Colluding on a Price Increase

Abstract

Adopting a general spatial framework, we analyse collusion concerning a price increase between two firms. We find that any variable affects the sustainability of collusion in the same way it affects the competitive profits.

Appendix

In this appendix, we show that the result would be qualitatively the same if the price increase was a percentage of the competitive price rather than a fixed amount. Consider the collusive price schedule. Firms collude to increase the price of a percentage equal to w. It follows that the collusive price schedule is: $p_h^C = p_h^N(1 + w)$ for all $h \in S$. Hence, the collusive profits are: ${\Pi^C} = \int_{{h \in {S^1}}} {[p_h^C - T(a,h)]f(h)dh} = {\Pi^N} + w\int_{{h \in {S^1}}} {p_h^Ndh}$. The deviation profits are: ${\Pi^D} = \int_h {[p_h^C - T(a,h)]f(h)dh} = {\Pi^C} + w\int_{{h \in {S^2}}} {p_h^Nf(h)dh}$. Due to symmetry, we have: $\int_{{h \in {S^2}}} {p_h^Nf(h)dh} = \int_{{h \in {S^1}}} {p_h^Nf(h)dh}$. Therefore, we can write: ${\Pi^D} = {\Pi^N} + 2w\int_{{h \in {S^1}}} {p_h^Nf(h)dh}$. Using the deviation profits, the collusive profits and the punishment profits into (1), we can write the critical discount factor as follows: $\delta * = \frac{{{\Pi^N} + 2w\int_{{h \in {S^1}}} {p_h^Nf(h)dh} - {\Pi^N} - w\int_{{h \in {S^1}}} {p_h^Ndh} }}{{{\Pi^N} + 2w\int_{{h \in {S^1}}} {p_h^Nf(h)dh} }} = \frac{w}{{1 + 2w - \Gamma }}$, where $\Gamma = \frac{{\int_{{h \in {S^1}}} {T(a,h)f(h)dh} }}{{\int_{{h \in {S^1}}} {p_h^Nf(h)dh} }}$. However, note that the competitive profits are a decreasing function of $\Gamma$, as the competitive profits must decrease with the transportation costs and must increase with the equilibrium competitive price. Therefore, we can conclude that the critical discount factor is inversely related to the competitive profits, as shown for a fixed amount increase of the price.