaller than its rectangular counterpart. This pattern



Figure 5 Stimulus for Study 5: Five Circles of Increasing Area to ElicitArea Ratio Judgments

Figure 6 Contextual Graphics Added to Figure 5 to Increase the Sa-lience of the Secondary or Adjusting Dimension

also led to the mean quantity purchased being higherfor the round tub (mean � 1.44) versus the rectangulartub (mean � 1.21, F(1,73) � 4.71, p � 0.05).

This study provides external validity to our resultsin Studies 1–3 that support Propositions P1 and P2.We now turn to testing propositions P3 and P4.

4.3. Propositions P3 and P4: Use of SecondaryDimension

We have proposed that consumers will compare sec-ondary (less salient) dimensions to adjust the initialarea comparison; and that the exponent �, which liesbetween 0 (no adjustment) and 1 (full normative ad-justment), depends on the relative salience of the pri-mary and the secondary dimensions. In Study 5, weestimate � in the two ‘‘salience’’ conditions, usingsimple contextual graphics to manipulate salience ofthe adjusting dimension and to show support for thepropositions. In Study 6, we use a more realistic sa-lience manipulation to show that the salience-depen-dent adjustment can have a strong effect on consum-ers’ reservation prices.

4.3.1. Study 5: Salience Dependence of the Ex-ponent. The literature is consistent in reporting thesize effect—when figures of the same shape but dif-ferent sizes are compared, perceived size increases ata lower rate than actual size, with perceived to actualratios following a power law. (Croxton and Stein1932, Ekman 1958, Stevens and Guirao 1963, Teght-soonian 1965). Given, for example, five circles of in-creasing area, as in Figure 5, the judged area ratiosRL,S(judged) of any of the larger circles to the smallestcan be expressed in terms of the true area ratiosRL,S(true).

(��1)/2R � R , or (3)L,S( judged) L,S(true)

ln[R ] � (� � 1)/2 ln[R ]. (4)L,S( judged) L,S(true)

The five circles can be used to provide four sets of

true and judged ratios, from which � can be estimat-ed by regression. Noting that Li � wi for a circle andthus RL,S(true) � (LL/LS)2, this is consistent with ourequation (2), R1,2 � I1,2∗(A1,2)�, with L � 1, and S � 2,and our theory provides more insight into the un-derlying mechanism. Specifically, the arrangement ofthe circles in a row encourages the initial comparisonIL,S to be made on the dimension perpendicular to therow. The secondary AL,S is then based on the dimen-sion parallel to the row, and the magnitude of theadjusting exponent � depends on the salience of thatdimension relative to the primary dimension. Thisimplies that if we modify the figure to draw moreattention to the dimension parallel to the row, weshould increase �. Such an effect would providestrong support for our theory. A simple way to in-crease the salience of the secondary (underused) di-mension is to provide contextual cues that draw at-tention to the dimension, e.g., one or more linesparallel to it (see Figure 6).

HYPOTHESIS 3. The value of the exponent in the ‘‘sizeeffect’’ is greater when the context increases the salience ofa secondary dimension.

Method. Figures 5 and 6 were used as stimuli, withthe addition of ‘‘100%’’ written beneath the smallestcircle. Subjects (n � 84) were shown one of the twofigures and asked to give the area of the larger fig-ures relative to the smallest, with the smallest as-signed 100%.

Results. Equation (4) was estimated for each con-dition by OLS. In the low salience condition, � wasestimated as 0.62 (s.e. � 0.02, R2 � 0.89), and in thehigh salience condition, as 0.86 (s.e. � 0.02, R2 �0.90). This provides strong support for Hypothesis 3.

However, because each subject provided several



data points the independence assumption of OLS wasviolated, leading to a possible underestimation of var-iance. A somewhat ad hoc method to determinegroup parameters would be to estimate individualcoefficients for each subject and average them acrosssubjects to produce group coefficient means.7 A moreformal approach to estimating the mean rate ofchange in each group, used in the econometric anal-ysis of panel data, where there are typically manysubjects but few data points for each subject, is a ran-dom coefficient model (Hsiao 1992, Judge et al. 1985).The slope is treated as a random variables across sub-jects, and we estimate its mean and variance usingtwo-stage EGLS.8

With this model, the estimated mean � for the lowsalience condition was 0.602 (s.e. � 0.043) and for thehigh salience condition 0.832 (s.e. � 0.046). While theestimated errors have increased over the OLS errorsas expected, the Wald statistic for testing coefficientequality is W � 3.36 (p � 0.07), consistent with Hy-pothesis 3.

