Optimal choice and consumption of cost cap tariffs: theory and empirical evidence
 1.2k Downloads
 2 Citations
Abstract
Cost cap tariffs are payperuse tariffs for which costs cannot exceed a predefined cost limit. They were recently introduced to telecommunications markets, but were previously also applied in the insurance industry as deductibles or in the rental industry as day rates. This paper develops and empirically validates a consumer surplus model that explains the optimal consumption pattern under cost cap tariffs and the conditions under which cost cap tariffs are chosen over pure payperuse and flat rate tariffs by a rational consumer. We find that cost cap tariffs are an optimal tariff choice only if the level of uncertainty is sufficiently high. Our theoretical predictions are supported by survey data.
Keywords
Tariff choice Consumer surplus model Cost cap tariff Payperuse tariff Flat rate tariff Tariff bias Service industries Telecommunications Tariff biasJEL Classification
D11 D12 M311 Introduction
A cost cap tariff is a twopart tariff that is a hybrid between a pure payperuse tariff (where costs accrue with usage) and a flat rate (where costs are independent of usage). The cost cap tariff is a pure payperuse tariff until the costs reach an upper limit. At this cost level, the tariff effectively becomes a flat rate because any further consumption is not charged to the consumer. Thus, the consumer might pay less, but never more than this upper limit.
Cost cap tariffs were first introduced in addition to existing payperuse plans and flat rates in the German mobile communications market in 2009. Provider O2/Telefonica introduced this new tariff type for voice communication in 2009 with a price of 0.15 €/min and a monthly cost cap of 60 €. At the time, this pricing was such that the minute price of the cost cap tariff exceeded the common market price for pure payperuse tariffs (0.09 €/min) while the cost cap was set at the level of comparable flat rate plans. The introduction of the cost cap tariff led to a clear increase in customers for O2 (Briegleb 2009). As a consequence, several competitive virtual mobile network operators soon followed with similar offers. While these mobile network operators are still offering cost cap tariffs, the initial provider O2 decided to no longer advertise its cost cap tariff. Thus, the long run effects on consumer choice and providers’ profits are an open research question. Furthermore, the use of cost cap tariffs is not limited to the telecommunications market. Insurance services often include deductibles which behave in the same fashion as a cost ceiling of consumer payment. Also rental services, e.g., for car or bike sharing, often include a timedependent rate which is covered by predefined day rates. Table 1 illustrates some selective examples for different industries. Thereby, note that cost cap tariffs are offered both by the same company in addition to its payperuse and flat rate tariffs as well as in response to flat rate and/or payperuse tariffs by competitors.
Exemplary tariff structures in different industries (September 2013)
Payperuse  Cost cap  Flat rate  

Telecommunications  
Tariff  O2 Loop  O2o  O2 Blue Allin 
Minute price p  0.09 €/min  0.15 €/min  
Cost cap c  50 €/month  
Fixed fee f  39.99 €/month  
Tariff  ntv go  simyo 9 Cent Tariff  klarmobil AllnetFlat 
Minute price p  0.06 €/min  0.09 €/min  
Cost cap c  39 €/month  
Fixed fee f  24.85 €/month  
Car rental  
Tariff  Drive Now  car2go  Enterprise 
Hour price p  0.31 €/min  14.90 €/h  
Day rate c  59 €/day  
Fixed fee f  49 €/day  
Bike rental  
Tariff  DB Call a Bike  DB Rental Station  
Hour price p  0.08 €/h  
Day rate c  15 €/day  
Fixed fee f  12.70 €/day  
Fitness studio  
Tariff  flexifit Vienna  euroGym Vienna  
Hour price p  0.10 €/min  
Day rate c  15 €/day  
Fixed fee f  14 €/day 
In this article, a simple model of consumer surplus is developed that seeks to explain consumers’ rationale in choosing a cost cap tariff over payperuse plans and flat rates. We focus on the realistic case where the marginal price of the cost cap tariff is not smaller than the marginal price of the payperuse tariff and where the cost cap level is at or above the price level of the flat rate. Otherwise, the cost cap tariff would clearly dominate both the payperuse plan and the flat rate. Moreover, to disentangle the pure pricing effect of tariffs from other effects that may influence a consumers tariff choice, such as tariff biases or brand effects, we assume that consumers base their consumption decision solely on prices and their (uncertain) preference for the service offered.
