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by Frederick V. Wough, Director
Division of Agricultural Economics
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Agriculture Handbook No. 128
UNITED STATES DEPARTMENT OF AGRICULTURE
Agricultural Marketing Service
Washington, D. C.
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CONTENTS
Page
Introduction .......................... .............................. 1
Frequency distribution .................................................. 2
Cumulative frequencies .................................. ............... 6
Lorenz curve ..... .................... .............. ............. ........ 8
Trends ................. ..................... ....... ...... .. ..... .... 10
Cycles ..................................... ............................. 14
Seasonal variation ....................... ............................... 18
Simple regression ............................. .......................... 22
Comparison of time series ............................................... 32
Multiple regression ..................................................... 34
Joint (3dimensional) regression ........................................ 42
Use of isoquants to study joint regression .............................. 44
Indifference curves ..................................................... 46
Linear programming ............ ....... .................................... 48
Linear programming in three dimensions .................................. 52
Averages ................................................................ 54
Elasticity .............................................................* 56
Method of determining most profitable output ............................ 58
Differentiation ......................................................... 60
Deriving a marginal curve from an average curve ......................... 62
Roots of a polynomial ....................... ............. ............. 66
Solution of simultaneous equations ...................................... 68
Issued July 1957
This publication supersedes Graphic Analysis in Economic Research,
Agriculture Handbook No. 84.
For sale by the Superintendent of Documents, U. S. Government
Printing Office, Washington 25, D. C. Price 40 cents
GRAPHIC ANALYSIS IN AGRICULTURAL ECONOMICS
By Frederick V. Waugh, Director
Division of Agricultural Economics,
Agricultural Marketing Service
INTRODUCTION
Since Handbook Number 84, "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 agricultural economists (and also many nonagricultural
economists) feel that a handbook of this kind meets a real need.
When our supply of the original handbook became exhausted, we decided to rework it
and get out a new handbookthe present volumerather than simply republishing the old
one. I wish I could thank here everyone who has made 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 land 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. They are Richard J. Foote, Hyman Weingarten,
and James R. Donald.
Graphic analysis has a long and honorable history in agricultural economics re
search. 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 should be used right along with the newer and fancier gadgets. In my opinion,
there is an unfortunate tendency, especially among socalled econometricianss," 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 numbers 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 consider
ing. To do this, he must understand economic theory and he must also be able to see
the nature 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 misused. So has any kind of sta
tistical analysis that can be named. Whatever tools are used, there is no substitute
for sound 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 1949
The economist often deals with averages. For example, he may be analyz
ing 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 understanding about the degree of variation around 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 would
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 agricultural 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 around 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 socalled "frequency distribution" of prices paid by farmers through
out the United States for laying mash in September, 1949. The average price
is a little more than $4.50 a hundred pounds. The range is from $3.40 to
about $6. This range is, of course, due to many things, including geograph
ical differences that reflect freight rates and differences in the ingredients
used in the mash.
Each of the bars in this diagram shows the number of dealers reporting
prices paid by farmers within the several ranges shown in the table beneath
the diagram. I have drawn a smooth curve representing a judgment as to the
general nature of the observed distribution. Note that I have not bent the
curve to make it go through the midpoint of each bar. Rather, I have drawn it
smooth, to show the general nature of the distribution.
This distribution appears to be fairly near what is commonly called a
"normal distribution." When this is true, the arithmetic average, the median,
and the mode all come at about the center of the distribution of prices. This
is somewhat of an accident. We .shall see in the next diagram a case in which
the curve is far from normal in the technical statistical sense.
1/ Unless otherwise specified, diagrams referred to in the text are those on
the facing page, the data for which are given in the table beneath the chart.
 2 
3 4 5 6 7
PRICE PAID ($ PER CWT.)
U. S. DEPARTMENT OF AGRICULTURE NEG. 397657(3) AGRICULTURAL MARKETING SERVICE
Figure 1
Laying mash: Frequency distribution of prices paid by farmers per hundredweight
as reported by feed dealers, September 1949
Price :Dealers reporting
Dollars :Number
3.375 3.624 .................: 12
3.625 3.874 .................: 136
3.875 4.124 .................: 302
4.125 4.374 .................: 652
4.375 4.624 .................: 808
4.625 4.874 .................: 715
4.875 5.124 .................: 486
5.125 5.374 .................: 183
5.375 5.624 .................: 82
5.625 5.874 .................: 16
5.875 6.124 .................: 3
Total ....................... 3,395
Data supplied by B. R. Stauber, Agricultural Estimates Division, Agricultural Marketing Service.
 3 
FREQUENCY DISTRIBUTION
Laying Mash: Prices Reported by Feed Dealers, September 1949
NO.
800
600
400
200
Corn Acreages in Sample Area Segments Enumerated
in 12 Southern States, June 1956
Actually, the socalled "normal curve" is a rather unusual
phenomenon in economic research, although it may frequently apply
fairly well when we consider residuals unexplained by a statisti
cal analysis. The agricultural 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 chart 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 1956. The survey covered
623 segments of a sample area. The figures were tabulated to
show how many of these segments reported corn acreage of 0 to 19,
how many reported from 20 to 39, from 40 to 59, and so on. The
figures are shown in the table below the diagram.
As in the preceding chart, the height of the bars indicates
the number of segments reporting corn acreages within the indi
cated ranges. I again have drawn a smooth curve, attempting to
describe the general nature of this distribution. The curve is
intended to run approximately through the midpoint of the top of
each bar. This smooth curve is an estimate of the actual distri
bution of corn 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 49 acres is far
from either the median or the mode. The largest number of seg
ments reported less than 20 acres of corn; many reported no corn
acreage at all.
This is an extreme case of skewness, but the agricultural
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 ina 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 summary of the ob
served facts in the sample. The smooth curve is an estimate of
the distribution in the statistical universe.
For some purposes the statistician may want to fit some form
of mathematical curve to the data in the histogram. However, it
is a good idea to draw the histogram and graphic curve first, be
fore 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.
4
..... 
FREQUENCY DISTRIBUTION
Corn: Acreage in 623 Segments of a Sample Area, 12 Southern States, July 1956
NO. SEGMENTS
111111 1111 I I
I I mmmtttt~trrrt
I
300
I'll1 ..'
2 A25 M = = ..= =i =ti
M 1 1 1 1 1 1 1 1 1 1 11 1 1 I II 1 1 1 1 11.
v/v II II II II I I I I IIII I I II IIII II liii I Ii I I
150
75
300
ACRES
U. S. DEPARTMENT OF AGRICULTURE NEG. 397857 (3) AGRICULTURAL MARKETING SERVICE
Figure 2
Allcorn acreage: Frequency distribution in sample area segments
enumerated, 12 southern States, June 1956
Acreage Segments : Acreage Segments :: Acreage Segments
Acres Number :: Acres Number :: Acres Number
019 ....: 308 : 120139 ...: 10 :: 240259 .... 1
2039 ....: 115 : 140159 ...: 6 : 260279 .... 2
4059 .... 61 :: 160179 ...: 8 :: 280299 ....: 1
6079 ....: 47 :: 180199 ...: 9 :: 300319 ....: 1
8099 ....: 25 :: 200219 ...: 5 :: 320339 .... 3
100119 ...: 19 :: 220239 ...: 1 :: 340 and over .: 1/1
: : Total ...... 623
1/ 456 acres.
Data supplied by Walter Hendricks, Agricultural Estimates Division, Agricultural Marketing Service.
5
LI K I ' '
0 
0
100
400
 . . . . 
I" *
tJ I K E E EE E EE.... .... ... E...E
rI T
vll ~ . . . .
I
V//i;I.I H
ho^S ___  _ _   
E/A . i^ ^ Z ^   
? $ l e T _ _
OftqL60.LT I I I 1 1I 4M..LL1 .1.61 ll6"I II I
225
200
CUMULATIVE FREQUENCIES
Percentages of Families With Incomes
Below Stated Levels
Economists and statisticians are concerned with many
kinds of frequency distributions. The particular distri
bution shown on the diagram refers to percentages of
families with various incomes. In this case we have shown
a cumulative frequency curve. Thus, 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 cumulative 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 $6,000, a larger class interval was used.
With unequal class intervals, it is awkward to draw and
use the ordinary type of frequency chart, and the cumula
tive chart is preferred.
In this case there was no problem of drawing a free
hand curve to fit the cumulative frequencies. The plotted
data all lie almost exactly along the freehand line we
have drawn.
Several mathematical functions have been proposed
and used to describe the distribution of incomes. Some
of these, like the Pareto curve, are purely empirical.
Others, like the Gibrat curve, are based upon logical
considerations. It is obvious that no mathematical curve
could fit the data much better than the freehand curve we
have drawn. In fact, the freehand curve probably fits
the data on the left hand side of the diagram better than
would a mathematically fitted Pareto curve.
 6 
CUMULATIVE FREQUENCY
Percentage of Families With Incomes Below
Specified Levels, United States, 1954
PERCENT
80
60
40
20
0
0 2 4 6 8 10 12 14
INCOME ($ THOUSANDS)
U. S. DEPARTMENT OF AGRICULTURE INEG. 397757(3) AGRICULTURAL MARKETING SERVICE
Figure 3
Families: Percentage with personal income below specified
levels, United States, 1954
Income level :Distribution
Dollars Percent
Under:
1,000 ...................... 2.7
2,000 ...................... 9.9
3,000 ......................: 20.0
4,000 ......................: 34.2
5,000 ......................: 50.6
6,000 ...................... 65.3
7,500 ..................... : 79.9
10,000 ...................... 91.4
15,000 ......................: 96.5
Survey of Current Business. U. S. Dept. Commerce. June 1956. p. 15.
