Question

Forty years ago, a gallon of gas cost $1.20. Today, a gallon of gas costs $2.60. Suppose that the gas price increase has been entirely due to inflation.

a. Calculate the annual inflation rate.

b. Today, you plan to make a cash purchase for a new car. The Model K costs $24,000 and you estimate the car will last 10 years and require 600 gallons per year. The Model M costs $28,000 but gets better mileage, so it will only require 400 gallons per year. The cars are identical in all other respects, and will both be worthless after 10 years. Assume that gas prices will rise by the rate of inflation (that you calculated in part a). If the nominal interest rate is 9% per year, which car should you purchase? Show all calculations and explain carefully.

Show all work. Label and clearly explain your answer. Work needs to be written out using formulas. You may use Excel to check your work.

Answer 1

a]

ending value = beginning value * (1 + inflation rate)^{number of years}

$2.60 =  $1.20 * (1 + inflation rate)⁴⁰

inflation rate = ($2.60 / $1.20)^1/40 - 1

inflation rate = 1.9518%

b]

Model M has a higher initial cost, but provides cost savings in later years. Hence, Model M should be purchased if the present value of cost savings is higher than the incremental initial cost.

Present value = future value / (1 + interest rate)^{number of years}

Cost savings in each year = 200 gallons * price per gallon

Price per gallon in each year = current price * (1 + inflation rate)^{number of years}

Present value of cost savings = (200 * $2.60 * (1 + 1.9518%)¹ / (1 + 9%)¹)) + (200 * $2.60 * (1 + 1.9518%)² / (1 + 9%)²)) + (200 * $2.60 * (1 + 1.9518%)³ / (1 + 9%)³)) + .............................. + (200 * $2.60 * (1 + 1.9518%)¹⁰ / (1 + 9%)¹⁰​​​​​​​))

Present value of cost savings = $3,666.95

Higher initial cost of Model M = $28,000 - $24,000 = $4,000

As the present value of cost savings is lower than the higher initial cost of Model M, it is not worth buying Model M.  

You should purchase Model K.