MATHEMATICS USED IN
We could give Equation 1A.3 an economic interpretation. For example, if we makeYthe labor costs of a firm andXthe number of labor hours hired, then the equation could record the relationship between costs and workers hired. In this case, there is a fixed cost of $3 (when X¼0, Y¼$3), and the wage rate is $2 per hour. A firm that hired 6 labor hours, for example, would incur total labor costs of $15½¼3þ2ð6Þ ¼3þ12. Table 1A.1 illustrates some other values for this function for various values ofX.
2.Y is anonlinear function of X. This case covers a number of possibilities, including quadratic functions (containingX2), higher-order polynomials (con- taining X3, X4, and so forth), and those based on special functions such as logarithms. All of these have the property that a given change inXcan have different effects onYdepending on the value ofX. This contrasts with linear functions for which any specific change inXalways changesYby a precisely predictable amount no matter whatXis.
To see this, consider the quadratic equation
Y¼ X2þ15X (1A:4)
Yvalues for this equation for values ofXbetween3 andþ6 are shown in Table 1A.1. Notice that asXincreases by one unit, the values ofYgo up rapidly at first but then slow down. WhenXincreases from 0 to 1, for example, Y increases from 0 to 14. But whenXincreases from 5 to 6,Yincreases only from 50 to 54. This looks like Ricardo’s notion of diminishing returns—as X increases, its ability to increaseYdiminishes.2
T A B L E 1 A . 1
Values of X and Y for Linear and Quadratic Functions
LINEAR FUNCTION QUADRATIC FUNCTION
Y¼ f(X) Y¼ f(X)
X ¼3þ2X X ¼ X2þ15X
3 3 3 54
2 1 2 34
1 1 1 16
0 3 0 0
1 5 1 14
2 7 2 26
3 9 3 36
4 11 4 44
5 13 5 50
6 15 6 54
2Of course, for other nonlinear functions, specific increases inXmay result in increasing amounts ofY(consider, for example,X2þ15X).
GRAPHING FUNCTIONS OF ONE VARIABLE
When we write down the functional relationship betweenXandY, we are sum- marizing all there is to know about that relationship. In principle, this book, or any book that uses mathematics, could be written using only these equations. Graphs of some of these functions, however, are very helpful. Graphs not only make it easier for us to understand certain arguments; they also can take the place of a lot of the mathematical notation that must be developed. For these reasons, this book relies heavily on graphs to develop its basic economic models. Here we look at a few graphic techniques.
A graph is simply one way to show the relationship between two variables.
Usually, the values of the dependent variable (Y) are shown on the vertical axis and the values of the independent variable (X) are shown on the horizontal axis.3 Figure 1A.1 uses this form to graph Equation 1A.3. Although we use heavy dots to show only the points of this function that are listed in Table 1A.1, the graph represents the function for every possible value ofX. The graph of Equation 1A.3 is a straight line, which is why this is called alinear function. In Figure 1A.1,Xand Ycan take on both positive and negative values. The variables used in economics generally take on only positive values, and therefore we only have to use the upper- right-hand (positive) quadrant of the axes.
Linear Functions: Intercepts and Slopes
Two important features of the graph in Figure 1A.1 are its slope and itsintercepton theY-axis. TheY-intercept is the value ofYwhenXis equal to 0. For example, as shown in Figure 1A.1, whenX¼0,Y¼3; this means that 3 is theY-intercept.4In the general linear form of Equation 1A.2,
Y ¼aþbX
theY-intercept will beY¼a, because this is the value ofYwhenX¼0.
We define theslopeof any straight line to be the ratio of the change inYto the change inXfor a movement along the line. The slope can be defined mathemati- cally as
Slope¼Change inY Change inX ¼DY
DX (1A:5)
where theD(‘‘delta’’) notation simply means ‘‘change in.’’ For the particular func- tion shown in Figure 1A.1, the slope is equal to 2. You can clearly see from the dashed lines, representing changes inXandY, that a given change inXis met by a change of twice that amount in Y. Table 1A.1 shows the same result—as X increases from 0 to 1,Yincreases from 3 to 5. Consequently
3In economics, this convention is not always followed. Sometimes a dependent variable is shown on the horizontal axis as, for example, in the case of demand and supply curves. In that case, the independent variable (price) is shown on the vertical axis and the dependent variable (quantity) on the horizontal axis. See Example 1A.1.