7
LORENZ CURVE
Percentage of Families in the United States with Personal
Incomes Below Stated Amounts and Percentage of Total
Personal Income Obtained by These Families, 1951
Lorenz curves are often used to analyze the distri
bution of incomes. The chart presented here uses a Lorenz
curve for this purpose. Obviously such curves could 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 th, xaxis and the yaxis are plotted as
percentages of the total. In this case, for example, the
U. S. Department of Commerce figures indicate that in 1954
2.7 percent of the families in the United States had per
sonal 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 in
comes. 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 inequality is be
coming 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 unequal. 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 40 percent
of the families received 19 percent of the income. Read
ing 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 ob
tained by the lower 41 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 CURVE
Families Ranked by Personal Income, United States, 1954
PERCENT OF INCOME
80
60
40
20
0 20 40 60 80 100
PERCENT OF FAMILIES
U. S. DEPARTMENT OF AGRICULTURE NEG. 398657 (3) AGRICULTURAL MARKETING SERVICE
Figure 4
Families: Percentage with personal income below specified amounts and percentage
of total personal income obtained by these families, United States, 1954
Percentage of
Incomes less than
Families : Income
Percent Percent
$1,000 .......................... 2.7 0.2
2,000 ..........................: 9.9 2.1
3,000 .........................: 20.0 6.4
4,000 ..........................: 34.2 14.8
5,000 .......................... 50.6 27.2
6,000 ..........................: 65.3 40.7
7,500 ..........................: 79.9 57.0
10,000 ..........................: 91.4 73.3
15,000 ..........................: 96.5 83.7
ao .................... .......: 100.0 100.0
Survey of Current Business. U. S. Dept. Commerce. June 1956. p. 15.
 9 
TRENDS
Population Trends in the United States
by Decades, 18001950
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 number of farm workers. In
such cases he will want to disregard minor fluctuations due to errors in the
data or to temporary disturbances. He will also generally want to disregard
cycles or other shorter term movements in the data if they exist. He is con
cerned 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 1800. 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 immi
gration. If we plot the data for each year, or decade, we can usually draw a
smooth curve running nearly through the points we have plotted. In this case,
departures from the curve could well be due to errors in estimating the popu
lation. It should 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 population
even though we know that the Bureau of the Census official figure of
150,697,761 persons for 1950 (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, amount of possible error in his
estimates.
Instead of drawing a freehand curve, the statistician could, of course,
fit some kind of mathematical function, such as a logistic curve. Our advice
would be to draw a freehand curve first. In this case, it is doubtful if any
mathematical function would give a better description of the trend than our
freehand line. A mathematical curve might have some advantage when comparing
trends in population in several different countries. If the same type of
function were fitted in each case, results could be summarized in a few sta
tistical measurements.
A practical application of trends is in forecasting. This always involves
an extrapolation beyond the range of the data. Extrapolation of trends is
dangerous whether it is done from a freehand curve or from a curve that has
been fitted mathematically. For example, before the 1950 census data were
available (so that we did not have the last observation on the diagram), many
population experts drew an Sshaped curve indicating that the rate of growth
had started to flatten. When this type of curve was extrapolated it suggested
that the population would become stationary, or even decrease, by 1960 or
1970. Such an extrapolation now looks doubtful in view of the census figure
for 1950.
 10 
Figure 5
Population: United States, by decades, 18001950
Year Population Year Population
Millions :: Millions
1800 ................... : 5.3 :1880 ................. : 50.2
1810 ...................: 7.2 :1890 .................: 62.9
1820 ................... 9.6 1900 ................. 76.0
1830 ................... 12.9 :: 1910 ................. 92.0
18 ...................: 17.1 :: 1920 ................. 105.7
1850 ...................: 23.2 :: 1930 ................. 122.8
1860 ...................: 31.4 :: 19 0 .................: 131.7
1870 ..................: 38.6 :: 1950 ................. 150.7
Eureau of the Census.
Volume of Agricultural Marketings
It is easy enough to draw a chart representing the
growth of population, because population tends to grow
at a steady rate. It may 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 many cases, how
ever, 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 agricul
tural marketing. This index is plotted on the accom
panying diagram for each year from 1910 through 1956.
Quite evidently there has been an upward trend in
agricultural marketing. However, this trend has not
been steady. For example, the chart shows that from
1925 to 1935 marketing did not increase but fluctuated
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 ac
count 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 thirddegree 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, espe
cially when they exhibit such irregularities as are shown
in this chart. In thinking about the future trend of the
volume of agricultural marketing, we must also remember
that it will be affected by such things as acreage allot
ments, marketing quotas, and the Soil Bank.
 12 
U. S. DEPARTMENT OF AGRICULTURE NEG. 131557 (3) AGRICULTURAL MARKETING SERVICE
Figure 6
Farm marketing and home consumption: Index numbers of volume, 191056
[194749=1001
Year Volume Year Volume Year Volume
1910 ..........: 58 : 1926 ..........: 73 : 1942 ..........: 90
1911 .........: 61 :: 1927 ..........: 73 :: 1943 .........: 94
1912 ..........: 62 :1928 ..........: 74 :: 1944 ........... 99
1913 ..........: 61 :: 1929 ..........: 74 ::1945 ..........: 99
1914 .......... 61 : 1930 .......... 72 :196 ..........: 97
1915 .......... 64 : 1931 ..........: 73 : 1947 ..........: 100
1916 ..........: 64 :: 1932 ..........: 71 :: 1948 ..........: 97
1917 ..........: 62 :: 1933 ..........: 72 :: 1949 ..........: 103
1918 ..........: 67 :: 1934 ..........: 71 : 1950 ..........: 99
1919 ..........: 67 :: 1935 ..........: 66 : 1951 ..........: 101
1920 ..........: 64 : 1936 ..........: 71 : 1952 ..........: 104
1921 ..........: 65 :: 1937 ..........: 74 :1953 ..........: 108
1922 ..........: 67 :: 1938 ......... : 76 :1954 ..........: 108
1923 ..........: 69 :: 1939 ..........: 79 :: 1955 ..........: 112
1924 ..........: 72 ::1940 ..........: 80 ::1956 1/ .......: 114
1925 ..........: 70 : 1941 ..........: 82
1/ Preliminary.
Agricultural Marketing Service.
 13 
TRENDS
Volume of Agricultural Marketings
75
50 ttL
1910
1920 1930 1940 1950 1960
1956 DATA ARE PRELIMINARY.
CYCLES
Cattle on Farms, January 1
Many important economic time series tend to fluctu
ate more or less regularly up and down around 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 are especially important in agriculture. The
very nature 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 sub
stantially, 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 ways 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 downturn. In the diagram shown here, data for these in
dividual cycles are plotted on the same scale, beginning
with the year of the low point in inventories in each
instance.
The several cycles of numbers of cattle are remarkably
similar. One handicap in this visual scheme is that each
cycle is of a different length. Similarity between cycles
would appear even closer if the cycles were telescoped into
a uniform length.
A good statistician knows that history seldom repeats
itself exactly. Cycles vary in length and in amplitude.
A knowledge of past trends, and of past cycles, gives some
perspective to the present. Often it suggests the general
direction of changes in the immediate future. But the
wideawake economist will be looking for factors that may
make the current cycle different from the others.
 14 
CYCLES
Cattle on Farms, By Cycles
MIL. HEAD IrI 1 1 I i I I i1
90
80
70
60
. 191228
T I I lJ I I t ,
1 3 5 7 9 11 13 15 17 19
YEARS*
YEAR OF CYCLES, BEGINNING FROM LOW IN NUMBERS ON FARMS.
1956 DATA ARE PRELIMINARY.
U.S. DEPARTMENT OF AGRICULTURE NEG. 131757 (3) AGRICULTURAL MARKETING SERVICE
Figure 7
All cattle and calves: Number on farms January 1, 18961956
Year Number Year Number Year Number Year Number
Millions :: : Millions :: Millions : Millions
1896 .......: 49.2 1912 .....: 55.7 : 1928 ...... 57.3 :: 1944 .....: 85.3
1897 .......: 50.4 :: 1913 .....: 56.6 :: 1929 .....: 58.9 :: 1945 .....: 85.6
1898 .......: 52.9 :: 1914 .....: 59.5 :: 1930 ..... 61.0 :: 1946 .....: 82.2
1899 .......: 55.9 :: 1915 .....: 63.8 1931 .....: 63.0 :: 1947 .....: 80.6
1900 .......: 59.7 :1916 .....: 67.4 : 1932 .....: 65.8 :: 1948 .....: 77.2
1901 .......: 62.6 :: 1917 .....: 71.0 : 1933 .....: 70.3 :1949 .....: 76.8
1902 .......: 64.4 :: 1918 .....: 73.0 : 1934 .....: 74.4 : 1950 .....: 78.0
1903 .......: 66.0 : 1919 .....: 72.1 : 1935 .....: 68.8 : 1951 ..... 82.1
1904 ...... : 66.4 : 1920 .....: 70.4 :1936 ...... 67.8 :: 1952 .....: 88.1
1905 .......: 66.1 : 1921 ..... 68.7 : 1937 .....: 66.1 : 1953 .....: 94.2
1906 .......: 65.0 :: 1922 .....: 68.8 : 1938 .....: 65.2 :: 1954 ....: 95.7
1907 .......: 63.8 : 1923 .....: 67.5 : 1939 ....: 66.0 : 1955 .....: 96.6
1908 ......: 62.0 :: 1924 ...: 66.0 :: 1940 ..... 68.3 :: 1956 1/ ..: 95.2
1909 .......: 60.8 : 1925 .....: 63.4 : 1941 ....: 71.8
1910 .......: 59.0 : 1926 .....: 60.6 : 1942 ..... 76.0
1911 .......: 57.2 :: 1927 .....: 58.2 :: 1943 .....: 81.2
1/ Preliminary.
Agricultural Marketing Service.
 15 
I 1 17111 rVI'
18961912 I II_
riE~i II I i1 III 1 FE I I
I
194956 /' 
I II I!_ I 1. 193849
SI 92838
y' a ] 7 ,? ', l:_i i i 1   
;II I I;J $   
i^ . ^ ,F . l T'*  1 1  
III 1 1111
I r OlX t I 00 I I
^
Trlil I I Ill+ttf
! I1 1 1
~fi~f~~
HogCorn Price Ratio to Hog Slaughter
The agricultural 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
developingwhether, for example, it will be shorter or
longer than average.
