LOYALTY AND DURABILITY: EVIDENCE FROM THE OPENING OF THE TOOTHPASTE …€¦ · LOYALTY AND...

LOYALTY AND DURABILITY: EVIDENCE FROM THEOPENING OF THE TOOTHPASTE TUBES

LEMIN WU AND YUHENG ZHAO

Abstract: This paper shows that durable producers under monopolistic competi-tion tend to overextend the durability to sell more service at a time. The distortionarises because customers are not committed to the same brand for the next purchase.A testable hypothesis is that firms with larger market shares should choose shorterdurability. We test the hypothesis with data from the Chinese toothpaste industry,where—as the theory predicts—we find a positive correlation between market shareand the size of opening of the toothpaste tubes.

Keywords: Market Structure, Optimal durability, ToothpasteJEL Classification: L13, L15

Lemin Wu (corresponding author): Peking University, [email protected] Zhao: Peking University.

EVIDENCE FROM TOOTHPASTE TUBES 2

1. Introduction

How monopoly distorts the price is basic to every principle course of economics,but a question of no less significance has remained controversial: does monopoly alsodistort quality? Would a manager like his product to be more or less durable becauseof the market power his firm enjoys?

Any serious scholar trying to answer the questions has to start with the bench-mark Independence Theorem. Swan (1970) showed that the optimal durability isthe same under pure monopoly as under perfect competition. However, limited bymodeling strategies, Swan was unable to check what would happen under monopo-listic competition, situations that we cannot simply extrapolate on with a theoremthat only applies to the polar cases (Schmalensee, 1979, p.194). In this paper, wefill in the gap and show a surprising result: firms under monopolistic competitiontend to distort durability upward; but the distortion is gone under monopoly andperfect competition. Swan’s result holds as a special case in our general theory,which addresses the whole spectrum of market structures.1

Monopolist competition distorts the durability because firms in that situationare never guaranteed an old customer’s commitment to future purchases. A cus-tomer may switch to other brands after the product at hand expires. When sellingdurables, the seller wishes the customer buy as much as possible. But if Toyota hasno way to persuade customers into buying two cars at a time, all it can do is to makethe car more durable and advertise on the length of service the car promises. Tothe extent that customers evaluate a durable product by the service flow it yields,Toyota gets to sell the customer more service by extending the durability. Whereasincentives under monopoly and perfect competition are different. A monopolist inSwan’s model knows for sure that the customer will come back again. Perfect com-petition means no profit from extra sales. Therefore, in the two polar cases, firmswill stick to the level of durability that minimizes the cost of service per unit, i.e.the social optimal durability.

As far as we know, ours is the first paper that links the choice of durability withthe imperfect loyalty of customers. Previous researchers neglected how loyalty prob-lem affects durability because they are stuck with conventional models that begin byspecifying demand functions. Demand functions, however, implicitly assume awaybrand switches. To study the stochastic switches, we borrow from the Marketing

1Bulow (1986) and Gul (1987) studied how collusion among oligopolists affects the durability. Theirresults cannot generalize to monopolistic competition, but our theory applies to both oligopoly andmonopolistic competition.


literature2 and build a simple first-order Markov brand choice model. Assuming thebrand switch as a stationary Markov process, we characterize a firm’s market statuswith a keeping rate, 𝑘, and a pulling rate, 𝑝. The keeping rate is the proportion ofthe old customers that will continue buying the firm’s product after using up theproduct at hand; the pulling rate is the proportion of the competitors’ customersthat will switch to the firm at the next purchase. The firm’s market share thenbecomes a function of the two rates and the durability, the inverse of which is thefrequency of repurchase and switch.

If our theory is valid, then firms with a larger market share will be less inclinedto overextend the durability. Most often, a larger share of the market implies ahigher loyalty from the customers. For firms with loyal customers, lengtheningdurability squeezes out its own future sales more than grabs its competitor’s share.Hence our theory predicts a testable hypothesis: the negative correlation betweendurability and market share.

Empirical test of the relationship between market status and durability choicehas been embarrassingly scarce.3 In an ideal world, the test could be conducted byregressing the durability of a dozen competing firms’ products on their respectivemarket shares. What prevents such a regression are three difficulties. First, toclearly define what durability is and measure it, if not impossible, is hard in mostcases. The tire of a car expires sooner than the engine. It takes arbitrary weighting ofthe parts to determine the “durability” of the whole car, and measuring the durabilityof all the parts would be another quagmire. Second, firms often differentiate theproduct lines to fill in various market niches. If Cadillac is more durable thanChevrolet4, it is confusing whose durability should be General Motors’ durability.Third, durability often reflects how technically advanced a firm is, but it is themanagers’ market-driven intention that economists seek to identify. The technicalcondition becomes a large noise that the researchers cannot afford with the smallsample size, limited by the number of firms within a typical industry. If the technicalconditions are correlated with market share, the estimate will be even biased ifwithout satisfactory instruments.

