1 Chapter 6 Costs SOLUTIONS TO END-OF-CHAPTER QUESTIONS THE NATURE OF COSTS 1.1 The ships that return to Asia are half or completely empty; therefore, the cost of having more merchandise in them is almost zero. Hence, it is less expensive to ship wine to Asia. 1.2 Carmen receives $75 of consumer surplus if she attends the Outside Lands Festival. She gains $50 consumer surplus attending the Aquarium. Her opportunity cost of going to the Aquarium is $ a. The incremental cost is the marginal cost to a corporation of an additional flight. Some of these marginal costs are the costs of fuel, flight attendants (if they are paid for each flight), landing and taking off fees, etc. b. The marginal opportunity cost of an executive s flight is the price the company could have earned from leasing the jet to someone else. 1.4 The opportunity cost of a resource is the value of the next best alternative use of that resource. Thus, the opportunity cost of the pipe is $10 because that is the amount that it could otherwise be sold for. A sunk cost is a past expenditure that cannot be recovered. The pipe s sunk cost is $9 because this the original cost of the pipe, but this amount cannot be recovered. 151
2 152 Perloff/Brander, Managerial Economics and Strategy, Second Edition SHORT-RUN COSTS 2.1 Nicolas average fixed cost is given by AFC = $8/q. The diagram will show a constantly and smoothly declining function (similar to the one in the solution to 2.3 that follows). His friend s reasoning is incorrect in two ways. First, it fails to account for other costs associated with listening to music (like the opportunity cost that might be the value of getting homework done, going to the movies, or some other activity). His friend also fails to account for the benefit side of the decision. As Nicolas listens to more and more music, it is likely that the benefit he receives from listening to another song will decline. 2.2 C = VC + F; AC = AVC + AF = VC/q + F/q. By going to one supermarket, a consumer can lower the fixed cost F, thus also lowering the average cost, AC. 2.3 a. AFC = 10/q. MC = 10. AVC = 10. AC = 10/q 10.
3 Solutions Manual Chapter 6: Costs 153 b. AFC = 10/q. MC = 2q. AVC = q. AC = 10/q q. See the figure below. c. AFC = 10/q. MC = 10 8q 3q 2 q 2. AVC = 10 4q 2 q 2. AC = 10/q 10 4q 2 q 2. See the figure below. 2.4 TC = q, AFC = 900/q, AVC = 5, ATC = 900/q + 5, MC = 5.
4 154 Perloff/Brander, Managerial Economics and Strategy, Second Edition a. See the graphs below. b. Note that ATC becomes smaller as the production increases, so there is an advantage to having only one big music publisher. This is because the fixed cost is spread over a larger production, although the MC is constant and equal to 5. c. Revenue publishing cost = ( ) = 4,500 2,400 = 2,100. The publisher is willing to pay up to 2,100 pfennigs to the composer.
5 Solutions Manual Chapter 6: Costs C = q, MC = AVC = AC = 1 for q less than or equal to 80 per day. C = (q 80) = 1.5q 40, MC = 1.5, AVC = AC = /q for all q greater than 80 per day. See figures below. 2.6 The total cost of building a one-cubic-foot crate is $6. It costs four times as much to build an eight-cubic-foot crate, $24. In general, as the height of a cube increases, the total cost of building it rises with the square of the height, but the volume increases with the cube of the height. Thus the cost per unit of volume falls. 2.7 See the figure below. Suppose L is measured in person-hours (number of workers times the hours they work), then the production function is Q = 4L. Then Then VC = 8L = 8(Q/4) = 2Q. AVC = MC = Since α + β < 1, the production function exhibits decreasing returns to scale. If capital is fixed at 100, then short-run output is q 10L 0.32 (100) q 10L q 131.8L
6 156 Perloff/Brander, Managerial Economics and Strategy, Second Edition The amount of labor required to produce output q is q L L ( q ). Thus, the firm s cost function is C = 10L + 20 C = 10(0.0075q) (100) C = 10(0.0075q) , its variable cost of production is VC = 10(0.0075q) 3.125, average variable cost is AVC = 10(0.0075) q 2 q 2.125, and marginal cost is MC = 3.125(10)(0.0075)(0.0075q) MC = 0.234(0.0075q) These can be graphed in Microsoft Excel. 2.9 A variable cost (VC) is a cost that changes as the quantity of output changes. For example, if output is 550, then VC = A firm s cost is the sum of a firm s variable cost (VC) and fixed cost (F):
7 Solutions Manual Chapter 6: Costs 157 C = VC + F C = = A firm s marginal cost (MC) is the amount by which a firm s cost changes if the firm produces one more unit of output. The marginal cost is MC = VC q MC = q 0.33 MC = The average fixed cost (AFC) is the fixed cost divided by the units of output produced: 600/550 = Average variable cost (AVC) or variable cost per unit of output is the variable cost divided by the units of output produced: 37.71/550 = Similarly, if output increases to 600, then VC = 39.97, C = , MC = 0.04, AFC = 1.00, and AVC = LONG-RUN COSTS 3.1 a. See panel (a) of the following figure. b. See panel (a) of the figure. The firm chooses labor-machine technology. c. See panel (b) of the figure. 3.2 If the firm were minimizing its cost, the extra output it gets from the last dollar spent on labor, MPL / w 50/ , should equal the extra output it derives from the last dollar spent on capital, MP / r 200/1, Thus, the firm is not minimizing its costs. It would do better if it used relatively less capital and more labor, from which it gets more extra output from the last dollar spent.
