(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:*

- Market failures arise from inadequate assignment of property rights
- Final allocations are independent of the initial assignment of rights.

**The “tax” model:**

Profit = q * P(q) – C (q) – v(s)where:

q = quantity of output (of the main product)Max Profit with respect to q & s:

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)

d(Profit)/dq = q(dP/dq) + P – dC/dq = 0(Important note: These should all be(firm maximizes profit by producing where marginal revenue = marginal cost)

andd(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)

**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(where:S- s)

everything is the same as above exceptu(

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

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