7 ^ n

\4

ANALYTICAL

TOOLS

for Measuring

Demand

é>f Richard J. Foote and Karl A. Fox

Agriculture Handbook No. 64

UNITED STATES DEPARTMENT OF AGRICULTURE

AGRICULTURAL MARKETING SERVICE

Washington, D. C. January 1954-
PREFACE

Considerable progress has been made in recent years in developing

methods to be used in analyzing the factors that affect prices and consuniption of individual commodities. This handbook discusses certain methods

vhich appear to be of value for this purpose. Seme of them are jrelatively

standardized; others were developed oily recently. These latter have been

applied in only a few cases. In most instances, exaniples are included ^ich

indicate specific ways in which these techniques can be used.

The handbook is designed mainly to acquaint research workers in agriciiltural economics and related subjects with seme of the recent developments

in the field. No attempt was made to cover all new developments that apply,

although many of the more important elements in analysis of demand are

touched upon. Use of the handbook presianes a general knowledge of the

theories of price and demand and of the standard techniques of regression

analysis. Some of the sections also presume a knowledge of college algebra

and some of the notes in the Appendix, of calculus. But the conclusions are

presented in nonmathematical terms, so that, except for certain developmental

or explanatory sections, the handbook as a whole can be used by those not

acquainted with higher mathematics.

Most of this material was presented at a series of seminars held dxiring

1951-53 for staff members of the former Bureau of Agricultural Economics

working In this field. Suggestions made at the seminars were Incorporated and

the material was brought up to date to include later developments. Certain

sections were prepared by other staff members; these axe indicated by footnotes. Helpful suggestions were received from various members of the staff.

i

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Washington 25, D. C. - Price 50 cents
COHTERrS

Page

Introduction 1

Diagrammatic niodels of the supply-demand structure 1

Statistical considerations in setting up the analysis 2

Choice of period of time ] 7

Years to be included [ j

Use of prices at the local market, wholesale or retail

level j

Choice of the quantity variable !!.!!!!!!!!!!! 8

Choice of the dependent variable 8

Choice of the demand shifter 9

Use of arithmetic or logarithmic variables 9

Use of actual data or of first differences 10

Use of total or of per capita data H

use of deflated data U

Alternative methods that allow for changes over time 11

Measuring demand for substitute (competing) or completing

commodities 12

Conamoer goods I3

Some statistical problems I5

Producer goods I9

Statistical considerations in interpreting the completed

analysis 25

Considering the unexplained residuals 25

Considering the natxxre of the statistical coefficients 26

The use and interpretation of tests of significance 26

Allowing for disturbances or for errors in the data 29

Forcing perfect correlation to obtain a reversible

demand ctirve • 33

Validity of an estimate from a multiple regression

equation 35

A survey of demand elasticities based on single^equation

analyses 36

Reasons for different elasticities of demand 37

Problems involved in interpretation 37

Coefficients obtained 36

The simultaneous equations approach 39

Specifying the econcmic model k2

Pi^oblems of identification U3

Problems of estimation kk
Page

Applicability of single-equation methods to analyses

designed to measure the elasticity of demand for

farm products ^5

Examples involving the use of more than a single equation ^9

Simultaneous-eq\xation systems for pork, beef, and

export crops if9

A system of simultaneous difference equations relating

to the feed-livestock economy « 52

A 10-region model of the feed-livestock economy 57

The technique of linear programning 6k

Demand analyses based on purchases by individiial consumers 68

Regression analyses based cai national aggregate data

by months 68

Regression analyses based on aggregates for selected

groups of families 68

Quasi-descriptive studies of consumers with specified

characteristics 68

Regression analyses based on purchases of individual

families by months • • • • • 69

Demand analyses based on sales frcai individual stores 70

Studies relating to specified fruits 70

Comments regarding specific analyses ••• 71

Comments regarding exx>erimental design «••••• • Jk

Literature cited • • • • • • 76

Appendix • 79

ii
ANALYTICAL T(X)LS FOR MEASURING DEMAND

by

Richard J. Foote, Agricultural Economic Statistician, and Karl A. Fox, Chief,

Statistical and Historical Research Branch

Agricultural Marketing Service

INTRODUCTION

The measurement of demand is a conç)licated subject. Competent analysis

requires three things. First, the economist must have a thorough knowledge

of the economic factors that affect the commodity and obtain adequate data

on vhlch to base the analyses. Second, he must understand economic theory

In general. Third, he must be able to use modem techniques of analysis.

This report is mainly concerned with techniques of analysis. But the

first section discusses a way to make a preliminary survey of the causeand-effect relationships to be expected. Here is where the researcher brings

to bear his knowledge of the commodity and his understanding of theory*

The report then discusses some of the techniques that can be used to

test the assumed model and to give quantitative estimates, or forecasts.

Specific examples are cited when needed to indicate how a particular technique shoiild be used.

DIAGRAMMATIC MODELS OF THE SUPPLY-DEMAND STRUCTURE

Diagrams that show the flow of commodities from producer to consvuner,

in terms either of physical products or of marketing channels, or both, have

teen in use for many years. Similar diagrams that show the economic forces

or relationships which affect a given commodity or group of commodities have

been developed in the last several years by staff members of the former Bureau

of Agricultural Economics. Four such charts are shown in figures 1 to U,

These were taken from Fox (gO) l/, Hermie (^), and Armore (2). Similar

charts have been prepared for a nuniber of other commodities. Charts of this

kind are closely related to the "path coefficient" diagrams which were used

by Sewall Wright in the eaurly twenties. When such diagrams are formalized

into a set of equations which indicate the variables involved in each of

several supply and demand relationshipsj they become the "models" discussed

in the monographs of the Cowles Commission for Research in Economics.

Such diagrams are usefxil in several ways: (l) They help the analyst

in thinking through the basic factors and relationships involved, (2) they

aid in preparing a logical writeup of the econcmic structure of the industry,

and (3) they help the reader to follow fairly complex relationships and

1/ Numbers in parentheses refer to Literature Cited, p. 72.
- 2 -

discussions. Any statistical analyses that are run shoiild be based directly

on the relationships indicated in the diagrazas except for such modification

as may be indicated in the diagrams after the research work is under way.

Figures 1 and 2 indicate the kind of statistical questions that can be

discussed in terms of such diagrams. Each shows the demand-supply structure

for a certain type of perishable crop. It is assumed that each crop marketed

under the plan diagramed in figure 1 will be sold in a single outlet -

watermelons make a good example - and that each crop marketed under the plan

diagramed in figure 2 will be sold in part in the fresh market and in part in

processed form, that the farm or local market price is identical in the two

outlets, and that the retail price in either form is not significantly

affected by the retail price or consimiptlon of the other form. This situation may apply approximately to consimçtion of table grapes, which axe used

in fresh form and also for making wine and other alcoholic beverages. Each

diagram is based on the assumption that the total supply will not be affected

by the price during the harvesting and marketing seasons.

If the purpose of an analysis is to estimate the expected price associated with given values for such variables as size of crop and consumer

income, a satisfactory answer in the case illustrated by either diagram may

be obtained by a least-squares regression analysis with price dependent and

other variables independent. Such an equation will give unbiased estimates

of the elasticity of demand and other structiJiral coefficients if, and only

if, the supply and consuiiç)tion for a given period are not affected significantly by the price during the corresponding marketing season. If figure 1

and the assimçtions noted above apply, these conditions will be met approximately. In this instance, retail price ordinarily would be considered to be

determined by consumption (or production) and consumer income. In the case

illustrated in figure 2, single equations could be used to measure the

interrelaticMishlps among consimiption, price, and income in either the fresh

or processed market, given the amount that moved throixgh each of these

outlets. However, a simultaneous system of equations would be needed to

estimate the relative proportion of the crop that could be expected to move

through each outlet in any given year. Such a system would at the same time

yield a measure of the different price and income elasticities of demand

prevailing in each outlet.

Conditions under which single equations or simultaneous equations shouH

be used are discussed in greater detail in a later section, and this subject

is discussed fully in Fox (20).

STATISTICAL CONSIDERATIONS IN SETTING UP THE ANALYSIS

A niimber of statistical decisions must be made before an analysis deal*

ing with the factors that affect price or consimiption of a given commodity

can be run. In this field, many decisions must be based mainly on Judgment.

Alternative methods exist, but in many cases methodological experts have not

agreed as to the best procedure to be followed. But in certain cases
- 3 -

DEMAND AND SUPPLY STRUCTURES

FOR PERISHABLE CROPS

Supply Predetermined: Single Morkei

DISPOSABLE

CONSUMER INCOME

MARKETING I

I SYSTEM

! I

UNHARVESTED |

PRODUCTION

WEATHER; ECONOMIC

INFLUENCE PRIOR

TO HARVEST HARVESTING

COST

ARROWS SHOW DIRECTION OF INFLUENCE. HEAVY ARROWS INDICATE MAJOR PATHS OF INFLUENCE

WHICH ACCOUNT FOR THE BULK OF THE VARIATION IN CURRENT PRICES. LIGHT SOLID ARROWS INDICATE DEFINITE BUT LESS IMPORTANT PATHS; DASHED ARROWS INDICATE PATHS OF NEGLIGIBLE,

DOUBTFUL, OR OCCASIONAL IMPORTANCE

U.S.DEPARTMENT OF AGRICULTURE NEG.48930-X BUREAU OF AG R I C U LT U R A k ECON CM I C S

Figure 1.- If production is not affected by price during the marketing

season and if all of a crop is sold in a single outlet, in general a singleequation analysis vill give an unbiased estimate of the elasticity of demand.
- k -

DEMAND AND SUPPLY STRUCTURES

FOR PERISHABLE CROPS

Supply Predetermined: Two Independenf Markets

CONSUMPTION,

FRESH

WEATHER;

ECONOMIC

INFLUENCES

PRIOR TO

HARVEST

PRODUCTION

CONSUMPTION,

PROCESSED

DISPOSABLE

CONSUMER

INCOME

ARROWS SHOW DIRECTION OF INFLUENCE. HEAVY ARROWS INDICATE MAJOR PATHS OF INFLUENCE

WHICH ACCOUNT FOR THE BULK OF THE VARIATION IN CURRENT PRICES. LIGHT SOLID ARROWS INDICATE DEFINITE BUT LESS IMPORTANT PATHS; DASHED ARROWS INDICATE PATHS OF NEGLIGIBLE,

DOUBTFUL, OR OCCASIONAL IMPORTANCE

U.S.DEPARTMENT OF AGRICULTURE NEG.48931-X BUREAU OF AGRICULTURAL ECONOMICS

Figure 2.- The expected farm price associated with given values of size

of crop and consuioer income frequently can be estimated "ty a single-equation

analysis even for crops used in multiple outlets. But a system of simultaneous equations would be needed to estimate the relative proportion of the

crop that will move throufii eau^h outlet and to yield unbiased estimates of

the different price and inccme elasticities of demand prevailing in each.
STRUCTURE OF PRICE-MAKING FORCES FOR WOOL

Production of Wool

in

Importing Countries

Stocks of Wool

in

Importing Countries

Stocks of Raw Wool

Stocks of Top

Stocks of Top,

Noils, and Raw Wool

Stocks of Wool Yarn

Stocks of Wool Yarn

Stocks of Wool

Fabrics

Stocks of Wool

Fabrics

Stocks of Finished

Wool Goods

Stocks of Finished

Wool Goods

Consumer Stocks

of Wool Goods

Rate of Change in

Consumer Income

Current Consumer

Income

Prices in Markets

of

Exporting Countries

Production of Wool

in

Exporting Countries

Stocks of Wool

in

Exporting Countries

Stocks of Raw Wool L Demand by

. \, United States

Farm Price ./ Dealers ^. Sä r /\ ^^y'S t

Stocks of Raw Wool Demand by y V /^ Raw Wool Prices

Importers at Boston / \ / ^

Demand by Wholesalers

and Retailers

Wholesale Price

of

Finished Goods

Demand by Consumers

Retail Price

of

Finished Goods

U.S.DEPARTMENT OF AGRICULTURE

DOTTED LINES REFLECT ANTICIPATIONS

NEG. 48015 BUREAU OF AGRICULTURAL ECONOMICS

Figure 3-- The demand for wool at the farm level is derived from the

combined demands of the many processors and users of wool. Domestic prices

tend to exceed world prices by approximately the amount of our tariff, but

prices rise and fall with changes in world supply and demand.
6 -

ECONOMIC FACTORS AFFECTED BY A

CHANGE IN THE YIELD OF COTTONSEED OIL

COTTONSEED OIL YIELD PER

TON OF COTTONSEED PROCESSED COTTONSEED HULL

YIELD PER TON

OF COTTONSEED

CRUSHED

SUPPLY OF

COTTONSEED

OIL

SUPPLY OF FATS

AND OILS USED

IN FOOD OTHER

THAN COTTONSEED OIL, BUTTER,

AND LARD

SUPPLY OF

FOOD FATS AND

OILS OTHER

THAN BUTTER

AND LARD

DISPOSABLE

INCOME

SUPPLY OF

LARD

AVERAGE PRICE

OF FOOD FATS

AND OILS OTHER

THAN BUTTER

AND LARD

SUPPLY OF

BUTTER*

PRICE OF

COTTONSEED

OIL

VALUE OF

COTTONSEED

OIL YIELD

VALUE OF PRODUCTS OBTAINED

PER TON OF

COTTONSEED

PROCESSED

PRICE OF

COTTONSEED

RECEIVED BY

FARMERS

PERCENTAGE OF

COTTONSEED

CROP, LESS USE

FOR PLANTING,

SOLD TO MILLS

*BUTTER DID NOT HAVE A STATISTICALLY SIGNIFICANT EFFECT BEFORE WORLD WAR II

BUT IT äAY BE OF MORE IMPORTANCE CURRENTLY.

VALUE OF

COTTONSEED

HULL YIELD

VALUE OF

COTTONSEED

MEAL YIELD

VALUE OF

COTTONSEED

LINTER YIELD

PROCESSING AND

MARKETING

MARGINS AND

OTHER CHARGES

PRICE OF

COTTONSEED

MEAL

PRICES OF COMMERCIAL FERTILIZER USED ON

COTTON

PRICE

OF

CORN

U. S. DEPARTMENT OF AGRICULTURE NEG. 48872-X BUREAU OF AGRICULTURAL ECONOMICS .

Figure U.- The effects of an increase in the yield of cottonseed oil per

ton crushed on total returns received "by farmers takes place through tvp

channels. One affects the quantity of oil obtained per ton of seed crushed

and therefore directly affects its value; the other affects the price of the

oil hy increasing its total supply* The net effect on the price received l3y

farmers for cottonseed must allow for both aspects.
- 7 -

particular conditions that are known to prevail with respect to the catmnodity

studied will indicate a preference for one method over another. In this

section, the more important decisions that must be made are listed, and some

of the considerations that indicate a preference for one method over another

exe discussed.

Choice of Period of Time

Prior to about 1945, most analyses of the factors affecting the price or

consumption of a given commodity were based on annual data for either calendar or crop years. In certain instances some other time period may be

preferable. For example, analyses of the factors that affect prices of com

were run separately for the periods November44ay and June-September, with

allowance in the latter period for the effects of new-crop supplies of oats

and barley and for the expected size of the new com crop. (See Foote (15).)

The period chosen should be long enough to average out the effects of irregular or nonmeasurable factors and short enough to insure that a homogeneous

set of factors will be operating. Pearson and Vial (39) give monthly analyses for many livestock products. They show that the effects of production

and stocks on price, for example, differ greatly from month to month and

give reasons for expecting this to be so.

Years to be Included

In general, as many years as possible should be included in an analysis

unless important changes in institutional or nonmeasurable factors are

believed to have affected the relationships. "Abnormal" years, such as those

in which price ceilings or rationing were in effect, should be omitted from

the analysis but the calculated values should be checked to see that they are

in line with expectations. If a sufficiently long series is available, the

postwar years could be omitted from an analysis to see whether the prewar

relationships hold in the postwar period. If they do, the analysis can be

used with greater confidence. If this is done, the ceuLculated values for the

postwar years should be obtained in all cases. Except in extreme cases,

decisions regarding the years to be included should be made before running

the analysis.

Use of Prices at the Local Market, Wholesale or Retail Level

Ret€Lil prices should be used in an analysis designed to measure consumer

demand at the retail level. In measuring domestic demand for items used to

a large extent by industry, such as fats and oils or cotton, a wholesale

price mi^t be preferred. Farm or local market prices then can be determined

by a simple correlation between farm and retail or ^rtiolesale prices. Such a

procedure can be expected to give improved results \rtienever marketing margins

change abnormally. If this method is used, the error of estimate for farm

prices will be a function of the error of estimate in the equation that

explains retail or wholesale price and the error in the equation that ccmnects farm price with the retail or wholesale price.

286234 0-54-2
- 8 .

Choice of the Qxiantlty Variable

The type of quantity variable chosen will depend mainly cm the purpose

of the analysis. If the analysis is designed to measure factors that affect

dcmestic demand^ domestic consumption should be used. If price is used as

the dependent variable, total supply of the commodity plus supplies of

closely competing commodities (either separately or combined) probably

should be used as the supply variable, except when consumption is bigihly

correlated with production. If exports are subsidized, supplies available

for domestic consumption and carryover may be the applicable supply variable.

To increase our knowledge of the effects of changes in production on

the several alternative outlets, a set of first-difference analyses can be

run, with production as the independent variable and with each of the components of the corresponding disposition as dependent variables in a series

of simple regression analyses. Year-to-year changes in prodxiction are

exactly equal to the sum of year-to-year changes in (l) domestic consumption,

(2) net foreign trade, and (3) the net change in inventories. These utilization groups could be further subdivided if the data permitted and the problem required it. The technique consists of calculating the simple leastsquares regression of each distribution category separately upon the

production variable, using first differences of actual data in each case.

When this is done, the sum of the slopes ("b" values) of these regression

equations equals 1.0 and the sum of the constant terms ("a" values) equals

0. 2/ Such a set of analyses indicates the change in domestic consimiption,

net exports, and stocks, respectively, that woxild be associated on the

average with a given change in production. If changes in exports and stocks

are negligible, the problem of isolating the domestic demand cxirve is simpler

than \Aien major changes occur in these outlets. For most commodities, domestic demand is less elastic than total demand, the difference depending to a

considerable extent on the average proportion of changes in production that

is represented by changes in dcmestic consumption.

Choice of the Dependent Veorlable

If an analysis is designed primarily for use in forecasting and it is

assumed that no change in structure has taken place over the period included

in the analysis or for which the forecast is made, a single equation, using

as the dependent variable the item for ^ich a forecast is desired, will be

appropriate. Marschak (31, pp. 1-8) discusses a simple example which illustrates the effect of changes in structure on the type of statistical analyses needed under specified conditions. If the analysis is designed mainly to

estimate the elasticity of demand or other structural coefficients and a

s ingle-equation approach is applicable (see p.^35!), the independent variables

should be those that are "predetermined," as by weather, by economic forces

in en earlier period, or by broad economic forces, such as those determining

total ccmsumer income.

2/ A proof of this is given in Appendix note 1.
- 9 -

Annual production, or production plus carry-in, of many crop and

livestock products could "be used as an independent variable, according to

this criterion. If consumption and production are almost identical, as with

some perishable commodities, or are very highly correlated, consumption

could be used as an independent variable. If prices are set by Government

action, as by a support program, price would be used as the independent

variable and consumption as the dependent vfitriable in an analysis designed

to measure the elasticity of demand in the dcanestic market.

Choice of the Demand Shifter

Disposable income is the commonly accepted demand shifter for most

consumer goods. The Federal Reserve Board index of industrial production is

used for items such as certain inedible fats and oils which are to a considerable extent industrial raw materials. Particularly for durable or semidurable goods, some analyses indicate that the change in income from the

preceding year as well as the current level of income is important. (See

Atkinson (U).) Changes in distribution of income also may be important for

certain items. Different kinds of demand shifters are needed for producers*

goods. For exançle, in an analysis of the factors that affect prices of

com, prices of livestock, and the number of animal units fed were used as

separate demand shifters. (See Foote (15).) Animal units are significant

partly because of the importance of diminishing returns in livestock feeding.

In general, the use of a retail or wholesale price series as a demand shifter

is frowned upon, but this use appeared to be justified on theoretical grounds

in analyzing the price of com. Attempts to develop satisfactory shifters

for storage and export demand have been largely unsuccessful. The 4-curve

equilibrium approach can be used to indicate why a given volume of exports

or imports will occur. However, this implies a system of simultaneous

equations. 3/

Use of Arithmetic or Logarithmic Variables

Linear (arithmetic) and logarithmic equations are the principal functional forms used in commodity analysis. For many caamodities, linear

equations yield elasticities which, when translated into total value-supply

curves, are more consistent with theory at the extremes than those given by

logarithmic equations, hj Logarithmic equations have the mechanical advantage of yielding ccmstant elasticity curves and in many analyses they appear

to fit the data better than arithmetic relationships. In general, logarithmic equations should be used when the relationships between the variables

are believed to be multiplicative or the relations are believed to be more

stable in percentage than in absolute tenas or both. If the relationship

¿/ This approach is discussed in Thcmsen and Foote (U5, PP- 232-2liO), based

largely on material developed by Wells (51) •

U/ In most cases no data are available for these extreme values and there

is little interest in them.
. 10 -

between the variables is believed to be additive, arithmetic variables should

be used. Differences between calculated values from arithmetic and logarithmic equations may be fairly small in the interwar period, but an attempt

to project the analyses to high postwar price and income levels may reveal

strong reasons for selecting the logarithmic form. The choice between

arithmetic or logarithmic variables has nothing to do with the decision to

run the analysis in terms of actual data or first differences. If logarithmic variables appear logical in a first-difference analysis, they would

be equally logical in an analysis based on actual data (deviations from the

mean). Use of certain other types of curves, some of which are hará to

describe mathematically, has been proposed, particularly in the analysis of

supply response. In general, the type of functional form to be used should

be decided before the analysis is run. Except in a few special cases, the

form of the equation cannot be detennined from the data.

