Utility: As with the material on preferences, this section is quite abstract and often difficult for students to understand.  It is crucially important to make the connection back to indifference curves and to point out that an indifference curve is simply a level curve of the utility function.  Again, a discussion of the cases of perfect substitutes and perfect complements and the form of their utility functions is often quite helpful.

 

The other critical part of this section is the connection between marginal utilities and the marginal rate of substitution.  There are a variety of methods to illustrate why the slope of the indifference curve is equal to the ratio of marginal utilities.  A simple demonstration (not a proof per se) is to remind students that marginal utility (MU) can be used to quantify the change in utility for any given size change in consumption of the good.  For example, if MU₁ is +3, then for every one-unit change in consumption of good 1, utility increases by 3.  Then, if the change in good 1 is denoted by d1 or something similar, the overall change as we move along an indifference curve can be written as du = d1*MU₁ + d2*MU₂.  Finally, since du = 0 as long as we stay on the same indifference curve, d1*MU₁ + d2*MU₂ = 0 which can be rearranged to show d2/d1 = - MU₁ / MU₂.