\[ \require{color} \definecolor{bcOrange}{RGB}{189,97,33} \newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \newcommand{\bm}[1]{\boldsymbol{\mathbf{#1}}} \]

Trade Policy in the Shadow of Power

Quantifying Military Coercion in the International System

1 September 2019

Brendan Cooley

Ph.D. Candidate

Princeton University

bcooley (at) princeton.edu

Power and Preferences

Anarchy and Inference

  • Anarchy \(\implies\) powerful governments can impose policies on others by force
  • Bargaining critique: policies reflect shadow of power
    • Brito and Intrilagator (1985), Fearon (1995), Art (1996)
  • Question: Do observed policies reflect preferences of the governments that adopted them or constraints of anarchy?

Trade Policy: What Do Governments Want?

  • Heterogeneity in desired protectionism
    • Grossman and Helpman (1994), Gawande, Krishna, and Olarreaga (2009)
  • Mercantilism induces conflicts of interest between governments
    • Gunboat diplomacy: powerful impose openness abroad (Findlay and O’Rourke 2007)

Trade Policy: Does It Matter?

  • Tariffs small, aggregate policy-induced trade frictions large
    • ~order of magnitude larger than tariffs (Cooley 2019) (barriers)
  • Welfare effects of trade frictions
    • Autor, Dorn, and Hanson (2013), Costinot and Rodríguez-Clare (2015), Goldberg and Pavcnik (2016)

Anarchy and Inference: Approach

Ideal Experiment

  • Exogenously manipulate governments’ military capacity
  • Attribute changes in trade policy to latent effect of military coercion

A Coercive International Political Economy

\[ \left\{ \bm{\theta}, \bm{M} \right\} \rightarrow \bm{\tau} \]

  • \(\bm{\theta}\) – structural parameters (preferences and power projection)
  • \(\bm{M}\) – governments’ military endowment (expenditure)
  • \(\bm{\tau}\) – governments’ trade policies

Estimation: \(\left\{ \bm{\tau}, \bm{M} \right\} \rightarrow \bm{\theta}\)

Counterfactuals: \(\left\{ \bm{\theta}^\prime, \bm{M}^\prime \right\} \rightarrow \bm{\tau}^\prime\)

  • Shadow of power
    • Effect of changing military endowments on trade policy
  • Liberal/commercial peace
    • Effect of changing preferences for openness on military strategies

Agenda

  1. Model
  2. Preliminary Results
  3. Identification

Model

\[ \left\{ \bm{\theta}, \bm{M} \right\} \rightarrow \bm{\tau} \]

Coercive International Political Economy

  • Governments indexed \(i \in \left\{ 1, ..., N \right\}\)

Sequence

  1. (\(\Gamma^{\bm{m}}\)) Governments set military strategies
    • allocations of effort over potential wars
  2. (\(\Gamma^{\bm{\tau}}\)) Governments make trade policy announcements
  3. (\(\Gamma^{\bm{a}}\)) Wars
    • Winners choose trade policies for vanquished governments
  4. (\(h(\bm{\tau})\)) Economic (general equilibrium) consequences of trade policies

Payoffs

\[ G_i \left( h(\bm{\tau}), \bm{a} \right) \]

(\(h(\bm{\tau})\)) Economy

\[ \bm{\tau} = \begin{pmatrix} \tau_{11} & \cdots & \cdots & \cdots & \tau_{1N} \\ \vdots & \ddots & \vdots & \iddots & \vdots \\ \vdots & \cdots & \tau_{ij} & \cdots & \vdots \\ \vdots & \iddots & \vdots & \ddots & \vdots \\ \tau_{N1} & \cdots & \cdots & \cdots & \tau_{NN} \end{pmatrix} \]

(\(h(\bm{\tau})\)) Economy

\[ \bm{\tau} = \begin{pmatrix} \tau_{11} & \cdots & \cdots & \cdots & \tau_{1N} \\ \vdots & \ddots & \vdots & \iddots & \vdots \\ \vdots & \cdots & \color{bcOrange} \tau_{ij} & \cdots & \vdots \\ \vdots & \iddots & \vdots & \ddots & \vdots \\ \tau_{N1} & \cdots & \cdots & \cdots & \tau_{NN} \end{pmatrix} \]


