1 Short-Run Cost Function
1
14.01 Principles of Microeconomics, Fall 2007
Chia-Hui Chen
October 15, 2007
Lecture 13
Cost Functions
Outline
1. Chap 7: Short-Run Cost Function
2. Chap 7: Long-Run Cost Function
Cost Function
Let w be the cost per unit of labor and r be the cost per unit of capital. With
the input Labor (L) and Capital (K), the production cost is
w × L + r × K.
A cost function C(q) is a function of q, which tells us what the minimum cost
is for producing q units of output. We can also split total cost into fixed cost
and variable cost as follows:
C(q) = F C + V C(q).
Fixed cost is independent of quantity, while variable cost is dependent on quan­
tity.
1
Short-Run Cost Function
In the short-run, firms cannot change capital, that is to say,
r × K = const.
Recall the production function given fixed capital level K in the short run (refer
to Lecture 11) (see Figure 1). Suppose w = 1, the variable cost curve can be
derived from Figure 1. Adding r × K to the variable cost, we obtain the total
cost curve (see Figure 2). Average total cost is
AT C =
TC
FC + V C
rK
wL(q; K)
=
=
+
.
q
q
q
q
With the definition of the average product of labor:
q
APL = ,
L
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1 Short-Run Cost Function
2
40
35
30
Q
25
20
15
10
5
0
0
1
2
3
4
5
L
6
7
8
9
10
9
10
Figure 1: Short Run Production Function.
50
45
40
TC
35
C
30
25
20
15
10
VC
5
0
0
1
2
3
4
5
q
6
7
8
Figure 2: Short Run Cost Function.
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OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
YYYY].
1 Short-Run Cost Function
3
we can rewrite AT C as
AT C =
rK
w
+
,
q
APL
in which the average variable cost is
VC
wL(q; K)
w
=
=
.
q
q
APL
Likewise, we rewrite the marginal cost:
MC =
dT C
dV C
dL(q)
w
w
=
=w
= ∂q =
.
dq
dq
dq
M PL
∂L
In Lecture 11, we discussed the relation between average product of labor and
marginal product of labor (see Figure 3). We draw the curves for AV C and
10
9
8
7
6
5
AP
4
3
2
1
0
MP
0
1
2
3
4
5
L
6
7
8
9
10
Figure 3: Average Product of Labor and Marginal Product of Labor.
M C in the same way (see Figure 4). The relation between M C and AV C is:
• If
M C < AV C,
AV C decreases;
• if
M C > AV C,
AV C increases;
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OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
YYYY].
2 Long-Run Cost Function
4
30
25
C
20
MC
ATC
15
10
5
0
AVC
0
1
2
3
4
5
L
6
7
8
9
10
Figure 4: Average Cost, Average Variable Cost, and Marginal Cost.
• if
M C = AV C,
AV C is minimized.
Now consider the total cost. Note that the difference between AT C and AV C
decreases with q as the average fixed cost term dies out (see Figure 4). The
relation between M C and AT C is:
• If
M C < AT C,
AT C decreases;
• if
M C > AT C,
AT C increases;
• if
M C = AT C,
AT C is minimized.
Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT
OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
YYYY].
2 Long-Run Cost Function
5
5
4.5
4
3.5
k
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
L
3
3.5
4
4.5
5
Figure 5: Isoquant Curve.
2
Long-Run Cost Function
In the long-run, both K and L are variable. The isoquant curve describes the
same output level with different combination of K and L (see Figure 5). The
slope of an isoquant curve is
−M RT S = −
M PL
.
M PK
Similarly, the isocost curve is constructed by different (K, L) with the same cost
(see Figure 6). The isocost curve equation is:
rK + wL = const,
therefore, it is linear, with a slope − wr .
Now we want to minimize the cost rK +wL subject to an output level Q(K, L) =
q. This minimum cost can be obtained when the isocost curve is tangent to the
isoquant curve (see Figure 7). Thus the slopes of these two curves are equal:
M RT S =
M PL
w
= .
M PK
r
Now consider an increase in wage (w). The slope of the isocost curve increases
(see Figure 8), and the firm use more capital and less labor. The firm’s choice
of input moves from A to B in the figure.
The expansion path shows the minimum cost combinations of labor and
capital at each level of output (see Figure 9).
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OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
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2 Long-Run Cost Function
6
7
6
5
K
4
3
2
1
0
0
1
2
3
4
5
6
7
L
Figure 6: Isocost Curve.
10
9
8
7
K
6
5
4
3
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 7: Minimize the Cost Subject to a Output Level.
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OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
YYYY].
2 Long-Run Cost Function
7
10
9
8
7
K
6
5
4
B
3
A
2
1
0
0
1
2
3
4
5
L
6
7
8
9
10
Figure 8: The Change of Cost Minimized Situation.
12
11
10
9
K
8
Expansion Path
7
6
5
4
3
2
2
3
4
5
6
7
L
8
9
10
11
12
Figure 9: Expansion Path.
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OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
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2 Long-Run Cost Function
8
Example (Calculating the Cost.). Given the production function
2
2
q = L3 K 3 .
In the short run,
3
CSR (q; K) = rK + w
q2
,
K
where K is fixed.
In the long run, according to the equation
M PL
w
= ,
M PK
r
we have
K
w
= .
L
r
Then the expansion path is
w
L.
r
Substituting this result into the production function, we obtain
K=
3
r 1
L = q 4 ( )2 ,
w
3 w 1
K = q 4 ( )2 .
r
Hence, the long-run cost function is:
3
CLR (q) = wL + rK = 2q 4
√
wr.
Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT
OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
YYYY].