2Markov brand choice has been a mature literature in Marketing since the 1960s. For examples,see Kuehn (1962), Ehrenberg (1965), Morrison (1966), Blattberg and Sen (1976), Givon (1984),Ehrenberg and Ehrenberg (1988), Colombo and Morrison (1989), and Rust and Zahorik (1993).3Despite the abundance of theoretical work, Avinger Jr (1981) is the only empirical study relatesdurability choice to market structure. However Avinger’s evidence is mostly narrative and does notfit our purpose.4Both are General Motor’s sub-brands.


Figure 1.1: The diameter of the tube opening is positively correlated withthe market share of the toothpaste companies.

To address the difficulties, we measured the size of the opening of the tooth-paste tubes, and plotted it against the market shares of the toothpaste producers(figure 1.1). Although toothpaste is not treated as a durable in national accounting,its durable nature warrants our analysis. We measured the size of the opening be-cause it affects how much toothpaste a user squeezes out and thereby influences thedurability of a piece of toothpaste given its content. The positive relationship be-tween market share and the size of opening indicates a negative correlation betweenmarket share and durability.

The unique dataset avoids the three difficulties altogether and provides a cleantest that would be hardly possible in other scenarios. First, measuring the size ofthe opening takes nothing other than a vernier calibre. Second, in my data, themajority of producers stick to a firm-specific size of opening across product lines inspite of the variety of flavors and weights. A 40-gram piece of Colgate has the samesize of opening as a 200-gram piece of Colgate. But across the firms, the variationis enormous. Third, the size of the opening of the tubes has nothing to do withtechnical constraints. Managers can pick whatever size they see fit, and have thetubes produced accordingly.

In our 2006 sample, which consisted of 17 Chinese toothpaste producers, thosefirms with a larger share of market chose a bigger opening, that is, shorter durability(figure 1.1 left). The p-value is smaller than 0.5%. One percentage point of increasein the market share is associated with more than one percentage point of increasein the cross-sectional area of the opening.


By shortening the durability, a bigger opening pushes the customers into usingthe next toothpaste sooner. Sales will be promoted if the customers stick to the samebrand, otherwise, the benefit will spill over to competing firms. As some customersbuy Crest after using up Colgate, Crest free rides on Colgate’s decision to enlargethe opening, and vice versa. How serious the spillover is depends on the loyalty ofthe customers. If the customers are fully committed to the brand they are using,the firm can harvest every extra dollar that a big opening brings about. But if noone buys the same brand as the last she uses, there is no point manipulating at all.

The size of the opening is thus positively correlated with the market share,because the firms with larger market shares usually enjoy higher loyalty of thecustomers. Without the intention to switch, the customers are subject to moreexploitation manifest in a shorter durability.

The toothpaste case adds a twist to our theory. Toothpaste customers who arewilling to pay more for a heavier toothpaste will never pay more for the extendeduse brought about by a smaller opening. It is different from the car industry wherethe customer value the length of use directly. As a result, while firms in the carindustry overextend the durability, the toothpaste companies enlarge the opening tocut the durability. The twist aside, when it comes to the testable hypothesis—thecorrelation between loyalty and durability—the logic is the same: the firms withlarger market shares overextend durability less in usual contexts and cut durabilitymore in the toothpaste case, hence the negative correlation between market shareand durability.

The interaction of loyalty and durability applies to almost every industry ofdurable goods and services so long as customers switch across the brands. Even abarber would recommend leaving longer hair to more loyal clients.

Our research belongs to a huge literature on durable goods. According to Wald-man (2003), durable goods theory thrives on three classical contributions, namely,Swan’s Independence Theorem (Swan, 1970, 1971; Sieper and Swan, 1973), whichstates the equivalence of optimal durability between perfect competition and mo-nopoly; Coase conjecture (Coase, 1972; Bulow, 1982, 1986), which points out thetime inconsistency problem that a durable monopolist faces; and Akerlof’s lemonmarket paper (Akerlof, 1970), which introduces informational asymmetry. Our workfollows Swan’s line. As Schmalensee (1979) shows, Swan made seven assumptionsto prove the Independence Theorem. Followers built on his work by relaxing theassumptions. It turns out the independence still holds after the introduction ofproduct differentiation (Swan, 1972), transaction cost (Parks, 1974), and diminish-ing return to scale (Auernheimer and Saving, 1977; Abel, 1983). Other researchers


found otherwise by introducing maintenance (Schmalensee, 1974), replacement (Su,1975), consumer heterogeneity (Basu, 1988), dynamics (Muller and Peles, 1990),uncertainty (Goering, 1993; Goering and Read, 1995), and imperfect substitutabil-ity between old and new products (Waldman, 1996a; Hendel and Lizzeri, 1999).However, no one has studied the optimal durability in the context of brand switch,a signature characteristic of monopolistic competition.

