Econ HW 1

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Feb 20, 2024

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Ilana Berger, Jessica Gonsalves, Sankalp Khare, James Lee, Cesar Valiente Professor Kirsten Daniel Microeconomics for Management Group Homework # 1 Problem # 1 Demand for a firm’s product is: P = 80 - 3Q. The firm’s cost equation is: C = 200 + 20Q. a) Determine the firm’s optimal quantity and price. R = 80Q – 3Q 2 MR = MC MR = 80 – 6Q MC = 20 Plugging into optimal quantity realization equation : 80 – 6Q = 20 60 = 6Q Q = 10 ← optimal quantity Plugging Q(10) into price equation: P = 80 – 3(10) P = 80 – 30 P = $50← optimal price b) Suppose that demand changes to P = 110 - 3Q. Determine the new optimal quantity and price. Explain why the results differ from those in part a. R = 110Q – 3Q 2 MR = MC MR = 110 – 6Q ; MC = 20 Plugging into optimal quantity realization equation : 110 – 6Q = 20 90 = 6Q Q = 15 ← optimal quantity

Plugging Q(15) into price equation: P = 110 – 3(15) P = 110 – 45 P = $65 ← optimal price The results are different because the x-intercept in part b) is greater than in part a). The fixed component of price increased by $30 and marginal revenue increased by $30 as well which caused a shift of the demand curve upwards and to the right. This caused the marginal revenue and marginal cost to intersect at the higher quantity of 15 rather than at 10. As a result, the demand is greater for the same price.

Problem # 2 The college and graduate-school textbook market is one of the most profitable segments for book publishers. A best-selling accounting text – published by Old School Inc (OS) – has a demand curve: P = 150 - Q, where Q denotes yearly sales (in thousands) of books. The cost of producing, handling, and shipping each additional book is about $40, and the publisher pays a $10 per book royalty to the author. Finally, the publisher’s overall marketing and promotion spending (set annually) accounts for an average cost of about $10 per book. a) Determine OS’s profit-maximizing output and price for the accounting text. P= 150 – Q R= 150Q – Q 2 MR = 150 – 2Q Cost of producing, handling, and shipping each additional book = $10 Royalty to author per book = $10 MC = $40 + $10 = $50 Plugging into optimal quantity realization equation : MR = MC 150 – 2Q = 50 100 = 2Q Q = 50 ← optimal quantity Plugging Q(50) into price equation: P = 150 – 50 P = $100 ← optimal price b) A rival publisher has raised the price of its best-selling accounting text by $15. One option is to exactly match this price hike and so exactly preserve your level of sales. Do you endorse this price increase? (Explain briefly why or why not.) Due to the price increase of OS’s rival, OS’s demand curve shifts to the right. This higher demand allows OS to charge higher prices than it used to. However, in order to maintain its price competitiveness and the higher demand, OS should not increase $15 to match its rival’s price. Instead, the company needs to find a sweet spot where it can maintain both price competitiveness and higher demand. Here is the calculation for the new price: P= 165-Q R= 165Q-Q 2 MR= 165-2Q MC= 50 Q= 57.5 P= $107.5 per book ← optimal price Thus, the price must increase by $7.50 rather than by $15. MR and MC intersect at a higher quantity because the demand curve shifted right and upwards, and so the price must only increase by only $7.5 since sales were constant.

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c) To save significantly on fixed costs, Old School plans to contract out the actual printing of its textbooks to outside vendors. When this happens, OS expects to pay a somewhat higher printing cost per book (than in part a) from the outside vendor (who marks up price above its cost to make a profit). How would outsourcing affect the output and pricing decisions in part a? If the actual printing of textbooks is contracted to outside vendors, then Old School should plan to pay a higher printing cost per book. This will lead to a high marginal cost and the MC and MR will end up intersecting at a lower optimal quantity than before. Thus, in order to achieve maximum profit, OS should reduce its expected production quantity and increase the price. Problem # 3 Night Timers is a small company manufacturing glow-in-the-dark products. One of the hottest items the engineering department has developed is adhesive tape that can be applied to walls and floors. Night Timers' chief engineer anticipates that the product will be sold in ten-foot rolls. At present, the company's maximum production capacity is 140,000 rolls per year. The engineer believes the cost function to be described by: C = $50,000 + .25Q. (The high fixed costs represent development cost and tooling to prepare coating equipment). Night Timers' president seeks to establish a price that maximizes profit (since she is the chief stockholder). She thinks that the firm should be able to sell at least 125,000 rolls of tape per year. a) If Night Timers plans to sell 125,000 rolls per year, what is the necessary price if the firm is to break even? What if it can only sell 100,000? Break-even price for 125,000 rolls: Q BE = Fixed Cost / Price - Average Variable Cost 125,000 = 50,000 / Price - 0.25 Price = (50,000/125,000) + 0.25 Price = 0.40 + 0.25 Price = $0.65 Necessary price for the firm to break even selling 125,000 rolls per year = $0.65 Break-even price for 100,000 rolls: Q BE = Fixed Cost / Price - Average Variable Cost 100,000 = 50,000 / Price - 0.25 Price = (50,000/100,000) + 0.25 Price = 0.50+0.25 Price = $0.75 Necessary price for the firm to break even selling 140,000 rolls per year = $0.75

