Using Gr ph
and Formulas
LEARNING OBJECTIVE
j1 VI .
t""
'J
e of 9
or-o 10 ' .1ul
G raphs a re used to illu strate key eco n om ics id eas. Graphs ap pear no t just in econom­
ics textb o ok s but also on Web sites an d in n ewspap er and ma gazine arti cles that di s­
cuss even ts in busin ess an d eco no mics . Why th e heavy use o f graph s? Beca use th ey
se rve two usefu l p ur poses: (l ) T he y sim p lify eco no m ic ideas, an d (2) they m ake the
id eas mo re co ncrete so t he y can be a ppl ied to re al-worl d pr obl ems. Eco no m ic a nd
busin ess issue s can be co m p lica ted, but a graph can hel p cu t th ro ugh comp lica tio n s
an d h ighl ight th e key relations h ips needed to und erstand th e issue. In th a t sens e, a
gra ph ca n be like a stree t m ap .
Fo r examp le, sup po se yo u ta ke a bus to New York C it y to see th e Em p ire State
Building. After ar riv ing at th e Port Author ity Bus Term ina l, yo u will proba bly use a map
simi lar to th e one sh own below to fin d your way to the Empi re State Bu ilding.
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t,
e
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CIS
\
C HAP TER 1
I
Economics: Foundations and Models
25
Map s are very familiar to just about everyone, so we don't usually th ink of them as
simplified versi ons of reality, bu t th ey are. This ma p do es not sh ow much mo re
~l11~he streets in th is par t of New Yo rk City an d some of the m ost imp ortan t bu ildings.
-o:nnam es, addresses, and telep hone numbers of the people wh o live an d work in the
a aren't given. Almo st no ne of th e stores and build ings those peo ple wo rk and live in
ar~ shown eithe r. The ma p doesn't tell which streets allow curbside pa rk ing and which
don't- In fact, the map tells alm ost n ot hin g about th e mess y reality of life in thi s section
of New Yo rk City, excep t how the streets are laid out, wh ich is the essentia l information
you need to get from the Po rt Authori ty to the Empire State Bui ld ing .
Think abou t so meo ne who says, " I know how to get around in the city, but I just
can't figur e out how to read a map ." It certa inly is poss ible to find you r destination in a
city witho ut a m ap, but it's a lot easier with one. Th e same is true of using graphs in eco­
nomic s. It is possi ble to arrive at a solu tion to a real-world p rob lem in economics and
business witho ut using graphs, b u t it is usu ally a lo t easier if yo u do use them.
Often , the difficu lty studen ts have with graphs and formulas is a lack of fam iliari ty.
With pr actice, all the graphs and formulas in this text will become fam iliar to you. Once
you are fam iliar with th em , you will be able to use them to an alyze prob lems that would
other wise seem very d ifficult. Wha t follow s is a bri ef review of how graphs and formu las
are used.
bei
Gra phs of One Variable
Figure lA -l displays values for market shares in the U.S. au to mobile marke t, using two
commo n types of graph s. Market sha res show the percentage of industry sal es
accoun ted for by differe nt firms. In this case, th e in for mation is for groups of firm s: the
"Big Three"-Ford, Gen eral Motors, and Dai mlerChrys ler -as well as Japanese firms ,
Europe an firms, an d Ko rean firm s. Panel (a) d isplays the in for m ation on market sha res
as a bar graph, where th e ma rke t share of eac h group of fir ms is rep resen ted by the
Shares of
the U.S:
automobile
market
60%
Korean firms
56 .2%
4.3%
European firms "
\
6.5%
40
30
20
Japanese firms ­
Big Three
33 .0%
56 .2%
10
o
Big
Three
Japan ese
firms
European
firms
Korean
firms
(a) Bar graph
(b) Pie chart
Figure 1A- l
Va lues for an economicvariable areoften displayed as a bar graph or as a pie chart. In
thiscase, panel (a) shows market share data for the U.S.automobile industry as a bar
graph. where the market share of eachgroup of firms is represented bythe height of
-
its bar. Panel (b) displays the same information as a pie chart, with the market share
of each group offi rms represented by the size of its slice of the pie.
So urce: "Au to Soles," m,ll Street [our nal, Ma rch 1, 2007.
