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Lecture 2: Preferences, Utility Functions, and Indifference Curves

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For the first three weeks of this class, we’ll be analyzing the classic problem of scarcity and choice: people have unlimited wants, but limited resources. Therefore, one fundamental problem of economics is what the “optimal” thing to do with those resources is.

The most familiar problem of scarce resources to most people is money. You have a certain amount of money, which you can spend on various different things, or save. How should you decide how to divide your money between competing needs? How much should you spend on food, on clothing, on apps, on boba tea? This is the model of consumer choice, and it underpins almost every microeconomic model. After all, if we’re modeling anything in an economy, someone has to be the end consumer; and that person’s decision process is going to drive everything else that happens in the model.

Our analysis of consumer choice will comprise three modules:

As we develop this economic model, we will develop from first principles the techniques of multivariate calculus used to solve this kind of optimization problem.

Additional resources for these modules

If you’d like to watch videos of people explaining these concepts, I’ve included links in these notes to the relevant videos from the Intermediate Microeconomics Video Handbook (IMVH) from UCSD. Note that you have to be logged in for the links to work; if a link takes you to a page with yellow buttons to buy or access the video handbook, just log in and then click the link again.

If you want another textual treatment, check out Hal Varian’s book Intermediate Microeconomics: A Modern Approach. Week 1 corresponds to the chapters titled “Preferences” and parts of “Utility;” week 2 corresponds to the rest of “Utility” and “Budget Constraints”; and week 3 covers the material from “Choice.”

Note that there are some notational differences between the three treatments. For example, I treat the MRS (which we’ll get to next class) as a positive number, while both IMVH and Varian treat is as a negative number. And while both Varian and I use $m$ to represent money income, the IMVH uses $I$. These kinds of notational differences are common in economics; on homeworks and exams, though, please use my notation.

OK – let’s get to the economics!

Choice spaces in general, and commodity bundles in particular

Relevant IMVH video: C1a

There are lots of kinds of choices people make: where to go to college, what kind of career to go into, who to marry. Economists look at all of these choices, and they have serious economic importance.

The first step in analyzing any model of choice is to define the choice space: that is, what the relevant universe of possible choices we’re talking about is. If you’re thinking about where to go to college, the relevant choice space is the set of all colleges and universities. If you’re thinking about what kind of career to go into, you’re probably thinking about a whole career path or trajectory of decisions.

One possible choice space is the choice of quantities of goods: for example, when you go to the grocery store, you can fill your basket with any combination of hundreds of items from the shelves. The combination you check out with is variously called “basket of goods,” a “consumption bundle,” or a “commodity bundle.” We can think of it as a vector, or a list of numbers, indexed one for each good. A vector representing a bundle of $n$ types of goods may be written as \(X = (x_1, x_2, x_3, ..., x_n)\) where $x_i$ is the amount of the $i^\text{th}$ good. Formally, we write this choice space as $\mathbb{R}_+^n$. The $\mathbb{R}$ indicates that each element of the vector is a real number. The “+” means we’re only considering (weakly) positive numbers: you can’t buy a negative number of apples at the store. The $n$ indicates the number of goods. For more on this notation, see this Wikipedia page.

For example, if “good 1” is apples, “good 2” is bananas, and “good 3” is cherries, then the vector \(X = (6,2,10)\) would represent $x_1 = 6$ apples, $x_2 = 2$ bananas, and $x_3 = 10$ cherries, and the choice space of all possible combinations of these fruits would be $\mathbb{R}_+^3$.

When we’re looking at any consumer choice, it’s generally important to also consider the time in which the choice is being made. Generally, we think of the resources a consumer has as being measured in some kind of time period: for example, a weekly allowance or yearly income. So when we’re looking at the decisions to be made with that income, we’re thinking about decisions within that time period. Hence, it’s most precise to think about 6 apples per week, or per some other time period; though for the purposes of this class we will generally just use the shorthand of saying 6 apples. Technically, economists refer to “6 apples” as a stock, and “6 apples per week” as a flow. For example, you can think of a store has having a stock of 200 apples, and a flow of 20 sales of apples per day.