We find that cost cap tariffs should never be chosen over payperuse or flat rate plans if consumers are certain about their preferences and consequently in their demand. In this case, the cost cap tariff is always dominated by either the payperuse plan or the flat rate. However, if consumers have uncertainty in their preferences, the cost cap tariff may generate a higher (expected) consumer surplus than a payperuse plan or a flat rate. This holds true even though consumers are charged more at the margin than in a payperuse tariff and the maximum possible bill amount is higher under a cost cap compared to a flat rate tariff. This is because the cost cap tariff does not only provide cost insurance in case of high demand (like a flat rate), but also cost flexibility in case of low demand (like a payperuse tariff). The main research questions that are addressed are (1) under which conditions are cost cap tariffs potentially chosen (over payperuse plans and flat rates)? and (2) what is the impact of demand uncertainty on tariff choice?
Finally, our theoretical predictions are compared to empirical data that was collected in a survey among a representative sample of German mobile telephony customers. We find a good model fit, both with respect to the predictions for expected telephony usage and for expected consumer surplus under a given tariff. However, we also find a systematic tariff bias, which is in line with previous research.
The remainder of this article is structured as follows. Next, Sect. 2 discusses the related literature on tariff research. Thereafter, Sect. 3 presents the consumer surplus model and the results on customer choice in the case where the customer has certainty about his preferences. Section 4 considers the optimal consumption pattern under cost tariffs when consumers have uncertainty about their preferences. Subsequently, the conditions under which it is optimal to choose cost cap tariffs over payperuse and flat rate plans are developed in Sect. 5. Finally, in Sect. 6, the consumer surplus model is evaluated empirically.
2 Related literature
Related literature on tariff choice modeling
References  Approach  Tariff structures  Uncertainty  

Analytical  Empirical  PU  FR  CC  2PT  3PT  
Oi (1971)  \(\bullet \)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Hayes (1987)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Danaher (2002)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Essegaier et al. (2002)  \(\bullet \)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Sundararajan (2004)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Schulze et al. (2005)  \(\bullet \)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Albers and Skiera (2006)  \(\bullet \)  \(\bullet \)  
Lambrecht et al. (2007)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Iyengar et al. (2008)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Schlereth and Skiera (2012)  \(\bullet \)  \(\bullet \)  
This research  \(\bullet \)  \(\bullet \)  \(\bullet \)  \(\bullet \)  \(\bullet \)  
Tariff structures:  Payperuse (PU), flat rate (FR), cost cap (CC), twopart (2PT), threepart (3PT) 
Most of the earlier literature which applied analytical tariff modeling focused on the impact of transaction costs (Sundararajan 2004) or capacity constraints (Essegaier et al. 2002) on a provider’s profit. Thereby, consumers’ tariff choice and consumption decision were addressed with rather simple consumer surplus models. In contrast, empirical research applied more sophisticated models (e.g., Albers and Skiera 2006) to explain observed tariff choice including certain irrationalities, such as tariff biases (Schulze et al. 2005) or brand effects (Iyengar and Jedidi 2012).
The papers that are most related to ours, but do not address the cost cap tariff, are Lambrecht et al. (2007) and Iyengar et al. (2008). More specifically, Lambrecht et al. (2007) develop an empirical model for uncertainty under threepart tariffs (consisting of a fixed fee, a usage allowance, and payperuse price for the usage that exceeds the allowance) by incorporating future usage shocks occurring prior to the consumption decision in the tariff choice stage. Within their model, uncertainty is a key driver for threepart tariff choice. Iyengar et al. (2008) derive the optimal consumption level for threepart tariffs under certainty first, and then allow for small variations of this level ex post. Thereby, both papers propose a similar consumer surplus model with the intention in mind to explain observed tariff choice by estimating the models’ underlying parameters. Schlereth and Skiera (2012) demonstrated that such models can be also used to predict tariff choice of innovative tariffs. The predictions of these models are, therefore, the result of a choice model that has been calibrated by actual tariff decisions, including consumers’ biases and irrationalities.