This chart'summarizes the history of the hogcorn
price ratio and hog slaughter since 1920. By using a
chart of this kind, the agricultural economist cannot
only get some understanding 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 hogcorn price ratio to move in cycles of about 4 or
5 years in length. The length of the cycle can be mea
sured 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 hogcorn 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 irreg
ular 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 
Figure 8
Hogs: Number slaughtered and hogcorn price ratio, 192056
Slaughter Hogcorn Slaughter Hogcorn Slaughter Hogcorn
Year of price Year of price Year of price
hogs ratio 1/ hogs ratio hogs ratio ~
:Millions :: : Millions :: : Millions
1920 ..... 61.5 9.8 : 19332/ : 79.7 10.4 1946 .... 76.1 12.6
1921 .....: 61.8 13.6 : 1934 ....: 68.8 7.0 1947 .... 74.0 13.6
1922 .....: 66.2 14.4 :: 1935 ... : 46.0 11.6 1948 .... 70.9 13.0
1923 .....: 77.5 8.7 : 1936 ..: 58.7 13.0 1949 .... 75.0 15.7
1924 .....: 76.8 8.2 :: 1937 **. : 53.7 11.1 :: 1950 ....: 79.3 13.7
1925 .....: 65.5 11.4 1938 ....: 58.9 16.0 :1951 ....: 85.5 12.4
1926 .....: 62.6 17.0 :: 1939 ... : 66.6 13.3 : 1952 ....: 86.6 11.0
1927 .....: 66.2 12.7 :1940 ...: 77.6 9.2 1953 ....: 74.4 15.0
1928 .....: 72.9 9.9 :1941 .... 71.4 14.2 :1954 ....: 71.5 15.0
1929 .....: 71.0 10.9 :1942 .... 78.5 16.5 :: 1955 ....: 81.1 11.8
1930 .....: 67.3 11.4 :: 1943 ....: 9.2 13.6 :: 1956 3/ .: 85.5 11.1
1931 ....: 69.2 11.7 :1944 .... 98.1 11.6
1932 .....: 71.4 12.3 :: 1945 .... 71.9 12.8
1/ Number of bushels of corn required to buy 100 pounds of live hogs at local markets, based on
average prices received by farmers for hogs and corn. Annual average is straight average of monthly
ratios. 2/ Includes those slaughtered for Government account. 3/ Preliminary.
Agricultural Marketing Service.
17 
SEASONAL VARIATION
Monthly Production of Pork and Prices Received
by Farmers for Hogs, 195056
Many businesses are seasonal in nature. A depart
ment store has a lot of extra business just before
Christmas and Easter. Hotels in Florida are full in
midwinter; while resorts in Maine do a big business in
summer. The production and marketing of many agricul
tural 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 sea
sonal 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 particular diagram is concerned with produc
tion of pork and prices received by farmers for hogs. Each
monthly figure was first expressed as a percentage of the
12month moving average. Then the average percentage was
computed for each month. For example, the production of
hogs in January was 130 percent of the 12month 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 midsummer, soon after produc
tion 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, 195056
PERCENT,
140
120
100
PtIrod
! !:1d
I I I1
lucti
ttlI
II I
F M A M J J
MONTH
*PERCENTAGE OF 12MONTH MOVING AVERAGE
U. S. DEPARTMENT OF AGRICULTURE
A S O N D
1956 DATA ARE PRELIMINARY.
397457(3) AGRICULTURAL MARKETING SERVICE
Figure 9
Production of pork and price received by farmers for hogs: Percentage of
12month moving average, by months, average 195056 2/
Month :Product' on : Price
Percent Percent
January .........................: 124 95
February .......................: 95 97
March ...........................: 105 97
April ...........................: 92 100
May .............................: 87 104
June ........................ : 87 105
July ............................: 78 106
August ....................... 81 108
September ....................... 88 106
October .................... 106 101
November ........................ 123 91
December ........................ 134 90
1/ Production under Federal inspection.
Data revised and brought up to date from Breimyer, Harold F. and Kause, Charlotte A.
Seasonal Market for Meat Animals. U. S. Dept. Agr. Agr. Handb. 83. 1955. pp. 4, 32.
19 
Charting the
80
60
40
i f I I~m I I
IIIgll ............
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 gradually, reflecting changes in
methods of producing and marketing.
A case in point is the accompanying diagram showing
the seasonal patterns of broiler placements and market
ings. 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 marketing.
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 periods194143
and 195355together with the average for each period.
The picture for these two periods is strikingly different.
Back in 194143 there was a sharp peak in broiler chick
placements toward the end of the year, and placements
were fairly stable from January through July.
By 195355 the seasonal swings in the broiler indus
try had become considerably different. The industry was
geared to reach a peak in placements in early summer, so
that the bulk of the marketing came in midsummer when
the demand for broilers is high. Placements then fell
off rather regularly until they reached a low point in
September 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 marketing about 10 weeks later. This is not
all. By studying seasonal diagrams, members of the in
dustry can usually anticipate with some degree of accu
racy 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 marketing
over a period much longer than 10 weeks.
 20 
SEASONAL VARIATION
Broiler Chick Placements*
% OF NOV. 1954 1t11 Ii1 % OF NOV.
 1954
S Average 194143 Average
1130
20 14940219 100
10 9 I I I I 1 I
JAN. JULY DEC. JAN. JULY DEC.
MONTHLY INDEX OF AVERAGE WEEKLY RATE (NOVEMBER 1954 = 100).
ABASED ON PLACEMENTS IN THE DELMARVA AREA.
BASED ON PLACEMENTS IN 11 STATES (1953). 13 STATES (1954). AND 22 STATES (1955).
U.S. DEPARTMENT OF AGRICULTURE NEG. 397557(3) AGRICULTURAL MARKETING SERVICE
Figure 10
Broiler chick placements: Index numbers of average weekly rate, by months, 194143 and 195355
[November 1954=100]
S 1941 : 1942 : 193 : Average :: 1953 : 1954 1955 Average
Month I I : I *i 2/ : 2/ 2/
January ....: 27.0 36.2 41.4 34.9 :: 108.5 120.1 101.9 110.2
February ....: 27.9 34.5 35.2 32.5 :: 110.2 123.3 130.6 121.4
March .......: 28.3 34.7 33.1 32.0 :: 119.6 129.8 132.2 127.2
April .......: 28.0 27.3 35.8 30.4 :: 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 31.4 :: 115.6 124.2 149.3 129.7
July ......... 27.9 34.2 31.3 31.1 :: 103.4 123.5 145.0 124.0
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
November ....: 35.6 38.5 49.2 41.1 :: 11.6 00.0 126.0 113.5
December ....: 49.3 48.3 43.0 46.9 :: 116.0 92.4 134.1 114.2
1/ Based on placements in the DelMarVa 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, Poultry and Egg Situation, PES183, May 1956. pp. 3640.
Service, Poultry and Egg Situation, PES183, May 1956. PP. 364o.
 21 
SIMPLE REGRESSION
Corn Yields Related to Nitrogen
A socalled "dot chart" is one of the handiest tools
of economic analysis. The agricultural economist ordi
narily must find the relation between two 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 experi
ments to determine the relation of corn yields to appli
cations 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 64.6 bushels was obtained with no nitrogen at all.
The second dot indicates that 90.4 bushels were obtained
when 40 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 maximum yield is apparently obtained with an appli
cation of about 200 pounds of nitrogen to the acre and
that additional applications neither increase nor de
crease 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 line. This could, however, be a statis
tical 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 appli
cation. 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 
YIELD
150
100
SIMPLE REGRESSION
Corn: Yield Per Acre in Relation to Applications of Nitrogen
BU.)
S
I1
I r niii I I I If I I I I I f iJ I I I Ifi I I I I ] 111 111
50
200
NITROGEN (LB.)
300
U. S. DEPARTMENT OF AGRICULTURE NEG. 397957 (3) AGRICULTURAL MARKETING SERVICE
Figure 11
Corn: Yield per acre by specified quantity of nitrogen applied, Ontario, Oregon
Nitrogen Yield of :Nitrogen :Yield of
applied :corn :: applied corn
Pounds :Bushels :: Pounds Bushels
0 .........: 64.6 :: 160 ......... 146.8
40 .........: 90.4 :: 180 .........: 141.2
80 ......... 118.2 :: 200 ......... 147.1
100 ......... 132.4 : 240 ......... 145.8
120 ......... 140.7 :280 ......... 147.4
140 ......... 141.0 :: 320 .........: 143.8
Paschal, J. L., and French, B. L. A Method of Economic Analysis Applied to Nitrogen Fertilizer
Rate Experiments on Irrigated Corn. U. S. Dept. Agr. Tech. Bull. 1141. 1956. p. 16.
23 
. .I .I ..l l ..1 1.11 1.1 1 1 1. .11 1 1.
100
I
Weekly Food Expenditures of Families, 1955
Economists have long been interested in the effect
of family income on expenditures for food. Many specific
studies have established two facts: (1) that the average
highincome family spends more for food than does the
average lowincome family and (2) that the average high
income family spends a smaller proportion of its income
for food than does the average lowincome 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 agri
culture. 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 56. 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 rela
tion. 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 ex
penditures 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 in
comes of over $10,000 a year. However, the report does
not show the average income of these families. Therefore,
it is not possible to plot this dot precisely. If we had
plotted the dot at the lower limit of this class inter
val, it would have deviated widely from the curve shown.
This chart confirms one of the two facts given in
the opening paragraph, namely that the average high
income family spends more for food than does the low
income family. A different chart would be needed to
demonstrate that the average highincome family spends
a smaller proportion of its income for food than does
the average lowincome family.
 24 
SIMPLE REGRESSION
Housekeeping Families of Two or More Persons: Weekly
Food Expenditures in Relation to Annual Income, 1955
FOOD EXPENDITURE ($)
40
20
0
0 2 4 6 8 10
DISPOSABLE INCOME ($ THOUSAND)
U. S. DEPARTMENT OF AGRICULTURE NEG. 398157(3) AGRICULTURAL MARKETING SERVICE
Figure 12
Housekeeping families of two or more persons: Weekly food expenditures by
specified income groups, United States, fprng 1955
Annual income
after taxes Food expenditures
Dollars Dollars
Under 1,000 ...................: 11.69
1,000 1,999 .................: 16.60
2,000 2,999 .................: 22.55
3,000 3,999 .................: 27.00
4,000 4,999 .................: 30.27
5,000 5,999 .................: 33.03
6,000 7,999 .................: 36.14
8,000 9,999 .................: 39.21
10,000 and over ................: 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
survey of tobacco consumption. 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 prac
tice of putting these data into the calculating machine
and assuming that the relationship was linear, we would
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 middleaged groups of around
40 to 45 years. Younger and older smokers apparently
consume fewer cigarettes. One might, of course, specu
late on the reasons for this. It is not primarily a
matter of income. These data have been tabulated sepa
rately by income groups and each income group exhibits
a similar curve 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 xaxis. 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 statis
tician should always warn his readers when this is the
case. Otherwise, the chart would give the overexagger
ated impression that the youngest and oldest age groups
smoked practically no cigarettes at all. This is not
true. The youngestaged 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 yaxis, as this scale is not apt to be misleading.
