Price Increase When Demand Shifts – Semi Rigorous Proof

Previously I have written about pricing for recessionary times and how CPGs and other businesses are realizing increase in profits despite drop in revenues. As more and more price sensitive customers switched to private labels and other low cost options, premium brands responded by raising prices. The claim is this price increase delivers higher profit than a price cut to gain back customers because once the price sensitive customers moved out those that continue to prefer the brand are less price sensitive. The claim is just that if it is not formally proved. With the recent changes in bottled water prices I made an attempt at proof. But it was one example based on one data point and is not really a proof.  Here is another attempt.

Let us take the Starbucks as example. I recently made the same claim on price sensitivity of Starbucks customers. Let us assume there are only two types of Starbucks customers one is price sensitive and the other is relatively less price sensitive. Each with linear demand curve:

q1 =  a  – b * p   (demand curve for price sensitive customers)

q2 = c  – d * p  (demand curve for  brand conscious customers)

Mathematically it is easy to show that the latter curve is steeper than the previous. Each demand curve yields a different profit maximizing price p*, the second curve’s is higher than that of the first (again proof exists in textbooks). Any price higher or lower than p* will yield lower profit (hence the name profit maximizing price). Let us call the p* for demand curves 1 and 2 as p1 and p2.

If Starbucks can find out who is who and can separate them then they can charge different prices. This is called Third degree price discrimination. But when a customer walks into one of their stores Starbucks has no way of finding whether she is of type 1 or 2. They are also attracted by the higher volume by combining the customer segments so to them the combined demand curve will be

q = q1 + q2 =  (a+c) -(b+d)p

This has its own profit maximizing price which is between the profit maximizing prices of the two demand curves.  Let us call this p3. For this demand curve any price different from p3 will yield lower profit than p3.

In other words, p1<p3<p2

With the down economy the price sensitive customers simply stopped coming to Starbucks, thereby revealing who the brand conscious customers are. In other words, the demand curve became

q = c – d*p

If Starbucks continued to price at the previous profit maximizing price of p3 (which is lower than  p2), its profit will be ower than what would it have been if it were to price at p2 (the profit maximizing price for the demand curve).

This proof can be extended to account for many different demand curves and non-linear demand curves.

Hence it makes sense for Starbucks to increase its price when the demand curve shifts.