## The Marginal Rate of Substitution and Marginal Utility

Consider point A in Figure 4.5. The marginal rate of substitution at point A equals 2 because the slope of the indifference curve at that point is –2:

In words, this means that in return for 1 more latte, Sarah is willing to give up 2 burritos. At point B, the marginal rate of substitution is 0.5, which implies that Sarah would sacrifice only half of a burrito for 1 more latte (or equivalently, she would sacrifice 1 burrito for 2 lattes).

This change in the willingness to substitute between goods at the margin occurs because the benefit a consumer gets from another unit of a good tends to fall with the number of units she already has. If you have already amped up on caffeine by drinking all the lattes you can stand but haven’t eaten all day, you might be willing to forgo lots of lattes to get an extra burrito.

Another way to see all this is to think about the change in utility (ΔU) created by starting at some point on an indifference curve and moving just a little bit along it. Suppose we start at point A and then move just a bit down and to the right along the curve. We can write the change in utility created by that move as the marginal utility of lattes (the extra utility the consumer gets from a 1-unit increase in lattes, MUlatte) times the increase in the number of lattes due to the move (ΔQ), plus the marginal utility of burritos (MUburritos) times the decrease in the number of burritos (ΔQburritos) due to the move. The change in utility is

ΔU = MUlattes × ΔQlattes + MUburritos × ΔQburritos

where MUlattes and MUburritos are the marginal utilities of lattes and burritos at point A, respectively. Here’s the key: Because we’re moving along an indifference curve (so utility is constant at every point on it), the total change in utility from the move must be zero. If we set the equation equal to zero, we get

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0 = ΔU = MUlattes × ΔQlattes + MUburritos × ΔQburritos

Rearranging the terms a bit will allow us to see an important relationship:

Notice that the left-hand side of this equation equals the negative of the slope of the indifference curve, or MRSXY. We now can see a very significant connection: The MRSXY between two goods at any point on an indifference curve equals the inverse ratio of those two goods’ marginal utilities:

∂ The end-of-chapter appendix uses calculus to derive the relationship between the marginal rate of substitution and marginal utilities.

In more basic terms, MRSXY shows the point we emphasized from the beginning: You can tell how much people value something by their choices of what they would be willing to give up to get it. The rate at which they give things up tells you the marginal utility of the goods.

This equation gives us a key insight into understanding why indifference curves are convex to the origin. Let’s go back to the example above. At point A in Figure 4.5, MRSXY = 2. That means the marginal utility of lattes is twice as high as the marginal utility of burritos. That’s why Sarah is so willing to give up burritos for lattes at that point—she will gain more utility from receiving a few more lattes than she will lose from having fewer burritos. At point B, on the other hand, MRSXY = 0.5, so the marginal utility of burritos is twice as high as that of lattes. At this point, she is less willing to give up burritos for lattes.

As we see throughout the rest of this chapter, the marginal rate of substitution and its link to the marginal utilities of the goods play a key role in driving consumer behavior.