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Economics

Economics 2150A/B

Prof

Fall

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Chapter 5
Cost Curves
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.
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 TC TVC AFC AC MC AVC TFC
1 100 40 60 100 40 40 60
2 110 50 30 55 10 25 60
3 120 60 20 40 10 20 60
4 180 120 15 45 60 30 60
5 250 190 12 50 70 38 60
6 330 270 10 55 80 45 60
2. A firm produces a product with labor and capital, and its production function is
described by Q = LK. 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
MP w
L =
MP K r
K 2
=
L 1
K = 2L Substituting into the production function yields
Q = LK
Q = L(2L)
L = Q
2
Plugging this into the expressionKfoabove gives
Q
K = 2 2
Finally, substituting these into the total cost equation results in
Q Q
TC = 2 +2
2 2
TC = 4 Q
2
TC = 8Q
and average cost is given by
TC 8Q
AC = =
Q Q
8
AC =
Q
3 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 (proof required):
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 for all positive quantities d. The long-run marginal cost curve lies below for some positive quantities
and above the 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.
4 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 (proof required):
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 for all positive quantities
d. The long-run marginal cost curve lies below for some positive quantities
and above the for some positive quantities
ANSWER: a
The equation of the AC curve is AC(Q) = TC(Q)/Q = 1000Q /Q = 1000Q -(1/. 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.
2 3
5. A firm’s long-run total cost curve is TC(Q) = 40Q − 10Q + Q , and its long-run marginal
cost curve is MC(Q) = 40 − 20Q + 3Q . 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 t2e AC curve will be2the point at which it intersects the marginal cost curve,
i.e.40 −10Q + Q = 40 − 20Q + 3Q . 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). 6. A firm produces a product with labor and capital as inputs. The production function is
described by Q = LK. 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
Q
25
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. TC STC(Q), w =1, r = 4
TC(Q), w =1, r = 4
20
STC(Q)
TC(Q)
5
Q
25 100
d) 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
Q
25/4 25
7. A firm produces a product with labor and capital. Its production function is described
by Q = L + K. 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.
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 long-
run 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
TC STC(Q) &
TC(Q)
STC(Q)
5
TC(Q)
Q
5
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
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) &
STC(Q) w = 1, r = 2 TC(Q)
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. STC(Q), w = 2, r =1
TC
STC(Q) &
TC(Q)
STC(Q) = STC(Q), w = 2, r =1
5
TC(Q) = TC(Q), w = 2, r = 1
10
5 Q
8. Consider a production function of two inputs, labor and capital, given by Q = (√L +
√K) . 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 cost-
minimizing 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
MP w
L =
MP K r
1/2 1/2 −1/2
L + K L 2
1/2 1/2 −1/2
L + K K 1
K = 4
L
K = 4L
Plugging this into the total cost function yields 2
Q = L +(4L) 1/2
Q = 3L 1/2
Q = 9L
Q
L = 9
Inserting this back into the solutionKfoabove gives
4Q
K =
9
b)
TC = 2 Q + 4Q
9 9
TC = 2Q
3
c)
TC 2Q
AC = = Q
Q 3
2
AC =
3
d) When Q ≤ 9 the firm needs no labor. If> 9 the firm must hire labor. SettiK = 9
and plugging in for capital in the production function yields
Q = L +92 1/2
Q 1/2= L +3
L1/2= Q1/2−3
2
L = Q 1/2−3
Thus,
2 2
L =

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