Use of Actual Data or of First Differences

First-difference equations have been used to some extent by price

ancLLysts for many years but they have come Into prominence only recently.

In equations designed primarily for use in forecasting, first differences

are used when emphasis is placed on measuring the factors that affect yearto-year change rather than oa deviations from a long-term average. The

recent interest in the use of first-difference equations has arisen partly

because of the obvious inapplicability of a prewar average to the postwar

period.

First differences also are used in: (l) Cases where strong trend

factors tend to overshadow the effects of economic variables (as in the stuc(y

of factors that affect production of milk) ; (2) cases where the failure to

allow for common trends would imply a relationship that does not actually

exist (for example, the upward trend in consumption of evaporated milk and

the downward trend in the price ratio of evaporated milk to fluid milk yields

a pseudo "elasticity of substitution" of I.76); (3) cases in which two variables are more closely correlated with an unrealized third factor than with

each other (for example, the coefficient of determination between prices of

tankage and of com is 0.75 when based on actual data for 1926-42 and

19I+7-49, and 0.02 when baaed cm first differences for the san« years and for

oats and com, for which a hi^i degree of correlation would be expected, the

coefficient of determination based on actual data is O.96 and on first differences 0.87); and (h) for certain technical reasons such as the reduction

of correlation between the ind.ependent variables and of the serial correlation in residuals.

First differences may be hard to use in connection with certain longrange forecasts and in scaoe types of analyses of relative benefits to be

obtained from specified Government programs when no level for a preceding

year can be assumed. The chief criterion for choosing between first differences and actual values (other than those given previously) for forecasting purposes is the standard error of the forecast from each equation form
. 11 .

for the relevant values of the independent variables. If the residuals from

an "actual value" equation are serially correlated to the extent of 40.5 or

greater, the standard error of estimate will generally he smaller for the

first-difference equation. The degree of intercorrelaticaa between the

independent variables also affects the relative forecasting value of firstdifference and actual value equations.

Use of Total or of Per Capita Data

To avoid confusing the time trend for population with that due to other

effects, per ccqpita data probably should be used whenever they are applicable. For certain items, such as the consumption of cigarettes, care must be

used in deciding what population groups to include in the population series.

Use of Deflated Data

At least two problems are involved in the use of deflated data:

(l) This approach assumes a one-to-one relationship between the original

series and the deflator, whereas certain series are known to swing more

widely during a business cycle than other series; and (2) deflation of a

value index by a price index with fixed weights cannot yield a quantity index

expressed in "constant" dollars of the base period.

But Prest {kl, pp. 33-3Ö) and Stone (kk, pp. 287-313) both used deflated

income series in their demand aneúLyses, which were based on carefxil theoretical formulations, although Stone concluded that it does not follow from a

statistical point of view that the deflated series will lead to more reliable

results. He says: "It is to be expected that both will lead to more or less

the same, but not identical, results." Prest concurs in this view. In any

analysis, it is important that the variables included be consistent. For

exanqple, if indxxstrial production is used as a demand shifter in an analysis

of consuxnption, prices should be deflated; if an income series is used as a

demand shifter, both income and price coxild be deflated or both could be

used in a nondeflated form.

Alternative Methods That Allow for Changes Over Time

Tinbergen {k6, pp. 1-29, 193-209) indicates that time concepts enter

into an analysis in four ways: - (l) Certain variables may affect the dependent variable with a certain time lag; (2) if the effect is greater or less as

the length of time varies, separate measurements might be made of the longand short-time effects; (3) the relative importance of the long- and shorttime effects may be affected by the "horizon," that is, the period over which

producers, dealers, and consumers look ahead, which, for example, will affect

the willingness of dealers and farmers to store a large crop; and (k) time

may enter into an analysis as a measure of sources of continuous systematic

variation that were not introduced explicitly into the equation. Time lags

are handled in the equation by including lagged values of the variable; for

exançle, using prices in each of 3 preceding years in a supply-response

analysis.
. 12 -

The importance of the systematic variation may be measured in one of two

ways: (l) Time may be included as a separate variable in the analysis and,

if its partial regression coefficient fails to differ significantly from

zero, it may be omitted; (2) time may be omitted from the initial analysis,

in vhich case the final residuals should be checked to see whether they

exhibit a trend over time. If they do, another variable which explains

them can be added or, if such data are not available, time may be used as a

catch-all variable. As with deflaticMi, it is hard to indicate definitely

\diich method is to be preferred.

If first differences are used, the constant value in the equation

represents the linear effects of time as, if different from zero, it ixaplies

that there would be some change in the dependent variable from the preceding

year even if there were no changes in the independent variables. The statistical significance of the constant value can be tested by use of the standaixl

error of the function at the point at which each of the Independent variables

is equal to zero. Formulas for this coefficient for analyses based CHI two or

three variables are given by Foote (lU, pp. k, 6). This test may be biased,

as residuals from first-difference analyses tend to be negatively correlated

so that the formulas given by Foote tend to imderestimate their variance.

MEASURING DEMAND FOR SUBSTITUTE (COMPETING) OR

COMPLETING COMMODITIES

Schultz (lj2) proposed comparing the coefficients of variation for the

price and supply ratios for two cooipeting commodities to measure the degree

of substitution. If the two items were nearly perfect substitutes, the price

ratios would be nearly constant, whereas the supply ratios would be expected

to vary considerably. If the two items were conoplementary, the price ratios

would vary more than the supply ratios. Thus, the ratio of the coefficients

of variation for the two ratios would vary between zero and one for substitute commodities and for complementary goods it would be greater than one.

Peters and Van Voorhis {kO, pp. 78-79) raised an objection to this test

as follows:

"In spite of the fact that this measure has received

considerable attention from statistical workers, the

authors have doubts of its value. For the mean may

be distorted by a padding of all the scores. Consider

the series of scores: 0, 3, 8, 12, 15> 20, 25, 29; and

the series 20, 23, 28, 32, 35, kO, U5, 1^9. The mean of

the first array is Ik and that of the second is 3^. The

coefficient of variation of the first is 68 while that

of the latter is only 28. Nevertheless the variabilities of the two distributions are precisely the same,

the distortion in coefficients of variaticai being due

solely to the padding of the scores in one of the

arrays."
- 13 -

In view of these objections, the Schultz test is not recommended. The

methods subsequently discussed appear to be better.

Consimer Goods

Suppose we are dealing with a consumer-demand equation of the following

form:

qj^ = a^ + b^^p^ + b^gPg + c^y, (l)

where g^ is the quantity l>ought of commodity 1; £ and g^ are prices paid for

commodities 1 and 2 respectively; and jr is consumer income •

For practical pxxrposes, the two commodities may be considered as

(1) independent, (2) competing, or (3) completing, depending upon whether

-12 ^® respectively (l) zero, (2) positive, or (3) negative. 5/ When two

commodities "conç)ete," as in case (2), and the price of commodity 2 rises,

people will consume more of commodity 1. In equation (l), K^ is always

negative and c^ is almost always positive (except in -Ule case of so-called

"inferior goods").

Suppose that the consumer-demand equation for commodity 2 is

q2 = ög + ^^21^1 + ^22^2 "^ ^2^* ^^^

One would logiccOly expect b to have the same sign as b , 6/ or to equal

zero when b^ equaled zero. This means only that if ccramodity 2 competes

with commodity 1, commodity 1 competes with commodity 2.

Equations (l) and (2) are written appropriately for an individual

consxjmer, who adjusts his purchases (3- and 3^) to the market prices (£, and

2^) which he must talice as given. But these equations generally are not appropriate for deriving statistical demand curves for the market as a whole.

For many agricultural commodities, the market supply in a given year is

determined by weather and by the eccMiomic influences that are operating when

decisions as to planting or breeding are made. When this is the case, supplies of the two commodities are among the "independent" variables in the

5/ A definition based on utility theory, called to the author*s attention

by Frederick V. Wau^, is given in Appendix note 2.

6/ According to the utility theory, bp^ should equal b^g. See Appendix

note 2.
market-demand functions, and current market prices adjust to the given

supplies. The market-demand equation may be written as

Pi = Al + BiiQi + Bi^2 -»- C^Y (3)

P2 = Ag + B21Q1 + B2^2 - " ^2^^ (^)

in which the letters have the same meanings as applied to market averages as

did those in equations (1) and (2) as applied to individual consumers. The

signs that indicate demand relationships in equation (3) are as follows:

1. Independent in demand: B-\ry = 0

2. Competing in demand: Bi2<0; (that is, an increased supply

of commodity 2 depresses the price of commodity l);

3. Coggpleting in demand: Bif>>0; (that is, an increased supply

of ccfflimodity 2 increases the price of ccmmodity l).

In general, B21 should have the same sign as B12 (s^^ footnote 6),

or equal zero when B^Q equals zero.

If the variables Q, and Q^ are strictly "predetermined" (that is, are

—j. —í¿

not affected significantly by price during the marketing season), equations

(3) and (k) are "structural equations" in the interpretation of the Cowles

Commission; that is, they are true demand equations. Production or supply

(production plus carry-in) will meet this conditicm in many cases. If consumption equals production, as with some highly perishable commodities, it

is also a predetermined variable. If consunçtion does not equal production

(because of small or moderate variations in stocks) but is highly correlated

with production, its use as a predetermined or independent variable probably

is still satisfactory for most practical purposes.

In some problems, we want to ascertain not only whether two coaxmodities

are competing, but whether they are perfect substitutes. In equation (3),

this would imply that B^-^ = Bjjg. Kuznets and Klein (3}^) tested the hypothesis that domestic lemons (Q^) and imported lemons (Q2) vei^ not perfect

substitutes during a certain period. In their analysis the coefficients

equivalent to Bjj^ and B12 ^e^^ foxind to be:

Bii = -0.0196, and B12 « -0.0172.

(.0026) (.OOU9)

in \Aiich the figures in parentheses are standard errors of the regression

coefficients. The difference between B^ and Bj^ is nonsignificant; that is.
- 15 -

It could have occurred by chance even though the camnodlties were, in fact,

perfect substitutes on a poimd for pound basis. Therefore, the authors

concluded that domestic and imported lemcxis could be combined into a single

total supply variable.

It appeeurs that the ratio B-] g/B-] i^ would be a reasonable measure of the

closeness of competition or substitution between two commodities, if a poxond

of one was equivalent to a pound of the other in a given end use. The ratio

would range from 1 for perfect substitutes down to zero for commodities that

were independent in demand. For exaniple, en analysis expressing the price

of beef (Pj^) as a function of the consuioption of beef (QQ^) and the consumption of pork (Qg) yielded the following coefficients:

B3JL = -0.1^33^^ and B^g = -0-1929, or ^12 = 0A1^5.

^11

The results indicate that pork competes with beef but that it is far from a

perfect substitute for it.

Examples of demand equations for red meats, chicken and tvirkey, and

western and eastern apples showing the effects of supplies of competing

commodities upon the price of a given commodity may be found in appendix

notes 3 and k. An exploratory analysis of competition among different cuts

and grades of beef is discussed in appendix note 5*

Some Statistical Problems

Problems in Interpreting Logarithmic Coefficients

If two commodities are about equal in terms of total consumption, their

logarithmic regression coefficients should stand in about the same ratio as

their arithmetic coefficients. Otherwise, to test the closeness of substitution by means of the ratio ^^2^1' ^* ^^ necessary to multiply the

ratio of the logarithmic coefficients, fiÍ2 by the ratio Qi . Based on

Bii Q2

the definition of elasticity of demand, it can be shown that

Thus, in equation (îO of appendix note 3, âi > -^^-^ -PP^^«^ *° ^«^^^ ^^ *'°"'

6 pounds and âa » "^1^^°^ applies to other meats, i¿ al»ûut

286234 O - 54 - 3
16

120 pounds. For this analysis.

^12 » -.6$ 6 0.065,

Bii -.50*120

vhlch suggests that lanib Is highly differentiated from other meats in consumer

demand.

Apparently Nonireverslble or Asynnetrlcal Coqpetltlon •

In parallel analyses of retail prices of beef and pork, the following

equations and standard errors of the regression coefficients were obtained:

?Y, » -1.08 (ib -OM <¿! + 0.88y

(.11)^ (.07K (.06) (5)

8/

p . -0.02 Œ- -1.16 a + 0.90y.

"^ (.13)^ (.08)^ (.07) (6)

Equation (Ç) iiqpUes that the B\xpgly of pork has a highly significant

competitive Impact upon the price of beef. But equation (6) shovs no significant lflq[>act of the siipply of beef upon the price of pork!

Possible partial explanations are: (l) As the supply of beef varies less

from year to year than the supply of pork^ It Is harder to establish significant regression coefficients upon ^ than upon ^. (2) The true regression

of ^ \q>on 2^ might be as hl^ as -0.30^ €UKi a value as far off as -0.02 could

still occur by chance about 2 or 3 times In 100. (But a value of -0.02 would

have been about as likely If the true regression had been 40*30 Instead of

-0,30.)

Both of these "possible explanations" are negative. Based on the statistical results alone ^ the most plausible conclusion vould be that the coiiq;>etltlon between beef and pork Is asyamietrlcal—a large supply of pork depresses

the price of beef, but a large supply of beef affects the price of pork very

little. Yet a research worker could hardly recoomend that an administrator

charged with supporting the price of hogs pay no attention to the possible

effects of a record large run of cattle.

Equations (3) aM (If) could be fitted simultaneoxisly, Inq^slng the condition that Bp;i a B22 cmâ thereby forcing a symmetrical result, consistent

with utility theory. When this was done for the analyses of beef and pork,

the following results were obtained:

7/ In equation (3)^ ^ Includes veal euid lamb as well as pork.

8/ In equation (6), gj^ Includes veeú. and lamb as well as beef.
- 17 -

P^ - -1.09 q^ - 0*32 (^ + 0.89 y (5-1)

Pp - -0-32 (jb . 1.16 <^ + 0.95 y. {6.1)

But this approach has Its dangers and shoxOd not be attempted unless these

equations also are fitted separately, for concwirative purposes, without this

special constraint.

Reconciliation of Denand Equations for Individual Cooçeting Comaodities

vith an Aggregative Demand Equation for the Group

Retail demand equations for each of two competing cormnodities are given

by equations (3) and {k). S\;©pose an aggregative analysis in terms of an

index number of the two prices and the sum of the two quantities also is computed • Here

^t » (^1^1 + V2)' - A^ »t (<il + Q2) + Vl> (7)

in ygbioh ïl + S2 " ^ ^^® *1^® weights used in confuting the index numbers of

prices. For given values of Qj^, Qg, and Y, the wei^rted sum of equations (3)

and {k) ought to equal, at least approximately, equation (7):

wi?!« W3A1 + v^BiiQi + ^1^12^ + ^l^lï (3-1)

^2^0" ^2^^ + ^^ZL^l + ^2^22^ WgCgï. {k.l)

In particular, the weighted sum of the four B's should (approximately) equal

It-

»t » (^1^11 + VgBgg) + (w^B^g + WgBgi). (8)

The terms "Wj^B^" and "îî^22 ' ^^^^s^* *^® direct influences of each

supply iipon the price index via its own price. The terms "w^B^" and "wJBpi"

represent the indirect influences of each supply upon the price index via the

price of the cooQ>eting commodity* Unless the commodities are perfect substitutes, the sum of the direct influences should exceed the sum of the indirect

Influences•

Furthermore, the sum of the direct influences is the weighted average

own-price flexibility of the two comeiodities. In general, this s\mi is smaller

than B^, the price flexibility of the groxjp. That is, on the average, the

demands for individual members of a competing group of cooamodities are more

elastic than the demand for the group as a whole.

Two applications of this approach were made, one to meat animals and the

other to meats at retail. In the analysis based on prices received by farmers,

Ij. » -1.266; s\im of direct influences « -I.O89; sum of indirect influences
. 18 .

» -0.l60; total effect, ßupplleB of four meate, « -l.ZkS, vhich ia almost

identical with Bf In the analysis based on prices at retail, B^. « -1.50;

B\m of direct influences « -l»068j sum of Indirect influences « -0,332j total

effect, stxpplies of four meats, « -1«1«00. Results of the retail analysis in

particular indicate that, on the average, demands for individual meats are

significantly more elastic than the demand for all meat as an aggregate.

Some Observations on Completing Ooods

Some coomodities, like edible fats and oils and sugar, are used almost

exclusively as ingredients in combinations of foods--in ^Aiich they often account for a relatively small part of the total cost. At any given time, consxmgption of such commodities tends to be related to consxm^tion of other

foods by a set of technical coefficients—number of teaspoonf\ils of sugar per

cup of coffee, ounces of butter-pl\is-margarine per pound of bread, ounces of

salad oil per \init of scú^d vegetables, emd so forth. If the demand for

sugar-using foods increases with consumer income, the demand for sugar also

will appear to increase with income, even though no consumer actively values

sugar for its own sake.

Consider the following two demand equations for sugar:

q^ « ai + b^Pe + c^y, and (9)

m

q^ =- ag + b2Pe + Cgy + d S w^q^ , (lO)

i»l

in vblch the last term of equation (lO) is an index of consimq^tion of sugarusing foods weighted by the quantity of sxigar (wj^) customarily used with a

unit of each food. Both price and income elasticities of demand for sugar

(given a certain consuiqption level for sugar-using foods) would be es^ected

to be very small so that, in geneiral, b2 would be considerably smaller than

b. and Cg would be considerably smaller than c, •

Other exaxQ)les of coiQ>leting goods may be fovmd in rayon and cotton in

the veiy short run (a few weeks or months). If the proportion of rayon in a

rayon-cotton blend is inflexible, an increased price for cotton, passed

through into the finished product, may curtail consumption of both cotton and

rayon. During a longer period and for the textile market as a whole, the competitive relation between rayon and cotton would predominate. Completing relationships also may exist among different types of tobacco used in a standard

blend.

Measurement of Coiqpetition when Supplies are not PredeteradLned

If marketings of two commodities during a given season are influenced by

their current prices, an "identification" problem of the type analyzed in

Cowles ConmisBion literature exists. In some cases, four equations would need

to be fitted simultaneously--two demand curves and two supply curves. If the
- 19 -

tvo ccamnodities coopete In demand, the two dememd curves would be identical

with equations (3) and (if). The two conmodities might or might not conçete

for productive resources on the supply side.

In 19'«'1, Wells (51) worked out an eight-equation model to meastire the

incidence of tariffs on two competing products. The eijtht equations included

domestic and foreign demand and supply functions for each commodity. The

coefficients of these equations were estimated on a Judgment basis, and an

equilibrium solution was found for all prices and quantities under specified

initiaúL conditions. A change in teiriff rates on either commodity would move

the entire system to a different equilibrixxm position, which could be derived

readily from the model. The same model could be applied to changes in transportation rates within a given country, with two commodities produced and

consuBied in each of two regions.

Pure Theory of Related Demands

Schultz (U2, pp. 569-65'i-) discusses some theoretical criteria for related

demands on the assung;)tion that consumers are ''rational.'' Statistical resxilts

seldom are as neatly symmetrical as is implied in these theories of rational

behavior. Also, the numerical difference between the "Hotelling conditions"

and the "Slutsky conditions," which receives much attention in Schultz's two

chapters, is small for commodities that acco\int for less than 10 percent or

so of disposable income. (Pork and beef, two of the major foods, account for

only 2 or 3 percent each of disposable income.) Slutsky's condition allows

specificeOly for the fact that a drop in the price of a ccMnmodity increases

the real income of its consumers. Hbtelling's condition lunçs together the

direct effect of a change in price upon consumption (income remaining constant) and the indirect effect restilting from the change in real income which

the change in price entails. (See appendix note 2.)

Producer Goods 2/

Certain theoretical concepts involved in the analysis of the relations

between col^peting goods are the same whether the comnodities are used by producers or by consumers. Other concepts differ some\Aat. The following methods were developed chiefly to study coinpetition among the minor feed grains—

oats, barley, and sorghum grains--with each other and with com. Similar

methods could be applied with only minor modifications to certain problems

that involve consumer goods•

The following methods were used:

1. The ratio between the price of the minor feed grain and the price of

com was expresBeá as a function of the supply ratio of the two feeds and

certain related variables. If the two items were perfect substitutes, their

price ratio would equal a constant and the regression coefficient on the

9/ Most of the material in this section is taken from Meinken (36).
- 20 .

supply ratio vould equal zero* Thus, the size of the regression coefficient

is a measTire of the degree to which the minor feed grain cannot be substituted

for corn. Certain limitations in the use of this approach to study the degree

of substitutability are discussed in Morrissett (37)^

2. The price of the minor feed grain vas expressed as a function of the

price of com, the supply ratio, and the same related variables as used in

method 1 above. Here com is thought of as the dominant price-making factor

in the feed-grains market Coefficients for the other variables would be interpreted in the same way as in method 1. In general, the coefficient of

multiple determination will be considerably hi^ier for this method, but the

iioplled standard erxx)r of estimate for the price of the minor feed grain

should be identical for the two analyses. Neither method yields good forecasting fozimilas, as the price of com must be known (or estimated) in each

case. The methods are primarily of value in studying the degree of substitutability among related items and the factors that affect price ratios.