Linear function An equation that is represented by a straight- line graph.
Slope
The direction of a line on a graph; shows the change inYthat results from a unit change inX.
4One can also speak of theX-intercept of a function, which is defined as that value ofXfor whichY¼0. For Equation 1A.3, it is easy to see thatY¼0 whenX¼ 3/2, which is then theX-intercept. TheX-intercept for the general linear function in Equation 1A.2 is given byX¼ a/b, as may be seen by substituting that value into the equation.
Intercept
The value ofYwhenX equals zero.
Slope¼DY DX¼53
10¼2 (1A:6)
It should be obvious that this is true for all the other points in Table 1A.1.
Everywhere along the straight line, the slope is the same. Generally, for any linear function, the slope is given by b in Equation 1A.2. The slope of a straight line may be positive (as it is in Figure 1A.1), or it may be negative, in which case the line would run from upper left to lower right.
A straight line may also have a slope of 0, which is a horizontal line. In this case, the value ofYis constant; changes inXwill not affectY. The function would be Y¼aþ0X, orY¼a. This equation is represented by a horizontal line (parallel to theX-axis) through pointaon theY-axis.
Interpreting Slopes: An Example
The slope of the relationship between a cause (X) and an effect (Y) is one of the most important things that economists try to measure. Because the slope (or the related concept of elasticity) shows, in quantitative terms, how a small (marginal) change in one variable affects some other variable, this is a valuable piece of information for
F I G U R E 1 A . 1
Graph of t h e Line ar Fu nctionY¼3 þ 2X
10
5 3
ⴚ5
ⴚ10
ⴚ10 ⴚ5 0 1 5 10
Y-axis
⌬Y
⌬Y⌬X
Y-intercept ⌬X
X-intercept
Slope ⴝ 5ⴚ3 1ⴚ0 2
ⴝ ⴝ
X-axis
TheY-intercept is 3; whenX¼0,Y¼3. The slope of the line is 2; an increase inXby 1 will increaseYby 2.
building most every economic model. For example, suppose a researcher discovered that the quantity of oranges (Q) a typical family eats during any week can be represented by the equation:
Q¼120:2P (1A:7)
WherePis the price of a single orange, in cents. Hence, if an orange costs 20 cents, this family would consume eight oranges per week. If the price rose to 50 cents, orange consumption would fall to only two per week.5 On the other hand, if oranges were given away (P¼0), the family would eat 12 each week. With this sort of information, it would be possible for an agricultural economist to assess how families might react to factors such as winter freezes or increased imports of oranges that might affect their price.
Slopes and Units of Measurement
Notice that in introducing Equation 1A.7, we were careful to state precisely how the variablesQandPwere measured. In the usual algebra course, this issue does not arise becauseYandXhave no specific physical meaning. But in economics, this issue is crucial—the slope of a relationship will depend on how variables are measured. For example, if orange prices were measured in dollars, the same behavior described in Equation 1A.7 would be represented by:
Q¼1220P (1A:8)
Notice that at a price of $0.20, the family still eats eight oranges per week. With a price of $0.50, they eat only two. The slope here is 100 times the slope in Equation 1A.7, however, because of the change in the wayPis measured.
Changing the way that Q is measured will also change the relationship. If orange consumption is now measured in boxes of 10 oranges each, andPrepresents the price for such a box, Equation 1A.7 would become:
Q¼1:20:002P (1A:9)
This equation still says that the family will consume eight oranges (that is, 0.8 of a box) each week if each box of oranges costs 200 cents and two oranges (0.2 of a box) if each box costs 500 cents.
Notice that, in this case, changing the units in which Qis measured changes both the intercept and the slope of this equation.
Because slopes of economic relationships depend on the units of measurement used, they are not a very convenient concept for economists to use to summarize behavior. Instead, they usually use elasticities, which are unit-free. This concept is introduced in Chapter 3 and then used throughout the remainder of the book.
5Notice that this equation only makes sense forP560 because it is impossible to eat negative numbers of oranges.
