Question

Suppose the marginal cost for mineral water production in a small isolated country is 20 + Q, and the demand for mineral water is P = 80 – 2Q, where P is the dollar price and Q is the tons of mineral water produced. Suppose the processing procedure in mineral water production generates pollution, which incurs damage to the environment described by a marginal function of MEC = Q.

Graph the above problem, clearly showing and list the equations.

a) D, MPC, MEC, and MSC

b) Natural equilibrium

c) Socially optimal level of consumption

d) Deadweight loss - shade in or label by vertices

e) Show how to find the optimal per-unit tax

Answer 1

(a)

From demand function:

When Q = 0, P = 80 (Vertical intercept) & when P = 0, Q = 80/2 = 40 (Horizontal intercept).

From MPC function:

When Q = 0, MPC = 20 (Vertical intercept).

From MEC function:

When Q = 0, MEC = 0 (Vertical intercept).

From MSC function [where MSC = MPC + MEC = 20 + Q + Q = 20 + 2Q]

When Q = 0, MSC = 20 (Vertical intercept).

(b)

Natural equilibrium occurs when Demand = MPC.

80 - 2Q = 20 + Q

3Q = 60

Q = 20

P = 20 + 20 = 40

In above graph, natural equilibrium is at point A where D & MPC intersect with price P0 (= 40) and quantity Q0 (= 20).

(c)

Socially optimal consumption occurs when Demand = MSC.

80 - 2Q = 20 + 2Q

4Q = 60

Q = 15

P = 20 + (2 x 15) = 20 + 30 = 50

In above graph, efficient outcome is at point B where D & MSC intersect with price P1 (= 50) and quantity Q1 (= 15).

(d)

When Q = 20, MEC = Q = 20.

Deadweight loss = Area ABC = (1/2) x MEC x Change in Q = (1/2) x 20 x (20 - 15) = 10 x 5 = 50

(e)

When Q = 15, MEC = Q = 15.

Per-unit Pigouvian tax = Area ABE = 15.