b) The marketing manager forecasts demand for the tape to be: Q = 350,000 - 200,000P. Find the firm's profit-maximizing output and price. Demand Equation: Q = 350,000 – 200,000P Price Equation: 200,000P = 350,000 – Q P = (350,000 – Q) / 200,000 P = 1.75 – 0.000005Q Revenue Equation: R = P*Q = 1.75Q – 0.000005Q 2 Marginal Revenue (MR): MR = 1.75 – 0.00001Q Marginal Cost (MC): MC = 0.25 Profit is maximum when Marginal Cost = Marginal Revenue 0.25 = 1.75 – 0.00001Q Q = (1.5) / 0.00001 = 150,000 rolls Q = 150,000 rolls ← optimal quantity Since the maximum limit capacity is 140,000 rolls, we take the maximum capacity to find out the optimal price. Plugging Q =140,000 in the price equation: Price(P) = 1.75 – (0.000005 * 140,000) = 1.75 – 0.70 = $1.05 P = $1.05 ← optimal price c) If the demand forecast in part b is realized in the first year of production, should the company consider expanding capacity? Explain. We believe that the company should consider expanding because they can maximize their profit if they were to have 150,000 rolls in production as seen in the calculations from part b. However, with their current production of 140,000 rolls, their profit maximization is limited.

Problem # 4 The University Eye Institute in upper New-York state is a state-of-the-art ophthalmology center that specializes in a sophisticated laser surgery to correct myopia. Current annual volume is 1000 operations. A major customer of the center is the United Health Insurance system. United currently sends the University Eye Institute 200 patients per year or 20% of the total. United pays $2,500 per operation as does every payor. The United Health Insurance Company is satisfied with the quality and service provided by the University Eye Institute and has proposed that they send the Center an additional 100 patients (operations) per year. United proposes that the fee be reduced to $2,000 for the additional 100 patients and for the prior 200 patients. Assume the fee paid by payers other than United Health remains the same. a) What is the marginal revenue per patient if the proposal is accepted? Volume Q = 1000 operations Price of Operation = $ 2500 Total Revenue = Price x Quantity Total Revenue = Initial price of operation x Initial total number of patients TR = $2500 X 1000 = $2,500,000 Marginal Revenue per patient in case the proposal is accepted: New Price of operation - Patients sent by United: $2000 Price of operation - Patients not sent by United: $2500 Total of patients sent by United: 300 Total of patients not sent by United: 800 Total number of patients: 1100 New Total Revenue: TR= ($2000 x 300) + ($2500 x 800) TR = $2,600,000 Marginal Analysis: Difference in total revenue = $2,600,000 - $2,500,000 = $100,000 Difference in number of total patients = 1100 – 1000 = 100 Marginal Revenue per patient = $100,000/100 Marginal Revenue per patient = $1000 b) What is the marginal cost per patient if the proposal is accepted? Here are some cost data to help you answer part b. Volume 1000 per year 1100 per year Average Total Cost $2,125 $2,100

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Total Initial Cost = Average Cost per patient x Number of patients Total Initial Cost = $2125 x 1000 Total Initial Cost = $2,125,000 New Total Cost after the Proposal is Accepted = Average Cost per Patient x Number of Patients $2100 x 1100 = $2,310,000 Difference in Costs (Marginal Cost) = $185,000 Difference in Number of Patients = 1100 – 1000 = 100 Marginal Cost per patient = 185,000 / 100 = $1850 c) Would you recommend that the proposal be accepted or not? Why? We recommend not accepting the proposal since after applying the marginal analysis optimization we find that the marginal cost per patient would be higher than the marginal revenue per patient by $850. This means that the university will incur a loss in operating every additional patient sent by United health in case the proposal is accepted.

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