Ch. 8 Comparative-Static Analysis of General-Function Models

1

Ch. 8 Comparative-Static Analysis of General-Function

Models • 8.1 Differentials• 8.2 Total Differentials• 8.3 Rules of Differentials (I-VII)• 8.4 Total Derivatives• 8.5 Derivatives of Implicit Functions• 8.6 Comparative Statics of General-

Function Models• 8.7 Limitations of Comparative

Statics

2

011

1)9

01

1

1

1)8

01

1

1)7

1006)

1005)

4)

xoperator with derivative partiallim)3

2)

such thatother affect the w/oitselfby can vary each x

another one oft independen all are xvariables)1

167 p. s,Derivative Partial 4.7

00

111

10

111

i1

2

2

btb

tT

Gtb-b

-b)d(taG)t(IT

btb

t)b(C

Gtb-b

a-bdG)-t)(Ib(C

btbY

Gtb-b

GI a-bdY

) t ; (d tY; dT

) b ; (a b(Y-T); a C

G I C Y

fyx

ΔxΔy

Δx),x,Δx,xf(xΔxΔy

),x,,xf(xy

*

o

*

*

o

*

*

o

*

Δx

n

n

3

8.1Differentials

8.1.1 Differentials and derivatives8.1.2 Differentials and point

elasticity

4

8.1.1 Differentials and derivatives

Problem: What if no explicit reduced-form solution exists because of the general form of the model? Example: What is Y / T when

Y = C(Y, T0) + I0 + G0

T0 can affect C direct and indirectly thru Y, violating the partial derivative assumption

Solution:• Find the derivatives directly from the

original equations in the model.• Take the total differential • The partial derivatives become the

parameters

5

TG ITYCTYCYT

G ITG ITY C Y

G ITY C Y

G ITY

G ITY CY

TY CC

G I C Y

bbYT

bGIbTaY

GIbTabYY

G I b(Y-T) a Y

b(Y-T) a C

G I C Y

0000*

0**

0

0000000**

000**

000**

000

0

00

00

00

00

00

,,,, :problem?)7

Then

dependentmutually )T and arguments (C var exog. ,,,)6

,5)

exists equil. assuming,,Y4)

dataparameter no,3)

,2)

1)

If

1)4

Then

tindependenmutually varsexog. and parameters1

)4

3)

dataparameter 2)

1)

If

6

xxfy

xxxfxfxf

xf

dxxfdy

xxxfy

xxxfy

xfxy

xxfxy

xyxfydx

dx

0

01001

1

0

)8

)7

:is y ofion approximat seriesTaylor order -firstA

alityproportion offactor theis where

aldifferenti)6

0 finite as)5

)4

)3

0 then 0 as)2

derivativelim)1

alsDifferenti 1.8

Differential: dy & dx as finite changes (p. 180)

7

fi·nite Mathematics. a.Being neither infinite nor infinitesimal. b.Having a positive or negative numerical value; not zero. c.Possible to reach or exceed by counting. Used of a number. d.Having a limited number of elements. Used of a set.

8

Difference Quotient, Derivative & Differential

f(x0+x)

f(x)

f(x0)

x0 x0+x

y=f(x)

x

y

x

f’(x)

f’(x0)x

x

A

C

D

B

9

Overview of Taxonomy - Equations: forms and functions

Primitive Form

FunctionSpecific

(parameters)General

(no parameters)

Explicit(causation) y = a+bx y = f(x)

Implicit(no causation) y3+x3-2xy = 0 F(y, x) = 0

10

Overview of Taxonomy – 1st Derivatives & Total Differentials

Differentiation Form

FunctionSpecific

(parameters)General

(no parameters)

Explicit(causation)

Implicit(no causation)

bydx

d

)(

dx )(

xfdx

dy

xfdy

xy

yx

dx

dy

dxyxdyxy

23

22

02223

2

2

22

y

x

y

x

F

F

dx

dy

dxF

Fdy

11


• From partial differentiation to total differentiation

• From partial derivative to total derivative using total differentials

• Total derivatives measure the total change in y from the direct and indirect affects of a change in xi

12


• The symbols dy and dx are called the differentials of y and x respectively

• A differential describes the change in y that results for a specific and not necessarily small change in x from any starting value of x in the domain of the function y = f(x).

