The theory of the firm:
Profit maximization
Lectures in Microeconomic Theory
Fall 2010, Part 3
07.07.2010
G.B. Asheim, ECON4230-35, #3
Profit maximization
1
Assume price-taking beh. in
output and factor markets.
Profit function :  (p )  max py such that y  Y
y
where p  ( p1 ,  , pn ) is a vector
of prices for inputs and outputs
Short run  - function :  (p, y n )  max py such
y
that y  Y and y n  yn
With one product:
Profit function :  ( p, w )  max pf ( x)  wx
x
where p is output price and
w  ( w1 ,  , wn ) are factor prices
07.07.2010
G.B. Asheim, ECON4230-35, #3
2
Conditions for profit maximization
Profit function :  ( p, w )  max pf ( x)  wx
x
First order conditions : p
f ( x  )
 wi for i  1,  , n
xi
 f 2 ( x  ) 
h  0 for all vector s h
Sec. order cond.s : h
 x x 
 i j 
Problems with
first-order
approach:
07.07.2010



Differentiability?
Interior solution?
Does a maximum exists?
G.B. Asheim, ECON4230-35, #3
3
1

No maximum;
profit can be
increased by
increasing scale y

No interior solution
y
Y
x
y
Y
x
Y

Kink at maximum x
07.07.2010
4
G.B. Asheim, ECON4230-35, #3
Factor demand function
y
x( p, w )  arg max pf ( x)  wx
x

Properties: (Comp. statics)
xi (tp, tw )  xi ( p, w )
xi
 0 for i  1,  , n
wi
xi x j
for i  1,  , n

w j wi
x
A Cobb-Douglas technology
Production function : f ( x)  x a
Output supply function
y ( p, w )  f ( x( p, w ))
07.07.2010
a0
First - order condition : pax a 1  w
Sec. - order cond. : pa ( a  1) x a  2  0
a
1
1
 w  a 1
Factor demand function : x ( p, w)   
a
 ap 
 w  a 1
Output supply function : y ( p, w)  f  x ( p , w)    
 ap 
1
 - function :
 1  a  w  a 1
py ( p, w)  wx ( p , w)  w
 
 a  ap 
5
G.B. Asheim, ECON4230-35, #3
What happens when the output price increases?
py  w x  py   w x
y
py  w x  py   w x
( y, x)
p y   y   w x  x
 p y   y    w x  x
( y, x)
( y , x)
If w   w , then
 p  p y  y  0
x
p  p implies y   y 
07.07.2010
G.B. Asheim, ECON4230-35, #3
Weak axiom of
profit maximization
6
2
Profit function
 ( p, w )  max pf ( x)  wx
x
Properties: (No assumptions about the technology needed).
(1) Non - decreasing in p , non - increasing in w.
Profit function: Proof of property (1)
Assume p  p and wi  wi for all inputs .
Let ( y , x) be  - maximizing at ( p , w ) ,
so that  ( p , w )  py  wx
Let ( y,x ) be  - maximizing at ( p, w),
so that  ( p , w )  p y   wx 
By  - maximizati on :  ( p, w )  py   w x  py  w x
Since p   p and wi  wi for all inputs , py  wx
Hence :  ( p , w )   ( p , w)
 py  wx   ( p , w)
(2) Homogene ous of degree 1 in ( p, w ) :
A Cobb-Douglas technology
Production function : f ( x)  x a
a0
First - order condition : pax a 1  w
a
1
Sec. - order cond. : pa (a  1) x a 2  0
1
 w  a 1
Factor demand function : x ( p, w)   
a
 ap 
 w  a 1
Output supply function : y( p, w)  f x( p, w)    
 ap 
1
 - function :
 1  a  w  a 1
py ( p , w)  wx ( p, w)  w
 
 a  ap 
 (tp, tw )  t ( p, w ) for all t  0.
 ( p, w )
 ( p, w)
Profit function: Proof of property (2)
Let ( y , x) be  - maximizing at ( p, w ),
so that  ( p, w )  py  wx
However, if  ( p, w )  py  wx  py  w x,
then tpy  twx  tpy  t wx 
Hence, ( y,  x ) is  - maximizing at (tp, tw ).
(3) Convex in ( p, w )
Profit function: Proof of property (3)
Let ( y,  x ) be  - maximizing at ( p , w ).
Let ( y ,x ) be  - maximizing at ( p , w ).
Let ( y, x) be  - maximizing at ( p, w)  t ( p, w )
 (1  t)( p, w).
By  - maximizati on :
t  py   w x  t  py  wx   t ( p, w )
(1  t ) p y   w x   (1  t )  py  w x   (1  t ) ( p , w )
 ( p, w)
Adding these two inequalities together
 ( p, w)  t ( p, w)  (1  t) ( p , w )
Slope : y ( p, w)
(4) Continuous in ( p, w )
07.07.2010
7
G.B. Asheim, ECON4230-35, #3
Hotelling’s Lemma  ( p, w)
 ( p, w )  py ( p, w )
Proof of Hotelling’s Lemma
Suppose y  is a  - maximizing output at ( p  , w  ).
Define the function

