GRAPHIC

ANALYSIS

ECONOMIC RESEARCH

I by Frederick V. Waugh, Director

Y Division of Agricultural Economics

AGRICULTURAL HANDBOOK No.84

UNITED STATES DEPARTMENT OF AGRICULTURE

Agricultural Marketing Service

Washington, D. C. June 1955
CONTENTS

Page

1

Introduction

k

Frequency distributions

6

Cumulative frequencies

10 Trends

Ccles ^'

20 Seasonal variation

Simple regression. •• •

Comparison of time series -^

I4ultiple regression ^

k2 Joint regression •

kk Linear programming •

Ud Indifference ciirves

Averages 50

Elasticity ^2

Differentiation 5^

Deriving a marginal curve from an average curve • • 5^

Roots of a polynaaial ^

Solution of simultaneous eqviations 62

For sale by the Superintendent of Documents, U. S. Government

Printing Office, V/ashingbon 25, D. C. -Price UO cents
GRAPHIC ANÁLISIS IN ECONOMIC RESEARCH

By Frederick V. Waugh, Director,

Division of AgricultureüL Economics,

Agricxxltural Marketing Service 1/

IRERODUCTION

Graphic analysis can be a powerful tool in economic research• Economic

statisticians have allowed this tool to become rusty in recent yeeurs* They

have been fascinated with nev tools—with new inathematical theories and

with wonderful electronic computing machines« The new tools are good, too.

We do not sxaggest that any analyst throw away his mathematicail textbooks

or discard his calculating machine. On the other hand, we do suggest that

all economists and statisticians should be familicu: with graphic analysis,

and should use it along with the other tools. The pinrpose of this handbook is to review seme of the graphic tools that can be useful. It is

not a complete cat€LLog. Rather, it is a collection of examples covering

a variety of analyses.

In the two decades from 1920 to 19^^ graphic analysis was very popular, especially among agricultural economists« Some auialysts \ised graphics

—and graphics €ÚLone—to measure demand curvea, supply cxirves, cost curves,

and input-output relationships. But sometimes they misused graphic

methods and got spurious results. Malenbaxmi and Black, 2/ in 1937^ warned

against the indiscriminate use of graphic analysis. Without doubt, there

vas merit in these warnings; some practitioners of the graphic arts had

done slipshod, unreliable work.

Perhaps i>artly because of its misuse, graphic analysis seems to be

under an eclipse today. Many of the recent books on econometrics—that

Is, books that deal with the use of economic theory, mathematics, and

statistics to measxn:^ the influence of economic forces—are filled with

high-powered mathematics and a few resxilts worked out on calcxilating

machines, but use little or no graphic analysis. A notable exception is

Tinbergen. 2/

In the Division of Agricultural Economics, we never have abeuoidoned

graphic analysis, althoxigh we, too, have been gxiilty of some neglect.

Maoy of our research bulletins would be more readable, and more exicura^te,

If ve had used more diagrams cmd probably somewhat less detedLled results

ground out of the calculating machines. Often there can be a fictitious

accuzacy in resxilts computed on the machine to six or eight ''significant

figures." Even the modem electronic computer cannot give results that

are any more accxirate than the numbers we put into it. The economist

often must use data that are reliable only to two or three signlficcint

figures.

l/ Several members of the staffs of the Agricultuwú. Marketing Service and

the Agricultural Research Service provided matericúL used in this report.

The author thanks especieüJy Richard J. Foote and Hyman Weingarten for preparing the material for publication.

2/ Medenbaum, I. W., and Black, J. D. The Short--Cut Graphic Method.

Quart. Jour. Econ. 52:66-112, illus. 1937-

All/ Tinbergen, J. Econometrics. Transi, from the Dutch by H. Ryken Van

out. 258 pp., illus. Philadelphia, Pa. 1951.
Graphic analysis is easy and flexible- Some have said that it is

too easy, and too flexible. It can go wild if it is not combined with

sound economic theory and with an understanding of probable errors. But

so can any method of analysis. Especially when the economist mechanically

extrapolates a trend into the future, he is Junrping blindfolded into the

wild blxie yonder, unless he understands the forces that made the trend

in the past and are likely to shape it in the future, the fanciest mathematical function computed to six decimals is as likely to lead him astray

as is the easy, flexible graphic method. Ease and flexibility are desirable features of any method. Would anyone seriously argue that difficulty

cmd inflexibility are virtues?

Two statisticians in the former Bureau of Agricultural Economics

were closely identified with the development of graphic analysis. One

of them, Louis Bean, \ised graphics almost excliisively. The other,

Mordecai Ezekiel, xxsed graphic methods to supplement machine ccnnputation* k

Hé often computed certain linear functions Qn_the machine, and then

used graphic methods to secure a better fit by making use of curvilinear

relations. This is a sound procedinre if used with judgment and in

moderation.

Graphic methods can and should be used to see whether mathematical

results appear to fit the data. This approach has been followed in

certain research studies issued in the last few years by the Bureau of

Agricultural Economics and its successor, the Agricultxiral Marketing

Service. ¿/ Too often, nowadays, econometricians simply assume that

economic relationships are linear in arithmetic or logarithmic terms

and never bother to test whether the assimiption is correct. Actually,

few economic relationships are likely to be strictly linear. Graphics

can often provide more meaningful relations than those given directly

by machine computation. In spite of some of the publicity, not even

the newest calculating machine can think. The economist must do it.

Graphic analysis C€üI help him.

But, as I e€3 it, the greatest value of graphics is in making a

quick preliminary analysis of a problem to determine which variables to

include €uid what kinds of mathematiceúL fuoictions to use. Here the

econometrician is guided both by logical theory and by enrpiricism.

Graphics c€ui help him think through the problem. It can also help >^^wi

choose functions that reeilly describe the data. Then he may well choose

to compute a mathematiceú. function by least-squares or otherwise.

This handbook presents some typical exainplee of diagrams that are

usefxil. The collection is, of course, far from coos>lete. In fact, no

collection will ever be complete, because new kinds of diagrams and

h/ See Ezekiel, Mordecai. Methods of Correlation Analysis. Ed. 2.,

531PP-> illus. New York. 19lfl.

¿/See, for exaa^le, U* S. Dept. Agr. Tech* Bulletins IO68, p. 29; 1070,

. 10-13; 1080, p. 28, 52, 63; and II05, p* lH, 57, 75, 89^

- 2 .
modifications of old diagrams are invented almost every day. Our piirpose is only to encourage econometricians to experiment with graphics—

not as their only tool of research, but as one that will help them do

better work*

Like any other scientist, the econometrician tries to \inderstand

relationships between variables. He tries to explain variations in production, distribution, costs, prices, and profits. He cannot do this by

logic alone. Pure theoiy is not enough. Nor does he find the answer by

stating economic theory in mathematical terms which he can manipulate

according to rules. Econometrics is the measurement of economic relationships. Theory must be tested and quantified.

To test and quantify economic theory, we must work with statistics.

Often we can use statistics that are published regularly by public agencies or by private industry. Sometimes we will need special statistics

obtained from suarveys or from experiments. In euiy case, statistics are

essential. But raw statistics alone axe not worth much--just as pure

theory alone is of little value. Theory and statistics must be combined

if the research is to be worthwhile. Graphic analysis can be useful in

this combination—especially in the preliminary stages.

- 3 -
FREQUENCY DISTRIBUTIONS

Yields of Com in Iowa. 1866-19^3

The economist studies many kinds of variables. He is concerned with

variations in prices, incomes, rates of consxamption, production, and many

other things* Ordinarily the economist wants to explain why certain

things have varied in the past and what sort of changes can be expected

in the future. Most of the diagrams in this report illustrate how to go

about explaining why certain things have varied. However, we start with

a diagram which shows simply how much variation there has been in the past

without explaining why.

We know, of course, that much of the variation in yields of com has

been due to weather. We haven't learned as yet to control the weather.

We may, however, e^qpect that weather conditions in the future will vary

about as they have in the past. With this in mind, what variations can

we expect in yields of corn?

The diagram on page 5 6/ shows the frequency distribution of yields

of com in Iowa in the period I866-1943. To make the data more nearly

comparable over this long period, yields in recent years were adjusted

downward to allow for estimated effects of the xise of hybrid seed. This

was a major factor in increasing yields during the late 1930's and the

early ISÄO's.

The shaded area in the diagram (sometimes called a histogram) shows

the actual nxmiber of years in which the yield was from I5.O to 19.9

bushels, from 20.0 to 24*9 bushels, and so on. The irregularities in the

steps would probably have differed slightly if data for fewer years or

for other yeare had been shown* We can get a better idea of what we might

expect generally by drawing a smooth cxirve such as the one shown on the

diagram. Such curves can be computed mathematically from well-known formulas. In this case, the graphics indicate that the cxirve is skewed to

the left. The analyst can easily see that a normal cxirve will not describe the variation. This is often the cetse with economic data. The

statistician cannot safely assume that his frequency distributions exe

normal. He woxild do well to draw freehand graphic curves before trying

any sort of mathematical fit. In many cases the graphic curve will

satisfy 6LL1 needs. When comparing several distributions and in certain

other cases, mathematical curves and coefficients have merit.

As the more recent data used in this chart were adjusted downward

to allow for the effect of the use of hybrid seed, the average level of

yields sviggested by the chart is not applicable currently. For exanq?le,

yields per harvested acre in 19^3-52, a period when nearly all of the com

grown in Iowa was from hybrid seed, averaged 50 bushels per acre cooipared

with a yield of kO bushels in 192O-29, a period having equally normal

weather but when practically no hybrid seed was used. The chart suggests

an average close to that in the 1920-29 period. The chart can be \ised,

however, to indicate the probable variation in yield cirovind some ai>propriate average.

6/ Unless otherwise specified, diagrams referred to in the text are those

on the facing page, the data for which are given in the table beneath the

chart.

- 4 -
FREQUENCY DISTRIBUTIONS

Corn : Yield Per Harvested Acre, Iowa, 1866-1943*

NO. YEARS: L¥.-•J^ " "" "~ ^ "~

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10 15 20 25 30 35 40 45 50 55 60

YIELD (BU.)

:¥ ADJUSTED FOR ESTIMATED EFFECTS OF HYBRID SEED

U.S. DEPARTMENT OF AGRICULTURE NEC. 1311-55(1) AGRICULTURAL MARKETING SERVICE

rigore 1

Com: frequency distribution of years classified by yield per

harvested acre, lova, I866-I9U3

Yield per harvested acre Freqiiency of years

Bushels

:

RiBber

15.0-19.9 : 2

20.0-24.9 : 1

25.0-29.9 5

30.0-34.9 ! 15

35.0-39.9 23

40.0-44.9 24

45.0-49.9 7

50.0-54.9 1

Foote, Richard J., and Bean, Louis H. Are Yearly Variations in Crop Yield Really Randoa? Agr. Econ. Research.

3:23-30, iUus. 1951.

5 -
CUMULATIVE FREQUENCIES

Percentages of All Families and Unattached Indlviáiíals

With Incomes Belov Stated Levels

Economists and statisticians are concerned with many kinds of frequency distributions. The particvilar distribution shown on the diagram refers

to percentages of families with various incomes. In this case we have

shown a cumulative frequency cxirve. Thus, instead of showing the percentage of families with incomes from 0 to $1,000, from $1,001 to $2,000, etc.,

we have shown the percentage with incomes below $1,000, below $2,000, etc.

This sort of cumulative frequency curve for incomes is sometimes known as

a Lorenz curve.

The cxmiulative frequency curve, or ogive, has some advantages over

the more usual noncumulative frequency curve. It can be used whatever

the "class intervals" may be. For example, in this case, class intervals

of $1,000 were used for the part of the curve from 0 to $5,000. For

incomes above $5,000, the class interval was $2,500. With unequal class

intervals, it is awtofard to draw and use the ordinary type of frequency

chart, and the cumulative chart is preferred.

In this case there was no problem of drawing a freehand curve to

fit the cumulative freqiaencies. The plotted data all lie aúmost exactly

6Q.ong the freehand line we have drawn.

Several mathematiccuL functions have been proposed and used to describe the distribution of incomes. Some of these, like the Peireto curve,

are purely empiricetl. Others, like the Gibrat curve, are based upon

logicflüL considerations. It is obvious that no mathematical curve could

fit the data much better than the freehand curve we have drawn. In fact,

the freehand curve probably fits the data on the left hand side of the

diagram better than would a mathematically fitted Pareto curve.

- 6 .
PERCENT

CUMULATIVE FREQUENCIES

Percentage of All Families and Unattached Individuals

With Incomes Below Specified Levels, 1950

3 4 5 6 7

INCOME ($ THOUSANDS)

8 9 10

U. S. DEPARTMENT OF AGRICULTURE NEC. 1312-55(1) AGRICULTURAL MARKETING SERVICE

YUgure 2

All families and unattached Individuáis: Percentage vlth inccnes belov

specified levels, United States, 19^0

Inccme level DistrilmtiQii

Dollars

Under:

1,000

2,000

3,000

if,000

5,000

7,500

10,000

Percent

1.6

22.1

39.2

56.6

71.0

88.5

Goldsmith, Selma, et. al. Size Distribution of Income Since the Mid-thirties. Review Econ. and Statis. 361^*. 195^.

- 7 -
Farms Ranked by Total Value of Products Sold

We have Just discussed the Lorenz curve • We nov consider a somewhat

different type of curve which is often useful*

The figures given below the diagram show that the lowest 20 percent

of the farms got OA percent of the income, the lowest kO percent got 3*^

percent, €uid so on. These figures eure plotted on the diagram with a

smooth cuarve drawn through them that gives an estimate of the percent of

the totul value of farm products obtained by any given percent of the

farms. For exanrple, the lowest 95 percent of the farms received about

68 percent of the income. We ceui turn this around (reading the diagram

upside down) and say that the top 5 percent of the farms got about 32

percent of the income.

On a diagram of this kind, if all farms had received the same income,

the observation would have faOlen along the straight dotted line. Thxis,

the degree of curvature is a meeisure of the inequality of income distribution .

