Question

For the below questions, consider a consumer that consumes two goods, x and z with the following utility function. U with bar on top equals x to the power of 0.25 end exponent z to the power of 0.75 end exponent. Suppose initial values for income and the prices of goods x and z are y = 100, Px =5 , and pz=15 respectively, then the price of good x falls to px=2 . What is the magnitude of the Total Effect? A)T E=12.5 B)T E=-12.5 C)T E=7.5 D)T E=-7.5

Answer 1

Correct option is (C).

U = x^0.25z^0.75

Utility is maximized when MUx/MUz = Px/Pz

MUx = $\partial$ U/$\partial$x = 0.25 x (z/x)^0.75

MUz = $\partial$ U/$\partial$z = 0.75 x (x/z)^0.25

MUx/MUz = (0.25/0.75).(z/x) = z/3x = Px/Py

Initially, Px/Py = 5/15 = 1/3

z/3x = 1/3

x = z

Substituting in budget line,

100 = 5x + 15z

100 = 5x + 15x = 20x

x = 5

z = 5

New price ratio = 2/15

z/3x = 2/15

6x = 15z

Substituting in new budget line,

100 = 2x + 15z

100 = 2x + 6x = 8x

x = 12.5

Total effect = 12.5 - 5 = 7.5