Steps to Linear/Perfect Substitutes Utility Maximization
U(x1, x2) = 3x1 + x2
1. Find the MRS and the price ratio
MRS =
2x1 + x2 = 10
Price ratio =
2. Determine which good the consumer will choose
 They will always choose all of one good and none of the other (the only exception is
when the MRS equals the price ratio)
 There are 3 ways to do step 2, choose the one that works best for you:
o Sketch a graph:
o Analyze (write MRS and price ratio into sentences):
MRS =
Price ratio =
o Compare utility levels:
If I bought all good one I could afford ________units
My utility level would be _________ utils
If I bought all good two I could afford ________ units
My utility level would be _________ utils
The consumer will choose the good that brings more utility
3. Write demand functions for both goods
 Demand function for the good the consumer will not choose:

Demand function for the good the consumer will choose (Hint—They will buy all of this
good, so how much can they afford?):
Steps to Leontief/Perfect Complements Utility Maximization
U(x1, x2) = min [x1/2, x2/3]
1. Set interior of min function equal
U(x1, x2) = min [x1/2, x2/3]  x1/2 = x2/3
2. Isolate x2
3. Plug isolated x2 into budget constraint and simplify  x1
4. Substitute x1* into isolated x2 and simplify  x2*
Note on Leontief utility functions:
Leontief utility functions mean that goods one and two must always be consumed in a fixed
ratio. This ratio can be found by looking at the denominators inside the min function.
i.e. U(x1, x2) = min [x1/2, x2/3] means that for every two units of good 1, the consumer wants 3
units of good 2
With a monotonic transformation: U(x1, x2) = min [6x1, 4x2] 
U(x1, x2) = (min [6/12x1, 4/12x2]) * 12 
U(x1, x2) = 12 min [x1/2, x2/3]
The consumer still wants two units of good 1 for every three units of good 2. The only thing that
changes because of the monotonic transformation, is the number assigned to the utility. Without
the monotonic transformation U(2, 3) = 1. With the monotonic transformation U(2, 3) = 12.