# Mathematical Model For (E-) Book Sales

It seems to be a no-brainer that with more books on the market, an author will see higher revenues. I wanted to know more about how the sales rate varies with the number of books. So I did what I always do when faced with an economic problem: construct a mathematical model. Even though it took me several tries to find the right approach, I’m fairly confident that the following model is able to explain why revenues grow overproportionally with the number of books an author has published. I also stumbled across a way to correct the marketing R/C for number of books.

The basic quantities used are:

• n = number of books
• i = impressions per day
• q = conversion probability (which is the probability that an impression results in a sale)
• s = sales per buyer
• r = daily sales rate

Obviously the basic relationship is:

r = i(n) * q(n) * s(n)

with the brackets indicating a dependence of the quantities on the number of books.

1) Let’s start with s(n) = sales per buyer. Suppose there’s a probability p that a buyer, who has purchased an author’s book, will go on to buy yet another book of said author. To visualize this, think of the books as some kind of mirrors: each ray (sale) will either go through the book (no further sales from this buyer) or be reflected on another book of the author. In the latter case, the process repeats. Using this “reflective model”, the number of sales per buyer is:

s(n) = 1 + p + p² + … + pn = (1 – pn) / (1 – p)

For example, if the probability of a reader buying another book from the same author is p = 15 % = 0.15 and the author has n = 3 books available, we get:

s(3) = (1 – 0.153) / (1 – 0.15) = 1.17 sales per buyer

So the number of sales per buyer increases with the number of books. However, it quickly reaches a limiting value. Letting n go to infinity results in:

s(∞) = 1 / (1 – p)

Hence, this effect is a source for overproportional growth only for the first few books. After that it turns into a constant factor.

2) Let’s turn to q(n) = conversion probability. Why should there be a dependence on number of books at all for this quantity? Studies show that the probability of making a sale grows with the choice offered. That’s why ridiculously large malls work. When an author offers a large number of books, he is able to provide list impression (featuring all his / her books) additionally to the common single impressions (featuring only one book). With more choice, the conversion probability on list impressions will be higher than that on single impressions.

• qs = single impression conversion probability
• ps = percentage of impressions that are single impressions
• ql = list impression conversion probability
• pl = percentage of impressions that are list impressions

with ps + pl = 1. The overall conversion probability will be:

q(n) = qs(n) * ps(n) + ql(n)* pl(n)

With ql(n) and pl(n) obviously growing with the number of books and ps(n) decreasing accordingly, we get an increase in the overall conversion probability.

3) Finally let’s look at i(n) = impressions per day. Denoting with i1, i2, … the number of daily impressions by book number 1, book number 2, … , the average number of impressions per day and book are:

ib = 1/n * ∑[k] ik

with ∑[k] meaning the sum over all k. The overall impressions per day are:

i(n) = ib(n) * n

Assuming all books generate the same number of daily impressions, this is a linear growth. However, there might be an overproportional factor at work here. As an author keeps publishing, his experience in writing, editing and marketing will grow. Especially for initially inexperienced authors the quality of the books and the marketing approach will improve with each book. Translated in numbers, this means that later books will generate more impressions per day:

ik+1 > ik

which leads to an overproportional (instead of just linear) growth in overall impressions per day with the number of books. Note that more experience should also translate into a higher single impression conversion probability:

qs(n+1) > qs(n)

4) As a final treat, let’s look at how these effects impact the marketing R/C. The marketing R/C is the ratio of revenues that result from an ad divided by the costs of the ad:

R/C = Revenues / Costs

For an ad to be of worth to an author, this value should be greater than 1. Assume an ad generates the number of iad single impressions in total. For one book we get the revenues:

If more than one book is available, this number changes to:

R = iad * qs(n) * (1 – pn) / (1 – p)

So if the R/C in the case of one book is (R/C)1, the corrected R/C for a larger number of books is:

R/C = (R/C)1 * qs(n) / qs(1) * (1 – pn) / (1 – p)

In short: ads, that aren’t profitable, can become profitable as the author offers more books.

For more mathematical modeling check out: Mathematics of Blog Traffic: Model and Tips for High Traffic.