SLOPE AND MORE
Answers, Explanations, and Elaborations
The slides of "Slope and More" take you from the basic concept of a slope to the application of that concept (and of a few others) to the macroeconomy. The overall goal of the exercise is to encourage the student to think of "math" and "the real world" at the same time. Offered below are answers, explanations, and elaborations.
1. The slope is defined as the RISE over the RUN.
That's right; it is. And you should observe that the RISE and the RUN can be expressed in physical dimensions (6 inches and 10 inches) or in any other dimensions appropriate to the application, such as an extra $6 of consumption spending brought about an extra $10 of income. The slope itself is a "pure" number: 6 inches divided by 10 inches is simply 0.6. Graphically, a slope is the ratio of the two legs (Vertical-to-Horizontal) of a right triangle. That is, the slope of the hypontenuse is the ratio of the vertical leg to the horizontal leg.
2. What is the slope of the roof of this garage?
Here you need only visualize the relevant right triangle. Its horizontal leg extends from the eave of the garage to the midpoint of the garage and therefore measures 10 feet. The slope of the roof, then, is 6 feet/10 feet, or 0.6.
3. How tall is this evergreen tree?
If you know the RISE and the RUN, you can find the SLOPE. Or, if you know the RUN and the SLOPE, you can find the RISE. Either way, the defining relationship is: SLOPE = RISE/RUN. The RUN, in the form of the tree's shadow, is given as 25 feet. The SLOPE, indicated by the relative lengths of the legs of the right triangle, is 0.6. We can write, then, that 0.6 = RISE/25. Multiplying both sides by 25, we see that RISE = 15. The evergreen tree is 15 feet tall.
4. What is the slope of this line?
Now we've gone from garages and trees to conventional grahpics and symbology. Your high-school math teachers and college math professors label the vertical axis with a lower-case "y" and the horizontal axis with a lower-case "x"; they use an "m" for the slope of a line and a "b" for its vertical intercept: y = mx + b. We are given the coordinates of two points on this line, and so we can determine both the RISE and the RUN. The RISE is 26-14, or 12; the RUN is 40-20, or 20. The SLOPE of the line, then, is RISE/RUN, which is 12/20, or 0.6.
To find the vertical intercept, we simply write the equation of the line, substituting the coordinates of either point for the x and y and substituting 0.6 for the slope, m.
y = mx + b
14 = 0.6(20) + b
14 = 12 + b
b = 14 - 12
b = 2
Equivalently: y = mx + b
26 = 0.6(40) + b
26 = 24 + b
b = 26 - 24
b = 2
5. How far does this luggage fall?
The hope here is that you will look at problem 5 and "see" problem 4. Now we've gone from conventional graphics and symbology to a conveyor belt and luggage. But the analytics are the same. Treat the vertical distance that represents "the distance that the luggage will fall" as the vertical intercept of the straight line traced out by the conveyor belt.
We are given both the vertical and the horizontal separation between the two pieces of luggage on this conveyor belt, and so we can determine both the RISE and the RUN. The RISE is 12; the RUN is 20. The SLOPE of the conveyor belt, then, is RISE/RUN, which is 12/20, or 0.6.
To find the vertical intercept, we simply write the equation of the conveyor belt, substituting the coordinates for either piece of luggage and substituting 0.6 for the slope, m.
|y = mx + b
21 = 0.6(20) + b
21 = 12 + b
b = 21 - 12
b = 9
|Equivalently:||y = mx + b
33 = 0.6(40) + b
33 = 24 + b
b = 33 - 24
b = 9
Is there an easier way to get the answer? Sure. Notice that both RUNS in the picture are given as 20 feet. One of the corresponding RISES is given as 12 feet. The other RISE has to be 12 feet as well. Just subtract the second 12-foot RISE from the given distance of 21 feet to get 9 feet, which is the distance that the luggage will fall.
6A. What is the Marginal Propensity to Consume?
This problem uses a straight line (which looks a lot like that conveyor belt) to describe consumer behavior as it is affected by changes in income. C (consumption spending) is a linear function of Y (income). Note the specific form of the function (C = a + bY) with Y being measured along the horizontal axis and C being measured along the vertical axis. The other symbols take their meaning from the role they play in the equation. The stand-alone term "a" is the vertical intercept: the coefficient "b" is the slope.
We are given the coordinates of two points on this consumption equation, and so we can determine both the RISE and the RUN. The RISE is 33-21, or 12; the RUN is 40-20, or 20. The SLOPE of the line, then, is RISE/RUN, which is 12/20, or 0.6.
6B. How much would people spend on consumption goods (C) even if their incomes (Y) fell temporarily to zero?
