Economics: Understanding Slope of Linear Functions

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We've been looking at the relationship between two variables, price and quantity, for Bob, who's making a decision about how many hamburgers to buy each week as the price changes. And we looked at how Bob's behavior changed when his income changed. I want to look at something else now, and that is the sensitivity of Bob's demand for hamburgers to changes in the price. What happens to the number of hamburgers that Bob buys each week when the price changes? And I bring this up because I want to discuss how economists think about the slope of a line.
Let's look at the line that we were analyzing before, Bob's original demand curve for hamburgers. We saw that when the price of hamburgers fell by 0.50; that is, from $2.00 down to $1.50 per hamburger, that Bob's quantity demanded increased from five hamburgers per week up to six hamburgers per week. This was represented by the slope of the line. Let's show how we calculate the slope.
The slope, remember, is the rise over the run, the change in the vertical axis variable or the y-axis variable, in this case, the change in price, divided by the change in quantity demanded, the change in the x-axis variable, in this case, the x-axis variable was the quantity of hamburgers that Bob purchased each week as the price changed. Now, let's calculate the slope on this original curve. The slope of this curve is going to be given by the rise over the run or the change in the y-axis variable divided by the change in the x-axis variable, and in this case, that's going to given by the change in the price of hamburgers divided by the change in the quantity of hamburgers that Bob buys each week.
Now, let me plug in my numbers here. The change in price, in this case, is a movement from $2.00 per hamburger down to $1.50 per hamburger. So, let's plug that in. The new price is $1.50 and the old price was $2.00. The price has actually fallen, and the change in quantity is given here by the movement from Bob buying five hamburgers a week to Bob buying six hamburgers a week. So the new number of hamburgers is six minus the old number is five, and that gives us the change in quantity. Let's actually solve this equation, and we get -0.50 divided by one hamburger, and the slope, therefore, is -0.50.
So Bob is relatively sensitive; that is, when the price falls by 0.50, he'll buy an extra hamburger each week. Let's suppose now we have a different relationship between price and quantity for Bob. And let's suppose that in this different relationship, when the price falls from $2.00 to $1.50 per hamburger, Bob increases his quantity, not just to six hamburgers per week, but all the way out to ten hamburgers per week. Now, this is an entirely different relationship for Bob. This is an entirely different relationship between price and quantity, so I would have to re-draw the demand curve to represent new combinations of price and quantity, and I'll do that using green. I'll show this new relationship with a green curve that has on it the two points that I've been talking about - $2.00 and five hamburgers and $1.50 and ten hamburgers.
In this new relationship, Bob's demand curve is flatter; that is, the slope is smaller than the slope on the red curve. The curve is flatter. Let's actually calculate the slope of the green demand curve, if this were Bob's demand curve and I've labeled this D' to represent a new relationship between price and quantity for Bob. If D', the green curve, were Bob's demand curve, what would the slope be? Well, let's calculate the slope. And I'll use prime to remind us that we're talking about this new curve, the green curve.
The slope on this curve is the rise over the run, and in this case we're dealing with the same rise, the same 0.50 change as before, only now we have a much bigger run. Bob increases his demand for hamburgers, not just by one hamburger, but by five, so in this case my change in quantity is much bigger, and I can call this Deltax', that is, a bigger change in the x-axis variable. So the slope is going to be the change in price divided by this new change in quantity. The change in price is still going to be 0.50, but the change is quantity is now much bigger. Let's plug in the numbers and calculate. The change in price, as before, is -0.50, the movement from $2.00 to $1.50. The change in quantity now is the new quantity is 10 minus the old quantity of 5, gives us a change in quantity equal to five hamburgers. So the new slope is -0.10. That is, when the price changes by 0.10 on average, Bob adds an extra hamburger to his weekly consumption. Bob's demand is more sensitive on the green line than it is on the red line, and that's why economists watch the slope of curves. It is the slope of curves that tells us something about the sensitivity of one variable to changes in another.
Now, one word of warning here. The slopes of curves depend entirely on how you are measuring the variables. Here, we're measuring the price of hamburgers in dollars, but if we were measuring the price of hamburgers in cents instead, the slope might change. That is, if this were instead of $1.00, one penny, then Bob wouldn't be buying any hamburgers until we were way, way up this axis. So if we change the way we measure the price, the slope is going to change. And if we change the hamburger measure from burgers to boxes of burgers or bags of burgers or half hamburgers, or parts of hamburgers, you can change this slope in all kinds of ways. The slope of a curve depends completely on the units in which you measure the variables. And that's why economists like measures that are called "elasticities." Elasticities don't depend on the units in which you measure variables. Elasticities are based on percentage changes, and whether the price of a hamburger goes from $1.00 to $2.00, a doubling of the price, or whether it goes from 100 cents to 200 cents, it's all the same. Economists are interested in measures that don't depend on units, but depend instead on percentage changes, and when we get into our discussion of elasticity in a later lecture, you will see how economists put this measure to work.
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Understanding the Slope of a Linear Function Page [2 of 2]