A utility operating natural-gas pipeline must decide how to sell its fixed capacity. It could sell a portion of its capacity with long-term contracts. A long-term contract specifies that the buyer (the natural gas companys customer) will purchase a certain amount of space at a certain price. The long-term contract rate is currently $1800 per standard cubic feet. If long-term contracts are not signed, then the company can sell the pipelines on the spot market. The spot market price is volatile, but the expected future spot price is around $2200. In addition, spot market demand is volatile: sometimes the company can find customers, other times it cannot on a short-term basis. Lets consider a specific gas-pipeline route that serves the Northeast Coast on a specific date. The companys capacity on that route is 60,000 cubic feet. The company expects that the spot market demand is normally distributed with mean 40,000 and standard deviation 18,000. On average it costs the company $300 to operate and maintain a unit of the pipeline (i.e. the cost is per cubic feet)

a. Suppose that the company relied exclusively on the spot market, i.e., it signed no long-term contracts. What would be the companys expected profit?

b. Suppose that the company relied exclusively on long-term contracts. What would be the companys expected profit?

c. Suppose that the company is willing to use both the long-term contract arrangement and the spot market. How many units of capacity should the company sell with long-term contracts to maximize profit? Hint: To be able to answer this part, define D and Q very carefully. Then, find the implied unit over-stocking and the unit under-stocking costs

d. Compute the expected profit of scenario (c) and make comparisons with the strategies laid-out in (a) and (b).