Marginal utility theory was devised as a proof of the law of demand. Its value for us, however, is perhaps mostly as a method of analysis. The theory is based on a few underlying ideas:
1) utility is the power to satisfy a want
2) marginal utility (the add'l utility for an add'l unit consumed) diminishes as consumption of the item increases
3) consumers want to maximize utility
These beginning points lead to the conclusion that people will consume a mix of goods such that the m.u. per one dollar of expenditure is the same for all goods consumed.  That is the MUa/Pa = Mub/Pb, etc.  This mathematical statement is called the consumer optimum condition.

To demonstrate its validity, consider the following hypothetical example:
a person, having a fixed budget, has consumed a certain amount of A and a certain amount of B.  The exact quantities don't matter.  Say the consumer has run out of money and the situation is as follows: the marginal utility of A, for the last unit, is 40 and the price is $5; and the m.u. of B is 50, and the price is $5.

Is the consumer achieving max. satisfaction?  No! Since the MUa/Pa<MUb/Pb, the consumer has blown it -- consumed too much of A and not enough of B.  Given a chance to correct the situation, the person will give back some of A in order to buy more of B.  For a very small incremental change, the consumer will lose 8 utils per $1 for each reduction in A, and will gain 10 utils per $1 for each increase in B.  As soon as they begin to do that, the MU of A  will rise and the MU of B will fall. How do we know that?  Because the more of an item you consume the lower will be the MU for the last unit.  To show what will happen when the consumer revises the game plan, we can increase the MU of A by an arbitrary small amount, and reduce the MU of B by a small amount.  Let's make it 42/$5 for A, and 48/$5 for B.  We are moving toward and equality!  (This is not the end of it, still more of B must be consumed before we achieve an equilibrium.  The point is that when the value on the right exactly equals that on the left, there is no more room for improvement.) When we choose the new values for MU, i.e., 42 and 48, we don't change them by huge amounts, otherwise we introduce "indivisibilities", which would stand in the way of a smooth and clear outcome.  It would have been ok to use 41 and 49 instead, we are not worrying about the details.

 That is the gist of it.  We have demonstrated the validity of the optimum condition.  We finish up by next taking a case where there is an equilibrium, then we disturb it with a price decrease for, say, good B.  That sets off  a correction process like above, where, in the end, a new equilibrium is attained in which more B is bought (so, the law of demand is upheld: lower price of B causes higher quantity demanded).

Class adjourned.