Lecture 4: Modeling Different Types of Preferences with Utility Functions

Click here for the quiz on this reading.

The story so far:

In lecture 2 we established that you could model consumer preferences using utility functions. We introduced the mathematical object of multivariate utility functions, and said that the level sets of such utility functions represented a consumer’s indifference curves.
In lecture 3 we analyzed three types of derivatives of utility functions: the marginal utility of each good, and the marginal rate of substitution, which is the magnitude of the slope of an indifference curve.

Today we’ll wrap up our analysis of preferences and utility by introducing and analyzing a few specific forms of utility functions. In the next few weeks, we’ll explore how these functional forms give rise to different kinds of consumer behavior, especially whether the consumer views two goods as complements (goods which an agent wants to consume together, like peanut butter and jelly, sugar and tea, or tennis balls and tennis racquets), substitutes (goods which can fulfill the same purpose, like different flavors of jelly, different kinds of tea, or different brands of tennis balls) or independent goods (neither complements nor substitutes, like tea and tennis balls).

Before we get to specific functional forms, though, let’s analyze two properties that preferences might exhibit: monotonicity and convexity.

Monotonicity

A preference ordering over bundles of goods exhibits monotonicity if more of any good is always better: that is, if bundle $A$ contains more of all goods than bundle $B$ (e.g. $A$ lies above and/or to the right of $B$ in good 1 - good 2 space), then you must like $A$ at least as much as $B$: that is, $A \succeq B$.

For example, think about your preferences over pizza (good 1) and soda (good 2). If you think about your lifetime consumption of pizza, it might make sense to model this as if you’d always like more pizza and more soda, meaning your preferences are monotonic. These preferences might be represented by a utility function like the one below, because no matter which bundle $X$ you start from, increasing either pizza or soda will move you into the green “preferred region:”

See interactive graph online here.

Now think about your consumption in a single meal. The first few slices of pizza and soda might taste great, but after a certain number of slices and cans of soda you might actually start to feel sick. Since consuming more beyond that point would actually make you feel worse, your preferences would be nonmonotonic:

See interactive graph online here.

You might notice that this utility function has a global maximum at (4 slices of pizza, 2 sodas), indicating that this combination of pizza and soda that gives you the most utility. Consuming any more or less of either good would decrease your utility. We call this the “satiation point” or “bliss point.”

In truth, most goods probably look like this if you look out far enough: that is, even over a lifetime, there’s some amount of pizza and soda you could have that would be just too much. For this reason, we sometimes call the region below and to the left of the satiation point the “economic region;” so one way of constraining preferences to be monotonic is to assume that we’re considering small enough quantities of each good that satiation isn’t an issue.

Strict vs. Weak Monotonicity

We say that preferences are strictly monotonic if any increase in any good strictly increases utility; that is, $MU > 0$ for all goods at all bundles, not just $MU \ge 0$. If preferences are strictly monotonic, it means their marginal utilities are never zero. This also implies that the indifference curves cannot be “thick:” even the slightest increase in a good will increase your utility, and therefore move you to a higher indifference curve.

By contrast, if a utility function is weakly monotonic, its marginal utilities might be zero or positive. For example, the Pfizer COVID-19 vaccine has a dose of 0.3 mL, and the Moderna vaccine has a dose of 0.5mL. Suppose that a clinic’s “utility function” is just the total number of usable doses it can obtain from various quantities of vaccines. It would make sense, then, that a vials of Pfizer vaccine containing 1 mL and 1.1 mL would yield the same utility, since each contains enough for just three doses, with a bit left in the vial. Mathematically, the total number of usable doses given $x_1$ mL of Pfizer vaccine and $x_2$ mL of Moderna vaccine would be $\text{Usable doses} = \text{trunc}(x_1/0.3) + \text{trunc}(x_2/0.5)$ where the function “trunc($x$)” means “$x$, rounded down to the nearest integer.” This utility function would have flat portions for any $x_1$ that is not divisible by 0.3, and any $x_2$ that is not divisible by 0.5, and “jump” by one util at any point where it gets enough to provide an additional dose. Its indifference curves would be correspondingly “thick,” because they would be indifferent between any quantities of vaccine that would give them the same number of usable doses:

See interactive graph online here.

