Question: A company plans to purchase a new machine to improve its productivity in the next 5 years. A standard machine can satisfy the current demand; it costs $100,000 and generates an annual profit $25,000. Meanwhile, the company foresees that with X% probability, after 2 years (i.e., starting from Year 3) the demand may be increased to a level beyond the capacity of the standard machine, and hence the company may lose the additional business opportunity valued at annual profit $11,000. There are two additional choices now. Choice 1 is to buy a premium machine that costs $120,000 and can satisfy all future demand. Choice 2 is to buy an optional upgrade package with the standard machine that allows for the company, if needed, to swap to the premium machine after 2 years by paying $20,000; the price of the upgrade package is Y.

(A) Between the standard (without upgrade package) and premium machines, do a breakeven analysis with respect to X. Let X0 be the break-even probability.

(B) For the case ofX

(C) For the case of X>X0.Then the standard machine is less profitable than the premium machine. So the company needs to consider two options: to buy the standard machine with the upgrade package, or to but the premium. What is the highest price of the upgrade package, as a function of X, so that the company is willing to buy the package?