Prof. Jay Bhattacharya
Spring 2001
Econ 11Lecture 12
1
Spring 2001
Econ 11Lecture 12
1
The Theory of the Firm II
•
Last lecture we covered:
– production functions
• Today:
– Cost minimization
– Firm’s supply under cost minimization
– Short vs. long run cost curves
Spring 2001
Econ 11Lecture 12
2
Firm Objectives
•
Profit maximization: firms choose that level
of output and that combination of inputs
which yields the highest level of profits
•
Cost minimization: firms choose that
combination of inputs to produce a certain
level of output at minimum cost.
•
Cost minimization should hold even for
nonprofit firms.
Spring 2001
Econ 11Lecture 12
3
Cost Minimization—
Mathematical Approach
(
)
L
K
F
Q
t
s
rK
wL
L
K
,
.
.
,
min
=
+
(
)
(
)
L
K
F
Q
rK
wL
G
,
−
−
+
=
λ
Spring 2001
Econ 11Lecture 12
4
First order conditions
(
)
0
,
0
0
=
−
=
∂
∂
=
⇒
=
∂
∂
−
=
∂
∂
=
⇒
=
∂
∂
−
=
∂
∂
L
K
F
Q
G
MP
w
L
F
w
L
G
MP
r
K
F
r
K
G
L
K
λ
λ
λ
λ
λ
L
K
MP
MP
w
r
=
⇒
Spring 2001
Econ 11Lecture 12
5
Input demand equations
•
Solving the three first order conditions
simultaneously yields two input demand
equations:
•
Plugging these back into the total input
expenditure identity (Expend. =
wL
+
rK
)
yields the minimum total cost curve:
(
)
Q
r
w
L
L
,
,
*
=
(
)
Q
r
w
K
K
,
,
*
=
(
)
Q
r
w
C
wL
rK
TC
,
,
*
*
=
+
=
Spring 2001
Econ 11Lecture 12
6
Cost Minimization—
Graphical Approach
(
)
L
K
F
Q
,
=
L
K
Decreasing
costs
r
w
slope
−
=
K
L
MP
MP
slope
−
=
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Prof. Jay Bhattacharya
Spring 2001
Econ 11Lecture 12
2
Spring 2001
Econ 11Lecture 12
7
Firm Optimization
•
Tangency Condition
– increase use of an input until its marginal
product is equal to its real price
– if MP
L
/MP
K
> w/r then increasing use of L
(while decreasing use of K) will decrease costs
– if MP
L
/MP
K
< w/r then decreasing use of L
(while increasing use of K) will decrease costs
•
Technology Constraint
Spring 2001
Econ 11Lecture 12
8
Example Problem
•
For CobbDouglass Production Q = K
a
L
b
,
what are the input demand functions for K
and L?
What is the minimum cost function?
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 Fall '07
 Tendall,M
 Economics, Cost curve

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