It was the opinion of the analyst that few youngsters
below 15 years of age smoke a significant number of cig
arettes per day.
 26 
18
15 20 25 30 35 40 45 50 55 60
AGE (YEARS)
U. S. DEPARTMENT OF AGRICULTURE NEG. 398257(3) AGRICULTURAL MARKETING SERVICE
Figure 13
Cigarettes: Average daily consumption by males who smoke regularly, by age, 1955
Age Consumption
Years Number
1824 ..................... 19.1
2534 ..................... 21.9
3544 ..................... 22.9
4554 ..................... 22.5
5564 .....................: 20.3
65 and over .................. 17.6
Smoking Survey. Bureau of the Census. 1955.
27 
SIMPLE REGRESSION
Males Who Smoke Regularly: Cigarette Smoking in Relation to Age,1955
CIGARETTES (NO. PER DAY)
22 ::
21
21 ::    s  
20
19
L
%
I I F
F
L J I I
Onion Prices Related to Production
Section A of the chart in this example is a dot
chart showing the relation between onion production and
prices in the years 193956. You will note that the ob
servations are scattered all around the diagram and that
s~ne of the highest prices occurred in the years of
medium 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 deal
ing with time series. It quickly shows up any trend in
the data and gives a rough idea at least of the slope of
the curve.
In this particular case, section A suggests that we
consider the relation of yeartoyear 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 explana
tion is far from perfect. For example, if we had used
the curve in section B to estimate expected changes in
prices we would have been over 60 cents too low in 1952
and 80 cents too high in 1953.
A more accurate way of studying the relation between
onion prices and production is discussed on page 40.
 28 
SIMPLE REGRESSION
Onions, Commercial Crop: Production and Average
Price Per 50lb. Sack Received by Farmers
PRICE 1($ 11) PRICE ($) 4
A. ACTUAL DATA 1.00 1 B. YEARTOYEAR
2.50 i CHANGES
2 i li l;bil iifh millllf
1.500
.50 I tII I50I= ^=1= '!I
2.00
x5 J1'45 '49 5 '564 
'42
1.50
I '495 '44
',41 .I 48 '44 .50 ,
055 1450
1.00 35 40 45 50 15 10 5 0 5 10 1
195'46 DATA ARE PRELIMINARY.
W.50',l 0 j I 1.00 1 1 ,
Onions, commercial crop: Prouction an average price per 50poun
sack received by farmers, 193956
Change PRODUCTION (MIL. SACKS)from
sack received by farmers, 193956
preceding year in preceding year in
Year o Price Year o Price :
: Produc Pc e : Produc :
: tion Price : : : tion : Price
Million Million :: : Million Million
sacks Dollars sacks Dollars :: sacks Dollars sacks Dollars
1939 **** 36.6 0.45   1948 ... 42.5 1.32 5.8 .76
1940 .... 32.9 .70 3.7 0.25 :: 1949 ... 38.8 1.47 3.7 .15
1941 .... 31.2 1.10 1.7 .40 :: 1950 ...: 45.8 .87 7.0 .60
1942 .... 38.9 .99 7.7 .11 1951 ... 39.4 1.67 6.4 .80
1943 .... 31.3 1.68 7.6 .69 :: 1952 ... 39.8 2.31 .4 .64
1944 .... 47.9 1.20 16.6 .48 :: 1953 ... 49.8 .68 10.0 1.63
1945 37.7 1.69 10.2 .49 1954 ... 43.6 1.07 6.2 .39
1946 .... 50.4 .89 12.7 .80 1955 ...: 42.8 1.18 .8 .1
1947 .... 36.7 2.08 13.7 1.19 1956 ... 49.4 1.30 6.6 .12
Agricultural Marketing Service.
Agricultural Marketing Senrice.
 29 
Shifts in Demand for Beef and Pork
Simple regression is often useful in analyzing problems that are more complicated
than those we have just considered. Actually, 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 2variable 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 capital consumption of beef and pork on the xaxis. 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 capital 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 postWorld War II years 194756 than in the prewar years 192541. 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 192531. It dropped somewhat in the
period 193241 and remained about the same in 194752. But in 195356 it seems to have
been substantially lower. Shepherd et al showed two regression lines for pork, but
their data ran only through 1952. On our uptodate 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 urbani
zation of the country and a more even distribution of incomes.
This diagram could serve as a connecting link between simple regression and multi
ple 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 equa
tion.
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
PRICE (t PER LB.)*
35
80 32 A. BEEF B.PORK
30 ___ :F  35_  
30 3
31
70 5 084 26 0 7
60 OF I 31 47
25 548 28 192531
50 26 41 53 49
192541 a H H Y
I95 DT R ELN51
54 2 193241
40 F r 41
40 r ,1953s56 ni 194752
55' 40
40 50 60 70 80 40 50 60 70 80
CONSUMPTION (LB. PER CAPITAL)
*AVERAGE RETAIL PRICE PER POUND AS COMPILED BY AMS DIVIDED BY INDEX
OF PER CAPITAL DISPOSABLE PERSONAL INCOME (194749 = 100).
1956 DATA ARE PRELIMINARY.
U. S. DEPARTMENT OF AGRICULTURE NEG. 398057(3) AGRICULTURAL MARKETING SERVICE
Figure 15
Beef and pork: Retail price per pound and consumption per capital, 192556
Beef :Pork, excluding lard:: Beef :Pork, excluding lard
Yea :Price of: Consump: Price : Consump Year ce of: ConsumpY Price Consump
: Cents Pounds Cents Pounds : Cents Pounds Cents Pounds
1925 ...: 59.7 58.6 60.5 65.8 1941 ... 56.0 60. 43.9 67.4
1926 ... 59.7 59.4 63.3 633 1942 3/. 49.7 60.4 42.6 62.8
1927 ...: 63.0 53.7 59.9 66.8 193 .: 45.9 52.5 39.2 77.9
1928 ...: 71.0 48.1 56.0 69.9 1944 .o ..9 33.9 8.5
1929 .. 71.1 49.0 55.0 68.7 1945 38.6 58.6 33.4 65.7
1930...: 74.2 8.2 59.6 646.7 6. 4.8 .
1931 .. 72.1 4.9 .0 67.4 194. 65.3 68.6 58.6
1932 ...: 79.0 46.0 49.5 69.7 1 72.8 62.3 4.6 66.8
1933 ..: 73.1 .50.8 47.3 2/68.7 1949 67.1 63.1 9.7 66.8
1934 .. 70.2 2/55.2 56.6 62.2 :: 1950 ... 7 62.6 46.1 68.2
1935 ..: 82.2 52.2 73.9 47.7 1951 .. 74.6 55.3 45.9 70.9
1936 ... 68.4 57.3 54.4 1952 70.9 61.4 42.7 71.4
1937 ... 73.0 4.4 62.2 55.0 :: 1953 ... 54.5 76.5 45.3 62.6
1938 ... 70.2 53.6 59.9 57.4 1954 .. 54.1 79.0 46.1 59.2
1939 ... 67.8 53.9 51.0 63.9 1955 ... 51.2 .9 37.2 65.9
1940 .4 54.2 41.5 72.4 1956 48. 84.2 33.9 66.8
1/ Retail price as computed by Agricultural Marketing Service divided by index numbers of dispos
able personal income (194749=100). 2/ Consumption less use under Federal programs. 3/ War years
omitted from diagram.
Agricultural Marketing Service.
 31
COMPARISON OF TIME SERIES
Food Prices, Consumer Incomes, and Volume
of Farm Marketings
Economists often work with time series; that is, with
records of prices, production, and consumption over a
period of time. When studying relations between time
series, particularly if several variables are involved,
it is a good practice to plot each series before drawing
dot 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 marketing 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 occurred over a period of time.
In this case it is clear that there is high correla
tion between the food price index and per capital disposable
income. In fact, the relationship is so pronounced that
it tends to overshadow the effect of per capital food mar
ketings. We might notice, too, that during the war years
from 1941 to 1945 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 capital disposable income would be reduced by de
flating each series (for example, by dividing each of
these by the consumer price index for all commodities).
Such a computation would also reduce the magnitude of the
gyrations to more nearly correspond to those for per capital
marketing of food. The sharp rise in marketing 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 yeartoyear
changes in the variables.
In some cases, a comparison of time series will in
dicate a timelag between changes in one variable and
changes in another. We saw previously 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
% OF 194749 I I
Per capital disposable income
I I I I I I I I I 1 4 1 i 1 1 i
Ti mr1 i17 z I
w I h Iv I1 1 ImL 'z
I ii "iii 111"
Per capital farm marketing
and home consumption 
I I I I I I I I I I I I I { { II
a' P I 4r 4 I I
*
%
t
125
100
75
50
25
160*I414
iiizi
Retail foodprice index
Retail foodprice index
I II I I I I I I I
Ii I I ~w~lt~HttCCH
1t 1 t 1 1111 1 1 1 1
1920 1930 1940 1950
1956 DATA ARE PRELIMINARY.
U.S. DEPARTMENT OF AGRICULTURE NEG. 132657 (3) AGRICULTURAL MARKETING SERVICE
Figure 16
Price and marketing of food and disposable income: Index numbers, 192056
[194749=100]
: Food : :: Food
: Farmnmarket Disposable :: : Farm market: Disposable
Year : Retail :ings and home: income : Year : Retail :ings and home: income
: price : consumption : per capital :: : price : consumption : per capital
: per capital : :: : per capital
1920 ..... 83.6 88 52.8 :: 1939 .....: 47.1 89 43.5
1921 ..... 63.5 88 41. 190 .....: 47.8 91 46.5
1922 ..... 59. 91 43.7 : 1941 .52.2 93 56
192 ..... 61.4 49.8 1942 ..... 61.3 101 04
1924.....: 60.8 49.3 19 .... 68.3
61 28 9 168 105o
192
1925 ..... 65.8 9 51.4 :: 1944 ..... 67.4 109 .6
1926 ..... 8.0 90 52.6 1945 ..... 68.9 86.8
192 ..... 6 90 52.1 1946 ..... 79.0 106 91.0
192 ..... 64.8 52.7 : 1947 .....