-26
II
PAR
TI
l
Introduction
Sales 7.5
(mill ions of
7.4
automobiles)
7.3
Sales 8.0
(millions of
automobiles) 7.0
... ;
6.0
7.2
7.1
5.0
7.0
4.0
69
6.8
3.0
6.7
2.0
6.6
1.0
65
0 .0
2000
0.0
2001
2002
2003
2004
2005
(a) Time-series graph with truncated scale
The slashes (If) indicate the scale on the
vertical axis is truncated. which means that
some numbers are omitted. The numbers
jump from 0 to 6.5.
Figure 1A-2
2006
L..-
2000
~
200 1
2002
~
2003
2004
2005
_
2006
(b) Time-series graph where the scale is
not truncated
TIme-Serie Graphs
Bothpanels present time-series graphs of Ford Motor Company's worldwide sales during each year from 2000-2006. Panel (a) has a truncated scale on the vertical axis, and
panel (b) does not.Asa result, the fluctua tions in Ford's sales appear smaller in panel (b) than in panel (a).
Source: Ford Motor Company, A:rnuar Report, va r ious years.
heigh t of its bar. Pan el (b ) displays the same in fo rma tion as a pie chart, with the market
share of each group of firms represen ted by th e size of its slice of th e pie .
In fo rm at ion o n econom ic var iab les is also ofte n d ispl ayed in time-series graphs.
T irne-ser ies grap hs are displayed on a coordinate gri d. In a coordinate gr id , we can m ea­
sure the value of one variable alo n g th e vertical ax is (or y-axis) , and the valu e of ano ther
vari able alo ng the horizo nt al ax is (or x-ax is). The point wh ere the vert ical axis inte rsects
the h orizon tal axis is called the origin. At the or igin, the value of both variables is zero.
The poin ts on a coo rd inate grid represent values of the two variables. In Figure l A-2, we
m easure th e numb er of au tomo biles a n d tru cks so ld wo rldwide by th e Ford Mo tor
Co mpa ny on the vertical axis , and we measure time on the horizon tal axis. In time ­
ser ies graphs, the he ight of the line at each date shows the valu e of th e variable m easur ed
on the vertical axis. Both p an els of Figur e lA -2 show Ford 's wo rldwide sales durin g each
year fro m 2000 to 2006 . T he differenc e between panel (a) and panel (b) illust ra tes the
imp ortance of the scal e used in a time -ser ies graph. In panel (a ), the sca le o n the ver tical
axis is tru ncated , whic h m eans th at it does no t sta rt with zero. T he slashes (II) near the
botto m of the axis in dicate that the scale is truncated. In p anel (b) , the scale is not trun ­
cate d . In panel (b ), th e declin e in Ford's sales since 200 0 appears smalle r than in panel
(a). (Tech n ically, the horizon tal ax is is also tru n cat ed because we sta r t with the year
2000, not the year 0.)
Gra phs of Two Variables
We often use graph s to sh ow th e rela tionship between two vari ables. For example, sup­
pose you are int erested in the relationship between th e price of a pepperoni pi zza and
the quantity of pizzas so ld per week in the small town of Bryan, Texas. A gra ph showing
the relatio nsh ip between the price of a good and th e q ua n tity of the good dem and ed at
each price is called a demand curve. (As we will discuss later, in drawing a demand curve
for a good, we have to hold constant any va riab les o the r th an price th at mi gh t affect the
-
CHAPTER 1
Price $16
(dollars
per pizza)
15
Economics: Foundations and Models
Figure 1A-3
Price
(dollars per pizza)
Quantity
(pizzas per week)
Points
$15
50
A
14
55
B
13
60
C
12
65
0
11
70
E
:
..
. .
~
The figure shows a two-dimensional grid on
which we measure the price of pizza along the
vertical axis (or y-axis) and the quantity of
pizza sold per week along the horizontal axis
(or x-axis). Each point on the grid represents
one of the price and quantity combinations
listed in the table.By connecting the points with
:A
14
i
a line, we can better illustrate the relationship
between the two variables.
....
,.
......
....i
.
13
12
11
oL\
I
50
55
60
65
Demand
curve
70
75
Quantity
(pizzas per week)
As you learned in Figure 1A-2, the slashes (If) indicate the scales on the axes are
truncated, which means that numbers are omitted: On the horizontal axis numbers
jump from 0 to 50, and on the vertical axis numbers jump from 0 to 11
willingness of consumers to buy the good.) Figure lA-3 shows the data you have col­
lected on price and quantity. The figure shows a two-dimensional grid on which we
measure the price of pizza along the y-axis and the quantity of pizza sold per week along
the x-axis. Each point on the grid represents one of the price and quantity combinations
listed in the table. We can connect the points to form the demand curve for pizza in
Bryan, Texas. Notice that the scales on both axes in the graph are truncated. In this case,
truncating the axes allows the graph to illustrate more clearly the relationship between
price and quantity by excluding low prices and quantities.
