Question

D Question 6 4 pts As in question 5, your utility function over the goods X and Z takes the following form: You want to maximize your utility subject to your budget constraint. Assume again that the price of X is $3 per unit and the price of Z is $6 per unit, but that owing to a new government welfare program, the total income you have to spend on X and Z is $900 instead of $720. The consumption bundle that will now maximize your utility subject to your budget constraint is X 300 and Z-0 (enter only numbers in the blanks, and round to the nearest integer if necessary)

Answer 1

The utility function of two goods X and Z is

U(X,Y) = X^0.5 +Z^0.5 ..........(i)

The marginal utility of good X,

dU/dX= 1/2 *1/X^0.5

and marginal utility of good Z,

dU/dZ= 1/2*1/Z^0.5

Price of good X (P_X )= $3/unit and price of good Z (P_Z )= $6/unit

Under utility maximization condition,

    dU/dX/dU/dZ= P_X /P_Z

or, 1/2 *1/X^0.5 /1/2*1/Z^0.5 = 3/6 =1/2

or, (Z/X)^0.5 = 1/2

or, Z/X=1/4 or, X=4Z

Again income (I) that is required to be spent on goods X and Z due to new government welfare= $900

the new budget constraint equation is,

   XP_X +ZP_Z = 900

or, 3X+6Z=900

   or, 18Z=900 or, Z=50 and X= 4Z=4*50=200

Therefore, the new consumption bundle to maximize the utility is X =200 and Z=50