Lagrange Method
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Lagrange Method. Lagrange Method. Why do we want the axioms 1 – 7 of consumer theory? Answer: We like an easy life!. By that we mean that we want well behaved demand curves. Let’s look at a Utility Function: U = U( ,y) Take the total derivative:.

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Lagrange Method

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Lagrange method

Lagrange Method


Lagrange method

Lagrange Method

  • Why do we want the axioms 1 – 7 of consumer theory?

  • Answer: We like an easy life!

By that we mean that we want well behaved demand curves.


Lagrange method

Let’s look at a Utility Function: U = U(,y)

Take the total derivative:

For example if MUx = 2 MUy = 3


Look at the special case of the total derivative along a given indifference curve

Look at the special case of the total derivative along a given indifference curve:

dy

dx


Lagrange method

y

x

  • Taking the total derivative of a B.C. yields

  • Px dx + Py dy = dM

  • Along a given B.C.dM = 0

  • Px dx + Py dy = 0


Lagrange method

y

Equilibrium

x

=>Slope of the Indifference Curve

= Slope of the Budget Constraint


Lagrange method

We have a general method for finding a point of tangency between an Indifference Curve and the Budget Constraint:

The Lagrange Method

Widely used in Commerce, MBA’s

and Economics.


Lagrange method

u2

u1

y

u0

Idea: Maximising U(x,y) is like climbing happiness mountain.

x

y

But we are restricted by how high we can go since must stay on BC - (path on mountain).

x


Lagrange method

u2

u1

y

u0

So to move up happiness Mountain is subject to being on a budget constraint path.

x

Maximize U (x,y) subject to Pxx+ Pyy=M


Lagrange method

= 0

= 0

= 0

Known: Px, Py & MUnknowns: x,y,l

3 Equations: 3 Unknowns: Solve


Trick

Note:

Trick:

U

But:


Lagrange method

= 0

= 0

= 0

Known:Px, Py & MUnknowns:x,y,l

3Equations:3Unknowns:Solve


Lagrange method

Notice:

U = x2 y3

<=> Slope of the Indifference Curve

Recall Slope of Budget Constraint =

Slope of IC = slope of BC


Lagrange method

Back to the Problem:

 + 

But

But  + 


Lagrange method

Back to the Problem:

 + 

But

But  + 


Lagrange method

So the Demand Curve for x when U=x2y3

If M=100:


Lagrange method

Recallthat: U = x2 y3

Let: U = xa yb

For Cobb - Douglas Utility Function


Lagrange method

Note that: Cobb-Douglas is a special result

In general:

For Cobb - Douglas:


Why does the demand for x not depend on p y

Why does the demand for x not depend on py?

Share of x in income =

In this example:

Constant

Similarly share of y in

income is constant:

So if the share of x and y in income is constant => change in Px only effects demand for x in C.D.


Lagrange method

Constraint

Objective fn

So l tells us the change in U as M rises

Increase from U1 to U2

Increase M

 in objective fn

 in constraint


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