• The derivative (dy/dx) is the quotient of two differentials (dy) and (dx)

• f '(x)dx is a first-order approximation of dy

dxxfdyxfy )(')(

13


• “differentiation”– The process of finding the differential (dy)

• (dy/dx) is the converter of (dx) into (dy) as dx 0

– The process of finding the derivative (dy/dx) or• Differentiation with respect to x

dx

dy

dx

dyDerivative

dxdx

dydyalDifferenti

14

8.1.2 Differentials and point elasticity

• Let Qd = f(P) (explicit-function general-form demand equation)

• Find the elasticity of demand with respect to price

1,1

arg

%

%

dd

d

d

d

d

dd

ifinelasticifelastic

functionaverage

functioninalm

PQ

dPdQ

PdP

QdQ

P

Q

15

8.2 Total Differentials

• Extending the concept of differential to smooth continuous functions w/ two or more variables

• Let y = f (x1, x2) Find total differential dy

22

11

dxx

ydx

x

ydy

2211 dxfdxfdy

16

8.2 Total Differentials (revisited)

• Differentiation of U wrt x1

U/ x1 is the marginal utility of the good x1

• dx1 is the change in consumption of good x1

tconsxx nx

U

dx

dU

tan...112

11

2

211

...dx

dx

x

U

dx

dx

x

U

x

U

dx

dU n

n

17

8.2 Total Differentials (revisited)

Total Differentiation: Let Utility function U = U (x1, x2, …, xn)

nn

dxx

Udx

x

Udx

x

UdU

22

11

11

2

211

...dx

dx

x

U

dx

dx

x

U

x

U

dx

dU n

n

To find total derivative divide through by the differential dx1 ( partial

total derivative)

18

8.2 Total Differentials• Let Utility function U = U (x1, x2, …, xn)

• Differentiation of U wrt x1..n

U/ xi is the marginal utility of the good xi

• dxi is the change in consumption of good xi

nn

dxx

Udx

x

Udx

x

UdU

22

11

• dU equals the sum of the marginal changes in the consumption of each good and service in the consumption function

19

8.3 Rules of differentials, the straightforward way

Find dy given function y=f(x1,x2)

1. Find partial derivatives f1 and f2 of x1 and x2

2. Substitute f1 and f2 into the equationdy = f1dx1 + f2dx2

20

8.3 Rules of Differentials (same as rules of derivatives)

Let k is a constant function; u = u(x1); v = v(x2)

• 1. dk = 0 (constant-function rule)• 2. d(cun) = cnun-1du (power-function rule)• 3. d(u v) = du dv (sum-difference

rule)• 4. d(uv) = vdu + udv (product rule)• 5. (quotient rule)

2v

udvvdu

v

ud

21

8.3 Rules of Differentials (I-VII)

6.

