$($Machine replacement$)$ A specialized machine essential for a company$$s operations costs $$10,000$

and has operating costs of $$2,000$ the first year. The operating cost increases by $$1,000$ each

year thereafter. We assume that these operating costs occur at the end of each year. The annual

interest rate is $10\%.$ Assume that due to its specialized nature, the machine has no salvage

value.

Compute the present values of the costs for $2-$year replacement, $4-$year replacement, and $5-$year

replacement. Then find the best replacement policy among them.

$($Optional question:$)$ How long should the machine be kept until it is replaced by a new identical

machine? In other words, what is the best replacement policy? Since this question is quite

challenging, there will be no bonus marks for this optional question. Please refer to Luenberger$$s

Investment Science for discussions. This optional question will not be tested on the midterm or

final examination.

Hint: This is a perpetual annuity problem. The cash flow lasts forever. For instance, the cash

flow of the $2-$year replacement policy is

$(10000,2000,13000,2000,13000,2000,13000,2000,13000,...).$