Homework 1, Solutions Managerial Economics: Eco 685. a. To get the wage per hour, we can divide total wages by total hours hours = 50 $

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1 Homework 1, Solutions Managerial Economics: Eco 685 Question 1 a. To get the wage per hour, we can divide total wages by total hours. wage per hour = total wages total hours = $100, hours = 50 $ hour. (1) Notice how the units match up: we are looking for dollars per hour. Keeping track of units will be helpful in this class. b. Since the plane tickets costs $300 2 = $600, we have: hours = $ $ hour Notice again the units match up: = 12 hours. (2) $ $ hour = $ hours $ = hours, (3) and hours are what we are looking for. c. Ralph chose to play video games so the economic profit of the game must be positive: it was better than the next best alternative. We have: Econ π( game ) = Revenue Opportunity Costs > 0 (4) Revenue > Opportunity Costs = $ 2 hours = $140 (5) hour Ralph must get at least $140 worth of entertainment value, otherwise he would have worked. d. Here Ralph chose the next best alternative which is working. So the economic profit of taking the class was negative. Econ π( class ) = Revenue Opportunity Costs < 0 (6) Revenue < Opportunity Costs = $2, hours week 1 $ 8 weeks 50 = $6,000 (7) hour

2 The class is worth less than $6,000 to Ralph, because this is the opportunity cost of the class and Ralph said no. Question 2 a. The behavior that is incentivized here is to spend as little time as possible with each call. This saves on costs (less workers need to be hired) if the cost of extra pay for each worker is less than the savings from fewer workers. Regardless, less time per call means less satisfied customers, which means less repeat business. That is, the policy does not address the revenue side. Indeed, the article says that the managers discovered that the call center employees were giving one word answers and hanging up. b. A bonus based on the call center s profits would work best. Another method would be to base pay on both time spent and customer satisfaction surveys. c. Profit based bonuses create only weak incentives. In a call center with hundreds of employees, a single employee s efforts do not have a large effect on profits. Weighting both customer satisfaction and time per call is difficult because it is hard to determine what weight will maximize profits. Question 3 a. This is the marginal product. The marginal product of labor is the additional production from an additional worker. Here an additional worker produces 8 loan applications. b. Marginal expenditure. This is the cost of the labor input. c. Marginal product. A gallon of fuel produces 10.2 additional miles traveled. This is the marginal product of fuel. d. Marginal revenue product. Each worker brings in on average about $25 in revenue per hour. e. Marginal revenue product. One unit of professors brings in $160,000 in revenue for the university. Question 4 a. The opportunity cost of building on the water is $80 million. Thus, building on the water generates revenue of $250 million but has an accounting cost of $200 and an opportunity cost of $80 million. We must also add as an opportunity cost the value of the next best option. Not building has a value of zero and building in Overtown has a value of $250 million less the $200 million to build the stadium and $60 million to 2

3 acquire the land, so the value of the Overtown option is -$10. So the value of the next best option is zero. Thus: Economic Profit (water) = $250 $200 $80 $0 = $30 (8) For Overtown, the accounting costs are building the stadium and buying the land. The opportunity cost is the value of the next best option. which is the value of not building, $0. Economic Profit (Overtown) = $250 ($200+$60) $0 = $10 (9) Finally, for not building, the next best option is Overtown, so: Economic Profit (No Build) = $0 $0 $10 = $10 (10) Note that the option with positive economic profits has the same value, but opposite sign, as the next best option. b. From the previous problem, the accounting profits are $50 million for the water, -$10 for Overtown, and $0 for not building. c. The correct decision is not to build. The accounting profits of building on the water are positive, but the economic profits are not because of the opportunity costs. d. This statement is incorrect. The waterfront land is not free. It has an opportunity cost, which is its value as a park. Question 5 a. Neither. When marginal revenue product or marginal product is zero, additional revenue/production is zero but the input has positive cost. Therefore, the firm is getting negative profits from the last unit of this input. Profits will increase by reducing the amount of the input. b. True for both. When marginal product is zero, an additional input an additional input generates zero extra production. Any more input would reduce production. Therefore production is maximized when marginal product is zero. When marginal product is zero, so is marginal revenue product since MRP = MP MR. c. True for marginal revenue product. At this point, the last unit of input generates just enough revenue to cover its costs. Any more input would generate less revenue than the cost of the input, reducing profits. It is not true for marginal product. If the price is greater thanone then ME = MP < MR MP = MRP. The last input is profitable, so we can increase profits further by increasing the input. If the price is less than one, 3

4 then ME = MP > MR MP = MRP, so the last unit of the input is not profitable. We need to reduce the input to increase profits. d. Neither. If MRP is less than marginal expenditure, an extra input reduces profits. The input costs more than the revenue it brings in. If MRP = MP MR < ME then marginal product is positive but reduces profits. e. True for marginal product. This is the definition of marginal product. Question 6 a. We maximize profits which are total revenues less total costs: π = max (50+100W $7 12 ) W W2 $49W (11) = max W $350+$651W 7 2 W2 (12) The maximum is where the slope or derivative equals zero: π W = $651 $2 7 W = 0, (13) 2 $7W = $651, W = 93. (14) b. We maximize profits with 93 pounds of wax. To compute the maximum profit, we plug 93 into the profit equation. π(93) = $7 ( ) 932 $49 93, (15) π(93) = $30, (16) c. Recall we have two ways of computing the marginal revenue product. If we allow fractional workers (part timers) and are considering whether or not to increase W by a small amount, then we want to use the derivative, which measures the change in total revenue with a very small increase in W. In this case: MRP = TR W = $700 7W (17) 4