  • \(\color{bcOrange} \tau_{ij}\): trade barrier government \(i\) imposes on imports from \(j\)
  • \(\color{bcOrange} \tau_{ij} \color{black} - 1\): ad-valorem tariff equivalent
  • \(\tau_{ij} = 1 \implies\) free trade
  • \(\tau_{ii} = 1\) for all \(i\)

(\(h(\bm{\tau})\)) Economy

\[ \bm{\tau} = \begin{pmatrix} \color{bcOrange} \tau_{11} & \color{bcOrange} \cdots & \color{bcOrange} \cdots & \color{bcOrange} \cdots & \color{bcOrange} \tau_{1N} \\ \vdots & \ddots & \vdots & \iddots & \vdots \\ \vdots & \cdots & \tau_{ij} & \cdots & \vdots \\ \vdots & \iddots & \vdots & \ddots & \vdots \\ \tau_{N1} & \cdots & \cdots & \cdots & \tau_{NN} \end{pmatrix} \]


\[ \color{bcOrange} \bm{\tau}_1 = \begin{pmatrix} \tau_{11} & \cdots & \cdots & \cdots & \tau_{1N} \end{pmatrix} \]


  • Government 1’s trade policy

(\(h(\bm{\tau})\)) Economy

(\(h(\bm{\tau})\)) Economy

(\(h(\bm{\tau})\)) Economy

\[ \color{bcOrange} \bm{\tau} \color{black} = \begin{pmatrix} \tau_{11} & \cdots & \cdots & \cdots & \tau_{1N} \\ \vdots & \ddots & \vdots & \iddots & \vdots \\ \vdots & \cdots & \tau_{ij} & \cdots & \vdots \\ \vdots & \iddots & \vdots & \ddots & \vdots \\ \tau_{N1} & \cdots & \cdots & \cdots & \tau_{NN} \end{pmatrix} \]


\[ \color{bcOrange} h(\bm{\tau}) \rightarrow \begin{pmatrix} w_1 \\ \vdots \\ w_N \end{pmatrix} = \bm{w} \]

(\(h(\bm{\tau})\)) Economy

\[ h(\bm{\tau}) \rightarrow \begin{pmatrix} w_1 \\ \vdots \\ w_N \end{pmatrix} = \color{bcOrange} \bm{w} \]


  • \(V_i( \color{bcOrange} \bm{w} \color{black})\): welfare of representative consumer in \(i\)
  • \(X_{ij}(\color{bcOrange} \bm{w} \color{black})\): \(i\)’s imports of goods from \(j\)
  • Rents (tariff revenue):

\[ r_i(\color{bcOrange} \bm{w} \color{black}) = \sum_j (\tau_{ij} - 1) X_{ij}(\color{bcOrange} \bm{w} \color{black}) \]

Government Objective

\[ G_i = V_i( \color{bcOrange} \bm{w} \color{black})^{1 - b_i} r_i(\color{bcOrange} \bm{w} \color{black})^{b_i} \]

(\(h(\bm{\tau})\)) Economy

\[ h(\bm{\tau}) \rightarrow \begin{pmatrix} w_1 \\ \vdots \\ w_N \end{pmatrix} = \bm{w} \]


  • \(V_i(\bm{w})\): welfare of representative consumer in \(i\)
  • \(X_{ij}(\bm{w})\): \(i\)’s imports of goods from \(j\)
  • Rents (tariff revenue):

\[ r_i(\bm{w} \color{black}) = \sum_j (\tau_{ij} - 1) X_{ij}(\bm{w}) \]

Government Objective

\[ G_i = V_i(\bm{w})^{ \color{bcOrange} 1 - b_i \color{black} } r_i(\bm{w})^{\color{bcOrange} b_i \color{black} } \]

  • \(\color{bcOrange} b_i = 0\): government maximizes consumer welfare
  • \(\color{bcOrange} b_i = 1\): government maximizes rents
  • Free Trade Counterfactual

