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Berkeley MBA 201A - Problem Set 3

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MBA 201A—Economic Analysis for Business DecisionsProblem Set #3UNIVERSITY OF CALIFORNIAHAAS SCHOOL OF BUSINESSMBA 201A—Economic Analysis for Business DecisionsFall 2009 Professor Catherine WolframProblem Set #3Doing these problems is optional. The solutions to these questions will be posted by on Thursday, September 24th and discussed in section Friday, September 25th. As always, the educational value of these exercises will be maximized if you attempt to answer these questions before you look at the answers.Question 1Your esteemed boss Hugo has just leased a new nut-grinding machine that has thecapacity to produce 100 units of hugonuts per day. The daily lease cost is $405. The market price for a unit of hugonuts is $120, and no one expects this to change in the foreseeable future. When running the machine, you observe that daily total costs follow the pattern: C(Q)=405+20Q+5Q2, where Q is the number of hugonuts produced and C(Q) is in $. a) Hugo has decided to minimize his total costs. How much should he produce? What are his total profits?C increases with Q and reaches a minimum when Q is just above 0, say at 0.01, at which point profits = 1.2-(405+.2+.005) ≈ -404.b) Hugo is back in the shop. He understands the importance of amortizing the daily lease cost of the machine over a large production run. His brother-in-law the VP of marketing has convinced Hugo of the importance of dominating the market and getting as much market share as he can with his existing machine. Hugo decides to follow this friendly advice, believing that not only will he make great heaps of money but he will also achieve much more pleasant conditions at the next gathering with his in-laws. How much does he instruct you to produce in order to maximize sales? What are his total profits (i.e. his total revenues minus total costs)?Running at capacity of 100, TC=$52,405. TR=100*$120=$12,000. Profits=TR-TC= -$40,405c) You notice Hugo has sprouted a few more gray hairs. Coincidentally, the marketing VP was called away to investigate a potential new client in Tierra del Fuego. Hugo has given the matter more thought, and instructs you toMBA 201a Fall 2009—Prof. Wolframminimize the average cost of production. How much do you produce? What are your profits?You have a choice of two methods to answer this question. At its minimum, you know AC is equal to MC. If the marginal cost of producing the next unit were below average cost, then AC would still be falling. If the marginal cost of producing the next unit were above the average cost, then AC would be increasing. So AC must reach a minimum where AC=MC. AC is found by dividing TC by Q, yielding AC=405/Q + 20 + 5Q. MC is found by taking the derivative of TC with respect to Q, yielding MC=20+10Q. Setting MC=AC and solving for Q you find that Q=9. At this point profits=90 (=1080-990). Alternatively, AC reaches a minimum where its derivative with respect to Q (AC’(Q) = -405/Q2 +5) is zero. This occurs at Q=9.d) Hugo is looking a bit better. He almost smiles now, especially when he shows everyone those spectacular post-cards of the fog and ice his brother-in-law keeps sending from Tierra del Fuego. You spot your opportunity and recommend that the time has come to maximize profits. Hugo is feeling so good, that he forgives you your MBA and follows your advice. How much doyou produce? What are your profits?As long as an additional unit of production adds more to revenues than to costs, profits (the difference between revenues and costs) are increasing. For a price-taking firm, maximum profits are achieved when price equals marginal cost. In this case, Hugo can sell each additional unit at a price of $120. MC=20+10Q. Setting 20+10Q=120, you find that profits are maximized by producing 10 units. At this point your profits are $95 (=10*$120-(405+200+500)). Hugo is impressed that your strategy has delivered greater profits than the alternatives of minimizing costs, maximizing market share, or minimizing average costs.Question 2You own the only carrot juice bar in Berkeley, which appears to be a valuable franchise. The daily demand you face for carrot juice is Q = 100−P . The total daily cost of operating the juice bar is TC = 100+10Q. (The daily fixed cost of operating is 100. That is an avoidable fixed cost if the juice bar is not open).a) What is your firm’s marginal revenue schedule. (Remember that marginal revenue (MR) is the change is revenue associated with a one unit change in quantity, not a one unit change in price)? How much should you produce in order to maximize profits? How much will your daily profit be?2MBA 201a Fall 2009—Prof. WolframQ = 100-P →P = 100 – Q → Total Revenue = P(Q)xQ = 100Q – Q2 → Marginal Revenue = 100 – 2Q. (Recall that you can get this either by taking the derivative of revenue with respect to quantity, or by remembering the rule that MR has the same intercept as the demand curve and twice its slope).MC = 10 (take the derivative of the TC function with respect to Q). Setting MR = MC implies 100 – 2Q = 10 → Q = 45 → P = 55.Profits are Total Revenue – Total Costs = 55x45 – (100+10x45) = 1925.b) You have brought in a hot-shot production consultant with a Stanford MBA who explains to you that your cost function is not what you thought. She says that your cost function is actually a bit more complicated. Your daily fixed cost is indeed 100, but your marginal cost depends on the quantity youproduce: MC=10 for Q ≤ 20MC=8 for 20<Q ≤ 50 MC=6 Q>50If this is so, how much should you produce in order to maximize profits?How much will your daily profit be?Note that for Q≤20, MR>MC (MC=10 and MR≥100-2x20 = 60). At Q=20,MR=60 and at Q=50, MR=0, so at some 20<Q<50, there is a Q forwhich MR=MC. We know MC=8 in this whole range. Setting 100-2Q=8→ Q=46 →P=54→ Profits = 54x46 -(100+10x20+8x26)=1976.Note that in the last step, we are calculating variable costs by summing up the marginal costs for all of the units we have produced. This is akin to calculating variable costs by summing the area under the MC curve, as we have done in class.c) Finally, you bring in a Berkeley-Haas MBA, who is confident without an attitude! She explains that your cost function is actually TC = 100 + Q2, so your marginal cost rises as you produce more, MC = 2Q. She also says thatthe demand function you face is actually Q = 20 − P. If this is so, how much should you produce in order to


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