lagrange multiplier presentation.odp

8/13/2019 lagrange multiplier presentation.odp

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QARSAM ILYASROLL NO 7

TOPIC LAGRANGE MULTIPLIERBRIEF INTRODUCTION, EXAMPLE AND

ECONOMIC APPLICATION


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INTRODUCTION

"What could be more fundamental to economic theory

than the idea of constrained optimisation? Even the

most elementary definitions of economics are based on

solving problems of scarcity and choice - on satisfying

unlimited wants with limited resources. The Lagrange

technique provides a tool for solving such problems"


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WHAT ARE LAGRANGEMULTIPLIERS ..........

Constraints are used in the formation of a Lagrangian Equation, an

equation used to maximize some objective given constraints. In the

Lagrangian function, the constraints are multiplied by the variable ,

which is called the Lagrangian multiplier.

Lamba is the marginal value associated with relaxing a constraint.

Since this value is not expressed or contracted upon in a market, it is

often called the shadow value or shadow price of the constraint.


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WHAT ARE LAGRANGEMULTIPLIERS.........

Lagrange !"#$%"$er& are a e#'() !&e) *(r

!"#$+ar$a"e -a"-!"!&.

I# -($ne& #'e !&e (* Der$+a#$+e& an) #'e

#e-'n$!e& !&e) #( &("+e L$near

Pr(gra$ng

- /! a n!er


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DIAGRAMMATICALLY


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LAGRANGE THEORM

The values of x*, y* and * which

maximise the function L(x,y, ) will necessarily

provide the solution x*,y* which

maximises f(x,y) subject to g(x,y) = c.

Let f and g satisfy Lagranges Theorem, and f will

have a minimum or maximum subject to

the constraint g(x,y)=c.

To find the minimum or maximum of f while

satisfying the constraint.

F

Solving

x(x,y)= gx(x,y)

fy(x,y)= gy(x,y)

f (x,y)= the objective

function

g(x,y)=constraint


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THE SIGNIFICANCE OF LAGRANGE

MULTILIER

T'$& -an e e0#en)e) #( (re #'an 1 +ar$a"e& an)-(n#ra$n#&.

C(na$n# -an e &("+e) 2$#'(!# a&&$g$n$ngn!er$-a" +a"!e #( 3.

= dM \dK

= change in M resulting from a 1-unit increase

in k


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EXAMPLE.......

GO4ERNMENT USES TAXES ASLAGRANGE MULTIPLIER

f(x) be the total net effect on the

population's happiness.

total gas consumption g(x)

Constraint g(x)=c


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CONTINOUED........

Tax is of $20\Gallon

f(x) + (-20) g(x)

By adjusting the size of the tax (the Lagrange

multiplier), the government can indirectly adjust total

consumption g(x) until it is at the desired level, g(x)=c


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LAGRANGE MULTIPLIER INECONOMETRICS

It was first used in econometrics by R. P.

Byron in 1968 and 1970 in two articles

T. S. Breusch and A. R. Pagan published in

1980 an influential exposition of

applications of the LM test to modelspecification in econometrics.


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CONTINOUED...........

To perform an LM test only estimation of the parameterssubject to the restrictions is required.

The LM testing principle has found wide applicability tomany problems of interest in econometrics.


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LM TEST FOR SERIALCORRELATION

The test belongs to the class of asymptotic (large sample)

tests known as Lagrange multiplier (LM) tests. Unlike the

Durbin-Watson statistic for AR(1) errors, the LM test may be

used to test for higher order and is applicable whether or not

there are lagged dependent variables.


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LM TEST

Yt = 1+ 2X2t + 3X3t + t .......... (1)

and we suspect a second order autoregressive scheme:t= 1t-1+ 2t-2+ t .............. (2)Then the mode coud !e written as:

Yt = 1+ 2X2t + 3X3t + 1t-1+ 2t-2+ t .............. (3)


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LM TEST

I* 2e e$a#e) #'e e!a#$(n a&,

Y# 5

6

1X

1#

8X

8#

#........... 9:;

1+ar$a"e 2$#' ' )egree& (* *ree)( a&

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lagrange multiplier presentation.odp