Convexity

A preference ordering over bundles of goods exhibits convexity if, when considering two bundles between which you are indifferent, you would prefer having any convex combination of those two bundles to either of the bundles themselves.

A convex combination is just a fancy way of saying “weighted average.” For example, if bundle $A = (a_1, a_2)$ and bundle $B = (b_1, b_2)$, then bundle $C = (c_1,c_2)$ would be a convex combination of $A$ and $B$ if $\begin{aligned} c_1 &= t a_1 + (1-t) b_1\\ c_2 &= t a_2 + (1-t) b_2 \end{aligned}$ for some $t$ between 0 and 1. As shown in the following graph, $C$ is located on a segment connecting bundles $A$ and $B$:

See interactive graph online here.

Visually, preferences are convex if, for any two points on the same indifference curve, a segment connecting those two points passes only through the “preferred region,” so any convex combination of the two points must be preferred to either of the points themselves. For example, the preference ordering represented by the utility function $u(x_1,x_2) = x_1 \times x_2$ is convex. Both $A = (20,30)$ and $B = (60,10)$ lie along the indifference curve for $U = 600$, as shown below. Drag bundle $C$ left and right to confirm that any convex combination of A and B yields a utility higher than 600:

See interactive graph online here.

By contrast, we say that preferences are concave if any convex combination of $A$ and $B$ is always dispreferred to both $A$ and $B$. For example, the preference ordering represented by the utility function $u(x_1,x_2) = x_1^2 + 4x_2^2$ is concave. Both $A = (10,35)$ and $B = (50,25)$ lie along the indifference curve for $U = 5000$, as shown below. Drag bundle $C$ left and right to confirm that any convex combination of A and B yields a utility lower than 5000:

See interactive graph online here.

Intuitively, convex preferences mean that you have a “taste for variety,” and they arise in a wide variety of applications. For example, convex preferences explain why you might prefer to smooth your income over your lifetime, rather than have some years of being extremely rich and others in which you’re extremely poor; or why you might choose to eat different foods throughout the week, rather than always eating the same thing.

Some common points of confusion

While monotonicity makes a lot of intuitive sense, wrapping your head around convex preferences the first time you encounter them is a little challenging. In particular, there are two common mistakes students make, and which are worth noting.

First, it’s tempting to interpret convex preferences as meaning that any bundle with an equal amount of goods is preferred to a bundle with unequal amounts of goods: for example, that bundle $C = (2,2)$ is always preferred to bundle $A = (1,3)$ and $B = (3,1)$. But convex preferences only say that if you’re indifferent between A and B, then you must prefer C. They say nothing about what happens if you’re not indifferent between $A$ and $B$. For example, suppose “good 1” is dinners at your favorite restaurant, and “good 2” is a chalupa from Taco Bell. (I’m assuming those aren’t the same thing.) You might like variety, so you might not want to eat at your favorite restaurant every night of the week. But that doesn’t necessarily mean that you’re indifferent between (3 dinners, 1 chalupa) and (1 dinner, 3 chalupas); and it’s easy to imagine that you might prefer 3 nice dinners and one chalupa to 2 of each.

Second, keep in mind that it’s the preferences which are convex or nonconvex, not the utility function modeling those preferences. This is because a convex set and a convex function refer to slightly different things. It’s an easy slip-up to make to call the utility function itself convex (I’m sure I’ll do it in lecture from time to time!), and not really a big deal at this level; but if and when you go to graduate school, you’ll want to be as precise as possible in your language.

“Well-Behaved” preferences and the “Law of Diminishing MRS”

In this lecture we’ve seen a lot of different characteristics that preference orderings might have. In practice, we’ll often focus on a subset of preferences that we call “well-behaved,” which have the following characteristics:

Strictly monotonic: If $X$ has more of at least one good than $Y$, then $X \succ Y$.
Strictly convex: If $A \sim B$, and $C$ is a convex combination of $A$ and $B$, then $C \succ A$ and $C \succ B$.
Continuous: the preferences may be represented by a utility function that is continuous
Smooth: the preferences may be represented by a utility function that is everywhere continuously differentiable

In Econ 50 this quarter, we will pretty much always be looking at continuous and smooth preferences; Econ 50Q will look at some slightly more esoteric utility functions.

