I don’t think I understand the distinction you’re drawing, however. I said that this equation:

D = a + bP

represents potential outcomes. You disagree, if I read you correctly. You say instead that this equation represents potential outcomes:

D(p) = a + bp for every P=p

which I don’t understand. How are these not equivalent?

Perhaps you mean the equation I wrote only applies to observables, hence “the consumer does not observe “potential price P(D), nor can he determine the “potential demand” D(P)” and “economists wrote it… using observed variables in the equations.”

Suppose that price could take only one of two values, P_0 or P_1. Then we read off quantity demanded from the demand equation at these two prices: it is D_0 = a + bP_0 or D_1 = a + bP_1. The people in the model, as it were, can decide how much they’d like to purchase in the each possible world. D_0 and D_1 are potential outcomes, and the causal effect of changing price from P_0 to P_1 on quantity demanded is their difference. This is exactly the Neyman-Rubin notion of potential outcomes.

That is, I would say that the demand equation tells us the quantity demanded at any price, not just at observed prices. Is this where we disagree?

Thanks, Chris.

]]>We agree conceptually on the “schedule” interpretation of structural equations, but we differ on notation. The modern notation is:

D := a + bP

S := c + dP

Where D, S and P are observed variables and := is an assignment operator, to be distinguished from algebraic equality. It means: An agent (possibly Nature)

observes P and, accordingly, assigns D the value a + bP.

[In your example, each consumer observes the realized price P and decides how much to purchase. At any given time, the consumer does not observe “potential price P(D), nor

can he determine the “potential demand” D(P)”]

If we wish to translate this assignment process to potential outcome notation, we get (according to the First Law):

D(p) = a + bp for every P=p

But there is no real need to translate.

We can leave it the way economists wrote it (eg. Haavelmo 1943) and the way we think

about it naturally, namely, using observed variables in the equations, and interpreting the equality sign as an assignment operator. This is precisely the role that arrows play in Sewall Wright’s path diagrams.

The notation := (for assignment) is standard in computer science and, as far as I know, was first used in economics by Nancy Cartwright (cant find the first reference)

]]>Many interesting comments here, thanks for this post. If you will humour me, I want to take up just one of your points, since it is one that comes up often and we touched on it recently on twitter.

You write:

“The term “potential outcome” is a late comer to the economics literature of the 20th century, whose native vocabulary and natural primitives were functional relationships among variables, not potential outcomes. The latters are defined in terms of a “treatment assignment” and hypothetical outcome, while the formers invoke only observable variables like “supply” and “demand”.”

That’s not correct. Supply and demand are not observable, they encode potential outcomes are causal concepts, not merely functional relations.

Here is a quick primer on the basic model:

There are many firms and many consumers, all of whom take the price of some good as parametric. At any price P, each consumer decides how much to purchase. The sum of those purchases is D(P). This function has a causal interpretation: if the price is changed to P’, then the causal effect on units demanded is [ D(P’) – D(P) ]. Put another way, this schedule represents a continuum of potential outcomes, one for each potential price P. Of course, all but one of these potential outcomes are not observed.

Similarly, at every potential price P, each firm decides how much it wants to produce, and the sum of production is supply, S(P). Supply is likewise a causal object representing a continuum of potential outcomes.

Finally, an equilibrium condition is imposed: through some unmodeled emergent process, the realized price, P*, satisfies D(P*) = S(P*). A linear version is then,

D = a + bP

S = c + dP

D = S.

Note this is not a cyclical relation, it’s not something like `consumers choose Q (which affects P) and firms choose P (which affects Q),’ despite the impression some econometrics textbooks give.

A well-known (relatively) modern treatment of this sort of model, cast explicitly in terms of potential outcomes, can be found in Angrist, Graddy, and Imbens (2000):

https://academic.oup.com/restud/article/67/3/499/1547484

but it is important to emphasize that adding the jargon “potential outcomes” doesn’t change the core (causal) concepts, which go back to the 19th century.

On Twitter you suggested that this model can be represented graphically using an “=” operator, which I haven’t seen before and would be interested in hearing more about.

Thanks, Chris.

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