determining consumers’ discount rates with field studiesmela/bio/papers/yao_mela_chen_zh… · to...

Journal of Marketing ResearchVol. XLIX (December 2012), 822–841

© 2012, American Marketing AssociationISSN: 0022-2437 (print), 1547-7193 (electronic) 822

SONG YAO, CARL F. MELA, JEONGWEN CHIANG, and YUXIN CHEN∗

Because utility/profits, state transitions, and discount rates areconfounded in dynamic models, discount rates are typically fixed for thepurpose of identification. The authors propose a strategy of identifyingdiscount rates. The identification rests on imputing the utility/profits usingdecisions made in a context in which the future is inconsequential, theobjective function is concave, and the decision space is continuous. Theythen use these utilities/profits to identify discount rates in contexts inwhich dynamics become material. The authors exemplify this strategyusing a field study in which cell phone users transitioned from a linearto a three-part-tariff pricing plan. They find that the estimated discountrate corresponds to a weekly discount factor (.90), lower than the valuetypically assumed in empirical research (.995). When using a standard.995 discount factor, they find that the price coefficient is underestimatedby 16%. Moreover, the predicted intertemporal substitution pattern anddemand elasticities are biased, leading to a 29% deterioration in modelfit and suboptimal pricing recommendations that would lower potentialrevenue gains by 76%.

Keywords: dynamic structural model, identification, forward-lookingconsumers, pricing, three-part tariff, discount rate

Determining Consumers’ Discount Rateswith Field Studies

Consumers often face situations in which they mustchoose either to engage in consumption in the presentor wait to consume in the future. A rich stream ofrecent literature has adopted dynamic structural models tostudy intertemporal consumption, yielding deep insights

*Song Yao is Assistant Professor of Marketing (e-mail: [email protected]), and Yuxin Chen is the Polk Brothers Pro-fessor in Retailing and Professor of Marketing (e-mail: [email protected]), Kellogg School of Management, Northwest-ern University. Carl F. Mela is the T. Austin Finch Foundation Profes-sor of Business Administration, Fuqua School of Business, Duke Uni-versity (e-mail: [email protected]). Jeongwen Chiang is Professor of Mar-keting, China Europe International Business School (e-mail: [email protected]). The authors thank seminar participants at Columbia University,Duke University, Northwestern University, Peking University, Universityof Pennsylvania, CEIBS Telecom Academic Forum 2011, China IndiaConsumers Insights Conference 2010, the Eighth Triennial InvitationalChoice Symposium, Marketing Science Conference 2011, NBER SummerInstitute IO Workshop 2011, UTD Forms Marketing Conference 2011; thereview team; Hunt Allcott; Eric Anderson; Ron Goettler; Michael Grubb;Wes Hartmann; Praveen Kopalle; Robin Lee; Harikesh Nair; Aviv Nevo;K. Sudhir; and Florian Zettelmeyer for their feedback and Hong Ke andJason Roos for their invaluable research assistance. The authors are listedin reverse alphabetical order. Peter Rossi served as guest editor and Jean-Pierre Dubé served as guest associate editor for this article.

into consumer behavior (see, e.g., Erdem and Keane 1996;Hendel and Nevo 2006; Rust 1987; Sun 2005). Althoughthe use of structural models to study dynamic consump-tion behavior is increasingly ubiquitous, the identificationof these models is problematic (Rust 1994). To identifyconsumer utility functions, it is often necessary to assumeor fix the discount factor at a given value. In contrast, wedevelop a dynamic structural model to identify and mea-sure consumer discount rates using field data. The identi-fication strategy relies on the intuition that we can imputeconsumers’ utility when their current decisions are inconse-quential for their future decisions, the objective function isconcave, and the decision space is continuous. Then, con-ditioned on the estimated utilities under this static setting,it becomes possible to identify discount rates when futureoutcomes become material to these consumers’ decisions.We outline a proof of identification and use a Monte Carlosimulation to show the sampling properties of the identifi-cation approach.The identification strategy has general applications in

contexts in which researchers have repeated observationsof firms or consumers making continuous decisions underboth static and dynamic settings. These include observa-tions of consumers’ reward points usage decisions in loy-alty programs with expiration dates, usage consumptiondecisions for Internet or cell phone data access plans witha monthly fixed allowance that does not roll over, firms’

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Consumers’ Discount Rates with Field Studies 823

markdown pricing for end-of-season clearance inventory,and sellers’ pricing decisions on perishable goods suchas ticket sales. In auctions, for example, consumers mayface posted prices for some goods and auctions for others.When consumers face identical items that are auctionedoff sequentially, the intertemporal trade-off is consequen-tial and affects the bidding behavior (e.g., Zeithammer2006). The posted price context in which consumers makestatic decisions can then be used to infer preferences. Like-wise, in the context of firms’ markdown pricing of sea-sonal goods such as swimwear and skis for which styleschange each year, intertemporal substitution between thefinal markdown period and the future is minimized in thatperiod; thus, the pricing decision in final periods becomesstatic and can be used to infer firms’ objective functions.What these contexts have in common is that there aremultiple observations of decision making in periods thatare independent of dynamics. The repeated observations ofstatic data provide information needed for identifying thepreferences, which can then be used in the dynamic con-texts to infer discount rates.To exemplify this approach, we apply it to consumer

cell phone minute usage data during a field study thatinvolves switching consumer pricing plans. In our data,consumers were initially under a linear “pay-per-minute”plan; thus, usage decisions involved no intertemporal trade-offs because usage early in the month had no bearing onprices paid later in the month. Subsequently, the cell phoneservice provider switched consumers to a nonlinear three-part tariff plan.1 The plan switch induced intertemporalsubstitution trade-offs as consumers’ minute usage deci-sions early in the month had consequences for the ratesthey faced later in the month. The results indicate that con-sumers have weekly discount factors in our data of approx-imately .9 with a 95% confidence interval of (.78, .97).2

The .995 discount factor, which is typically assumed in theliterature, implies that a consumer would accept 1 minuteand 1 second at the end of the month as a substitute foran equally priced minute at the beginning of the month. Incontrast, we compute that consumers value a minute moreclosely to 1 minute and 30 seconds at the end of the month.We organize the remainder of the article as follows:

Next, we provide an overview of the relevant literatureto differentiate the current study from previous research.Then, we detail our identification strategy, and to exemplifyour identification strategy, we detail our modeling con-text. Next, we report a Monte Carlo simulation to demon-strate the validity of the identification strategy and thenapply the approach to a field setting. Subsequently, wepresent and discuss the results and sensitivity analyses.We conclude with some managerial implications and futureresearch directions.

1A three-part tariff plan contains three components: an access fee, acertain amount of allowance minutes, and a marginal price if the con-sumer’s usage exceeds the allowance within a billing cycle. As a result,the consumer may be subject to extra fees when the consumption exceedsthe allowance (overage) and overpays if the consumption falls below theallowance (underage). Because of the allowance and the high marginalprice, a consumer must decide how to allocate his or her consumptionacross time within a billing cycle, intending to avoid overage and underageso as to maximize total utility.

2The implied weekly discount rates are approximately 1.11 with a 95%confidence interval of (1.03, 1.28).

RELEVANT LITERATURE

Given that discount rates are not typically identified, sev-eral approaches have emerged to contend with the prob-lem, including (1) assuming a fixed value for the discountrate, (2) functional identification using structural assump-tions and/or estimation using exclusion restrictions, and(3) experimental approaches. Table 1 provides an overviewof a sample of these approaches and their resulting dis-count values converted to their weekly equivalents. Table 1makes it apparent that discount rates vary considerablyacross studies: The mean weekly discount factor is .98 witha standard deviation of .032. The corresponding weekly dis-count rates average 2.09% with a large standard deviationof 3.66%. In short, there is no clear consensus regarding thevalue of discount factors, partially because discount ratesare typically not identified.First, most studies assume or fix the discount factors

to certain values, typically between .995 and 1.0. For thepurpose of identification, it is also a common practice toassume the discount factor is the same across consumers,which might be, in some instances, a strong assumption(Frederick, Loewenstein, and O’Donoghue 2002).A second approach to the identification of discount rates

includes the imposition of structure on the model, such asassuming the distribution of the model errors, individu-als knowing the state transition probability, and no unob-served heterogeneity (e.g., Goettler and Clay 2011; Hotzand Miller 1993). However, these identification assump-tions may be difficult to validate in some contexts (e.g.,homogeneous consumers, full information of state transi-tion probability for new products or markets). Moreover,the data needed to achieve parametric identification areoften prohibitive. Repeated decisions are necessary for eachcombination of states to attain parametric identification.When the state space increases and is continuous, the like-lihood of observing multiple observations at each com-bination of states quickly becomes negligible. A relatedstream of research in the descriptive (nonstructural) litera-ture imputes discount factors from data wherein consumersface trade-offs between immediate returns/costs and futureflows of returns/costs (e.g., the current capital cost of amore energy-efficient car with its future gasoline costs).Examples of this approach include Allcott and Wozny(2011), Busse, Knittel, and Zettelmeyer (2012), Dubin andMcFadden (1984), Harrison, Lau, and Williams (2002),Hausman (1979), and Warner and Pleeter (2001). As in thestructural literature, it is typically necessary to invoke cer-tain restrictions to identify the discount factor. For example,to quantify the future operating cost of a durable appliance,it is necessary to assume that the consumer has perfect fore-sight about future prices of electricity. As a result, there isno uncertainty about the future state transition.A third identification strategy is to rely on an exclusion

restriction argument (Chung, Steenburgh, and Sudhir 2011;Fang and Wang 2012; Lee 2011). Exclusion restrictionsinvolve specifying a set of exogenous variables that do notaffect current utility but do affect state transitions. Accord-ingly, variation in these exogenous variables affects futureutilities through their impact on the state transition but donot have an effect on current utility. By exploring howchoices are made in light of changes in future utility whencurrent utility remains fixed, the utility and the discount

824 JOURNAL OF MARKETING RESEARCH, DECEMBER 2012

Table 1EXAMPLE DISCOUNT RATES IN EMPIRICAL STUDIES

Choice Discount Weekly Discount Implied WeeklyStudy Domain Approacha Factor Period Factor Interest Rate (%)

Rust (1987) Bus engine replacement F .9999 Month → 1 0002Hotz and Miller (1993) Children/sterilization E 065 (.68) Year 0992 083Erdem and Keane (1996) Laundry detergent F .995 Week 0995 050Ackerberg (2003) Yogurt E 098 (.02) Week 098 2004Hartmann (2006) Golf F .99 Day 0932 7029Gordon (2009) Personal computers F .98 Month 0995 047Kim, Albuquerque, and Camcorders F 1.0 Minutes 100 0

