Q:  u(x,y)=3x+4y, I=12, Px=1, and Py=1.  What is the optimal bundle?

A:  We have discussed 2 ways to go about solving this problem.

The first way is to compare MUx/Px to MUy/Py:
Since MUx=3 and MUy=4 we have MUx/Px=3/1=3 and MUy/Py=4/1=4.
Therefore, regardless of the consumption bundle, this person would always want to consume
more good y and less x since MUy/Py>MUx/Px.  However, once this person is spending no
money on x (that is x=0) and all of their money on y (Y=I/Py) they cannot buy more y and less
x, so this is where they will end up.  The optimal bundle is (0,12).    Notice that if the two ratios
are equal then the person doesn't care about buying more of x and less y, or less x and more y.
This is the case where
the BC is the same as the optimal IC, and the person would be equally
happy anywhere on the BC.  Therefore, you could simply pick any point on the BC as the optimal.

The second way to solve this problem is to recognize that the indifference curves are linear.  As
we talked about in class, the optimal bundle will be at one of the two corners of the BC.
Therefore to find the optimal we simply need to compare the utility for the all x no y bundle to
the utility for the all y and no x bundle.  Formally, we evaluate u(I/Px,0) and u(0,I/Py).  The utility
of the all x bundle is u(I/Px,0) = u(12,0) = 3*12+4*0 = 36.  The utility of the all y bundle is u(0,I/Py)
= u(0,12) = 3*0+4*12 = 48.  Since 48>36, (0,12) is preferred to (12,0).  Hence (0,12) is the optimal
bundle.  Notice that if the utility of the two corners is the same then the person is indifferent between
them.  Since the IC is linear and has these two points and the BC is linear has these two points, the
BC is the same as the optimal IC, and the person would be equally happy anywhere on the BC.
Therefore, you could simply pick any point on the BC as the optimal.