Price-Quantity Competition under Strategic Uncertainty

We consider the market for a homogeneous good in which two firms simultaneously decide on both the price and the production level of the good. Firms have mean-variance preferences and they hold probabilistic conjectures about the actions of the other firm. We show that a pure-strategy equilibrium may exist in this setup, unlike in the standard version of simultaneous price-quantity competition.
We calculate the symmetric pure-strategy equilibrium numerically and we analyze how it depends on the degree of risk aversion and the amount of uncertainty in the conjectures. We find that the more risk averse the firms are, the less they produce and the higher price they ask in equilibrium. Aggregate production exceeds market demand for low degrees of risk aversion but as firms become more risk averse, they will not serve the whole market in equilibrium. Our results show that firms react differently to price uncertainty than to output uncertainty. When price uncertainty increases, firms charge a higher price and they produce less. In contrast, higher output uncertainty leads to a lower price whereas production levels may increase as well as decrease.