Most of the papers cited above are decades old. Modern research effort hasswitched to Coase conjecture, the second line of the literature—how manipulatingthe durability might alleviate seller’s commitment problem5 (see Waldman, 1993,1996b; Choi, 1994; Fishman and Rob, 2000; Esteban and Shum, 2007; Goering,2010). Our paper shows that consumers’ commitment problem—that is, consumersare not committed to buying from the same seller next time—is no less importantto durability choice than sellers’ commitment problem. While the seller’s commit-ment has received enormous research, consumers’ commitment problem was nevermentioned. This paper presents a simplest model of how consumer commitmentmatters for durability choice, leaving embellishments for future research.

The rest of the paper proceeds as follows. The next section gives the model.The third section shows how the size of tube opening affects the speed of use, andcorrelates with the market share. The last section concludes.

2. Theory

One of our major interests is by how much the firms deviate from the mostefficient length of durability. There are two ways to define the efficiency. The first iscost efficiency. Cost efficiency is achieved by choosing the durability that minimizesthe average cost per unit of service. The second is demand efficiency. Demandefficiency is achieved when the chosen durability maximizes the customers’ utilitygiven the service the durables deliver. The literature has focused on cost efficiencyonly, but in many real-world scenarios such as the case of toothpaste, customersmay prefer a 200g tube to a 50g tube, and a 8mm opening to a 6mm opening, thendemand efficiency is relevant.

Below we will present a cost efficiency model and a demand efficiency model.Despite the difference of assumptions, both models predict the over-extension ofdurability, and the impact of customers’ loyalty on the distortion.

5The sellers’ failure to commit to a small production plan for tomorrow makes customers reluctantto buy today.


2.1. Cost Efficiency Model. Suppose a Markov Lighting Company produces bulbs.The bulbs are such an expensive and long-term investment that every time a cus-tomer buys bulbs, she buys only one. Assume the durability of the bulb is publiclyknown and verifiable.

Assumption 1. The cost of the bulb is twice differentiable, increases with the dura-bility, 𝑐′(𝑑) > 0. The average cost per unit of service, 𝑐(𝑑)

𝑑 , has a lowest point 𝑑*,such that 𝑐′(𝑑*)𝑑* − 𝑐(𝑑*) = 0. The average cost is U shaped. It decreases with 𝑑

when 𝑑 < 𝑑*, and increases with 𝑑 when 𝑑 > 𝑑*.

Definition 1. The level of durability is cost efficient if the average cost per unitof service is the lowest, that is, 𝑑* = arg min

𝑑

𝑐(𝑑)𝑑 .

Assumption 2. Customers do not have preference over durability 𝑝𝑒𝑟𝑠𝑒. They onlycare about the amount of service the durable goods yield.

In Swan’s model, assumption 2 means that the price the customers are willingto pay depend on the total amount of service that the durable goods contain, thatis, the inverse demand for the bulbs 𝑃 (𝑑,𝑄) = 𝑃 (𝑑𝑄)𝑑. Suppose the cost functionis linear to the quantity, 𝐶(𝑑,𝑄) = 𝑐(𝑑)𝑄, then the monopolist’s profit will be

𝜋 = 𝑃 (𝑑,𝑄)𝑄− 𝐶(𝑑,𝑄)

= [𝑃 (𝑑𝑄)𝑑− 𝑐(𝑑)]𝑄

=

[︂𝑃 (𝑑𝑄) − 𝑐(𝑑)

𝑑

]︂𝑑𝑄

Obviously, the profit-maximizing durability is the level that minimizes the av-erage cost, 𝑐(𝑑)

𝑑 . Such is the simplest version of Swan’s Invariance Theorem.As market structure deviates from monopoly and perfect competition, ignor-

ing the customers’ switches becomes impossible. To characterize the switches, weassume

Assumption 3. Customers switch across the brands following a Markov process.

Suppose Markov Lighting is able to keep 𝑘 proportion of the old customers. If𝑘 = 0.5, half of the old customers return to buy from Markov again after their lastMarkov bulb dies out. Meanwhile, Markov Lighting pulls 𝑝 proportion of customersfrom the competitors when these customers use up their bulbs and start looking forreplacement.

