Basic Math in Economics - เอกสารประกอบการสอนวิชา คณิตเศรษฐศาสตร์เบื้องต้น

.320

(Introductory Mathematical Economics) ( 2/50)

1

.

255

1

1 ........................................................................................................................................62.1

2.1.2 (MEMBERSHIP IN A SET)............................................................................................. 9

2.2 (RELATION AND FUNCTION) ...........................................................9 2.2.1 (ORDERD PAIRS AND CARTESIAN PRODUCT) ...................... 10

2.3

2.3.3 (EXPONENTIAL LOGARITHM FUNCTION) .............................. 36

3.1

3

3.1.3. : 1 .................................................. 51 4 BASIC MATRIX ALGEBRA MATRIX ALGEBRA

..104

5.1.1 (SLOPE) ........................................................................................... 104 5.2DERIVATIVE.....................................................................................................................................106

5.2.1 (CONTINUITY) ....................................................................................................... 106

2 .........................................................................7 (THE CONCEPT OF SETS)..........................................................................8 2.1.1 (SETS)..................................................................................................................................... 8

2.1.3 (RELATIONSHIPS BETWEEN SETS) ............................................................. 9

2.2.2 (FUNCTION)...................................................................................................................... 12 (FUNCTION AND TYPES OF FUNCTION) ...........................32 2.3.1 (CONSTANT FUNCTION) ........................................................................................ 32 2.3.2 (POLYNOMIAL FUNCTION) ............................................................................. 32

2.3.4 ..................................................................................................................... 40 2.4. (INVERSE FUNCTION)..............................................................................40

3 (STATIC EQUILIBRIUM ANALYSIS) ...................................42 .....................................................................................43 3.1.1 (A SIMPLE MACRO ECONOMIC MODEL)............................. 43.1.2. IS-LM ...................................................................................................................... 48

............................................................................................................................................................66

5 ..............103 5.1 N (POLYNOMIAL OF DEGREE N) ..............................................

(DIFFERENTIATION / DERIVATION) / PROCESS OF OBTAINING THE

2

5.2.1 SMOOTH FUNCTION .................................................................................................................... 107

5.3

LOBAL OCAL EXTREMUM CONCEPT..................................................................................5.3.2 1 (FIRST-DERIVATIVE TEST) ................................................. 116

7.1

7.3

7.4

7.4.2 DIFFERENTIAL............................................................................................................ 159

CONVEXITY AND CONCAVITY , MAXIMA AND MINIMA ..................................................115

5.3.1 G V.S. L 115

5.3.3 2 (SECOND-DERIVATIVE TEST) ............................................. 119 5.3.3 (CURVATURE OF A GRAPH) ............................................................................... 120

6 ().....1236.1 ()............124

6.1.1 ................................................................................................... 124 6.1.2 ...................................................................................... 129

6.2 ..............................................................................131 6.2.1 (COMPETITIVE MARKET)............................................................................. 131 6.2.2 (MARKET POWER: MONOPOLY) .............................................................................. 134 7 1 .............143 FUNCTIONS BETWEEN EUCLIDIAN SPACES .......................................................................144

7.1.1 (GEOMETRIC REPRESENTATION OF FUNCTION) ......................... 145 7.1.2 GEOMETRIC REPRESENTATIONS OF FUNCTIONS OF SEVERAL VARIABLES......................................... 146 7.1.3 LEVEL CURVES FOR Z = F(X,Y)..................................................................................................... 147

7.2 PARTIAL DERIVATIVE .............................................................................................................149

7.2.1 (ECONOMIC INTERPRETATION)........................................................ 151 THE CHAIN RULE ....................................................................................................................152

THE TOTAL DIFFERENTIAL OF A FUNCTION OF SEVERAL VARIABLES ........................157

7.4.1 TOTAL DIFFERENTIALS.................................................................................................................. 158

7.4.2 THE TOTAL DERIVATIVES ............................................................................................................. 160

7.5 IMPLICIT FUNCTIONS.............................................................................................................163

3

1 )....

8.2

8.3

(COMPETITIVE IRM INPUT HOICES 6

)....

10

8 (......................................................................................................................................................164

8.1 THE DIFFERENTIAL (OPTIMIZATION CONDITION) .......................165 8.1.1 / (EXTREME VALUE OF A FUNCTION OF TWO VARIABLES) ........................................................................................................................................... 166

QUADRATIC FORMS...............................................................................................................169

............................182 8.3.1 PRICE DISCRIMINATION................................................................................................................. 182 8.3.2 F C :COBB-DOUGLAS TECHNOLOGY)............................................................................................................. 188.3.3 1 (PROBLEM OF A MULTIPRODUCT FIRM)........... 188 8.3.4 1 (MULTIPLANT FIRM PROBLEM) ............. 191 9 ( 1 ......................................................................................................................................................195

9.1 ( 1 )......196 9.1.1 2 1 : OPTIMIZATION WITH EQUALITY CONSTRAINTS :TWO VARIABLES AND ONE EQUALITY CONSTRAINT ................................................................................ 197

9.2 .......................................207 9.3 ( MINIMIZE TOTAL EXPENDITURE UNDER THE LEVEL OF UTILITY) .......................................................................211

9.4 .........................................................................2129.5 (LEAST COST COMBINATION OF INPUTS) (CONDITIONAL INPUT DEMAND) .............215 9.6 (PROFIT MAXIMIZATION)

(UNCONDITIONAL INPUT DEMAND) ............................................................................218 10 INTEGRATION INTEGRATION ...................225.1 (INDEFINITE INTEGRALS)...................................226 10.1.1 (ANTIDERIVATIVE) (INTEGRATION) ...................................... 226 10.1.2 BASIC RULE OF INTEGRATION..................................................................................................... 226 10.1.2 INITIAL-VALUE PROBLEMS.......................................................................................................... 230

4

10.2 (DEFINITE INTEGRALS) ..........................................231 10.2.1 A DEFINITE INTEGRAL AS AN AREA UNDER THE CURVE ............................................................... 232 10.2.2 (SOME PROPERTIES OF DEFINITE INTEGRAL)........ 234.3 IMPROPER INTEGRAL

10

10

ONSUMER AND PRODUCER SURPLUS10.4.6. FIRST DEGREE PRICE DISCRIMINATION PERFECT PRICE DISCRIMINATION .............................. 242 10.4.7. ............................................................................................................... 243 10.4.8 : (RANDOM VARIABLES)........................ 244

........................................................................236

.4 INTEGRATION ..............................................23810.4.1. (RECOVERING TOTAL COST FROM MARGINAL COST).................................................................................................................................. 238 10.4.2. (TOTAL PROFIT FUNCTION) (MARGINAL PROFIT FUNCTION) ........................................................................................................................................... 238 10.4.3. ........................................................................ 239 10.4.4. ........................................ 240 10.4.5. (C ) .............. 240

5

2

unctions)

(Non-linear Function) ion)

(Exponential and Logarithm Functions) (Inverse Function)

: 1 (Relations and F- - - (Linear Function) - - (Polynomial Funct- - -

7

2.1 (The Concept of Sets) 2.1.1 (Sets) A set is simply a collection of distinct objects. These objects may be a group of (distinct) numbers, persons, food items, or something else : Chiang (2005) 2 1. (Enumeration) 2. (Description) 1 S 3 2, 3, 4 S Enumeration I (all positive integers) I Enumeration I Enumeration I (Description) : J (All real numbers) 2 5 A finite set An infinite set

8

2

2.1.2 (Membership in a set) (epsilon) (is an element of)

2.1.3 (Relationships between Sets) 2 3 1: 2:

2.2 (Relation and Function)

(Ordered Set) (Ordered Set) 2 Ordered Pairs 3 Ordered Triples 4 Ordered Quadruple 4 (Ordered Pairs) . 320

9

2.2.1 (Orderd Pairs and Cartesian Product) xy (Cartesian coordinate plane) Cartesian Plane 2 (Cartesian Product Direct Product) 5 x y

10

6 ( ){ }, 2x y y x= ( ){ },x y y x

( ){ }, 2x y y = x x y x y (One-to-One Relationship One-to-One Mapping) ( ){ },x y y x x y

11

x y 1 x y (One-to-Many Relationship One-to-Many Mapping)

2.2.2 (Function) y x (Function) y x y x y x ( y equals f of x) x (Argument) (Independent Variable) y (Value of Function) (Dependent Variable) x (Domain) y x (Images) (Range) x y 1 (One-to-Many Relationship One-to-Many Mapping) (One-to-One Relationship One-to-One Mapping)

12

function (a) (b) 7 (Total cost:C) (Total output:Q) 150 7QC = + 100 : (Domain) (Range)

13

Powerpoint Set

14

2.3 (Function and Types of Function)

(Constant Function)

Range 1

8 (National income Model) Investment) Autonomous investment

2.3.1

( )0I ( 100

2.3.2 (Polynomial Function)