Bronnenberg (2010)Goettler and Clay (2011) Online grocery service E 0974 (.001) Week 0974 2067Dubé, Hitsch, and Chintagunta (2010) Video game consoles F .9 Month 0976 2049Dubé, Hitsch, and Jindal (2010) HD DVD players P .7 Year 0993 069Hartmann and Nair (2010) Razors and blades F .998 Week 0998 020Chung, Steenburgh, and Sudhir (2011)b Sales force compensation E .95 Month 0988 1020Fang and Wang (2012)b Mammography exams E 072 (.09) Two 0997 0032

080 (.03) Years 0998 021Ishihara (2010) Video games E 0885 (.006) Week 0885 12099

Overall mean (SD) .980 (.032) 2.088 (3.663)Mean of fixed discount rates (SD) .983 (.026) 1.825 (2.824)Mean of estimated discount rates (SD) .973 (.037) 2.932 (4.972)

aF in the “Approach” column indicates an assumed fixed value for the discount factor. The study labeled P estimates the discount rate using experimentaldata. E indicates that the discount parameter is estimated by functional restrictions and/or the use of exclusion restrictions. Standard errors of the estimatesare in parentheses.

bChung, Steenburgh, and Sudhir (2011) and Fang and Wang (2012) also consider hyperbolic discounting. We only report their results of exponentialdiscount factors. Fang and Wang (2012) use two specifications in their estimation; thus, we report two discount factors. Chung, Steenburgh, and Sudhir(2011) obtain the discount rate using grid search, so there is no sampling error to report. Note that the grid search approach yields estimated parameterdistributions that are conditionally marginal with respect to the discount rate, which can lead to inefficient estimates.

factor can potentially be identified together.3 However, suchexclusion restrictions are often unavailable in field data orare difficult to validate.To alleviate these concerns, Dubé, Hitsch, and Jindal’s

(2010) recent work uses experimental conjoint analysis datato identify a dynamic model in the context of durable goodsadoption. In particular, Dubé, Hitsch, and Jindal manipu-late consumers’ beliefs about state transitions by informingthem of alternative future market situations in the experi-ments. As a result, the authors are able to identify utilityand discount factors. This approach is most similar to oursin that it uses data rather than invoking assumptions to inferdiscount rates. Although an important step forward, it isoften difficult to replicate dynamic choices in lab settingsbecause of demand artifacts and contracted durations. Forexample, Dubé, Hitsch, and Jindal consider annual bud-get trade-offs in a lab experiment that lasts one session.It would therefore be desirable to supplement this researchusing a field context with choices made in practice and overextended periods.

3In a discrete choice context, Magnac and Thesmar (2002, p. 807, Equa-tion 8) specify an exclusion restriction in which exogenous state vari-able(s) do not affect the “current value function” (defined as the differencebetween the expected values of [1] choosing a specific alternative followedby the outside option and behaving optimally thereafter, and [2] choosingthe outside option now and in the next period, and behaving optimallythereafter). This exclusion restriction, coupled with the normalization ofthe future value function of the outside option to zero, enables identifica-tion of the current value function and the discount factor (Proposition 4,p. 809). We thank Robin Lee for the insight regarding this identificationstrategy.

Following a similar logic, we advance the research indynamic structural models by identifying discount factorsusing field data. Our identification strategy is to first iden-tify consumers’ utilities and the distribution of randomconsumption shocks using data that have no dynamicsinvolved. Then, we further recover their discount factorwhen the dynamic structure was exogenously imposed.Using the static data to pin down inferences regarding utilityand the distribution of random shocks, the discount rate canthen be inferred from the dynamic decision-making context.Our contributions are fourfold. First, we show that it

is possible to recover discount rates by supplementingdynamic decision data with static decision data whenthe decision variables are continuous and the objectivefunction is concave. Second, we provide an empiricalillustration using field data to exemplify this identifica-tion strategy. Third, we explore the potential for biasedparameter estimates in this sample as a result of misspec-ifying discount rates as well as the potentially subopti-mal marketing decision making. Finally, our research alsoadvances the empirical literature on the nonlinear pricingof telephony or Internet services (e.g., Grubb and Osborne2012; Iyengar, Ansari, and Gupta 2007; Lambrecht, Seim,and Skiera 2007; Lambrecht and Skiera 2006; Narayanan,Chintagunta, and Miravete 2007). Most previous empiricalstudies are based on aggregate usage data, limiting theirability to investigate consumers’ intertemporal substitutionin consumptions.4 Because our data are at the disaggregate

4Grubb and Osborne’s (2012) study, which focuses on plan choice underuncertainty and learning, is a notable exception. Our emphasis lies insteadon the allocation of minutes used within the month.


level, we are able to evaluate the trade-off of consumptionsacross time and the corresponding managerial implicationsfor the firm’s pricing strategy.

IDENTIFICATION OF DISCOUNT FACTORS WITHSTATIC DATA

In this section, we formalize our logic that the discountfactor can be identified when (1) the utility and randomshock distribution are known, (2) the objective functionis concave, and (3) the controls are continuous. We focusthe discussion on the finite horizon case, which is consis-tent with the empirical application detailed subsequently.In Appendix A, we extend the logic of the identificationstrategy to the infinite horizon with continuous controls anddiscuss our conjecture about the case of discrete actions.In this finite horizon case, at any given period of t < T,

we can write the optimization problem of consumption asfollows:

maxxt

u4xt3 st1 Ít5+T∑

k = t + 1

Äk − tEu[xk3g4sk − 11xk − 151 Ík

]1(1)

where u4·5 are the utility function (or profit function); x. arecontinuous control variables (e.g., the minutes of cell phoneconsumption in our application); s. are the state variable(the cumulative minutes consumed in our application); Í.are the random shock (the expectation is taken over the ran-dom shock); Ä is the discount factor; and g4·1 ·5 is the statetransition equation such that st + 1 = g4st1xt5, ∀ t (in ourcase, g4st1xt5 = st + xt5.As discussed in Rust (1994, p. 3090), under weak

regularity conditions, the optimal decision rule è∗1 =

8Ñ∗1 1 Ñ

∗2 1 0 0 0 1 Ñ

∗T9 exists and can be computed using back-

ward induction:

Ñ∗T4sT1 ÍT5 = argmax

xT

u4xT3 sT1 ÍT5

Ñ∗T − 14sT − 11 ÍT − 13è

∗T5 = argmax

xT − 1

u4xT − 13 sT − 11 ÍT − 15

+ ÄEu6Ñ∗T3g4sT − 11xT − 151 ÍT7

000

Ñ∗1 4s11 Í13è

∗25 = argmax

x1

u4x13 s11 Í15

+T∑

k = 2

Äk − 1Eu[Ñ∗k 3g4sk − 11xk − 151 Ík

]0

(2)

In particular, because the utility is concave in x. and thelinear combination of concave functions are also concave,

Table 2THREE-PART TARIFF PLANS

Access Fee Allowance Marginal Number of Average Linear Rate(CNY) (Minutes) Price (CNY) Enrollees Percentage Before Switch (SD)

98 450 040 284 50035 .28 (.10)128 600 040 111 19068 .27 (.09)168 800 040 83 14072 .27 (.07)218 1100 036 50 5092 .25 (.07)288 1500 036 21 2086 .24 (.07)388 2500 030 15 2031 .28 (.04)

backward induction leads to one unique optimal level of x.for each period, beginning with the terminal period. If wetake the first-order condition of Equation 1 with respectto xt, ∀ t < T, we have the Euler equation:

ux4x∗t 3 st1 Ít5+ ÄEug

[x∗t + 13g4st1x

∗t 51 Ít + 1

]= 01

where ux and ug are the partial derivatives with respect to xand g, respectively.Because the Euler equation is linear in Ä, given that the

utility and the distribution of Í are known and x∗. are theunique and observed optimal consumption levels, the dis-count factor Ä can then be uniquely identified from the data.Next, we apply this identification strategy to show it is pos-sible to recover the discount rates from a finite sample andto explore the implications of these rates for inference andmanagerial policy decisions.

ILLUSTRATIVE APPLICATION

Consumer Usage Data and Carrier Tariff Structure

A major mobile phone service provider in China sup-plied the data we use to illustrate the sampling prop-erties of our identification strategy. The data cover theperiod from September 2004 to January 2005. The dataprovider accounted for more than 70% of the marketshare of Chinese mobile phone service market during thattime. Initially, this firm used only linear pricing schedules(i.e., consumers were billed on a pay-per-minute basis).In November 2004, the firm offered three-part tariff plansto a randomly selected set of consumers. The firm dividedthese consumers into multiple groups according to theirpast usage volumes. The firm then offered each group arespective three-part tariff plan. A consumer could choosethe three-part tariff plan or remain on the original pay-per-minute plan.Tariff structure. Table 2 depicts the pricing structure of

the most popular three-part tariff plans, covering 90% ofthe consumer base. Consumers who enroll in one of thelisted plans are allowed a fixed number of free callingminutes in a given calendar month by paying the monthlyaccess fee. When a given consumer places or receives acall, the minutes of the phone call are deducted from theallowance and the consumer does not need to pay for thatusage. However, when the monthly allowance is exhausted,the consumer is billed the marginal price for each minuteof usage beyond the allowance. There is no “rollover” forthese plans (i.e., unused allowance minutes cannot be car-ried over to next month). At the beginning of next month,the consumer’s allowance of free minutes is replenishedafter paying the new month’s access fee. The consumers


have different linear rates before the switch. The mean lin-ear rate is .27 with a standard deviation of .09.Usage data. For the first four months (from Septem-

ber 2004 to December 2004), we observe each consumer’saggregate monthly minute usage and expenditures. How-ever, in the last month (January 2005), we observe call-levelconsumer records, including the starting time, duration,and expense of each telephone call. The data also includesome demographic information, including the age, gen-der, and zip code of each consumer. Table 3 summarizesthe consumers’ average usage levels (normalized by theirallowance level) and demographic information. The aver-age usage under both the linear and three-part tariff plansare close.Overage and underage. Underage occurs when con-

sumers do not use all the allowance at the end of a month.In this case, consumers are overpaying in the sense that theyhave been charged for minutes they do not use. In compar-ison, overage occurs when usage exceeds the allowance. Inthis case, consumers again overpay inasmuch as a plan withmore allowance minutes normally has a lower average priceper allowance minute (Iyengar, Ansari, and Gupta 2007). Asa result, consumers who strategically manage the minutesshould evidence less underage or overage.In Figure 1, we plot the histogram of the ratios of min-

utes used to minutes allowed for the last month of data. The

Table 3SUMMARY STATISTICS

M SD Min Max

Monthly usage under linear 092 046 002 2022plan/allowance level

Monthly usage under three-part 096 035 001 1063tariff/allowance level

Female 016 — 0 1Rural residency 041 — 0 1Age (years) 36018 7045 19 58New customer (enrolled less 016 036 0 1

than 12 months)

