来源：中博 | 更新：2016/8/29 0:00:00 | 关键词：F5经典题型解析
Jewel Co is setting up an online business importing and selling jewellery headphones. The cost of each set of headphones varies depending on the number purchased, although they can only be purchased in batches of 1，000 units. It also has to pay import taxes which vary according to the quantity purchased.
Jewel Co has already carried out some market research and identified that sales quantities are expected to vary depending on the price charged. Consequently, the following data has been established for the first month：
（a）Calculate how many batches Jewel Co should import and sell.
（b）Explain why Jewel Co could not use the algebraic method to establish the optimum price for its product.
（b）Therefore Jewel Co should import and sell four batches （4，000 units） of headphones since at this point it will make the greatest profit： $14，400 for the month.
（b）The algebraic model requires several assumptions to be true. First, there must be a consistent relationship between price （P）and demand （Q）, so that a demand equation can be established, usually in the form P = a-bQ. Here, although there is a clear relationship between the two, it is not a perfectly linear relationship and so more complicated techniques are required to calculate the demand equation. It also cannot be assumed that a linear relationship will hold for all values of P and Q other than the five given.
Similarly, there must be a clear relationship between demand and marginal cost, usually satisfied by constant variable cost per unit and constant fixed costs. The changing variable costs per unit again complicate the issue, but it is the changes in fixed costs which make the algebraic method less useful in Jewel's case.
The algebraic model is only suitable for companies operating in a monopoly and it is not clear here whether this is the case，but it seems unlikely, so any 'optimum' price might become irrelevant if Jewel's competitors charge significantly lower prices. Other more general factors not considered by the algebraic model are political factors which might affect imports, social factors which may affect customer tastes and economic factors which may affect exchange rates or customer spending power. The reliability of the estimates themselves -for sales prices, variable costs and fixed costs - could also be called into question.
Swim Co offers training courses to athletes and has prepared the following breakeven chart：
(a)State the breakeven sales revenue for Swim Co and estimate, to the nearest $10,000, the company‘s profit if 500 athletes attend a training course.
(b)Using the chart above,explain the cost and revenue structure of the company.
(a)The breakeven sales revenue for Swim Co is $90,000. The company‘s profit, to the nearest $10,000, if 500 athletes attend the course is $20,000 ($140,000 - $120,000). (From the graph, it is clear that the precise amount will be nearer $17,000, i.e. $140,000 - approximately $123,000.)
(b)Cost structure From the chart, it is clear that Line C represents fixed costs, Line B represents total costs and Line A represents total revenue.
Line C shows that initially, fixed costs are $20,000 even if no athletes attend the course. This level of fixed costs remains the same if 100 athletes attend but once the number of attendees increases above this level, fixed costs increase to $40,000.
Line B represents total costs. If 100 athletes attend, total costs are $40,000($400 per athlete).Since $20,000 of this relates to fixed costs, the variable cost per athlete must be $200. When fixed costs step up beyond this point at the level of 200 athletes, total costs obviously increase as well and Line B consequently gets much steeper. However, since there are now 200 athletes to absorb the fixed costs, the cost per athlete remains the same at $400 per athlete($80,000/200), even though fixed costs have doubled.
If 300 athletes attend the course, total cost per athlete becomes $300 each ($90,000/300).Since fixed costs account for $40,000 of this total cost, variable costs total $50,000, i.e. $166﹞67 per athlete. So, economies of scale arise at this level,as demonstrated by the fact that Line B becomes flatter.
At 400 athletes, the gradient of the total costs line is unchanged from 300 athletes which indicates that the variable costs have remained the same. There is no further change at 500 athletes;fixed and variable costs remain steady.
As regards the revenue structure, it can be seen from Line A that for 100每400 athletes the price remains the same at $300 per athlete. However, if 500 athletes attend, the price has been reduced as the total revenue line becomes flatter. $140,000/500 means that the price has gone down to $280 per athlete. This was obviously necessary to increase the number of attendees and at this point, profit is maximised.