Introduction to Macroeconomics

1 Appendix. Working With Numbers and Graphs


  1. Working With Numbers
    1. Percentage Change
    2. Marginal Change and Analysis
    3. Elasticity
  2. Working With Graphs
    1. Relating Two Variables
    2. Graphical Representation
    3. Graphing an Equation
    4. The Intercept of a Curve
    5. The Slope of a Linear Curve
    6. Nonlinear Relationships
    7. Shifting and Rotating a Curve

1. Working With Numbers

  1. Percentage Change

    Sales of apples increased by 1 million bushels last year. Sales of oranges increased by only 500,000 bushels. Which market had the greatest increase? If you're only interested in the absolute change then the answer appears to be apples. But you can't always compare apples and oranges. If the market for apples was 4 times the size of the market for oranges, then the apple market would had the smaller proportional increase. This is revealed by calculating changes as percentages rather than in terms of often noncomparable physical measures.

    The conventional measure of percentage change is to divide the change in the variable by its starting value, not the average or ending value:

    Percentage Change = (ending value - starting value) * 100
                                  starting value

    We are frequently interested in decomposing the change in the product of two variables. For example, gross domestic product represents the average level of prices (P) in an economy times total physical output (Q). If gross domestic product increases by 5 percent we usually want to know how much of that increase can be attributed to a change in prices and how much to a change in output (an increase in the latter is much preferred to an increase in the former). A simple approximation is:

    Percent Change in (P x Q) = (Percent Change in P) + (Percent Change in Q)

    Similarly, the percent change in a ratio, such as total output (Q) divided by the total labor force (L), or per capita output, is:

    Percent Change in (Q / L) = (Percent Change in Q) - (Percent Change in L)

  2. Marginal Change and Analysis

    A marginal change is a very small increase or decrease in the quantity of some variable. Marginal analysis relates to the effect that a small (usually 1 unit) change in one variable has on another variable. For example, in microeconomics we often ask how much will operating costs increase if we increase production by 1 unit (marginal cost). In macroeconomics we might ask how much our personal consumption expenditures will increase if we receive one more dollar of income (marginal propensity to consume).

  3. Elasticity

    The concept of elasticity represents a combination of percentage change and marginal analysis. Like the slope of a curve, elasticities indicate how much one variable changes when a second variable changes. The unique feature of elasticities is that the changes are measured as percentage changes rather than in the units of the variable measured. Elasticity relates to the effect that a one percent (rather than 1 unit) change in one variable has on another variable.

    In microeconomics we often ask how a change in the price of a good will affect the demand for that good. The problem we run into is that the relationship between a change in X and a change in Y depends on the units of measurement of both X and Y. For example a one dollar change in the price has a very different influence on consumer demand than a one peso change, or a one yen change. Moreover, a one dollar change in price has a very different influence on the demand for oranges than automobiles. To relieve ourselves of the need to worry about the units of measurement (dollars or pesos or yen on oranges or automobiles) we simply ask what percentage change in Y (demand) results from a 1 percent change in X (price).

    There are several ways to measure elasticity. Here we will consider only the simple measurement of arc elasticity (using calculus you could derive a point elasticity):

    Elasticity of Y with respect to X = (Y2 - Y1) / Y1 = percentage change in Y
    		                    (X2 - X1) / X1   percentage change in X

    subscript 1 = starting value
    subscript 2 = ending value

    Elasticity - the percentage (or proportionate) change in one variable (e.g., quantity demanded or quantity supplied) brought about by a one percent change in another variable (e.g., price or income).

    While understanding measures of elasticity is critically important in microeconomics, references to elasticities are infrequent in macroeconomics.

2. Working With Graphs

You are probably most familiar with the use of graphs to display magnitudes (e.g., bar graphs), shares (pie charts), and time trends (line graphs). In this course we will make more demanding use of graphs to demonstrate our macroeconomic models. This appendix provides a guide to how graphs are constructed and can be used to interpret economic models.

  1. Relating Two Variables

    We will use graphs to illustrate the relationship between two or more variables. For example, how does consumption expenditure relate to income? How much does investment change if there is a change in the real interest rate? How much will gross domestic product change if there is an increase in government expenditures?

