## The MRS and Optimal Choice

Again, recall that for a given utility function u(x1,x2) the MRS is given by

MRS(x1,x2) = - u1(x1,x2) / u2(x1,x2),

where u1(x1,x2) and 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 both x1*>0 and x2*>0 then

MRS(x1*,x2*)=-p1/p2 if and only if (x1*,x2*) 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 x1*=0 or x2*=0) then MRS(x1*,x2*)=-p1/p2 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:

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(x1*,x2*)=-p1/p2 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:
• MRS(x1*,x2*)=-p1/p2 never holds. This is the case for example if u(x1,x2)=x1+x2 and p1=1 and p2=2, as MRS(x*)=-1 and -p1/p2=-1/2.
• x1*<0 or x2*<0.