7. d(uvw) = vwdu + uwdv + uvdw

dwdvduwvud

22

Rules of Derivatives & Differentials for a Function of One Variable

dxnxdxnxxdx

d

dxdccdx

d

nnnn 11 )'2)2

00)'10)1

dxxgdxxfxgxfd

xgxfxgxfdx

d

)'3

)3

23

Rules of Derivatives & Differentials for a Function of One Variable

dxxgxfdxxgxfxgxfd

xgxfxgxfxgxfdx

d

)'4

)4

dxcnxdcxbcnxcxdx

db

cdxdcxaccxdx

da

nnnn 11 )'5)5

)'5)5

24

xg

dxxgxfdxxgxf

xg

xfd

xg

xgxfxgxf

xg

xf

dx

d

2

2

)'6

)6

Rules of Derivatives & Differentials

for a Function of One Variable

25

8.3 Example 3, p. 188: Find the total differential (dz) of the function

dy

xdx

x

yxdz

xx

y

yx

y

x

x

yy

zx

yxx

x

xyxx

x

xy

x

xx

x

y

x

x

xx

z

dyy

zdx

x

zdz

x

y

x

xz

x

yxz

23

2222

34

22

2222

22

22

22

2

2

1

2

2)6

2

1

222 )5

2

22

4

442

2

4

2

42

22)4

)3

22)2

2 )1

26

8.3 Example 3 (revisited using the quotient rule for total differentiation)

dxx

yxdy

x

dxx

yxxdy

x

x

yxdxdxxdyxx

yxdxdxxdyxdxxx

xdxyxdydxxx

xdyxyxdxxx

yxd

32

4

2

4

2

224

2224

24

22222

2

2

2

14

42

4

2

4224

1

44224

1

4)()(24

1

)2()()(22

1

2

27

8.4 Total Derivatives

• 8.4.1 Finding the total derivative

• 8.4.2 A variation on the theme• 8.4.3 Another variation on the

theme• 8.4.4 Some general remarks

28

8.4.1 Finding the total derivative from the differential

11

221

1

1

1

2211

22

11

21

)4

by sidesboth dividingby found is

example,for ,y wrt of derivative totalpartial The

...)3

)2

:yin changes partial theof sum the toequal isdy aldifferenti Total

1)

Given 5,-184 pp. 1.4.8

dx

dxf

dx

dxff

dx

dy

dx

x

dxfdxfdxfdy

dxx

ydx

x

ydx

x

ydy

,x,,xxfy

nn

nn

nn

n

29

8.4.3 Another variation on the theme

derivative totalpartial)6

)5

aldifferenti total)4

,)3

,)2

,,,)1

3.4.8

21

21

21

21

21

2

1

21

uuxuxcv

vuxx

vuxx

fhfgf§u

§y

du

dvff

du

dxf

du

dxf

du

dy

dvfdufdxfdxfdy

vuhx

vugx

vuxxfy

30

2

2121

22121

22211

221

2212211

32

2121

123§

§,10

§

§

12310

12310

12310

12310

435),(

vffv

yfuf

u

y

dvvffdufufdy

dvvfdufdvfudufdy

dvvdufdvudufdy

dvvdudxdvududxdxfdxfdy

vuxvuxxxfy

ucvc

8.4.3 Another variation on the theme

31

8.5 Derivatives of Implicit Functions

• 8.5.1 Implicit functions• 8.5.2 Derivatives of implicit

functions• 8.5.3 Extension to the

simultaneous-equation case

32

8.5.1 Implicit functions

• Explicit function: y = f(x) F(y, x)=0 but reverse may not be true, a relation?

• Definition of a function: each x unique y (p. 16)

• Transform a relation into a function by restricting the range of y0, F(y,x)=y2+x2 -9 =0

33

8.5.1 Implicit functions

• Implicit function theorem: given F(y, x1 …, xm) = 0

a) if F has continuous partial derivativesFy, F1, …, Fm and Fy 0 and

b) if at point (y0, x10, …, xm0), we can construct a neighborhood (N) of (x1 …, xm), e.g., by limiting the range of y, y = f(x1 …, xm), i.e., each vector of x’s unique y

then i) y is an implicitly defined function y = f(x1 …, xm) and ii) still satisfies F(y, x1 … xm) for every m-tuple in the N such that F 0 (p. 195)

dfn: use when two side of an equation are equal for any values of x and y

dfn: use = when two side of an equation are equal for certain values of x and y (p.197)

34

8.5.1 Implicit functions• If the function F(y, x1, x2, . . ., xn) = k is an

implicit function of y = f(x1, x2, . . ., xn), then

where Fy = F/y; Fx1 = F/x1

• Implicit function rule

• F(y, x) = 0; F(y, x1, x2 … xn) = 0, set dx2

to n = 0

0...21 21

nn xxxxy dFdxFdxFdyF

35

8.5.1 Implicit functions• Implicit function rule

y

xdxdx

n

xxxxy

xxxxy

F

F

x

y

dx

dy

dx

dFdxFdxFdyF

dFdxFdxFdyF

nx

nn

nn

1

21

21

10.