M i c r o Q u i z 1 A . 1
Suppose that the quantity of flounder caught each week off New Jersey is given by
Q¼100þ5P(whereQis the quantity of floun- der measured in thousands of pounds andPis the price per pound in dollars). Explain:
1. What are the units of the intercept and the slope in this equation?
2. How would this equation change if flounder catch were measured in pounds and price measured in cents per pound?
Changes in Slope
Quite often in this text we are interested in changing the parameters (that is, a and b) of a linear function. We can do this in two ways: We can change theY-intercept, or we can change the slope. Figure 1A.2 shows the graph of the function
Y¼ Xþ10 (1A:11)
This linear function has a slope of1 and aY-intercept ofY¼10. Figure 1A.2 also shows the function
Y¼ 2Xþ10 (1A:12)
We have doubled the slope of Equation 1A.11 from1 to2 and kept the Y-intercept atY¼10. This causes the graph of the function to become steeper and to rotate about theY-intercept. In general, a change in the slope of a function will cause this kind of rotation without changing the value of itsY-intercept. Since a linear function takes on the value of itsY-intercept whenX¼0, changing the slope will not change the value of the function at this point.
Changes in Intercept Figure 1A.3 also shows a graph of the function Y¼ Xþ10. It shows the effect of changes in the constant term, that is, theY- intercept only, while the slope stays at1. Figure 1A.3 shows the graphs of
Y¼ Xþ12 (1A:13)
and
Y¼ Xþ5 (1A:14)
All three lines are parallel; they have the same slope. Changing theY-intercept only makes the line shift up and down. Its slope does not change. Of course, changes
KEEPinMIND
Marshall’s Trap
In Chapter 1, we described how the English economist Alfred Marshall chose to put price on the vertical axis and quantity on the horizontal axis when graphing a demand relationship. This decision, although sensible for many economic purposes, has posed nightmares for students for more than a century because they are used to seeing the ‘‘independent variable’’ (in this case, price,P) on the horizontal axis. Of course, it is easy to solve Equation 1A.7 forPas:
P¼605Q (1A:10)
This equation even has an economic meaning—it shows the family’s ‘‘marginal willingness to pay’’ for one more orange, given that they are already consuming a certain amount. For example, this family is willing to pay 20 cents per orange for one more orange if consumption is eight per week. But making price the dependent variable is not the customary way we think about demand, even though this is how Marshall graphed the situation. It is usually far better to stick to the original way of writing demand (i.e., Equation1A.7), but keep in mind that the axes have been reversed, and you need to think carefully before making statements about, say, changing slopes or intercepts.
in theY-intercepts also cause theX-intercepts to change, and you can see these new intercepts.
In many places in this book, we show how economic changes can be repre- sented by changes in slopes or in intercepts. Although the economic context varies, the mathematical form of these changes is of the general type shown in Figure 1A.2 and Figure 1A.3. Application 1A.1: How Does Zil- low.com Do It? employs these concepts to illustrate one way in which linear functions can be used to value houses.
Nonlinear Functions
Graphing nonlinear functions is also straight- forward. Figure 1A.4 shows a graph of
Y¼ X2þ15X (1A:15)
F I G U R E 1 A . 2
Ch a ng es i n t h e S l op e o f a L i n ea r F un c t i on Y
X Y ⴝ ⴚX ⴙ 10 (slope ⴝ ⴚ1) Y ⴝ ⴚ2X ⴙ 10 (slope ⴝ ⴚ2) 10
5
0 5 10
When the slope of a linear function is changed but theY-intercept remains fixed, the graph of the function rotates about theY-intercept.
M i c r o Q u i z 1 A . 2
In Figure 1A.2, theX-intercept changes from 10 to 5 as the slope of the graph changes from1 to 2. Explain:
1. What would happen to the X-intercept in Figure 1A.2 if the slope changed to5/6?
2. What do you learn by comparing the graphs in Figure 1A.2 to those in Figure 1A.3?
for relatively small, positive values ofX. Heavy dots are used to indicate the specific values identified in Table 1A.1, though, again, the function is defined for all values ofX. The general concave shape of the graph in Figure 1A.4 reflects the nonlinear nature of this function.
