Lecture 1: Welcome to Econ 51

Click here for the quiz on this reading.

Welcome to Econ 51! I’m excited to get to see you in person soon.

In our first lecture we’ll briefly go over the syllabus and do a quick review of consumer theory. These lecture notes are meant to help you with that, but you might want to review the Econ 50 textbook if you’re rusty on this stuff!

Purpose and Outline of the Course

Econ 50 was largely concerned with how individuals make choices, and how those choices are reflected in markets. But by and large, individual agents didn’t engage with other individual agents; they only responded to market prices. It was a very anonymous model in many ways.

Econ 51 is all about interactions between agents:

Weeks 1-3: Trade Interactions

We’ll spend the first three weeks talking about people trading with one another, and how specialization and trade can lead to everyone being better off.

The takeaway from the first three weeks of class is that if you just let everyone do what’s best for themselves, you end up at an efficient (if not equitable) outcome: nobody can do better without someone else doing worse.

On the Monday of Week 4, we’ll have a Checkpoint exam on the material from these first three weeks.

Weeks 4-8: Strategic Interactions

The next four weeks will be spent on game theory, which is the branch of economics devoted to strategic interactions between agents. We’ll discuss everything from interpersonal relationships to oligopoly to auctions.

One interesting feature of these kinds of models, that we’ll get to in the very first week, is that everyone doing what’s best for themselves may not lead to an efficient outcome – indeed, in the case of the famous “Prisoners’ Dilemma,” it may lead to the least efficient outcome.

We’ll have two Checkpoint exams on this material to help solidify it: one on the Monday of Week 7, and one of the Monday of Week 9.

Weeks 9-10: Interactions with Asymmetric Information

We end with some really fun models of interactions between someone who wants someone else to do something for them, but can’t directly contract for them to do it, or lacks some information about the other person. In particular, we’ll look at incomplete contracts and the principal-agent model, in which one person (the “principal”) tries to incentivize another (the “agent”) to do something for them, even though they can’t perfectly observe the agent’s actions.

Here the key insight will be the role of information: how do you operate in a world where you can’t fully observe what someone else does, or what their preferences are? What can you do to incentivize them to do what you want them to do, or to reveal key information to you?

The final exam will be held on Thursday, December 11, from 3:30-6:30pm. It will be cumulative, but focus on the material from the last few weeks of class.

On Thursday we’ll be introducing the notion of the Edgeworth Box, which is the model that got me truly interested in economics. To make the most of our time with that model, we’ll need to review the theory of consumer preferences, and how we model them with utility functions. We’ll also talk about preferences over time, which may be new to some of you.

Preferences

In order to analyze consumer choice, we need to know something about what the consumer’s preferences are: specifically, how she feels about her options. We’ll 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, which we’ll call $A$ and $B$. There are three possibilities for how the agent feels about $A$ and $B$:

She might strictly prefer $A$ to $B$, which we’ll write $A \succ B$ (or $B \prec A$)
She might strictly prefer $B$ to $A$; that is, $B \succ A$ (or $A \prec B$)
She might be indifferent between $A$ and $B$, which we’ll write $A \sim B$

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 $x = 3$.

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

Preferences are complete: for any possible pair of choices $A$ and $B$, the agent knows whether $A \succ B$, $A \sim B$, or $A \prec B$. Put another way, there is no way that we could ask her how she felt about $A$ and $B$ and she would reply, “I don’t know!” (Note that “I don’t know” is different than “I’m indifferent between the two.”)
Preferences are transitive: if there are three options $A$, $B$, and $C$ such that the agent prefers $A$ to $B$, and also prefers $B$ to $C$, then it must be the case that she prefers $A$ to $C$. Furthermore, if her preferences are strict — that is, $A \succ B$ and $B \succ C$ — then it must be the case that she strictly prefers $A$ to $C$ ($A \succ C$). Again, the first statement is a little like saying that for any three real numbers $x$, $y$, and $z$, if $x \ge y$ and $y \ge z$, then $x \ge z$; and the second statement (about strict preferences) is similar to the proposition that if $x > y$ and $y > z$, then $x > z$.