1

2

21

21

|

such that 0dxLet

...

0...

36

8.5.1 Deriving the implicit function rule (p. 197)

0)5

)4

0)3

),()2

0),,()1

22112211

2211

2211

21

21

dxFdxFdxfdxfF

dxfdxfdy

dxFdxFdyF

xxfy

xxyF

y

y

37

8.5.1 Deriving the implicit function rule (p. 197)

,)11

)10

0)9

0)8

0)7

0)6

2

22

1

11

222

111

222111

22112211

y

y

y

y

yy

yy

F

F

x

yf

F

F

x

yf

dxFfF

dxFfF

dxFfFdxFfF

dxFdxFdxfFdxfF

38

Implicit function problem:Exercise 8.5-5a, p. 198

• Given the equation F(y, x) = 0 shown below, is it an implicit function y = f(x) defined around the point (y = 3, x = 1)? (see Exercise 8.5-5a on p. 198)

• x3 – 2x2y + 3xy2 - 22 = 0• If the function F has continuous partial

derivatives Fy, F1, …, Fm

• ∂F/∂y =-2x2+6xy ∂F/∂x =3x2-4xy+3y2

39

Implicit function problemExercise 8.5-5a, p. 198

• If at a point (y0, x10, …, xm0) satisfying the equation F (y, x1 …, xm) = 0, Fy is nonzero (y = 3, x = 1)

• This implicit function defines a continuous function f with continuous partial derivatives

• If your answer is affirmative, find dy/dx by the implicit-function rule, and evaluate it at point (y = 3, x = 1)

• ∂F/∂y =-2x2+6xy ∂F/∂x =3x2-4xy+3y2 • dy/dx = - Fx/Fy =- (3x2-4xy+3y2 )/-2x2+6xy

• dy/dx = -(3*12-4*1*3+3*32 )/(-2*12+6*1*3)=-18/16=-9/8

40

8.5.2 Derivatives of implicit functions

• ExampleIf F(z, x, y) = x2z2 + xy2 - z3 + 4yz = 0, then

yzzx

zxy

F

F

y

z

z

y

432

4222

41

8.5 Implicit production function

• F (Q, K, L) Implicit production function K/L = -(FL/FK) MRTS: Slope of the isoquant

Q/L = -(FL/FQ) MPPL

Q/K = -(FK/FQ) MPPK (pp. 198-99)

42

Overview of the Problem –8.6.1 Market model

• Assume the demand and supply functions for a commodity are general form explicit functionsQd = D(P, Y0) (Dp < 0; DY0 > 0)Qs = S(P, T0) (Sp > 0; ST0 < 0)

• where Q is quantity, P is price, (endogenous variables) Y0 is income, T0 is the tax (exogenous variables)no parameters, all derivatives are continuous

• Find P/Y0, P/T0 Q/Y0, Q/T0

43

Overview of the Procedure -8.6.1 Market model

• GivenQd = D(P, Y0) (Dp < 0; DY0 > 0)Qs = S(P, T0) (Sp > 0; ST0 < 0)

• Find P/Y0, P/T0, Q/Y0, Q/T0

Solution: • Either take total differential or apply implicit function rule • Use the partial derivatives as parameters• Set up structural form equations as Ax = d, • Invert A matrix or use Cramer’s rule to solve for x/d

44

8.5.3 Extension to the simultaneous-equation case

• Find total differential of each implicit function

• Let all the differentials dxi = 0 except dx1

and divide each term by dx1 (note: dx1 is a choice )

• Rewrite the system of partial total derivatives of the implicit functions in matrix notation

45


22

212

2

21

1

2

211

12

2

11

1

1

22

22

2

21

1

2

11

12

2

11

1

1

2212

1211

0)6

0)5

0)4

0)3

0),,()2

0),,()1

dxx

Fdxdy

y

Fdy

y

F

dxdxx

Fdy

y

Fdy

y

F

dxx

Fdy

y

Fdy

y

F

dxx

Fdy

y

Fdy

y

F

xyyF

xyyF

46


• Rewrite the system of partial total derivatives of the implicit functions in matrix notation (Ax=d)

