/A 1

GRAPHIC

ANALYSIS

AGRICULTURAL ECONOMICS

Agriculture Handbook No. 128

UNITED STATES DEPARTMENT OF AGRICULTURE

Agricultural Marketing Service

Washington, D. C.

■■■■■■■■■■■■■■■■■■■■■I
COHPIENTS

Page

Introduction • 1

Frequency distribution 2

Cumulative frequencies • 6

Lorenz curve 8

Trends 10

Cycles ...*.. Ik

Seasonal variation l8

Simple regression 22

Comparison of time series 32

Multiple regression ^ 3k

Joint (3HÜmensional) regression ,, k2

Use of isoq\mnts to study joint regression kk

Indifference curves k6

Linear programming , ^ \ 48

Linear programming in three dimensions 52

Averages • ^k

Elasticity , 55

Method of determining most profitable output 58

Differentiation , 60

Deriving a marginal curve from an average curve 62

Roots of a polynomial , ^ 55

Solution of simiiltaneous equations 68

Issued July I957

This publication supersedes Graphic Analysis in Economic Research^

Agriciature Handbook No. 8k,

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

Printing Office, Washington 25, D. C. - Price kO ôents
GRAEBIC ANALYSIS IN AGRICULTURAL ECONOMICS

By Frederick V, WaiJgh, Director

Division of Agricultural Economics,

Agricultural Marketing Seorvice

INTRODUCTION

Since Handbook Number Sh^ "Graphic Analysis in Economic Research," was issued in

June 1955^ I have received many suggestions for improvement. Also, I have been pleased

to see that a large number of agricoaltural economists (and also many non-agricultural

economists) feel that a heuidbook of this kind meets a real need.

When oi2r supply of the original handbook became exhausted, we decided to rework it

and get out a new handbook—the present volume—rather than simply republishing the old

one. I wish I could thank here everyone who has waáe a useful suggestion. But the

list would be too long. I would like especially to thank the staffs of the Departments

of Agricultural Economics and Rural Sociology of the letnd grant colleges. In answer to

my request, practically all of these departments sent useful suggestions, many of which

have been used. Some of the principal suggestions are acknowledged at various places

in this handbook. I am also happy to acknowledge the important help of three members

of the Division of Agricultural Economics. íRiey are Richard J. Foote, Hyman Weingarten,

and James R. Donald.

Graphic analysis has a long and honorable history in agricultiiral economics research. Back in the 1920's, it was perhaps the principal research tool used by the

former United States Bureau of Agricultural Economics and by the land grant colleges.

In recent years less attention has been given to graphics. Agricultural economists

and statisticians have become intrigued with new mathematical methods and with improved

calculating machines. These are powerful tools and definitely have a place in economic

research. Nevertheless, my own view is that graphic analysis is an indispensable tool

which shoTild be used right along with the newer and fancier gadgets. In my opinion,

there is an unfortunate tendency, especially among so-called " économetricians," to be

satisfied with a purely mechanical analysis. Good research is not simply a matter of

recording various statistical series on tape, feeding them into an electronic computer,

and getting back a lot of nxmibers computed to 6 or 8 "significant" figures. First, a

capable economist must understand the data he uses. He often must work with estimates

which are significant to only 2 or 3 digits. Second, he must have a thorough and basic

understanding of the nature of the relationships between the variables he is considering. To do this, he must understand economic theory and he must also be able to see

the nat\ire of the relationship shown empirically. In many cases the economist cannot

assume linear relationships, for example.

As I see it, the greatest value of graphics in economic research is in making a

quick, preliminary analysis of a problem to determine which variables to use and the

general nature of the relationships. For many practical purposes, the graphic analysis

alone is fully satisfactory. In other cases, the economist will want more precise

measures. In these cases, he will want to fit some sort of mathematical function to

the observations. I believe that graphics is an indispensable tool for choosing the

sort of function to fit. I believe that the neglect of graphics has frequently led

economists and statisticians to choose inappropriate kinds of functions.

To be sure, graphic analysis has sometimes been misixsed. So has any kind of statistical analysis that can be named. Whatever tools are used, there is no substitute

for so\ind judgment and common sense. Without this, the economist is going to get into

trouble anyway. If he has reasonably good Judgment, I believe that his best approach

to economic research will be a combination: Using graphics in the preliminary analysis

of a problem, then more elaborate mathematical methods to pin down results with greater

precision.
FREQUENCY DISTRIBUTION

Number of Dealers Reporting Various Prices Paid by

Farmers for Laying Mash, September l^k-S

The economist often deals with averages. For example, he may be analyzing the average price received by farmers for wheat or the average price paid

by farmers for some item used in production or in farm family living.

He must remember that averages often cover up important information. To

understand the meaning of the data he uses, the economist needs always to have

some \mderstanding about the degree of variation aroxmd the average. In some

simple cases he knows this in a general way by observation. If he were told

that the average height of men in a large group was 5 feet 10 inches, he woiild

not expect many of them to be less than 5 feet or more than 7. But many

economists work with data obtained from various sources. They often know

little about the kind and amount of dispersion to expect. Probably economists

and agricultxoral statisticians ought to do more work on this subject. And

they should publish their findings so others could judge the reliability of

averages and the variation to expect arotind them.

B. Ralph Stauber of the Agricultural Estimates Division, Agricultural

Marketing Service, supplied the data used to draw the accompanying diagram. 1/

It is a so-called "frequency distribution" of prices paid by farmers throxaghout the United States for laying mash in September, 19ÍÍ-9. The average price

is a little more than $1|..50 a hundred pounds. The range is from $3.4o to

about $6. This range is, of course, due to many things, including geographical differences that reflect freight rates and differences in the ingredients

used in the mash.

Each of the bars in this diagram shows the number of dealers reporting

prices paid by farmers within the several ranges shown in the table beneath

the diagram. I have drawn a smooth curve representing a judgment as to the

general nature of the observed distribution. Note that I have not bent the

cu3rve to make it go throx;igh the midpoint of each bar. Rather, I have drawn it

smooth, to show the general nature of the distribution.

This distribution appears to be fairly near what is commonly called a

"normal distribution." When this is true, the arithmetic average, the median,

and the mode all come at about the center of the distribution of prices. This

is somewhat of an accident. We .shall see in the next diagram a case in which

the curve is far from normal in the technical, statistical sense.

1/ 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.

- 2 -
FREQUENCY DISTRIBUTION

Laying Mash: Prices Reported by Feed Dealers, September 1949

Kin HFAIFR S — — — "-~ — —

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4 5 6

PRICE PAID($ PER CWT.)

U. S. DEPARTMENT OF AGRICULTURE NEG. 3976-57(3) AGRICULTURAL MARKETING SERVICE

Figure 1

Laying mash: Frequency distribution of prices i)aid by farmers per hundredweight

as reported by feed dealers, Septender 19ÍÍ-9

Price Dealers reporting

Dollars

3.375 - 3.62if

3.625 - 3.8?^^

3.875 - ^-12^

4.125 - 4.374

4.375 - 4.624

4.625 - 4.874

4.875 - 5.124

5.125 - 5.374

5.375 - 5.624

5.625 - 5.874

5.875 - 6.124

Total

Nuniber

12

136

302

652

808

715

486

183

82

16

3

3,395

Data supplied by B. R. Stauber, Agricultural Estimates Division, Agricultural Marketing Service.

- 3 -
Corn Acreages in Sample Area Sega^nts Enimerated

in 12 Southern States^ Jime 19^5

Actxmlly^ the so-called "normal ciirve" is a rather unusiml

phenomenon in economic research^ although it my frequently apply

fairly veil when we consider resid\aals unexplp-ined by a statistical analysis. The agriciiltural economist often works with initial

data which are skewed. He may need to know something about the

nature and degree of such skewness. One of the best ways to find

out is to plot the frequency distribution.

^The facing char«t is based upon data supplied by Walter

Hendricks of the Agricultural Estimates Division, Agricultural

Marketing Service. It summarizes the results of a survey of corn

acreage made in 12 southern States in 195^. The survey covered

623 segments of a sample area. The figures were tabvilated to

show how many of these segments reported corn acreage of 0 to 19^

how many reported from 20 to 39^ from kO to 59^ s-nd so on. The

figures are shown in the table below the diagram.

As in the preceding chart, the height of the bars indicates

the nuira>er of segments reporting corn acreages within the indicated ranges. I again have drawn a smooth curve, attempting to

describe the general nature of this distribution. The curve is

intended to rvin approximately through the mid-point of the top of

each bar. This smooth curve is an estimate of the actual distribution of com acreages in the southern States. The variations

may be due to errors in sampling and in reporting.

In this case, the average reported acreage of corn was about

49. However, the curve is so badly skewed that k9 acres is far

from either the median or the mode. The largest nimber of segments reported less than 20 acres of corn; many reported no corn

acreage at all.

This is an extreme case of skewness, but the agricultxiral

economist often must use data that are noticeably skewed. He

should be aware of the kind of distribution he is dealing with.

He can inform himself of this in-a few minutes by the sort of

analysis we have shown.

The shaded area represented by the bars in the diagram is

often called a histogram. The histogram is a simmary of the observed facts in the sample. The smooth curve is an estimate of

the distribution in the statistical xiniverse.

For some purposes the statistician may want to fit some form

of aa'Uiematical c\arve to the data in the histogram. However, it

is a good idea to draw the histogram and graphic curve first, before deciding what sort of mathematical curve to try. Don't try

to fit a normal curve or a Poisson curve to any old data. Look at

them first.

^ h ^
Cot

FREQUENCY DtSftfilPlON

n: Acreage in 623 Segments of a Sample Area, 72 Southern States, July 1956

NO. SEGMENTS

j

I

±

300

225

150

75

°;

U. S. DEPA

3¡

E»l M M 1 M 1 M M

KM 1 M M M I 1 M 1 .^ .^ .__ „,„ _ _^ —«■■■-

K^ 1 T

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$%%;?- ■■ ■■ %^%/u ■■ ■■ íí»m^ ■■ ■■ %%!%í^^«HH ■■■

w/yy///jy/.'¿'^w. ■■■

'/MV/y/z/M'/y;:' ■■

) 100 200 300 400

ACRES

RTMENT OF AGRICULTURE NEG. 3978-57(3) AGRICULTURAL MARKETING SERVICE

Figure 2

All-com acreSige: Frequency distribution in saiaple area segments

émanerated, 12 southern States, June 195^

Acreage ; Se^pnents : : Acreage Segtoents : : Acreage : Segments

Acres KuBà>er : : Acres : Number : : Acres : Kuffitoer

0-19 308 ; : 1^-139 ... 10 i i 2lfO-259 : 1

20-39 ..- 115 \ : lUo-159 ... ! 6 :' i 260-279 : 2

iiO-59 .... 61 \ : 160-179 ... i 8 i : 280-299 : 1

60-79 ....' kl ; : 180-199 ... ! 9 i : 300-319 .... : 1

80-99 ....' 25 : : 200-219 ... • 5 i *: 320-339 .... : 3

100-119 ...< 19 : : 220-239 ... 1 : : 3^0 and over .

: Total

; 1/1

*: 623

1/ 456 acres.

Data sxï^lied by V^lter Hendricks, Agricult\aral Estimates Division, Agricult\iral Marketing Service,

- 5 -
CUMUIATTVE FREftUESKiCIES

Percentages of Families With Inccanes

Below Stated Levels

Economists and statisticians are concerned vith many

kinds of frequency distributions. The particular dist^rihution shown on the diagram refers to percentages of

families with various incomes. In this case we have shown

a cumulative frequency curve. Thxas, instead of showing

the percentage of families with incomes from 0 to $1,000,

from $1,001 to $2,000, and so on, we show the percentage

with incomes "below $1,000, below $2,000, and so on.

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 $6,000. For

incomes above $6,000, a larger class interval was used.

With unequal class intervals, it is awkward 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 frequencies. The plotted

data all lie almost exactly along the freehand line we

have drawn.

Several mathematical functions have been proposed

and used to describe the distribution of incomes. Some

of these, like the Pareto curve, are purely empirical.

Others, like the Gibrat ciorve, are based upon logical

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
CUMULATIVE FREQUENCY

Percentage of Families With Incomes Below

Specified Levels, United States, 1954

PERCEN

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INCOME

U.S. DEPARTMENT OF AGRICULTURE

8 10 12 14

: ($ THOUSANDS)

INEG. 3977-57(3) AGRICULTURAL MARKETING SERVICE

Figure 3

Families: Percent€ige with personal income below specified

levels. United States, 195^

Income level : Distribution

Dollars Percent

Under:

1,000 2.7

2,000 9.9

3,000 : 20,0

4,000 : 3^.2

5,000 : 50.6

6,000 : 65.3

7,500 : 79.9

10,000 : 91.4

15,000 : 9^ ^5

Survey of Current Business. U. S. Dept. Commerce. -June 1956. p. 15-

- 7 -
LORENZ CURVE

Percentage of Faitiilies in the United States with Personal

Incomes Belov Stated Amounts and Percentage of Total

Personal Income Obtained by These Families^ 19^4

Lorenz curves are often used to analyze the distribution of incomes. The chart presented here uses a Lorenz

ciarve for this purpose. Obviously such curves coiild be

used to analyze any kind of distribution, such as those

shown in the first two charts in this book.

The distinctive feature of the Lorenz curve is that

the data on both tht- x~axis and the y-axis are plotted as

percentages of the total. In this case, for example, the

U. S. Department of Commerce figxn-es indicate that in 195^

2.7 percent of the families in the United States had personal incomes of less than $1,000. These families obtained

0.2 percent of the total personal income. The table below

the chart shows similar percentages for families with

incomes below $2,000, below $3>000, and so on. Each pair

of percentages is plotted on the chart and indicated by

a dot.

The dotted line on the diagram indicates what the

distribution would be if all families got the same incomes. The area between the dotted line and the curve

is a measure of income inequality. If we should plot a

series of such curves over a period of years, changes in

this area would indicate whether this ineqimlity is becoming more or less.

Note that the Lorenz curve can be used to plot data

that are grouped by any kind of class interval, whether

equal or imequal. As in the preceding chart, the first

six groups of families are classified into income ranges

of $1,000. Above $6,000, the class intervals" are wider.

Another distinctive feature of the Lorenz curve is

that it can be read up, down, or sidewise. Reading up,

for example, we might estimate that the lowest ^0 percent

of the families received 19 percent of the income. Reading down we would find that the top 10 percent of the

families got 30 percent of the income. Reading from left

to right, we find that 20 percent of.the income was obtained by the lower 4l percent of the families. Reading

from right to left we see that 20 percent of the income

was obtained by the top 5 percent of the families. These

are just a few illustrations of the many uses of this

ingenious form of curve.

8 -
LORENZ

Families Ranked by Personal

CURVE

Income, United States, 1954

PERCENT OF INCO 1

ME ^

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4^^ T

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OÜ 4* 1/

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on ^ -.4 ^

ZU / ^z -

~^ ^w —3 ^^

z. ^2

n 4-*

"0 20 40 60 80 100

PERCENT OF FAMILIES

U.S. DEPARTMEKT OF AGRICULTURE NEC. 3986-57(3) AGRICULTURAL MARKETING SERVICE

Fig\:ire If

Familles: Percentage with personal incoae below specified amounts and percentage

of total personal income obtained by these families. United States, 195^

Incomes less than ;

Percentage of—

Families Income

Percent Percent

$1,000 2.7 0.2

2,000 9.9 2.1

3,000 20.0 e.k

U,000 3^.2 14.8

5,000 50.6 27.2

6,000 : 65.3 i^O.7

7,500 : 79.9 57.0

10,000 : 91.^ 73.3

15,000 : 96.5 83.7

^ao . • • : 100.0 100.0

Survey of C\irTent Business. U. S. Dept. CoBunerce. June 1956. p. 15'

- 9 -
TRENDS

Population Trends in the United States

b^; Decades, lB00-19$0

The economist is often concerned with time trends. He wants to find out

how some variable has been increasing or decreasing over a period of several

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

the United States or the rate of decline in the nuntoer of farm workers. In

such cases he will want to disregard minor fluctuations due to errors in the

data or to teiirporary disturbances. 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 18OO. It is a simple matter to draw a freehand

curve describing the trend. Ordinarily, at least, population does not change

abruptly except by major wars, serious epidemics, or a sharp increase in immigration. If we plot the data for each year, or decade, we can usually draw a

smooth curve nmning nearly through the points we have plotted. In this case,

departures from the curve coiad well be due to errors in estimating the population. It sho\ild be noted that even official estimates may not warrant the

naive faith in their accuracy that sometimes prevails. We, as economists, are

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 reluctant to

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

even though we know that the Bureau of the Census official figure of

150,697,761 persons for I950 (or that for any other year) may not be exact.

