Advanced Microeconomics
(LECTURE 3:production theory III)
Ye Jianliang
lecture3 for Chu Kechen Honors College
Cost Minimization
? Content,
? Definitions
? Properties of cost function
? WACM
? Some forms of cost functions
lecture3 for Chu Kechen Honors College
1.Definitions
? One production,cost function is
? The optimal solution x(w,q),is the
conditional factor demand function,
? Question1:calculate the conditional factor
demand function of C-D tech,and CES tech,
0
(,) m i n
., ( )
x
cq
s t f x q
?
??
?
w w x
lecture3 for Chu Kechen Honors College
1.Definitions
? Recall the cost minimization condition,let
x>0,then set Lagrangian,
? We got,
see the fig,
? What is?
(,) ( ( ) )fq??? ? ?x w x xL
()f? ?? ??wx
?
lecture3 for Chu Kechen Honors College
2.Properties of cost function
? Proposition1,c(w,q) is homogeneous of
degree 1 in w,and non-decreasing in q,
? Proposition2,c(w,q) is concave function of
w,
? Proposition3,x(w,q) is homogeneous of
degree 0 in w,
? Proposition4:if V(q) is convex,then is x(.) if
V(q) is strictly convex,x(.) is single point,
lecture3 for Chu Kechen Honors College
2.Properties of cost function
? Proposition5,(Shephard’s lemma) if x(w,q)
is single point,then
? Proposition6,is
symmetric negative semi-definite,and
see the fig,
? Proposition7,if f(.) is HD1,c(.) and x(.) is
HD1 too,if f(.) is concave,c(.) is convex in q,
(,) (,)wq c q??x w w
2(,) (,)wwD q D c q?x w w
(,) 0wDq ??x w w
lecture3 for Chu Kechen Honors College
3.WACM
? Weak Axiom of Cost Minimization (WACM),
if xs,xt are in Y,and choice by firm under
price ws and wt,then, we can get,
?t t t sw x w x
? ? ?w x 0
lecture3 for Chu Kechen Honors College
3.WACM
x2
xA
x1
xB
x2
xA
x1
xB
lecture3 for Chu Kechen Honors College
3.WACM
x2
xA
x1
xB
x2
xA
x1
xB
VI VO
lecture3 for Chu Kechen Honors College
4.Some forms of cost functions
? If,corresponding price,
and is the limited factor (fixed assets),
while is the variable factor,The
total cost is
(,)vf?x x x (,)vf?w w w
f z?x
(,,)v v fq?x x w x
(,,) (,,)f v v f f fc q S T C S C V F C q? ? ? ? ?w x w x w x w x
(,,) /,(,,) /
/,(,,) /
f v v f
f f f
S A C c q q S A V C q q
S A F C q S M C c q q
??
? ? ? ?
w x w x w x
w x w x
lecture3 for Chu Kechen Honors College
4.Some forms of cost functions
? Envelope theorem,an objective function
depend on some parameter,
? Then,
see the prove,
( ) m a x (,)xM a f x a?
()
( ) (,)
x x a
d M a f x a
d a a ?
??
?
lecture3 for Chu Kechen Honors College
4.Some forms of cost functions
? LAC and SAC,
? So
See the fig,
( ) m i n (,)
f
fL A C q S A C q? x x
()
(,)()
ff
f
q
SA C qdLAC q
dq q ?
?
?
? xx
x
lecture3 for Chu Kechen Honors College
4,Duality
? Given the technology,we can obtain the
cost function,are the cost function contains
the same information of the technology
(production function)?
? If the answer is,yes”,then the cost
minimization behavior will indicate the
technology of the firm,
lecture3 for Chu Kechen Honors College
Assignment
? Textbook,
? ex.5.3,ex.5.8,ex.5.11,ex.5.12,ex.5.17
lecture3 for Chu Kechen Honors College
Definitions
x2
x1
x(w,q)
w
{, (,) }cq? ? ?n - 1+x w x w
( ) {, ( ) }v q f q? ? ? ?n - 1+xx
()Qq
( ( ) )fq? x w,
lecture3 for Chu Kechen Honors College
Properties of cost function
x2
x(w,q)
w
(,)wDqxw
lecture3 for Chu Kechen Honors College
Prove of envelope theorem
? Let be the solution of the problem,then
? Differentiating both sides,we have,
? And x* maximize f,so
? Then
()xa?
( ) ( ( ),)M a f x a a??
( ) ( ( ),) ( ) ( ( ),)d M a f x a a x a f x a a
d a x a a
? ? ?? ? ?? ? ?
? ? ?
( ( ),) / 0f x a a x?? ? ?
()
( ) (,)
x x a
d M a f x a
d a a ?
??
?
lecture3 for Chu Kechen Honors College
LAC and SAC
q
AC AC
q q*
lecture3 for Chu Kechen Honors College