2

2

2

2

2

1

2

2

1

2

2

1

1

1

1

1

1

2

1

1

2

2

1

2

2

1

1

1

2

2

2

2

2

2

2

1

1

2

1

2

2

2

1

1

1

2

2

2

2

1

2

1

1

1

1

1

1

2

2

1

1

1

1

1

0)12

0)9

)110)8

0)10)7

x

F

dx

dydx

dy

y

F

y

F

y

F

y

F

x

F

dx

dydx

dy

y

F

y

F

y

F

y

F

x

F

dx

dy

y

F

dx

dy

y

F

dx

dy

y

F

dx

dy

y

F

dx

dy

y

F

dx

dy

y

F

x

F

dx

dy

y

F

dx

dy

y

F

47

7.6 Note on Jacobian Determinants• Use Jacobian determinants to test the

existence of functional dependence between the functions /J/

• Not limited to linear functions as /A/ (special case of /J/

• If /J/ = 0 then the non-linear or linear functions are dependent and a solution does not exist.

0

2

2

1

22

1

1

1

2212

2111

y

F

y

Fy

F

y

F

xyxy

xyxyJ

48


• Solve the comparative statics of endogenous variables in terms of exogenous variables using Cramer’s rule

2

2

1

2

2

1

1

1

1

1 1

y

F

x

F

y

F

x

F

Jdx

dy

49

8.6 Comparative Statics of General-Function Models

• 8.6.1 Market model• 8.6.2 Simultaneous-equation

approach• 8.6.3 Use of total derivatives• 8.6.4 National income model• 8.6.5 Summary of the

procedure

50

Overview of the Problem –8.6.1 Market model

• Assume the demand and supply functions for a commodity are general form explicit functionsQd = D(P, Y0) (Dp < 0; DY0 > 0)Qs = S(P, T0) (Sp > 0; ST0 < 0)

• where Q is quantity, P is price, (endogenous variables) Y0 is income, T0 is the tax (exogenous variables)no parameters, all derivatives are continuous

• Find P/Y0, P/T0 Q/Y0, Q/T0

51

Overview of the Procedure -8.6.1 Market model

• GivenQd = D(P, Y0) (Dp < 0; DY0 > 0)Qs = S(P, T0) (Sp > 0; ST0 < 0)

• Find P/Y0, P/T0, Q/Y0, Q/T0

Solution: • Either take total differential or apply implicit function rule • Use the partial derivatives as parameters• Set up structural form equations as Ax = d, • Invert A matrix or use Cramer’s rule to solve for x/d

52

General Function Comparative Statics:

A Market Model (8.6.1)

