beamerCompStat2-13

7/27/2019 beamerCompStat2-13

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The Implicit Function Theorem (IFT): key points1 The solution toanyeconomic model can be characterized as the level set

corresponding to zero ofsomefunction

1 Model: S=

S(

p;

t)

,D=

D(

p)

,S=

D;p=

price;t=

tax;

2 f(p; t) =S(p; t)D(p) =0. Level Set:{(p, t) :f(p; t) =0}.2 When you do comparative statics analysis of a problem, you are studying

the slope of the level set that characterizes the problem.

1 Intuitive comp. stat. question: changet, what happens top?

2

Intuitive idea:t; how muchpis needed to keep on LS?3 Math answer is the slope ofLS

3 The implicit function theorem tells you

1 when this slope is well defined

2 if it is well-defined, what are the derivatives of the implicit function

4 Its an extremely powerful tool1 explicit functionp(t)could be nasty; no closed form

E.g., : f(p; t) =tp15 + t13 + p95p=0; whatsp(t)?2 dont need to knowp(t)in order to know dp(t)

dt .

3 can compute dp(t)

dt from partials off(

;

).

() December 5, 2013 1 / 21
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Level Sets locally well-defined diffable functions

() December 5, 2013 2 / 21
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Implicit function theorem (single variable version)

Theorem: Givenf : R2 R1,f C1 and(, x) R2, if f(,x)x =0,

nbdsU

of,Ux

ofx& aunique g: U

Ux

,g C1

s.t. U

,

f(,g()) = f(, x)i.e.,(,g())is on the level set off through(, x)

g() = f(,g())

f(,g())

xto slide 4 to slide 8

Trivial Proof of the second line:

f(,x) := f(,g()) = 0

d(f,g())

d =

f(,g())

+

f(,g())

x

dg()

d = 0

dg()d

= f(,g())

f(,g())

x

Note that the following isnottrue: if f(,x)

x

=0,

nbdU of, & and aunique

functiong: U R,g C1 s.t. U,f(,g())=f(, x). to slide 4() December 5, 2013 3 / 21
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U andUx to slide 3

0

x g()

(,x)

U

Ux

Note that

There aretwo C1

functionsg1

andg2

mappingU

to R s.t.fori= 1,2, for U f(,gi()) =f(, x).but only one of these mapsU into some nbd ofx.

() December 5, 2013 4 / 21
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Implicit function theorem examples I

Example: Illustrates why its called theimplicitfunction theorem

closed-formexplicitfunction relating and xdoesnt exist

tis time,pis an equilibrium price that depends on t; assumep>0

1 f(p; t) =tp15 + t13 + p95p2

ft =p

15

+ 13t12

; f

p=15 tp14

+ 95 p94

1/2 1

p3 if fp=0, dpdt =

p15 + 13t12

15 tp14 + 95 p941/2 1

p

Also true (if you are a mathematician but not an economist):

1 if ft=0, dtdp=

15 tp14 + 95 p941/2 1p

p15 + 13t12

() December 5, 2013 5 / 21
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Implicit function theorem examples II

Example: (slightly less dramatic)

Illustrates thatU needs to be chosen carefully: cant be too big1 f(x;) =x2 +21=0 ; LS0= {(x;) :f(x;) =0}.2 f(x;)

=2; f(x;)

x =2x

3 if f(x;)

x =0,then dx

d=

df(x;)

d df(x;)

dx =

/x.

Suppose=0.9,x=

10.81 0.44.Use IFT to estimate effect of a 100% increase in to 1.8

dxd = 0.9/0.44 2;x(1.8) 0.44 + (20.9) 1.36

Clearly, noxexists such that(x;+ 0.9) LS0conclude: hazardous to base policy decisions on comp stat analysis

dis unlikely to be small enough for the theorem to be applicable.