All too often we do not even take the trouble to understand what the publisher

has to say about the known, or estimated, amoimt of possible error in his

estimates.

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

fit some kind of mathematical function, such as a logistic curve. Our advice

woiad be to draw a freehand curve first. In this case, it is doubtful if any

mathematical fimction would give a better description of the trend than oxor

freehand line. A mathematical curve might have some advantage when comparing

trends in population in several different countries. If the same type of

function were fitted in each case^ results coiild be summarized 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 censixs data were

available (so that we did not have the last observation on the diagram), many

popiaation experts drew an S-shaped cxirve indicating that the rate of growth

had started to flatten. When this type of curve was extrapolated it sviggested

that the population would become stationary^ or even decrease, by i960 or

1970. Such an extrapolation now looks doubtf\il in view of the census figure

for 1950.

10
TRENDS

U. s. Population By Decades, 1800-1950

125

100

75

50

25

PERSCNS """ " "" iiiiiiiiiiiiiiiiiiiiiiiiiiiiy^

: « :: : " :" " ^i^z zp

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- I 4Í " J^

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j r

^

^^J Ü — «-.«---«.~"

^

ilr

. ? ""

J ÍÍ 2

1800 1820 1840 1860 1880 1900 1920 1940

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

Figure 5

Population: United States, by decades, 180O-I95O

Year [ Population \ ; Year Popvilation

: Millions : Millions

1800 : 5.3 ! ! 1880 50.2

1810 Î 7.2 ; : 1890 62.9

1820 • 9.6 ; : 1900 76.0

1830 12.9 : : 1910 92.0

I8Í4-O 17.1 : : 1920 105.7

1850 23.2 i 1930 122.8

i860 31.i^ :' : 19^1-0 131.7

1870 38.6 ! ': 1950 ...: 150.7

Bureau of the Censtis.

- n -
Volijme of Agricultural tfeirketings

It is easy enough to draw a chart i-^presenting the

growth of population^ because population tends to grow

at a steady rate* It inay be upset a little sometimes by

such things as wars, depressions, and epidemics. But in

spite of such factors, estimates of population tend to

lie fairly close to a smooth curve. In metny cases, however, the agricultural economist must work with data that

do not lie along any sort of smooth trend.

A case in point is the index of volume of agricultural marketings. This index is plotted on the accompanying diagram for each year from I9IO through I956.

Quite evidently there has been an upward trend in

agricul.tural marketings. However, this trend has not

been steady. For example, the chart shows that from

1925 to 1935 marketings did not ^increase but fluctixated

around an index of about 70.

In a case of this kind the agricultural economist

needs to use a good deal of Judgment in drawing a trend

line. The first edition of this handbook exhibited these

same data with a graphic trend. The trend shown in that

edition of the handbook was properly criticized by

several able economists. I have redrawn it to take account of their criticisms.

In some cases the statistician may want to compute

a mathematical trend. Before doing so, he would be well

advised to draw a freehand trend to indicate what sort

of mathematical function should be used. In this case,

for example, a straight line would not adequately describe

the trend of the data. A third-degree parabola might give

a fair fit but would be an arbitrary sort of trend and an

extremely bad one to extrapolate into the future. In

general, the economist would do well to avoid a parabolic

trend. Logically, they seldom make economic sense.

Of course, one of the main reasons for fitting a

trend is to get some idea of the current direction of the

series and perhaps its probable direction in the future.

However, it is dangerous to extrapolate trends, especially when they exhibit such irregularities as are shown

in this chart. In thinking about the future trend of the

volume of agricultural marketings, we must also remember

that it will be affected by such things as acreage allotments, marketing quotas, and the Soil Bank.

- 12 -
TRENDS

Volume of Agricultural Marketings

% OF 1947-49

100

50

1910 1920 1930 1940 1950 1960

7956 DATA ARE PRELmiNARY.

U. S. DEPARTMENT OF AGRICULTURE NEC. 1315-57(3) AGRICULTURAL MARKETING SERVICE

Figure 6

Farm marketings and home consiimption: Index niimbers of volume, I9IO-56

[19^7-^9=100]

Year Volume Year Volimie Year Volume

1910

1911

1912

1913

1914

1915

1916

1917

1918

1919

1920

1921

1922

1923

192ÍÍ

1925

58

61

62

61

61

6k

6k

62

67

67

6k

65

67

69

72

70

1926 : 73

1927 : 73

1928 : 7k

1929 : 74

1930 : 72

1931 : 73

1932 : 71

1933 : 72

193l^ 71

1935 : 66

1936 : 71

1937 : 7^^

1938 : 76

1939 : 79

19*«3 : 80

19^1 82

19^*2 ....: 90

19lt3 9h

19^^^ .... : 99

19^5 .... : 99

19^6 .... : 97

19i>7 ...... : 100

1948 .... : 97

19i»9 ....: 103

1950 ....: 99

1951 ....: 101

1952 : lOlf

1953 108

1954 : 108

19?5 : 112

1956 1/ ... ....: 114

1/ Preliminary.

Agricultural Marketing Service.

-13 -
CYCLES

Cattle on Farms ^ Janiaaiy 1

Many importsmt econcanic time series tend to fluctuate more or less regularly up and down aroimd a trend

line. Such fluctuations often have important economic

implications. For example, the business cycle is of

great interest to economists and has been studied in

great detail by many competent economic theorists and

statisticians.

Cycles axe especially important in agriculture. The

very natiire of agriculture tends to generate cyclical

movements. Take cattle, for example. When cattle prices

are 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 after prices become unprofitable. Then the

reverse happens and herds are gradually decreased.

One of the best vays to forecast the probable behavior

of a current cycle from that of previous cycles is to break

the total series into individual cycles. In 1956, the most

recent cycle in numbers of cattle on farms was beginning

a downtuCTi. In the diagram shown here, data for these individiial 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 imiform 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 past cycles, gives some

perspective to the present. Often it suggests the general

direction of changes in the immediate future. But the

wide-awake economist will be looking for factors that may

make the current cycle different from the others.

- lU -
CYCLES

Cattle on Farms, By Cycles

MIL.

90

80

70

60

50

HEAD "1"' " """T" " 1—

1 1 1 1 1

, 1^ HI 1 1 i ^

^ ^ ^ >

f IOAO-«;AT"J Il 1 11^'^ki / -— K.? — y

r~ y <, 11938-49 1 ~~" — ^z y*^ s L„ /• t^ ^r *><

F —

-—

p z r

f— y •H \

/'^ 11928

1 38-1

__ .^'^ ^^ *«=<.^ ~~~ L> "2^ JÍ^ '^fc, 1 "~" ?^- > ^' rS"--f• ^

~ -^ ,^ ;»> ^

fT-JT ^^ J1912-28 ^ ¿?: ]2ip^ "-*,te -y ..^

—

1^' * -

^

^T -4-^4- 3

0á ^

'k ^ ^;zp 1 1 M#l l_

1896-1912

T

ÏÎ

.;<V r"^

'Jà

cm ^ ^

- — !^ !^ " ¿=: =i — - r-=-

11 13

YEARS*

15 17 19

YEAR Q¥ CYCLES, BEGINNiNG FROM LOW IN HUÍáBERS OH FÁRÍÁS.

7956 DATA ARE PRELIMINARY,

U.S. DEPARTMENT OF AGRICULTURE NEC. 1317-57(3) AGRICULTURAL MARKETIKG SERVICE

All cattle and calves: Nunà)€r on farms January 1, I896-1956

1/ Preliminary

Agricultural Marketing Service.

-15 -
Hog-Corn Price Ratio to Hog Slaiagfater

The agriciiltural economist usually 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 he shorter or

longer than average.

This chart'siamDO&rizes the history of the hog-corn

price ratio and hog slaughter since 1920. By using a

chart of this kind^ the agriciatural economist cannot

only get some \anderstanding of past cycles, he can get a

fairly clear idea of the current situation and probable

developments in the next year or so.

Note that there is a tendency for hog slaughter and

the hog-corn price ratio to move in cycles of about k or

5 years in length. The length of the cycle can be measured from one peak to the next or from one trough to the

next. Note that there is a lag between changes in the

price ratio and changes in hog slaughter. This lag is

from 1 to 2 years in length. It is indicated by the

dotted lines connecting peak years of price ratio with

peak years of slaughter. Thus, information on recent and

current hog-corn price ratios gives some indication of

what is likely to happen to hog slaughter 1 or 2 years in

the future.

Most economic cycles are not regular. They vary in

length and in amplitude. Sometimes mathematicians are

tempted to fit some type of curve that assumes perfect

regularity. For example, a sine curve. Actually, such

mathematical curves rarely fit the data well. If for any

reason the economist should want to smooth out the irregular variations shown in this chart, he would do well to

draw smooth curves that describe the general nature of

movement of the two lines. However, for most purposes

he would do just as well to leave the chart as it stands.

16 -
CYCLES

Hog-Corn Price Ratio and Hog Slaughter

1920 1925 1930 1935 1940 1945 1950 1955

7956 DATA AHE PRELÍMWARY.

U. S. DEPARTMENT OF AGRICULTURE NEC. 13V«-57 (3) AGRICULTURAL NtAWtf Til*« SERVICE

Figure 8

Hogs: NuBiber slatjghtered and hog-com price ratio, 1920-56

Year

1920

1921

1922

1923

192if

1925

1926

192?

1928

1929

1930

1931

1932

Slaughter

of

hogs

Hog-com

price

ratio 1/

Millions

61.5

61.8

66.2

77.5

76.8

$5.5

62.6

66.2

72.9

71.0

67.3

69.2

71.^

9.8

13.6

ikX

8.7

8.2

11.4

17.0

12.7

9.9

10.9

11-4

11.

12.

.7

.3

Year

1933 2/

1934 ..

1935 ••

1936 ..

1937 .•

1938 ..

1939 ..

19^0 ..

19^1 ..

19^^-2 ..

19^3 ..

19MV ..

19^5 ..

Slau^ter

of

hogs

Hog-corn

price

ratio 1/

Millions

79.7 10.4

68.8 7.0

kS.O 11.6

58.7 13.0

53.7 11.1

58.9 16.0

66.6 13.3

77.6 9.2

71.4 14.2

78.5 16.5

9'5.2 13.6

98.1 11.6

71.9 12.8

Year

19AV6 ..

19^7 ..

19^ ..

19^9 ..

1950 ..

1951 ..

1952 ..

1953 ..

1954 ..

1955 ..

1956 3/

Slaa^ter

oaf

hogs

Hog-corn

price

ratio 1/

Millicaas

76.1

7^.0

70.9

75.0

79.3

85.5

86.6

Ik.k

71.5

81.1

85-5

12.6

13.6

13.0

15.7

13.7

12.4

11.0

15.0

15.0

11.8

11.1

1/ Number of bushels of corn required to louy 100 po\mds of live hogs at local markets, based on

average prices received by farmers for hogs and corn. Anniml average is straight average of monthly

ratios. 2/ Incl-udes those slaughtered for Government account. 3/ Preliminary.

Agricultural Marketing Service. „_

-17 -
SEASONAL VARIATION

Monthly Production of Pork and Prices Received

by Farmers for Hogs^ 1950-56

îteny businesses are seasonal in nature. A dej^irtlaent store has a lot of extra business just before

Christmas and Easter. Hotels in Florida are full in

mid-vinter; \Aiile resorts in Maine do a big business in

sianaer. The production and marketing of laany agricultural products is strongly seasonal. This is simply a

reflection of seasonal changes in the weather.

The farmer, the processor, and the distributor all

are concerned with the seasonal movement of production,

marketing, and prices. An understanding of these seasonal movements help determine the most profitable time

to sell.

The accompanying chart is a typical example of a

simple analysis of average seasonal variations in recent

years. This particiJlar diagram is concerned with production of pork and prices received by farmers for hogs. Each

monthly figure was first expressed as a percentage of the

12-month moving average. Then the average percentage was

cosrputed for each month. For example, the production of

hogs in Januaiy was 130 percent of the 12-month moving

average centered on January. We simply plot the average

percentage for production and prices on the chart, as

shown. In general, the seasonal low point in prices is

in the late fall and early winter months when production

is at a seasonal high. Prices then usually rise and

reach a seasonal high in mid-sumn^r, soon after production has reached its seasonal low point.

This kind of analysis, of course, shows only what

the average seasonal variation has been in past years.

More detailed studies would be required to explain why

the seasonal swings in production and prices vary from

year to year.

18 -
SEASONAL VARIATION

Monthly Production of Pork and Prices Received by Farmers

For Hogs, 1950-56

PERCENT^ II -I - II

1I^VJ Ac\ : II .1 I«I

^pßt- ^4 L

-I - -.;? I

i-orNm l>r ^^ 2^± 1 1 ZUIVM M M M M 1 M 1 1 M M M M 1 1 r. •f^^ M M 1 M 1 M ( 1 M IJ^I M M 1 1 M ! 1 J

: Jl ~ V5~ \ w

SI II :¿í- L ^

•*?'"'V '•' ,< ^ ,^ M

100 ;;:Spiíss5S5::^f:::"::::::"

Ü ' ^

jl! ^^IBWi^i

iÊ.^

o A ^rSí SN* mú'^^ y --: : : ^/r _ i ^? 5 p*

^^ 1^1 1

~ Prociuc ion

X/N OU 1 1

Àr\

^^ J F M AM J J

Mor

* PERCENTAGE OF 12-tAONTH tAOVÍNG AVERAGE

U.S. DEPARTMENT OF AGRICULTURE

A S O N D

JTH

Ï956 DATA ARE PRELIMINARY,

NEG. 3974-57(3) AGRICULTURAL MARKETING SERVICE

Figure 9

Production of pork and price received by fanners for hogs: Percentage of

12-2nonth moving average, by months, average 1950-5^ 1/

Month Product* on Price

Percent Pèrceat

January ..

February .

March ....

April

May

June

July

August ...

September

October ..

November .

Decemiber .

124

95

105

92

67

87

78

81

88

106

123

13^

95

97

97

100

lOl*

105

106

108

106

101

91

90

l/ Production under Federal inspection.

Data revised and brought up to date from Breimyer, Harold F. and Kause, Charlotte A, Charting the

Seasonal Market for Meat Animals. U. S. Dept. Agr. Agr. Handb. 83. 1955. W' ^^ 32.

-19 -
Broiler Chick Placements and Marketings

In many cases the seasonal swings in production and

prices do not remain fixed over a long period of time.

Rather^ they change gradiially, reflecting changes in

methods of producing and marketing.

A case in point is the accompanying diagram showing

the seasonal patterns of broiler placements and marketings. This diagram was taken from a recent report by

Martin Gerra^of the Agricultural Marketing Service.

Important changes have been occurring in the broiler

industry which have affected the timing of marketings.

To study such changes in timing, it is often desirable

to plot the data for each year separately. We show here

the data for each year within two short periods—19^1-^3

and 1953-55--together with the average for each period.

The picture for these two periods is strikingly different.

