Eastern Economic Journal, 2009, 35, (248–263) r 2009 EEA 0094-5056/09 www.palgrave-journals.com/eej/
Mathematical Miscalculations and Monopoly Pricing Strategies Bryan C. McCannon Wake Forest University, Department of Economics, 123 Carswell Hall, Box 7505, Winston-Salem, NC 27109, USA. E-mail:
[email protected]
Economic models focus on prices that announce a cost for one unit. Often prices assigning a cost for multiple units are chosen. For either, any quantity can be purchased at a constant per unit price. I incorporate the experimental phenomenon of mathematical miscalculations to explain the use of both linear pricing strategies. When calculating the per unit price, an error may lead a consumer to either purchase mistakenly or not purchase a unit when she should. These two consequences distinguish the two. Furthermore, I show that announcing a price for multiple units, in certain environments, outperforms non-linear prices. Eastern Economic Journal (2009) 35, 248–263. doi:10.1057/eej.2008.18 JEL: D4; L1; L2 Keywords: consumer mistakes; mathematical miscalculations; multi-unit price; non-linear price; price discrimination
INTRODUCTION Economic models are dominated by the implicit assumption that the price of a good takes the form of a cost for one unit of the good.1 Such a price, which I refer to as a single-unit price, is linear in that multiple units may be purchased at the constant per unit price. Non-linear prices, on the other hand, allow a firm to vary the per unit price with the quantity purchased to discriminate between heterogeneous consumers. There exists a third form to the announcement of a price. Often, multiple units of a good are quoted a cost. For example, a good may have a quoted price ‘‘two for eight dollars’’ where one unit can be purchased for four dollars. I refer to such an announcement as a multi-unit price. Multi-unit prices are linear and thus one might be inclined to think that this is equivalent to a single-unit price, both allow any number of units to be purchased at a constant per unit price. The question I pose is why do multi-unit prices exist? I show in a model with a monopolist selling units of a good to a heterogeneous population of consumers that there exist environments where each of the three pricing strategies arise. The innovation introduced to generate this result is the following. When confronted with a multi-unit price a consumer, who does not want to buy the recommended quantity, must complete a mathematical division problem to determine the price she must pay for fewer units. This allows for the possibility of a miscalculation. The possibility of such an error has two potential consequences. She may buy a unit when it is in her best interest not to do so. Alternatively, she may not purchase a unit when she should have. These two consequences provide a reason for the monopolist to prefer each of the two linear pricing strategies to the other.
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Consider the following example. Suppose there are one hundred consumers of either type A or type B. There are 70 A consumers who value one unit at $4.10 and two units at $8.10 (so the second unit has a value of $4). The 30 B consumers value one unit at $2 and two at $3. The multi-unit price of ‘‘two for $8’’ encourages A consumers to buy two units. B consumers, though, do not prefer to make a purchase, but with a probability of 0.1 they make a miscalculation and buy a unit. With this pricing strategy a profit of $572 is generated. The monopolist cannot use a single-unit price to extract any more of A consumers’ surplus. A higher per unit price results in fewer sales.2 The monopolist could lower its per unit price to $2 and use a single-unit pricing strategy to sell a unit to all B consumers, which results in a profit of $340. Finally, it could select a non-linear price that charges $8.10 for two units and a high price for one unit (any price above $4 discourages A consumers from buying a unit). This, though, generates only $567 in profit. Thus, in this example multi-unit pricing is best. Alternatively, if there were 30 A consumers and 70 B consumers the monopolist would prefer all the B consumers to purchase two units of the good. The non-linear price schedule of ‘‘two for $3 and one for $2.50’’ (any price for one greater than $2 discourages single-unit purchases) generates a profit of $300, which is more than the $268 generated from the multi-unit price. The example highlights the main result of the model. If one type of consumer has preferences such that the marginal utility of both the first and second unit of the good exceeds the marginal utility the other type of consumers place on the first (which I refer to as dominant preferences), then multi-unit pricing has an advantage. The monopolist selects a per unit price at the level those with the dominant preferences receive for the second unit. This amount is greater than the other type is willing to pay for one. Some of these consumers, though, mistakenly purchase a unit. The additional sales generated by a multi-unit price make such a pricing strategy optimal. It is required that there is a sufficient amount of each type of consumer. If the population consists primarily of one type of consumer it is best for the monopolist to select a pricing strategy that extracts as much surplus as possible from multiple purchases made by this type. Additionally, if the probability of a miscalculation is too small the additional profit generated from mistaken consumers does not compensate for the surplus that could be extracted from a non-linear price schedule. Finally, the dominant preferences allow for miscalculations to increase the monopolist’s profit by selling the former two units and collecting mistaken purchases from the latter. Without a consumer with such preferences a non-linear price schedule effectively separates the consumers and extracts more surplus. The paper is organized as follows. The standard model is presented in the next section while the subsequent section introduces the extension incorporating mathematical miscalculations. Further sections study miscalculations when the monopolist selects linear prices and consider non-linear pricing. Empirical data of multi-unit prices in use is summarized in the penultimate section and the final section concludes. The proofs of the results are given in the Appendix.