Preferences over quantities

The preferences framework is broadly applicable to any choice someone might make: not only which combinations of goods to consume, but where to go to college, or what to major in, or where to work, or even whom to marry. 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.

First, let us be precise about what we mean by a consumption bundle. A consumption bundle, or “bundle” for short, is a combination of different quantities of goods, indexed by good. For example, if “good 1” is apples, “good 2” is bananas, and “good 3” is cantaloupes, then we might describe a combination of 4 apples, 3 bananas, and 6 cantaloupes as “bundle A,” and write it as the vector $A = (4, 3, 6)$ More generally, we might say that a “generic” bundle $X = (x_1, x_2, x_3)$ contains $x_1$ units of good 1, $x_2$ units of good 2, and $x_3$ units of good 3; in other words, each variable $x_i$ represents some quantity of good $i$. If Chuck “prefers bundle $A$ to bundle $B$,” therefore, it means that he prefers consuming the combination of goods $(a_1, a_2, a_3)$ to the combination $(b_1, b_2, b_3)$.

In this course, as in Econ 50, we’ll try to boil things down to their simplest possible case – so to analyze consumer choice, we’ll look at a fundamental tradeoff between two goods. If a consumer is making a choice between two “bundles” that contain quantities of two goods, every option can be represented by a point in a Cartesian plane, with the quantity of good 1 on the horizontal axis, and the quantity of good 2 on the vertical axis. We will call this diagram good 1 - good 2 space. The diagram below shows the general representation of two bundles in good 1 - good 2 space; move the bundles around to solidify your understanding.

See interactive graph online here.

As a shorthand, we will sometimes write “good 1 - good 2 space” as $\mathbb R_{+}^2$; that is, the set of all vectors $(x_1,x_2)$ such that $x_1 \ge 0$ and $x_2 \ge 0$. We don’t allow ourselves to think of negative quantities of goods: you can’t consume $-3$ apples! On the other hand, we’ll always allow for continuous quantities: we’re fine with consuming $\pi$ apples, or for that matter apple pie.

(Apologies. Moving on.)

Indifference Curves

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 $A$. We can, in fact, 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.) 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:

Indifference curves cannot cross. If they did, points along one indifference curve would lie in the preferred set of the other indifference curve; but since by definition the agent likes all points along an indifference curve the same amount, this is a contradiction.
Indifference curves do not move. Every point has an indifference curve passing through it; so if you change your consumption bundle, it appears that the indifference curve moves. For example, suppose you start out with 10 units of good 1 and 40 units of good 2; and someone gives you 30 more units of good 1. You can illustrate this change by dragging the point $A$ in the diagram above from $(10,40)$ to $(40,40)$. When you do so, it appears as if the indifference curve is moving to the right; however, that’s not the case! It’s just that the indifference curve passing through your initial bundle of $(10,40)$ is lower than the indifference curve passing through $(40,40)$: indeed, $(40, 40)$ was in your preferred set to $(10,40)$, because you enjoyed getting the additional 30 units of good 1.

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.

The Marginal Rate of Substitution (MRS)

The slope of the indifference curve has a special meaning: it’s the rate at which a person is just willing to exchange good 2 for good 1 — what we call the marginal rate of substitution, or MRS.

For example, suppose “good 1” is apples and “good 2” is bananas, and further suppose the bundles $X = (10,24)$ and $Y = (12,20)$ lie along the same indifference curve for someone. This means that if they currently have bundle A, and someone offered them apples in exchange for 4 of their bananas, they would be just willing to accept the offer. In other words, their MRS between goods 1 and 2 at this point is approximately 2 bananas per apple.

See interactive graph online here.

Note that the MRS is measured in units of good 2 per units of good 1: in this case, bananas per apple. In fact, this is the same units as the price ratio we saw in the last chapter. Fundamentally, both the MRS and the price ratio measure a tradeoff between good 1 and good 2. The price ratio measures the cost of an additional unit of good 1, in terms of the amount of good 2 you have to give up to get it; the MRS measures the benefit of an additional unit of good 1, in terms of the amount of good 2 you’re willing to give up to get it.