Slopes of Lines
Once you have plotted the data in Figure lA-3, you may be interested in how much the
quantity of pizza sold increases as the price decreases. The slope of a line tells us how
much the variable we are measuring on the y-axis changes as the variable we are measur­
ing on the x-axis changes. We can use the Greek letter delta (6) to stand for the change
in a variable. The slope is sometimes referred to as the rise over the run. So, we have sev­
eral ways of expressing slope:
51 ope =
27
Change in value on the vertical axis
Rise
Change in val ue on the horizontal axis
Run
Figure lA-4 reproduces the graph from Figure lA-3. Because the slope of a
straight line is the same at any point, we can use any two points in the figure to calcu­
late the slope of the line. For example, when the price of pizza decreases from $14 to
$12, the quantity of pizza sold increases from 55 per week to 65 per week. Therefore,
the slope is:
Slope =
6Price of pizza
($12 - $14)
-2
i1Quantity of pizza
(65-55)
10
-0,2,
28
I
I
I I
I
PAR T 1
I
In troduction
Fig ure lA-4
COlculating the Slape ot a Line
We cancalculate theslope ofa line as thechange
in thevalue of thevariableon they-axis divided
by the change in thevalue of the variableon the
x-axis. Because the slope of a straight line is
constant,wecan use anytwo points in the figure
to calculate the slope of the line. For example,
when the price of pizza decreases from $14 to
$12, the quantity of pizza demanded increases
from55perweek to65perweek. $0, theslope of
this line equals -2 divided by lO,or -0.2.
Price
(dolla rs
per pizza)
':: ~ • •..•.. .
14
r ..
:r .
- ~ Demand
.curve
L-,-·------'--- -----'-- -----'- ---- -- '-------'
!
o
50
55
60
65
70
75
Quantity
(pizzas per week)
T he slo pe of this line gives us some insig ht into how respo nsive co nsume rs in Bryan,
Texas, are to cha nges in the price of pizza. The larger the value of the slope ( igno ring the
negative sign ), the steeper the line will be, which in d icates that not m any ad d itional piz­
zas ar e sold wh en the pr ice falls. The sm aller the value of the slope, the flatter the line
will be, whi ch in d icates a greater increase in pizz as sold wh en the price falls.
Taking into Account More Than Two Variables
on a Graph
T he dem and curve gr aph in Figu re lA -4 shows the-relations hip between the p rice of
pizza and the quantity of piz za sold, but we know tha t the quantity of any good sold
de pends on mo re than just th e price o f the good. Fo r example, th e quan tity of pizz a sold
in a given week in Bryan, Texas, can be affected by such other variables as the price of
hamburgers, whether an advert ising cam pai gn by loc al pizza p arl ors has begu n that
week, and so on. Allowing th e values of any other variables to change will cause the posi ­
tion of the demand curve in the graph to change.
Sup pose, for example, th at the demand curve in Figure lA -4 was d rawn holding the
price of hamb urgers constan t at $1.50 . If the price of hamburgers rises to $2.00, then
some consumers will sw itch from b uying ha m burgers to buy ing pizz a, and m o re pizzas
will be so ld at ever y p rice. The result on the gr aph will be to shift the lin e representing
the demand curve to the right. Sim ilarly, if the p rice of hamburgers falls from $1 .50 to
$ 1.00, so me co nsumers will switch fro m buyin g pizza to bu ying hamburgers, and fewer
pizza s will be so ld at every pr ice. T he resu lt on th e graph will be to sh ift the line repre­
sen ting th e dem and curve to th e left.
