The MRS and Optimal Choice
Again, recall that for a given utility function
u(x_{1},x_{2}) the MRS is given by
MRS(x_{1},x_{2}) = 
u_{1}(x_{1},x_{2})
/
u_{2}(x_{1},x_{2}),
where
u_{1}(x_{1},x_{2}) and
u_{2}(x_{1},x_{2}) denote the partial
derivatives of the utility function with respect to the first and the
second argument, respectively.
If the utility function is "nice", i.e., it is
monotone and has
convex weakly preferred sets,
and if (x_{1}*,x_{2}*)
is not on the boundary (that is
both
x_{1}*>0 and x_{2}*>0
then
MRS(x_{1}*,x_{2}*)=p_{1}/p_{2}
if and only if (x_{1}*,x_{2}*) is the optimal
consumption choice.
i.e., the MRS at the optimal choice x* is equal to the slope of the
budget line.
On the other hand, if the optimal choice is
on the boundary (that is
either
x_{1}*=0 or x_{2}*=0)
then
MRS(x_{1}*,x_{2}*)=p_{1}/p_{2} need
not hold (i.e., it is possible that the equality holds at an optimal
consumption choice x*, but typically the MRS and the
slope of the budget line will not be the same).
This leads to the following question:
How can we find an optimal consumption choice?
The procedure is the following:
 First make sure that the utility function is "nice" (convex
weakly preferred sets, and monotonicity). If you are not specifically
asked to show this in a particular homework problem, you can leave
this step out.
 Use the
MRS(x_{1}*,x_{2}*)=p_{1}/p_{2}
condition. This gives you one equation for
x_{1}* and x_{2}*.
Use the budget line
p_{1}x_{1}* + p_{2}x_{2}* = m as the
second equation. Now solve for
x_{1}* and x_{2}*.
If the solution has the property that
x_{1}*, x_{2}*>=0 then you are done.
 Otherwise, if the previous step does not work then you must have
a boundary optimum. In other words,
there are two candidates
(m/p_{1},0) and (0,m/p_{2}). The one with the
higher utility is the optimum. If both have the same utility, then
both are optimal. Also note that the previous step might not work for
the following two reasons:
 MRS(x_{1}*,x_{2}*)=p_{1}/p_{2}
never holds. This is the case for example if
u(x_{1},x_{2})=x_{1}+x_{2} and
p_{1}=1 and p_{2}=2, as
MRS(x*)=1 and p_{1}/p_{2}=1/2.
 x_{1}*<0 or x_{2}*<0.
