A Californian student consumes Internet services (I) and books (B). Her preferences are represented by a Cobb-Douglas utility function:

U(I,B) = I1/4B1/4
The prices of each good is $2 and the student has an income of $200. Over the course of the past year, the price of internet services has risen to $4, but the price of books has remained the same. The government has decided provide this student with additional money to compensate for the higher price of internet services. In order to determine the transfer the government has three consultants who have made the following suggestions:
Consultant A: The student's income should be increased by a percentage found using a consumer price index (CPI).
Consultant B: The additional income should allow the student to get her initial level of utility.

a. Find the consumer's optimal bundle before the increase in price occurs.
b. Find the consumer's optimal bundle after the increase in price occurs with income still at $200.
c. Find the amount of the transfer implied by consultant A.
d. Is the student necessarily better or worse off than before from such a transfer implied by consultant A? Explain why.
e. Is the transfer implied by consultant B more or less than the amount implied by A? Explain.
What is the precise dollar amount implied by consultant B?

a. MRS = B/I
MRT = 1
B = I
2B + 2I = 2I + 2I = 200 I* = 50, B* = 50
b. MRS = B/I
MRT = 2
B = 2I
2B + 4I = 4I + 4I = 200 I*= 25, B* = 50
c. $100 (additional amount needed to buy 50 units of internet at $4 instead of $2 )
d. Better. See notes/book/etc. for graph. Consumer would be better with additional cash by moving to a new bundle with less I and more B.
e. Less, the "true COLA" is less than the Laspeyres COLA.
f. First find utility level in a.
I = B = 50 U(50,50 ) = 7.071
Find MRS = MRT condition
MRS = B/I same as before b/c using same utility function
MRT = 2 using the new prices of $4 and $2
B = 2I
Find Optimal Bundle on Original IC
The two conditions we have are
B = 2I and B1/4I1/4 = 7.071
Now solve:
BI = 2500 (2I)I = 2500 I2 = 1250 I* = 35.36, B* = 70.82
Find Income
Y = 2B + 4I = 2(70.82 ) + 4(35.36 ) = $282.84
Thus, he will need $82.84