Before the main content:

• I am creating an R Community on Google Groups. You can join the group using this form.
• With all the funding cuts to PhD studies, I’d appreciate if you can donate here.

In Economics we often use reduced-form models, which are simplified ways to describe a process. I tend to see the criticism to this as unfair, as in physics they do the same, and so many models in physics assume “spherical objects in a vacuum” as a simplification to make real-world problems mathematically solvable.

With reduced form models we are not interested in how faithful those ressemble reality but how useful these are. As an example in support of reduced form models, consider the two extreme cases of models to answer “how to get from 37 Old Queen St to St. James Park underground station?”:

1. Person 1 draws a quick sketch like this on an old receipt

      _ Embassy
     |   
  ---
 |
Underground

1. Person 2 built a very detailed 1:10000 architectural model of St. James neighbourhood and shows you the tiny streets, cars, etc.

For the question, person 1 sketch is sufficient.

The Armington model of trade sits between these two extremes. It considers a world of \(N\) countries, each country \(i\) has a fixed labour endowment \(w_i\) and there is an endogenous labour wage. Income is given by \(Y_i = w_i L_i\). Trade flows are given by \(X_{ij}\) (and so the sales share is \(\gamma_{ij} = X_{ij} / Y_i\)). There is a productivity shift \(\x_i\), a trade cost $_{ij}, and an elasticity of substitution \(\varepsilon\).

The strong assumption in this model is a known and homogenous elasticity of substitution. In this model countries can substitute between Apple and Android phones even when many would argue that those phones compete in different markets.

The model imposes a market clearing condition \(w_i L_i = \sum_{j=1}^N \lambda_{ij} w_j L_j\).

There are many values \(\lambda_{12}, \lambda_{13}, \ldots, \lambda_{1N}\) that balance the market clearing condition but in this model the solution is a gravity-type equation that leads to ${ij} = (i / ({ij} w_i)^{}) / sum{l=1}^N (l / ({lj} w_l)^{}). More about gravity-type equations can be read in Gravity for Undergrads by Prof. Dr. Yoto V. Yotov.

Consider, for example, the EU increase on low value imports from the UK described [https://hboltd.co.uk/uk-eu-shipping-changes-2026-customs-overhaul/]. In this model, instead of a product-by-product specific tariff have an initial UK-EU average trade cost of \(\tau_{ij}\) that increased to \(\tau_{ij}’\). The hat form of this is \(\hat{\tau_{ij}} = \tau_{ij}’ / \tau_{ij}\).

After the trade cost change the market clearing would be \(w_i’ L_i’ = \sum_{j=1}^N \lambda_{ij}’ w_j’ L_j’\), and then

\[ \hat{w_i} \hat{L_i} = \frac{\sum_{j=1}^N \lambda_{ij}’ w_j’ L_j’}{\sum_{j=1}^N \lambda_{ij} w_j L_j} \]

\[ \hat{w_i} \hat{L_i} = \frac{\sum_{j=1}^N X_{ij}’}{w_i L_i} \]

\[ \hat{w_i} \hat{L_i} = \sum_{j=1}^N \gamma_{ij} \hat{X_{ij}} \]

Similarly, $ = .

The new hat equations are useful, for example, to determine the effect of the change to the trade cost over wages \(\hat{w_i}\). The hat market clearing means that \(\hat{w_i} = \sum_{j=1}^N \gamma_{ij} \hat{X_{ij}}\) (labour is exogenous), and therefore \(\hat{w_i} = \sum_{j=1}^N \gamma_{ij} \hat{\lambda_{ij}} \hat{w_j}\).

Merging the hat market clearing with the hat gravity equation, we get

\[ \hat{w_i} = \sum_{j=1}^N \frac{\gamma_{ij} \hat{w_j} / (\hat{\tau_{ij}} \hat{w_i})^{\varepsilon}}{\sum_{l=1}^N \gamma_{lj} \hat{w_l} / (\hat{\tau_{lj}} \hat{w_l})^{\varepsilon}} \]

From the initial equilibrium shares \(\lambda_{ij}\) and \(\gamma_{ij}\) and a fixed \(\varepsilon\) (for example, estimated using USITC data), we have a system of \(N \times N\) equations that we can solve to get \(\hat{w_j}\) and then recover \(\hat{\lambda_{ij}}\). In other words, we simplified an identification problem by solving the differences by resorting on a sufficient statistic instead of estimating all trade costs.