Question

Janice Doe consumes two goods, X and Y.   Janice has a utility function given by the expression:   U = 4X0.5 Y0.5 .

So, MUX = 2Y0.5 and MUY = 2X0.5 . The current prices of X and Y are 25 and 50, respectively. Janice

X0.5  Y0.5 currently has an income of 750 per time period.

1. Write an expression for Janice's budget constraint.

Calculate the optimal quantities of X and Y that Janice should choose, given her budget constraint

Answer 1

a) Current Price of X, P_x = 25

Current Price of Y, P_Y = 50

Current income, M = 750

Budget constraint, M = P_XX+ P_YY

or, 750 = 25X + 50Y

Hence, Janice's budget constraint is, 750 = 25X + 50Y

At equilibrium, MRS = P_X/ P_Y

MU_x / MU_Y = P_X/ P_Y

or, 4 * 0.5X^-0.5Y^0.5 / 4 * 0.5X^0.5Y^-0.5 =  P_X/ P_Y

or, Y / X = P_X/ P_Y .... [from here we can say that the optimal condition will be, Y/X = 25/ 50=> 2Y = X]

or, P_YY = P_XX

Now, 750 = P_XX+ P_YY

or, 750 = 2P_XX

or, X = 750 / (2*25) = 750 / 50 = 15

and, 750 = P_XX+ P_YY = 2P_YY

or, Y = 750 / 2P_Y = 750 / (2*50) = 7.5

Hence, given her budget constraint, the optimal quantities of X and Y will be, X^* = 15 and Y^* = 7.5