ORIGINAL ARTICLE: Wagner, U., & Akbari, K. (2023). Refining the freeloading and no purchase behavior in pay as you wish pricing. Review of Managerial Science, 4. https://doi.org/10.1007/s11846-023-00678-1
Article Abstract: Pay as you wish (PAYW) pricing offers a radical shift from posted pricing schemes. Modeling consumer behavior under PAYW pricing promises insights into conditions under which PAYW is profitable. Firstly, this paper extends an established model that builds on inequity-averse consumers and models their behavior in PAYW as well as the seller’s profits. The paper uses a comprehensive approach to describe consumers with low fairness concerns and points to a new segment of consumers who were not considered in previous PAYW models. They are characterized by a decision not to buy a good under a PAYW pricing policy, even if they can get it for free, and are not strongly averse to advantageous inequity. Secondly, the paper discusses the profitability of PAYW with a suggested price when the seller’s ability to suggest high prices is limited. Thirdly, the paper incorporates the effect of disadvantageous inequity aversion on PAYW with a minimum price. Finally, the paper offers guidelines on how a seller should choose the optimal pricing policy.
This blog post aims to explain the methodology used to derive the figures in the academic paper “Refining the Freeloading and No Purchase Behavior in Pay As You Wish Pricing” through the use of Wolfram Mathematica. It will offer a detailed walkthrough of the computational steps involved in modeling consumer behavior within the Pay As You Wish (PAYW) pricing framework. The blog post will be structured into three main sections:
1
.
Introduction to the Article: This section will provide an overview of the academic paper, focusing on its exploration of consumer behavior in Pay As You Wish (PAYW) pricing models.
2
.
The Dynamic Tool to Determine the Optimal Pricing Scheme: Here, the discussion will center around how Mathematica was used as a dynamic tool for identifying the most effective pricing strategy, considering various consumer behaviors and market dynamics.
3
.
The Figures in the Article Built in Mathematica: The final section will detail how the visual elements and figures in the paper were constructed using Mathematica.

Introduction to the Article

Over the last decade, there has been increasing interest in participatory pricing schemes that give the consumer power over pricing decisions (Spann et al., 2018). Pay-as-you-wish (PAYW) pricing is probably the most disruptive and extreme form, giving the buyer (almost) no limit on how to set the price. Under PAYW, customers are offered a product or service and are free to choose any price they want, including zero (obtaining the good for free; Kim et al., 2009)Previous examples from practice have shown that sellers can operate profitably using PAYW (Natter & Kaufmann, 2015). As an example, the Viennese restaurant Der Wiener Deewan has run profitably on PAYW prices for over 15 years (Riener & Traxler, 2012). However, many sellers try to implement PAYW but fail to do so on a permanent basis, going bankrupt or switching to fixed prices. For example, the US bakery chain Panera Bread experimented with PAYW several times, only to realize that its products could not be sold under PAYW conditions. Previous empirical research has established several factors that support the profitability of PAYW: as an example, consumers must be sufficiently generous and fair-minded to pay significant prices, and the seller has to operate with low variable costs (Gerpott, 2017).However, while empirical research offers an invaluable case-by-case perspective on conditions under which PAYW can be profitable, striving for generalizations calls for microeconomic modeling. In addition, sellers apply variants of pure PAYW, such as setting suggested or minimum prices. Modeling PAYW promises to support the seller in finding the optimal suggested and minimum price and give guidelines on which pricing scheme to choose. In this article, we build on the work by Chen et al. (2017) and give guidelines on the optimal prices and pricing schemes for a variety of circumstances. We build on inequity averse consumers (Fehr & Schmidt, 1999) and assume a customer utility of​
​The consumption utility
(r
i
)
describes the customer’s benefit from consuming the good, disregarding any transaction utility (Thaler, 1983). The term
β
i
max
p
i
-
p
f
i
,0
captures disutility from disadvantageous inequity, namely if the actual price (
p
i
) is above the consumer’s perceived fair price (
p
f
i
). This term becomes relevant only if the seller sets the price. Higher
 
