## Friday, April 26, 2013

### Book prices on Amazon

Question:  A bookseller has two royalty schemes. An author can accept a 35% royalty for books priced as low as \$0.99.  The author can accept a royalty of 70% but the book must be priced at \$2.99 and above.   Under what circumstances should the author select a book price between  \$1.00 and \$2.98 inclusive?   When should the author choose the \$2.99 & 70% royalty option over a \$0.99 & 35% royalty option?

A price increases above  \$0.99 will lead to a decrease in books sold.  The gain to the author is 0.35 x Revenue so if the price increase leads to a decrease in revenue the author will keep the price of the book at \$1.00.

Let’s illustrate with numbers.

The author will sell 200 copies at \$1.00.  Revenue is \$200 (\$1.00x200).

The author sets the price at \$1.10.  Book sales drop.  If revenue is above \$200 the author is better of setting prices at \$1.10 than \$1.00.  If revenue falls below \$200 the author should keep the price at \$1.00.  The cutoff quantity is obtained from

\$1.10 x Q =\$200

or

\$200/1.10=Q or Q=181.81.

If book sales fall below 181.81 the author will keep price at \$1.00 rather than raise it to \$1.10.  Note that (200-181.81)/181.81 is equal to 0.10, which is the size of the price increase.    The author will only increase the price of the book if the quantity sold is inelastic with respect to price.

Changing the price to \$2.99 is another story because the author’s share of revenue doubles to 70%.    When is an author indifferent between selling 200 copies of a book at \$1.00 compared to Q copies of a book at \$3.00.  Indifference occurs when

\$3.00 x Q x 0.75 =\$200 x 0.35

or

Q=\$70/(\$3.00 x 0.75)=31.1111

Note that \$200 x 0.35 and 31.11 x \$ 3 x 0.75 are both \$70.

If an increase in book price from \$1.00 to \$2.99 leads to a decrease in book sales from 200 to below 31 the author will be better off keeping the book price at \$1.00.  I suspect there are a lot of \$2.99 books on sale at Amazon and relatively few books priced between \$0.99 and \$2.99.

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