32

polynomial degree Polynomial

1 Polynomial Function

Degree n functionn=0 n=1 n=2 n=3 n=1

=3 n=4

n=1

n=2

n

33

9 (Total Cost: TC)

TC a bQ= + n 1

2 (Quadratic Function)

0 0

x 1 Quadratic formula (Completing the

square) (Root of Function) x 1 ( ) 0f x y= = Cubic Fuction degree

3 3

34

2 42

b b aa

Quadratic formula c 10

11

2( ) 8 16 0f x x x= + = 2( ) 8 16 0f x x x= + = 12 2( ) 11 18f x x x= +

35

13 2( ) 8 20f x x x= +

x (Factoring the function)

x x

2.3.3 ential Logarithm Function)

lem)

15 t bt b

= == =

2 14

4 4x +

(Expon

(Growth Prob

(Exponential Function)

3 2

2

( )( )

t

t

y gy f

36

16 t b

y f t b= == =

e Natural exponential function

( )( ) 2

t

t

y g

e (: Irrational number)

ty e=

2.71828...e =

e

Natural Exponential Functions and the Problem of Growth

1( ) 1f m = +

m

m

37

111 21

f + = 21(2) 1 2.25f =

3

4

(1)

2

(4) 1 2.441414

=

= + 1(3) 1 2.370373

f = 1f

= +

= + =

e

m

1 m lim ( ) lim 1m m

e f mm

=

A (Initial Principal) t (Continuous-compounding process) 1 t

t

t

m t .

+ t t

38

( ) 1mtrV m A m= +

1

m

1( ) lim 1rtw

m

rt

mV m A whereww r

Ae

= +

=

= Error! Objects cannot be created from

(Logarithm) (Inverse function of

xponential)

y==

editing field codes. e

log

t

e

y et

39

2 .3.4

(Power Function)

2.4. (P) (X)

1 21 2

b

b b

y Axy Ax x==

(Inverse Function)

3 2x p= (1) (Marginal Revenue) 1 Demand Function 2 Inverse Demand unction F

40

( ) 2 3y f x x= = +

2( )y f x x= =

41

3 (Static Equilibrium Analysis)

(Static Equilibrium Analysis) : 3

- ltaneous

- (Simple Macroeconomic Model)

- Simu- - (Break-even Point) - - - - -

- - (Elasticity)

- IS-LM

42

3.1

3.1.1 (A Simple Macro economic Model) (Closed Economy)

1

(Closed Economy) (National Income) (Total spending) (Definitional equation)

(3.1) (Gross Domestic Product)

C (Consumption spending) (Investment spending: I)

Y I

( )C

0C C bY= + (3.2)

0C b

43

( )I

0I I= (3.3)

0I

(1)

(the equilibrium level of DP)

(3.5)

(3.5)

(3.6) (3.5) (3.6)

(Exogenous Variables) (Parameter variables)

(3.4) 4 G 0I I= I I= 1

0( )I I= 1I I=

44

11 b (Keynesian Investment

Multiplier)

1 2 (National Income) (Total spending) (3.7) (3.7)

45

0( )G G=

( )T tY=

(Exogenous variable)

(3.8) eY = (3.9)

eC = (3.10)

1 (Comparative tatic) (3.9) .10)

S(3

0G 1G

46

(Multipler: k)

3

d

I (3.11)

ures) ( )

Y =

dY Y TT tyC C bYI

= == +=

0

0

0

0

G GX XM mY

===

Y, C, M (Autonomous Expendit 0 0 0 0, , ,C I G X

47

(B) (G)

.1.2. IS-LM 3 (A

(a market for money) (Quantity of money supplied :

3 market for good) M 0

DM fY r= ) ,f ) (Quantity of money

emanded :d (Positive constants)

(Conditional Equation)

(3.12)

0

0

0

0

.........................

d

d

Y C I GY Y TT tyC C bYI I erG G

fY r M

= + += == += =

=

(3.13)

48

(Endogenous variables) (3.13) (Exogenous variables) (3.13) hree levels of Autonomous spending) (the constant stock of

( 6 ) ,r)

/( / Equilibrium

the good market requires that planned investment and planned saving be equal) (3.13) (Y,r)

LM (LM Curve)/ / (L for liquidity

and for money is sometimes known; M for the quantity of oney supplied).

I

(3.14)

(tmoney) (Y IS (IS Curve)in (preference, as the demm S

49

LM (3.13) (r) (Y)

(3.15)

(3.14) 5) (

(3.16)

(3.17)

IS LM 2 IS LM

*Y ) (3.1 ( *Y ) (3.14)

50

2 2 (Endogenous Variables) 3 . . .

(Clearing Market

ondition)

8)

(3.18) (Conditional Equation) ? ehavioral Equations)

r

(3.19)

3 : 1 .1.3.

123 C (3.1

(B (A decreasing lineafunction of P) 2

51

(An increasing linear

function of P)

(3.20)

1 (Conditional equation)

dQ a bP a b= > P (3.21) (Partial Equilibrium) () (General Equilibrium)

.21) 3 3 Simultaneous

m 3 21

(Elimination of Variables)

d sQ Q=

( , 0) 2 (Behavioral equations) sQ c d= +

(3Equation syste

3 System of

Linear Equations

52

(Excise Tax) (Excise Tax)

2 . Specific Tax

1 ()

. Ad Valorem Tax2 ()

(Comparative Static Analysis) 2

53

1 t (Specific Tax) 1.1

1.2

54

2 x 2.1

2

.2

55

(3.19) (3.19) (P) (Q) a 3.20) (3.20) (P) (Q) -c -b d () (Elasticity)

(3.22)

(

56

57

(3.23)

(Absolute V

(Price Elasticity of Demand)

alue)

1pE < 1pE > 1pE =

P =

PQ

Q b

58

(Total Revenue: TR

TR = (3.24)

Inverse Demand Function P = (3.25)

TR = (3.26)

(3.26) 1) ..................................... 2) Q

Inverse Demand Function (3.25)

)

P (25) TR (24)

3) ................................... TR (3.26)

and) (Individual Demand) . n nq a b P= + 1 n P (the horizontal sum of the individual demand curves)

(Market Demand and Market Supply)

(Behavioral Equations) 2 (Market Dem

(Market Demand)

1 1 1 1dq a b P= +

2

n n P

2 2 2dq a b P= +

d

60

61

! (Market Supply) (Individual Supply) 1 1 1 1sq c d P= + 2 2 2 2sq c d P= + . j sj j jq c d P= + P 1 j P (the horizontal sum of the individual supply curves)

1 A 3

A

1 6 10 16 2 4 8 13 3 2 6 10 4 0 4 7 5 0 2 4

3 (Individual Demand function) (Market Demand function)

(Market Supply)

62

A

A A A A

2 I J

I

)

)

)

) (Identical Agent Homogenous Agent)

, 1, 2,...,

, 1, 2,...,

di i isj j j

Q P iQ a b P j J

= + == + =

[63]

3

400 2dmQ P=

8SiQ P= +

) 10 ) 56

.1) 10 .2)

[64]

(Break-Even Point) 1 300 10 . . 1 . 1 30

2 ( )

S C MPC MPS

Yd 100 -80 200 -60 2.1 S C 2.2 Break-Even income 2.3 I= 50+0.1Y C I 2.4

[65]

4 Basic Matrix Algebra Matrix Algebra

Basic Matrix Algebra Matrix Algebra : 4 Basic Matrix Algebra

- Terminology Matrix Algebra - Special Matrices - Matrix

, , Matrix, Matrix (Scalar) - Matrix Algebra - Inverse - Determinants - Inverse Matrix Determinants - The Determinants and Non-singularity - (Cramers Rule)

[66]

Powerpoint: Basic Matrix Algebra and its applications in economics4

4 Powerpoint Powerpoint (.421) ...

[67]

5

: 4

- (Quadratic Function) - (Non Linear Function) - (Slope)

- (Rule of Differentiation) (Non differentiable Function)

- Convexity Concavity - - (Maxima - Minima) - (Inflection Point)

- (Derivatives)

-

- (Derivatives) (Marginality) - - (Elasticity)

[103]

5.1 n (Polynomial of Degree n)

a x

n n = 1

(linear function) (5.1)

polynomial of degree n 20 1 2 ...y a a x a x= + + + + nn y a b= + x

( )1 1,x y ( )2 2,x y (5.1)

y

x

(slope) (slope) y x 1

1 (Marginal Cost)

1 (Marginal Utility) (Linear function) (slope)

5.1.1 (C F q=

)

( )u U x=

[104]

= 2

( ) 20 1 2f x y a a x a x= = + +

y

x

x

3h

x 2x h = x 1x h = slope ( y x

0x T

A )

/

x ( )f x x (Derivative of f at x ) ( )f x x ( )'f x , ( )df x

dx ,

x , , ( )x 'y dydx D f

1 ( ) 22 4f x x= + ( )'f x ( )' 4f x x=

( )f x '

[105]