THE MODEL There are two types of consumers, denoted A and B, who may buy a good sold by a monopolist. Normalize the size of the population to unity and let aA[0, 1] be the q fraction of the population that is of type A. Let u i denote the benefit a consumer of type i receives from buying a quantity q of the good. The good is indivisible so Eastern Economic Journal 2009 35
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consumers buy quantities that are non-negative integers. Furthermore, utility is zero with no purchases and preferences exhibit diminishing marginal utility. Label the consumers so that u1AXu1B. The total payoff received by a consumer of type i is uqi P where P is the total amount spent on goods. A monopolist, unable to distinguish between the consumers, sets a price for the product. Assume that the cost of production is zero. For simplicity assume that each consumer’s utility function is strictly increasing over the first two units and each is sufficiently concave so that the monopolist never finds it optimal to sell three or more units to any consumer, but it is profitable to sell two units to either consumer. Hence, I focus on pricing to sell at most two units to each consumer. I solve for the subgame perfect Nash equilibrium.3 Let mi denote the increase in utility to a consumer of type i if consuming a second unit. Rather, mi ¼ u2i u1i and, from the assumption of diminishing marginal utility, u1i >mi. This implies that u1AXu1B>mB. The preferences can take one of three general cases. In Case I, A consumers receive a greater marginal utility from both the first and second unit than B consumers receive from their first. In this case I refer to A consumers as having dominant preferences. Hence, Case I is defined by u1A>mAXu1B>mB. In Case II, A consumers receive a greater utility than B consumers from both the first unit each consumes and the second unit, but now the benefit B consumers receive from one unit exceeds that added by A consumers from their second unit, u1AXu1B>mA>mB. Finally, in Case III, B consumers value the second unit more than A consumers, u1AXu1B>mBXmA. As will be shown, the pricing decisions of the monopolist and the impact of mathematical miscalculations differ in each of the cases. A monopolist has three general pricing strategies available. It can sell the good by announcing a single-unit price where a cost is quoted for each unit desired. Also, the monopolist can announce a multi-unit price. For example, the monopolist may announce that two units of the good can be purchased for a price P. A consumer may buy two units for P or one unit for P/2. Both are linear pricing strategies. Alternatively, a third pricing strategy the monopolist can use is to set a price for two units as well as a price for one unit. This is referred to as a non-linear price schedule since the price for two need not be twice the price of the single unit.4 Assume that resale markets are not possible (or sufficiently costly) so that non-linear pricing is feasible.
MATHEMATICAL MISCALCULATIONS Mathematical miscalculations are a frequent problem that individuals face. They are unavoidable and common. Experimental evidence supports this as a problem. Rubinstein et al. [2001] conduct experiments where subjects are given simple math problems to be solved. Their goal is to study the mental process of task switching. They show that the frequency of mistakes ranges from only 2.3 to 5.6 percent when subjects perform addition and subtraction problems. The rate of miscalculations jumps to 10.7 to 13.0 percent when they must complete simple division problems. It is this experimentally relevant phenomenon, mathematical miscalculations, which I incorporate into the consumer purchasing behavior to study a monopolist selecting among various pricing strategies. To incorporate mathematical miscalculations I use the frequency of error that arises if individuals must do a division problem. It is up to the consumer to do the Eastern Economic Journal 2009 35
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mathematical exercise to determine the per unit price if the monopolist selects a multi-unit price. While there is no mistake in what is the price of two units, there is a chance that a consumer makes a miscalculation determining whether or not to buy one unit. As a consequence, a mathematical miscalculation may either induce a sale when it is in the consumer’s best interest not to purchase a unit of the good or cause a consumer not to purchase a unit when she should do so. To model such mistakes, let l be the probability that a consumer makes a mathematical miscalculation that changes her behavior. Since the size of the population is normalized to unity l also represents the proportion of the population that will make such an error. Thus, if the monopolist announces a price ‘‘two units for P’’ and, if the consumer prefers to buy only one unit, she buys one unit with probability 1l while with probability l she makes a miscalculation and believes it is best not to buy the good. Furthermore, if it is her best response to purchase zero units of the good a consumer does so with probability 1l, while with probability l she buys a unit. I make a few assumptions regarding the mechanism for which consumers make mistakes. First, at extremely high prices (those with a per unit price above u1A) consumers do not mistakenly purchase a good. This eliminates the opportunity for the monopolist to charge a price beyond the level that any consumer would be willing to pay and make sales. Without this assumption the monopolist would simply announce a very high multi-unit price and rely only on mistaken sales, which would occur with a probability strictly greater than zero, and earn an indeterminately large profit. Second, consumers only make miscalculations when doing division problems. Even though errors are made in all types of mathematical problems, miscalculations in division problems are much more common. Similarly, the miscalculation only occurs if deciding whether to buy one or zero units. While it is possible for a consumer to make a mistake when determining whether to buy one or two units this simplifies the analysis and maintains the main result, although it narrows the admissible values of l under which multi-unit pricing is optimal. Furthermore, the probability of a miscalculation is assumed to be not too large. Specifically, I assume that lpu2B/u1A. Finally, a mistake is made when implementing a best response, not when calculating which is the optimal purchase for a consumer. Thus, a mathematical miscalculation is an exogenous probability that if a consumer’s best response is to purchase zero units one is bought and the probability that if a consumer’s best response is to buy one unit of the good none are purchased. Similar setups occur in other literatures. For example, in the refinement literature, Young [1993] uses mistakes as a way to generate the use of conventions. Specifically, player j makes a mistake with probability aje where aj is player specific and e is common. Kandori et al. [1993] study long-run equilibrium in 2 2 games with mutations. They assume that a player chooses her best response with a probability 12e and randomizes over the two strategies, playing each with probability e. Therefore, the model presented here can be thought of as a simplification of these models where mistakes occur only by consumers, occur with the same frequency by all consumers, and are generated by mistaken division problems when a multi-unit pricing strategy is used. Similar assumptions are made in evolutionary games. Hehenkamp [2002] studies an evolutionary model of price competition with consumers sluggishly learning prices. With a positive probability they randomly select a firm to buy from. Hehenkamp models random learning in a similar manner that miscalculations occur in this model. The literature on public goods giving also acknowledges the importance of mistakes. Palfrey and Prisbey [1996; 1997] Eastern Economic Journal 2009 35
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attempt to extract the random ‘‘noise’’ of mistakes from individual’s feeling of ‘‘warm-glow.’’ They remark, ‘‘the anomalies [in experimental data] might be cause for a serious reexamination of the theory, as they signal trouble for current economic models’’ [p. 829]. This paper is a modest attempt to do so.