The table in Figure lA-5 show s the effect of a change in the price of hamburge rs on
th e quanti ty of pizza dem an ded . For exam ple, su ppose at first we are on th e line labeled
Demand cur vej' If the pri ce of pizza is $14 (poi n t A ), an increase in the price of ham­
burge rs fro m $1.50 to $2.00 inc reases the quantity o f pizzas de manded from 55 to 60 per
week (po in t B) an d shifts us to Dem and curve 2 , O r, if we start on Demand curve j and the
price of pizza is $ 12 (po int C), a dec rease in the price o f ham burgers from $ 1.50 to $1.00
decreases the quantity o f pizzas dem ande d from 65 to 60 per week (p oint D ) and shifts
us to Demand curve3• By shiftin g the demand curve, we have take n in to accou nt the
effect of ch anges in th e value o f a th ird variable-the price of ham burgers, We will use
th is techniqu e of sh ifting curves to allo w for the effects of ad d itio nal variables man y
times in thi s book,
Positive and Negative Relationships
We can use graphs to show th e rela tionships be twee n an y two var iables. Sometimes
th e rela tionshi p betwee n t he va riab les is nega ti ve, m eanin g th at as o n e variable
increa ses in value, the other va riable decreases in value. Th is wa s the case with the
Ec onom ic s: Found ati o ns and Mode ls
CHAPTER 1
29
Fig ure lA-5
auantlty (pizzas per week)
Price
(dollars pe r pizza)
When the Price of
Hamburgers = $1.00
When the Price of
Hamburgers = $1.50
When the Price of
Hamburgers = $2 .00
$15
45
50
55
14
50
55
60
13
55
60
65
12
60
65
70
11
65
70
75
Showing
Three Variables
o,:!-a Graph
The demand curve for pizza shows the
relationship between the pnce of pizzas and
the quantity of pIZZa, demanded, holding
constant other factors that mIght affect the
wllililgness oj consumers to buy pizza. If the
pnce of pIZZa is $14 (point A), an increase 10
the price of hamburgers from $1.50 to $2.00
increases the quantity of pizzas demanded
from 55 [0 60 per week (pomr B) and shifts us
[0 Demand curve? Or, if we start on Demand
curvel and the pnce of pizza IS $12 (point C). a
decrease in the pnce of hamburgers from
$ 1.50 to $l.00 decreases the quantity of pIZZa
demanded from 65 [Q 60 per week (poin t D)
and shifts us to Demand curve).
Price $ 16
(dollars
per pizza) 15
14
13
12
11
Demand
. c u rv~3
10
Demand ' [)ema~ <:! '
curve,
, cu.rv.e.2
9
c .". . ,_ ----'--_ -----'_ _ -----'-_ _
o
45
50
55
L -_
- L -_
60
_
65
- ' -_
- - - ' -_
70
-------'
75
80
auantity
(pizzas per week)
price of p izza and the quan tity of pizzas dem anded . T he relation sh ip between two
variables can also be positive, meaning th at the values of both variables inc rea se or
decrease togeth er. For example, when the level of to tal inco m e-o r disposable personal
income-receIved by h ou sehol ds in th e Uni ted States in creases, the lev el of total
co nsumption spending, whi ch is sp end in g by households on goo ds and services, also
increases. Th e table in Figure l A-6 shows the values for in come and co n sump t io n
spen d in g fo r the years 200 3- 2006 ( the values are in billions of do llars). Th e graph
Comsumption
spending
(billions of
dollars)
Year
Disposable Person al Income
(billions of dolla rs)
Consum plion Spend ing
(billions of dollars)
2003
$8 ,163
$7,704
200 4
8,682
8,2 12
20 05
9 ,036
8,74 2
2006
9,523
9,269
Figure 1A-6
••
• •
9,000
8,500
8,000
7,500
7,000
L ,<-,-----'- _
$8,000
----'-_
8,200
--'-_
8,400
--l.._ - - l . ._
8,600
8,800
'.'
In a positive relationship between twoeconomic
variables, as one variable Increases, the other
variable also increases. This figure shows
the posuive relationship between disposable
personal mcorne and consumption spending.As
disposable personal mcorne III [he United States
has increased,so has consumpnon spending.
Source: U.S Department of Commerce, Bureau
of Economic i\nolysis
$9,500
o
::
--'-_
---'-_
-'---_
g,OOO
9,200
9400
Disposable personal income (billions of dollars)
-----l
9.500
a
30
PART 1
Intro duction
plot s th e data from th e ta ble , w ith nati on al in co me measu red alo ng the h orizontal axis
an d consu mption sp endin g me asu red alo ng the ve rtica l ax is. No tice th at th e four
points do not all fall exactly o n the line. This is often th e case with real-world data. To
exa m ine th e relatio ns hip between two vari ab les, economists often use the stra igh t line
that be st fit s th e d ata.