3. The price of the feed grain was expressed as a function of the individual supplies of each of the con^eting grains and certain related factors.

These eqxxations are similar to equations (3) and (4) on page 1^, except that

different "demand shifters" were used.

The third approach was coiqplicated by the fact that the basic demand

equation for feed grains is believed to be of a logarithmic type, whereas the

effects of the separate sxzpply factors is believed to be additive. An iterative method was used which permits the combination of these two types of

relations. Supply factors considered for the svmner period dioring ^Aiich newcrop oats and barley are mcürketed in volume but before the new crop of com

is available were: New-crop supply of oats, new-crop supply of barley, July 1

stocks of com, and the forecast of the new com crop. As an initicúL step,

these variables were added together on a tonnage basis ana used as a compoBite

supply variable in the logarithmic analysis, together with two demand factors

--animal units fed and prices of livestock and livestock products. The price

of the feed grain was then adjusted for the effect of the demand factors and

a linear analysis was run, using this adjusted price against the separate

components of supply. These con^onents were then weighted by the respective

regression coefficients obtained from this analysis to obtain a second approximation to the conqposite siipply variable. This variable then was used, along

with the factors measuring demand, in a second approximation for the logarithmic analysis. The process was continued for two or three iterations until

the results became stable. This method is described in detail in Foote (15).

If two competing comoiodities were perfect substitutes, their prices would

move perfectly together« Changes in factors of demand would result in identical price responses for each and a changed s\xpply of one would affect the

price of each by the same amount. In multiple regression analyses of the type

described in method 3 above, with price as the dependent variable, the regression coefficients for the demand variables in both analyses woxild be identical

and the regression coefficients (or weights) for the supply variables in each

analysis also would be identical. If the price of one was expressed as a
- 21 -

ratio to the price of the other, the ratio vould always be 1, regairdless of

the ratio between the supplies of the two coomoâltles. If the price ratio

were related to the supply ratio In a linear regression analysis, the constant

would be 1 and the regression coefficient for the supply ratio would be 0,

thus giving rise to an Infinitely large elasticity of substltixtlon.

But svippose a given quantity of one resulted In a 10 percent larger output than the same quantity of the other In any given end rise, regardless of

the level of substitution. Svq^rpose also that the demand functions had the

following form:

Pi - '^n^l -^12^2 (" )

T^^^ hl > ^12' ^12 ^' ^21 > ^2' ^ ' ^ " ^'^' ^11^22 " ^12' ^ ^1

equals 110 percent of Pg, regardless of the values of Qj^ ®^ Sg*

This Is clear from the exaiq^le that follows. Suppose the following coefficients are substituted In equations (11) and (12):

P^ « 1100 - llOQ^ -lOOQg (11.1)

Pg » 1000 - lOOQj^ -90.9102- (12.1)

Note that for these values, ^jj^ " -12* ^^ ^ these equations the given

values of Qh and Qg are substituted, price ratios are obtained as follows:

Ql 02 % ^1 ^2 £l

^ P2

1 1 1.0 890 809.09 1.10

12 .5 790 TI8.I8 1.10.

Similar results are obtained for any value of Q^ and Qg.

If this price ratio were related to the supply ratio in a linear regression analysis, the constant would always be 1.1 and the regression coeffldent again would be 0, thus giving rise to an infinitely large elasticity of

substitution. This is the comaon interpretation of the theory of substitute

goods.

But in certain cases the relative contribution of a substitute caanodlty

to the total output depends on the relative size of its consu^tlon or use.

For example, Henry and Morrison (30, p. hsk) say with regard to oats as a
- 22 -

substitute for com In feeding hogs: ''Ground oats are vorth about as much as

com per 100 pounds vhen forming a rather small part of the ration^ but vhen

fed in large amounts^ they are vorth much less than com, •«•" and ''Numerous

e^erlments have shown that oats have the highest value for pigs vhen groimd

oats form not over one-fourth of the ration* When thus fed to replace part of

the com In 20 trials with pigs In dry lot, the addition of ground oats Increased the rate of gain a trifle."

Pi

In such cases, the relationship between the price ratio (^) and the supply ratio (5=) results In a regression coefficient for the supply 3»tlo less

than 0, and an elasticity of substitution less than infinity. In the notation

used above, bn still would be greater than h^, b^ still could eq\aal bg^,

and bjjbpp fflust be greater than b^g to satisfy the requirement that the price

ratio varies inversely vlth the supply ratio. Suppose that the following

coefficients are substituted in equations (U) and (12):

Pi « 1100 -nOQi -lOOQg (11.2)

Pg « 1000 -lOOQi -IIOQ2 . (12.2)

2

Note that bjjbgg now is greater than b^

Substituting in the equations under these conditions, the following price

ratios Bxe obtained:

ill ^

02

Pi P2 Pi

P2

1 1 1.0 890 790 1.127

1 2 .5 790 680 1.162

1 3 .3 690 570 1.211 .

Pi Qi

Under the clrcinnstaiices, =- varies inversely vlth g-. 10/

1^ Ezekiel {lu, p. 179) found the ratio of the retail price of pork to the

retail price of beef to be significantly correlated (inversely) with the

ratio of the supply of pork to the supply of beef. Schultz (tó, p. 3Qk),

using the same data, determined the following demand equations for beef and

pork:

Pb " '^'^ -13.3Qb -^.3Qp + Ml

Pp - 68.8 - 5.H) -î-5Qp + .tól>
- 23 -

.oio+?í*^Í4*^r Pf"i^i^*ie8 ^e illustrateci through the use of input-output

«îi^îîr^ ^^ ^ ^^^* ^* ^ ^^^ "«^*' *^ vertical and horizontal axes Bimultaneousjjr measure input of feed and output of livestock product. For

slapUcity of presentation, livestock products are assumed to be measured in

units such that 1 pound of com yields 1 unit of product.

^. ^J^^ *^® ration is composed of 100 pounds of com, the output is 100 units.

If the ration is coa^osed of, say, 8o pounds of com and 20 pounds of a perfect substitute, the output is still 100 units, as any combination of com and

a perfect substitute will yield 100 units of product, so long as the total input is 100. In this case, the output function is the straight line A illustrated, intersecting the T and X axes respectively at 100. Under this condition, the demand functions for com and the perfect substitute would be

equivalent to those discussed in the first case referred to on page 20, and

the elasticity of substitution would be infinite.

If com were worth HO percent of some other svibstitute, at all levels

of relative supply, the output curve would intersect the Y axis at 100 and the

X axis at 90.91* as shown by the straight line B. The demand functions for

com and the substitute would be similar to that illvistrated by equations

(11.1) and (12.1). Again the elasticity of substitution would be infinite.

If com (or, conversely, the substitute) possessed varying values depending on the level of relative supply, the output function would be curvilinear

as illustrated by curve C. The demand functions for com and the substitute

in question would take the font shown by equations (11.2) and (12.2), aa^d the

elasticity of substitution would be less than infinity. The rate that productivity (or utility) increases or decreases per xmit of input for the substitute connodlty from any given level of relative supply determines the elasticity of substitution at that level.

in which Pjj and ïL are coaposite retail prices of beef and pork, respectively,

in cents per pound, Q. and Q^ are total consunçtion of federally inspected

beef and pork, respectively, in billions of pounds, and I is an index of payrolls lagged by 3 months. If I Is assumed to be constant, and given values

of ^ and Q. are substituted in these eqxiations, the following resxilts are

obtained:

Qb S ?b Pb Pp ^

% Pp

1 1 1.0 59.8 55.9 1.07

1 2 .5 55.5 ^'^ 1.15

1 3 .3 51.2 lK).9 1.25

It should be noted that in these equations b^ > b , b^ » b^^, and ^^j^

> bfg in the notation used here.

286234 0-54-4
2k

OUTPUT FUNCTIONS:CORN AND VARIOUS

SUBSTITUTES, FEED CONVERSION

OF I TO I ASSUMED FOR CORN*

Y-CORN

100

90 -

(A)

OUTPUT = INPUT OF Y + INPUT OF X

OUTPUT = INPUT OF Y + «CINPUT OF X, WHERE

oC IS A FUNCTION OF THE RELATIVE SIZE

OF X WITH RESPECT TO Y

10

-^

OUTPUT = INPUT OF Y +

90.91 INPUT OF X

10

M^

90 100

X-SUBSTITUTE

INPUTS OF CORN AND SUBSTITUTE EQUALS 100 AT ALL POINTS.

U. S. DEPARTMENT OF AGRICULTURE NEG.49140-X BUREAU OF AGRICULTURAL ECONOMICS J

Figure 5*- Functions A and B illustrate perfect substitutes for com, the

feeding value of the substitute grain in function A being equal to 1 pound of

com at all levels of substitution and in function B to O.909 pounds of com

at all levels of substitution. Function C shows the relation for a substitute

grain for which feeding values vary relative to com, depending on the level

of substitution.
- 25 -

The statistical analyses run by Meinken (¿6) and Foote (3¿) indicate that

the four feed grains compete with each other "but that they are not perfect

substitutes* This is consistent with available experimental evidence on feeding.

Knowledge of whether a substitute commodity has a constant or varying

marginal rate of substitution is important from a practical viewpoint. Farmers and feed manufacturers alike are interested in obtaining least-cost

rations while maintaining feeding efficiency. These usually can be approximated only if the manner in which one commodity substitutes for another is

known. Prices of the substitute feeds also would be needed. Heady and

associates (22,, 28) have for some years been conducting feeding experiments

designed to determine, among other things, substitution coefficients between

various feedstuffs. These coefficients enable farmers and feed manufacturers

to take advantage of changes in relative prices in determining least-cost rations.

STATISTICAL CONSIDERATIONS IN INTEKPBETING THE COMPLETED ANALYSIS

Considering^ the Unexplained Besiduals

In the 1920's and the early thirties, analysts expected to explain

practiceúLly all of the variation in the dependent variable and frequently

obtained multiple correlations of O.99 or above. Now it is expected that part

of the variation in the dependent variable will remain lanexplained. Factors

that cause unexplained residuals can be divided into three types:

1. Those due to errors in the data. The usual least-squares analysis

assumes that the independent variables are known without error. In forecasting, better results are obtained by the least-squares method than by any

other single-equation approach, regardless of the degree of error in the independent variables. But if interest is centered on measuring coefficients of

elasticity and particularly if a "reversible" equation is desired, use of a

"weighted" regression equation which allows for the relative errors in the

several variables may be preferred. This problem is considered in more detail

below.

2. Those due to the omission of certain variables. This may be because

the analyst fails to think of them, because no data are available, or because

in most years they are so minor as not to be worth including in the study.

This is the kind of random error which normally is assianed in a least-squares

analysis and it is the type of error allowed for in simultaneous-equation

"shock" models.

3. Those resulting from the use of wrong types of curves, incorrect lags,

and similar factors. Charts that Indicate the degree of partial correlation

help to find the relative Importance of the nature of the chosen curves in

causing residuals. Methods for preparing such charts are given in Foote (lj¿,pp.

11-15) and exaarples of their use are shown by Thomsen and Foote (V^^PP* 29O-297).

Residuals for years not Included in the analysis frequently are larger than

those for the years Included. The Increase in \mexplained variation may be

caused by the following factors: (1) Extrapolation beyond the range of data
- 26 -

Included in the analysis. Some tests designed to measure the Inçortance of

this are mentioned on page 35- (2) A change in the basic structure of the

relationships may have occurred. This might reflect a change In the nature of

the curves or the increased inçortance of some omitted factor. The charts that

indicate the degree of partial correlation mentioned previously help to Indicate

whetlierthe nature of the curves has changed. (3) The regression curves In the

analysis tend to adjust so far as possible to compensate for the types of errors

discussed in the preceding paragraphs. Deviations In the later years reflect

not only the true variability but also the extent to which the regressions were

warped from their true shape to keep the deviations for the years Included in

the analysis as small as possible. This factor was particularly Important in

connection with some of the analyses run by the graphic method in the early

thirties.

Residuals for years omitted because of price ceilings or rationing programs

should be checked for consistency with known conditions. Eesiduals for all

years should be approximately random when plotted against time. If the residuals appear to be serially correlated, a test should be made for this. Sometimes the addition of another variable, or a shift ta the use of first

differences, will reduce the degree of serial correlation In the residuals. An

example of the former is given in Armore (2, pp. 30 and 57-5Ö). The latter is

discussed in Cochrane and Orcutt (2).

Considering the Nature of the Statistical Coefficients

In general, the direction and general shape of the regression curves should

be consistent with expectations. "Wrong" signs or shapes may occasionally lead

to a revamping of the underlying theory but they are more likely to indicate the

need for a different approach In the analysis. Technical considerations may indicate that regression coefficients should be of a certain size. More confidence

can be placed in an analysis if the results are consistent with factors of this

sort. The constant value in a first-difference analysis should be consistent in

sign and approximate magnitude with expectations, based on known long-term trends

that affect the dependent variable, provided It differs significantly from zero.

The appropriate error term to use in connection with such a test Is the standard

error of forecast when the Independent variables equal zero* (See Foote (14, pp.

U, 6)

The Use and Interpretation of Tests of Significance

Yule and Kendall (^, p . U37) say, "it cannot be over-emphasised that

estimates from small sançles are of little value In Indicating the true value of

the parameter which Is estimated. ...Nevertheless, circumstances sometimes drive

us to base Inferences...on scanty data. In such cases we can rarely, if ever,

make any confident attençt at locating the value of a parameter within serviceably rxarrow limits. For this reason we are usually concerned, In the theory of

small samples, not with estimating the actual value of a parajneter, but In

ascertaining whether observed values can have arisen by sampling fluctuations

from some value given in advance." Tests of significance as commonly used are

designed to measure whether the observed value differs significantly from zero.

This is referred to as "the null hypothesis". In most cases, a test could equally
- 27 •

well be made as to whether the observed value differs significantly from some

other value.

Tests of significance may be made under any of the following conditions:

(1) With no previous knowledge; or (2) applied to a factor that is believed to be

uninçortant. In these cases, a nonsignificant value for a regression or correlation coefficient would indicate that the factor should be omitted from the

analysis. (3) Applied to a factor which for theoretical reasons is believed to

be important. A nonsignificant result in this instance does not indicate that

the factor is not important. It does indicate a need for further evidence to

prove beyond reasonable doubt that the factor is important. Therefore, such a

factor might be left in an analysis, particularly if its effect is believed to

be small. If repeated samples of 20 observations were drawn from a population

for which the true partial correlation was 0.3, nonsignificant results,based on

a 5-percent probability standard, would be obtained about three-fourths of the time,

Such a factor would not greatly affect the calculated value in most years, but

if interest were centered on the structural nature of the relationship, the analyst might wish to include it tentatively pending further evidence.

Requirements for Tests to be Valid

Yule and Kendall (^, pp. '*37-'^38) say, "Our results will be strictly true

only for the normal universe. ...It appears that, provided the divergence of

the parent from normality is not too great, the results which are given below as

true for normal universes €ü:'e true to a large extent for other universes. ...If

there is any good reason to suspect that the parent is markedly skew, e.g. U-or

J-shaped, the methods ...cannot be applied with any confidence." Nonpeirametrie

tests which are independent of the nature of the distribution from which the

sample was drawn are available in certain cases. But in general, the requirement

of normality is not very restrictive.

If tests of significance as applied to correlation measurements are to be

valid, the residuals should be randomly distributed. Certain modified tests

have been suggested for cases in which the residuals are serially correlated,

but, in general, it is preferable to use a "transformation", such as a shift to

first differences, to eliminate the serial correlation in the residuals.

Some Common Tests

1. Student's t is defined as the difference between any particular value

assumed to be the true value in the universe for the statistical measure examined

and the value determined from the sample divided by the standard error of this

coefficient. In common practice, the assumed value is taken as zero, so that t

is given as the sample value divided by its standard error. In such cases, the

null hypothesis is tested. However, t could be used equally well to test for

significant deviations from any other assumed value, such as a slope of-1 for a

supply-price regression coefficient. The t-test may be applied with reasonable

confidence to sample values that depart somewhat from normal in their distribution,

but it should not be used, of course, where other more appropriate tests are

available.

2. For saiiç)les of 20 or 30 observations, the standard error of a correlation

coefficient cannot be estimated with much reliability if the correlation in the
- 28 -

f'TríS'aHp 38Î-386'r^»«-^S! PP.-«. 286) provides ccn«nl«.t

^¿:î" iëûS mother Ule, partlaîT and «Itlpl. oorr.l.tlons differ

Significantly from zero.

^ Chi-sQuare Is used to detamlne whether a series of frequencies differ

slgni?icaSïîyf!^ a theoretical or expected set of frequencies. Chl-square Is

defined as

m

in which X is the observed and m is the expected frequency.

U The F-test is used to determine whether the assembly of data in various

classed defined by certain restrictions exhibit on the average greater differences beîJeen meters of two different classes than between members of the same

CIäSS If they do, it is concluded that the restrictions result in significant

sïïition of ihe data into at least two classes; that is that the means of at

least two classes differ. For example, the F-test was used to learn whether the

weiffhts differed significantly for the several supply components in the analysis

for feed grains discussed on page 20. Steps involved in this test are given in

Foote (1¿, pp. 37-39)-

The F-test also can be used in connection with a problem in which the value

of certain regression coefficients are assumed in advance. The following steps

are used in such cases:

1. Minimize the sum of squares, making no assumption about the coefficients.

The degrees of freedom attached to this sum of squares equals the number of

observations minus the total number of coefficients.

2. Minimize the sum of squares after assigning values to such coefficients

as axe to be tested.

3. Find the difference between the first and the second sums of squares.

The degrees of freedom attached to this difference equal the number of

coefficients to which a value was assigned.

U. Obtain the two mean squeœes by dividing the sum of squares obtained in

steps 1 and 3 "by the appropriate number of degrees of freedom.

5» F equals the ratio of the two mean squares. If it differs significantly

from zero, the sançle Indicates that the assigned values should not be used. If

it does not differ significantly from zero, we have no reason to doubt that the

sample came from a population for which the true values were equal to the

assigned values and the assigned values can be used.

Melnken (^ used this approach to determine whether related coefficients

in a series of regression analyses differed significantly from each other. It
- 29 -

also could be used in checking whether price and income elasticities differ from

month to month or in a prewar and postwar period.

Application of the F-test to a Problem in Regression Analysis

Suppose we have an equation of the following type:

C = a +hiP + hgY, (13)

in which C equals consumption, P equals price, T equals income, and all variables

are expressed in logarithms. We wish to test whether bi can be assumed to equal

-1, that is, that the price elasticity is equal to unity.

The usual least-squares regression analysis is run and the following value

computed: {TC^ -C SC) (1-E^). The degrees of freedom attached to this sum of

squares is N - 3^ when N is the number of years included in the analysis. The

following series next is obtÄined: C' = C + P. This is the value that C would

have in each year if bj^ were equal to -1 and Y and the unexplained residual were

at their given levels. A simple correlation between C and Y then would be

computed and the value ( rtJ* -C* SC*) (1-r^) obtainedT Obtain the difference

between these two sums of squares. The degrees of freedom attached to this sum

of squares equal 1, as a value was assigned to 1 parameter. F would be coiiç)uted

as described in steps k and 5 above and looked up in an F-table, using N-3 and

1 degrees of freedom.

In this case, Student's t could be used instead. The t-value would be

obtained as follows:

t = bj^ +1

=^1

and looked up in the usual table, using N-3 degrees of freedom. But if values

were to bè assumed for more than one of the parameters and a single test for all

of the assigned values simultaneously was desired, the F-test would be needed.

Allowing for Disturbances or for Errors in the Data

Most research analysts are inclined to appraise the success of a regression

analysis by the level of the siiiç>le or multiple coefficient of determination

{if or E^) obtained. But if errors of measurement are known to be present in

the data, or if some minor variables are known to be omitted from the regression

equation, a less-than-perfect correlation must be expected.

Consideration of the following cases illustrates the effects of errors in

the variables and disturbances (omitted variables) on the regression and

correlation coefficients. The model in each case illustrates the mechanism by

which the observed data were generated; the true regression coefficient is

bo; and x^ and 19 are true values of the variables expressed as deviations from

their respective means. Disturbances are indicated by the symbol d, and errors
- 30 -

in the variables by e^ and §2 . Results shown hold precisely only for large

samples•

Alternative Cases

1. Perfect correlation: Neither disturbances nor errors:

A A

Model: x^ = b^ X2

Proportion

of variance 2 '^8 -K 2 ^2

explained: p2 = ^og^S » ^o ^^ = 1

„^ L2 A2

r XI o 2X2

Least-squares

regression ^ A

coefficient: b = ^^1 ^ = ^o .

