Question

(a) Toyota motors sells two similar sized cars in Hong Kong. The Corolla (petrol engine) costsHK$ 210,000 while the Prius (petrol+electric engine) costs HK$ 305,000. The Corolla uses23Km/L, while the Prius uses 27Km/L of petrol. Assuming an interest rate of 5%, and a useful lifeof 10 years, and average annual usage of 20000Km, what is the break-even cost of petrol at whichthe two cars are economically equivalent. (b) Assuming that the vehicles can last for the same period, write the equation for the sensitivityof the break-even price of petrol to the annual usage?[Hint: assume that the annual usage is 20000X Km]

Answer 1

Solution

Step-by-step explanation:

(a) To determine the break-even cost of petrol at which the two cars are economically equivalent, we need to compare the present values of the costs of purchasing and running the two cars over 10 years.

The present value of the cost of purchasing the Corolla is HK$ 210,000, while the present value of the cost of purchasing the Prius is HK$ 305,000. Therefore, the difference in purchase price is HK$ 95,000.

Over 10 years, the Corolla will use 20000/23 = 869.57 liters of petrol per year, while the Prius will use 20000/27 = 740.74 liters of petrol per year. Assuming a petrol cost of P dollars per liter, the present value of the petrol costs for the Corolla is:

PV(Corolla) = (869.57 liters/year) x (P dollars/liter) x [(1 - (1 + 0.05)^-10)/0.05] = 6985.15P

Similarly, the present value of the petrol costs for the Prius is:

PV(Prius) = (740.74 liters/year) x (P dollars/liter) x [(1 - (1 + 0.05)^-10)/0.05] = 5937.92P

The difference in present value of petrol costs between the two cars is:

PV(Prius) - PV(Corolla) = 1047.23P

To find the break-even cost of petrol, we need to set the difference in present value of petrol costs equal to the difference in purchase price:

1047.23P = 95,000

Solving for P, we get:

P = HK$ 90.66 per liter

Therefore, if petrol costs HK$ 90.66 per liter or more, the Prius is economically equivalent to the Corolla over 10 years.

(b) To write the equation for the sensitivity of the break-even price of petrol to the annual usage, we can use the same approach as in part (a), but replace the annual usage with 20000X, where X is a variable representing the annual usage in kilometers. This gives:

PV(Corolla) = (20000X/23) x (P) x [(1 - (1 + 0.05)^-10)/0.05] = 869.57XP x [(1 - (1 + 0.05)^-10)/0.05]

PV(Prius) = (20000X/27) x (P) x [(1 - (1 + 0.05)^-10)/0.05] = 740.74XP x [(1 - (1 + 0.05)^-10)/0.05]

The difference in present value of petrol costs between the two cars is:

PV(Prius) - PV(Corolla) = (740.74XP - 869.57XP) x [(1 - (1 + 0.05)^-10)/0.05]

Simplifying, we get:

PV(Prius) - PV(Corolla) = -128.83XP x [(1 - (1 + 0.05)^-10)/0.05]

To find the sensitivity of the break-even price of petrol to the annual usage, we differentiate the above equation with respect to X:

d(PV(Prius) - PV(Corolla))/dX = -128.83 x [(1 - (1 + 0.05)^-10)/0.05]

Simplifying, we get:

d(PV(Prius) - PV(Corolla))/dX =