5.2 (Differentiation / derivation) / process of obtaining the derivative

smooth continuous smooth and continuous

.2.1 (Continuity) 5 f a 3 1. ( )

f a

2. ( )limx a

f x . 3 ( ) ( )lim

x af x f a =

2 0.6x = ( )g x

( )0.6f = 1. 2. ( )

0.6limx

f x ( )

0.6limx

f x ( )0.6limx f x+ ( )f x

0.6 3

0.6x =

( )f x = 2x 2x <

1x +

y

2x

x

[106]

5.2.1 Smooth Function / (smooth)

x

f a ( ) ( ) ( )'

0limh

f a h f af a

h+ =

x a h= + x a ( )'f a 0h

( )x a

f ax a

= , ( ) ( )' lim x a f x f a

f

a a

x x x x (continuity) (Necessary)

[107]

3 ( ) 2 1f x x= +

:

x (all differentiable functions are )

(continuity) (differentiable)

continuous) (But not all continuous functions are differentiable

( )0f c or f c : (f is continuous) f c or ( )1 'f c : (f is continuously differentiable) ( )0c c ( )1c 'c (Rule of Differentiation) 1. f(x) = c c ( ) =f x 2. g(x) =cf(x) c ( ) =f x . f(x) = nx n ( ) =f x 3

[108]

4. f(x) = ( ) ( )U x V x U V x

( ) =

f x

4 f(x) = 232 x ( )f x

5 f(x) = 13 22 3 5+ +x x x ( )f x 5. f(x) = ( ) ( )U x V x U V x

( ) =f x

6 ( )7 18 100f (x) x 2x=

6. f (x)V(x)

= U V x

( ) =f x

U(x)

[109]

2 36 4 3f (x) x x x= + ( )f x 7

tion) Chain Rule

7. f(x) =U(V(x)) U(V(x)) V(x)

(Composite Func

( )f x

8 2 3f (x) (x 4x )= ( )f x 9 2f (x) (1 x ) 1 2x= ( )f x (Derivatives of Inverse Function)

f (Increasing and Continuous function) ,b] g f f(x) 0 x (a,b) g(x) 0 y y = f(x)

[a

[110]

10 y = f(x) = 3x dxg '(y) /dy

t Differentiation)

(Implici

11 2if 2xy y+ dy x ydx

= +

12

2 2dy if 6x 2y 5y 16xdx

+ = 3

[111]

(Derivative of Logarithmic

Function and exponential Function)

xy b= x = yx e= y =

(Natural logarithm) ln Log ln

yln xe

==

xyxy

==

. f(x) =

ln

ln

ln x ( )f x 8

. 9 y ln V(x)= ( )f x 0. 1 V(x)y e= ( )f x

[112]

11. xa where a 0 and a 1> ( )f x y =

12.

ay log x= ( )f x

(x) e

13 f '(x) if f

3x 1=

14

15

2f '(x) if f (x) ln(1 2x x )= + +

2x12f '(x) if f (x) (1 x)(1 e )(3 x)= + +

[113]

2

23

2 3xf '(x) if f (x) (x 1)1 2x= 16

(Second and Higher Derivatives) f 1 (First derivative) 2

f(x) f lih 0

f '(x h) f '(x)mh

+ 2 f(x) x (Second derivative of f(x) with respect to x)

3

17

f(x) x f(x) x f(x)

4 1

4 3 3y x x 6x= +

'''f (x)

[114]

5.3 Convexity and Concavity , Maxima and Minima

/ (Local max/ Local min)

5.3.1 Global V

(1) (2)

(3)

.S. Local Extremum Concept

a, b , c d

a (Constant Function) b (Strictly Increasing Function)

! (Strictly Increasing Function) f(x) < f(y) x y x < y

[115]

C

d

5.3.2 1 (First-Derivative Test) y = f(x) (Local Max/ Local Min)

1 2 1. 2.

(d)

0x x=

(Smooth Function) (Necessary Condition) (Local max/ Local min)

[116]

x (Critical point) (Stationary Point) !

f (x) 0 =

f (x) 0 = (Sufficient Condition)

(Necessary Condition) f (x) 0 = a

f (x j) 0 = = a x = j f(j) (Sta x = j b

f(x) 1

1 1 f(

tionary Value) x = j (Inflection Point)

f(x)

x) 0x x= 0 ( 0(x ) 0 = ) 0f (x )

(-)

1. 1 (+)

f

0x x= 0x x=

[117]

2. 1 (+) (-)

3. 1

0x x= 0x x=

0x x= 0x x= 18 /

19 (Average-Cost Function)

3 2y f (x) x 12x 36x 8= = + +

2AC f (Q) Q 5Q 8= = +

[118]

5.3.3 2 (Second-Derivative Test) 2 (Interpretation of the Second Derivative)

1 ( ) 2 ( )

(x ) f(x) (x ) < 0

(x ) f(x) (x ) < 0

(x ) > 0 >0 > 0 0 < 0 < 0

f (x) f (x)f

> 0 0

f 0 f > 0 0f 0

f 0 0f (x )

0f (x ) 0f (x )

0f (x ) 0f (x )

0f (x ) 0f (x )

[119]

a

5.3. (Curvature of a Graph)

2 Strictly Convex/ Convex

f(x) Strictly Convex M N f(x) MN MN

f(x) Strictly Concave M N f(x) MN MN f(x) x

b

3 Strictly Concave/Concave

f(x) x

[120]

2 x Strictly Concave function x Strictly Convex function !

.2 2 1 f(x)

0f (x )

0f (x )

1 0x x= 0 ( 0f (x ) 0 = )

1. 2.

(Necessary Condition) (Sufficient ondition)

(Maximum) (Minimum)

0f (x )

C

First-order necessary Second-order necessary Second-order sufficient

[121]

20 ( )4 21f (x) x 8x8=

21

42x 3g(x) x

4 2=

22

4f (x) x=

[122]

6

) ()

(Maximize Profit): - -

- - (Lump-sum Tax) - (Profit Tax) - (Excise Tax)

-

(

: 3 -

[123]

6.1 ()

(Optimization) ( a limitation of resources) (The essence of the

ilable)

6.1.1

optimization problem is to choose the BEST alternative ava

[124]

? 1. 2. 3.

1.

(Necessary Condition) (Sufficient

ndition)

(Maximum) (Minimum)

Co

First-order necessary S d-order necessary econSecond-order sufficient

[126]

6.1.2 (Firms Maximization) (Total Revenue)

q p (q)

Q* Q

1. 2.

(Total Cost) C

FONC: SOSC:

[129]

TR TC MC MR

[130]

6.2

6.2.1 (Competitive Market) Concept

s, many sellers) 2. (Homogeneous product)

formation llers)

(Free entry and perfect mobility)

1. (Many Buyer

3. (Perfect inamong buyers and Se4.

ONC: ONC:

FS

[131]

1 Q

60 (Shut down price?)

2 uglas technology

2100C Q= +

Constant-returns Cobb Do

4w = 1 12 2Q L K= L K

r=1

[132]

48

(Market Demand) 294 /dQ p= (Short un Profit)

R

(Long-run equilibrium) Concept

1. 2.

=

[133]

6.2.2 (Market Power: Monopoly) ? - - - - :

(Average Revenue:AR)

(Marginal Ravenue:MR)

[134]

(Total Revenue:TR) (Total Cost: TC) = Q ONC:

: Rule of Thumb for Pricing

P

F SOSC:

(Measuring Monopoly Power) = MC Q* (Searching for the optimal price) MC P MC Rule of Thumb

! Abba Lerner .. 1934 Lerners Degree of Monopoly Power

[135]

( )P MCLP= =

he Lerner Index 0 1 Lerner Index

1. Curve) (Firms Demand Curve)

T L

L (Elasticity of Market Demand

2. . ( ) L ( )

.

.

.

[136]

(Taxation)

P a bQ= TR= (Tot Cost)

TC=cQ+f

1

al

(Lump sum Tax) (Total Profit after Tax) = FONC:

OSC:

FONC

S

[137]

Q* P* (T) Neutral Tax 2 (Profit Tax) t (Total Profit after Tax)

= ONC:

OSC:

F S

FONC

[138]

Q P* * *.t

eutral Tax ! 3 (Excise Tax)

(Total Profit after Tax)

N

t (Specific Tax) =

ONC:

OSC:

Q* P*

F S FONC

[139]

Distortion Tax (Comparative Static) ()

*Qt

=

*Pt

=

*

t

=

TF S

*Q = t otal Tax Revenue =ONC:

OSC:

*.t Q =

Tax Revenue

[140]

(Specific Tax) t

T =

1

100 0.01P Q= Q () P (1) (2) (3

() 50 30,000C Q= + 10

)

[141]

2:

P a bQ= TC=cQ+f (Sale Tax) t

t

. P* Q*

T tPQ= ( )

. FONC SOSC

* .

