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 *

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. *