(\(\Gamma^{\bm{a}}\)) Wars

Optimal Post-Conquest Policies

\[ \bm{\tau}_i^{j \star} = \argmax_{\bm{\tau}_i} \quad G_j(\bm{\tau}_i ; \tilde{\bm{\tau}}_{-i}) \]

(\(\Gamma^{\bm{a}}\)) Wars

Optimal Post-Conquest Policies

\[ \color{bcOrange} \bm{\tau}_i^{j \star} \color{black} = \argmax_{\bm{\tau}_i} \quad G_{\color{bcOrange} j} \color{black} (\bm{\tau}_{\color{bcOrange} i} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

  • \(\color{bcOrange} \bm{\tau}_i^{j \star}\): trade policies in \(\color{bcOrange} i\) imposed by government \(\color{bcOrange} j\) post-conquest

(\(\Gamma^{\bm{a}}\)) Wars

Optimal Post-Conquest Policies

\[ \color{bcOrange} \bm{\tau}_i^{j \star} \color{black} = \argmax_{\bm{\tau}_i} \quad G_{j}(\bm{\tau}_{i}; \tilde{\bm{\tau}}_{-i}) \]

  • \(\color{bcOrange} \bm{\tau}_i^{j \star}\): trade policies in \(j\) imposed by government \(i\) post-conquest

Conquest Values

\[ G_j(\color{bcOrange} \bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

Contest Function (Dyadic) (Multilateral Version)

\[ \chi_{ji}(\bm{m}) = \frac{ \rho_{ji}(\bm{W}; \bm{\alpha}) m_{ji} }{ \rho_{ji}(\bm{W}; \bm{\alpha}) m_{ji} + m_{ii} } \]

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

Contest Function (Dyadic)

\[ \color{bcOrange} \chi_{ji}(\bm{m}) \color{black} = \frac{ \rho_{ji}(\bm{W}; \bm{\alpha}) m_{ji} }{ \rho_{ji}(\bm{W}; \bm{\alpha}) m_{ji} + m_{ii} } \]

  • \(\color{bcOrange} \chi_{ji}\): probability \(\color{bcOrange} j\) successfully conquers \(\color{bcOrange} i\)

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

Contest Function (Dyadic)

\[ \chi_{ji}( \color{bcOrange} \bm{m} \color{black}) = \frac{ \rho_{ji}(\bm{W}; \bm{\alpha}) \color{bcOrange} m_{ji} \color{black} }{ \rho_{ji}(\bm{W}; \bm{\alpha}) \color{bcOrange} m_{ji} \color{black} + \color{bcOrange} m_{ii} \color{black} } \]

  • \(\chi_{ji}\): probability \(j\) successfully conquers \(i\)
  • \(\color{bcOrange} \bm{m}\): military effort
  • \(\color{bcOrange} m_{ji}\): effort \(j\) dedicates toward threatening \(i\)
  • \(\color{bcOrange} m_{ii}\): effort \(i\) dedicates toward self-defense

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

Contest Function (Dyadic)

\[ \chi_{ji}(\bm{m}) = \frac{ \color{bcOrange} \rho_{ji} \color{black} (\bm{W}; \bm{\alpha}) m_{ji} }{ \color{bcOrange} \rho_{ji} \color{black} (\bm{W}; \bm{\alpha}) m_{ji} + m_{ii} } \]

  • \(\chi_{ji}\): probability \(j\) successfully conquers \(i\)
  • \(\bm{m}\): military effort
  • \(m_{ji}\): effort \(j\) dedicates toward threatening \(i\)
  • \(m_{ii}\): effort \(i\) dedicates toward self-defense
  • \(\color{bcOrange} \rho_{ji} \color{black} \in [0, 1]\): loss of strength gradient

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

Contest Function (Dyadic)

\[ \chi_{ji}(\bm{m}) = \frac{ \rho_{ji} (\color{bcOrange} \bm{W} \color{black} ; \bm{\alpha}) m_{ji} }{ \rho_{ji} ( \color{bcOrange} \bm{W} \color{black} ; \bm{\alpha}) m_{ji} + m_{ii} } \]

