Derive the demand curve good X for the utility function U X

Derive the demand curve good X for the utility function U = X^2Y^3. Show your work.

Solution

U = X² Y³

The budget constraint is: I = P_xX + P_yY

Where, P_x = Price of good X

           P_y = Price of good Y

         I = Income

Then, MU_x = 2XY³

            MU_y = 3X²Y²

For utility maximization

MU_x/P_x = MUy/P_y

2XY³ /P_x = 3X²Y² /P_y

Solving for Y

2XY³ / 3X²Y² = P_x / P_y

2Y/3X = P_x / P_y

2Y = 3XP_x / P_y

Y = 3XP_x / 2P_y = 1.5X(P_x/P_y)

Substituting the value of Y into budget constraint

I = P_xX + P_yY

= P_xX + P_y[1.5X(P_x/P_y)]

= P_xX +1.5XP_x

= 2.5XP_x

Then, X = I / 2.5P_x   (This is the demand curve for good X)