8 158 Perloff/Brander, Managerial Economics and Strategy, Second Edition 3.3 The price of labor (w) is now 25 percent cheaper. Assuming that the price of capital (r) does not change and remains the same, the isocost faced by producers becomes less steep ( w/r before the subsidy as compared to 0.75w/r after the subsidy). Assuming a strictly convex technology, at the optimum, the producer will use less of both workers and capital than before for the same output. However, because labor is now cheaper relative to capital (relatively cheaper), the firm will now employ relatively more labor than capital. 3.4 From the information given and assuming that there are no economies of scale in shipping baseballs, it appears that baseballs are produced using a constant return to scale, fixedproportion production function. The corresponding cost function is C(q) = (w + s + m)q, where w is the wage for the time period it takes to stitch one baseball, s is the cost of shipping one baseball, and m is the price of all material to produce a baseball. Because the cost of all inputs other than labor and transportation are the same everywhere, the cost difference between Georgia and Costa Rica depends on w + s in both locations. As firms choose to produce in Costa Rica, the extra shipping cost must be less than the labor savings in Costa Rica. 3.5 Because the two flavorings are perfect substitutes, we can draw a linear isoquant. Assuming the alcohol-based flavoring produces more flavor per ounce than the nonalcoholic flavoring (we can use less of the alcohol-based flavoring), we draw our isoquant with a MRTS >1. If the cost of each flavoring is equal, the isocost curve has a slope of 1, and the producer will buy only the alcohol-based flavoring. After the tax, the alcohol-based flavoring will be so much more expensive that the new isocost curve s slope will likely have an absolute value greater than the MRTS, causing the producer to buy only nonalcoholic flavoring. 3.6 Firms will maximize profit by producing where the ratio of the marginal product of labor to the marginal product of capital is equal to the ratio of the input prices: MP L w. MP r
9 Solutions Manual Chapter 6: Costs 159 The ratio of marginal products equals MPL MP 0.5L 0.5L L. In the United States, the ratio of input prices equals 1, so when producing optimally L = L. 1 To produce 100 units of output in the United States, 100 = L 0.5 L 0.5 L = 100 and = 100. The cost of producing 100 units of output in the United States is C = 10(100) + 10(100) = 2,000. In Mexico, the ratio of input prices equals 0.5, so when producing optimally L = L. To produce 100 units of output in Mexico, 100 = (2) 0.5 () = 1.41 = and L = The cost of producing 100 units of output in Mexico is C = 5(141.42) + 10(70.71) = $1, The cost function for Google is C = F + cq. The marginal cost in this case is constant and equal to c. (If you know calculus, this can be found by taking the derivative of the cost function.) Average cost can be found by dividing the cost function by q, which gives AC = (F/q) + c. Google enjoys economies of scale as long as the AC curve is falling. Because this AC function is always declining, Google will enjoy economies of scale anytime it increases output.
10 160 Perloff/Brander, Managerial Economics and Strategy, Second Edition 3.8 If it takes two units of labor (at a wage of w per unit) and one unit of labor (at a price of r per unit) in a fixed proportion to produce a unit of the good, then the cost associated with producing one unit will be 2w + r and the cost of producing q units will be (2w + r)q. In the case of two inputs that are perfect substitutes with identical marginal productivities (as implied by the q = L + production function), then the firm will simply use whichever input is cheapest, and its cost will be the price of that input multiplied by the number of units produced. For example, if labor is cheaper than capital, then the total cost of production will be C = wq. 3.9 When the long-run curve is sloping downward, the short-run curve touches the long-run curve to the left of its minimum. When the long-run curve is upward sloping, the short-run curve touches the long-run curve to the right of its minimum. At the minimum of the longrun curve, the short-run curve touches the long-run curve at its minimum. THE LEARNING CURVE 4.1 Learning by doing is where the productive skills and knowledge that workers and managers gain from experience lowers the average cost of production. Workers who are given a new task perform it slowly the first few times they try, but their speed increases with practice. Managers may learn how to organize production more efficiently, discover which workers to assign to which tasks, and determine where more inventories are needed or where they can be reduced. Engineers may optimize product designs by experimenting with various production methods. For these reasons, the average cost of production tends to fall over time, and the effect is particularly strong with new products. 4.2 A firm s learning curve, which shows the relationship between average cost and cumulative output (the sum of its output since the firm started producing), is AC r a bn, where AC is its average cost; N is its cumulative output; and a, b, and r are constants. If r = 0, then AC = a + b. If r = 0, then average cost does not fall with cumulative output (average cost remains constant with cumulative output), so there is no learning by doing. If average cost decreases with output, then r < 0, which is a characteristic of learning by doing. If N = 0, then AC = a. Therefore, a represents the average cost of production before any learning by doing. 4.3 a. The total cost is $20 per unit multiplied by 50 units in year 1 (total of $1,000) and $40 per unit multiplied by 40 units in year two ($1,600) for a total over the two years of $2,600. A total of 60 units are produced so AC = $2,600/60 = $43.33.