Monotonicity, Convexity, and the “Law of Diminishing MRS”

The above analysis was completely general: preferences over quantities can have a wide variety of attributes. However, economists are often interested in a subset of preferences which are characterized by two specific properties, monotonicity and convexity. Briefly and imprecisely, we say that preferences are strictly monotonic if a consumer feels that “more” of any good is always “better,” and that preferences are strictly convex if a consumer has a taste for “variety.”

The most important thing to know is that if preferences are both strictly monotonic and strictly convex, then it turns out that all indifference curves will be downward-sloping curves that are “bowed in” toward the origin. (The downward-sloping aspect follows from strict monotonicity; the “bowed in” follows from strict convexity.) This shape of an indifference curve is sometimes referred to as the “standard” shape, and is the one most economists would sketch if they were asked to draw a generic indifference curve.

Based on this geometry, each indifference curve will be getting flatter as you move down and to the right. This is sometimes known as the law of diminishing MRS. Intuitively, it suggests that the more you get of one good, the fewer other goods you are willing to give up to obtain even more of that good. It’s probably not surprising that this is related to the law of demand, for reasons that we’ll see in a few chapters.

Utility Functions

Having established what preferences are, we now need to figure out a way to model them mathematically in order to bring preferences into a quantitative model.

One approach is to think of preferences is to imagine that every possible choice is associated with a certain amount of “utility,” and that we prefer choice A to choice B if the amount of utility associated with choice A is greater than the amount associated with choice B.

For choices over consumption bundles, we could imagine a utility function, $u()$, which would take as inputs a bundle of goods $(x_1,x_2)$, and assign a number (in “utils”, or units of utility) to the resulting happiness. We could then say that if we compare bundle $A = (a_1,a_2)$ with bundle $B=(b_1,b_2)$, the preference relationship follows directly from comparing the number of “utils” associated with bundle A and bundle B: $A \succ B \iff u(a_1,a_2) > u(b_1,b_2)$ $A \sim B \iff u(a_1,a_2) = u(b_1,b_2)$ $A \prec B \iff u(a_1,a_2) < u(b_1,b_2)$ 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 $x \ge z$.)

Utility functions and indifference curves

We defined an indifference curve by saying that if $X$ lies along the same indifference curve as $A$, then the consumer is indifferent between $X$ and $A$ (formally, $X \sim A$).

Using our utility function approach, we would say that every bundle along an indifference curve produces the same number of utils: that is, if $X$ lies along the same indifference curve as $A$, then $u(x_1,x_2) = u(a_1,a_2)$.

Therefore, we can define an indifference curve through a bundle $A$ mathematically as the level set of the utility function, for the utility generated $\text{Indifference curve through }A = \{(x_1,x_2)\ |\ u(x_1,x_2) = u(a_1,a_2)\}$

For example, one simple utility function we’ll see a lot is $u(x_1,x_2) = \sqrt{x_1x_2}$ The following diagram shows a plot of this utility function in 3D space, as well as a map of its level sets. If you drag the point around, you can see that through any given point, there is an indifference curve, which corresponds to the level set of the utility function passing through that point:

See interactive graph online here.

Importantly, note that even though an indifference map only shows a few indifference curves, every point has a utility, and therefore a corresponding indifference curve. Moving the point in diagrams like this might seem like you’re “shifting” an indifference curve, but in fact all you’re doing is highlighting a different one. Bear this in mind as you play with these graphs!

Equivalent utility functions and the interpretation of “utils”

While it’s clear that assigning some real number of “utils” to every consumption bundle is useful, and allows us to plot indifference curves mathematically, we should 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 at $(20,20)$.

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!

Any two utility functions that rank bundles in the same way must also generate the same 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.

