ch08Solution manual

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Chapter 8 Cost Curves Solutions to Review Questions 1. What is the relationship between the solution to the firm’s long-run cost-minimization problem and the long-run total cost curve? The long-run total cost curve plots the minimized total cost for each level of output holding input prices fixed. In other words, for a given set of input prices, the long-run total cost curve represents the total cost associated with the solution to the long-run cost minimization problem for each level of output. 2. Explain why an increase in the price of an input typically causes an increase in the longrun total cost of producing any particular level of output. When the price of one input increases, the isocost line for a particular level of total cost will rotate in toward the origin. Assuming the isocost line was tangent to the isoquant for the firm’s selected level of output, when the isocost line rotates it will no longer touch the original isoquant. In order for an isocost line to reach a tangency with the original isoquant, the firm would need to move to an isocost line associated with a higher level of cost, i.e. an isocost line further to the northeast. 3. If the price of labor increases by 20 percent, but all other input prices remain the same, would the long-run total cost at a particular output level go up by more than 20 percent, less than 20 percent, or exactly 20 percent? If the prices of all inputs went up by 20 percent, would long-run total cost go up by more than 20 percent, less than 20 percent, or exactly 20 percent? If the price of a single input goes up leaving all other input prices the same and the level of output constant, total cost will rise but by a smaller percentage than the increase in the input price. This occurs because the firm will substitute away from the now relatively more expensive labor to the now relatively less expensive other inputs. So, if the price of labor rises by 20% holding all other input prices constant, total cost will rise by less than 20%. If the prices of all inputs go up by the same percentage, total cost will rise by exactly that same percentage. So, if input prices rise by 20%, total cost will also rise by 20%. 4. How would an increase in the price of labor shift the long-run average cost curve? An increase in the price of labor would result in a long-run total cost curve that lies above the initial long-run total cost curve at every quantity except Q  0 . Since AC  TC / Q , increasing

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total cost will raise average cost at every quantity except Q  0 . Therefore, the long-run average cost curve will shift up. 5. a) If the average cost curve is increasing, must the marginal cost curve lie above the average cost curve? Why or why not? b) If the marginal cost curve is increasing, must the marginal cost curve lie above the average cost curve? Why or why not? a) When MC  AC , average cost is increasing, and when MC  AC , average cost is decreasing. So, if the average cost curve is increasing it must lie below the marginal cost curve. b) If the marginal cost curve is increasing, it may lie above or below the average cost curve. The only determining factor here is whether or not marginal cost lies above or below average cost. If it lies above, average cost will be increasing and if it lies below, average cost will be decreasing. Knowing that marginal cost is increasing or decreasing tells us nothing about average cost. 6. Sketch the long-run marginal cost curve for the “flat-bottomed” long-run average cost curve shown in Figure 8.11. TC MC AC

Q When average cost is falling, marginal cost will lie below average cost, and when average cost is increasing, marginal cost will lie above average cost. Over the flat-bottomed portion where average cost is neither increasing nor decreasing, marginal cost and average cost will be equal. 7. Could the output elasticity of total cost ever be negative? The output elasticity of total cost, when simplified can be written as MC  TC ,Q  AC Since AC  TC / Q , and since TC and Q must always be positive, AC will always be positive. Marginal cost, MC, represents the change in total cost associated with an increase in output. When output increases, total cost must always rise for a given set of input prices, implying that MC is also always positive. Therefore, the output elasticity of total cost must always be positive. Copyright © 2014 John Wiley & Sons, Inc.

Chapter 8 - 2

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

8. Explain why the short-run marginal cost curve must intersect the average variable cost curve at the minimum point of the average variable cost curve. Because fixed cost does not change, marginal costs reflect the change in variable costs. Thus, as with the relationship between any average and marginal, if average variable cost is decreasing, marginal cost must be below average variable cost, and if average variable cost is increasing, marginal cost must lie above average variable cost. This implies marginal cost will intersect average variable cost at the minimum of average variable cost. 9. Suppose the graph of the average variable cost curve is flat. What shape would the short-run marginal cost curve be? What shape would the short-run average cost curve be? If the average variable cost curve is flat, average variable cost is neither increasing nor decreasing. Marginal cost will therefore be equal to average variable cost and the marginal cost curve will therefore also be flat. Since average fixed cost is always declining, and since average total cost is the vertical sum of average variable and average fixed costs, average total cost must also be declining at all levels of Q if average variable cost is constant. Graphically, average total cost will be declining and asymptotic to the average variable cost curve. 10. Suppose that the minimum level of short-run average cost was the same for every possible plant size. What would that tell you about the shapes of the long-run average and long-run marginal cost curves? The long-run average cost curve is the envelope to the short-run average cost curves associated with each level of output. If each of these short-run average cost curves has the same minimum point, the long-run average cost curve will be a horizontal line tangent to all of these minimum points. Because the long-run average cost curve will be flat, long-run average cost is neither increasing nor decreasing, and the long-run marginal cost curve will also be flat and equal to long-run average cost. 11. What is the difference between economies of scope and economies of scale? Is it possible for a two-product firm to enjoy economies of scope but not economies of scale? Is it possible for a firm to have economies of scale but not economies of scope? Economies of scale refer to a situation when average total cost for a single product declines as the level of output for that product increases. These economies of scale might occur, for example, because workers can specialize in tasks as the level of output increases and the workers’ productivity may increase. Economies of scope refer to efficiencies that arise when a firm produces more than one product. In particular, economies of scope exist if one firm producing N products does so at a lower total cost than N separate firms producing the same quantities of each product individually. The notion of economies of scale can actually be applied to a multi-product firm as well. We can use this extension to further refine the distinction between economies of scale and scope. Suppose a firm is producing N products, with output levels measured by Q1 , Q2 ,K , QN . If it operates with economies of scale, the total cost of production will rise by less than 1% when

