# Is there an economically sound way to decide between buying media or a streaming subscription?

How can a consumer (like myself) decide on whether they should pay for a music streaming service based on economic reasoning?

Assumptions:

• The budget allows paying for a € 10/Month music subscription with what practically is a near infinite amount of songs.
• For 90% of the music, previews or ad-supported versions can be streamed for free. The streams do not offer a high-quality listening experience, but they are good enough to discover and prelisten
• The user would, without a subscription, buy 10 albums a year for € 9 each.
• The purchased albums can be played infinitely and passed on to family members.
• The user listens to music 1 hour a day when listening to purchased albums, 1,5 a day when paying for streaming
• 80% of the time the user listens to music they listened to before or will listen to again

The numbers are just examples, but key is that streaming services offer near infinite libraries, whereas purchased personal libraries are always limited to a budget.

• Who would want to listen to the same 10-15 albums in a given year (cost €90-135) when the alternative is to listen to any album, anywhere (cost €120)? – M3RS Sep 12 '17 at 12:58
• It's only 10-15 albums in the first year, as a user's collection grows over time. Personally I don't think I ever bought that many albums in a year. Because of assumption 2, people are not limited to listening to only what they know already. – kslstn Sep 13 '17 at 19:28

Partly your question relates to more general questions like "buy versus rent a house", or "buy versus lease a machine". Under neoclassical assumptions of competition, full information, etc, you can imagine that arbitrage would make these options equivalent for the average of the population (or the representative agent). In practice, heterogeneous individuals might prefer one option than the other.

Having said this, I think the study that is more pertinent for your example is "The economics of subscriptions", by Glazer and Hassin (1982). They evaluate whether it is welfare enhancing for a service provider (in their case, a publisher issuing a monthly magazine), to offer both subscriptions and individual sale of issues, rather than one of them alone. Adapting their terminology and notation to the case of music, define:

• $$\bar p$$: average value to a consumer of an album. Might vary among consumers.
• $$m$$: number of albums available with subscription per year.
• $$t$$: price of each album by itself (assume identical). This might include the transaction cost of buying the albums.
• $$c$$: cost of subscription. (Assume no re-selling is allowed, an important disclaimer!).

# Subscription

In this setting, the expected value of a consumer's surplus obtained by purchasing a subscription is

$$m\bar p -c$$

# Individual purchases

Here, we assume that you can choose your favorite albums. Hence, for a given population of albums, you will choose the $$n$$ that give you the most surplus (your favorite artists). For simplicity, I assume you do you know the universe of albums (To add more complexity, you could instead assume you know a section of it, and hence a subscription could surprise you. I leave this example to you.)

The surplus from buying the best $$n$$ albums (subset denoted $$\Theta$$) is

$$\sum_{i \in \Theta} (\hat p + \epsilon_i - t)$$

where $$\epsilon_i$$ is the value above $$\bar p$$ of the $$i$$-th album you choose to buy. The above is equal to:

$$n(\hat p - t) + \sum_{i \in \Theta}\epsilon_i$$

Hence, the valuation hinges on this comparison:

$$m\bar p -c \text{ vs } n(\hat p - t) + \sum_{i \in \Theta}\epsilon_i$$

# Analysis

Since you chose the best albums, $$\sum_{i \in \Theta}\epsilon_i$$ is large. However, these albums are also included in the subscription. Hence, define the non-negative function $$X$$ as the combined value of all the albums in the subscription that you would not buy outside of it. This is:

$$X = (m - n)\bar p - \sum_{i \in \Theta}\epsilon_i$$

Notice that $$\lim_{n \rightarrow m} X = 0$$.

The comparison is now:

$$X \text{ vs } c - nt$$

Thus, you prefer the subscription when:

• the number of albums available under subscription is high
• the albums you would not buy are very valuable
• the cost of subscription is low
• the cost of individual albums is high
• the value of your best albums is low (if some $$\epsilon <0$$, then such album is below average)
• the effect of $$n$$ depends on the value distribution (the distribution of $$\epsilon$$). Clearly, at some point you are buying bad albums for the same price, so you rather buy the subscription.
• notice that if you do not know all the albums, then there is a stronger case for a subscription as you might discover some very good albums which value might even be higher than those you would buy otherwise. Here, a modelling of such distribution becomes crucial.
• the above does not include a risk-loving valuation from risk (i.e. discovering good music!). Such consumer would then more likely favour subscription.
• Wow! The number of albums is virtually endless (more than a person can listen even by sampling) and the value distribution is unknown. But because of the endless supply, there should be enough great music for ϵi to stay high, if only the user can find the right music. A very strong case for subscriptions. That said, maybe people's minds limit them in accepting an album as great. There's nothing like your first album of what became your favourite band, right! And listening to it over and over again could make it more valuable over time (something I experience even with mediocre albums). – kslstn Sep 13 '17 at 19:06
• The equation ignores that while paying for a subscription for decades users may grow emotionally attached to a large amount of music that at some point in the future may no longer be available in the subscription. Price increases could also lead to a situation in which a user is no longer able to pay for the subscription that has their favourite songs, leaving them with only costs and no music. This risk is impossible to calculate. The solution may be hedging against such scenarios by paying for subscriptions only part of the time and then buying the favourite albums in the remaining time. – kslstn Sep 13 '17 at 19:25
• @kslstn There are many many things not included. As always with modelling, you have to choose what to include and what not to include. – luchonacho Sep 13 '17 at 19:50

I don't know if it is an economically sound solution, but a person could run experiments with alternating every few months between buying and subscribing for a couple of years. They'd self-report satisfaction with the music they listen to as well as the amount they pay for albums. Based on the assumption that the music they love will always be available in the subscription and that the subscription price will remain the same, they could then decide what works best for them.