EconS 301
Review Session #6 – Chapter 8: Cost Curves
8.12. Consider a production function with two inputs, labor and capital, given by Q = (√
. 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 costminimizing quantity of capital depends on quantity Q.
Starting with the tangency condition we have
MPL w
=
MPK r
⎡⎣ L1/ 2 + K 1/ 2 ⎤⎦ L−1/ 2 2
=
⎡⎣ L1/ 2 + K 1/ 2 ⎤⎦ K −1/ 2 1
K
=4
L
K = 4L
Plugging this into the total cost function yields
Q = ⎡⎣ L1/ 2 + (4 L)1/ 2 ⎤⎦
Q = ⎡⎣3L1/ 2 ⎤⎦
2
2
Q = 9L
Q
L=
9
Inserting this back into the solution for K above gives
4Q
K=
9
b) Find the equation of the firm’s long-run total cost curve.
⎛ Q ⎞ 4Q
TC = 2 ⎜ ⎟ +
⎝9⎠ 9
2Q
TC =
3
1
c) Find the equation of the firm’s long-run average cost curve.
TC ⎛ 2Q ⎞
AC =
=⎜
⎟ Q
Q ⎝ 3 ⎠
2
AC =
3
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. = 9).
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
Q = ⎡⎣ L1/ 2 + 91/ 2 ⎤⎦
2
Q1/ 2 = L1/ 2 + 3
L1/ 2 = Q1/ 2 − 3
L = ⎡⎣Q1/ 2 − 3⎤⎦
2
Thus,
2
1
⎪⎧ ⎣⎡Q 2 − 3⎦⎤
L=⎨
⎪⎩0
if Q > 9
if Q ≤ 9
e) Find the short-run total cost curve, and graph it along with the long-run total cost
curve.
2
1/ 2
when Q > 9
⎪⎧2 ( Q − 3) + 9
TC = ⎨
when Q ≤ 9
⎪⎩9
Graphically, short-run and long-run total cost 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
Q
2
20
25
30
f) Find the associated short-run average cost curve.
⎧ 2 ( Q1/ 2 − 3)2 + 9
⎪
if Q > 9
TC ⎪
Q
=⎨
AC =
Q ⎪9
if Q ≤ 9
⎪Q
⎩
8.14. 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.
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
w r
K = Q/7. So the firm’s total cost is wL + rK = wQ / 4 + rQ / 7 = [ + ]Q .
4 7
w r
+ . Note that this average cost
4 7
curve is independent of Q and is simply a straight line.
The average cost curve is LRAC = TC / Q =
8.19. 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 followsL
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.
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
3
MPL w
=
MPM m
KM 5
=
KL 2
5L
M=
2
Plugging these into the production function yields
⎛ 5L ⎞
Q = L(5L) ⎜ ⎟
⎝ 2 ⎠
25L3
Q=
2
2Q
L3 =
25
1/ 3
⎛ 2Q ⎞
L=⎜
⎟
⎝ 25 ⎠
Substituting into the tangency condition results above implies
1/ 3
⎛ 2Q ⎞
K = 5⎜
⎟
⎝ 25 ⎠
and
1/ 3
5 ⎛ 2Q ⎞
M= ⎜
⎟
2 ⎝ 25 ⎠
b) Find the equation of the firm’s long-run total cost curve.
1/ 3
1/ 3
1/ 3
⎛ 2Q ⎞
⎛ 2Q ⎞
⎛ 5 ⎞⎛ 2Q ⎞
TC = 5 ⎜
⎟ + 5⎜
⎟ + 2 ⎜ ⎟⎜
⎟
⎝ 25 ⎠
⎝ 25 ⎠
⎝ 2 ⎠⎝ 25 ⎠
1/ 3
⎛ 2Q ⎞
TC = 15 ⎜
⎟
⎝ 25 ⎠
c) Find the equation of the firm’s long-run average cost curve.
1/ 3
TC 15 ⎛ 2Q ⎞
AC =
= ⎜
⎟
Q Q ⎝ 25 ⎠
d) Suppose that the firm is required to produce Q units of output, but that its capital is
50). Show how the cost-minimizing
fixed at a quantity of 50 units (i.e.
quantity of labor depends on the quantity Q. Show how the cost-minimizing
quantity of materials depends on the quantity Q.
Beginning with the tangency condition
4
MPL w
=
MPM m
KM 5
=
KL 2
5L
M=
2
Setting K = 50 and substituting into the production function yields
⎛ 5L ⎞
Q = L(50) ⎜ ⎟
⎝ 2 ⎠
Q = 125 L2
L=
Q
125
Substituting this result into the tangency condition result above implies
Q
M = 125
2
Q
M=
20
e) Find the equation of the short-run total cost curve when capital is fixed at a quantity of
50) and graph it along with the long-run total cost curve.
50 units (i.e.
5
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.
5
SRTC
120.00
Total Cost
100.00
80.00
60.00
40.00
20.00
LRTC
0.00
0
1000
2000
3000
Q
f) Find the equation of the associated short-run average cost curve.
Short run average cost is given by
TC
AC =
=
Q
2
Q
+ 50
5
Q
6
4000
5000