Discussion. A normatively correct judgment is in-dicated by � � 1. We observe the usual ‘‘size effect’’bias in area comparison judgment, with the magni-tude of the bias in the low-salience condition com-parable to that reported in the literature. When therelative salience of the secondary dimension is in-creased contextually, the bias is reduced, in accor-dance with Propositions P3 and P4.

In the final study, we investigate the implicationsof the salience-based adjustment for consumers’ res-ervation prices in a more complex managerial settingand challenge the theory further with another manip-ulation.9

7It is interesting to note that for individuals, the model fits verywell. Only 7 of 84 subjects had R2 values less than 0.97 when Equa-tion (4) was fit to each subject’s data individually. The interpretationis that the majority of the variance in the pooled OLS regression isa result of differences in the coefficients between subjects, ratherthan of the errors within subjects.8We also allow for heteroscedasticity across individuals, which apriori seems reasonable, and subsequent specification tests confirm.9We also replicated the effect of this study directionally in a match-ing task. Subjects were asked to match a square to the series ofcircles. In one condition the square was blank, and in another it wasfilled with parallel lines to increase the salience of the adjustmentdimension.

4.3.2. Study 6: Salience-based Adjustment—DoesDescription of Sizes Affect Prices? Pepperoni’s, aneighborhood pizzeria had to decide how to labeltheir ‘‘small,’’ ‘‘medium,’’ and ‘‘large’’ sizes on theirmenu, and how to price the different sizes. Shouldsizes be given as diameter dimension (e.g., 8 inches,10 inches) as per the prevailing practice? Should theydepart from convention and provide actual pizza ar-eas, or perhaps display sample pizzas? How wouldthe different presentations affect customers’ reserva-tion prices?

Pepperoni’s is prepared to encourage customers topurchase larger sizes by giving them a quantity dis-count. The implication of the established size effectbias for Pepperoni’s, however, is that customers willnot fully appreciate how much extra they are gettingin the medium and large pizzas compared to thesmall pizza and thus underestimate the value of anyquantity discount. A menu that states only the di-ameters is providing information on only one dimen-sion, cuing the sole use of that dimension to make atwo-dimensional judgment (size of the pizza). Be-cause there is nothing in the stimulus to draw atten-tion to a second dimension, the relative salience ofthe orthogonal secondary dimension is low—specifi-cally, lower than when an actual pizza is presented.This implies less adjustment, a larger size-effect bias,and a smaller perceived quantity discount for a givenprice.

Salience in this study is manipulated via the man-ner in which information on the three sizes of pizzais presented: using information on one linear dimen-sion (e.g., ‘‘8-inch diameter’’) or graphic information(a figure of diameter 8 inches displayed). Formally, �,which captures the salience and relative use of thesecond dimension, should be greater in the graphiccase than in the linear dimension case. PropositionsP3 and P4 imply

�linear � �graphic (5)

Given this pattern of �, it follows that:

HYPOTHESIS 4. The rate of increase of Reservation Pricewith pizza size when Graphic Information is provided willbe greater than when Linear Dimension Information is pro-vided.



In contrast to the previous study, where contextualgraphics were used to increase the salience of the ad-justing dimension and hence reduce the bias, in thisstudy we use numeric diameter information to reducethe salience of the adjusting dimension and predictthis will increase the bias. Because our dependentvariable is reservation price, and the relation betweenperceived area and reservation price may include anexpected quantity discount, we include a third con-dition where we present true area information (e.g.,‘‘50 square inches’’). Providing true area informationeliminates the judgment bias so that any expectedquantity discounts can be observed directly. In termsof the model, the salience of the secondary dimensionis effectively high—in fact, it is the same as the sa-lience of the primary dimension. Propositions P3 andP4 imply:

�graphic � �area � 1. (6)

Given this pattern of �, it follows that:

HYPOTHESIS 5. The rate of increase of Reservation Pricewith pizza size when Area Information is provided will begreater than when Graphic Information is provided.