Mathematical properties of utility functions: marginal utility and the MRS

By modeling utility using multivariable functions, we can assign economic meaning to the mathematical properties of the function:

The partial derivatives of the utility function may be interpreted as reflecting the “marginal utility” of each good: that is, $\begin{aligned} \text{Marginal utility of good 1 }&= MU_1 = {\partial u(x_1,x_2) \over \partial x_1} {\text{utils} \over \text{units of good 1}}\\ \text{Marginal utility of good 2 }&= MU_2 = {\partial u(x_1,x_2) \over \partial x_2} {\text{utils} \over \text{units of good 2}}\\ \end{aligned}$
The magnitude of the slope of the level set is the marginal rate of substitution. By the implicit function theorem, this is the ratio of the marginal utilities: $MRS(x_1,x_2) = {MU_1 \over MU_2}$

While we can simply apply the implicit function theorem here, it’s good to think about the economic interpretation of the MRS as the ratio of the marginal utilities.

First, note that the marginal utilities are rates: they represent the change in utility per change in consumption of the good. Thus, if you gain 3 units of good 1, your utility increases by approximately $3MU_1$.

Imagine you’re at some point $A$ on an indifference curve, and exchange some goods to arrive at some other point $C$ on the same indifference curve. Specifically, let’s say you give up $\Delta x_2$ units of good 2 in exchange for $\Delta x_1$ units of good 1. This movement is shown in the following two graphs:

See interactive graph online here.

Losing the $\Delta x_2$ units of good 2 decreases your utility by $\textcolor{#d62728}{\text{Utility loss from A to B }= \Delta x_2 \times MU_2}$ However, gaining the $\Delta x_1$ units of good 1 increases your utility by $\textcolor{#31a354}{\text{Utility gain from B to C }= \Delta x_1 \times MU_1}$ Note, however, that you end up with the same amount of utility (since $C$ is on the same indifference curve as $A$). Therefore, you know that the utility loss from giving up the good 2 must exactly equal the utility gain from the additional units of good 1: $\Delta x_2 \times MU_2 = \Delta x_1 \times MU_1$ or, cross multiplying, ${\Delta x_2 \over \Delta x_1} = {MU_1 \over MU_2}$ As $A$ and $B$ get closer and closer together, this approaches the instantaneous rate of change along the indifference curve, or $MRS = {MU_1 \over MU_2}$

Examples of utility functions

While economists use a wide range of utility functions, we’ll just use a few main classes of functional forms in this course:

Perfect substitutes: $u(x_1,x_2) = ax_1 + bx_2$
Cobb-Douglas: $u(x_1,x_2) = a \ln x_1 + b \ln x_2$
Quasilinear: $u(x_1,x_2) = a \ln x_1 + bx_2$

There are a few others that will pop up from time to time (especially other forms of quasilinear utility functions) but we’ll be able to do almost everything we need to do with those three functional forms. Let’s remind ourselves of what kinds of preferences they represent.

Perfect Substitutes

Some goods can always be used in place of one another, though not necessarily in a 1:1 ratio; we call these perfect substitutes.

For example, suppose you’re getting drinks for a party, and all you care about is the total amount of soda you buy. Suppose two-liter bottles of soda are “good 1” and one-liter bottles of soda are “good 2.” In this case, no matter how many bottles you already have, you would view a two-liter bottle of soda as a “perfect substitute” for 2 one-liter bottles of soda. Therefore, if you got one util per liter of soda, your utility function would be $u(x_1,x_2) = 2x_1 + x_2$ You would be indifferent between any two bundles that yielded the same total amount of soda. For example, the bundle (10 two-liter bottles, 10 one-liter bottles) would give you a total of 30L of soda. You would have this same amount of soda if you had 15 two-liter bottles, or 30 one-liter bottles, or any combination of those. Your utility function and indifference map would look like this:

See interactive graph online here.

Marginal utilities and the MRS

The central feature of perfect substitutes is that the MRS is constant: no matter how many units of each good you have, you’re always willing to trade them at the same rate.