[142]

7 1

1

- (Partial Differentiation) - (Second-order Partial Derivative) - (Differentials) - (Total Differentials) - (Total Derivative) - Implicit Functions

- (Partial Market Equilibrium) - (Multipliers)

- -

-

[143]

(Comparative Static)

1( )y f x=

One-variable calculus (Production function) (Cost function) (Profit function)

1

Multivariable

t es

efinition

(Utility Function) (Demand Function)

1( ,..., )ny f x x=

calculus

7.1 Func ions between Euclidian Spac D A function from a set A to B is a rule that assigns to each object in A, ne and only one object in B. In this case, we write 1

x y x y= + omain: ange:

:f A Bo

2 1: R R ff 2 2( , )DR

1 1:f R R 1

( )g xx

= Domain: Range: X 1 domain f(X) 1 f(X)

Range

nction f X X)

f (fu

[144]

2 unctionhe constant elasticity demand function

nR R F 1 1 2 1 1 2( , , )q f p p y k p p y = = T

The firms production function

( )

1 1 2 2

1 2

1 2min ,x x

q input =

1 2

1 1 2 2 tanb

a a a

a x a x Linear

kx x Cobb Douglas

outputc c

k c x c x Cons t Elasticity of Substitution

= + =

+

he firms production function that uses three inputs to produce two outputs f x x x q f x x x

q q f x x x f x x x F x x x

= == = =

.1.1 (Geometric representation of function) , its all that counts.

(Increasing function) ecreasing function)

Demand negative ope Demand (Diminishing marginal

roduct)

q

q

=qFunction k mR R T

( ) ( )( ) ( )( ) ( )

1 1 1 2 3 2 2 1 2 3

1 2 1 1 2 3 2 1 2 3 1 2 3

, , , ,

( , ) , , , , , , ,

q

q 3 2:F R R

7The picture is not just worth a thousand words; sometimes

1R 1R

(D slp

[145]

7.1.2 Geometric Representations of Functions of Several Variables ( , , )y f x y z= 3

(x,y,z) 3

. x=a .

px qy rz m+ + =

. 2 2( , )z f x y x y= = +

. 2 2 2 2x y z+ + =

[146]

Budget Constraint y L K : Cobb-Douglas (L) (K)

( , )y f L K=

7.1.3 Level Curves for z = f(x,y)

3 x y f(x,y) = c z=c plot z= c xy Level Curve z=c Level Curve

( , )f x y c=

[147]

4 Level Curve 2 2( , )z f x y x y= =

5 Level Curve

Isoquant Level curve Level curve y=10, y=20 Isoquant 1 Cartesian Plane

: Level Curves for a Production Function

2

2 4

!

[148]

7.2 Partial Derivative

1 2( , ,..., )ny f x x x=

ix y (i=1,2,...,n) 1 2, ,..., nx x x

ix 1x ix y

y 1x

1 0x

110

1 1

limx

y yf

x x

1xy1x

with respect to 1x ) y (The partial derivative of y

6 1 2 1 1 2 2( , ) 3 4y f x x x x x x= = + +

1

yx 2

yx

2 2

7 ( , ) ( 4)(3 2 )y f u v u u v= = + + y

u

yv

[149]

8 2(3 2 ) /( 3 )y u v u v= + yu

yv

9

( )1 2 1 2, loU x x x x= g

xMU yMU

ive of the following function

3 6

1. Compute all the partial derivat s a) 4x

2

2 3 xy y xy x + b) xy d) 2 3x ye + c)

e)x yx y+ f)

23 7x y x y 2. Compute the partial deriv

nt Elasticity of Subative of the Cobb-Douglas function

stitution (CES) production function

) 1 21 1 2

a aq k x x= and of the Consta

( 1 1q k c x = 2 2 ba a ac x + assuming that all the parameters are positive

[150]

7.2.1 (Economic Interpretation) : Marginal Product

1

F K L= 2 (L) (K)

L=L* K=K*

( )y f x= ( )f x

Q ( , )

* *( , )F K LK

K K

K =1

K * *( , )F K L

kMP

1 Marginal product of capital ( )

* *( , )F K LL

LMP

1 Marginal product of labor ( )

10 Cobb-Douglas production function Q=

3 14 44Q K L=

L= 625 K = 10,000

10

LMP KMP

[151]

ti , interpret ( )* *,T x y

x 1.1 If ( , )T x y is the temperature func on

2 13 39Q L K= 1.2 Consider the production function

a) What is the output when L = 100 and K = 216? b) Use the marginal analysis to estimate Q(998,216) and Q(1000, 217.5) c) Use the calculator to compute these two value of Q to three decimal places

stim 1.3 The demand function

and compare these value with your e ate in b.11 12 1

1 1 1 2a a bQ K P P I= is called a constant elasticity

demand function a) Compute the three elasticities ( own price, cross price, and income), and show that they are all constants b) What are reasonable ranges for the four parameters in

Composite function Composite

function

1Q

7.3 The Chain Rule

t ? Definition:im

Let and be two functions. Suppose that B, the age of f , is the subset of C , the domain of g . Then, the composition of f with g,

:f A B :g C D

:f A D , is defined as the function ( )( )g f =xD

g D

: The composition of f with g

[152]

z x y ( , )F x y= z

x y t ( ), ( )x f t y g t= =

), ( ))z F f g t( (t=

11 (Direct Substitution) Chain rule

1 2log( )y x x= + 1x t= 22x t= dydt

12

0dz

at tdt

= if 2

2 22

5 31 1

2tt xy

1x t y t w ew y+= = + = + = + z

[153]

Higher-Order Derivative

ix 1 2 n 1 2 n

partial derivative i

fx

Continuously Differentia

f ( , ,..., )y f x x x= , ,...,x x x

ble Function y x= der

f

1C

ivative x=0 (J) J

x J f is continuously differentiable on J

x ( )f x

( )f x x J is twice differentiable on J

x is twice continuously differentiable on J

( )( ) ( )f x f x = f( )f x f 2C

( )*

i

fx = x

i is differentiable at f *x ( )*i

fx x

1C *x

f is continuously differentiable on J at

( )*i

fx x differentiable

j i

fx x

i jx x second order partial derivative of f 2

j i

fx x

2f

2f

i ix x

2

j i

fx x

i j Cross partial derivatives or mixed partial derivatives 2ix

[154]

Second-order Derivative and Hessians

13 4 44Q K L= all second derivatives

3 1

function 2 second order partial erivative 4 n second order partial derivative Matrix

d

2n n n (i,j) ( )2 *

j i

fx x

x Matrix Hessian Matrix

2D f =x 14 ( ) 21 21 2 1, x xy f x x x e += = 1 2,f f Hessian Matrix

[155]

Second Partial Derivative

15 Cobb-Douglas function

: (YoungTheorem) ( )1,....,= ny f x x J

2C nR

( ) ( )2 2

j i i j

f fx x x x = x x

( , , ) a b c , , ,A a b and c are positive K L T AK L T= F Marginal Products the econd-order partials

S

Noted: Youngs Theorem Hessian Matrix Symmetric matrix

[156]

7.4 The Total Differential of A Function of Several Variables differential

( )dy f x dx= the differential y x he tangent to the function at x) y x

)

(t

( ) (y f x x f x = +

x ( )0x y

( )y f x x x = + 0

dy dx dy dx differential of x differential of y

( )0x ( )f x

16 23 7 5y x x= + dy

Remark

1.

3. 2.

[157]

dy y=f(x) Differentiation Differentiation

dydx

dy Differentiationdx

with respect to x

.4.1 Total differentials differential

S Y i=

S Saving I Interest rate

S (ds) y i (dy,di) Total differential

U

7 Concept S ( , )

Y national Income

1S C dS =

: nf R R 1 2( , ,..., )nU U x x x=

d =

[158]

7.4.2 differential 1 2( , )y f x x=

Rule:1 ule:2 ule:3

0dk =R ( ) =nd cUR ( ) =d U V Rule:4 ( )d UV Rule:5

= =

UdV

Rule ( )d:6 U V W

2x x+

18 2

19

=Rule:7 ( ) =d UVW 17 5 3y = 21 2 21 13y x x x= +

1 2212

x xyx+=

[159]

20

.4.2 The Total Derivatives

f

Total Derivative

Channel Map y x w

1 2 33 (2 1)( 5)= +y x x x

7

( , ) ( )y f x w where x g w= = y x w w y (1) (Indirect Effect) g (2) (Direct Effect) f partial derivative

[160]

y xf g

f

w

o l derivative total differential x wdy f dx f dw T ta = + dw

x wdy dx dwf fw dw= +

d dw=

Total derivative dy The total d dw

ifferentiation of Y with

respect to w

21 Total derivative dy dw

Y=f(x,w)=3 ( ) 2 4x w where x g x w w = = + + 2 2 C

( ) ( )y f x x here x g w x h w= = =

22 U=U(C,S)

SS=g(C)

( )1 2 1 2, ,w w dy = dw

[161]

f

2

f h

y 1x w f g

x

23

=L(t)

