An analysis of three pricing strategies for digital music products

Abstract

Digital music albums can be sold either as singles or as a full album, which is not an option for selling physical music products. This paper investigates three pricing strategies for selling digital music albums. The first (strategy S) sets a unified unit price and all songs are sold at this price. The second (strategy F) determines a bundling price for selling the whole album. The third (strategy M) mixes strategies S and F. It offers two prices: one is for a unit song and the other for the whole album. We develop mathematical models for this problem and design enumeration-based iteration algorithms to solve these models for optimality. Through a database consisting of 243 groups of numerical results, we establish a decision tree model, a popular data mining technique, to explain which strategy should be employed under various conditions.

Keywords:

• Revenue management
• digital music product
• pricing
• bundling

Acknowledgements

We thank the Editor-in-Chief, Associate editor, and three anonymous referees for their constructive comments that greatly improved the content and expositions of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Figure 1 is captured from the Netease Cloud App, a popular digital music seller/platform in mainland China.

2 Note that the licensing fee could be a sunk cost at the pricing stage, which implies that the unit fee could play any role in the setting of songs’ prices. We use $l=0$ to cover such case. Also, one may consider the licensing fee of the album could be a fixed cost. Denote the fixed licensing fee as $F.$ One can use $F/N$ to replace $l$ to cover such case.

3 We show a procedure for estimating the probability distribution of $y$ in Appendix E.

4 We assume that a customer demanding $y$ songs will be lost if $V\left(y,v\right)<yp.$ Certainly, a customer with $V\left(y,v\right)<yp$ and $V\left(y-i,v\right)\ge \left(y-i\right)p$ ($\forall i=1,2,\dots ,y-1$) may purchase other quantity of songs. Our model covers such cases through considering a full market segmentation scheme in which market share of each segmentation is determined by a distribution law.

5 For a given $Q^S,$ Property 1 excludes the case with $l>(1-{\beta }^{Q^S})\overline{v}/\left[Q^S(1-\beta )\right],$ which implies that the unit price $p_{Q^S}^S$ will not cover the customer needing $Q^S$ unit songs because $p_{Q^S}^S>l$ should be held. Thus, this case should be dropped and then move to the case with $Q^S+1$ songs (see Step 3 of Algorithm S). Certainly, if $l>(1-{\beta }^{Q^S})\overline{v}/\left[Q^S(1-\beta )\right]$ is established for all $Q^S=1,2,\dots ,N,$ then strategy $S$ is infeasible for this album.

6 Similar with Property 1, Property 2 excludes the case with $Nl>(1-{\beta }^{Q^F})\overline{v}/\left(1-\beta \right).$

7 Still two other discrete distributions defined over a finite set of non-negative integers could be used to characterise the distribution of $y:$ Hypergeometric and Negative Binomial distributions. However, two parameters should be estimated for either of these two distributions. It thus could bring more errors when they are used to estimate probability $\mathrm{Pr}(y=i).$ We also conduct some numerical experiments using the Hypergeometric and Negative Binomial distributions to show the robustness of the results established in the Poisson and Binomial distributions. See Appendix F.

8 The procedures of calculating ${\mathrm{\Pi }}_{}^{SR},$ ${\mathrm{\Pi }}_{}^{FR},$ and ${\mathrm{\Pi }}_{}^{MR}$ are shown in Appendixes B, C, and D, respectively. Under the same settings of model parameters, we compare the expected profits of using our theoretically optimal prices and using real prices. Though there may exist other performance indications determining songs’ prices in reality, the comparison results of the expected results could be helpful for verifying the values of our models and algorithms.

Additional information

Funding

This paper is partially supported by the National Natural Science Foundation of China (grant nos. 72171073, 71771127 and 72188101) and the Fundamental Research Funds for the Central Universities (grant no. JS2021ZSPY0033).