Substituion Effect Question

[Akash]

Charlie's utility function is xA*xB. The price of apples used to be $1 and the price of bananas used to be $2, and his income used to be $40. If the price of apples increased to 8 and the price of bananas stayed constant, the substitution effect on Charlie's apple consumption reduces his consumption by

(a) 17.50 apples.
(b) 7 apples.
(c) 8.75 apples.
(d) 13.75 apples.
(e) None of the other options are correct.

Doubt when we calculate compensated income, is it 110 or 120?

[Dr. Strange]

Compensated income is coming out to be 113.13 using Hicksian compensation and substitution effect is 12.95 apples.

 Using Slutsky theorem, compensated income is 180 and  substitution effect is coming 8.75 apples.

[Akash]

Thanks for the reply. But how did you calculate compensated income? Please explain

[Dr. Strange]

For slutsky method we have to keep income in such way that he can just buy that previous bundle
so previous bundle was 20,10 and new prices are 8,2 so income should be 20* 8 +10*2 =180

For Hicksian method , compensated income should be such that he remains in same utility level.
Earlier for bundle 20,10 utility was 200
now lets new income be m
new bundle = m/16, m/4 (you  can find this by maximising utility with budget constraint 8a+2b<=m)
new utility = m^2/64 =200
m=113.95 approx

[Akash]

That clears it up completely. Thanks a lot!