Method. Subjects (n � 136) were informed thatthere was a new pizzeria that was opening on cam-pus. They were told that the pizzeria was decidingon its product offerings, and was interested in as-sessing consumers’ reactions, including the price theywere willing to pay for various pizzas.

We used a 2 (shape—circle, square) � 3 (Salienceof Adjusting Dimension—Low Linear Dimension In-formation, Medium Graphic Presentation, and HighArea Information) � 3 (Size: 50, 100, 150 sq. inches)mixed design with the first two factors administeredbetween subjects and the last within subjects. Thus,each subject made three price judgments: one each forthe small, medium, and large pizzas, with the orderof elicitation counterbalanced (ascending or descend-ing). In the linear dimension information condition,the diameter of the pizza was provided (8 inches,11.25 inches, and 13.75 inches). On the other hand, inthe high salience area information condition, the di-mensions were not provided, and only the area of the‘‘pizza’’ was given (50, 100, and 150 sq. inches). In the

graphic information condition, a white sheet of card-board represented each pizza, and no dimension orarea information was provided. Here, ‘‘Pizzas’’ wereplaced in a row, in order of size, in the center of arectangular table. Subjects, sitting all around the ta-ble, estimated the maximum price they were willingto pay for each of the three pizzas. There were a num-ber of other questions to increase the credibility ofthe cover story used and also questions on demo-graphics, motivation level, and knowledge of mea-sures. The task took approximately 20 minutes, afterwhich subjects were debriefed and dismissed.

Results. A majority of the subjects were aware ofthe correct formula for calculating the area of asquare (n � 125, or 92%) and circle (n � 114, or 84%).Knowing the correct formula did not affect price es-timates (all ps � 0.50). Handedness (n � 128, or 94%,reported being right-handed) and gender (females �111, or 82%) also did not affect price estimates. Re-ported motivation and interest levels were acceptable(means � 4.49 and 4.57, respectively, on a seven-pointscale).

(a) Impact of Pizza Description on Reservation Price forDifferent Sizes. Figures 7 and 8 depict the mean RPsof the six conditions.10 The graphic condition is con-sistently lower than the two numeric conditions, sug-gesting that numeric information generally biases res-ervation prices upwards, although the variable ofinterest—the rate of change of reservation price withsize in the graphic condition—is between the rate ofchange in the two numeric conditions. Interestingly,the mean reservation prices in each condition are verynearly linear with size and have positive intercepts.The pattern is as if subjects are expecting a quantitydiscount that is implemented as a fixed fee and a con-stant marginal price. Appendix A describes how weuse this observation and our psychophysical model toderive a relation between the true area and the res-ervation price, and how to use this relation to test thehypotheses.

10Note, however, that the different description conditions (numericlinear dimension, graphic, numeric area dimension) may introducesystematically different area judgments for reasons other than theprocess under investigation, and that the relation between perceivedarea and reservation price may further distort the effect.



Figure 7 Mean Reservation Prices for Small, Medium, and Large Cir-cular Pizzas

Note. The mean reservation price across the three information conditionssuggest that consumer price expectations can be captured with a linearmodel of true area.

Figure 8 Reservation Price Patterns for Square Pizzas Are Similar toThose of the Circular Pizzas Shown in Figure 7

Table 1 Results of Study 6

Info Condition

Mean Coefficients

Linear Numeric(e.g., Diameter 8�)

Intercept(s.e.)

Slope (b)(s.e.)

Graphic (Figure)

Intercept(s.e.)

Slope (b)(s.e.)

Areal Numeric(e.g., 50 sq. in.)

Intercept(s.e.)

Slope (b)(s.e.)

CirclesSquaresCircles � Squares

14.9 (2.41)17.9 (4.07)16.3 (2.21)

27.7 (2.33)22.1 (2.13)25.3 (1.63)

8.9 (2.08)3.73 (3.05)6.6 (1.81)

32.3 (3.03)28.4 (3.58)30.5 (2.31)

13.0 (2.56)14.3 (3.58)13.6 (2.16)

31.7 (3.92)47.1 (4.85)39.2 (3.26)

(b) Estimation. As in Study 5, we use two-stageEGLS in a random coefficient framework to estimatethe mean value of the coefficients given in Equation(4) across subjects, in each of the information andshape conditions, and for circles and squares togeth-

er. We then test for the ordering that the salience ofthe adjusting dimension should impose on the mar-ginal RP.

