- 224 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Lagrange Method' - alika-tyler

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

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:

For example if MUx = 2 MUy = 3

y given indifference curve:

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

y given indifference curve:

Equilibrium

x

=>Slope of the Indifference Curve

= Slope of the Budget Constraint

The Lagrange Method

Widely used in Commerce, MBA’s

and Economics.

u between an Indifference Curve and the Budget Constraint:2

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

u between an Indifference Curve and the Budget Constraint:2

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

= 0 between an Indifference Curve and the Budget Constraint:

= 0

= 0

Known: Px, Py & M Unknowns: x,y,l

3 Equations: 3 Unknowns: Solve

= 0 between an Indifference Curve and the Budget Constraint:

= 0

= 0

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

3Equations:3Unknowns:Solve

Notice: between an Indifference Curve and the Budget Constraint:

U = x2 y3

<=> Slope of the Indifference Curve

Recall Slope of Budget Constraint =

Slope of IC = slope of BC

So the Demand between an Indifference Curve and the Budget Constraint:Curve for x when U=x2y3

If M=100:

Recall that: between an Indifference Curve and the Budget Constraint:U = x2 y3

Let: U = xa yb

For Cobb - Douglas Utility Function

Note that: between an Indifference Curve and the Budget Constraint:Cobb-Douglas is a special result

In general:

For Cobb - Douglas:

Why does the demand for x not depend on between an Indifference Curve and the Budget Constraint: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.

Constraint between an Indifference Curve and the Budget 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

Download Presentation

Connecting to Server..