  • \(\chi_{ji}\): probability \(j\) successfully conquers \(i\)
  • \(\bm{m}\): military effort
  • \(m_{ji}\): effort \(j\) dedicates toward threatening \(i\)
  • \(m_{ii}\): effort \(i\) dedicates toward self-defense
  • \(\rho_{ji} \in [0, 1]\): loss of strength gradient
  • \(\color{bcOrange} \bm{W}\): dyadic geography

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-j}) \]

Contest Function (Dyadic)

\[ \chi_{ji}(\bm{m}) = \frac{ \rho_{ji} (\bm{W}; \color{bcOrange} \bm{\alpha} \color{black}) m_{ji} }{ \rho_{ji} ( \bm{W}; \color{bcOrange} \bm{\alpha} \color{black}) m_{ji} + m_{ii} } \]

  • \(\chi_{ji}\): probability \(j\) successfully conquers \(i\)
  • \(\bm{m}\): military effort
  • \(m_{ji}\): effort \(j\) dedicates toward threatening \(i\)
  • \(m_{ii}\): effort \(i\) dedicates toward self-defense
  • \(\rho_{ji} \in [0, 1]\): loss of strength gradient
  • \(\bm{W}\): dyadic geography
  • \(\color{bcOrange} \bm{\alpha}\): structural parameters

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-j}) \]

Contest Function (Dyadic)

\[ \color{bcOrange} \chi_{ji} (\bm{m}) \color{black} = \frac{ \rho_{ji} (\bm{W}; \bm{\alpha}) m_{ji} }{ \rho_{ji} (\bm{W}; \bm{\alpha}) m_{ji} + m_{ii} } \]

Reservation Values

\[ \color{bcOrange} \chi_{ji} (\bm{m}) \color{black} G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \color{bcOrange} \left( 1 - \chi_{ji} (\bm{m}) \right) \color{black} G_j(\tilde{\bm{\tau}}) - c_j \]

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

Contest Function (Dyadic)

\[ \chi_{ji} (\bm{m}) = \frac{ \rho_{ji} (\bm{W}; \bm{\alpha}) m_{ji} }{ \rho_{ji} (\bm{W}; \bm{\alpha}) m_{ji} + m_{ii} } \]

Reservation Values

\[ \chi_{ji} (\bm{m}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\bm{m}) \right) G_j(\tilde{\bm{\tau}}) - \color{bcOrange} c_j \]

  • \(\color{bcOrange} c_j\): war cost

(\(\Gamma^{\bm{a}}\)) Wars

Conquest Values

\[ G_j(\bm{\tau}_i^{j \star} \color{black} ; \tilde{\bm{\tau}}_{-i}) \]

Contest Function (Dyadic)

\[ \chi_{ji} (\bm{m}) = \frac{ \rho_{ji} (\bm{W}; \bm{\alpha}) m_{ji} }{ \rho_{ji} (\bm{W}; \bm{\alpha}) m_{ji} + m_{ii} } \]

Reservation Values

\[ \color{bcOrange} \underline{G}_{ji}(\bm{m}) \color{black} = \chi_{ji} (\bm{m})G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\bm{m}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

  • No war \(\implies G_j(\tilde{\bm{\tau}}) \geq \color{bcOrange} \underline{G}_{ji}(\bm{m})\) for all \(j, i\)
  • Proposition C1: \(\exists\) \(\left\{ c_j^\star \right\}_{j \in \left\{1,..., N \right\}}\) such that no war occurs if \(c_j > c_j^\star\) for all \(j\).

(\(\Gamma^{\bm{\tau}}\)) Trade Policy Announcements

\[ \color{bcOrange} \underline{G}_{ji}(\bm{m}) \color{black} = \chi_{ji} (\bm{m}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\bm{m}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

Constrained Optimal Policies

\[\begin{equation} \begin{split} \max_{ \tilde{\bm{\tau}}_i } & \quad G_i(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \end{split} \end{equation}\]

(\(\Gamma^{\bm{\tau}}\)) Trade Policy Announcements

\[ \color{bcOrange} \underline{G}_{ji}(\bm{m}) \color{black} = \chi_{ji} (\bm{m}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\bm{m}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