T.-2

2, Distvirbances (due to ondtted variables) but no errors in the data:

Model: x^ = b^ X2 + dj

The disturbances, d, are assumed to be uncorrelated with Xp»

Proportion of

variance ,2 0 2

Least-squares

regression ^ ^

coefficient: b = ^ 1^ = "bo.

i» Disturbances, and also random errors in the dependent variable;

Model: x^ = (x^ + e^) = b^ X2 + d, or

^1 = ^o ^ •" (^ " «1^

Proportion of

• r2 =

Do -^ d + ei

variance ^ 2 ^2

explained: " r2 = o ^^ < 1
- 31 •

Least-square8

regression ^ A y\ y\

coefficient: b = sC^l -^Jl) ^ = ^ ^1 /2 = bo

As errors of measurement are assumed random, j ii 52 = 0,

^^ Random errors In Independent variables; no distúrbemeos and no errors

in the dependent variable;

Model: x^ = "^o (^ "^ ^2) = ''^o ^ "* ^o ®2

Proportion

of variance . 2 0 2

explained: r = ^o ^^ < 1

a,25:Ç^2 +>,2 y^ 2

^o ^ ^o ^^2

Least-squares

regression A A /VA

coefficient: b = z^l (^ -h ^2) = r^l ^2 ^^ b .

S(X2 + eg) 2 2:222 + 21622 ^

In case U, b is biased toward zero: its absolute value in a large sample will

be smaller than that of the true regression coefficient, b^» In contrast,

neither errors in the dependent variable nor disturbances which are uncorrelated with Xg produce any bias in the least-squares regression coefficient,

¿* Plßtur^ancps ma randop errors in totla Yftrlal?le§:

/\ .A . Model: ^1 "* ®1 " "^o ^^2 *' ®2^ "*" ^' ^^

A A

^1 = ^o^ + (d - ej^ + b^ e2)

Proportion

of variance , 0 «C" 2

explained: r2 = ^o S^ < 1

^o^ SÍ2^ -^ ^¿L^ -^Se^"^ + b^ ^e^^

Least-squares

regression ^ ^ A A

coefficient: b = ^(^1 H- ^1)(^ -^ ^2) = ^^1 ^ z b^

r^Xg +e2^ s X2 + ^«2

provided the errors and disturbances are uncorrelated with each other or with

the true values of Xi and J2 • As in case k, the least-squares regression

coefficient is biased toward zero.

286234 0-54-5
- 32 -

The inçllcatlons of the five cases may be summarized as folloifs:

1. In general, the unexplained variation In a regression analysis comes

from a combination of disturbances (omitted variables or factors) and errors

In the data. Each additional source of random error or disturbance tends to

reduce the correlation coefficient.

2. Random errors In the dependent variable do not bias least-squares

regression coefficients; random errors In the Independent variables bias

regression coefficients toward zero.

3. If we have a basis for estimating the degree of error In each Independent

variable, we can correct the least-squares regression coefficients for bias

simply: Instead of using the observed sum of squares ( Xp ), we use this sum

minus the estimated error component ( fig ) • The original bias depends on the

proportion of the observed variation in x^ which is due to errors of measuremyent.

p h. The above correction will also increase the coefficient of determination,

r . A correction for errors in the dependent variable would also increase r^,

but would not affect the regression coefficient.

5. Disturbances have real causes. They can be reduced only by identifying some of these causes and including them in the regression analysis. Thus,

if any disturbances are believed to exist, the adjustment for errors should

not be large enough to produce perfect correlation between the corrected

variables.

Exançles of Regression Equations Adjusted for Estimated

Errors of Measurement

The notation used in these examples is obvious. The retail price of eggs is

indicated by P^^^^; the farm price of turkeys, by P^^^s ©tc. Consumer income

is represented by ^. Numbers in parentheses under ihe regression coefficients

are their standcurd errors.

1. Retail price of eggs;

a. Unadjusted:

TJ""^ = - 0.010 . 1.83 q^ + 1.2U y (lU)

(oMf (0.15)

R^ =0.80 : r 2 =o.U8 : r 2 ^Q.SO

P^*y py.q

S = 0.029

b. Adjusted (Assuming random measurement errors of 1 percent in

production of eggs and disposable Income, and O.5 percent in prices of eggs.

eírorr¡í^7T/^^f' ^"^ P^^^uction of eggs are SMLII, the assumed measurement

errors account for I5 percent of the observed variance in production, but only
- 33 -

1.5 percent for income and 0.2 percent for price.)

P (^^ = - 0.010 - 2.3U q^ + I.3I* y (1**)

* ioMf (0.13)

«a' =0.87 : r^¿j^ =0.61. : r^^^^^ =0.87

? = 0.023

1. Farm price of turkeys;

a. Unadjusted;

P. (^) = 0.02U - IM q^. + O.9U y (15)

^ (0.26) "^ (0.21)

B2 =0.86 ; r^¿^ =0.78 ; r^/^^ =0.68

S = 0.0U6

b. Adjusted (Assuming random measurement errors of 2 percent in the

price of turkeys, 2.5 percent in production of turkeys, and 1 percent in disposable income. The assumed variance resulting from errors in the data for

production amounted to nearly 10 percent of the observed variance, but only about

1 percent for price and income.)

p^(^) = 0.028 - 1.58 q^. + 0.95 y (15)

(0.20) ^ (0.16)

E^2 = 0.92 ; T^¿y = 0.87 ; rpy2^ = 0.8O

s„ = 0.03U

The errors of measurement in the above examples were based on Judgments of

commodity and price specialists* If they are approximately correct, they are

helpful in these ways: (1) They indicate that a third to a half of the variance

unexplained by the unadjusted equation may come from errors in the data. There

is no point in trying to explain this part of the variation with additional real

variables or unusual manipulations of data. But this variability need not be

ascribed to irrationality of consumers or other unknown causes; (2) By getting

a better approximation to the true regression, better estimates of such things

as costs of alternative price-support programs can be made. These are ultimately

determined as **universe" values—the totality of price-support purchases and

costs—and not as sample values subject to eirror.

Forcing Perfect Correlation to Obtain a Reversible Demand Curve

This can be thought of as a special case of adjustment for errors in the

data. It involves the assumption that there are no disturbances (or omitted

variables) and that the correct functional form (such as arithmetic or
- 3»^ -

logarithmic) has "been chosen. Under these assumptions the only reason for

less-than-perfect correlation is the presence of random errors in each variable.

The relative level of error in each variable (that is, the percentage of the

observed variance that is due to random errors) is estimated, and the equation is

solved mathematically for the absolute level of error that oust be assumed in

order to produce perfect correlation.

The following example relates to the estimation of domestic demand for all

food at retail:

1. Unadjusted ;

P (r) = - o.OOU - 2.00 qf + 0.91 y (l6)

^ (0.U9) (0.08)

E2 = 0.86 : T^¿^ = 0.51 : rfy.^ = 0.86

q^ = - 0.000 - 0.25 P^.^^^ + 0.25 7 (17)

(0.06) (0.05)

^ = 0.^ : ^qp.y = 0.51 : rfy^^ = 0.65

If there is a single demand curve for all food at retail, the price flexibility

at any point of the curve is exactly equal to the reciprocal of the elasticity

of demand at that point. However, equations (16) and (I7) give (1) a price

flexibility of -2.0 and (2) a demand elasticity of -O.25. The reciprocal of the

latter figure is -U.O, which is twice as large as the first estimate of price

flexibility. Is there a value somewhere "between these two figuires which represents a best estimate of the price flexibility? This is given (on certain

assumptions) by the adjusted equations.

2. Adjusted:

a. First assumption (Errors as percentage of observed variance: Prices

of food 3 percent; income 12 percent; consunç)tion of food 50 percent. Perfect

correlation requires errors only 39 percent as large as these):

P^^^^ . a^ - 2.97 qf + 0.95 y (I6.I)

^f = ag - 0.3U p^(^) -I- 0.32 y (17-1)

2

S. = 1 in each case; demand elasticity is exact reciprocal of price

flexibility, except for rounding.

^« Second assumption (Assumed relative errors in price and income same

as before; error in consumption of food estimated at 25 percent of observed

variance. Perfect correlation requires measurement errors only 51 percent
35

as large as these):

p^(r) = ai2 - 2.70 q^ -^ 0.92 y (l6.2)

<lf = 3122 • °-37 Pf^^^ + O.3U y (17.2)

2

P = 1 In each case; demand elasticity Is exact reciprocal of price

flexibility.

This method is not recommended for general use; but when such rigid

assumptions seem Justified, it should give the best (statistical and Judgmental)

estimate of the true reversible demand curve, if one exists«

Validity of an Estimate from a Multiple Begresslon Equation ll/

Only the sinçlest case is considered here - a linear equation in three

variables :

^1 = * ^ ^2.3 ^2 ^ ^13.2 ^3 • (^^^

Under what conditions does this equation give a valid estimate of X.?

1. There must be a significant ^scatter" between Ig and X - that is, r

must differ significantly from 1, This is not a matter of sampling error, A

correlation of O.99 may differ significantly from 1 so far as sampling is concerned. It is solely a matter of errors of observation. A statistician should

know enough about his data to Judge whether the observed scatter is larger than

could halppen as a result of errors in Xp and Xo. If not, the scatter is nonsignificant, and the equation is worthless.

2. Usually the equation is invalid in the case of extrapolation beyond

observed values of X2 and Xo. This can be tested by drawing a slnçle scatter

diagram for Xg and X-^* If they are highly correlated, the observations lie within

a narrow ellipse. The values of X^, associated with combinations of Ip and X^

which lie within this ellipse can be estimated from the equation. Unless we are

willing to extrapolate, we cannot estimate the value of X^ associated with any

combination of Xp and X^ lying outside the ellipse. A chi-square test described

In Waugh and Been (50)or Armore and Burtis (37 pp. 7*9) can be used to measure

the degree of extrapolation Involved in equations having more than 2 Independent

variables.

3. The error of a forecast of X is composed of two parts: First, the

standard error of estimate; and second the error associated with the regression plane.

11/ This section was prepared by Frederick V. Waugh, Director, Agricultural

Economics Division, Agricultural Marketing Service.
- 36 -

Xi - a + b3^2.3 ^2 " ^13.2 ^3 •

The first of these errors is constant; the second varies with the particular

values of X and X . The error of the regression plane is least at the center of

the ellipse mentioned above. We could draw a series of ellipses around this

center; as we moved away from the center, each successive ellipse would connect

combinations of X^ and X^ for which the error of the regression plane would be

equal. And each successive ellipse would indicate a larger error. The standard

error of a forecast as described by Ezeklel (12, pp. 3^2-3^^) allows for both

types of errors. It applies exactly only for a set of Independent variables

identical to that included in the original analysis. However, it may be assumed

to hold approximately for other values that lie within this range.

A SURVEY OF DEMAND ELASTICITIES

BASED ON SINGLE-EQUATION ANALYSES

Most research analysts are familiar with the term, "elasticity of demand",

which was popularized, if not invented, by Alfred Marshall. Elasticity of demand

is the ratio of the percentage change in consumption of a coimnodity to the

associated percentage change in its price. In mathematical notation it is written

veuriously as

^ = 44~ * -T~ ^''^^''^ ^^ equivalent to ( ^ji-)- (A^)J

d (log pj

Henry L. Moore, a pioneer in the field of price analysis, was Interested in

the reciprocal of demand elasticity which he called "price flexibility7 Price

flexibility is the ratio of a percentage change in the price of the commodity to

the associated percentage changes in its consumption. But the term is also

applied to the ratio of percentage change in price to the associated percentage

change in whatever supply variable is used In the analysis (consumption, production, or total supply). Obviously, there is no logical reason why the percentage

relationship between price and production or price and supply should be the

exact reciprocal of the ratio between changes In consumption and changes in

price.

Terminology regarding regressions of price upon Income is not standardized.

Some analysts refer to the ratio of a percentage change in price to the corresponding percentage change in Income as "price flexibility with respect to Income."

The ratio of a percentage change in consumption to the corresponding percentage

change in Income, whether in a time-series analysis or a family budget study,

is usually referred to as "income elasticity of demand" or elasticity of demand

with respect to income.

Some writers apply the term, "income elasticity of demand", to the ratio of

a percentage change in retail expenditures to the corresponding percentage change

in consumer income. This use is confusing, as traditionally elasticity of

demand has been associated with price rather than value relationships. The
- 37 -

authors prefer to refer to a percentage relationship "between expenditures and

Income explicitly as **the elasticity of expenditures with respect to Income.*^

Reasons for Different Elasticities of Demand

The Immediate ohject of a statistical demand analysis Is the measurement of

relationships rather than an explanation of the particular values obtained. But

the analyst feels under pressure to rationalize the numerical results on either

a commonsense or a theoretical basis. It Is often said, for example, that the

demand for an Item which Is a trivial object of expenditure (that Is, one that

takes up an almost Infinitesimal fraction of total Income) Is likely to be highly

Inelastic. To support this, another factor that tends to make for Inelasticity

would have to be added, that Is, that the commodity has no close substitutes.

A cofflDDaodlty such as potatoes or onions, on this score, would have a less elastic

demand than a commodity such as beef, pork, or chicken which has several fairly

close substitutes. Numerical results from a group of analyses described In

Fox (18, 20)are generally In accord with this ass\unptlon (tables 1 and 2).

Whether two commodities are effective (economic) substitutes depends on consumer

attitudes toward them and not directly upon their nutritional and physical

attributes.

Many possible compllcatlais arise In Interpreting coefficients that are

Intended as elasticities of demand. For example, moderate changes In relative

prices might not cause measureable changes in relative consumption of two

commodities. But sharp increases in the price of one might lead to a substantial

and possibly cumulative or Irreversible shift from this commodity to the other.

Although the immediate cause was physical shortage of butter rather than its

high price, the change in relative consumption and in consumer attitudes toward

butter and margarine during World \íaT II is one of the most dramatic cases on

record.

Problems Involved in Interpretation

Some economists run an analysis in which price is made a function of the

production or supply of a commodity. They refer to the reciprocal of this

relationship as the elasticity of demand for the commodity. If the commodity

has more than one outlet (including storage) there is no necessary reason for

the elasticity of demand in any one of these outlets to equal the reciprocal of

"price flexibility with respect to production." An example of this is mentioned

in Fox (18, pp. 70-71). Changes in commercial stocks and in net exports of meat

tended to cushion the effects of a change in production of meat upon its retail

price. The reciprocal of the price-production relationship suggest a unit

"elasticity of demand"for meat. But the regression of consumption of meat upon

its retail price gives an elasticity of demand of around - 0.6. Year-to-year

changes in consixmption were highly correlated with changes in production

(îT = 0*9^)/ but were only 70 percent as large. Hence the elasticity of consumer

demand is only 70 percent as large as the reciprocal of the price-production

flexibility.

An elasticity of demand for "meat to store" and one for exports of meat from

this country could be calculated. An elasticity of supply for Imports of meat
- 38 -

(perhaps a separate one for each major country fr^om which meat Is Imported)

also would be Involved. Ordinarily elasticity of demand means the elasticity

of domestic consumption with respect to price. It is likely that the elasticities of demand for exports of meat and meat to store are greater than the

elasticity of demand for consumption*

The three major categories of utilization for wheat have very different

demand curves and elasticities. For example, the price elasticity of demand

for wheat for domestic food use is probahly not greater than -0.1. The demand

for wheat as a livestock feed is inelastic as long as the price of wheat is

considerably above the prices of feed grains. But if the price of wheat falls

to or slightly below the price of com on a pound-for-pound basis, the use of

wheat for feed is likely to increase tremendously. In other words, the demand

for wheat in the range of (say) 20 cents a bushel below the price of com to

5 or 10 cents above is highly elastic. The elasticity of demand for exports of

wheat has varied in the last 30 years. In the late twenties, when exports of

wheat from this country amounted to about a sixth of total world exports, and

exports to Europe amounted to 5 to 7 percent of European production, the

elasticity of demand for our wheat exports may have been substantially greater

than one. Dollar rationing as such—that is, the setting aside by other

countries of a specific dollar amount to be spent for our wheat—would imply

a unit elasticity of demand with respect to price. An elasticity of demand

for wheat for export under current conditions of extensive Import and exchange

regulations probably has little meaning.

The implication of this is that research workers should indicate

specifically the particular utilization, or set of utilizations, to which a

given coefficient of elasticity refers. The ratio of a percentage change in

total utilization of wheat to a percentage change in the price of wheat should

be a weighted average of the elasticity in each utilization group.

The value of statistical regression coefficients depends upon the extent

to which they enable us to act more intelligently and appropriately in specific

situations. An application that is frequently encountered is the question of

whether compensatory payments or purchase and diversion programs would be less

costly in supporting the price of perishable commodities. So far as cost to

the Public Treasury is concerned, the answer to this question turns largely on

the elasticity of demand for the commodity. If demand is unit elastic, this

tends to make the costs to Government of the two methods of price support

identical. If demand is highly inelastic, purchase and diversion will be less

expensive to the Government; If demand is laore than unit elastic, compensatory

payments will presumably be less exDenslve to Government as well as more

satisfactory to consumers. The reasons are discussed in detail in Fox (12).

The accuracy of statistical regression coefficients and their validity In the

particular context—time, place, and duration of time with which a projected

program is concerned—bear upon the quality of administrative actions and the

overall effectiveness of Government programs.

Coefficients Obtained

Numerous analyses based on data for 1922-lfl are presented in Fox (l8, 20).

results from these are summarized In tables 1 and 2.

an. -h^^ ^^"^^ ^""^^^ ^"^ ''^^^^ ^^^^^ *^^l®ß ^® ^ased were fitted by singleequation methods after considering the conditions under which each commodity was
- 39 -

produced and marketed. Commodities with complicated patterns of utilization

were treated partially or not at all.

The functions selected were straight lines fitted to first differences of

logarithms of annual data. In most cases, retail price was taken as the dependent

varlahle and per capita production and per capita disposal)le Income undeflated

as the major Independent variables. Per capita consumption was substituted for

production In some analyses.

The logarithmic form was chosen on the ground that price-quantity relationships In consumer-demand functions were more likely to remain stable In percentage than In absolute terms when there were major changes In the general price

level. First differences (year-to-year changes) were used to avoid spurious

relationships resulting from trends and major cycles In the original variables,

and for their relevance to the outlook work of the Bureau of Agricultural

Economics, which focuses on changes from one year to the next.

Results from these analyses. Including projections Into the post-World War

II period, are discussed In detail In the references noted above. Tables 1 and 2

are believed to be largely self-explanatory, at least when combined with the

discussions given In earlier parts of this handbook.

Most of th# demand elasticities for livestock products at the retail price

level range between -0.5 and -1.0 with respect to domestic consumption. If

demand elasticities at the farm price level are derived from the elasticities of

domestic food consumption with respect to retail price by dividing the elasticity

at the retail price level by the flexibility of farm price with respect to

retail price, they center around -0.5. But demand elasticities at the farm level

with respect to total supply or production are greater than elasticities derived from domestic consumption alone, as the Intacts of changes In production

upon farm price are softened by adjustments In commercial stocks, exports, and

Imports.

THE SIMULTANEOUS EQUATIONS APPROACH

A modem econometric Investigation consists of three major steps: (1) Specifying the model or system of relationships believed responsible for generating

the observed data (for example, prices and consvunptlon of a given commodity);

(2) establishing the Identlflability (that Is, measureablllty) of Individual

equations or coefficients In the model; and (3) statistical estimation of the

Identifiable equations or coefficients.

A single-equation least-squares analysis of demand Involves the same steps.

Implicitly If not explicitly. It assumes (1) that the demand function Is such

that one variable can be selected as dependent upon the others, and that all

residual errors or disturbances are concentrated In the dependent variable; (2)

that none of the Independent variables In the demand function are. In fact Influenced by or determined simultaneously with the dependent variable; (3) that the

disturbances In the dependent variable tend to be nonnally distributed and not

serially correlated.

In the 1920*8 and 1930's some price analysts were aware that these assumptions

were not always satisfied. Particular departures from this model were considered

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- kl -

Table 2, - Selected crops: Alternative estimates of elasticity of

demand at farm price level, Itoited States, 1922-lH l/

Commodity

:Least-squaxe6 coefficients using as the

: dependent variable

: Production 2/ : Price 3/

Fruits:

Deciduous :

Apples

Peaches kj

Cranberries 2/

: Percent

' -1.21

' -1.18

' -.57

Percent

-1.27

-I.U9

-.67

All deciduous -1.11 -1.1*7

Citrus :

Oranges

Grapefruit

Lemons:

Shipped fresh:

Summer '

Winter

: -0.58

: -.1*0

: -.29

-.61

-0.62

-.56

'.ko

-.72

All lemons • , -«^5 -.59

All citrus -0.69 -0.76

All fruits : -0.Ö2 -1.06

Potatoes, using as quantity variable: :

Production :

Consumption 6/ :

Sweetpotatoes :

Onions : :

Late summer :

-0.26

-.22

-.71*

-.28

-0.28

-.27

-1.30

All onions : -,39 -.41*

Truck crops for fresh market: :

Winter :

Spring :

Suznosr :

Fall :

-o,U5

1/ -.30

-.U2

-.in

1/

-0.88

-1.05

-.58

-.60

Calendar year : -.59 -.97

Hay t

Corn: :

1st analysis :

2d analysis :

•

-0.62

2/

-Û.72

-.52

-.79

1/ Bepresents elasticity of toted market demand in most cases; derived demand

for final domestic consujoption would typically be less elastic.

