1.1 Modeling Preferences
In this first 50Q section, let’s set the tone by taking a more formal approach to the topics from the “core” Econ 50 material. In particular, we’ll take the core material on modeling preferences with utility functions that’s introduced in Lecture 2 of Econ 50. (You should read the material for Lecture 2 before reading this.)
Most of the time, economists model preferences using utility functions, which assign higher numbers to stuff people like more. In Lecture 2, we asserted that you could use a utility function to represent preferences in a mathematically coherent way. As it turns out, however, there are lots of perfectly reasonable preferences one can have that can’t be represented with such a function. The goal of this lesson is to think about what properties a preference relation needs to have in order to be representable by a utility function. In doing so, as said by a former Q student, we’ll be “earning the right” to use utility functions in Econ 50.
Before we move on to the economics, here’s a quick review of mathematical notation we will be using throughout this book:
- $\forall$ = ‘for all’
- $\in$ = ‘is an element of’ or ‘is in’
- $\exists$ = ‘there exists’
- $F:A \to B$ = ‘F is a function that has domain A and codomain (range) B’
- $\mathbb{R}$ = ‘The Real Numbers’
- $\mathbb{N}$ = ‘The Natural Numbers’
- s.t. = ‘such that’
So the statement “$\forall x\in A,$ $\exists y \in \mathbb{R}$ s.t. $f(x)=y$” means “for all $x$ in the set $A$, there exists a real number $y$ such that $f(x)=y$.”
Preferences
As in the Econ 50 lecture, we will be ranking objects according to a preference relation $\succ_i$, which we read as “person $i$ prefers.” The set of alternatives we consider will live in a set called $A$, sometimes called the “choice space.” As an example, suppose you’re trying to decide where to grab dinner after class, and you can choose between Wilbur ($W$), Stern ($S$), and Lakeside ($L$). We would say $A = \lbrace W, S, L \rbrace$, and might write $L \succ_i S$ to say person $i$ prefers Lakeside to Stern. This is natural, as at Stanford, west campus = best campus. As in the Econ 50 lecture notes, we will use $\succeq$ to mean “prefers or is indifferent to,” and $\sim$ to mean “is indifferent to.”
Utility
Utility functions are a fundamental tool for communicating preferences in a compact and convenient way. While symbols like $\succ$ are very nice and general in what they can represent, it can be very helpful to put numbers to a problem.
Fundamentally, for whatever sets of alternatives we’re considering (and these can be relatively abstract, like bundles of goods or political candidates), the purpose of a utility function is to map elements of $A$ to real numbers in a way that makes more preferred elements of $A$ output higher numbers. Formally,
Definition (Utility Representation): We say a function $u:A \to \mathbb{R}$ represents a preference relation $\succ$ if $\forall a, b \in A$, we have $u(a) > u(b)$, and vice versa. That is, if we prefer option $a$ to option $b$, then $u(a)$ must be a larger number than $u(b)$, and vice versa.
The advantage of utility functions is their adherence to the structure of the real numbers, in particular, the ordering of the real numbers. It’s not obvious what comes first between $(6,5) \in \mathbb{R}^2$ and $(5,6) \in \mathbb{R}^2$, or between apples and bananas in the set of all fruits. The real numbers, however, have a clear ordering we can make use of in our analysis.
Notice that the definition above says nothing about whether utility function representations are unique for a given set of preferences. Indeed, they are generally not unique. The function $u$ where $u(W)=1$, $u(S)=2$, and $u(L)=3$ and the function $v$ where $v(W)=2$, $v(S)=5$, and $v(L)=12$ put the dining halls in the same order, and so are equivalent in how they represent preferences. Thus any monotonic transformation of a utility function – that is, a transformation that preserves order, like multiplying by 2 or squaring when outputs are positive – will represent the same underlying preferences.
We will therefore only consider ordinal utility functions in this class, meaning we will only care about the order of ranked objects rather than the actual numerical utility values. In some applications, the numerical value of utility can be considered relevant in and of itself, in which case we speak about cardinal utilities. Such utilities will not be relevant in this course.
Some Necessary Conditions
You saw in the Econ 50 reading that a utility representation cannot exist if preferences are not complete and transitive. Let’s make these ideas a bit more formal:
Definition (Complete): A preference relation $\succ$ on $A$ is complete if $\forall a, b \in A$, we have $a \succeq b$ or $b \succeq a$ (or both). That is, any two alternatives can be compared.
Definition (Transitive): A preference relation $\succ$ on $A$ is transitive if $\forall a, b, c \in A$, whenever $a \succeq b$ and $b \succeq c$, we have $a \succeq c$.
For instance, if a non-transitive preference relation ranked the dining halls as $S\succ W \succ L \succ S$, we would need to have $u(S)>u(W)>u(L)>u(S)$, which violates the ordering of the real numbers. Thus we have shown that there are examples of preference relations which cannot be represented by a utility function.