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

production of all outputs increases by 1%. If it operates with diseconomies of scale, the total cost of production will rise by more than 1% when production of all outputs increases by 1%. By contrast, economies of scope exist if it is less costly to have the outputs produced by one firm instead of by N firms, each specializing in the production of one of the outputs. Note that information about economies of scope does not tell us whether the firm has economies of scale. If a production process has economies of scope, there may not be economies of scale. Further, information about economies of scale does not tell us whether the firm has economies of scope. If a production process has economies of scale, there may not be economies of scope. 12. What is an experience curve? What is the difference between economies of experience and economies of scale? The experience curve represents the relationship between average variable cost and cumulative production volume over time. One would expect that as cumulative production volume increased, average variable cost would fall. Economies of scale refer to a situation when average cost declines as the level of output for that product increases within a given time frame. In general, economies of scale would occur if the average cost curve declined as the level of output increased. Economies of experience would occur if, as cumulative production volume increased, the average cost curve shifted downward for all levels of output. So, economies of scale refer to lower average costs that occur as output increases and economies of experience refer to lower average costs for all levels of output as cumulative production volume increases.

Solutions to Problems 8.1. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

The table is reproduced below. First, since fixed costs are independent of quantity, the entire TFC column can be easily filled in. Proceeding through the table row by row, for Q = 1 it is easy to see that TVC = TC – TFC = 80, and the rest of the row is similarly straightforward. For Q = 2, TC = TVC + TFC = 180, and the rest of the follows easily. For Q = 3, all we have is TFC = 20; thus, we cannot infer anything else. For Q = 4, TC = Q*AC = 380. It’s then possible to get TVC and AVC; however, we cannot find MC since we don’t know TC or TVC for Q = 3. For Q = 5,

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the important step is to use MC(5) = TC(5) – TC(4) to find TC(5) = 550. For Q = 6, the important step is TVC = AVC*Q = 720. Q 1 2 3 4 5 6

TC 100 180 380 550 740

TVC 80 160 360 530 720

TFC 20 20 20 20 20 20

AC 100 90 95 110 123.3

MC 80 80 170 190

AVC 80 80 90 106 120

8.2. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

It helps to rewrite this table adding an extra column for Total Fixed Costs at each level of output. The TFC for Q = 2 is just 2*30 = 60, and this is also the TFC value for every other output level. Then for Q = 1, we know TC = AC*Q = 100, TVC = TC – TFC = 40 and the rest is straightforward. Similarly we can fill in the rows for Q = 2, 3, 4, and 6. For Q = 5, we need to use the fact that MC(6) = TC(6) – TC(5) to infer TC(5) = 250. The rest is straightforward. Q 1 2 3 4 5 6

TC 100 110 120 180 250 330

TVC 40 50 60 120 190 270

AFC 60 30 20 15 12 10

AC 100 55 40 45 50 55

MC 40 10 10 60 70 80

AVC 40 25 20 30 38 45

TFC 60 60 60 60 60 60

8.3. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Q 1 2 3 4 5 6

TC 18 30 46 66 90 118

TVC 8 20 36 56 80 108

TFC 10 10 10 10 10 10

AC 18 15 46/3 66/4 18 118/6

MC 8 12 16 20 24 28

AVC 8 10 12 16 16 18

8.4. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

Q 1 2 3 4 5 6

TC 20 36 55 82 112 144

TVC 10 26 45 72 102 134

TFC 10 10 10 10 10 10

AC 20 18 18.33 20.5 22.4 24

MC 10 16 19 28 30 32

AVC 10 13 15 18 20.4 22.33

8.5. A firm produces a product with labor and capital, and its production function is described by Q = LK. The marginal products associated with this production function are Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

MPL = K and MPK = L. Suppose that the price of labor equals 2 and the price of capital equals 1. Derive the equations for the long-run total cost curve and the long-run average cost curve. Starting with the tangency condition, we have MPL w  MPK r K 2  L 1 K  2L Substituting into the production function yields Q  LK Q  L(2 L) L

Q 2

Plugging this into the expression for K above gives K 2

Q 2

Finally, substituting these into the total cost equation results in 

Q    2 2  Q  2

TC  2   

TC  4  

Q 

2

TC  8Q and average cost is given by AC  AC 

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8Q TC  Q Q 8 Q

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

8.6 A firm’s long-run total cost curve is

. Derive the equation for the

corresponding long-run average cost curve, Given the equation of the long-run average cost curve, which of the following statements is true: a. The long-run marginal cost curve lies below for all positive quantities b. The long-run marginal cost curve

is the same as the

for all positive

quantities c. The long-run marginal cost curve

lies above the

d. The long-run marginal cost curve

lies below

and above the

for all positive quantities

for some positive quantities

for some positive quantities

ANSWER: c The equation of the AC curve is AC(Q) = 1000Q. It is increasing in Q. Given the relationship between AC and MC curves, the fact that the AC curve is increasing means that the MC curve must lie above the AC curve.

