Assignment 5. Intermediate Micro, Spring Due: Thursday, April 10 th

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1 Assignment 5 Intermediate Micro, Spring 2008 Due: Thursday, April 0 th Directions: Answer all questions completely. Note the due date of the assignment. Late assignments will be accepted at the cost of 5 points per day, up until 2:30pm (beginning of class) on Tuesday, April 5 th. At that time I will return the graded assignments and post the answers on the web. You may turn in assignments to me after that time so that I can check your work for you, but please realize that you will not receive a grade for the assignment. You may work in a group consisting of up to 3 members for each group please turn in only set of answers and make sure all group member names are on that set of answers. All group members will receive the same grade. Monopoly (70 points) Suppose that a monopolist faces the following inverse demand curve, P (Q) = 65 5Q. The rm s total cost function is: T C = 2:4Q 3 9Q :5Q Qty. Price TR MR TC MC Pro t MC-using function MR-using function xxx 40 xxx : : : (6 points) Write down this monopolist s total revenue and marginal revenue functions solely as a function of quantity. We know that T R = P (Q) Q, and P (Q) = 65 5Q, so: T R (Q) = (65 5Q) Q = 65Q 5Q 2 We also know that if the inverse demand function is of the form P (Q) = a bq and the rm does NOT have to consider how much other rms produce, the MR function is: MR (Q) = a 2bQ. Plugging in the numbers from the inverese demand function in the example gives us: MR (Q) = 65 2 (5) Q = 65 0Q 2. (0 points) Fill in the table above, using the cost function and the inverse demand function above. Use the inverse demand function to ll in the price by plugging the quantity into 65 5Q. Use the MR and T R functions from part to ll in T R and MR. Use the T C function to ll in T C. To nd MC, calculate the increase in T C from one unit to the next. (For those of you who derived the MC function for the bonus and used the function, I have provided a separate column for you. Also, the MR numbers from the function are provided the ones in the main table are from subtracting the TR

2 from the next unit from the one preceding it.) For pro t, simply subtract T C from T R. The answers are in the table. 3. (6 points) What is this monopolist s pro t-maximizing price and quantity? (Use the table and assume the monopolist can only produce integer amounts.) The monopolist s pro t maximizing price and quantity are $45 and 4 units. We can look at the pro t column and see that pro t reaches its peak at a level of $24.4. Note: Technically, the pro t-maximizing quantity is 3: This is why, if you look at M R and MC using the functions, the MC > MR (29:7 > 25) at the pro t-maximizing quantity. However, because the rm can only produce integer amounts it chooses to produce a little more (4 units) rather than a little less (3 units) due to the slightly higher pro t from producing 4 units. The picture below shows a graph of the actual pro t function, and the fact that the pro t from producing 4 units is greater than the pro t from 3 units, even though MC > MR for 4 units. Price Quantity (6 points) Calculate the price elasticity of demand at the pro t-maximizing quantity. Recall that the formula for price elasticity of demand is: = Q P P Q We know that P = 45 and Q = 4. Since the inverse demand function is P (Q) = 65 function is found by solving for Q in terms of P : 5Q, the demand P = 65 5Q 65 + P = 5Q 3 0:2P = Q The term Q Q P is just the coe cient on the price of the good in the demand function, so P = 0:2. Plugging in the numbers we get: = 0: = 2:25 2

3 Some of you used the fact that at the pro t-maximizing quantity, MR = P + : If you used this method, then you would have (if you used the MR from the MR function): 25 = = = 9 4 = 9 9 = 4 Amazingly enough, 9 4 = 2:25. If you obtained the MR by subtracting the TR from the next unit from the one preceding it, then you would have found that the elasticity is (6 points) Calculate the Lerner Index at the pro t-maximizing quantity. The Lerner Index is simply the negative of the reciprocal of the, so if you found that the 9 was 4 then you would have that the Lerner Index is 4 9 0:44. If you found that the in part 4 was 3, then you would have had that the Lerner Index is 3 0:33. If we use the P MC P formula (using 2.7 as the MC, and not the MC from the function), we get: 45 2:7 45 = :57 This number is slightly di erent from the one obtained from the elasticity formula due to the fact that we are not quite at the pro t-maximizing quantity. If we were to use the true pro t-maximizing price and quantity (as well as the true MC from the MC function), we would have an elasticity at the pro t-maximizing price and quantity of: 2: This would give a Lerner Index of: 2: = 45: : : The price-cost margin would be: 45: = 0: So the numbers are truly equal only at the true pro t-maximizing price and quantity. 6. (6 points) Suppose that the demand for the monopolist s product becomes more elastic, so that the inverse demand function is now P (Q) = 65 2Q. Recalculate the price and the marginal revenue (for 0 through 6 units of the good) using this inverse demand function. The MR function is MR (Q) = 65 4Q. The table below shows the price and the MR (using the MR function) for each quantity level. I have also included the MC column from the table to aid in answering question 7. Quantity Price MR 0 65 xxx MC xxx