We know completeness and transitivity are necessary conditions on $\succ_i$ for utility representation existence, but are they sufficient? That is, do we need any other requirements to ensure we can write down a function $u:A\to \mathbb{R}$ that represents $\succ$ according to the representation definition above? Before reading on, see if you can come up with a list of criteria which would guarantee a preference relation has a utility representation.
When are we guaranteed to have utility functions?
In class, we will discuss two cases in which we are guaranteed the existence of a utility function for a given set of preferences. The first result is a bit more intuitive, the second will be more useful for the types of preferences we’ll be considering in class.
Theorem 1 (Existence of Utility Functions): If $\succ$ is a complete and transitive preference relation over a set $A$ where $\lvert A \rvert <\infty$ (that is, over a set with finitely many elements), then $\succ$ has a utility representation.
Finiteness of $A$ plays a key role here. The idea is pretty straightforward: by completeness and transitivity, we can line up the elements of $A$ in order of how much we like them. Say the order looks something like $a_1, a_2, a_3…, a_N$. We can just assign $u(a_1)=1, u(a_2)=2, …,u(a_N)=N$ to the utility values, and we’re left with a perfectly valid utility representation. Anytime we’re indifferent between two elements i and j, we can set the relevant $u(a_i)=u(a_j)$. We’ll go over a more careful proof of this theorem using a method called proof by induction in class.
While the case of finitely many alternatives is fairly intuitive, we will generally be dealing with infinite choice sets in this class. In particular, our usual choice space will be the positive quadrant of $\mathbb{R}^2$, representing (possibly fractional) amounts of 2 goods to be consumed. With that in mind, Theorem 1 will not always be the most helpful for us. Instead of restricting the choice space, we’ll guarantee utility existence by putting a third constraint on the preference relation.
Definition (Continuous Preferences): We say $\succ$ is continuous if, for every sequence $x_1, x_2, ... x_n, ... \in A$ and $y_1, y_2, ... y_n, ... \in A$ such that $\lim x_n =x \in A$ and $\lim y_n =y \in A$, and $x_n \succ y_n$ $\forall n \in \mathbb{N}$, we have $x \succ y$.
This definition basically says that if I move around a little bit in the choice space, I can’t suddenly have a huge change in what I prefer: if apples are good 1 and bananas are good 2, consuming $x_1 = 2$ apples and $x_1^\prime = 2.0001$ apples should give me about the same utility. For a utility function like $u(x_1,x_2) = x_1x_2$, this holds trivially: if we think about adding just a little of good 1, we get \(u(x_1 + \delta, x_2) = (x_1 + \delta)x_2 = x_1x_2 + \delta x_2\) As $\delta \rightarrow 0$, this approaches $x_1x_2$; therefore \(\lim_{\delta \rightarrow 0} u(x_1 + \delta, x_2) = u(x_1,x_2)\) However, think about a different kind of preference ordering: suppose that what matters most to us is how any apples I have; but for two bundles with the same number of apples, I prefer to have more bananas to fewer bananas. Formally, we would say that if bundle $X\in\mathbb{R}^2$ is $X=(x_1,x_2)$ and bundle $Y\in\mathbb{R}^2$ is $Y=(y_1,y_2)$, then:
- If $x_1>y_1$, then $X\succ Y$
- If $x_1=y_1$ and $x_2>y_2$, then $X\succ Y$
These are called “lexicographic” or “dictionary” preferences, because they act a bit like alphabetical order: specifically, we order bundles first by their first component, then by their second component, and so on like words in the dictionary (Aardvark comes before Azimuth, which comes before Baby). You can play around with these preferences in the graph below. Try dragging bundles $X$ and $Y$ around to see which is preferred to which:
Why are these preferences discontinuous according to the definition above? Suppose we have a sequence where $X_n=(2+\frac{1}{n}, 2)$ for all $n\in \mathbb{N}$, and we compare this to the bundle $Y=(2,6)$. Since $\frac{1}{n} > 0$ for all $n$, $X_n\succ Y \forall n$. However, notice that $\lim X_n=(2,2)$, so $\lim X_n \prec Y$, and thus these preferences are not continuous. Intuitively, we always preferred the $X_n$ to $Y$ for every finite $n$ since the $X$ bundles were to the right of $Y$, but our preferences suddenly “jumped” to preferring $Y$ in the limit. This sort of behavior angers the math gods who have very particular ideas about how infinities ought to behave – more on that in class! For now though, we have a theorem to keep us safe from such odd examples:
Theorem 2 (Continuity of Utility Functions): If $\succ$ is a complete, transitive, and continuous preference relation over a set $A$, then $\succ$ can be represented by a continuous utility function from $A \to \mathbb{R}$.
Theorem 2 is how we earn the right to use utility functions in Econ 50. Now you know that we can only do so because the preferences we’ll be considering are complete (all bundles are ranked), transitive (no cycles in preferences), and continuous (no sudden jumps in how happy we are for small changes in bundle).