1929 ..... 194 1 98 103.3
1930 ..... 62.4 48.8 :: 1949 .....: 100.0 98 101.9
1931 ..... 51.4 41.5 1950 ....: 101.2 96 109.
1932 ..... 42.8 31.4 :: 1951 .....: 112.6 97 118.3
1933 ..... 41.6 86 29.4 :: 1952 .....: 114.6 97 122.1
193 .....: 46.4 87 33.2 :: 1953 .... 112.8 97 126.7
1935 ..... 49.7 0 ::195 .....: 112.6 97 126.6
1936 ..... 50.1 .8 :1955 ....: 110.9 98 132.2
1937 .. 52.1 83 44.5 1956 .....: 111.7 100 137.6
1938 ..... 8.4 87 40.
Prices from Bureau of Labor Statistics, marketing of food from Agricultural Marketing Service,
and income from Department of Commerce.
33 
r: r'
Z
+t++iz~tt+tttt
iLfrr
IRR Sr4 q i. I I,,T 1" IT _j IILl. ,
I II T T e,, I I 1 4 1"
BI I I I
1II Il~i~l~l[ll I I I I I I IIQ~M~I
I I I I I I I I I I
LI
I
1
MULTIPLE REGRESSION
Per Capita Consumption 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 2variable regression, the statistician can put the data for a mul
tiple regression problem into the calculating machine and compute the answer by least squares.
However, in my 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 Department 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 socalled "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 capital income, together with the indexes of per capital food consumption. Each dot
shows the pair of indexes for a particular year. (The years 1942 through 1947 were excluded from
this analysis because food consumption was affected by such things as rationing and price con
trol.) 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 con
cerned 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 postWorld 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, to
gether 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 units above the regression line. This observation is plotted in section B with the coor
dinates 83.0 on the xaxis and +0.6 on the yaxis. 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 consumption. 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 satisfac
tory.
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 variablesincome and price. The
largest errors are about two index points in 1926, 1935, 1937, and 1951. For most years our es
timates are within one index point of the true figure. With the index of consumption varying
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. 24:386397, illus. 1929.
 34 
U. S. DEPARTMENT OF AGRICULTURE NEG. 398357 (3) AGRICULTURAL MARKETING SERVICE
Figure 17
Index numbers: Consumption of food, disposable income, and retail food price, 192241 and 194856
[194749=100]
S Per capital Price Per capital Price of
Year : Consumption *Disposable food Year Consumption :Disposable food
Sof food income / : / of food income i/: /
1922 .....: 89.0 61.0 83.0 : 1937 .....: 90.4 72.5 84.9
1923 .....: 90.9 68.3 84.2 : 1938 .....: 90.6 67.8 80.3
1924 .....: 91.5 67.4 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 ..... 98.9 100.1 98.2
1930 .....: 90.7 68.3 87.4 :: 1950 .....: 99.9 106.8 98.4
1931 ....: 90.0 64.0 79.1 : 1951 ..... 98.1 106.6 101.4
1932 .....: 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 .....: 104.0 118.4 95.9
1 Deflated by dividing by the Bureau of Labor Statistics Consumers' Price Index.
Agricultural Marketing Service.
35 
MULTIPLE REGRESSION
Food: Consumption Per Capita in Relation to Real Incohne Per
Capita and Real Food Price
CONSUMPTION(%OF 194749) DEVIATIONS
A0 B.
105 RELATION TO1 10 RELATION TO
REL TIO T IK I I IIIiI
100
95
90
85
80,
INCOME (%OF 194749) PRICE*(%OF 194749)
*DEFLATED BY DIVIDING BY THE CONSUMERS' PRICE INDEX.
1956 DATA ARE PRELIMINARY.
Price of Corn Related to Price of Livestock and
Supply of Feed Concentrates Per Animal Unit
The diagram illustrates an analysis of corn prices
(Xo) related to two independent variablesprices of
livestock and livestock products (Xl) 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 relationshipsperhaps to fore
cast prices of corn.
Section A of this chart shows corn prices and
prices of livestock and livestock products from 1936
through 1955. Before drawing the regression line, we
try to take account of X2. We draw several regressions
for subsamples of data, commonly called "drift lines."
Thus in 1948, 1949, and 1950, supplies of concentrates
were from 1.05 to 1.07 tons. We connect these observa
tions with a drift line. Similarly we connect the
observations for 1940, 1941, and 1942, when supplies
were 0.90 tons. After drawing all possible drift lines,
we draw a net regression line the slope of which repre
sents approximately an average of the slopes of, the
drift lines. In this case, a straight line 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
line we have drawn. If a nearly perfect fit were not
given by the dots around this line, the process of suc
cessive approximation would be used. Foote 4/ has shown
that when we use this method graphically based on linear
relationships, the slopes of the successive approxima
tions 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 amount
of imagination and some practice. But it often shows up
important relationships that are not brought to light by
grinding figures out of a computing machine.
4/ Foote, Richard J. The Mathematical Basis for the
Bean Method of Graphic Multiple Correlation. Jour. Amer.
Statis. Assoc. 48:778788. 1953.
 36 
MULTIPLE REGRESSION
Corn: NovemberMay Prices Received by Farmers
in Relation to Specified Factors
100 200 300 .65 .75 .85 .95 1.05 1.15
PRICES OF LIVESTOCK AND
PRODUCTS (% OF 191014)X1
SUPPLY OF FEED CONCENTRATES
PER ANIMAL UNIT (TONS)X2
U. S. DEPARTMENT OF AGRICULTURE NEG. 132757 (3) AGRICULTURAL MARKETING SERVICE
Figure 18
Corn: Price per bushel received by farmers and related variables, 193655
: Price received by : y : : Price received by :
farmers (NovemberMay): Supply :: : farmers (NovemberMay): Supplyof
feed con feed con
Period n Livestock centrates Periodn : Livestock centrates
beginnnn : p :: beginning :
Corn and per animal Corn and per animal
products /: unit 2/ : products /i unit 2
Cents Tons : : Cents Tons
1936 ......: 106 123 0.65 : 1946 ......: 138 278 0.99
1937 ......: 51 114 .89 :: 1947 ......: 220 305 .87
1938 ...... 44 108 .88 : 1948 ......: 120 285 1.05
1939 .....: 55 107 .87 : 1949 ......: 118 260 1.07
1940 ......: 58 122 .90 :: 1950 ......: 155 329 1.06
1941 ...... 74 159 .90 : 1951 ......: 166 318 1.01
1942 ...... 90 194 .90 : 1952 ......: 147 278 1.05
1943 ......: 112 196 .85 :: 1953 ......: 142 271 1.10
1944 ......: 107 206 .92 :: 1954 ......: 138 240 1.13
1945 ......: 115 215 .92 : 1955 ......: 121 224 1.19
i1 Index number, 191014=100. / Year beginning October.
Computed from data in Foote, Richard J. Statistical Analyses Relating to the FeedLivestock
Economy. U. S. Dept. Agr. Tech. Bull. 1070. 1953. p. 6.
37 
Yields of Corn in Illinois, 193455, Related to Reported
Condition on September 1 and a Time Trend
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. Stat
isticians have found that the farmers' reports as to current condition of crops is a fairly good
indication of the yield that would occur with 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, sug
gested 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 1934
through 1956 to the final estimate of harvested yield. It is easy to see that there is some posi
tive 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, 40 or 50 or 60 percent of normal, the yield tends to be low. But this is not the whole
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 1940 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
1934. 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 9.0 bushels. Thus, in section B we
indicate 9.0 for the year 1934. 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 1942. 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 1940'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 1956. Farmers reported a
condition 96 percent of normal. Section A indicates a yield of 58 bushels. Section B indicates
that we should add 9.5 bushels to account for the trend; thus giving us an estimate of 67.5 bush
els. Actually, the yield in 1956 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 ob
served 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*
DEVIATI
II
20
10
0
10
20
940 1948
1932
SEPT. 1 CONDITION (%OF NORMAL)
*ADJUSTED FOR TIME TREND.
1956 DATA ARE PRELIMINARY.
956
U. S. DEPARTMENT OF AGRICULTURE NEG. 398457(3) AGRICULTURAL MARKETING SERVICE
Figure 19
Corn: Condition September 1 and yield per harvested acre, Illinois, 193456
Year Condition field Year 'Condition Yield
Percent Bushels :: Percent Bushels
1934 .........: 48 21.5 : 1946 .........: 93 56.0
1935 .........: 77 38.5 :: 1947 .........: b2 39.5
1936 .........: 42 23.5 :: 1948 .........: 94 61.0
1937 .........: 89 48.0 :: 1949 .........: 95 54.0
1938 .........: 85 44.0 :: 1950 .........: 84 51.0
1939 .........: 94 51.0 :: 1951 .........: 88 56.0
1940 .........: 70 43.0 :: 1952 .........: 86 58.0
1941 .........: 85 53.0 :: 1953 .........: 81 54.0
1942 .........: 87 54.0 :: 1954 .........: 73 49.5
1943 .........: 79 50.0 :: 1955 .........: 80 56.0
1944 .........: 74 45.4 :: 1956 .........: 96 68.0
1945 .........: 77 46.5 ::
/ As a percentage of normal.
Data supplied by C. E. Burkhead, Agricultural Estimates Division, Agricultural Marketing Service.
 39 
Price of Late Onions Related to Production
and Disposable Income
The data for this diagram, taken from Shuffett, 5/
are expressed as first differences (i.e. yeartoyear
changes) in logarithms. The rationale of this may be
found in Shuffett's bulletin and need not concern us
here. Graphic analysis will handle logarithms and first
differences, as well as the unmanipulated data.
The main purpose of this diagram is to illustrate
successive approximations to the true regression lines.
We have already discussed the graphic determination of
the net regression lines. So far, we have tacitly
assumed 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 differ
ences 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 regres
sion of disposable income on price.