Constrained Optimal Policies

\[\begin{equation} \begin{split} \max_{ \tilde{\bm{\tau}}_i } & \quad G_i(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \\ \text{subject to} & \quad G_j(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \geq \color{bcOrange} \underline{G}_{ji}(\bm{m}) \color{black} \text{ for all } j \neq i \end{split} \end{equation}\]

(\(\Gamma^{\bm{\tau}}\)) Trade Policy Announcements

\[ \underline{G}_{ji}(\color{bcOrange} \bm{m} \color{black}) = \chi_{ji} (\color{bcOrange} \bm{m} \color{black}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\color{bcOrange} \bm{m} \color{black}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

Constrained Optimal Policies

\[\begin{equation} \begin{split} \max_{ \tilde{\bm{\tau}}_i } & \quad G_i(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \\ \text{subject to} & \quad G_j(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \geq \underline{G}_{ji}( \color{bcOrange} \bm{m} \color{black}) \text{ for all } j \neq i \end{split} \end{equation}\]

(\(\Gamma^{\bm{\tau}}\)) Trade Policy Announcements

\[ \underline{G}_{ji}(\color{bcOrange} \bm{m} \color{black}) = \chi_{ji} (\color{bcOrange} \bm{m} \color{black}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\color{bcOrange} \bm{m} \color{black}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

Constrained Optimal Policies

\[\begin{equation} \begin{split} \tilde{\bm{\tau}}_i^\star( \color{bcOrange} \bm{m} \color{black}) = \argmax_{ \tilde{\bm{\tau}}_i } & \quad G_i(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \\ \text{subject to} & \quad G_j(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \geq \underline{G}_{ji}( \color{bcOrange} \bm{m} \color{black}) \text{ for all } j \neq i \end{split} \end{equation}\]

(\(\Gamma^{\bm{m}}\)) Military Strategies

\[ \underline{G}_{ji}(\color{bcOrange} \bm{m} \color{black}) = \chi_{ji} (\color{bcOrange} \bm{m} \color{black}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\color{bcOrange} \bm{m} \color{black}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

Constrained Optimal Policies

\[\begin{equation} \begin{split} \tilde{\bm{\tau}}_i^\star( \color{bcOrange} \bm{m} \color{black}) = \argmax_{ \tilde{\bm{\tau}}_i } & \quad G_i(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \\ \text{subject to} & \quad G_j(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \geq \underline{G}_{ji}( \color{bcOrange} \bm{m} \color{black}) \text{ for all } j \neq i \end{split} \end{equation}\]

Optimal Military Strategies

\[\begin{equation} \begin{split} \max_{ \color{bcOrange} \bm{m}_i \color{black} } & \quad G_i \left(\tilde{\bm{\tau}}^\star( \color{bcOrange} \bm{m}_i \color{black} ; \bm{m}_{-i}) \right) \end{split} \end{equation}\]

\[ \color{bcOrange} \bm{m}_i = \begin{pmatrix} m_{i1} & \cdots & m_{iN} \end{pmatrix} \]

  • \(\color{bcOrange} m_{ij}\): military effort \(i\) dedicates toward threatening \(j\)

(\(\Gamma^{\bm{m}}\)) Military Strategies

\[ \underline{G}_{ji}(\bm{m}) = \chi_{ji} (\bm{m}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\bm{m}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

Constrained Optimal Policies

\[\begin{equation} \begin{split} \tilde{\bm{\tau}}_i^\star(\bm{m}) = \argmax_{ \tilde{\bm{\tau}}_i } & \quad G_i(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \\ \text{subject to} & \quad G_j(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \geq \underline{G}_{ji}(\bm{m}) \text{ for all } j \neq i \end{split} \end{equation}\]

Optimal Military Strategies

\[\begin{equation} \begin{split} \max_{ \bm{m}_i } & \quad G_i \left(\tilde{\bm{\tau}}^\star( \bm{m}_i ; \bm{m}_{-i}) \right) \\ \text{subject to} & \quad \sum_j m_{ij} \leq \color{bcOrange} M_i \end{split} \end{equation}\]

