Question

Facts: Your firm produces and leases vehicles while reimbursing customers for the cost of fuel of these vehicles. Each customer pays you $2,000 a year to operate a vehicle for ten years. Over the life of a vehicle, the customer drives 20 miles a day for 300 days a year for ten years before scrapping the vehicle (for 60,000 miles in total). The price of gas is now $2 a gallon. The steel vehicle consumes 0.04 gallons per mile (or 25 miles per gallon). So the firm pays $4,800 for fuel over the course of ten years. Since the cost of producing a vehicle is $10,000, the company’s profit per customer is $2,000*10 - $4,800-$10,000 = $5,200. You are designing the new version of the vehicle. You could leave the vehicle unchanged and have the company earn $5,200 per customer. Or you could use aluminum. Aluminum reduces weight but increases the cost of making the vehicle by $3 per pound of weight saving. The steel vehicle weighs 3000 lbs. You can only reduce the vehicle weight by 50% using aluminum. Assume that a 50% reduction in vehicle weight leads to a 50% reduction in the number of gallons consumed per mile. Represent this problem as a decision tree. (5 points) [Hint: There are no uncertainties in this decision tree. As a result, probabilities are not involved. You either choose aluminum and observe the consequences or choose steel and observe the consequences. Also remember that rectangles are only used to represent decisions and ovals are only used to represent uncertainties. It is not used to represent the payoffs at the end of the decision tree.] Suppose there is a 50% chance of gas prices increasing from $2 to $4 a gallon (and a 50% chance of no change): Represent this problem as a decision tree. (10 points). [Hint: This problem does introduce uncertainty. You must choose between steel and aluminum. While there is no uncertainty in your decision about steel and aluminum, there will be uncertainty --- once you make your decision --- about the outcome of the gas price uncertainty. ] The expected profit from using steel is the sum of (1) the probability gas prices increase multiplied by the payoff from using steel if gas prices increase and (2) the probability gas prices do not change multiplied by the payoff from using steel if gas prices do not change. What is the expected profit from using steel? [Note that the value associated with an uncertainty is a weighted average of the value of the possible outcomes of the uncertainty where each outcome is weighted by its probability of occurrence. So the value associated with an uncertainty is computed differently than the value associated with a decision is computed.] The expected profit from using aluminum is the sum of (1) the probability gas prices increase multiplied by the payoff from using aluminum if gas prices increase and (2) the probability gas prices do not change multiplied by the payoff from using aluminum if gas prices do not change. What is the expected profit from using aluminum? What is the expected profit if you make the decision (choose aluminum or steel) which has the maximum expected profit? . [Hint: This should be the maximum of the expected profit from using steel and the expected profit from using aluminum.]

Answer 1