LINEAR PRICING AND MISCALCULATIONS As stated, the monopolist, in this framework, has three pricing strategies from which to choose. I first focus on the two linear pricing strategies, single-unit and multi-unit pricing, to illustrate the effect mathematical miscalculations have on a market. Consider, first, Case I. In this case A consumers have dominant preferences valuing the first and second unit more than B consumers value the first. Consider the multi-unit pricing strategy of ‘‘two units for P.’’ Each consumer must decide whether to buy zero, one, or two units. She prefers one unit to zero if the payoff is greater, or rather, u1i Xp where p is the per unit price of the good sold by the monopolist.5 A consumer prefers to buy two units rather than one unit if the payoff is greater, or rather, u2i 2pXu1i p. This simplifies to require that ppmi. Thus, if selecting a multi-unit price, there exists intervals of the per unit price, separated by mi and u1i , where consumer behavior is unchanged throughout. For example, every per unit price selected by the monopolist between mB and u1B results in B consumers preferring to buy one unit and A consumers preferring to buy two units (since mA is greater than u1B in Case I). It follows that if a monopolist is to select a price in an interval the best is to choose the upper bound of the interval. Table 1 describes the per unit price, consumer behavior, and resulting profit if the monopolist selects the price at the upper bound of the interval in the four possible scenarios.6 Miscalculations decrease the monopolist’s profit in scenario [s] while they increase its profit in [m]. In the former the monopolist selects a per unit price to sell one unit to B consumers and two to A consumers. Miscalculations result in lost sales. As a result, a single-unit price is preferable. In the latter the monopolist selects a price that is intended to exclude B consumers and extract more surplus from A consumers. Some B consumers mistakenly purchase a unit of the good making a multi-unit pricing strategy a better decision for the monopolist. When, though, is a multi-unit pricing strategy the preferred linear pricing strategy? The price of ‘‘two for 2mA’’ results in single-unit sales to mistaken B consumers. If a significant proportion of the population of consumers is type B it would be better for the monopolist to reduce the price to encourage all B consumers to make a purchase. Furthermore, if A consumers are willing to pay more for one unit than the monopolist is charging them for two (u1AX2mA) then a single-unit price that sells one unit to them is an attractive strategy. Multi-unit pricing is better, then, if the gained sales from mistaken B consumers provides more profit than the lost revenue from A consumers. This requires that not too great of a proportion of the Table 1 Multi-unit pricing in Case I Scenario [2] [s] [m] [1]
Price (per unit)
Units to A
Units to B
mB u1B mA u1A
2 2 2 1l
2 1l l l
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Profit (multi-unit price) 2mB [2a+(1a)(1l)]u1B [2a+(1a)l]mA [a(1l)+(1a)l]u1A
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population of consumers is type A and that miscalculations are not too rare. Label the interval [aL, a- L] where 2mB lmA u1B lmA lðu1A mA Þ L a ¼ max ; ; ð2 lÞmA ð2 lÞmA u1B ð2 lÞmA ð1 2lÞu1A and a- L ¼ lmA/u1A(2l)mA. Hence, a must be in [aL, a- L]. Let 2mB ðu1A 2mA Þ u1B ðu1A 2mA Þ u1A 2mA ^ ; ; l ¼ max mA mA ðu1A 2u1B Þ mA ðu1A 2u1B Þ Thus, l must be greater than ^ l. Lemma 1 provides the result.7 Lemma 1. In Case I the multi-unit price two for 2mA is the optimal linear price when (1) if u1AX2mA then aA[aL, a- L], and lX^ l or (2) if u1Ao2mA then aXaL. Now consider Cases II and III. The important difference between the two and Case I is that A consumers no longer have dominant preferences, or rather, B consumers value one unit more than A consumers value their second. As a consequence, there no longer exists a per unit price where A consumers prefer to buy two and B consumers prefer to buy zero, which is the scenario where the monopolist strictly prefers a multi-unit price. Table 2 presents the per unit price, consumer behavior, and resulting profit in the four possible scenarios of prices in Case II. In contrast to the previous case in both scenarios [s1] and [s2] miscalculations strictly reduce the monopolist’s profit. Only if a majority of the population is type B in scenario [1] do miscalculations raise the monopolist’s profit, but, unless the probability of the mistake is great, the monopolist would prefer to lower the price and make the sales to the B consumers with certainty. Thus, single-unit pricing is preferred in Case II. Table 3 presents the per unit price, consumer behavior, and profit for Case III. Similarly, single-unit pricing is preferred in Case III. Lemma 2 states the result. Lemma 2. In Cases II and III each multi-unit price is at least weakly dominated.
Table 2 Multi-unit pricing in Case II Scenario [2] [s2] [s1] [1]
Price (per unit)
Units to A
Units to B
mB mA u1B u1A
2 2 1l 1l
2 1l 1l l
Price (per unit)
Units to A
Units to B
mB mA u1B u1A
2 1l 1l 1l
2 2 1l l
Profit (multi-unit price) 2mB [2a+(1a)(1l)]mA (1l)u1B [a(1l)+(1a)l]u1A
Table 3 Multi-unit pricing in Case III Scenario [2] [s2] [s1] [1]
Profit (multi-unit price) 2mA [a(1l)+2(1a)]mB (1l)u1B [a(1l)+(1a)l]u1A Eastern Economic Journal 2009 35
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What effect does mathematical miscalculations have on the market? Miscalculations have the potential of causing lost sales in all three cases. The only scenario where they provide the opportunity to increase sales is if one type, who makes up a sizeable proportion of the population, has dominant preferences. In this circumstance, the monopolist, pricing to sell them two units, selects a per unit price that the other type has a best response to buy none. Some, though, miscalculate and buy a unit. If either of these requirements fails to hold it is no longer advantageous to use a multi-unit price if selecting a linear price.
