Pollution tax, partial privatization and environment

Considering a differentiated mixed duopoly we show that when privatization and pollution tax are used together environmental damage will be non-monotone in the level of privatization, and optimal privatization is always partial privatization. Whether privatization will improve the environment or not depends on the public firm's concern for environment. If the public firm is unconcerned about environment, the socially optimal privatization will also damage the environment most. But when the public firm is concerned about environment, privatization will improve the environment. Generally, the relationship between optimal privatization and product substitutability is also non-monotone and inverted U-shaped.