Many mathematical functions have been used to study cumulative

frequencies of incomes and other economic variables. The Pareto curve

and the Gibrat curve are well Imown. Such mathematical functions are

especially useful when ccmpeüring several curves. But simple graphics

will help in emy case to choose an appropriate function.

- 8 -
CUMULATIVE FREQUENCIES

Farms Ranked by Total Value of Products Sold

SALES, PERCENT —

80 -

60

40

20

20 40 60 80 100

PERCENT OF FARMS

U.S. DEPARTMENT OF AGRICULTURE NEC. 1313-55(1) AGRICULTURAL MARKETING SERVICE

Figure 3

RelAtioDBblp between percentage of taxms and percentage of total value of products sold

Panas ranked by value of products Percentage of total value of products sold

Percent

Lovest:

20

ko

60

80

Percent

O.k

3.k

11.2

29.2

CoBçlled fron »Jïilt^ States Census of Agriculture, 1950.

- 9 -
TRENDS

Population Trends In the United States by Decades, l800-1950

The economist Is often concerned with time trends • He wants to ilnd

out how some variable has been Increasing or decreasing over a period of

severeúL years or decades. For example, he may be studying the growth of

population In the United States or the rate of decline In the number of

farm workers. In such cases he will want to disregard minor fluctuation»

due to errors In the data or to temporary dlsturbauices. He will also

generally want to disregard cycles or other shorter term movements In

the data If they exist. He Is concerned only with the gradual rate of

change In a variable In relation to time.

The chart shows the Bureau of the Census estimates of the population

In this country by decades since l600. It Is a slaiple matter to draw a

freehand curve describing the trend. Ordinarily, at least, i)opulatlon

does not change abruptly except by major wars, serloxis epidemics, or a

sharp Increase In Imnlgratlon. If we plot the data for each year, or

decade, we can usxially draw a smooth curve running nearly throvigh the

points we have plotted. In this case, departures from the curve could

well be due to errors In estimating the population. It should be noted

that even offIclaü. estimates may not warrant the naive faith In their

accuracy that sometimes prevails. We, as economists, 6ü:e probably as

much responsible as any other group of users of published data for the

Insistence upon the publishing of a single number (point estimate) to

represent, say, the population of the United States. We are reluctcuit

to accept a lower and upper estimate (interval estimate) of the actual

pop\ilatlon even though we know that the Bureau of the Census official

figure of 150,697,761 persons for 1950 (or any other year) may not be

exact. All too often we do not taise even the trouble to understand what

the publisher has to say about the known, or estimated, amount of

possible error in his estimates.

Instead of drawing a freehand curve, the statistician could, of

covirse, fit some kind of mathematical function, such as a logistic

curve. Our advice would be to draw a freehand curve first. In this

case, it is doubtful if etny mathematlc€tl function would give a better

description of the trend than our freehand line. A mathematical curve

might have some advantage when comparing trends in population in severed,

different countries. If the same type of function was fitted in each

case, results could be si mimar1zed in a few statistical measurements.

A practical application of trends is in forecasting. This always

involves an extrapolation beyond the range of the data. Extrapolation

of trends is dangerous whether it is done from a freehand curve or from

a curve that has been fitted mathematically. For example, before the

1950 census data were available (so that we did not have the last observation on the diagram), many population experts drew an S-shaped curve indicating that the rate of growth had started to flatten. When this type of

curve was extrapolated it suggested that the population would become stationary, or even decrease, by i960 or 1970. Such an extrapolation now

looks doubtful in view of the census figure for 1950.

• 10 -
TRENI

U. s. Population By Dec

DS

odes, I800-I950

MIL PERSONS ~"- ^ ^

2

/ 4 t

lOR - -- -

1 ¿0 "K

^

^^

1 ?

\r\r\ - -

^ ^

IÜÜ _ _ _ : ^^

á ^

jr

1^ ^

7C _ _ j 2

(0 ^ ^

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Ï?

KH - -

j l¡ ^

OU ^ ^ '^

^ ^

J N ^

OK --,.^^'^- ¿0 Z Al^^ét

ma ^^

i baa ^fc**'^^* f\ T* 1

1800 1820 1840 I860 1880 1900 1920 1940

U. S. DEPARTMENT OF AGRICULTURE NEC. 1314-55(1) AGRICULTURAL MARKETING SERVICE

Figure k

Population: United States, by decades, 1ÔOO-1950

Year Population i Year Population

Mil lions MilliOQB

1800 5.3 :

\ 1880 50.2

1810 7.2 : \ 1890 62.9

1020 9.6 : 1900 76.0

1830 12.9 1910 92.0

IOÍ4O ; 17.1 :

: 1920 105.7

1850 23.2 : : 1930 122.8

i860 31A 19^ 131.7

1870 38.6 : 1950 : 150.7

Bureau of the Census.

- 11
Volume of Agricultural Marketings

In most cases^ of course, successive observations by years or decades

when plotted on a chart do not lie exactly upon a smooth line. However,

it is usually possible to draw a freehand line or ciirve describing the

general trend.

This diagram shows an index of the volume of agriciiltural marketings

each year from 19IO through 1953. Obviously, the growth has not been so

steady as the growth of population. For example, marketings did not

increase from 1928 through 193?. This was due in part to a business

depression and low prices, and in part to two serious droughts. During

World War II, there was a remarkable expansion in agricultural marketings

to meet the needs of domestic and foreign markets. After 19^5> there was

a slow increase. We need a trend line that will describe these characteristics. The freehand line shown on the diagram perhaps describes them

fairly well.

Again, the statistician may want to use some mathematical function

to describe this trend. Before doing so he wo\ad be well advised to draw

first a freehand trend in order to indicate what sort of function should

be used. In this case, for example, a straight line would not adeqioately

describe the observed trend. A third-degree parabola might give a fair

fit, but would be an extraordinarily bad curve to extrapolate into the

future. In general, the economist wotild do well to avoid parabolas.

Logically, they seldom make cuay economic sense.

For most purposes, the freehand trend is as good as any mathematical

trend we might compute. An exception might be the problem of conrparing

trends in severcú. variables. Suppose we wanted to compaure the trend in

agricultural marketing with trends in agrictiltural output, the amo\mt of

fertilizer used, population, and so on. We might find some type of mathematical function that fits all the trends reasonably well. Then each

curve could be summarized by a few constants. It woxild be easy to compare one with another. But this sort of procedure often would cover up

some interesting and iioportcuit features of some of the trends. It is

6Q.ways a good idea to plot the data and to draw freehand cxirves first.

- 12 •
TRENDS

Volume of Agricultural Marketings

100

75

50

194 7-49: :

^ ^

~1 "^nFT 1

f

_i L

^ .JÍ J^' 4

ff LT ^

i^ ^m

^'

ty

-/

Vn

j

/

^-flT^

^> il

^i«0^ ii

-ir--^M 1 i 1^^ r" ^ f

^r 1 >r ^ i'Sf -iU ~ir"^5^ "~

(P^ .^J^ ^

JL ' "'^U f^^^

4MH n r ^ '

i f~ [—

h

_^

1910 1920 1930 1940 1950 I960

U.S. DEPARTMENT OF AGRICULTURE NEC. 1315-55(1) AGRICULTURAL MARKETING SERVICE

Figure 5

ran narketlcgB aod home consuiiptlon: Index nuBÉbers of volune, 1910-53

(1947-49=100]

Year VoluBe Year Volu»e : Year Volume

1910 ! 57 1925 ! 70 ! : 1940 80

1911 61 1926 : 73 : 19^1 82

1912 61 1927 73 : : 1942 90

1913 : a 1928 Ik : : 19^*3 94

191^» 61 1929 I 73 : : 1944 99

1915 6if 1930 72 : 19^5 99

1916 63 1931 73 : : 19^6 97

1917 62 1932 71 : : 19^7 100

191Ô 66 :: 1933 72 : : 19^^ 97

1919 67 :: 193^ 71 : : 1949 103

1920 6if 1935 ! 66 : 1950 99

1921 65 1936 71 : : 1951 101

1922 67 1937 Ik 1952 104

1923 69 1938 16 : : 1953 109

192Í* 72 1939 78 :

Agricultural Marketing Service.

- 13 -
CYCIES

Cattle on Farms, Jcuauary 1

Some economic data exhibit successive cycles of approximately the same

length covering periods of up to several years. This is especially true

in the case of some agricultural data such as those on cattle numbers.

When cattle prices €u:e high, farmers are likely to start breeding for

larger herds. It takes several years to increase the herds substantially,

and the increase ordinarily continues for some time €tfter prices became

unprofitable. Then the reverse happens and herds are gradually decreased.

V/hen the euinual data on cattle numbers are plotted, as shown in the

diagram, it is easy to see that there have been fairly regular ups and

downs. We have drawn a smooth curve through the data to describe these

ups and downs.

The statistician could, of course, fit some kind of mathematical

function to data of this kind. He would choose a curve that would allow

for an upward trend, and for the cycliccLl swings around the trend. But

it would teüce a very complicated mathematical curve to fit the data as

well as the freehand curve.

The main practical interest in cycles stems from the need for

forecasts. The farmer naturally wants to know where we are in the

current cycle—are cattle nxambers approaching a peak and when are they

likely to txnrn down? An ancdysis of peist histoiy will help him answer

such questions. But he will do well to give special attention to current

developments. For example, he will need to watch current trends in cattle

slaughter.

- 14 -
CYCLES

Cattle on Forms, Jon. 1

MIL. HEAD

100

1920 1925 1930 1935 1940 1945 1950 1955

J954 DATA ARE PRELIMINARY

U. S. DEPARTMENT OF AGRICULTURE NEC. 1316-55(1) AGRICULTURAL MARKETING SERVICE

Figure 6

All cattle azMi calves: Huniber on farBB January 1, 1920-^

Year Humber :

:

: Year Humber : i Year Humber

Milllonfl

:

Millions MllllonB

1920 70.U : i 1932 65.8 i i 19i»4 85.3

1921 68.7 : : 1933 70.3 : 1945 : 85.6

1922 : 68.8 : 193^ Ik.k : 19^6 82.2

1923 : 67.5 : 1935 68.8 : : 19^7 : 80.6

192if : 66.0 : 1936 : 67.8 : : 19^ 77.2

1925 : 63.U : : 1937 i 66.1 : : 19^9 76.8

1926 : 60.6 : : 1938 65.2 : : 1950 78.0

1927 : 58.2 : : 1939 66.0 : : 1951 82.0

1928 : 57.3 : : 19^ : 68.3 : 1952 87.8

1929 : 58.9 : : 19^1 : 71.8 : 1953 : 93.6

1930 : 61.0 : : 19i^2 : 76.0 : : 195^ 1/9^.7

1931 : 63.0 : : 19^3 : 81.2 :

1/ Preliminary.

Agricultural MBurketing Service.

- 15 -
Cattle on Farms by Cycles

One of the best ways to forecast the probable behavior of a ciirrent

cycle from that of previous cycles is to break the total series Into

Individual cycles. In 195lf, we were about halfway through the most recent

cycle In numbers of cattle on farms. In the diagram shown here, data for

these Individual cycles are plotted on the same scale, beginning with the

year of the low point in inventories in each instance.

The several cycles of numbers of cattle are remarkably similar.

One handicap in this visual scheme is that each cycle is of a different

length. Similarity between cycles would appear even closer if the cycles

were telescoped into a uniform length.

A good statistician knows that history seldom repeats itself exactly.

Cycles vary in length and In amplitude. A knowledge of past trends, and

of pa^t cycles, gives some perspective to the present. Often it suiggests

the general direction of changes in the immediate future. But the wideawake economist will be looking for factors that may make the current

cycle different from the others.

- 16 ^
CYCLES

Cattle on Forms, By Cycles

MIL. HEAD

1938-49

* YEAR OF CYCLES, BEGINNING FROM LOW IN NUMBERS ON FARMS.

1954 DATA ARE PRELIMINARY.

U. S. 0€PARTMENT OF AGRICULTURE NEC. 1317-55(1) AGRICULTURAL MARKETING SERVICE

Figure 7

1/ See tabulation on page 15 for data for later yearc

Agricultural Marketing Service.

All cattle and calves: Number on farms January 1, I896-I919 ly/

Year Number : Year Number

Millions Millions

1896 49.2 ; : I90Ô 62.0

1Ö97 50.4 : : 1909 60.8

IÔ96 52.9 : : 1910 59.0

1899 55.9 : : 1911 57.2

1900 59.7 : : 191¿ 55.7

1901 62.6 : 1913 56.6

1902 6k.h : 191** 59.5

1903 66.0 : : 1915 63.8

1904 66.U • : : 1916 61.k

1905 : 66.1 : : 1917 71.0

1906 : 65.0 : 1918 : 73.0

1907 : 63.8 : : 1919 : 72.1

-17
Hog Slaughter and the Hog^Corn Price Ratio

The agricultiiral economist iisually is not content with simply observing periodic movements in prices or in production. He wants to know

what causes the swings. And he especially wants to know how the current

cycle is developing—whether, for example, it will be shorter or longer

than average.

Of course, if all cycles were completely regular, all the statistician would have to do is to find some kind of curve, making etllowance

for trend ana for cyclic ups and downs. A projection of this curve

would be a forecast of what is likely to happen in the next few months

or years. The business-cycle cumlysts have lecurned from i)ainful experience that such mechanical forecasts are unreliable. Many of the cycles

in agriculture are more regxilar than those in business. Still there is

a good deal of variation in agricxatural cycles. To vmderstand what is

going on and what is likely to happen in the immediate futvire, the

agricultxiral economist analyzes the forces that have shaped the cycles

in the past and that appear to be influencing the current cycle.

The accompanying chart illustrates a sinqple analysis of the hog

cycle. The lower part of the chart shows the number of hogs slaughtered

each year for I92O-5U. The upper jpart of the chart shows the ratio

between hog prices and corn prices (that is, the nimaber of bushels of

corn required to buy 100 pounds of live hogs, based upon average prices

received by farmers for hogs aoid corn). A high hog-corn ratio indicates

the situation in which hog production is profitable; a low ratio indicates the opposite.