NON-LINEAR PRICING AND MISCALCULATIONS A monopolist may instead use a non-linear price schedule to engage in price discrimination where the price for two units, P2, need not be twice the price of one unit, P1. Each can be set to target a particular type of consumer. There are three possible outcomes of a non-linear price schedule: one type of consumer buys two units and the other buys one, denoted [2, 1] if A consumers buy two or [1, 2] if they buy one; one type buys two units and the other buys zero, denoted either [2, 0] or [0, 2]; or both types of consumers buy two units, [2, 2].8 Since the objective is to derive the conditions under which multi-unit pricing is optimal and Lemma 2 shows that multi-unit pricing is at least weakly dominated in Cases II and III, I limit attention to non-linear pricing in Case I. In Case I, mA>mB, which implies u2A>u2B. Some outcomes can be eliminated. First, there is no way for a price schedule to result in the outcome [0, 2]. If B consumers are willing to buy two units then A consumers are as well. Also, there is no non-linear price schedule that results in [1, 2]. To induce A consumers to prefer purchasing one unit to two the difference in the prices needs to be greater than the additional benefit received, P2P1>mA. To induce B consumers to prefer two units to one this difference must be less than the marginal gain, P2P1pmB. Since mA>mB, both cannot hold. Therefore, in Case I the monopolist has, in effect, the choice between selecting a schedule so that either [2, 1], [2, 0], or [2, 2] occurs. Consider the optimal non-linear price schedule that results in each of these outcomes. If the monopolist sets the prices so that the outcome [2, 1] occurs the amount it charges for one unit extracts all of the surplus from B consumers buying one unit and the price for two units is at the point where A consumers are indifferent between buying two at P2 and making another purchasing decision. A consumers may either buy one unit at P1 or two single units at 2P1. I assume the firm is unable to stop a consumer from buying a single unit twice. For example, a consumer may buy one and then immediately return to buy a second. This assumes that the firm does not have the technology to track each buyer’s consumption history. The effect of this second incentive compatibility constraint is that the price of two units cannot exceed twice the price of one. Since, in Case I, A consumers value the second unit more than B consumers value one (mAXu1B), the monopolist cannot charge A consumers more than u1B per unit. Thus, P1 ¼ u1B and P2 ¼ 2u1B. The monopolist may price so that the outcome [2, 0] occurs. It sets P2 ¼ u2A to extract all of the surplus from A consumers and sets P1 high enough so that they are not enticed to purchase only one unit. To induce the outcome [2, 2] a price P2 ¼ u2B encourages both A and B consumers to buy two, rather than zero, and a high price for one discourages single-unit purchases. Eastern Economic Journal 2009 35
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When does the monopolist prefer each of these outcomes? For an interior range of a a non-linear price schedule selling two units to each A consumer and one unit to each B consumer is best. Label this interval [aNL, a- NL] where aNL ¼ mB/u1B and a- NL ¼ u1B/u2Au1B. It follows that this interval is non-empty if u2Apu2Bu1B/mB. If the proportion of the population that is type A exceeds a- NL or if u2A>u2Bu1B/mB, then the monopolist receives more profit from selling them two units and not having the revenue constrained by pricing to sell single units. Thus, [2, 0] is optimal. Finally, if most of the consumers are type B then the monopolist wants to sell every consumer two units, which results in the outcome [2, 2]. Lemma 3. In Cases I if aA[aNL, a- NL], which is non-empty if u2Apu2Bu1B/mB, then the optimal non-linear price schedule is P1 ¼ u1B and P2 ¼ 2u1B. If aXa- NL or if u2A>u2Bu1B/mB and aXu2B/u2A, then P1>u1A and P2 ¼ u2A. Otherwise, P1Xu1B and P2 ¼ u2B. Multi-unit prices have the potential of gaining additional sales from mistaken consumers. In Case I it is shown that so long as the population of consumers is not composed primarily of one type and the probability of a miscalculation is not too small and the added profit from the multi-unit price is better than using a single-unit price. This result continues to hold if the monopolist may also select a non-linear price schedule.9 If most of the population of consumers is of one type the monopolist finds it best to select a non-linear price schedule that sells two units to this type of consumer. Otherwise, so long as the probability of miscalculation is not - where too small, the multi-unit price strategy is optimal. Label the interval [a, a] u1B lmA u2B lmA a ¼ max ; ; 0 ð2 lÞmA u1B ð2 lÞmA and a- ¼ lmA/u2A(2l)mA. Hence, multi-unit pricing requires that a is within [a, a]. Let 1 1 uB ðuA mA Þ u2B ðu1A mA Þ ~ l ¼ max ; mA ðu2A 2u1B Þ mA ðu2A u2B Þ Thus, l must be greater than ~ l.10 The following proposition states the equilibrium of the model allowing for all three pricing strategies in Case I. Proposition 1 In the subgame perfect Nash equilibrium strategy of the mono- and lX~ polist in Case I, if aA[a, a] l then the multi-unit price two for 2mA is the optimal price. If
mB u2 lmA u2B ; ap min 1 ; B uB ð2 lÞmA u2A then P1Xu1B, P2 ¼ u2B, while if
mB u1 u2 aX max aI ; 1 ; 2 B 1 ; 2B uB uA uB uA
then P1>u1A and P2 ¼ u2A. Otherwise, P1 ¼ u1B, P2 ¼ 2u1B. Eastern Economic Journal 2009 35
