Question

Pigouvian tax algebra problem.

The Pigouvian prescription says to fix an externality by setting a tax rate equal to marginal damages at the optimal quantity. When marginal external damages are constant, the ``at the optimal quantity'' part is redundant. But, when marginal external damages are changing with the quantity of the good, then you have to figure out the right quantity to determine the right tax rate. This problem illustrates this with an algebra example.

Consider a market where total private benefits are equal to . Total private costs are Total external damages (costs) are equal to

Graph this market, showing the supply curve, demand curve, private equilibrium and optimal allocation.* (Upload an image of your graph with your answers. Just draw this by hand and take a picture.)

And What is the optimal tax rate on the good?

Answer 1

Marginal benefit (MB) = dTB/dQ = 50 - 2Q

Marginal cost (MC) = dTC/dQ = 12.5 + 0.5Q

Marginal external damage (MED) = dTED/dQ = 5Q

In market equilibrium, MB = MC.

50 - 2Q = 12.5 + 0.5Q

2.5Q = 37.5

Q = 15

In optimal outcome, MB = MC + MED

50 - 2Q = 12.5 + 0.5Q + 5Q

7.5Q = 37.5

Q = 5

When Q = 5, Optimal unit tax (Pigouvian tax) = MED = 5 x 5 = 25

In following graph, market equilibrium is at point A where demand (D) intersects MPC (private supply) with equilibrium price P0 and output Q0 (= 15). Socially efficient outcome is at point B where demand (D) intersects MSC (social supply = MC + MED) with higher price P1 and lower output Q1 (= 5). Optimal tax rate is vertical distance BE, and the deadweight loss generated by externality is area ABC.

MSC MA @, so