Two oil companies are deciding how much oil to extract from their properties, which lie above the same underground reservoir. The faster that oil is extracted, the less total oil is extracted. Letting x denote the extraction rate for company X and y denote the extraction rate for company Y, we assume that the total amount of oil extracted is 1/(x + y) million gallons of oil. Of the total amount that is extracted, the share going to company X is x/(x + y), and the share to company Y is y/(x + y); that is, a company’s share depends on how fast it extracts compared with the other company. The price of oil is $100 per gallon. Each company chooses its extraction rate from the interval [1,10] in order to maximize the monetary value of the oil that it extracts. Find the Nash equilibrium extraction rates. (Note: You can assume that the payoff function is hill shaped.)
Two oil companies are deciding how much oil to extract from their properties, which lie above the same underground reservoir. The faster that oil is extracted, the less total oil is extracted. Letting x denote the extraction rate for company X and y denote the extraction rate for company Y, we assume that the total amount of oil extracted is 1/(x + y) million gallons of oil. Of the total amount that is extracted, the share going to company X is x/(x + y), and the share to company Y is y/(x + y); that is, a company’s share depends on how fast it extracts compared with the other company. The price of oil is $100 per gallon. Each company chooses its extraction rate from the interval [1,10] in order to maximize the monetary value of the oil that it extracts. Find the Nash equilibrium extraction rates. (Note: You can assume that the payoff function is hill shaped.)
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