Figure 1HISTOGRAM OF TOTAL USAGE VERSUS ALLOWANCE

.5

0

5

10

15

Per

cen

t o

f T

ota

l

1.0 1.5

Monthly Total Usage vs. Quota

average ratio is close to 1 (.96) under the three-part tariff,suggesting that consumers on average tend to avoid overageor underage. Yet this average behavior belies a large stan-dard deviation (.35). Thus, we next consider whether andhow users manage their minutes over the month to comportwith the allowance; to the extent this behavior changes asthe allowance becomes more salient, evidence is affordedfor the strategic use of minutes.Strategic minute usage within a month. We consider

some model-free evidence that consumers are strategic intheir usage of allowance minutes. This evidence is predi-cated on the notion that minute consumption changes as thedistance between minutes used and the allowance becomessmall; in particular, consumers begin to conserve minutesas the number of free minutes dwindles and the overagepotential increases.Dividing the last month of the data into five six-day peri-

ods, t = 11 0 0 0 15, we compute the ratio of cumulative min-utes used to the allowance for each consumer at the endof each six-day period.5 Figure 2 portrays a scatter plot ofthis ratio and its lag value for each period t = 21 0 0 0 15. Theline in this figure depicts a nonparametric function relatingthe ratio and its lag, and the gray band indicates the 95%confidence interval for this function.6 A key insight fromthis figure is that this function is concave when the cumu-lative usage is within the quota (the ratio in period t − 1 isless than 1). In contrast, when usage exceeds the quota (the

5To test the model’s sensitivity to the definition of the six-day period,we also consider the specifications of ten-, five-, and three-day periods.The implied weekly discount factors and price coefficients are statisticallyequivalent across these different specifications.

6We consider both spline and local regression methods, and the resultsare similar. Figure 2 presents the results from spline method.

Figure 2THE EFFECT OF ALLOWANCE ON MINUTE USAGE OVER TIME

Cumulative Minutes vs. Quota at t – 1 (t = 2, 3, 4, 5)

1.41.21.0.8.6.4.2.0

.0

.5

1.0

Cu

mu

lati

ve M

inu

tes

vs. Q

uo

ta a

t t 1.5


ratio in period t − 1 is greater than 1), the line is almostlinear. The concavity of the line before exceeding the quotasuggests that people decelerate usage as they approach thequota; that is, they begin to ration their minutes to avoidoverage. Moreover, those who are far from the quota appearto accelerate usage to avoid underage. In comparison, con-sumers who have already exceeded the quota do not decel-erate (or accelerate) their usage. Instead, they follow somerelatively stable usage rates, as might be expected if theyno longer faced an intertemporal trade-off in usage. Misraand Nair (2011) and Chung, Steenburgh, and Sudhir (2011)use similar methods to investigate the effect of quota ona salesperson’s allocation of efforts across time. They findanalogous patterns of dynamic effort allocation in salesforce due to quotas.To further elaborate on these insights arising from Fig-

ure 2, we consider how usage acceleration (deceleration)

Figure 3USAGE CHANGE OVER TIME

.5 1.0 1.5 2.0 2.5 3.0

Overage: Cumulative Usage/Quota>1.8<Cumulative Usage/Quota 1

.4<Cumulative Usage/Quota .6 .6<Cumulative Usage/Quota .8

.2<Cumulative Usage/Quota .4Cumulative Usage/Quota<_ .2

0 1 2 3

0 1 2 3 0 1 2 3

0 1 2 3

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

30

0

10

20

30

0 1 2 3

0

10

20

30

40

Per

cen

tag

eP

erce

nta

ge

Per

cen

tag

eP

erce

nta

ge

Per

cen

tag

e

Per

cen

tag

e

0

5

10

15

20

<_

<_<_

<_

changes as people approach their allowance/quota. Thisacceleration can be summarized by the statistic (Usageduring period t)/(Usage during period t − 1). This ratio isanalogous to the slope of the line in Figure 2. When theratio is 1, consumers are neither decelerating nor accel-erating use. When the ratio is greater than 1, usage isaccelerating, and when it is less than 1, usage is decel-erating. We compute this ratio for each person in eachperiod and then, in Figure 3, present a histogram of thisminute acceleration measure across users and periods con-ditioned on users’ distance to the quota. For example, theupper leftmost histogram shows the distribution of usageacceleration observations conditioned on consumer usageat time t − 1 being less than 20% of their allowance. Fig-ure 3 indicates that consumers’ usage decelerates as theyapproach their allowance. Furthermore, when the consumerreaches an overage situation (in which there is no longer


an intertemporal trade-off in usage), the slopes averageapproximately 1, indicating a stable usage rate. Theseobservations are consistent with Figure 2. Overall, we con-clude that there is some model-free evidence of strategicbehavior on the part of consumers.

MODEL

Utility Under the Linear Pricing Plan

In this section, we first specify the consumer utility forconsumption under a linear pricing plan and derive the opti-mal level of consumption. We then extend the analysis tothe case of the three-part tariff plan. Similar to Lambrecht,Seim, and Skiera (2007), we begin by assuming that con-sumer i derives utility from phone usage and the consump-tion of a composite outside product (numeraire). To bespecific,

uit4xit1 zit5 =(ditbxit −

12b

x2it

)+ zit1(3)

subject to zit = yi − pi0xit1

dit1b > 0(4)

where t = 1121 0 0 0 1T are the periods within a month;7 xitis the minutes of phone usage during period t; pi0 is thelinear price rate of consumer i before switching to the three-part tariff; zit is the consumption of the numeraire;8 yi isthe income; dit/b is the linear component of minute usageutility; and (−1/2b) is its quadratic effect.Consumer i chooses the optimal levels of phone usage xit

and numeraire consumption zit so as to maximize his or hertotal utility subject to the budget constraint. Substitutingthe budget constraint Equation 4 into Equation 3, we canrewrite the utility function as follows:

uit6xit1 z4xit57 =ditxitb

−x2it2b

+ yi − pi0xit0(5)

Solving the maximization problem of Equation 5 yields theoptimal consumption

x∗it =

01 if dit − bpi0 < 0

dit − bpi01 if dit − bpi0 ≥ 00(6)

The foregoing equation clarifies the interpretation of(1) b as the price sensitivity and (2) dit as the baselineconsumption level under the linear pricing plan as it rep-resents a fixed shift in the demand curve as well as theminute consumption level when pi0 = 0 (Lambrecht, Seim,and Skiera 2007). Following Narayanan, Chintagunta, and

7We use t to index periods within a month and Ò to index months.8The budget constraint in Equation 4 normalizes the price of the out-

side good to one, effectively turning it into the numeraire. The purposeof this normalization is twofold. First, it normalizes the marginal utilityof income (or numeraire) to one. Because we do not observe consumerchurning or variation in plan choices due to income effect in the data,the identification of the marginal utility of income is infeasible. Sucha normalization treatment for the purpose of identification is similar toNarayanan, Chintagunta, and Miravete (2007) and Ascarza, Lambrecht,and Vilcassim (2012). Second, it transforms the consumption of minutesinto a dollar metric in comparison to the numeraire, making its interpre-tation more meaningful.

Miravete (2007) and Lambrecht, Seim, and Skiera (2007),we further allow baseline consumption, dit, to be affectedby consumer characteristics, Di, and a random shock Íit :

dit = exp4D′

iÁ5+ Íit1(7)

where Á is a vector of parameters and Íit ∼ N401 Æ25 isexogenously i.i.d. across consumers and periods.9 Sourcesof the shock may include technical problems with theconsumer’s phone set or coverage that limit the phoneusage; unexpected events that require extra communica-tions with others; and so on. Although the random shocksare unknown to the researchers, the consumer observes theshocks at the beginning of each period, before decidinghis or her usage levels accordingly. Given that all con-sumers in the data set are experienced users, we assumethat the distribution of the shocks is known to the con-sumer.10 Summing optimal period consumptions within thesame month Ò yields the optimal total minutes consumedwithin a month as

qiÒ =T∑

t = 1

x∗it0

Utility under the three-part tariff plan. The three-part tar-iff plan can be described as the triple 8F1A1p9, where Fis the fixed access fee, A is the allowance amount, andp is the marginal price after the consumer exhausts theallowance. With regard to period utility and budget con-straint, at period t during a given month, consumer i has autility level

uit6xit1 z4xit53 sit1 Íit7 =ditxitb

−x2it2b

+ zit4xit51

subject to zit4xit5 = yi −C4xit5

C4xit5 =

(t − 1∑k = 1

xik + xit −A)pI∑t − 1

k = 1 xik + xit > A1

ift − 1∑k = 1

xik < A

pxit ift − 1∑k = 1

xik ≥ A

dit = exp4D′iÁ5+ Íit1

(8)

where sit is a vector containing state variables at periodt that include (1) cumulative usage up to period t − 1,∑t − 1

k = 1 xik and (2) period t (or the distance to the ter-minal period). Among these state variables, the cumula-tive usage is endogenous and the period t is exogenous.I∑t − 1

k = 1 xik + xit > A is an indicator, which takes the value of 1

9The exponential function ensures that, on average, dit > 0. A relatedconcern with the use of a normal distribution assumption for the ran-dom shocks is that the baseline demand, dit , may become negative. Oneapproach is to consider a truncated normal distribution. However, this iscomputationally costly. Thus, we instead assume that the magnitude of Íit(standard deviation) is small compared with exp4D

′iÁ5, so a normal dis-

tribution is a good approximation of a truncated normal distribution. Thisassumption is consistent with the estimation results.