4 7. (6 points) Find the pro t-maximizing price and quantity when the inverse demand function is P (Q) = 65 2Q. (Again, assume the monopolist can only produce in integer amounts.) Using the table from question 6 and the MC column from the beginning of the problem we see that when we produce 5 units we have MR > MC but when we produce 6 units we have MC > MR (much so). So this rm would produce 5 units and charge a price of $55. Alternatively, you could have calculated the pro t at each quantity level if you had done this you would see that the maximum pro t is $77:5, which occurs at the quantity and price pair of 5 units and $ (6 points) Calculate the price elasticity of demand at the pro t-maximizing price and quantity when the inverse demand function is P (Q) = 65 2Q. Recall that the formula for price elasticity of demand is: = Q P P Q We know that P = 55 and Q = 5. Since the inverse demand function is P (Q) = 65 function is found by solving for Q in terms of P : 2Q, the demand P = 65 2Q 65 + P = 2Q 3 0:5P = Q The term Q Q P is just the coe cient on the price of the good in the demand function, so P = 0:5. Plugging in the numbers we get: = 0: = 5:5 Some of you used the fact that at the pro t-maximizing quantity, MR = P + : If you used this method, then you would have (if you used the MR from the MR function): 45 = = = 2 = = 2 Amazingly enough, 2 = 5:5. If you found MR by subtracting one level of T R from the next, you would have found that = 55 8 = 6:875. 4

5 9. (6 points) Calculate the Lerner Index at the pro t-maximizing quantity when the inverse demand function is P (Q) = 65 2Q. The Lerner Index (LI) is found by:. Since = 5:5 (from part 8 above), the Lerner Index is: If you had the = LI = 6:875, then: = 5:5 = 2 = 0:8 = 6:875 0:45 If we use the P MC P formula, we get: 55 4:9 55 = 0:238 Again, this is slightly di erent from the elasticity formula due to the fact that we are not at the pro t-maximizing quantity. 0. (6 points) Does the monopolist have more market power when the inverse demand function is P (Q) = 65 5Q (relatively inelastic) or P (Q) = 65 2Q (relatively elastic)? How do you know? The monopolist has more market power when the demand curve is relatively inelastic (inverse demand function is P (Q) = 65 5Q). The Lerner Index for the relatively inelastic demand curve is 0:44, while the Lerner Index for the relatively elastic demand curve is 0:8.. (6 points) If you were a monopolist which of these demand curves would you rather face? Explain. Relatively elastic (inverse demand function is P (Q) = 65 2Q). Even though the monopolist has more market power when the demand curve is relatively inelastic, I would choose the relatively elastic demand curve (inverse demand function is P (Q) = 65 2Q). The reason for my choice? The maximum pro t that the monopolist earns under the relatively inelastic demand curve is $24:4. The maximum pro t that the monopolist earns under the relatively elastic demand curve is $77:5. As a pro t-maximizing monopolist, I would choose the demand curve that yields a higher pro t. Although one could argue that having more market power would make the monopolist less susceptible to competition, and that in the long run the relatively inelastic curve would be better because the monopolist would be protected from competition. Note: You might wonder why the relatively elastic inverse demand curve (with less market power) yields a higher pro t than the relatively inelastic inverse demand curve (with more market power). The reason is because the a term in both inverse demand functions is the same. Look at the picture below: 5

6 Price Qty. The red line is the 65 5Q and the black line is 65 2Q. In this case, even though the black line is more elastic and has less market power, it will lead to a higher pro t because it intersects the Price axis at the same point as 65 5Q (the one with more market power). To take it even further, if you told a monopolist that it could pick a demand curve, but that the demand curve had to intersect the price axis at 65, it would pick (from a pro t standpoint) the function P = 65, which is a perfectly elastic demand curve. The reason is that the marginal revenue is ALWAYS $65! So the monopolist doesn t get to set price, but never faces a declining marginal revenue curve, so it doesn t have to worry about the tradeo s from lower prices/more customers and higher prices/fewer customers. Now, the problem is that if a rm were facing a perfectly elastic demand curve it would unlikely be a monopolist. 2 Price-taker market (30 points) Currently, the price in the perfectly competitive market is $65. table: Quantity TC TFC TVC ATC AVC MC The following is a price-taking rm s cost MR TR Pro t (5 points) Given that the price is $65, how much should the rm produce if it wishes to maximize pro t? (Assume the rm must produce integer amounts.) Explain how you found your answer. The rm should produce 6 units. Since this is a perfectly competitive market, the MR for the rm is the same as the price, which is $65. Using the MR = MC rule, the rm would like to produce the quantity where the MC = 65. Since that is somewhere between 6 and 7 units, the rm chooses 6 units because MR > MC, while at 7 units MC > MR. I have included columns for MR, T R, and Pro t. 2. (5 points) What is the pro t at the pro t-maximizing quantity that you found in part? 6