In the case above, we had $MU_1(x_1,x_2) = 2$ $MU_2(x_1,x_2) = 1$ so $MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = {2 \over 1} = 2$ That is, no matter how many 2L and 1L bottles you have, you’re always willing to exchange 2 one-liter bottles for 1 two-liter bottle.

Note that this doesn’t necessarily require that the marginal utilities are constant: for example, suppose again that you’re buying soda for a party, and that your overall utility is the square root of the total number of liters: $u(x_1,x_2) = \sqrt{2x_1 + x_2}$ Here we have $MU_1(x_1,x_2) = \tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}} \times 2$ $MU_2(x_1,x_2) = \tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}}$ However, the MRS is still just 2: $MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = \frac{\cancel{\tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}}} \times 2}{\cancel{\tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}}}} = 2$

General formulation

The general formulation of a perfect substitutes utility function is generally presented as the linear function $u(x_1,x_2) = ax_1 + bx_2$ The MRS is therefore constant at $a/b$. If $a$ increases, you like good 1 more, so you’re more willing to give up good 2 to get good 1. As $b$ increases, you like good 2 more, so you’re less willing to give it up to get more good 1.

See interactive graph online here.

It should be clear that perfect substitutes represent a utility function that is monotonic (more is always better) but not strictly convex or concave: indeed, if you’re indifferent between any two bundles $A$ and $B$, then if $C$ is a convex combination of $A$ and $B$, $C$ lies on the same (linear) indifference curve as $A$ and $B$.

Cobb-Douglas

The Cobb-Douglas functional form was first proposed as a production function in a macroeconomic setting, but its mathematical properties are also useful as a utility function describing goods which are neither complements nor substitutes.

The general form of a Cobb-Douglas function over two goods is $u(x_1,x_2) = x_1^a x_2^b$ However, we will often transform this function by taking the natural log, which allows us to transform exponents into coefficients: $\hat u(x_1,x_2) = \ln(x_1^a x_2^b) = a \ln x_1 + b \ln x_2$ This can be particularly useful when performing linear regressions. It’s also much easier to find the MRS.

Marginal utilities and the MRS

Using the exponential form $u(x_1,x_2) = x_1^a x_2^b$ the marginal utilities are $\begin{aligned} MU_1(x_1,x_2) &= ax_1^{a - 1}x_2^b\\ MU_2(x_1,x_2) &= bx_1^ax_2^{b - 1} \end{aligned}$ so the MRS is $MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = {ax_1^{a - 1}x_2^b \over bx_1^ax_2^{b - 1}} = {ax_2 \over bx_1}$

Using the log form $u(x_1,x_2) = a \ln x_1 + b \ln x_2$ the math is even simpler: since the derivative of $k \ln x$ is just $k/x$, the marginal utilities are $\begin{aligned} MU_1(x_1,x_2) &= a/x_1\\ MU_2(x_1,x_2) &= b/x_2 \end{aligned}$ so the MRS is $MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = {a/x_1 \over b/x_2} = {ax_2 \over bx_1}$

Either way, the MRS is increasing in $a$ and decreasing in $b$; the more you like good 1 (or the less you like good 2), the more good 2 you’ll be willing to give up to get more good 1.

Its indifference map is familiar to you by now, as we’ve seen several examples of it. Try changing $a$ and $b$ to see how it affects the indifference map:

See interactive graph online here.

As you can see, at any given point, increasing $a$ (or decreasing $b$) causes the MRS to increase and the indifference curve to become steeper at that point. Conversely, decreasing $a$ (or increasing $b$) causes the MRS to decrease and the indifference curve to become flatter at that point.

Try choosing a pair of values for $a$ and $b$ and then doubling both of them: for example, look at $a = 2$ and $b = 3$, and then $a = 4$ and $b = 6$. You’ll see that this generates the same indifference map and MRS. We can use this fact to “normalize” functions of this form, as described in the next section.

