Q:  If u(x,y)=x²y, I=24, Px=2, and Py=2, what is the optimal bundle?

A:  First we notice that utility function is Cobb-Douglas.  
    Therefore we set the MRS = ratio of the prices.  
    This gives 2xy/x² = 2/2, which is the same as 2y/x=1 or after rearranging 2y=x.

   We also know the BC is 24=2x+2y. 
    Substituting in the fact that x=2y we can rewrite the BC as 24=2x+x which says that 24=3x or x=8.
    Plugging back in to the equation 2y=x we find that 2y=8 or y=4.

   Therefore the optimal bundle is (8,4)!