Using Slope in Economics
Introduction
The slope of a line is a numeric value that describes both its direction and steepness. Slope helps economists understand the relationship between two variables. For example, it can show the relationship between income and spending in an economy.
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Slope sign
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Provenance: Jeffrey Sarbaum, University of North Carolina at Greensboro
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
The sign of slope shows direction; a positive slope reflects an upward sloping line, and a negative slope reflects a downward sloping line.
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Slope magnitude
Provenance: Jeffrey Sarbaum, University of North Carolina at Greensboro
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
The magnitude of slope shows steepness. A larger magnitude indicates a steep slope. A lower magnitude indicates a gradual slope.
How do I calculate slope?
Slope is measured by the rise divided by the run. The slope of a line is a rate of change that tells us how much the variable on the vertical axis changes while the variable on the horizontal axis changes by one unit.
`\s\l\o\p\e = (\r\i\s\e)/(\r\u\n)`
Steps to calculate slope
- Subtract the starting value on the horizontal axis from the ending value on the horizontal axis. This is the run, the change in income.
- Subtract the starting value of on the vertical axis from the ending value on the vertical axis. This is the rise, the change in spending.
- Divide the result from step 2, the rise, by the result from step 1, This is the slope.
For example:
In 2008 U.S. households received tax rebates ranging from $300 to $1200. One study found that spending increased as shown below:
- The run is $1200 - $300 = $900.
- The rise is $400- $100 = $300
- The slope is $300/$900 = 1/3, showing that 1/3 of the additional income was spent. (In this case, the remainder of the rebate (2/3) was saved or used to pay off debt.)
In mathematics and in economics, symbols are often used to express terms. Using symbols is a way to generalize and shorten formulas. For example, the symbol "Δ" ("delta") is often used to express the concept of change, and the letter "X" is often used to represent a variable. When a variable such as X has different values, it will often be denoted with a subscript, such as X1 or X2.
Using symbols we can express the formula for the change in X as:
`X_2 - X_1 = \DeltaX`
Similarly, using symbols we can express the formula for the change in Y as:
`Y_2 - Y_1 = \DeltaY`
Now using symbols we can express the formula for slope as:
`\s\l\o\p\e = (\DeltaY)/(\DeltaX)`
The slope of a straight line is constant. No matter where on the line you calculate slope, you'll get the same answer. This is not true of a curve. There are two approaches to calculating the slope of a curve.
- Calculating slope between two points on a curve:
First pick two points on the curve, connect the points with a straight line, and then calculate the slope of that line by dividing the change in the variable on the Y-axis, by the change in the variable on the X-axis.
Provenance: Peter Schuhmann, University of North Carolina-Wilmington
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
- Calculating slope at a single point on a curve:
First pick one point on a curve, and draw a line that is tangent to the curve at that point. Then calculate the slope of the tangent line by dividing the change in the variable on the Y-axis, by the change in the variable on the X-axis.
Provenance: Peter Schuhmann, University of North Carolina-Wilmington
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Where is slope used in Macroeconomics?
In macroeconomics, slope is useful for examining how economic variables change in response to other economic variables.
For example, we can use slope to understand the relationship between the interest rate and money demand between two points.
- Subtract the starting value of price from the ending value on the horizontal axis. This is the change in X, or ΔX.
`\DeltaX = "Demand for Money at 7%" - "Demand for money at 5%" = `
`"$900 billion" - "$1,000 billion" = "-$100 billion"`
- Subtract the starting value of quantity from the ending value on the vertical axis. This is the change in Y, or ΔY.
`\DeltaY = "Interest rate 7%" - "Interest rate 5%" = "2%"`
- Divide the result from step 2 by the result from step 1 to obtain the slope.
`\s\l\o\p\e = "2%"/"-$100 billion" = "1%"/"-$50 billion"`
This slope is negative, indicating that a 1% increase in the interest rate causes a $50 billion decrease in the demand for money.
Where is slope used in Microeconomics?
In microeconomics, we calculate slope for a wide variety of things including demand, supply, production possibility frontiers, budget constraints, production functions, and cost functions.
Here is an example using the hypothetical business, "Surfzup":
- In 2013, the Surfzup surfboard corporation sold 1,000 surfboards at a price of $600 each.
- In 2014, Surfzup lowered the price of surfboards to $400 each and sold 1,400 surfboards.
These points are plotted on the "demand function" shown in the figure, with price (in dollars) on the Y-axis and quantity (in surfboards sold) on the X-axis. Using the coordinates of these two points, we can calculate the slope of the line as follows:
- Subtract the starting value of price from the ending value on the horizontal axis. This is the change in X, or ΔX.
`\DeltaX = "Quantity sold in 2014" - "Quantity sold in 2013" = "1,400" - "1,000" = 400`
- Next, subtract the starting value of quantity from the ending value on the vertical axis. This is the change in Y, or ΔY.
`\DeltaY = "Price in 2014" - "Price in 2013" = $400 - $600 = "-$200"`
- Finally, divide the result from step 2 by the result from step 1 to obtain the slope.
`\s\l\o\p\e = ("-200")/(400) = -1/2`
This slope is negative, indicating that the demand function is downward sloping.
✓ Final thoughts on slope
- Whatever line or curve we are dealing with, the operation for calculating its slope does not change. It is always rise over run, or ΔX/ΔY.
- Often economists go one step beyond slope, measuring the ratio of the percent changes --- a concept called elasticity. This calculation uses changes in each variable, but then divides by the absolute numbers involved to arrive at a percentage. For example, in the case of our hypothetical Surfzup corporation, the price changed by -40% and the quantity sold changed by 33%. The ratio --- the elasticity --- is 33%/-40% = -0.83.
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