Question

Question 4. Suppose there is a consumer that is trying to solve the following utility maximization problem where p_x =20 , p_y = 10, and m=100:

Max _^x,y 10 − (x−5) ² − (y−10) ²

s.t. p_xx+ p_y^y≤ m

(a) Use the Lagrangian method to solve the above utility maximization problem. That is, jump straight to setting up the Lagrangian and solving. (8 points)

(b) Are the demands you solved for in part a the utility maximizing values for x and y? If yes, explain. If no, what are the utility maximizing values for x and y and why did the Lagrangian not give you the correct answer? (6 points)

Answer 1

Lagrangian can be set as below

Z=10-(x-5)^2-(y-10)^2-@(100-20x-10y)

Finding FOC

dZ/dx=-2(x-5)-@(-20)=0

dZ/dy=-2(y-10)+10@=0

(x-5)/(y-10)=2

X=2y-20+5=2y-15

Budget constraint

100=20(2y-15)+10y

100=50y-300

y=8 & x=1

Ans B)

These numbers of x and y will not provide maximum utility because maximum utility is possible when x=5 & y=10 whee these basket is not affordable given our budget.

for given budget these values of x,y gives utility of -10 that is maximum possible for given parameters

Lagrangian don't give corner solitoon