1. A price-taking, crude oil marketing firm holds B barrels that it intends to market over the next two months. Let b1 and b2 be the amounts marketed in the first and second months, respectively, and assume that all barrels are sold, so the marketing constraint is B = b1 + b2. The firm can either sell all of it in the first month, in which case B = b1 and b2 = 0; or it can sell some in month 1, and store the remainder to market in month 2, so that both b1 and b2 are positive. Let p1 be the (known) price in the first month and p2 the expected future spot price in month 2 (so I have dropped the expectations operator to simplify notation). Any barrels that are sold include a marketing cost of cb2 where c is a constant and b is either b1 or b2. We assume that storage costs are zero for simplicity. And ?? = 1/1+r is the discount factor, with the discount rate, r being a market interest rate such as the rate on one-month Treasury bills. Note that 0 < δ < 1 since payments received in the future are worth less than payments received today.
We must also impose the following assumption, the need for which will be clear below: assume that ??1 > ????2. Since δ < 1, this is not as stringent an assumption as assuming just ??1 > ??2, but it implies essentially that the market is backwardated, with the month 1 price higher than the (discounted) month 2 price (note that since we are looking only one month into the future, δ will be close to 1, and saying ??1 > ??2 is almost the same thing as saying ??1 > ????2).
he firm solves the following maximization problem, subject to three constraints:
maxb1,b2 {??1??1 - ????12 + ??[ ??2??2 - ????22]}
??. ??. ?? = ??1 + ??2
??1 > 0
??2 ≥ 0
If you go on to study more economics or finance (or any other technical subject for that matter), you will learn how to solve maximization problems that include inequality constraints. Such problems often have "corner solutions" where one of the variables being maximized (or minimized) is actually zero when the function in question is at its maximum. Instead of teaching you any new methods, I will show you a shortcut. Remember, what we are doing is not just finding the optimal b1 and b2, but showing when that optimal b2 will actually be positive, indicating that the profit-maximizing strategy for the firm is to store some of the crude oil for sale in the second month.
a. Solve the firm's problem. That is, maximize profits subject to the inventory constraint and give an expression for the optimal b2 in terms of the prices and the parameters of the model. This was set up for you in class, and is fairly straightforward.
b. Now show under what conditions is ??∗ positive? Hint: look for a condition on the relative magnitude of the "spread" (really the discounted spread) between p1 and p2, because doesn't it make sense that a producer would allocate sales across two months according to the expected relative prices of crude between those two months?
c. Let the current spot price for crude oil on the first of the month be $50 per barrel and assume that the futures price for the front month (a.k.a. "prompt" month or "nearby" month) contract is $45 per barrel. Let the 30-day Treasury bill rate be 1.0%. Longhorn Marketing has crude oil inventory totaling B = 100 barrels. Marketing costs are 5 cents per barrel. Will Longhorn marketing choose to store crude at these prices?
2. I came up with this problem during the early periods of the Ukrainian crisis, and it has been assigned previously. For years, now, Russia has used its natural gas exports to Western Europe as a political tool. Ukraine has gotten involved in two unfortunate ways: first, it is also a consumer of Russian gas, and the Russian government has limited exports to Ukraine as a means of "punishing" the Ukrainian government. At the same time, most of the gas exported from Russia to Western Europe passes through the Ukraine. In light of the restricted supply to Ukraine, that country had actually taken some of the gas that was intended for customers in Western Europe, which obviously caused geopolitical tensions. This is a straightforward application of the Cournot model.
Assume that due to the conflict in Ukraine, the gas supply for the western European market evolves into a duopoly: the Russian company Gazprom (which is majority-owned by the Russian government) supplies gas by pipeline and Cheniere Energy, a U.S. LNG supplier. Both Gazprom and Cheniere set maximize profits by choosing production (or capacity) taking the production of the other producer as given. As such, we model the western European gas market as a Cournot duopoly, as in Chapter 12 of the Dahl text.
Let QG and QC be Gazprom's and Cheniere's choice of quantity, respectfully, and let their respective cost functions be ??G(??G) = 26??G and ??C(??C) = 32??C so that Gazprom's pipeline gas enjoys a cost advantage over Cheniere's LNG. The inverse demand function for gas in Western Europe is given by
?? = 200 - 3(??G + ??C)
a. Determine both firms' reaction functions.
b. Draw a diagram showing the reaction functions for the two terms as well as the profit maximizing output level for both firms.
c. What is the profit maximizing level of output for both firms and what is the profit maximizing price? What are profits for both companies?
d. Now assume that the U.S. federal government wants to assist the Ukrainian population in a way that also supports Cheniere in its competition with Gazprom. Assume that the federal government subsidizes Cheniere's costs such that the cost function now facing Cheniere is given by
CC(??C) = 20??C
What affect does this have on Cheniere's reaction function? What is the effect on the Cournot equilibrium output for both firms? What will be the new profit maximizing price of natural gas in Western Europe?