If the total market size is 1, and Markov Lighting’s market share is 𝑀 ∈ [0, 1],within ∆𝑡, there will be (1−𝑘)𝑀Δ𝑡

𝑑 customers switching out of Markov Lighting,


and 𝑝(1−𝑀)Δ𝑡𝑠 pulled in from the competitors, who are assumed to have all set the

durability at 𝑠.The balance of the inflow and outflow determines the equilibrium market share:

𝑀 =𝑝𝑑

𝑠(1 − 𝑘) + 𝑝𝑑

Proposition 2. The market share increases with the level of durability 𝑑, if 𝑘 < 1.To produce more durable bulbs is to sell more bulb service. The monopolist in Swan’smodel does not have the same incentive because it is implicitly assumed that 𝑘 = 1.

The profit that Markov Lighting earns per unit of time is then

𝜋 = [𝑃 − 𝑐(𝑑)]𝑀

𝑑

Under perfect competition, a firm has to choose the most cost efficient levelof durability, 𝑑*, to minimize 𝑐(𝑑)

𝑑 , and set the price equal to 𝑃 * = 𝑐(𝑑*)𝑑*. Forotherwise, if the firm sets a price above 𝑃 *, its competitors can charge a bit lessand drive the firm out of the market; if the firm sets a price below 𝑃 *, it will incura loss. Therefore,

Proposition 3. The firms’ optimal durability is cost efficient under both perfectcompetition and pure monopoly (𝑘 = 1).

Oligopoly and monopolistic competition are more complicated. Firms face thetradeoff between the promotion of sales and the cost efficiency of production. Let 𝑥be the average price, 𝑥 ≡ 𝑃

𝑑 ; and let 𝜇(𝑑) ≡ 𝑐(𝑑)𝑑 . Assumption 2 implies the keeping

rate 𝑘 and pulling rate 𝑝 both are functions of 𝑥. Therefore we can write the profitfunction into

𝜋 = [𝑥− 𝜇(𝑑)]𝑝(𝑥)𝑑

𝑠[1 − 𝑘(𝑥)] + 𝑝(𝑥)𝑑

As shown in appendix A.1, the first-order condition with respect to 𝑑 is

(𝑥− 𝜇)(1 −𝑀) = 𝑑𝜇′(𝑑)

The average cost function, 𝜇(𝑑) is U shaped and twice differentiable, 𝜇′′(𝑑) > 0.At 𝑑* where 𝜇′(𝑑*) = 0, the durability minimizes the average cost per unit of service,and is thus cost efficient. As is clear in the equation, 𝜇′(𝑑) = 0 if either 𝑥 = 𝜇, whichholds under perfect competition, or 𝑀 = 1, which holds for the monopolist in Swan’smodel.

Under oligopoly and monopolistic competition, the price is usually higher thanthe average flexible cost, and the market share is always smaller than one, i.e. 𝑥 > 𝜇


and 𝑀 < 1. Therefore 𝜇′(𝑑) > 0 and the optimal durability is larger than the costefficient level, 𝑑*.

Theorem 1. The firm’s optimal durability is cost efficient if the market is perfectcompetitive or if the firm’s market share is 100%. Otherwise the optimal durability islarger than the cost efficient level. Firms have incentives to overextend the durability.

The theorem is intuitive. Firms overextend durability in order to sell a bitmore service each time a customer visits the stores. The incentive is gone underperfect competition because doing so adds no profit. Neither will a monopolistdistort the durability, for the customer will come back for sure, so that there is nopoint manipulating the durability to postpone the customer’s guaranteed return.

The first-order condition also provides a testable hypothesis. Since 𝑑𝜇′(𝑑) in-creases with 𝑑 when 𝑑 > 𝑑*, the optimal durability increases with (𝑥− 𝜇)(1 −𝑀).

Corollary 4. For firms under oligopoly and monopolistic competition, the optimaldurability increases with the average markup but decreases with the market share.

Unfortunately, we do not have the markup data. So we will only test therelationship between durability and market share in section 3.

All the above results are derived under the Markov-process assumption. Violat-ing the assumption would require reconsidering the boundary of a market. Supposethere are ten villages along a circle as figure 2.1 illustrates. Each village has a cus-tomer and a firm. Each time a customer buys things, she randomly picks one of thethree firms that are closest to her village. For instance, customer 𝑏 will choose fromfirms 𝐴, 𝐵, and 𝐶 with equal probabilities; firm 𝐵 will never serve any customerother than 𝑎, 𝑏, and 𝑐. Although firm 𝐵’s share is 10% in the aggregate market, firmB will act as if he is operating on a market which consists of only three customers,𝑎, 𝑏 and 𝑐. On this market de facto, firm B’s keeping rate and pulling rate bothare 1

3 . Ignoring the difference of durability, the effective market share of 𝐵 will be𝑝

1−𝑘+𝑝 = 13 . A firm with 1

3 market share should choose a smaller durability than afirm with 1

10 market share. I call such a market as a localized market. Localizedmarket strengthens firms’ market power over a smaller set of customers. The bound-ary of the relevant market shrinks. Accordingly, firms will adjust the durability inresponse to the rise of the effective market share.

