9/16/2011
1
1
Cost minimisation
& cost curves
Chapters 20 and 21
Cost Minimization
A firm is a cost-minimiser 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 Cost-Minimization 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 Cost-Minimization 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 Cost-Minimization 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?