Financial Mathematics �08: Modeling the Market for a Diminishing Resource

January 10, 2009

California has long been a leader in U.S. efforts to increase the use of renewable energy sources and decrease consumption of fossil fuels. The California Renewables Portfolio Standard, established by the California Senate in 2002, aims to have renewable energy account for 20% of all sales by electric companies by 2010---perhaps the most ambitious such standard in the United States. Unfortunately, so long as drilling for oil remains cheaper than harvesting solar, wind, or other sources of renewable energy, most companies will not switch to alternatives.

Ronnie Sircar, a professor in the Operations Research and Financial Engineering Department at Princeton University, addressed that last problem in an invited talk, "Games with Exhaustible Resources," at the second SIAM Conference on Financial Mathematics and Engineering, held in November in New Brunswick, New Jersey. In joint work with Chris Harris of the Economics Department at Cambridge University and Sam Howison of the Oxford Centre for Industrial and Applied Mathematics, Sircar has studied the impact of diminishing oil reserves on both production and prices in markets governed by oligopolies---essentially the market situation currently in existence, in which a small number of competitive players control the oil market.

He began by using a simple continuous-time Cournot game to analyze the effects of exhaustibility on players' strategies. In the original "game," Cournot studied the sale of mineral water, whereas Sircar was modeling the market for oil---a product with an exhaustible supply.

"Cournot came up with the static model in 1838," Sircar says. "He was interested in the market for mineral water, which was an inexhaustible resource. You play the game once; there are n companies, and the more they bottle and bring to market, the [lower the] price they're going to get. If you flood the market, the price goes down. But nothing changes over time; the next day you go back and play the same game again, because it's an infinite resource. [But] with something like oil, things do change as reserves are depleted."

Sircar used a stochastic differential game to model price formation. Each player had a certain amount of reserve oil and chose a specific production rate. Where the stochasticity came in was in modeling the uncertainty in the measurement of oil reserves. "People argue about how much is left," he says. "There's a lot of uncertainty about how much oil is out there."

A main focus of this work, however, is what happens when a firm actually does exhaust its resources. In the model studied by Harris, Howison, and Sircar, a firm with no remaining oil to sell can switch to an alternative, or "backstop," technology, such as solar or wind power, at a given cost, which is typically much higher than the cost of producing oil. It cannot store such power, however, and must sell it immediately. The firm thus remains in the game---or the market---but in a disadvantaged position.

Those results, and the larger implications of exhaustible resources, are the first step of Sircar's work.

What we want right now, he explains, is to understand, in as simple a form as possible, the effects of impending oil exhaustion. "How that affects people's behavior, how these firms produce oil, how it affects market price---how exactly running out changes their behavior. But the next step is sort of the inverse problem: Policymakers want to change behavior; they want to encourage people to switch to solar, even though at the moment [they] don't have to."

It's at that next step that Sircar hopes to see his work eventually used. In the case of the California standard, for example, a tax could potentially force a partial conversion to renewable energy.

With switching to renewable energy sources more expensive than drilling for oil, Sircar says, it's hard to imagine people making the switch "without some sort of penalty or tax if they don't."

He illustrates the point with a modern-day example:

"In September and October of 2008, and probably still, every firm's stock is tanking as the market is going into freefall," he points out. Data show that companies in renewable energy were falling faster than most other companies. "With the price of oil down to $50/barrel, a lot of people came to the conclusion, �Oil is cheap---I'm done investing in these renewable energies.'"

"One of the things we would like to do," he continues, "is play with the cost function in our models, or introduce some sort of penalty scheme in this setup, and decide what has to be done so that firms are encouraged to switch to alternative energies before they have to. Before the last drop of oil comes out of the ground."---Michelle Sipics

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