# Thought experiments that exact hat algebra can and cannot compute

Among other things, I’m teaching the Eaton-Kortum (2002) model and “exact hat algebra” to my PhD class tomorrow. Last year, my slides said “this model’s counterfactual predictions can be obtained without knowing all parameter values by a procedure that we now call ‘exact hat algebra’.” Not anymore. Only some of its counterfactual predictions can be attained via that technique.

As I reviewed in a 2018 blog post, when considering a counterfactual change in trade costs (and no change in exogenous productivities nor population sizes), the exact-hat-algebra calculation requires only the trade elasticity and initial trade flows in order to solve for the endogenous proportionate wage changes associated with any choice of exogenous proportionate trade-cost changes.

In Section 6.1 of Eaton and Kortum (2002), the authors consider two counterfactual scenarios that speak to the gains from trade. The first raises trade costs to their autarkic levels (“dni goes to infinity”). The second eliminates trade costs (“dni goes to one”). Exact hat algebra can be used to compute the first counterfactual; see Costinot and Rodriguez-Clare (2014) for a now-familiar exposition or footnote 42 in EK (setting α = β = 1). The second counterfactual cannot be computed by exact hat algebra.

One cannot compute the “zero-gravity” counterfactual of Eaton and Kortum (2002) using exact hat algebra because this would require one to know the initial levels of trade costs. To compute the proportionate change in trade costs associated with the dni=1 counterfactual, one would need to know the values of the “factual” dni. The exact hat algebra procedure doesn’t identify these values. Exact hat algebra allows one to compute proportionate changes in endogenous prices in an underidentified model by leveraging implicit combinations of parameter values that rationalize the observed initial equilibrium without separately identifying them.

Exact hat algebra requires only the trade elasticity and the initial trade matrix (including expenditures on domestically produced goods). That’s not enough to identify the model’s parameters. (If these moments alone sufficed to identify bilateral trade costs, the Head-Ries index that only computes their geometric mean wouldn’t be necessary.) Thus, one can only use exact hat algebra to compute outcomes for counterfactual scenarios that don’t require full knowledge of the model’s parameter values. One can express the autarky counterfactual in proportionate changes (“d-hat is infinity”), but one cannot define the proportionate change in trade costs for the “zero-gravity” counterfactual without knowing the initial levels of trade costs. There are some thought experiments that exact hat algebra cannot compute.

Update (5 Oct): My comment about the contrast between the two counterfactuals in section 6.1 of Eaton and Kortum (2002) turns out to be closely related to the main message of Waugh and Ravikumar (2016). They and Eaton, Kortum, Neiman (2016) both show ways to compute the frictionless or “zero-gravity” equilibria when using additional data (namely, prices or price deflators). See also footnote 7 of Sposi, Santacreu, Ravikumar (2019), which is attached to the sentence “Note that reductions of trade costs (dij − 1) require knowing the initial value of dij.”