(\(\Gamma^{\bm{m}}\)) Military Strategies

\[ \underline{G}_{ji}(\bm{m}) = \chi_{ji} (\bm{m}) G_j(\bm{\tau}_i^{j \star}; \tilde{\bm{\tau}}_{-j}) + \left( 1 - \chi_{ji} (\bm{m}) \right) G_j(\tilde{\bm{\tau}}) - c_j \]

Constrained Optimal Policies

\[\begin{equation} \begin{split} \tilde{\bm{\tau}}_i^\star(\bm{m}) = \argmax_{ \tilde{\bm{\tau}}_i } & \quad G_i(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \\ \text{subject to} & \quad G_j(\tilde{\bm{\tau}}_i; \tilde{\bm{\tau}}_{-i}) \geq \underline{G}_{ji}(\bm{m}) \text{ for all } j \neq i \end{split} \end{equation}\]

Optimal Military Strategies

\[\begin{equation} \begin{split} \color{bcOrange} \bm{m}_i^\star \color{black} = \argmax_{ \bm{m}_i } & \quad G_i \left(\tilde{\bm{\tau}}^\star( \bm{m}_i ; \bm{m}_{-i}) \right) \\ \text{subject to} & \quad \sum_j m_{ij} \leq M_i \end{split} \end{equation}\]

Estimation and Identification

\[ \left\{ \bm{\tau}, \bm{M} \right\} \rightarrow \bm{\theta} \]

Observables and Unobservables

Data

  • \(\bm{\tau}\): policy barriers to trade (barriers)
  • \(\bm{M}\): military expenditure

Observables and Unobservables

Data

  • \(\bm{\tau}\): policy barriers to trade (barriers)
  • \(\bm{M}\): military expenditure

Observables and Unobservables

Data

  • \(\bm{\tau}\): policy barriers to trade (barriers)
  • \(\bm{M}\): military expenditure

Structural Parameters \[ \bm{\theta} = \left\{ \bm{b}, \rho(\bm{W}) \right\} \]

Observables and Unobservables

Data

  • \(\bm{\tau}\): policy barriers to trade (barriers)
  • \(\bm{M}\): military expenditure

Structural Parameters \[ \bm{\theta} = \left\{ \color{bcOrange} \bm{b} \color{black}, \rho(\bm{W}) \right\} \]

  • \(\color{bcOrange} \bm{b}\): governments’ preferences

Observables and Unobservables

Data

  • \(\bm{\tau}\): policy barriers to trade (barriers)
  • \(\bm{M}\): military expenditure

Structural Parameters \[ \bm{\theta} = \left\{ \bm{b}, \color{bcOrange} \rho(\bm{W}) \color{black} \right\} \]

  • \(\bm{b}\): governments’ preferences
  • \(\color{bcOrange} \rho(\bm{W})\): loss of strength gradient

Observables and Unobservables

Data

  • \(\bm{\tau}\): policy barriers to trade (barriers)
  • \(\bm{M}\): military expenditure

Structural Parameters \[ \bm{\theta} = \left\{ \bm{b}, \color{bcOrange} \rho(\bm{W}) \color{black} \right\} \]

Unobservables

\[ \bm{m}^\star(\bm{\theta}, \bm{M}) \]

Observables and Unobservables

Data

  • \(\bm{\tau}\): policy barriers to trade (barriers)
  • \(\bm{M}\): military expenditure

Structural Parameters \[ \bm{\theta} = \left\{ \bm{b}, \rho(\bm{W}) \right\} \]

Unobservables

\[ \bm{m}^\star(\bm{\theta}, \bm{M}) \]

Measurement Error \[ \tau_{ij} = \tilde{\tau}_{ij}^\star(\bm{m}^\star; \bm{\theta}) - \epsilon_{ij} \]

Estimation Problem

\[\begin{equation} \begin{split} \hat{\bm{\theta}} = \argmin_{\bm{\theta}} & \quad \sum_i \sum_j \left( \epsilon_{ij}(\bm{\theta}) \right)^2 \end{split} \end{equation}\]
  • Reformulate as mathematical program with equilibrium constraints (MPEC) (Su and Judd 2012)

Identification

Identification

Thank You

brendancooley.com

[email protected]