So far, our analysis is the same as in several pre
vious diagrams. We now proceed to make a second approxi
mation. 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 approxi
mation 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 approxi
mations 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. 3843. 1954.
 40 
MULTIPLE REGRESSION
Late Onions: AugustApril Prices Received by Farmers in Relation to Specified Factors
PRICE ($ PER CWT.)LOGARITHMS'X 0 DEVIATIONSX., I 1 I 11 1 I I I I 1
.50  SECTION A
'33 1 1 1
'3 .411 I
.25
34
0 4 '39
_ 3 35
.25 "'"
.50 i I I c
.20 .10 0 .10 .20
PRODUCTION (LBS.PER CAPITAL)
LOGARITHMS*X,
U. S. DEPARTMENT OF AGRICULTURE
.15 .10 .05 0 .05 .1'
DISPOSABLE INCOME ($ PER
LOGARITHMS*X 2
0 .15
CAPITAL)
*CHANGE FROM PRECEDING YEAR.
NEG. 132955(1) AGRICULTURAL MARKETING SERVICE
Figure 20
Late onions: Average price per 100 pounds received by farmers
and related variables, AugustApril average, 192841
: First difference of :: : First difference of
Actual logarithms :: Actual logarithms
Period Per capital Per capital Period Per capital Per capital
begin' :: :begin e.
ning :r Produc Price:P :Dispos i ning Produc Price Pr Dispos
: /: in '" :.... "abe "" tion I" _, ae able
Stion : Icome: :Prod income : / 2/ :I : : ron :incable
3/.. : : : : : tion :
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: .86 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 6.41 456 .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 :: 1940 :1.12 9.93 601 .105 .027 .042
1934 : 1.36 7.89 420 .026 .017 .037:: 1941: 2.08 9.47 748 .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.
41 
JOINT (3DIMENSIONAL) REGRESSION
Yield of Corn In Relation To Applications of
Nitrogen and Phosphoric Acid
Many problems of economic analysis can be handled by simple (2variable) 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
o = fl(xl) + f2(x2) + ..... fn(xn) (1)
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 (xl, x2, ....., Xn) (2)
A case in point is the relationship of crop yields to various dosages of fertilizer. We con
sidered 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 3dimensional 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 05). 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' (K20) held constant at
40 pounds to the acre. The applications of nitrogen (N) varied from 0 to 240 pounds. The appli
cation 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, P205 = 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 quite 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 phos
phoric acid. When the nitrogen application was increased to 240 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 phos
phoric acid were beneficial.
This is a 3dimensional dot chart. In principle, it is the same thing as the several
2dimensional dot charts we have looked at. We want to visualize a graphic, 3dimensional 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 observations
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. 441, 1956.
 42 
Figure 21
Corn: Yield per acre for given applications of phosphoric acid and nitrogen 1j
Nitrogen in pounds
Phosphoric
acid
acid0 h 80 : 160 : 240
Pounds Bushels Bushels Bushels Bushels Bushels
0 ..... 32.00 49.85 65.50 68.25 61.20
40 ..... 32.25 49.55 61.55 74.75 66.90
80 ..... 23.25 48.55 62.65 88.15 78.45
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
Variable Nutrients. Iowa Agr. Expt. Sta. Research Bull. 441. 1956. p. 815.
43 
JOINT REGRESSION
NEG. 425957(5) AGRICULTURAL MARKETING SERVICE
U.S. DEPARTMENT OF AGRICULTURE
_ L_
USE OF ISOQUANTS TO STUDY JOINT REGRESSION
Yield of Corn in Relation to Applications of
Nitrogen and Phosphoric Acid
Another technique for studying variation of a 3dimen
sional surface is similar to that used in surveying and
grading land. We can forget for the moment that the chart
refers to yield of corn. Suppose that the vertical axis
measures distances north and south, the horizontal axis
measures east and west, and the numbers written by the
dots on the diagram indicate the elevation of the land at
various points and determined by surveyor's transit. Any
one used to maps would recognize that the land is gradually
increasing in height as we move toward the right side of
the diagram, becoming steeper as we move toward the upper
right corner. You would also see that there are bumps and
hollows. In simple regression we smooth in only one dimen
sion. 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 con
tour 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 combina
tions of two independent variables which correspond to a
given value of the dependent variable. In the case illus
trated by the diagram, it is clear that yields increase
steadily with increases in nitrogen up to about 160 pounds
per acre and then begin to decline. The effects of addi
tional 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 threedimen
sional relationships is the "isometric projection." Those
who are not familiar with isoquants may find such projec
tions easier to visualize. But they are also harder to
read accurately.
 44 
JOINT REGRESSION
Corn: Yield Per Acre Related to Applications of Nitrogen
PHOSPHORIC ACID (LBS.)
and Phosphoric Acid*
100
5C
50
100 150
NITROGEN (LBS.)
200
250
WITH POTASH APPLIED AT 40 POUNDS PER ACRE.
NUMBERS ON POINTS REFER TO YIELD OF CORN IN BUSHELS.
U.S. DEPARTMENT OF AGRICULTURE NEG. 398557 (3) AGRICULTURAL MARKETING SERVICE
Figure 22
Corn: Yield per acre for given applications of phosphoric acid and nitrogen 1/
Nitrogen in pounds
Phosphoric
acid
ac 0 40 80 : 160 : 240
Pounds : Bushels Bushels Bushels Bushels Bushels
0 ..... 32.00 49.85 65.50 68.25 61.20
40 ..... 32.25 49.55 61.55 74.75 66.90
80 ..... 23.25 48.55 62.65 88.15 78.45
120 .....: 20.20 50.75 69.80 86.80 81.90
/ 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
Variable Nutrients. Iowa Agr. Expt. Sta. Research Bull. 441. 1956. p. 815.
45 
20.2 5'0.7 69.80 86.80 1 I I 81 .90
2 j1 48.55 62.65 88.15 781.5
U: I I 'k VI 1 1 ; A j1I
F i I I I I L I TI 1 1 'T I
S 49.55  61.55 '74.75 90
32.25
32.00 4 9.85 65.50 6  8.25 6 41.20
 i, i I I I Ir4 ] l
INDIFFERENCE CURVES
Beef and Pork, 194756
The first edition of the handbook included a diagram labeled "Indiffer
ence Curves" based on data for beef and pork. Two periods were shown, 192131
and 193241. The present chart shows a family of curves for the postWorld
War II period 194756.
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 indif
ference curves from any analysis of market data.
In any case, the economists and statisticians that I know would 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 capital consumption of beef was 84.2 pounds and of pork was
66.8 pounds. The slope of the line drawn through that point indicates a price
ratio of 1.42. In other words, in 1956 the consumer would have to give up
1.42 pounds of pork to get a pound of beef. After plotting for each year the
consumption of beef and pork, together with the price ratio, we note that the
ratio was not constant. It was high when pork consumption was high and beef
consumption was low. It was low when pork consumption was low and beef con
sumption was high. This is as we would expect. The light curves drawn
through this diagram are similar to the contour lines used in the diagram on
page 45 to analyze a problem of joint regression. In drawing such lines we
attempt to portray a smooth 3dimensional 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 1947 are almost the
same as those of the neighboring contour lines. The observed price ratio in
1947 was smaller than we would 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 consumer. 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 course, 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 assumptions in most statistical analyses. For
example, when deriving a demand curve from market data.
/ Waugh, Frederick V. A Partial Indifference Surface for Beef and Pork.
Jour. Farm Econ. 38:102112, illus. 1956.
 46 
INDIFFERENCE CURVES
Consumption of Beef and Pork Per Capita, 194756*
PORK (LBS.)
,S2
70 1f s1 x
,53
0 s  '   : : :
50
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 NEG. 398757 (3) AGRICULTURAL MARKETING SERVICE
Figure 23
Beef and pork: Ratio of beef price to pork price at retail and per capital consumption, 194756
Retail price per pound Consumption
Year Price
Beef Pork ratio : Beef Pork
Cents Cents : Pounds Pounds
1947 ...........: 61.8 55.5 : 1.11 : 68.6 68.6
1948 ........... : 75.3 56.5 : 1.33 : 62.3 66.8
1949 ........... : 68.4 50.6 : 1.35 : 63.1 66.8
1950 ........... : 75.4 50.3 : 1.50 : 62.6 68.2
1951 ........... : 88.2 54.3 : 1.62 : 55.3 70.9
1952 ........... : 86.6 52.1 : 1.66 61.4 71.4
1953 ........... : 69.1 57.4 1.20 : 76.5 62.6
1954 ...........: 68.5 58.3 : 1.18 : 79.0 59.2
1955 ...............: 67.7 49.2 1.38 : 80.9 65.9
1956 ........... : 66.0 46.6 1.42 84.2 66.8
Agricultural Marketing Service.
 47 
LINEAR PROGRAMMING
Combination of Two Farm Enterprises
Programming is the planning of economic activities to maximize income or to mini
mize costs. In some cases it is reasonable to assume that the inputoutput relation
ships are approximately linear. For example, if we know how much seed, labor, and
fertilizer is required to grow an acre of potatoes, we can assume that it will require.
about twice as much of each input factor to grow two acres of potatoes by the same
process. In a similar manner, if we know the amount of protein, calcium, and other
nutrients in a bushel of corn, there would be twice as much of each nutrient in two
bushels of corn. 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 amount of any resource. In an analysis of this kind it is conven
ient first to compute for each enterprise the proportion of available resources needed
to produce some arbitrary amount of net income. In this case we chose $10,000. For
example, to get a net income of $10,000 from beef cattle would require 4.63 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 SeptemberOctober labor. To get an income of $10,000 from fall cabbage
would require 2.17 times as much SeptemberOctober labor as the farmer has available.
If he used all of his SeptemberOctober labor on cabbage, his income would be $10,000
divided by 2.17, or $4,608. 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 pro
duction capital. Finally, in combinations that are mostly fall cabbage and only a
little beef cattle, the limiting factor is SeptemberOctober 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 91 percent of (1) the
available production capital and (2) the SeptemberOctober labor to produce fall cab
bage. The other 9 percent of these two limiting factors would be used for beef cattle.
To get an income of $10,000 from these combinations would require almost twice as much
of the two factors as are available. So the best the farmer could get with these two
enterprises would be an income of a little over $5,000.
