Answers to Chapter 3 Exercises
3.1. DRAM factory. You own and operate a facility located in Taiwan that manufactures 64-megabit dynamic random-access memory chips (DRAMs) for personal computers (PCs). One year ago you acquired the land for this facility for $2 million, and used $3 million of your own money to ﬁnance the plant and equipment needed for DRAM manufacturing. Your facility has a maximum capacity of 10 million chips per year. Your cost of funds is 10% per year for either borrowing and investing. You could sell the land, plant and equipment today for $8 million; you estimate that the land, plant, and equipment will gain 6% in value over the coming year. (Use a one-year planning horizon for this problem.) In addition to the cost of land, plant, and equipment, you incur various operating expenses associated with DRAM production, such as energy, labor, raw materials, and packaging. Experience shows that these costs are $4 per chip, regardless of the number of chips produced during the year. In addition, producing DRAMs will cause you to incur ﬁxed costs of $500k per year for items such as security, legal, and utilities. (a) What is your cost function, C(q), where q is the number of chips produced during the year?
Answer: The $5 million you originally spent for the land, plant, and equipment is a sunk expenditure and thus not an economic cost. However, there is a “user cost of capital” associated with the land, plant and equipment, based on its current market value of $8 million and your cost of funds and the rate of depreciation or appreciation of the asset over the planning horizon. Your (opportunity) cost of investing $8 million for one year is $800k, but these assets will appreciate by $480k over the year, giving a (net) user cost of capital of $320k. (The depreciation rate is 6%.) This is a ﬁxed cost of making DRAM’s, to which we must add the other ﬁxed costs of $500k to get a combined ﬁxed cost of $820k for the year. The variable costs are a constant $4 per chip, so the cost function is C(Q) = 820k + 4 Q, in the range of 0 < Q < 1m. (One could also report that C(0) = 0, by deﬁnition, and that C(Q) is inﬁnite for Q > 1m, since your maximum capacity is one million chips per year. Of course, in practice there would likely be a way to push production beyond “rated capacity,” at some cost penalty, but that is beyond the scope of this problem.) Assume now that you can sell as many chips as you make at the going market price per
chip of p.
(b) What is the minimum price, p, at which you would ﬁnd it proﬁtable to produce DRAMs during the coming year?
Answer: The average cost function is AC (Q) = 820k/Q + 4, again up to one million chips per year. This declines with Q, so the minimum AC is achieved at full capacity utilization. At one million chips per year, the ﬁxed costs come to $0.82 per chip, so average costs are $4.82 per chip. This is your minimum average cost, and thus the minimum price at which is makes sense to stay open for the year.
3.2. mp34u. Music Ventures sells a very popular mp3 player, the mp34u. The ﬁrm currently sells one million units for a price of $100 each. Marginal cost is estimated to be constant at $40, whereas average cost (at the output level of one million units) is $90. The ﬁrm estimates that its demand elasticity (at the current price level) is approximately -2. Should the ﬁrm raise price, lower price, or leave price unchanged? Explain. Answer: Optimal pricing implies m =
In this problem, we have
which is greater than 1/| 2| = .5. This tells us that the price/cost margin is too high, so a lower price would be optimal. Note that the margin depends on MC , not AC . 3.3. KindOfBlue jeans.
Two years ago, KindOfBlue jeans were priced at $72 and
121,000 units were sold. Last year, the price was lowered to $68 and sales increased to 132,000.
(a) Estimate the value of the demand elasticity.
Answer: Using the percent...
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