10Although this assumption is not material for the static model becausethe error is revealed before the usage decision, the assumption becomesimportant under the context of the three-part tariff when future shocksbecome relevant to current period consumptions.


if∑t − 1

k = 1 xik +xit > A and 0 if otherwise. Note that the fixedaccess fee F does not enter the period budget constraintbecause it is essentially a sunk cost. It does not affect theoptimal decision at period t as long as the choice is not acorner solution. Substituting the budget constraint to Equa-tion 8, we can rewrite the period utility as follows:

uit6xit1yi −C4xit53 sit1 Íit7 =(ditxitb

−x2it2b

)+ yi −C4xit50(9)

Similar to the period utility under the linear pricing plan,we assume that dit is affected by the random shock Íit andÍit ∼ N401 Æ25, where Íit is observed by consumer i at thebeginning of period t, before making the decision of minuteconsumption.With regard to total discounted utility, because a con-

sumer’s current minute consumption may affect his or herfuture marginal price, the consumer aims to maximize thetotal discounted utility by optimizing consumption overtime. In particular, we can write the total discounted utilityat period t ≤ T− 1 as follows:

Uit4uit1ui4t + 151 0 0 0 1uiT5≡ E(uit +

T − t∑k = 1

Äkui4t + k5

)1

where Ä ∈ 60117, representing the discount factor.11

We model the consumer’s minute usage decision as thedynamic optimization problem of a Markov decision pro-cess (MDP) such that the decision rule of minute usageof period t depends only on the then-current state vec-tor sit (Rust 1994) and the random shock Íit . To facili-tate the exposition, we first define xit = Ñit ≡ Ñit4sit1 Íit5 asthe decision rule of consumer i at period t, depending onthe state variables sit and random shock. We also defineèit ≡ 4Ñit1 Ñi4t + 151 0 0 0 1 ÑiT5 as a decision rule profile forthis MDP from period t onward; this profile includes a setof decision rules that dictate current and future consump-tions. In addition, we denote Vit4sit3èit5 as the expectedcontinuation utility at period t conditioned on sit and èit .Because of the finite horizon of this MDP, we can define Vitrecursively as follows: The expected utility of the terminalperiod T for a given siT is

ViT4siT3èiT5≡ EuiT6ÑiT1yi −C4ÑiT53 siT1 ÍiT71(10)

where

C4ÑiT5 =

(T − 1∑k = 1

xik +ÑiT −A)pI∑T − 1

k = 1 xik + ÑiT > A1

ifT − 1∑k = 1

xik < A

pÑiT ifT − 1∑k = 1

xik ≥ A1

(11)

11We also estimate a model with an additional hyperbolic discount fac-tor. We do not find strong support for the existence of hyperbolic discount-ing in our context. Such a result is consistent with Chevalier and Goolsbee(2005) and Dubé, Hitsch, and Jindal (2010). Although time-inconsistentpreferences and, thus, hyperbolic discounting exist (Angeletos et al. 2001),they may not be universal in all contexts.

where the expectation is taken over the random shock ÍiT.Then, we can recursively write the continuation utility func-tion Vit at period t < T as follows:

Vit4sit3èit5 = Euit[Ñit1yi −C4Ñit53 sit1 Íit

](12)

+Ä[Vi4t + 154si4t + 153èi4t + 155 � sit1 Ñit

]1

where

C4Ñit5 =

( t − 1∑k = 1

xik +Ñit −A)pI∑t − 1

k = 1 xik + Ñit > A1

ift − 1∑k = 1

xik < A

pÑit1 ift − 1∑k = 1

xik ≥ A1

(13)

where the expectation is taken over the random shockÍit . Furthermore, given sit, Íit , and Ñit4sit1 Íit5, the statetransition Ï4si4t + 15 � sit, Ñit5 is deterministic such that∑t

k = 1 xik =∑t − 1

k = 1 xik +Ñit , and the consumer is one periodcloser to the terminal period T.We further recursively define the optimal decision rule

profile è∗it ≡ 4Ñ∗

it1 Ñ∗i4t + 151 0 0 0 1 Ñ

∗iT5, beginning with the ter-

minal period as

Ñ∗iT = argmax

ÑiT

uiT[ÑiT1yi −C4ÑiT53 siT1 ÍiT

]1(14)

and we can recursively define the optimal decision rule ofperiod t < T as

Ñ∗it = argmax

Ñit

uit6Ñit1yi −C4Ñit53 sit1 Íit7(15)

+Ä[Vi4t + 154si4t + 153è

∗i4t + 155 � sit1 Ñit

]0

ESTIMATION

Static Decisions

Consumers make static consumption decisions under(1) the linear pricing plans and (2) the terminal period ofthe three-part tariff plan. We discuss the likelihood functionseparately for each case.Minutes usage under the linear pricing plans. For a given

month Ò under the linear pricing plans, we observe con-sumer i’s characteristics Di. In our specific application, Diincludes (1) age, (2) the consumer’s tenure with the firm,(3) gender, and (4) whether the consumer lives in a ruralor urban area.We also observe individual consumer monthly aggregate

usage qiÒ =∑

t x∗it .

12 As we show in Appendix B, while thereis no closed form for the distribution of qiÒ , the distributioncan be approximated by a normal distribution. As a result,we can write down the likelihood function of consumer ifor the minutes usage under linear pricing plans:

Li ·Linear =∏Ò

f4qiÒ �ì51(16)

12Note that for the linear pricing plans, we observe qiÒ only and not theindividual x∗it .


where f4·5 is the approximation density function of qiÒdetailed in Appendix B and ì ≡ 8Á1b1 Æ9 (i.e., the utilityparameters and the distribution of random shocks).Minutes usage in terminal period T under the three-part

tariff. In the terminal period T, the consumption becomesa static decision given that the allowance will be reset thenext month. Thus, we can solve the optimal minute con-sumption strategy Ñ∗

iT such that

Ñ∗iT =

diT − bp1 ifT − 1∑t = 1

xit + diT − bp > A

diT1 ifT − 1∑t = 1

xit + diT < A

A−T − 1∑t = 1

xit1 ifT − 1∑t = 1

xit + diT − bp≤ A

≤T − 1∑t = 1

xit + diT0

(17)

The first component of Equation 17 accounts for the situ-ation under which the consumer faces a positive marginalprice after her cumulative usage exceeds the allowance.The second component represents the situation in whichthe consumer’s cumulative usage is less than the allowanceand the marginal price is zero. The third component rep-resents the situation in which the cumulative usage underthe optimal Ñ∗

iT exceeds the allowance at a zero marginalprice but falls below the allowance with a positive marginalprice. We follow Lambrecht, Seim, and Skiera (2007) andset the optimal usage under such a situation at a mass pointÑ∗iT = A−

∑T − 1t = 1 xit.

According to Equation 17, the density of each observedconsumption level xiT conditioned on Ñ∗

iT can be written as

fT4xiT � Ñ∗iT1ì5

=

f4xiT = dit − bp5 ifT∑

t = 1

xit > A

f4xiT = diT5 ifT∑

t = 1

xit < A

Pr(T − 1∑

t = 1

xit + diT − bp≤ A ≤T − 1∑t = 1

xit + diT

)

ifT∑

t = 1

xit = A1

(18)

and the likelihood for consumer i in the terminal period T is

Li ·Terminal = fT4xiT � Ñ∗iT1ì50(19)

Dynamic Decisions

Under the three-part tariff, decisions become dynamicfor periods t < T. As the result of the dynamic nature ofthe consumption decision, the discount factor Ä enters thedata-generating process and thus the estimation. Becausethere is no closed form solution to the optimal decisionrule in the dynamic decision context, a likelihood functionbased on observed xit and è∗

it becomes infeasible. Instead,we implement a numerical approximation method to estab-lish a simulated likelihood function for estimation. This

approximation method contains two steps: (1) using theGauss-Hermite quadrature method to approximate the valuefunction Vit at a grid of state points and interpolating Vitat the remaining state points using regression and (2) sim-ulating the density for each observed xit using Vit from theprevious step. We elaborate each step next.Approximating and interpolating Vit4sit3è

∗it5. Using

backward recursion and simulation, it is possible to numer-ically evaluate the value function under the optimal strategyè∗

it specified in Equations 15 and 17. Specifically, begin-ning with the terminal period T:

1. Pick ns = 250 random draws of state points∑T − 1

k = 1 xik toconstruct a grid (i.e., the cumulative minute usage at thebeginning of period T).

2. For each grid point that we draw, conditioned on ì, thecontinuation value function ViT4siT5 is approximated asViT4siT5 using the Gauss-Hermite quadrature method withnr = 15 nodes (Judd 1998); on each of the nodes, the optimalminute consumption level x∗iT4siT1 ÍiT5 is calculated usingEquation 17.

3. For state points that are not drawn, in line with the valuefunctions obtained on the grid from the previous step, weuse a spline interpolation to approximate their values.

Then, for period t < T, we have the following backwardrecursion steps:

4. Pick ns = 250 random draws of state points∑t − 1

k = 1 xik toconstruct a grid (i.e., the cumulative minute usage at thebeginning of period t).

5. For each grid point, conditioned on ì, Ä, and the Vi4t + 15,the continuation value function Vit4sit5 can be approximatedas Vit4sit5 using the Gauss-Hermite quadrature method withnr = 15 nodes; on each of the nodes, we calculate the opti-mal minute consumption level x∗it4sit1 Íit5 using the follow-ing equation:

x∗it4sit1 Íit5 = argmaxxit

uit6xit1yi −C4xit53 sit1 Íit7

+ ÄVi4t + 154si4t + 15 � sit1xit506. For state points that are not drawn, in line with the continu-

ation functions obtained on the grid from the previous step,we use a spline interpolation to approximate their values.

Simulating the density of observed xit, fit4xit � sit1ì1Ä5.For each xit observed in the data and its corresponding statepoint sit, we use the following steps to simulate its density:

1. Draw nrdensity = 100 random shocks Íit .2. For each random draw of Íit and the observed sit , calculate

the optimal minute consumption by solving the followingequation:

x∗it4sit1 Íit5 = argmaxxit

uit6xit1yi −C4xit53 sit1 Íit7

+ ÄVi4t + 154si4t + 15 � sit1xit503. Using the calculated nrdensity = 100 optimal x∗it4sit1 Íit5, simu-

late fit4·5, the density of the observed xit, using the Gaussiankernel density estimator.

