Utility

Utility is an ordinal concept that allows a consumer to rank choices.

Consumers behave to maximize their utility. They have preferences, which is what they want, and they get as much as possible within a certain budget.

We make three assumptions about the preferences of consumers:

1. Completeness: Consumers do have an opinion on all possible choices.
2. Transitivity: $ A > B, B > C \implies A > C $
3. Non-satiation: More is always better.
   • Marginal utility is always positive.

A plot of indifference curves is a graphical map of preferences. The two axes on such a plot are two different goods, and a consumer is indifferent between any of the combinations of goods on an indifference curve.

1. Consumers prefer higher indifference curves.
2. Indifference curves must slope downwards due to non-satiation.
3. Indifference curves may not cross due to transitivity.
4. There is only one indifference curve through every possible combination of goods.

A utility function is a mathematical representation of the preferences of a consumer.

Consider the following utility function for an individual consuming good X and good Y:

$$ U = \sqrt{XY} $$

Marginal utility is the derivative of utility with respect to the quantity of a good. Marginal utility generally diminishes as the number of goods consumed increases. The marginal utility for good X in the example is:

$$ \frac{\partial U}{\partial X} = \frac{1}{2}\sqrt{\frac{Y}{X}} $$

We can see that this quantity decreases for increasing values of $X$.