Skip to main content

A mathematical reformulation of the reference price

Abstract

Reference prices have long been studied in applied economics and business research. One of the classic formulations of the reference price is in terms of an iterative function of past prices. There are a number of limitations of such a formulation, however. Such limitations include burdensome computational time to estimate parameters, an inability to truly account for customer heterogeneity, and an estimation procedure that implies a misspecified model. Managerial recommendations based on inferences from such a model can be quite misleading. We mathematically reformulate the reference price by developing a closed-form expansion that addresses the aforementioned issues, enabling one to elicit truly meaningful managerial advice from the model. We estimate our model on a real world data set to illustrate the efficacy of our approach. Our work is not only important from a modeling perspective, but also has valuable behavioral and managerial implications, which modelers and non-modelers alike should find useful.

This is a preview of subscription content, access via your institution.

References

  • Briesch, R. A., Krishnamurthi, L., Mazumdar, T., & Raj, S. P. (1997). A comparative analysis of reference price models. Journal of Consumer Research, 24(September), 202–214.

    Article  Google Scholar 

  • Bucklin, R. E., & Gupta, S. (1992). Brand choice, purchase incidence, and segmentation: an integrated approach. Journal of Marketing Research, 29(May), 201–215.

    Article  Google Scholar 

  • Chib, S., Seetharaman, P. B., & Strijnev, A. (2004). Model of brand choice with a no-purchase option calibrated to scanner panel data. Journal of Marketing Research, 41(2), 184–196.

    Article  Google Scholar 

  • Cunha, M., & Shulman, J. (2011). Assimilation and contrast in price evaluations. Journal of Consumer Research, 37(5), 822–835.

    Article  Google Scholar 

  • Grewal, D., Janakiraman, R., Kalyanam, K., Kannan, P. K., Ratchford, B. T., Song, R., et al. (2010). Strategic online and offline retail pricing: a review and research agenda. Journal of Interactive Marketing, 24(2), 138–154.

    Article  Google Scholar 

  • Guadagni, P., John, D., & Little, C. (1983). A logit model of brand choice calibrated on scanner data. Marketing Science, 2(Summer), 203–238.

    Article  Google Scholar 

  • Hardie, B. G. S., Johnson, E. J., & Fader, P. S. (1993). Modeling loss aversion and reference dependence effects on brand choice. Marketing Science, 12(Fall), 378–394.

    Article  Google Scholar 

  • Heath, T. B., Chatterjee, S., & Russo, K. (1995). Mental accounting and changes in price: the frame dependence of reference dependence. Journal of Consumer Research, 22(June), 90–97.

    Article  Google Scholar 

  • Heyman, J., & Mellers, B. (2008). Perceptions of fair pricing. In C. Haugtvedt, F. Kardes, P. Herr (Eds.), Handbook of Consumer Psychology.

  • Janiszewski, C., & Lichtenstein, D. R. (1999). A range theory account of price perception. Journal of Consumer Research, 25(March), 353–368.

    Article  Google Scholar 

  • Jank, W., & Kannan, P. K. (2005). Understanding geographical markets of online firms using spatial models of customer choice. Marketing Science, 24(Fall), 623–634.

    Article  Google Scholar 

  • Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decisions under risk. Econometrica, 47(2), 263–291.

    Article  Google Scholar 

  • Kalwani, M. U., & Yim, C. K. (1992). Consumer price and promotion expectations: an experimental study. Journal of Marketing Research, 29, 90–100.

    Article  Google Scholar 

  • Kalwani, M. U., Yim, C. K., Rinne, H. J., & Sujita, Y. (1990). A price expectation model of consumer brand choice. Journal of Marketing Research, 27(August), 251–262.

    Article  Google Scholar 

  • Kalyanaram, G., & Little, J. D. C. (1994). An empirical analysis of latitude of price acceptance in consumer packaged goods. Journal of Consumer Research, 21(December), 408–418.

    Article  Google Scholar 

  • Kamakura, W. A., & Russell, G. J. (1989). A probabilistic choice model for market segmentation and elasticity. Journal of Marketing Research, 26, 379–390.

    Article  Google Scholar 

  • Kamins, M. A., Dreze, X., & Folkes, V. S. (2004). Effects of seller-supplied prices on buyers’ product evaluations: reference prices in an internet auction context. Journal of Consumer Research, 30(March), 622–627.

    Article  Google Scholar 

  • Kannan, P. K., & Wright, G. P. (1991). Modeling and testing structured markets: a nested logit approach. Marketing Science, 10(1), 58–82.

    Article  Google Scholar 

  • Kopalle, P. K., Rao, A. G., & Assunção, J. L. (1996). Asymmetric reference price effects and dynamic pricing policies. Marketing Science, 15(1), 60–85.