With the simulated densities for all observed xit, we areable to write a likelihood function for consumer i such that

Li =∏t

fit4xit � sit1ì1Ä50

We use maximum likelihood estimation to estimate theparameters. The total likelihood function is

L =∏i

Li×Li ·Terminal×Li ·Linear0


Inference of the Utility Parameters

In this section, we provide an informal discussion of theinference of the utility parameters. Parameters that con-struct our model can be categorized into two sets. Thefirst set of parameters appear under both the linear pricingplan and the three-part tariff plan, including ì ≡ 8Á1b1 Æ9(i.e., the utility parameters and the distribution of randomshocks). The second set of parameters, Ä, only affect thedemand under the dynamic setting. In essence, the param-eters ì ≡ 8Á1b1 Æ9 are identified from choices under thelinear plan and the terminal period of the three-part tariff,where there are no dynamics involved. Conditioned on ì,we then recover the discount factor Ä.The consumption decisions under the linear plans and

the terminal period of the three-part tariff have no dynam-ics involved. In addition to individual consumption acrosstime, we further observe the following information underthe linear plan and the terminal period of the three-parttariff:

•Different linear prices across individual consumers;•Depending on whether a consumer has exhausted his or herallowance at the beginning of the terminal period, variation inmarginal prices across individuals; and•Variation in demographic characteristics across individuals.

The variation in consumption across and within consumersover time, conditioned on the variation in prices and demo-graphics, enables us to identify Á and b. Together, Á,b prices, and demographics determine the mean levels ofconsumptions of each consumer over time. The observeddeviations from these mean levels across consumers andtime identify Æ, the standard deviation of the randomshocks Íit .

Table 4MONTE CARLO SIMULATION, N = 50

Three-Part Linear Pricing Data and Three-Part TariffOne-Step Static Tariff Three-Part Tariff Without

with All Data Data Only Two-Step Data Only Without Last Period Data Last Period Data

Satiation d = 100 99058 (.79) 100055 (.87) 100055 (.87) 101010 (1.29) 100088 (1.03) 150055 (21.54)Price b = 1 094 (.18) 092 (.20) 092 (.20) 090 (.30) 091 (.28) 2011 (1.54)SD of shocks Æ = 05 051 (.03) 048 (.03) 048 (.03) 055 (.10) 055 (.08) 1002 (.50)Discount factor Ä = 09 088 (.10) — 088 (.14) 089 (.17) 092 (.15) 07140375

Notes: Bold values indicate significance at 95% level.


Three-Part Linear Pricing Data and Three-Part TariffOne-Step Static Tariff Three-Part Tariff Withoutwith All Data Data Without Last Last Period DataData Only Two-Step Only Period Data (Does Not Converge)

Satiation d = 100 99089 (0.66) 99059 (.83) 99059 (.83) 101000 (1.08) 100060 (.99) —Price b = 1 095 (.15) 1001 (.19) 1001 (.19) 093 (.25) 094 (.22) —SD of shocks Æ = 05 051 (.02) 051 (.02) 051 (.02) 054 (.10) 053 (.07) —Discount factor Ä = 09 091 (.07) — 089 (.11) 092 (.12) 091 (.11) —


MONTE CARLO SIMULATION

To evidence the identification approach detailed in thesection “Identification of Discount Factors with StaticData” and show that it is possible to recover discount ratesfrom sample data, we employ a Monte Carlo simulation.Using the same consumption model as in our empiricalillustration, we simulate three data sets with 50, 75, and 100customers. To be consistent with the empirical application,in this simulation, each customer makes consumption deci-sions over five periods. We set the satiation level parameterd = 100, the price sensitivity b = 1, the standard deviationof the random shock Æ = 05, and the discount factor Ä = 09.Each data set has both static data and dynamic data.We report the results in Tables 4, 5, and 6. From the

results, we observe the following:•Estimating utility with static and dynamic data (Column 1):Using the complete data, estimates of the discount factor andthe preference coefficients are statistically indistinguishablefrom their true values, well within the 95% confidence intervalof the true values.•Estimating preference parameters using only static data (Col-umn 2): Using only the terminal period of three-part tariff andthe linear pricing data, we assess whether preferences can beidentified without dynamic usage data. The results indicatethat the preference estimates are statistically indistinguishablefrom their true values, suggesting that the static data are suf-ficient to infer utility.•Estimating the preference parameters and discount factorsusing a two-step approach (Column 3): In this analysis, weillustrate the identification strategy using a two-step approach.Specifically, we estimate the preference parameters using lin-ear pricing data and the last period of the three-part tariff (as inthe preceding step). Conditioned on the preference estimates,in the second step, we then estimate the discount factor usingthe three-part tariff data, excluding the last period. The prefer-ence parameters and discount factor in this two-step approachare statistically equivalent to those estimated jointly in one



Three-Part Linear Pricing Data and Three-Part TariffOne-Step Static Tariff Three-Part Tariff Without

with All Data Data Only Two-Step Data Only Without Last Period Data Last Period Data

Satiation d = 100 100015 (.60) 100041 (.72) 100041 (.72) 100095 (.98) 100059 (.95) 133011 (25.80)Price b = 1 099 (.13) 099 (.15) 099 (.15) 095 (.21) 097 (.20) 2001 (1.21)SD of shocks Æ = 05 049 (.02) 049 (.02) 049 (.02) 052 (.09) 052 (.05) 070 (.55)Discount factor Ä = 09 089 (.06) — 089 (.08) 091 (.10) 091 (.08) 088 (.50)


single step (Column 1). However, the one-step approach ismore efficient than two-step approach (Column 3).•Estimating the preference parameters and discount rates usingonly a portion of the static data: Building on the insights weobtained from the previous two steps, this analysis further clar-ifies the role of static data in identification. Specifically, weestimate the model by pooling static data and dynamic datatogether. However, for the static data, we either use (1) thelinear pricing data or (2) the last period of the three-part tariffdata (rather than using both). The results appear in Columns 4and 5. We find that the preference and discount factor in bothcases are statistically equivalent to those estimated with allstatic data. This finding suggests that the discount factor canbe recovered when there are at least some static data to aug-ment the dynamic data. With more data, the estimate of thediscount factor becomes more efficient (Columns 4 and 5 vs.Columns 1 and 2).•In cases in which the preference and discount factor param-eter are identified, more observations under both static anddynamic settings lead to more efficient estimates (Columns1–5). Moreover, pooling static and dynamic data yields moreprecise estimates for utility (Column 1 vs. Column 3).•The model cannot recover the discount factor when no staticdata are available (Column 6).

– One of the simulations does not converge (samplesize = 75; see Table 5).

– When the model converges, the price coefficient and dis-count factor are insignificant, and the satiation level andrandom shock have greater sampling variance relative tothe identified models.

– In contrast to the case in which the discount factor isidentified, more data cannot improve the efficiency ofestimates.

•To assess potential parameter biases when the discount factoris misspecified to be .995, we reestimate our model using thisrate on the simulated data for N = 100. The satiation level dand random shock standard deviation Æ are still statisticallyindistinguishable from the true value (d = 99033, SE4d5 = 033;Æ = 047, SE4Æ5 = 002). However, the price parameter b is under-estimated (b = 053, SE = 018). The intuition behind the bias isas follows: The higher discount factor implies that consumersexcessively substitute future consumption for current within-allowance consumption. Given that future over-allowance con-sumption is costly, the model compensates by lowering pricesensitivity to generate the same level of overall utility for thefuture consumption occasion. Consequently, the smaller pricecoefficients imply that the overage charge has less impact onfuture utility; as a result, there is no need for the consumer tomake the trade-off between consumptions across time.

In conclusion, the simulations exemplify the validity of theproposed identification strategy and the potential for mis-specification of the discount rate to induce parameter bias.

RESULTS

In this section, we detail the results of our model estima-tion in the field context described in the “Illustrative Appli-cation” section. Whereas the foregoing simulation strategyillustrates the validity of our identification strategy and itsability to uncover the discount factor, the field applicationis useful in showing that (1) we can estimate discount fac-tors in an actual field setting and (2) the estimated rate isdifferent than commonly assumed in our context. Moreover,as we discuss in the “Managerial Implications” section, thisdifference can have a material consequence for the infer-ence of the utility parameters, and these biases in utilityestimates affect implied policy decisions of firms.Our sample includes the 284 consumers (50% of the

observations) who select the three-part tariff plan with amonthly access fee of 98 RMB (approximately $14), anallowance of 450 minutes, and a marginal price of .4 RMB(see Table 2). We first report the results of our base modeland then explore how these estimates are affected by theavailability of static data.13 As robustness checks, we con-clude this section by considering (1) alternative specifi-cations to our continuous specification of heterogeneity,(2) an alternative model of pricing dynamics wherein con-sumers account for minute usage when billed rather thanwhen used, and (3) potential selection bias.

Parameter Estimates

Table 7 reports the parameter estimates. Considering theutility estimates first, our price estimates imply that a userwould cut monthly consumption by approximately 11 min-utes for each unit price increase. We also find that ruralconsumers value the minute consumption marginally higherthan urban residents, probably due to their relatively lim-ited access to alternative communication methods such aslandlines and the Internet. We also find that female con-sumers have a higher baseline consumption rate than maleconsumers.Turning to the estimate for discount factor, the estimated

weekly discount factor is .9 with a 95% confidence inter-val of (.78, .97).14 The corresponding weekly interest rateis 1.11 with a 95% confidence interval of (1.03, 1.28).This discount factor is statistically lower than those typi-cally assumed in structural modeling studies (M = 0983; see

13A separate analysis using 83 consumers (access fee = 168 RMB,allowance of 800 minutes) suggests that our key results are not an artifactof the sample selected.

14We reparameterize the discount factors as Ä = exp4Ï5/61 + exp4Ï57during estimation.


Table 7ESTIMATES OF MODEL PARAMETERS

Estimate (SE)

SatiationConstant 4085 40135Price (cent) 2015 40025New customer −003 40025Age 004 40045Age2 −001 40045Rural residency 003 40015Female 006 40025SD of shocks 7046 420055Weekly discount factor 090 40055


Table 1). Placing this result in perspective, our estimatesindicate that a consumer values an immediate one-minutephone call at the beginning of the month the same as afuture call of one minute and 34 seconds at the end ofthe month. In contrast, a weekly discount factor of .995implies that a consumer values a one-minute phone call atthe beginning of the month to be equal to a one-minute andone additional second at the end of the month (under theassumption of a constant pricing rate). The implication inour model is that consumers would be hesitant to make thelatter trade but willing to make the former.15

The Effect of Static Data on Inference

As in the case of the simulation, we consider howstatic data affect inference in our sample of field data (seeTable 8). Overall, the findings are consistent with thosein our Monte Carlo simulation—namely, that it is difficultto infer discount factors without static data and that moreinformation improves the efficiency of our estimates.In particular, we note that the model cannot recover the

discount factor when no static data are available (Table 8,Column 5). Reflective of identification issues, when esti-mating the model on three-part tariff data without the lastperiod or linear data, the estimation algorithm takes a muchlonger time to converge even though fewer data are used inestimation (approximately 2.5 times less than the poolingcase). More important, the discount factor estimate has amuch larger variance (Column 5 in Table 8). The discountfactor takes the median of .95 and a standard error of .52.We implement two likelihood ratio tests for H0: Ä = 0 andH0: Ä = 1. In both tests, we cannot reject the null hypothe-ses; that is, we cannot determine the discount factor withoutthe static data.