By comparing the two parts of the chart it is easy to see that

changes in hog slaxighter lag a yecu: or two behind the hog-corn ratio.

The dotted lines connect some of the peaks and troughs of the two

curves.

By keeping a chart of this kind up to date, an economist c€ui get

a fairly accurate idea of prolAble developments in the next six-monthsto-a-year period.

- 18 ^
CYCLES

Hog-Corn Price Ratio and Hog Slaughter

PRICE RATIO

MIL. HEAD

1920 1925 1930 1935 1940 1945 1950 1955

1954 DATA ARE PRELIMINARY.

U. S. DEPARTMENT OF AGRICULTURE NEC. 1318-55(1) AGRICULTURAL MARKETING SERVICE

Figure 8

Hogs: Number slaughtered and hog-corn price ratio, 1920-51*

Year

• Slaughter

of

Hog-com ]

price \

ratio : Year

: Slaughter

: of

Hog-com

1 price

ratio :: Year :

:

Slaughter i

of :

Hog-cotm

price

ratio hogs i/ ; :

hogs

1/ ::

bogs :

y

: MnilonB ! Mil Hon« ;; Million«

1920 : 61,5 9.8 ; I 1932 ^ ; 71-1* 12.3

: t

:: 191*1* J 98.1 11.6

1921 61.8 13.6 : : 1933 2/ I 79.7 10.1* 1: 19»f5 : 71.9 12.8

1922 66.2 Ik.k : : 193^ 68.8 7.0 :: 191^ I 76.1 12.6

1923 77.5 8.7 : Î 1935 1*6.0 11.6 i: 19»*7 : 71*. 0 13.6

1924 76.8 8.2 : : 1936 58.7 13.0 :: 19W I 70.9 13.0

1925 65.5 11.1* : : 1937 53.7 11.1 :: 19»*9 : 75.0 15.7

1926 62.6 17.0 : : 1938 58.9 16.0 :: 1950 : 79.3 13.7

1927 66.2 12.7 : : 1939 ! 66.6 13.3 : : 1951 : 85.6 12.1*

1926 72.9 9.9 : : 19*^ Î 77.6 9.2 :; 1952 : 86.7 11.0

1929 • 71.0 10.9 : : 19**1 : 71.1* ll*.2 • • 1953 : 7l*.8 15.0

1930 : 67.3 11.U : : 191*2 : 78.5 16.5 :: 1951* 3/: 7l*.0 15.1*

1931 : 69.2 11.7 : : 191*3 95.2 13.6 : :

1/ Number of bushels of corn required to buy 100 pounds of live hogs at local markets,based on average prices received

by fanners for hogs and com. Annual average is straight average of monthly ratios. 2/ Includes those slaughtered for

Government account. 3/ Preliminary.

Agricultural Marketing Service.

- 19 -
SEASONAL VARIATION

Egg Prices > 1Q2'S>29 and 1943-^9

Many economic time series follow rather regular seasonal patternsmoving in 12-month cycles. This is especially true of many agricultural

time series. Production of some agricultural commodities naturally follows

a regular seasonal pattern, which results from the seasonal pattern of the •

weather and certain production practices related to it. This tends to

bring about an annual cycle in marketing and prices—especially in the

case of perishables.

The mathematicail statistician is sometimes tempted to make a mechanical analysis of seasonal variation. He may fit to the data a combination of a lineeur trend ana a sine curve—and he may feel satisfied with

a high correlation coefficient. But usually a simple graphic analysis

will show up some important facts that might otherwise escape the

researcher.

The first part of any analysis of seasonal pattern must be rather

mechanical. We must fit scane sort of trend and stxidy the monthly deviations from it. The diagram is based upon an average of deviations of

monthly egg prices from a 12-month moving average—an appropriate sort

of trend for our purposes. We have shown the average deviation for each

month of the yeax in two different ^-year periods.

The striking fact brought out by this diagram is that the seasonal

pattern in egg prices is changing. Low prices still occur in the spring,

high prices in the fall. But the seasonal swing is much less than it

was 20 years ago. And the -peak price comes earlier in the fall. These

changes chiefly are a reflection of new euid improved practices on the

farm. The trend toward a less pronounced seasonal pattern and toward

an earlier fall peak in egg prices is continuing. These facts are

obviously imi)ortant to farmers amd to storers of eggs. Without a graphic

analysis, such imiKDrteuit facts could be easily overlooked. It is not

iDçpossible, of course, to fit a mathematiceüL curve which allows for a

damping of the seasonal swing. But the point is that a siniple graphic

analysis shows what sort of curve is needed.

A graphic method of measuring shifts in the seasonal pattern over

time Is illustrated in figure 11.

- 20 -
SEASONAL VARIATION

Egg Prices, 1925-29 ond 1945-49

160

140

120

lOOffi

80

60

:ENT'^:: : :~:-:--:-:-: :- - -^

^ ^ -

^ ? ^

"i "

1

"^

"1

* t i

^ á Mi^U

^. ' ' ' * ^"3 '•»5

5r " -1945-49ÍÍ i -.L2%

^ j».^,rp___ _______ ^^-L,^ __^^ ^1925-29

^L. d C ^

afc

\' ±

JAN. MAR. MAY JULY SEPT. NOV.

PERCENTAGE OF 12-MONTH MOVING AVERAGE

U.S. DEPARTMENT OF AGRICULTURE NEC. 1319-55(1) AGRICULTURAL MARKETING SERVICE

Figure y

Eggs: Index numbers of seasonal variation of prices

received by farmers, 1925-29 and 1914-5-49 1/

Month 1925-29 191+5.49

January 123 102

February 99 90

March 77 87

April 73 86

May 76 87

June 76 90

July 81 97

August 88 103

September 101 110

October 118 117

November lUO 117

December ikQ IIÍ4

1/ Average of percentages of 12-month moving average.

Agricultural Marketing Service.

- 21 .
Monthly Production of Pork and Prices

Received by Farmers for Hogs

When dlsciisslng cycles, we observed that the economist \isually wants

to analyze the forces that have been responsible for upward and downward

swings. The same Is true with seasonal variation* Some fairly regular

seasonal movements are well-known, such as the low price of eggs In the

spring and the higher price dvirlng the late fall and early winter • However, no two years are Just alilse; sometimes the seausonal swing Is big,

sometimes It Is little.

The accompanying chart shows the average seasonal variation In the

production of pork and In prices received by farmers for hogs* The

seasonal low point In prices comes In November and December when production Is high. As pork production falls off In winter, prices go up

somewhat. The swing In price Is much less mcurked than the swing In production. This Is due essentially to storage, which evens out the supply

of pork to the consumer. In an average year, the price must rise enough

In summer to cover storage costs.

This chart, of course. Illustrates only the averaige seasonal variation for the postwar ye6ü:s. To be most usefxil as a guide to current

marketing operations, we would need to breeds: down these averages to show

how the seasonal variation In hog prices Is affected by different kinds

of seasonal patterns In production. In that way we could forecast more

accurately the seasonal price changes for a current yeeir.

- 22 -
SEASONAL VARIATION

Hogs: Monthly Production of Pork and Prices Received

by Formers for Hogs, Postwar Years

125

100

75

:ENT " : :::—:—"—:"

^C

(^ ^

Y - /

1 r

r ^

\ r

t t

5 r% • i

IÇ Price^. ^.--'' ""s I

\ .^ ^ ^ r ^

î-:r*i^r -r-^ nC^

:iï-r5;;--^^^^;^i^^-ÎÎîS=?5-::::::::::::::::-?-:-:^::::

iiiiiii ::::^5c=!!:::::::!5-: :::::::::::: III'^Îî

T ^r ^

^^^5|. w'^

11 1 1 "^^"""aL

JAN. MAR. MAY JULY SEPT. NOV.

^PERCENTAGE OF 12-MONTH MOVING AVERAGE

U. S. DEPARTMENT OF AGRICULTURE NEG. 1320-55(1) AGRICULTURAL MARKETING SERVICE

Figure 10

Hogs: Index nuinbers of seasooal variation of production of pork and

price received by farmers for hogs, postwar yeaxs

l'îonth Production 1/ Price

January 130 96

FebruELry 94 98

March 97 100

April 91 96

May 9^^ 96

June 97 99

July Ö5 106

August 79 108

September 78 m

October 99 104

November 120 95

December 136 91

1/ Excluding lard.

Breinyer, Harold F., and Johnson, Lucille W. Seasonallty in Marketings and Prices of Meat Animals. U. S. Agr. Mkt.

Service. Uvestock and Meat Situation, Nov.-Dec. 1952, pp. 12-17, illus. (Processed).

- 23 -
Trends In the Seasonal Pattern for ^gge

A convenient way to determine graphically whether there has been a

significant shift over time in the seasonal pattern is to plot the ratios

to trend for all yeeo's against time in a series of charts, iising a separate

chart for each month. In most cases, visual inspection indicates whether

the degree of seasonal variation is significant and whether it has changed

during a period of time. If most of the values for any given month are

consistently above or below a ratio of 1 by a fairly uniform amount for all

yeeurs, it can be assvmwd that a seasonal pattern prevails and that it has

not chcmged significantly during the period. If the charts do not indicate

clearly whether the seasonal pattern is significant and if no change over

time in the pattern is indicated, a mathematical test based on analysis of

variance described in Foote and Fox jj can be applied.

Such a chart for eggs for December is shown in this diagram. In this

it is clear that the nature of the seasonal pattern has shifted over time,

thus verifying the findings from the chart on page 21. In this case, a

linear trend might fit the ratios fairly well. If a linear trend appeared

to be applicable on the other 11 charts, the 12 trends could be fitted

simviltaneously by a mathematical method described by Foote emd Fox. The

advantage of this over a graphic fit is that the trends are fitted subject

to the condition that the constant values add to 1,200 and the regression

coefficients add to zero. Hence, the computed index nxmibers of seasonal

variation for the 12 months for each yeeur addvto 1,200. If a curvilinear

or irregular trend on at least some of the charts is indicated, all of the

trends can be fitted graphiceú.ly in such a way that the trend values for

the 12 months in each year add to approximately 1,200.

jJ Foote, R. J., and Fox, Karl A. Seasonal Variation: Methods of

Measurement and Tests of Significance. U.S. Dept. Agr. Agr. Handb. U8,

16 pp. 1952.

- 24 -
SEASONAL VARIATION

Eggs. Price Received by Formers os o Percenfoge of

12-Month Moving Average, December

PERCENT -it

ft 11 Tll 4Í i \

t\ A it ^ ïï T t

1 Rn <ii.i.jt ju-i - lOU *^f"\-------f--^

^t JC L it ^\.^^\ i 1

3f 3^lt I

ij t

of t -tí- \t A -i»- A —!i- I

t s Ä M PI A

^f^ X J^\í 1 Ji n lOK \ C.D _ S*

_p,<i._-.jt._^ i—*-4U Ax él --it

jtjs— Í 1i

£

iïïÏ JC""—jtl— il r 3*3^ Jt A Í 4 1

1 t

t ^\t>r iiXW 5 1IF XX

j

t JL JE 1 \

\ t t JL \l \ f

Ï 4^á- ir\r\ i ? lin

1925 1930 1935 1940 1945 1950

U. S. DEPARTMENT OF AGRICULTURE NEC. 1321-55(1) AGRICULTURAL MARKETING SERVICE

Figure 11

Eggs: Price received "by farmers as a percentage of

12-month moving average, December, 192U-53

Year Percentage Year Percentage

: Percent Percent

192k 151 : :' 1939 111

1925 IU9 : : 19^ •133

1926 158 19^1 118

1927 IU7 : : 19J^2 lli^

1928 : liiO : : 19i^3 123

1929 IU7 : 19i^U 123

1930 129 : : 19^5 126

1931 : I5U : I9UÓ 111

1932 : 172 : 19^7 122

1933 133 19^+8 in

193^ : 124 : : 19^9 102

1935 125 : : 1950 135

1936 132 1951 115

1937 126 1952 ]01

1938 135 : : 19:3 108

Agricultural Meurketing Service.

- 25 -
SIMPI£ REGRESSION

Relation of Corn Yields to Nitrogen

The so-called "dot chart" Is one of the handiest tools of economic

analysis- Any competent economic analyst draws dot charts and studies

them before putting numbers In a calculating machine*

This Is a fairly typical example of a dot chart. The construction

of the chart Is easy. For example, the data obtained by certain experiments In North Carolina Indicate that with 20 pounds of nitrogen per acre,

an average yield of UI.5 bushels of corn Is obtained. To plot this

observation we simply measure 20 units along the horizontal axis and

then 41.5 units upward and mark that point with a dot. Similarly, for

the other points on the diagram.

Through these points we draw a smooth ciurve to Indicate the response

of com yields to varying amounts of nitrogen. In this case this Is easy

to do because all of the observations (that Is, all the dots) He close

to a smooth curve. This Is because the data are averages obtadned from

controlled experiments. If we had used Individual data obtained from the

survey covering a number of farms in different parts of the State, with

varying soil and moisture conditions, the dots probably would not have

clustered so closely around the smooth curve.

Economists are, of course, partlculeurly concerned with Input-output

relationships. In agrlcvilture a great deal of research has been done on

such matters as the response of crop yields to fertilizer application €uid

on the response of animals to varying amounts ana kinds of feeds.

Mathematlclems have suggested certain mathematlcsú. formulas to measure

such responses. For example, specific formulas have been suggested by

Mltscherllch, Splllman, Cobb, 6üid Douglas. These mathematlcaLL formulas

are all based upon logical considerations and are of considerable Interest

to economists. Still, none of them may show accurately the relationship

which c€ui be seen easily In the dot chart. The economist should not forget logical considerations In drawing freehand Unes or curves. The

curve we have drawn In this case probably represents the relationship at

least as satisfactorily as emy of the mathematical functions. In addition, there may be some question about the logic of any of the proposed

mathematical functions when they are extrapolated far beyond the reuige

of the data. For example, none of the three mathematiccú. functions

mentioned would €L11OW for the i>ossiblllty that excessive amounts of

nitrogen might actually reduce corn yields.