8.7 A firm’s long-run total cost curve is

. Derive the equation for the

corresponding long-run average cost curve, Given the equation of the long-run average cost curve, which of the following statements is true: a. The long-run marginal cost curve lies below for all positive quantities b. The long-run marginal cost curve

is the same as the

for all positive

quantities c. The long-run marginal cost curve

lies above the

d. The long-run marginal cost curve

lies below

and above the

for all positive quantities

for some positive quantities

for some positive quantities

ANSWER: a The equation of the AC curve is AC(Q) = TC(Q)/Q = 1000Q1/2/Q = 1000Q-(1/2). This is a decreasing function of Q. Given the relationship between AC and MC curves, the fact that the AC curve is decreasing means that the MC curve must lie below the AC curve.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

8.8. A firm’s long-run total cost curve is TC(Q) = 1000Q − 30Q2 + Q3. Derive the expression for the corresponding long-run average cost curve and then sketch it. At what quantity is minimum efficient scale? AC 

TC 1000Q  30Q 2  Q 3  Q Q

AC  1000  30Q  Q 2 Graphically, average cost is

Average Cost

1200 1000 800 600 400 200 0 0

5

10

15

20

25

30

Q

Minimum efficient scale occurs where the average cost curve reaches a minimum, Q  15 for this cost function. 8.9. A firm’s long-run total cost curve is TC(Q) = 40Q − 10Q2 + Q3, and its long-run marginal cost curve is MC(Q) = 40 − 20Q + 3Q2. Over what range of output does the production function exhibit economies of scale, and over what range does it exhibit diseconomies of scale? From the total cost curve, we can derive the average cost curve, AC (Q )  40  10Q  Q 2 . The minimum point of the AC curve will be the point at which it intersects the marginal cost curve, i.e. 40  10Q  Q 2  40  20Q  3Q 2 . This implies that AC is minimized when Q = 5. By definition, there are economies of scale when the AC curve is decreasing (i.e. Q < 5) and diseconomies when it is rising (Q > 5). 8.10. For each of the total cost functions, write the expressions for the total fixed cost, average variable cost, and marginal cost (if not given), and draw the average total cost and marginal cost curves. a) TC(Q) = 10Q Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

b) TC(Q) = 160 + 10Q c) TC(Q) = 10Q2, where MC(Q) = 20Q d) TC(Q) = 10√Q, where MC(Q) = 5/√Q e) TC(Q) = 160 + 10Q2, where MC(Q) = 20Q a)

TFC = 0, AVC = 10, MC = 10.

MC = AC = 10

b)

TFC = 160, AVC = 10, MC = 10.

AC MC = 10

c)

TFC = 0, AVC = 10Q.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

MC

AC

d)

TFC = 0, AVC = 10

Q

.

AC MC

e)

TFC = 160, AVC = 10Q.

MC

AC

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

8.11. A firm produces a product with labor and capital as inputs. The production function is described by Q = LK. The marginal products associated with this production function are MPL = K and MPK = L. Let w = 1 and r = 1 be the prices of labor and capital, respectively. a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q. b) Solve the firm’s short-run cost-minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive the equation for the firm’s short-run total cost curve as a function of quantity Q and graph it together with the long-run total cost curve. c) How do the graphs of the long-run and short-run total cost curves change when w = 1 and r = 4? d) How do the graphs of the long-run and short-run total cost curves change when w = 4 and r = 1? a) Cost-minimizing quantities of inputs are equal to L = √Q √(r/w) and K = √Q / √(r/w). Hence, in the long-run the total cost of producing Q units of output is equal to TC(Q) = 10 + 2√(Qrw). For w = 1 and r = 1 we have TC(Q) = 2√Q. b) When capital is fixed at a quantity of 5 units (i.e., K = 5) we have Q = K*L = 5 L. Hence, in the short-run the total cost of producing Q units of output is equal to STC(Q) = 5 + Q/5. TC STC(Q) TC(Q) 5

25

Q

c) We have L = √Q √(r/w) and K = √Q / √(r/w). Hence, TC(Q) = 2√(Qrw) and STC(Q) = 5r + wQ/5. When w = 1 and r = 4 we have TC(Q) = 4√Q and STC(Q) = 20 + Q/5.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

TC

STC(Q), w =1, r = 4 TC(Q), w =1, r = 4

20 STC(Q) TC(Q) 5

25 d)

Q

100

When w = 4 and r = 1 we have TC(Q) = 4√Q and STC(Q) = 4Q/5.

TC STC(Q), w = 4, r = 1

TC(Q), w = 4, r = 1 STC(Q) TC(Q) 10

25/4

25

Q

8.12. A firm produces a product with labor and capital. Its production function is described by Q = min(L, K). Let w and r be the prices of labor and capital, respectively.

