Advanced Microeconomics
(lecture 1,production theory I)
Ye Jianliang(叶建亮 )
lecture 1 for Chu Kechen Honors College
Summary
? textbook,
– Varian,Hal R.,1992,Microeconomics Analysis,3rd ed,
– Mas-Colell,A.,M,Whinston,and J,Green,1995,Microeconomics
Theory,
? assignments,
– twice a week;
– Team work;
– Deliver on the class,
? examinations,
– Mid-term,by the assignments;
– Final-term,80% of the questions coming from the assignments,
lecture 1 for Chu Kechen Honors College
The Basic Framework of
Microeconomics
Technology Preference
Choice
(production)
Choice
(purchase)
Supply Demand
Market
Equilibrium
lecture 1 for Chu Kechen Honors College
Technology
? Contain,
–,production (possibilities) set” (PS) and
“production function”;
– Properties of the,PS”;
– Technical rate of substitution;
– Returns to scale,
lecture 1 for Chu Kechen Honors College
1,Production set
? Production plan (production vector,or
input-output vector),
? Production set Y,all technological feasible
y,
? Restricted production set Y (z),
– Some of yi in y are restricted on z,
– Short-run production set,
12(,,,)ny y y? ??????y
{, a r e t e c h n o l o g i c a l l y f e a s i b l e }n? ? ?Y y y
lecture 1 for Chu Kechen Honors College
1,Production set
? Input requirement set,
– All yi in y are negative,let them be –x (then x
is positive),and the rest yj to be q,
– So,and,and
– the input requirement set is,
I???x O???q () n?? ? ?y q,- x
( ) {, (,) }IV ?? ? ? ? ?q x q x Y
lecture 1 for Chu Kechen Honors College
1,Production set
? Transformation function,
and satisfied
? if and only if y is
technologically efficient,
then
,nT ? ? ?
{, ( ) 0 }n T? ? ? ?Y y y
( ) 0T ?y
y2
y1
{, ( ) 0 }n T? ? ? ?Y y y
{, ( ) 0 }T ?yy
lecture 1 for Chu Kechen Honors College
1.Production set
? Production function,
– If O = 1,and I = n - 1,then,
– means,
? Isoquant,
? Question1,calculate the PS,RS,TF,PF,
Isoquant of Cobb-Douglas technology and
Leontief technology,
( ) w h e n (,) 0q f T q? ? ?xx
( ) = {, (,) }f q q q? ? ?x x Y
( ) { ( ), (,) }Q q f q q? ? ? ?x x Y
lecture 1 for Chu Kechen Honors College
2.Properties of PS,
? Y is nonempty,we have something to do,
? Y is close,Y contain it’s boundary,
? No free lunch,
See the fig,
? Free disposal,
See the fig,
t h e a r r a y,a n d m e a n s nn? ? ?y y y Y y Y
{ 0 }n???Y
n?? ? ?yY
lecture 1 for Chu Kechen Honors College
2.Properties of PS,
? Additive (free entrance),
? Convexity,
See the fig,
? Proposition1,if Y is convex,so is V(q),
? Proposition2,if V(q) is convex,f(x) is quasi-
concave,
,a n d,t h e n ??? ? ? ?y Y y Y y y Y
,a n d,t h e n ( 1 ),h e r e [0,1 ]? ? ???? ? ? ? ? ?y Y y Y y y Y
lecture 1 for Chu Kechen Honors College
3.Technical rate of substitution
? Marginal rate of transformation (MRT) of
good j for good i at y and T(y)=0,
? T(y)=0,we got,
See the fig,
( ) /
( ) /
j
ji
i
TyM R T
Ty
???
??
y
y
12
12
( ) ( ) 0TTd y d y
yy
?? ??
??
yy
lecture 1 for Chu Kechen Honors College
3.Technical rate of substitution
? one production,the marginal rate
technical of substitution at x*,
? As x changes,we got technical rate of
substitution
( ) /
()
( ) /
j
ji
i
fx
M R TS
fx ?
? ???
??
x
x
x x=x
( ) /
( ) /
j
ji
i
fxTR S
fx
???
??
x
x
lecture 1 for Chu Kechen Honors College
3.Technical rate of substitution
? The elasticity of substitution,the
curvature of the isoquant,
? Question2,calculate the,TRS” and the
“elasticity of substitution” of Cobb-
Douglas technology and CES technology,
( ) / l n ( )
0
/ l n ( )
j j j
i i i
x x x
x x x
ji
ji ji ji
d
TR S TR S d TR S
?
?
? ? ?
?
lecture 1 for Chu Kechen Honors College
4.Returns to scale
? Nonincreasing returns to scale,
? Nondecreasing returns to scale,
? Constant return to scale,
,,[0,1 ]aa? ? ? ?y Y y Y
,,1aa? ? ? ?y Y y Y
,,0aa? ? ? ?y Y y Y
lecture 1 for Chu Kechen Honors College
4.Returns to scale
? Proposition 3,Y is constant returns to
scale if Y is both,additive” and,convexity”,
? Proposition 4,single production,if and
only if f(.) is homogenous of degree 1,Y is
constant returns to scale,
lecture 1 for Chu Kechen Honors College
4.Returns to scale
? Why we assume that Y is non-increasing
(or decreasing for usual) returns to scale?
? Suppose Y is decreasing returns to scale,
and it’s PF f(x),now we introduce new
input z,and difine a new PF,
F(.) is homogenous of degree 1,
F z z f z?(,x ) ( x / )
lecture 1 for Chu Kechen Honors College
4.Returns to scale
? Homogeneous and homothetic tech,
– Homogeneous of degree k,
– Homothetic function,
? Elasticity of scale,
( ) ( ) 0kf t t f t? ? ?xx
( ) ( ( ) ) f g h?xx
i s H D 1,i s a p o s s i t i v e m o n o t o n i c f u n c t i o n, hg( x ) (, )
( ) ( ) q t f t? x
1
l n ( ) ( )
ln t
d q te
dt ?
?x
lecture 1 for Chu Kechen Honors College
Assignment
? Textbook,ex.1.1,ex.1.3,ex.1.7,ex.1.9
lecture 1 for Chu Kechen Honors College
Properties of PS,
y2
y1
y2
y1
Free
lunch
lecture 1 for Chu Kechen Honors College
Properties of PS,
y2
y1
{, }nY y y y ?? ? ? ?
yY?YY?
lecture 1 for Chu Kechen Honors College
Properties of PS,
y2
y1
{, }nAAY y y y ?? ? ? ?
AyY?
ABY Y Y??
ByY?
{, }nBBY y y y ?? ? ? ?
lecture 1 for Chu Kechen Honors College
Technical rate of substitution
y2
y1
{, ( ) 0 }y T y ?
12MRT?
()Ty?
{, ( ) 0 }y T y ?
lecture 1 for Chu Kechen Honors College