    Article  Google Scholar 

  • Kopalle, P., Kannan, P. K., Boldt, L. B., & Arora, N. (2012). The impact of household level heterogeneity in reference price effects on optimal retailer pricing policies. Journal of Retailing, 88(1), 102–114.

    Article  Google Scholar 

  • Lattin, J. M., & Bucklin, R. E. (1989). Reference effects of price and promotion on brand choice. Journal of Marketing Research, 26, 299–310.

    Article  Google Scholar 

  • Liu, W., & Soman, D. (2008). Behavioral pricing. In F. R. Kardes, C. P. Haugtvedt, & P. Herr (Eds.), Handbook of consumer psychology. Mahwah: Erlbaum.

    Google Scholar 

  • Mazumdar, T., & Papatla, P. (1995). Loyalty differences in the use of internal and external reference prices. Marketing Letters, 6(March), 111–122.

    Article  Google Scholar 

  • Mazumdar, T., Raj, S. P., & Sinha, I. (2005). Reference price research: review and propositions. Journal of Marketing, 69(October), 84–102.

    Article  Google Scholar 

  • Rajendran, K. N., & Tellis, G. J. (1994). Contextual and temporal components of reference price. Journal of Marketing, 58(January), 22–34.

    Article  Google Scholar 

  • Van der Vaart, A. W. (1998). Asymptotic statistics. New York: Cambridge University Press.

    Google Scholar 

  • Winer, R. S. (1986). A reference price model of brand choice for frequently purchased products. Journal of Consumer Research, 13(Spring), 250–256.

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank Hovannes Abramyan, Erik Johannassen, Steven J. Miller, Praveen Kopalle, Brian Ratchford, Mike Taylor, and Chi Kin (Bennett) Yim for helpful comments on an earlier draft of this manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kevin D. Dayaratna.

Technical appendix with mathematical details

Technical appendix with mathematical details

With our utility function for brand j at time t:

$$ {u_{{j,t}}} = {\beta_j} + {\beta_p}{p_{{j,t}}} + \left\{ {\matrix{ {\left( {{\pi^{{t - 1}}} \cdot {p_{{j,1}}} + \left( {1 - \pi } \right)\sum\limits_{{i = 1}}^{{t - 1}} { \cdot {\pi^{{i - 1}}}{p_{{j,t - i}}}} - {p_{{j,t}}}} \right){\beta_g}\quad if\;{r_{{j,t}}} > {p_{{j,t}}}} \\ {\left( {{\pi^{{t - 1}}} \cdot {p_{{j,1}}} + \left( {1 - \pi } \right)\sum\limits_{{i = 1}}^{{t - 1}} { \cdot {\pi^{{i - 1}}}{p_{{j,t - i}}}} - {p_{{j,t}}}} \right){\beta_l}\quad if\;{r_{{j,t}}} \leqslant {p_{{j,t}}}} \\ }<!end array> } \right. $$

we can go can go a step further and use the Heaviside step function H to avoid having to impose the above inequality restrictions:

$$ \matrix{ {{u_{{j,t}}} = {\beta_j} + {\beta_p}{p_{{j,t}}} + \left( {{\pi^{{t - 1}}} \cdot {p_{{j,1}}} + \left( {1 - \pi } \right)\sum\limits_{{i = 1}}^{{t - 1}} { \cdot {\pi^{{i - 1}}}{p_{{j,t - i}}}} - {p_{{jt}}}} \right)\beta_g^{{H\left( {{\pi^{{t - 1}}} \cdot {p_{{j,1}}} + \left( {1 - \pi } \right)\sum\limits_{{i = 1}}^{{t - 1}} { \cdot {\pi^{{i - 1}}}{p_{{j,t - i}}}} - {p_{{j,t}}} - \varepsilon } \right)}} \cdot } \hfill \\ {\beta_l^{{H\left( { - \left( {{\pi^{{t - 1}}} \cdot {p_{{j,1}}} + \left( {1 - \pi } \right)\sum\limits_{{i = 1}}^{{t - 1}} { \cdot {\pi^{{i - 1}}}{p_{{j,t - i}}}} - {p_{{j,t}}}} \right)} \right)}}.} \hfill \\ }<!end array> $$

with an arbitrarily small ε ∼ 0.000001 to accommodate for the inequality constraints. The choice of ε does not matter as long as it is non-zero within machine precision.

