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9/18/2010
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Cost minimisation
Chapters 20 and 21
Cost Minimization
A firm is a costminimizer if it produces any given
output level y
0 at smallest possible total cost.
c(y) denotes the firm’s smallest possible total
cost for producing y units of output.
c(y) is the firm’s total cost function.
when the firm faces given input prices w =
(w
1
,w
2
,…,w
n
) the total cost function will be written
as
c(w
1
,…,w
n
,y).
The CostMinimization Problem
Consider a firm using two inputs to make one
output.
The production function is
y = f(x
1
,x
2
).
Take the output level y
0 as given.
Given the input prices w
1
and w
2
, the cost of an
input bundle (x
1
,x
2
) is w
1
x
1
+ w
2
x
2
.
The CostMinimization Problem
For given w
1
, w
2
and y, the firm’s cost
minimization problem is to solve
subject to:
min
,
x
x
w x
w x
1
2
0
1 1
2 2
f x
x
y
(
,
)
.
1
2
The CostMinimization Problem
The levels x
1
*(w
1
,w
2
,y) and x
1
*(w
1
,w
2
,y) in the least
costly input bundle are the firm’s conditional
demands for inputs 1 and 2.
The (smallest possible) total cost for producing y
output units is therefore
c w
w
y
w x
w
w
y
w x
w
w
y
(
,
,
)
(
,
,
)
(
,
,
).
*
*
1
2
1 1
1
2
2 2
1
2
Conditional Input Demands
Given w
1
, w
2
and y, how is the least costly input
bundle located?
And how is the total cost function computed?
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Isocost Lines
A curve that contains all of the input bundles that
cost the same amount is an isocost curve.
E.g., given w
1
and w
2
, the $100 isocost line has
the equation
w x
w x
1 1
2 2
100
.
Isocost Lines
Generally, given w
1
and w
2
, the equation of the
$c isocost line is
that is:
Slope is  w
1
/w
2
.
x
w
w
x
c
w
2
1
2
1
2
.
w x
w x
c
1 1
2 2
Isocost Lines
c’
w
1
x
1
+w
2
x
2
c”
w
1
x
1
+w
2
x
2
c’ < c”
x
1
x
2
Slopes = w
1
/w
2
.
The y’Output Unit Isoquant
x
1
x
2
All input bundles yielding y’ units
of output.
Which is the cheapest?
f(x
1
,x
2
)
y’
The CostMinimization Problem
x
1
x
2
f(x
1
,x
2
)
y’
x
1
*
x
2
*
At an interior costmin input bundle:
(a)
and
(b) slope of isocost = slope of isoquant
f x
x
y
(
,
)
*
*
1
2
w
w
TRS
MP
MP
at
x
x
1
2
1
2
1
2
(
,
).
*
*
A CobbDouglas Example
A firm’s CobbDouglas production function is
Input prices are w
1
and w
2
.
What are the firm’s conditional input demand
functions?
y
f x
x
x
x
(
,
)
.
/
/
1
2
1
1 3
2
2 3
9/18/2010
3
A CobbDouglas Example
At the input bundle (x
1
*,x
2
*) which minimizes
the cost of producing y output units:
(a)
(b)
y
x
x
(
)
(
)
*
/
*
/
1
1 3
2
2 3
and
w
w
y
x
y
x
x
x
x
x
x
x
1
2
1
2
1
2 3
2
2 3
1
1 3
2
1 3
2
1
1 3
2
3
2
/
/
(
/
)(
)
(
)
(
/
)(
)
(
)
.
*
/
*
/
*
/
*
/
*
*
A CobbDouglas Example
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This note was uploaded on 06/12/2011 for the course ECONOMICS 3291 taught by Professor Professorsnamespublishedtheyarethesoleowners during the Three '11 term at University of New South Wales.
 Three '11
 professorsnamespublishedtheyarethesoleowners
 Economics

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