Solution

1. Start by deriving the expenditure function:

   U = x^0.25 * y^0.75

   U = (0.25*M/Px)^0.25 * (0.75*M/Py)^0.75

   U = (0.25/Px)^0.25 * (0.75/Py)^0.75 * M

   M = U * (Px/0.25)^0.25 * (Py/0.75)^0.75

   Now find the original level of utility. Demand for x is 0.25*100/1 = 25; demand for y is 0.75*100/1 = 75. Utility, U1, is:

   U1 = 25^0.25 * 75^0.75

   U1 = 57

   Inserting U1 and new prices into the expenditure function gives M3, the amount of income needed to achieve the original utility at the new prices:

   M3 = 57 * (2/0.25)^0.25 * (1/0.75)^0.75

   M3 = 119

   The CV is $119-$100 = $19.

2. To find the income and substitution effects, start by calculating the total change in x from the initial equilibrium to the equilibrium following the price increase. From above, the initial value of x, call it x1, is 25. After the price change, the new value of x, call it x2, will be:

   x2 = 0.25*100/2 = 12.5.

   The total change in x is:

   x2 - x1 = 12.5 - 25 = -12.5

   To decompose the total change into the income and substitution effects, calculate the amount of x, call it x3, that would be demanded if the individual had been compensated for the price change (that is, calculate the x demanded if M had been equal to $119 and Px is $2):

   x3 = 0.25*119/2 = 14.9

   The difference between x3 and x1 is the substitution effect:

   x3 - x1 = 14.9 - 25 = -10.1

   The difference between x2 and x3 is the income effect:

   x2 - x3 = 12.5 - 14.9 = -2.4