8/ See King, R. A., and Freund, R. J. A Procedure for Solving a Linear Programming
Problem. N. C. Agr. Expt. Sta. Jour. Paper 503, 18 pp. 1953. (Processed.) This study
lists 9 different inputs needed to carry on each of 6 different enterprises.
 48 
LINEAR PROGRAMMING
Combination of Two Farm Enterprises
BEEF CATTLE* FALL CABBAGE*
SFall Land
4 4llill
3 Production capital 3
24 Spring land
2
Sept.Oct. labor
E I 1
Nov.Dec.labor
JulyAug. Iabor.
0 ";Ef ",., .1 0
0 20 40 60 80 100
RESOURCES USED FOR FALL CABBAGE (PERCENT)
PROPORTION OF RESOURCES REQUIRED TO PRODUCE $10,000 NET INCOME.
U. S. DEPARTMENT OF AGRICULTURE NEG. 133155(1) AGRICULTURAL MARKETING SERVICE
Figure 24
Beef cattle and fall cabbage: Proportion of available resources required to produce
$10,000 net income for a farm, North Carolina
Proportion required
Resource
Beef cattle :Fall cabbage
Land:
Spring ........................: 4.63 0
Fall ..........................: 4.63 0.80
Production capital .............. 3.78 1.80
Labor:
JulyAugust ...................: 0 1.08
SeptemberOctober ............. 0 2.17
NovemberDecember ............. 0 1.22
King, R. A. and Freund, R. J. A Procedure for Solving a Linear Programming Problem. N. Ca. Agr.
Expt. Sta. Jour. Paper 503. 1953. (Processed.) p. 13.
49 
The MinimumCost Dairy Feed
Here is another diagram that is useful in linear programming.
In this case, we want the least cost combination of feeds that will
meet stated requirements. The prices of several feeds are given;
also such requirements as total 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
result is shown on the table and plotted on the chart.
We then consider combinations of two feeds that will meet two
requirementsthose 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 45 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 (1) the line joining the two dots slopes downward to
the right and (2) it crosses the 45degree 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 combi
nation 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 would
indicate that the cost would be reduced by substituting this feed
for one of those in the combination. If the combination of gluten
and middlings not only meets the requirements for total digestible
nutrients and for protein, but also meets all other requirements,
the combination we have found is the final answerthat 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. 2/
The one drawback to this kind of diagram is that we can con
sider only combinations of two feeds meeting two nutritive require
ments. 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 dimensionsin other words, how we can find the least expen
sive combination of three feeds meeting three nutritive requirements.
2/ Waugh, Frederick V. The MinimumCost Dairy Feed. Jour. Farm
Econ. 33:299310, illus. 1951.
 50 
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 Total digestible
: protein nutrients
Corn ............................: 0.136 0.441
Oats ............................: .187 .375
Milo maize ......................: .203 .495
Bran ............................: .321 .423
Middlings .......................: .332 .436
Linseed meal ....................: .400 .272
Cottonseed meal .................: .464 .268
Soybean meal ....................: .504 .286
Gluten ..........................: .412 .395
Hominy ..........................: .158 .448
Waugh, Frederick V. The Minimum
1951.
Cost Dairy Feed. Journal Farm Economics. 33:299307, illus.
 51 
LINEAR PROGRAMMING IN THREE DIMENSIONS
The MinimumCost Broiler Feed
The preceding diagram illustrated a method of finding the leastcost 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 dif
ferent 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
3dimensional 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 num
bered 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 nonfiber 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 1.63 percent of the require
ments of nonfiber. 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 corre
sponds 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 nonfiber. The object of our programming is to get a feed which will have
100 percent of each of these requirementsand also 100 percent, or more, of the other
requirements which are not considered here. To get 100 percent of all these require
ments 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 13. 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 13. (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 would 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 combi
nation.
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 non
fiber. 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 pur
poses perhaps it is enough 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 
Figure 26
Broiler ration: Percentage of nonfiber, productive energy
and protein met by $1 worth of feed, specified feeds
umbr Percentage of required nutrients
Number :
on : Feed : :re
Productive
chart : : Nonfiber : er Protein
energy
Percent Percent Percent
:Meal:
1 : Soybean, 44 percent ............: 1.63 1.4 3.6
2 : Linseed, solvent ...............: 1.30 .8 2.2
3 : Cottonseed, expeller ........... 1.33 1.3 2.8
4 : Gluten .........................: 1.24 1.1 2.6
5 : Meat and bone scrap ..............: 1.70 1.6 4.0
6 : Meat scrap .......................: 1.33 1.4 3.4
7 : Fish meal, menhaden ..............: .69 .7 2.0
8 : Buttermilk, dried ................: .32 .3 .4
9 : Corn distillers solubles, dried ..: 1.35 1.5 1.8
10 : Corn .............................: 1.75 2.1 .8
11 : Milo .............................: 1.57 1.9 .8
12 : Wheat, standard middlings ........ 1.74 1.4 1.4
13 : Hominy feed, yellow .............. 1.83 1.7 1.0
14 : Barley ...........................: 1.75 1.6 1.0
53 
LINEAR
PROGRAMMING.
3Dimensional,
Analysis of
Broiler
Feeds i I ;il
U. S. DEPARTMENT OF AGRICULTURE
_ _
IqEG. 426057 (5) AGRICULTURAL MARKETING SERVICE
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 194048. A practical
question is whether it would be 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 returns 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 40 billion in the first
period and 50 billion in the second. Costs of storage,
handling, and any loss in quality would have to be de
ducted 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 con
cave 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 aver
ages. The economiststatistician 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 numbers from the diagram,
copy them on a piece of paper, add them, divide by two,
and put the average back on the diagram. The simple arith
metic average of any two points on any curve can be located
graphically by the graphic method explained here.
 54 
TOTAL RETURNS '40
($ M IL.)'
AVERAGES
Eggs: Gross Profit From Storage *
 '41 
.=k" 4zL
500
0
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 NEG. 1334 55(1) AGRICULTURAL MARKETING SERVICE
Figure 27
Eggs: Production, price per dozen received by farmers, and total returns, 194048
Pi :l iiTotal
Year : Production ce returns
Million
Billions Cents dollars
1940 ...................: 39.7 36 1,188
1941 ...................: 41.9 35 1,225
1942 ...................: 48.6 30 1,230
1943 ..................: 54.5 22 990
1944 ...................: 58.5 16 784
1945 ...................: 56.2 20 940
1946 ...................: 56.0 21 987
1947 .................. : 55.4 21 966
1948 ...................: 54.9 22 1,012
1/ Adjusted for estimated effect of disposable income on price.
Data derived from Figure 92 in Thomsen, Frederick L., and Foote, Richard J. Agricultural Prices.
New York. 1952. p. 431.
55 
, .
e^
'%48
'46
Average 4 III1I;II 3
47
11000
ELASTICITY
Coefficient of Elasticity of Demand
Many economists have trouble with coefficients of elasticity.
They are frequently concerned with the elasticity of demandmore
precisely, with the elasticity of consumption with respect to
price. The diagram shows how this can be measured graphically.
The curved line on the diagram represents an assumed demand
curve for eggs. The scales for consumption 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 small squares on the
grid, this equals 17.75 divided by 32.25. Either computation in
dicates an elasticity of 0.55.
This piece of graphics comes from Alfred Marshall. 10/ It
derives from the definition of elasticity = . Note
thaA c+d as b
that dqa c+d and (by similar triangles) c+d b Also
dp a+b a+b b
p=a, and q=c. So   = a = a/b.
dp q b c
Some economists have found the concept of elasticity so dif
ficult that they have used "are 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 curvesbut to any curve. When we speak of the elasticity
of demand we (usually) mean the elasticity of consumption with re
spect to price. But we might want the elasticity of cost of pro
ducing potatoes with respect to the amount 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. 1948. First published 1920.
 56 
ELASTICITY
Demand For Eggs
CONSUMPTION (NO. PER CAPITA)q
800
400 :
0 b
0 20 40 60 80 100
RETAIL PRICE (< PER DOZ) p
U.S. DEPARTMENT OF AGRICULTURE NEG. 133555(1) AGRICULTURAL MARKETING SERVICE
Figure 28
Eggs: Consumption per capital associated with given retail price per dozen
Consumption Price
Number Cents
777 ..................... 20
621 ..................... 30
530 ..................... 40
469 ................... .. 50
424 .....................: 60
390 ..................... 70
362 ..................... 80
Based on an assumed elasticity of demand coefficient of 0.55. See Foote, Richard J. and Fox,
Karl A. Analytical Tools for Measuring Demand. U. S. Dept. Agr. Agr. Handbook 64. 1954. p. 40.
 57 
METHOD OF DETERMINING MOST PROFITABLE OUTPUT
Applications of Nitrogen on Corn
The chart on page 23 shows, based on experimental results, the relation
ship between observed 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 per
sons who read the first edition of this handbook suggested 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 QA at the bottom of the chart shows the number of bushels of
corn required to pay for various amounts of nitrogen when one bushel of corn
will buy 10 pounds of nitrogen. This would be the case, for example, 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 inputoutput 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 QA
now just touches the inputoutput curve. Then that edge of the triangle will
be parallel to line OA and we draw the line indicated.
At this point the farmer would buy about 160 pounds of fertilizer to the
acre and would expect to get a yield of about 146 bushels. If he used 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 could be made, of course, with any assumed 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 inputoutput curve. In this par
ticular case, however, there would be little change in the most profitable
rate of nitrogen application unless the price of nitrogen were greatly in
creased or the price of corn were greatly decreased. This is because the
inputoutput function curves vary sharply in the neighborhood of the point
that is most profitable with the price assumption shown on the chart. Assuming
that our inputoutput function is approximately right, applications of much
more than 160 pounds to the acre would obviously be unprofitable since the
maximum corn yield is apparently attained with a fertilizer application of a
little over 160 pounds of nitrogen.
This general type of chart is useful in analyzing a wide variety of eco
nomic problems. It is not limited to fertilizer applications, but applies
just as well to such problems as the most profitable amount of concentrates to
feed to dairy cows, or the most profitable amount of labor to be used in any
operation whether on the farm or in marketing.