Finally, we’re going to make the seemingly extremely unrealistic assumption that all goods may be infinitely divisible: so we’ll talk about buying 1.6 apples, having $\frac{2}{3}$ of a child, or even buying $\pi$ slices of pie. Of course, this doesn’t make any sense if you’re thinking about putting 1.5 apples in your shopping cart, and it’s hard to say that someone buying $\pi$ slices of pie is a rational choice This is the kind of joke you’re going to have to get used to in this class. I apologize in advance.! But if you think of someone buying 48 apples per month, it might make sense to think of them buying about 1.6 apples per day. Regardless, for the purposes of this course, every number you see may be infinitely divisible. (This will also allow us to use calculus, which relies on functions being continuously differentiable.)

Good 1 - good 2 space

So, we’ve gone from choice in general to choices of vectors of $n$ goods. Now let’s get even simpler, and think of choices of just two goods: that is, the choice space $\mathbb{R}_+^2$. This model represents the most basic fundamental tradeoff a person can face: the choice of how to divide one’s resources between two competing goods.

It also has a distinct advantage, which is that we can plot a commodity bundle with just two good in a Cartesian plane:

See interactive graph online here.

This is the fundamental kind of graph we’ll be looking at to analyze any kind of tradeoff faced by an economic agent. It shows the quantity of good 1 on the horizontal axis, and the quantity of good 2 on the vertical axis. Therefore, any point in this graph represents a bundle of quantities of these two goods. You can drag the points around the space; try it out!

Preferences

Now that we’ve established a choice space, we can start to think about how a consumer will go about choosing an optimal bundle. Let’s build up our theory of preferences from a foundation of bilateral comparison: that is, let’s initially consider two potential choices any economic agent could make, such as the choice between bundles $A$ and $B$ in the graph above. There are three possibilities for how the agent feels about $A$ and $B$:

We can combine these to describe potential relationships between choices: in particular, we can say that someone weakly prefers $A$ to $B$, which we’ll write $A \succsim B$, if $A \succ B$ or $A \sim B$. This is a little like saying $x \ge 3$ if either $x > 3$ or $x = 3$; intuitively, you’d say something like “you like $A$ at least as much as you like $B$.”

Assumptions of rational choice

Our theory of rational choice will rely on two fundamental assumptions about preferences:

The IMVH video C1b discusses these properties; in that video they also talk about monotonicity, convexity, and continuity, which we’ll get to on Monday.

Indifference curves: representing preferences in good 1-good 2 space

This preferences framework is broadly applicable to any choice space; however, in the particular choice space for quantities of goods, in which $A$ and $B$ represent potential consumption bundles, we can define some additional concepts related specifically to the quantities of the goods.

Our assumption that preferences are complete means that for any bundle $A$, every other bundle in the choice space is either preferred to $A$, dispreferred to $A$, or indifferent to $A$. This means we can shade every point in good 1 - good 2 space with a color representing this relationship. In the graph below, the curve passing through bundle $A$ represents all the bundles in the choice space for which the agent is indifferent between that bundle and bundle $A$.

See interactive graph online here.

We call this the indifference curve for bundle $A$, and define it more formally as \(\text{Indifference curve for }A = \{B \in \mathbb R_{+}^2 | A \sim B\}\) This indifference curve partitions the choice space into those bundles which are preferred to $A$, and those which are dispreferred to $A$: \(\begin{aligned} \textcolor{#2ca02c}{\text{Preferred set to }A} &= \{B \in \mathbb R_{+}^2 | B \succ A\}\\ \textcolor{#d62728}{\text{Dispreferred set to }A} &= \{B \in \mathbb R_{+}^2 | A \succ B\} \end{aligned}\) (Note: As shown, the preferred set in this case lies above and to the right of the indifference curve; this is often, but not always, the case; we’ll talk about that later on.) You can toggle the visibility of these sets using the checkboxes in the diagram above. Furthermore, if you drag bundle $B$ around, you can see which bundles are preferred, dispreferred or indifferent to bundle $A$.

