1. Assume that there are only two countries in OPEC, Saudi Arabia and Venezuela, that they produce the same grade of crude oil, and there are many other producing countries that produce the same grade of crude oil.

Assume that the inverse demand curve for crude oil in the global market is given by

P = 300 - 5Q!

and the world supply curve is give by

P = -50 + 2Q!

where Q is measured in million barrels per day.

Saudi Arabia is the low cost producer in OPEC, with total cost function given by

C_SA(Q_SA) = 500 + 2Q²_SA

Venezuela's cost function is

C_v(Q_v) = 800 + 4Q²_v

a. First, assume that the market for crude oil is competitive and that the two OPEC members are just two of the competitive producers that individually maximize profits (so it's as if there is no OPEC at all). What is the market price for crude oil, the profit maximizing level of output for both countries, and their respective profits?

b. In the short run, would the two countries choose to produce positive output of crude oil or shut down?

c. Now assume that OPEC "gets teeth" and is able to restrict production among its members. For simplicity, we will make the additional assumption (which was once almost fairly realistic, but no longer is) that because OPEC's production costs are so low compared to other countries, it is a monopoly producer. OPEC therefore faces the demand curve given at the beginning of the problem (the supply curve is not relevant here). In this case, show the following: (i) output of both countries is a constant fraction of total production, (ii) output is greater for both countries than under the competitive case, (iii) the profit-maximizing price exceeds MR, and (iv) profits are positive for both. All three of these require you to maximize joint profits for OPEC (using the multiplant monopoly model).

2. This question addresses the dominant firm model of global oil production, which is discussed in Dahl, pp. 174 - 178. In the above problem, we demonstrated how a monopoly provider with more than one supply source will allocate production among those sources. To do that, we effectively assumed that OPEC controlled the entire world market. In reality, this is not true, since there are many oil-producing countries that do not belong to OPEC. Here, we model the global oil market as being divided between OPEC and the "fringe" producers. OPEC is treated as a dominant firm that maximizes its own profits. The profit maximizing output by OPEC then determines the market price, which is taken as given by the fringe producers, who are assumed to behave competitively. We will model the fringe as a single entity as well, but one that sets output such that the world price set by OPEC equals the fringe's marginal cost of production.

We are also going to take a different approach to this problem and solve a simple symbolic problem first, then plug in values for the parameters to get numerical solutions.

The world demand for oil is given by

P = α - βQ

where Q (measured in million barrels per day) is the total quantity demanded regardless of supply source (OPEC or fringe): Q = Q_OPEC + Q_F.

Fixed costs are assumed to be zero. The marginal cost functions are given by

MC_OPEC = A + BQ_OPEC

MC_F = D + EQ_F

a. What is the inverse demand function facing OPEC (which takes into account the fringe's "response") as a function of the parameters of the model (note that the method for doing so is shown in the text)?

b. Derive an equation that can be used to determine the profit maximizing level of output for OPEC (note: you don't need to solve for this output level, just set up the equation; you will solve it numerically below).

c. Now assume the following parameter values:

Let A = 1 B = 2
D = 2
E = 2
α = 50
β = 1

Verify that using these parameter values, both OPEC and the fringe will supply the market.

d. At the profit maximizing output level for OPEC and price, will the fringe produce? If so, what are profit maximizing output levels for OPEC and the fringe? If not, what are those values for OPEC as a monopoly producer?

e. What are the joint profits for the members of OPEC and the joint profits for the members of the fringe? Assume that fixed costs are zero.