When all four of these conditions are met, all indifference curves will be downward-sloping curves that are “bowed in” toward the origin: that is, as you move down and to the right along an indifference curve, the MRS will be continuously decreasing, so the indifference curve will be getting flatter. This is sometimes known as the law of diminishing MRS. It asserts that the more you get of one good, the fewer other goods you are willing to give up to obtain even more for 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.

Testing for “well-behaved” preferences using calculus

If a utility function is smooth and continuous, we can calculate its marginal utilities and MRS using calculus. From those we can determine whether it’s monotonic and convex.

Determining if preferences are monotonic

When we looked at monotonicity, the indifference curve showing pizza and soda over a lifetime used the utility function $u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}$ This utility function has the marginal utilities $\begin{aligned} MU_1(x_1,x_2) &= \tfrac{1}{2}x_1^{-{1 \over 2}}x_2^{1 \over 2}\\ MU_2(x_1,x_2) &= \tfrac{1}{2}x_1^{1 \over 2}x_2^{-{1 \over 2}} \end{aligned}$ Because these are both positive for all values of $x_1$ and $x_2$, this utility function represents preferences which are monotonic.

By contrast, the utility function we used to plot the utility function for pizza and soda in a singe meal was $u(x_1,x_2) = 24 - (4 - x_1)^2 - 2(2 - x_2)^2$ which has the marginal utilities $\begin{aligned} MU_1(x_1,x_2) &= 8 - 2x_1\\ MU_2(x_1,x_2) &= 8 - 4x_2 \end{aligned}$ Because these change sign (at $x_1 = 4$ and $x_2 = 2$ respectively), this utility function represents preferences which are non-monotonic.

Determining if monotonic preferences are convex

If preferences are monotonic, the indifference curve will be downward sloping. In that case, we can tell if preferences are also strictly convex by examining what happens to the MRS as you move down and to the right along an indifference curve. Specifically, if $MU_1(x_1,x_2) > 0$ and $MU_2(x_1,x_2) > 0$ for all $(x_1,x_2)$, a sufficient condition for preferences to be strictly convex is if both ${\partial MRS(x_1,x_2) \over \partial x_1} \le 0$ and ${\partial MRS(x_1,x_2) \over \partial x_2} \ge 0$ with at least one of these being strict. That is, the MRS is decreasing along a downward-sloping indifference curve if it decreases when $x_1$ increases and $x_2$ decreases, since moving down and to the right along a downward-sloping indifference curve means you’re simultaneously increasing $x_1$ and decreasing $x_2$.

For example, in our analysis of convexity, we asserted that the utility function $u(x_1,x_2) = x_1x_2$ represented convex preferences. We can see that its MRS is $MRS = {MU_1 \over MU_2} = {x_2 \over x_1}$ which is decreasing in $x_1$ and increasing in $x_2$. By contrast, the concave preferences described by the utility function $u(x_1,x_2) = x_1^2 + 4x_2^2$ have the MRS $MRS = {MU_1 \over MU_2} = {2x_1 \over 8x_2}$ which is actually increasing in $x_1$ and decreasing in $x_2$; i.e., getting steeper, not flatter, as you move down and to the right along an indifference curve.

Now that we have those properties well in hand, we can analyze some specific utility functions. In fact, you’ve already seen these in the first problem set – but let’s now take the time to understand what kinds of real-world preferences may be represented by these kinds of mathematical functions.

Perfect substitutes

Let’s start with an easy one: 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 $B$.

The Cobb-Douglas Utility Function

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 we’ve a while to go before we get to that result…

Cobb-Douglas utility functions with many goods

The Cobb-Douglas utility function can easily be extended to any number of goods; for example, $u(x_1,x_2,x_3) = x_1^ax_2^bx_3^c$ or $u(x_1,x_2,x_3) = a \ln x_1 + b \ln x_2 + c \ln x_3$ Again, we can normalize this so that the sum of the exponents or coefficients is 1.