Back in 19^1-^3 there was a sharp peak in broiler chick

placements toward the end of the year, and placements

were fairly stable from January through Jioly.

By 1953-55 the seasonal swings in the broiler industry had become considerably different. The industry was

geared to reach a peak in placements in early summer, so

that the bulk of the marketings came in mid-summer when

the demand for broilers is high. Placements then fell

off rather regularly imtil they reached a low point in

Septeinber or October.

An analysis of this kind can be of great practical

value to broiler producers and distributors. The weekly

reports on placements alone are good indications of

probable marketings about 10 weeks later. This is not

all. By studying seasonal diagrams, members of the industry can usually anticipate with some degree of accuracy the changes in placements that might be expected

several weeks ahead. This, together with the lag of

10 weeks between placement and marketing, gives at least

a general indication of probable changes in marketings

over a period much longer than 10 weeks.

20
SEASONAL VARIATION

Broiler Chick Placements^

7o OF NOV 1954" 7o OF NOV A-kVs

1 1 I

IT T 195^ i V -t

i-klLl A^ / 1% AH - S :CT ON A..J ivin -- -Z^,^'\-:LSECTION B° ou ,4y -u.^^^^^

Y"

L _o- -^^^^- 2tÄ^ X y

CA - .- ,1^1 , nn i'^"^^^ -

j L ou j3r±x '^^ JuJ^ N^%- "ç 7

AZ^JL liá^ A \ L I?ÍL / jfir w/ ^ A -Ä

4ok:::A-7/^^ft ill il o / i*-V .. rj* j u *X \ y'Average 941.43/9 ,^^ ]^ i«^^2S^r---. Jtgî liíü TT ^Ç :^"^"2^ 1953-55

¿s JL £îî- -- ^ í- ^ î7>^l L0 W^TÂïÎc /WA { / it . fcLi5y5-JÈî^ S /Jw 4 y \ . ÎSx JT

1 ou

OA M i 1T'M/X^? i Wn^m 1 1 1iuVTV>Î>L l/JJ'^i'I y_^/ MIM nn' ' ^ IH-T ÏL: 1 1 1 M M 1 J.k

.--EÍ- \^ -lt.—Jfl

^-rr T x=5c-+.¿>- H.JTTO'^^ M M 1 1 T/ M M M M M J\M M\l\ 1 // M ' I ' I

1 M M 1 1 1/ M M 1 1 7SM AM M 1 1 1 1/M 1 l-V.'«' '^1 t ^ir^l 1954 1 l'fi 1 >r VTiir <r iv:?^:^^ _TT T7 .7 OA "1942; V r 1 nn - 1 M 1 A U-3flJí¡L. ZU ^ ^ .QJ 1941 [^^ ï

ï^ë S^SÍ^

. V ^ X-

- Ak- ^ %

- Jfc^Z x

in on : ±3E Í:

JAN. JULY DEC. JAN. JULY DEC.

^MONTHLY INDEX OF AVERAGE WEEKLY RATE (NOVEMBER 1954 = 100).

^ BASED ON PLACEMENTS IN THE DEL^MAR-VA AREA.

^ BASED ON PLACEMENTS IN H STATES (1953), 13 STATES (1954), AND 22 STATES (1955),

U.S. DEPARTMENT OF AGRICULTURE NEC. 3975-57(3) AGRICULTURAL MARKETING SERVICE

Figure 10

Broiler chick placements: Index numbers of average weeiay rate, "by months, 19^+1-43 and 1953-55

[Noveniber 195^=100]

19ifl : 19^2 19*^3 : Average : : 1953 i 1954 i 1955 : Average Month i/ 1/ 1/ : ^ : : ^ :

2/ : 2/

: ^

January 27.0 36.2 iH.lt 3^.9 : : 108.5 120.1 101.9 110.2

February ...• 27.9 3it.5 35.2 32.5 : 110.2 123.3 130.6 121.14-

March • 28.3 3^-7 33.1 32.0 : : 119.6 129.8 132.2 127.2

April 28.0 27.3 35.8 30.if : : 121.5 133.8 144.2 133.2

May ; 27.6 27.1 32.9 29.2 : : 119.5 127.7 147.8 131.7

June 26.8 33.7 33.7 31.if : : 115.6 124.2 149.3 129.7

July 27.9 3í^.2 31.3 31.1 : : 103.4 123.5 145.0 12l(-.0

August 26.2 30.1 28.3 28.2 : : 93.9 115.2 128.6 112.6

September ... 22.7 27-3 27.7 25.9 : : 91-2 109.4 115.4 105.3

October : 27.2 29.6 38.6 31.8 : : 98.8 97.7 120.0 105.5

Noveniber .... : 35.6 38.5 49.2 41.1 : : 114.6 100.0 126.0 113.5

December .... : ^9.3 i^8.3 lv3.0 k6.9 : : 116.0 92.4 134.1 114.2

1/ Based on placements in the Del-Mar-Va area.

2/ Based on placements in 11 States (1953)> 13 States (1954), and 22 States (1955).

Gerra, Martin J. Seasonal Changes in Broiler Chick Placements and Marketings. U. S. Agr. Mkt.

Service, Poiatry and Egg Situation, PES-183, May 1956. pp. 36-40.

- 21
SIMPLE REGRESSION

Corn Yields Related to Nitrogen

A so-called "dot chart" is one of the handiest tools

of economic analysis. The agricultural economist ordinarily must find the relation between tvo variables.

Before putting numbers in a calculating machine, he should

almost always draw a chart like the one on the opposite

page.

This particular chart gives the results of experiments to determine the relation of corn yields to applications of nitrogen fertilizer. Each dot shows the yield

obtained from some amount of fertilizer. For example,

the dot to the left of the chart indicates that a yield

of 6k.6 bushels was obtained with no nitrogen at all.

The second dot indicates that 90-^ bushels were obtained

when ko pounds of nitrogen were used. In this case, the

several dots all lie fairly closely along a smooth curve

which we have drawn freehand. This curve can be taken

as an estimate of current yields to be expected from

varying applications of nitrogen.

Note that in this case two alternative extensions

of the curve are drawn at the right hand side of the

diagram. The solid line is horizontal, indicating that

the maximimi yield is apparently obtained with an application of about 200 pounds of nitrogen to the acre and

that additional applications neither increase nor decrease yields. The dotted curve suggests that yields

begin to decrease with applications significantly above

200 pounds. So far as the observed statistics go, the

dotted curve appears to fit the data slightly better

than the solid liue. This could, however, be a statistical accident. In deciding which extension to choose,

the economist must consult with technical experts on

soils and crops. Also, he should consider the results

of other experiments.

It is possible to use a chart of this kind to

determine the most profitable rate of fertilizer application. This matter is discussed in more detail on

page 58* At present our sole purpose is to illustrate

a graphic method of determining a simple regression.

22 -
(Zo

SIMPLE REGRESSION

rn\ Yield Per Acre in Relation to Applications of Nitrogen

YIELD ÍBUJ~

150

100

'

.,,

_ _„ ._ «^ _____ _ _ _ i_ ..._ 1...r-.

— — — — —

k

^

P" «««

1 ~ ^ =^^ i^ —^ —

^ ,_, /

J

/r

/

>

À

r

•yf

i\ j

/

"

À

F

C ^

^

1 1

L"

50

(

U. s. DEPA

tl , _j _ _J — _

RTMENT C)F AGRICULTl

100 200 300

NITROGEN (LB.)

JRE NEC. 3979-57(3) AGRICULTURAL MARKETING SERVICE

Figure 11

Corn: Yield per acre by specified quantity of nitrogen applied, Ontario, Oregon

r

1

Nitrogen

applied

Yield of :

com :

Nitrogen

: applied :

Yield of

com

r

Poimds Bushels : : Po\mds Biishels

0 64,6 \ : 160 ; 146.8

i^ IfO 90.i+ : : l80 141.2

j 80 ! 118.2 :* 200 ! 147.1

100 132.4 : : 240 ; 145.8

120 : l40.7 : : 280 147.4

i lii-0 : l4l.O : : 320 \ 143.8

Paschal, J. L., and French, B. L. A Method of Economic Analysis Applied to Nitrogen Fertilizer

Rate Experiments on Irrigated Com. U. S. Dept. Agr. Tech. B\ill. Il4l. 1956. p. 16.

- 23 -
Weekly Food Expenditures of Families^ 1935

Economists have long "been interested in the effect

of family income on expenditures for food. Many specific

studies have established two facts: (l) that the average

high-income family spends more for food than does the

average low-income family and (2) that the average highincome family spends a smaller proportion of its income

for food than does the average low-income family.

We need rather precise estimates of the relation of

family income to food expenditures in order to analyze

some of the principal economic problems confronting agriculture. In my opinion^ a great deal of confusion exists

about such terms as "the income elasticity for food." We

shall discuss elasticity in general on page 5^. For the

present^ it is necessary only to note that this is a

simple regression showing the average or expected weekly

food expenditures associated with various levels of

family income. It is a gross relation^ not a net relation. It is a relation between expenditures and income^

not between quantities and income.

Each dot on this diagram shows the average income

of a group of families and the average weekly food expenditures of the same group. The dots all lie close to

the smooth curve we have drawn. We have not plotted the

last observation shown in the table, which normally would

have gone some place on the extreme right of the chart.

This represents food expenditures of families with incomes of over $10,000 a year. However, the report does

not show the average income of these families. Therefore,

it is not possible to plot this dot precisely. If we had

plotted the dot at the lower limit of this class interval, it would have deviated widely from the curve shown.

This chart confirms one of the two facts given in

the opening paragraph, namely that the average highincome family spends more for food than does the lowincome family. A different chart would be needed to

demonstrate thkt the average high-income family spends

a smaller proportion of its income for food than does

the average low-income family.

2k
Houseket

Food Ei

SIMPLE REGRESSION

Bp\nQ Fomilies of Two or More Persons: Weekly

Kpendifures in Relafion fo Annual income, 1955

FOOD E;KPENDI TURE 1$)

Af\ 4U mm^ 00" '*

dC^

gl«a

Í•^

i

^<''

. ÊL^

^0^

0 ^

OA -

^ ^

zU : ~ ^5"

- ^^^ .^*'

^(*^

/*. ,.

t

U. S. DEPARTMENT

Dl

OF AGRICUU

I

SPOSAB

"URE

4

LE INC(

t

DME ($

NEC. 3981-

)

THOl

-57(3)

8

iSAND)

\GRICULTURAL

10

MARKETING SERVICE

Figure 12

Housekeeping families of two or more persons: Weekly food expend!t\ires by

specified income groups, United States, Spring 1955

Annual income

after taxes Food expenditures

If

Dollars

^ Under 1,000

1,000 - 1,999

2,000 - 2,999

3,000 - 3,999

4,000 - ii,999

5,000 - 5,999

6,000 - 7,999

8,000 - 9,999

10,000 and over

Dollars

11.69

16.60

22.55

27.00

30.27

33.03

36.lif

39.21

52.44

Food Expenditures of Households in the United States.

Rpt. U. S. Dept. Agr. 1956. p. 4.

- 25 -

Household Food Consumption, 1955, Prel.
Cigarette Smoking Related to Age

This is a very simple dot chart based upon a recent

siirvey of tobacco consimrption. It shows how the average

daily consumption of cigarettes is related to age of

cigarette smokers.

The main reason for showing this particular chart

is to emphasize that regression curves can take almost

any shape* If we had followed the reprehensible practice of putting these data into the calculating machine

and assimaing that the relationship was linear, we wouild

have come to the conclusion that there is practically

no relation between age and smoking habits. Actually,

there seems to be a decided relation, but a relation

that is distinctly curvilinear. By far the highest rate

of consumption is in the middle-aged groups of around

^0 to ^5 years. Younger and older smokers apparently

consume fewer cigarettes. One might, of coxirse, speculate on the reasons for this. It is not primarily a

matter of income. These data have been tabiilated separately by income groups and each income group exhibits

a similar ciirve in relation to age. A possible reason

is that young people get the habit rather slowly and

they may turn to cigars and pipes as they get older.

Anyway, whatever the reason, the evidence in the chart

is rather clear.

Note the wavy lines representing the x-axis. This

is a warning that the scale does not start with zero.

In this case, if we started both scales with zero, the

curve would not have shown up so well. But the statistician should always warn his readers when this is the

case. Otherwise, the chart would give the over-exaggerated impression that the youngest and oldest age groups

smoked practically no cigarettes at all. This is not

true. The yoimgest-aged group smokes an average of 19-9

cigarettes a day, and the highest consumption of any

group is only 22.9 cigarettes. A wavy line is not used

on the y-axis, as this scale is not apt to be misleading.

It was the opinion of the analyst that few yoimgsters

below 15 years of age smoke a significant number of cigarettes per day*

- 26
SIMPLE REGRESSION

Males Who Smoke Regularly: Cigarette Smoking in Relation to Age,l955

CIGARETTES (NO. PER DAY)

22

21

20

19

18

15 20 25 30 35 40 45 50 55 60

AGE (YEARS)

*^ U. S. DEPARTMENT OF AGRICULTURE NEC. 3982-57(3) AGRICULTURAL MARKETING SERVICE

Figure 13

Cigarettes: Average dsiily consianption by males who smoke regiilarly, by age^ 1955

Age : Consimiption

Years number

l8-2it 19.1

SS-S^t : 21.9

33-kh : 22.9

h3-3^ ': 22.5

55-6i^ : 20.3

65 and over «,..••.. • •••«••e»« : 17.6

Smoking Svrvey. Bioreau of the Census. 1955-

- 27 -
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 1939-56. You will note that the observations are scattered all aro\md the diagram and that

Bcme of the highest prices occurred in the years of

inedium to large production. Also, some 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, we have

drawn a line from each observation to each succeeding

observation. This is generally a good practice in dealing with time series. It quiciay 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

and in production. This relation is shown in section B.

It appears that changes in production give a fairly good

indication of expected changes in prices. The explanation is far from perfect. For example, if we liad used

the curve in section B to estimate expected changes in

prices we would have been over 60 cents too low in 1952

and 80 cents too high in 1953.

A more acc"urate way of studying the relation between

onion prices and production is discussed on page hO.

- 28
SIMPLE REGRESSION

Onions, Commercial Crop; Production and Average

Price Per 50-lb. Sock Received by Farmers

"'" 'V 11104- "'" <"

i:A. ACTUAL DATA::

iW-7- 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 no--5,, Jl"- J. YEAR-TO-YEAR-

^"^^^ !•! ii^^~^ ~^.52 THANnF^ V-nr\ 1^ V^C0

^^•43- - ^ J -

•40ÍW — — - »' '^ K."'5S ~ 56--- —

M9 Í ^É| •

C

S '44

rn ' % - -|Ä -.ou ■■ ; n .

^? ^ ^

^1,"«'

1 nn .^J -I.UU "^r--

^ %

'^^S

1 izr\ - -- - .--

- '•-'0 _ _.53

,L 1 ' , 30 35 40 45 50 -15 «10 -5 0 5 10 15

PRODUCTION (MIL. SACKS)

7956 DATA ARE PRELIMINARY,

U. S. DEPARTMENT OF AGRICULTURE NEC. 1324-57(3) AGRICULTURAL MARKETING SERVICE

Figure ik

Onions, commercial crop: Production and average price per 50-pound

sack received by farmers, 1939-56

Change from " Change frcan

Production

[preceding year in--*,.

P-oduotion

.preceding year in~

Year : Price

: Produc-

. Year Price

• Produc- : „ ^

: tion Price .

tion : ^^^^^

Million Million : Million Million

sacks Dollars sacks Dollars : ; sacks Dollars sacks Dollars

1939 .... 36.6 0.i^5 ««.i. —.«•• • ': 19ít8 ... 42.5 1.32 5.8 ..76

19^ : 32.9 .70 -3.7 0.25 : : 191*9 ...