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EVIDENCE AND PREDICTIONS The theoretical model derives multi-unit pricing under three conditions. Each is shown to be necessary to generate the result. First, the preferences of the consumers must fall under Case I, or specifically, u1A>mAXu1B>mB. Second, the population of - Third, potential consumers must be sufficiently heterogeneous, or rather, aA[a, a]. the frequency of mistakes cannot be too small, lX~l. Empirical data could be collected to provide evidence as to whether or not these three conditions are met. The first condition requires that the consumers making the bulk of the purchases not only value the first unit of the good, but also continue to place a high value on the second unit — the marginal utility does not diminish too rapidly. To better understand whether or not this is a reasonable condition I collected prices from a variety of retail stores. For each good with a multi-unit price the type of good, the price, and the quantity were recorded. A total to 770 observations were collected. The Appendix describes in more detail the data collection method. Table 4 presents the breakdown of the types of goods that receive a multi-unit price. As one can see, the sample is dominated with food products (81.7 percent). The food items that make up the observations are staple goods frequently consumed. Furthermore, they are storable.11 Both of these characteristics match the requirements in Case I. If the good is storable, frequently used, and a staple then one would expect the marginal utility of the second unit to, when consumed, bring a somewhat equivalent benefit to the consumers as the first unit. The second condition could be empirically tested with sales data. The proportion of sales that are the multiple, suggested quantity must be sufficiently less than one and sufficiently greater than zero. A statistical test of this ratio would either reject this condition or provide evidence of its validity. Evidence from the data collection provides support for the potential for miscalculations. Rubinstein et al. [2001] show that mathematical miscalculations occur at a 10.7 to 13.0 percent rate when individuals must complete division problems. In their experiment these division problems always had whole number solutions. Presumably, solutions that are not integers would increase the rate of miscalculation. Table 5 provides a description of the pricing announcements. While most of the observations have whole number announcements for the price of the multiple units, the per unit price is often not an integer. Even if one includes per unit prices that end in $0.50 as ‘‘simple’’ division problems, 20.8 percent of the observations are prices that must be calculated down to the penny or fraction of a penny. Furthermore, 26.2 percent of the observations are prices where the denominator in the division problem is an integer not equal to two, which presumably would have been an easier division problem. Thus, given the evidence
Table 4 Goods with multi-unit prices Beverages Dessert Fruits and vegetables Chips and nuts Sauces/condiments Personal hygiene Cereal and crackers Frozen pizza and prepared meals Meat Eastern Economic Journal 2009 35
13.5% 12.6% 10.4% 7.7% 6.5% 6.0% 5.3% 5.2% 5.2%
Paper products Dairy Pet food Office supplies Pasta Bread Misc. food Other non-food
3.5% 2.6% 2.6% 2.5% 2.3% 2.3% 4.5% 6.4%
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N=770 91.9%
Per unit price (p) Whole number Ends in 0.50 Ends in whole penny Ends in fraction
29.5% 49.7% 12.5% 8.3%
Quantity 2 3 4 Othera
73.8% 11.7% 7.5% 6.6%
a The ‘‘other’’ includes 30 goods quoted in a quantity of five, 10 goods in a quantity of six, six goods with a quantity of 10, three with a quantity of seven, and one good with a quantity of eight and of 12.
presented it seems likely that for at least a significant proportion of the goods the multi-unit price is being set on goods where many consumers have dominant preferences and the price itself is set to encourage frequent miscalculations. The theoretical model employs assumptions where empirical data may be used to test the validity of the model. The premise of the argument is that miscalculations of the price lead to type I errors by the consumers. Data on items brought to the checkout line and then refused (due to the recognition of the true per unit price) and returns could provide a test of the model. One would predict that refused sales and returns would be comprised of consumers who purchased a single unit of the good due to the miscalculation, learned of her error, and rectified it. If such data were collected the model would require that the proportion of returns that are single-unit purchases would not be statistically different from one. How might a firm recognize the potential for multi-unit prices to increase its profit? A monopolist, deciding on whether or not to select a multi-unit price, can look at sales data from a single-unit price. If most of the sales are currently made in a quantity of two, then the monopolist could expand sales by quoting a multi-unit price inducing a miscalculation-driven purchase by currently untapped demand. Also, goods with a high volume of sales and those with expiration dates that are further in the future would be candidates for multi-unit prices. Both of these characteristics would likely be those of goods where the marginal utility for the second unit remains high for many consumers. Finally, the phenomenon relies not only on the occurrence of mistakes, but also the inability to rectify it through, for example, returns. The results would continue to hold if the cost associated with returning the item to the store exceeds the lost utility from the error. Hence, a monopolist would find multi-unit pricing profitable on goods with low price tags where the gain from returning the good is small for each individual consumer.