In passing we might note that the curve indicates both decreasing

average returns and decreetsing marginal returns throughout the range of

observations. The graphic derivation of meurginal curves is described

on page 56 .

. 26 -
SIMPLE REGRESSION

Corn: Yield and Quantity of Nitrogen Applied Per Acre *

YIELD IBU.)

' inn \\J\J —

^^ 4 ^

*-

^^^^

7^ ^^^

1 \j ^F^

^^

f^ u^ ^^' —

<j\J RO j^^-i^

, ¿

y^

^'^

0^4 dO^ I]

Q . _-._«__«_«

"Ó 20 40 (

U. S. DEPARTMENT OF AGRICULTURE

so 80 100 120 140 160 180200

NITROGEN (LBS.)

AVERAGES FROM EXPERIMENTAL DATA

NEC. 1322-55(1) AGRICULTURAL MARKETING SERVICE

Figure 12

Com: Yield per acre by specified quantities 01 nitrogen applied

nitrogen applied Yield of com

Pounds Bushels

0 : 24.5

20 hl.5

i+0 52.1

60 61.h

80 73.5

120 86.0

160 92.8

160 9^.0

Johnson, Paul R;, Alternative Functions for Analyzing a Fertilizer-Yield Relationship. Jour. Farm Econ. 35:519-529.

1953.

- 27 -
Food Expenditures Related to Incomes Based on

Averages frcm Survey Data

The first chart on sixnple regression \xsed averages obtained from experimental data. In cases of this kind, it may be possible to control

most of the factors affecting the dependent variable. Because of this

control, the observations may lie close to a smooth curve.

The economist can seldom make experiments. He usually has to relyon data obtained from surveys or from published time series.

This chart shovs the relationship of food expenditinres to per capita

incomes as indicated by two surveys. Here again the observations clxister

closely €u:ound the two smooth curves we have drawn. This does not

necessarily indicate a high correlation between the food expenditures

of individual families and the incomes available to those families.

Actually there is a great variation in food expenditure within a group

of families getting the same income. This variation is covered up in

the averages. This is etll right for our purpose, assuming that we want

to estimate the average food expenditure for families with any given

income. In this case, food expenditure is the dependent variable (that

is, the variable we are trying to estimate).

When using Bxxcvey data of the kind indicated here the economist

relies heavily upon large numbers of observations. The two c\irves

shown on the diagram appear to be reasonably accurate. This is indicated

by the fact that the observations lie very close to the smooth curves.

The analysis shows that each income group spent more for food in 19^1

than in I935.

- 28 -
SIMPLE REGRESSION

Food Expenditures and Disposable Income Per Capita *

FOOD EXPENDITURE ($)

200

100 200 300 400 500 600 700 800 900

DISPOSABLE INCOME($)

,000

AVERAGES FROM SURVEY DATA

U.S. DEPARTMENT OF AGRICULTURE NEC. 1323-55(1) AGRICULTURAL MARKETING SERVICE |

Figure 13

ExpendltvireB far food, per capita, by specified income groups, 1935 and 19ltl

Disposable income

Expenditures

for

food

Per

consumer

unit

Per capita

1935

i/

19^1 1935

1/

1941

Dollars Dollars Dollars Dollars Dollars

Under 500 113 12? 69 91

500 - 999 2^*2 293 104 130

1>000 - 1,U99 370 kk6 132 lb?

1,500 . 1,999 502 529 154 179

2,000 . 2,999 679 73^ 179 206

3,000 - 4,999 982 1,008 209 247

5,000 and over 3,270 2,027 344 354

1/ Most of the data used in income and expenditure studies of the National Resources Committee relate to year beginning July 1935. Son» data, however, cover calendar year 1935» Data shewn here were derived from the studies.

Burk, Marguerite C. A Study of Recent Relationships Between Income and Food Expenditures, /vgr. Econ. Research.

3:87.97, illus. 1951.

29 -
Onion Prices Related to Production

Section A of the chart In this example Is a dot chart shoving the

relation between onion production and prices In the years 1939-52• You

will note that the observations are scattered all around the diagram ana

that some of the highest prices occxirred In the years of medium to large

production. Also, scane of the lowest prices occurred In years of low

production.

This does not Indicate a positively sloping demand curve. It Indicates only that both prices and production Increased during the period

studied. To get a rough Idea of the relation between production and

prices, ve have drawn a line from each observation to eetch succeeding

observation. This Is generally a good practice In dealing with time

series. It quickly shows up any trend In the data and gives a rough

Idea at least of the slope of the curve.

In this particular case, section A suggests that we consider the

relation of year-to-year changes In prices €tnd in production. This

relation Is shown In section B. It appears that changes In production

give a talrly good Indication of e^gpected changes In prices. The

explanation Is far from perfect. For example. If we had used the curve

In section B to estimate expected changes In prices we would have been

almost 40 cents too high In 19^5 and about U5 cents too low In 1952.

A more accurate way of studying the relation between onion prices

and production Is discussed on page 40.

- 30 -
SIMPLE REGRESSION

Onions, Commercial Crop: Production and Average

Price Per 50-lb. Sack Received by Farmers

PRICE ($): PRICE ($) .

L 1 kj LL _ ' Pi. LI± A ATTIIAI HATA ^ü R ^ îrÈAR.TO-YtAR O sn r7 1 1

.w ^^.^ 4/ -,. -.

¿.OU 1 1 1 I.UÜ ^w^ ¡s' rUAKK^FC " 1 '52 . .^^. ...lût. , V«nMl>IV7CD -

1 )

'47 ^ '43^^ 52

O nn ^ Rn _ _^ jk^ . , jL 2.ÜÜ ^. .0Ü ••_ ^^^^

-. N', \43 ."^i ^V- 140 »^^

'v* ^ ^ S ^ÍL I ^ ^!- 5" 1 e ^ S "* ^^ '^I "ñ:^ Q I ^ ^

I.DU ""^~ "^^X 5 ' S J J

> ^. • 4fl ^ ^Ä ^ . 1 i^

b\_ ^V- S\ ^s V V ...

1 nn _St-=-=^b^ h-Ss,- "^"^ - RH- '^^ -M .0Ü jr *: ^: s: L-.OU I~" A ^ k

t 4^ <•_ ?-46 1 ' ' ^ ^

V •50 - 48^, ^

-•40^* ^ ^ ^

RH N inn ^ Il

.DU ^1 -I.UU

•39

r% ^

30 35 40 45 50 -15-10 -5 0 5 10 15

PRODUCTION (MIL. SACKS)

U. S. DEPARTMENT OF AGRICULTURE NEC. 1324-55(1) AGRICULTU RAL MARK ETINC SERVICE

Figure Ik

Onions, cooBterclal crop: Production and average price per 30-pound

sack received by farmers, 1939-52

Year Production Price

Change from preceding year InProductlon Price

Million Minion

sacks Dollars sacks Dollars

1939 36.6 O.U5 —

19^ 32.9 .70 - 3.7 0.25

19U1 31.2 1.10 - 1.7 .40

19^2 3Ô.9 .99 7.7 -.11

19^3 31.3 1.68 - 7.6 .69

19Mf k7.9 1.20 16.6 -.hd

I9i^5 37.7 1.69 -10.2 M

19^6 50.U .89 12.7 -.80

1947 36.7 2.08 -13.7 1.19

19^^ U2.5 1.32 5.8 -.76

19^9 38.8 I.U7 - 3.7 .15

1950 U5.Ö .87 7.0 -.60

1951 : 39.^ 1.67 - e.k .80

1952 : 39.Í* 2.25 0 .58

Agricultural Marketing Service.

- 31 -
Yields of Corn In Michigan and SeptemW-1 CondltloBThe Agricultural Marketing Service estixnates the probable production

of maxiy of the principal crops several months before they are harvested*

Such estimates are based on returns obtained from farmers concerning the

acreage planted and also on the farmers* Judgment of "condition as a percent of normal." The reported condition of crops is one of the best

available indications of the probable yield, assuming average weather conditions between the time of the report and the harvest of the crop*

The Division of Agricultural Estimates makes extensive use of dot

charts and graphic analysis to interpret reported condition* One of the

simplest types of graphic analysis xised is that shewn in the accompanying

illustration* In this case we have plotted the September 1 condition and

the final yield for each year from 1944 through 1953* The 10 observations

lie fairly close to the curve we have drawn* Assiiming that observations

of the future will continue to cluster fairly closely around this cinrve,

it could be used to estimate yields of com in Michigan in future years.

In this case there may be some doubt about the curvatiare of the

regression relationship* Conceivably, it could be a straight line rather

than a cvurve which is concave upward. If we drew a straight line

running approximately through the observations for 19^9 and 19^7^ the

deviations for the years 1952 and 1953 would be greater than ftom the

curve we have drawn* However, this might reflect a net upward trend in

yields of corn*

The way in which such charts are used by the Crop Reporting Board

in making its estimates is described in detail in a publication entitled

"The Agricultural Estimating and Reporting Services of the United States

Department of Agriculture." 8/

8/ U* S. Dept. Agr* Mis* Pub* 703> 266 pp*, illus* 191*9.

- 32 -
SIMPLE REGRESSION

Corn: Sept. 1 Condition and Yield Per Acre, Michigan

50

25

60 70 80 90

SEPT. 1 CONDITION (%OF NORMAL)

0 (BU.) [^ '

'

__

'J -Ír

9-

,

;T P

À' ^

'4

•^ 1_ ^^

^

H

~~~

. .

0> -^~ '^ i

""

•-

^ ^ 18 •50

'A 4 ^

•4;r— . -46 «

f ^ "~

^ 0 ^

iP '""'

1 1

j 1 ~

1

'

j ^^ ^J _^ ^^ ^^ __ .^_ __ ^^ ^^ —J

100

U. S. DEPARTMENT OF AGRICULTURE NEC. 1325-55(1) AGRICULTURAL MARKETING SERVICE

Figure 1^

Com: Condition September 1 and yield per acre, Michigan, 19^4-:)3

Year Condition 1/ Yield

Percent Bushels

V9kk 68 32.0

19^5 ; 77 35.0

19U6 6k 28.0

19U7 60 27.5

19U8 Qk 39.3

19^*9 9J ^9-0

1950 87 38.5

1951 3> U1.5

1952 9^ 50.0

1953 39 ^5.5

Agricultural Marketing Service.

- 33 -
COMPARISON OF TIME SERIES

Food Prices> Consumer Incomes, and Volume of Food Marketings

Economists often work with time series; that Is, with records of

prices, production, and consumption over a period of time. When studying

relations between time series, particularly If several variables are

Involved, It Is a good practice to plot each series before drawing dot

chfiorts such as the ones we have Jxist discussed.

Suppose, for example, that we were trying to discover the factors

which affect retail food prices* Two of the factors that would doubtless come to mind are consumer Incomes and the volume of food marketings.

Before rushing to the calculating machine or even drawing a dot chart.

It would be a good Idea to plot each series as we have done In this

diagram and to stxady the changes which have occurred over a period of

time.

In this case It Is clear that there Is high correlation "between the

food price Index €uid per capita disposable Income. In fact, the relationship Is so pronounced that It tends to overshadow the effect of per

capita food marketings. We might notice, too, that during the war years

from 19^1 to 19if5 the relationships do not seem to be the same as In other

years.

Coc^^arlsons of these three time series sviggest that the correlation

between the average price Index for food auid per capita disposable Income

would be reduced by deflating each series (for exainple, by dividing each

of these by the consumer price Index for all commodities ) • Such a computation would 6Ú.S0 reduce the magnitude of the gyrations to more nearly

correspond to those for per capita marketings of food. The sharp rise

in marketings of food during the war years and subsequent decline, which

appears to have taücen place Independent of changes in the other series,

suggests that the war years be omitted from the cuialysis. If a chart of

this sort indicates pronounced trends in one or more variables, it suggests that the analysis might yield improved results if it were based

on year-to-year changes in the vsüriables.

In some cases, a comparison of time series will Indicate a tlmelag

between changes in one variable emd changes in another. We saw previoxisly that changes in slaughter of hogs occur several months after a

change in the rtitio of hog prices to those for corn.

- 3h -
«w^ ^^ r

•oo

COMPARISON OF TIME SERIES

d Prices, Consumer Incomes and Volume of Food Marketings

Vo ur iv^/ -^v 1

_

_ .Per caoita disoosable income

1 1 1 1 1 1 1 1 1 1 1 1 1 .

lOR - VI 1

~~~

icO _ 1 1 1 1 1 1 A^ ^^ "

. Per caoita marlcetinas of food . &x "~

1 A 1

^^

_¿

i ^^ #^ — — — — >::

N ^ ^ l-J r-^ 1 _ Z

100"■■ " " ~T^c — -^^títssr'*' — — -

(^

Sî>^

a^l^k H-/ iT

75

50

25!

1^ s IËÎ ^ H

^?

i H '1 ,-*' y

»^ # K

L

1 JF Í

T

i • # "1?S H

Y ^

TT

• 'J>^g7~ Retail food-nriff» ¡nr^Av f^U^Jr r •i ^r -, . . ,

f

1 1 1 1 1

\ jy

sr^t-., i-ii«•ir'^y,\ _j^ML 1y ""

,V" il^ \^w. iyjr' V«6-Í7

r

T

_ji.

