Question

Do some research online and find 3 cars you are thinking of buying (ranging from low budget, to mid-budget, to one that is your dream car). Find their prices and how many miles per gallon they get

Car A: $26,793 28MPG

Car B: $39,735 17MPG

Car C: $161,139 13PMG

Suppose that you plan on using the car for 100,000 miles . Also let’s assume that all the cars have about the same overall cost of maintenance (just to simplify so you don't have to figure that into your calculations).

• The major uncertainty that you need to entertain is the price of gas in the future. You guess that there are roughly 4 options, given peak oil production and a tailing-off of global oil resources within the next 40 years (gas is a finite resource, in other words): 1) gas at $3 per gallon; 2) at $4; 3) at $5; 4) at $8. This is your partition, your states of affairs.

• The courses of action you may take are the three choices you have for your cars.

• The utilities you assign to your options are the respective price of each car plus the cost of gas for a “lifetime.”

a) Draw a table that lists the states of affairs across the top, and the choices/cars down the left side.

b) Work out the cost of gas for each car over its lifetime given the respective mpg, then fill in the utility of each choice (gas

plus cost of car) in each state of affairs. [40pts]

c) According to the table, is there an option that “dominates” the others? (Remember, in this example domination is about the lowest overall price.) In other words, is there a state of affairs in which one choice of car has costs lower than all the others AND in no other state of affairs does that choice of car cost more than any of the others? Briefly explain your answer and what that answer means.

d) Go back to the table you made in b). Attach the following probabilities to the gas prices over the lifetime of your car: 1) Pr($3)=50%; 2) Pr($4)=40%; 3) Pr($5)=8%; 4) Pr($8)=2%. Compute the expected value of each choice with these probability assignments, and then assess whether there is a value that dominates the others. Briefly explain your answer and what it means.

Answer 1

a) Table with states of affairs on top and choice of car on the side.

	Gas price per gallon
	$3	$4	$5	$8
Car A
Car B
Car C

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b) Table with costs of car inclusive of gas over lifetime:

	Gas price per gallon
	$3	$4	$5	$8
Car A	$37,507.29	$41,078.71	$44,650.14	$55,364.43
Car B	$57,382.06	$63,264.41	$69,146.76	$86,793.82
Car C	$184,215.92	$191,908.23	$199,600.54	$222,677.46

Formula used: Car cost = Buying price of car + (100,000/Mileage of car)*Gas price

100000/Mileage of car is the number of gallons of gas that the car will need which in turn multiplied by the price per gallon will give the total fuel cost.

Example calculation: For Car A with gas price $3, cost = 26793+100000/28*3 = 26973 + 10714.29 = 37507.29

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c) Car A is the option that dominates the others. For all states, i.e. for all possible prices of gas for car A, the costs are lower than all the costs for the other cars. Even for highest gas price of $8, car A will cost a total of $55,364.43 which is lower than all the costs of the other cars in every scenario of the gas price.

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d)

	Gas price per gallon				Expected value
	$3	$4	$5	$8
	p = 0.5	p = 0.4	p = 0.08	p = 0.02
Car A	$37,507.29	$41,078.71	$44,650.14	$55,364.43	$39,864.43
Car B	$57,382.06	$63,264.41	$69,146.76	$86,793.82	$61,264.41
Car C	$184,215.92	$191,908.23	$199,600.54	$222,677.46	$189,292.85

Expected value of each car choice is the sum of probability of each state * cost of each state.

For example: Expected value of car A: 37507.29*0.5 + 41078.71*0.4 + 44650*0.08 + 55364.43*0.02 = 39864.43

The expected value (cost) of car A is the lowest. Hence it dominates the other cars. With the current estimated probabilities of the states of gas price, car A is going to have the cheapest cost. This is was expected because even with the highest gas price, car A was already in a dominant state over the other cars irrespective of the gas price.

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Thanks. Kindly comment if any additional clarifications are needed.