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.
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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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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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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
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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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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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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
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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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MC
AC
d)
TFC = 0, AVC = 10
Q
.
AC
MC
e)
TFC = 160, AVC = 10Q.
MC
AC
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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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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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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
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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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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
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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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Chapter 8 - 19
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.
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
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.
Copyright © 2014 John Wiley & Sons, Inc.
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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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Chapter 8 - 22
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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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Chapter 8 - 23
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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
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Chapter 8 - 24
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.
Copyright © 2014 John Wiley & Sons, Inc.
Chapter 8 - 25
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.
Chapter 8 - 27
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?
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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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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
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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Chapter 8 - 30
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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
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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
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
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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