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a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q and input prices, w and r. b) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive the equation for the firm’s short-run total cost curve as a function of quantity Q. Graph this curve together with the long-run total cost curve for w = 1 and r = 1. c) How do the graphs of the long-run and short-run total cost curves change when w = 1 and r = 2? d) How do the graphs of the long-run and short-run total cost curves change when w = 2 and r = 1? a) The inputs are complementary and the cost-minimizing firm uses them in proportions 1:1. Hence, we have TC(Q) = Q(w + r). b) If w = r = 1, then TC(Q) = 2Q. In the short run, it is impossible to produce more than 5 units. This is because min(L,5) cannot be any greater than 5. To produce Q  5 units, we set L = Q. With w = r = 1, this implies STC(Q) = 15 + Q. (10 is the fixed cost of the indivisible input, 5 is the fixed cost of labor, and Q is the variable cost of labor.) The diagram below shows TC(Q) and STC(Q) for Q  5 when w = r = 1. TC

TC(Q)

STC(Q)

5

5 c)

Q

For w = 1 and r = 2 we have TC(Q) = 3Q and STC(Q) = Q + 10 for Q not larger than 5.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

TC(Q), w = 1, r = 2

TC

TC(Q) STC(Q), w = 1, r = 2

10 STC(Q) 5

Q

5 d)

For w = 2 and r = 1 we have TC(Q) = 3Q and STC(Q) = 2Q + 5 for Q not larger than 5.

TC

TC(Q), w = 2, r = 1 TC(Q) STC(Q), w = 2, r = 1 STC(Q)

15

10 5

Q

8.13. A firm produces a product with labor and capital. Its production function is described by Q = L + K. The marginal products associated with this production function are MPL = 1 and MPK = 1. Let w = 1 and r = 1 be the prices of labor and capital, respectively. a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q when the prices labor and capital are w = 1 and r = 1.

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b) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5), and w = 1 and r = 1. Derive the equation for the firm’s short-run total cost curve as a function of quantity Q and graph it together with the longrun total cost curve. c) How do the graphs of the short-run and long-run total cost curves change when w = 1 and r = 2? d) How do the graphs of the short-run and long-run total cost curves change when w = 2 and r = 1? a) With a linear production function, the firm operates at a corner point depending on whether w < r or w > r. If w < r, the firm uses only labor and thus sets L = Q. In this case, the total cost (including the fixed cost) is wQ. If w > r, the firm uses only capital and thus sets K = Q. in this case, the total cost is rQ. When w = r = 1, the firm is indifferent among combination of L and K that make L + K = 10. Thus, we have TC(Q) = Q. b) When capital is fixed at 5 units, the firm’s output would be given by Q = 5 + L. If the firm wants to produce Q < 5 units of output, it must produce 5 units and throw away 5 – Q of them. The total cost of producing fewer than 5 units is constant and equal to $5, the cost of the fixed capital. For Q > 5 units, the firm increases its output by increasing its use of labor. In particular, to produce Q units of output, the firm uses Q – 5 units of labor, for a cost of Q – 5, and 5 units of capital, for a cost of 5. Thus, STC(Q) = Q – 5 + 5 = Q STC(Q) & TC TC(Q)

STC(Q) 5

TC(Q) 5

Q

c) In the long run, since w < r, the firm produces its output entirely with labor. Thus, TC(Q) = Q, just as in part (b). In the short-run, with capital fixed at 5 units, the firm’s output would be given by Q = 5 + L. If the firm wants to produce Q < 5 units of output, it must produce 5 units of output and throw away 5 – Q of them. It can produce this output using its fixed stock of 5 units of capital and no labor. The total cost of producing Q < 5 units of output when the price of capital is $2 per unit is $10. For Q > 5 units, the firm increases its output by increasing its use of labor. In particular, to produce Q units of output, the firm uses Q – 5 units of labor, for a cost of Q – 5, and 5 units of Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

capital, for a cost of 10. Thus, STC(Q) = (Q – 5) + 10 = Q + 5. Notice that when K = 5, w = 1, and r = 2, the STC curve strictly lies above the TC curve. This is because K = 5 is never an optimal capital choice for the firm when w = 1 and r = 2. As a result the firm’s total costs are always higher in the short run than they are in the long run. TC

STC(Q) & TC(Q)

STC(Q) w = 1, r = 2 10

STC(Q) 5

TC(Q) w = 1, r = 2

5

Q

d) The total cost curve is the same as in part (b), i.e. TC(Q) = Q. This is because the cheaper input (in this case capital) continues to have a price of $1 per unit. In the short run, with capital being fixed at 5 units, the cost of producing Q < 5 is $5. To produce more than Q units, the firm uses Q – 5 units of labor at a total cost of 2(Q – 5) = 2Q – 10. It also uses 5 units of capital at a total cost of 5. Thus, for Q > 5, STC(Q) = 2Q – 10 + 5 = 2Q – 5.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

TC

STC(Q), w = 2, r =1 STC(Q) & TC(Q)

STC(Q) = STC(Q), w = 2, r =1 5

TC(Q) = TC(Q), w = 2, r = 1 10 Q

5

8.14. Consider a production function of two inputs, labor and capital, given by Q = (√L + √K)2. The marginal products associated with this production function are as follows:

Let w = 2 and r = 1. a) Suppose the firm is required to produce Q units of output. Show how the costminimizing quantity of labor depends on the quantity Q. Show how the cost-minimizing quantity of capital depends on the quantity Q. b) Find the equation of the firm’s long-run total cost curve. c) Find the equation of the firm’s long-run average cost curve. d) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 9 units (i.e., K = 9). e) Find the short-run total cost curve, and graph it along with the long-run total cost curve. f ) Find the associated short-run average cost curve. a)

Starting with the tangency condition we have MPL w  MPK r  L1/ 2  K 1/ 2 L1/ 2

 L1/ 2  K 1/ 2 K 1/ 2

Plugging this into the total cost function yields

Copyright © 2014 John Wiley & Sons, Inc.



