Four Models of Utility Maximization

This page is intended to very briefly summarize 3 different examples of modeling utility maximization in increasingly more complex decision-making scenarios. It also includes a fourth case as general summary of all applications of utility maximization models.

A Word on Models: Consumer Theory and utility maximization are models of consumer decisionmaking behavior. They are not descriptions or explanations. They do not tell us how people actually behave or actually decide or actually reason. Rather, they tell us how a perfectly, economically rational consumer ought to reason in order to guarantee maximum total benefit for themselves.

Case 1: A Free Good

This is a model of an economically rational consumer deciding how much of a good to consume if a good is given away with no explicit cost (no price) that the consumer must pay. In this case an "economically rational" consumer (one who is both seeking to maximize the benefit from consuming the good and who can prioritize their preferences) must decidie how much of a free good to consume in order to maximize their total utility.

Maximization Condition. The utility maximization condition in this case is:
Total utility is maximum when the marginal utility of the last unit of the good consumed is equal to zero (MU =0).

Explanation.  As a consumer continues to consume a good one unit at a time the consumer will continue to consume additional units of the good as long as each new unit provides some additonal benefit, no matter how small. Due to the law of diminishing marginal utility, after some point the consumer is aware that each new unit consumed provides less benefit that the previous unit consumed. Thus, when the last unit consumed  provides no additional (net) benefit the consumer knows the next unit will provide negative utility (it will harm her), and the consumer has acheived the maximum total benefit from consuming all previous units of the good.

Case 2: A Standard Market Good with a Price

This is the standard model of an economically rational consumer deciding how much of a good to purchase at a single price. There is no budget constraint in this version of this model.
Maximization Condition. In this case the maximization condition is:
Total utility is  maximum when the marginal untility  of the last unit purchased is equal to the price of the good. (MU = P) or (MU - P = 0) or MU/P = 1.

Case 3: Two Market Goods with Prices

This is the standard model of an economically rational consumer deciding how much to purchase of two market goods with different prices within a budget .

Maximization Condition. For the two good case, the maximization condition is:
Total utility is maximum when MU1/P1 = MU2/P2 and the budget is exhausted.

Case 4: The General Case

This is just the summary of a general principle that applies to all decisions.

Maximization Condition. An economically rational person will maximize their total utility from a decision when the marginal utiity equals the marginal cost.  The marginal cost  is the additional cost incurred by consuming one additional unit of a good.

Depending upon the specific situation the marginal cost may be reflected in different ways. In the situation of a market good (case 2) the marginal cost is simply the price of the good. In Case 3 the marginal cost is the value (MU/P) of the good not chosen. This is also the opportunity cost of the chosen good. When there is not an explicit price, nor a specific good that will not be chosen, the marginal cost can be evaluated as the "willingness to pay" (WTP) for the good. Thus, all of these situations are equivalent and the general principle of utility maximization  can be summarized as:
Total utility of the a decsion is maximized when
marginal utility = marginal cost = price = opportunity cost = willingness to pay.