2/ Ulis coefficient has traditionally been callad elasticity of demand in

single-equation euoalyeis.

3/ In traditional terminology, this is the reciprocal of »»price flôxibiZlty* '

According to simultaneous equations theory, this coefficient is an unbiased

estimate of the true demand elasticity in a uniequatlonal complete model.

kJ Excluding Callfoxiila.

5/ Based on data for I922-36.

6/ Elasticity measured at retail prices.

2/ Nonsignificant at 5-percent level.

0/ May reflect demand for all feed concentrates as an aggregate.

2/ More nearly reflects demand for com given constant supplies of all other

grains and feed concentrates.
- te -

in some detall. The more Important of these theoretical discussions are mentioned

In Fox (20). As late as I9U2 opinion differed as to which variable should he

placed lñ"the dependent position In a least-squcires analysis of demand, although

Ezeklel followed a causal principle which In many practical cases led to the

same choice as that Indicated hy modem econometric theory* But the probing and

questioning of the s Ingle-equation, least-squares model continued on a plecemead

basis until 19'^3*

In that year, Haavelmo (22,)put forward a general theory of econometric

analysis which Included the traditional single-equation model as a special case.

This general theory successfully Integrated the three problems of specification,

Identification, and estimation for systems of simultaneously detennined variables.

However, the new theory was stated in a relatively new terminology; it presximed

a knowledge of matrix algebra; and it involved "maximum-likelihood estimation'*

rather than least squares. All of these elements were barriers to understanding,

and hence to acceptance and application, of Haavelmo's new synthesis.

Haavelmo recognized these difficulties in a second publication Issued in

19l^U:

*Ve believe that, if economics is to establish itself as a reputable

quantitative science, many economists will have to revise their

ideas as to the level of statistical theory and technique and to

the amount of tedious work that will be required, even for modest

projects of research.** (26, p. llU)

This statement has an unduly ominous ring. But it is true that the basic logic

of the simultaneous-equations approach must be understood in order to use and

interpret single-equation analyses with proper discrimination.

The three major problems of (1) specification, (2) identification, and

(3) estimation are considered in turn.

Specifying the Economic Model

An economic model consists of a set of relationships between observed

variables and a set of assumptions concerning the nature of variations in the

data that are not explained by the systematic relationships but are caused by

residual errors or disturbances, or both. The relationships between observed

variables are called "structural equations" in the Cowles Commission literature.

^S^?^^ (¿i)mentlons four types of structural equations: (1) "Behavior equations",

Which describe a certain type of economic decision taken by a certain category

or economic agents." These Include the demand and supply curves of economic

theory; (2) institutional equations," wMch describe behavior patterns set

Dy law or rule; (3) "technical equations," which express the physical relation

between input and output in production; and (1^) "identities," which "should be

^iff!; 1 ^! d^^ivlng directly from the definitions of the variables through the

principles of economic accounting."

Pon.,,!!!^ ^^itary elements are Involved in the specification of a model,

consumers, marketing agencies, and producing areas must be aggregated even when
-U3 -

there may te sizable variations within these categories. The specification of

a model also requires the classification of variables Into "exogenous" and

"endogenous" groups. Exogenous variables are those which represent forces outside the confines of the economic system (for example, weather) whereas endoggenous variables are those determined within the system of economic forces In a

narrow sense (as prices or consumption). The nuníber of structural equations In

a complete model must be equal to the number of endogenous variables whose values

are to be explained by the equation system. No attempt Is made to develop

equations explaining the exogenous variables; they Influence the endogenous variables but are assumed to be not Influenced by them. Structural equations also

may Include values of endogenous variables for preceding time periods. Lagged

endogenous variables are similar to exogenous variables In that they Influence

current values of the endogenous variables but are not affected by them. For this

reason the lagged endogenous and the exogenous variables are often referred to

collectively as predetermined variables.

The en\nneratlon of structural equations, the selection and classification

of variables, and the choice of variables and functional forms In each equation

collectively define the economic properties of a model. If each structural

equation expressed a true and exact relationship, knowledge of the coefficients

In each equation would enable us to find the precise values of all the endogenous

variables that would result from a given set of values of the predetermined

variables. As a matter of practice, we do not know the values of the coefficients

and our ability to ascertain them depends upon the mathematical properties of the

equations as well as the statistical properties ascribed to the unexplained residuals. Only the mathematical properties of the model are Involved In the problem of Identification.

Problems of Identification

If a structural equation contains only one nonlagged endogenous variable,

Its parameters or coefficients are Identifiable. In this case, a least-squares

equation expressing the endogenous variable as a fimctlon of the predetermined

variables will yield appropriate estimates of the parameters. Such an equation

Is called a "unlequatlonetl coniplete model."

Suppose, however, that two endogenous variables appear In each of two

equations. For example, suppose a set of price-quantity observations were

generated by the following model:

Demand: Q = a + bP -i- u (19)

Supply: q = ,^>^ +pP + v, (20)

In which u and v are regarded as random disturbances. From a statistical viewpoint we have no basis for distinguishing between these two equations. Both

contain the same variables and both have random disturbances. It can be shown

that the least-squares regression of g upon P Is, In an Indefinitely large sample,

B - b <^v^ - (b^B) r^v ^u ^7 -^ßS^? . (21)

ö;^-2r^vi<- ^u'
- kk -

If the demand curve has not shifted (that is, If 0;; = 0), B = h; If the supply

curve has not shifted (that is, if ^ = O), B = ß . But in general B wi^J;^;«* ^

equal either of the structural coefficients and neither of them are identiflahle.

If equations (19) and (20) constitute the true and complete model there is, in

general, no wa^r ty which b and ^ can he estimated from the data.

The situation is different if each equation includes a different predetermined variahle. Suppose that the structural equations have the following

form:

Demand: Q = a+hP+cY + u (22)

Supply: Q = a +ßp + >Z + V, (23)

where u and v again are random disturbances, Y might represent consumer income

and Z,"livestock nunibers on January 1. If ^ and P are expressed as functions

of the predetermined variables and the disturbances and each of the variables is

expressed in deviation form, we obtain the "reduced form" of the model as follows:

The coefficients of the reduced-form eqioations may be estimated by leaat squares.

The structural coefficient ß can be estimated as the ratio of the coefficients

of 2 iû equation {2k) and (25); b can be estimated as the ratio of the coefficients

of z. Khoving £ and b, we can derive c and ^ from the coefficients of equation

(2U7; a and a can be estimated from the other coefficients, as means of Q, P, Y,

and Z for the sample are known* The model consisting of equations (22) and (23)

is Just identified; That is, our information is sufficient to make a unique

estimate of each structural parameter or coefficient. In other models it may be

possible to identify some of the structural equations but not others. Still

other models may be "overidentified. " In this case there are two or ir^re ways

of deriving a ¿ven equation, and the various possible individioal estimates must

be reconciled or averaged by a maximum likelihood method.

Identification, then. Is a matter of mathematics or logic which precedes

statistical estimation. When statistical considerations are Introduced additional

difficulties arise. These Include the situation In which two variables are

logically distinct but happen to be highly correlated during the period of

observation. The parameters or coefficients still could be accurately determined

In an Infinite sample, but the standard errors of parameters estimated from a

small sample may be greatly Increased by such Intercorrelatlon.

Problems of Estimation

The preceding two sections have considered only the economic and mathematical properties of a simultaneous-equations model. The problems of statistical

estimation are basically more difficult than those that arise In specifying Its

economic and mathematical properties.
-U5 -

As noted above, the nuinber of structural equations in a complete model must

"be equal to the number of endogenous variables whose values are determined within the model* Each structural equation may be written as a function of the

endogenous and predetermined variables which appear In It, and of a random

disturbance.

The problem of statistical estimation Is to Identify values of the

coefficients of the structural equations which maximize the probability of obtaining

the observed values of the endogenous variables on the basis of an assumed (usually

a normal) probability distribution of the disturbances. The theoretically Ideal

method for obtaining such values of the coefficients Is that of maximum likelihood, using all of the Information contained In the structural equations.

Vihen the number of endogenous veurlables Is large, the calculations Involved

In maximum likelihood estimation are laborious and expensive. This In Itself

would encourage the use of less expensive, even If less accurate, methods of

computation. If the Investigator does not require estimates of the structural

coefficients but Is content to obtain forecasts of the endogenous variables on

the basis of given values of the predetermined variables and no changes In

structure have taken place, a least-squares estimating equation, using a specified endogenous variable In the dependent position, Is Indicated. (See Marschak,

(Hi PP* 1"3) Such an equation gives the best linear unbiased estimate of this

variable, given the specified values of all predetermined variables. This will

not In all cases be the most efficient possible method of estimation, as Information afforded by the structural equations Is disregarded.

But It can be shown that when the structural equations In the system are

Just Identified, the Information contained In a set of least-squares equations

of the type Just described, one for each endogenous variable, Is equivalent to

that contained In the structural equations themselves. In this case It Is

possible to make two single, straightforward, and unique transformations. One

transforms structural equations Into least-squares equations, each containing one

endogenous vetrlable; the other transforms the least-square estimates of

coefficients back to estimates of structural coefficients. This approach Is called

the **method of reduced forms." Most applications of the simultaneous-equations

approach. Including that made by Glrshlck and Haavelmo (£2,), have used this

method.

Applicability of SIngle-Equation Methods to Analyses

Designed to Measure the Elasticity of Demand

for Farm Products

On page U3 it was noted that If a structural equation contains only one

nonlagged endogenous variable. Its parameters or coefficients are Identifiable

and that a least-squares equation expressing the endogenous variable as a

function of the predetermined variables will yield appropriate estimates of the

parameters.

Consider the following rather typical consumer-demand function for an

agricultural food product:

X3^(t) = b^X2(t) ^ o^X^it) + dj^(t) (26)
- k6 -

where X^^ = retail price;

X = per capita consumption;

X = per capita disposable income;

"3

d = random disturbance;

b , c = true values of structural coefficients,

""o o

In order to show that a single least-squares equation involving these variables

will give unbiased estimates of the structural coefficients in this function, we

must show that the explanatory variables--consiuiiption and disposable income—are

in some sense predetermined, that is, that their values result from economic

decisions made before harvest and from the action of exogenous or noneconomlc

variables such as weather. In addition, we must assume that the veoriance in retail price that is not explained by variations in consumption and disposable

Income is duo to a random disturbance, representing the effect of minor variables

omitted from the analysis, which is uncorrelated with the Independent variables. If

this is true, consumption and disposable Income may be regarded as predetermined

variables. As the disturbances are assumed to be reflected only in retail

price, the demand function must be fitted with price in the dependent position.

The discussion that follows is intended to show that demand functions for

laany farm products probably meet the specification of the uniequatlonal complete

model. It cannot be shown affirmatively that the disturbances in a given case

are uncorrelated with the explanatory variables, because the disturbances by

definition are not directly observable. Thus it is possible that a noneconomlc

variable, such as summer teaaperature, which affects consumer demand for lemons,

will be correlated with disturbances that eirise from minor economic factors.

Similarly, it is possible that production of apples, although determined by

weather and by economic influences before harvest, will somehow be correlated

with nonmeasurable disturbances in the demand function for apples during the

subsequent marketing season. But there is no reason to expect that the disturbances will be dependent (in a probability sense) upon the variables in question.

From a commonsense viewpoint it appears that the nonanswerable question

as to whether the disturbances are distributed independently of the explanatory

variables can be disregarded unless there are special reasons for assuming the

contrary. The answerable question as to whether certain variables entering

into demand functions for farm products are predetermined in a logical sense,

or nearly enough so to be used as explanatory variables without leading to

seriously biased estimates of elasticities of demand is considered only briefly

here. It is discussed in detail in Fox (20,22).

In ascertaining whether the single-equation approach can be used in deriving estimates of the elasticity of demand and similar coefficients, the

following questions must be answered.

1. Is consumption a predeterailned variable? One possibility is that

consumption of a perishable commodity is precisely equal to its production and

that production itself is a predetermined variable. It is evident that a variable

determined """^ ^^^^^ ^^ ^ predetermined variable may be regarded as pre-
- 1*7 -

A second cas© is that in which consumption of an item may differ from its

predetermined production because of variations in exports, imports, or stocks,

but in which consiamption is highly correlated with production. If the disturbance or unexplained residual in the relationship between cons\imption and production is random, it appears that the degree of bias in the least-squares estimate

of b^ in equation (26) will be fairly small.

But this situation could also be handled by a structural approach, as

follows :

Assume a true demand function,

p =ßq + u (27)

and a consumption-production relationship,

q ="* + V, (28)

in which £, a, and £ arm, respectively, price, consumption, and production, all

in deviation form; u and v are random disturbances uncorrelated with £ which is

a predetermined variable.

If we try to estimate gby fitting the least-squares regression of £ on g^,

we obtain

1> = Spq =ßSq -l-Equ = ß-CV zu -i-Zvu (29)

^q2 2:q2 Vq2

As rifzu = 0 under our assumption, this is equivalent to

b =0+ luA^ =ß+b^^öiL ^ (30)

Ö^q2 £5^2

If the disturbances u and v are not correlated^ b is an unbiased estimate of g^ .

If <S^ is small relative to (f , the bias in b will be small, as it seems unlikely

-V "11

that the regression of disturbances in equation (27) upon disturbances in

equation (28), b , will be as large in absolute value as the structural coefficient,

â , relating £ SÏ !•

In the present model, ß might be estimated as follows:

By least squares, fit the reduced-form equations,

p = dz and (31)

q = cz. (32)

As d is an estimate ot^^ and c is an estimate of ^, the estimate of the

structural peo^ameter ß is given by d/c. The precision of this estimate will tend

to increase as the size of the sample increases. This estimate should be a
. k8 .

statistically consistent one even if £ contains random measurement errors as

well as the effects of disturbances.

2. Is production predetermined? For many products this question can be

answered on logical grounds. The production of a crop is the product of planted

acreage, which was influenced "by economic and other considerations at or before

planting time, and yield, which in most cases is largely a fxmction of weather.

For livestock products, the question might be approached on a partly statistical

basis: What part of the observed variance in production can be explained by

(1) variables whose vfidues were ax:tuâlly known before the beginning of the current

period, (2) variables whose values, though not known in advance, must clearly

have been determined before the current period, (3) exogenous variables, such

as weather and disease, and (1^) errors of measurement. If by such a procedure

we can explain 95 percent or so of the observed variation in production, we may

conclude that, for practical purposes, production is a predetermined variable.

The residual variation sets an upper limit to the possible endogenous or Jointly

determined element in production.

3« l£ disposable incomo significantly effected by price or consumption of

the given commodity? Disposable income is determined by a vast conçlex of

economic decisions, some of which may be influenced by current prices of an

Individual commodity or group of commodities. But gross Investment and Government

expenditures in the current year are mainly predetermined variables, and these

are considered to be lergely responsible for major changes in disposable Income.

Hogs - a major agricultural commodity-account (as pork) for less than 3 percent

of all consumer expenditures. It is Improbable that the back-effects from

consumption and prices of pork upon disposable income are great enough to rise

above the level of measurement error in the income series.

Consumer-demand functions for many farm and food products approximately

meet these requirements. For these items single least-squares equations can

be used In deriving coefficients of elasticity. Other potentially simultaneous

commodity structures may be broken down Into a set of individual least-squeœes

equations if these relationships operate in sequence (perhaps in terms of time

units shorter than a year). The 1^-equatlon model of the feed-livestock econonçr,

discussed on p. 53 , is an example of this.

However, the demand-supply structures for export crops, including fats

and oils, for milk and dairy products, and for some fruits and vegetables with

two or more major outlets involve two or more simultaneous equations.

Structural coefficients estimated from single-equation demand functions fitted

for such commodities are likely to be unreliable. Serious experimentation is

needed to see whether reliable measures of elasticity can be derived, despite

basic data and small saDÇ)le limitations, by means of the simultaneous-equations

approach. Anthony S. Pojko has developed some 3-equatlon models to explain the

demand and price structure for individual dairy products. Price elasticities

estimated from single-equation models were found to be substantially lower than

corresponding estimates from the 3-equatlon model. Besults of this research are

to be published m the 1953 Proceeding Issue of the Journal of Farm Economics.
.U9 .

EXAMPLES INVOLVING THE USE OF MORE THAN A SINGLE EQUATION

The examples that follow were selected because they represent unusual

approeches to the problem under consideration* Some of these analyses are concerned as much with supply as with demand, but the approach that was used appears

to have wide application in related fields.

Simultaneous-Equation System for Pork,

Beef, and Export Crops

Pork

Suppose we are interested in estimating a consumer-demand equation for pork,

using calendar-year data« Farmers can alter average weights and numbers of hogs

sold for slaughter somewhat during a given calendar year in response to prices

of hogs during that year even though slaughter of hogs is largely determined by

number of j^igs saved and sows bred during the previous year. Similarly, meatpackers can alter their year-end inventories of pork in response to current-year

influences. If we as sume, therefore, that consumption of pork is determined

simultaneously with prices of pork, we are led to a 2-equation model, as follows:

The structural equations are:

Demand: p=iq-»-cy + u (33)

D

Supply: q =3P +>z+ v, (3'^)

where £ is price, j^ is disposable income, £ is consumption, and ^ is an estimate

of production based wholly on predeteimined variables; u and v are random

disturbances. Each equation is Just identified. If the variables are in logarithmic form, b is the elasticity of demand and §^ the elasticity of supply.

The reduced-form equations are:

P

each of which can be fitted by least squares. As Q^ will be calculated as the

ratio of the coefficients of ^ in the two equations, our ability to establish a

value of P that differs significantly from zero depends among other things upon

whether the net regression of ^ upon ^ in equation (36) significantly differs

from zero.

If we argue that equation (33) may be fitted by least squares without

significant bias, we are arguing that g is equal to, or close to, zero. If § = 0,

the reduced-form equations become:

p = cy + (X) z -h (u + I) (35.1)

b b
- 50-

q = O.y +'Yz + ( ), (36.1)

from which the structural equations are

Demand: p=lq + cy+u (33.1)

b

Supply: q ='Y z + V. (3'^.1)

For pork, with about 95 percent of the variation In hog production predetermined, the following results were obtained.

The reduced-form equations are:

p = -0.9581 z + 0.9707 y + u; B^ =0.92 (35.2)

(0.1091) (0.1026)

q = 0.8370 z -0.06IH y + v; E^ =0.91. (36.2)

(O.O67O) (0.0629)

The two structural equations, assuming J^O, are:

Demand: p = -I.IUU7 q + 0.897*^ y + u (33.2)

Supply: q = -0.0660 p + 0.7738 z + V. (3i*.2)

The least-squares demand function Is

p = -1.16 q + 0.90 y + u; P^ =0.97 (33.3)

(0.07) (0.06)

and the "supply" function, assuming J =0, Is

q = O.8U03 z + v; r^ =0.90. (3*^.3)

(O.O67O)

Differences between the structural demand coefficients and those of the

least-squares demand function are small In relation to the standard errors of the

latter. The supply elasticity, §.> is negative and does not differ significantly

from zero.

Beef

The potential response of production of beef to current-year Influences, and

particularly to prices of beef and cattle, is greater than that of production of

pork. Using a 2-equation model similar to that for pork, we find that the supply

elasticity, _g , does not differ significantly firom zei^s. Assuming J =0, we obtain

the following structural demand function:

p = -0.96 q + 0.82 y -O.U3 z + u. (33.1^)

The least-squares demand function, using ^ directly and ignoring a possible

simultaneous-supply fxinction, is
- 51 -

p = -1^06 q + 0.88 y -0.52 z + u; E^ =0.95. (33-5)

(0.12) (0.06) (0.09)

Although the differences between equations (33.U) and (33.5) are larger than In

the case of pork, they do not exceed one standard error of the least-squares

regression coefficients.

All Farm Products: Separating the Effects of Domestic

and Export Demand

Assume that the domestic demand function for all farm products Is given

by

p^«a+bq^ + cy , (37)

and the export demand by

P^ =-t + pq^ +'Ye, (38)

e G

In which a and ^ are domestic consumption and exports respectively; j^ Is

domestic Income, and e Is a measure of foreign demand which, under free-trade

conditions, might be simply the total Income of foreign consumers. Let us

assume also that total disappearance (3^ "*" £© ^ ^ ^ ^® ^ predetermined variable.

Then the equilibrium price (E.^ = ]^ = E^) for any given combination of c^^, j^ and

e is

P ' C^^) * (UJ) s ^ i-^) 7 * i^) >. (39)

The following equation, which applies to the index of prices received by

farmers for all commodities^ involves approximately the same variables as does

equation (39)*

(Uo)

log (prices received » 2«Ô12 -

I.65Ö

(0.273) log (physical volume of farm marketings) +

1.241

(0.102) log (disposable income) +

0.ll»2

(0.035) log (value of agricultural exports). ^\^23k « 0.97.

This equation should he subject to interpretation in terms of separate domestic

and foreign demand ctirves.

