**When to buy at Amazon**

Amazon just raised their 'free shipping' threshold to 35$, a few weeks before the holiday shopping season. This simple 'entropic optimization' approach, which utilizes Amazon's wish list to time-prioritize purchases remains valid, but requires an increased level of procrastination. What also caught my attention is Amazon's Kindle rental models. Beyond the initial sunk costs, virtual products are high margin, with negligible holding cost, besides an infinite, instantaneously replenish-able inventory. They are also scratch/damage proof. The only long-term downside to providing a renting option appears to be faulty pricing. If we price too low, we may turn many potential buyers into renters, and a high price may discourage potential renters. Let's look at a couple of (real) Kindle rentals for which I laboriously pulled data while watching Sachin Tendulkar's 199th cricket test match.

**Kindle Book 1 (Undergrad Math textbook)**

The minimum rental period is 60 days (50$), and the maximum (apparently) is around 360 days (140$), with the marginal price held approximately constant. We pay 30 cents for every extra rental day beyond the minimum period.

Kindle Book 1: Price Versus Rental Days |

The cost (snapshot at the point of observation) of purchasing a permanent copy was 200$. If we plot the percentage price discount versus the rental period expressed as percentage of a year, we can see that the discount varies between 25% and 70% of the full cost. Approximately linear model employed for this book. Here, we can rent the book for an entire year without paying the full price.

Price Discount Percentage versus Rental Percentage-of-Year |

**Kindle book 2 (Advanced forecast-modeling textbook)**

The second example is a bit more interesting. The content is technically far more sophisticated compared to Book 1, but the target market is different, and the number of (paper) pages is far lower, and so is the price. In both instances, the cheapest rental can be purchased at less than half the full price. There are roughly three different marginal prices employed within a rental period that varies between a minimum of 30 days (~$15) and a maximum of 365 days (~$35, also the full Kindle price). The corresponding breakpoints occur (roughly) near the 90-day, and 180-day rentals, respectively. If we restrict our attention to this rental time period, the price is concave, with the marginal prices decreasing as the rental period increases. It is preferable to simply buy the book rather than rent it for close to a year.

Kindle Book 2: Piece-wise linear rental pricing |

**Optimization Models**

**Seller**: How would optimization scientists go about determining these marginal rental prices? Suggestions welcome. Perhaps ideas from analytical rental models for other products (cars, houses, equipment ...) can be used as a starting point to figure out this "information rental" model. Perhaps the pricing model can be initialized using historical rental data gathered for similar books. This being an online retail sales model, we can dynamically and frequently update these models or their parameters to maximize performance metrics.

**Buyer:**From a user-perspective, if we can assign a value for owning a permanent copy, and have an informal mental model of the temporal utility of a rental as T(x), then (for example), we could solve some variation of this single-decision problem in 'x':

Maximize Value V = T(x)/f(x) (⇒ Maximize log T - log f, optionally)

l ≤ x ≤ u

to determine an optimal 'rent versus buy versus walk-away' decision based on our willingness-to-pay. Assuming a 1:1 mapping between 'f' and 'x', so we could transform any price range limit into an equivalent (l, u) bound on 'x'.

A simple way to solve this problem is to enumerate the values of V for all rental days using a spreadsheet.

Like thousands of Indian immigrants in the U.S, I came on an assistantship, carrying a couple of hundred bucks in my pocket that represented a big chunk of my parent's savings. I actually felt rich when I discovered that the take-home monthly income from my research assistant-ship after tuition fees turned out to be more than what my engineer dad was earning after decades of dedicated service in Nehruvian India. Until I saw the prescribed textbook prices, that is. Buying overpriced books was simply out of the question when the few copies in the library were already taken. There was no Amazon then, and it would've been amazing to have a rental option like this, especially when continued funding is dependent on maintaining good grades.

For example, if we only cared about a textbook for portions of a semester (our total planning horizon), and our total price budget is 'pmax', then we can informally solve a multiple-period version of the above optimization model to come up with a best

to determine an optimal 'rent versus buy versus walk-away' decision based on our willingness-to-pay. Assuming a 1:1 mapping between 'f' and 'x', so we could transform any price range limit into an equivalent (l, u) bound on 'x'.

A simple way to solve this problem is to enumerate the values of V for all rental days using a spreadsheet.

**Renting digital textbooks over a semester**Like thousands of Indian immigrants in the U.S, I came on an assistantship, carrying a couple of hundred bucks in my pocket that represented a big chunk of my parent's savings. I actually felt rich when I discovered that the take-home monthly income from my research assistant-ship after tuition fees turned out to be more than what my engineer dad was earning after decades of dedicated service in Nehruvian India. Until I saw the prescribed textbook prices, that is. Buying overpriced books was simply out of the question when the few copies in the library were already taken. There was no Amazon then, and it would've been amazing to have a rental option like this, especially when continued funding is dependent on maintaining good grades.

For example, if we only cared about a textbook for portions of a semester (our total planning horizon), and our total price budget is 'pmax', then we can informally solve a multiple-period version of the above optimization model to come up with a best

*waiting strategy*and rent for one or more time periods ("quiz time") which maximizes our total T(x) and also keeps us within our "knapsack" like price budget for the semester. This policy of "rent as needed" may work well with book rentals having a constant marginal price. On the other hand it may be worthwhile renting fewer times for an optimally longer duration if the price is concave in the length of the rental, as it is in our second instance.
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