Investigating the Strategic Nature of Supply Functions in Oligopoly

We show that supply functions cannot be classified as either strategic complements or substitutes according to the twofold criterion advanced by Bulow et al. (1985). This is because while the slope of the best reply is univocally positive, this is not the case with the sign of the cross derivative of marginal profit. We first show this discrepancy in the original Klemperer and Meyer (1989) setting, and then in a linear-quadratic model of differentiated duopoly. We further confirm and strengthen our result by proving that the game in supply functions is neither supermodular nor submodular.