There are a few important things to understand about indifference curves:

For this reason, we often show an (unmoving) indifference map, which shows the indifference curve passing through various bundles. Now, as you move the points around, you can see that the indifference curves don’t move. However, it’s a little harder to determine all the time what the preference relationship between the two bundles is. If they’re separated by an indifference curve, it’s clear which one is preferred; but if they lie between the same two indifference curves, we can’t tell from this diagram which one will be preferred.

See interactive graph online here.

“Goods” and “bads”

Note that we’ve drawn the indifference curves such that moving up and to the right – that is, having more of each good – is preferred. But this need not be the case! Some things – like risky investments, or terrorist attacks, or long wait times in airports, or pollution – are always “bad.” If you would prefer having less of something to having more of it, then we say that commodity is a “bad.”

When we’re analyzing a tradeoff between two goods or two bads, the indifference curves will always slope downward: in the case of two goods, utility is increasing as we move up and to the right (increase our consumption of the goods), and in the case of two bads, utility is increasing as we move down and to the left (decrease our consumption of the goods). On the other hand, if we’re dealing with one good and one bad, the indifference curves will be upward sloping: for example, if good 1 is “good” and good 2 is “bad,” then our utility will increase as we move up and to the left in good 1 - good 2 space.

See interactive graph online here.

Note that a “bad” may be converted into a “good” by re-casting it as its lack. For example, you could plot the TSA’s problem as balancing “passengers screened per hour” vs. “days with no terrorist attack.” These measure the same tradeoff, but cast the tradeoff as being between two good things rather than two bad things.

Utility functions

We’ve established what preferences are; but in order to bring preferences into a quantitative model, we need to figure out a way to model them mathematically.

Suppose that instead of just saying “$A$ is preferred to $B$,” we said “$A$ brings me more utility than $B$.” This implies that $A$ and $B$ each give some “level of utility” which can be compared. We’ll only have the squishiest notion of what “utility” might be. We’ll measure it in a made-up unit call “utils.” It’s derived from the moral and ethical school of philosophy known as utilitarianism, which is much more complicated and nuanced than just “we can quantify happiness using utils.” If you’d like, you can read more about it here.

In any case, this approach asserts that we can associate any bundle $X = (x_1,x_2,…,x_n)$ with some utility level $U$: that is, there is some mapping, which we can write $u()$, such that \(u(x_1,x_2,...x_n) = U\) We could then say that bundle $A = (a_1,a_2)$ is preferred to bundle $B=(b_1,b_2)$ if and only if it yields a higher utility number: \(A \succ B \iff u(a_1,a_2) > u(b_1,b_2)\) \(A \sim B \iff u(a_1,a_2) = u(b_1,b_2)\) \(A \prec B \iff u(a_1,a_2) < u(b_1,b_2)\) Because we’re assigning every bundle a real number in utils, we immediately get completeness and transitivity, because the set of real numbers itself is complete and transitive. (That is, you can compare any two numbers, and if $x \ge y$ and $y \ge z$ then $x \ge z$.)

However, to do this, we have introduced a new kind of mathematical object: the multivariate function. Let’s take a little pause from the economics now, and get acquainted with what some of the properties of such a function are.

Multivariate (vector) functions

Many people describe a function as “black box” in which something goes in (an input), is transformed, and comes out as something else (an output). For a univariate function, both the input and output are scalars (i.e. numbers).

A multivariate function takes a vector of inputs and returns either a real number or a vector as output. As we described above, a vector is an ordered array of numbers: for example, $(4,3)$ or $(2,1,2)$. For now, let’s concentrate on a specific, simple kind of multivariate function: one that takes an ordered pair $(x,y)$ and returns a number $z$.

For example, $f(x,y) = x^2 + y^2$ is a multivariate function: if you input the vector $(4,3)$ into the function you get the real number $25$ out: \(f(4,3) = 4^2 + 3^2 = 16 + 9 = 25\)

Plotting a multivariate function with two inputs

To plot a univariate function, we use a two-dimensional graph: that is, if $y = f(x)$, we have one axis representing the value of $x$, and a second representing the value of $y$.

