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Economics 2301. Lecture 34 Multivariate Optimization. Second Total Differential of Bivariate Function. Example of Second Total Differential. Second Order Conditions. - PowerPoint PPT Presentation
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Economics 2301
Lecture 34
Multivariate Optimization
Second Total Differential of Bivariate Function
21122
2222
111
22
22111
1
22112
22121211
21
2
aldifferenti totalsecond
get the wefunctions, as terms and constants as terms
thegby treatin aldifferenti total theof derivative total theTaking
.
is aldifferenti our total ,function bivariate For the
dxdxfdxfdxf
dxx
dxfdxfdx
x
dxfdxfyd
fdx
)dx,x(xf)dx,x(xfdy
),xf(xy
ii
Example of Second Total Differential
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2)ln()ln(d
is aldifferenti totalsecond The
)ln()ln(
is aldifferenti totalThe
)ln()ln(
dxdxxx
dxx
xdx
x
xy
dxx
xdx
x
xdy
xxyLet
Second Order Conditions
If the second total differential evaluated at a stationary point of a function f(x1,x2) is negative for any dx1 and dx2, then that stationary point represents a local maximum of the function.
If the second total differential evaluated at a stationary point of a function f(x1,x2) is positive for any dx1 and dx2, then that stationary point represents a local minimum of the function.
Deriving the second order conditions
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2
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get we, gsubtractin and adding
by aldifferenti totalsecondour of square theCompleting
dxf
ffdx
f
fdxf
dxf
ff
dxf
fdxdx
f
fdxfyd
fdxf
Deriving the second order conditions
.
if and negative is if ly,equivalent or, 0
if and negative is if and
of any valuesfor negative is aldifferenti totalsecond The
.
if and positive is if ly,equivalent or, 0
if and positive is if and
of any valuesfor positive is aldifferenti totalsecond The
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fdxdx
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Understanding the 2nd Order condition
well.as
southwest)-northeast as(such directions possibleother all
inbut south),-north andwest -(east directions basic twothe
in only not be)may case theas or valley, (hillion configurat
of typesame with thesections cross l)dimensiona-(two yield
illquestion win surface that theensure tois derivative partial
latter by this played role The . derivative partial cross
on the alsobut south),-(north and west)-(east by shown
directions basic twoin the slide)next (figure point arond
ion configurat surface with thedo tohave which , and
ononly not hinges ofsign the, function, For the
xy
yx
yyxx
2
f
TT
A
ff
zdf(x,y)z
Understanding the 2nd Order Conditions
0
0
dy
dx
0
0
dy
dx
0
0
dy
dx
y
0x
x0
Ty
y0 Tx
0
0
dy
dx
Maximum drawing
A
Ty
Tx
x
y
z
dx>0,dy>0
Our Slice of the cone
dx>0,dy>0dx<0,dy<0
A
2nd Order conditions for Bivariate Function
point.
stationary at this minimum local a reachesfunction then the
and0 if and
point stationary thehas function theIf
point.
stationary at this maximum local a reachesfunction then the
and0 if and
point stationary thehas function theIf
2
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21
2
2112212221112111
21
),x(xf),x(x)f,x(xf ),x(xf
),x(x),xf(xy
),x(xf),x(x)f,x(xf ),x(xf
),x(x),xf(xy
********
*2
*1
********
*2
*1
Example
minimum. a have We
41660125
4,012,05
ConditionsOrder 2nd
0,1
01244
0455
:ConditionsOrder First
465.245100,
212
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122211
*2
*1
212
211
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Multiproduct Monopolist
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227055
27055
now isfunction revenue Our total
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55
functions revenue average to theseConverting
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:products twofor the
functions demand following thefaces monopolistOur
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QQP
QQP
PPQ
PPQ
Multiproduct Monopolist Cont.
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278Q
us gives This
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03455
:ConditionsOrder First
3370255
227055:Profit
C :Cost Total
227055 :Function Revenue
*2
*1
212
211
21222
211
2221
2121
222
211
2221
21
21222
211
Qand
QQQQQQ
QQQQQQQQQQCR
QQQQ
QQQQQQR
Multiproduct Monopolist Cont.
maximized. are Profits
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3,06,04
:ConditionsOrder Second
06370
03455
:ConditionsOrder First
212
22211
122211
212
211
Saddle Point
saddle. a like looksfunction thebecause
named so point, saddle a called ispoint stationary the
case In this .0 and 0 example,for negative, isother
theand positive is derivative partialorder -2nd one that is
conditions sufficient by these coverednot y possibilit One
2211 ff
Figure 10.4 A Saddle Point
Example of Saddle Point
t.saddlepoin a have We
41660125
4,012,05
ConditionsOrder 2nd
0,1
01244
0455
:ConditionsOrder First
465.245100,
212
22211
122211
*2
*1
212
211
2122
212121
fff
fff
xx
xxf
xxf
xxxxxxxxfLet
Example 2 from last Lecture
5.35.10128
01650288get we
1, into ngSubstituti .502get we2,equation From
0248
01688
:ConditionsOrder First
21628420
*2
*11
11
12
212
211
212221
21
xandxx
or).x(x-
.xx
xxx
y
xxx
y
xxxxxxyLet
Example 2 Continued
point. inflectionor t saddlepoin bemay
It have. what wesurenot are We
satisfied.not conditionsOrder 2nd
8643248
8,04,082
122
2211
122211
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Graph of Previous Slide
http://www.compute.uwlax.edu/calc2D/output/2112/