Determining Cause and Effect
Wh en we gra p h th e relat ionsh ip bet ween two va riab les, we oft en wa n t to d raw con­
cl usi o ns about wheth er ch a ng es in one variab le ar e caus ing chan ges in th e other
var iable. Doing so, however, ca n lead to in co rr ec t co ncl usio ns. Fo r example, suppose
you gra ph th e number o f homes in a neighbo rho od that have a fire bu rning in the
fireplac e and the numbe r of le aves on trees in th e neighborhood . You wo uld get a
relatio ns h ip like that show n in panel (a) o f Figure lA- 7: T he m o re fires burning in
the neig h bo rho od, th e fewer lea ves the trees have. Can we draw th e conclusion from
thi s graph that using a firepl ace cau ses tr ees to lose th eir leaves? We k no w, of course,
tha t such a co nclusio n wo uld be inc orrect. In sp ring and sum me r, th ere are relatively
few fireplac es bein g used, and th e tr ees are fu ll o f leaves. In th e fall , as tre es begin to
los e th eir leaves, firepl aces ar e used more frequently. And in winter, many firepl aces
are bein g used an d m an y trees have lost all the ir leaves. The reason th at the gr aph in
Figure lA-7 is m islead ing abo ut cau se and effect is that th ere is o bv io usly an omi tted
varia ble in th e an aly sis-the se ason o f the ye ar. An o m itted va ri able is one that
affects oth er variables , and its om ission ca n lead to false conclusions about cause and
effect.
Altho ug h in our exa m ple the o mitted variable is o bvious , there are m any debates
abou t caus e and effect where the existence of an om itt ed vari able has not been clear.
For ins tance, it has been kn own for many year s th at people who smoke cigar ette s suffer
fro m hi gher ra tes o f lu ng can cer th an do non smoke rs. For so me time, tob acco compa­
nie s and some scien tists ar gued th at th er e was an omitted variable- pe rhaps psycho­
logical temperament-tha t m ade som e peop le mo re likely to smoke and m ore likely to
develop lun g canc er. If thi s omitted variable exist ed, then the finding th at smokers were
Number of
leaves on
trees
Rate at
which
grass
grows
o
Number of fires
in fireplaces
(a) Problem of omitted variables
Figure 1A-7
o
Number of lawn mowers
being used
(b) Problem of reverse causation
Determ ning Cause and Elfe , i
Using graphs to draw conclusions about cause and effect can be hazardous. In pane!
(a), we see that there are fewer leaves on the trees in a neighborhood when many
homes have fires burning in their fireplaces. We cannot draw the conclusion that the
fi res cause the leaves to fall because we have an omitted variable-the season of the
year.In panel (b), wesee that more lawn mowers are used in a neighborhood during
times when thegrass grows rapidly and fewer lawn mowers are used when thegraSS
grows slowly. Concluding that using lawn mowers causes the grass to grow faster
would be making the error of reverse causality.
C HAP TER 1
i
I
Ec o no m ic s: Fo und ati ons and Models
more likely to devel op lung cancer wo uld not have been evide nce th at smoking caused
lung cancer. In th is cas e, however, n earl y all sc ien tists eventu ally co nclu d ed tha t th e
omitted variable d id no t exist an d th at , in fact , sm ok ing doe s cause lung ca ncer.
A related p ro blem in determ inin g ca use and effect is kn own as reverse causality. The
error o f rever se ca usa lity occ u rs whe n we con clud e that cha nges in va ria ble X ca use
changes in variable Ywhen , in fact, it is actually changes in va ria ble Yt ha t cause cha nges
in variable X. For example, pa ne l (b ) of Figur e l A-7 plot s the n umbe r of lawn mowers
being used in a ne ighbo rhood agai ns t th e rate at which gra ss on lawns in the neighbo r­
hood is growing. We co uld conclude from this graph that usin g lawn mowe rs causes the
grass to grow faster. We kn ow, h ~wever, tha~ in reality, the cau sality is in .the other direc­
tion: Rapidly growin g grass dunng the sp n ng and summer caus es th e Increased use of
lawn mowers. Slowly grow ing grass in the fall or winter or during per iod s of low rainfa ll
causes decreased use of lawn mowers .
Once again, in o ur examp le, the potential error of reverse cau salit y is obv ious . In
many econ omic deb ates, however, ca use and effect can be more difficult to dete rmi ne.