«PY^y^ J I»' ^•^ .•?• — /

l^

?.. j

I* r'r

1920 1930 1940 1950

U. $. DEPARTMENT OF AGRICULTURE NEC. 1326-55(1) AGRICULTURAL MARKETING SERVICE

Figure 16

Price and marketing of food and disposable income: Index numbers, 1920-53

[19U7-49=100]

Retail

price of

food

Per capita *

: Year

Retail

price of

food

Per capita

Year Marketing ] Disposable ' Marketing ' Disposable

of food income ; of food ; income

1920 ! 83.6 90 52.5 \ i 1937 52.1 83 44.5

1921 63.5 86 ifO.7 : : 1938 U8.U 87 40.7

1922 59.^ 91 U3.U : : 1939 U7.I 89 ^3.3

1923 61.4 93 U9.4 : : 19^0 U7.8 90 46.2

192if : 60.8 93 i*8.9 : : 19^1 52.2 93 55.6

1925 : 65.8 87 50.9 : : 19^2 61.3 101 69.9

1926 68.0 89 52.1 : : 19^3 68.3 105 78.2

1927 65.5 Ö9 51.6 : 19H 67.U 109 85.7

1928 61f.8 90 52.3 : : 19^5 68.9 108 87.2

1929 65.6 88 5^.7 : : 19^6 79.0 106 90.7

1930 62.U 86 UÔ.3 19^7 95.9 lOU 95.0

1931 51.i^ 87 Ul.O : ; 19^ lOU.l 98 103.7

1932 U2.8 85 30.9 : : 19^*9 100.0 98 101.3

1933 Ui.6 85 29.1 : : 1950 101.2 97 109.5

193^ U6.U 86 33.0 : : 1951 U2.6 97 117.7

1935 ^9.7 79 36.7 : 1952 UU.6 98 120.8

1936 50.1 8U 1*1.7 : 1953 : 112.5 101 125.4

Prices from Bureau of Labor Statistics, marketings of food from Agricultural Marketing Service, and income from

Department of Conmerce.

35 -
MLIIiriPIE REGRESSION

Price of Corn Related to Prices of Livestock and Uvestock Products and

SuTyply of Feed Concentrates Per Animal Unit

The graphics of slnrple (2-varlable) regression Is quick and easy. The

graphics of multiple (several-variable) regression Is naturally more ccrapllcated. But, as Bean 2/ showed, graphics can handle these problems, too.

In multiple correlation we try to estimate the expected value of some

dependent variable from given values of two or more Independent variables.

We aissume an additive relation, such as:

^o = ^1 (^l) + ^2 (X2)

€uid our problem is to estimate these fiinctlons.

The diagram illustrates an analysis of com prices (XQ) related to two

independent variables—prices of livestock and livestock products (Xi) and

supplies of feed concentrates per animal unit (Xg)* We knov from theory and

from general observation that high livestock prices tend to be associated

with high prices of com. We also know that large supplies of feed concentrates tend to be associated with low prices of corn. But we want to quantify

these relationships—perhaps to forecast prices of corn.

Section A of this chart shows corn prices and prices of livestock and

livestock products from I936 through 1951* The dots are not clustered

closely eiround any smooth curve--indicating that the simple (2-variable)

correlation is rather low. Before drawing the regression line, we try to

take accoTint of X2. We draw severcú. regressions for subsamples of data,

conmonly called "drift lines." Thus in 1948, 19^9^ and 1950, supplies of

concentrates were from I.03 to I.06 tons. We connect these observations

with a drift line. Similarly we connect the observations for I9U0, 19^1,

and 19^2, when supplies were O.90 tons. After drawing all possible drift

lines, we draw a net regression line the slope of which represents ai>proximately an average of the slopes of the drift lines* In this case, a

straight line happens to be satisfeu^tory. In many caaes, a ciorve woxild

be indicated.

Section B shows how the residuals (dei)artures from the first regression line) are related to Xg. These residuals are clustered closely

around the regression line we have drawn. If a nearly perfect fit were

not given by the dots €uround this line, the process of successive approximation would be used. Foote 10/ has shown that when we use this method

graphically based on linear relationships, the slopes of the successive

approximations tend to converge toward the value that would be obtained

had we fitted a mathematiccO. regression line by the method of least squares.

Graphic multiple regression requires a fair amount of imagination cmd

some practice. But it often shows up important relationships that are not

brought to light by grinding figures out of a computing machine.

9/ Bean, Louis H. A Simplified Method of Graphic Curvilinear CorrelationJour. Amer. Statis. Assoc. 2^1:386-397, illus. I929.

10/ Foote, Richard J. The Mathematical Basis for the Bean Method of

Graphic Multiple Correlation. Jour. Amer. Statis. Assoc. 1^8:778-788. 1953-

- 36 -
MULTIPLE REGRESSION

Corn: November-May Prices Received by Farmers in Relation to Specified Factors

PRIC65U PER BU.) -X^NQ DEVIATION^-X L/tVI/nll Vy 1^ ^ "TV Q ^

..1 1 1 ±

2t>ü ::: SECTION A - — _ — _____>__ 1 ^\^\ €* rà A. 1 UU SE<:T ON B—

/- OAn - - __,2__ RA ^ -

¿UU "i DÜ<í:3¡^. .,,

,^ .J, ^!>*" iiiü-i. .'40 ? -51 'x^ •!- "Mi A

^ ^• ^•so- îs^liï A . '

150 "= _,¿5^.:_i^ 0 -"-""":í¡tíi:::::::::

J_L-,^ p;^:::::::

h-36-i~ -¿1a•^5~w=H?H~h-^ h—H—fj-^-:-iSv;=:^--

100 +-—^^JLi- _-l_l—u -50 H-:::;:Í:;?;;-::

"^^ îki^.z-^tz^.z >J\J -•51-1-1?*—-»',

.37' zip "^ '50- 1 )

JLU^^JV} '48

KH ÎÎC---1- ---

.1 r\r\ _ OVJ i^ 1 UU

/> -

100 150 200 250 300 .65 .75 .85 .95 1.05

PRICES OF LIVESTOCK AND SUPPLY OF FEED CONCENTRATES

PRODUCTS (%OF 1910-14 )-X , PE R ANIMAL UNIT (TONS)-X j

U. S. DEPARTMENT OF AGRICULTURE NEC. 1327-55(1) AGRICULTURAL MARK ETING SERVICE

Figure 17

Com: Price per bushel received by feimiers and related variables, I936-51

Price ]received by farmers (Nov.-May}

Period

beginning Com ; Livestock and

products 1/

: Supply

per

of feed concentrates

animal unit 2/

Cents Tons

1936 • 106 123 0.6;

1937 51 nU .09

I93Ö hh 108 .88

1939 55 107 .37

I9ÍK) 58 122 .90

191^1 7^ 159 .90

19^*2 90 19Í* .90

19i^3 112 196 .85

19i^ 107 206 o'l

19^5 115 215 0

19i^ I3Ö 278 • '^J

19i^7 : 220 305 .86

19i^ 120 285 l.Oif

I9U9 118 258 1.06

1950 155 327 1.03

1951 : 167 318 .97

1/ Index number, 1910-1^*^100. 2/ Year beginning October.

CoŒputed from data in Foote, Richard J. Statistical Analyses Relating to the Feed-Livestock Economy. U. S. Dept.

Agr. Tech. Bui. 1070. 1953. p. 6.

- 37
Production of C^nfornla Bartlett Pteara and Aiigust^TCondltlon

The condition of California Bartlett pears as reported by farmers on

August 1 gives a good indication of probable production. Hovrever, in this

case use of a multiple regression analysis helps to refine the relationship. A reported condition of Ö0 percent in recent years indicates a production considerably higher than we would have expected from the same condition figure 5 or 10 years ago.

Section A of the diagram is a simple dot chart showing the relation

of August 1 condition to production. As in the case of several other diagrams, we have drawn some light lines covering the observations in chronological order* Then we have drawn the indicated net regression line.

Section B of the diagram shows the net trend in production. The

residual (that is, the difference between actual and estimated production

ir Section A) was plotted for each year in succession. Then a line that

fits the points approximately was drawn to indicate the net rate of increase

over the 10-year period.

The production of CeLLifornia Bartlett pears, like that of other fruit

crops, dei)ends upon many things including the number and ages of trees,

changes in production techniques, and similar factors. Many of these

factors are not fully reflected in the reported condition. It is, therefore, essentiaú. when deeOlng with ceirtain crops to consider whether factors

not measured by condition have changed consistently over time. If so, a

net trend, as drawn here, measures the increase or decrease that we would

expect after allowing for the effects of changes in condition throughout

the period.

In handling problems of this type, a number of approaches can be

used. For example, deviations from the line or curve in section A may

be expressed as percentages of production before using them to measure

the trend in section B.

. 38.
MULTIPLE REGRESSION

èartiett Pears: Aug. ï Condition and Production, California

PRODUCTION (MIL. BU.)-X DEVIATIONS-Xo,

I I I MM I I h'

40 50 60 70 80 90

AUG. 1 CONDITION

(% OF NORMAL)-X,

U. S. DEPARTMENT OF AGRICULTURE

1944 '46 '48 '50 '52

YEAR-Xj

NEC. 1328-55(1) AGRICULTURAL MARKETING SERVICE

Figure 18

Bartlett pears: Cnnriition August 1 and prcxiuction, California, I9M1-53

Year ': Condition 1/ Production

Million

Percent bushels

19U4 59 9.2

19^5 83 12.3

19k6 68 11.2

19^7 : 77 12.3

19i^8 : 56 9.i^

19i^9 : 86 1U.3

1950 69 12.7

1951 Ih 13.0

1952 : 79 lif.5

19ÍJ3 : 60 10.3

1/ Percentage of normal.

Agricultural Marketing Service.

- 39 -
Price of Late Onions Related to Production

ana Disposable Income

The data for this diagram, taken from Shuffett, 11/ are expressed as

first differences (i.e. year-to-year changes) In logarithms. The rationale

of this may be foiind In Shiiffett's bulletin cmd need not concern us here.

Graphic analysis will handle logarithms and first differences, as well as

the unmanlpulated data.

The mcdLn purpose of this diagram Is to Illustrate successive approximations to the true regression lines. We have cOready discussed the

graphic determination of the net regression lines • So far, we have tacitly

assumed that one ai^proximatlon is enough. But In many cases the statistician

should try two or more successive approximations.

The original data (here they are the first differences of logarithms)

€ü:e plotted BLB in the regression charts we have already discussed. The

black dots in section A show the joint scatter of production and price.

The heavy line is o\ir first approximation to the net regression of production on price. (Drift lines were drawn, but have been erased to keep

from cluttering up the chart.) Deviations from this line were then plotted

as heavy dots in section B. The solid line through these heavy dots is the

first approximation of the net regression of disposable Income on price.

So far, our analysis is the same as in severed previous diagrams. We

now proceed to make a second approximation. The deviations from the solid

line in section B are now plotted as circles in section A. The dashed line,

drawn through these circles, is our second approximation to the net regression of production on price. Then the deviations from this dashed line are

plotted as circles in section B. A dashed line, drawn to fit these circles,

is our second approximation to the net regression of disi>osable income on

price.

This process can be continued to get as many approximations as needed.

If done correctly, the successive approximations will converge to the true

(least squares) regressions. Ordinarily two or three approximations are

enough.

n/ Shuffett, D. Milton. The Demand and Price Structure for Selected

Vegetables. U. S. Dept. Agr. Tech. Bui. 1105, pp. 38-li3. 19514-.

- IK) .
i

lofe Onions. August-Ap

PRICE ($PER. CWT.) -LO

^AULTIPLE REGRESSION

ril Prices Received by Formers in Relation to Specified Factors

GARITHMS-XQ DEVIATIONS-X Q ^

>v1

.0\J •^0 j^5^ <; »;*C^T rf^KI A - f\r\ -CC^Tir\KI D >tLI UN A"- .OU ScCTlUN b"

i\ •'"

T;-^^ 4- 41. v^^4:'4i

OR '-^ P Jl- oc \z2 '4 0t,^ '

.¿0 ^'^M — .¿0 '39 V |C -^ ^'

•29 lÎ"^ i^^

NM ?JJ^ Q ^ ^ ^

iTp »34 K 34Œ.

Ay .. --.-.MÎJÉL -.

1 M i 1 M M 1 I I 1 Ä M M M M M M M LVf ! 1 yjH I M 1 ,^^í—.

T'39 W 1 1 -^..i -^ f"RTT^-

1 ri^ •36.«_ .'32 ^^^ 0 •35

'3 1 1'^ • *' él --à^ '^8 Ql"i-- ^^^^

-9R "¿0 ^^^30A^^ 0 Ss^: ^ -•.¿0 -9«^ -^T

5ft 1 ^

-R/^ _;>^js_ 0Ü • 5^_^ O USED FOR SECOND APPROXIMATION — -

. . ~

^ '32• N v^ > Il

1

-.20 -10 0 .10 .20 -.15 -.10 -.05 0 .05 .10 .15

PRODUCTION (LBS.PER CAPITA) DISPOSABLE INCOME ($ PER CAPITA)

LOGARITHMS*-X, LOGARITHMS*-X ^

ECHANGE FROM PRECEDING YEAR

U. S. DEPARTMENT OF AGRICULTURE NEC. 1329-55(1) AGRICULTURAL MARKETING SERVICE

Figure 19

Late onions: Average price per 100 pounds received by fanners and related

variables, August-April 1920-Ul

Price

y

: Per capita :

Price

.1 /

Per capita

: Production : Disposable : Production Disposable

Period

begin-

: 1 /

incane

; Period '

; beglntl L/ income

! : First : First I : First : : First : First : First

: xdiffer- : :differ-: , :differ-: :differ- :differ- :differ- n-ÍT|g

: Actual :ence of Actual

; 2/

:ence of: A<=^^ :ence of:

¿/ : loga- :

, iiiiïg Actual :ence of . Actxial

. 2/

:ence of Actiial

3/

:ence of

\ : loga- : loga- : : loga- : loga- : loga-

: rrithms trithms : rrithms : :ritbms :rithms :rithms

: Dollars Pounds Dollars : Dollars Pounds Dollars

1928 : 2.5^ 6.65 •>»— 658 — ; i 1935 1.18 -.06166 8.20 .01673 467 .04607

1929 • 1.30 -.29089 9.08 .13527 663 .00328 : : 1936 .86 -.13738 9.23 .05139 534 .05822

1930 \ .82 -.20013 9.75 .03091 557 -.07565 : : 1937 1.30 .17944 8.36 -.04299 532 -.00163

1931 \ 2.02 .3915^ 6.U1 -.1821Í* U56 -.08690 : : 1938 ' 1.06 -.08863 8.59 .01178 509 -.01919

1932 .5^* -.57296 8.75 .13515 347 -.11863 : : 1939 '

.88 ..08083 10.57 .09009 546 .03047

1933 : 1.28 .37^^ 7.58 -.0623^ 386 .04626 : : 1940 • 1.12 .10474 9.93 -.02713 601 .04168

193"* 1.36 .02633

5

7.89 .OI7UI 420 .03666 : : 1941 2.08 .26884 9.47 -.02060 7tó .09503

1/ Excludes quantities produced in market gardens for sale in nearby cities prior to 1939.

