Advanced Microeconomics
(lecture 2,production theory II)
Ye Jianliang
lecture 2 for Chu Kechen Honors College
profit maximization
? Contain,
– Some definition
– Isoprofit curve
– Demand function and it’s properties
– supply function and it’s properties
– Profit function and it’s properties
lecture 2 for Chu Kechen Honors College
1.Some definition
? Basic assumption,
– Y is changeless,no technical improvement,
– The firm is the price taking,
? Profit function,
– When it is one production,
( ) m a x {, }y y Y? ??pp
( ) (,) m a x ( )p p f?? ? ? ?xp w x - w x
lecture 2 for Chu Kechen Honors College
2,Isoprofit curve
? The profit,
? We then got the
isoprpfit curve,
? Then
and is negative
semidifinite,
(,) ( )p p f??? ? ? ? ?w x w x
q
x
()pf ?? ? ?x w x =
/ p?
s lo p e / p? w
()qf? x
x*
()fp???x w /
2 ()Df ?x
lecture 2 for Chu Kechen Honors College
2.Isoprofit curve
? Weak Axiom of Profit Maximization (WAPM)
if ys,yt are in Y,and choice by firm under
price ps and pt,then, we can
get,or
?s s s tp y p y
)0?s t s t( p - p ) ( y - y 0? ?? ?py
价格上升，最优产量也上升
lecture 2 for Chu Kechen Honors College
2,Isoprofit curve
y1
y2
x
q
y1
y2
x
q
lecture 2 for Chu Kechen Honors College
2,Isoprofit curve
y1
y2
x
q
y1
y2
x
q YI YO
lecture 2 for Chu Kechen Honors College
3.Demand function
? factor demand function (set),
? proposition5,it’s homogenous of degree
zero,
X { ( ), (,) }V q p q p?? ? ? ?x w x w
(,)px = x w
lecture 2 for Chu Kechen Honors College
3.Demand function
? One production,regular p as 1,let x(,w)
be the profit maximization choice (function
means x is single point under w) of input
factor vector under factor price w,then
must hold,
(, ) 0 ( )(,) f?? ? ? ? ?? xwxw
2
2(, ) 0 ( ) is s y m m e tr ic n e g a ti v e d e f in it e,Df?? ??
? 2 x x
注意：这两条式子成立
的条件是 one
production
lecture 2 for Chu Kechen Honors College
3.Demand function
? When,differentiate with
respect to w,we get,
or
? Substitution matrix is a symmetric
negative matrix,
– 1,
– 2,
( (,) )f??x w w
2 ()D f D I??x (,w) x (,w) 21[ ( ) ]D D f ??x (,w) x (,w)
Dx(,w)
0Td d d D d??w x w x (,w) w
//i j j ix w x w? ? ? ? ?
lecture 2 for Chu Kechen Honors College
3.Demand function
? Are the profit maximization give us the
sufficient information about the production
behavior? Such as (1) factor demand
function; and (2)yeild (supply) function?
? Proposition6:demand for factor i is,
or (,)(,)
i
i
xp w???? ?w (,) (, )wpD ???xw
lecture 2 for Chu Kechen Honors College
4.supply function
? Supply set,
? Proposition7,y(.) is homogenous of
degree 1,
? Proposition8,If Y is convex for any p,
then is y(p),if Y is strictly convex,y(p) is a
single point,
{, }???y( p ) y Y p y = ( p )
lecture 2 for Chu Kechen Honors College
4.Supply function
? Proposition9,(Hotelling’s lemma) y(p) is
single point,then (recall prop.6)
? The supply substitution matrix
is symmetric positive matrix,
? And (why?) see the fig,
()???y ( p ) p
2 ()DD ??y( p ) p
0D ??y ( p ) p
lecture 2 for Chu Kechen Honors College
5.Profit function
? Proposition10,is homogenous of
degree 1,
? Proposition11,is convex,
()? p
()? p
lecture 2 for Chu Kechen Honors College
2.5 profit function
? LeChatelier Principle,
something is limited in
short-term,but not long-
term,definite as z,
and,let,
if,
()s z? p,
( ( ) )L z? p,p
( ) ( ( ) ) ( ) 0Lszz??? ? ? ?p p,p p,
( ) 0???p
( ) 0D ? ? ?? ? ? ?SLp y ( p ) y ( p )
2 ( ) 0D D D? ? ?? ? ? ?SLp y ( p ) y ( p )
y(p)
p
yL(p)
yS(p)
p*
lecture 2 for Chu Kechen Honors College
Assignment
? Textbook,ex.1.7,ex.2.2,ex.2.5,ex.3.4,
lecture 2 for Chu Kechen Honors College
Supply function
y1
Dy(p)
p
{, ( ) 0 }y T y ?
y2
y(p)
lecture 2 for Chu Kechen Honors College