1 ch. 8 comparative-static analysis of general-function models 8.1differentials 8.2total...
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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
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8.1Differentials
8.1.1 Differentials and derivatives8.1.2 Differentials and point
elasticity
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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
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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.
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Difference Quotient, Derivative & Differential
f(x0+x)
f(x)
f(x0)
x0 x0+x
y=f(x)
x
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x
f’(x)
f’(x0)x
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A
C
D
B
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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
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Overview of Taxonomy – 1st Derivatives & Total Differentials
Differentiation Form
FunctionSpecific
(parameters)General
(no parameters)
Explicit(causation)
Implicit(no causation)
bydx
d
)(
dx )(
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xfdy
xy
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F
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dy
dxF
Fdy
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8.1.1 Differentials and derivatives
• 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
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8.1.1 Differentials and derivatives
• 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 )(')(
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8.1.1 Differentials and derivatives
• “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
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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
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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
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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
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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)
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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
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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
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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
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8.3 Rules of Differentials (I-VII)
6.
7. d(uvw) = vwdu + uwdv + uvdw
dwdvduwvud
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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
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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
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xg
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xg
xfd
xg
xgxfxgxf
xg
xf
dx
d
2
2
)'6
)6
Rules of Derivatives & Differentials
for a Function of One Variable
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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
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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
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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
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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
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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
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dvfdufdxfdxfdy
vuhx
vugx
vuxxfy
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2
2121
22121
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221
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2121
123§
§,10
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dvvfdufdvfudufdy
dvvdufdvudufdy
dvvdudxdvududxdxfdxfdy
vuxvuxxxfy
ucvc
8.4.3 Another variation on the theme
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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
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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
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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)
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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
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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...
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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
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8.5.1 Deriving the implicit function rule (p. 197)
,)11
)10
0)9
0)8
0)7
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dxFdxFdxfFdxfF
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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
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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
![Page 40: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/40.jpg)
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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
![Page 41: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/41.jpg)
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)
![Page 42: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/42.jpg)
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
![Page 43: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/43.jpg)
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
![Page 44: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/44.jpg)
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
![Page 45: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/45.jpg)
45
8.5.3 Extension to the simultaneous-equation case
22
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![Page 46: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/46.jpg)
46
8.5.3 Extension to the simultaneous-equation case
• Rewrite the system of partial total derivatives of the implicit functions in matrix notation (Ax=d)
2
2
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![Page 47: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/47.jpg)
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
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![Page 48: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/48.jpg)
48
8.5.3 Extension to the simultaneous-equation case
• Solve the comparative statics of endogenous variables in terms of exogenous variables using Cramer’s rule
2
2
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2
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1
1
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1 1
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Jdx
dy
![Page 49: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/49.jpg)
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
![Page 50: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/50.jpg)
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
![Page 51: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/51.jpg)
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
![Page 52: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/52.jpg)
52
General Function Comparative Statics:
A Market Model (8.6.1)
0 )4
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![Page 53: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/53.jpg)
53
General Function Comparative Statics: A Market Model
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![Page 54: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/54.jpg)
54
General Function Comparative Statics: A Market Model
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dTSQdPdS
dYDQdPdD
![Page 55: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/55.jpg)
55
General Function Comparative Statics: A Market Model
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![Page 56: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/56.jpg)
56
General Function Comparative Statics: A Market Model
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![Page 57: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/57.jpg)
57
General Function Comparative Statics: A Market Model
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sderivativeendogenoustheforequationsSolve
dY
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DSDS
![Page 58: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/58.jpg)
58
General Function Comparative Statics: A Market Model
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![Page 59: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/59.jpg)
59
Market model comparative static solutions by Cramer’s rule
consumedquantitymequilibriuindecreaseacausestaxesinincreaseAn
paidpricesmequilibriuinincreaseancausesTtaxesinincreaseAn
J
TS
PD
J
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consumedquantitymequilibriuinincreaseancausesincomeinincreaseAn
paidpricemequilibriuinincreaseancausesYincomeinincreaseAn
J
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![Page 60: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/60.jpg)
60
Market model comparative static solutions by matrix inversion
consumedquantitymequilibriuindecreaseacausewilltaxesinincreaseAn
paidpricesmequilibriuinincreaseancausewillTtaxesinincreaseAn
J
TS
PD
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paidpricemequilibriuinincreaseancausewillYincomeinincreaseAn
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![Page 61: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1Differentials 8.2Total Differentials 8.3Rules of Differentials (I-VII) 8.4Total Derivatives](https://reader030.vdocuments.mx/reader030/viewer/2022032805/56649ee15503460f94bf1d8c/html5/thumbnails/61.jpg)
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