CONCLUSION In practice, many types of linear pricing strategies are used. Economic models tend to focus on prices that take the form of a cost for each single unit purchased. More Eastern Economic Journal 2009 35
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complicated pricing strategies that announce a cost for more than one unit, which I refer to as multi-unit pricing, has not been mentioned in the literature. In this paper, I provide a reason for the existence of both single-unit and multi-unit pricing strategies. The mechanism that differentiates the two linear pricing strategies is the occurrence of mathematical miscalculations illustrated in experimental psychology. In an environment where a monopolist is selling units of an indivisible good to an unobservably heterogeneous population of consumers who make mathematical miscalculations there exists situations where the monopolist prefers single-unit pricing to multi-unit pricing to guard against lost sales and other scenarios where multi-unit pricing is preferred to single-unit pricing to take advantage of the miscalculations. Multi-unit pricing is shown to be the preferable pricing strategy for the monopolist if one type of consumer, who does not constitute too small nor too large of a proportion of the population of consumers, has dominant preferences. In such a scenario, the monopolist prices to sell two units to this type of consumer. Because this consumer type has dominant preferences the price per unit is greater than the other type is willing to pay for one. Some of these consumers, though, mistakenly purchase a unit. Since they do not make up too much of the population of consumers, the monopolist does not have the incentive to lower the price to sell all of them a unit of the good. Hence, multi-unit pricing is the preferable pricing strategy. If any of these conditions is violated the monopolist no longer prefers to use a multi-unit price. The model presented considers the monopolist pricing to sell zero, one, or two units only of the good. In the scenario where multi-unit pricing is the optimal pricing strategy one type of consumer purchases the recommended quantity of two. This eliminates the possibility of lost sales due to the miscalculation. In practice, multiunit prices may state a quantity other than two (e.g. four for ten dollars). This suggests that the quantity used in the multi-unit price is the amount the bulk of the consumers are willing to purchase at the per unit price. Hence, the goal of a multiunit pricing strategy is not just to gain sales, but also to eliminate lost sales. Future work could expand on the occurrence of mathematical miscalculations in more complicated environments that the one presented here. For example, if multiple firms are competing for the sales the reduced market power might cause the firms to act differently. Secondly, the model considers a large number of consumers who take one of two types. This results in a demand curve that makes four ‘‘steps’’ as illustrated in Tables 1–3. Multi-unit pricing either expands out or contracts in each step. A more general model would expand the number of types of consumers creating a more standard demand curve. The choice among linear pricing strategies should remain similar, but is left for future investigation. Also, the miscalculations only occur if a division problem is required. An extension to include addition problems would distinguish single-unit pricing from linear pricing schedules. Additionally, the results I derive depend on mistaken consumers. Such reliance always leaves one exposed to critiques that if the consumer is able to eliminate the mistakes (e.g. using a calculator) then the phenomenon no longer occurs. Future work could explore explanations for multi-unit pricing without requiring errors. Furthermore, mathematical miscalculations are one example of a mistake that can be made by a consumer that firms react to for their advantage. Future work could incorporate other psychological phenomenon expressed in the experimental literature to market activity. Testing behavior in experiments might provide a more accurate model of mistakes. Also, the work presented here creates a simple model of Eastern Economic Journal 2009 35
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miscalculations and the experimental results referenced do not necessarily replicate decisions made in markets. Finally, empirical evidence of miscalculations from actual market activity could provide valuable insight into the form that such mistakes take in practice and what responses firms have to these errors.
APPENDIX This appendix first provides the proofs of the results in the text. Proof of Lemma 1 Consider Table 1. It follows immediately that (1) selecting a per unit price at the upper bound of each interval results in a greater profit than any other per unit price within the interval, (2) a single-unit price of u1B in [s] results in a greater profit than any multi-unit price in [s], and (3) a multi-unit price of two for 2mA in [m] results in a greater profit than any single-unit price in [m]. Therefore, the multi-unit price of two for 2mA is the optimal linear price schedule if the profit generated, [2a þ (1a)l]mA, is greater than 2mB, (1 þ a)u1B, [a(1l) þ (1a)l]u1A, and au1A. Simplifying, the first requires that aX2mBlmA/(2l)mA (which is less than one since mA>mB), the second requires aXu1BlmA/(2l)mAu1B (which is less than one since mAXu1B), the third requires aXl(u1AmA)/(2l)mA(12l)u1A so long as l>u1A2mA/2u1AmA (which is non-negative only if u1AX2mA), and the fourth requires ap(lmA/u1A(2l)mA) ¼ a- L only if u1AX2mA (otherwise there is no constraint on a). Define
