Dimensional analysis in economics
Economics as the science of confusing stocks with flows
This post is an introduction to another series tangentially related to my other interests. It’s about the mathematical structure of economic models. I will be posting about it Fridays, and more current or topical news Wednesdays.
When Trump announced his tariffs April 2, the shocked responses of people I follow fell into two basic categories. People in finance asked “what will this mean for markets?” People in economics asked “what is he thinking?”1 As a mathematician my first thought was “What does this even mean?”
What was being proposed does not originate in conventional economic theory. There was a rapid effort to reverse engineer the numbers on the silly posters, and then to understand the kind of thinking they could be based on. It was quickly observed that what Trump called “foreign tariffs” were in fact trade deficits (of goods, relative to US goods imports). The proposed US tariffs were 50% of this, with an absolute minimum of 10%. FT Alphaville had an explainer up very quickly.
What is jaw-dropping to me, a mathematician, is that someone could confuse a price (the tariff) with a quantity (the trade deficit). Let alone equate them in some way. Of course they interact, but they have different units, different dimensions, and cannot simply be interchanged. So I was relieved to see the formula the “theory” is based on2:
(x for exports, m for imports) contains parameters \eps and \phi that someone had chosen to multiply to 1 … but they still have dimensions and can make the units on either side of the equation correspond.
The rest of the ideas are still stupid economically, financially and politically. But at least they compute.
The reference in the subtitle is due to reconstruction-era economist Michal Kalecki, who lamented the juxtaposition of quantities (stocks) and rates (flows: change of quantities over a period of time). It used to be economists at least distinguished prices from both!
Thank you for reading this far. What is this all leading to? Well, my mathematical work on dimensional analysis in economics. If you are meticulous in distinguishing prices, quantities, and flows, you notice a structure in economic models that mathematicians call symplectic. In economics3 the budget constraint is at the center of any model, and it defines not only the working of the model but also the prices, quantities, values and periods under consideration—the very form of the model. Treating this structure rigorously takes care of most model bookkeeping, which can become involved when we consider changes not only to quantities but to coordinates in which these quantities are measured4. Over the coming weeks I hope to convince you that economic models that are concerned with equilibrium and no-arbitrage can be characterized by this budget constraint and a single generating function to describe their dynamics.
Same goes for psychologists—and should matter for members of Congress to whom the executive branch is accountable. But I digress.
Apparently it appears in a book by Trump’s trade advisor Peter Navarro, and is the first response of LLMs when asked about optimal tariffs.
The dismal science, remember.
Think of calculating an inflation index: through time the underlying basket of goods and their quality drifts.