15This estimate corresponds to a high annual interest rate of 22,640%.Consumers evidence high levels of impatience in some settings. Warnerand Pleeter (2001) find that annual discount factors vary between 0 and.58 (annual interest rate > 61× 10147%) in a field study by observinghow agents trade off current and future payoffs. In experimental set-tings, researchers also find that discount factors may approach zero (e.g.,Kirby and Marakovic 1995; Madden et al. 1997; Petry and Casarella1999). For a more detailed discussion, see Frederick, Loewenstein, andO’Donoghue (2002).

Robustness Checks

In this section, we consider several robustness checksto assess how inference regarding the discount rate in oursample might be affected by the interval used for aggregat-ing the data, our assumptions about heterogeneity, and thetiming of the payment. We then discuss selection effects.Data aggregation. We consider the potential effect of

temporal aggregation of data on our model estimates. Thisrobustness check demonstrates the effect of data aggre-gation on the identification of discount rates and prefer-ences (ten-day, five-day, and three-day). The five-day andthree-day results are similar to the results reported here;therefore, we focus on the ten-day results. As indicated inColumns 6 and 7 of Table 8, we cannot recover prefer-ences and discount rates using the ten-day intervals unlesswe use both the static and dynamic data. These results sug-gest that (1) by reducing the number of observations andthe concavity evidenced in Figure 2 (from which we inferintertemporal trade-offs in demand), data aggregation leadsto a decrease in the efficiency of the parameter estimatesand (2) this problem can be addressed to some degree byusing the static data.Heterogeneity. As another robustness check, we explore

the effect of our parametric assumption regarding con-sumer utility on the inferred discount factor. To do this,we estimate the model consumer by consumer (by com-bining static and dynamic data), thereby allowing all usersto have different preferences and discount factors. We findthe median discount factor across consumers to be .87(SD = 026), similar to the level estimated under the homoge-nous model.16 We also consider a latent class mixturemodel. This analysis yielded two segments. The first, com-prising 85% of the consumers, evidenced a discount rateof .89 (SE = 004), while the smaller segment rate was esti-mated to be .92 (SE = 005). Overall, it appears that ourdiscount factor estimates are robust to our specification ofheterogeneity.Payment timing. Note that Equations 5 and 9 presume

that consumers account for minutes at the time they aretransacted rather than when they are billed. To explore thisassumption, we consider some model-free tests. Specifi-cally, when the “mental accounting” of the expense is con-current with the billing date and the pricing plan is linear,the usage decision becomes dynamic in the early periods of

16Recognizing the potential for small sample problems arising frommaximum likelihood estimation in our panel (we have limited observa-tions per customer for inference), we use simulated data to explore thispotential. Specifically, we generate data using the parameters estimatedfrom our homogeneous model. In one case, we generate 1000 static obser-vations per consumer. In another case, we use 3 static observations (asobserved in the data) to supplement the dynamic data. We then proceed toestimate our model consumer by consumer for both simulated data sets.The estimates from the larger sample are statistically equivalent to thosein the current study (price Mdn = 2017, SD = 010; discount Mdn = 091,SD = 011). We further find a small degree of small sample bias for thethree-observation case. The price coefficients are slightly overestimated(Mdn = 2099, SD = 079) and the discount factors are slightly underesti-mated (Mdn = 084, SD = 028); but they are both still statistically equivalentto those in the current study (2.15 and .90). The simulation suggests thatthe small bias is not sufficient to change our conclusion that the static dataenable identification of discount rates. This is consistent with our MonteCarlo simulation results.


Table 8ROBUSTNESS CHECK: IDENTIFICATION

Three-Part Linear Pricing Data Three-Part Tariff All Data withStatic Tariff and Three-Part Tariff Three-Part Tariff Data Only with Ten-DayData Data Without Last Period Without Last Ten-Day Period PeriodOnly Two-Step Only Data Period Data Definition Definition

Constant 4092 (.19) 4092 (.19) 5015 (.52) 5007 (.45) 3099 (1.26) 9012 (1.26) 7089 (.81)Price (cent) 2021 (.05) 2021 (.05) 2003 (.14) 2013 (.09) 1050 (1.00) 1073 (1.41) 2009 (.10)New consumer −005 (.03) −005 (.03) −001 (.11) −002 (.10) 010 (.10) −002 (.08) −004 (.07)Age 002 (.05) 002 (.05) 006 (.24) 005 (.23) 011 (.30) 002 (.14) 001 (.10)Age2 −0004 (.06) −0004 (.06) −009 (.26) −005 (.21) −001 (.29) −005 (.15) −005 (.16)Rural residency 004 (.01) 004 (.01) 003 (.04) 001 (.02) −003 (.08) 004 (.07) 005 (.03)Female 004 (.02) 004 (.02) 0091 (.046) 011 (.05) 021 (.39) 033 (.08) 010 (.09)SD of shocks 7075 (2.28) 7075 (2.28) 7050 (3.81) 7059 (2.91) 9091 (3.60) 9058 (4.32) 9099 (3.90)Weekly discount factor — 089 (.09) 092 (.13) 092 (.11) 095 (.52) 085 (.55).


the month. This is because the consumption xit in period twill affect the total payment in period T, but not the periodin which the usage occurs. This implies a downward trendin optimal consumption within the month because futurepayments are discounted more steeply in earlier periods.If the payment only materializes in period T, the utilitiesin periods t and T become

uit =ditxitb

−x2it2b

+ yi

uiT =diTxiTb

−x2iT2b

+ yi − pi0T∑

k = 1

xik0

The optimization problem of period t is

x∗it = argmaxxit

uit + ÄT − tEduiT0

Taking first-order condition to solve x∗it , we can show that

x∗it = dit − bÄT − tpi00

This implies that, on average, the minute usage shoulddrop over time if the payment only materialized in the util-ity function of period T. This observation suggests somemodel-free tests of whether consumers budget for minutesat the point of consumption or at the payment:

•Model-free evidence 1: Although our data for the first fewmonths do not include call-by-call observations under the lin-ear pricing plan, sales agents contacted consumers in month 3to offer the three-part tariff option. The timing of the call foreach consumer was random, and the consumers had no fore-sight of the call. Importantly for our immediate concern, datawere collected regarding the total minute consumption underthe linear plan before the offer. These data enable us to calcu-late average weekly minute consumption across consumers upto the week of the switch. If the consumers account for pay-ments at the terminal period T, usage rates over time shoulddecrease on average. Figure 4 is a box plot that depicts theaverages of weekly usages of consumers who switched duringthe same week. It indicates no downward trend in consump-tion. This pattern is inconsistent with a mental accounting ofexpense at the time of billing.•Model-free evidence 2: In the bottom right-hand panel of Fig-ure 3, when consumers exceed the quota, the three-part tar-iff becomes a linear plan. If there is a downward trend in

consumption, we should likewise see the histogram skewedtoward 0, reflective of usage deceleration. However, accordingto Figure 3, the consumption rate is centered on 1, which isalso inconsistent with the mental accounting of expense at thetime of billing.•Model comparison: In addition to these model-free tests, wefurther consider a third test based on the three-part tariff data.We estimate a dynamic model similar to the base model butwith the payment only entering in period T. The results indi-cate that the price sensitivity is not significant and the discountfactor, though similar to our estimate in the current study, hasa large standard error. Model fit deteriorates as measured byBayesian information criterion (BIC) (9470.12 vs. 9511.30)and mean absolute percentage error (MAPE) (.14 vs. .23).

Accordingly, the data seem to be more consistent with the“mental accounting” of payment at time of consumptionthan with payment at time of billing.Selection. Our data include only consumers who switch

price schedules. Although the static and dynamic data aresufficient to identify the preference and discounting, asshown in the Monte Carlo simulation, to the extent a con-sumer’s adoption decision could be endogenous, the esti-mates in this field application may not be generalizedto the overall population due to potential selection bias.To explore the potential selection effects, we estimate the

Figure 4AVERAGE USAGE OVER WEEKS

432

90

100

110

Ave

rag

e M

inu

tes

Usa

ge

120

130

1

Week of the Switch


model separately for consumers who chose two differentthree-part tariff plans (450 minutes plan vs. 800 minutesplan). Table 9 presents the results.Of note, the discount factors for consumers selecting

each plan are statistically similar (.90 vs. .88). In contrast,the preference estimates are different. In particular, thosewho chose the 450-minute plan have a lower baseline con-sumption (a smaller constant estimate) and lower price sen-sitivity. This is intuitive because (1) consumers with lowerbaseline consumption needs are more likely to enroll in alower allowance plan and (2) consumers with lower pricesensitivity are more likely to enroll in a lower allowanceplan because they care less about overage charges.This robustness check merely shows that the estimates

are conditional on a set of consumers that self-select into apricing scheme that induces a forward-looking incentive (asin Goettler and Clay 2011). However, one key implicationof this exercise is that the findings are consistent with theconjecture that selection plays a greater role in consumptionutility than discounting.

MANAGERIAL IMPLICATIONS

The dynamics of consumer behavior may have substan-tial impact on firm strategic decisions and revenues. Inthis section, we build on the broad literature pertaining toplan choice over months for telephony or Internet services(e.g., Grubb and Osborne 2012; Iyengar, Ansari, and Gupta2007; Lambrecht and Skiera 2006; Lambrecht, Seim, andSkiera 2007; Narayanan, Chintagunta, and Miravete 2007)to explore (1) the implications of within-month pricing pol-icy on carrier revenue and (2) how discount rates affectthese policies.

