A manufacturer decides to sell products at a per-unit price of $150. The per-uni

Question

A manufacturer decides to sell products at a per-unit price of $150. The per-unit material cost was $50 a year ago when the material cost index was 100. The current material cost index is 125. The products are being made by a machine that currently costs $100,000 and produces a 1000 units during its lifetime after which it needs to be replaced. The machine can be replaced by a larger one that produces proportionally larger number of units within its lifetime. However, the larger machine costs more: The power-sizing exponent for the machine cost is ½.

(a) What is the current profit (or loss) for 1000 units produced?
(b) By what factor x should a new machine be larger than the current machine so that the manufacturer can breakeven?
(c) What is the profit (or loss) for 2000 units produced if the new machine is twice the size of the current machine (i.e. when x = 2)?

Explanation / Answer

(a) when 1000 units are produced then, profit=no.of units produced*price of each unit-cost of production of each unit*no.of units produced - the cost of the machine since material cost index was 100 a year ago when material cost was $50 and the current material cost index is 125..there fore current cost= 125*50/100=62.5 hence profit=1000*$150 - $62.5*1000 - $100,000= -$12500 or a loss of $12500 (b)at the breakeven point profit is zero therefore let the machine larges by a factor of x then also cost of machine B=cost of machine A*((size of B/size of A)^power sizing exponent)) here power sizing component(p)=1/2 therefor cost of machine B=$100,000*(x^1/2) therefore profit=1000*$150 - $62.5*1000 - $100,000*(x^1/2) now for breakeven profit=0 therefore 1000*$150 - $62.5*1000 = $100,000*(x^1/2) on solving for x we get x=0.7656 (c) given new machine is twice the size of current machine put this in the formula -- cost of machine B=cost of machine A*((size of B/size of A)^power sizing exponent)) we get cost of machine B= $100,000*(2^1/2)=141421.3562 hence profit/loss=2000*$150 - $62.5*2000 - $141421.3562= $33,578.6438

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