: 38.8 1.47 -3.7 .15

19^1 ...• : 31.2 1.10 -1.7 .ko : : 1950 ... : 45.8 .87 7.0 -.60

19^2 .... : 38.9 .99 7.7 -.11 : : 1951 ... : 39-4 1.67 -6,k .80

19^3 .... 1 31.3 1.68 -7.6 .69 : : 1952 ... : -39.8 2.31 .h .6k

19^ ^7.9 1.20 16.6 -.48 : : 1953 ...

• 49.8 .68 10.0 -1.63

19i^5 .... 37.7 1.69 -10.2 .li9 : : 1954 ... 43.6 1.07 -6.2 .39

19lf6 .... 50.if .89 12.7 -.80 : : 1955 •••

: 42.8 1.18 -.8 .11

19^7 .... . 36.7 2,08 -13.7 1.19 : ! 1956 ... ! 49.4 1.30 6.6 .12

AgriculturaJ. Marketing Service.

29
Shifts in Demand for Beef and Pork

Simple regression is often useful in analyzing problems that are more coniplicated

than those we have just considered» Actxaally, the demand for "beef and the demand for

pork are each affected by a number of different variables. Still, it is possible to

discover certain basic relationships by simple 2-variable regressions.

The diagram facing this page is based upon an ingenious analysis by Shepherd,

Purcell, and Manderscheid. 2/ They first allowed for the effects of population growth

by using data on per capita consimiption of beef and pork on the x-axis. They also,

allowed for changes in the general price level by deflating beef prices and pork

prices. This was done by dividing retail prices by an index of per capita disposable

personal income. While this^is not the usual method of deflation, it seems to work

well in this case.

In the original report all data were plotted, including those for the war years.

As the authors point out, the relationship between consumption and prices in the war

years was abnormal because of price controls and meat rationing. In order to simplify

the diagram, I have not shown the data for the war years. However, the figures are

shown in the table.

This chart suggests that we have several different regressions in each section

instead of just one. It seems quite clear, for example, that the demand for beef was

higher in the post-World War II years I947-56 than in the prewar years 1925-41. Thus,

it seems desirable to draw two lines through the data rather than the usual single

line.

Now, looking at the pork data in the right hand part of the diagram, we find that

the demand was apparently highest in the period 1925-31« It dropped somewhat in the

period 1932-41 and remained about the same in I947-52. But in 1953-56 it seems to have

been substantially lower. Shepherd et al showed two regression lines for pork, but

their data ran only throu^ 1952. On our up-to-date chart, it seemed appropriate to

me to draw a third line.

The Iowa publication discusses in detail the reasons for a rising demand for beef

and a falling demand for pork. Among the principal reasons are the increasing urbemization of the country and a more even distribution of incomes.

This diagram could serve as a connecting link between simple regression and multiple regression. A study of the two parts of this diagram indicates, for example, that

beef prices are affected not only by the supply of beef but also by the supply of pork.

Also, pork prices are affected by the supply of beef as well as by the supply of pork.

You can see this, for example, if you look at the years 1934 to 1937 in the left hand

part of the diagram. In these years the price of beef was higher than indicated by the

regression line. Apparently this was because the droughts of 1934 and 1935 severely

reduced the supplies of pork in this period. Thus, if we wanted to get a more complete

explanation of changes in the prices of beef and pork, we would need to consider more

than two variables. This would take us into multiple regression, which is the subject

of the next several diagrams. In fact, we might need to consider more than one equation.

2/ Shepherd, Geoffrey S., Purcell, J. C, and Manderscheid, L. V. Economic Analysis

of Trends in Beef Cattle and Hog Prices. Iowa Agr. Exp. Sta. Res. Bull. 405, 1954,

p. 737.

- 30 -
SIMPLE REGRESSION

Shifts in Demand for Beef and Pork

PRirF í t PPP 1 R »*T

1

'

n^ rt

iii- S

80 Í

-äi-3r.- i\.BEEC II D OriDI/'JI 2=ir"" TTS ^ CI -!-»• v^ r\fx

3 ÎT ^^ -51-^^

35^ °"^-iy 33 V^ \r

70 I

l^-=lL- 438 ¿?7 S I:--481 ^.

=^-^i Vtf34 •s S ^y-^ Sit ¡ s 47 . o

Í7 56Z:

S 1 LI M 261 1 M M 1

439--^ i«5s. / 9i \ X ' .0.4-

2^^^! -^X-\ --36 : \ rJ 27 ■■■

.27:'-- .49 50. S dí- 5 J.' «L ^ ¿"o«. :S---- i 1 .7^' •^34fÍ5 ^ 30

OU ^ "ito^ s J

1

' nvi 1

%ir.TjT-"

9'»'^ '^ - S : s ¿

4" 38' "1 •IS - ' ?fl 1 925-31 -- -Í¿ ^ ^ I 2^ JL OfCLLJiifl .1

^.i' '' ^ ¿í - ^ V.-^^v \48 29VI

'\n - ii__41_ A SS*" 2

ou -

SI- - - ^ 111briir-' d--

:: 9:>5-'4lh^\- ¿_ II S Mn'M"323:~-

V 55' jl^S. 2 " » 50^ Jht-hi3^

s 56 ^tttl T^Hh5i

'1932-i

>2;

54'! I\ l%l IH-M.U/

A rv V

~-

1 1 1 IMIii-Ét:^ 4U - 1953-56 J^^=n—n^i 1947-i

W-55- -3 _40 S.' '^

L ¡^ J 56- - -' 5 -I 30 -

- ._ — U- Í

4C

U. S. DEPART

) 5

WENT 0

0 6

*A\

OF PE

19

F AGRICl

0 :

'ERAGE

R CAPIT

56 DAr>\

JLTURE

70 8'C

CONS

RETAIL PR

A DÍSPOSAB

ARE PREL

)

»UMPTJ

ICE PER

LE PERSC

IMINARY.

40 50 60 70 80

ON (LB. PER CAPITA)

POUND AS COMPILED BY AMS DIVIDED BY INDEX

)NAL INCOME (7947-49 = 100).

NEC. 3980-57(3) AGRICULTURAL MARKETING SERVICE

Figure 15

Beef and pork: Retail price per potmd and consumption per capita, 1925-5^

Year

Beef

Price of:

choice :

cuts i/:

Consimrption

Pork, excluding lard

Price

i/

ConsomptionYear

Beef

Price of:

choice :

cuts l/;

Consumption

Pork, excluding lard

Price Constmption

1925

1926

1929

1930

1931

1932

1933

I93Í

1935

1936

1937

1938

1939

19Ç0

Cents

59

59

63

71

7h

72

79-0

73.1

70.2

Ô2.2

68.4

73.0

70.2

67.8

63.it

Pounds

58.6

59.»t

48.1

49.0 uè.2

50.8

2/55.2

|752.2

53.6

53.9

55.2

Cents

60.5

63.3

59.9

56.0

55.0

59.6

57.0

Í9.5

%:i

m

62.2

59.9

51.0

il.5

Pounds

65.8

63.3

66.8

66,1 m

67.it

69.7

2/68.7

W62.2

m

55-0

57.it

63.9

72.4

19ÍH ..

19it2 3/

IMaiI

m

19k9

1950

1951

1952

1955

1956

Cents

56.0

Í9.7

it5.9

ItO.O

38.6

56.7

65.3

72.8

67.1

68.7

7it.6

70.9

5^.5

5it.l

51.2

48.0

Potinds

60.0

60.it

52.5

5it.9

58.6

60.8

68.6

3

1

6

62.

63.

62,

61,

76,

19.

.5

.0

8iÎ!2

Cents

43.9

it2.6

39.2

33.9

Í0.8

58.6

5lt.6

Í9.7 Ité.l

it5.9

it2.7

45.3

it6.1

37.2

33.9

Pounds

67.it

62.8

77-9

7Ô.5

65.7

7Í.9

68.6

66.8

66.8

68.2

70.9

71.4

62.6

59-2

65.9

65.8

1/ Retall price as coinputed ty Agricultural Marketing Service divided by index nvmibers of disposable personal income ( 19it7-49=100). 2/ Consumption less use \inder Federal programs. 3/ War years

omitted from diagram.

Agricult\aral Marketing Service.

31 -
COMPARISON OF TIME SERIES

Food Prices > Consumer Incomes^ and Vol-ume

of Farm Marketings

Economists often work with time series; that is^ with

records of prices^ production, and consimiption 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 charts such as the ones we have just 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 marketings for food. 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 study the changes which

have occxorred over a period of time.

In this case it is clear that there is high correlation between the food price index and 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 I9U1 to 19^5 the relationships do not seem to be the

same as in other years.

Comparisons of these three time series suggest that

the correlation between the average price index for food

and per capita disposable income would be reduced by deflating each series (for example, by dividing each of

these by the cons\mier price index for all commodities).

Such a computation would also 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 taken place independent of changes in the other series,

suggests that the war years be omitted from the analysis.

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 variables.

In some cases, a comparison of time series will indicate a timelag between changes in one variable and

changes in another. We saw previoiisly that changes in

slaughter of hogs occur several months after a change in

the ratio of hog prices to those for corn.

32 -
COMPARISON OF TIME SERIES

Food Prices, Consumer Incomes and Volume of Farm Marketings

«7. OF 1947-49 1 1 1 M 1 M M 1

125

100-

. 1 M 1 1 1 1 1 1 1 1 II 1 1

Per cemita disoosable income >

Il ••^ *'

<

?• t^ t ,.J

].J

' Per capita farm mar ketinqs >•

II r ^ J N. 1p^ ><UU^

r — ana nome consumpnon '^^ |

^ b ^ /\i . Mi BHi .

p- ,« M »

-1 _ MB ». M »1 i. 1. .

saÍ4-, -^4-! ^^ h- _.

75

50

25

19

u. s. DEPA

~?

^ rit

—

N H s

— — —

H

jt —-làrr ~

=5 Cü ^

c:3 — — —

> H S ^ h H Si? ^/

H

V

i.*' #

2 # r<

1

1 r

.1^ i1* ,v

T -i

.• f ^'

V fU •v. j ^^N^i

^wy^ Retail food-orice index

a. ^wf n'^ f

1

^J^ «

. 1 isl tí iflrî \

^

Hks; i

^* 2 I* r \ 1w «r

i*^ r ^^</ #^ i*' ^•, ^# i'

'

\ ,1 i*^ 1 LJ ,^ l' —1 •1 I*

20 1930 1940 1950

1956 DATA ARE PRELIMINARY

RTMENT OF AGRICULTURE NEG. 1326 - 57 ( 3 ) AGRICULTURAL MARKETING SERVICE

Figiore l6

Price and marketing of food and disposable Income: Index numbers, I92O-56

[19^7-^9=100]

Year

Food

Retail

price

t--Farm-market¡^

:ings and home:

: consimiption :

; per capita ;

Disposable

income

per capita

Food

Year Retail

price

: Farm market-:

:ings and home;

: consumption :

: per capita ;

Disposable

income

per capita

1920

1921

1922

192?

1925

1925

1926

1927

1926

1929

1930

1931

1932

1933

I93Í

1935

1936

1937

1938

83.6

63.5

59.Í

61A

60.8

65.8

68.0

64.8

65.6

62.if

5lA

Í-2.8

i+1.6

kS.k

^9.7

50.1

52.1

kd.k

88

88

91

89

90

90

p

U

86

87

§3

87

52.8

Í-1.0

ïi:^

^9.3

51.5

52.6

52.1

52.7

Í-8.8

41.5

31.Í

29.ii33.2

41.8

44.5

40.8

1939

1940

1941

1942

]IU

1945

1946

1947

194Ô

1949

1950

1951

1952

1953

1954

1955

1956

47.1

47.8

52.2

61.3

68.3 67.Í

6Ô.9

79.0

95.9

104.1

100.0

101.2

112.6

114.6

112.8

112.6

110.9

111.7

89

91

93

101

105

109

108

106

104

9§

^^96

97

97

97

100

u

^3.5

^6.5

56.?

70.4

Ô5.6

86.8

91.0

9^.7

103.3

101.9

109.8

118.3

122.1

126.7

126.6

132.2

137.6

Prices from Bureau of I^abor Statistics, marketings of food from Agricultural Marketing Service,

€ind income from Department of Commerce.

- 33
MULTIPLE REGRESSION

Per Capita Consiimption Related to Deflated

Per Capita Income and Food Prices

Multiple regression has been used to analyze a wide variety of economic problems. This is

because most economic variables (such as prices and rates of consumption) are influenced by a

number of different factors. Ordinarily the economist cannot conduct controlled experiments

allowing only one of these factors to vary. Rather, he must try to unscramble market data in

order to separate out the influence of each of several variables. Whenever the influences of the

separate variables can be added together, the problem can be studied by multiple regression.

As in the case of simple 2-variable regression, the statistician can put the data for a multiple regression problem into the calculating machine and compute the answer by least sqimres.

However, in lay opinion there are many advantages to a graphic analysis of such problems. Such

analyses were made popular by Louis H. Bean 3/ and have been used widely in the Deparianent of

Agriculture and in the State colleges.

A case in point is the relation of food consumption to income and food prices. If we can

accurately measure these relationships, we have the basis for determining the so-called "income

elasticities" and "price elasticities" for food consumption. James P. Cavin recently made a

mathematical analysis of the data presented here. I shall illustrate how the data can be analyzed

graphically.

In section A of this chart I have first plotted the data to show for each year indexes of

deflated per capita income, together with the indexes of per capita food consumption. Each dot

shows the pair of indexes for a particular year. (The years 19^^-2 throxigh 19^7 were excluded from

this analysis because food consumption was affected by such things as rationing and price control.) If we were to draw a line representing the simple relationship between food consumption

and income, we would doubtless draw a curve which would be steepest at the left hand side of the

chart, and which would become less steep as we move from left to right. However, we are not concerned with this simple relationship. We want a regression line which will be our best estimate

of what the index of food consumption would have been if food prices had remained constant. An

examination of the dots in section A and the price data in the table indicate a general tendency

for food consumption to be reduced when food prices are high and to be increased when food prices

are low. The regression line in section A is drawn with this in mind. For example, it is drawn

considerably higher than the dots for the post-World War II period when real (or deflated) food

prices were higher than in the prewar period.

Section B attempts to explain how food consumption was related to the level of real food

prices. Specifically, each dot shows for some year the deflated food prices for that year, together with the deviation above or below a regression line in section A. For example, take the

first year, 1922. The index of deflated food prices was 83.0, and the dot for 1922 in section A

is 0.6 unii:^ above the regression line. This observation is plotted in section B with the coordinates 83.0 on the X-axis and +0.6 on the y-axis. Similarly, for each other year. After.these

dots are properly located in section B, we draw the regression line indicating the net effect of

deflated food prices on food consimiption. Then our Job is done unless a further study of the data

suggests a need for making some adjustment. In this case the two lines seem reasonably satisfactory.

The deviations from the regression line in section B indicate the amount of error that is

made in estimating food consumption from the two independent variables—income and price. The

largest errors are about two index points in 1926, 1935^ 1937> and 1951. For most years our estimates are within one index point of the true figure. With the index of consumption vaiying

between 87.8 and 104.0, this amount of error seems reasonably small.

3/ Bean, Louis H. A Simplified Method of Graphic Curvilinear Correlation. Jour. Amer. Statis.

Assoc. 2if: 386-397^ illus. I929.

