The theory of the firm:
Cost minimization
Lectures in Microeconomic Theory
Fall 2010, Part 4
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G.B. Asheim, ECON4230-35, #4
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Cost minimization — Motivation

An alternative way to consider the behavior of a
profit-maximizing firm that takes output and
factor prices as given.

A way to analyze the behavior of a profitmaximizing firm that takes factor prices as
given, but can influence output prices.
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G.B. Asheim, ECON4230-35, #4
Cost minimization
Assume price-taking behavior in factor markets.
Cost function : c ( w ,y )  min wx
x
such that f ( x)  y
Short - run cost function : c ( w ,y , x f )  min wx
x
such that f ( x)  y and x  ( x v , x f )
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1
Conditions for cost minimization
Cost function : c ( w ,y )  min wx such that f ( x)  y
x
f ( x  )
 0 for i  1,  , n
xi x
2
f (x )  y
First order conditions : wi  
wi

wj
f (x  )
xi
f ( x  )
x j
Problems with
first-order
approach:



07.07.2010
Differentiability?
Interior solution?
Global minimum?
x1
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G.B. Asheim, ECON4230-35, #4
Conditional factor demand function
x  ( w , y )  arg min wx s.t. f ( x)  y x2
x

Properties: (Comp. statics)
xi (tw , y )  xi ( w , y )
xi
 0 for i  1,  , n
wi

x j
xi

for i  1,  , n
w j wi
x1
Cost function for the Cobb-Douglas tech.
Production function : f ( x1 , x2 )  Ax1a x2b
x1 , x2
w1  a xy1
w2   b xy2
Ax1a x2b  y
w1 a x2

w2 b x1
b
x1* ( w1 , w2 , y )  A
 a 1b
x2* ( w1 , w2 , y )  A
 a1 b
c( w1 , w2 , y )  A
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a b 1
Cost fn. : c( w , y )  min w1 x1  w2 x2 s.t. Ax1a x2b  y
 a1 b
 aw2  ab a1b

 y
 bw1  a
 bw1  ab a1b

 y
 aw2 
 
a
b
b
a b


a b1  b1  abb
y A x1
b
  ab ab w1ab w2ab y ab
a
a
b
1
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G.B. Asheim, ECON4230-35, #4
What happens when a factor price increases?
w x  w x
x2
w x  w x
w x  x  0
( x1, x2 )
 w x  x  0
( x1, x2 )
( x1, x2)
If wj  wj for all j  i , then
wi  wixi  xi  0
wi  wi implies xi  xi
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x1
Weak axiom of
cost minimization
G.B. Asheim, ECON4230-35, #4
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2
Cost function
c ( w , y )  min wx s.t. f ( x)  y
x
Properties: (No assumptions about the technology needed).
Cost function: Proof of property (1)
Assume wi  wi for all inputs .
Let x be cost minimizing at ( w , y ) ,
so that c( w , y )  wx and f ( x)  y
Let x be cost minimizing at ( w , y ) ,
so that c( w , y )  wx and f ( x)  y
By cost minimization : c(w , y )  wx  wx
(1) Non - decreasing in w.
Cost function for the Cobb-Douglas tech.
Production function : f ( x1 , x2 )  Ax1a x2b
Since wi  wi for all inputs , w x  wx  c ( w, y )
Cost function: Proof of property (2)
Let x be cost minimizing at ( w, y ),
so that c( w , y )  wx and f ( x)  y
Adding these two inequalities together
w2  b xy2
c ( wi , w i , y )
(3) Concave in w
c (w, y )  tc (w , y )  (1  t )c(w , y )
Ax1a x2b  y
w1 a x2

w2 b x1
b
c(tw , y )  tc ( w , y ) for all t  0.
Cost function: Proof of property (3)
Let x be cost minimizing at (w  , y ) where
w   tw  (1  t )w .
By cost minimization :
twx  twx  tc(w , y )
(1  t )w x  (1  t )wx  (1  t )c (w , y )
w1  a xy1
x1* ( w1 , w2 , y )  A
 a 1b
x ( w1 , w2 , y )  A
 a1b
*
2
Hence, x is cost - minimizing at ( tw, y) .
Let x be cost minimizing at ( w, y ).
x1 , x2
(2) Homogene ous of degree 1 in w :
However, if c( w, y )  wx  w x, then twx  twx
Let x be cost minimizing at (w , y ).
ab1
Cost fn. : c ( w, y )  min w1 x1  w2 x2 s.t. Ax1a x2b  y
Hence : c (w, y)  c(w, y )
c( w1 , w2 , y )  A
 a1 b
 aw2  a b a1 b