In contrast to the stated literature, we derive a consumer surplus model with the intention to explain the theoretically optimal tariff choice. Within this model, tariff choice relies on the preferences of a representative consumer, and not on an exogenous demand or observed choice. This modeling approach has the distinct advantage that the demand for tariff usage is derived endogenously and depends also on the chosen tariff and the pricing structure. In other words, given the same preferences, a consumer will exert a different consumption pattern under a cost cap tariff than under a payperuse or flat rate plan, because prices are different. This endogenous change in demand should be taken into account when choosing a tariff.
Thereby, we assume that consumers have uncertainty about their preferences ex ante (Kridel et al. 1993). This assumption is driven by the belief that consumers are uncertain about the realization of their preferences during the runtime of their contract when choosing a tariff, e.g., due to unforseen changes in their habits. Thus, uncertainty in consumption is driven by the uncertainty in preferences and not by external shocks as similarly stated by Hayes (1987). Consequently, the optimal consumption is uncertain as well. Therefore, even small levels of uncertainty do not simply imply kinks in the demand function as in Iyengar, Jedidi, and Kohli; instead, they may cause actual discontinuities in the demand functions (cf. Moffitt 1986, p. 320), because demand is shifted to a different tariff segment. In contrast to external shocks, these discontinuities in the demand function are tariff dependent.
Moreover, for the most part the present paper abstracts from any bias or other irrationality in tariff choice. Instead, a fully rational consumer is assumed, who may, however, face uncertainty about his preferences (demand). Since we abstract from any bias (or irrationality), we can provide insights under which condition a cost cap tariff should be chosen, even though it offers a higher variable rate than the concurrently offered payperuse tariff and a higher cost ceiling than the concurrently offered flat rate tariff.
Finally, it is worth mentioning that, with the exception of Krämer and Wiewiorra (2012), cost cap tariffs have not received academic attention before. Krämer and Wiewiorra conduct an empirical investigation of the flexibility effect in tariff choice in which they also consider the cost cap tariff. They highlight that there might exist a “cost cap bias”, by which customers favor cost caps over payperuse tariffs and flat rates even if the tariffs yield the same economic costs. In our empirical analysis, we can confirm such a cost cap bias, which exists over and beyond the rational choice of cost cap tariffs.
3 Tariff choice and consumption under certainty
As noted above, we restrict our analysis to the interesting case where all parameters are nonnegative and \(c_{\rm CC} > b_{\rm FR}\) and \(p_{\rm CC} > p_{\rm PU}.\) Moreover, we assume that \(\beta_1 > p_{\rm CC},\) which ensures that the optimal consumption levels under all tariffs are positive.
A consumer is thus indifferent between a PU and an FR tariff if and only if \(u_{\rm PU}(n^*_{\rm PU}) = u_{\rm FR}(n^*_{\rm FR}).\) Solving this equation for the preference parameter \(\beta_1\) yields: \(\beta_{1}^{*} = 2 \beta_2 \frac{b_{\rm FR}}{p_{\rm PU}} + \frac{p_{\rm PU}}{2}.\) Thus, for every \(\beta_1 < \beta_1^*,\) the consumer prefers the PU tariff over the FR tariff and vice versa.
Thus, irrespective of which consumption candidate is optimal under a CC tariff, both are consumer surplus dominated by the optimal consumption plans under either the FR tariff or the PU tariff.
Proposition 1
(CC choice under certainty) A rational consumer would never choose a cost cap tariff over a flat rate or payperuse tariff if he has certainty about his preferences.
4 Tariff consumption under uncertainty
In the following, we relax the assumption that the preferences are known with certainty and assume that \(\beta_1\) is a realization of the random variable \(B_1\) that is distributed according to the probability density function \(f_{B1}(\beta_1)\) and the corresponding cumulative distribution function \(F_{B1}(\beta_1).\) As discussed above, we assume uncertainty in preferences exante (see Hayes 1987; Kridel et al. 1993), which also has an effect on optimal tariff choice. The only assumptions that are made about \(F_{B1}\) are that (1) \(F_{B1} (p_{\rm CC})=0,\) which ensures a positive consumption level for all \(\beta_1\) under all tariffs and (2) that \(0<F_{B1} (\beta_1^{**})<1,\) which ensures that both consumption candidates of the CC tariff, \(n1_{\rm CC}\) and \(n2_{\rm CC},\) are chosen with positive probability. Otherwise, the same logic as under certainty would apply and the CC tariff would never be chosen by a rational consumer.