0 )4

0 )3

T and ,Y P, of functions are S D, :Parameters

T,Y:Exogenous

P Q, :Endogenous

.continuous derivative All

commodity. on the tax theis T and income is Y Where

)0;0(,)2

)0;0(,)1

:becommodity afor functionssupply and demand Let the

0002

0001

00

00

00

//0

//0

0

0

Q) – , TPS(), T(P, Q; YF

Q) – , YPD(), T(P, Q; YF

SSTPSQ

DDYPDQ

TPs

YPd

53

General Function Comparative Statics: A Market Model

0//

0//

0//

0//

0*

0*

0*

0*

0

0

0

0

0

0

)8

)7

leftonvarsendog.onlyPut

0)6

0)5

;(4)&(3)equationsofaldifferentitotaltheTake

dTdP ,dYdP ,dTdQ ,dYdQ Find

0),()4

0),()3

dTSQdPdS

dYDQdPdD

QddTSPdS

QddYDPdD

QTPS

QYPD

TP

YP

TP

YP

54


01

1)10

.det

0

0

1

1)9

);()8(&)7(

)8

)7

//

/

/

0

0

0/

/

/

/

0//

0//

0

0

0

PP

P

P

T

Y

P

P

TP

YP

DSS

DJ

JacobiantheofsigntheCalculate

dT

dY

S

D

Qd

Pd

S

D

dAxformatmatrixinequationsPut

dTSQdPdS

dYDQdPdD

55


/

0

0/

/

/

0

0/

/

00

0

0

0/

/

/

/

0

0

0

0

1

1)13

01

1)12

dT var

)11(

0

0

1

1)11

TP

P

Y

P

P

T

Y

P

P

S

dT

QddT

Pd

S

D

D

dY

QddY

Pd

S

D

anddYsexogenouswrt

equationofsderivativetotalpartialtheTake

dT

dY

S

D

Qd

Pd

S

D

56


;0

111)13

12

01

1)12

dT var

)11(

0

0

1

1)11

0

0/

////

/

0

0/

/

00

0

0

0/

/

/

/

0

0

0

dY

QddY

PdD

DSDS

equationsofmatricesAtheinverting

bysderivativeendogenousofvectorstheforSolve

D

dY

QddY

Pd

S

D

anddYsexogenouswrt

equationofsderivativetotalpartialtheTake

dT

dY

S

D

Qd

Pd

S

D

Y

PPPP

Y

P

P

T

Y

P

P

57


;;0)15;0)14

01

1

)0;0()0;0(

;13

0

111)13

//

//

0//

/

0

//

/

/

////

0

0/

////

00

00

0

PP

YP

PP

Y

PP

P

P

TPYP

Y

PPPP

DS

DS

dY

Qd

DS

D

dY

Pd

DSS

DJ

SSDD

sderivativetheofsigntheCalculate

sderivativeendogenoustheforequationsSolve

dY

QddY

PdD

DSDS

58


0)18;0)17

01

1

)0;0()0;0(

;16

0111)16

//

//

0//

/

0

//

/

/

////

0

0/////

00

00

0

PP

TP

PP

T

PP

P

P

TPYP

TPPPP

DS

SD

dT

Qd

DS

S

dT

Pd

DSS

DJ

SSDD

sderivativetheofsigntheCalculate

sderivativeendogenoustheforequationsSolve

dT

QddT

Pd

SDSDS

59

Market model comparative static solutions by Cramer’s rule

consumedquantitymequilibriuindecreaseacausestaxesinincreaseAn

paidpricesmequilibriuinincreaseancausesTtaxesinincreaseAn

J

TS

PD

J

TS

PSPD

dT

Qd

J

TS

J

TS

dT

Pd

consumedquantitymequilibriuinincreaseancausesincomeinincreaseAn

paidpricemequilibriuinincreaseancausesYincomeinincreaseAn

J

YD

PS

JPS

YD

PD

dY

Qd

J

YD

J

YD

dY

Pd

)'16

)()'15

0

0

)16;0

1

10

)15

.)'14

.)()'13

00

)14;010

1

)13

0

00

0

00

0

0

0

0

0

0

0

0

60

Market model comparative static solutions by matrix inversion

consumedquantitymequilibriuindecreaseacausewilltaxesinincreaseAn

paidpricesmequilibriuinincreaseancausewillTtaxesinincreaseAn

J

TS

PD

dT

Qd

J

TS

dT

Pd

dT

QddT

Pd

T

S

P

D

P

SJ

consumedquantitymequilibriuinincreaseancausewillincomeinincreaseAn

paidpricemequilibriuinincreaseancausewillYincomeinincreaseAn

J

YD

PS

dY

Qd

J

YD

dY

Pd

dY

QddY

Pd

Y

D

P

D

P

SJ

)'18

)()'17

0)18;0)17;0111

)16

.)'15

.)()'14

0)15;0)14;0

111)13

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00

61

8.7 Limitations of Comparative Statics

• Comparative statics answers the question: how does the equilibrium change w/ a change in a parameter.

• The adjustment process is ignored

• New equilibrium may be unstable

• Before dynamic, optimization

Ch. 8 Comparative-Static Analysis of General-Function Models

Documents

Transcript of Ch. 8 Comparative-Static Analysis of General-Function Models