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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By that we mean that we want well behaved demand curves.
Take the total derivative:
For example if MUx = 2 MUy = 3
=>Slope of the Indifference Curve
= Slope of the Budget Constraint
The Lagrange Method
Widely used in Commerce, MBA’s
Idea: Maximising U(x,y) is like climbing happiness mountain.
But we are restricted by how high we can go since must stay on BC - (path on mountain).
So to move up happiness Mountain is subject to being on a budget constraint path.
Maximize U (x,y) subject to Pxx+ Pyy=M
Known: Px, Py & M Unknowns: x,y,l
3 Equations: 3 Unknowns: Solve
Known:Px, Py & MUnknowns:x,y,l
U = x2 y3
<=> Slope of the Indifference Curve
Recall Slope of Budget Constraint =
Slope of IC = slope of BC
Let: U = xa yb
For Cobb - Douglas Utility Function
For Cobb - Douglas:
Share of x in income =
In this example:
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.
So l tells us the change in U as M rises
Increase from U1 to U2
in objective fn