Normalizing a Cobb-Douglas utility function

As we discussed earlier, it’s often possible to normalize a utility function by making its relevant coefficients (or in this case, exponents) sum to 1. In this case that means raising the utility function to the power $1/(a+b)$: $\hat u(x_1,x_2) = \left[x_1^a x_2^b \right]^{1 \over a + b} = x_1^{a \over a + b} x_2^{b \over a+b}$ or $\hat u(x_1,x_2) = x_1^\alpha x_2^{1 - \alpha}$ where $\alpha = {a \over a + b}$ Of course, you do this in the log form as well, to get $u(x_1,x_2) = \alpha \ln x_1 + (1-\alpha) \ln x_2$

For example, you can normalize the function $u(x_1,x_2) = x_1^3x_2^2$ by raising it to the power of 1/5: $\hat u(x_1,x_2) = \left[x_1^3 x_2^2\right]^{1 \over 5} = x_1^{3 \over 5} x_2^{2 \over 5}$ which would be $x_1^\alpha x_2^{1 - \alpha}$ for $\alpha = {3 \over 5}$.

We can therefore plot the utility function just using the single parameter $\alpha$ to express the relative weight the agent places on good 1:

See interactive graph online here.

This is handy because it allows us to summarize an agent’s preferences over two goods with a single parameter. It’s also particularly important for the Cobb-Douglas utility function, because it will turn out when we analyze market behavior that this normalized $\alpha$ will be the fraction of a consumer’s income they spend on good 1. But that’s a topic for next week…

Quasilinear utility functions

One class of utility functions of particular interest to economists model preferences in which the marginal utility for one good is constant (linear) and the marginal utility for the other is not. That is, the utility function might be written as $u(x_1,x_2) = v(x_1) + x_2$ The marginal utilities are therefore $\begin{aligned} MU_1(x_1,x_2) &= v^\prime(x_1)\\ MU_2(x_1,x_2) &= 1 \end{aligned}$ so the MRS is $MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = v^\prime(x_1)$ It’s easy to show that this utility function is strictly monotonic if $v^\prime(x) > 0$, and strictly convex if $v^{\prime \prime}(x_1) < 0$; that is, if good 1 brings diminishing marginal utility.

Some examples of quasilinear utility functions are: $\begin{aligned} u(x_1,x_2) &= a \ln x_1 + x_2 & \Rightarrow & MRS(x_1,x_2) = {a \over x_1}\\ u(x_1,x_2) &= a \sqrt{x_1} + x_2 & \Rightarrow & MRS(x_1,x_2) = {a \over 2\sqrt{x_1}}\\ u(x_1,x_2) &= ax_1 - bx_1^2 + x_2 & \Rightarrow & MRS(x_1,x_2) = a - 2bx_1 \end{aligned}$

The key feature here is that the MRS only depends on good 1. Therefore, the indifference curves are all parallel transforms of each other:

See interactive graph online here.

How should we interpret this utility function? One intuitive way of thinking about it is that the two goods each have diminishing marginal utility, but that one diminishes a lot faster than the other; so that for the purposes of the analysis we’re conducting, one of the goods (in this case, good 2) might as well have a constant marginal utility.

One common use of a quasilinear utility function is when we’re thinking about one good in isolation, or more precisely in comparison to “all other goods.” In this case we can let “good 2” be what’s sometimes called a composite good. In a market setting, we often let good 2 be “dollars spent on other goods,” in which case we can interpret the MRS as your willingness to pay for good 1 (i.e. your willingness to give up a certain number of dollars on other things.) For small purchases, it makes sense to think that each dollar you spend has a constant marginal utility, while each unit of the good might have diminishing marginal utility; hence the use of a quasilinear utility function.

For instance, think about a situation in which you have €1000 in cash and want to buy some boxes of tic tacs for, say, a dollar each. The first, second, and third box of tic tacs probably exhibit dramatically diminishing marginal utility. However, the utility difference between your 1000th dollar, your 999th dollar, and your 998th dollar is probably not nearly as great. Therefore, it makes sense to treat your utility from money linearly.