^ Production divided by November 1 civilian population.

3/ Disposable incane at annual rates divided by November 1 civilian population.

Shuffett, D. Milton. The Demand and Price Structure for Selected Vegetables. U. S. Dept. Agr. Tech. Bui. IIO5.

1954. p.43.

41
JOmT REGRESSION

Ascorbic Acid In Snap Beans Related to Storage Time and Temperature

Sometimes the graphic method of multiple correlation Is criticized

as being too flexible. Actually, It Is not flexible enough to describe

at all accurately scMne of the common relationships In physical and economic science» Any linear multiple regression, whether It Is determined

by mathematical computations or by graphics, assimies that the effects of

two or more Independent variables can be added together to estimate the

dependent variable- Joint regression assîmes a more general relationship between the variables. In the 3-variable case, we need to determine a smooth 3-dimenslonal surface describing how one variable changes

in relation to two others.

One technique for doing this is similar to that used in surveying

and grading land. We can forget for the moment that the chart refers to

snap beans. Suppose that the vertical axis measured disteuicès north emd

south, the horizonteil axis measvired distances east euad west, and the

numbers written by the dots on the diagram indicated the elevation of

the land at various points as determined by a sxirveyor's transit. Anyone \ised to maps wovild recognize that the land is level at the left side

of the diagram and that it slopes rather steeply at the right side, going

down hill as we go \ip on the diagram. He would eú.so see that there are

bumps and hollows. If a landscape gardener were going to Improve this

plot of land, he would smooth out the siuiace. In simple regression we

smooth in only one dimension. Here we aire smoothing in two dimensions.

We can describe the general lay of the land by a series of smooth contour lines.

Of course, we are not dealing here with land and contour maps. However, the genereú. problem of Joint regression is that of determining a

series of isoquants. Whatever the three variables may be, an IsoquEuit

will show the combinations of two Independent variables which correspond

to a given value of the dependent veiriable. In the case Illustrated by

the dia^am, it is clear that storage time has little or no effect upon

ascorbic acid in snap beans if they axe held at a temperature of 0^ F.

Regardless of length of storage, the beans contain nearly l6 percent of

acid. As the temperatiure increases above 0, however, storage effects

ascorbic acid more and more. At 20^ F, for exaaiple, snap beans lose ascorbic acid very rapidly. The isoquant labeled 13 indicates that the

beans will contain roughly 13 milligrams of acid per 100 grams with

storage for a very short time at 20 degrees, with storage of h weeks at

about llf degrees, with storage of 6 weeks at 11 degrees, or with storage

of 8 weeks at 8 degrees.

With a little practice anyone can draw Isoquants graphically, as we

have in this diagram, that give at least a general indication of the relationships involved. If the researcher wants to fit mathematicaúL functions, the diagram should sxaggest the kind of function to use. In this

case, the series of isoquants look something like a spircuL staircase,

with the stairs beconming steeper as they go down. The formula for a

spiral staircase is simple enough mathematically if anyone cared to fit

it.

Another technique which is sometimes used to study three-dimensional

relationships is the "isometric projection." Those who are not familiar

with isoqixants may find such projections easier to visualize. But they

are also harder to read accurately.

- 42 -
SI

TIM

Toj:> Beans.

JOINT REGRESSION

Ascorbic Acid Concentration Related to Specified Factors*

t IIM brORACE (NO. WKS.

_ 1 1 1 1 1 1 1

5

f"-

14 , 13 1

al5_ V í¿. s

11

i^

1 1^

8

15.5 ±\ ^ \^^ ..' s i

• 4 IC

9

V

8 .74^6 ^ skZ

J s¡ >.x \^ —^ ^

^

S, ^

s [^ •^

-5.3^

1 V \

k

N V s s S 5

Sw rv

\

^

^

>

S. ^s s k

s ^ s w^ s ^ s

1 > V s. s, s w. S, ^

s

N

S S

6

•15 4 ^ ^lÀ Â > i M3.7 ^^i > V s s s"^7.0^

I \

^^ S "s ^

s ^ *s ^

\

\

V

s ^ s SV

N s

s», >

w

>i w s

^

s^ S

s* ^s

^

V

s ^ s Sr

4

»1 C 7 ^ ., „ ^ k i

I

L 1 .á 1 s ^

^ w^ S 1 n o-J \

k 1 s; "s ^

s

' 9.2-f

TV i s L s ^ "^ k s

S N V s ^ S k

\

^

Sw

N V S

2 15.0 _4 LI . i i 1 4.9 i^ -^

11.44 i IV s 1 'S

\ \ sk

_v N s V

/N- T

-^

V

16.:J v^

''Ó 5 10 15 20

TEMPERATURE(°F)

* NUMBERS ON POINTS AND LINES REFER TO ASCORBIC ACID IN MILLIGRAMS PER 100 GRAMS

U. S. DEPARTMENT OF AGRICULTURE NEC. 1330-55(1) AGRICULTURAL MARKETING SERVICE

Figure 20

Snapbeans: Average concentration of eiscorblc acid per 100 grams,

by temperature and time In storage

Time

:

Concentration with

In degrees

storage temperature

Fahrenheit

In

Btcrage

: 0 ! 5

•

2

:

:

10

t

:

•

20

Weeks : Milligrams Milligrams Milligrams Milligrams

0 16.2 — ...

2 15.0 llf.2 14.9 ll.U

k ! 15.7 16.0 1^.3 9.2

6 15.i* U.l* 13.7 7.0

8 15.5 13.2 12.3 5.3

Snedecor, George W. Application of the Theory of Experimental Design in Biology. Proc. of Int. Statls. Inst.

3:^*^52. 19lf7.

^3 -
LINEAR PROGRAKMING

Ccmblnatlon of ÜVo Farm Enterprises

Programming is the planning of economic activities to maximize income or to minimize costs. In some cases it is reasonable to assxmie that

the input-output relationships are approximately linear. For exainple, if

we know how much seed, labor, and fertilizer is required to grow an acre

of potatoes, we can assume that it will require about twice as much of

each input factor to grow two acres of potatoes by the same process. In

a similar manner, if we know the amount of protein, cedcium, and other

nutrients in a bushel of com, there woxald be twice as much of each nutrient in two bushels of com. These are linear relationships, and in

cases of this kind we can estimate the optimum program by a technique

known as lineeur programming.

The data on this chart show two possible farm enterprises in North

Carolina and six input factors. 12/ To be feasible a ccMbination of inputs must not reqxiire more than the avedlable amoxmt of any resource.

In an analysis of this kind it is convenient first to compute for each

enterprise the proportion of available resources needed to produce some

arbitrary amount of net income. In this case we chose $10,000. For

example, to get a net income of $10,000 from beef cattle would require

h.63 times as much spring land as the farmer has available. The left

scale of the chart represents the proportions of available resoiorces

needed to get a net income of $10,000 from beef cattle. The right scale

shows the proportion of available resources needed to produce $10,000 of

net income from fsOl cabbage. If we had to clioose one or the other of

these enterprises, the choice should be fall cabbetge, since the highest

dot on the right scale is lower than the highest dot on the left scale.

The limiting factor for fall cabbage is September-October labor. To get

an income of $10,000 from feOl cabbage woxad require 2.17 times €is much

September-October labor as the farmer has available. If he used all of

his September-October labor on cabbage, his income would be $10,000 divided by 2.17, or $4,608. This is better than he coxild get from beef

cattle 6ü.one.

However, this farmer could raise his income by combining beef cattle

with fall cabbage. Each of the six lines drawn across the diagram show

the proportion of some resource needed for various combinations of beef

cattle and fall cabbage. The limiting factor for any combination is indicated by the top line at that point on the horizontal scale. A combination that is mostly beef cattle has as its limiting factor fall land.

With combinations including tó to 91 percent fall cabbage, the limiting

factor is production capital. Finally, in combinations that are mostly

fall cabbage and only a little beef cattle, the limiting factor is September-October labor. The mini max point (that is, the lowest of the

maximum points for any combination) indicates that the most profitable

combination of these two enterprises would use about 91 percent of (l)

the available production capital and (2) the September-October labor to

produce fall cabbage. The other 9 percent of these two limiting factors

would be used for beef cattle. To get an income of $10,000 from these

combinations would require almost twice as much of the two factors as are

available. So the best the farmer could get with these two enterprises

would be an income of a little over $5,000.

12/ See King, R. A., and Freund, R. J. A Procedure for Solvimr a Linear

Programming Problem. N. C. Agr. Expt. Sta. Jour. Paper 503? lÖ^p! 1953

S'^r^'JÍ^^ .^^l ^*^ ^^""^^ 9 different inputs needed to can? on eadi of 6 different enterprises. I«
LINEAR PROGRAMMING

Combmotion of Two Farm Enterprises

BEEF CATTLE*ErrLLLLi|| I I I I I I I I | | iFALL CABBAGE*

Fall Land

20 40 60 80 100

RESOURCES USED FOR FALL CABBAGE (PERCENT)

PROPORTION OF RESOURCES REQUIRED TO PRODUCE S10.000 NET INCOME.

U. S. DEPARTMENT OF AGRICULTURE NEC. 1331-55(1) AGRICULTURAL MARKETING SERVICE

Figure 21

Beef cattle and fall cabbage: Proportion of available resources required to produce

vlO,000 net income for a farm, llorth Carolina

rr.-p:.rticn required

Resource

Beef cattle Fall cabbage

Land:

Spring U.63 0

Fall ^.03 0.80

Production capital 3.7Ô 1.30

Labor:

July-August 0 1.08

SeptemDer-October 0 2.17

November-December 0 1.22

King, R. A. and Freund, R. J. A Procedure for Solving a Linear Programming Problem. N. Ca. Agr. Expt. Sta. Jour.

Paper 503. 1953- (Processed.) p. I3.

- 45 -
The Minimum-Cost Dairy Feed

Here Is another diagram that Is usefiil in linear programming. In this

case, we want the leeust cost combination of feeds that will meet stated requirements. The prices of several feeds are given; also such requirements

as totea digestible nutrients and protein. We first compute the proportion

of each requirement that could be supplied by $1 worth of corn, $1 worth of

oats, etc. The net result is shown on the table and plotted on the chart.

We then consider combinations of two feeds that will meet two requirements—those for total digestible nutrients and for protein. For $1 we could

buy any combination lying along a straight line joining two dots. We have

drawn such a line showing combinations of gluten €uid middlings. A balanced

ration would lie on a line through the origin having a slope of k^ degrees.

The point at which this line cuts the line connecting the points for gluten

and middlings indicates a ration mostly of gluten with a small amoiint of

middlings^ It can be shown that this combination will meet the two requirements at less expense than either feed alone. This is tr\xe because (l) the

line Joining the two dots slopes downward to the right and (2) it crosses the

k^'-áegree line. If these two conditions were not met, it would be less

expensive to meet the two nutritive requirements from a single feed. Also,

this combination is less expensive than emy other combination of two feeds

that would meet the two nutritive conditions. This is because no dot lies

above the line (extended by dashes) Joining the dots for gluten and middlings.

If there were a dot above this line it would indicate that the cost would be

reduced by substituting this feed for one of those in the combination. If

the combination of gluten and middlings not only meets the requirements for

total digestible nutrients and for protein, but also meets all other requirements, the combination we have found is the final answer—that is, it will

meet all requirements at less expense than any other possible combination

of feeds. This example is discussed in more detail in €Ln article published

in 1951. 12/

In each of these cases, we have shown only combinations of two enterprises. In lineeur programming we need to study other pairs. This can be

done quickly by the graphic method. This method is more difficult when we

consider combinations of three enterprises, and becomes impossible when we

consider more than three. In such cases we need to use the so-called

"simplex technique." lU/ However, even when we use the simplex technique,

the first two or three steps should be done graphically. The diagrams

shown here are similar to those used by Dorfman. 13/

12/ Waugh, Frederick V. The Minimum-Cost Dairy Feed. Joior. Farm Econ.

33:299-310, lllus. 1951.

Ik/ This is described in Dantzlg, George B., Maximization of a Linear

Function of Variables Subject to Linear Inequalities and Application of

the Simplex Method to a Transportation Problem, ajid Dorfman, Robert, Application of the Simplex Method to a Game Theory Problem. In Koopmans,

TJailing C, ed. Activity Analysis of Production and Allocation. Cowles

Commission for Research in Economics Monogr. 13, pp. 339-373. New York. 1951.

13/ Dorfman, Robert. Mathematical, or "Linear," Programming: A Nonmathematlcca Exposition. Amer. Econ. Rev. U3:797-825, lllus. 1953.

- 46 .
LINEAR PROGRAMMING

The Minimum-Cost Dairy Feed

.5

.4

.3

.2

.1

GESTIRI F PPOII-lMJti llllllliiiiiiiiiiiiiiiii m i i i i i u

- II II 1 Ml Mill M M M M 1 U- - - — :: l^tZZ

Soybean meal ""_, i?"*

I 1r t-rr'H" 1 1 1 1 1 -

i ''.% ^-^^ '^

Cottonseed meal .4i_ S ^^^

^

lili Tt- !S j^ -r -H-h -^^ 3"

Linseed meal ->- G uten.^í!

^^^^ S

-^^ -^n-Midd inas _^^ Bran^^i^j 1 1 1 1 "^j'

ÉÉ ^ V " " ~J" ~ ~

^ ^ VI -,^^ :s + +

^^ '^ ' V AAilo mniTo jp ^ "Oats ' ^ » • •« muixe —

--,¿ J-- ~"5± " —

¿^ • nominy*- —

-,-*' ^»"' 1 '^

^ ^

3^^

3^^

.äZ

I .2 .3 .4 .5

TOTAL DIGESTIBLE NUTRIENTS*

.6

^PROPORTION OF REQUIREMENTS IN S1.00 WORTH OF FEED.