    The common characteristic in each of these questions is that a change in an independent variable X produces a predictable change in a dependent variable Y. This relationship can usually be represented with an algebraic equation. For example, consider a simple linear relationship:

    Y = 10 + 0.75 * X

    In this equation the dependent variable Y is a function of the independent variable X. When X equals 0, Y equals 10. For each 1 unit (marginal) increase in X, Y increases by 0.75. Or, for each $1 increase in national income, household consumption expenditures increase by 75 cents.

    Dependent Variable - a variable that depends on the value of the independent variable(s). The left-hand side variable in a equation.

    Independent Variable - a variable that is not influenced by the dependent variable. The right-hand side variable(s) in an equation.

  2. Graphical Representation

    The conventional graphical representation of the relationship between two variables uses the Cartesian coordinate system with the value of the independent variable X measured along the horizontal axis and the value of the dependent variable Y measured along the vertical axis.

    The Graph Plotting Area

    Figure 1A-1. The Graph Plotting Area

    The intersection of the horizontal x-axis and the vertical y-axis is called to origin where the values of both X and Y are equal to zero. In this course the values of the economic variables we will deal with are usually positive so we will primarily be concerned with the upper right quadrant where the values of both the dependent and independent variables are zero or positive (i.e., non-negative).

  3. Graphing an Equation

    Again consider the algebraic equation, Y = 10 + 0.75 * X, that we presented above. We can demonstrate the relationship between X and Y with a table (also called a "schedule" in some economics texts):

    X 0 10 20 30
    Y 10 17.5 25 32.5

    We can also graph the equation to provide a visual aid that illustrates the relationship between the two variables. A line showing the X and Y pairs is referred to as a curve, whether or not it is straight or curved (a straight line curve is called a linear curve and a curved line is a nonlinear curve). A curve can also be referred to as a function where Y is a function of X.

    Graphing an Equation

    Figure 1A-2. Graphing an Equation

  4. The Intercept of a Curve

    The point where the curve crosses the vertical axis is referred to as the intercept. The intercept represents the value of the dependent variable (Y) when the independent variable (X) is equal to zero.

    For example, suppose you quit your job so that your income (X) goes to zero. Does your consumption spending (Y) necessarily also equal zero? You still need to eat. This is often referred to as the "subsistence" level of consumption expenditures.

  5. The Slope of a Linear Curve

    Evaluating the slope of a curve is equivalent to performing marginal analysis. The slope of a curve tells you how much one variable changes when there is a change in a related variable.

    The relationship between two variables can be positive or negative. On a graph a positive, or direct, relationship is depicted by a curve that slopes upward and to the right of the origin (i.e., Y increases as X increases). A negative, or inverse, relationship slopes downward from left to right (i.e., Y declines as X increases).

    Positive Slope - curve slopes upward and to the right of the origin. Also referred to as a direct relationship. As the value of X increases, the value of Y increases.

    Negative Slope - curve slopes downward from left to right. Also referred to as a inverse relationship. As the value of X increases, the value of Y decreases

    The slope of a curve can be determined from a graph by dividing the vertical change in the Y variable (the "rise") by the horizontal change in the X variable (the "run").

         Slope = Rise = change in Y variable (vertical axis)
                  Run   change in X variable (horizontal axis)

    The slope can easily be calculated from a table. Consider the table presented earlier where Y is a function of X:

    X 0 10 20 30
    Y 10 17.5 25 32.5

    If X increases from 0 to 10, Y increases by 7.5. The slope is the change in Y (7.5) divided by the change in X (10), or 0.75. We get the same result when going from X equals 10 to X equals 20, and from 20 to 30 because the relationship is linear.

    The slope can also be quickly determined from the corresponding linear equation:

    Y = 10 + 0.75 * X

    With a linear equation the slope is equal to the number (more commonly referred to as a "coefficient") that the X value is multiplied by, or:

    Dependent Variable = Intercept + Slope * Independent Variable

  6. Nonlinear Relationships

    In most economic models in this course we assume a linear or straight-line relationship between variables. However, we also consider some nonlinear relationships, such as in the next chapter on opportunity cost and the Production Possibilities Curve.