Let us assign the following values: g^ » 100; y - 100j e = lOOj B = loo.
-52 -

During the 1930«8 (and again in 19^9-50) ä^ averaged about 10 percent of g^, and

domestic sales (ä^) averaged 90 percent of «i^. Finally, we can assign some

reasonable values to b, ¿, c, and >, based on collateral evidence. We shall

assume a domestic price flexibility of -2 vith respect to ä^ «^d 1-5 with respect,

to y; also,a price flexibility of -1 with respect to q and 1 with respect to e.

The resulting arithmetic slopes are b = 2.22; ß = 10;^ç = 1.5 andl = 1.0.

Substituting in equation (39) we have

or

P^ = A -1.817 q^ + 1-230 y + 0.182 e.

The assumed set of structural coefficients Is reasonably consistent with equation

IkO). If the assumed value of b Is dropped to -2,0 (which means a domestic

price flexibility of -1.8), the'"correspondence Is still closer:

Pt = A -- (j^-) (f~) qt ^t ^-^ ^-tr) (-^) yy -^^ ^i[2 (% ^ (39.2)

or

P^ = A -1.667 q^ + 1.250 y + 0.167 e .

Two coefficients of this equation are almost Identical with those of equation (UO)

and the third Is within one standard error. It Is evident that equation (1^0) Is

consistent with the assumed structural coefficients of the domestic and export

demand functions when these are substituted In equation (39).

A System of Simultaneous Difference Equations Relating to

the Feed-Live s took Econongr

Demand and supply functions for some agricultural commodities ln5>ly an Inner

mechanism which, In the absence of other factors such as weather, would generate

an endless series of price and production observations for successive years on the

basis of specified Initial conditions. A simple system of this sort that Is

familiar to agricultural economists Is the "cobweb" model described by Ezeklel (11).

For any given commodity, there Is some Interest In knowing whether this Inner,

or endogenous, mechanism Is essentially a stabilizing factor, or whether It tends

to perpetuate cycles of considerable, and possibly everlncreaslng^magnitude.

The Cyclic Structure of a Two-Equation Model

The slnçlest simultaneous equation system of this type would Involve the

following equations. In which each variable Is expressed In terms of deviations

from some mean or normal value:

Demand equation: P+ = ^ q-t ^^^^

Supply equation: q^^^ = B p. . (^^2)
- 53 -

A mathematical method exists "by which the nature of the inner mechanism can "be

determined from the coefficients of the equations. Essentially this Involves

finding the nature of the roots of an equation formed from these coefficients.

In this instance, the root of the equation is hB. 32/ If this is negative, as will

always he the case for this set of equations, one-period oscillations will occur.

When bB is less than 1, the oscillations gradually drop down to zero; when bB is

greater than 1, the oscillations become larger and larger; when bB equals 1, the

oscillations are of constant magnitude.

Research by commodity analysts in the Agricultural Marketing Service

indicates that for most agricultural commodities analyses run in terms of logarithms with prices at the farm level will yield coefficients for b between -0.7

and 3»5 (see tables 1 and 2) and for B between 0.1 and 0.5. Table 3 shows data

for the first 5 years for price and quantity for systems of equations similar to

(Ul) and (U2) and having coefficients of specified size within these ranges. As,

for these equations, b Is a price flexibility and B is the elasticity of supply

(when eill variables are expressed in logarithms), the case in which b = -2 and

B = 0.5 is equivalent to one in which the elasticity of demand is equal to the

elasticity of supply. In such cases, as pointed out by Ezekiel (11), the

oscillations ore of constant magnitude.

A Four-Equation Model of the Feed-Livestock Economy

A system of simultaneous linear-difference equations was developed to

study the effect of support programs for com on prices of com, production of

livestock products, and returns to farmers from com and livestock over a series

of years. These equations can be used in sequence to follow through the effects

of changes in feed supplies or consumer income on livestock production and on

feed and livestock prices or to study the effects over time of specified support

programs for com.

Ig/ The way in which bB fits into the system is clear from the following:

Pi = !> ^1

42 " ^ ?!

Pg = b qg = bB p^ .

Extending this, we see that

2

p^ = (bB ) p. and, in general,

p^= (bB)*"SiThe same "multiplier" (bB)would appljr to year-to-year changes in both prices

and quantity.
-5^ -

The following variables were used in the system of equations:

C - Price received l)y farmers for com, which is assumed to be

" representative of the general price level for all feeds, in

cents per "bushel.

S - Supply of all feed concentrates, in million tons.

A - Number of grain-consuming animal units fed annually, in millions.

L - Price received "by farmers for livestock and livestock products,

index numbers (I9IO-IUIOO).

g - Production of livestock and livestock products for sale and

home consumption, index numbers (1935-39=100)•

I - Personal disposable income, in billion dollars.

Table 3.- Successive values of price and quantity for a

2-equation simultaneous lineeü* difference system

with specified coefficients l/

* Coefficients

»

Time period

: b = -0.7 :

! B = .1 :

: bB = -.07 :

b = -0.7 : b = -3.5 :

B = .5 : B = .1 :

bB = -.35 : bB = -.35 :

b = -2.0 !

B = .5 :

bB =-1.0. :

: b = -2.5

B = .5

bB = -1.25

: ^t : pt : > * • <it : Pt : ^t : Pt : • • • •

It : i^t : • « ^ ; h

1 :

2 :

3 !

h ;

5 :

: 1.00 -0.70

-.07 .05

.00 - .00

1.00 -0.70 1.00 -3.50

-.35 .2U -.35 1.22

.12 - .09 .12 - .»+3

-.OU 03 -,0k .15

.02 - .01 .02 - .05

1.00 -2.00

-1.00 2.00

1.00 -2.00

-1.00 2.00

1.00 -2.00

1.00 -2.50

-1.25 3.12

1.56 -3.91

-1.95 1^.88

2.kk -6.10

1/ Computed from unrounded data, p^ and q^^ are expressed in terms of

deviations from some mean or normal value.

All of the equations were fitted by single-equation least-squares

techniques, using logarithmic data for approximately the crop years beginning

1922 through I9U2. The use of single equations in fitting is permissible because

the variables to the right of the equality sign in each equation are assigned to

be either exogenous or lagged values of endogenous variables. In the following

equations, A equals the change in a particular item from the preceding year. The

numbers in parentheses underneath the regression coefficients sure their respective

standard errors.

AC = 0.00373 - 2.36ÛS -H 1.9ltó A + 1.13a L

(0.2U) (0.57) (O.I8)

R"" = 0.91

AA = -0.092 + 0.2ll*^S - 0.185 C 13/ -H 0.207 L 13/ E^ = 0.86

(o.oUo) (0.032) (0.036)

(1^3)

{hk)

11/ Prices apply to the calendar year in which the crop year begins.
- 55 -

AQ =0/)0369 + 0.56aî^A r^ = O.7I (U5)

(Ô.090)

Û L = 0.00578 - 2.O8AQ + I.U5M R^ « 0.96 (i^6)

(0.25) (0.08)

A^ Is assuined to be a direct function of A^, as feed fed per animal unit had the

wrong sign and had no statistically significant effect on sales of livestock

products. Therefore, equation {k6) can "be rewritten as:

AL = -0.00190 - l.l-í^A + I.U5AI. (1^6.1)

If this is done, equation (^5) can \>e omitted.

The economic and statistical aspects of these equations are discussed in

Foote (15 )>

Mathematical anedysis similar to that described for the 2-equatlon model

Indicates that this system probably will involve one-period oscillations and that

the fluctuations will tend to increase in amplitude. The algebraic computations

required are discussed in detail by Foote (16,17).

Application to Analyses of Specified Corn-Loan Programs

In the analysis of the effects of specified loan programs for com on the

feed-livestock economy, the following general approach was used: (1) Certain

assumptions were made regarding year-to-year changes in the exogenous variables,

feed production and disposable Income; (2) the effects of these on prices of com

and livestock, on production of livestock, and on total returns to farmers, within

the assumed loan-program framework, were measured on a year-to-year basis;

(3) results for the several loan programs and those arising without a loan program

were compared, both by years and in total for a nimber of years. This method

appears preferable to one in which results for specified loan programs are coxi5)ared

with actual prices for a given period. Many factors other than those allowed

for in the model affect actual prices, so that the differences between results

under a given loan program and actual prices reflect not only those due to the

loan program as such, but also the effects of all factors not allowed for in the

model. Under the system described above, the differences that are found are

mainly due to the type of loan program assumed. But the results will depend to

some extent on the particular pattern chosen for the exogenous factors, the

starting levels for the various items, the particulsir equations used, and the

length of time for which results are computed. All of these must be chosen as

logicedly and as realistically as possible*

The following sequence was asstimed in analysing year-to-year changes in the

several variables involved: (1) Prices of livestock for November to May are

determined by the number of animal units fed on farms in the preceding year and

current disposable income from November to May; (2) prices of com for November

to May are determined by the number of animal units in the preceding year, newcrop supplies (production plus stocks) of feed, and prices of livestock from

step 1; (3) animal units fed are determined by new-crop supplies of feed and prices

of com and livestock for the calendar year in which the October to September

corn-marketing year begins. Prices for the calendar year were obtained by weighting

prices for November to May and June to October by the proportion of the sales

for the calendar year obtained from each period; (U) prices of livestock from
-56 -

T ,m^ fO Octol)er are determined by the number of animal units from step 3 and

cS^ SS^sLu moo« for J»ne to October; (5) prlc« of corn ^- Jun. to

October are determined by the number of animal units from step 3, the level of

prÍcefor^ from step U, and the supplies of feed used in step 2.

As all of the analyses were based on first differences of logarithms, the

results are given in the fonn of percentage changes from the preceding year.

Actual levels for any given year were obtained by applying these changes to the

computed level in the preceding year. By this method, the study was continued

year by year for as long as desired.

For each of the two periods within the corn-marketing year, four alternative

price levels must be computed: A price equivalent to the loan rate, the release

price, a computed price based on nonloan supply only-, and a computed price based

on total supplies. With a loan program in effect, prices are assumed to average

at or above the loan equivalent. If the loan-equivalent price is above the

computed price based on nonloan supply only, the loan-equivalent price prevails.

Under such circumstances loan stocks will, in general, increase. If the computed

price, based on nonloan supply only, is between the loan-equivalent price and

the release price, the coi^uted price will prevail and loan stocks will not

change. If loan stocks exist and the release price is between the computed price

based on total supply and the computed price based on nonloan supply only, the

release price will prevail. In general, uuader such circumstances, loan stocks

will be reduced during the year. If the computed price based on total supply

is between the release price and the computed price based on nonloan supply only,

computed prices based on total supply will prevail. Under such circumstances,

loan stocks are reduced to zero by the end of that year. On the worksheets, the

four prices were listed side by side and the prevailing price was circled.

These circled prices were used in computing calendar-year average prices for use

in the animal-units equation and in computing marketing-yeeur average prices to

ascertain the value of the crop.

To find the magnitude of the changes in loan stocks, the supply consistent

with the loan rate at the end of the season was computed. If K is used to represent the parity index, S" is the supply that is consistent with the loan rate,

and all variables are expressed in logarithms, the equation is as follows:

AS" = 0.00158 -O.ltóUuK + 0.822AA + 0.U79AL, (1^3.1)

This is obtained by transposing equation (U3) and substituting AK for ÛC

and AS" for 45^. The supply consistent with the release price at the end of the

season also is needed in those years in which the release price prevails from

June through October. This can be obtained in a similar way.

If S", in million tons, is smaller than the nonloan supply available, the

differences represent the quantity by which loan stocks will be increased. If

the release price prevails from June through October, the supply consistent with

the release price, in million tons, will be larger than the nonloan supply available, and the difference represents the quantity by which loan stocks will be

reduced. As noted above, loan stocks will be reduced to zero in those years in

which the computed price based on total supply is between the release price and

the computed price based on nonloan supply only.
- 57 -

A similar approach could "be used in analyzing the effects of specified loan

programs for other crops.

A 10-region Model of the Feed^Livestock Economy ik/

Agricultural economists long have used statistical demand functions and

estimating equations based on production and average prices for the country as a

whole. Such aggregation disregards information concerning regional differences

in prices, production, and consunçtion of the commodities involved. Many data

exist on prices and production "by regions. States, and in some cases even smaller

geographic units. Marketing agencies and commodity specialists use this information in various, though generally informal,ways.

Agricultural outlook work has "been a major undertaking of the Department of

Agriculture for almost 30 years. The fruitfulness of this line of work probably

can be increased if more accurate account is taken of regional differences in the

situatiOEB that confront the different coimnodities. In addition, special problems

center around changes in costs of transportation — the major factor in geographical

price differentials— and their incidence upon consumers, producers, and transportation agencies. Incidence problems often are discussed piecemeeúL in terms of

two or three shipping points competing for a particular market. But it is hardly

possible by informal or qualitative methods to predict the effects of changes in

a single freight rate, or in a limited nvimber of freight rates, upon all producing

areas and markets in the national econonçr. The simplified model of the livestock

feed economy in this country which is presented here emphasizes the factors that

determine prices and consumption of feed and the pattena of Interregional trade

in feed grains and other feed concentrates.

Data and Assunçtions

Ten regions were defined in rough accordance with differences In the types

of livestock and production of feed that are emphasized« The model includes a

demand function for feed in each of the 10 regions and a structure of freight

rates or transportation costs between all possible pairs of regions.

Demand functions.- Regional demand functions for feed were based upon a

statistical analysis of demand for the country as a whole. This analysis indicated

that a 1-percent increase in the supply of feed concentrates per grain-consuming

anima^ unit was associated on the average with a 2-percent decrease in the average

farm price of com; also, that a 1-percent increase in prices of grain-consuming

livestock and livestock products was associated on the average with about a

1-percent increase in the farm price of com. This demand function was converted

to arithmetic form in such a way that these price flexibilities were realized for

approximately the actual values of United States farm prices of livestock and

feed and production of feed concentrates in the 19^9-50 feeding year. Specifically,

this arithmetic demand function is as follows:

Ijjy A detailed discussion of this example is given in Fox (21).
. 58 .

Pe = 2.6873 -3.50 (jp) +0.0135 P^, (^'^^

where P = United States average price of com, in dollars per bushel;

—c

Q^, = Total quantity of feed concentrates available for feeding, in million

tons;

N = Number of grain-consuming animal units fed during the preceding season,

"" in millions and

p. = Index of prices of grain-consximing livestock and llveßtock products. il United States (average in 1950 = 100)

As prices of livestock show considerable geographical differences, an index

of prices of grain-consuming livestock and livestock products was calculated for

each of the 10 regions, using 1950 data. For convenience, the constant term in

each regional demand curve is adjusted to include the specified value of livestock

prices. This leaves a demand function in each region which includes as variables

only prices of feed (com) and consumption of feed per animal unit. Basic data

on these variables are presented in table k.

As an equilibrium condition, total consumption of feed in the country as a

whole must equal the supply available for feeding. Hence, the demand functions

must be further transformed in such a way that total consumption of feed in each

region is made dependent upon the price of feed. These demand functions are

given in table 5. A necessary condition for solution of this "geographical

equilibrium** system is that the set of regional prices must cause the sum of the

regional estimates of feed consumption to equal the total supply available for

feeding in the country as a whole.

Freight rates.- Equilibrium prices also must be consistent with the structure

of freight rates among regions. Freight rates are a special problem because the

model mathematically implies that production and consujnption of feed in each

region are concentrated at a single point. In this analysis, freight rates were

estimated on the basis of data on freight charges by mileage blocks, from the

1950 Interstate Comnerce Commission waybill sample. The relationships used for

estimating freight rates between regions are as follows:

Xjj = 5.6 +0.0168 M^j (U8.1)

Xjj = 5-6 +0.022U M^j (U8.2)

^ij " 5-^ -^0.0280 M^j (lrô.3)

where X^^ = freight rate on com from region i to ¿, in cents per bushel; and

Mij= distance in miles between centers of production of grain-consuming livestock

in regions 1 and ¿.

Equation (U6.1) was used when the ICC data indicated relatively low freight

rates for corn in the railroad territories involved; equation (lf8.2) was used
- 59 -

Table 4«- Basic data on price and nimber of livestock and on supply of feed available,

by regions. United States, specified years

Price of

[ Supply of feed ;

concentrates ! Grain-consuming

••

: Feed supply per

Region 1 livestoek» available. ! animal units fed. : animal unit.

1950 i/ ; year beginning

; October 1949 2/

year beginning

: October 1949

: year beginning

: October 1949

•

1,000 tons Thousands Tons

Northeast i 116.5 7,U5 17,602 0.4230

Com Belt ; 97.0 55.136 62,005 .8892

lake : 90.2 18,395 20,216 .9099

Northern Plains : 91.8 18,430 16,893 1.0910

Appalachian : 102.2 8,636 14,184 .6089

Southeast : 111.1 4,571 8,981 .5090

Delta : 101.0 3,176 6,364 .4991

Southern Plains : 102.8 5,735 9,472 .6055

Mountain : 99.6 3,296 3,638 .9060

Pacific : 110.3 2,735 6,766 .4042

United States : 100.0 127,555 166,121 .7678

i/ Index numbers (United States average in 1950 a lOO). Regional average price of each major

livestock product weighted by its appropriate contribution to total grain-consuming animal units

in the region.

2/ Available for livestock feeding after eliminating non-feed uses and changes in end-of-year

stocks«

Table 5«- Data relating to regional demand functions for feed concentrates under conditions

applying approximately to the year beginning October 1949 l/

'Price per bashel of feed dependent: Consumption of feed dependent:

Region :

Intercept ' Price in Intercept [ Change in

(price if absence of (consumption if [ consumption per

feed supply , interregional , price ; dollar change

is zero) trade is zero) in pnce

Dollars Dollars Million tons Ällion tons

Northeast : 4.26 2.7Ô 21.4229 -5.0289

Com Belt : 4.00 .88 70.8022 -17.714S

Lake : 3.90 • 72 22.5517 -5.7757

Northern Plains : 3.93 •11 1S.954^V -4.8263

Appalachian ; li.07 1.94 16.4834 -4.0524

Southeast î 4.19 2.41 10.7454 -2.5659

Delta Î 4.05 2.30 7.3665 -1.8182

Southern Plains : 4.08 1.96 11.0292 -2.7062

Mountain : 4.03 .80 4.1903 -1.0394

Pacific : 4.13 2.76 6.0746 -1.9330

United States j 4.04 1.35 191.6206 -47.46CÍÍ

1/ All prices are per bushel of com or an equal weight of other feeds. For simplicity,

prices and freight i^tes for com are taken as representative of average prices and freight

rates for all feed grains and byproduct feeds.
- 60 .

when an intermediate level of freight rates was indicated; and equation (^«^8.3)

was used when the data suggested relatively high freight rates on com, as well

as for all paths over which actual shipments of com were ixnprohahle or rare.

Feed production and livestock numbers>- Production of feed and numbers of

livestock are used as predetermined variables in the explanation of feed prices,

although the latter is strictly Justified only during the early part of the feeding

year. In this case the numher of grain-consuming livestock in each region during

the feeding year ended September 1950 was used. Actual data on 19^9 production

and carry-in stocks of feed were adjusted as follows: (1) Nonfeed utilization of

feed grains in each region, as in food products or for seed, was sub s tracted from

the total supply of feed grains; (2) Exports of feed grains were allocated as

nearly as possible to the regions of origin; (3) Carryover stocks of feed grains

in each region at the end of the year were sub s trac ted from the beginning supply.

The remaining (and by far the largest) quantity of feed grains, plus the total

production of byproduct feeds, was assumed to be available for feeding domestic

livestock during the 19*^9*50 feeding year. The resulting figures on both a total

and a per animal-tinit basis are shown in table k.

It is evident that the model presented here is schematic and Involves many

simplifying assumptions. The effects of relaxing some of these are discussed

later.

Equilibrium Solutions

The problem attacked in this section may be stated as follows: Given (1)

the demand function for feed in each region and (2) the structure of transportation

costs between regions, to find the equilibrixam values of prices and consumption

of feed in each region and the net quantities of feed shipped over each interregional path as a result of any specified set of regional values of feed production, livestock numbers, and livestock prices. This problem was solved for

several different sets of initial conditions, illustrating the effects of changes

in freight rates, changes in regional production of feed, and changes in the

regional distribution of grain-consuming livestock.

A necessary requirement for equilibrium is that no region can increase its

revenue by changing its pattem of consumption or shipment. If one region ships

to another, the prices must differ by the cost of transportation between the two.

If two surplus regions ship to the same deficit region, the difference between

equilibrium prices in the surplus regions will equal the differences between their

freight rates to the deficit region. Thus, an equilibrium solution for the whole

system involves a precise structure of regional prices bound together by specific

rrelght rates (except for regions which prove to be self-sufficient under the

given conditions).

nr. _^'"j®^f®*ically> working out an equilibrium solution in a 10-region system

^TttL,^ laborious. Fortimately, intuition or Judgment enables us to move rather

directly toward the equilibrium arrangement.

of An Approximation to Actual Conditions in IQliQ^SQ.^ If the data on supplies

winT ^ T ''''^^ ^"^ *f^"^ "" "^ examined, it appears that four regions probably

will be surplus regions, or "sources", and that the other six probably will be deficit
- 61 -

Ï!5Î^''^\?Î* "destinations•" In general it appears likely that a given surplus

region will ship to deficit regions in which it has a freight advantage over other,

usually more distant, surplus regions.