( , , )Q Q K L t=K=K(t) L

( ),x g u v= ( )1 2, , ,y f x x u v= ( )1

2 ,x h u v=

dy = du

dydv

= 24 Total derivative dz

dy

z f x y x xy y where x g y= = + = =

( ) 2 2, 2 , ( ) 3y

[162]

25 Total derivative ( ), ,z f x y t where x a bt and y c dt= = + = +

26 partial total derivative

wu

wv

2w x= a bxy cux u vy u

where

+ += +=

7.5 Implicit Functions

Explicit Function y x

( )y f x=

4( ) 3y f x x= =

43 0y x =

( )1, ,..., 0mF y x x =

implicit function

( )1,..., my f x x=

[163]

8

( 1 )

( 1

: 3 ( 1 )

- ( 1 ) - Third Degree Price Discrimination - Multiple-Plant Firm - Multiple-Product Firm

)

[164]

[165]

8.1 The Differential (Optimization Condition) 1 (Optimization

onditions for problems with a single choice variables) derivative

e ntials

irst Order Necessary Condition

FONC

c Diff re

F ( )z f x=

SOSC:

( )z f x= differential (Maximum) (Minimum)


8.1.1 / (Extreme Value of a Function of wo Variables) T

( , )z f x y= First-Order Necessary Condition

1

0 dx dy dz =

2

[166]

(Necessary Condition) (Sufficient Condition) 2 2 [b]

otal differential econd-Order Total differential

1 dz

econd-Order Sufficient Condition

1 2 [a]

Second-order tS

x ydz f dx f dy= +

2d z

3 25z x xy y= + 2d z S

( , )z f x y=

[167]

( , )z f x y= / ( ),z f x y=

(Maximum) (Minimum)


2 / 3 2 28 2 3 1z x xy x y+ + +

=

3 /

22 x yz x ey e e= +

[168]

8.2 Quadratic Forms polynomial 1

0 1 ...n

na a x a x+ + + 1

2 33 4 2x x y xy+ polynomial 1 term term (The sum of exponents each term is uniform) polynomial Form

in 4 9x y z

2 24 3x xy y + 2 22 7x xy yw w+ +

Sec rond-Order Total Diffe ential as a Quadratic Form

2

2 2 2xx xy yyd z f dx f dxdy f dy= + +

u

ab =h

===

=

2d z

[169]

q

positive definite

q Non-positive negative semidefinite q negative definite

Matrix Form

Matrix q determinant

positive definite

negative definite

q q ( 0)> q Nonnegative ( 0) positive semidefinite

( 0) ( 0)<

2 22q au huv bv= + +

A

a hh b = A

q q

[170]

a a= determinant First leading principle minor of A

h

h b second leading principle minor of B a

2 2 22xx xy yyd z f dx f dxdy f dy= + +

positive definite

4 25 3 2q u uv v q positive negative definite ?

2d z

2z negative definite d

2= + +

5 definiteness matrix 3 44 6 = B

[171]

Quadratic Forms x n Definition: A quadratic form on Rn is a real-valued function of the form

Q(x , x , ... , x ) = .(1) in which each term is a monomial of degree two quadratic form x 2 Matrix form (1) =

1 2 n ij i ji j

a x x

( )Q T 6 two-dimentional quadratic form

Matrix form

three-dimentional quadratic form Matrix form Definiteness of Quardratic Forms

2 211 1 12 1 2 22 2a x a x x a x+ +

2 2 211 1 22 2 33 3 12 1 2 13 1 3 23 2 3a x a x a x a x x a x x a x x+ + + + +

quadratic form x = 0 rm positive definite

2y ax= 0a > 0y 0

fo 0a < 0y 0 x = 0 quadratic form

egative definite two-dimentions

=

N

( )1 1 2,Q x x 2 21 2 0x x+ 1Q positive definite ( )2 1 2,Q x x = 2 21 2 0x x Q negative definite 2

( )3 1 2,Q x x = 2 21 2x x ( ( )3 1,0Q = +1 = -1 ) ( )3 1,0Q 3Q indefinite (4Q 1) ( )2 2 21 1 2 22 0x x x x+ + 1 0

2,x x = 2x x+ = 0 1 2,x x 4Q positive semidefinite

( )5 1 2,Q x x = ( )21 2 0x x +

[172]

0 1 2,x x 0 5Q negative semidefinite

efinition D : Let A be an symmetric matrix , then A is n n

(a) positive definite if T 0> for all 0 in nR (b) positive semidefinite if for all T 0 0 in (c) negative definite if

nR T 0< for all 0 in nR

(d) negative semidefinite if T 0 for all 0 in (e) indefinite if for some

nR T 0> in and nR 0< for some other in

Principal Minors of a Matrix

nR

matrix n n matrix k k

matrix n k n k matrix k-order principal submatrix of determinant k k

principal submatrix

ipal minor ofprinc 7 matrix 3 3

=

hird-order principal minor

econd-order principal minor

11 12 1321 22 2331 32 33

a a aa a aa a a

T

S

Definition

First-order principal minor

: matrix n n matrix k kk n k order leading principal

atrix determinant the order leading principal inor

thk

thk

n

submm

[173]

order leading principal submatrix k k leading principal minor k 8 Matrix 3 3 leading principal minors 3 11 1211

21 22

,a a

aa a

11 12 1321 22 2331 32 33

a a aa a aa a a

: symmetric matrix n n ) positive definite n leading principal minor ) negative definite n leading principal minor (a

(b 1 2 10, 0, 0A A A< > < etc.

(c) indefinite

9 Matrix (a) 1 2 30, 0, 0, 0A A A A> > > > A (b) 0, 0,A A A< > < >

4 4 4

1 2 3 40, 0A )

A(c 1 2 3 40, 0, 0, 0A A A A> > = < A (d) 1 2 3 40, 0, 0, 0A A A A< < < < A

1. definiteness Sym etm ric matrics :

(e) (a) 1 1

2 11 2 02 4 50 5 6

(b) 4 5 (f) 1 10 0

3 4

02

1 1 0

) (g) 3 44 6

1 0 3 00 2 0 53 0 4 00 5 0 6

(c

(h) 2 23 4 7q u uv v= +

[174]

(i) ) 1)

23 2 35 4 2u u u u+ +

. definiteness

3

2 27 3q u uv v= + (j 2 28 3q uv u v=

2 26 5 2q xy y x= (k(l) 21 1 2 1 3 23 2 4q u u u u u= + 2(m) 2 2 24 6 4 7q u uv uw v w= +

22 2 21 2 3 1 2 26 3 2 4q u u u u u u u= + +

2 2 22 3 6 8 2q u v w uv uw vw= + +

[175]

Objective functions with more than two variables ( )1 2 3, ,Z f x x x= F.O.N.C : S.O.S.C

( )2d Z d dZ= =

H = All leading principal minor =

= 1H

2H 3H =

Z is a if maximumminimum 1 2 3 1 2 3

0, 0, 00, 0

H H HH H H

< > < > 0, > >

[176]

10 Extreme Values Z = 2 2 2 21 1 2 2 1 3 32 4x x x x x x x+ + + + +

11 Extreme Values

Z = 3 2 21 1 3 2 2 33 2 3x x x

x x x + +

3

[177]

n-variable case: ( 1 2, ,...Z f x x x= )n .O.N.CF :

dZ =

2

0ii

fx

= i 1,2,...i n= S.O.S.C:

2 u quadratic form

d Z

H =11 12 1

21 22 2

1 2

n

n

n n nn

f f ff f f

f f f

"

All principal minors H , 2H , nH Z maximum Z minimum

values check Extreme values (maximum) (minimum)

2

Extreme

1. Z = 1 2 1 2 2 3 33 3 4 6x x x x x x x+ + + ( )2 2 2

2 2

2. Z = 1 2 329 x x x + + 3. Z = 2 2 21 3 1 2 2 3 2 33x x x x x x x x+ + + +

( )2x y w w x y + 4. Z = 2e e e e+ +

[178]

( )22 2 2x y w we e e x e y+ + + 5. Z = 6. Z = 4 2 26 3x x xy y+ + 7. Z = 2 26 2 10 2 5x x y x y + + + 8. Z = 2 3xy x y xy+ 9. Z = 34 23 3x x y y+ 1 . 0 F = 212 2 26 3 4 10 5x xy y yz z x y z+ + +

( ) ( )2 2 22 2 22 3 x y zx y z e + ++ + 11. F = Second-order conditions in relation to concavity and convexity:

Concept Convex and Concave Function

convex set U

f concave U nR 1 2,x x ( )0,1t

f convex U convex set U

nR 1 2,x x ( )0,1t

[179]

Theorem 1: (Linear Function) f linear function f Convex Concave Strictly Convex concave heorem 2:

T (Negative of a Function) f concave (convex) function oncave)

heorem 3:f convex (cT f g concave (convex) function g = f+g g concave (convex) function :

U Convex set f oncave function Hessian matrix

2f C nR ( )2Df x

x ( )2

C negative semidefinite U f convex function Hessian matrix f x positive definite

8

D 12 ( )1, 3f x y x x y= +

concave

4 2 2 4y x y+ convex 13 ( )1,f x y xy= convex concave

[180]

( ) 1 14 41,f x y x y= 14 convex concave

15 ( )1, a bf x y x y= convex concave

. concave convex )

1 ( ) 43 5 lnxf x e x= + (a x

) ( ) 2 21, 3 2 3 4 1f x y x xy y x y= + + + (b(c) ( ) 41, , 3 5 lnxf x y z e y z= + (d) )1, , a b cf x y z Ax y z= , , 0a b c > (

1. ( 1 ) 1 / ( ),z f x y=

(Maximum) (Minimum)

First-order necessary

Seco dnd-or er sufficient

[181]

( )1 2, ,..., nz f x x x= 2 /

(Maximum) (Minimum) First-order necessary

Second-order sufficient

8.3

8.3.1 Price Discrimination Concept

rice Discrimination 3 P1. 2.

irst Degree Price Discrimination Second Degree Price iscrimination

3.