Proposition 2
(CC consumption) A rational consumer of a cost cap tariff will never (expect to) consume exactly at the level at which the cost cap becomes binding.
5 Tariff choice under uncertainty
5.1 General case
Equation (32) reveals that the expected consumer surplus of a CC tariff may exceed that of a PU tariff if (1) the marginal price of the CC tariff, \(p_{\rm CC},\) is not much larger than the marginal price of the PU tariff, \(p_{\rm PU}\) and (2) if high values of \(\beta_1,\) i.e., \(\beta_1 > \beta_1^{**}\) are sufficiently likely. Again, the first condition ensures that the first, negative summand of Eq. (32) is not too small, whereas the second condition ensures that the second, positive summand is rather large because \(E[B_1] \gg {\hat{E}}.\)
Notice that in order for the CC tariff to dominate both the FR and the PU tariff, the consumer must face a sufficiently high probability for low and high values of \(\beta_1.\) Evidently, the CC tariff must additionally be reasonably priced in comparison to the FR and PU tariff.
5.2 Example
To conclude, the example demonstrates that CC tariffs may indeed present an optimal tariff choice for a rational and riskneutral consumer under certain parameter conditions. In particular, these depend on a consumer’s level of uncertainty about his preferences, which, by Eqs. (19) and (24), directly translates into demand uncertainty.
Proposition 3
(Choice of CC tariffs under uncertainty) A (reasonably priced) cost cap tariff may be chosen over a flat rate and a payperuse tariff by a riskneutral consumer in the presence of a sufficiently high demand uncertainty.
6 Empirical evaluation
In the following, an empirical evaluation is presented to demonstrate the applicability of the previously developed consumer surplus model. To this end, the consumer surplus model is evaluated based on survey data. We contracted with a professional marketing research agency to conduct a survey online with a sample that is representative of the population of German mobile telephony users. A total of 122 respondents completed the survey, which consisted of two parts.
In the first part, respondents had to imagine that they use a PU tariff for mobile telephony and that this is the only tariff type available to them. In a repeated openended question design (Miller et al. 2011; Schulze et al. 2005), the respondents had to estimate their expected average monthly mobile telephony usage (in min) under a PU tariff, for each of the following minute prices (in €/min) \(p_{\rm PU}=\{0.40, 0.20, 0.10, 0.05\}.\) As will be described in detail below, the four consumption tuples \((E[N_{\rm PU}], p_{\rm PU})\) that are obtained in this part of the survey are used to estimate a respondent’s individual preference parameters \(E[B_1]\) and \(\beta_2,\) which are then employed to calibrate the individual consumer’s surplus function.
Tariffs to be evaluated by respondents
PU  CC  FR  

Base fee (b)  0 (0) €  0 (0) €  20 (25) € 
Minute price (p)  0.10 (0.13) €/min  0.12 (0.15) €/min  0.00 (0.00) €/min 
Cost cap (c)  0 (0) €  25 (30) €  0 (0) € 
6.1 Calibration of the individual consumer surplus function
We then evaluate our model in two different ways. First, given the expected usage under a given tariff (derived from the calibrated consumer’s surplus function) is compared to the reported average usage under each given tariff. Second, the expected consumer’s surplus (also derived from the calibrated consumer’s surplus function) is used as a predictor for the reported tariff rating.
6.2 Evaluation of the usage prediction
 \(D_{t,{\rm CC}}:\)

Dummy variable identifying a cost cap tariff
 \(D_{t,{\rm FR}}:\)

Dummy variable identifying a flat rate tariff
 \(D_{t,{\rm HighPrice}}:\)

Dummy variable identifying the price level (1 = high or 0 = low) of each tariff
 \(\zeta_{i}:\)

Fixedeffect of subject \(i.\)
 \(\epsilon_{i,t}:\)

Error term
Furthermore, the coefficient in our regression in Table 4 is significant and with 0.901 close to one, indicating a good prediction quality (Iyengar et al. 2008, p.201). However, the theoretical model has a slight tendency to overestimate the reported usage. This is particularly true for the reported usage under the tariffs with a high price level. Moreover, we find that respondents tend to overestimate their usage under an FR tariff. This socalled overestimation effect is well known from previous research (see, e.g., Nunes 2000). Finally, the regression model explains 80.6 % of the total variance, indicating a good model fit.