    A nonlinear curve actually curves. the slope of a nonlinear curve is different at every point along the curve. The slope at a given point is determined by drawing a straight line that is tangent to that point on the curve. A tangent line is one that just touches the curve without crossing it.

    Consider the probable nonlinear relationship between time spent studying and your expected exam grade. Let's say it's a multiple choice exam with five possible answers to each question. If you don't study at all you should expect to make a grade of 20 (the probability of guessing the right answer on each question). If you study for one hour you might increase your grade from a 20 to a 38. If you study a second hour you may raise your grade an additional 15 points. Study a third hour and boost your grade another 13 points to a 66. Each additional hour you study yields a declining marginal benefit. If you study 7 hours you could max your expected grade out at an 89. Studying beyond 7 hours and your expected grade starts to decline as you sacrifice sleep and start to forget all the good stuff you learned when you first started. This relationship between studying and expected grade is illustrated in the following graph:

    Studying and Grades

    Figure 1A-3. Studying and Grades

    In this graph the slope starts out positive but declines as the number of hours spent studying increases. Beyond 7 hours of studying the slope turns negative as brain cells start dying from sleep depravation and caffeine poisoning. You should recognize that when the curve is nonlinear it becomes difficult to accurately determine the slope from a graph. As mentioned earlier, the preferred method is to derive it from an equation if it is available. For a nonlinear curve this takes some simple calculus. The studying-grade graph was drawn using the following equation:

    Y = 20 + 19 * X - 1.3 * X2

    The slope is calculated by evaluating the first-order derivative with respect to X, which in this case is equal to 19 - 2.6X. Thus, at the point X = 4, the slope is equal to 8.6. The second-order derivative will tell you if the slope is increasing or decreasing. Since the second order derivative is equal to -2.6, the slope is decreasing as X increases. OK, calm down. We won't be using calculus in this class. But the lesson that should be learned is that if you want to do advanced economics you must learn the basics of calculus.

  7. Shifting and Rotating a Curve

    When we draw a graph that relates one (dependent) variable, Y, to another (independent) variable, X, we make two important assumptions:

    1. Any other variable that affects the value of the dependent variable, Y, remains unchanged, and
    2. The relationship between Y and X remains unchanged.

    Let's consider a simple aggregate spending function:

    Aggregate Spending = Government Spending + Consumer Spending

    Suppose that consumer spending is a function of income in that every $1.00 increase in income leads to a $0.75 increase in consumer spending, i.e., when income goes up families spend 90 percent of the increase and save 10 percent. This relationship between a change in consumer spending and a change in income is called the marginal propensity to consume. The aggregate spending function can then be rewritten:

    Aggregate Spending = Government Spending + 0.75 • Total Income

    To start let's assume that government spending is fixed at $10 billion at the start of the year when Congress passes the annual budget. The aggregate spending equation can be rewritten:

    Aggregate Spending = $10 + 0.75 • Total Income

    The graph of this relationship is identical to that in Figure 1A-2 above with aggregate spending (Y) on the vertical axis and total income (X) on the horizontal axis. This graph assumes that government spending remains unchanged. Now let's say that in the middle of the year the government votes to increase spending to $15 billion. This represents a violation of our first assumption, that any variable other than national income that affects the value of the dependent variable, aggregate spending, remains unchanged. At every level of total income aggregate spending increases by $5 billion. The curve shifts up by $5 billion as shown in Figure 1A-4.

    Shifting a Curve

    Figure 1A-4. Shifting a Curve

    Our model and graph assumes that every $1.00 change in income leads to a $0.75 change in consumer spending. What happens when this relationship changes? This would represent a violation of our second assumption, that the relationship between Y and X remains unchanged. If consumer attitudes change such that they now spend $0.90 out of every $1.00 increase in income, the aggregate spending equation now becomes:

    Aggregate Spending = $10 + 0.90 • Total Income

    Now we can see what happens in Figure 1A-5. The intercept remains unchanged but the line rotates up. The slope of the line increases from 0.75 to 0.90.

    Rotating a Curve

    Figure 1A-5. Rotating a Curve

File last modified: May 1, 2003

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