For exampl e, changes in the m on ey supply, or the total amoun t of m oney in the econ­
omy, tend to occur at the sam e time as cha ng es in the total amoun t of in com e people in
the econ omy earn. A famo us debate in eco nom ics was about whe the r th e changes in the
money su pply ca us ed the cha nges in total in come or wheth er the changes in total
income caused th e ch an ges in th e m oney sup ply. Each side in the d eb ate accused the
oth er side of co m m itt ing the error of rever se cau salit y.
t
i
r
J
e
Are Gra phs of Economic Relationships Always
Straight Lines?
The graphs o f relatio ns hips between two economic vari ables that we have d rawn so far
have been straig h t line s. T he relationship between two variables is linear wh en it can be
represented by a st raight line . Few economic relationships are actually line ar. Fo r exam­
ple, if we carefully plot dat a o n the p rice of a product and the quantity dem anded at each
price, holding con stan t other varia bles that affect the quantity demanded , we will usu ­
ally find a curved- or nonlinear-relationship rather than a linear relati onship . In pr ac­
tice, however, it is ofte n useful to app roxim ate a nonlinear relationship with a lin ear rela­
tionship. If th e relatio ns h ip is reason abl y close to being linear, th e ana lysis is not
significantly affected. In add itio n , it is easie r to calculate the slope o f a straig ht line, an d
it also is easier to calcu late th e area und er a st raigh t line . So, in th is text book, we often
assume th at the relationsh ip between two economic variables is linear even when we
know that thi s assumption is not p recisely correct.
Slopes of Nonlinear Curves
ng
ISS
ter
In some situa tio ns, we need to take into account the nonlinear nature of an econo m ic
relation ship . For exa m p le, panel (a) o f Figure lA- 8 shows th e hypothetical rela tionshi p
benveen Apple 's to tal cos t o f pr od uc ing iPod s and the qu antity o f iPod s pr odu ced. The
relation ship is curved , rathe r than linear. In this case, the cost o f p rod uc tion is increas ­
ing at an increasing ra te, whic h often happens in ma nu facturi ng. Put a d ifferent way, as
we move up th e curve , its slo pe becomes larger. (Rem embe r tha t with a stra ight line , the
slope is always constant. ) To see thi s effect, first rememb er th at we calculate the slope of
a curve by dividing the ch an ge in the variable on the y-axi s by the cha nge in the variable
on the x-axis. As we m ove from point A to point B, the qu anti ty pro d uced increases by
1 million iPo ds, wh ile the total cost of production increases by $50 mi llion . Farther up
the curve, as we move from point C to point D, the change in quantity is the same-l
millio n iPod s-but the cha nge in the total cost of production is n ow much lar ger: $250
millio n. Because the cha nge in the y variable has in creased, whi le th e change in the x
variable has rem aine d the same, we know that the slop e has incr eased.
To me asure th e slo pe of a n on linea r curve at a pa rt icular point. we m ust mea sure
the slope of th e tangent line to th e curve at that poi nt. A tange n t lin e will only to uch the
curve at that po int. We can m easure the slo pe o f the tan gent line just as we wo ul d
31
32
PART 1
Introd uc tion
Total cost of
production
(millions of
dollars)
Total cost of
production
(mill ions of
dollars)
Total cost
Total cos t
o
$ 1,000
$900
~Y=
250
750
750
350
300
o
350
A
275
L ,/ - -- - ' -4
3
-
- --
-
8
-----'--­
9
o
L,-
3
-
-
-
-
-
4
Figure 1A-8
8 e- pe 0
'on1l1"9 r
..If
-
-
-----'-8
-
-
­
9
auantity produced
(m illions per month)
auantity produced
(m illions per month)
(a) The slope of a nonlinear curve is not constant
-
(b) The slope of a nonlinear cu rve is measured by the
slo~e of the tangent line
-
e
The relationship between the quantity of iPods produced and the total cost of
production is curved, rather than liner. In panel (a), in moving from point A to point
B, the quantity produced increases by l million iPods, while the total cost of
production increases by $50 million. Farther up the cure, as we move from point Cto
point D, the change in quantity is the same- l million iPods- but the change in the
total cost of production is now much larger: $250 million.
Because the change in the y variable has increased, while the change in the x variable
has remained the same, we know that the slope has increased. In panel (b), we
measure the slopeof the curve at a particular point by the slope of the tangent line.
