RATES—AVERAGE AND INSTANTANEOUS
This section is useful for understanding the mathematics of the average and incremental IRS income tax rates discussed in Chapter 9 ’ It is not necessary for understanding Chapter 9 or anything else in this book.
Suppose I were to take a walk for 25 minutes. I’m walking along a marked track, so I know exactly how far I’ve walked at all times. Every few seconds, I write down how long I’ve been walking and how far I’ve walked. After I’ve finished my walk, I produce the graph shown in Figure 1.7 ’ interpolating my data as described above.
At the end of 25 minutes, I’ve walked about 2,200 feet. As the graph shows, I started off walking at a good pace and then I slowed down. From about 8 minutes to about 13 minutes, I hardly moved at all. Then I started walking faster and faster, until the end of my walk. The total distance traveled (2,200 feet) divided by the total time spent (25 minutes) is called the average rate of distance traveled. On this graph,
Figure 1.7 Graph of my walk—distance versus time.
it will be measured in feet per minute. “Rate” usually refers to something changing over time, so when I say “rate of distance traveled,” the phrase “with respect to time” is implied. When the horizontal axis of a graph is something other than time, the definition of rate must be spelled out carefully.
The rate of distance with respect to time is such a common number that it has been given its own name, speed. If the distance is measured in feet and the time is measured in minutes (these are called the units of measurement), then the speed is measured in feet per minute. Speed can of course also be measured in miles per hour, inches per second, and so on. Since my speed was varying over the course of my walk, dividing the total distance traveled by the total time spent gives me the average speed:
If I draw a straight line connecting the start of my walk (time = 0, distance = 0) to the end of my walk (time = 25, distance ~ 2,200), this line represents how my walk would have been graphed had I walked at a constant speed, identically the average speed (Figure 1.8).
Mathematically, the property of a line describing the change in the vertical axis over the length of the line divided by the change in the horizontal axis over the length of the line is called the slope of the line. When a graph is showing distance on the vertical axis and time on the horizontal axis, the slope is the speed.
If I were to draw a line from my position at, say, time = 10 to time = 20, then the slope of this line would be my average walking speed between times 10 and 20.
Finally, I would like to be able to mathematically describe my speed at different times during the walk. Graphically, this means that I want to know the slopes of lines “rangent to” my graph at arbitrary points (just touching the graph at these
Figure 1.8 Graph of my walk showing average speed.
Figure 1.9 Graph of my walk showing several instantaneous speeds.
points). A few examples of this are shown in Figure 1.9 . Mathematically, this is a topic in differential calculus, which we certainly aren’t going to get into.
Fortunately, what I’m describing is very easy to picture intuitively. If I ask, for example, “How fast was I walking at time = 16?” then mathematically I am asking, “What is the slope of the line tangent to the curve at time = 16?”
In Figure 1.9 . you see that I am asking about the line that’s “bumping up” to and just touching the curve at time = 16. The answer is, at time = 16, I was walking 54 feet per minute. You can also see that I was walking a little slower than this at time = 5 and a good bit faster than this at time = 25.
Also, if you can picture drawing a tangent line to the curve at about time = 10, you will see that the line would be horizontal, that is, have zero slope, which means zero speed, which also means standing still. At time = 10, I may have paused to tie a shoelace.
When dealing with graphs where the horizontal axis is something other than time, some of the terminology changes a bit. You’ll see this when I present an IRS tax curve in Chapter 9 . In that case, the horizontal axis will be taxable income and the vertical axis will be tax owed to the IRS. I will be interested in tax rate. Since both of the axes have units of dollars, the tax rate has the units of dollars per dollar. Since these units can be converted to pennies per penny or Swiss francs per Swiss francs, without the resultant number changing, a tax rate is a dimensionless quantity.
The average tax rate will look, on the graph, just the same as the average speed on the graphs in this chapter. However, since there is no time axis in the tax curve, the term “instantaneous tax rate” would be inappropriate. Instead, we will use the term “incremental tax rate.”
If Figures 1.7-1.9 represented an IRS tax curve instead of a description of my walk, then the straight line of Figure 1.8 would be my average tax rate, and the last dotted line of Figure 1.9 would be my incremental tax rate. Don’t worry if these terms seem strange to you; I’ll start from the beginning and go through them all in Chapter 9 .