Corollary 5. When a market becomes more localized, the optimal durability willbecome smaller, and the correlation between durability and the nominal market sharewill blur.


Figure 2.1: Localized markets along a circle

Localized market arises when a mature market develops many niches. Smallbrands strategically avoids the competition of large brands by targeting customerswith special needs. The effective market share of Lincoln Limousine can be evenlarger than Toyota Corolla.

Theorem 1 states that firms with 100% market share have no incentives todistort durability. The full market share ensures there be no switch across brandswithin the industry. Yet consumers may still switch from cars to bicycles, and thiswill induce the monopolist to distort durability too. Swan’s independence actuallyrelies on all consumers having perfect foresight, and the second-hand transactionsfree of cost—two of the seven assumptions Schmalensee (1979) listed as crucial to theindependence theorem. Only then would the monopolists refrain from manipulatingthe durability. Discussion of why the assumptions are crucial will be left to theconcluding remarks. For now, to the extent that real-world consumers sometimesmiscalculate, and that second-hand market is anything but costless, we claim thatthe consumer’s commitment problem is universal to all durable producers in varyingdegrees. The consumers’ failure to commit subjects themselves to the distortion ofdurability away from the social optimum.

2.2. Demand Efficiency Model. In the real world, customers sometimes havepreference over durability per se, which violates assumption 2. This section willcheck whether the results we get in the above cost efficiency scenario carry over tothe demand efficiency scenario.


Now assume that the average cost, 𝜇 ≡ 𝑐(𝑑)𝑑 , is constant, but the firms are not

at liberty to choose infinitely large durability because customers have a bliss point,𝑑*, in their preference over durability.

Assumption 4. The keeping rate and pulling rate are now functions of both theaverage price per unit of service, 𝑥, and the durability, 𝑑. There exists a bliss levelof durability, 𝑑*. The keeping rate 𝑘(𝑥, 𝑑) and the pulling rate 𝑝(𝑥, 𝑑) increases when𝑑 < 𝑑* and decreases when 𝑑 > 𝑑*.

Definition 6. The level of durability is demand efficient if the keeping and pullingrates are the largest given the average cost, i.e.

𝑑* = arg max𝑑

𝑘(𝑥, 𝑑) = arg max𝑑

𝑝(𝑥, 𝑑), ∀𝑥

The profit of the firm is

𝜋 = (𝑥− 𝜇)𝑝(𝑥, 𝑑)𝑑

𝑠[1 − 𝑘(𝑥, 𝑑)] + 𝑝(𝑥, 𝑑)𝑑

=𝑥− 𝜇

1 + 𝑠𝑑1−𝑘(𝑥,𝑑)𝑝(𝑥,𝑑)

If 𝑥 > 𝜇, to maximize 𝜋 is to minimize 𝑠𝑑1−𝑘(𝑥,𝑑)𝑝(𝑥,𝑑) . If without 𝑠

𝑑 , the term would

be minimized when 1−𝑘(𝑥,𝑑)𝑝(𝑥,𝑑) is the smallest, which is achieved when 𝑑 = 𝑑*. But

the presence of 𝑠𝑑 means the firm faces the tradeoff between the demand efficiency

and the sales-promoting effect of lengthened durability. Therefore if 𝑥− 𝜇 > 0 and𝑘 < 1, the optimal durability is larger than the demand efficient level.

Proposition 7. When customers have preference over durability per se, the firm’soptimal durability is demand efficient if the market is perfect competitive or if theold customers are committed to always buying from the firm. Otherwise, the optimaldurability is larger than the demand efficient level.

3. Evidence

In this section, we will show that consumers squeeze out more toothpaste ata time with tubes of larger openings, and that the size of the opening is positivelycorrelated with the market share.

3.1. How do consumers squeeze toothpaste? We recruit ten male and tenfemale Peking University students to an experiment. We ask each of them to squeezetoothpaste out of three tubes of different size of opening in random order. The threetubes are from different brands, but have the same volume, material and feel of


Figure 3.1: The frequency of the weights of the squeeze-out

touch. To make sure the paste is also the same despite the difference of containingtubes, we first squeeze the toothpaste tubes empty and then use an injector to fillthe tubes with the same paste.

The subjects squeeze the toothpaste onto replaceable toothbrush heads thatare attached to a shaft. After every squeeze, the toothbrush head is taken off andimmediately weighed on a laboratory balance. Then a new toothpaste head will beused for the next squeeze. The standardized toothbrush head is 2.81 grams. Takingit off the scale reading, we can easily calculate the weight of the squeezed toothpaste.