The arrangement that emerges is simple and logical in most respects. The

Com Belt acts as a basing point for four deficit regions—the Northeast, Southeast,

Appalachian, and Delta. Prices in these regions are equal to the Com Belt price,

plus freight. The Lake region supplies part of the requirements of the Northeast,

and its price is equal to the price in the Northeast, minus freight. The northern

Plains region serves as a basing point for the Pacific and southern Plains regions,

and the Mountain region serves as an auxiliary source of supply for the Pacific

region. The only additional item needed is a link connecting these two systems.

In this case, it tums out that the Com Belt, in equilibrium, would import some

feed from the northern Plains and would reexport an equivalent quantity to other

regions. Hence, the link is provided by the fact that the price in the Com Belt

is equal to that in the northern Plains, plus freight.

The technique used to obtain the equilibrium solution for this arrangement

was as follows: (i) it was assumed that the arrangement Just described did

apply; (2) using the demand functions in table 5. the consumption of feed in each

of the 10 regions was calculated on the basis of the assumed price in the Com

Belt (in this case $l.lfO per bushel). But total consumption of feed at this price

would not exhaust the available supply^ As the assumed rigid structure of price

differentials makes total consumption of feed in this country as a whole a linear

function of the price in the Com Belt, the adjustment in price that would be

necessary to equate total consumption of feed with the available supply can be

calculated immediately. This adjustment gives the equilibrium prices in each

region, assuming the specified arrangement, and the equilibrium rates of feed

consumption.

A comparison of the regiCÄial consumption estimates with the production

estimates in table k gives at once the net imports or exports of each region.

In conjunction with the assumed arrangement, the precise quantity of feed shipped

over each interregional path can be determined. These figures are shown in table 6.

A Forecasting Model.- The usefulness of the 19lf9-50 equilibrium model for

the purpose of forecasting changes in regional prices of feed and in interregional trade are now explored. Suppose that the 19*^9-50 model represents the

actual situation in the year Just past, and that numbers and prices of livestock

are substantially \uichanged during the current year. In July of the current year,

well before harvesttime, published forecasts of production of feed grains are

available. They indicate changes, region by region, from the previous year.

If the new production forecasts are substituted into our model, we should obtain

forecasts of the corresponding regional prices and consumption of, and net trade

in, feeds, if sufficiently accurate, these forecasts should be of considerable

value to farmers, marketing, transportation, and processing agencies, Grovemment

price-support and procurement agencies, and others.

But the July production forecasts are subject to error. They are estimates

of the production of feed grain that would be expected if growing conditions

during the remainder of the season (after July 1) were average. In some years,
- 6a

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-. 63 -

weather may cause marked Improvement or deterioration in crop prospects after

July 1. Hence ve are interested in the accuracy with which the pattern of

equililDrium forecast in July anticipates the pattern that emerges after harvest.

For illustration, the pattern of equilibrium for ^^191^9-50" was compared with

patterns reflecting differences from I9U9 production of feed grains by regions,

"based on (1) July 19*^7 production forecasts and (2) December I9U7 production

estimates. The July I9Û7 forecasts indicated a reduction of 18 million tons below reported production in 19^9^ and the December I9U7 estimates indicated a

further reduction of 6.5 million tons.

The regional price differentials were identical in all three cases. The

July forecasts implied a price increase of 38 cents a bushel, and the December

estimates indicated a further rise of lU cents a bushel, com equivalent.

But the net trade positions of the different regions showed striking changes.

The July forecasts suggested that the Com Belt would have a small surplus and

that the northern Plains would ship much larger quantities than in 19^9-50. The

December estimates indicated that the Com Belt would shift to a net deficit

basis and that shipments from the northern Plains would be only moderately above

those for 19*^9-50. The Appalachian region would shift to an almost selfsufficient basis. Under the ass\imed structure of freight rates, even after it

had shifted to a deficit position, the Com Belt would reship large quantities of

corn received from the northern Plains. A reduction of freight rates out of the

northern Plains by 1 to 7 cents a bushel on various freight paths would be required to eliminate this forwarding role of the Com Belt.

The Indicated volume of net trade declined much more sharply than did

production of feed. Compared with 19^^9-50^ the July forecasts implied a lU-percent drop in production of feed and a 29-percent drop in interregional shipments

of feed. The December estimates showed a further drop of 6 percent in production

and 21 percent in shipments. Total net trade evidently depends to a considerable

extent upon the regional distribution of feed production in a given year as well

as upon its general level.

Problems Involved in the Application of This Approach

In the livestock-feed economy, adjustments should be made for the following:

1. Allowance for short-run responses of livestock production to changes

in supplies and prices of feedJ

2. Allowance for longer run effects of increases in freight rates on production

of livestock and feed;

3. Recognition that each livestock product has a geographical price-production-consumption equilibrium of its own;

k. Allowance for less-than perfect substltutablllty between different feed

concentrates;

5. Allowance for possible differences in elasticities of demand for feed

(a) by regions and (b) by classes of livestock;

6. Allowance for the existence of price-dependent demands for storage and

export of feed;

7. Removal of the assumption that production of feed in a large region is

concentrated at a single point; and

8. Allowance for seasonal elements in the equilibrliun price structure.
. 6k .

For example, the South uses its own grain production early in the crop year, because of storage prohlems, and gets most of its "imports" of feed late in the

crop year»

It remains to be seen whether a useful compromise can be effected between

the complexities of a full description of reality and the need for a model

sufficiently aggregated so that it can be manipulated and interpreted without

undue expense. A partial answer might be found in terms of the nature of policy

decisions, either private or Governmental, which might be made on the basis of an

equilibrium analysis or on the basis of forecasts from such a model»

It should be ençhasized that the most important and most conçlex of the

commodity structures in our agriculture was purposely chosen. Livestock and feed

products account for approximately 60 percent of total cash receipts from farm

marketings in this country, and livestock products account for about an equal

percentage of total retail expenditures for food. There may be many sinçler and

more appropriate applications of spatial equilibrium analysis to other farm

products. Some unpublished analyses along these lines have been made by the

Bureau of Agricultural Economics for oranges, potatoes, and celery. When the

number of shipping points to be considered is relatively small, it may be feasible to use specific freight rates between given sources and destinations instead

of relying upon some partly arbitraxy procedure of estimation.

The 10-region model of the livestock-feed economy presented here is an

exploratory ventxxre. The potentialities of spatial equilibrium models for economic forecasting and outlook work are far from clear, both as to the degree of

precision that may be obtained and the commodities for which such an approach

may be helpful. Such models appear to have considerable value (1) for teaching

principles to students of transportation and marketing and (2) for showing to

policy officials the central features and tendencies of a geographically extended

demand and supply structure which might be affected by their decisions. These

expository uses probably are enough to Justify considerably more experimentation

with spatial equilibrium models based on actual data for specified commodity

groups.

The Technique of Linear Programming 1^/

Linear programming may be the key to many economic problems, including

problems in farm management and in agricultural marketing. But most of the

literature on the subject is abstract and mathematical. Few practical applications

have yet been made and most of these concern military programs, so the results

are classified. A partial exception is the recently published monograph by

Chames, Cooper, and Henderson. They point out that "linear programming is concerned with optimal planning of Interdependent activities subject to a complex

of restrictions" (¿, p^3 ). An example which involves the highest profit combination for certain mixes of nuts is described. The first 10 pages, which give

15/ This section was prepared by Frederick V. Waugh, Director, Agricultural

Economics Division, Agricultural Marketing Service. Further details regarding

this example are given by Waugh (Ug).
- 65-

an insight into the basic approach of the method, can he understood by research

workers who are not versed in mathematics*

The general idea can be illustrated by the example of dairy feeds. Relevant

information concerning the following is summarized in a convenient form in table 7<

1. Required amounts of total digestible nutrients (TDN), digestible

protein (DP), calcium (Ca) and phosphorus (p) in a 2Ú-percent

dairy feed, according to Henry and Morrison (30).

2. Amounts of each of these nutrients in 100 pounds of several feeds,

and

3. Average prices of these feeds for 191^9-50, wholesale at Kansas City.

Table 7*" Proportion of requirements in a 2U-percent

dairy feed supplied by $1 worth of each feed

Feed : Total

: digestible

: nutrients

Nutritive factor

Digestible

protein Calcium : Phosphorous

Weight

1/

Com : O.UUl

Oats : .375

Milo : .U95

Bran : .U23

Middlings : .U36

Linseed meal ....: .272

Cottonseed meal..: .268

Soybean meal ....: .286

Gluten : .395

Hominy : .kkS

0.136

.187

.203

.321

• 332

.Uoo

Mk

.50U

.kl2

.158

O.OUO 0.168 0.U17

.170 .201 .397

,066 .206 .i*59

.312 .900 ,U67

.176 .U3U MO

.511 .336 .262

.268 .513 .281

.335 .238 .270

.879 .U71 .385

.U12 Ml .39^

1/ Proportion of 100 pounds that can be bought for $1.00, or the reciprocal of

the price.

First, we shall concentrate upon the four nutritive factors.

Assumptions

The following assumptions are required:

1. A dairyman is satisfied with any combination of feeds

that supplies all the nutritive requirements.

2. A combination is acceptable if it supplies more than

the required amounts of some elements.

3. The listed feeds can be bought at exactly the quoted prices.
. 66 .

k. No other feeds are available.

5, A negative quantity of any feed cannot "be "bought.

The Problem

What feeds, and how much of each should he buy to satisfy all requirements

at the minimum cost?

The Analysis

1. The least expensive source of a single nutrient.- Take total digestible

nutrients, for example. The largest number in the first column in table 7 is

O.U95, indicating that $1 worth of milo will buy O.U95 times the TDN requirement.

The required amount of TDN could be obtained from 1/0.1^95 dollars = $2.02 worth

of milo. This is the least expensive source of TMí»

Before going on, we note two points:

a. Sooa one feed will always supply a single nutrient at less expense than

any possible combination of two or more feeds. In this case, straight milo will

supply TDN at less expense than any combination, such as milo and bran, or com

and middlings.

b. If the least expensive soiurce of one nutrient happens to supply or oversupply the requirements of another nutrient, then that single feed is the least

expensive source of both nutrients together. In this example, $2.02 worth of milo

does not completely supply any of the other requirements. But suppose that the

price of middlings were reduced enough so that middlings became the least expensive source of TEW. This would Increase all the figures in the middling row.

Then if we bought enough middlings to supply the TDN requirements, we would also

supply enough DP and P. Thus, if the price of middlings were reduced, it alone

would become the least expensive source of TDN, DP, and P together. Any combination of two or more feeds that would supply these three nutrients would cost more.

2. The least expensive source of a pair of nutrients>- A graphic analysis

of this problem can be found in Waugh (^2, p . 30U). It leads to three criteria:

a. A feasible ccaabination is one that does not involve negative purchases

of any feed. A nonfeaslble way to supply TEN and DP would be to buy a large

positive quantity of middlings and a large negative quantity of oats. But we

rule out such solutions.

b. A necessary combination of two feeds is one that supplies two nutrients

at less expense than either single feed. Thus, gluten alone supplies TMÎ and DP

at less expense than any feasible combination of linseed meal and gluten, so that

the latter would not be a necessary combination.

c. A graphic test Is given to determine the least-cost> feasible, and

necessary combination of two feeds meeting two requirements.

Suppose we have found the least-cost, feasible, and necessary combinations

Ox two feeds supplying the requirements of TDN and DP. In this case it is about
- 67 -

35 cents worth of middlings, and about $2.lU worth of gluten. Then:

d. No combination of three or more feeds will supply these two nutrients

at less cost and

e* If the least-cost, feasible, and necessary combination supplying TKÎ

and DP hapx>ens also to supply requirements of some other nutrient it is the

least-cost, feasible, and necessary combination for all three nutrients.

In this case it happens that 35 cents worth of middlings and $2.llf worth

of bran supply enough Ca and P, so it is defined as a complete combination, as

far as nutrients are concerned. But this combination weighs only 97 pounds.

The feed manufacturer may have another requirement—a combination that would

meet all four nutritive requirements and also weigh 100 pounds.

In this case he could find a combination of two feeds that would do this.

He could Just meet the requirements for DP and weight with 39 cents worth of

bran and $2.12 of gluten. (Use of this additional requirement would necessitate

redrawing the diagram used in the graphic analysis.)

3. The general case.- The graphic criteria have been put into mathematical

terms and extended to define the least-cost, feasible, necessary combination

of m Inputs to supply n requirements (m^n). One procedure Involves the

following steps. Find the single input that will supply any one requirement at

least-cost; test to see if it happens to supply other requirements also. If

not, find the combination of two Inputs which is the least-cost, feasible,

necessary way of meeting any pair of requirements. Again test whether it supplies

other requirements in full. If not, proceed to 3^ ^^ or more inputs. At some

point, it will be found that some combination of £ inputs exactly meets £ requirements and also happens to supply or oversupply all other requireraents.

The analysis is over. We have found the least-cost, feasible, necessary

combination of Inputs meeting all n requirements.

General Observations

The feed problem is a very simple case. But it Illustrates a problem that

iß basic to most economic research. In general, our resources are limited. We

search for ways of using existing resources to maximize real net income or something like it. Or we search for ways of meeting stated objectives at the least

cost or something like it.

Research workers at the University of North Carolina are using linear

programming to find the most profitable combination of enterprises on certain

types of farms. Railway Age (l) discusses the minimization of cost in distributing empty freight cars. Presumably many problems in agricultural marketing

essentially Involve linear programming, liiere are two main difficulties: (1)

Most economic problems are big—they Involve many Inputs and many outputs. Here

the electronic computer could be the answer; (2) It is ordinarily difficult or

even impossible to get an accurate statement of either input-output relations

or requirements (objectives). In case of dairy feeds, we probably know fairly

well the average nutritive content of most common feeds. But the requirements

may not be fully stated.
. 68 .

EEMAKD ANALYSES BASED ON PUBCEASES BY INDIVIEUAL CONSUMEES

For many years, commodity analysts have used family-budget data applying to

a single unit of time In measuring elasticity of income. Such data cexinot l?e

used to measure elasticity of price as, in general, only an average price for the

period is published.

Since October 191^9 the United States Department of Agriculture has published

monthly and quarterly reports of household purchases of fresh citrus fruits,

frozen and canned juices, and dried fruits (]il,|iÔ). These reports are based on

figures obtained under contract ^rom the Market Research Corporation of America.

The monthly reports relate to U-week periods and show for each product the total

quantity bought by householders, average price paid, percentage of all families

that purchased, average number of purchases made during the period by buying

families, and the average quantity bought per purchase, based on a sample of

approximately ii,200 families blownup to a United States total basis. The quarterly

reports relate to 13-week periods and show similar data by geographic regions

and type of retail outlet. Several research projects that would make use of the

data on which these reports are based are being conducted or considered by the

Agricultural Marketing Service in cooperation with the University of California.

Some of these studies are outlined below.

Regression Analyses Based on National Ag/^re^ate Data by Months

George M. Kuznets of the University of California has run a number of

analyses, using both arithmetic and logarithmic data, from the published monthly

reports on citrus. Considering the short period of time Involved, some of these

turn out surprisingly well. The "own-price" elasticities of demand for frozen

orange concentrate and canned single-strength Juice turn out to be highly significant. There appears to be significant competition between fresh oranges and

frozen orange concentrate. The evidence for conçetltlon between fresh oranges

and canned Juice or between canned and frozen orange Juice is not conclusive.

The analyses for fresh oranges are weak. Panel data on prices of fresh oranges

are given in cents per dozen, with no adjustment for size. Use of retail prices

published by the Bureau of Labor Statistics gave better results, but the price

elasticities were still barely significant. For the studies discussed below,

data on the price of fresh oranges will be expressed in terms of cents per pound.

Based on outside information the prices per pound of oranges appear to be relatively

uniform regardless of size. Thus, statistical analyses based on them should be

more reliable than those based on price per dozen.

Regression Analyses Based on Aggregates for Selected Groups of Families

A major purpose of these studies is to ascertain the relative elasticities

Of demand for fresh oranges and for frozen Juice and the ultimate potential

demand for frozen Juice. Separate regression analyses will be made for panel

ramilles who regularly use frozen Juice and for those who seldom or never use

frozen Juice,

Quasi-Descriptive Studies of Consumers with Specified Characteristics

f^nifl'^SL^^] analyzed household purchases of citrus products by 5OO urban

families from November I9U8 through October I9U9. In addition to those aspects
.69 -

normally covered "by a one-period "budget study, he determined the frequency of

purchase of the various items by income groups and size of family throughout

the period of the study. For example, he found that less than half of the

families who bought frozen orange Juice concentrate on a trial basis became

regular purchasers. But about half of those who became regular buyers of frozen

Juice discontinued purchases of fresh oranges or canned Juice, or both. Total

purchases of all citrus products increased when families became regular users of

frozen Juice. Frozen Juice is now used much more widely that was the case during

the time period covered by the Ogren study. A similar study covering a longer

time period and possibly a larger sample is planned. Questions the study will

attempt to answer include: (1) What is the typical pattern of consumption of

frozen Juice for a family from the time purchases first start until family

consumption levels off on a flat plane, and what happens to consumption of other

citrus products; and (2) What part of the rapid increase in consumption of frozen Juice represents new consumers and what part represents increased consumption on the part of habitual users?

Regression Analyses Based on Purchases of Individual Families by Months

So far as we know, this type of study has not been attempted previously.

If successful, it would open up a wealth of information to commodity analysts.

The unit of observation would be purchases of a single family within a single

month. The large n\anber of observations would permit a high degree of stratification and the use of relatively conçlex equations. But individual consumers

are known to behave erratically, so that low correlations would be expected from

such a study. Whether the large number of observations would more than offset

the low correlations could be determined only by experimentation. Several pilot

studies are proposed which would serve to test the feasibility of the full study

discussed here. Preparation of punch cards from data available in the files

of the Market Research Corporation of America has been completed. These cards

are to be used for the pilot studies and certain of the other studies discussed

above, but they also contain all of the information needed for the full study

as discussed here.

Strata

Separate analyses are proposed for rural and for urban consumers within

each of 5 geographic regions. Small towns in which frozen Juice is not readily

available would be included in the rural classification. Separate analyses also

are proposed for summer and winter, as California navel oranges differ essentially

from California Valencias. Thus 20 separate strata would be considered

(5x2x2).

Items to be Considered

(1) California-Arizona fresh oranges, (2) Florida, Texas, and unspecified

oranges, (3) fresh grapefruit, {k) frozen orange Juice, (5) canned orange Juice,

(6) canned grapefruit Juice, (?) canned orange-grapefruit blend, (8) canned

tomato Juice, (9) canned pineapple Juice, (10) Juice equivalent of fresh lemons,

canned lemon Juice, shelf-pack lemonade base, and frozen lemonade base.
- 70-

Analyses to be Run

Separate equations for each strata are proposed with consumption of each

of the first 6 Iteics In turn as dependent variables. Thus, 6 equations would

be detemilned for each strata. The following would appear as Independent variables In each equation: (l) Prices for each of the 10 coimnodlty groups

(Intercorrelatlons do not appear to be unduly high), (2) Income per famlly(based

on annual data), (3) number In family, Including boarders (based on annual data),

(k) percentage of stores within the area (weighted by voliane of business) that

handle frozen orange Juice, (5) month within the season (this would be a **0-l

variable" such that an additive factor for each month representing a composite of

those factors normally associated with seasonal variation would be determined).

Each equation would contain between l6 and 19 variables, the number depending

on the number of months In the winter or summer season.

A problem In connection with such analyses would be f^.idlng the prices that

confronted consumers for those Items they did not buy In any given month. Prices

that confronted nonbuylng families could be estimated from the prices paid by

buying families. As an alternative, data on retail prices from the Bureau of

Labor Statistics might be used for all families.

The major purpose of these studies is to estimate elasticities and crosselasticities of demand for a complex group of competing items. In this instance,

the problem is complicated by the fact that certain of the competing items have

been available for only a few years and that availability of and demand for

these items has shifted upward and probably is still doing so. Thus conventional time-series analyses are inapplicable. Whether the proposed approach

will provide a feasible solution is yet to be determined.

I3EMAND ANALYSES BASED ON SALES FROM INDIVIDUAL STORES

Experiments in retail stores have been used for many years to learn the

effect on sales of various innovations, such as prepackaging or packaging in

various sizes or various types of containers, or to determine relativo sales

of varying qualities or degrees of ripeness of specified commodities. But

only a few such studies have tried to ascertain the effect of changes in prices

on sales. Studies along this line that have come to our attention are noted

below.

Studies Relating to Specified Fruits

Apples

In a study by Cravens (8, £). 19 retail food stores in Detroit were visited

or each of 50 shopping days in the winter of 1950-51- Data were obtained on

sales, prices, and display space for apples and other fruits. Grade ins-oections

were made of the apples. Prices of apples and the size of the apnle display

were the chief factors affecting day-to-day changes In the volume^of sales of

apples relative to sales of all fruit. A 1-percent increase in the relative

price of apples was associated with a 1-percent decrease in the relative

quantity of apples sold. The appearance of the apples also affected sales.
- 71 -

The effect of price on sales was greater in médium-sized and large stores than

in small stores.

