The light bulbs currently made by a manufacturer currently last 1 year.People place a value of $2.42 on one year's worth of light from a light bulb,and the market rate of interest is 10%.The manufacturer is considering a quality improvement that would make its light bulbs last 3 years.Should the manufacturer make the quality improvement?

Question

Quizplus · Accepted Answer

To determine if the quality improvement is worth it, we need to calculate the present value of 3 years' worth of light from a light bulb, discounted at the market rate of interest (10%). The present value (PV) of $2.42 received at the end of each year for 3 years is calculated as follows:$$ PV = \frac{2.42}{(1+0.10)^1} + \frac{2.42}{(1+0.10)^2} + \frac{2.42}{(1+0.10)^3} $$$$ PV = \frac{2.42}{1.10} + \frac{2.42}{1.21} + \frac{2.42}{1.331} $$$$ PV = 2.20 + 2.00 + 1.82 = 6.02 $$However, since the question implies a comparison to the initial $2.42 cost for one year, we need to adjust our understanding. The correct interpretation involves comparing the cost of improving the bulb to the present value of buying new bulbs each year for three years. The calculation provided was incorrect in the context of the question's intent, which is to find the break-even cost of the improvement.The correct calculation for the break-even point, considering the error in the initial explanation, should focus on the value of extending the bulb's life to 3 years from a cost perspective, not the present value of future payments. The correct approach is to find the maximum cost of improvement that makes the 3-year bulb equivalent in present value terms to buying a new bulb each year, under the given interest rate.Given the mistake in the calculation process, the correct answer hinges on understanding that the improvement should be worthwhile if the cost is less than the combined present value of purchasing new bulbs annually for three years, taking into account the 10% interest rate. The correct present value calculation for the three years of light (without the mistake of summing future values of $2.42 directly) would indeed suggest that the improvement is worth it if the cost is less than the present value of buying bulbs annually, adjusted for the interest rate. Given this, the correct rationale is that the manufacturer should proceed with the improvement if the cost is less than what consumers would otherwise pay (in present value terms) for replacing the bulb each year over three years, considering the time value of money. The option suggesting a threshold cost below which the improvement is justified reflects the need to compare the improvement cost against the present value of future savings from not having to replace the bulbs as frequently.

The Light Bulbs Currently Made by a Manufacturer Currently Last