Exercise 6

Consider a firm that also generates carbon emissions as a byproduct of its business-as-usual activity. The level of carbon emissions z will depend on how much output x the firm produces. Namely, suppose carbon emissions and output are related according to the following equation: z(x) = x, > 0. Moreover, environmental and health damages from carbon emissions (in $) are given by the equation D(z) = z, where > 0. Output is produced with total cost function c(x) = x^2 . Finally, output is sold in the market at price p.

1. Determine the profit-maximizing level of output the firm produces.

2. Determine the level of output a benevolent social planner would choose in this setup - i.e. the level of output that accounts for the social marginal cost of production instead of only the private marginal cost.

3. Suppose the government imposes an ad-quantum tax of t dollars per unit of output in order to correct the externality. What will be the amount of the optimal tax that implements the planner's solution?