Question

Q7. Ivy lives in a world where there are only two goods, X and Y. Her utility function per period is given by: U = 30x – 0.5x2 + 40y - y2

(c) Show that at the point of utility-maximization in (b) the ratio of marginal utility of the two goods equals their price ratio.

(d) Provide a one-sentence interpretation for the value of the Lagrangian multiplier obtained in (b).

(e) At what income level is Ivy’s utility of money maximized?

(f) Using the answers from earlier parts, calculate Ivy’s income elasticity of demand for good X and Y.

Answer 1

(c). maximize U = 30X - 0.5X² +40Y - Y²

subject to budget constraint : P_X.X + P_Y.Y = M

solving through lagrange function,

maximize U = 30X - 0.5X² +40Y - Y² + ( M - P_X.X - P_Y.Y )

dU/dX = 30 - X -P_X = 0

(30 - X)/P_X =

dU/dY = 40 - 2Y - P_Y = 0

(40 - 2Y)/P_Y =

dU/d = M - P_X.X - P_Y.Y = 0

M = P_X.X + P_Y.Y

From the value of , we get (30 - X)/(40 - 2Y) = P_X/P_Y

as we know MU_X = 30 - X and MU_Y = 40 - 2Y

Therefore, at the point of utility maximization, MU_X/MU_Y = P_X/ P_Y

(d). The value of shows the equi-marginal principle.

(e). dU/dM = = 0

Therfore, (30 - X)/P_X = = 0

X = 30

   (40 - 2Y)/P_Y = = 0

Y = 20

Ivy's utility of money will be maximized at M= 30P_X + 20P_Y

(f). Income elasticity is as follows:

e_M,X = dM/dX X/M = P_XX/M

e_M,Y = dM/dY Y/M = P_XY/M