Table of Contents

Economy (Consumption)

Consumer’s Problem

\[\begin{equation*} \label{eq:consumer} \begin{split} \max & \quad U_i = Q_i^{\nu_i} S_i^{1 - \nu_i} \\ \text{subject to} & \quad P_i Q_i + P_i^s S_i \leq w_i L_i \end{split} \end{equation*}\]

CES Preferences over Tradable Varieties

\[ Q_i = \left( \int_{[0,1]} \alpha_{h(\omega)}^{\frac{1}{\sigma}} q_i(\omega)^{\frac{\sigma - 1}{\sigma}} d \omega \right)^{\frac{\sigma}{\sigma - 1}} \]

Tradable Price Index

\[ P_i = \left( \int_{[0,1]} \alpha_{h(\omega)} p_i(\omega)^{1 - \sigma} d \omega \right)^{\frac{1}{1 - \sigma}} \]

Expenditure on Tradables

\[ E_i^q = \nu_i I_i + D_i = P_i Q_i^\star \]

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Economy (Production)

Costs

\[\begin{equation} \label{eq:c} c_i = w_i^{1 - \beta} P_i^{\beta} \end{equation}\]
  • \(w_i\) - cost of labor
  • \(P_i\) - cost of composite intermediate good
    • equivalent to composite consumption good

\[ E_i^x = \beta X_i \]

Local Prices

\[ p_{ii}(\omega) = \frac{c_i}{z_i(\omega)} \]

Technology (Frechet)

\[ F_i(z) = \text{Pr} \left\{ z_i(\omega) \leq z \right\} = \exp \left\{ - T_i z^{-\theta} \right\} \]

Economy (Equilibrium)

Total Expenditure on Tradables

\[ X_i = \underbrace{E_i^q + E_i^x}_{E_i} - D_i \]

Trade Shares

\[ \Omega_{ij}^\star = \left\{ \omega \in [0,1] \left. \right\vert p_{ij}(\omega) \leq \min_{k \neq j} \left\{ p_{ik} \right\} \right\} \]

\[ \lambda_{ij}(\boldsymbol{w}) = \frac{1}{E_i} \int_{\Omega_{ij}^\star} p_{ij}(\omega) q_i \left( p_{ij} (\omega) \right) d \omega \]

Market Clearing

\[ X_i = \sum_{j=1}^N \lambda_{ji}(\boldsymbol{w}) E_j \]

Economy (Equilibrium in Changes)

\[ \hat{r}_i = \frac{1}{r_i} \left( E_i \hat{E}_i(\hat{\bm{w}}) - \sum_j X_{ij}^{\text{cif}} \hat{X}_{ij}^{\text{cif}}(\hat{\bm{w}}) \right) \] \[ \hat{w}_i = \frac{1}{\nu_i w_i L_i} \left( \sum_j \left( (1 - \beta) X_{ji}^{\text{cif}} \hat{X}_{ji}^{\text{cif}}(\hat{\bm{w}}) \right) + (1 - \nu_i) r_i \hat{r}_i(\hat{\bm{w}}) \right) \] \[ \hat{E}_i(\hat{\bm{w}}) = \frac{1}{E_i} \left( E_i^q \hat{E}_i^q(\hat{\bm{w}}) + E_i^x \hat{E}_i^x(\hat{\bm{w}}) \right) \] \[ \hat{x}_{ij}(\hat{\bm{w}}) = \left( \hat{\tau}_{ij} \hat{w}_j^{1 - \beta} \hat{P}_j(\hat{\bm{w}})^\beta \right)^{-\theta} \hat{P}_i(\hat{\bm{w}})^{\theta} \] \[ \hat{P}_i(\hat{\bm{w}}) = \left( \sum_j x_{ij} \left( \hat{\tau}_{ij} \hat{w}_j^{1 - \beta} \hat{P}_j(\hat{\bm{w}})^\beta \right)^{-\theta} \right)^{-\frac{1}{\theta}} \] \[ 1 = \sum_i y_i \hat{w}_i \]