FD

[182]

Third- Degree Price Discrimination

Third-Degree Price Discrimination 2

16 (Total Revenue)

2

( ) ( ) ( )1 1 2 2 3 3TR R Q R Q R Q= + + iR i ( Total Cost )

1 2 3( )C C Q Q Q Q Q= = + +

1 Objective Function

= 2 FONC

1

2

........(1)

===

........(2)

3 ........(3) 1, 2 3

3

MC MR MR MR= = = ..(4) 1 2

ii iR PQ=

iMR = 4

[183]

di ............ 3 SOSC * * *1 2 3, ,Q Q Q

* ONC

===

* *1 2 3, ,P P P F Hessian Matrix

11 12 13

21 22 23

31 32 33

= == == =

H

=

Check all leading principal minors

1

2

H

H

= = =

3H = =

[184]

17 3 1 163 4P Q= 2 2105 5P Q=

3 375 6P Q=

20 15C Q= +

3

2 FONC ........(1)........(2)

1 Objective Function

12 =3 ........(3) =

=

2*Q =

=

*1*

QQ

==

3

*

1*

2*

3

PP

==

P

[185]

3 SOSC Hessian Matrix

13

21 22 23

33

= = == = == = =

11 12

31 32

H

=

Check all leading principal minors

1

2

3

H

H

H

= = = = =

.3.2 (Competitive Firm Input hoices: Cobb-Douglas Technology) Cobb-Douglas roduction Function

8CP

Q L K = = w = r

[186]

1 (Objective Function)

=

2 FONC L

K

==

(The Economic interpretation)

Hessian Matrix

22

3 Check SOSC

11

21 = 12

= =

=

H

=

heck all leading principal minors C

H1

2

== = H

[187]

.3.3 1 (Problem of a Multiproduct Firm) 18 2 (Exogenous Variable)

TR =

2

8 2 21 1 22 2C Q QQ Q= + + Profit Function

TR TC = =

C

hoice Variables

[188]

First Order Necessary Condition (FONC) 1

2

==

*1Q = *2Q = Check Second Order Sufficient Condition (SOSC)

= H

1H = 2H = 2d .......................definite 19 2 1 2

24015

Q P PQ P P

2 +1 1=

2 1 2

..............................

= +

.......

[189]

function P

Q Cramers rule

TR =

2

(Profit Function)

First Order Necessary Condition (FONC)

1 2,Q Q

1 2,P P 1 2,Q Q

1 2,P P

1

2

PP==

2 21 1 22 2C Q QQ Q= + +

Choice Variables

1

=

2 =

*1Q = *2Q =

[190]

Check Second Order Sufficient Condition (SOSC)

=H

1H = 2H =

2d .......................definite

P lem)

1

8.3.4 1 (Multiplant Firm rob

( )i i iTC C q= i = 1,2,,n

i

iq

TR =

Q = ( )*TR P Q Q= (Exogenous

Q

P Variable)

P ( )P Q =

[191]

First Order Necessary Condition

i = i = ,2,,n

1

Hessian Matrix

1 2

H

=

1 1 0H R C = < 2H =

:

[192]

20 2

2:

1) 1 2 (AR)

(MR)

2)

1: ( ) 21 1 110C Q Q=

( ) 21 2 220C Q Q=

700 5P Q= 1 2Q Q Q= +

1 2, , ,totalQ Q Q P 1 (Profit Function) Choice Variables

[193]

2 First Order Necessary Condition (FONC) 1 = 2 =

*1Q = *2Q = 3 Check Second Order Sufficient Condition (SOSC)

= H

1H = 2H =

2d .......................definite

[194]

9

( 1 )

( 1 ) : 3

- Lagrange Multiplier

- - -

-

[195]

9.1 ( 1 )

Choice Variables (Independent) 2 ()

1 2 950Q Q+ =

1Q 2Q

( Constrained Optimum value) (Free Extremum) ( Constrained Extremum)

1Q 2Q1 2 950Q Q+ =

[196]

(The optimal allocation of scarce resources) (Optimal) optimization ( scarce resource) optimization (with constraint) Constrained Optimization problem (Prototype)

9.1. Constraints :Tw

x x x

( )( )( )( )( )( )

1 2

1 2

1 1 2 1

2 1 2 2

1 2

1 1 2 1

2 1 2 2

max ( , ..., )

, ,...,

, ,...,

, ,...,...............................

, ,...,

, ,...,

, ,...,...............................

nn

n

n

n

n n n

n

n

imize f x x xwhere x x x R must satisfy

g x x x b

g x x x b

g x x x b

h x x x c

h x x x c

==

( )1 2, ,...,n n nh x x x c=

1 2 1 : Optimization with Equality o variables and One Equality Constraint

1 U = 1 2 12+

1 2,x x 1 2

2 2 1 4

[197]

1 2 .. 2

1. (Substitution Method) 2. Lagrangian Method

bstitution Method) : (Substitution Method) Budget Constraint 1

1 2 60

(Su

1x 2 1

U =

1

dUdx

FONC: =

1 2

Constraint

2x explicit function 1x Lagrange-

ultiplier Method M

[198]

Lagrange Multiplier f h 2 1C

( )1 2max ,imize f x x St. ( )1 2,h x x c=

( ) ( ) ( )1 2 1 2 1 2, , , ,L x x f x x c h x x = + 1 (FONC)

( )( )

( )1 2, 0c h x x = =

1 1 1 21

2 2 1 22

, 0

, 0

L f h x xxL f h x xxL

= = = =

Lagrangian function

ONC:

L = F

[199]

2

. 6Max z xySt x y

=+ =

1 Lagrangian function 2 FONC:

3

1 2 1 2

1 2 1 2

( , ). ( , ) 4 16

Max U x x x xSt h x x x x= + =

=

grangian function 2 FONC:

1 La

[200]

4

21 2 1 2( , )Max f x x x x=

2 21 2 1 2. ( , ) 2 4 3St h x x x x= + =

1 Lagrangian function 2 FONC:

6

[201]

n 1 : Optimization with Equality s and One Equality Constraint Constraints : n variable

1 2

1 2

( , ,..., ). ( , ,..., )

n

n

f x x xSt g x x x c=

n 2 : Optimization with Equality Constraints : n variables and two Equality Constraint

1 2

1 2

1 2

( , ,..., ). ( , ,..., )

( , ,..., )

n

n

n

f x x xSt g x x x c

h x x x d==

[202]

Co

n k : Optimization with Equality nstraints : n variables and k Equality Constraint

1 2( , ,..., )nf x x x

. ( , ,..., )( , ,..., )

...............................

n

n

St h x x x ch x x x c

1 1 2 1

2 1 2 2

1 2( , ,..., )k n kh x x x c=

2 (Second-Order Sufficient Condition)

==

1 2( , ,..., )nf x x x

. ( , ,..., )St h x x x c=

.....................( , ,..., )k n kh x x x c

1 1 2 1

2 1 2 2( , ,..., )..........

n

nh x x x c=

1 2 =

he Lagrangian T

[ ] [ ][ ]

1 2 1 2 1 1 1 1 2 2 1 1 1 2

1 2

( , ,..., ) ( , ,..., ) ( , ,..., ) ( , ,..., ) ...