7 Just nd T R T C = = (0 points) Given the rm s costs above, what is the lowest possible price this rm could see in the market and still choose to continue to produce rather than to shutdown? Explain how you know. The minimum of the AV C (based on the fact that the rm can only produce integer amounts) is $4:5. By the shutdown rule, if the price were to fall below this level the rm would not continue to produce in the short-run. However, the rm would produce a positive quantity at prices above $4:5. Note: Technically, the minimum of AV C is slightly lower than this note that the MC at 4 units is $39, which is less than the AV C at 4 units which is $4:5. Since the MC < AV C, the AV C must still be falling. However, when we look at 5 units the MC is greater than the AV C ($49 > $43), so the AV C must be rising. So the AV C reaches its minimum somewhere between 4 and 5 units and $39 and $ (0 points) Assume that this is a constant-cost industry. What would the price of the good have to be in order for the rm and market to be in long-run equilibrium? Explain. For a perfectly competitive rm to be in long-run equilibrium, the rm must be earning zero economic pro ts. This occurs when the price in the market equals the minimum of the AT C. Since the minimum of the AT C is $53, if the price in the market is $53 then the rm will be in long-run equilibrium. The assumption of the constant-industry is needed because IF rms enter and exit the industry (due to economic pro ts or losses) then the total market output will change. If the total market output changes and we are NOT in a constant-cost industry, then the cost curves will shift. If the cost curves shift, then we cannot answer this question because we would not have enough information on the rm s costs when the price changes from $65 to $53. Note: Technically, the minimum of AT C is slightly lower than this note that the MC at 5 units is $49, which is less than the AT C at 5 units which is $53. Since the MC < AT C, the AT C must still be falling. However, when we look at 6 units the MC is greater than the AV C ($59 > $54), so the AV C must be rising. So the AV C reaches its minimum somewhere between 5 and 6 units and $49 and $53. 3 Bonus (5 points) In problem number 2, the monopolist s total cost function is: T C = 2:4Q 3 9Q :5Q Find the monopolist s average total cost function and marginal cost function. The AT C function is simply found by T C Q. T C Q = 2:4Q3 9Q :5Q + 40 = 2:4Q 2 9Q + 66: Q Q The MC function can be found by two ways. respect to Q. This gives: d 2:4Q 3 9Q :5Q + 40 dq The easiest way is to di erentiate the T C function with = 7:2Q 2 38Q + 66:5 The much more di cult way is to use the T C (Q) and T C (Q + h) method that is similar to the one we used to derive the MC function in class. T C (Q) = 2:4Q 3 9Q :5Q + 40 T C (Q + h) = 2:4 (Q + h) 3 9 (Q + h) :5 (Q + h)

8 Expanding the terms, we get: T C (Q + h) = 2:4 Q 3 + h 3 + 3Qh 2 + 3Q 2 h 9 2Qh + Q 2 + h :5 (Q + h) + 40 Distributing the constants through we get: T C (Q + h) = 2: 4Q 3 + 2: 4h 3 + 7: 2Qh 2 + 7: 2Q 2 h 38Qh 9Q 2 9h :5Q + 66:5h + 40 Now, we know that T C = T C (Q + h) T C (Q), so: 2: 4Q 3 + 2: 4h 3 + 7: 2Qh 2 + 7: 2Q 2 h 38Qh 9Q 2 9h :5Q + 66:5h :4Q 3 9Q :5Q + 40 Cancelling out terms we get: T C = 2: 4h 3 + 7: 2Qh 2 + 7: 2Q 2 h 38Qh 9h :5h Now, the change in quantity, Q, is just Q + h Cancelling out the h we get: Q = h, so: T C Q = 2: 4h3 + 7: 2Qh 2 + 7: 2Q 2 h 38Qh 9h :5h h T C Q = 2:4h2 + 7:2Qh + 7:2Q 2 38Q 9h + 66:5 Again, we want h to be a very small number, essentially zero, so our T C Q T C Q = 7:2Q2 38Q + 66:5 becomes: when we plug in h = 0. Technically, we are taking the limit as h approaches 0 (or lim h!0 ), but plugging in zero works for us. Notice that you get the exact same answer regardless of which way you nd the MC function (either through taking the derivative or nding the change in T C using the Q to Q + h method). However, the derivative method takes one line of paper, while the alternative method takes at least one sheet of paper. 8