One question of interest is how well a linear modelcaptures the relation between reservation price andtrue area for individuals. While R2 cannot be inter-preted as a measure of variance explained in the ran-dom coefficient model, the question can be addressedby calculating a separate regression for each individ-ual, and then calculating a total R2 using the resultingresiduals. In each of the three information conditions,this gives an R2 value of 0.99. For all but four of the136 individuals in the study, R2 is greater than 0.96,indicating the sequence of reservation prices for mostindividuals is very nearly linear. The linearity, ofcourse, carries through to the means across individ-uals, as can be seen in Figures 7 and 8. We also es-timated the relation (17) including a quadratic term.For four of the six conditions the quadratic term wasnonsignificant (p � 0.5). For two of the conditions thequadratic term was significant at p � 0.05. The max-imum effect size of the significant quadratic termsover our data range was 12% of the effect of the linearterm in one case and only 8% in the second. Thus,we take the linear model with nonzero intercept asthe most appropriate structure for the relation be-tween true dimensions and perceived price, and de-rive how � should enter into the model, in AppendixB. Estimated coefficients and their standard errors(which indicate that all coefficients are strongly sig-nificant) are given in Table 1.

(c) Only Linear Dimensions. As argued, the slopesin the Linear Numeric and Graphic conditions (Table1) show that for both shapes, subjects were on aver-



age prepared to pay more for a ‘‘marginal squareinch’’ of pizza when they saw the actual pizza thanwhen they were given linear dimensions. A Wald testof the differences for circles or squares does not, how-ever, achieve significance. For the combined circle andsquare data, the mean slope in the numeric case (25.3)is significantly less than in graphic case (30.5). Theoverall Wald statistic testing for equal coefficients isW � 14.45 (distributed � ; see Green, p. 215), a2

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strong rejection. For the slope coefficient alone, W �3.33 (p � 0.08), supporting Hypothesis 4. The mag-nitude of the effect is striking: consumers are willingto pay 20% more per unit of pizza if they can see thesizes as compared to when they are given only thediameters on a menu.

The two conditions in which subjects were giventhe true area in square inches (eliminating the per-ceptual bias and hence attempting to tease out theexpected quantity discount) produced unusual re-sults (see Figures 7 and 8 and Table 1). When subjectswere told that the pizzas were circular, the marginalprice was the same as for the graphic condition, arejection of Hypothesis 5. However, when subjectswere told that the pizzas were square, both the res-ervation prices and the marginal prices were muchhigher, supporting Hypothesis 5.

Discussion. Not surprisingly, subjects expect aquantity discount as size increases. However, whenpresented with only the linear dimensions, as in mostpizza parlors, customers will apparently expect agreater quantity discount than when they can see theactual pizzas, as would be expected given Proposi-tions P3 and P4, and the relatively lower salience ofthe second dimension when only information on onedimension is presented.11

One point to note is that for all sizes, the largestreservation price is obtained when the area of the piz-

11When the true area was presented in square inches to get an in-dication of the expected quantity discount without perceptual bi-asing, reservation prices were much higher for squares. The anom-alous effects between circles and squares may be occurring becausewhen told the pizzas were square, some subjects interpreted ‘‘100sq. inches’’ as ‘‘100 inches square.’’ The support for Hypothesis 5is thus questionable, and we leave further investigation of thisanomaly to future research.

za is provided—a practice followed by very few, ifany, pizzerias. These results apply equally to baker-ies, which sell cake ‘‘sheets’’ by giving linear dimen-sion info or by showing sample sheets but seldom byproviding the actual area. Our results also suggestthat Pepperoni’s should present pizza information inarea terms and that they would get greater sales fromtwo small pizzas versus a larger one of the samearea.12

5. ConclusionsThis paper examined how people simplify an areajudgment task. We proposed that a single linear di-mension, chosen by virtue of its salience, would beoverweighted in an area comparison judgment. Fur-thermore, we argued that the extent of use of the sec-ondary dimension relative to the primary dimensiondepends on the relative salience of the secondary di-mension to the initially used primary dimension. Thisrelative salience was contextually manipulated: usingvisual cues that enhanced its being noticed (lineswithin and outside the figure), and semantic descrip-tors that reduced or increased its usage (e.g., diame-ter information versus area information). This simpleand parsimonious account of how people judge areaswas demonstrated across contexts, experimental pro-cedures, and tasks and was used to reconcile contra-dictory effects noted in the visual perception litera-ture.