U. S. DEPARTMENT OF AGRICULTURE N EG. 1332-55(1) AGRICULTURAL MARKETING SERVICE

Figure 22

Dedry feed: Proportion of the reqiiirements for protein and total

digestible nutrients s\q)plied "by $1 worth of each feed

; Proportion supplied

Feed

Digestible Total digestible

protein nutrients

Com 0.136 O.khl

Oats .187 .375

Milo maize .203 .495

Bran .321 .423

Middlings .332 .U36

Linseed nkeal .400 .272

Cottonseed meal .k6h .268

Soybean meal .504 .286

Gluten .U12 .395

Hominy .158 .448

Waugh, Frederick V. The Minimum Cost Dairy Feed. Journal Fann Econonica. 33: 299-307, illua. 1951-

- U7 -
INDIFFERENCE CURVES

Beef and Pork

Economists have often discussed the possibility (or the impossibility)

of deriving a set of indifference curves from market data. But they have

seldom tried it. The pure economic theory of indifference curves is important. But in the practical analysis of such problems as cost-of-living

indexes and the incidence of taxes, we need reasonably accvirate lines or

curves derived from market data. In some cases, at least, I believe that

ve can obtain approximate indifference functions by a slurple graphic method.

This is particularly true if the income elasticities for two goods are

approximately equal. I hope to discuss this problem in more detail elsewhere and to Justify the method illustrated by this example.

In each of the sections of the acconipsuiying diagram, each x (which

looks like a short cross-line) indicates the combination of beef and pork

bought per capita in one year. The heavy line drawn through the x shows

combinations that could have been bought with a given expenditure. Taike

1932, for example. (See section B.) A typical consumer, buying the average

per capita amounts, purchased k6.0 pounds of beef and 69.? pounds of pork.

The price of beef was I.596 times the price of pork. So with the same

expenditure he could have bought 1 less pound of beef and 1.596 more pounds

of pork--or 10 pounds less of beef and I5.96 more of pork» The fact that

he chose to buy tó.O pounds of beef and 69.7 pounds of pork shows that he

preferred this combination to the others on the heavy straight line. An

indifference curve must, therefore, be tangent ^to the heavy straight line

at the point marked with an x, and, similstrly, with the other lines and

points on the diagram. Also, we know that no two indifference curves can

cross one another. Our problem, then, is to draw a set of curves in each

section meeting two conditions: (l) Each curve must be (approximately)

tangent to the heavy straight line at the point marked x, and (2) no pair

of curves can cross one another.

The family of curves shown on the diagrams were drawn graphically

following the two rules stated above. The fit is excellent in almost all

years. It is not perfect, as is the case of most statistical work. The

slopes and curvatures must be approximately as we have drawn them. Otherwise they would conflict either with the statistics or with the logical

conditions that must be met. We assume that the consumer prefers larger

amounts of meat to smaller amounts. Thus, the least preferred combinations are those nearest to the lower left part of each diagram; the most

preferred are at the upper right.

The analyses based on data for I92I-3I and I932-I4.I and a similar one

based on 1948-53 indicate that important shifts have taken place in the

relative demand for these items over time. Within any of the three periods, indifference curves can be drawn that are approximately tangent to

the indifference lines at the point for which an observation is available

and that meet the theoretical requirements. However, if the data from

the three periods are combined in a single chart, such lines cannot be

drawn. A research project currently underway in the Agricultural

Marketing Service will attempt to explain why this is so.

- h8 .
INDIFFERENCE CURVES

Consumption of Beef ond Pork Per Capita*

PORK (LBS.) A 1921-31 B. 1932-41

40

BEEF (LBS.)

"^SLOPES OF THE HEAVY LINES ARE PROPORTIONAL TO THE PRICE RATIOS

SEE TEXT FOR METHOD OF DRAWING CURVES.

U.S. DEPARTMENT OF AGRICULTURE NEC. 1333-55(1) AGRICULTURAL MARKETING SERVICE

Figure 23

Beef and pork: Ratio of beef price to park price at retail

and per capita consumption^ 1921-41

: Retail price

: per pound

i :

t

Consumption ]

:

t

: Retail price

: per pound

t

î :

CoQSUBption

•

Year mce ,

ratio ;

: Year

X •

—, Price

* ratio t

«

Beef : Pork

: I

Beef Pork : : Beef :

: :

Pork : Beef

:

1 Pork

: Cents Cents Pounds : : Cents Cents Pounds Pounds

1921 29.3 28.1 1.0U3 54.7 63.9 i : 1932 : 24.9 15.6 1.596 46.0 69.7

1922 2T.7 26.8 1.03U 58.3 64.6 : : 1933 : 21.5 13.9 1.546 50.8 69.8

1923 • 28.8 25.3 1.138 58.8 73.2 : : 1934 î 23.3 18.8 1.239 63.0 63.6 192Í* 29.5 25.3 1.166 58.7 73.0 : : 1935 t 30.5 27.4 1.113 52.5 47.7

1925 30.7 31.1 .987 58.6 65.8 : : 1936 ! 28.6 26.9 1.063 59.7 54.4

1926 ^.h 33.3 .9*^3 59.4 63.3 : : 1937 I 32.5 27.7 1.173 54.4 55.0

1927 32.8 31.2 1.051 53.7 66.8 : : 1938 : 28.7 24.5 1.171 53.6 57.4

1928 37.U 29.5 1.268 48.1 69.9 : : 1939 : 29.5 22.2 1.329 53.9 63.9

1929 ; 39.2 30-3 1.294 49.0 68.7 : : 19»^ : 29.5 19*3 1.528 54.2 72.4

1930 Î 36.2 29.1 1.244 48.2 66.1 : : 1941 I 31.5 24.7 1.275 60.0 67.4

1931 :

Î

30.0 23.7 1.266 47.9 67.4 :

Agricultural Markietljig Service.

1*9-
AVERAGES

Gross Profit from Storage

In economic ancOysls we often want to compute the average of two or

more points on a curve.

In this diagram the curve represents total returns to growers from

sales of varloiis amounts of eggs. In deriving these figures, allowance

was made for the effect of disposable Income on prices of eggs. The

prices shown cure those that might have been expected with Income at Its

average level for the period 191*0-48. A practical question Is whether

It would be profitable to store up the surplxis In periods of large production and to sell It In periods of small production.

Suppose we produced kO billion eggs In one period and 50 billion In

another period. The ret\u:Tis for each period would be shown on the curve.

The average for the two periods would be halfway between these two

points. This average Is Indicated by the dot at the midpoint on the

straight line Joining the appropriate points on the cvrve. In this case

It Indicates a moderate gross profit from storage. That Is, the gross

Income from selling k^ billion eggs In each period would be greater than

the average Income from selling kO billion In the first period and 50

billion In the second. Costs of storage, handling, and any loss In

quality would have to be deducted In order to determine whether net returns would be larger from storage.

It Is esBy to see that there will be a gross profit from storage

If, and only If, the returns curve Is concave downward. The degree of

cxirvature Is an Important Indication of the possible amount of gross

profit.

Of coxirse, this Is only one of the many \ises of averages. The economist-statistician often wants to compute average prices, average cost,

average yield of a crop, and so on. When working with graphic diagrams,

such averages can be conputed graphically with little time or trovible.

There Is no need to read the numbers from the diagram, copy them on a

piece of paper, add them, divide by two, and put the average back on the

diagram. The slxnple arithmetic average of any two points on any cxxrve

can be located graphically by the graphic method explained here.

- 50 -
AVERAGES

Eggs: Gross Profit From Storage *

($MIL.)

1,000

500

L RETURNS, "0^ T""í

^^S^L .) ::¿^^!:: ::: :^ ::: ^ ^s;^

1- =i^^ /- "^^^ 'Añ aß W 1

~ Average '4 3I n^wj ^

A-J ^C

Si '44

SL

^F

1

1

35 40 45 50 55 60

TOTAL PRODUCTION (BIL. EGGS)

iifi RETURNS ADJUSTED FOR ESTIMATED EFFECT OF DISPOSABLE INCOME ON PRICE.

U.S. DEPARTMENT OF AGRICULTURE NEC. 1334-55(1) AGRICULTURAL MARKETING SERVICE

Figure 2k

Eggs: Production, price per dozen received by farmers, and total returns, 191+0-^*8

Year \ Production

Price Total

returns

1/

Î Billions Cents Million dollurs

l9i»o 39.T 36 1,186

19»*1 Ul,9 35 1,225

19^*2 kQ.6 30 1,230

19»*3 5k.5 22 990

l9Mf ! 58.5 16 iQh

19»*5 ! 56.2 20 9^*0

19»^ : 56.0 21 987

19^7 ; 55.k 21 966

19i»Ô ! ^k.9 22 1,012

1/ Adjusted for estimated effect of disposable Income on price. .. ^ ^ TOCO

Data derived from Figure 92 in Thomsen, Frederick L., and Foote, Richard J. Agricultural Prices. New York. iy>¿.

p. hn,

- 51 -
ELASTICITY

Coefficient of Elasticity of Demand

Many economists have trouble with coefficients of elasticity. They

are freqviently concerned with the elasticity of demand—more precisely,

with the elasticity of cons\aniptlon with respect to price. The diagram

shows how this can be measured graphically.

The curved line on the diagram represents an assigned demaxid curve

for eggs. The scedes for consxanptlon and prices wovdd not need to be

shown. They are unimportant, because the coefficient of elasticity is

Invariant to changes in scale provided that the axes start at the origin.

Suppose we want the coefficient of elasticity at the point (p=a, q=c).

We draw the indicated straight line tangent to the demand cvurve at that

point. The elasticity in question is -a/b. For this example, this

equals -35.5 divided by 6U.5 based on the scales shown. In terms of

small squares on the grid, this equals 17.75 divided by 32.¡25- Either

computation indicates an elasticity of -0.55'

This piece of graphics comes from Alfred Marshall. 16/ It derives

from the definition of elasticity »^ = -^ ' -^. Note that -^ =

- -^- , and (by similar triangles) -^^ = ~— Also p=a, and q=c.

So 4^ • -2- = ^ • -2> = -a/b.

dp q b c

Some economists have found the concept of elasticity so difficxilt

that they have used "arc elasticity," or the "average elasticity of a

curve." If the graphic approach to elasticity is \ised, there is little

need for such concepts. The elasticity coefficient shown here is exact

and easy to compute.

We shoxild note that the concept of elewticity applies not only to

demand curves—^but to any curve. When we speak of the elasticity of demand we (usually) mean the elasticity of constmqption with respect to

price. But we might want the elasticity of cost of producing potatoes

with respect to the amoimt of fertilizer used, for example. Whatever

the curve, we can measxire its elasticity at any point, iising the same

graphics as shown here.

16/ Marshall, Alfred. Principles of Economics. Ed. 6, pp. 102-103.

New York. 19^8. First published 1920.

- 52 -
ELASTICITY

Demand For Eggs

CUNbUMPTION (NO. PER CAPITA) -Q^

^ 4r-

—

QOO oUU - ^s»<^"^^^ ^%5à

T ^»^^ s

c ^sjs

w ^

4.no ^ ^ ^^ ** ^ ^ HUU j

^«s. '^"^^••j^L- C j ^ ^•^ n 1

^s

^ ^^

^^^

^^s^

"^v^

1 U ^^ 0 -— ..^ j . *> ^^^

0 20 40

RETAIL PRICE (<

U. S. DEPARTMENT OF AGRICULTURE

60 80 100

PER DOZ)-p

MEG. 1335-55(1) AGRICULTURAL MARKETING SERVICE

Figure 25

Eggs: Consumption per capita associated with given retail price per dozen

Consuinption Price

Number Cents

777 20

621 30

530 40

h69 50

424 60

390 70

362 80

Based on an assumed elasticity of demand coefficient of -0.55. See Foote, Richard J. and Fox, Karl A. Analytical

Tools for Measuring Demand. U. S. Dept. Agr. Agr. Handbook 64. 1954. p. 4o.

- 53
DIFFERENTIATION

Some statist!clans and economists find calculus a difficxilt subject.

Differential calculus is relatively easy if you do it graphically. The

differential at any point on a curve is simply the slope of a tangent

drawn at that point. The tangent can be drawn easily with a transparent

straightedge. The differential, ^ , is the slope of this tangent.

In figure 26, the slope of the straight line is 5-8 (that is, y increases 5.8 units for each increase of one unit of x). In figure 27>

the slope is -0.002 (that is, y decreases 0.002 vmits for each increase

of one unit of x). Thus, in figure 26 -J^ = 5-8, and in figure 2?, -7^

= -0.002. ^ ""

These differentials can be read most easily by drawing the dotted

lines shown on the diagrams. These dotted lines are drawn pairallel to

the tangent and throxigh the origin (the point x=0, y=0). To draw these

parallel lines, place one side of a right triangle along the original

curve, place a straight-edge along another side of the triangle, and then

slip the triangle along the straight edge. With a little practice it is

very easy to draw p8Lra3JLel lines.

The slope of the tangent is the same as the slope of the dotted

parallel line. It is measured by the height of the dotted line corresponding with one \mit on the x-axis. In figure 26 it is 5*8. In

figure 27 it would not be possible to read the height of the dotted line

corresponding to one unit on the x-axis. So we read the height corresponding to 1,000 units. It is -2. So the slope is -2/1,000 or -0.002.

Graphic differentiation is qxiick and easy. It is important in any

sort of marginal analysis.

We have not given data for these charts as the curves are purely

hypothetical and exe shown merely to illustrate the method.

