Profitable collusion on costs: a spatial model

Abstract

This paper uniquely demonstrates the scope for profitable collusion on transport costs under delivered pricing. In addition to being profitable, such collusion is shown to be more stable than price collusion and harder to detect as it presents to authorities as continued Bertrand price competition. Such collusion generates endogenous duopoly locations outside the quartiles with less stable but more profitable collusion happening toward the endpoints. These results emerge in both the traditional model of inelastic demand and an extended model of elastic demand.

Appendices

Appendix 1: the expression of $\delta^{*}$ for inelastic demand

$$\delta^ * = \frac{{2L_{1}^{2} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t}^{2} + 3L_{1}^{2} \bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} - L_{1}^{2} \bar{t}^{2} - 2L_{1} L_{2} \bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} + 2L_{1} L_{2} \bar{t}^{2} - L_{2}^{2} \bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} - 3\bar{t}^{2} L_{2}^{2} - 2L_{1} \bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} - 2L_{1} \bar{t}^{2} + 2L_{2} \bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} + 2L_{2} \bar{t}^{2} }}{{L_{1}^{2} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t}^{2} - L_{1}^{2} \bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} - 2L_{1} L_{2} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t}^{2} + 2\bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} L_{1} L_{2} - L_{2}^{2} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t}^{2} - L_{2}^{2} \bar{t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} + 2\bar{t}^{2} L_{2}^{2} }}$$

Appendix 2: deriving unit transport cost under competition with elastic demand

Due to the symmetry, we impose $L_{2} = 1 - L_{1}$ and $t_{2} = t_{1}$ to $\partial \pi_{1}^{C} /\partial t_{1}$, where $\pi_{1}^{C}$ is shown in (15). This yields: $\frac{{\partial \pi_{1}^{C} }}{{\partial t_{1} }} = \left( { - \frac{4}{3}L_{1}^{3} + \frac{3}{2}L_{1}^{2} - \frac{1}{2}L_{1} + \frac{1}{12}} \right)t_{1} - L_{1}^{2} + \frac{{L_{1} }}{2} - \frac{1}{8} { \gtrless }0$ if $t_{1} { \gtrless } - \frac{{3\left( {8L_{1}^{2} - 4L_{1} + 1} \right)}}{{2\left( {16L_{1}^{3} - 18L_{1}^{2} + 6L_{1} - 1} \right)}}$. It can be shown that $- \frac{{3\left( {8L_{1}^{2} - 4L_{1} + 1} \right)}}{{2\left( {16L_{1}^{3} - 18L_{1}^{2} + 6L_{1} - 1} \right)}} > 0.9$ for $L_{1} < 1/2$. Thus, the optimal unit transport costs for both firms are: $t_{1}^{C} = t_{2}^{C} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t}$.