2 1

K 4 L K  4L

Chapter 8 - 18

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Q   L1/ 2  (4 L)1/ 2

2

2

Q   3L1/ 2 Q  9L Q L 9

Inserting this back into the solution for K above gives K

4Q 9

b) 4Q  Q   9  9 2Q TC  3 TC  2 

c) TC  2Q   Q  3 2 AC  3 AC 

Q

d) When Q  9 the firm needs no labor. If Q  9 the firm must hire labor, setting K  9 and plugging in for capital in the production function yields 2

Q   L1/ 2  91/ 2 Q1/ 2  L1/ 2  3 L1/ 2  Q1/ 2  3 L   Q1/ 2  3

2

Thus,   Q 12  3  L   0 e)

2

if Q  9 if Q  9

 2  Q1/ 2  3 2  9

when Q  9

 9

when Q  9

TC  

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Graphically, short-run and long-run total costs are shown in the following figure. 25.00 SRTC

Total Cost

20.00 15.00 10.00 5.00

LRTC

0.00 0

5

9 10

15

20

25

30

Q

f)  2  Q1/ 2  3 2  9

 TC  AC   Q  9  Q 

Q

if Q  9 if Q  9

8.15. Tricycles must be produced with 3 wheels and 1 frame for each tricycle. Let Q be the number of tricycles, W be the number of wheels, and F be the number of frames. The price of a wheel is PW and the price of a frame is PF. a) What is the long-run total cost function for producing tricycles, TC(Q,PW, PF)? b) What is the production function for tricycles, Q(F,W )? a) Each tricycle requires the purchase of three wheels at price PW and one frame at price PF. Thus, TC(Q, PW, PF) = Q(3PW + PF). b) Three wheels and one frame are perfect complements in production. Thus the production function is Q(F, W) = min{F, (1/3)W}. Notice that (F, W) = (1, 3) yields Q = 1, (F, W) = (2, 6) yields Q = 2, etc. 8.16. A hat manufacturing firm has the following production function with capital and labor being the inputs: Q = min(4L, 7K )—that is it has a fixed-proportions production function. If w is the cost of a unit of labor and r is the cost of a unit of capital, derive the firm’s long-run total cost curve and average cost curve in terms of the input prices and Q.

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The fixed proportions production function implies that for the firm to be at a cost minimizing optimum, 4 L  7 K and both of these equal Q. Therefore, L = Q/4 and K = Q/7. So the firm’s w 4

r 7 w r The average cost curve is LRAC  TC / Q   . Note that this average cost curve is 4 7

total cost is wL  rK  wQ / 4  rQ / 7  [  ]Q .

independent of Q and is simply a straight line. 8.17. A packaging firm relies on the production function Q = KL + K, with MPL = K and MPK = L + 1. Assume that the firm’s optimal input combination is interior (it uses positive amounts of both inputs). Derive its long-run total cost curve in terms of the input prices, w and r. Verify that if the input prices double, then total cost doubles as well. Since we can assume an interior solution, the tangency condition must hold. Therefore the K w rK  . This means L  1  . Substituting this back into L 1 r w rK 2 Qw , so K  the production function, we see that Q  . w r Qr  1. The total cost curve is then TC  wL  rK = 2 wrQ  w. If we This implies that L  w

optimal bundle must be such that

substitute 2w and 2r in the place of w and r respectively, we get TC2 = 2 ( 2 w)( 2r )Q  ( 2 w)  4 wrQ  2 w  2 * TC , so total cost does indeed double when input prices double. 8.18. A firm has the linear production function Q = 3L + 5K, with MPL = 3 and MPK = 5. Derive the expression for the 1ong-run total cost that the firm incurs, as a function of Q and the factor prices, w and r. As we saw in Chapter 7, linear production functions usually have corner solutions. In this case, the firm will use only labor if MRTS L , K 

Similarly, it will use only capital if

w r  . 3 5

If the firm does use labor, then it will use L  capital it will use K 

w w r , or  r 3 5

Q with a total cost of wQ/3. Similarly if it uses 3

Q with a total cost of rQ/5. 5 w r 3 5

Therefore, the firm’s total cost curve can be expressed as TC  min{ , }Q.

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8.19 A firm uses two inputs: labor and capital. The price of labor is

and the price of

capital is . The firm’s long-run total cost is given by the equation Based on this equation, which change would cause the greater upward rotation in the long-run total cost curve: a 10 percent increase in or a 10 percent increase in ? Based on your answer, is the firm’s production operation more capital intensive or labor intensive? Explain your answer. A 10 percent increase in r would cause the TC curve to rotate upward more than a 10 increase in w. (In fact, a 10 percent increase in r would cause a (4/5)*10 = 8 percent increase in TC for any positive level of output Q, while a 10 percent increase in w would cause only a (1/5)*10 = 2 percent increase in TC for any positive level of output Q. The fact that total costs are more responsive to a change in the price of capital than to a change in the price of labor, suggests that the firm’s production operation is more capital intensive than labor intensive. 8.20. When a firm uses K units of capital and L units of labor, it can produce Q units of output with the production function Q = K√L. Each unit of capital costs 20, and each unit of labor costs 25. The level of K is fixed at 5 units. a) Find the equation of the firm’s short-run total cost curve. b) On a graph, draw the firm’s short-run average cost. a)

From the production function we see that Q  5 L , so the amount of labor required to

produce Q is given by L 

Q2 . The short run total cost function is 25

 Q 2 C  25 L  20 K  25    20(5)  100  Q 2 .  25 b)

8.21. When a firm uses K units of capital and L units of labor, it can produce Q units of output with the production function Q = √L + √K . Each unit of capital costs 2, and each unit of labor costs 1.