Figure 3.1 shows the weights of the squeeze-out. The subjects squeeze out moretoothpaste when the opening is larger. The 60 observations allow us to estimate theeffect of the opening on the weight of the squeeze-out. To fix the variation betweensubjects, we run a fixed effect panel-data regression with the following econometricmodel:

𝑦𝑖𝑗 = 𝑐 + 𝛼𝑖 + 𝛽𝑀𝐷𝑖𝑀 + 𝛽𝐿𝐷𝑖𝐿 + 𝜖𝑖𝑗


where 𝑦𝑖𝑗 is the weight of the paste that subject 𝑖 squeezes out of tube 𝑗, (𝑗 ∈{𝑆,𝑀,𝐿}). 𝛼𝑖 is the subject fixed effect; 𝐷𝑖𝑀 and 𝐷𝑖𝐿 are the dummies for mediumopening and large opening.

Given the structure of the panel data, the estimates in table 1 are equivalentto the average of weights for each size of opening: the average weight is 1.21 g forthe small opening (7.20 mm), 1.21 g +0.13 g = 1.34 g for the medium opening(7.60 mm), and 1.21 g +0.32 g = 1.53 g for the large opening (8.02 mm). Thestandard errors of the fixed-effect estimation are much smaller than those of OLS—the twenty subjects all squeeze more with larger-opening tubes. The pattern isobvious, significant and robust.

Table 1: The size of opening increases the amount of toothpaste squeezed

Dependent Variable: the weight of the squeezed toothpaste (gram)Regressor OLS Fixed Effect

Middle opening dummy 0.134*** 0.134***(0.074) (0.023)

Large opening dummy 0.316*** 0.316***(0.074) (0.023)

Constant 1.209*** 1.209***(0.053) (0.016)

Fixed Effect (Volunteer) No Yes

𝑅2 0.241 0.838 (within)

Observations 60 60

Notes: *** denotes significance at the 99% level.

Before the experiment, we thought that a larger opening makes the squeeze-outthicker, but consumers will respond by squeezing a shorter piece to “target” a certainvolume of the squeeze-out. However, if you divide the weights by the cross-sectionalarea of the opening (14𝜋𝑑

2), you will find that the implied length of the squeeze-out actually increases with the size of the opening. It means that the subjects aretargeting not the volume but the shape of the squeeze-out! This explains why thesize of the opening has such a huge impact on the weight of the squeeze-out.

In our 2006 sample, the smallest opening is 6.43mm wide and the largest is8.52mm wide. Calibration with the above estimates predicts that people will squeeze


out 70% (quadratic fit) ∼ 90% (linear fit) more toothpaste using the latter than usingthe former. Such will make a big difference to how fast a piece of toothpaste is usedup.

3.2. How does the size of the opening correlate with the market share?Having collected as many toothpaste as we can, we sampled 17 brands in 2006, and19 brands in 2013. Though not exhaustive, our samples cover all the major brandsin Chinese toothpaste industry. We measured the size of the tube openings in bothyears. The size of opening is positively correlated with the market share in bothyears. Figure 1.1 plots the relationship.

We take the market share data from the industry reports. However, the marketshare data of 2013 is not available yet, hence we use the 2010 data as a closeapproximate. The error might cause attenuation bias, which means the true effectis perhaps larger than our estimates. The data only provides the market shares ofthe top 6 brands in 2006. The 6 brands had taken 83% of the whole market in total.The 2010 data includes the market shares of the top 11 brands, which sums up to93.3%. For the purpose of regression, we assume each of the rest brands had 1% ofthe market.

Table 2: The correlation between market share and the size of the opening

Dependent Variable: the diameter of the tube opening (mm)Regressor (1) (2) (3) (4) (5)

Market Share 4.71*** 3.52* 3.71* 1.82** 3.05***(1.47) (1.66) (2.09) (0.79) (0.83)

Constant 7.08*** 7.13*** 7.11*** 7.42*** 7.26***(0.14) (0.16) (0.14) (0.08) (0.08)

p-value 0.006 0.051 0.097 0.035 0.0002

𝑅2 0.41 0.23 0.18 0.24 0.28

Observations 17 17 16 19 36

Notes: (1) OLS for 2006; (2) OLS for 2006 under least favorable assumptions; (3) OLSfor 2006 with Colgate removed; (4) OLS for 2013; (5) Random effect. *,** and *** de-note significance at the 90%, 95% and 99% levels respectively. The 𝑅2 within group forthe random effect regression is 0.002, its 𝑅2 between group is 0.47. 0.28 is the 𝑅2 overall.


