Question

Light emitting diode (LED) light bulbs have become required in recent years, but do they make financial sense? Suppose a typical 60-watt incandescent light bulb costs $.53 and lasts for 1,000 hours. A 15-watt LED, which provides the same light, costs $3.80 and lasts for 12,000 hours. A kilowatt-hour is 1,000 watts for 1 hour. Suppose you have a residence with a lot of incandescent bulbs that are used on average 500 hours a year. The average bulb will be about halfway through its life, so it will have 500 hours remaining (and you can’t tell which bulbs are older or newer).

If you require a return of 9 percent, at what cost per kilowatt-hour does it make sense to replace your incandescent bulbs today? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and round your answer to 6 decimal places, e.g., 32.161616.)

Answer 1

Solution:
	Break-even cost	0.004009
Working Notes:
	Let $C be the cost per Kilowatt- hour be the break even cost
	Particulars	60 watt incandescent light bulb	LED
		1Bulb x 60 (Watts Each) = 60 Watt	1LED x 15 (Watts Each) = 15 Watt
	(converting watt in into Kilowatt) 1 Kilowatt = 1000 watts	60 Watts / 1000 = 0.06 kw (Kilowatt)	15 Watts / 1000 = 0.015 kw (Kilowatt)
		0.06 kw x 500 hrs per year = 30 kwh (Kilowatt/Hour)	0.015 kw x 500 hrs per year = 7.5 kwh (Kilowatt/Hour)
a.	Per Annum Operating Cost	30 kwh x $C	7.5 kwh x $C
	(kilowatt usage x cost per kilowatt $C)
I	Cost Per CFL/bulb	$0.53	$3.80
	Life (as per use 500 hrs a year)	1000/500 = 2 year	12000/500=24 year
	Cost of Capital	9%	9%
II	Cumulative PVF @ 9 %
	2 year	1.75911118600
	24 year		9.70661176900
	Equivalent Annual Cost ( a + (I/II) )	[$C 30 + (0.53/ 1.759111186)]	[$C 7.5 + ($ 3.80/ 9.706611769)]
Notes:	Equivalent Annual Cost = Per Annum Operating Cost + (Cost Per CFL or bulb /Cumulative PVF @ 9%)
	Cumulative PVF @ 9% for 1 to 2th is calculated = (1 - (1/(1 + 0.09)^2) ) /0.09 =1.759111186
	Cumulative PVF @ 9% for 1 to 24th is calculated = (1 - (1/(1 + 0.09)^24)) /0.09 =9.706611769
	Now to get the Value of Break Even cost per Kilowatt-hour we equivalent these both EAC
	$C 30 + (0.53/ 1.759111186) =$C 7.5 + ($ 3.80/ 9.706611769)
	$C 30 - $C 7.5 =   ($ 3.80/ 9.706611769) - (0.53/ 1.759111186)
	$C 22.50 = 0.09019721386
	$C = 0.09019721386/22.50
	$C =0.004009
	Hence break-even cost per kilowatt-hour =   0.004009
Please feel free to ask if anything about above solution in comment section of the question.