The Relationship of Price and Demand
For this article, I am going to discuss how we model the relationship between price and demand. Most people have an intuitive understanding that when the price of a good increases, the demand will decrease. Conversely, when the price of a good decreases, the demand will increase. The change in demand according to a change in price is called the price elasticity of demand. In its simplest form, this relationship can be expressed with a linear price-response function, as seen in Figure 1.
For our examples, let us assume that your company produces a widget that typically sells at an average price of $10. It is assumed that widget demand will drop to 0 at $20.
The linear price-response function is just a straight line. At a price of $0, the demand is at its maximum. At some price ($20 in our example), the demand drops to zero. And the linear price-response function assumes that demand changes at a constant rate over all prices. This function should look familiar to anyone who has taken an introductory economics course. It is an incredibly simple approach to modeling how demand changes in response to price.
Unfortunately, the pursuit of simplicity can often require concessions in other criteria. In this case, simplicity requires one to sacrifice some realism with the model. The constant slope of the function implies that demand changes at the same rate along the entire price range for any given price change. For example, this function posits that a 20-cent price decrease will increase demand a set amount whether that price decrease is from $20.00 to $19.80 or from $10.00 to $9.80.
However, this oversimplification is quite unrealistic. If you had your product labeled with a sell price of $20 (which will have a demand of 0), people are not going to start lining up outside your shop because you dropped the price to $19.80.
If a customer is used to paying $10 for a comparable widget, reducing your $20 starting price by $1 or $2 or $5 will probably not stimulate much additional demand. On the other hand, if your starting price is $10, then you may be able to stimulate additional demand by reducing your price by 20¢, 10¢, or even 5¢. Customers are much more sensitive to price changes when your starting price is near a competitor’s price. Generally speaking, we should expect competition to be strongest in the middle of the typical market prices. Thus, we would expect price changes there to have the largest impact on consumer behavior.
A linear price-response function can give you a simplistic understanding of how demand changes in response to price changes. However, this function should not be recklessly applied across all possible prices for a product. At best, a linear price-response function can help you approximate how demand will change in response to price changes over narrow price ranges. This does not mean that this function is useless. It is a great way to communicate the concept of elasticity from a theoretical perspective. But it is very important to keep the limitations of this linear function in mind when using it.
In reality, the elasticity of a given product can change quite dramatically for different price ranges. A much better model of the relationship between price and demand is shown in Figure 2.
The sigmoid price-response function effectively models a range of demand responses across a large price range. Research indicates that it effectively models price responses across many markets. The function shows that demand responds little to price changes when the starting price is very low. Consider the fact that consumers would still very much desire the product if a seller increased the price from $1 to $3. Even though you would be increasing the price by 200%, demand would remain very high because consumers are used to paying around $10 for this widget.
However, as the starting price approaches the market price, consumers become much more sensitive to price changes. Demand drops much more sharply when the price increases from $6 to $8 than from $4 to $6, and demand drops even more dramatically when the price increases from $8 to $10.
The function in Figure 2 is symmetrical on either side of $10, so we see the rate of demand change in reverse order as we continue increasing price. The demand decrease from $10 to $12 is very dramatic, the demand decrease from $12 to $14 is less so, and a price change from $14 to $16 decreases the demand very little. (The thinking is that as the price increases past the normal range of market prices, the remaining customers exhibit less response to prices. Perhaps they are very loyal customers. Perhaps they are ignorant of what the market price for your widget is. In either case, they are less price sensitive than most of your market, so the slope of the curve changes.)
There are several variations of the sigmoid function, depending upon what your purpose is. (For example, one that is commonly used in pricing is the logit price-response function.) So the next time that somebody pulls out a linear demand curve, you can be prepared to show them a better, more accurate way.
Phillips, Robert L. Pricing and Revenue Optimization. Stanford (Calif.): Stanford University Press, 2005.