Problem
Heterogeneous Externalities (Need the math steps! No need to explain your answer!)
There are two types of farmers. Those whose land is flat and far away from any bodies of water (L), and those whose land is sloped towards proximal bodies of water (H). Their application of herbicides contaminates ground and surface water as a function of the chemical runoff and proximity to the surface waters. The L-type farmers impose total external damages from herbicide use equal to the 10QL where QL is the quantity of herbicide the L-type farmer applies. And the H-type farmer imposes total external damages of 28QH where QH is herbicide quantity applied by the H-type farmers. Both types of farmers have demand for chemical herbicides that reduce damage to their crops. The herbicide demand for L-types is 200 - 2QL and for the H-types is 408-4QH. Suppose the herbicide is sold into a competitive market at a price of $20 per unit.
Suppose there are equal numbers of each type of farmer.
A. What is the magnitude of the optimal Pigouvian tax imposed on each type of farmer if the social planner can impose heterogeneous taxes?
B. What is the efficient quantity of production for both types of farmers?
C. What is the optimal (second-best) uniform tax the social planner can impose?
D. What are the quantities produced by each type of farmer when the uniform tax is imposed? Who produces too much relative to the efficient level, and who produces too little?
E. In words of math, explain how the optimal uniform tax would change if there twice as many H-type farmers as L-type farmers.