Use the method of Lagrange to solve the following problem fo

Use the method of Lagrange to solve the following problem for x_1* & x_2* : Objective is to maximize Q where U(x, y) = x^alpha y^beta and the constraint is: m = P_xx + P_yy.

Solution

Ans:

Setting up the Lagrangian expression:

L = U(XY) + (I – P_X*X – P_Y*Y)

Taking the partial derivatives of the equations and setting them to 0, we find that:

(1)      L/X = X^-1Y – P_X = 0

(2)      L/Y = XY^-1 – P_Y = 0

(3)      L/ = I – P_X*X – P_Y*Y = 0

On dividing 1/2, we get that:

X^(-1) Y^ / X^ Y^(-1) then (/ ) (X^-1 Y^)/ (X^. Y^(-1)) = Px/Py then    

Y / X = Px/ Py . Here all the terms are constants.

Thus, Py. Y = (/) * (Px .X) => Py. Y = (1-/) * (Px .X)

On substituting I – P_XX –  (1-/) * (Px .X) = 0

On solving this further we get that,

X = (I/P_X)

Similarly for good Y we have that :

Py . Y = ( )/ (1-) * (Px. X) or Px. X = (1-)/ * (Py . Y)

On substituting thsi equation into teh budget line equation , we get that I= (1-)/ * (Py.Y) + (Py.Y)

On solving the equation , we get : Y = (I/P_Y)