This year at Carleton I’ve gotten to teach one of my favorite parts of multivariable calculus, the multivariable chain rule. Despite its scary-sounding name, the multivariable chain rule seems to capture a fundamental principle about how the world works, a principle I call “narratives add.” I’ll walk you through how the multivariable chain rule works (even if you know zero calculus) and then show you how it plays out in two real-life examples: whether to raise the minimum wage and how strict to be about pandemic regulations.
The multivariable chain rule
The chain rule in calculus tells you what happens when a cause has an effect… which in turn causes another effect. A diagram might look something like this, with quantity y depending on quantity x, and quantity z depending in turn on quantity y, like the links of a chain:
We use fraction notation, “dy/dx”, to explain what size effect changing quantity x would have on quantity y: “dx” is supposed to represent a little change in x, “dy” is supposed to represent the resulting change in y, and “dy/dx” is their ratio. The “chain rule” just says that if a change dy results in a change dz, then the ratio dz/dx is just the product of the two ratios (dz/dy) and (dy/dx).
It gets more complicated when there can be more than one reason for a change in some quantity. Here’s a scenario in which quantity z depends on y, which depends on x as before, but z also depends on some other quantity w. Meanwhile, both x and w depend on another quantity, t:
If quantity t changes, two sets of things happen. First, the change in t gives rise to a change in x, which changes y, which will have some effect on z. Second, the change in t changes w, which has another effect on z. So how do you know what the total effect on z is?
The Multivariable Chain Rule:
To find the ratio dz/dt for how much changing t has an effect on z, find all the paths from t to z in the diagram, for each path multiply all the change ratios along that path, and then add up the results.
In other words, each path is like a story you could tell about how t affects z, a kind of artificial “narrative”, and the right way to combine the narratives is to add up what each one tells you individually.
(There’s a little bit of subtlety to the notation, in that for quantities that depend on more than one other quantity, like how z depends on both y and w, we write the change ratios ∂z/∂y and ∂z/∂w with a curly symbol “∂” instead of “d”. That different symbol is meant to remind us that ∂z/∂y means “the ratio of a change in z as a result of a change in y, assuming w doesn’t change.” That’s pretty unrealistic: outside of artificial experiments, usually if y has changed it’s because x and t have changed as well, which means w has changed too. But the cool thing is that you can do those experiments to find out what the values of ∂z/∂y and ∂z/∂w are, and then the multivariable chain rule tells you how to use those to calculate the change in z when y and w both change.)
Example 1: Should we increase the minimum wage?
When people talk about whether or not to increase the minimum wage, there are two main narratives I hear:
- Increasing minimum wage means that people who are earning minimum wage (of which there are many) will be earning more money. So if we want to make life better for those who are worst off, we should raise the minimum wage.
- Increasing minimum wage means that some businesses that offer minimum-wage jobs will no longer be able to afford to offer as many, so some minimum-wage workers will lose their jobs or have to cut back on hours. So if we want to make life better for those who are worst off, we shouldn’t raise the minimum wage.
This isn’t a case of different people valuing different things: both narratives are making claims about what will increase minimum-wage workers’ take-home pay. So how do we reconcile their opposite conclusions — which narrative is the right one? I argue that that’s the wrong question — rather than one being right and the other wrong, the narratives add.
Here’s what that looks like. We have three quantities we’re interested in: the income (I) of minimum-wage workers, which depends on both the minimum wage (M) and the number of minimum-wage job hours available (H). Increasing either M or H while keeping the other the same would increase I, so the ratios ∂I/∂M and ∂I/∂H are both positive. However, increasing M would decrease H, so dH/dM is negative.
There are two paths from M to I: One that goes directly, with positive change ratio ∂I/∂M (indicating what would happen to I if only M changes, and not H); and one that goes indirectly, with change ratios dH/dM and ∂I/∂H, whose product is negative. These correspond to the two narratives, the first claiming a positive direct effect (higher wages means higher income), and the second claiming a negative indirect effect (higher wages means fewer hours means lower income). The multivariable chain rule says, if you want to find the total effect changing M has on changing I, add the results from each narrative, in this case one positive result and one negative result:
dI/dM = (∂I/∂M) + (∂I/∂H)(dH/dM)
What matters, then, isn’t which narrative is the right one — they’re both necessary to make the equation correct! What matters is whether the total is positive or negative; that’s what tells you whether increasing the minimum wage will have the desired effect of increasing minimum-wage income. My general impression from economic data is that the negative ratio dH/dM tends to be small enough that the total (∂I/∂M) + (∂I/∂H)(dH/dM) is still positive, so increasing minimum wage is a net benefit. But that’s a belief I hold lightly — I can imagine circumstances changing so that dH/dM becomes large enough that the negative effect drowns out the positive one, for example if the minimum wage became so high that no business could afford to pay its workers.
Example 2: Should we close the beaches in a pandemic?
As state and local governments struggle with determining what the right rules should be for different types of gatherings, I noticed a pattern of public health officials arguing that too-strict rules about outdoor gatherings could actually be making the pandemic spread faster. It’s a case again of two competing narratives:
- If we increase the strictness of outdoor gatherings (S), then there will be less gathering outdoors (O) and the rate (R) at which the pandemic is spreading will decrease.
- If we increase the strictness of outdoor gatherings (S), then people will move their gatherings indoors, and the increase in indoor gathering (I) will increase the pandemic spread rate (R).
Here’s how that looks in a diagram:
The right way to reconcile these two narratives is to write down the ratios for how each change in S propagates to a change in R and then add the results:
dR/dS = (∂R/∂O)(dO/dS) + (∂R/∂I)(dI/dS).
We now know that outdoor gatherings contribute very little to the spread of the pandemic: ∂R/∂O is much, much smaller than ∂R/∂I. So even if increasing the strictness of outdoor regulations eliminates more outdoor gatherings than get moved indoors, the positive (∂R/∂I)(dI/dS) term could still be the larger one and the overall effect could be to make the pandemic worse.
Both of these examples are more complicated in real life than the way I’ve presented them. People don’t earn minimum wages in a vacuum: there’s a lot else going on that affects how many jobs there are at what pay levels (see example 2), and workers’ income has its own effects on the economy and how much businesses can afford to pay in wages.
Meanwhile, politicians deciding how strict to make their regulations have to balance the competing phenomena that stricter regulations might increase the public perception (P) of how severe the pandemic is, while also making it more likely that people will disregard the rules (D). Here might be a more complete model of the situation:
The point isn’t necessarily to write down the 100%-correct model the first time, but rather to understand how to grow your understanding over time as you encounter new perspectives and seek to integrate them with your own. Ignoring narratives that don’t fit your conclusions isn’t the answer, but neither is throwing up your hands in defeat when there are two sides to an issue. You can combine all the narratives, weigh their contributions, and act based on a more complete picture of reality.
How about you — what’s another example of an issue where people talk past each other with competing narratives? What are the different quantities involved, and how do they depend on each other? What does the multivariable chain rule tell you about how to add up the contributions from all the narratives? I’m curious to hear even your partial thoughts in the comments!