3h -
MULTIPLE REGRESSION

Food: Consumption Per Capita in Relation to Real /ncoitie Per

Capita and Real Food Price

CONSUMPTION^%OF 1947-49) DEVIATIONS

80,

RELATION TO

PRICE

'50 65 80 95 HO 125 72 76 80 84 88

S4-H53-K

.51_

92 96 100

INCOME (%OF 1947-49) PRICÉ*(%OF 1947-49)

^DEFLATED BY DIVIDING BY THE CONSUMERS' PRICE INDEX,

7956 DATA ARE PRELIMINARY,

U.S. DEPARTMENT OF AGRICULTURE NEC. 3983-57(3) AGRICULTURAL MARKETING SERVICE

Fig\n*e 17

Index nuidbers: Consumption of food, disposable income, and retail food price, 1922-41 and 19^+8-56

[19^7-^9=100]

Per capita Price of I

food

: Per capita Pri ce of

Year • Consumption ;Disposable Year \ Consumption [Disposable food

; of food ; income l/ i/ ;

; of food ; income l/ i/

1922 ... .: 89.0 61.0 83.0 • ': 1937 ..... 90*4 72.5 8k.9

1923 ... 90.9 68.3 84.2 : : 1938 : 90.6 67.8 80.3

1924 ... .: 91.5 67.il- 83.2 : : 1939 ..... : 93.8 73.2 79.3

1925 ... .: 90.9 68.5 87.7 : : 1940 : 95.5 77.6 79.8

1926 ... 92.1 69.6 89.9 : : 1941 : 97.5 89.5 83.0

1927 ... ..: 90.9 70.2 88.3 :

1928 ... ..: 90.9 71.9 88.4 : : 1948 99.1 100.6 101.3

1929 ... ..: 91.1 75.2 89.5 : : 1949 : 96.9 100.1 98.2

1930 ... ..: 90.7 68.3 87.4 : : 1950 : 99.9 106.8 98.lt

1931 ... ..: 90.0 64.0 79.1 : : 1951 : 98.1 106.6 101. li1932 ... ..: 87.8 53.9 73.3 : : 1952 ..... : 100.4 107.6 101.0

1933 ... ..: 88.0 53.2 75.2 : : 1953 101.5 110.8 98.6

1934 ... ..: 89.1 58.0 81.1 : : 1954 101.4 110,3 98.1

1935 ... ..: 87.3 63.2 84.7 : : 1955 : 102.8 115.5 96.9

1936 ... ..: 90.5 70.5 84.5 : 1956 ..... : 104.0 II8A 95.9

1/ Deflated by dividing by the Bureau of Labor Statistics Consumers^ Price Index.

AgriciAltural Marketing Service.

- 35 -
Price of Corn Related to Price of Livestock and

Supply of Feed Concentrates Per Animal Unit

The diagram Illustrates an analysis of corn prices

(XQ) related to two Independent variables—prices of

livestock and livestock products (X^) and supplies of

feed concentrates per animal unit (X2)• We know from

theory and from general observation that high livestock

prices tend to be associated with high prices of corn*

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 com.

Section A of this chart shows corn prices and

prices of livestock and livestock products from 1936

through 1955* Before drawing the regression llne^ we

try to take accoxmt of X2. We draw several regressions

for subsamples of data^ commonly called "drift lines."

Thus In 1948^ 19^9^ and 1950, supplies of concentrates

were from I.05 to I.07 tons. We connect these observations with a drift line. Similarly we connect the

observations for 19^^ 194l> sind 19^2, when supplies

were O.9O tons. After drawing all possible drift lines,

we draw a net regression line the slope of which represents approximately an average of the slopes of the

drift lines. In this case, a straight Une happens to

be satisfactory. In many cases, a curve would be

Indicated.

Section B shows how the residuals (departures from

the first regression line) are related to X2. These

residuals are clustered closely around the regression

Une we have drawn. If a nearly perfect fit were not

given by the dots around this line, the process of successive approximation would be used. Foote kj 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 mathematical regression line by

the method of least squares.

Graphic multiple regression requires a fair amoxxnt

of Imagination and some practice. But It often shows up

Important relationships that are not broiight to light by

grinding figures out of a computing machine.

h/ Foote, Richard J. The Mathematical Basis for the

Bean Method of Graphic Multiple Correlation. Jour. Amer.

Statls. Assoc. tô:778-788. 1953-

36
MULTIPLE REGRESSION

Corn.- November-May Prices Received by Farmers

in Relation to Specified Factors

PRICE (C PER BU.) - X

200

100

SECT

TT

ON A.

DEVIATION - Xoi 1

IT :r

nr

inn . ccmr^Ki R

•36 '47

CA Z „ J!

^"^^Si '-ío '54 »

**'^fc >] 1 '40 •c') '53 / "i"^

i^r"! il \y 1 1 ./ / 1

f\^ w '^"^ íHBPTí V / ^ J 1 H M «1 M Il •Spi"^, 1T Al 3t - p - - -

3Ö / ji\^^'44 yi] ****!»

'3/|41^fTj'5] ^^ *^^i

,1 Mb',' <¡T s

-42 ^--'4á-r'49

inn ^

100 200 300

PRICES OF LIVESTOCK AND

PRODUCTS (%OF 1910-14)-X,

U. S. DEPARTMENT OF AGRICULTURE

.65 .75 .85 .95 1.05 1.15

SUPPLY OF FEED CONCENTRATES

PER ANIMAL UNIT (T0NS)-X2

NEC. 1327-57(3) AGRICULTURAL MARKETING SERVICE

Figure l8

Com: Price per bushel received by farmers and related variables, 1936-55

Price received by- Price received by Supply of

feed concentrates

farmers (November-May) Supply of

. feed con- ;

centrâtes '

: Period

fanners (November-May)

Period . Livestock Livestock beginning - Com : and per ajilmal *

: beginning Com and ; per animal

¡products 1/ unit 2/ : products 1/ unit 2/

: Cents Tons : Cents Tons

1936 106 123 0.65 : ': 19l^6 ' 136 278 0.99

1937 51 114 .89 : : 19^7 220 305 .87

1938 : hh 108 .88 : : 19^ 120 285 1.05

1939 55 107 .87 : : 19^9 118 260 1.07

19ÍK) 58 122 .90 : : 1950 155 329 1.06

19^1 : 7h 159 .90 : : 1951 : 166 318 1.01

19í^2 90 19l^ .90 : : 1952 Ikl 278 1.05

19i^3 ! 112 196 .85 : : 1953 11+2 271 1.10

19i<-')- ! 107 206 .92 : : 195^^ 138 2li.0 1.13

19ÍV5 ; 115 215 .92 : : 1955 121 22if 1.19

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

Computed from date in Foote, Richard J. Statistical Analyses Relating to the Feed-Livestock

Economy. U. S. Dept. Agr. Tech. Bull. 1070, 1953» p. 6.

- 37 -
Yields of Corn in Illinois, 193^-^^^ Related to Reported

Condition on September 1 and a Time TreriS

The Agricultural Marketing Service estimates the probable production of many of the principal

crops several months before they are harvested. Such advance estimates of probable production are

based in part upon the judgment of farmers concerning "the condition of the crop as a percentage

of normal." It is unnecessary here to explain in detail the concept of normal production. Statisticians have found that the farmers' reports as to current condition of crops is a fairly good

indication of the yield that would occur \T±th average growing conditions during the rest of the

growing season.

The Division of Agricultural Estimates makes extensive use of dot charts in graphic analysis

to interpret reported condition and to estimate probable yields. The accompanying chart, suggested by C. E. Burkhead of the Agricultural Estimates Division, Agricultural Marketing Service,

illustrates how this can be done in the case of corn yields in Illinois. Section A of the chart

is a scatter diagram relating the reported condition as of September 1 of each year from 193^

through 1956 to the final estimate of harvested yield. It is easy to see that there is some positive correlation between reported condition and final yield. When farmers report a condition of

90 to 95 percent of normal, the final yield tends to be high. When they report a low condition

of, say, ko or 50 or 60 percent of nonnal, the yield tends to be low. But this is not the who2-e

story shown in the chart. When dealing with time series, it is always a good idea to label each

dot to indicate the year, as we have done in this case. Notice that the dots for the early years

are all in the lower top part of the scatter. In other words, a reported condition of 80 percent

of normal today indicates a higher yield of corn than would have been suggested 20 years ago.

The solid, straight line drawn through this scatter is an estimate of the relation we might

have expected between condition and harvested yield at about the middle of the period studied;

that is, from around 19^0 through 1950. To make a good estimate of corn yields today we need to

consider not only the average relationship between reported condition and harvested yield for the

whole period, but also a "net trend"; that is, the trend in yields after allowing for the average

relationship shown in section A of the chart.

This is a problem in multiple regression. The corn yield is the dependent variable; that is,

the variable we are trying to estimate. In this case there are two independent variables which

are useful in estimating yield. The first of these is reported condition and the second is time.

The effect of time is shown in section B. Here we have plotted for each year the deviation of the

actual yield from the regression line shown in section A. Take the first year in the series—

193^. The regression line indicates a yield of 30.5 bushels. The actual harvested yield was

21.5 bushels. So, there was a deviation (or "residual") of -9.O bushels. Thus, in section B we

indicate -9.O for the year 193^« Similarly, for each of the other years in the series. When

these dots are plotted, it is apparent that there was a definite net trend. It rose sharply from

1934 to about 19^2. Then leveled off until about 1950. Since 1950 it has again risen sharply.

Probably the sharp increase in the early years of the series was due mainly to the introduction of

hybrid corn. The effect of this began to peter out in the 19^0's. Since about 1950 there has

been a new upward trend, probably due to increased use of fertilizer.

The use of an analysis of this kind can be illustrated by data for I956. Farmers reported a

condition 96 percent of normal. Section A indicates a yield of 5Ö bushels. Section B indicates

that we should add 9*5 bushels to account for the trend; thus giving us an estimate of 67.5 bushels. Actually, the yield in 195^ turned out to be 68.0 bushels. In this case the two regression

lines would have given us a good forecast of corn yields in Illinois', somewhat more accurate than

we should expect in an average or typical year.

One of the difficulties with time series of this kind is that of extrapolating the net trend

shown in section B. Each year we make a forecast we have to extrapolate beyond the range of observed data. We don't really know what the net trend in Illinois corn yields will be in the

future and have to do some guessing.

- 38 -
MULTIPLE REGRESSION

Corn: September 1 Condition and Yield Per Harvested Acre, Illinois^

YIELD(BU.)- " T

1 1

j 1 7n CV-TT r^hÈ A -- ii^-_

j

48

- -- 1 5? \

A o .'^'^ •I '^5j i: ÖU — " " . L\4^2:,;\ ^^'^ _|3^^:^^ „ï'

ii. iv • i>'^^ üt

iqn • ^W-«í^^- ^6 <>.49. OU - - -- - —^'l^ T^^ ^

^fl, 50.^ 39.

^_, ^^L U ^ 37

f^\ ^é!Í^ T,-45. X 1 1

>< n :i „ Éf- : „, -> - .; ^R

4U — ;p*= 40 I

^^ ^. " -^ -is ¿^ 41

^n -.--,;;?- -

.¿f^

34

"^ V " 1JL_

90 4^' ^ -.

"^40 50 60 70 80 90

DEVIATIONS 1

1 1 J

t 1

^AN 1 M 1 M lcCi^Ti/^fc.1 D 1 M M 1 1 ^IX u.

in -- - --, ^ lU ~ i

-iU ^L

-« -'^'

¡nr^>—: ^-_^^<L ._

0 -::::::::¿?í:nr^^-S

-HK^'^ it J L_

4 1- ^z^^L^ :

in -J2- -

lU

on - - zu

1932 1940 1948 1956

SEPT.! CONDITION (%OF NORMAL)

Tin ADJUSTED FOR TIME TREND,

1956 DATA ARE PRELIMINARY^

U. S. DEPARTMENT OF AGRICULTURE NEC. 3984-57(3) AGRICULTURAL MARKETING SERVICE

Figure 19

Com: Condition September 1 and yield per haorvested acre, Illinois, 193^-56

Year ;

Condition

1/

; Yield ; ; Year : Condition

i/ ; Yield

Percent Bushels : : Percent Bushels

193^ k8 21.5 ': • 19h6 i 93 56.0

1935 77 38.5 : : 19^7 : 02 39.5

1936 k2 23.5 :

: 19^8 : 9k 61.0

1937 89 iiâ.O : : 19^*9 : 95 ^k.O

1938 : 85 kk.O : 1950 : 84 51.0

1939 : 9^ 51.0 : : 1951 : 88 56.0

19^0 : 70 ItS.O : : 1952 : 86 58.0

19^1 .•. : 85 53-0 : : 1953 : 81 5^.0

19^2 87 5i^.0 : : 195^ : 73 49.5

19^3 : 79 50.0 : : 1955 : 80 56.0

19^4 : Ih k3.h : : 1956 96 68.0

19^5 : 77 k6.3 :

1/ As a percentage of normal.

Data supplied by C. E. Burkhead, Agricultural Estiniates Division, Agricultural Marketing Service.

- 39 -
Price of Late Onions Related to Production

and Disposable Income

The data for this diagram^ taken from Shuffett^ ¿/

are expressed as first differences (i.e. year-to-year

changes) in logarithms. The rationale of this may he

found in Shuffett^s bulletin and need not concern us

here. Graphic analysis will handle logarithms and first

differences^ as veil as the xinmanipulated data.

The main purpose of this diagram is to illustrate

successive approximations to the true regression lines.

We have already discxissed the graphic determination of

the net regression lines. So far, we have tacitly

ass"umed that one approximation 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) are plotted as 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 our 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 several 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 disposable 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.

5/ Shuffett, D. Milton. The Demand and Price Structure

for Selected Vegetables. U. S. Dept. Agr. Tech. Bull.

1105, pp. 38-^3. 195^-

40
MULTIPLE REGRESSION

Late Onions. August-April Prices Received by Farmers in Relation to Specified Factors

PRICE {$ PER CWT )-LOGARITHMS*.Xo DEVÏATIONS-X ^ ,

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

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

LOGARITHMS*-X, LOGARITHMS*-X ^

^ CHANGE FROM PRECEDING YEAR,

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

Figure 20

Late onions: Average price per 100 pounds received by farmers

and related variables^ August-April average, 1928-41

Actual First difference of :

logarithms :

•period"

1begin-_

', ning ;

Actual First difference of

logarithms

Period Per capita Per capita Per capita Per capita

beginning Price

1/

Production

1/2/

Income

3/

Price

^^°'' : income ;

Price

i/

Production

.1/2/

Income

3/

Price

. ^^^'^ ; income

Dol. Lb. Dol. • Dol. Lb. Dol.