 y
 bw1  a
 bw1  ab a1 b

 y
 aw2 
 
a
b
b
a b

  ba  ab w1ab w2ab y a b
a
a
c( wi , w i , y )
(4) Continuous in w
wi
wi
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G.B. Asheim, ECON4230-35, #4
Shephard’s Lemma
c ( wi , w i , y )
Proof of Shephard’s Lemma
Slope : xi ( wi, w i , y )
c( wi , w i , y )
Suppose x  is cost minimizing factor vec tor at (w  , y ).
Define the function
g (w )  c (w , y )  wx
Then g (w  )  0 and
g ( w  ) c( w  , y ) 

 xi  0 for i  1, , n
wi
wi
c( wi, w i , y )
c (w , y )  wx * (w , y )
wi
wi
Assume differentiability.
Invoke concavity.
Cost function for the Cobb-Douglas tech.
Production function : f ( x1 , x2 )  Ax1a x2b
x1 , x 2
w1  a xy1
w2  b xy2
Ax1a x2b  y
w1 a x2

w2 b x1
b
 aw2  a b a 1b
 bw  y
 1a
a b
1
 1  bw 
x*2 ( w1 , w2 , y )  A a b  1  y ab
 aw2 
c ( w1 , w2 , y )  A
c ( w , y )
x (w, y ) 
wi

i
a  b 1
Cost fn. : c ( w, y)  min w1 x1  w2 x2 s.t. Ax1a x2b  y
x1* ( w1 , w2 , y )  A
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b
Sl
Slope
: xi ( wi, w i , y )
c( wi, w i , y )
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a b1  b1  abb
y A x1
b

 a1 b
 a1 b
 
a
b
b
a b


a 1b  1b  a b b
y A x1
b
  ab a b w1a b w2a b y ab
a
a
b
1
xi ( w , y )  2 c ( w , y )

0
wi
wi2
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G.B. Asheim, ECON4230-35, #4
More on Shephard’s Lemma
c ( wi , w i , y )
Assume differentiability.
Slope : xi ( wi, w i , y )
c ( wi , w i , y )
c ( wi, w i , y )
wi
c ( w , y )
xi ( w , y ) 
wi
wi
Invoke Young’s theorem.

xi ( w , y )  2 c( w , y )  2 c( w , y ) x j ( w , y )



w j
w j wi
wi w j
wi
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3
An identity
x( p, w )  x  ( w , y ( p, w ))
xi ( p, w ) xi ( w , y ( p, w )) y ( p, w )

y
p
p
xi ( p, w )
xi ( w , y ( p, w ))
p

Hence :
y ( p, w )
y
p
xi ( p, w ) xi ( w , y ( p, w )) xi ( w , y ( p, w )) y ( p, w )


w j
w j
y
w j

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xi (w , y ( p, w ))

w j
xi ( p , w ) y ( p , w )
p
w j
y ( p , w )
p
Puu’s equation
G.B. Asheim, ECON4230-35, #4
10
Applying Puu’s equation

Profit maximizing response greater than cost
minimizing response when a factor price changes?
xi ( p, w ) xi ( w , y ( p, w ))


w j
w j
 2 ( p , w )  2 ( p , w )
pwi
w j p
by Hotelling’s
Lemma.
The response if the factor’s own price changes (i  j):
 2 ( p , w )
p 2
xi ( p, w ) xi ( w , y ( p, w ))  2xi(p(,w
, 2y ( p, w )) y ( p, w )
w ) 
 


w
w
xi (pw, w
)

