Question

A firm rents light bulbs to consumers. The firm covers the cost of the light bulb and replacing it every time it burns out. The interest rate is 14%. The cost of producing a light bulb is √N where N is the number of years the light bulb lasts.
What is the firm's present value cost of providing a lightbulb to a consumer and replacing it everytime it burns out forever? Using light bulbs that last 10 years.

Answer 1

We can model this problem as follows:

		1/1.14 ^ time
Time	Cost of Bulb	Discount Factor
0	3.162278	1
1		0.877193
2		0.769468
3		0.674972
4		0.59208
5		0.519369
6		0.455587
7		0.399637
8		0.350559
9		0.307508
10	3.162278	0.269744
11		0.236617
12		0.207559
13		0.182069
14		0.15971
15		0.140096
16		0.122892
17		0.1078
18		0.094561
19		0.082948
20	3.162278	0.072762

Here the first bulb is provided in year 1, and then at teh end of year 10 and 10 years after that every year.

The discount factor for teh cost of bulb (which let's assume remains constant strangely) at the end of Y10 is 0.269744, which is the same as (1/1.14)^10

This discount factor the bulb provided at teh end of 20 years woudl be 0.072762, which is (1/1.14)^20 or 0.269744^2

In other words, the rate of interest between two cash flows (Y0, Y10, Y20 and so on) can be thought of as

1-0.269744 = 0.730256 per period (of 10 years)

So the cost of providing these bulbs in perpetuity for the firm is

3.162278 + PV of perpetuity of 3.162278 at interest rate of 0.730256

= 3.162278 + 3.162278 / 0.730256

= 7.492647