1928 • 2.54 6.65 658 ...• ... ——— • ': 1935 ! 1.18 8.20 467 •-0.062 0.017 0.046

1929 : 1.30 9.08 663 --0.291 0.135 0.003 : : 1936 : .06 9.23 534 -.137 .051 .058

1930 : .82 9.75 557 -.200 .031 -.076 : : 1937 : 1.30 8.36 532 .179 -.043 -.002

1931 : 2.02 6.1+1 1+56 .392 -.182 -.087 : : 1938 : 1.06 8.59 509 -.089 .012 -.019

1932 : .54 8.75 347 -.573 .135 -.119 : : 1939 : .88 10.57 546 -.081 .090 .030

1933 : 1.28 7.58 386 .375 -.062 .046 : : 19^ : 1.12 9.93 601 .105 -.027 .042

1934 : 1.36 7.89 420 .026 .017 .037 : : igi^i : 2.08 9.^7 748 .269 -.021 .095

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

2/ Production divided by November 1 civilian population,

3/ Disposable income 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. Bull. 1105. 1954. p. 43.

i^l -
JOIHT (3-DIMENSIONAL) REGRESSION

Yield of Corn In Relation To Applications ^

Nitrogen and Phosphoric Acid

Many prohlems of economic analysis can he handled hy simple (2-variahle) regression. We have

considered several examples of problems that can be handled by this technique. Many other economic

problems can be analyzed rather well by the use of multiple regression. However, the use of a

multiple regression is limited to problems in which the effects of several variables can be added

to one another^ except where special transformations of the data are made, such as the use of

logarithms. In mathematical terms multiple regression is limited to the analysis of problems that

can be stated in the form

XQ = fi(xi) + f2(x2) + fn(^n) (l)

Actually, many important problems in economic research cannot be handled satisfactorily by

such an additive function. In many cases we must consider the more general relation

Xo = F (xi, X2, , Xn) (2)

A case in point is the relationship of crop yields to various dosages of fertilizer. We considered on page 22 the relationship of corn yields to a single variable, the application of

nitrogen. In this case the applications of potash and phosphoric acid were held constant. While

this sort of analysis tells us something about response to nitrogen, researchers want to know the

response to various combinations of nitrogen, potash, and phosphoric acid. Many experiments have

been conducted in which all three of these have been varied. The 3-cLiniensional diagram facing the

page shows how we can analyze the combined effects of two independent variables at a time. In

this case we consider the combined effects of nitrogen and phosphoric acid (P2 O5). The data are

taken from a recent report of the Iowa Agricultural Experiment Station. 6/

In this case we have plotted the data with the application of pota;'! (K2O) held constant at

40 pounds to the acre. The applications of nitrogen (N) varied from 0 to 2^1-0 pounds. The application of phosphoric acid varied from 0 to 120 pounds. The location of each hatpin on the base of

the diagram indicates a combination of nitrogen and phosphoric acid. The height of the hatpin

above the base indicates the yield of corn. For example, one of the hatpins is located at the

point corresponding to N = 0, P2O5 = 0, and the height of this hatpin corresponds to a yield of

32.00 bushels of corn to the acre. Similarly, each of the other hatpins shows the yield obtained

by some combination of nitrogen and phosphoric acid. The relationship between these combinations

of fertilizer applications and corn yield is q.uite apparent when one looks at a diagram of this

kind. It is easier to see it in the original diagram than in the photograph. The highest corn

yields were obtained by a combination of about 160 pounds of nitrogen and about 80 pounds of phosphoric acid. When the nitrogen application was increased to 2ÍÍ-0 pounds, yields were definitely

reduced. Also, there is some indication of a reduction in yield when phosphoric acid is increased

to 120 pounds. When no nitrogen is used, applications of phosphoric acid tended to decrease the

yield. As increased amounts of nitrogen were used, applications of a considerable amount of phosphoric acid were beneficial.

This is a 3-dimensional dot chart. In principle, it is the same thing as the several

2-dimensional dot charts we have looked at. We want to visualize a graphic, 3'-<iisiensional surface

which describes the general nature of the relationship. Such a surface could be constructed

either graphically or by fitting some proper form of mathematical function. Before choosing a

mathematical function, however, the researcher would do well to sketch in a smooth regression

surface, such as the one shown on the diagram. Such a sketch will show that any satisfactory

mathematical function would permit an inverse relation between corn yields and phosphoric acid

when nitrogen applications are low, and a positive relation between corn yields and phosphoric

acid when nitrogen applications are high. None of the usual formulas used to describe the results

of fertilizer application do this. Therefore, they will not fit these particular obsearvations

well. If other experiments should produce similar results, we ought to either look for another

mathematical formula, or else be satisfied with the results we can get from graphic analysis.

6/ Brown, William G., Heady, Earl 0., Pesek, John T., and Stritzel, Joseph A. Production

Functions, Isoquants, Isoclines and Economic Optima in Corn Fertilization for Experiments with

Two and Three Variable Nutrients. Iowa Agr. Exp, Sta. Res. Bull, hkl, 1956.

- Í4.2 -
JOINT REGRESSION

U.S. DEPARTMENT OF AGRICULTURE NEC. 4259-57(5) AGRICULTURAL MARKETING SERVICE

Figure 21

Corn: Yield per acre for given applications of phosphoric acid and nitrogen l/

Nitrogen In pounds

Phosphoric .

acid ; 0 ! )J0 80 i 160 2lK)

Poxinds Î Btishels Bushels Bushels Bushels Bushels

0 : 32.00 ^9.85 65.50 68.25 61.20

1+0 ': 32.25 1+9.55 61.55 7lt.T5 66.90

80 ': 23.25 46.55 62.65 88.15 78.1+5

120 : 20.20 50.75 69.80 86.80 81.90

1/ With potash at kO pounds per acre.

Brown, William G., Heady, Earl 0., Pesek, John T., and Stritzel, Joseph A. Production Functions,

Isoquants, Isoclines and Economic Optima in Com Fertilization for Experiments With Two and Three

Variable Nutrients. Iowa Agr. Expt. Sta. Research Bull, kkl, 1956. p. 815.

-1*3 -
USE OF ISO^UAHTS TO STUDY JOINT REGRESSION

Yield of Corn in Relation to Applications of

Nitrogen and Phosphoric Acid

Another technique for studying variation of a 3-cLi^^nsional surface is similar to that used in suarveying and

grading land. We can forget for the moment that the chart

refers to yield of corn. Suppose that the vertical sixis

measures distances north and south, the horizontal axis

measures east and west, and the numbers written hy the

dots on the diagram indicate the elevation of the land at

various points and determined by surveyor*s transit. Anyone used to maps would recognize that the land is gradually

increasing in height as we move toward the right side of

the diagram, hecomdng steeper as we move toward the upper

right corner. You woxild also see that there are "bumps and

hollows. In simple regression we smooth in only one dimension. Here we are 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 general problem of joint regression

is that of determining a series of isoquants. Whatever the

three variables may be, an isoquant will show the combinations of two independent variables which correspond to a

given value of the dependent variable. In the case illustrated by the diagram, it is clear that yields increase

steadily with increases in nitrogen up to about l60 pounds

per acre and then begin to decline. The effects of additional phosphoric acid are less for small applications of

nitrogen than for large applications of nitrogen. Yields

can be increased substantially by high level applications

of both nitrogen and phosphoric acid, although the maximum

combination, as indicated by this diagram, is reached by

the use of perhaps 175 pounds of nitrogen and 75 pounds

of phosphoric acid.

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 mathematical functions, the

diagram should suggest the kind of function to use. Another

technique which is sometimes used to study three-dimensional relationships is the "isometric projection." Those

who are not familiar with isoquants m.ay find such projections easier to visualize. But they are also harder to

read accirrately.

- kk -
JOINT REGRESSION

Corn: Yield Per Acre Related to Applications of Nitrogen and Phosphoric Ac/

PHOSPHORIC ACID (LBS.) •

j*

1 1

20.20 \ 1 i \ L. V69".8Ó \ 86.80 , . \ i

F

81.90 J

-

I ' i V * \ T \ \ T 1 /

1

TOO

:

i \ L \ ! V J

\ \ \ \ \ J

f~

À

r

' > \ V /

r

i \

^ À À

/

'

\ \ i

r

< IK 4 4R .55. z !.<-C \ V 8i .1

L

5 /

r 1 1

78.45- V \ / A 1

50

1 ' \

^ i T / T

i 1 \ \ > U^

Ir i V \ ^^ 1 \ S 0^

^

1 1 1 —1 S - 2 / V y

^

j 1 _ -4 >.55- SI.

Ï

ce )

I

4.75-

k i

1—

4À6.90.

32.25 L 1 1 V J /

^

1 \ T /• -

~- • V [> ^

' > L ^ /*

1 s ^ ^

r^

32.00

I

1 -... S ^ ^

f / k ^^ ^ 1- 49.í

1

si - -65 5 [) * — ^

A1 on i ... »_

-^.o...

i.1 u ^

0

U. S. DEPAR

50 100 150 200 250

NITROGEN (LBS.)

*WITH POTASH APPLIED AT 40 POUNDS PER ACRE,

NUÍÁBERS ON POINTS REFER TO YIELD OF CORN IN BUSHELS.

TMENT OF AGRICULTURE N EG. 3985 - 57 ( 3 ) AGRICULTURAL MARK EJING SE RVICE

Figure 22

Corn: Yield per acre for given applications of phosphoric acid and nitrogen l/

Nitrogen in poionds

Phosphoric

acid 0 ii-0 80 160 214-0

Pounds Bushels Bushels Bushels Bushels Bushels

0 32.00 1+9.85 65.50 68.25 61.20

40 \ 32,25 ^9.55 61.55 7^^.75 66.90

80 23.2Í5 i^8.55 62.65 88.15 78.Í+5

120 : 20.20 50.75 69.80 86.80 81.90

1/ With potash at 40 pounds per acre.

Brown, William G., Heady, Earl 0., Pesek, John T., and Stritzel, Joseph A. Production Functions,

Isoquants, Isoclines and Economic Optima in Corn Fertilization for Experiments With Two. and Three

Variahle Nutrients. Iowa Agr. Expt. Sta. Research Bull. 441. I956. p. 815.

-U5 -
INDIFFERENCE CURVES

Beef and Pork^ 1914.7-^6

The first edition of the handbook included a diagram labeled "Indifference C"U3rves" based on data for beef and pork. Two periods were shown, 1921-31

and 1932-^1. The present chart shows a family of curves for the post-World

War II period 1947-56.

This is doubtless the most controversial diagram in the handbook. I

tried to justify such a diagram in a recent paper. 7/ But not all economists

accept my justification. Some of them think it is impossible to derive indifference curves from any analysis of market data.

In any case, the economists and statisticians that I know woiild agree

that a diagram such as the one presented here gives a satisfactory explanation

for changes in the ratio of beef prices to pork prices. Three observations

are plotted for each year. For example, the x marked '56 indicates that in

1956 the per capita consumption of beef was 84.2 pounds and of pork was

66.8 poimds. The slope of the line drawn through that point indicates a price

ratio of 1.42. In other words, in 195^ the consimier would have to give up

1.42 pounds of pork to get a pound of beef. After plotting for each year the

consuiirption of beef and pork, together with the price ratio, we note that the

ratio was not constant. It was high when pork consxmption was high and beef

consumption was low. It was low when pork consxomption was low and beef consumption was high. This is as we would expect. The light curves drawn

through this diagram are similar to the contoiir lines used in the diagram on

pa>ge 45 to analyze a problem of joint regression. In drawing such lines we

atteiEpt to portray a smooth 3-dii^iensional surface, changing gradually and

slowly. The slope of these curves is to be about the same as the slope of the

neighboring heavy straight lines indicating observed price ratios. You will

note that the observed price ratios for each year except 194? are almost the

same as those of the neighboring contour lines. The observed price ratio in

1947 was smaller than we wotild have estimated. This may be an indication that

the postwar demand for meat did not become stabilized until after 1947.

Certainly this family of curves does not tell us anything about the total

welfare or level of living of the typical American consvuner. To analyze this

we would have to study his entire expenditure pattern. However, in a sense at

least, I believe the diagram does represent a "partial indifference surface"

for beef and pork alone, indicating the amount of satisfaction obtained from

these two commodities. This assumes, of coinrse, that the satisfactions from

beef and pork are independent of satisfactions obtained from other goods and

services. Perhaps we have to take this assumption with a grain of salt. But

don't forget that we make assxmiptions in most statistical analyses. For

example, when deriving a demand curve from market data.

7/ Waiigh, Frederick V. A Partial Indifference Surface for Beef and Pork.

Jour. Farm Econ. 38:102-112, illus. 1956.

46 -
INDIFFERE^

Consumpf/on of Beef and

4CE CURVES

Pork Per Cap/fa, 1947-56*

PORK VLD5.)

"S ^ I

, SA. V X Í

\- ^ S \ \ ^r- t r \ ^ S ^ ^^ > 5 s, V \

^ S ^ \ ^r ÎS ^

V \S .52 N^ v^ ^ ^

k "^u ^ % " s V TH " "5 ' t U\ \ N^ _ -S-- S --

/U ^^ \^P a ^ A " \ vSif°, K \

.1

,1 - \ fs^ > -^ \r-- çQ-^. -56,

L^ 48C" ^J\ fe~^jio49 . - wS^ N V K

5fc ^ iki "S ^^ S- C - w- \- \ s V. ^L,/53_ ^ ^

^^ \ -\ ^Li S ^ S^ \ ^^ St \ s AH S----\ ..S >L.54 .:...%. OU Ï—^^ . \ \ ^^ ^

\ ^^ - ^.^ s ^ ^< \ S^_ > s s >

r ^ S_ s s; s ^c^ «s: s. \ s s ^ \ s

^ »«r A

f ^n _— ::— :

50 60 70 80 90

BEEF (LBS.)

SLOPES OF HEAVY LINES ARE PROPORTIONAL TO THE PRICE RATIOS AT RETAIL.

SEE TEXT FOR METHOD OF DRAWING CURVES.

1956 DATA ARE PRELIMINARY.

U.S. DEPARTMENT OF AGRICULTURE NEC. 3987-57(3) AGRICULTURAL MARKETING SERVICE

Figxrre 23

Beef and pork: Ratio of beef price to pork price at retail and per capita consianption, 1947-5^

r-"- - Retail price per pound ]

Price

ConstiBiption

i Year ;

Beef Pork '

ratio Beef ; Pork

Cents Cents ! Poimds Pounds

, 19^7 ..... 61.8 55.5 1.11 ! 68.6 68.6

r 1948 75.3 56.5 ! 1.33 ! 62.3 66.8

19it9 68.1+ 50.6 : 1.35 ! 63.1 66.8

1950 i' 75-4 50.3 ! 1.50 Î 62.6 68.2

1951 88.2 5'».3 ! 1.62 : 55.3 70.9

, 1952 86.6 52.1 1.66 ! 6l.lt 71.**

1953 69.1 ^T.k 1.20 : 76.5 62.6

X95h 68.5 58.3 ! 1.18 : 79.0 59.2

. 1955 67.7 49.2 : 1.38 ! 80.9 65.9

1956 : 66.0 1*6.6 : X.k2 : 81f.2 66.8

Agricultural Marketing Service.

-hl -
LINEAR PROGRAjyiMING

Combination of Two Farm Enterprises

Programming is the planning of economic activities to maximize income or to minimize costs. In some cases it is reasonable to assume that the input-output relationships are approximately linear. For example, if we know how much seed, labor, and

fertilizer is required to grow an acre of potatoes, we can assimie 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, calcium, and other

nutrients in a bushel of corn, there would 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 linear programming.

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

six input factors. 8/ To be feasible a combination of inputs must not require more

than the available amoTmt 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 amoxmt 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 k.63 times as

much spring land as the farmer has available. The left scale of the chart represents

the proportions of available resources 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 fall cabbage. If we had to choose one or the other of these

enterprises, the choice should be fall cabbage, 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 fall cabbage

would require 2.17 times as 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 $¿i-,6o8. This is better than he could get from beef cattle alone.

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 46 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 minimax 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 9I 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 wo\ü.d 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.

8/ See King, R. A., and Freimd, R. J. A Procedure for Solving a Linear Programming

Problem. N. C. Agr. Expt. Sta. Jour. Paper 503^ I8 pp. 1953- (Processed.) This study

lists 9 different inputs needed to carry on each of 6 different enterprises.

- 1^8 -
LINEAR PROGRAMMING

Combination of Two Farm Enterprises

BEEF CA'rTLt' x FALL CABBAet^

f ^^

À §^ Fa 1 Land _ > 4 ,. "ï^ >

4

**'"*.^

^

^^^^

*^---^>^^

^ .Producti

W feat ^ , o

on capital y5h ^^^^ O

i # "^Sj **"*N^ "^ ^

^5a

Sorina land ^.^ '*^i^ **• ^^

r\ ^^ ^ *'"*"^^ ^-^"^^ r\ 2 *^^s ^ '^^^ «. "* "^ ^'f" "*"* =B ^

'^

Sh ^^"^v "^ 1 "^ ^ Sept.-Oct. labor i^> ^^ *^S 1

1 ^' P"*^^^ ^S "^^J' ^

t KIA. ,. r\^^ 1 1 1 \\"^^^ 1 Mill ;,v, rrr^rf^.'i i i Lw-tri i i ii ii ii -.^—'•"'^— N^-'"""'l *^w.

i\î ^-""^"^ II L — — —^-4_-L-C —

"^C ^^J^+--rLJ-J=ata*SM-r! MINIMI iVliilv-Aiin Inhor 1 1 M 1 II ! PSwl II 1

w ^^ . —äH^-"*-rrri i ii ii ii i M i i ii ii i i. . . . ,,,,,, M 1 1 i 1 i; M r^ini

f

''Ó 20 40 60 80 lOO'

RESOURCES USED FOR FALL CABBAGE (PERCENT)

PROPORTION OF RESOURCES REQUIRED TO PRODUCE $10,000 NET INCOME.

