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Lagrange Method

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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

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.

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 & MUnknowns: 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:

Recallthat: 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