The Incidental Economist

In a comment, steve (not co-blogger Steve) reminded me of a very good post by Scott Sumner that illustrates the endogeneity of prices with respect to quantity. It turns out I had read it, but I’m glad to be reminded of it.

So what do we know about prices? We know that if the price falls because supply increases, then consumption will increase, and if the price fell because demand fell, then consumption will decrease. In other words we know that if the price (or interest rate or exchange rate) changes, we can predict with 50% confidence that quantity will increase, and 50% confidence that quantity will decrease. So that’s progress, I guess.

This ambiguity is precisely why Sumner advises to never reason from a price change. The problem is price and quantity are both related to supply and demand factors. Thus, knowing only that prices changed one can’t draw a conclusion about quantity without knowing something about supply or demand. Doing so is like trying to infer reading comprehension from foot size. Both are related to age, among other things. In a word, price is endogenous. Sumner could have just as easily said, “Never reason from a change in an endogenous variable.”

As Angrist and Krueger described in a Journal of Economic Perspectives paper I summarized recently, the earliest known application of instrumental variables was to address the endogeneity of prices in estimating supply and demand elasticities of flaxseed.

If the demand and supply curves shift over time, the observed data on quantities and prices reflect a set of equilibrium points on both curves. Consequently, an ordinary least squares regression of quantities on prices fails to identify—that is, trace out—either the supply or demand relationship. P.G. Wright (1928) confronted this issue in the seminal application of instrumental variables: estimating the elasticities of supply and demand for flaxseed, the source of linseed oil. Wright noted the difficulty of obtaining estimates of the elasticities of supply and demand from the relationship between price and quantity alone. He suggested (p. 312), however, that certain “curve shifters”—what we would now call instrumental variables—can be used to address the problem: “Such additional factors may be factors which (A) affect demand conditions without affecting cost conditions or which (B) affect cost conditions without affecting demand conditions.” A variable he used for the demand curve shifter was the price of substitute goods, such as cottonseed, while a variable he used for the supply curve shifter was yield per acre, which can be thought of as primarily determined by the weather. …

Wright (1928, p. 314) observed: “Success with this method depends on success in discovering factors of the type A and B.” … Wright’s econometric advance went unnoticed by the subsequent  literature. Not until the 1940s were instrumental variables and related methods rediscovered and extended.

References

Angrist, Joshua and Alan Kreuger. (2001). “Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments,” Journal of Economic Perspectives, 15(4), 69-85.

Wright, Phillip G. (1928). The Tariff on Animal and Vegetable Oils. New York: MacMillan.

Having just spent too many hours on this, I’m posting it so I won’t forget it. Maybe it’ll save some other folks some time too.

I’ve learned that, up to a constant, consumer surplus is the integral of the demand function. That’s wrong. It’s negative the integral of the demand function. The reason has to do with the fact that what we call the demand function–that which translates price into quantity–is not what we draw on a graph. Thanks to Alfred Marshall, the custom is to draw the inverse demand function, with price on the vertical axis and quantity on the horizontal axis.

To show formally that consumer surplus is the negative of the integral of demand (or that demand is negative the derivative of consumer surplus), one can use the rule for integrating inverses. To do so, let

• Q denote quantity and P denote price,
• Q = D(P) be the demand function,
• and the equilibrium quantity and price be Q₀ and P₀, respectively.

Then, consumer surplus (CS) is

where the first line is the definition of consumer surplus, the second follows from the rule of integrating inverses and the facts that Q=D(P) and P=D^-1(Q), and the third holds if PD(P) is zero when P is infinity.

This is important for the very reason I spent hours trying to find or work out the above explanation. If one is working with discrete choice models (logit, nested logit, conditional logit, etc.) and wishes to derive the expression for consumer surplus, one might do so by recalling two things:

1. The functional form of consumer surplus is the log of the denominator of the discrete choice probability model, which is also the demand function. That is, consumer surplus has a log-sum of exponentials form.
2. The derivative of consumer surplus is negative the demand function, i.e. the discrete choice model.

Recalling point 1 and not 2, which is what I did, one is tempted to put the wrong sign on consumer surplus. That leads to very incorrect (exactly backwards) results. One is tempted to scrutinize one’s code for bugs. But the problem isn’t the code, it’s Alfred Marshall’s graphing convention.