To plot a multivariate function that takes a pair $(x,y)$ as its input and returns $z$ as its output, we can create a three-dimensional graph showing the inputs as an $(x,y)$ point on a Cartesian plane, and the output as a vertical $z$ value. In this way, a plot of a multivariate function is a surface plot, like the one below:

See interactive graph online here.

Level sets of a multivariate function with two inputs

Relevant IMVH video: A2a

Another object of interest is the set of all points which have a certain height, or “level.” These are called level sets; the level set for some level $z$ can be expressed mathematically as \(\text{Level set for }z = \{(x,y)\ |\ f(x,y) = z\}\) which we read as “$z$ is the set of all vectors $(x,y)$ such that $f(x,y) = z$.” Visually, if we imagine a horizontal plane with a height of $z$, the level set for $z$ is where that plane intersects the surface plot of the function $f(x,y)$:

See interactive graph online here.

To see how the level set is generated in this way, click the “Show 2D Projection” box in the diagram above. This draws the level set on the plane.

Contour maps of a multivariate function with two inputs

Drawing 3D graphs is hard, and often unimportant after you’ve gotten an initial intuition for what’s going on in three dimensions. We’ll therefore generally visualize the three-dimensional function in our normal two-dimensional space by thinking about a projection of the level sets onto the $x-y$ plane.

We can do this for any number of values of $z$, of course, and repeating the process for a number of values of $z$ gives us a contour map of the function. This is just like a topographical map of a mountain, which shows the sets of points which share an altitude, displayed on a paper map.

See interactive graph online here.

Utility functions and indifference curves

Relevant IMVH video: C1g

Now that we have the mathematical apparatus of multivariate functions, we can use them plot indifference curves and analyze preferences.

Let’s think about a simple utility function: \(u(x_1,x_2) = \sqrt{x_1x_2}\) For example, the bundle $(x_1,x_2) = (40,10)$ would have a corresponding utility of $u(40,10) = \sqrt{40 \times 10} = 20$.

The set of all points with that utility is any combination of $(x_1,x_2)$ such that \(\begin{aligned} u(x_1,x_2) &= 20\\ \sqrt{x_1x_2} &= 20\\ x_1x_2 &= 400\\ x_2 &= {400 \over x_1} \end{aligned}\) Therefore, this indifference curve goes through points like $(5,80)$, $(10,40)$, $(20,20)$, etc. Try dragging the blue dot in the right-hand graph below to see how the indifference curve through various different points changes:

See interactive graph online here.

We can also compare any two bundles to see which one has a higher utility. Because any bundle has some amount of utility, it has an indifference curve passing through it; so a bundle on a higher indifference curve corresponds to a higher utility level, and is therefore preferred:

See interactive graph online here.

The hardest thing for most students to get their head around is the fact that even though we only draw a few indifference curves, every point in good 1-good 2 space has an indifference curve passing through it. Hopefully this exposition helps you understand it now; but keep it in mind as we move forward, and the models get much more complicated!

Do we need to take utils seriously to use utility functions?

Absolutely not!

While it’s clear that assigning some real number of “utils” to every consumption bundle is useful, it’s important to pause and ask ourselves whether it’s something we can actually do in a philosophically coherent way. After all, we don’t want to build an entire theory of consumer behavior on top of a mathematically convenient but false assumption!

We can first note that the cardinal value of “utils” has no meaning, any more than the “10” that represents the maximum volume on most amplifiers.

However, we’re not interested in cardinal values: we’re only interested in utility functions insofar as they can represent ordinal preferences. That is, we only need the utility function to be able to tell us whether we prefer bundle A or bundle B, not by how many utils we prefer bundle A to bundle B.

For example, we previously looked at the utility function $u(x_1,x_2) = \sqrt{x_1x_2}$. According to this utility function, $u(40, 10) = 20$, $u(10,10) = 10$, and $u(20,20) = 20$. Therefore, according to that utility function, $(40,10)$ is preferred to $(10,10)$ and generates the same utility as $(20,20)$.

Let’s compare this utility with a utility function which gives twice as many utils to every bundle: that is, $\hat u(x_1,x_2) = 2\sqrt{x_1x_2}$. This utility function would assign 40 utils to $(40, 10)$ and $(20,20)$, while assigning 20 utils to $(10,10)$. But it would rank all three bundles in exactly the same way!