2mB lmA u1B lmA lðu1A mA Þ a ¼ max ; ; ð2 lÞmA ð2 lÞmA u1B ð2 lÞmA ð1 2lÞu1A L
Suppose u1AX2mA. It follows that aLpa- L if
2mB ðu1A 2mA Þ u1B ðu1A 2mA Þ u1A 2mA lmax ; ; mA mA ðu1A 2mB Þ mA ðu1A 2u1B Þ
¼ ^l
Therefore, if u1AX2mA the multi-unit price is the optimal linear price if aA[aL, a- L] and lX^ l. Instead, suppose u1Ao2mA. As a consequence, the fourth constraint does not bind. Hence, each of the first three constraints on a must be less than one. The third is less than one if l(u1AmA)/(2l)mA(12l)u1Ap1, or simplified, u1A2mAplu1A, which holds. Therefore, if u1Ao2mA the multi-unit price is the optimal linear price if aXaL. & Proof of Lemma 2 It suffices to show that each of the multi-unit prices in Table 2 (3) is weakly dominated by another price in Case II (III). The monopolist’s profit in [2] is not a function of l. Thus, the single-unit price of mB [mA] weakly dominates in Case II (III). The profit in [s1] and [s2] are both decreasing in l. Thus, the single-unit price of u1B and mA (mB), respectively, weakly dominates. Finally, in [1] if aX1/2 the payoff is decreasing in l so that, again, the price of one for u1A weakly dominates the announcement two for 2u1A. Instead if ao1/2 the profit in [1] is increasing in l. At a ¼ 0 it reaches its peak of lu1A. As shown in the proof of Proposition 1 a non-linear price of P1>u1A and P2 ¼ min{u2A, u2B} results in both A and B consumers buying two units generating a profit of min{u2A, u2B}. Therefore, so long as lpmin{u2A, u2B}/u1A, such a price is dominated. It is Eastern Economic Journal 2009 35
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assumed that lpu2B/u1A. If min{u2A, u2B} ¼ u2A then, since u2A ¼ u1A þ mA, u2A/ u1A>1Xl. & Proof of Lemma 3 As shown, the monopolist selects P1 and P2 so that either [2, 1], [2, 0], or [2, 2] occur. Consider, first, [2, 1]. A consumers’ individual rationality and incentive compatibility constraints are IRA : u2A P2 X0 ð1Þ
ICA1 : u2A P2 Xu1A P1 ICA2 : u2A P2 Xu2A 2P1
The monopolist chooses the greatest price that satisfies (1). Thus, P2 ¼ min{u2A, P1 þ mA, 2P1}. B consumers’ individual rationality and incentive compatibility constraints are IRB : u1B P1 X0 ð2Þ
ICB1 : u1B P1 4u2B P2 ICB2 : u1B P1 4u2B 2P1
The monopolist chooses the greatest price that satisfies (2). Thus, P1 ¼ min{P2mB, u1B} so long as P1>mB.12 Since P1pu1B, P1 þ mApu1B þ mA ¼ u2A(u1Au1B)pu2A. As a consequence, P2au2A. Since mA>mB in Case I, either P1 ¼ P2mB or P2 ¼ P1 þ mA, but not both. Therefore, if P1 ¼ P2mB then P2 ¼ 2P1 so that P1 ¼ mB and P2 ¼ 2mB. Instead, if P1 ¼ u1B then in Case I P2 ¼ min{P1 þ mA, 2P1} ¼ 2u1B, which gives the monopolist a greater payoff since u1B>mB. Finally, P1>mB, as was required. As a result, if it prices so that [2, 1] occurs it does so with P1 ¼ u1B and P2 ¼ 2u1B. Now suppose that the monopolist prices so that [2, 0] is the outcome. A consumers’ constraints are the same as in (1). Hence, P2 ¼ min{u2A, P1 þ mA, 2P1}. B consumers must be unwilling to make any purchase, CB1 : u2B P2 o0 ð3Þ
CB2 : u2B 2P1 o0 CB3 : u1B P1 o0
Thus, P1>max{u2B/2, u1B} ¼ u1B and P2>u2B. Since no single-unit sales are to be made P1 can be set high enough so that it does not constrain P2, P1 þ mAXu2A and 2P1Xu2A. Thus, P1>max{u1B, u1A, u2A/2}. Since u1AXu1B and 2u1A>u2A this simplifies P1>u1A so that P2 ¼ u2A>u2B. Finally, suppose the monopolist prices so that [2, 2] is the outcome. A consumers’ constraints are the same as in (1). Hence, P2pmin{u2A, 2P1, P1 þ mA}. B consumers’ individual rationality and incentive compatibility constraints are IRB : u2B P2 X0 ð4Þ
ICB1 : u2B P2 Xu2B 2P1 ICB2 : u2B P2 Xu1B P1
Hence, P2pmin{u2A, u2B, 2P1, P1 þ mA, P1 þ mB} and since mA>mB and u2A>u2B in Case I, P2 ¼ min{u2B, 2P1, P1 þ mB}. Again, P1 can be set high so that it does not constrain P2, or rather, 2P1Xu2B and P1 þ mBXu2B, which reduces to P1Xmax{u2B/ 2, u1B} ¼ u1B. Thus, P1Xu1B and P2 ¼ u2B. Eastern Economic Journal 2009 35
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The [2, 1] outcome generates a profit of u1B þ au1B in Case I, while [2, 0] generates a profit of au2A. The former is greater if apu1B/u2Au1B ¼ a- NL. The outcome [2, 2] generates a profit of u2B. This is less than the profit in [2, 1] if aXmB/u1B ¼ aNL. The interval [aNL, a- NL] exists if u2Apu2Bu1B/mB. If u2A>u2Bmin{mA, u1B}/mB then the [2, 0] outcome generates a greater profit than the [2, 2] outcome if aXu2B/u2A. & Proof of Proposition 1 From the proof of Lemma 1 there are five outcomes to consider if the monopolist selects a linear price: the single-unit prices of mB, u1B, and u1A and the multi-unit prices of two for 2mA and two for 2u1A. From Table 1 they generate a profit of 2mB, (1 þ a)u1B, [2a þ (1a)l]mA, au1A, and [a(1l) þ (1a)l]u1A, respectively. From Lemma 3 there are three outcomes to consider if the monopolist selects a non-linear price. They generate a profit of au2A, (1 þ a)u1B, and u2B, respectively. Since u2i >u1i >mi, au2A>au1A and u2B>2mB. Also, [a(1l) þ (1a)l]u1A is greater than au1A only if ao1/2. If ao1/2 the profit reaches its peak when a ¼ 0, lu1A. Since it is assumed that lpu2B/u1A, this is less than the profit u2B. Therefore, consider the outcomes that generate the profits (1 þ a)u1B, [2a þ (1a)l]mA, au2A, and u2B. The profit [2a þ (1a)l]mA is greater than the other three if aXu1BlmA/(2l)mAu1B, aplmA/u2A(2l)mA, and aXu2BlmA/(2l)mA. Rather,
u1B lmA u2B lmA lmA a ¼ max ; ; 0 pap 2 ¼a 1 ð2 lÞmA uB ð2 lÞmA uA ð2 lÞmA Such a value is possible if
u1B ðu1A mA Þ u2B ðu1A mA Þ lXmax ; mA ðu2A 2u1B Þ mA ðu2A u2B Þ
