Equilibrium existence in price-quantity games: a sunk cost paradox

Nonexistence of a pure-strategy Nash equilibrium is a notorious problem in price-quantity games. What drives this problem is the presence of spillover demand, i.e., demand coming from competitors' unserved customers. We argue that such demand spillovers may stem from a strong implicit assumption that costs associated with obtaining a product are sunk and do not affect consumers' future payoffs. We relax this assumption by considering a more general class of cost functions. This is shown to admit a pure-strategy equilibrium that coincides with the Bertrand price equilibrium.