Lagrange Method Lagrange Method Why do we want

Lagrange Method • Why do we want the axioms 1 – 7 of consumer

Let’s look at a Utility Function: U = U( , y) Take the total

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

y x • Taking the total derivative of a B. C. yields Px dx

y Equilibrium x => Slope of the Indifference Curve = Slope of the Budget

We have a general method for finding a point of tangency between an Indifference

y u 0 u 1 u 2 Idea: Maximising U(x, y) is like climbing

y u 0 u 1 u 2 So to move up happiness Mountain is

=0 =0 =0 Known: Px, Py & M Unknowns: x, y, l 3 Equations:

Notice: U = x 2 y 3 <=> Slope of the Indifference Curve Recall

So the Demand Curve for x when U=x 2 y 3 If M=100: Px

Recall that: U = x 2 y 3 Let: U = xa yb For

Note that: Cobb-Douglas is a special result In general: For Cobb - Douglas:

Why does the demand for x not depend on py? Share of x in

Constraint Objective fn So l tells us the change in U as M rises

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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: For example if MUx = 2 MUy = 3

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

y x • Taking the total derivative of a B. C. yields Px dx + Py dy = d. M • Along a given B. C. d. M = 0 Px dx + Py dy = 0

y 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.

y u 0 u 1 u 2 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

y u 0 u 1 u 2 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 =0 =0 Known: Px, Py & M Unknowns: x, y, l 3 Equations: 3 Unknowns: Solve

Trick: Note: But: U

=0 =0 =0 Known: Px, Py & M Unknowns: x, y, l 3 Equations: 3 Unknowns: Solve

Notice: U = x 2 y 3 <=> Slope of the Indifference Curve Recall Slope of Budget Constraint = Slope of IC = slope of BC

Back to the Problem: + But +

So the Demand Curve for x when U=x 2 y 3 If M=100: Px x. D 10 4 8 5 5 8 2 20

Recall that: U = x 2 y 3 Let: U = xa yb For Cobb - Douglas Utility Function

Note that: Cobb-Douglas is a special result In general: For Cobb - Douglas:

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.

Constraint Objective fn So l tells us the change in U as M rises Increase from U 1 to U 2 Increase M in objective fn in constraint