Usage Prediction and Intertemporal Substitution Pattern

Biased price effects. To assess the potential bias in modelestimates arising from specifying the commonly employeddiscount factor of .995 rather than using the estimated dis-count factor, we reestimate the model by fixing the dis-count factor to .995. While there is little impact on mostestimates, we find the price coefficients to be underesti-mated (have smaller absolute magnitudes) by 16%. Theprice coefficient becomes 1.81 (vs. 2.15 in Table 7), with astandard error of .03. This underestimation of price effects

Table 9ROBUSTNESS CHECK: SELECTION

Plan 450 MinutesPlan 800 (Estimates in theMinutes Current Study)

Constant 5005 (.33) 4085 (.13)Price (cent) 2091 (.04) 2015 (.02)New customer −010 (.05) −003 (.02)Age 005 (.11) 004 (.04)Age2 −002 (.05) −001 (.04)Rural residency 010 (.03) 003 (.01)Female 019 (.07) 006 (.02)SD of shocks 8085 (2.01) 7046 (2.05)Weekly discount factor 088 (.06) 090 (.05)


is consistent with the findings of the Monte Carlo simula-tion we performed.Biased forecasts. To ascertain how well the model fits

the data and resulting intertemporal substitution pattern, wecalculate the in-sample MAPE, BIC, and mean percentageerror (MPE) across time under both the estimated discountrate and the assumed weekly discount factor of .995. TheMAPE measures a model’s overall accuracy of fitting thedata, while the MPE indicates bias in model predictions.Tables 10 and 11 depict the results.According to Table 10, the fit under the .995 discount

factor is universally worse than the fit under the estimatedcoefficients across time as measured by both MAPE andBIC. To develop a better sense of why the higher dis-count factor performs more poorly, we next turn to theMPE. Based on Table 11, there is no obvious forecast-ing bias from using the higher discount rate when sum-ming across all periods, and yet aggregating across timeobscures the patterns in intertemporal substitution. Whensetting the discount factor at .995, the demand in the firstfour periods is underestimated, and the demand in the ter-minal period is overestimated. This bias occurs becauseconsumers are more impatient than what Ä = 0995 implies.As a result, in the early periods, when the allowance hasnot been exhausted, impatient consumers consume morethan predicted under the .995 discount factor. Furthermore,consumers are more price sensitive than the .995 discountfactor case implies (recall that the price coefficient is under-estimated under the .995 discount factor). As a result, con-sumers in overage (roughly coincident with the last period)evidence lower consumption than predicted under the .995discount factor.

Table 10MAPE COMPARISON

MAPE

Estimated Discount Factors (BIC: 9470.12)First four periods .14Final period .16Monthly aggregate .14

Discount Factor Set to .995 (BIC: 9499.95)First four periods .17Final period .21Monthly aggregate .18

Notes: Likelihood ratio test Õ2 = 37042, H0: Ä = 0995 is rejected.

Table 11MPE COMPARISON

MPE

Estimated Discount FactorsFirst four periods .02Final period → 0Monthly aggregate .02

Discount Factor Set to .995First four periods −009Final period .02Monthly aggregate −003


Although the difference between the estimated dis-count factor (M = 090, with a 95% confidence interval of40781 09755 and .995 seems negligible, the effect on theforecast results are highly significant. This is because thejoint distributions of all of the coefficients differ consider-ably under the two scenarios. The inflated discount factor.995 also causes biased estimates of preference, especiallydownward-biased price sensitivity (2.15 vs. 1.81).

Elasticities

To ascertain how consumers’ minutes usage varies underalternative allowance and the marginal price levels, wecompute their monthly minutes demand changes for boththe estimated discount factor and .995. Table 12 presentsthe results. The elasticities in Table 12 suggest that the .995discount factor leads to an underestimation of the effect ofallowances and price on usage (i.e., users are not as pricesensitive as it implies when the discount factor is set to.995). If consumers were to actually have a discount factorof .995, they would be more forward looking than they areunder the actual discount rates we estimate. As a result, themore forward-looking consumers implied by .995 shouldconserve minutes so as not to pay overage in later periods.Because they do not actually conserve minutes, the modelwith a .995 discount factor needs to rationalize the observedoverage. It does so by estimating a relatively lower sensi-tivity to price and allowance; a lower price and allowancesensitivity means that consumers do not mind paying over-age as much and have lower elasticities.

Alternative Pricing Schedule

According to our communication with the data provider,its process of picking the three-part tariffs in this field studyis ad hoc: There was no price optimization consideration.As a result, the focal three-part tariff is not likely to beoptimal for the firm in terms of maximizing its revenue.To access the potential for revenue improvement, we cre-ate a grid of alternative allowance and price levels. Foreach combination of allowance and price, we calculate the

Table 12DEMAND ELASTICITIES

Demand Elasticity Demand Elasticitywith Regard to with Regard toPrice Overall Price After Overage

Estimated discount −007 4−0091−0055 −019 4−0211−0165factors

Discount factor −006 4−0071−0045 −013 4− 0141−0115set to .995

Demand Elasticity withRegard to Allowance

Estimated discount 024 40211 0265factors

Discount factor set 018 40151 0195to .995

Notes: 95% confidence intervals are in parentheses. Bold values indicatethat the biases are significant (p > 005).

percentage of revenue change at the end of the month.17

Table 13 reports the results.18

Table 13 includes the current plan (allowance = 450 min-utes, price = RMB040). Surrounding the current plan, eachcolumn from left to right represents a 2-cent change inthe marginal price for minutes in overage and each rowfrom top to bottom stands for a 25-minute change in theallowance. A lower allowance enhances the possibility ofoverage; and a moderately decreased price tends to increasethe consumption level under the overage situation. Theoptimal combination of allowance and marginal price is400 minutes and .36 RMB. The revenue of the firm wouldincrease by 1.11% under this alternative pricing schedule.To the extent that similar exercises can be implementedacross all groups of consumers, the revenue increase wouldbe considerable.As shown previously, under the discount factor of .995,

the model may lead to biased estimates of coefficients andelasticities. To determine whether such biases may lead toinaccurate policy recommendations, we recreate the samegrid but calculate the revenue changes using the estimatesunder the .995 discount factor. As Table 14 indicates, withthe .995 discount factor, the model generates notably dif-ferent pricing plan recommendations. Because the assumeddiscount factor is much higher, to enhance consumers’ like-lihood of overage, the allowance levels would be muchlower. As a result, under the .995 discount factor, the opti-mal allowance level becomes 325 instead of 400. Moreover,because the price sensitivities are underestimated, the opti-mal price would be higher. This effect leads to the optimalprice changes from .36 to .38. In short, the firm sets itsallowance too low and its marginal price somewhat high,thereby overcharging its consumers when using the stan-dard practice of setting discount rates.The predicted revenue gains also significantly differ

between the two scenarios, Ä = 090 vs. Ä = 0995. For eachgrid point, we calculate the predicted revenue differencebetween the two scenarios. As in Figure 5 shows, the per-centage differences can be substantial. By implementingthe pricing plans as suggested by the model with Ä = 0995,the firm would forgo revenue gains. For example, withan allowance of 400 and a price of .36 (where Ä = 090),the firm’s revenue improvement would be 1.11% (95%confidence interval (.80%, 1.43%)). However, if the firm

17Note that we do not model plan choice because there are no planchoice data available. To ensure that the presented price/allowance changesdo not lead to consumers leaving the company or opting for a differentplan, we calculate consumer welfare for each point on the grid as mea-sured by consumers’ total discounted utility. We then compare it withthe welfare level under the original plan. None of the welfare changesis significantly different from zero; thus, we do not believe that the rec-ommended policies will result in substantial plan switching even thoughthese changes benefit the firm. Furthermore, the company had a significantmarket share, and there was no cell phone number portability in Chinauntil October 2010 (ChinaTechNews.com 2010). Consequently, we con-clude that (1) the new price structures in Table 12 are unlikely to havecaused large consumer churning and plan switching and (2) the effect ofcompetitive response is likely to be modest.

18To account for sampling error when computing the revenue changes,we sample 100 sets of values from the estimates distribution. We thencalculate the revenues using these sampled estimates. The table presentsthe median values.


Table 13REVENUE PERCENTAGE CHANGE UNDER ALTERNATIVE PRICING SCHEDULES (Ä = −090)

Marginal Price

Allowance (Minutes) .34 .36 .38 .40 .42 .44 .46

300 −011 −006 −005 −003 −009 −015 −008325 005 019 027 020 007 −001 −005350 005 023 030 036 029 025 018375 089 093 098 072 069 055 040400 092 1011 097 096 083 077 061425 082 095 094 080 073 068 051450 001 027 022 0 016 011 007475 −018 012 −015 −022 −030 −033 −051500 −021 −014 −023 −025 −033 −048 −061

Notes: The best alternative is in boldface.

Table 14REVENUE PERCENTAGE CHANGE UNDER ALTERNATIVE PRICING SCHEDULES (Ä = 0995)

Marginal Price

Allowance (Minutes) .34 .36 .38 .40 .42 .44 .46

300 059 079 092 072 061 055 039325 065 080 096 088 077 073 064350 048 057 071 094 083 080 075375 043 047 054 068 058 044 039400 011 023 033 049 044 040 036425 008 014 019 021 021 018 017450 −011 −007 −002 0 −001 −005 −008475 −022 −021 −017 −023 −031 −034 −038500 −039 −036 −033 −041 −046 −051 −057

Notes: The best alternative is in boldface.

used Ä = 0995, it would predict that the revenue couldonly improve by .23% (95% confidence interval (−005%,.67%)). In comparison, with Ä = 0995, the firm would adoptthe “optimal” plan with an allowance of 325 and a priceof .38 and expect a revenue improvement of .96% (95%

Figure 5REVENUE PREDICTION DIFFERENCES: Ä = 090 VERSUS

Ä = 0995

.36

.38

.40

.42.44

.46

450

500 .34

400

350

300

–.5

.0

.5Differen

ce (%)

Allowance (minutes)

Price (R

MB)

confidence interval (.66%, 1.23%)). However, the actualrevenue improvement would only be .27% (95% confidenceinterval 4−001%1 053%55. Such a suboptimal pricing levelwould reduce the potential gains by 76% relative to theoptimal pricing level.