Visually, any two utility functions that rank bundles in the same way must also generate the correct indifference curve through any consumption bundle. In other words, as long as a utility function results in the correct indifference map, it doesn’t matter what numerical “level” each of the indifference curves has. Here’s the utility function $\hat u(x_1,x_2) = 2\sqrt{x_1x_2}$ plotted, along with its indifference map. We can see that it produces a lot more “utils” from the left-hand graph than the figure above, but the indifference curve through any given point is exactly the same:

See interactive graph online here.

According to the first utility function $u(40,10) = 20$; according to the second utility function, $\hat u(40,10) = 40$. So the bundle $(40,10)$ gives twice as many utils as it did before! However, the new utility function doubles the utility of every bundle. This means that all the bundles which were previously giving utility of 20 are now giving utility of 40; so the set of all bundles yielding the same utility as $(40,10)$ — that is, the indifference curve passing through $(40,10)$ — doesn’t change. Intuitively, this is true for the same reason that it doesn’t matter whether a contour map shows the altitude for each contour line in feet or meters; all that matters is that each contour line shows the set of points which share the same altitude.

More generally, if $\phi(U)$ is a strictly increasing function in $U$, then applying $\phi()$ to any utility function doesn’t change the preference ordering represented by that function. We call such an application a positive monotonic transformation of the utility function. For example, consider the utility function $u(x_1,x_2) = x_1x_2$, and transform it according to the function $\phi(u) = 2u^3 + 4$. Therefore, \(\hat u(x_1,x_2) = 2u(x_1,x_2)^3 + 4 = 2x_1^3x_2^3 + 4\) To find equation of the indifference curve through the point $(1,2)$ using $u(x_1,x_2)$, we would first establish that \(u(1,2) = 1 \times 2 = 2\) The expression for the indifference curve is therefore \(x_1x_2 = 2\) If we do the same thing using $\hat u(x_1,x_2)$, we would establish that \(\hat u(1,2) = 2 \times 1^3 \times 2^3 + 4 = 20\) So the expression for the indifference curve is \(\begin{aligned} 2x_1^3x_2^3 + 4 &= 20\\ 2x_1^3x_2^3 &= 16\\ x_1^3x_2^3 &= 8\\ x_1x_2 &= 2 \end{aligned}\) which is exactly the same expression. Since this is true for any point $(x_1,x_2)$, we can see that these two functions have exactly the same indifference curve through any point; and therefore represent the same preferences.

Normalizing utility

It will often be useful to use positive monotonic transformations to “normalize” a utility function. For example, let’s take the utility function \(u(x_1,x_2) = ax_1 + bx_2\) where $a>0$ and $b>0$. This is a linear utility function (what we’ll introduce in on Monday as “perfect substitutes”) in which each unit of good 1 yields $a$ utils and each unit of good 2 yields $b$ utils. If we multiply this by $1/(a+b)$, we get the transformed utility function \(\hat u(x_1,x_2) = \tfrac{a}{a+b}x_1 + \tfrac{b}{a+b}x_2\) By construction, the coefficients on $x_1$ and $x_2$ now sum to 1: \({a \over a+b} + {b \over a+b} = {a + b \over a+b} = 1\) Let’s write \(\alpha = {a \over a+b}\) Therefore \({b \over a + b} = 1 - {a \over a + b} = 1 - \alpha\) So our new utility function becomes \(\hat u(x_1,x_2) = \alpha x_1 + (1-\alpha) x_2\) In this formulation, we can interpret $\alpha$ as being the relative weight this utility function places on good 1, with $1 - \alpha$ being the relative weight it places on good 2. This allows us to express an agent’s preferences in terms of the single variable $\alpha$ rather than two variables $a$ and $b$.

See interactive graph online here.

For more on this topic, and to see some examples, see the IMVH video C1d.

Summary and next steps

In this lecture we have:

Next time we will dive a little deeper in to the characteristics of utility functions and indifference curves. See you then!

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