The Slope of a Nonlinear Function
Because the graph of a nonlinear function is, by definition, not a straight line, it does not have the same slope at every point. Instead, the slope of a nonlinear functionat a particular pointis defined to be the slope of the straight line that is tangent to the function at that point. For example, the slope of the function shown in Figure 1A.4 at pointBis the slope of the tangent line illustrated at that point. As is clear from the figure, in this particular case, the slope of this function gets smaller asXincreases. This graphical interpretation of ‘‘diminishing returns’’ to increasing X is simply a visual illustration of fact already pointed out in the discussion of Table 1A.1.
F I G U R E 1 A . 3
C hanges i n t h e Y-I nte rcept of a Line ar F unctio n Y
X Y ⴝ ⴚX ⴙ 10
Y ⴝ ⴚX ⴙ 12 Y ⴝⴚX ⴙ 5 10
12
5
0 5 10 12
When theY-intercept of a function is changed, the graph of the function shifts up or down and is parallel to the other graphs.
A P P L I C A T I O N 1 A . 1 How Does Zillow.com Do It?
The web site Zillow.com (founded in 2006) provides esti- mated values for practically every residential home in the United States. Because this amounts to more than 70 million homes, there is no way that the company can study the details of each house as a traditional real estate appraiser might. Instead, the company uses public data on homes that recently sold together with statistical techniques to estimate a relationship between the price of a house (P) and those characteristics of a house that can be obtained from public sources (such as the number of square feet,X).
A Simple Example
For example, Zillow might determine that houses in a parti- cular area obey the relationship:
P¼$50,000þ$150X (1)
This equation says that a house in this location costs $50,000 (for the lot, say) plus $150 for each square foot. So, a 2,000 square foot house would be worth $350,000, and a 3,000 square foot house would be worth $500,000. Figure 1A
shows this linear relationship. Using this relationship, Zillow can predict a value for every house in its database.
Location, Location, Location
One factor that Zillow must pay close attention to is the location of the houses it is pricing. As any real estate agent will tell you, location is often all that matters in a home price.
Hence, it would not be appropriate to estimate a relation- ship such as Equation 1A.1 for the entire United States or even for a fairly large city. Instead, it must narrow its focus on localities where the square foot value of a house might reasonably be expected to be constant. In especially desir- able locations, houses might sell for $500 per square foot or more, and lots would cost much more than $50,000.
What Zillow Can’t See
A second problem with the Zillow estimates is that actual house prices may depend on factors about which Zillow has no information. For example, real estate databases may have no information about whether a house has a nice view or not. If having a view would raise a typical lot price by $100,000, for example, the relationship for houses with views should be the one shown by the upper line in Figure 1A. Zillow would systematically underestimate the values of such houses.
How Accurate Is Zillow?
The advent of the Zillow web site has raised a lot of questions about how accurate its estimates really are. The company admits that it cannot value what it cannot see (such as whether a house has a fancy interior), but it defends its estimates as providing a good starting place for home- buyers. Independent evaluations of Zillow pricing conclude that its estimates are within 10 percent of a home’s actual sales price about 70–80 percent of the time. Hence, the site does seem to provide useful information and a chance for home voyeurs to take a peak at really expensive homes ( Zillow also provides aerial views).
TOTHINKABOUT
1. How should Zillow decide the size area over which it will estimate Equation 1A.1?
2. Will Zillow put traditional real estate appraisers out of business?
FIGURE 1ARelationship between the Floor Area of a House and Its Market Value
House value (dollars)
150,000 350,000 500,000
House with view
House without view
50,000
Floor area (square feet)
0 2,000 3,000
Using data on recent house sales, Zillow.com can calculate a relationship between floor area (X, measured in square feet) and market value (P). The entire relationship shifts upward by
$100,000 if a house has a nice view.
Marginal and Average Effects
Economists are often interested in the size of the effect thatXhas onY. There are two different ways of making this concept precise. The most usual is to look at the marginal effect—that is, how does a small change inXchangeY?For this type of effect, the focus is onDY/DX, the slope of the function. For the linear equations illustrated in Figure 1A.1 to Figure 1A.3, this effect is constant—in economic terms, the marginal effect ofXonY is constant for all values ofX. For the nonlinear equation graphed in Figure 1A.4, this marginal effect diminishes asXgets larger.