Now that we have a closed-form utility function incorporating the reference price, we can use our function to estimate a model of purchase choice and incidence using a nested logit formulation (Kannan and Wright 1991; Bucklin and Gupta 1992). The probability that an individual chooses brand j in category B, Pr t (jB), is:

$$ \matrix{ {{{\Pr }_t}\left( {j \cap B} \right) = {{\Pr }_t}\left( {j\left| B \right.} \right){{\Pr }_t}(B)} \\ { = \frac{{\exp \left( {{u_{{j,t}}}} \right)}}{{\sum\limits_{{k = 1}}^K {\exp \left( {{u_{{k,t}}}} \right)} }}\frac{{\exp \left( {{\alpha_0} + {\alpha_1}C{V_t}} \right)}}{{1 + \exp \left( {{\alpha_0} + {\alpha_1}C{V_t}} \right)}}} \\ {{{\Pr }_t}(j \cap B) = \frac{{\exp \left( {{u_{{j,t}}}} \right)}}{{\sum\limits_{{k = 1}}^K {\exp \left( {{u_{{k,t}}}} \right)} }}\left( {\frac{{\exp \left( {{\alpha_0} + {\alpha_1}\log \left( {\sum\limits_{{k = 1}}^K {\exp \left( {{u_{{k,t}}}} \right)} } \right)} \right)}}{{1 + \exp \left( {{\alpha_0} + {\alpha_1}\log \left( {\sum\limits_{{k = 1}}^K {\exp \left( {{u_{{k,t}}}} \right)} } \right)} \right)}}} \right)} \\ }<!end array> $$

where α 0 is an intercept parameter and α 1 is a coefficient corresponding to category value. Assuming latent segments where ψ s is the fraction of customers that belong to a particular segment s, in our finite mixture model formulation we have:

$$ {{\Pr}_t}\left( {j \cap B} \right) = \sum\limits_{{s = 1}}^S {{\psi_s}\frac{{\exp \left( {{u_{{j,t,s}}}} \right)}}{{\sum\limits_{{k = 1}}^K {\exp \left( {{u_{{k,t,s}}}} \right)} }}\left( {\frac{{\exp \left( {{\alpha_{{0,s}}} + {\alpha_{{1,s}}}\log \left( {\sum\limits_{{k = 1}}^K {\exp \left( {{u_{{k,t,s}}}} \right)} } \right)} \right)}}{{1 + \exp \left( {{\alpha_{{0,s}}} + {\alpha_{{1,s}}}\log \left( {\sum\limits_{{k = 1}}^K {\exp \left( {{u_{{k,t,s}}}} \right)} } \right)} \right)}}} \right)} . $$

Therefore, our likelihood function, which we conduct maximum likelihood estimation on, is:

$$ L(j) = \prod\limits_{{i = 1}}^I {\prod\limits_{{j = 1}}^J {\prod\limits_{{t = 1}}^T {\sum\limits_{{s = 1}}^S {\left( {\matrix{ {{\psi_s}{{\left( {\frac{{\exp \left( {{u_{{j,t,s}}}} \right)}}{{1 + \sum\limits_{{k = 1}}^{{K - 1}} {\exp \left( {{u_{{k,t,s}}}} \right)} }}} \right)}^{{{y_{{i,j,t}}}}}}{{\left( {\frac{{\exp \left( {{\alpha_{{0,s}}} + {\alpha_{{1,s}}}\log \left( {\sum\limits_{{k = 1}}^{{K - 1}} {\exp \left( {{u_{{k,t,s}}}} \right)} } \right)} \right)}}{{1 + \exp \left( {{\alpha_{{0,s}}} + {\alpha_{{1,s}}}\log \left( {\sum\limits_{{k = 1}}^{{K - 1}} {\exp \left( {{u_{{k,t,s}}}} \right)} } \right)} \right)}}} \right)}^{{{x_{{i,t}}}}}}.} \hfill \\ {\left( {\frac{1}{{1 + \exp \left( {{\alpha_{{0,s}}} + {\alpha_{{1,s}}}\log \left( {\sum\limits_{{k = 1}}^{{K - 1}} {\exp \left( {{u_{{k,t,s}}}} \right)} } \right)} \right)}}} \right)} \hfill \\ }<!end array> } \right)} } } } $$

where y ijt  = 1 if household i brand j is bought at time t and is 0 otherwise and \( {x_{{i,t}}} = \sum\limits_{{j = 1}}^J {{y_{{i,j,t}}}} . \)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dayaratna, K.D., Kannan, P.K. A mathematical reformulation of the reference price. Mark Lett 23, 839–849 (2012). https://doi.org/10.1007/s11002-012-9192-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11002-012-9192-3

Keywords

  • Reference price
  • Logit choice models
  • Logistic regression
  • Non-iterative estimation
  • Heaviside step function
  • Maximum likelihood estimation
  • Finite mixture models
  • Misspecified models