Q:  If u(x,y)=min(2x,y) and I=12, Px=1 and Py=1 what is the optimal bundle?

A:  Since the utility function is of the form min(Ax,By) we cannot use the equation MRS=Px/Py.  (This equation says that the slope of the IC and the slope of the BC are the same which is not true in this case.)  With these preferences we have to use the fact that the optimal will be at the corner of the IC.  As we talked about in class the corner of the IC will be where Ax=By, or in this problem where 2x=y.  This equation, together with the BC, 12=x+y, gives us two equations to solve simultaneously.  Substituting into the BC we have 12=x+2x or 12 =3x.  Therefore x=4, and since 2x=y we have y=8.  The optimal bundle is (4,8)!