 58 
U.S. DEPARTMENT OF AGRICULTURE NEG. 398857 (3) AGRICULTURAL MARKETING SERVICE
Figure 29
Corn: Yield per acre by specified quantity of nitrogen applied, Ontario, Oregon
Nitrogen Yield of :: Nitrogen :Yield of
applied corn :: applied corn
Pounds Bushels :Pounds Bushels
0 ......... 64.6 :: 160 ......... 146.8
40 .........: 90.4 : 180 .........: 141.2
80 ......... 118.2 :: 200 ......... 147.1
100 .........: 132.4 :: 240 ......... 145.8
120 ......... 140.7 :: 280 ......... 147.4
140 .........: 141.0 :: 320 ......... 143.8
Paschal, J. L., and French, B. L. A Method of Economic Analysis Applied to Nitrogen Fertilizer
Rate Experiments on Irrigated Corn. U. S. Dept. Agr. Tech. Bull. 1141. 1956. p. 16.
 59 
METHOD OF DETERMINING MOST PROFITABLE OUTPUT
Corn: Yield Per Acre In Relation To Applications of Nitrogen
YIELD (BU.)
100
50
0
400
NITROGEN (LBS.)
DIFFERENTIATION
Some statisticians and economists find calculus a
difficult subject. Differential calculus is relatively
easy if you do it graphically. The differential at any
point on a curve is simply the slope of a tangent drawn
at that point. The tangent can be drawn easily with a
transparent straightedge. The differential, y is
the slope of this tangent.
In figure 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 figure 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,  = 5.8, and in figure 31,
dx
d = 0.002.
dx
These differentials can be read most easily by draw
ing 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 tri
angle, 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
xaxis. 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 xaxis. 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
0 1
U. S. DEPARTMENT OF AGRICULTURE
2 3 4
VALUE OF X
NEG. 133655(1)
AGRICULTURAL MARKETING SERVICE
Figure 30
DIFFERENTIATION
VALUE OF Y
3
2
0
1
1 dy/dx=
0.002
2
0 200 400 600 800 1,000
VALUE OF X
U. S. DEPARTMENT OF AGRICULTURE NEG. 133755(1) AGRICULTURAL MARKETING SERVICE
Figure 31
 61 
DERIVING A MARGINAL CURVE FROM AN AVERAGE CURVE
Marginal Returns
Often the economist has a demand curve showing esti
mates of average prices corresponding with a range of
quantities sold. His problem may call for an analysis of
marginal returns (or marginal expenditures of consumers).
The easiest way to do this is to find graphically several
points on the returns curve.
Robinson 11/ explained the geometry of this. Briefly,
dR dp
total returns are R = pq. We want = p + q. We
dq dq
can take any point on the demand curve, such as point A in
our diagram (32 pounds, at an average price of 41 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 de
mand curve (that is, at 41 cents). This parallel cuts a
perpendicular dropped from A at point B, and the price
equivalent of B measures the marginal returns correspond
ing to the quantity sold at point A on the demand curve.
Here marginal returns are 10.5 cents when 32 pounds per
capital 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 curve.
11/ Robinson, Joan. The Economics of Imperfect Compe
tition, p. 30. London. 1933.
 62 
MARGINAL RETURNS
Chicken Meat
40
B=Marginal return
of 10.5 cents
0 10 20 30 40 50 60
CONSUMPTION (LBS. PER CAPITAL) q
U. S. DEPARTMENT OF AGRICULTURE NEG. 133855(1) AGRICULTURAL MARKETING SERVICE
Figure 32
Chicken meat: Price per pound at retail associated with given levels of consumption per capital
Price Consumption
Cents Pounds
59.1 ..................... 20
49.9 .................... 25
43.5 .................... : 30
38.8 ....................: 35
35.1 .................... 40
32.1 ....................: 45
29.7 ....................: 50
Regression coefficient based on the reciprocal of an assumed 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 64. 1954. p. 40.
Igr Igr Ha db o II. I9 4 p. III40. III I II I IIII I+ L
 63 
Marginal Costs
A marginal cost curve can be obtained froa 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 mar
ginal curve will be above the average curve. To find
the marginal cost at point A in the diagram, we erect a
perpendicular line at point A and draw a tangent to the
average cost curve at this point. We also draw a hori
zontal line from point A to the cost axis and note the
point at which this line cuts the axis. We then draw a
line through 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 unit of output are about $0.237.
Marginal costs, as indicated by B, are $0.292. As in the
preceding example, several points on the marginal curve
can be located as a basis for drawing the entire curve.
If xy is given as a fraction of x, as in these ex
amples, we always can compute
dxy = y + x
dx dx
by this process no matter what x and y represent.
 64 
MARGINAL COSTS
Cost of Land and Fertilizer for Varying Inputs of Fertilizer
AV. COST OF OUTPUT ($ PER UNIT) Y
.4 0 114 i.. ,,,f
.35 1 B=Marginal cost
: of 0.292 dollars
.30
.25
.20 I
.15 ,
0 1 2 3 4 5 6 7 8 9 10 11 12
INPUTS OF FERTILIZER ($)X
U. S. DEPARTMENT OF AGRICULTURE NEG. 133955(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 Cost per unit of output for
Total
Land Fertilizer utut Land Fertilizer Total
Dollars Dollars Dollars Dollars Dollars
10 1 47 0.213 0.0213 0.2343
10 2 51 .196 .0392 .2352
10 3 56 .178 .0536 .2316
10 4 62 .161 .0645 .2255
10 5 64 .156 .0781 .2341
10 6 67 .149 .0895 .2385
10 7 68 .147 .1030 .2500
10 8 69 .145 .1159 .2609
10 9 70 .143 .1287 .2717
10 10 : 64 .161 .1562 .3172
10 11 48 .208 .2294 .4374
Black, John D. Production Economics. New York. 1926. pp. 317318.
 65 
ROOTS OF A POLYNOMIAL
x3 1.2240 x2 + 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 poly
nomial in a handbook dealing with graphic analysis. Roots of polynomials are
used mainly in highpowered 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 con
nection 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 rootsand 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=O, the polynomial equals
0.0183; so we plot y= 0.0183 corresponding to x=O. If x=0.1, the polynomial
equals 0.0074; so we plot y=0.0074 corresponding to x=0.1. We proceed to com
pute several points on the curve, y= x3 1.2240x2 + 0.3695x 0.0183. When
we have enough points, we draw a curve through them. Wherever this curve
crosses the xaxis, it indicates a real root. In this case, the roots are
approximately 0.06, 0.38, and 0.78. The canonical correlation is approxi
mately equal to the square root of 0.78.
We could locate any of these roots more exactly by blowing up the part of
the diagram near the root. Thus, we could draw a new diagram for the part of
the curve between x=0.76 and x=0.80, plot the curve on a blownup 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 third
degree 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 hand
book.
12/ Tintner, Gerhard. Econometrics. New York. 1952. Taken from equations
(l1 on p. 119, letting x= 2
 66 
ROOTS OF A POLYNOMIAL
Y = X3 1.2240X2+ 0.3695X 0.0183
VALUE OF Y
.01
0O
0~
0o.061
.01
.02
zizmzi
I.. . .. .i.I I I. . .I.. .
! ! !~1 j ! ! 1 1 1 1 !!A !.
..... .....l ..... ......................t
IIII M19 111 111111MII
04 F i
0 1. .2
U. S. DEPARTMENT OF AGRICULTURE
.3 .4 .5 .6
VALUE OF X
.7 .8 .9
1.0
NEG. 134055(1) AGRICULTURAL MARKETING SERVICE
Figure 34
Values of a thirddegree polynomial, y, at specified levels of x 1/
x y
0 ...................... 0.0183
.1 ...................... .0074
.2 ...................... .0146
.3 ......................: .0094
.4 ...................... .0023
.5 ......................: .0146
.6 ...................... .0212
.7 ...................... .0164
.8 ...................... .0059
!/ y = x3 1.2240 x2 + 0.3695 x 0.0183.
Data compiled using equations (18) as a basis and
New York. 1952. p. 119.
67
letting x = X2. Tintner, Gerhard. Econometrics.
I I 1 I 1 1 I 1 11711771 11 I 1 1711771 11 1 1 1 1 1 17 11 III 1_1
~vv 
 
,0.781i:
t,,,lll:ll
UT7T11 I 1 1 171 1
. .r..I I 1 il l l i
SOLUTION OF SIMULTANEOUS EQUATIONS
Determining the Supply Below 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 com
putation. Statisticians often must solve two or more equations simultaneously. Various
methods of solution are available, including the popular GaussDoolittle 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 GaussDoolittle 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 curve 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 solutionexcept in
the extreme case where the two lines are identical or parallel, where there are infi
nitely 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 equations of any degree, solutions are real wherever the curves cross one another;
otherwise they are imaginary.
Gustafson, in an unpublished manuscript 14/, 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 calcu
lus. However, a method is outlined by which approximate rules can be obtained by
carrying out certain essentially arithmetic operations. One of the necessary computa
tions 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 curves
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 31.0 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 feasi
ble 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 Equatioqs. Math. Tables and Other Aids to Computations,
5:246248, illus. 1951.
14/ Gustafson, Robert L. Optimal Storage Rules for Grains, unpublished manuscript,
1957.
 68 
SOLUTION SIMULTANEOUS EQUATIONS
Constants Used in Obtaining Optimal Storage Rules*
VALUE OF K
U. S. DEPARTMENT OF AGRICULTURE :EG. 398957(3) AGRICULTURAL MARKETING SERVICE
32 
iK=31.24 .57L
SValue of k as computed
305 .56
Curve plotted from tabular values
28
0 31 .2 .6 .8 1.0
VALUE OF L
*SEE TEXT FOR METHOD OF OBTAINING CURVES.
U.S. DEPARTMENT OF AGRICULTURE NEG. 98957(3) AGRICULTURAL 31.RKETING 18 SERVICE
Figure 35
Values of k and L as computed in alternative ways
Tabular method
Value of k as computed
From formula
Given value of k i Computed value of L
29.5 1.02 30.66
30.5 .56 30.92
31.5 ..27 31;09
32.5 .i10 31.18
Gustafson, Robert L. Optimal Storage Rules for Feed Grains. Unpublished manuscript. 1957.
9tU. S. GOVERNMENT PRINTING OFFICE : 1957 O 430756
 69 
