Question

Question 3. Jane consumes X-rays(X) and fancy yogurts(Y) with the following preferences U(X,Y) = X 0.2Y 0.8. X-rays cost $10 each and fancy yogurts cost $20 each. The consumer has $100 to spend. (a) What is the optimal bundle of X and Y that optimizes Jane’s utility? (b) Jane is super happy to find out that the price of fancy yogurts(Y) drop to only $10! What is her new optimal bundle of X and Y? (c) Decompose the change in her demand for goods X and Y between (a) and (b) into the substitution and income effec

Answer 1

Utility is maximized when MUX/MUY = Px/Py

MUX = U/X = 0.2 x (Y/X)^0.8

MUY = U/Y = 0.8 x (X/Y)^0.2

MUX/MUY = (0.2/0.8) x (Y/X) = Y / 4X = Px/Py

Y = 4X.(Px/Py)

4X.Px = Y.PY

(a)

Y = 4X.(10/20) = 2X

Substituting in budget line,

100 = 10X + 20Y

10 = X + 2Y

10 = X + 2 x 2X = X + 4X = 5X

X = 2

Y = 2 x 2 = 4

(b)

Y = 4X.(10/10) = 4X

Substituting in new budget line,

100 = 10X + 10Y

10 = X + Y

10 = X + 4X = 5X

X = 2

Y = 4 x 2 = 8

(c)

With initial bundle, U = 2^0.24^0.8 = 2 x 2^0.8 = 3.48

To find substitution effect (SE), we keep U unchanged and plug Y = 4X in utility function:

X^0.2(4X)^0.8 = 3.48

X^0.8X^0.2(4)^0.8 = 3.48

X x 3.03 = 3.48

X = 1.15

Y = 4 x 1.15 = 4.6

For good X:

Total effect (TE) = 2 - 2 = 0

SE = 1.15 - 2 = - 0.75

Income effect (IE) = TE - SE = 0 + 0.75 = 0.75

For good Y:

TE = 8 - 4 = 4

SE = 4.6 - 4 = 0.6

IE = 4 - 0.6 = 3.4