The slope of the tangent line at point B is 75, and the slope of the tangent line at
point C is 150.
th e slo pe of any st raig h t lin e, In pan el (b) , the ta ngent line at poin t B ha s a slop e
eq ual to :
c.Cost
c.Qua nt ity
75
1
75 .
The tan gen t line at poi nt C has a slope eq ual to:
c.Cost
c.Qua ntity
=
150
1
150.
Once again, we see that the slope of the curv e is larger at poin t C than at poin t B.
Formulas
We have just seen tha t graphs are an im port an t economic too l, In th is sectio n, we will
review severa l useful fo rm ulas and show how to use th em to sum marize data an d to cal­
culate imp or tant relations hips,
C HAP TER 1
I
Economics: Fo und a tio ns and Models
33
Formula for a Percentage Change
One importan t form ula is the pe rcen ta ge change. The percentage change is the cha n ge in
some economi c variable , usually fro m one p er iod to the next, expressed as a percenta ge.
An important ma croecono m ic m easure is th e real gross domestic product (GDP ). GDP
is the value of all th e final goo ds an d ser vices produced in a country during a yea r.
"Real" GDP is corrected for the effects of in flation . 'Wh en economists say that the U.S.
economy grew 3.3 percent du rin g 2006, they mea n that real GDP was 3.3 percent hig h er
in 2006 than it was in 200 5. T he form ula for ma king thi s calculation is:
[
GDP2006 - GDP 200S )
x 100
GDP 200S
or, mo re gen erally, for any two per iods:
Percentage c ha nge =
Val ue in th e seco nd pe rio d - Va lue in th e first pe riod
Val ue in the first per iod
x 100.
In this case, real GD P was $ 11,049 bill ion in 2005 an d $ 11,4 15 bill ion in 2006 . So, th e
growth rate of th e u.s. eco nomy du ring 2006 was:
(
$11,415 - $11,049J
x 100 = 3.3%.
$11,0 49
Notice tha t it didn't matter th at in usin g th e form ul a, we ign ored the fact that GDP is
measured in billions of do lla rs . In fact, w hen calc ulating percentage changes, the uni ts
don't matter. The percen tage increase fro m $11 ,049 billion to $11,415 billion is exactly
the same as the percen tage increase from $ 11,049 to $11 ,415.
Formulas for the Areas of a Rectangle and a Triangle
Areas that form rect an gles a nd trian gles on graphs can have important eco no m ic mean­
ing. For example, Figure 1A-9 shows the demand curve for Pepsi. Suppose th at th e price
is currently $2.00 and th at 125,000 bottles of Pepsi are sold at that p rice. A firm 's total
revenue is equ al to the amo unt it receives from selling its prod u ct, or th e quantity sold
multiplied by the pr ice. In th is case, total revenue will equal 125,000 bottles times $2 .00
pe r bottle, or $250 ,000 .
Th e form ula for the area of a rectangle is:
Area of a rect an gle = Base x Height
Price of
Pepsi
(dollars
per bottle)
Figure 1A- 9
-Showing a Firm's rotol Revenue
on a Graph
The area of a rectangle is equal to its base
multiplied by its height.Total revenue is equal
to quantity multiplied by price, Here, total
revenue is equal to the quantity of 125,000
bottles times the price of $2,00 per bottle,
or $250,000, The area of the green-shaded
rectangle shows the firm's total revenue,
$2 .00
Total.
Revenue
Dema nd curve
for Pe psi
o
125 ,000
Quantity
(bottles per month)
34
PAR
TI
l
In t ro d u c t io n
Price of
Pepsi
(dollars
per bottle)
Figu re 1A- 10
.
:
.. . . ..:'
The area of a triangleis equal to .K multiplied
by its base multiplied by its height. The area of
the blue-shaded triangle has a base equal to
150,000 - 125,000, or 25,000, and a height
equal to $2.00 - $1.50, or $0.50. Therefore, its
area equals X x 25,000 x $0.50, or 56,250.
Area = 1/ 2 X base x
height = 1/2 x 25,000
/
/
$2.00
x $0.50
=$6,250
1.50
Demand curve
for Pepsi
o
125,000
150,000
Quant ity
(bottles per month)
In Figur e lA-9 , the green-shaded rectangle also represents the firm's total revenu e because
its area is given by the base of 125,000 bo ttles mu ltiplied by the price of $2.00 per bottle.
