6.4 Conditions for Calculus to Work
If you think about all the graphs of MRS vs. MRT, it’s clear that the solution to an optimization problem involves a tangency between the constraint (thus far, a PPF) and the level sets of the objective function (i.e., indifference curves) only under certain circumstances:
- The utility function must be monotonically increasing - this avoids a satiation point within the constraint
- At the left corner of the constraint, the MRS must be greater than the MRT - this avoids a corner solution at $x_1 = 0$
- At the right corner of the constraint, the MRS must be less than the MRT - this avoids a corner solution at $x_2 = 0$
- The MRS must smoothly descend along the constraint, and/or the MRT must smoothly increase, without any discontinuities - this avoids a solution at a kink
If all of these conditions are met, then the optimal condition will be characterized by a tangency condition. If they are not, you need to use the logic of the “gravitational pull” argument to determine the optimum.
Intuitively, there are some decisions we make in life in which we are carefully balancing the marginal costs and benefits of doing a little bit more of something. There are other decisions in which we proceed until there is no more marginal benefit, or in which we don’t produce or consume any of a good, or in which we face a shift (like a change in technology) that generates a discontinuity in the decision-making process. As economists, we want to know the modeling techniques available to us to analyze a broad range of real-world situations.