Q: If u(x,y)=x^{2}y, 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/x^{2} = 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)!