We will see in later chapters th at areas that are triangles can also have economic sig­
nificance. The form ula for the area of a trian gle is:
Area of a triangle =
!.2
x Base x Height.
The blu e-shaded area in Figure 1A-1O is a tr iangle . The base eq uals 150,000 - 125,000, or
25,000. Its height equals $2.00 - $ 1.50, or $0.50. Therefore, its area equals Y; x 25,000 x
$0.50, or $6,250. Notice tha t the blue area is a t riangle only if the demand curve is a
straight line, o r linear. No t all demand curves are linear. However, the form ula for the
area o f a triangle wi ll usua lly still give a good approximation , even if the demand curve is
not linear.
Summary of Using Formulas
Yo u will enco un ter severa l other formulas in this book. Whenever you must use a formula,
you should follow these steps :
~
Make sure you understand the econom ic con cept that th e form ula represents .
Make sure you are using the correct formula for the problem you are so lving.
Make sure that the number you calculate using the formula is econom ically reaso n­
abl e. For example, if you are usin g a form ula to calcu late a fir m's rev en ue and your
answer is a neg ative n umber, you know you m ade a mistake somewhere.
LEARNING OBJECTIVE
II
•
•
rt....' ,Iilo.'I"/<.oI r..I· (,.,-....
Vlsil www .rnvec onlob.c orn to om. !e te th9se e xerc ise s
c ntlne oo o ge r Insta nt feedba c k
Problems a nd Applications
l A.l The following table gives the relationship between
the price o f cus tard pie s and the num ber of pies
Jacob b uys per week.
" . pa ges 24-34 .
PRICE
QUANTITY OF PIES
$3 .00
2.00
5.00
6
7
July 9
4
July 16
6.00
3
8
5
July 30
1.00
4.00
WEE K
July 2
July 23
Augu st 6
I
C HAP TER 1
a. Is the relationshi p between the price of pies and
the number o f pies Jacob buys a po sitive rela ­
tionship or a neg at ive relationship?
b. Plot the data from th e table on a graph similar to
Fig ure lA-3. Dr aw a stra igh t line that best fits
th e point s.
c. Calcula te the slope of th e lin e.
lA.2 T he follow ing tabl e gives informa tio n o n the qu an­
tity of glasses of lemonade deman ded on sun ny and
overcast days. Plot th e da ta fro m the tabl e on a
grap h similar to Figure l A-5. Draw two straig h t
lines represen ting th e two demand curves-one for
sunny days and one fo r overcas t days.
PRICE
(DOllARS PER
GLASS)
QUANTITY
(GLASSES Of
LEMONADEPER DAY)
WEATHER
$080
0.80
0 70
0 70
0.60
0.60
0.50
0.50
30
10
40
20
50
30
60
40
Sunny
Overcast
Sunny
Overcast
Sunny
Overcast
Sunny
Overcast
lA .3 Using the informa tion in Figure l A-2, calculate th e
pe rcen tage ch an ge in au to sales from on e yea r to
the next. Between wh ich years did sales fall at th e
fastest rate?
lA.4 Real GDP in 198 1 was $5,292 billion. Real GD P in
1982 was $5, 189 b illion . Wh at was the percent age
cha nge in real GDP from 1981 to 1982? Wh at d o
econ omists call the percenta ge change in real GD P
fro m one year to the next?
lA.5 Assu m e that the d em and curve fo r Pepsi pa sses
through the following two point s:
PRICE PER
BOTTLE Of PEPSI
NUMBER OF BOTTLES
Of PEPSI SOLD
$250
1.25
100.000
200,000
35
lA.6 Wha t is th e are a of the blu e tr ian gle show n in th e
following figure?
Price of
Pepsi
(per
two-liter
bottle)
$2.25
1.50
Demand curve
for Peps i
o
115,000
175,000
Quantity of two-liter
'b o tt les of Pepsi
sold per week
lA.7 Calcul ate th e slope of the to tal cos t cur ve at point A
and at point B in the followin g figu re.
Total cost of
production
(millions of
dollars)
Total
cos t
$900
700 .
300 .
175
a. Draw a grap h with a lin ear demand curve th at
pass es throu gh th ese two poin ts.
b. Show on th e graph the areas rep resent in g to ta l
revenue at each pr ice . Give the valu e fo r total
re ven ue at each price.
\
?
Ec ono mic s: Fou nd cn ons and Models
o
5
7
12
14
Quantity produced
(millions per month)