Question

An upstream Elf steel mill is producing steel, s , while creating an externality, x , affecting a downstream Hobbit fishing industry negatively. The steel is sold in a competitive market at a per unit price of p_s = 200. The steel firm’s cost function is c_s = 10s² + 4x² − 24x so that producing the externality reduces the cost to the steel mill up to a point. The fishing industry produces fish, f , sold in a competitive market at a per unit price of p_f = 200. The fishing industry has a cost function c_f(f, x) = 5f² + 8x.

a) Set up the profit maximization problem for the Elf steel mill (which does not consider its impact on the Hobbit fishing industry) and find the profit maximizing levels of s and x.

b) Given the chosen Elf level of x (the level of x you found in question a)), set up the fishing industry’s profit maximization problem and solve for the profit maximizing level of fish, f.

The Hobbits get increasingly angry with the Elves and their externalities and demand that the wizard Gandalf, a famous economist, should come and resolve the argument. Gandalf, in his infinite wisdom, assumes command of the situation and decides on the levels of s, f and x with the aim of maximizing the sum of profits from both industries.

C) What levels of s, x and f will Gandalf pick?

Answer 1

(1) In presence of negative externality, Social Marginal Cost (SMC) > Private Marginal Cost (PMC).

The reason is, SMC = PMC + External Cost (which is the cost arising due to pollution).

However, note it that PMC is the Supply curve itself (but SMC is not the private supply curve).

(2) Graphically this is shown below. Socially efficient point is where SMC intersects the Demand curve at point E. Since SMC is higher than PMC, the SMC curve lies to the left of PMC (Supply) curve.

c.