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7/28/2019 Notes Chapter 6 7
1/20
COMPARATIVE STATICS AND THE
CONCEPT OF DERIVATIVE
Chapter 6
Alpha Chiang, Fundamental Methodsof Mathematical Economics
3rd edition
7/28/2019 Notes Chapter 6 7
2/20
Nature of Comparative Statics
concerned with the comparison of different equilibriumstates that are associated with different sets of values ofparameters and exogenous variables.
we always start by assuming a given initial equilibrium
state. Then ask how would the new equilibrium comparewith old.
can either be qualitative (direction) or quantitative(magnitude)
problem under consideration is essentially one of findingrate of change
concept of derivative takes significance differentialcalculus
7/28/2019 Notes Chapter 6 7
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Rate of Change and the Derivative
0 0
( )
( ) ( )
y f x
f x x f xy
x x
0 00 0lim lim(6 3 ) 6x x
y
x x xx
Difference quotient:
Derivative: y = f(x) = 3x2-4
Notation: f(x), f
7/28/2019 Notes Chapter 6 7
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Skipped Topics
The Derivative and Slope of the Curve
The Concept of Limit
Digression on Inequalities and Absolute Values
Limit Theorems
Continuity and Differentiability of a function
7/28/2019 Notes Chapter 6 7
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RULES OF DIFFERENTIATION
AND THEIR USE IN COMPARATIVE STATICS
Chapter 7
Alpha Chiang, Fundamental Methodsof Mathematical Economics
3rd edition
7/28/2019 Notes Chapter 6 7
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Rules of Differentiation for a
Function of One Variable
( )
0 0 '( ) 0
y f x k
dy dk f x
dx dx
1
( ) n
n n
y f x x
dx nx
dx
1
( ) n
n n
y f x cx
dcx cnx
dx
Constant Function Rule:
Power Function Rule:
Generalized Power Function Rule
7/28/2019 Notes Chapter 6 7
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RULES OF DIFFERENTIATION INVOLVING
TWO OR MORE FUNCTIONS OF THE SAME VARIABLE
[ ( ) ( )] '( ) '( )d
f x g x f x g xdx
[ ( ) ( )] ( ) '( ) ( ) '( )d f x g x f x g x g x f xdx
2
( ) '( ) ( ) ( ) '( )
[ ]( ) ( )
d f x f x g x f x g x
dx g x g x
Sum-Difference Rule
Product Rule
Quotient Rule
7/28/2019 Notes Chapter 6 7
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Relationship between Marginal Cost and
Average Cost Functions
2
( )
( ), 0
( ) [ '( ) ( ) 1 1 ( )
'( )
0
( ) ( )0 '( )
C C Q
C QAC Q
Q
d C Q C Q Q C Q C Q
C QdQ Q Q Q Q
for Q
d C Q C Qiff C Q
dQ Q Q
Book example:
3 212 60C Q Q Q
7/28/2019 Notes Chapter 6 7
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RULES OF DIFFERENTIATION INVOLVING
FUNCTIONS OF DIFFERENT VARIABLES
'( ) '( )
via g via f
dz dz dy f y g ydx dy dx
x y z
z y z
y x x
Chain Rule: If we have a function z=f(y)where y is in turn a function of
another variable x, say y=g(x)then the derivative of z with respect to x is
equal to the derivative of z with respect to y, time the derivative of y with
respect to x:
7/28/2019 Notes Chapter 6 7
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Examples of Chain Rule
2Example 1: If 3 , where 2 5, then
6 (2) 12 12(2 5)
z y y x
dz dz dyy y x
dx dy dx
3
2 2
Example 2: If -3, where , then
1(3 ) 3
z y y x
dzx x
dx
7/28/2019 Notes Chapter 6 7
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Examples of Chain Rule
2 17
2
17 2
16 2 16
Example 4:
( 3 2)
3 23 2
17 (2 3) 17( 3 2) (2 3)
z x x
y x xz y and y x x
dz dz dyy x x x x
dx dy dx
7/28/2019 Notes Chapter 6 7
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Examples of Chain Rule
'( ) '( )
L L
dR dR dQf Q g L
dL dQ dL
MRP MR MPP
Example 4: Given a total revenue function of a firm R=f(Q)
where output Q is a function of labor input L, orQ = g (L),
derive the marginal revenue product of labor
7/28/2019 Notes Chapter 6 7
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Inverse Function RuleIf a function y = f(x) represents a one-to-one mapping, i.e. if the function is such that
a different value of x will always yield a different value of y, the function f will have an
inverse function x = f-1(y).
This means that a given value of x yields a unique value of y, but also a given value
of y yields a unique value of x.
The function is said to be monotonically increasing: if 1 2 1 2( ) ( )x x f x f x Practical way of ascertaining monotonicity: if the derivative f(x) always adheres
to the same algebraic sign.
Examples:
1 15 5
5 25 5
5 /
1
y x dy dx
x y dx dy
dx
dy dy dx
5 4
4
5 1
1 1
5 1
y x x dy dx x
dx
dy dy dx x
7/28/2019 Notes Chapter 6 7
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PARTIAL DIFFERENTIATION
1
1 2
1 1 2 1 2
1 1
10
1 1
( , ,..., )
( , ,..., ) ( , ,..., )
lim
n
n n
x
y f x x x
f x x x x f x x xy
x x
y yf
x x
7/28/2019 Notes Chapter 6 7
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PARTIAL DIFFERENTIATION
Techniques of Partial Differentiation: Just hold (n-1)
independent variables constant while allowing one variable to
vary.
2 21 2 1 1 2 2
1 1 2
1
2 1 2
2
( , ) 3 4
6
8
y f x x x x x x
yf x x
x
y
f x xx
Example 1
Example 2
2
( , ) ( 4)(3 2 )
(3 2 ) /( 3 )
y f u v u u v
y u v u v
7/28/2019 Notes Chapter 6 7
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Applications To Comparative-static
Analysis: Market Model
( , 0) [ ]
( , 0) [ ]
Q a bP a b demand
Q c dP c d Supply
a cP
b d
ad bcQb d
7/28/2019 Notes Chapter 6 7
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Applications To Comparative-static
Analysis: Market Model
2
2
1 ( )
( )
1 ( )
( )
P P a c
a b d b b d
P P a c
c b d a b d
Four partial derivatives:
0 0P P P P
anda c b d
Conclusion:
7/28/2019 Notes Chapter 6 7
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Applications To Comparative-static
Analysis: National Income Model
0 0
0 0
( ) ( 0;0 1)
( 0;0 1)
Solution :
1
Y C I G
C Y T
T Y
I GY
7/28/2019 Notes Chapter 6 7
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Applications To Comparative-static
Analysis: National Income Model
0 0
Solution :
1
I GY
0
0 0
2
Government expenditure multiplier:
10
1
Non-income-tax multiplier:
0
1Income tax rate multiplier:
( )0
(1 ) 1
Y
G
Y
I GY Y
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