Mathematical Restatement of Coase’s Theorem
(with apologies to Baumol and Oates)

The discussions during and after class convinced me I should simplify this, so here is a restatement of the problem and the proofs.  (This time there are no consumers to complicate matters—the payments are directly to and from the government.)

Two conjectures:

  1. Market failures arise from inadequate assignment of property rights
  2. Final allocations are independent of the initial assignment of rights.
Two approaches to controlling effluents:

The “tax” model:

Profit = q * P(q) – C (q) – v(s)
where:
q = quantity of output (of the main product)
P(q) = demand function for q (price is a function of q)
C(q,s) = cost function for q (depends on both q and s)
v(s) = effluent charge or excise tax on effluents (for example, 10 cents per pound of s)
Max Profit with respect to q & s:
d(Profit)/dq = q(dP/dq) + P – dC/dq = 0

(firm maximizes profit by producing where marginal revenue = marginal cost)

and

d(Profit)/ds = - dC/ds – dv/ds = 0

(firm maximizes profit by setting MC of producing the effluent to the per unit tax on the effluent)

(Important note:  These should all be partial derivatives.  I have written them as d()/x to insure that they can be read in HTML.)

Now consider a second approach where the firms have the right to continue producing without an excise tax, but they have the option to sell this right to the government.

Profit = q * P(q) – C (q) + u(S - s)
where:
everything is the same as above except

u(S - s) = payment made to the firm for each unit of effluent it no longer produces; S is the starting level (fixed by regulation or whatever).

Since S is a constant, all the derivatives remain unchanged so long as the per unit tax v is equal to the per unit subsidy u.  (There is a sign change to reflect the fact that v flows in the opposite direction from u.)  Therefore it appears that the second conjecture is correct—each firm will produce the same levels of q and s under either approach.

But…maybe the long-run situation will be different.  Under the first approach, the marginal firms will pay v(s) to the government and this might cause some of them to leave the industry.  Under the second approach, the marginal firms will receive u(S – s), and stay in business with increased profit.  If this is true, the total amount of effluent is not the same under the two approaches.  So, was Coase correct in his second conjecture?

Yale Braunstein
created: October 4, 2000
updated: August 1, 2002