Perfect Complements (Q only)

Students in Econ 50Q, please read this page on perfect complements. Others may read it for interest, of course; but 50Q students will have homework and test questions based on this material.

Constant Elasticity of Substitution (CES) family of utility functions

The utility functions described above are actually all members of a more general “family” of utility functions called constant elasticity of substitution (CES) functions. These may be written via the formulation $u(x_1,x_2) = \left(\alpha x_1^r + (1 - \alpha)x_2^r\right)^{1 \over r}$ A little math shows that the MRS of this utility function is $MRS = {\alpha \over 1 - \alpha} \left( {x_2 \over x_1}\right)^{1 - r}$ There are two parameters in this utility function:

$\alpha$, as usual, measures how much the agent likes good 1, relative to good 2. This is a number between 0 and 1; when $\alpha = {1 \over 2}$, the utility function weighs the two goods equally. The higher $\alpha$ is, the more the agent likes good 1; this is illustrated by a higher MRS, or a higher willingness to give up good 2 to get good 1. If $\alpha = 1$, it means the agent doesn’t care about good 2 at all, and the utility function just becomes $u(x_1,x_2) = x_1$. (When $\alpha = 0$, likewise, the utility function becomes $u(x_1,x_2) = x_2$.)
$r$ measures the complementarity or substitutability of the two goods: when $r > 0$, the goods are substitutes, and when $r < 0$, the goods are complements. It affects how quickly the MRS changes as you move down and to the right along an indifference curve. The more substitutable goods are, the less quickly the MRS changes, and the “shallower” the indifference curves are.

You can check to see that the marginal utilities for this are positive, so the preferences are monotonic. For any $r < 1$ the MRS is decreasing as you move down and to the right (i.e., as $x_1$ increases and $x_2$ decreases), so the preferences are also convex. For any $r > 1$ the MRS is increasing as you move down and to the right, so the preferences are concave.

Try playing with $\alpha$ and $r$ in the diagram below to see how the indifference map changes:

See interactive graph online here.

Note that when $r$ is an extremely large negative number, the indifference curves approach the L-shaped curves of the perfect complements utility function; when $r = 0$, the indifference curves resemble those of a Cobb-Douglas utility function; and when $r = 1$, the indifference curves are linear like a perfect substitutes utility function. In fact, if you compare the MRS of those utility functions, you can confirm that this is the case. It also illustrates that there is a wide range of preference that are complements but not perfect complements (with $-\infty < r < 0$) and substitutes but not perfect substitutes (with $0 < r < 1$). For students in 50Q, one of the problems in Homework 1 has you do this derivation; others may do it if they like.

Quasilinear family of utility functions

Another “family” 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.

Summary and next steps

Characteristics of preferences:

Preferences are monotonic if having more of any good increases your utility; that is, if $MU_1$ and $MU_2$ are both always positive.
Preferences are convex if, given any two points $A$ and $B$ such that $A \sim B$, you prefer to have a combination of $A$ and $B$ to having either $A$ or $B$ themselves.

Functional forms:

Perfect substitutes: $u(x_1,x_2) = a x_1 + b x_2$
Cobb-Douglas: $u(x_1,x_2) = x_1^ax_2^b$ or $u(x_1,x_2) = a \ln x_1 + b \ln x_2$
Perfect complements (Q only): $u(x_1,x_2) = \min \{ax_1,bx_2\}$
Constant Elasticity of Substitution (CES): $u(x_1,x_2) = \left(ax_1^r + bx_2^r\right)^{1 \over r}$
Quasilinear: $u(x_1,x_2) = v(x_1) + x_2$

This concludes our analysis of preferences and utility! Next we’ll move on to constrained optimization: how does a consumer maximize their utility, subject to a budget constraint? So, in lecture 5 we’ll discuss budget constraints; and in lecture 6 we’ll see how to use the method of Lagrange multipliers to solve the constrained optimization problem. It’s a busy week…stay on top of things!

Reading Quiz

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