() December 5, 2013 6 / 21
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Implicit function theorem examples III

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.81.5

1

0.5

0

0.5

1

x

() December 5, 2013 7 / 21
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Implicit function theorem (intermediate version)

Theorem: Givenf: Rn+1 R1,f C1 and(, x) RnR1, if f(,x)x =0,

j, nbdsUj

ofj,Ux

ofx& aunique g: Uj

Ux

,g C1

s.t.j Uj

,

f(,g()) = f(, x)i.e.,gputs us on the level set of f containing(, x)

dg()

dj= f(,g())

j

f(,g())

xor in vector form

g() = f(,g())/ fx(,g()). to slide 3

This is a straightforward extension of the previous theorem. Example:1 Model: S= S(p; t),D=D(p; y),S= D;t=tax,y=income;

2

f(p; t,y) =S(p; t)D(p; y),3 If f

p=0, can use IFT to compute p= (pt,py)

4 Can use IFT to compute p= (pt,py)5 Once you have gradient, can get any directional derivative

6 Use IFT to approx impact of anycombinationof paramchanges() December 5, 2013 8 / 21
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Equilibrium price as a function of tax and income

P

Q

p(t, y)

p(t, y)

D(p, y)

D(p, y)

S(p, t)

S(p, t

)

() December 5, 2013 9 / 21
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The zero level set ofS(p, t)D(p,y)

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

10

0.5

1

1.5

2

2.5

3

tax

Zero Level set of Supplydemand model

income

equilibrium

price

() December 5, 2013 10 / 21
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Implicit function theorem (multivariate version) to slide 13

Theorem: Givenf: Rn+m Rm,f C1 &(, x) RnRm, ifdet (Jfx(, x)) =0, U of,Ux ofx&!g:U Ux,g C1 s.t. U,

f(,g()) =f(, x) i.e.,g puts us on level set off containing(, x)

g1

()1 g1

()ng2()1

g2()n...

. . ....

gm()1

gm()n

= Jfx(,g())1

f1

(,g())1 f1

(,g())nf2(,g())

1 f2(,g())n

.... . .

...fm(,g())

1 fm(,g())n

where Jfx(,g())is the Jacobian off treatingas parameters

() December 5, 2013 11 / 21
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Multivariate IFT: graphical representation

0

00

0

dx

1

dx2

0

00

0

dx

1

dx2

00

0

x1x1 x2x2

d

d

d

Level set of f1 corresponding to 0 Level set of f2 corresponding to 0

The tangent plane to the 0-level set of f1 The tangent plane to the 0-level set of f2

The two tangent planes combinedThe two tangent planes: enlarged

1

f: R1+2 R2 , i.e., two endog variables x, one exog1 xis in the horizontal plane;on vertical plane

2 (, x) LS1 LS2 is a soln, i.e., belongs to both zero level sets3 as changes,x changes to keep vector in LS1 LS24 (,x)slides along the groove in the bottom right panel

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Example: apply IFT to first order conditions I

Problem

max

L,K

(L,K) = pLK

wL

rK Soln:(L,K)

In this case, the level set LS0 we stay on is the FOC, where

FOC(w, r; L,K) =

LK

=

pL1KwpLK1 r

= 0

Match concepts: general expression for IFT vs this example to slide 11

fis FOC i.e., = (L(L,K),K(L,K))xis(L,K).

is(w, r).

Jfx(, x)is Jacobian of (L,K)w.r.t.(L,K), orH, the Hessian ofJf(, x)is Jacobian of (L,K)w.r.t.(w, r).g()is

L(w, r)K(w, r).

() December 5, 2013 13 / 21
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Example: apply IFT to first order conditions IIPlug in all the pieces...

Jfx(, x)is Jacobian of (L,K)w.r.t.(L,K),

Jfx(, x) = H(L,K) = 2(L,K)

L2

2(L,K)

LK2(L,K)LK

2(L,K)K2

Jf(, x)is Jacobian of (L,K)w.r.t.(w, r).

Jf(, x) = Jw,r = 2(L,K)Lw

2(L,K)Lr

2(L,K)

Kw

2(L,K)

Kr

g()is

L(w, r)K(w, r)

is implicitly defined but not explicitly.