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

Figure 2k

Beef cattle and fall cabbage: Proportion of available resoiirces reqiiired to produce

$10,000 net income for a farm. North Carolina

Proportion reqiiired

Resource

Beef cattle Fall cabbage

" Land;

^ Spring

^ Fall

1i Production capital .

|i Labor:

July-August

September-October

November-December

k.63

h.63

3.78

0

0

0

0

0.80

1.80

1.08

2.17

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. 13-

'; - k9 -
The Minimum-Cost Dairy Feed

Here is another diagram that is useful in linear programming.

In this case^ we vant the least cost combination of feeds that will

meet stated requirements. The prices of several feeds are given;

also such requirements as tota^l 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^ and so on. The net

resiilt is shown on the table and plotted on the chart.

We then consider combinations of two feeds that wi3,l 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 and middlings. A balanced ration would lie on a line

through the origin having a slope of ^5 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 amount

of middlings. It can be shown that this combination will meet the

two requirements at less expense than either feed alone. This is

true because (l) the line joining the two dots slopes downward to

the right and (2) it crosses the if5-degree 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 any 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 wo\;ad

indicate that the cost wo\ild 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

an article published in 1951« 9/

The one drawback to this kind of diagram is that we can consider only combinations of two feeds meeting two nutritive reqmrements. Yet, in many practical cases we want to study combinations

of three or more feeds meeting three or more nutritive requirements.

The next diagram will show how these principles can be extended to

three dimensions—in other words, how we can find the least expensive combination of three feeds meeting three nutritive requirements.

9/ Waugh, Frederick V. The Minimum-Cost Dairy Feed. Jour. Farm

Econ. 33:299-310, illus. 1951-

50
LINEAR PROGRAMMING

The Minimum-Cost Dairy Feed

.1 .2 .3 .4 .5

TOTAL DIGESTIBLE NUTRIENTS*

* PROPORTION OF REQUIREMENTS IN $1.00 WORTH OF FEED.

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

Figure 25

Dairy feed: Proportion of the requirements for protein and total

digestible nutrients supplied "by $1 worth of each feed

Proportion supplied

Feed

Digestible

protein

Total digestible

nutrients

Corn

Oats •

Milo maize

Bran

Middlings ......

Linseed meal ...

Cottonseed meal

Soybean meal ...

Gluten

Hominy

0.136

.187

.203

.321

.332

.if00

.h6k

.i+12

.158

O.il-iH

.375

.if 23

.1^36

.272

.268

.286

.395

Waugh, Frederick V. The Minimum Cost Dairy Feed. Journal Farm Economics. 33:299-307, illus.

1951.

- 51 -
LINEAR PROGRAMMING IN THREE DIMENSIONS

The Minimuin-Cost Broiler Feed

The preceding diagrajn illustrated a method of finding the least-cost combination

of two feeds meeting two nutritional requirements. In many practical cases^ of course^

we are concerned with combinations of more than two ingredients meeting more than two

requirements.

In the case of "broiler feeds, for example, poultry nutritionists consider about

20 different requirements which must be met by any satisfactory mixed feed. This makes

a complete graphic analysis impossible because we cannot graphically portray 20 different dimensions. We can, however, work with at least three dimensions at once, as we

have already seen in the chart on page 43. In that chart we were working with a

3-dimensional regression surface. Here our problem is quite different. But we are

again working in three dimensions and can use the same kind of box to plot our data.

In this case each hatpin represents some feed ingredient. The hatpins are numbered at the base. For example, the hatpin numbered 1 represents the first feed shown

in the table; that is, soybean meal. The head of each hatpin shows the percentage of

the required amounts of protein, productive energy, and non-fiber that could be bought

by one dollar's worth of some ingredient. For example, hatpin number 1 shows that one

dollar's worth of soybean meal would buy 3.6 percent of the requirements of protein,

1.4 percent of the requirements of productive energy, and I.63 percent of the requirements of non-fiber. Similarly, for the other hatpins. Each is identified by the same

number as in the table.

Now, look at the wire which runs diagonally across the diagram. This wire corresponds to the 45 degree line in the chart on page 51. Any point on this wire would be

a balanced ration; that is, it would have equal percentages of protein, productive

energy, and non-fiber. The object of our prograoaming is to get a feed which will have

100 percent of each of these requirements—and also 100 percent, or more, of the other

requirements which are not considered here. To get 100 percent of all these requirements would take us far outside the scope of this diagram. However, we want to find a

combination of feeds which will get us as far up on the wire as possible.

The optimum in this case is a combination of feeds 5, 10, and I3. These are meat

and bone scrap, corn, and hominy. To show that this is the optimum combination, we have

laid a plexiglass plane on top of the hatpins representing feeds 5, 10, and I3. (For

$1 we could buy any combination of these three feeds lying on the plane represented by

the plexiglass.) A balanced combination of these three feeds wo\ad be at the point

where the plane is cut by the wire. At that point, we can get almost 2 percent of each

requirement for $1. In other words we can get 100 percent of each requirement for a

little more than $50. It is fairly easy to see graphically, even in a photograph, that

this combination of feeds can get us higher up on the wire than could any other combination.

One further point should be noted. The hatpin near the top of the wire does not

represent a feed ingredient. It simply represents the point at which an imaginary feed

would have two percent protein, two percent productive energy, and two percent nonfiber. There is no such feed. It is plotted here only to locate the wire which is

needed to find a balanced mixture.

The geometry of this problem could be discussed in more detail. For present purposes perhaps it is enoiigh to indicate that for the solution to be an optimum the plane

must slope downward in both directions. It does so in this case.

- 52 -
LINEAR

PROGRAMMING

3-Dimensionol

Analysis of

Broiler

Feeds

U. s. DEPARTMENT OF AGRICULTURE NBG. 4260-57 (5) AGRICULTURAL MARKETING SERVICE

NiJiriber

on

chart

1

2

3

h

5

6

7

8

9

10

11

12

13

li+

Figure 26

Broiler ration: Percentage of non-fiber, productive energy

and protein met by $1 worth of feed, specified feeds

Feed

Percentage of required nutrients

Non-fiber

Meal :

Soybean, kk percent »..

Linseed, solvent

Cottonseed, expeller

Gluten

Meat and bone scrap

Meat scrap

Fish meal, menhaden

Buttermilk, dried

Corn distillers solubles, dried

Corn

Milo

Wheat, standard middlings

Hominy feed, yellow

Barley

Percent

1.63

1.30

1.33

1.2if

1.70

1.33

.69

.32

1.35

1.75

1.57

1.7^

1.83

1.75

Productive

energy

Percent

l.k

.8

1.3

1.1

1.6

l.k

.7

.3

1.5

2.1

1.9

l.k

1.7

1.6

Protein

Percent

3.6

2.2

2.8

2.6

k.O

3.k

2.0

.k

1.8

.8

l.k

1.0

1.0

- 53
AVERAGES

Gross Profit from Storage

In economic analysis 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 various amounts of eggs. In

deriving these figures, allowance was made for the effect

of disposable income on prices of eggs. The prices shown

are those that might have "been expected with income at

its average level for the period 1940-48. A practical

question is whether it wo\ild he profitable to store up

the surplus in periods of large production and to sell it

in periods of small production.

Suppose we produced 40 billion eggs in one period

and 50 billion in another period. The retvirns 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

curve. In this case it indicates a moderate gross profit

from storage. That is, the gross income from selling

45 billion eggs in each period would be greater than the

average income from selling 4o billion in the first

period and 50 billion in the second. Costs of storage,

handling, and any loss in qimlity would have to be deducted in order to determine whether net returns would

be larger from storage.

It is easy to see that there will be a gross profit

from storage if, and only if, the returns curve is concave downward. The degree of curvature is an important

indication of the possible amount of gross profit.

Of course, this is only one of the many uses 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 computed graphically with little time or trouble.

There is no need to read the nimibers from the diagram,

copy them on a piece of paper, add them, divide by two,

and put the average back on the diagram. The simple arithmetic average of any two points on any curve can be located

graphically by the graphic method explained here.

- 54 -
TOTAL

($ MIL.;

DPTIIDMQ

AVERAGES

Eggs: Gross Profit From Storage *

r\ k 1 v/ IX 1 -w */ ¡aasSS ..1 Sa».^--

( ^t0^^^ /i- '^^

, ^0^ /. ^^ '48 1 nnci /^ t ^ *- - - ■■ ^^^LA '46 1 ,uuu Average t— "" "" "" •43^%^' .

-^ 47 *s

-t- *Srf4 it ^^^

^r

_j

son - sJ\J\J

1

f\

" 35 40 45 50 55 60

TOTAL PRODUCTION (BIL. EGGS)

:¥ RETURNS ADJUSTED FOR ESTIMATED EFFECT OF DISPOSABLE INCOME ON PRICE.

U. S. DEPARTMENT OF AGRICULTURE NEC. 1334-55(1) AGRICULTURAL MARKETING SERVICE

Figure 27

Eggs: Production, price per dozen received by fanners, and total returns, 19kO~k8

Total

Year : Production Price

1/

returns

i/

Million

: Billions Cents dollars

19^0 ... i 39.7 36 1,188

19^^! ... : i+1.9 35 1>225

19^1-2 ... : kQ.6 30 1,230

19^3 ... : 5i^.5 22 990

I9^if ... : 58.5 16 leh

19^5 ...

: 56.2 20 9^0

19^6 ... : 56.0 21 987

19^7 ... 55.i^ 21 966

1948 ... : 54.9 22 1,012

1/ Adjusted for estimated effect of disposable income on price.

derick L.

- 55 -

Data derived from Figure 92 in Thomsen, Frederick L., and Foote, Richard J. Agricultural Prices.

New York. 1952. p. ^31.
EIASTICITY

Coefficient of Elasticity of Deiaand

Maoiy economists have trouble with coefficients of elasticity.

They are frequently concerned with the elasticity of demand—more

precisely, with the elasticity of consiamption with respect to

price. The diagram shows how this can be measured graphically.

The curved line on the diagram represents an assiimed demand

curve for eggs. The scales for consimiption and prices would 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 curve at that point. The elasticity in

question is -a/b. For this example, this equals -35-5 divided by

64.5 based on the scales shown. In terms of sioall 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. 10/ It

derives from, the definition of elasticity m = -3 • -£-. Note ^ / dp q

that -, and (by similar triangles) ^"v" = —-• Also dq,:

dp

. = _

p=a. smd q=c.

a-fb a+b

dp q b c ^

Some economists have found the concept of elasticity so difficult that they have used "arc elasticity," or the "average

elasticity of a curve." If the graphic approach to elasticity is

used, there is little need for such concepts. The elasticity

coefficient shown here is exact and easy to compute.

We should note that the concept of elasticity applies not only

to demand curves—^but to any curve. When we speak of the elasticity

of demand we (usually) mean the elasticity of consumption with respect to pric^. But we might want the elasticity of cost of producing potatoes with respect to the amotint of fertilizer used, for

example. Whatever the curve, we can measure its elasticity at any

point, using the same graphics as shown here.

10/ Marshall, Alfred. Principles of Economics. Ed. 8, pp. 102-

103. New York. 19^8. First published 1920.

56
ELASTIC

Demand For

CONSUMPTION (NO. PER CAPITA)

:iTY

^ggs

-aH

4rSN ^ A

orvA ^-- ». oUU "*->. IT T- ^'^^\ d ^sï**

'VL

"^i ^

^^^ L

1 5fc

_i *^ 2 ' ^ *-• Ann - -1 - !^^ * jtli M ^J» 4UU ^r "j ^S^ ^"'^ ••ji.É u. C Ij ^ ^ ""^'Ji

^s^

''s^

"^s

^v

^^

0 .2-; ^-j ; mJSLm mlm ^^

0 20 40

RETAIL PRICE (<

U. S, DEPARTMENT OF AGRICULTURE Ï

60 80 100

PER DOZj-p

^EG. 1335-55(1) AGRICULTURAL MARKETING SERVICE

Figurée 28

Eggs: Consiaiiption per capita associated with given retail price per dozen

Constaaption Price

NiÄttber Cents

777 20

621 30

530 ko

h69 50

h^k 60

390 70

362 80

Based on an assumed elasticity of demand coeffic

Karl A. Analytical Tools for Measuring Demand. I

;ient of -0.55.

J. S. Dept. Agr.

See Foote, Richard J. and Fox,

Agr. Handbook 6k, 1954. p. ko.

- 57 -
METHOD OF DETERMINING MOST PROFITABLE OUTPUT

Applications of Nitrogen on Com

The chart on page 23 shows, based on experimental results, the relationship between obsearved yields of corn and the amount of nitrogen applied per

acre. The discussion of that chart promised that we would discuss later a

graphic analysis of the most profitable rate of fertilizer application. The

chart facing this page is an attempt to carry out that promise. Several persons who read the first edition of this handbook sxiggested that this type of

graphics be included. One of the men who suggested this was George G. Judge

of Oklahoma Agricultural and Mechanical College. Professor Judge suggested

the general type of chart shown on the facing page.

The dots and the heavy curve shown on this chart are the same as those

shown on page 23, except for a change in the scale for corn yields. The

straight line OA at the bottom of the chart shows the nimber of bushels of

corn required to pay for various amounts of nitrogen when one bushel of corn

will buy 10 pounds of nitrogen. This woiold be the case, for exemple, if a

farmer could sell corn for $1.50 a bushel and could buy nitrogen for 15 cents

a pound.

The most profitable rate of nitrogen application with the given price

ratio is determined by drawing a tangent to the input-output curve parallel to

line OA. To do this, we lay one edge of a transparent triangle along line OA,

place a straightedge along one of the other sides of the triangle, and slip

the triangle along the straightedge until the edge that was touching line OA

now just touches the input-output curve. Then that edge of the triangle will

be parallel to line OA and we draw the l^ne indicated.

At this point the farmer would b\jy about l60 pounds of fertilizer to the

acre and would expect to get a yield of about 1^6 bushels. If he xised more

fertilizer than this, it would cost him more than the value of the additional

corn. If he used less fertilizer, the saving in fertilizer would be less than

the value of the additional corn.

A similar analysis coiold be made, of course, with any asstmied prices of

corn and nitrogen. ' Changes in the price ratio would change the slope of line

OA. Therefore, they would change the slope of the tangent and would move the

point of contact between the tangent and the input-output curve. In this particular case, however, there would be little change in the most profitable

rate of nitrogen application unless the price of nitrogen were greatly increased or the price of corn were greatly decreased. This is because the

input-output function curves vary sharply in the neighborhood of the point

that is most profitable with the price assumption shown on the chart. Assuming

thlt our input-output function is approximately right, applications of much

more than l6o pounds to the acre would obviously be unprofitable since the

maximimi corn yield is apparently attained with a fertilizer application of a

little over l60 pounds of nitrogen.

This general type of chart is useful in analyzing a wide variety of economic problems. It is not limited to fertilizer applications, but applies

just as well to such problems as the most profitable amount of corx©eBtrates to

feed to dairy cows, or the most profitable amoimt of labor to be used in any

operation whether on the farm or in marketing.

