Question

A machine will reduce labor costs by $3,000 per month. It has a lifespan of 2 years, and no salvage value. Maintenance costs start at $50 per month, and increase by $50 each month. (So for months 1,2,3,4 costs are $50, $100, $150, $200.) Assuming a nominal annual discount rate of 6%, what’s the most you should pay for such a machine?

Answer 1

The most you should pay for the machine considering the given parameters and discount rate of 6% is approximately $61,434.87.

Step-by-step explanation:

To find the most you should pay for the machine, you can calculate the present value of the cost savings and expenses over its lifespan.

First, calculate the total cost savings over the two years:

Labor cost savings per month: $3,000

Total months in 2 years: 24 months

Total cost savings over 2 years: $3,000/month * 24 months = $72,000

Next, calculate the total maintenance costs over the two years using the increasing maintenance costs formula for an arithmetic series:

Total maintenance costs = (n/2) * (first term + last term)

Where:

• n is the number of terms and the first and last terms are the first and last monthly costs.

Using the formula:

n = 24 (number of months)

first term = $50

last term = $200

Total maintenance costs = (24/2) * (50 + 200)

Total maintenance costs = 12 * 250 = $3,000

Now, calculate the net savings over two years:

Net savings = Cost savings - Total maintenance costs

Net savings = $72,000 - $3,000 = $69,000

Finally, calculate the present value of the net savings using the formula for present value:

PV = FV / (1 + r)^n

Where:

• PV is the present value
• FV is the future value
• r is the discount rate
• n is the number of periods

Using the values:

FV = $69,000

r = 6% or 0.06

n = 2 years

PV = $69,000 / (1 + 0.06)^2

PV = $69,000 / (1.06)^2

PV ≈ $61,434.87

Thus, the most you should pay for the machine is approximately $61,434.87.