Specifically, we developed a psychophysical modelof area estimation and tested it across five experi-ments and a field study. The model proposed thatconsumers simplify the complex area estimation andcomparison task by making an initial comparisonbased on the most salient dimension between two fig-ures and adjust this comparison by incorporating theless salient secondary dimension, with the degree ofadjustment dependent on the relative salience of thedimensions. This relative salience was manipulateddifferently using visual and verbal cues. Across stud-

12In another study (not reported), 33 of 43 subjects did prefer twosmall pizzas over one large one of the same area.



ies, we found support for all four propositions of themodel.

Study 1 used process measures to determine thatconsumers can commonly recognize their area com-parison strategy as overly relying on a primary di-mension and that they can report relative usage oflinear dimensions that are consistent with saliencemanipulations. Studies 2 and 3 used different meth-ods and measures to test Propositions P1 and P2.Next, in a field experiment (Study 4), we showed ex-ternal validity for Studies 1–3. Studies 5 and 6 testedPropositions P3 and P4 by manipulating relative sa-lience of the secondary dimension in different ways.

The main contributions of this article are to (i) de-velop a single model based on behavioral assump-tions that can explain size and shape effects; (ii) makea contribution to the cognitive psychology literature,by not only reconciling conflicting results (specificallythe square-circle controversy) but also predicting neweffects (e.g., that presentation format influences thesize-effect bias); (iii) showing that salience of dimen-sions can be manipulated through the shape of thefigure, and also through orientation and descriptionof the shape; (iv) show context (orientation of shape)related and information presentation (description ofshape) related influences on shape and size judgmentbiases; (v) demonstrate economic implications of thejudgment biases on two important managerial vari-ables: prices people are willing to pay, and purchasequantity; and (vi) add to the marketing literature thattests behavioral theories with experimental data us-ing mathematical models.

5.1. Managerial Relevance and Areas for FutureResearch

This research has implications for package design,pricing, communication, and shelf space. We showedthat the shape of a container affects estimated sizeand through this affects purchase quantity. This hasdirect implications for package design.

Specifically, perceived area implications may applyto the surface area of packages as seen on retailshelves. If consumers want to buy the ‘‘biggest’’ boxof detergent within a price range and do not perusethe unit price information (Dickson and Sawyer 1986),

then the shelf facing that is perceived to have the larg-est surface area will be chosen. Thus, tall rectangularboxes have two advantages over square ones of equalvolume: more fit on a shelf facing and greater per-ceived volume. The variation in shapes of cereal boxessuggests that this is not conventional wisdom in thepackaging industry.

The experiments showing the difference in percep-tions of area of a square as a function of the salienceof the diagonal versus side suggest that displays us-ing such shapes would benefit from an arrangementwhere the box is placed on its corner rather than onits side. We also have strong evidence for other man-agerially relevant effects of area perception. Contex-tual presentation can be used as a managerially ac-tionable tool. Clever graphics can lead to a perceptionof a larger package. If certain dimensions of a pack-age are larger than others (rectangular cartons), thesedimensions could be made salient with the use ofgraphics like double sided arrows or lines throughthe longest dimension. Or, their manner of presenta-tion in a store layout, advertisement, or Web layout,could be managed appropriately. For example, in apizzeria, where visual information about the sizes ofdifferent pizzas is presented, it is clear that simplyshowing the ‘‘larger’’ pizza is inadequate for consum-ers to appreciate the increase in size between a small,medium, and large pizza. Additional informationabout the size of the larger pizzas should allow piz-zerias to charge a higher price for their larger pizzas(i.e., offer a smaller volume discount).