g ( p)   ( p, w  )  py   w  x

p
p
 wx ( p, w )
 ( p, w)
 ( p, w)
Slope : y ( p, w)
p
g ( p)
p
p
Then g ( p  ) 
p
 ( p  , w  )
 y  0
p
Assume differentiability.
y ( p, w ) 
 ( p, w )
p
 xi ( p, w ) 
 ( p, w )
wi
07.07.2010
Invoke convexity.
Production function : f ( x)  xa
a0
First - order condition : pax a 1  w
Sec. - order cond. : pa( a  1) x
a 2
a
1
0
1
 w  a 1
Factor demand function : x( p , w)   
a
 ap 
 w  a 1
Output supply function : y( p , w)  f  x( p , w)    
 ap 
1
 - function :
 1  a  w  a 1
py ( p , w)  wx( p , w)  w
 
 a  ap 
y ( p, w )  2 ( p, w )

0
p
p 2

xi ( p, w )  2 ( p, w )

0
wi
wi2
8
G.B. Asheim, ECON4230-35, #3
More on Hotelling’s Lemma
y ( p, w ) 
A Cobb-Douglas technology
 ( p, w )
p
 ( p, w)
 ( p, w)
 ( p, w)
Slope : y ( p, w)
p
 xi ( p, w ) 
 ( p, w )
wi
p
Invoke Young’s theorem.
x ( p, w )
y ( p, w )   ( p, w )  2 ( p, w )


 i
wi
wi p
pwi
p
2

x j ( p, w )
xi ( p, w )  2 ( p, w )  2 ( p, w )



w j
w j wi
wi w j
wi
07.07.2010
G.B. Asheim, ECON4230-35, #3
9
3
The LeChatelier principle

Long-run response greater than short-run response?
Suppose one output and that all input prices are fixed.
Short - run profit :  S ( p, z )
where z is some input fixed in the short - run.
L
Long
- run profit
fi :  L ( p )   S ( p, z ( p ))
Let p  be some output price, and
let z   z ( p  ) be the optimal long - run demand at p  .
h( p )   L ( p )   S ( p, z  )   S ( p, z ( p ))   S ( p, z  )  0
07.07.2010
10
G.B. Asheim, ECON4230-35, #3
h( p )   L ( p )   S ( p , z  )
h( p  )  0
h( p  )  0
yL ( p ) 
h( p  )  0
p
p
By Hotelling’s Lemma:
d L ( p  )  S ( p  , z  )

 yS ( p  , z  )
dp
p
By Hotelling’s Lemma:
dy L ( p ) d 2 L ( p  )  2 S ( p  , z  ) y S ( p  , z  )



dp
dp 2
p 2
p

07.07.2010
G.B. Asheim, ECON4230-35, #3
11
Profit function: Proof of property (1)
Assume p  p and wi  wi for all inputs .
Let ( y , x) be  - maximizing at ( p, w ),
so that  ( p, w )  py  wx
Let ( y , x) be  - maximizing at ( p, w ),
so that  ( p, w )  py  w x
By  - maximizati on :  ( p, w )  py  w x  py  w x
Since p  p and wi  wi for all inputs , py  w x
Hence :  ( p, w )   ( p, w )
07.07.2010
 py  wx   ( p, w )
G.B. Asheim, ECON4230-35, #3
12
4
Profit function: Proof of property (2)
Let ( y , x) be  - maximizing at ( p, w ),
so that  ( p, w )  py  wx
However, if  ( p, w )  py  wx  py   w x,
th n tpy  twx  tpy   tw x
then
Hence, ( y, x) is  - maximizing at (tp , tw ).
07.07.2010
13
G.B. Asheim, ECON4230-35, #3
Profit function: Proof of property (3)
Let ( y , x) be  - maximizing at ( p, w ).
Let ( y , x) be  - maximizing at ( p, w ).
Let ( y , x) be  - maximizing at ( p, w )  t ( p, w )
 (1  t )( p, w ).
By  - maximizati on :
t  py  w x  t  py  wx   t ( p, w )
(1  t ) py  w x  (1  t ) py   w x  (1  t ) ( p, w )
Adding these two inequalit ies together
 ( p, w )  t ( p, w )  (1  t ) ( p, w )
07.07.2010
14
G.B. Asheim, ECON4230-35, #3
Proof of Hotelling’s Lemma
Suppose y  is a  - maximizing output at ( p  , w  ).
Define the function
g ( p )   ( p, w  )   py   w  x  
g ( p)
p
Then g ( p  ) 
07.07.2010
p
 ( p  , w  )
 y  0
p
G.B. Asheim, ECON4230-35, #3
15
5