A Californian student consumes Internet services (I) and books (B)

Her preferences are represented by a Cobb-Douglas utility function of the type: U(I,B) = I.5B.5. Initially Y = 100,
PI = PB = 1. Lately, however, because of the electricity shortage, the price of the Internet services has increased to 2. The government has decided to give a transfer to the student so that she can recover her initial welfare. In order to determine the transfer the government has hired three consultants who have made the following suggestions:
Consultant A: The transfer should allow the student to buy her initial bundle.
Consultant B: The transfer should allow the student to get her initial level of utility.
Consultant C: The government should give her a transfer of 20.
a. Using the expenditure function, find the amount of the transfer implied by consultant A.
b. Find the amount of the transfer implied by consultant B.
c. Determine whether the consumer is better or worse off from Consultant C's suggestion than before the price increases.

a. One trick when the utility and prices are symmetric for the two goods is to realize that the solution will be where the quantities are equal. Thus X = 50, Y = 50 will be the consumption bundle. So she will need $50 more to consume that bundle.
b. With X = 50, Y = 50, she will have a utility level of 50. At the new prices, in order to achieve that much utility we need to minimize the constraint subject to this level of utility. In other words, the problem is
minimize 2X + Y
subject to X.5Y.5 = 50
We can use the Lagrangian, or in this case I suppose brute force is easier.
By the (utility) constraint
Y = 2500/X
Substitute into the expenditure expression and take the derivative:
2 – 2500/X2 = 0
Solve for X = 35.2 and Y = 70.5 (appx.). The expenditures are then $141. So she needs $41 to bring her back to her initial level.
c. Since she is not getting the $41 needed, she will be worse off.