¼ ~l
The payoff au2A is greater than the other three if aXu1B/u2Au1B, aXlmA/u2A(2l)mA, and aXu2B/u2A. Rather, aXmax{a- I, u1B/u2Au1B, u2B/u2A}. The payoff u2B is greater than the other three if apmB/u1B, apu2BlmA/(2l)mA, and apu2B/u2A. Rather, apmin{mB/u1B, u2BlmA/(2l)mA, u2B/u2A}. Finally, the payoff (1 þ a)u1B is greater than the other three if apu2BlmA/(2l)mAu1B, apu1B/u2Au1B, and aXmB/u1B. Rather, a2
mB ; min u1B
u2B lmA u1B ; ð2 lÞmA u1B u2A u1B
which exists if lpu2Bu1B2mAmB/mA(u1BmB) and u2Apu2Bu1B/mB. & The data were collected at 10 retail outlets. The retail outlets can be grouped into three categories: discount retailers, supermarkets, and convenient stores. The three discount retailers, K-Mart, Target, and Wal-Mart, were included because they carry a wide variety of goods. The multi-unit prices at these retailers are predominantly found at the supermarkets. Prices were collected at four supermarkets: IGA, Jubilee, Wegmans, and Weis. The final category of retailers, convenient stores, includes two stores that sell gasoline, Casey’s and Sunoco, and a pharmacy, Eckerd. All of the retailers are located in the Elmira, NY area except Casey’s and IGA, which are located in St. Joseph, IL. The prices from the Elmira retailers were collected between July 19 and 22 July 2005 and Target was visited 25 April 2007, while the St. Joseph Eastern Economic Journal 2009 35
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retailers were visited on 27 June 2005. The dates were selected to include no holiday and the different locations were chosen to control for regional effects. At each retail outlet each good with a multi-unit price was recorded. The quantity, cost, and type of each good was noted. Also, it was recorded as to whether the good was a store brand, frozen, canned (if it was a fruit or vegetable), pre-packaged, or had a multi-unit price printed on the package by the manufacturer. Goods with multiple varieties with the same price were counted as one data point while a good that came in various package sizes at different prices were recorded as separate data points. Thus, for example, if a store offered both Coke and Diet Coke in 12-pack cases at a price of two for six dollars this counted as one observation. If they also offered 2-liter bottles a second observation was recorded. Instead, if every flavor/ variety received a separate observation the distribution of product types presented in Table 4 would be even more skewed towards the items most frequently observed; beverages, chips, cereals, etc. often came in many flavors. It should be pointed out that a good may show up more than once as a data point if it had a multi-unit price at more than one retail outlet.
Notes 1. I thank Kaushik Basu, Thomas Gresik, Matt O. Jackson, Alan Kirman, Eric Maskin, J. Tomas Sjostrom, Gil Skillman, and Daniel Spulber for their helpful comments. Also, I thank the participants at the 2005 Bolzano Summer School in Bolzano, Italy and the 2004 Midwest Economic Theory meetings at Washington University. 2. This assumes that the monopolist sells single units to a consumer that prefers to buy only one. A nonlinear price that discourages single-unit sales with a very high price is equivalent to banning single-unit sales and is shown to generate a lower profit in this example. 3. Formally, one can think of this as an extensive-form game where the monopolist selects a strategy from the set fði; si Þ; ½ð1; s1 Þð2; s2 Þg8si 2Rþ 8i¼1;2 where si is the cost for i ¼ 1 or 2 units. Thus, (1, s1) is a single-unit price, (2, s2) is a multi-unit price and [(1, s1)(2, s2)] is a non-linear price. Given the strategy chosen each consumer simultaneously selects an action from {0, 1, 2}. 4. Specifically, by pricing strategy I mean three subsets of the strategy set fð1; s1 Þg8s1 , fð2; s2 Þg8s2 , and f½ð1; s1 Þð2; s2 Þg8s1 ;s2 . 5. Assume that, when indifferent, a consumer buys the larger quantity. 6. The label [2] represents the scenario where both types prefer to buy two units. In [s] and [m] the monopolist will be shown to prefer single-unit pricing and multi-unit pricing respectively. [1] represents the monopolist pricing to sell only one unit to A consumers. 7. Lemma 1 focuses on the environment where the multi-unit price two for 2mA is the optimal linear price. If ao1/2; a multi-unit price of two for 2u1A is also a profitable linear price. As will be shown, a non-linear price of P1Xu1B and P2 ¼ u2B encourages both to buy two units and extracts all surplus from B consumers. Therefore, since lpu2B/u1A it is always better to use this non-linear price schedule. 8. A non-linear price schedule that results in [1, 0], [0, 1], or [0, 0] is never optimal. Since u2i 4u1i 40 the monopolist can sell more units and increase its profit. 9. Since multi-unit pricing is at least weakly dominated in Cases II and III, a non-linear pricing strategy does at least as well, if not better, as a linear price schedule. 10. It was assumed that lpu2B/u1A. Thus, a feasible value of l (lA[~l, u2B/u1A]) exists so long as the utility does not drop off too fast for A, or rather, mA/u1A4u2B/mA1, and ~l is less than one, which requires that mA/u1A4u2B/u2A. 11. Only 33.75 percent of the fruits and vegetables sampled where fresh and non-packaged. 12. Owing to the tie-breaking rule that a consumer, if indifferent between two choices, selects to buy more rather than less, if min{P2mB,u1B} ¼ P2mB then P1 ¼ P2mBe for a small value of e. Since, in equilibrium min{P2mB,u1B} ¼ u1B this revision is avoided.
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