- 5h '
DIFFERENTIATION

VALUb Of T

Lrf x-r'

^

p^ I

>^ 1—

on -

^ ?^ r^ - 1 ' ¿u ^

H ^~

/ ^

^

'

/, í^

^

/

^^

^

^' n 1 ^

i< ^

^ ^ «^

^ *** ^

^

lO -

^ y

1 \j >

/^ ^ 1^

VÍr^'' ^

-,r ^

^ * ^

^

i^

^ ^ ,^ ~ " " " " \^

^^ #^ *

"'J

^

X* \ dy/dx=5.8

1 1 1 1 1 1 1

—1 -H

, tf ^

V

n . r*" ^. 1 1 1 1 M 1 1

\J T

0

U. S. DEPAR

1

TMENT OF AGRICULTURE

12 3 4

VALUE OF X

NEC. 1336-55(1) AGRICULTURAL MARKETING SERVICE

Figure 26

DIFFERENTIATION

VALUE OF Y *^

— ^». ^ .. S..

O !« = ;;":^5;

o -«Sifc.^_ ¿ : : : —.--a-^^ç^ »*.«

"****» '

11 : — ^ ^ * fc * 5^^ ^

* *

0 i^s"""":""""""""-

* * ^ j

^ * * » dv/dx =

-1 — :^=-.,,--

1 U^y^UA1 FPT .,____ -0.002

f

t

.p ._. ±_. ±:±ï^;J : ±„

"Ó 200 400 6

VALU

U. S. DEPARTMENT OF AGRICULTURE

00 800 1,000

E OF X

NEG. 1337-55(1) AGRICULTURAL MARKETING SERVICE

Figure 27

- 55 -
DERIVING A MARGINAL CURVE FROM AN AVERAGE CURVE

Marginal Returns

Often the economist has a demand curve shoving estimates of average

prices corresponding with a range of quantities sold. His problem may

call for an analysis of marginal returns (or marginal expenditures of

consumers). The easiest way to do this Is to find graphically several

points on the returns curve.

Robinson 17/ explained the geometry of this. Briefly, total returns are R = pq. We want ^ = P + °P q* We can take any point on

dq dq ,

the demand curve, such as point A In our diagram (32 pounds, at an average price of Ifl cents), and draw a tangent to the curve at that point.

We then draw a line parallel to the tangent such that It cuts the price

axis at the price indicated by the point on the dememd curve (that is,

at kl cents). This parallel cuts a perpendicular dropped from A at point

B, and the price equivalent of B measures the marginal returns corresponding to the qxiantity sold at point A on the demand curve. Here meurglnal returns are 10.5 cents when 32 pounds per capita are sold.

This is a simple process and can be done in five seconds. With a

little practice you can quickly locate several points on the marginal

3ret\ims curve, and then draw the whole c\arve.

17/ Robinson, Joan. The Economics of Imperfect Competition, p. 30,

London. )ndon. 19^^.33*

. 56 ^
MARGINAL RETURNS

Chicken Meat

PRICE UPER LB.)-p

10 20 30 40 50 60

CONSUMPTION (LBS. PER CAPITA) -q

U.S. DEPARTMENT OF AGRICULTURE NEC. 1338-55(1) AGRICULTURAL MARKETING SERVICE

Figure 28

Chicken meat: Price per pound at retail associated with given

levels of consunption per capita

Price Consumption

Cents

59.1

38.8

35.1

32.1

29.7

Pounds

20

25

30

35

IK)

i*5

50

Regression coefficient based on the reciprocal of em assumed elasticity of demand coefficient of -1.33- See

'oote, Richard J. and Fox, Karl A. Analytical Tools for Measuring Demand. U. S. Dept. Agr. Agr. Handbook 6k. 195^. p. Uo.

- 57 -
Marginal Costs

A marginal cost curve ccui be obtained from a curve of average costs

by the same graphic procedure as that Just explained for marginal returns. This process Is Illustrated In the diagram* In this Instance,

the marginal cxirve will be above the average curve. To find the marginal

cost at point A In the diagram, we erect a perpendicular line at point A

and draw a tangent to the average cost curve at this point. We also

draw a horizontal Une from point A to the cost axis and note the point

at ii^ch this Une cuts the axis. We then draw a line throiagh this point

that Is parallel to the tangent. The cost at which this line cuts the

perpendicular Une Is the marginal cost for the Input represented by

point A.

In the example used here, we show average costs of land and fertilizer per \inlt of output for given Inputs of fertilizer applied to an

acre of land. Point A applies to slightly more than $6 worth of fertilizer. For this amount, average costs per \inlt of output are about

$0.237» Marginal costs, as Indicated by B, are $0.292. As In the preceding example, several points on the marginal curve c€tn be located as a

basis for drawing the entire curve.

If xy is given as a fraction of x, as in these examples, we always

can compute

d X dx

"by this process no matter vb&t x and y represent.

- 58 -
MARGINAL COSTS

Cost of Land and Fertilizer for Varying Inputs of Fertilizer

AV. COST OF OUTPUT ($ PER UNIT) -Y

.40

.35

.30

.25

.20

.15

B=Marginai cost

of 0.292 dollars

0 1 23456789 10 II 12

INPUTS OF FERTILIZER ($)-X

U.S. DEPARTMENT OF AGRICULTURE NEC. 1339-55(1) AGRICULTURAL MARKETING SC3VICE

Figure 29

Total output of a given crop ana cost per unit of output for given inputs

Cost of total input

Total

output

Cost per unit of output ífor -

Land Fertilizer land Fertilizer Total

Dollai's Dollars Dollars Dollars Dollars

10 1 hi 0.213 0.0213 "' 0.2343

10 2 51 .196 .0392 .2352

10 3 56 .178 .0536 .2316

10 k 62 .161 .06k5 .2255

10 5 6k .156 .0781 .23^1

10 6 67 .IU9 .0895 .2385

10 7 68 .1U7 .1030 .2500

10 8 69 .1^5 .1159 .2009

10 9 70 .1U3 .1287 .2717

10 10 6k .161 .1562 .3172

10 11 hS .208 .229^ Mlh

Black, John D. Production Economics. New York. 1926. pp. 317-31Ö.

- 59 -
ROOTS OF A POLYNOMIAL

x3 . 1.221*0 x2 + 0-3695 X . 0-0183 - 0

It may seem strange to discxiss the roots of a polynomial in a handbook dealing with graphic analysis. Roots of polynomials are used mainly

in high-powered mathematical studies dealing with such things as canonical regression, component analysis, and cyclical variation. But graphics

can help, even in these studies.

We have included this diagram to illustrate the use of graphics in

connection with more elaborate mathematical techniques. The particular

polynomial is taken from Tinter. I8/ Tinter was dealing with a problem

of canonical regression. The largest root of the above equation indicates the squared correlation coefficient. We shall not bother to explain

how the equation was obtained. We are concerned only with computing its

roots—and especially its largest root.

The roots of a polynomial are values of x which satisfy the equation.

There are many mathematical tricks for discovering such values of x. But

the graphic method illustrated here is practical and easy.

We simply plot several values for x. Thixs if x=0, the polynomial

equals -O.OI83; so we plot y=-O.Ol83 corresponding to x=0. If x-0.1,

the polynomial equals 0.007^; so we plot y=0.007^ corresponding to x=0.1.

We proceed to compute several points on the cxarve, y»x3 - 1,221^0 x2 + O.3695X

- 0-0183. When we have enough points, we draw a curve through them.

Wherever this curve crosses the x-axis, it indicates a real root. In

this case, the roots are approximately O.O6, O.38, and O.78. The canonical correlation is approximately equal to the square root of O.78.

We could locate any of these roots more exactly by blowing up the

part of the diagram near the root. Thus, we could draw a new diagram

for the part of the curve between x=0.76 and x=0.80, plot the cvirve on a

blown-up scale, and compute the largest root more accurately. This

could be repeated vintil we obtciined as many significant figures as

wanted.

As a guide to the paxts of the curve that must be plotted, we know

that there must be as many roots as the degree of the c\irve. Here we

have a third degree polynomiCLL, so we know that there must be three

roots. Once we have located them, our job is finished. Sometimes we

have multiple roots (that is, two or more roots at a single point) or

imaginary roots. These also can be located by graphic means but these

topics 6u:e beyond the scope of this handbook.

18/ Tinter, Gerhard. Econometrics. New York. 1952. Taken from

equations (18) on p. 119> letting x=X^

- 60 -
ROOTS OF A POLYNOMIAL

y = X^- 1.2240X'- 0.3695X - 0.0183

VALUb Oh Y

f¿ ^^

^ "S^

n 1 Tí2 - Ai- •

.U 1 7 ^t /

Jf \

If

_T V T

M- k^ ^

o '~L j*

/I

yj l*^ T?^- — —" — " " " /J — —"- — — — —— —

y V

j -0.06 0.38 \ ^f "0.78

- n 1 jt s #

-.U II s 3

# \ ^

r ' ^ 2

f~ S J\ (\o 'X- § ^

-. Ufc N N^*^ - — —

~

- 04 - - --

"0 .1 .2 .3 .4 .Í

VALUE

U.S. DEPARTMENT OF AGRICULTURE

) .6 .7 .8 .9 1.0

: OFX

NEC. 1340-55(1) AGRICULTURAL MARKETING SERVICE

Figure 30

Values of a third-degree polynomial, y, at specified levels of x 1/

X y

0 \ -0.0183

.1 .007U

.2 .01U6

.3 .009U

M \ - .0023

.5 - .01U6

.6 ! - .0212

.7 \ - .016»*

.8 ; .0059

1/ y = x^ - 1.22140 x^ + 0.3695 X - O.OIÔ3.

Data conipiled using equations (18) as a basis and letting x -K . Tintner, Gerhard. Econometrics. New York. 1952.

p. 119.

- 61 .
SOLUTION OF SIMULTANEOUS EQUATIONS

Supply and Demand Curves for Winter Tomatoes

We end this handbook with another vise of graphics as an aid to

mathematical computation. Statisticians often must solve two or more

equations simiataneously. Various methods of solution are available,

including the popular Gaixss-Doolittle technique. But the equations can

also be solved graphically. The diagram illustrates only the solution

of pairs of equations. It is possible to solve any number of equations

graphically by a process very similar to the Ga\iss-Doolittle method. But

we shall not explain the procedure here. 19/

To solve any pair of eqxxations, we substitute several successive

values of x in each eqxiation, compute the corresponding values of y,

plot the value of y corresponding to each value of x, and draw a smooth

c\arve through the observations. When these operations axe performed for

each equation, this gives us a pair of curves. Wherever the two curves

cross one another, there is a solution of the two equations.

Any pair of linear equations will have one, and only one, real solution—-except in the extreme case where the two lines are identical or

parallel, where there are infinitely many or no solutions, respectively.

Quadratic equations have \xp to foxir solutions to a pair of equations,

depending on how they are situated one to another. For equations of any

degree, solutions are real wherever the curves cross one another; otherwise they are Imaginary.

In the linear equations shown here we have a supply curve and a

demand curve for winter tomatoes. 20/ The scales in this chart refer to

logaurithms. The siq)ply c\arve shows the quantity of tomatoes that will

be imported with given prices. It is q\iite steep, indicating that large

changes in prices axe required to have much effect on imports. The demand curve shows the relation between domestic prices and the qxoantity

imported. It is highly elastic, reflecting the high degree of competition between domestic auad iniported tomatoes. The values in 1952 of the

other factors that affect imports of tomatoes and their prices have been

combined with the constant velues in the respective equations to give

these two eqxiations that show directly the simultaneous relations between

imports and price. Since none of the coefficients in these equations

differs significantly from zero, little confidence should be placed in

their econcHnic meaning. However, they serve as a good example of the

simultaneous solution of a pair of eq\iations.

Linear relations are shown for this example because these are common in economic analysis. Graphic solutions, however, axe much more useful for approximating simultaneous solutions for more complicated equations.

12/ The method is described in Maxfield, John E. and Waugh, Frederick Vo

A Graphic Solution to Simultaneous Linear Eqxaations. Math* Tables and

Other Aids to Computations, 5:2U6-2lf8, illus. 1951.

20/ This analysis is taken from Shuffett, D. Milton, The Demand and

Price Struction for Selected Vegetables. U. S. Dept. Agr. Tech. Bui.

1105, pp. 107-108. 1954.

- 62 -
SOLUTION OF SIMULTANEOUS EQUATIONS

Supply and Demand Curves for Winter Tomatoes *

PRIttl» KbR bUJ-LOGARITHMS'^r

I

-1—1—\— 1 . 1 . . . . 1 1 1 1 1 1 1 1

/

1— 7

ft --

^^^ ^^^ ir

\J " "7^ — "

1 1 1 1 /

L_ ^ ^

I 1 ! I T T

^

r^

- 09 - Q = 0.049+0.27P à /

".Vfc 3r 1 1 ... / >

/—

>L J

r~

\ T

-04 - V ^z •w^t X "^ 3iT- ^

-.06;

-.08-

- lO :

P =-0.073-0.180^ NHr

' z

—1

J

1 1^' 1 ^ ;ï r-^

^

E

r

"~^

«

- — ^ c —1 — i

-^K.^ — ^ - ;:

— — — —'

À r s ^ =dS

/

>

A

-:02 Ô .02 .04 .06

IMPORTS (LBS. PER CAPITA) -LOGARITHMS °

WHEN THE OTHER FACTORS THAT AFFECT PRICES AND ItAPORTS ARE AT THE

SAME LEVEL IN RELATION TO PRECEDING YEAR AS IN 19S2

Ö CHANGE FROM PRECEDING YEAR

U. S. DEPARTMENT OF AGRICULTURE NEC. 1341-55 ( 1) AGRICULTURAL MARKETING SERVICE

Figure 31

Winter tomatoes: Associated prices received by farmers and quantities

imported, change from preceding year

Demand equation S\jpply equation

Price 1/ Quantity ; Quantity 1/ Price

Logarithm Logarithm : Logarithm Logarithm

-0.071 -0.02 : 0.022 -0.10

- .073 0 ; .027 - .08

- .077 .02 ; .033 - .06

- .080 .OU \ \ .038 - .OU

- .084 .06 ; \ .OU - .02

1/ When other variables that affect prices or imports, respectively, are at the same level in relation to the preceding year as in 1952.

Shuffett, D. Milton. The Demand and Price Struct\ire for Selected Vegetables. U. S. Dept. Agr. Tech. Bui. 1105- 195^.

PP- 107 and 108.

63