Preferences over time

We’ll conclude by thinking about a particular kind of tradeoff: the tradeoff between the present and the future. This will be review for some of you, but new for anyone who didn’t take Econ 50 in the Spring (unless you’ve seen it in another class).

We can use “good 1 - good 2” space to analyze tradeoffs between two goods: for example, you have €20 in your pocket to spend on apples and bananas; your choice is how to allocate that money between the two goods.

Many economic decisions come down to allocating resources across time: that is, between current and future uses. If you have that €20, should you spend it all on apples and bananas, or should you save some of it for your future self?

As with any tradeoff, any decision comes with an opportunity cost; and the “good 1 - good 2” model we have can easily be adapted to model the tradeoff between present and future consumption. For the purposes of this analysis, we’ll just think of a two-period model, where “good 1” is “consumption now” and “good 2” is “consumption in the future.” However, just as “good 1 - good 2” model could be extended to any number of goods, so too can we think of this as a simplification of a more general model in which we consider decisions made over a larger number of time periods.

Technically, one could have any preferences over consumption in different time periods, just as one could have any preferences over any goods. However one models these preferences, though, there’s a fundamental breakdown between how one feels about consumption within a time period, and how one thinks about comparing consumption across time periods.

One elegant (but almost certainly flawed) way of thinking about these preferences is to assume that the way you feel about money within a time period is independent of the time period: that is, consuming $€100$ today gives you exactly as much utility today as consuming $€100$ tomorrow would give you tomorrow. In other words, there is some “indirect utility” function $v(c)$ which relates monetary consumption $c$ (in dollars) with happiness (in utils) within a given time period.

However, your present self doesn’t value current consumption and future consumption equally: if you’re impatient, or present-biased, you might value current your utility over your own future utility. Thus, your overall utility, as viewed from the present, might have a form something like $u(c_1,c_2) = v(c_1) + \beta v(c_2)$ where $\beta < 1$ represents the amount by which you “discount” future utility.

In fact, we’ve already seen a number of utility functions which have this general form. In particular, if $v(c) = \ln c$, then intertemporal preferences may be represented by the Cobb-Douglas utility function $u(c_1,c_2) = \ln c_1 + \beta \ln c_2$ The MRS of this function would be $MRS = {c_2 \over \beta c_1}$ As a first approximation of modeling preferences over time, this has a lot going for it:

it is decreasing in $c_1$: the more money you are consuming in the present, the less you’re willing to give up future consumption to increase present consumption.
it is increasing in $c_2$: the more money you are consuming in the future, the more you’re willing to give up future consumption to increase present consumption.
it is decreasing in $\beta$: the more patient you are, the less you’re willing to give up future consumption to increasing present consumption (or, more naturally, the more you’re willing to give up present consumption to increase future consumption)

While this works for the metaphor of two periods, for a long time economists took this a step further, postulating that people would discount a future stream of payments exponentially: $u(c_1,c_2,...,c_T) = v(c_1) + \beta v(c_2) + \beta^2 v(c_3) + \cdots \beta^{T-1}v(c_T)$ This goes beyond saying that you’re just present-biased: it hypothesizes that “rational” people have time-consistent preferences. That is, it says the way you compare consumption today and consumption tomorrow is exactly the same way as you compare consumption, say, 1000 days from now with consumption 1001 days from now. A large literature in behavioral economics has pretty well disproved this hypothesis, and in its placed offered a range of other ways to think about how people make intertemporal choices.

But as with all our utility functions, realism is not our goal: we’re trying to model the kind of tensions that exist in evaluating how someone might think about distributing their consumption across time, and for those purposes the metaphor of a two-period model with a “patience” parameter $\beta$ works just fine.

What to do before the first class

Before the first class, please read the course syllabus and take the quiz on Canvas using the link below.

Please also fill out the section preferences form as soon as possible, and definitely before the first class; sections start in Week 1, and we’ll assign people to sections no later than Tuesday. Don’t worry, you can easily change sections!

See you on Tuesday morning at 10:30am in Bishop Auditorium!

Reading Quiz

That's it for today! Click here to take the quiz on this reading.