Mixed spatial duopoly, consumers' distribution and efficiency

- We model a mixed spatial duopoly with non-uniform consumers' distributions.
- We prove the existence of equilibrium when the consumers' distribution is logconcave.
- We assess the efficiency properties of the non-cooperative equilibrium.
- Unlike the case of uniform distribution, equilibrium is typically inefficient.
- The inefficiency may increase as preferences become more concentrated.

We solve a mixed spatial duopoly with a generic log-concave consumers' distribution. We show that the sub-game perfect equilibrium in prices and locations exists and is generally inefficient, so that the efficiency in the standard uniform distribution case stands out as an exception. Notable examples show that the inefficiency may increase as the distribution becomes more concentrated.