## The Marginal Rate of Substitution

Indifference curves are all about tradeoffs: how much of one good a consumer will give up to get a little bit more of another good. The slope of the indifference curve captures this tradeoff idea exactly. Figure 4.5 shows two points on an indifference curve that reflects Sarah’s preferences for food in a month. She lives on lattes and burritos. At point A, the indifference curve is very steep, meaning Sarah will give up multiple burritos to get one more latte. The reverse is true at point B. At that point, Sarah will trade multiple lattes for one more burrito. As a result of this change in Sarah’s willingness to trade as we move along the indifference curve, the indifference curve is convex to the origin (the middle bends toward the origin).

Figure 4.5: Figure 4.5 The Slope of an Indifference Curve Is the Marginal Rate of Substitution
Figure 4.5: The marginal rate of substitution measures the willingness of a consumer to trade one good for the other. It is measured as the negative of the slope of the indifference curve at any point. At point A, the slope of the curve is –2, meaning that the MRS is 2. This implies that, for that given bundle, Sarah is willing to trade 2 burritos to receive 1 more latte. At point B, the slope is –0.5 and the MRS is 0.5. At this point, Sarah is only willing to give up 0.5 burritos to get another latte.

This shift in the willingness to substitute one good for another along an indifference curve might seem confusing. You might be more familiar with thinking about the slope of a straight line (which is constant) than the slope of a curve (which varies at different points along the curve). Also, how can preferences differ along an indifference curve? After all, a consumer isn’t supposed to prefer one point on an indifference curve over another, but now we’re saying the consumer’s relative tradeoffs between the two goods change as one moves along the curve. Let’s address each of these issues in turn.

First, the slope of a curve, unlike a straight line, depends on where on the curve you are measuring the slope. To measure the slope of a curve at any point, draw a straight line that just touches the curve (but does not pass through it) at that point but nowhere else. This point where the line (called a tangent) touches the curve is called a tangency point. The slope of the line is the slope of the curve at the tangency point. The tangents that have points A and B as tangency points are shown in Figure 4.5. The slopes of those tangents are the slopes of the indifference curve at those points. At point A, the slope is –2, indicating that at this point Sarah would require 2 more burritos to forgo 1 latte. At point B, the slope is –0.5, indicating that at this point Sarah would require only half of a burrito to be willing to give up a latte. Second, although it’s true that a consumer is indifferent between any two points on an indifference curve, that doesn’t mean her relative preference for one good versus another is constant all along the line. As we discussed above, Sarah’s relative preference changes with the number of units of each good she already has.

115

### FREAKONOMICS

Do Minnesotans Bleed Purple?

People from Minnesota love their football team, the Minnesota Vikings. You can’t go anywhere in the state without seeing people dressed in purple and yellow, especially during football season. If there is ever a lull in a conversation with a Minnesotan, just bring up the Vikings and the conversation will spring to life. What are the Vikings “worth” to their fans? The answer to that question is obvious: The Vikings are priceless. Or are they?

There has often been discussion that the Vikings would leave the state because the owners were unhappy with their stadium. Two economists carried out a study to try to measure how much Minnesota residents cared about the Vikings.* The authors looked at the results of a survey that asked hundreds of Minnesotans how much their household would be willing to pay in extra taxes for a new stadium for the team. Every extra tax dollar for the stadium would be one less the household could spend on other goods, so the survey was in essence asking for the households’ marginal rate of substitution of all other goods (as a composite unit) for a new Vikings stadium. Because it was widely perceived that there was a realistic chance the Vikings would leave Minnesota if they didn’t get a taxpayer-funded stadium, the way some other NFL teams did, these answers were also viewed as being informative about Minnesotans’ marginal rate of substitution between all other goods and the Minnesota Vikings.

Jim Mone/AP Photo

It turns out that fan loyalty does know some limits. The average household in Minnesota was willing to pay \$571.60 to keep the Vikings in Minnesota. Multiplying that value by the roughly 1.3 million households in Minnesota gives a total marginal value of about \$750 million. In other words, Minnesotans were estimated to be willing to give up \$750 million of consumption of other goods in order to keep the Vikings. That’s a lot of money—imagine how much people might pay if the Vikings could actually win a Super Bowl.

Alas, stadiums aren’t cheap. Officials estimate that a new stadium will cost about \$1 billion. That might explain why a law passed by the state legislature in 2012 finally gave the Vikings the stadium they wanted but only agreed to provide \$500 million of funding, with the team responsible for providing the remaining money. Construction began in 2014 and the new Vikings Stadium is scheduled to host the Super Bowl in 2018.

The Los Angeles Vikings? Don’t even bring up this possibility in conversation with a Minnesotan, unless, of course, you have a check for \$571.60 that you’re ready to hand over.

*John R. Crooker and Aju J. Fenn, “Estimating Local Welfare Generated by an NFL Team under Credible Threat of Relocation,” Southern Economic Journal 76, no. 1 (2009): 198–223.

An interesting feature of this study is that it measures consumers’ utility functions using data on hypothetical rather than actual purchases. That is, no Minnesotan had had to actually give up consuming something else to keep the Vikings around. They were only answering a question about how much they would pay if it came time to actually make that choice. While economists prefer to measure consumers’ preferences from their actual choices (believing actual choices to be a more reliable reflection of consumers’ preferences), prospective choices are sometimes the only way to measure preferences for certain goods. An example of this is when economists try to measure the value of abstract environmental goods, such as species diversity.

116

marginal rate of substitution of X for Y (MRSXY)

The rate at which a consumer is willing to trade off one good (the good on the horizontal axis X) for another (the good on the vertical axis Y) and still be left equally well off.

Economists have a particular name for the slope of an indifference curve: the marginal rate of substitution of X for Y (MRSXY). This is the rate at which a consumer is willing to trade off or substitute exactly 1 unit of good X (the good on the horizontal axis) for more of good Y (the good on the vertical axis) and feel equally well off:

(A technical note: We put a negative sign in front of the slope so that we can refer to MRSXY as a positive number. The slope of the indifference curve will be a negative number on its own because the curve slopes downward.) The word “marginal” indicates that we are talking about the tradeoffs associated with small changes in the composition of the bundle of goods—that is, changes at the margin. It makes sense to focus on marginal changes because the willingness to substitute between two goods depends on where the consumer is located on her indifference curve.

Despite an intimidating name, the marginal rate of substitution makes sense. It tells us the relative value that a consumer places on obtaining one more unit of the good on the horizontal axis, in terms of the good on the vertical axis. You make this kind of decision all the time. Whenever you order off a menu at a restaurant, choose whether to fly or drive home for the holidays, or decide what brand of jeans to buy, you’re evaluating relative values. As we see later in this chapter, when prices are attached to goods, the consumer’s decision about what to consume boils down to a comparison of the relative value she places on two goods and the goods’ relative prices.