Aside from deciding on what package shape andgraphics to have, it is also important how packageinformation is communicated to consumers. Notenough companies are focusing attention on this.Again, a simple example of this is fast-food pizza par-lors where some parlors give size dimensions in di-ameter, whereas others paste up a picture of the dif-ferent sized pizzas. Our research suggests thatconsumers understand the size increase more readilywhen presented with pictures of the various sizesversus when merely presented with the diameters.The best strategy, however, is to give the actual areanumerically where consumers are willing to paymore than under linear dimension or graphic descrip-



tions. This should be particularly relevant for thosebusinesses that rely on takeout or order-in usingmenus. Such pizzerias should not rely on providingmere diameter or side information about sizes, be-cause consumers are unable to apply these accuratelyto compute overall size.

Our results also suggest that offering a discount ontwo small pizzas may be perceived as more favorablethan a discount on one large pizza of the same totalarea, which may go some way to explaining the suc-cess of 2-for-1 pizza outlets. This may also suggestthat ‘‘2-for-1’’ or ‘‘buy 1, get 1’’ promotions may beparticularly effective, compared to economicallyequivalent discounts on larger packages.

However, we need to exercise caution before apply-ing these findings directly to the field. The tasks herewere all done in a within-subjects domain, and themodel proposed is a model of comparisons. It is un-clear whether the model will translate into contextswhere consumers do not make choices between dif-ferent container sizes/shapes, but decide on a buy/no buy decision (i.e., make a monadic purchase de-cision for a single product).

Future research could examine whether there aresystematic differences in the quantity that consumerspurchase as a function of the shape of the productpackaging: e.g., ‘‘Are there differences in purchasequantity of bottles versus cans versus tetra-packsused to market drinks?’’ One may expect that whenpeople are trying to limit consumption or when con-sumption level is expected to be fixed, then therewould be an inverse relationship between perceivedsize and purchase quantity, but when utility is mono-tonically related to amount consumed, then the rela-tionship would be direct. For example, in the productcategory of beer, bottles are more elongated thancans. Does this suggest that fewer bottles comparedto cans will be purchased per purchase occasion incertain situations and the reverse in other situations?

Furthermore, future research could examine the to-tal volume that is purchased when people buy manysmall-sized packages (e.g., single serving packs ofchips, cans of soda) versus single larger-sized pack-ages (party-sized bags or large soda bottles)? If largersizes are underestimated, this suggests that control-

ling for desired consumption level, a larger absolutequantity would be purchased when larger containersare purchased as compared to smaller containers.

Furthermore, the results in this paper have impli-cations for the amount people are willing to pay as afunction of the elongation of a package. Do moreelongated packages charge a higher price? Are na-tional brand names more likely to be in more elon-gated packages, compared to package shapes thatminimize package wastage? Furthermore, this raisesthe issue of how package shape and size affect like-lihood of purchase. Based on these studies, one wouldexpect that controlling for price, more elongatedshapes would be perceived as better value for money.

Another issue that is raised by our studies iswhether there are ancillary effects of package aes-thetics, i.e., are some shapes and ratios more pleasingto the eye than others? There is plenty of anecdotalevidence that the ratio of ( � 1.618) was used frompre-Pythagorean times and common during the timeof Euclid, with continuing use by Fibonacci (Herz-Fischler 1987). Its almost mystical properties havegained it the name of the ‘‘divine proportion,’’ or the‘‘Golden ratio,’’ and it is believed to have special es-thetic properties, with famous paintings and archi-tectural monuments (e.g., the Parthenon) based onsuch a ratio (Ghyka 1977). An interesting applicationof the perceived-size effects noted in this paperwould be to assess whether an elongation proportionof brought with it additional attitudinal benefits.As such, this would extend the perceptual distortionliterature into an affective domain.

Will the effects noted in the paper translate fromaerial views to frontal views? If they do, they mayhave implications for store layout and and be a start-ing point for those interested in examining routechoice. Future research could also examine the role ofindividual factors in moderating area perception bi-ases. Do area perception biases reduce for peoplewho have a high need for cognition? Or may theyincrease? How do individual differences in visualversus verbal ability affect the manner in which peo-ple use visual cues? Are those with a high visual im-agery ability less or more likely to be prone to theshape and size biases? If so, would the effects be



moderated by cross-cultural factors as countries differin the visual/semantic aspect of their script (e.g., Chi-nese versus English)? Further research on contextualfactors can also investigate the effect of ‘‘framing’’ onarea perception. Does a circle framed in a square looksmaller or larger than the other way round?