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a) The level of K is fixed at 16 units. Suppose Q ≤ 4. What will the firm’s short-run total cost be? (Hint: How much labor will the firm need?) b) The level of K is fixed at 16 units. Suppose Q > 4. Find the equation of the firm’s shortrun total cost curve. a)

Even if the firm hires zero units of labor, with K fixed at 16 it can still produce up to Q =

0  16 = 4 units of output. So for Q  4 , L = 0 is the cost-minimizing choice of labor and the

short-run total cost function is just the cost of capital: C = rK + wL = 2(16) + 1(0) = 32. b) For Q > 4, the firm needs to hire positive amounts of labor, according to Q  L  16 or L = (Q – 4)2. So for Q > 4, the short-run total cost function is C(Q) = rK + wL = 2(16) + 1(Q – 4)2 = 32 + (Q – 4)2. 8.22. Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1, and m = 2, where m is the price per unit of materials. a) Suppose that the firm is required to produce Q units of output. Show how the costminimizing quantity of labor depends on the quantity Q. Show how the cost minimizing quantity of capital depends on the quantity Q. Show how the cost-minimizing quantity of materials depends on the quantity Q. b) Find the equation of the firm’s long-run total cost curve. c) Find the equation of the firm’s long-run average cost curve. d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50 units (i.e., K = 50). Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how the cost-minimizing quantity of materials depends on the quantity Q. e) Find the equation of the short-run total cost curve when capital is fixed at a quantity of 50 units (i.e., K = 50) and graph it along with the long-run total cost curve. f) Find the equation of the associated short-run average cost curve. a)

Equating the bang for the buck between labor and capital implies MPL w  MPK r KM 5  LM 1 K  5L

Equating the bang for the buck between labor and materials implies

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

MPL w  MPM m KM 5  KL 2 5L M 2 Plugging these into the production function yields  5L   2

Q  L(5 L)  25L3 2 2Q L3  25

Q

 2Q   25

1/ 3

L

Substituting into the tangency condition results above implies  2Q K  5   25

1/ 3

and 5  2Q M   2  25

1/ 3

b)  2Q TC  5    25  2Q TC  15    25

1/ 3

 2 Q  5   25 

1/ 3

   5    2     2

2Q 25

1/ 3

1/ 3

c) TC 15  2Q AC     Q Q  25 d)

1/ 3

Beginning with the tangency condition

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

MPL w  MPM m KM 5  KL 2 5L M 2 Setting K  50 and substituting into the production function yields  5 L   2

Q  L(50)  Q  125 L2 L

Q 125

Substituting this result into the tangency condition result above implies Q M  125 2 Q M 20 5

e)

In the short run, TC  5

Q Q  50  2 125 20

TC  2

Q  50 5

Graphically, short-run and long-run total cost curves are shown in the following figure.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

SRTC

120.00 Total Cost

100.00 80.00 60.00 40.00 20.00

LRTC

0.00 0

1000

2000

3000

4000

5000

Q

f)

Short run average cost is given by AC 

TC  Q

2

Q  50 5 Q

8.23. The production function Q = KL + M has marginal products MPK = L, MPL = K, and MPM = 1. The input prices of K, L, and M are 4, 16, and 1, respectively. The firm is operating in the long run. What is the long-run total cost of producing 400 units of output? First, notice that if the firm uses L it must necessarily use K and vice versa; there is no point using a positive amount of one of these inputs and zero of the other. Thus, there are three possible solutions to the long-run cost minimization problem: (i) interior, using positive amounts of K, L, and M; (ii) a corner with K = L = 0 and M > 0; or (iii) a corner with M = 0 but positive amounts of both K and L. Using approach (i), we equate MPK/r = MPM/s and MPL/w = MPM/s to get K = 16 and L = 4. Using the production constraint then yields M = Q – 64. Total cost using this approach will be CKLM(Q) = 16(4) + 4(16) + 1(Q – 64) = Q + 64. Using approach (ii), we have K = L = 0 and the input demand for M comes from the production constraint: M = Q. Total cost will be CM(Q) = Q. Using approach (iii), M = 0 and the tangency condition between K and L yields MPL/w = MPK/r, or K = 4L. Combined with the production function, we get the input demand functions K  2 Q and L  12 Q . Total cost will be CKL(Q) = 16



1 2

 



Q  4 2 Q  1 0  16 Q .

Comparing the three approaches, it is easy to see that CM(Q) < CKLM(Q) for all values of Q. Hence, a cost-minimizing firm will never use K, L, and M simultaneously; it could produce the same output at less cost by just using M. Furthermore, CM(Q) < CKL(Q) only for Q < 256. So for Q = 400, the firm should set M = 0 and, following approach (ii), set K = 40 and L = 10. Total cost will be CKL = 320.