β
i
s indicate that customers have stronger opposition when the seller is over-privileged, that is, charges high prices. Correspondingly, the term
γ
i
max
p
f
i
-
p
i
,0
captures disutility from advantageous inequity: that is, if the price is below the consumer’s perceived fair price. Higher values of
γ
i
correspond to a stronger aversion to advantageous inequity, that is, characterizing more fair-minded customers. Thus, the higher
γ
i
, the more the customer dislikes being over-privileged.The perceived fair price is defined as ​
,
0<=
λ
i
<=1
​where c describes the seller’s costs and
λ
i
describes the seller’s equitable share of the total surplus. By assumption, the consumer knows the seller’s costs and compares them to her consumption utility. When the costs are higher than the consumption utility
r
i
, the consumer’s fair price is equal to the seller’s costs, as the consumer knows that lower prices will cause a loss for the seller. When the consumption utility is higher than the costs, taking the good creates a positive surplus (
r
i
-c>0
). The fair price implies that the consumer is willing to split this surplus proportionally (i.e.
λ
i
:(1-
λ
i
)
). Previous research has shown that the fair price is relevant to both PAYW (e.g. Kim et al., 2009) and traditional pricing schemes (Koschate-Fischer et al., 2016). The parameter
λ
i
describes the generosity of the consumer. The higher
λ
i
, the stronger the consumer’s conviction that she should pay the seller in order to be fair. Using this model of consumer behavior, we derive optimal prices, profits, the share of freeloaders, and more in the article. For more information on the theory, check it out here: https://doi.org/10.1007/s11846-023-00678-1

The dynamic tool to determine the optimal pricing scheme

The tool allows us to assess optimal prices, market coverage, the size of the freeloader segment (Segment I ), and the profits under all four pricing schemes for all parameter combinations. For example, please refer to Fig. 2 in the paper and the replication below.
The code for this tool is available in Online Resource D, accompanying the article, and is also shared as an executable CDF file. This report will further elaborate on this code, providing a comprehensive explanation of its functionalities and applications.
The initial step will involve setting up the model based on the demand and profit equations detailed in the paper. This setup will encompass all pricing schemes, focusing on establishing the critical consumption utilities that demarcate the thresholds for purchase and non-purchase decisions. Additionally, the model will address the penetration rate, indicating the proportion of customers who decide to purchase, and will also consider the segment of freeloaders.
In[]:=
(*Generalmodelparameters*)​​ϕ[r_]:=1;(*Thedemandisuniformlydistributedinthemodel*)​​SetOptions[$FrontEndSession,PrintPrecision2]​​​​(*PAAP*)​​profitPAAP[β_,λ_,c_]:=
2
(1-c)
(1+βλ)
4(1+β)
;(*e.g.,ThisisFormula(4)inthepaper*)​​pricePAAP[β_,λ_,c_]:=c+(1+βλ)/(2(1+β))(1-c);(*e.g.,ThisisFormula(3)inthepaper*)​​rstar[β_,λ_,c_]:=pricePAAP[β,λ,c]+
(-c+pricePAAP[β,λ,c])β(1-λ)
1+βλ
;​​penetration[β_,λ_,c_]:=
1
∫
rstar[β,λ,c]
ϕ[r]r;​​penetrationPAAP[β_,λ_,c_]:=
1-c
2
;​​​​(*PAYW*)​​profitPAYW[ω_,λ_,c_,γ_]:=
2
(1-c)
λ(1-ω)
2
-c(1-cγ)ω;​​EpricePAYW[λ_,c_]:=c+
λ
2
2
(1-c)
;​​penetrationPAYW[ω_,c_,γ_]:=(1-ω)(1-c)+ω(1-cγ);​​freeloaderPAYW[ω_,c_,γ_]:=ω(1-cγ);​​​​(*PAYW-SP*)​​m[ω_,c_,γ_]:=cωγ/((1-c)(1-ω));​​pricePAYWSP[ω_,c_,γ_,λ_]:=
λ+(1-λ)c
m[ω,c,γ]>
1
2
λ+(1-λ)c
m[ω,c,γ]≤
1
2
&&λ≤
2
3
(1+m[ω,c,γ])
c+2λm[ω,c,γ](1-c)/(3λ-2)
m[ω,c,γ]≤
1
2
&&λ>
2
3
(1+m[ω,c,γ])
0
True
;​​gλ[ω_,c_,γ_,λ_]:=
(4-3λ)/2+2m[ω,c,γ]
m[ω,c,γ]>
1
2
(4-3λ)/2+2m[ω,c,γ]
m[ω,c,γ]≤
1
2
&&λ≤
2
3
(1+m[ω,c,γ])
2(m[ω,c,γ])^2/(3λ-2)+1
m[ω,c,γ]≤
1
2
&&λ>
2
3
(1+m[ω,c,γ])
0
True
;​​profitPAYWSP[ω_,c_,γ_,λ_,z_]:=(1-ω)(1-c)^2λ/2(z+(1-z)gλ[ω,c,γ,λ])-cω(1-cγ);​​​​pfmax[c_,λ_]:=c+λ(1-c);​​pii[ω_,c_,γ_,λ_]:=Min[pricePAYWSP[ω,c,γ,λ],pfmax[c,λ]];​​freeloaderPAYWSP[ω_,c_,γ_,z_,λ_]:=zω(1-cγ)+(1-z)ω(1-pii[ω,c,γ,λ]γ);​​penetrationPAYWSP[ω_,c_,γ_,z_,λ_]:=(1-z)(1-ω)12
Min[pricePAYWSP[ω,c,γ,λ],pfmax[c,λ]]
∫
c
ϕ[r]r+​​
1
∫
Min[pricePAYWSP[ω,c,γ,λ],pfmax[c,λ]]
ϕ[r]r+z(1-ω)(1-c)+freeloaderPAYWSP[ω,c,γ,z,λ];​​​​(*PAYW-MP*)​​k[ω_,λ_,β_]:=
1
λ≤
1
2+β
(ω(1+βλ))/((2+β(1+ω))λ-1+ω)
λ>
1
2+β
0
True
;​​pricePAYWMP[ω_,c_,λ_,β_]:=c+(1-c)k[ω,λ,β]λ;​​profitPAYWMP[ω_,c_,λ_,β_]:=​​
((1-c)^2λ(1-λ))/(1+βλ)
λ≤
1
2+β
((1-c)^2λ/2)((λ(2+β)+2ω(1-λ)-1)/((2+β(1+ω))λ-1+ω))
λ>
1
2+β
0
True
;​​rplus[ω_,c_,λ_,β_]:=Min[(pricePAYWMP[ω,c,λ,β]+​​(β(1-λ)(pricePAYWMP[ω,c,λ,β]-c))/(1+λβ)),1];​​​​rpmin[ω_,c_,λ_,β_]:=(pricePAYWMP[ω,c,λ,β]-(1-λ)c)/λ;​​penetrationPAYWMP[ω_,c_,λ_,β_]:=(1-ω)
Min[rpmin[ω,c,λ,β],1]
∫
rplus[ω,c,λ,β]
ϕ[r]r+​​
1
∫
Min[rpmin[ω,c,λ,β],1]
ϕ[r]r+ω
1
∫
rplus[ω,c,λ,β]
ϕ[r]r;
Following the setup of the model, we use the definitions to create an interactive ‘Manipulate’ environment in Mathematica. This environment not only enables dynamic exploration of the model but also emphasizes key outcomes, including the highest values of prices, profits, penetration rates, and the size of the freeloader segment. For a more in-depth understanding of the coding process and the specific styling choices made, the post directs readers to the comments accompanying the code, where additional context and explanations are provided.