Policy Problem in Changes

Government Welfare

\[ \hat{G}_i(\hat{\bm{\tau}}; b_i) = \hat{V}_i \left( \hat{h}(\hat{\bm{\tau}}) \right)^{1 - b_i} \hat{r}_i \left(\hat{h}(\hat{\bm{\tau}}) \right)^{b_i} \]

Optimal Post-Conquest Policies

\[\begin{equation*} \begin{split} \max_{\hat{\bm{\tau}}_j} & \quad \hat{G}_i(\hat{\bm{\tau}}_j; \hat{\tilde{\bm{\tau}}}_{-j}) \\ \text{subject to} & \quad \hat{\tau}_{jj} = 1 \end{split} \end{equation*}\]

Constrained Policy Problem in Changes

\[\begin{equation*} \begin{split} \max_{ \hat{\tilde{\bm{\tau}}}_i } & \quad \hat{G}_i(\hat{\tilde{\bm{\tau}}}_i; \hat{\tilde{\bm{\tau}}}_{-i}) \\ \text{subject to} & \quad \hat{G}_j(\hat{\tilde{\bm{\tau}}}) - \hat{G}_j(\hat{\bm{\tau}}_i^{j \star}) + \hat{c} \left( \chi_{ji}(1; \bm{0}_{-j, -i}, \bm{m}) \right)^{-1} \geq 0 \quad \text{for all } j \neq i \end{split} \end{equation*}\]

Multilateral Contest Function

\[ \chi_{ij}(\bm{a}, \bm{m}) = \frac{ a_{ij} \rho_{ij}(\bm{W}; \bm{\alpha}) m_{ij} }{ \sum_k a_{kj} \rho_{kj}(\bm{W}; \bm{\alpha}) m_{kj} } \]

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Sample Countries

iso3 Country Name
AUS Australia
BRA Brazil
CAN Canada
CHN China
EU European Union
IDN Indonesia
JPN Japan
KOR Republic of Korea
MEX Mexico
ROW Rest of World
RUS Russian Federation
TUR Turkey
USA United States of America

Policy Barriers to Trade, 2011

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Welfare Under Free Trade

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Mathematical Program with Equilibrium Constraints

Equilibrium Constraints

\[\begin{align*} \nabla_{\hat{\bm{x}}_i} \mathcal{L}_i^{\hat{\bm{x}}}(\hat{\bm{x}}_i; \bm{m}, \bm{\lambda}_i^{\bm{\chi}}, \bm{\lambda}_i^{\hat{\bm{w}}}, \bm{\theta}_m) &= \bm{0} \quad \text{for all } i \\ \lambda_{ij}^{\bm{\chi}} \left( \hat{G}_j(\hat{\bm{w}}; \bm{\theta}_m) - \hat{G}_j \left( \hat{\bm{\tau}}_i^{j \star}(\bm{\theta}_m) \right) + \hat{c} \left( \chi_{ji}(\bm{m}, \bm{\theta}_m) \right)^{-1} \right) &= 0 \quad \text{for all } i, j \\ \hat{\bm{w}} - \hat{h}(\hat{\tilde{\bm{\tau}}}) &= \bm{0} \\ \nabla_{\bm{m}} \mathcal{L}_i^{\bm{m}}(\bm{m}; \bm{\lambda}^{\bm{m}}, \bm{\theta}_m) &= \bm{0} \quad \text{for all } i \\ \lambda_i^{\bm{m}} \left( M_i - \sum_k m_{jk} \right) &= 0 \quad \text{for all } i \end{align*}\]

Estimation Problem

\[\begin{equation*} \begin{split} \min_{\bm{\theta}_m, \bm{m}, \bm{\lambda}^{\bm{\chi}}, \bm{\lambda}^{\bm{w}}, \bm{\lambda}^{\bm{m}}} & \quad \sum_i \sum_j \epsilon_{ij}(\bm{\theta}_m)^2 \\ \text{subject to} & \quad g(\bm{1}, \bm{\theta}_m, \bm{m}, \bm{\lambda}^{\bm{\chi}}, \bm{\lambda}^{\bm{w}}, \bm{\lambda}^{\bm{m}}) = \bm{0} \end{split} \end{equation*}\]

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