CONCLUSION

Due to the ability to capture the trade-off between long-term and short-term goals, the application of dynamic struc-tural models to consumer and firm decision making hasbecome increasingly widespread. However, dynamic struc-tural models face a fundamental identification problem—namely, that the preference, the state transition, and thediscount factor are confounded and become difficult toidentify simultaneously. If the rate is misspecified, infer-ences about agent behavior might be misleading, and theimplied policies for improving agent welfare might besuboptimal.We advance the literature on identifying discount fac-

tors by using field data to measure them. Specifically, weestimate a dynamic structural model using consumers’ cellphone usage data. The data contain observations of con-sumers’ cell phone consumptions under both static anddynamic settings. Using the static data, we first identifyconsumers’ utilities and the distribution of random con-sumption shocks. Conditioned on the identified utilities andrandom shocks, we then recover the discount factor usingthe dynamic data. The identification strategy proposed mayhave more general applications in contexts beyond the cell


phone pricing strategy we consider herein. The identifica-tion of discount factors is possible if researchers observeconsumers making continuous decisions under both staticand dynamic settings. Such contexts may include loyaltyprograms, markdown pricing during clearance and seasonalsales, some finite duration auctions, and perishable goodssuch as ticket sales.Findings suggest that the discount factor in our spe-

cific context (.90) is significantly below those commonlyassumed in the literature (.995). As a consequence, priceeffects are underestimated in our application. Moreover, thehigher rate leads to a mistaken presumption that more min-utes would be saved for later use, leading to a 29% increasein the MAPE in model fit. The attendant consequences forpricing policy are notable, leading to pricing recommenda-tions that are generally too high and would lower potentialrevenue gains by 76%.The inherent complexity of dynamic structural models

often requires simplifications that correspondingly repre-sent future research opportunities. Our model is no excep-tion. First, our study focuses on a specific consumptioncontext with a small focal group of consumers over a spe-cific duration. Therefore, the results may not generalize toother contexts involving different consumers, decisions, ordecision durations. Thus, more research is necessary to gen-eralize the degree to which discount rates are an inherenttrait or the degree to which they are context dependent.For example, it would be fruitful to consider how discountrates might vary in practice when considering different con-texts of intertemporal consumption; do consumers invokethe same level of patience when making choices over yearsas they do when making decisions over days?Second, two potential sources of selection bias exist in

our field study. The first arises from the firm’s choices ofconsumers to participate in the plans. Per our discussionwith the firm’s managers, consumer selection was random-ized, so this form of selection bias is not germane. The sec-ond selection bias arises from the consumer’s decision ofwhether to adopt the three-part tariff plan that was offered.The decision to adopt the plan can be correlated with bothpreferences and discounting behavior. As such, our esti-mates should be interpreted conditional on plan selection(as in Goettler and Clay 2011), limiting the generalizabil-ity of estimated rates to the overall consumer populationof the company. As a result, another area of interest isto extend our research into the domain of plan choice tobroaden insights regarding the distribution of discount ratesacross the population. Third, our identification strategy isbased on a dynamic model with continuous controls. Wediscuss the set identification conjecture for discrete deci-sions in Appendix A. A more formal investigation on theidentification on discrete decision dynamic models wouldbe a challenging but significant extension.In summary, the current study is the first (to our knowl-

edge) to provide field study–based evidence regarding thenature of discount rates that obviate the need for structuralassumptions or exclusion restrictions to identify discountrates. Consistent with Dubé, Hitsch, and Jindal (2010) andIshihara (2010), we find evidence that discount rates aresubstantially lower than those used in practice and that thisdifference is material from a policy perspective. Given thewidespread use of dynamic models in marketing and eco-nomics, we hope our analysis will spark future work to

pin down how such intertemporal trade-offs are made inpractice.

APPENDIX A: THE IDENTIFICATION OF THEDISCOUNT FACTOR

Appendix A details the identification of the discountfactor when the utility and random shock distribution areknown. We extend the discussion of finite horizon case inthe “Identification of Discount Factors with Static Data”section to a more general infinite horizon case. We showthat, conditioned on the utility and random shock distribu-tion being identified by the static data, the discount factoris identified. We also provide a conjecture about the iden-tification of discrete choice dynamic decision process withfinite horizon.

Infinite Horizon with Continuous Control

For the infinite horizon case, at any given period of t,we can write the optimization problem of continuous con-trol as

maxxt

u4xt3 st1 Ít5+�∑

k = t + 1

Äk − tEu6xk3g4sk − 11xk − 151 Ík71(A1)

where x. are the continuous control; s. is the state variable;Í. is the random shock and the expectation is taken over therandom shock; and g4·1 ·5 is the state transition equation.Because the backward induction is no longer an option

for the infinite horizon case, we need to instead establishthat the sequence of xt1 ∀ t converges to a unique limit,which is the optimal decision rule of Equation A1. To showthat the solution to the optimization problem exists andis unique, we first rewrite the dynamic problem of equa-tion A1 in the form of the Bellman’s equation as (we dropthe subscript t):

V4s1 Í5 = maxx

u4x3 s1 Í5+ ÄEV6g4s1x51 Í′71(A2)

where V4·5 is the value function.Next, we check the Blackwell’s sufficient conditions to

show that Equation A2 is a contraction mapping with mod-ulus Ä (Stokey and Lucas 1989, p. 55):

1. (Monotonicity) Take V1 ≥ V2. Then,

TV1 = maxx

u4x3 s1 Í5+ ÄEV16g4s1x51 Í′7

≥ maxx

u4x3 s1 Í5+ ÄEV26g4s1x51 Í′7

= TV20

2. (Discounting) For a real number a ≥ 0,

T4V+ a5 = maxx

u4x3 s1 Í5+ Ä8EV6g4s1x51 Í′7+ a9

= maxx

u4x3 s1 Í5+ ÄEV6g4s1x51 Í′7+ Äa

= TV+ Äa0

Because both conditions are satisfied, V is a contractionmapping and there is a unique solution to the optimiza-tion problem. Denote the corresponding optimal policy ruleas Ñ∗. It follows that at any period t, x∗t = Ñ∗4st1 Ít5 must beoptimal. Examining Equation A1, this means that for anyperiod t, x∗t must solve

maxxt

u4xt3 st1 Ít5+ ÄEu6xt + 13g4st1xt51 Ít + 171


which implies the Euler Equation of the optimization prob-lem in Equation A1:

ux4x∗t 3 st1 Ít5+ ÄEug6x

∗t + 13g4st1x

∗t 51 Ít + 17 = 01

where ux and ug are the partial derivatives with respect tox and g, respectively. Note that the Euler equation can alsobe derived using the first-order condition of Equation A2combined with envelope theorem. Given that the utility andthe distribution of Í are known, and x.* are the uniqueand observed optimum, the discount factor Ä can then beuniquely identified from the data.

Finite Horizon with Discrete Choice

The identification of discrete choice dynamic decisionprocess is beyond the scope of the current study. Instead ofproviding a formal argument, we conjecture that set iden-tification of the discount factor is possible with parametricrestrictions on the utility shock. First, for the static dis-crete choice model, the distribution of random shock inthe utility function is normally unidentified. It is a com-mon practice to impose additional assumptions, including(1) one alternative’s utility is normalized to 0 (e.g., the out-side option) and (2) distribution assumptions regarding therandom shock (e.g., probit and logit assume the distributionand normalize the standard deviation to fixed values). Giventhe assumptions, the utility parameters may be identified.Second, given the additional assumptions and the utility

identified in the static period, we can write the optimizationproblem of an agent as

maxat ∈ A

u4at3 st1 Ít5+T∑

k = t + 1

Äk − tEu6ak3g4sk − 11 ak − 151 Ík71(A3)

where A is the set of available alternative actions, at is theaction chosen in period t, st is the state, Ít is the randomshock and the expectation is taken over the random shock,and g4·1 ·5 is the state transition function such that st + 1 =g4st1xt51 ∀ t.

Now we define the Bellman equation based on Equa-tion A3 for t < T:

Vt4st1 Ít5 = maxat ∈ A

u4at3 st1 Ít5+ ÄEVt + 16g4st1 at51 Ít + 170(A4)

Conditional on the assumed distribution of random shocks,normalized outside option, identified utility, and observedaction a∗t , EVt + 16g4st1 a

∗t 51 Ít + 17 is known. The observed

at must satisfy

a∗t = argmaxat ∈ A

Vt4st1 Ít5(A5)

= 8at 2 u4at3 st1 Ít5+ ÄEVt+16g4st1 at51 Ít + 17

≥ u4a′t3 st1 Ít5+ ÄEVt + 16g4st1 a′t51 Ít + 171 ∀ a′t ∈ A93

that is, a∗t results in the highest value function amongall alternatives. However, the optimal decision rule a∗t =Ñ∗4st1 Ít3 Ä5 may not be injective with respect to Ít and Ä.For example, for two different discount factors, Ä 6= Ä′, thereare two different sets of values of Ít and Í′

t such that a∗

satisfies Equation A5, though the two sets of Ít and Í′t can

overlap. As a result, for a given pair of observed (st, a∗t 5,

both Ä and Ä′ satisfy Equation A5. In contrast, it is possible

for Ä to be set identified. For example, both Ä and Ä′ arein that set. When there are more observations, potentially,the identified set of Ä becomes smaller and the inferencepertaining to Ä becomes more accurate. In contrast, whenstatic data are not available, parametric restrictions on therandom shock and the utility and discount factor are notsufficient for identification in a nonparametric sense (e.g.,Magnac and Thesmar 2002; Rust 1994).

APPENDIX B: THE DISTRIBUTION OF MONTHLYMINUTES USAGE qiÒ UNDER LINEAR PRICING PLAN

Because we only observe the monthly minute consump-tion qiÒ = ètx

∗it but not each respective x∗it , we need to find

the likelihood of qiÒ . As discussed in Equation 6, the opti-mal minutes usage at period t, x∗it , may take two values,0 (if dit − bpi0 ≤ 0) and dit − bpi0 (if dit − bpi0 > 0). Becausedit = exp4D′

iÁ5+ Íit and Íit ∼ N401 Æ25, x∗it follows a normaldistribution N6exp4D′

iÁ5−bpi01 Æ27 that is truncated at zero.

Thus, the density of x∗it is

f4x∗it5 =1ÆÔ

(x∗it −Ìi

Æ

)/[1−ê

(−Ìi

Æ

)]1

where Ìi = exp4D′

iÁ5− bpi00(B1)

The monthly minute consumption qiÒ = ètx∗it can then be

written as the summation of a series truncated normal ran-dom variables with the same truncation at zero. Althoughthere is no closed form for the distribution of qiÒ , if theoccurrence of zero minute consumption for any period isnearly zero 6Pr4x∗it > 05→ 11 ∀ t7, qiÒ can be approximatedwell by a normal density function whose mean is T×Ìiand the variance as T× Æ2g.

19 Intuitively, although x∗it is atruncated normal random variable, if Pr4x∗it > 0) is nearlyone, the truncation becomes moot and the distribution of x∗itcan be approximated well by a normal distribution. Thesummation of a series of i.i.d. normal random variablesis also normally distributed. We implement a Monte Carlosimulation using this approximation. The parameters arerecovered with reasonable accuracy. As a robustness checkto this approximation, we also use a Kernel estimator tocompute the density in the Monte Carlo simulation (Härdleand Linton 1994). The results are similar to the ones usingthe approximation, but the Kernel estimation is much morecomputationally demanding.

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