How do people make such judgments? Theoreti-cally, the limits to the effects noted in this paper needto be established. Can biases in perception of size bereduced through adequately motivating consumers?What about biases in perceptions of shape? Our workleads us to suspect that if there is a simple algorithmavailable for making area comparisons from linearjudgments, high motivation can remove the bias; oth-erwise, the bias remains. These are important issuesfrom the point of consumer welfare, as they informwhether these biases can be reduced through appro-priate incentives. Can consumer education help inthis regard, or must regulators insist on more prom-inent displays of quantity information on packages?

AcknowledgmentsThis research was partially funded by Grant DAG96/97.BM48, awarded to Robert Krider, and by GrantDAG94/95.BM56, awarded to Priya Raghubir, by theResearch Grants Commission, Hong Kong, throughthe Hong Kong University of Science and Technology.We express our thanks to Brian Ratchford, an Asso-ciate Editor, and ‘‘Reviewer A’’ for their constructivecriticism and encouragement through the review pro-cess. The authors also thank seminar participants atHKUST, the Berkeley Marketing Science Conference,the Hong Kong ACR, Willem Burgers, and CarolynYoon for helpful comments, as well as Iris Chow andCharmaine Leung for assistance in developing stim-uli.

Appendix 1

The reservation price is modeled as a linear function of perceivedarea, which in turn is a power function of true area. Because ourmeasures are of reservation price and true area, and our hypothesisis on the rate of increase of reservation price with true area, we takea first order approximation of the combined function, which is bothconvenient and adequate for our purposes. First, we derive the re-lation between the true area and the estimated area from our prop-osition. For ease of exposition, we assume L � w.

2Let A �� true area �� KLt

13where K is a geometric constant dependent on shape.

Let A �� perceived area.e

Let Rms be the criteria ratio (see Equation (2)) between the me-dium and small sizes. We interpret this ratio as being the ratiobetween estimated areas, that is,

1��A (medium) � R A (small) � (L /L ) A (small), (1)e ms e m s e

A (medium) A (small)e e� . (2)1�� 1��L Lm s

The true area At(med) � KL , implying2m

[At(med)/K](1��)/2 � (L )(1��)/2.2m (3)

Substituting into the denominator of Equation (4):

A (medium) A (small)e e� . (4)(1��)/2 (1��)/2[A (med)/K] [A (small)/K]t t

This ratio is constant for all sizes for any one individual, but ofcourse varies across individuals i. Define this individual ratio con-stant as Ci, which captures individual differences in converting eye-balled graphic sizes to numbers. Substituting Ci for the LHS ofEquation (4) and rearranging, we write

Ae(small) � Ci[At(small)/K](1��)/2. (5)

Define: � � (1 � �)/2. Then

Ae(small) � CiK��[At(small)]�. (6)

And, in fact, for any size, we have the following relation betweentrue and perceived areas.

Ae � CiK��A .�t (7)

Next, assume customers will pay a fixed price, f , and a per-per-ceived-square-inch price, m, for pizza:

RP � f � mAe. (8)

Or in terms of true area,

RP � f � mCiK��A .�t (9)

A Taylor expansion of RP � g(At) to first order

RP � a � b∗At (10)

gives a marginal fee per true square inch

b � �mCiK��A ,��10 (11)

where A0 is the true area value about which the function is ex-panded. Unless A0 is chosen very large, b is increasing in �, and

13For a square when the linear dimension is the side, the value ofK is one. For other figures and linear dimensions, K is typicallysomewhat less than one. For a circle when the linear dimension isdiameter, for example, K is �/4.



hence in the adjustment exponent �. This suggests that if the ex-ponent varies across information conditions, we can detect it in theempirical relation between reservation price and true area by run-ning regressions for each of the information conditions

RPinfo � ainfo � binfoAt, (12)

and measuring slopes. The intercept reflects that the expected fixedfee may be different for pizzas displayed in the three different in-formation presentation conditions (real prices of different size piz-zas show a nonzero intercept, consistent with a fixed fee plus con-stant marginal price structure). The parameter of interest, binfo,includes the perceptual bias in judging size increases, and anyquantity discount consumers expect. In summary, Hypotheses 4and 5 imply, respectively,

b � b , (13)linear graphic

b � b . (14)graphic area

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Pizzas: π or square? psychophysical biases in area estimation

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