We estimate a simplest econometric model, regressing the diameter of the open-ing on the market share of the brand:

𝑑𝑖 = 𝑐 + 𝛽𝑀𝑖 + 𝜖𝑖

The effect of market share on the size of opening is obvious and significant.In 2006, one percentage point increase in the market share corresponds to 0.047

mm increase in the diameter, or 1.3% increase in the cross-sectional area of theopening (column 1, table 2). The 𝑝-value is 0.6%. The largest opening is 76% biggerthan the smallest opening in terms of the cross-sectional area. The relationship isstill significant if we assume the market shares in the least favorable way6 insteadof assigning 1% to all the brands outside the top 6 (column 2). To make surethe correlation should not be driven by extreme observations, we remove Colgate,which happens to have both the largest share of the market and the largest sizeof the opening. The pattern remains (column 3). Seven years after 2006, thecorrelation is weaker—perhaps due to attenuation bias—but still significant at 0.95%

level (column 4). Column 5 treats the 36 observations together as a panel datato enlarge the sample size. Now the p-value of the random-effects model is assmall as 0.02%. We use the random-effect model not because we have reasons tobelieve the independence of market share with the brand-specific error (neither dowe have reasons to believe otherwise), but because a fixed-effect model would beinappropriate. Each brand has only two observations. There is too little variationto exploit for the fixed-effect model.

Our preferred regression is the OLS with 2006 data (column 1). In that year, nobrand was found to differentiate over the size of opening, despite the large variationacross the brands. For example, we measured the opening of six Colgate tubes in2006. The tubes contain different volumes of toothpaste, ranging from 40g to 200g;their flavors are different—strawberry, mint, lemon and etc.; so are the ingredients;even the designs of the openers vary. But they all share the same size of opening. Incontrast, the data of 2013 is messier. We found at least four brands that have differ-ent sizes of opening for different products. For these brands, we take the average ofthe measured diameters across the sampled products. Whereas, it is the 2006 datathat has addressed the second empirical headache discussed in the introduction—the

6The 6th brand, Hei-mei (Black Girl), has 6% of the market. The least favorable assumption is tohave the brands with the smallest opening take as large a share as possible in the rest 17% of themarket. So we assume that among the brands outside the top six, An-li (Amway, 6.43mm) has 6%,Shi-wang (Lion King, 6.58mm) has 6%, Sheng-jie-lan (S.Synergy, 6.82mm) has 5%, and all the restsmall brands have 0%.


differentiation over durability—and provided the cleanest evidence that we had setout to look for.

3.3. Ruling out the alternative explanations. We explained the correlationbetween market share and the size of the opening with the Markov brand switchmodel. Alternative explanations might exist beyond our knowledge but we can atleast rule out some obvious ones:

(1) Size as a luxury: The correlation arises because toothpastes of largerbrands target more wealthy people, who prefer larger openings. Testablehypothesis: the diameter of opening is positively correlated with the price.

(2) Design of proportion: The correlation arises because tubes of larger vol-ume have larger openings on average. Since the larger brands tend to selllarger volumes, the size of opening appears to be correlated with marketshare. Testable hypothesis: the diameter of opening is positive correlatedwith the volume of the toothpaste.

(3) Extended subconscious impression: The small brands who cannot af-ford advertisement shrink the openings to extend the time of use, so thatthe users may see the brand every day for a longer time, which may leave adeeper subconscious impression.

We rule out the first and second alternative explanations by regressing thediameter on the price and volume of the toothpaste. The samples are 34 pieces oftoothpaste from 16 brands that we bought from the same shop in 2013. Buyingfrom the same shop controls for the shop-specific effect on the price.

As table 3 shows, the estimates of the effect of volume and price are insignificant;and if anything, the signs are opposite to what the alternative explanations predict(column 2, 3, and 4). Those expensive large tubes on average have smaller sizeof opening than the cheap small tubes. In contrast, the effect of market sharestands out significant and positive when regressed on with volume and price together(column 5).

We also have reasons to believe the subconscious impression hypothesis is triv-ial. In 2006, Colgate operated a sub-brand, San-xiao (Three Smiles). Three smilesis an extremely tiny brand, and never advertises on its Colgate background. If sub-conscious impression is so important as to explain the correlation between marketshare and the size of opening, Three Smiles should choose a small size of openingas its market status demands. It turns out Three Smiles has exactly the same sizeof opening as Colgate, the largest size in the whole industry—we would have been


Table 3: Ruling out the alternative explanations

Dependent Variable: the diameter of the tube opening (mm)Regressor (1) (2) (3) (4) (5)

Market Share 1.87*** 1.76***(0.54) (0.53)

Volume (g.) -3.9E-3 -4.5E-3*(2.6E-3) (2.3E-3)

Price (yuan) -0.022(0.017)

PriceVolume (yuan/g)

