The MRS and Optimal Choice
Again, recall that for a given utility function
u(x1,x2) the MRS is given by
MRS(x1,x2) = -
u2(x1,x2) 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 (x1*,x2*)
is not on the boundary (that is
x1*>0 and x2*>0
if and only if (x1*,x2*) is the optimal
i.e., the MRS at the optimal choice x* is equal to the slope of the
On the other hand, if the optimal choice is
on the boundary (that is
x1*=0 or x2*=0)
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
condition. This gives you one equation for
x1* and x2*.
Use the budget line
p1x1* + p2x2* = m as the
second equation. Now solve for
x1* and x2*.
If the solution has the property that
x1*, x2*>=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/p1,0) and (0,m/p2). 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:
never holds. This is the case for example if
p1=1 and p2=2, as
MRS(x*)=-1 and -p1/p2=-1/2.
- x1*<0 or x2*<0.