6.3 Evaluation of consumer surplus prediction
6.4 Robustness of prediction
Fixed and mixedeffects OLS regressions of model predictions on reported values
Fixedeffects OLS regression  Mixedeffects OLS regression  

(1a)  (2a)  (1b)  (2b)  
Average usage  Tariff rating  Average usage  Tariff rating  
Expected usage \((\gamma_{1})\)  0.901*** (0.147)  0.840*** (0.092)  
Expected consumer surplus \((\gamma_{2})\)  0.058*** (0.006)  0.046*** (0.007)  
High price dummy \((\gamma_{3})\)  −26.903*** (7.062)  −0.162 (0.116)  −27.677*** (6.271)  −0.236*** (0.068) 
Cost cap dummy \((\gamma_{4})\)  9.354 (6.390)  0.381** (0.146)  16.830** (6.181)  0.589** (0.207) 
Flat rate dummy \((\gamma_{5})\)  73.069*** (10.214)  2.509*** (0.149)  81.276*** (12.981)  2.754*** (0.227) 
Constant  −1.168 (20.123)  2.284*** (0.165)  12.995 (9.052)  2.872*** (0.164) 
Observations  6 × 121  6 × 121  6 × 121  6 × 121 
# of independent variables  125  125  245  245 
df  601  601  481  481 
\(R^{2}\)  0.806  0.536  0.763  0.265 
7 Conclusions
This paper has developed a consumer surplus model under cost cap tariffs by which the optimal consumption and choice under this new tariff type can be determined. Under a reasonable set of assumptions, we find that cost cap tariffs are only an optimal tariff choice for those consumers that face considerable uncertainty about their future demand, such that both relatively low and relatively high consumption levels are considered feasible. In this case, cost cap tariffs provide an insurance against extraordinary high costs (like a flat rate), but also cost flexibility in case of low demand (like a payperuse tariff). Therefore, consumers are willing to accept a higher marginal price compared to a payperuse tariff as well as a cost cap which is priced above the fixed fee of a flat rate. It was demonstrated that the model is useful for predicting the actual tariff usage and rating. The proposed model may, therefore, serve as a benchmark for future empirical research.
However, some limitations apply. The present model merely considers the rational choice of a riskneutral consumer. However, our empirical results suggest that the rational tariff choice is systematically biased, e.g., due the abovementioned insurance effect (Lambrecht and Skiera 2006), the flexibility effect (Krämer and Wiewiorra 2012), the overestimation effect (Nunes 2000), or brand effects (Schlereth and Skiera 2012). The detailed empirical investigation of the extent of a bias for cost cap tariffs, in particular, in relation to the wellknown flat rate bias, therefore, seems to be a fruitful avenue for future research. Furthermore, for expositional clarity, our consumer surplus model assumes a quadratic functional form, as in Lambrecht et al. (2007) and Iyengar et al. (2008). However, alternative functions might be more appropriate to describe consumer surplus as discussed by Skiera (1999) and Albers and Skiera (2006). Thus, future work should address the implications and suitability of alternative modeling approaches.
Notes
Acknowledgments
Financial support by Deutsche Forschungsgemeinschaft through Project KR 3889/21 is gratefully acknowledged.
References
 Albers, Sönke and Bernd Skiera. 2006. Einsatz einer erweiterten Form der ConjointAnalyse zur empirischen Schätzung von Zahlungsbereitschaftsfunktionen für die mengen und zeitbezogene Preisdifferenzierung. In Quantitative Marketingforschung in Deutschland: Festschrift zum 65. Geburtstag von Klaus Peter Kaas, ed. Thorsten Posselt and Christian Schade, 125–144. Berlin: Duncker & Humblot.Google Scholar
 Armstrong, J. Scott, and Fred Collopy. 1992. Error measures for generalizing about forecasting methods: empirical comparisons. International Journal of Forecasting 8(1): 69–80.CrossRefGoogle Scholar
 Briegleb, Volker. 2009. O2 mit mehr Kunden und Gewinn. http://heise.de/857876. Accessed 06 Jan 2011.