Figures

The upcoming section will discuss the three main figures (Figures 3 to 5) from the original article, which are instrumental in illustrating the results derived from the model. Each figure provides a unique visual representation of key findings and data analyses pertinent to the study.

Creating Figure 3

Figure 3 in the paper serves as the optimal prices figure. It displays the optimal prices for each pricing scheme at specific values of λ. This figure provides insights into the optimal prices determined for a specific parameter constellation under different λ values. To accurately replicate the appearance of this figure as seen in the paper, specific styling options were employed in Mathematica.
The initial step involves defining the functions that will be plotted, specifically focusing on the optimal prices for the different pricing schemes. In the PAYW scheme, this involves calculating the expected value of the fair price, as there is no defined ‘optimal price’ from the seller’s perspective.
We identify specific ‘interesting’ values in the model, which are represented as vertical lines in the plots to aid in visual analysis.
In this part of the analysis, we define vertical lines for the profit figure (Figure 4) as well. Since the same parameters are assumed for both figures, these definitions are made beforehand. This approach allows for the evaluation of expressions that highlight specific points of interest, particularly at points where the profits from different pricing models—such as PWYW-SP, PAAP, PAYW-SP, and PAYW-MP—intersect or are equivalent.
We set specific constants for plotting the prices. This setup is chosen for demonstration purposes and represents just one of many possible configurations.
Next, we configure the style settings for the curves to ensure they are visually distinct and clear in the graphical representation.
Finally, we proceed to plot.