To find the vertical intercept, we simply write the equation, substituting the coordinates of either point for Y and C and substituting 0.6 for the slope, b.
C = a + bY
21 = a + 0.6(20)
21 = a + 12
a = 21 - 12
a = 9
Equivalently: C = a + bY
33 = a + 0.6(40)
33 = a + 24
a = 33 - 24
a = 9
This is the level of consumer spending we would expect to see even if income has (temporarily) fallen to zero.
7A. Find the MPC for this economy and write the equation that describes consumption behavior.
The MPC is the Marginal Propensity to Consume, which is simply the slope of the consumption equation. To get the slope, we need to find two points on the equation whose coordiantes are known. The intercept is one such point: Y = 0; C = 6. The other point is the intersection of the consumption line with the 45-degree line: Y = 15; C = 15 (Remember that the 45-degree line has a slope of 1.0, which means that, starting from the origin, the Y-distance and the C-distance to any point on that line are are the same). So, now we visualize a right triangle with one acute vertex touching the C-intercept and the other touching the intersection with the 45-degree line. The vertical leg of the triangle, the RISE, is 15-6, or 9; The horizontal leg of the triangle, the RUN, is 15. The SLOPE is the RISE over the RUN: 9/15 = 0.6.
Now we know both the vertical intercept (a = 6) and the slope (b = 0.6) and can write the equation describing consumption behavior:
C = a + bY
7B. How much do people spend on consumption goods (C) when total income (Y) is 35?
This question calls your bluff: Show me you can actually make use of this equation relating C to Y. You can simply substitute 35 for Y and solve for C:
C = 6 + 0.6Y
7C. How much do they save?
There are two ways to calculate the level of saving. The easiest is simply to recognize that saving is what's left of your income after you're through spending. That is, S = Y - C; S = 35 - 27 = 8
Alternatively, you could write the saving equation by observing the general form: S = -a + (1 - b)Y, and then evaluating for an income of Y = 35:
S = -a + (1 - b)Y
7D. How much would the investment communtiy have to spend for the economy to be in equilibrium with Y = 35?
An income-expenditure equilibirum requires that income be equal to expenditures. For this wholly private economy, we can write the equilibrium condition as Y = C + I. We know that for an income of 35, consumption spending is 27. So we can write: 35 = 27 + I. Therefore I = 35 - 27 = 8.
Alternatively, we can recognize that for a wholly private economy, it is always true that Y = C + S. That is, your income has to be equal to the part of it that you spend plus the part of it that you don't spend. We can now write the equilibrium condition as C + S = C + I. Subtracting consumption spending from both sides gives us the alternative equilibrium condition for a wholly private economy: S = I. This means that if S = 8, then that amount needs to be borrowed and spent by the investment community for the economy to be in macroeconomic equilibrium.
(By the way, if we were dealing with a mixed economy, we would write as our accounting identity: Y = C + S + T. That is, your income has to be equal to the part of it that you spend plus the part of it that you don't spend plus the part that you don't even see--because the government took it as taxes before you were given your (after-tax) income. In this case, the alternative equilibrium would be S + T = I + G.)
8A. What is the equilibium level of income?
We're shown a wholly private economy (no government spending; no taxes) in which consumption spending is given by the equation C = 30 + 0.6Y and investment spending is 34. The equilibrium level of income is the income that satisfies the equilibrium condition Y = C + I. So, we simply make use of all we know about C and about I and solve for Y:
Y = C + I
8B. How much are people saving?
There's an easy way and a hard way to get this answer--and even the hard way is pretty easy. First, if we know that the economy is in equilibrium and we know that investment is 34, then saving must be 34, too. So, S = 34.
Alternatively, we can write the saving equation and evaluate it for an income of Y = 160:
S = -a + (1 - b)Y
8C. How much are they spending on consumption goods?
Again, there's an easy way and an almost-as-easy way. First, if people are earning Y = 160 and saving S = 34, then they must be spending the difference: C = 126. That is, we simply make use of the identity: Y = C + S.
Alternatively, we can write the consumption equation and evaluate it for an income of Y = 160:
C = 30 + 0.6Y
8D. Can you say whether or not the labor force is fully employed?
We can't say. We take the so-called "going wage" as the wage rate that cleared the labor market during some earlier period when the economy was in good macroeconomic health. That going wage is still going--even if the demand for labor has fallen (and the economy is in recession). In recessionary conditions, the income of Y = 160 is equal to going wage rate times the quantity of worker-hours currently demanded at that wage rate.
Only if the current demand for labor happens to intersect the given supply of labor at the going wage (a possible but not very likely circumstance, according to Maynard Keynes) would the corresponding income (Y = WN) be full-employment income.