Solution to Homework Set Managerial Economics Fall 20 Conceptual and Computational Questions 2. What is the maximum amount you would pay for an asset that generates an income of $50,000 at the end of each of five years if the opportunity cost of using funds is 9 percent? The maximum you would be willing to pay for this asset is the present value, which is P V = 50, 000 + 0.09 = $583, 447.69. + 50, 000 ( + 0.09) 50, 000 50, 000 50, 000 + + + 2 ( + 0.09) 3 ( + 0.09) 4 ( + 0.09) 5 3. Suppose that the total benefit and total cost from an activity are, respectively, given by the following equations: B(Q) = 50 + 28Q 5Q 2 and C(Q) = 00 + 8Q. (a) Write out the equation for the net benefits. Net benefits are N(Q) = 50 + 20Q 5Q 2. (b) What are the net benefits when Q =? Q = 5? Net benefits when Q = are N() = 50 + 20 5 = 65 and when Q = 5 they are N(5) = 50 + 20 5 5(5) 2 = 25. (c) Write out the equation for the marginal net benefits. Marginal net benefits are MNB(Q) = 20 0Q. (d) What are the marginal net benefits when Q =? Q = 5? Marginal net benefits when Q = are MNB() = 20 0 = 0 and when Q = 5 they are MNB(5) = 20 0 5 = 30. (e) What level of Q maximizes net benefits? Setting MNB(Q) = 20 0Q = 0 and solving for Q, we see that net benefits are maximized when Q = 2. (f) At the value of Q that maximizes net benefits, what is the value of marginal net benefits? When net benefits are maximized at Q = 2, marginal net benefits are zero. That is, MNB(2) = 20 0 2 = 0.
4. A firm s current profits are $550,000. These profits are expected to grow indefinitely at a constant annual rate of 5 percent. If the firm s opportunity cost of funds is 8 percent, determine the value of the firm: (a) The instant before it pays out current profits as dividends. The value of the firm before it pays out current dividends is ( ).08 P V firm = $550, 00 0.08 0.05 = $9.8million. (b) The instant after it pays out current profits as dividends. The value of the firm immediately after paying the dividend is ( ) P Vfirm Ex-Dividend.05 = $550, 00 0.08 0.05 = $9.25million. 7. It is estimated that over 90,000 students will apply to the top 30 M.B.A. programs in the United States this year. (a) Using the concept of net present value and opportunity cost, explain when it is rational for an individual to pursue an M.B.A. degree. The net present value of attending school is the present value of the benefits derived from attending school (including the stream of higher earnings and the value to you of the work environment and prestige that your education provides), minus the opportunity cost of attending school. As noted in the text, the opportunity cost of attending school is generally greater than the cost of books and tuition. It is rational for an individual to enroll in graduate when his or her net present value is greater than zero. (b) What would you expect to happen to the number of applicants if the starting salaries of managers with M.B.A. degrees remained constant but salaries of managers without such degrees increased by 5 percent? Why? Since this increases the opportunity cost of getting an M.B.A., one would expect fewer students to apply for admission into M.B.A. Programs. 8. Jaynet spends $20,000 per year on painting supplies and storage space. She recently received two job offers from a famous marketing firm one offer was for $00,000 a year, and the other was for $90,000. However, she turned both jobs down to continue a painting career. If Jaynet sells 20 paintings per year at a price of $0,000 each: 2
(a) What are her accounting profits? Her accounting profits are $80,000. These are computed as the difference between revenues ($200,000) and explicit costs ($20,000). (b) What are her economic profits? By working as a painter, Jaynet gives up the $00,000 she could have earned under her next best alternative. This implicit cost of $00,000 is in addition to the $20,000 in explicit costs. Since her economic costs are $20,000, her economic profits are $200,000 - $20,000 = $80,000. Problems and Applications 4. Tara is considering leaving her current job, which pays $56,000 per year, to start a new company that manufactures a line of special pens for personal digital assistants. Based on market research, she can sell about 60,000 units during the first year at a price of $20 per unit. With annual overhead costs and operating expenses amounting to $3,60,000, Tara expects a profit margin of 25 percent. This margin is 6 percent larger than that of her largest competitor, Pens, Inc. (a) If Tara decides to embark on her new venture, what will her accounting costs be during the first year of operation? Her implicit costs? Her opportunity costs? Accounting costs equal $3,60,000 per year in overhead and operating expenses. Her implicit cost is the $56,000 salary that must be given up to start the new business. Her opportunity cost includes both implicit and explicit costs: $3,60,000 + $56,000 = $3,26,000. (b) Suppose that Tara s estimated selling price is lower than originally projected during the first year. How much revenue would she need in order to earn positive accounting profits? Positive economic profits? To earn positive accounting profits, the revenues per year should greater than $3,60,000. To earn positive economic profits, the revenues per year must be greater than $3,26,000. Additional discounting exercises. A prospective house owner is interested in taking up a 30 year fix rate mortgage of CHF 500 000. This type of mortgage offers a fix yearly interest rate and will be payed off within 30 years. Each year the borrower pays the amount X to the lender and interest on the remaining dept D t at time t is I t. Thus, X I t is then deducted from 3
the dept. How big does X have to be, if the yearly interest rate is 6%, and the mortgage has to payed back in 30 years? Write D t as a function of X: D 0 = 500 000 D = ( + i)d 0 X D 2 = ( + i)d X = ( + i) 2 D 0 ( + i)x X... D t = ( + i) t D 0 X Solve D 30 = 0 for X: ( t ) ( + i) s. 0 = ( + i) 30 D 0 X ( 29 ( + i) s ) ( 29 ( ) ) s ( + i) 30 D 0 = X( + i) 29 + i = X( + i) 29 ( = X ( + i)30 + i ( + i) i = X ( + i)30 i X = id 0 ( + i) 30 ( + i) 30 = 36 324.46. ( + i) 30 ) +i 2. A potential student of a Fachhochschule faces the following two choices: Either he takes a job right away which pays him CHF 60 000. in the first year. Given his education level he can expect his wage to increase by % each year. Alternatively, he could go to Fachhochschule. If he does, he earns nothing for 4 years. After Fachhochschule, he can get a job which also pays CHF 60 000. in the first year, but he can expect the wage to increase by 2% each year. The student is 20 years old and will work until the end of the year in which he turns 65. (a) What should the student choose to do, if he can lend and borrow money at an interest rate of 3%? 4
The present value of the income for taking a job right away is 45 ( ).0 t P V = CHF 60 000 ( ( ).0 t ( ) ).0 t = CHF 60 000 t=46.0 46 ( ).0 t = CHF 60 000.0 46 = CHF 60 000.0.0 46 = CHF 60 000.0 = CHF 60 000 46.0 46.0 45 = CHF 836 96.9. For going to Fachhochschule first, it is 45 P V = CHF 60 000 t=4 = CHF 60 000 4 ( = CHF 60 000 4.02 t 4 t 4 ( ).02 t ) ) 42 (.02 ( ).02 t = CHF 60 000 42.02 42 4 42.02. = CHF 60 000 42.02 42.02 45 = CHF 845 968.97. The young man should therefore choose to go the Fachhochschule for four years. (b) What is the rate of lending and borrowing that makes the student indifferent between the two options? Complete the Excel worksheet hw.xls to compute the wages for the different years in one column. In another column calculate the present value (PV) of future wages depending on the interest rate. Solve for the interest rate that equates the PV of the two options. See fachhochschule sol.xls 5