Jg() = Jf(, x)Jf(, x)

J

L(w, r)K(w, r)

= 2(L,K)L2 2(L,K)LK2(L,K)

LK2(L,K)

K2

1 2(L,K)Lw

2(L,K)

Lr2(L,K)Kw

2(L,K)Kr

= 2(L,K)

L22(L,K)LK

2(L,K)LK

2(L,K)K2

1 1 00 1

() December 5, 2013 14 / 21
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What some other people do

Many economists dont apply IFT directly; two differences

they totally differentiate each line of fseparately

they end up a different place from where the IFT ends up

The IFT gives a formula for the Jacobianof the implicit functg

Total differentiators end up with an expression fordifferentialof g

Relationship between Jacobian and differential is as it always is

Recall: for everynnmatrix, uniqueL: Rn Rn.dx=Jg(;x)d

Summary:IFT route ends up at: Jg(;x)Total differentiation route ends up at: dx=Jg(;x)d

() December 5, 2013 15 / 21
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IFT via total differentiationmax

L,KpLKwL rK

Requirement: stay on the zero level set of f, where

f(w, r; L,K) =LK

=

pL

1

K

wpLK1 r

= 0

Totally differentiate:

0 = LL

dL+LK

dK+Lw

dw

0 = K

L dL+KK dK+

Kr dr

Move exog variables to right hand side; put in matrix form:

LL

LK

K

L

K

K

dL

dK=Lw

0

0 K

r

dw

dr = 1 0

0 1dw

drInvert

dL

dK

=

LL

LK

KL

KK

1 1 0

0 1

dw

dr

() December 5, 2013 16 / 21
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Apply IFT to an equilibrium system

Question: (from ARE prelim, 2008): A monopolist sells in two countries: 1 and

2. It produces a good at a constant marginal cost ofcand cannot produce

more than a total ofQunits. Country 1 imposes a per unit tax of units on the

good sold in that country. Assume that the constraint binds.

1 For a given value of, show how the equilibrium in the two countries are

determined.

2 Show how these equilibrium prices change as increases.

() December 5, 2013 17 / 21
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Answer to ARE prelim, 2008 question

Answer: Assume that inverse demand curves are concave in price.

The monopolists optimization problem is

maxp1,p2

(p1 )D1(p1) +p2D2(p2) c(D1(p1) + D2(p2))s.t. D1(p1) + D2(p2) Q

The Lagrangian is

L(p1,p2,;) = (p1 )D1(p1) +p2D2(p2) c(D1(p1) + D2(p2))+ (QD1(p1)D2(p2))

Assuming capacity constraint binds, the first order conditions are

Lp1 = D1(p1) + (p1 c) D1(p1) = 0Lp2 = D2(p2) + (p2 c) D2(p2) = 0L = QD1(p1)D2(p2) = 0

Solution is a triple(p1,p2,)which solves this system,given.() December 5, 2013 18 / 21
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The monopolists optimization problem, graphical

Q

Marginal revenue

D1(p1)D2(p2)

MR2()

c

MR1(, )

MR1(, )

c + ()

c+ ()

() December 5, 2013 19 / 21

L D ( ) + ( ) D ( ) 0
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Lp1 = D1(p1) + (p1 c) D1(p1) = 0Lp2 = D2(p2) + (p2 c) D2(p2) = 0L = QD1(p1)D2(p2) = 0

The Hessian of the Lagrangian w.r.t. endog vars is:

HLp, =

2D1(p1 ) +(p1c)D1 (p1) 0 D1(p1)0 2D2(p2) +(p2c)D2 (p2) D2(p2)

D1(p1 ) D2(p2) 0

HL =D1(p1)0

0

.

Hence from the implicit function theorem, we have

dp1d

dp2d

dd

= HL1(p,)

D1(p1)0

0

= HL1

(p,)

D1(p1)0

0

() December 5, 2013 20 / 21

Its straightforward to check that the determinant of HL(p ) is
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It s straightforward to check that the determinant ofHL(p,) is

D2(p2 )2

2D1(p1 ) + (p1 c) D1 (p1 )

+D1 (p1 )2

2D2 (p2 ) + (p2 c) D2 (p2 )

>0

Applying Cramers rule, we have

dp1d

= det

D1(p1 ) 0 D1(p1)0 2D2(p2 ) + (p2 c) D2 (p2 ) D2(p2)0 D2(p2) 0

det(HL(p,))

=D1(p1 )D2(p2 )2det(HL(p,))>0dp2d

= det

2D1(p1 ) + (p1 c ) D1 (p1 ) D1(p1 ) D1(p1)0 0 D2(p1)

D1(p2 ) 0 0

det(HL(p,))

= D2(p1 )D1 (p2 )2

det(HL(p,))

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