( , ,..., )n n n

k k k n

nx x x f x x x c h x x x c h x x x

c h x x x

= + + +

irst Order Necessary Condition: FONC

L

F

1

Lx

2

Lx

[203]

[204]

..

n

Lx

1

L L

1..

n

L

Second Order Sufficient Condition: SOSC order Hessian Matrix B

1

( ) ( )1 * 1 *0 0 h h

x x" "# %

1 1h h

x x" "# % #

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1

* *1

1 * * * * * *11 1

1 * * * * * *1

0 0

, ,

, ,

n

k kn

kn

kn n n nn

g gH

L L

h h L L

=

x x

x x

x x x x

# # % #" "

# % #" "

the last n-k leading principle minors H n Choice Variables k

1 2

1 1 1 1 2

2 2 1 2 2

0 x x

x x x x x

x x x x x

h h

H h L L

h L L

=

Choice Variables (n) (k) the last n-k = leading principle minors Matrix

Choice Variables (n) (k) the last n-k = leading principle minors Matrix

(Local Max) the last (n-k) leading principal minor determinant (Alternate in Sign) determinant (k+n)x(k+n) (Local Min) the last (n-k) leading principal minor determinant

1 2

1 1 1 1 2

2 2 1 2 2

0 x x

x x x x x

x x x x x

h h

H h L L

h L L

=

1 2 3

1 1 1 1 2 1 3x x x x x x xh L L L

H = 2 2 1 2 2 3

3 3 1 3 2 3 3

0 x x x

x x x x

x x x x x x x

h h h

h L L L

h L L L

2x x x

1 2

2 1 1 1 1 2

2 2 1 2 2

0 x x

x x x x x

x x x x x

h h

H h L L

h L L

=

1 2 3

1 1 1 1 2 1 3

2 2 1 2 2 2 3

3 3 1 3 2 3 3

3

0 x x x

x x x x x x x

x x x x x x x

x x x x x x x

h h h

h L L LH H

h L L L

h L L L

= =

( 1)n

( 1)k

[205]

5 cal Max Local Min

6 2 Local Max Local Min

in

1 Lo

7 3 Local Max Local M

[206]

9.2

(The optimal allocation of scarce resources)

St.

( , ) ( , 0)x yU U x y U U= > x yxP yP B+ =

1 Lagrangian Function 2 First-Order Condition (FONC): (Economic Interpretation) 1 1 2 3

[207]

2 Indifference Curve

Lagrange multiplier

[208]

3 Second-Order Sufficient Condition 9.1 Bordered Hessian

H =

9.2 Check the last n-k leading principle Minor n = the last leading princip Minor

k = Check le

H =

FONC 2d U

definite dB = 0

[209]

8 1 12 2( , ) 10U x y x y=

m 1 xP

2 yP x y

[210]

9.3 ( Minimize Total ure Under the Level of Utility)

y x y

tion

Expendit 0U

, )(U U x= E=

Lagrangian Func 1 Lagrangian Function 2 First-Order Condition (F ): ONC 3 Second-Order Sufficient Condition

3.1 Bordered Hessian

H

=

3.2 Check the last n-k leading principle Minor n = k =

Check the last leading principle Minor

[211]

H =

FONC

def

9.4

inite dB = 0

2d U

( , )Q q L K=

TC wL rK c= + =

L K c

Lagrangian Multiplier Method

unction 1 Lagrangian F 2 First-Order Condition (FONC):

[212]

(Economic Interpretation)

3

Second-Order Sufficient Condition


[213]

H =

Check the last n-k leading principle Minor

n = k = Check the last leading principle Minor

3.2

H =

FONC definite dB = 0

2d U

[214]

(Least Cost Combination of Inputs) (Conditional Input Demand)

2 (L) (K) w r

9.5

TC =

F L K=

Lagrangian Multiplier Method for the cost minimization roblem 1

0Q ( , )Q

p Lagrangian Function

2

First-Order Condition (FONC):

(Economic Interpretation)

[215]

3 Second-Order Sufficient Condition


H

=

3.2 Check the last n-k leading principle Minor

Check the last leading principle Minor

n = k =

[216]

H =

FONC definite dB = 0

The Expansion Path (Comparative Static Analysis) ( Isoquants ) ( Least Cost combination between L* and K*) Isoquant slope Isoquant slope Isocost (Expansion Path)

Isoquants Strictly Convexity ( 2) (Expansion Path) 1 Cobb-Douglas Production Function

2d U

Q AL K = 1

[217]

L* K*

9.6 (Profit Maximization) (Unconditional Input Demand)

9.5 L* K* ( Profit Maximization) L K Objective function

( )P AL K wL rK =

PQ wL rK = Q AL K = Q AL K () Lagrangian Multiplier Method 1 Lagrangian Function Endogenous Variables / Choice variables

[218]

2 First-Order Condition (FONC): 3 Second-Order Sufficient Condition


H

=

3.2 Check the last n-k leading principle Minor n = k = Check the last leading principle Minor

H =

[219]

Optimization 1. y2Q x=

x y 6 4

2. ( ),U x y xy=

1) x y 1 20 80 2) x y 4 20 160

3. Firm 2 1 2,x x

1 2 10.01 2 20Q x x x= + 1000 30 40 1 % () 1 1) 2)

4. 2

11252

x p= , 230y p= 92 22 2 6 2TC x y x xy= + + +

5. 2 x,y

1 12 22Q x y= + (x,y) 4 5 100

1) 20

[220]

2)

220

6.

3)

M wx py= +

4,1,2

( ),U U x y=

U is Differentiable 7. 3 1 2,x x 3x

60 1 2 3, ,U x x x= UM

8. B

U x x x x x x= = 3( )1 2 3 1 2 3, , 1 2, ,p p p U

1) 3 2) (Sufficient condition) 3) Homogenous function 9. a)

( ) ( ) 2 22 2, 2 x yf x y x y e = + b) 2

2,

2 2 21 2 3 1 2 34 2 3 1y x x x x x x= + c) ( ,L x y z) 2 2z x y x x y z= + + d)

e) Maximize :

2xyw e xy= ( )2 22 2 221

2x y xx y xe e + + ( ),x y R

[221]

10. Utility function ( )1 2 1 ba b aU x x x b a = + subject to 2

1i i

ip x M

== a,b

1) 2) () 3) 11. 1 21 2Y AX X = W R 12. ( ) 1 21 2 1 2, , , ... nn nU x x x Ax x x =

1 2, ,..., np p p M 1x 13. 2

1 2,Q Q 1 1100p Q= 2 280p Q= 1)

2)

3) ()

14.

( )1 26C Q Q= +

, ,x y z ( ) 21 1200 100C x x= + , ( ) 22 1200 300C y y y= + + ,

2000

( )3 200 10C z z= +

[222]

15. M

( ), , log log logU x y z a x b y c z= + + , ,a b c 1 2 3, ,P P P , ,X Y Z

1) 3 2) Comparative Static Analysis

3) , ,X Y Z 3 ( )

16. Q AK L =

K , L r w 1) K 2) 3) Comparative static analysis exogenous

endogenous 17. CES Function ( ) 11 21p p pQ A x x = + 1 2,x x 1 2,p p 1) Cobb-Douglas function 2) Demand function

M 3) Check second order condition

18. Neo-classic (Two period consumption model) Microeconomics maximize utility 2 utility function n

0p

( )1 10

, lt

tt t t

iU c c c+ +

==

t tM r

[223]

Comparative static analysis endogenous exogenous ( Hint : ( )11 tt t

cc Mr

++ =+ ) 19. Utility function ( )ln ln 24U c N= +

N

c P w M 1) Constraint 2) Utility maximize subject to constraint in 1) 3) Demand for c ( c*) Supply function ( N*)

4) 3) Homogenous 5) Let * . Show that ( )* *ln ln 24U c N= + * 0UM > and

*

0Up

> ( ) ( ) ( ), ln 1 ln 2 2 3f x y x y x y= +

6) ( ) ( ), ln 3 2f x y x y= + 7) ( ) ( ), 1 x y 8f x y x e y= + : 8) 1

0, 0x y> > 2 2 23 3 ; : 2 3x y z yz sub x y z+ + + + + =

9) ( ) ( )( )ln ; : 1 1 4xy sub x y+ + = 10) ( )

1 1

ln ; :n n

i i i ii i

x sub p x y= =

= 11) 0

( ) ( )2 24 2 8; : 2 1 8x y sub x y+ + =

[224]

10 Integration Integration

Integration Integration : 2 Integration

- Notation and Terminology - (Rules of Integration) - (Definite Integrals)

Integration - (Total Revenue Function)

(Marginal Revenue Function) - (Total Cost Function) (Marginal

Cost Function) - (Total Profit Function) (Marginal Profit Function) - - - (Consumer Surplus)

- (Producer Surplus) - First Degree Price Discrimination Perfect Price Discrimination -

[225]

[226]

10.1 (Indefinite Integrals) (Marginal Utility) X partial derivative x Marginal Cost Marginal Utility Marginal Cost

10.1.1 (Antiderivative) (Integration) F(x) f(x) f(x) F(x) F(x) integral antiderivative f(x) f(x) x

( ) ( )f x dx F x C= + ( ) ( )dF x f xdx = integral sign f(x) integrand ( integrated) (Antiderivative)

( ) +F x C

( ) f x dx

10.1.2 Basic Rule of Integration Rule I: (the power rule) Rule II: (the exponential rule)

x xe dx e c= +

11 ( 11

n nx dx x c nn

+= + + )

[227]

Rule III: (the logarithmic rule) Rule IIa: Rule IIIa: Rule of operation Rule IV: (Integral of a sum) Rule V: (Integral of a multiple) Rule Involving Substitution Rule VI: (the substitution rule) Rule VII: (Integration by parts)

vdu uv udv=

( ) ( ) ( )duf u dx f u du F udx

= = c+

( )( )kf k dx k f x dx=

( ) ( )( ) ( )f x g x dx f x dx g x dx+ = +

( )( ) ( ) ( )

( ) ( )ln 0

ln 0

f xdx f x c f x

f x

or f x c f x

= + > +

( ) ( ) ( )f x f xf x e dx e c = +

1 lndx x cx

= +

1 (Indefinite Integral-Antiderivative) 1. xdx =

2. 31 dxx

xdx =

4. ( )4 23 5 2x x d+ x

x x x dx =

6. x3 2( )x x xe e e d + =

=

3. =

5.

