Behavior and Choice: EC 102

Consumer preferences represent measures of subjective well-being: I prefer bundles which make me happier, provide me with more satisfaction. Economists use a unique term - utility - to measure the numerical representation of this well-being. It can be used to make the statements “I prefer bundle \(a\) to bundle \(b\)” and “Bundle \(a\) gives me 80 units of utility¹ while bundle \(b\) gives me 41 units of utility” equivalent. As such, a bundle which gives a higher level of utility should be preferred to a bundle which gives a lower level of utility.

Let’s be real: people in the real world do not do this. We do not assign arbitrary and fictional numerical values to bundles of goods we are considering buying. Nevertheless, utility can be a valuable tool in modeling consumer choice.² Importantly, utility is an ordinal concept: since utility values are fictionally assigned, absolute utility values matter less than relative utility values. The example of equivalent statements above could have just have easily read “I prefer bundle \(a\) to bundle \(b\)” and “Bundle \(a\) gives me 4,907 units of utility while bundle \(b\) gives me 4,096 units of utility.” The use of an ordinal concept, such as utility, is appropriate when modeling consumer preferences precisely because consumer preferences themselves are presented in ordinal preference rankings.

One advantage of the concept of utility is how it allows us to visualize a consumer’s relative preferences for available consumption bundles, as we see in the graph below. If there is one good considered, the graph would show units of the good (\(x\)) on the x-axis, and utility on the y-axis. If there are two goods in the consumer’s consumption bundles, then we can represent good \(x\) on the x-axis and good \(y\) on the y-axis, then assign utility values to each bundle on the graph³.

It captures the rate of change between a change in units of \(x\) and a change in the consumer’s utility. If, for example, an additional slice of pizza increases utility by 20, then the marginal utility of that slice is \(\frac{\Delta utility}{\Delta x} = \frac{20}{1} = 20\text{.}\) If a given bundle has two goods in it, \(x\) and \(y\text{,}\) we can define both the marginal utility of \(x\) (\(MU_x\)) - the additional utility from receiving another unit of \(x\) - and the marginal utility of \(y\) (\(MU_y\)) - the additional utility from receiving another unit of \(y\text{.}\)

Assumption A3 captures the simple concept that having more of a good is preferred to having less than a good. If more units increases utility, and A3 holds, then marginal utility must be positive: \(MU_x = \frac{+}{+} = +\text{.}\) How realistic is the monotonicity assumption? In most scenarios, it is probably fair to say that having more of something is better than having less: more money, more pairs of shoes, more books. But there may be times when having an additional unit makes you unambiguously worse off. Can you think of an example?⁴

What about assumption A4? Diminishing marginal utility captures the concept that as you acquire more units of a good, additional units are worth less utility. The first slice of pizza you eat and the third slice of pizza you eat will both increase your utility. But the first slice will likely increase utility by more than the third slice, since you are hungrier! If A4 holds, then as \(x\) increases, the marginal utility of \(x\) decreases.

Since utility translates consumer preferences to numerical values, economists often represent this translation with a utility function: a mathematical relationship between the number of goods consumed and the associated utility value. If a consumer considers how many bicycles he owns, given by \(b\text{,}\) then his utility function could be given by \(u(b) = \sqrt{b}\text{.}\) Or, if someone considers how many dollars she earns, given by \(x\text{,}\) we may want to represent the utility she receives from those earnings by \(u(x) = x^2\text{.}\)⁵

Once we graph a utility function, we can interpret its slope as marginal utility! Since marginal utility describes the rate of change in utility given a change in the number of units consumed - \(MU_x = \frac{\Delta utility}{\Delta x}\) - given a specific function or graph, this rate of change is exactly the slope of the function. Marginal utility itself can be expressed as a function. For example, \(MU_x = \frac{1}{x}\) captures A3 everywhere (\(MU\) is always positive) and A4 everywhere (as \(x\) increases, the value of \(MU_x = \frac{1}{x}\) always diminishes.)⁶

Since there are an infinite number of mathematical functions to choose from, what makes for a good utility function? As we mention above, the use of a utility function is valid only if assumptions A1 and A2 are satisfied. Moreover, our choice of function reflects how closely we adhere to assumptions A3 and A4. A utility function satisfies monotonicity (A3) if and only if the function is increasing. This holds because increasing function have a positive slope ( = MU). Similarly, a utility function satisfies diminishing marginal utility (A4) if and only if the function is concave. This holds because concave functions have a slope ( = MU) which is decreasing.

Practice Problems: Subsection 2.7.2