How To Estimate A Demand Curve

Demand is a downward sloping curve with price on the y-axis and quantity on the x-axis. You have historical sales data. Seems like you should plot the prices against the quantities and fit a line to it, right?

Well, not quite - there are some important factors to take into account.

The Problem With Price-Quantity Plots

The problem with fitting a line to a price-quantity plot is that those price, quantity pairs probably didn't happen in a vaccum.

For example, say that you sell a stylish new brand of sneakers, and everybody loves them. Demand for these sneakers explodes, so you raise the price.

What does the price-quantity plot look like in this case? Upward sloping. The demand curve increased, but your supply curve stayed the same. The price-quantity plot would actually trace your supply curve, not your demand curve.

As another example, say that you sell flatscreen televisions, and the technology to produce them gets cheaper and cheaper every year. Assuming that the demand for flatscreens is rather stable, then you'll be forced to lower prices as competitors enter the market.

In this case, fitting a line to the price-quantity plot actually works: Supply shifted while demand stayed the same, so the plot accurately traces your downward sloping demand curve.

Price-Quantity Plots: it can be hard to tell if you're measuring supply or demand.

If both supply and demand increase at the same time, then you could end up producing more at the same price, creating a flat price-quantity plot!

In general, there will be both supply and demand factors at play, and it's not always obvious how to interpret a plot of price and quantity.

How To Fix It

You fix this problem with price-quantity plots with more data and a modeling technique called instrumental variables. Here we'll focus on applying the fix to demand, but the same process can be used to address supply curves as well.

Collect More Data

If all the data you have is price and quantity, then you'll need some more information.

Because quantity demanded depends on factors other than price, you must collect data on those other factors to make sure that you don't attribute every change in quantity to a change in price. The major factors you'll want to include are likely related to customer income, tastes and preferences, customer expectations about the future, and the prices of related goods.

Next, you'll need to collect data on supply.

Collecting data on supply may seem counterintuitive, but we'll use it to estimate the demand curve just like we did in the flatscreen television example: we'll use historical shocks to supply to trace out the curve of demand. Relevant supply data can include technology changes, supplier prices, tax and subsidy changes, forces of nature, and changes in the number of competitors.

Model With Instrumental Variables

The nice thing about having supply data is that it allows us to trace out the demand curve, and we can do this with an econometric tool called instrumental variables (IV). There are some details to IV, but the intuition is rather simple:

Find a variable that would shift the supply curve and not the demand curve. This is called an instrument, and you already found these by gathering data on supply.
Observe what happens when the instrument changes. When the instrument changes, this is a supply shock. Observing how price and quantity change in response to these supply shocks creates a trace of the demand curve.

A nice example of this is in a study done about the Fulton Fish Market in New York City. In order to estimate the demand curve for the fish market, the researcher used weather as an instrumental variable: when the weather was bad, fewer fish were caught and the supply curve shifted. Price and quantity then changed according to the trace of demand.

There are more details to the mechanics of IV, but the key idea is that it allows you to properly estimate a demand curve. It's also a very common and accessible tool - software implementations of regression will all have a way to specify instrumental variables.

Other Ways To Estimate Demand Curves

It's worth noting that the IV method is mostly applicable for a specific, but common situation: A situation where you passively observe price and quantity changes over time. More proactive ways to estimate demand include asking customers directly and running controlled pricing experiments.

Asking customers directly about their willingness to pay may be convenient, but it can also be unreliable. Customers may not know how much they're willing to pay, or they may not want to tell you. It's also more difficult to think in terms of how much you would hypothetically pay rather than being confronted with a real situation and then making a decision.

Pricing experiments are more reliable than asking customers directly. You may do this in a lab-like setting where you invite representative customers into a simulated environment with fake money. You can also run focused A/B tests in the real world to observe how customers respond in actual purchasing situations. Either way, these experiments will likely give you a better estimate than asking customers to self-report.

Conclusion

The simple relationship between price and quantity for demand curves can make it seem seductively easy to estimate. However, fitting a line to price-quantity data can be misleading because it doesn't account for the dynamic effects of supply and demand. By collecting more data and using instrumental variables, you can take these effects into account and estimate demand curves more accurately. Lastly, other methods like pricing experiments and customer interviews can also be useful tools when trying to understand the shape of demand.