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8.24. The production function Q = KL + M has marginal products MPK = L, MPL = K, and MPM = 1. The input prices of K, L, and M are 4, 16, and 1, respectively. The firm is operating in the short run, with K fixed at 20 units. What is the short-run total cost of producing 400 units of output? With K fixed at 20 units, the production function becomes Q = 20L + M. Thus, L and M are perfect substitutes. Since MPL/w = 1.25 > MPM/s = 1, the marginal product per dollar spent on labor is always higher than that on materials. So the cost-minimizing input combination is M = 0 with L solving 400 = 20L + 0, or L = 20. The short run total cost is C = 16(20) + 4(20) + 1(0) = 400. 8.25. The production function Q = KL + M has marginal products MPK = L, MPL = K, and MPM = 1. The input prices of K, L, and M are 4, 16, and 1, respectively. The firm is operating in the short run, with K fixed at 20 units and M fixed at 40. What is the short-run total cost of producing 400 units of output? With K fixed at 20 and M fixed at 40, the production function becomes Q = 20L + 40. To produce 400 units, the firm needs to hire labor until 400 = 20L + 40, or L = 18. The short-run total cost is C = 16(18) + 4(20) + 1(40) = 408. 8.26. A short-run total cost curve is given by the equation STC(Q) = 1000 + 50Q2. Derive expressions for, and then sketch, the corresponding short-run average cost, average variable cost, and average fixed cost curves. STC (Q)  1000  50Q 2 STC (Q) 1000 SAC (Q)    50Q Q Q AVC (Q)  50Q 1000 AFC (Q)  Q Graphing SAC (Q ) , AVC (Q) , and AFC (Q) yields

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

1200.00

SAC

1000.00 Cost

800.00 600.00 400.00

AVC

AFC

200.00 0.00 0.00

5.00

10.00

15.00

20.00

Q

8.27. A producer of hard disk drives has a short-run total cost curve given by STC(Q) = K + Q2/K . Within the same set of axes, sketch a graph of the short-run average cost curves for three different plant sizes: K = 10, K = 20, and K = 30. Based on this graph, what is the shape of the long-run average cost curve? 3 3 Cost

2 2

SAC10

1

SAC20

SAC30

1 0 0

15

30

45

60

Q

Since each of these short-run average cost curves reaches a minimum at an average cost of 2.0, the long-run average cost curve associated with these short-run curves will be a horizontal line, tangent to the bottom of each of these curves, at a long-run average cost of 2.0. 8.28. Figure 8.18 shows that the short-run marginal cost curve may lie above the long-run marginal cost curve. Yet, in the long run, the quantities of all inputs are variable, whereas in the short run, the quantities of just some of the inputs are variable. Given that, why isn’t short-run marginal cost less than long-run marginal cost for all output levels?

Copyright © 2014 John Wiley & Sons, Inc.

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With some inputs fixed, it is likely that the fixed level is not optimal given the firm’s size. Therefore, it may be more expensive to produce additional units in the short run than in the long run when the firm can employ the optimal, i.e., cost minimizing, quantity of the fixed input. 8.29. The following diagram shows the long-run average and marginal cost curves for a firm. It also shows the short-run marginal cost curve for two levels of fixed capital: K = 150 and K = 300. For each plant size, draw the corresponding short-run average cost curve and explain briefly why that curve should be where you drew it and how it is consistent with the other curves.

The SRAC curves are shown below. Each curve must satisfy two requirements- (i) the SRAC curve must be tangent to the LRAC curve at the output level at which the SRMC curve for that particular plant size intersects the LRMC curve; and (ii) the SRAC curve must reach a minimum at the output level at which it intersects its own SRMC curve. For the plant size of 300 this is easily achieved by just drawing a curve tangent to the LRAC curve at its minimum point, since this is also the point at which the LRMC and the corresponding SRMC curves intersect (at the output level Q = 5). For a plant size of 150, these two points must be kept in mind and the curve must be drawn carefully to comply with both. First, SRAC is tangent to LRAC at Q = 2, where LRMC intersects SRMC for K = 150. Second, SRAC reaches its minimum where it intersects SRMC, near Q = 2.4.

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SRAC, K=150

SRAC, K=300

8.30. Suppose that the total cost of providing satellite television services is as follows:

where Q1 and Q2 are the number of households that subscribe to a sports and movie channel, respectively. Does the provision of satellite television services exhibit economies of scope? Economies of scope exist if TC (Q1 , Q2 )  TC (Q1 , 0)  TC (0, Q2 )  TC (0, 0) In this case TC (Q1 , Q2 )  1000  2Q1  3Q2 TC (Q1 , 0)  1000  2Q1 TC (0, Q2 )  1000  3Q2 TC (0, 0)  0 So, economies of scope exist if

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(1000  2Q1  3Q2 )  (1000  2Q1 )  1000  3Q2 3Q2  1000  3Q2 0  1000 which is certainly true. Thus, in this case the cost of adding a movie channel when the firm is already providing a sports channel is less costly (by $1000) than a new firm supplying a movie channel from scratch. Economies of scope exist for this satellite TV company. 8.28. A railroad provides passenger and freight service. The table shows the long-run total annual costs TC(F, P), where P measures the volume of passenger traffic and F the volume of freight traffic. For example, TC(10,300) = 1,000. Determine whether there are economies of scope for a railroad producing F = 10 and P = 300. Briefly explain.

