Price Optimization from Data

Question

How can I find the optimal price that maximizes profits, given past sales data? I thought I could do this, but I've been running into problems.

Data:

Price, Quantity Sold, Unit Cost
$90, 1100, $10
$100, 1000, $10

I can find elasticity (-0.82) and I've been reading some things that say that you can find the optimal price, but I've only been able to tell the direction the price should be moved. This answer says use the Lerner condition, but that just shows if elasticity < -1 decrease and if elasticity > -1 increase price. And this article says what is much harder to do (although there are structural techniques to do so) is to work out what the optimal price should be

edit to add: I can find the linear equation for demand based on these 2 data points and find the point that maximizes revenue. I also found a demand function equation Qd=Qh*((P/Ph)^Ed), where Qd is the predicted Quantity Demand, Qh is observed Quantity, Ph is observed Price, Ed is Elasticity of Demand, and P is the independent variable price. But are any of these the best method to use?

You can of course optimize price (in general). But I hope you have more data than that... — Hack-R, Feb 21 '16 at 00:46
It's just a simple example. Can you link to any information on price optimization — Adam12344, Feb 21 '16 at 00:51
There's so much... I would suggest you start by understanding optimization in general. There's no magic about price. You can optimize practically anything with appropriate data, price is just one application. https://en.wikipedia.org/wiki/Price_optimization https://en.wikipedia.org/wiki/Mathematical_optimization — Hack-R, Feb 21 '16 at 00:56
Thanks for the links. I know how to do optimization in general, but I haven't seen much specifics on the models used (other than the simple ones above) — Adam12344, Feb 21 '16 at 01:07
Is it true in general that you can't extrapolate much from data? If you have pricing data from $5-$10 you couldn't tell if the optimal price is $12.50? — Adam12344, Feb 21 '16 at 01:11
You can extrapolate somewhat with a given degree of uncertainty if you have rich data, but it's infinitely better to interpolate. As for examples, you'll find a ton of studies on scholar.google.com and there are lots of textbooks and chapters of textbooks on the topic. Price Theory by Milton Friedman is one. There's really a wealth of material. Here's a theoretical paper from Stanford http://web.stanford.edu/~yyye/convexpricingms.pdf — Hack-R, Feb 21 '16 at 01:12
The Lerner condition described in the answer you have linked to in your question actually yields an exact optimal price, not just a direction. — Giskard, Feb 22 '16 at 12:09
@denesp The optimal price derived from that would be different when measuring from different price points, right? So if in addition to the data above I found at P=110, Q=900. My optimal price calculation would be different if I measured the elasticity from 90 and 100, or 100 and 110, or 90 and 110. Is there a way to use more than 2 data points? Or would you just use the min and max price points? Is this way better than trying to model the demand curve from data points? Thanks — Adam12344, Feb 23 '16 at 17:53
@Adam12344 I am not sure I understand your comments. (Consider editing your question to clarify.) A lot depends on how you model your demand function. Unless you specifically modell a demand function which has constant price elasticity price elasticity will not be constant and you can set a $p$ for which $\epsilon(p) = -1$. If price elasticity is constant and it is not equal to $-1$ then you have no revenue maximizing price but using the Lerner condition you may still have a profit maximizing price. — Giskard, Feb 23 '16 at 18:46

score 0 · Answer 1 · answered Feb 22 '16 at 11:58

One way can be to start from the extreme. For example, when price set $1000, zero sales; $500 two sales on the higher price side equation. To the other, price $10, 10,000 sales; price $50, 300 sales. In this way, come out with a figure that maximizes revenue.

Price Optimization from Data

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