-1.97 -2.49(2.23) (1.97)

Constant 7.38*** 8.08*** 7.73*** 7.65*** 8.21***(0.07) (0.38) (0.17) (0.16) (0.41)

p-value 0.0015 0.15 0.21 0.38 0.0032

𝑅2 0.27 0.06 0.05 0.02 0.36

Observations 34 34 34 34 34

Notes: *,** and *** denote significance at the 90%, 95% and 99% levels respectively.

surprised to find such a small brand picking this size, had we not known the con-nection between Three Smiles and Colgate. The coincidence of the size of openingis more compatible with our theory: being a sub-brand, Three Smiles’s external-ity is internalized by Colgate, so that the managers of Three Smiles face the sameincentive of enlarging the opening as the managers of Colgate.

Managers of toothpaste companies do not have to comprehend our theory beforethey act as the theory predicts. Imagine a counselor advises the manager to enlargethe opening, and argues that a larger opening will make the customers buy the nexttoothpaste sooner. Managers of Colgate gladly take the advice, but managers ofsmall brands are likely to say, “The next purchase? Customers no longer buy fromus. Why bother?”

The correlation is not a result of data mining. Neither does our explanationcome from trail and error. Instead, one of the authors worked out the theory first.The theory predicts the correlation between market share and the size of the opening


of the toothpaste tubes. Curious about the predictive power of economic reasoning,he collected the samples and was amazed to find how closely the reality matches thetheory.

4. Concluding Remarks

This paper shows how customers’ switch across the brands influences the op-timal durability firms choose. Since customers are not committed to buying fromthe same brand, firms overextend the durability in hope of selling a longer flow ofservice at a time. The theory predicts a negative correlation between market shareand the chosen durability. We test the hypothesis with data from the Chinese tooth-paste industry, where we found that enlarging the openings will speed up the use ofthe toothpaste—thus shortening the durability—and that larger brands make tubesof bigger openings. The correlation between market share and the size of openingprovides clean evidence of the impact of market status on the durability choice. Itaddresses the three major difficulties that have prevented empirical research in thedurable goods literature.

We deviate from Swan’s independence theorem by relaxing the assumption ofcostless second-hand market. The perfect second-hand market makes the customersanonymous: it no longer matters who actually makes the purchase; whoever needs touse the durable can rent it from the market without transaction cost. The firms thushave nothing to gain by manipulating the durability—why would they care aboutthe buyer’s next purchase if she bought but not kept it? The buyers in Swan’s modelare essentially a group of scalpers.

It is hard to find more trivial research than studying the opening of the tooth-paste tubes. Equally rare are experiments cheaper than having a score of boys andgirls to squeeze toothpaste. But just as a straw shows which way the wind blows,and a falling leaf indicates the coming of autumn, we peep through the tube, andas the Chinese saying goes, “see spot, see leopard”.

Appendix A. Proofs and Derivations

A.1. The optimality condition with respect to durability. The profit of thefirm is

𝜋 = [𝑥− 𝜇(𝑑)]𝑝(𝑥)𝑑

𝑠[1 − 𝑘(𝑥)] + 𝑝(𝑥)𝑑


Take derivative with respect to 𝑑:

𝜕𝜋

𝜕𝑑= −𝜇′(𝑑)

𝑝𝑑

𝑠(1 − 𝑘) + 𝑝𝑑+ (𝑥− 𝜇)

𝑝[𝑠(1 − 𝑘) + 𝑝𝑑] − 𝑝2𝑑

{𝑠(1 − 𝑘) + 𝑝𝑑}2

= −𝜇′(𝑑)𝑝𝑑

𝑠(1 − 𝑘) + 𝑝𝑑+ (𝑥− 𝜇)

{︂𝑝

𝑠(1 − 𝑘) + 𝑝𝑑− 𝑝2𝑑

[𝑠(1 − 𝑘) + 𝑝𝑑]2

}︂= −𝜇′(𝑑)𝑀 + (𝑥− 𝜇)

[︂𝑀

𝑑− 𝑀2

𝑑

]︂= 𝑀

[︂−𝜇′(𝑑) + (𝑥− 𝜇)

1 −𝑀

𝑑

]︂The first-order condition with respect to 𝑑 is 𝜕𝜋

𝜕𝑑 = 0, i.e.

(𝑥− 𝜇)(1 −𝑀)

𝑑= 𝜇′(𝑑)

Since [𝑥−𝜇(𝑑)] and 𝑀(𝑑) both are concave, their product 𝜋(𝑑) = [𝑥−𝜇(𝑑)]𝑀(𝑑)

is concave at the point where the first-order condition is satisfied. Therefore thefirst-order condition characterizes the maximum of 𝜋(𝑑).

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