(
,
(
,
))
x  ( w , y(wp,j w ))
x
y
p


p
w
w j
j
 i
   2 ( pi,w )y i
wi
wi
wi
xi ( p 2, w ) y ( p , w )
p
xi 2( w , y ( p, w
))
p
w j
2
2


y ( p, w)( p , w )
since  p(pw,iww) j  w(ip,pw ) and
 0.
2
p p
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Long-run and short-run cost functions
In the long run : c ( w , y )  wx  ( w , y )
Long - run total cost  LTC  c( w , y )
Long - run average cost  LAC  c (w , y ) / y
Long - run marginal cost  LMC  c(w, y ) / y
x  ( x v , x f ) and w  ( w v , w f )
In the short run : c ( w , y , x f )  w v x v ( w , y , x f )  w f x f
Connection between long - run and short - run costs :
c ( w , y )  w v x v ( w , y )  w f x f ( w , y )  c ( w , y , x f ( w , y ))
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4
More on short-run cost functions
Short - run total cost  STC  c( w , y , x f )
Short - run average cost  SAC  c( w , y , x f ) / y
Short - run variable cost  SVC  w v x v ( w , y , x f )
Short - run average var.
var cost  SAVC  w v x v ( w , y , x f ) / y
Short-run Cobb-Douglas cost function
Production function : f ( x1 , k )  x1a k 1 a
Short - run cost fn. : c(w , y, k )  min w1 x1  w2 k s.t. x1a k 1 a  y
x1 , x2
x1* ( w1 , w2 , y, k )  yk a 1 a
1
c( w1 , w2 , y, k )  w1 yk a 1 a  w2 k
1
Short - run average cost  w1


y
k
Short - run average variable cost  w1
Short - run average fixed cost 
Short - run marginal cost 
y
k
1 a
a
1 a
a

w2k
y
Fixed cost  FC  w f x f
w2 k
y
w1
a

y
k
1 a
a
Short - run fixed average cost  SAFC  w f x f / y
Short - run marginal cost  SMC  c ( w , y, x f ) / y
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G.B. Asheim, ECON4230-35, #4
Relationship between AC and MC
dAC d  c ( y )  1 
c( y )  1

   c( y ) 
  MC  AC 

dy
dy  y  y 
y  y
p
MC
AC is decreasing
if and only if
MC < AC.
AC
y
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G.B. Asheim, ECON4230-35, #4
Geometry of costs I
y TC
p
TC
MC
AC
Y
y
x
y TC
y
p
TC
AC
MC
Y
x
y
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y
15
5
Geometry of costs II
y TC
p
TC
MC
Y
AC
Long-run
x
y
y TC
y
p
TC
MC
AC
Y
Short-run
x
y
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y
16
G.B. Asheim, ECON4230-35, #4
Geometry of costs III
y TC
TC
p
MC
AC
Y
x
y
y
p
y TC
TC
AC
MC
Y
x
y
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y
17
Cost function: Proof of property (1)
Assume wi  wi for all inputs .
Let x be cost minimizing at ( w , y ),
so that c ( w , y )  wx and f ( x)  y
Let x be cost minimizing at ( w , y ),
so that c( w , y )  w x and f ( x)  y
By cost minimizati on : c( w , y )  w x  w x
Since wi  wi for all inputs , w x  wx  c ( w , y )
Hence : c ( w , y )  c ( w , y )
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6
Cost function: Proof of property (2)
Let x be cost minimizing at ( w , y ),
so that c( w , y )  wx and f ( x)  y
However, if c (w , y )  wx  w x, then twx  tw x
Hence, x is cost - minimizing at (tw , y ).
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Cost function: Proof of property (3)
Let x be cost minimizing at ( w , y ).
Let x be cost minimizing at (w, y ).
Let x be cost minimizing at ( w , y ) where
w   tw  (1  t ) w .
By cost minimizati on :
tw x  twx  tc (w , y )
(1  t ) w x  (1  t )w x  (1  t )c (w , y )
Adding these two inequalit ies together
c (w , y )  tc (w , y )  (1  t )c (w , y )
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Proof of Shephard’s Lemma
Suppose x  is cost minimizing factor vec tor at (w  , y ).
Define the function
g (w )  c (w , y )  wx 
Then g (w  )  0 and
g (w  ) c(w  , y )

 xi  0 for i  1,  , n
wi
wi
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7