5 MRP (10) = $700 $7 10 = $630 (18) So the marginal revenue product is $630, which is greater than the marginal expenditure of $49. Using the 10th pound of wax would generate $630 of revenue but cost only $49. Therefore, we should definitely use the 10th pound of wax. d. We have: Q = max W W 1 2 W2 (19) Q W = 100 W = 0, W = 100. (20) e. Workers produce something, just not enough to cover the cost of hiring them. Therefore, it is not profit maximizing to hire these workers. Question 7 a. We can compute the claims filed with two workers as follows: MP 3 = Q 3 Q 2 3 2, (21) 25 = 30 Q Q 2 = 5. (22) Using the same equation, we can calculate: MP 5 = Q 5 Q = = 20, (23) Since all claims have the same price, marginal revenue is constant and is found via: MRP 3 = MR MP 3, (24) $100 = MR 25 MR = $4. (25) Hence: MRP 5 = MR MP 5 = $4 20 = $80. (26) 5

6 For the next input level, which we will call x: MRP x = MR MP x, (27) $40 = $4 MP x MP x = 10. (28) Now to calculate the number of workers here, we can use: MP x = Q x Q 5 x 5, (29) 10 = x 5, (30) x 5 = 2 x = 7. (31) Next: MRP 10 = MR MP 10 = $5 5 = $25. (32) Further: MP 10 = Q 10 Q , (33) 5 = Q , (34) 15 = Q Q 10 = 105. (35) Table 1 summarizes the results: 6

7 Input (L) Claims Filed (Q) Marginal product Marginal Revenue Product 2 5 NA NA $ $ $ $20 Table 1: Insurance claim production. b. We are looking for where MRP = ME = 20. This occurs at 10 worker hours. Question 8 (REQUIRES TUESDAY S NOTES) Some answers refer to the Production function table: Washes (W) Fuel (F) Table 2: Production (Q), in billions of airline seat miles. Fuel is in billions of gallons and engine washes has units of 10K washes per year. a. For the marginal product of fuel, we hold the washes column fixed and compare production across rows. We are interested in the extra production from adding an extra unit (billion gallons) of fuel, and so we don t want to also change the number of engine washes, which would prevent us from isolating the effect of an extra unit of fuel. For example, at 1.41B gallons fuel and K engine washes, from the production function table 2, MP F = Q F = = (36) Add an extra billion gallons of fuel adds about 68.6 billion passenger seat miles. The rest of the numbers are: 7

8 Washes (W) Fuel (F) NA NA NA NA NA Table 3: Marginal product of labor. Now for marginal product of an engine wash, we hold fuel fixed and compare across columns. For 1.41B gallons of fuel and K engine washes, we have: MP w = Q w = = (37) An extra 10K engine washes results in 2.19B extra passenger seat miles traveled. The extra engine washes allows the planes to travel farther on the fixed 1.41B gallons of fuel, since the engine washes saves on fuel economy. The rest of the numbers are: Washes (W) Fuel (F) NA NA NA NA NA NA NA NA NA NA Table 4: Marginal product of fuel. Finally, to compute the MRTS, we divide the marginal products. For example, at 8

9 1.43B gallons fuel and K engine washes: MRTS = MP F = = (38) MP w 2.22 An extra billion gallons of fuel is times more productive as an extra 10K engine washes. Of course, we also need to know the relative costs. The above table seems to indicate fuel is a more productive input, so we should use fuel. However, we also must calculate the costs. It could be that fuel is more than 30 times more expensive, favoring engine washes. Here are the rest of the numbers. Washes (W) Fuel (F) NA NA NA NA NA 1.43 NA NA NA NA NA NA NA NA NA Table 5: Marginal rate of technical substitution. b. From table 2, at 1.43 billion gallons of fuel and K engine washes, we are a little short of 98.1 billion airline seat miles. So K engine washes is more than enough, so either between 0.10 and 0.11 washes or 0.11 are acceptable answers. c. The marginal product of fuel is constant. This is from the relationship miles = mpg F which is constant returns to scale in F. However, this is not quite correct in the real world. Holding the planes constant, fuel economy tends to diminish as more fuel per plane adds weight to the plane, decreasing fuel economy. Probably this data shows up as constant returns because the number of planes is changing (recall marginal product holds all other inputs constant). d. Diminishing marginal product. For example, with fuel fixed at 1.43B gallons, from table 4 the marginal product diminishes from 2.22 B mile-seats per 10K engine washes, when engine washes are 0.1, to 1.94 when engine washes are e. We can calculate the price ratio using: P F = $1.56 = $31.2. (39) P w $0.05 9

10 From table 5, we see that at 1.49B gallons of fuel and K engine washes, the MRT S is which is the closest value to 31.2 (0.06 difference). This combination is closest to optimal. f. The units for the marginal product is the change in output for a change in input. So the marginal product of fuel has units of billion seat miles per billion gallons of fuel which is the same as seat miles per gallon. At the optimum of K engine washes, the seat miles per gallon are (table 3). g. The MRTS has units of w/ F. So 1/MRTS = F/ w, which is the change in fuel for a 1 unit change in engine washes, holding production constant. Since 1/MRTS = 1/31.14 = 0.032, in fact Southwest saves 0.032B gallons of fuel per 10K engine washes. Since the optimum is K engine washes, we have: Savings = K washes B gallons 10 K washes = B gallons (40) So we have that Southwest saved 3.85 million gallons of fuel. Note: this calculation assumes the MRTS is constant for all engine washes. In fact, the marginal product is declining as noted above. So the first few engine washes were actually more productive and yielded more fuel savings. So this estimate will typically be a low-end estimate. In fact, Southwest saved about 8 million gallons. 10