The phrase “exact hat algebra” is used by trade and spatial economists far more often than it is clearly defined. In “Spatial Economics for Granular Settings” (September 2023), Felix Tintelnot and I aim to make clear that exact hat algebra is a means of conducting comparative statics, not a means of calibrating model parameters.
The phrase “exact hat algebra”, which I discussed in a 2018 blog post, is an extension of “hat algebra”. The latter phrase, per Alan Deardorff’s glossary entry, refers to “the Jones (1965) technique for comparative static analysis in trade models.” Jones (1965) presents local comparative statistics that leverage minimalist neoclassical assumptions (e.g., the Rybczynski theorem). By contrast, exact hat algebra delivers global comparative statics by exploiting full knowledge of the supply and demand curves (which is simple when these are constant-elasticity functions).
Both of these techniques are ways of presenting the comparative statics of a theoretical model. Exact hat algebra is not about identification or estimation per se. As Felix and I stress (page 7):
We emphasize the distinction between using the comparative statics defined by equations (5)-(7) to compute counterfactual outcomes and fitting the model’s parameters. Because equations (5)-(7) show that computing counterfactual outcomes only requires knowing the model’s parameters up to the point where the model delivers the shares lkn/L and ykn/Y, others have used the phrase “exact hat algebra” to refer to both rewriting the equations in hats and calibrating combinations of model parameters to rationalize observed shares. In fact, the system of equations defines counterfactual outcomes regardless of how one estimates or calibrates the parameters of the baseline equilibrium. The key question is how to fit the model’s parameters to data.
In our paper, Felix and I show how to fit a model of bilateral commuting flows in a variety of ways: regressing flows on observed bilateral covariates, using matrix approximations such as a rank-restricted singular value decomposition, or calibrating pair-specific cost parameters to replicate the shares observed in the raw data. These different methods produce different predictions about counterfactual outcomes because they produce different baseline equilibrium shares. But exact hat algebra defines the comparative statics of the model (with a continuum of individuals) for each of these parameterizations.