Question

Tempe is considering replacing its fleet of gasoline powered cars with electric cars. The manufacturer of the electric cars claims a significant cost savings over the life of the fleet if Tempe chooses to pursue the conversion. If the manufacturer is correct, the city will save about $1.5 million. However, some critics argue that the technology is faulty and will result in significantly higher costs – estimated at a loss of $700,000 over the life of the fleet compared to gasoline-powered cars. A third possibility, suggested by independent analysis of an engineering firm last year is that the conversion will breakeven. A consultant hired by the city to assess the conversion program estimates the probabilities of the three outcomes – savings, loss, breakeven – at 30%, 30%, 40%, respectively.

The city has the opportunity to implement a pilot program that involves renting a small number of electric cars for three months and running them under typical conditions. This program would cost the city $75,000. The city consultant believes the pilot program results will be useful for estimating the value of the conversion, but not conclusive. She has prepared the following table of probabilities based on the experience of other cities after 3 months of usage. For example, given that a conversion actually results in savings of $1.5 million, the conditional probabilities that the pilot will indicate that the city saves money, loses money, and breaks even are 60%, 10% 30%, respectively.
Indication of outcome from pilot program Actual outcome over the life of the fleet
Savings Loss Breakeven
Indicates Savings 60% 10% 40%
Indicates loss 10% 40% 20%
Indicates breakeven 30% 50% 40%
Questions:
a. Assuming the city wishes to use expected payoff as the criterion for decision making, use Precision Tree to construct and solve a neatly labeled decision tree for Tempe's decision problem. Express the optimal strategy erbally and state the corresponding optimal expected payoff.
b. Does your recommended strategy from part (a) change if the city is risk averse with a risk tolerance parameter of $1 million (with an exponential utility function)? Express the optimal strategy verbally and state the corresponding optimal expected payoff and the certainty equivalent.

Answer 1

To construct the decision tree, we start with the initial decision to implement the pilot program, which has a cost of $75,000. The optimal strategy is to implement the pilot program, and the corresponding optimal expected payoff is $450,000.

Step-by-step explanation:

To construct the decision tree, we start with the initial decision to implement the pilot program, which has a cost of $75,000. From there, we branch out to the three possible outcomes: savings, loss, and breakeven.

Each outcome has its own probability and associated costs or savings. We then calculate the expected payoffs for each decision path by multiplying the probabilities and payoffs at each branch and summing them up.

The decision tree helps us visualize the different scenarios and their expected outcomes.

The optimal strategy, based on expected payoff, is to implement the pilot program.

This is because the expected payoff for implementing the pilot program is positive, which means that on average, it will result in a net gain for the city.

The corresponding optimal expected payoff is the sum of the expected payoffs for each decision path, which can be calculated by multiplying the probabilities and payoffs and summing them up.

Optimal expected payoff = (0.30 * 1,500,000) + (0.30 * -700,000) + (0.40 * 0) = $450,000