11 Solutions Manual Chapter 6: Costs 161 b. If the firm produces 60 units in year 1, then its average cost in year two will fall by $20 to $30 (10% or $5 decline for every 10 units over 20 produced in year 1). This means that total cost in the first year is 60 units times $50 (or $3,000). In year 2, the additional 40 units are produced at only $30 each (so $1,200). This is a total cost of $4,200 and an average cost of $4,200/100 = $42. c. The cost of the additional 40 units in year 1 is only $1,680 on average because of the lower costs in year Assume the average cost of production is AC r s a b N 1 b2m. Holding a, b 1, b 2, and s constant, this firm exhibits learning by doing if r > 0 because r b1 N average cost will decrease with N because 0. Similarly, holding a, b 1, b 2, and s N constant, this firm exhibits learning by doing if s > 0 because average cost will decrease s b2 M with M because 0. M THE COSTS OF PRODUCING MULTIPLE GOODS 5.1 Referring to Figure 6.7 in the chapter, a restriction on the size of a store might restrict output to the left of the minimum point in the long-run cost curve, q 2. (This is likely because stores in other countries are larger than in the United ingdom.) Thus, in the long run, the stores will not be able to take advantage of economies of scale and economies of scope to lower average fixed costs and total average costs of carrying a large amount of any single item, or a wider range of items. 5.2 Economies of scope is the situation in which it is less expensive to produce goods jointly than separately. Diseconomies of scope is the situation in which it is less expensive to produce goods separately than jointly. A measure of the degree to which there are economies of scope is SC = C( q1,0) C(0, q2) C( q1, q2), C( q, q ) 1 2 where C(q 1,0) is the cost of producing q 1 units of the first good by itself, C(0,q 2 ) is the cost of producing q 2 units of the second good by itself, and C(q 1,q 2 ) is the cost of producing both goods together. If the cost of producing the two goods separately is the same as producing them together, then SC is zero. If it is cheaper to produce the goods jointly, SC is positive. If SC is negative, there are diseconomies of scope, and the two goods should be
12 162 Perloff/Brander, Managerial Economics and Strategy, Second Edition produced separately. In this example, SC is positive, so Laura experiences economies of scope. 5.3 Economies of scope is the situation in which it is less expensive to produce goods jointly than separately. Diseconomies of scope is the situation in which it is less expensive to produce goods separately than jointly. A measure of the degree to which there are economies of scope is SC = C( q1,0) C(0, q2) C( q1, q2), C( q, q ) 1 2 where C(q 1,0) is the cost of producing q 1 units of the first good by itself, C(0,q 2 ) is the cost of producing q 2 units of the second good by itself, and C(q 1,q 2 ) is the cost of producing both goods together. If the cost of producing the two goods separately is the same as producing them together, then SC is 0. If it is cheaper to produce the goods jointly, SC is positive. If SC is negative, there are diseconomies of scope, and the two goods should be produced separately. For example, if a firm produces fuel and heating oil in fixed proportions, then it does not produce heating fuel without producing gasoline (or, alternatively, it does not cost any extra to also produce gasoline when producing heating fuel). Similarly, the firm does not produce gasoline without producing heating fuel (or it does not cost any extra to also produce heating fuel when producing gasoline). Therefore, the firm s measure of economies of scope is positive and equal to 1 because C(q 1,0) = C(0,q 2 ) = C(q 1,q 2 ). 5.4 According to the Mini-Case, Carey et al. (2015) finds evidence that outpatient surgeries generate economies of scope and therefore should be considered for provision in an acute-care facility. Cohen and Morrison Paul (2011), on the other hand, find large diseconomies for drug abuse treatments, so those services should be offered in different facilities.
13 Solutions Manual Chapter 6: Costs 163 MANAGERIAL PROBLEM 6.1 An isocost line shows all the combinations of inputs that require the same total expenditure. For the firm to be indifferent between using the wafer-handling stepper technology and the stepper technology, the isocost line must pass through the input combinations for both technologies, as shown by isocost line C 4. The slopes of the isocost lines equal the wage divided by the price of capital (multiplied by minus one). Because isocost line C 4 is flatter than isocost line C 2, the absolute value of the wage-cost of capital ratio on isocost line C 4 is less than that ratio on isocost line C 2. Because isocost line C 4 is steeper than isocost line C 3, the absolute value of the wage-cost of capital ratio on isocost line C 4 is greater that ratio on isocost line C 3. SPREADSHEET EXERCISES See the associated Excel files.