 Danaher, Peter J. 2002. Optimal pricing of new subscription services: analysis of a market experiment. Marketing Science 21(2): 119–138.Google Scholar
 Dubin, Jeffrey A., and Daniel L. McFadden. 1984. An econometric analysis of residential electric appliance holdings and consumption. Econometrica 52(2): 345–362.CrossRefGoogle Scholar
 Essegaier, Skander, Sunil Gupta, and Z. John Zhang. 2002. Pricing access services. Marketing Science 21(2): 139–159.CrossRefGoogle Scholar
 Hanemann, Michael W. 1984. Discrete/continuous models of consumer demand. Econometrica: Journal of the Econometric Society 52(3): 541–561.Google Scholar
 Hausmann, Jerry A. 1985. The econometrics of nonlinear budget sets. Econometrica 53(6): 1255–1282.CrossRefGoogle Scholar
 Hayes, Beth. 1987. Competition and twopart tariffs. The Journal of Business 60(1): 41.CrossRefGoogle Scholar
 Hyndman, Rob J., and Anne B. Koehler. 2006. Another look at measures of forecast accuracy. International Journal of Forecasting 22(4): 679–688.Google Scholar
 Iyengar, Raghuram, and Kamel Jedidi. 2012. A conjoint model of quantity discounts. Marketing Science 31(2): 334–350. doi: 10.1287/mksc.1110.0702.
 Iyengar, Raghuram, Kamel Jedidi, and Rajeev Kohli. 2008. A conjoint approach to multipart pricing. Journal of Marketing Research 45(2): 195–210.CrossRefGoogle Scholar
 Kennedy, Peter. 2008. A guide to econometrics. Malden: Blackwell Publishing.Google Scholar
 Krämer, Jan, and Lukas Wiewiorra. 2012. Beyond the flat rate bias: the flexibility effect in tariff choice. Telecommunications Policy 36(1): 29–39.CrossRefGoogle Scholar
 Kridel, Donald J., Dale E. Lehman, and Dennis L. Weisman. 1993. Option value, telecommunications demand, and policy. Information Economics and Policy 5(2): 125–144.CrossRefGoogle Scholar
 Lambrecht, Anja, Katja Seim, and Bernd Skiera. 2007. Does uncertainty matter? Consumer behavior under threepart tariffs. Marketing Science 26(5): 698–710.Google Scholar
 Lambrecht, Anja, and Bernd Skiera. 2006. Paying too much and being happy about it: existence, causes, and consequences of tariffchoice biases. Journal of Marketing Research 43(2): 212–223.Google Scholar
 Leland, Hayne E., and Robert A. Meyer. 1976. Monopoly pricing structures with imperfect discrimination. The Bell Journal of Economics 7(2): 449–462.Google Scholar
 Klaus M. Miller, Reto Hofstetter, Harley Krohmer, and Z. John Zhang. 2011. How should consumers’ willingness to pay be measured? An empirical comparison of stateoftheart approaches. Journal of Marketing Research 48(1): 172–184.Google Scholar
 Moffitt, Robert. 1986. The econometrics of piecewiselinear budget constraints: a survey and exposition of the maximum likelihood method. Journal of Business & Economic Statistics 4(3): 317–328.Google Scholar
 Murphy, Michael M. 1977. Price discrimination, market separation, and the multipart tariff. Economic Inquiry 15(4): 587–599.CrossRefGoogle Scholar
 Nunes, Joseph C. 2000. A cognitive model of peoples usage estimations. Journal of Marketing Research 37(4): 397–409.Google Scholar
 Oi, Walter Y. 1971. A disneyland dilemma: twopart tariffs for a mickey mouse monopoly. The Quarterly Journal of Economics 85(1): 77–96.Google Scholar
 Schlereth, Christian, and Bernd Skiera. 2012. Measurement of consumer preferences for bucket pricing plans with different service attributes. International Journal of Research in Marketing 29(2): 167–180.CrossRefGoogle Scholar
 Schulze, Timo, Karen Gedenk, and Bernd Skiera. 2005. Segmentspezifische Schätzung von Zahlungsbereitschaftsfunktionen. Zeitschrift für betriebswirtschaftliche Forschung 57(August): 401–422.Google Scholar
 Skiera, Bernd. 1999. Mengenbezogene Preisdifferenzierung bei Dienstleistungen. Wiesbaden: Gabler.Google Scholar
 Sundararajan, Arun. 2004. Nonlinear pricing of information goods. Management Science 50(12): 1660–1673.CrossRefGoogle Scholar
Copyright information
This article is published under license to BioMed Central Ltd. Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.