Diminishing returns and diminishing marginal effects amount to the same thing.
Sometimes economists speak of the average effect of Xon Y. By this, they simply mean the ratio Y/X. For example, as shown in Chapter 6, the average productivity of labor in, say, automobile production is measured as the ratio of total auto production (say, 10 million cars per year) to total labor employed (say, 250,000 workers). Hence, average productivity is 40 (¼10,000,000 250,000) cars per year per worker.
F I G U R E 1 A . 4
G r a p h of t h e Q u a d r a t i c Fu n c t i o nY¼ X2þ15X Y
60
50
40
A
B
30
20
10
0 1 2 3 4 5 6
X
⌬Y
⌬X Tangents: slope ⴝ
Y Average: chord slope ⴝ X
The quadratic equationY¼ X2þ15Xhas a concave graph—the slopes of the tangents to the curve diminish as X increases. This shape reflects the economic principle of diminishing marginal returns. The slope of a chord from the function to the origin shows the ratioY/X.
Marginal effect
The change inYbrought about by a one unit change inXat a particular value ofX. (Also the slope of the function.)
Average effect The ratio ofYtoXat a particular value ofX. (Also the slope of the ray from the origin to the function.)
Showing average values on a graph is more complex than showing marginal values (slopes). To do so, we take the point on the graph that is of interest (say, pointAin Figure 1A.4 whose coordi- nates areX¼4,Y¼44) and draw the chordOA.
The slope of OA is then Y=X¼44=4¼11—the average effect we seek to measure. By comparing the slope of OAto that of OBð¼54=6¼9Þ, it is easy to see that the average effect ofXon Yalso declines as X increases in Figure 1A.4. This is another reflection of the diminishing returns in this function. In later chapters, we show the relationship between marginal and average effects in many dif- ferent contexts. Application 1A.2: Can a ‘‘Flat’’ Tax Be Progressive? shows how the concepts arise in disputes about revising the U.S. personal income tax.
Calculus and Marginalism
Although this book does not require that you know calculus, it should be clear that many of the concepts that we cover were originally discovered using that branch of mathematics. Specifically, many economic concepts are based on look- ing at the effect of a small (marginal) change in a variable X on some other variableY. You should be familiar with some of these concepts (such as marginal cost, marginal revenue, or marginal productivity) from your introductory eco- nomics course. Calculus provides a way of making the definitions for these ideas more precise. For example, in calculus, mathematicians develop the idea of the derivativeof a function, which is simply defined as the limit of the ratioDY/DXas DXgets very small. This limit is denoted asdY/dXand is termed the derivative ofY with respect toX. In graphical terms, the derivative of a function is identical to its slope. For linear functions, the derivative has a constant value that does not depend on the value ofXbeing used. But for nonlinear functions, the value of the derivative varies, depending on which value of X is being considered. In economic terms, the derivative provides a convenient shorthand way of noting themarginaleffect ofXonY.
Perhaps the most important use for calculus in microeconomics is to study the formal conclusions that can be derived from the assumption that an economic actor seeks to maximize something. All such problems reach the same general conclusion—that the dependent variable,Y, reaches its maximum value (assuming there is one) at that value ofXfor whichdY/dX¼0. To see why, assume that this derivative (slope) is, say, greater than zero. ThenYcan not be at its maximum value because an increase inXwould, in fact, succeed in increasingY. Alternatively, if the derivative (slope) of the function were negative, decreasingXwould increaseY.
Hence, only if the derivative is 0 canXbe at its optimal value. Similar comments apply when one is seeking to find that value of X which yields a minimum value forY.
M i c r o Q u i z 1 A . 3
Suppose that the relationship between grapes harvested per hour (G, measured in pounds) and the number of workers hired (L, measured in worker hours) is given byG¼100þ20L:
1. How many additional grapes are harvested by the 10th worker? The 20th worker? The 50th worker?
2. What is the average productivity when 10 workers are hired? When 20 workers are hired? When 50 workers are hired?