Question:

　　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：

　　Required：

　　(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.

　　Answer:

　　(a)

　　(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.