TC(10, 300) = 1000 while TC(10, 0) + TC(0, 300) = 500 + 400 = 900. Thus TC(10, 300) > TC(10, 0) + TC(0, 300) so economies of scope do not exist at this output level. 8.29. A researcher has claimed to have estimated a long-run total cost function for the production of automobiles. His estimate is that TC(Q, w, r) = 100w−½r½Q3, where w and r are the prices of labor and capital. Is this a valid cost function—that is, is it consistent with long-run cost minimization by the firm? Why or why not? 100Q3 r TC  w This TC function implies that for a fixed Q and r , increasing w would lower long-run total cost. If the firm were minimizing cost in the long run, by using the optimal combination of K and L , it would not be possible to reduce total cost when w is increased. As Figures 8.3 and 8.4 in the text illustrate, when one input price increases, the total long-run cost will increase. Therefore, this long-run total cost function is not consistent with long-run cost minimization by the firm. 8.30. A firm owns two production plants that make widgets. The plants produce identical products and each plant (i) has a production function given by Qi = √KiLi, for i = 1, 2. The plants differ, however, in the amount of capital equipment in place in the short run. In Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

particular, plant 1 has K1 = 25, whereas plant 2 has K2 = 100. Input prices for K and L are w = r = 1. a) Suppose the production manager is told to minimize the short-run total cost of producing Q units of output. While total output Q is exogenous, the manager can choose how much to produce at plant 1(Q1) and at plant 2(Q2), as long as Q1 + Q2 = Q. What percentage of its output should be produced at each plant? b) When output is optimally allocated between the two plants, calculate the firm’s shortrun total, average, and marginal cost curves. What is the marginal cost of the 100th widget? Of the 125th widget? The 200th widget? c) How should the entrepreneur allocate widget production between the two plants in the long run? Find the firm’s long-run total, average, and marginal cost curves. a)

Essentially, the firm’s production function is Q  Q1  Q2  25 L1  100 L2  5 L1  10 L2

That is, the firm has two variable inputs, L1 and L2. The marginal products are MPL1 = 2.5(L1)–0.5 and MPL2 = 5(L2)–0.5. Using the tangency condition, we see that L2 = 4L1. Using the production functions at each plant, we see Q2  10 L2  20 L1  4Q1

Since total output Q = Q1 + Q2, we have Q2 = 4(Q – Q2) or Q2 = .8Q. Similarly, Q1 = .2Q. So the firm should produce 80 percent of output at plant 2 and 20 percent at plant 1. b) Combining the above tangency condition and the production constraint, we find the input demands are L1 = Q2/625 and L2 = 4Q2/625. Including the cost of capital, total cost is then C = 125 + (Q2/125). Average cost is AC = (125/Q) + (Q/125). Marginal cost is MC = 2Q/125. MC(100) = 1.6, MC(125) = 2, and MC(200) = 3.2. c) In the long run, the plants are identical so the entrepreneur should split production equally between the two plants (i.e. Q1 = Q2). Thus the total production function can be written as Q  Q1  Q2  2Q1  2 K1 L1

Again, we can view total output as depending on the choice of only two inputs. (Since the plants are identical the entrepreneur will hire equal amounts of capital at each plant; and similarly for labor.) Minimizing cost implies MRTS L1 , K1  w / r or K1 = L1. Input demands are then L1 = K1 = 0.5Q so that total cost is C = w(L1 + L2) + r(K1 + K2) = 2Q. Long-run average cost is AC = 2 and long-run marginal cost is MC = 2. 8.31 A railroad has two types of services: freight service and passenger service. The standalone cost for freight service is where equals the number of ton-miles of

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freight hauled each day and

is the total cost in thousands of dollars per day. The stand-

alone cost for passenger service is passenger-miles per day and

where

equals the number of

is the total cost in thousands of dollars per day. When a

railroad offers both services jointly its total is provision of passenger and freight service exhibit economies of scope?

Do the

The provision of passenger and freight service do not exhibit economies of scope. Economies of scope would be present if, for all positive values of Q1 and Q2, the total cost of offering both services together is less than the sum of the stand-alone costs of offering each service. However, in this case, the reverse is true: the sum of the stand-alone costs of freight and passenger services is 1,500 + Q1 + 2Q2 which is less than the total cost if both services are offered together, which is 2,000+ Q1 + 2Q2 8.32 Suppose that the experience curve for the production of a certain type of semiconductor has a slope of 80 percent. Suppose over a five-year period, cumulative production experience increases by a factor of 8. Input prices over this period did not change. At the beginning of the period, average variable cost was $10 per unit. This cost is independent of the level of output at any particular point in time. What is your best estimate of average variable cost at the end of this five-year period? The problem tells us that AVC is independent of output at any point in time and that factor prices did not change. It is plausible to conclude that the level of AVC over the five-year period is affected by changes in cumulative experience. If cumulative experience has increased by a factor of 8, this means that cumulative experience has doubled three times (from N to 2N, then from 2N to 4N, and then again from 4N to 8N). With a slope of 80 percent:  The first doubling reduced AVC to 80 percent of what they had been, i.e., from $10 to $8.  The second doubling reduced AVC to 80 percent of the new level, i.e., from $8 to (0.8)*$8 = $6.4  The third doubling reduced AVC to 80 percent of this new level, i.e., from $6.4 to (0.8)*$6.4 = $5.12

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Chapter 8 - 33