Notes Chapter 6 7

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


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


3/20

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


4/20

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


5/20

RULES OF DIFFERENTIATION

AND THEIR USE IN COMPARATIVE STATICS

Chapter 7

Alpha Chiang, Fundamental Methodsof Mathematical Economics

3rd edition


6/20

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


7/20

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


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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


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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:


10/20

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


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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


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'( ) '( )

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


13/20

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


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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


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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


16/20

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


17/20


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:


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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


19/20


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


20/20

Notes Chapter 6 7

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