- 58 -
METHOD OF DETERMINING MOST PROFITABLE OUTPUT

Corn: Yield Per Acre In Relation To Applications of Nitrogen

Yim n iRii 1 ^-.SMBI TICLU VDUw -r*""ir35 "

L\-L -t- -\t ±

: ^: _ ]È''-.: :

\w 1

^2- - . .¡ne dra>vn paral el to (t A

: :- -1^^ .

/^ ,2 1 r^n ^t

UU -y -"

/ - - >^

- /

A^ rr - -^

nitroaen when one bushel of corn

^yrr\ „- . . >uys 10 pouncs of nitrogen | |

\wi A

W ^^"1* "" *"

/[^ ai"* '* ""*'

1 •*' '*" ""

• e"'"'*'**'*

>st profitab e rate of nitrogen app ication

lili il 1 1 I 1 1 1 i i 1 > 1 < > 1 1 1 1 1

^..-as--"'"' Mc

-«^-*"" .Ml fS _-- "! -- _- . tJA 1 I 1 1 II 1 1 1 1 1 1 1 1 1 1 111 M 1 1 1 1 1 M

"Ô 80 160 240 320 400

NITROGEN (LBS.)

U.S. DEPARTMENT OF AGRICULTURE NEC. 3988-57(3) AGRICULTURAL MARKETING SERVICE

Figure 29

Corn: Yield per acre "by specified quantity of nitrogen applied^ Ontario, Oregon

Nitrogen

applied :

Yield of :

corn :

: Nitrogen

: applied :

Yield of

corn

Poimds : Bushels : : Pounds Bushels

0 \ 6k.e \ *: 160 \ 146.8

ko : 90.i^ : \ 180 : l4l.2

80 ! 118.2 :' : 200 : 1^7.1

100 : 132.4 : : 2lf0

: 280

145.8

120 : li^0.7 : 147.4

l40 ! lifl.O \ i 320 143.8

Paschal, J. L., and French, B. L. A Method of Economic Analysis Applied to Nitrogen Fertilizer

Rate Experiments on Irrigated Com. U. S. Dept. Agr. Tech. B\ill. Il4l. 1956. p. l6.

- 59 -
DIFFEREHTIATION

Scane statisticians and economists find calculus a

difficTxlt subject. Differential calculus is relativelyeasy if you do it graphically. The differential at aioy

point on a curve is siioply the slope of a tangent dravn

at that point. The tangent can be drawn easily with a

ây

transparent straightedge. The differential, "to"^ ^®

the slope of this tangent.

In figtore 30, 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 figozre 31, the slope is -0.002 (that is,

y decreases 0.002 "units for each increase of one unit of

x). Thus, in figure 30, -3^ = 5-8, and in figiore 31,

dx

dx

-0.002.

These differentials can he read most easily by drawing the dotted lines shown on the diagrams. These dotted

lines are drawn parallel to the tangent and through 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 straightedge along another side of the triangle, and then slip the triangle along the straightedge.

With a little practice it is very easy to draw parallel

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 unit on the

x-axis. In figure 30 it is 5*8. In figure 31 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 quick 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 are shown merely to illustrate

the method.

- 60 -
DIFFERENTIATION

VALUE OF Y

^'^ ^^^

r^rs 20 ' ____ - -^;?- ^^^

g^2^^

. ^

^

«^ g ^

^^^^ ^ ?^

^ "^ ^ ü*"

inly " -¿j^t ^^''___ _

V*^ ,*î "í! -/-^^"^ -4 -J,^ -^^ ^^^

x"^ ~> -^ -.- .

^^ " --'""

'^ étt.^ - ^ rlw/rlv-l^ SI ;^'"^ > ay/axsD.o

u^"" 1 1 1 1 1 1 1 0 -cl"«-«------. ^. 1 1 1 1 IJ ^ -.

VALUE OF X

U. S. DEPARTMENT OF AGRICULTURE NEC. 1336-55(1) AGRICULTURAL MARKETING SERVICE

Figure 30

DIFFERENTIATION

VAI

3

2

1

0

-1

-2

LJE OF Y I I -=-: ::::":::::::: "i: : "::~ :^: "":_ "

_!S^_ _ lt_ _. „ _ ^ _

* ** ^ ^ *• h

***** ^5l '" iil^^fc.

* sKi ^

'S * n "^

^"'^illBMB" "

* i *"

*"*'** *

* * S 5 ^

"*** !

*** Bi

_ :5a.^,^_ d>/dx =

..^ü^í»^. ^ _ _ r A A/^1

llllllllllllllllllllllllllllllin^liïHItHlitt^J^ffli

200 400 600 800 1,000

VALUE OF X

U.S. DEPARTMENT OF AGRICULTURE NEG. 1337-55(1} AGRICULTURAL MARKETING SERVICE

Flgxire 31

- 61 -
DERIVING A MARGINAL CURVE FROi AN AVERAGE CURVE

Marginal Retiarns

Often the economist has a demand c\irve showing estimates of average prices corresponding with a range of

qtiantities sold. His problem loay call for an analysis of

marginal returns (or marginal expenditures of consiimers).

The easiest way to do this is to find graphically several

points on the retoams curve.

Robinson 11/ explained the geometry of this. Briefly,

total returns are R « pq. We want -~- = p + ^^ q. We

can take any point on the demand cuarve, such as point A in

our diagram (32 pounds, at an average price of kl 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 demand ciirve (that is, at 4l cents). This parallel cuts a

perpendicular dropped from A at point B, and the price

equivalent of B measiores the inarginal returns corresponding to the q\xantity sold at point A on the demand curve.

Here marginal returns are 10.5 cents when 32 poimds 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 returns curve, and then

draw the whole cxarve.

11/ Robinson, Joan. The Economics of Imperfect Competition, p. 30. London. 1933.

62 -
MARGINAL RETURNS

Chicken Meat

PR CE It PER LB ) -D- *

1

^^ ^^

*^^ \^

I -'^Ne^Sl. '^''ííb -"^5iir aifc t A ~

À /\ mm, .. ...... «. .. .... .. .« .. ..->.... 40 ^^iti—r"ttr TT"" 'ZÄ _. _ : îfc

*^ 5 ¡A^

^. 3^ '"' í M L

-5i- ^fcSte X^.^v^ ^5 j^ 1^, ^ ^ ^

** Il on _ :J^ ,, , *^5|. zu _ Z^'7 ::: — ::—:: ^^^s:—

î^^- "^.^

4^ ^îi

,^ <^ D-iviarginai reiurn^

^ of 1 ).5 cents

n

0 10 20 30 40 50 60

CONSUMPTION (LBS. PER CAPITA) -q

U. S. DEPARTMENT OF AGRICULTURE NEC. 1338-55(1) AGRICULTURAL MARKETING SERVICE

Figure 32

Chicken meat: Price per poimd at retail associated with given levels of cons-umption per capita

Price Consumption

Cents Pounds

59.1 20

^9-9 25

h3.3 30

38.8 35

35.1 i+0

32.1 45

29.7 • 50

Regression coefficient based on the reciprocal of an ass\xmed elasticity of demand coefficient of

-1.33, See Foote, Richard J. and Fox, Karl A. Analytical Tools for Measuring Demand. U. S. Dept.

Agr. Agr. Handbook 6k. 195^. p. 40.

- 63 -
Marginal Costs

A marginal cost ctirve can 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 curve will be above the average cxurve. To find

the marginal cost at point A in the diagreun, 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 line from point A to the cost axis and note the

point at which this line cuts the axis. We then draw a

line thro\:igh this point that is parallel to the tangent.

The cost at which this line cuts the perpendicular line

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 unit 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 xmit 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

can 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 what x and y represent.

- 6k
MARGINAI

Cost of Land and Fertilizer for

AV. COST OF OUTPUT ($ PER UNIT]

COSTS

Varying Inputs of Fertilizer

\ • Y T"

"J" f'

ACi I- -„- ««: j

.4U »—m

w

.«^_ n_AA*l '^

'Il ^ _ ._ E-A/arginal cost -. -- ^p[..

•^^ - -in onri 1 of 0.292 d<altars JL

A1

-^

on -S .oU _ "^^ g IB P "" ^^ — — —- —

* *

^ p0

^ L. * '* ||P| —• '''*'' '

• * " ..^ É ** —^B^*

0 ^ u*''^~* ^ ^¿e***"" (Ji^!

__5^^--??!:__. ;:|AEEE::::::===============

on - -^..^-äS:!!- --

.ZU _

"

IC «.

0 12 3 4 5 6

INPUTS OF FER

U. S. DEPARTMENT OF AGRICULTURE

7 8 9 10 11 12

JILIZER($)-X

NEG. 1339-55(1) AGRICULTURAL MARKETING SERVICE

Figure 33

Total output of a given crop and cost per unit of output for given inputs

Cost of total input

Total

Cost per unit of output for—

Land Fertilizer output Land Fertilizer Total

Dollars 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 h : 62 .161 .0645 .2255

10 5 : 64 .156 .0781 .2341

10 6 67 .149 .0895 .2385

10 7 : 6e> ,147 .1030 .2500

10 8 : 69 .145 .1159 .2609

10 9 r 70 .1^3 .1287 .2717

10 10 : G\ .161 .1562 .3172

10 11 : 48 .208 .2294 .^37^

Black, John D. Production Economics. New York. 1926. pp. 317-318.

-65 -
ROOTS OF A POLYNCMEAL

x3 - 1.22^0 x^ + 0.3695 X - 0.0183 = 0

Some of my good friends, including Professor Charles H. Merchant, of the

University of Maine, think it is out of place to discuss the roots of a polynomial in a handbook dealing with graphic analysis. Roots of polynomials are

used mainly in high-povered mathematical studies dealing with such things as

canonical regression, component analysis, and cyclical variation. But graphics

can help, even in these studies. So other friends have induced me to leave

this piece in the handbook.

We have included this diagram to illustrate the use of graphics in connection with more elaborate mathematical techniques. The particular polynomial

is taken from Tintner. 12/ Tintner 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. Thus if x=0, the polynomial eqimls

-0.0183; so we plot y= -O.OI83 corresponding to x=0. If x=0.1, the polynomial

equals O.OOT^j so we plot y=0.007Íí- corresponding to x=0.1. We proceed to compute several points on the curve, y= x3 - 1.2214-0x2 + O.3695X - O.OI83. When

we have enough points, we draw a cu3rve 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 diagrajn near the root. Thus, we could draw a new diagram for the part of

the oxarve between x=0.76 and x=:0.8o, plot the curve on a blown-up scale, and

compute the largest root more accurately., This could be repeated until we

obtained as many significant figures as wanted.

As a guide to the parts of the curve that must be plotted, we know that

there must be as many roots as the degree of the curve. Here we have a thirddegree polynomial, 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 are beyond the scope of this handbook.

12/ Tintner, Gerhard. Econometrics. New York. I952. Taken from equations

(iHj on p. 119, letting x= \2.

- 66
ROOTS OF A POLYNOMIAL

y = X^- h2240X'+ 0.3695X - 0.0183

VALUE OF Y

ni ^2-- -k^ ---

t

• U1 J^ .....^ T

If"

X

n.--JL-.-....-.--.-.S..-.-- £L U—f^ X-j^SZ— "fv

r > ^ y^\ # V

JL .0.C6 C .38 ^^ -,' CJl

n 1 .Jt -- \

/

-.Ul TT S TT"

^

r 2

Ik JF

AO ----- - - S^ ¿

-.Uz N N ^

r\À

•""Ô 1. .2 .3 .4 .,

VALUE

U. S. DEPARTMENT OF AGRICULTURE

5 .6 .7 .8 .9 1.0

: OF X

NEC. 1340-55(1) AGRICULTURAL MARKETING SERVICE

Figure 3^1-

Values of a third-degree polynomial, y, at specified levels of x l/

X y

0 »•*•.••.•.......••. ; -0.0183

.1 ... .0074

.2 ... .01if6

.3 ... .0094

.h ... -.0023

.5 ... -.01if6

.6 ... -.0212

.7 ... -.0104

.8 ... .005S^

1/ Y = x3 - 1.22ÍÍ-0 x^ + 0.365?5 X - 0.0183.

Data compiled using equations (l8) as a basis and letting x = K^. Tintner, Gerhard. Econometrics.

New York. 1952. p. 119.

.67-
SOLUTION OF SIMULTANEOUS EQUATIONS

Determining the Supply Belov Which No Grain Should

Be Stored in An Optimal Storage Program

We end this handbook with another use of graphics as an aid to mathematical computation. Statisticians often must solve two or more equations simultaneously. Various

methods of solution are available, including the popular Gauss-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 Gauss-Doolittle method. But we shall not

explain the procedure here. 13/

To solve any pair of equations, we substitute several successive values of x in

each equation, compute the corresponding values of y, plot the value of y corresponding

to each value of x, and draw a smooth cxxrve through the observations. When these

operations are 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 up to four

solutions to a pair of equations, depending on how they are situated one to another.

For eqimtions of any degree, solutions are real wherever the curves cross one another;

otherwise they are imaginary.

Gustafson, in an unpublished manuscript l4/, outlines some methods for determining

storage rules which are optimal in terms of certain, economic criteria. These optimal

rules can be obtained exactly by mathematical solutions that involve the use of calculus. However, a method is outlined by which appîroximate rules can be obtained by

carrying out certain essentially arithmetic operations* One of the necessary computations requires the obtaining of a value for k, which represents the supply below which

no grain should be stored. To obtain this value of k, we must find a solution for two

curves which show the relation between k and another variable, L. One of these c\u:ves

is obtained by a tabular method by which values of L are computed for given values of

k. Results of this tabulation are shown by the curved line on the facing chart. The

second relation is a linear one in which value's of k are computed from the values of L

obtained by the tabular method. The formula used is shown on the chart. The desired

value for k is obtained from the intersection of these two curves. The chart shown

here suggests a value for k of approximately 31.O bushels per acre.

Gustafson suggests that instead of drawing the complete curves, the range within

which the solution lies be determined by inspection so that only a pair of values for

k and L respectively is involved. A greatly enlarged graph is then drawn, and the

respective points connected by a pair of lines. When this method was used, a value for

k of 31.01 bushels was given. In this instance a graphic solution was the only feasible method, since the mathematical formula for the curve plotted from tabular values

was not known.

13/ The method is described in Maxfield, John E. and Waugh, Frederick V. A Graphic

Solution to Simultaneous Linear Equations. Math. Tables and Other Aids to Computations,

5:2tó-2¿f8, illus. 1951,

14/ Gustafson, Robert L. Optimal Storage R\iles for Grains, unpublished manuscript,

1957.

- 68 -
SOLUTION SIMULTANEOUS EQUATIONS

Constants Used in Obtaining Optimal Storage Rules^

VALIJF OF K

' k -

^N _ - -.

5L :

^ - il». , '

QO I -Sifcr ÓÁ ^'^ç;""

11" " '^ c=:! .2^-ô.îszL "i:

- ■■«B -- - -"'**^^ -

- -_ ..-ikHHHaa^.^

"" ^ n

ouon - --::--::.-.:-.-: 5 *

ILij" *"' •• fc ta

IlCurve plotted from tabular values .

Z¡¡¡¡^^^ * ¡i ^

^^ *' "í* M " ! r

9ft :___:___:::-__:_.:::-::::-:: —

0 .2 .4 .6 .8 1.0

VALUE OF L

JEE TEXT FOR METHOD OF OBTAINING CURVES.

U. S. DEPARTMENT OF AGRICULTURE NEC. 3989-57(3) AGRICULTURAL MARKETING SERVICE ,

Figure 35

Values of k and L as camputed in alternative vays

Tabular method

Value of k as coiHputed

froBi fortnula

29.5

30.5

31.5

32.5

1.02

.56

.27

.10

30,66

30.92

31.09

31.18

Gustafson, Robert L. Optimal Storage Rules for Feed Grains. Unpublished manuscript. 195?.

•• t>y -• T^U. S. GOVERNMENT PRINTING OFFICE : 1957 O -430756