Creating Figure 4

In the creation of Figure 4, the focus is on the optimal profits, which are derived by incorporating the previously calculated optimal prices into the profit function. This figure is generated using a specific set of parameters along with various values of λ. To achieve a presentation that matches the one in the original paper, again, a range of styling options is applied. These stylistic choices ensure that the figure is both visually engaging and informative.
Again, the initial step involves defining the functions that will be plotted, specifically focusing on the optimal profits for the different pricing schemes.
Then, we add additional elements for the figure.
Then, we create the plot.

Creating Figure 5

For Figure 5, the objective is to determine the optimal pricing scheme across a wide range of parameter values. This figure explores a larger set of parameter combinations, although it does not specifically address absolute profit values but only which pricing scheme is optimal. It encompasses variations in β (beta), c, λ (lambda), and ω (omega), while maintaining γ (gamma) and z at constant values of 0.5.
​
The code for this process is extensive and involves several sections:
​
- Parameters are reset, and profit functions are redefined to accommodate the expanded parameter space.
- Lines connecting the optimal profit functions are defined to enhance the visual depiction of distinct regions.
- A table is created to evaluate functions at multiple β and c points, which are later used as the outer x and y axes in the plot.
- Creating the final plot.
​
This comprehensive approach allows for a thorough exploration of pricing schemes across diverse parameter combinations.
As outlined, we first reset the workspace.
Next, we redefine the profit functions.
The automatic delineation of optimal regions in the final figure is not feasible. Consequently, we manually create lines to distinguish these optimal regions. This is accomplished by identifying points where the relevant profit functions are equal. Our approach focuses on selecting only the relevant parts of the profit function to streamline computation.
A table is created to evaluate functions at multiple β and c points, which are later used as the outer x and y axes in the plot.
Finally, we create the plot. Each Function within Map generates a series of plots and overlays them using Show, with a wide variety of styling and labeling options. The final result is a grid of these plots, each accompanied by a legend explaining the different components.

References

Chen, Y., Koenigsberg, O., & Zhang, Z. J. (2017). Pay-as-You-Wish Pricing. Marketing Science, 36(5), 780-791. https://doi.org/10.1287/mksc.2017.1032
Fehr, E., & Schmidt, Kl. M. (1999). A Theory of Fairness, Competition, and Cooperation. The Quarterly Journal of Economics, 114(3), 817-868. https://doi.org/10.1162/003355399556151
Gerpott, T. J. (2017). Pay-What-You-Want pricing: An integrative review of the empirical research literature. Management Science Letters, 7(1), 35-62. https://doi.org/10.5267/j.msl.2016.11.004
Kim, J.-Y., Natter, M., & Spann, M. (2009). Pay what you Want: A New Participative Pricing Mechanism. Journal of Marketing, 73(1), 44-58. https://doi.org/10.1509/jmkg.73.1.044
Koschate-Fischer, N., Huber (née Stefan), I. V., & Hoyer, W. D. (2016). When will price increases associated with company donations to charity be perceived as fair? Journal of the Academy of Marketing Science, 44(5), 608-626. https://doi.org/10.1007/s11747-015-0454-5
Natter, M., & Kaufmann, K. (2015). Voluntary market payments: Underlying motives, success drivers and success potentials. Journal of Behavioral and Experimental Economics, 57, 149-157. https://doi.org/10.1016/j.socec.2015.05.008
Riener, G., & Traxler, C. (2012). Norms, moods, and free lunch - Longitudinal evidence on payments from a Pay-What-You-Want restaurant. The Journal of Socio-Economics, 41(4), 476-483. https://doi.org/10.1016/j.socec.2011.07.003
Spann, M., Zeithammer, R., Bertini, M., Haruvy, E., Jap, S. D., Koenigsberg, O., Mak, V., Popkowski Leszczyc, P., Skiera, B., & Thomas, M. (2018).
Beyond Posted Prices: the Past, Present, and Future of Participative Pricing Mechanisms. Customer Needs and Solutions, 5(1-2), 121-136. https://doi.org/https://doi.org/10.1007/s40547-017-0082-y
Thaler, R. H. (1983). Transaction Utility Theory. Advances in Consumer Research, 10(1), 229-232.

CITE THIS NOTEBOOK

Refining the freeloading and no purchase behavior in pay as you wish pricing​
by Udo Wagner and Karl Akbari​
Wolfram Community, STAFF PICKS, January 30, 2024
​https://community.wolfram.com/groups/-/m/t/3113079