The geometry of inconvenience and perverse equilibria in trade networks

Abstract

The structure bilateral trading costs is one of the key features of international trade. Drawing upon the freeness-of-trade matrix, which allows the modeling of N-state trade costs, we develop a ``geometry of inconvenience'' to better understand how they impact equilbrium outcomes. The freeness-of-trade matrix was introduced in a model by Mossay and Tabuchi, where they essentially proved that if a freeness-of-trade matrix is positive definite, then the corresponding model admits a unique equilibrium. Drawing upon the spectral theory of metrics, we prove the model admits nonunique, perverse, equilibria. We use this result to provide a family of policy relevant bipartite examples, with substantive applications to economic sanctions. More generally, we show how the network structure of the freeness of trade is central to understanding the impacts of policy interventions.

Figures (6)

• Figure 1: The complete graph $K_5$ (left) and the bipartite graph $K_{3,2}$. Every vertex also has a self-edge.
• Figure 2: The $K_{3,2}$ graph (left) along with its corresponding metric matrix $M_{\Phi^{K_{3,2}}}$ (center) and freeness of trade matrix $\Phi^{K_{3,2}}_{t}$ (right). The entries of the metric are given by the length of the shortest path between two vertices.
• Figure 3: The impact of one state leaving an antibloc on MT Stability. The vertices are $C_{5,1}$ (Bottom Left), $K_{4,2}$ (Top), $K_{3,3}$ (Bottom Right). Interior points are linear combinations of the networks at the vertices. Neither $K_{4,2}$ nor $K_{3,3}$ are MT Stable, one state leaving the trade network from the right hand side anti-bloc induces MT Stability in the $K_{4,2}$ case, but not the $K_{3,3}$ case. The diagrams A-D provide examples of the networks at the vertices.
• Figure 4: The impact of one state leaving an antibloc on MT Stability. The vertices are $C_{1,5}$ (Bottom Left), $K_{4,2}$ (Top), $K_{3,3}$ (Bottom Right). Interior points are linear combinations of the networks at the vertices. Neither $K_{4,2}$ nor $K_{3,3}$ are MT Stable, one state leaving the trade network from the left hand side anti-bloc does not induce MT Stability in the $K_{4,2}$ case, nor the $K_{3,3}$ case. The diagrams A-D provide examples of the networks at the vertices.
• Figure 5: The impact of universal trade barriers on MT Stability. The vertices are $D_6$ (Bottom Left), $K_{4,2}$ (Top), $K_{3,3}$ (Bottom Right). $D_6$ is the discrete metric, in which all states in the network erect barriers to trade with all others. Interior points are linear combinations of the networks at the vertices. Neither $K_{4,2}$ nor $K_{3,3}$ are MT Stable. However, by universal introduction of barriers to trade (approaching $D_6$) both can be made MT Stable. The diagrams A-D provide examples of the networks at the vertices.

Theorems & Definitions (5)

• Theorem 4.1
• Theorem 4.2
• proof
• Theorem 4.3
• proof