[228]

7.2( 2)y dy

y =

8. 2 15(1 )x x d+ x

xxe dx =

10. 1 ln xdxx =

=

9.

[229]

11. =

12. 2 3xe x dx x xdx

x x+

3 21x x+ =

14. ln( 2)2 4x dxx++

=

13. =

10.1.2 Initial-Value Problems

( ) ( )f x dx F x C= + Initial condition C

[230]

2 Initial-value problem a) F(x) 1( ) 2

2F x x = and F(0)=1/2

b) F(x) and F(1) = 5/12

10.2 (Definite Integrals)

2( ) (1 )F x x x =

( ) ( )f x dx F x C= + a b x ( a < b)

[ ] [ ]( ) ( ) ( ) ( )F b c F a c F b F a+ + = 1 definite integral of f(x) from a to b a lower limit of integration b upper limit of integration

( ) ( ) ( ) ( )bb

a a

f x dx F x F b F a= = 3 definite integral

( )10

0e d

[231]

411

t

t

d dxdt x +

( ) ( )22 3 21

2 1 6x x dx

10.2.1 A Definite Integral as an Area under the Curve

1 1 A f(x) f(x)

y

x a b

y=f(x)

A=?

[232]

[233]

2 y=f(x) [a,b] ( A) [a,b] 4 1 2 3 4, , ,x x x x A

A = [a,b] 3

3

[234]

4

10.2.2 (Some Properties of Definite Integral) Property I: The interchange of the limits of integration changes the sign of the definite integral: Property II: A definite integral has a value of zero when the two limits of integration are identical Property III: A definite integral can be expressed as a sum of a finite number of definite subintegrals as follows

( ) ( )b aa b

f x dx f x dx=

( ) ( ) ( ) 0aa

f x dx F a F a= =

( ) ( ) ( ) ( ) ( )d b c da a b c

f x dx f x dx f x dx f x dx a b c d= + + < <

[235]

5 Property IV: Property V: Property VI: Property VII: (Integration by part)

( ) ( )b ba a

f x dx f x dx =

( ) ( )b ba a

kf x dx k f x dx=

( ) ( ) ( )( )b b ba a a

f x g x dx f x dx g x dx+ = +

x b x bx bx a

x a x a

vdu uv udv= =

==

= ==

[236]

4 3

21x dx +

5 0

24

22

xdx

x

10.3 Improper Integral f a x b ( )F x ( )f x

( ) ( ) ( )b

a

f x dx F b F a=

[237]

3

20

1dx

x 21( )f x x= 0x = 1

a

dxx

(Closed Interval) 1 f x a

( ) lim ( )b

ba a

f x dx f x dx

= 2 f x a

( ) lim ( )b b

aa

f x dx f x dx=

3 f x R 0

0

( ) ( ) ( )f x dx f x dx f x dx

= +

6 21

2dx

x

[238]

7 1

1dx

x

10.4 Integration

10.4.1. (Recovering Total Cost from Marginal Cost) MC(Q) (the underlying total cost) TC(Q) = 8 MC(Q) = 3 + 2Q TC(Q) = What is the constant in a total cost function? It is fixed costs, so using a marginal cost function we are unable to recover the level of fixed costs that a firm faces.

10.4.2. (Total Profit Function) (Marginal Profit Function) (MP) MP=MR-MC q1 Profit =

[239]

9 MR= 50-2q MC= 10+q q=10

10.4.3. 1 2( ) 200 2MR t t= 2( ) 25 2MC t t t= + + 1. 4 2.

[240]

3. S(t) ( ) 1000 119S t t=

10.4.4. Yahoo

12( ) 0.3 0.1S Y Y

= 81 Yahoo

10.4.5. (Consumer and Producer Surplus) (consumer and producer surplus) (when the demand curve and supply curve are not linear)

[241]

Q = D(P) Q = S(P). P* Q* (Consumer surplus) CS (Producer surplus) PS CS = CS = PS = PS =

[242]

10 D(P) = 25 1

2 2P S(P) = 1 1

2 2P +

Welfare Effects of Price changes

30 12 2

P Welfare Effects of Tax distortion 4

10.4.6. First Degree Price Discrimination Perfect Price Discrimination 2247P Q= MC = 4+3Q CS

[243]

Perfect Price Discrimination

10.4.7.

(Net Investment) 1312t (capital stock) t = 0 25 (K) [1,3]

[244]

10.4.8 : (Random Variables) (Random variables) (a probability distribution) X a random variable x 1 (Fair die) X f(x) f(1)=1/6, f(2)=1/6, f(3)=1/6, f(4)=1/6, f(5)=1/6, f(6)=1/6 and f(i)=0 for any other value i.

( any discrete random variable X: DRV) f(x) 1. 2. x domain f(x) 3. X Z ( any continuous random variable ) Z [a,b] f(x) pdf

( ) 0f x ( ) 1

xf x =

( ) ( ) ( )t x

F x P X x f t

= =x<

[245]

1. 2. 3. X cont. r.v. pdf. t f(t), Distribution Fn. Cumulative Distribution Fn. of X 4. X continuous r.v. f(x) probability density x, (Expected value) x E(X) = 5. X continuous r.v. f(x) probability density x, (Variance value) x

( ) 0,f x for x < <

( ) 1f x dx

=

( ) ( ) ( )x

F x P X x f t dt

= =

x < <

( )x f x dx =

[246]

Uniform distribution f(x)

1) CDF Uniform distribution f(x)

2) E(x)

3) Var(x)

1 22.1 (The Concept of Sets)2.1.1 (Sets)2.1.2 (Membership in a set)2.1.3 (Relationships between Sets)

2.2 (Relation and Function)2.2.1 (Orderd Pairs and C2.2.2 (Function)

2.3 (Function and Types of Func2.3.1 (Constant Function)2.3.2 (Polynomial Function)2.3.3 (Exponential Logar2.3.4

2.4. (Inverse Function) 3 (Static Equilibrium Analysis)3.1 3.1.1 (A Simple Macro econo3.1.2. IS-LM3.1.3. : 1

4Basic Matrix Algebra Matrix Algebra 5 5.1 n (Polynomial of Degree n)5.1.1 (slope)

5.2 (Differentiation / derivation) / process o5.2.1 (Continuity)5.2.1 Smooth Function

5.3 Convexity and Concavity , Maxima and Minima5.3.1 Global V.S. Local Extremum Concept5.3.2 1 (First-Derivative 5.3.3 2 (Second-Derivative5.3.3 (Curvature of a Graph)

6 (- 6.1 (6.1.1 6.1.2

6.2 6.2.1 (Competitive Market)6.2.2 (Market Power: Monopoly)

7 1 7.1 Functions between Euclidian Spaces7.1.1 (Geometric representatio7.1.2 Geometric Representations of Functions of Several Vari7.1.3 Level Curves for z = f(x,y)

7.2 Partial Derivative7.2.1 (Economic Interpretation)

7.3 The Chain Rule7.4 The Total Differential of A Function of Several Variable7.4.1 Total differentials7.4.2 differential7.4.2 The Total Derivatives

7.5 Implicit Functions 8 8.1 The Differential (Optimization Condit8.1.1 / (Extreme V

8.2 Quadratic Forms8.3 8.3.1 Price Discrimination8.3.2 8.3.3 1 (Problem of8.3.4 1 (Mul

9 9.1 ( 1 )9.1.1 2 1 : Optimi

9.2 9.3 9.4 9.5 (L9.6 (Profit Maximization) 10Integration Integration Integration

10.1 (Indefinite Integr10.1.1 (Antiderivative) (I10.1.2 Basic Rule of Integration10.1.2 Initial-Value Problems

10.2 (Definite Integrals)10.2.1 A Definite Integral as an Area under the Curve10.2.2 (Some Propert

10.3 Improper Integral10.4 Integration 10.4.1. 10.4.2. (Total Profit Function) 10.4.3. 10.4.4. 10.4.5. (Consumer10.4.6. First Degree Price Discrimination Perfect Price10.4.7. 10.4.8 :

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