Oranges

In a study by Godwin {2k) sales of oranges associated with positive and

negative deviations in price of 5, 10, and Ip cents a dozen from the established

market price were analyzed. Thus 7 levels were tested. To permit the use of a

latin square, the "test was run in 7 large retail stores in a city in Kentucky.

By analysis of variance, the effect of price on sales was measured after allowing

for differences in the purchase patterns of customers among stores and seasonal

changes in purchase retes. Weekly data were taken over a 7-week period in

May-June 1952. Quality was held constant. Elasticity ranged between 0.7 and

1.5, with an average elasticity of I.I6, although a logarithmic (or constant

elasticity) curve apparently would have fitted the data equally well. A coefficient of determination between sales and price of 0.9U was obtained.

Fresh Fruits

A pilot study covering 2 stores in Denver, Colo, was conducted in the summer

of 1952 by the Technical Committee of the Western Regional Deciduous Fruit

Project. Data were collected by days during a 10-week period on volume of sales,

price, extent of display space, color, ripeness, condition, size, and method of

display for each fresh fruit (excluding watermelons) sold at any time during the

period. Daily sales of each item were divided by the value of total sales of

all fruits to eliminate factors associated with overall store traffic. The

latter 6 items were rated on a scale ranging from 1 to 9^ with most ratings

falling in the 3""7 range. Many color pictures were taken to ascertain whether

a uniform rating system could be developed. A major aim of this study was to

find the effect of individual factors of quality on sales, but it is realized

that price effects should be allowed for before determining these relationships.

A measure of cross-elasticities between the various fruits also is desired.

It is doubtful whether all of these desires can be met even with a substantial

Increase in the number of stores Included in the study. However, the analyses

given below indicate that some useful information probably could be derived from

studies of this type.

Comments Be^ardin^^ Specific Analyses

Three Items from the Denver study were analyzed on the assumption that they

would illustrate certain basic principles. In view of the limited amount of

data available from the pretest, the results should be considered only as

illustrative of the kind of Information that may be obtained from such studies

and the types of analyses that might be desirable.

A check was first made to deterailne whether weekend sales for specific

items differed significantly in their relation to sales of all fruit from such

sales on weekdays. If, for example, consumers tended to buy melons or grapes,

say, chiefly on weekends, significant differences would show up even after

dividing dally sales of the item by sales of all fruit. Weekend averages of

the "quantity divided by value" series tended to be higher than weekday averages

for both cantaloups and Thompson seedless grapes, but the differences were not
- 72 -

statistically significant when tested by analysis of variance. Graphic analyses

for both fruits Indicated that little was gained by analyzing these periods

separately. In the regression analyses discussed below, no differentiation was

made between weekdays and weekends*

In an attempt to determine which factors of quality appeared to be most

Inçortant, scatter diagrams were made between the "quantity divided by value"

series and price for cantaloups and Thompson seedless grapes. No single factor

appeared to be consistently associated with positive or negative deviations

from the apparent regression line on these charts. Hence, In the mathematical

analyses discussed below, the choice of variables was made largely on a Judgment basis. Certain variables were eliminated because they showed practically

no variation oyer the period for which data were available.

In each of the analyses discussed below, the dependent variable (X^) Is

the quantity of the Item sold divided by the value of sales of all fruits,

nxxmbers In parentheses are the standard errors of the respective regression

coefficients, and r^refers to the highest order partial correlation between the

dependent vetrlable and the 1 th Independent variable. A double asterisk Indicates that these coefficients differ significantly from zero at the 5-pôrcent

probability level and have the correct sign; a single asterisk Indicates that

they do not differ significantly from zero at the 5*pörcent probability level

but have the correct sign. In each case, reasons for the choice of the particular

variables and the Implications of the analysis are given Immediately following

the presentation of the results. No attempt was made to check these analyses

to learn whether curvilinear relationships other than those used would give

Inçroved results. It Is possible that Improvements would have been Indicated

by such a check.

1. Cantaloups - data for 59 days for each of the 2 stores, analysis

based on logarithms.

X - price of cantaloups, X - ripeness of cantaloups, I - maximum

dally temperature In Denver,

X = 2.k -0,68 X^ -0.26 X^ -0,23 X^ (k9)

^ (0.12) ^ (0,17)2 (O.U3)3

** r^ = -O.Ul^, «Tg = -O.lU, r^ = -0.05, **r = O.U9,

For cantaloups It was believed that ripeness might be the main quality factor.

Temperature was Included on the assumption that melon and Ice cream Is a

tempting dessert on extremely hot days.

The analysis yielded the wrong sign on temperature. Price appears to be

the dominant factor of the variables Included but ripeness also was of some

Importance.

2, Cantaloupes - average data for 10 weeks for each of the 2 stores,

analysis based on logarithms.
- 73 -

X^ - prie© of cantaloups, Xq • ripeness of cantaloups,

Xo = 2.1 -0.70 Xi +0.05 X2 Í50)

(O.lU) (OM)

**r^ = -0.76, r^ = 0.03, ••HfR = 0.77.

This was run primarily to compare results from an analysis "based on weekly

averages with one "based on daily data. The percentage of variation in sales

explained \>y price and hy all factors was considerably higher for the analysis

hased on weekly data, reflecting the tendency of random influences to average

out over the longer period. The regression coefficient on price was almost

identical for the two analyses. Effects of quality factors (ripeness) were

negligible in the weekly analysis, however, reflecting the small variability in

this factor when weekly averages were used.

3. Cantaloups - average data for 10 weeks for each of the 2 stores,

analysis "based on logeurithms.

X]^ - price of cantaloups, Xg - sales of cantaloups divided "by

value of all fruit sales in the preceding week.

Xn = 2.1 -0.70 X. +0.08 Xp (51)

(0.16) (0.18)

**r^ = -0.7U, rg = 0.11, ^m = 0.82.

This was run to learn whether the magnitude of sales in the preceding week had

a significant effect on sales in the current week. Ripeness was omitted, as

it had a nonsignificant effect in the preceding analysis. Sales in the preceding week had a negligible effect and the regression coefficient had the

wrong sign. This factor could be expected to be more important for certain

other fruits.

U. Thompson seedless ^apes - data for 59 days for one store, analysis

based on actual data.

Xn - price of Thompson seedless grapes, Xg - quantity of other

grapes sold divided by value of all fruit sales (zero values

Included), X3 • average of ratings for ripeness, color, condition

and size for Thompson seedless grapes.

Xo = 0.99 - 0.02 Xi - O.2I1 X2 - 0.001 X3 (52)

(0.006) (0.20) (0.01)

* rj^ = - 0.33, r2 = -O.15, ^rg = - 0.02, *^ = 0.33

Elasticity of demand at the means equals - O.55.

This is the first of two analyses designed to ascertain whether the effects

on sales of individual competing commodities were measurable. Prices of
- 7U.

Thonçson seedless and other grapes were almost Identical over the period, so

that the price for "other" grapes could not te used In the analysis. Quantity

of other grapes (divided by value of sales of all fruit) was used Instead.

Zero values were Included. As the logarithm of zero Is not specified, the

analysis was run In terms of arithmetic data. Use of such data In any case gives

a more readily Interpretable regression coefficient for Xg. The analysis Indicates that sales of 1 pound of other grapes tended on the average to reduce

sales of Thompson seedless grapes by one-fourth of a pound. Price of Thompson

seedless grapes was the chief factor affecting sales, quantity of other grapes

sold had some effect, and the effect of the Indicated variations of quality

of Thompson seedless grapes was negligible. As no one factor of quality was

believed to be of major Importance, an average of ripeness, color, condition,

and size was used.

5. Apricots - data for 28 days for one store, analysis based on

logarithms.

X-y - price of apricots, Xg - price of Elberta peaches, Xo - average

of ratings for ripeness, color, condition, and size for apricots.

x^ = 1.85 - 0.31 X, + 0.90 Xp-1.00 X^ (53)

0 (0.72) 1 (2.05) 2(0.57) 3

*ry^ = -0.09, ^2 = °-09, «r3 = - O.33, R = 0.U5.

This was designed to find whether Elberta peaches are a major competitor of

apricots. Prices of the two fruits were not higihly correlated over this period,

so that each could be used In the analysis. As In the case of grapes, an

average of ripeness, color, condition, and size was Included for apricots. An

average of these showed considerable variation for apricots (partly because of

the high correlation between the several factors of quality), but It would have

shown little variation for Elberta peaches. The analysis Indicates that a 1-

percent change In the price of peaches had about 3 times as much effect on sales

of apricots as did a 1-percent change In the price of apricots although the

standard errors of these coefficients are sufficiently large so that these

differences would not be statistically significant. All of the signs were

correct in this analysis but, partly because of the small number of observations

available, none of the coefficients differed significantly from sere. Of the

three factors, quality appeared to be most Important.

Comments Regarding Experimental Design 16/

Two criticisms of the experimental approach to the measurement of consumer

demand, similar to that used by Godwin, should receive attention Inasmuch as

they have been serious deterrents to Its use. A properly designed experiment

measures the Importance of each.

16/ This section Is based on a talk by Glenn L. Burrows, Mathematical

Statistician, Agricultural Marketing Service.
-75 -

First, with only a sample of stores cooperating In an area, there Is no

assurance that, with increased prices in test stores, consumers will not shift

their purchases to other stores in the same area. The validity of the pricequantity relationship suffers in direct proportion to the extent to which a shift

takes place, and the possibility of such shifts places practical limits upon

the range of price differentials for which valid results can he obtained. But

fortunately with increasing price differentials, any such shift would manifest

Itself in a decline in the nuniber of "buyers, a result that can be verified and

tested. Data presented by Godwin {2k) on the number of customers associated with

each price differential indicated that this was not a serious problem in his

study. An additional check on this point could be made by maintaining a simple

count, probably during only a sample of business hours, of purchasers of the

item per 100 customers in neighboring nontest stores. This problem does not

appear to be as serious as many fear, although conclusive evidence will have to

await further experimentation along these lines.

The second criticism is harder to deal with. It arises from the sluggish

response of purchases of consumers to changes in prices. If the test period is

too long, some of the advantages of this technique are lost because of adjustments in incomes, prices of competing commodities, and similar factors during

the test period. This problem has been at least partly solved with the Innovation of experimental designs incorporating provisions for the measurement of

carryover effects. This approach was used by Henderson (22) in a retail store

study but, unfortunately, the explanation of the carryover feature is Inadequate.

The method was described in detail by Cochran, Autrey, and Cannon (6), although

their explanation is not always easy to follow.

Even in cases in which the carryover effects are not Important, the

changeover, or rotational-type, experiments, of which the Godwin analysis Is an

example, yield more precise measures of the effects being studied. Store-tostore variability does not enter into experimental error as it would with the

usual compao^tlve type of experiment.

The following quotation from Kempthome (¿2), p, 7) indicates the area

within which designed experiments may be most useful, "The real distinction

between two of the applications of statistics, the design of experiments and

sample surveys /pf which analyses based on time series, in a sense, form a

part7> is that, in the design of experiments applied to a problem, the populations that are studied are formed by the experimenter in a specified way,

whereas, in a sample survey dealing with the same problem, the population under

study has arisen from a set of forces, the relation of which to the forces \inder

consideration is unknown.. ./^A surve£/can demonstrate the existence of associations between characteristics in the population, but...the existence of an

association between attributes X and Y in the population in no way suggest that

attribute X can be altered to a""specified value by altering attribute Y in a

particular way. In an experiment we determine whether altering attribute X has an

effect on attribute Y, and this is the knowledge that is necessary for any""actlon

program,. .Survey work can be very useful in cases in which a deductively

formulated theory exists and it is desired to estimate some parameters in the

theory. It is,however, difficult to visualize how a theory can be started without some experimentation on which to base the original abstractions."
- 76 -

Analysis of demand is concerned to a considerable degree with the estimation of parameters, such as coefficients of elasticity. But our knowledge of

behavior patterns for individual consumers is sufficiently limited that much

probably can be learned by the further use of designed experiments in retail

stores.

LITERATURE CITED

(1) Anonymous

1953• Operations Research in Distributing Empty Cars. Railway Age.

131^:73-7*^-

(2) Armore, Sidney J.

1953. The Demand and Price Structure for Food Fats and Oils. U. S.

Dept. Agr. Tech. Bui. IO68, 69 pp., illus.

(3) } and Burtis, Edgar L.

1950. Factors Affecting Consumption of Fats and Oils Other than Butter,

in the United States, U. S. Bur. Agr. Econ., Agrie. Econ.

Research. 2:1-9»

(U) Atkinson, L. Jay

1950. The Demand for Consumers* Durable Goods. U» S. Dept. Commerce,

Survey of Current Business. 30:5-10, illus.

(5) Chames, A., Cooper, W . W., and Henderson, A.

1953. An Introduction to Linear Programming. 7*^ pp., illus. New York.

(6) Cochran, W. E., Autrey, K. M., and Cannon, C. Y.

19*^1. A Double Change-over Design for Dairy Cattle Feeding Experiments.

Jour. Dairy Sei. 2U:937-951.

(7) Cochran, D., and Orcutt, G. H.

19*^9. Application of Least-square Regression to Relationships Containing

Autocorrelated Error Terms. Amer. Statis. Assoc. Jour. Uli^:32-6l

illus.

(8) Cravins, M. E., Jr.

1952* Selling Michigan Apples. Mich. Agr. Expt. Sta. Special Bui. 382,

30 pp., illus.

(9)

1952. Studies in Midwest Apple Marketing, I. Retail Merchandising of

Apples. Mich. Agr. Expt. Sta. Special Bui. 378, 75 pp., illus.

(10) Ezekiel, Mordacai

1933. Some Considerations on the Analysis of Prices of Competing or

Substitute Commodities. Econometrica. I:172-l80, illus.

1938. The Cobweb Theorem. Quart. Jour. Econ. 52:255-280-, illus.

(12) __^

19*^1. Methods of Correlation Analysis. Ed. 2., 531 pp., illus. New York.

(13) Ferber, Robert

19^9. Statistical Techniques in Market Research. 5U2 pp., illus. Hew York.

{Ik) Foote, Richard J.

1952. Calculation of Partial and Multiple Regression and Correlation

Coefficients - 3 to 5 Variables. U. S. Bur. Agr. Econ. I5 pp.

(processed.)

(15) ,

1953. Statistical Analyses Relating to the Feed-live stock Economy. U. S.

Dept. Agr. Tech. Bui. IO7O, kl pp., illus.
- 77 -

(16)

1953* A Four-equation Model of the Feed-livestock Economy and Its

Endogenous Mechanism. Jour. Farm Econ. 3^:kk-6l, illus.

(17)

1953. A Four-equation Model of the Feed-livestock Economy and Its

Endogenous Mechanism - A Correction. Jour. Farm Econ.,

35:613-615.

(18) Fox, Karl A.

1951. Factors Affecting Farm Income, Farm Prices, and Food Consumption,

U. S. Bur. Agr. Econ., Agr. Econ. Eesearch. 3:65-82.

(19)

(20)

(21)

(22)

1951. The Measurement of Price Support Cost. Jour. Farm Econ. 33:U70-lf8U.

1953. The Analysis of Demand for Farm Products. U. S. Dept. Agr. Tech.

Bui. 1081, 90 pp., illus.

1953. A Spatial-equilihrium Model of the Livestock-feed Economy in the

United States^ Econometrica. 21:5^7-566.

195^« Structural Analysis and the Measurement of Demand for Farm

Products. Review Econ. and Statis.,v. 36 (in press.)

(23) Girshick, Meyer A., and Haavelmo, Trygve

I9U7. A Statistical Analysis of the Demand for Food: Exançles of

Simultaneous Estimation of Structural Equations. Econometrica.

15:79-111-

{2k) Godwin, Marshall E.

1952. Customer Response to Varying Prices for Florida Oranges. Fla.

Agr. Expt. Sta. Bui. 508, 2k pp., illus.

(25) Haavelmo, Trygve

I9U3. The Statistical Implications of a System of Simultaneous

Equations. Econometrica. 11:1-12.

(26)

igkk. The Probability Approach in Econometrics. Econometrica, v. 12,

Supplement, II8 pp., illus.

(27) Heady, Earl 0., and Olson, Russell 0.

1952. Substitution Relationships, Resource Requirements and Income

Variability in the Utilization of Forage Crops. Iowa Agr.

Expt. Sta. Research Bui. 3901 pp. 861^-938, illus.

(28) , Woodworth, R., Catron, 0., and Ashton, G.

1953. Productivity and Substitution Coefficients in Pork Output.

Jour. Farm Econ. 35:3^1-35^> illus.

(29) Henderson, P. L.

1952. Application of the Double Change-over Design to Measure Carryover

Effects of Treatments in Controlled Experiments. Cornell

University, Methods of Research in Marketing Paper 3, 86 pp.

(Processed.)

(30) Henry, W. A., and Morrison, F. B.

I9U8. Feeds and Feeding, A Handbook for the Student and Stockman,

rewritten by Frank B. Morrison, Ed. 21, unabridged, 1,30? pp.,

illus. Ithaca, N. Y.
-73 .

Hermle, Albert M.

1951 • Prices of Apparel Wool. U. S. Dept* Agr. Tech. Bui. lOM, ko pp.,

lllue.

Kempthome, Oscar

1952. The Design and Analysis of Experiments, 63I pp., Illus. New York.

Koopmans, TJalllng

191^5. Statistical Estimation of Simultaneous Economic Relationships.

Amer. Stat. Assoc. Jour. kO:kk8'k66,

Kuznets, George M., and Klein, Lawrence R.

I9U3. A Statistical Analysis of the Domestic Demand for Lemons, I92I-I1I.

Giannlni Found. Agr. Econ. Rept. 8U, 112 pp., illujs. (Processed.)

Marschak, Jacol)

1953. Economic Measurements for Policy and Prediction. In Hood, Vm. C,

and Koopmans, TJalllng C, ed.. Studies in Econometric Method,

Cowles Commission for Research in Economics Monogr. lU, 32*^ pp.,

illus.

Melnken, Kenneth W.

1953. The Demand and Price Structure for Oats, Barley and Sorghum ,

Grains. U. S. Dept. Agr. Tech. Bui. 1C80, IO3 pp., illus.

Morrissett, Irving

1953. Some Recent Uses of Elasticities of Substitution - a Survey.

Econometrica. 21:Ul-62, illus.

Ogren, Kenneth E.

1951. Analysis of Household Purchases of Citrus Products by 5OO Urban

Families, November 19U8-October 19^9^ U. S. Bur. Agr. Econ.

51 pp., illus. (Processed.)

Pearson, Frank A., and Vial, Edmond E.

I9U6. Prices of Dairy Products and Other Livestock Products. I5U pp.,

illus. Ithaca, N. Y.

Peters, Charles C, and Van Voorhis, Walter R.

I9U0. Statistical Procedures and Their Mathematical Bases. 516 pp.,

illus. New York and London.

Prest, A. R.

19*^9. Some Experiments in Demand Analysis. Review Econ. and Statis.

31:33-1^9 •

Schultz, Henry

1938. The Theory and Measurement of Demand. 817 pp., illus. Chicago.

Snedecor, George W.

19^0. Statistical Methods. Ed. 3. U22pp., illus. Ames, Iowa.

Stone, Richard

19*^5. The Analysis of Market Demand. Royal Statis. Soc. Jour.

108:286-382, illus.

Thomsen, Frederick Lundy, and Foote, Richard Jay

1952. Agricultural Prices. Ed. 2. 509 pp., illus. New York.

Tinbergen, J.

1951. Econometrics. Transi, from the Dutch by H. Ryken Van 01st.

258 pp., illus. Philadelphia.

(kj) United States Agricultural Marketing Service.

19*^9-53. Consumer Purchases of Fruits and Juices, Washington, D* C.

(Processed.)
- 79 -

(»^8)

19^9-53 • Consumer Purchases of Fruits and Juices by Regions and Retail

Outlets. Washington, D. C. (Processed,)

{k9) Waugh, Frederick V.

1951. The Minimum-cost Dairy Feed. Jour. Farm Econ. 33:299-310, illus*

(50) f and Been, Richard 0.

1939. Some Observations About the Validity of Multiple Regressions.

Statis. Jour. 2:6-lU, illus.

(51) Wells, C. F.

19*^1. The Incidence of Tariffs on Competing Products. U* S. Bur. Agr.

Econ., Stat. and Agr. Ser. 2, 28 pp., illus. (Processed.)

(52) Yule, G. Udney, and Kendall, M. G.

1937. An Introduction to the Theory of Statistics. Ed. 11. 570 pp.,

illus. London.

APPENDIX

Note 1. Proof That if X Equals Y, Plus Yg and Simple Regressions Are

Run Between X and Yn and Yg } Respectively, the Sum of the Slopes

Will Equal 1 and the Sum of the Constant Values Will Equal 0

1. The following two simple regression equations are fitted:

Y^ = a^ + h^ X (5*^)

Yg = ag + hg X. (55)

Let the small letters, x, ^ and jg represent deviations of X, Y^, and

Yp from their respective means. From "basic regression formulas,

\x- ^ (56)

b = Í2 (57)

Also, the regression of x on Itself is

b = ^^ = 1 . (58)

IX 2

2^

Since X = y^^ + 72 , equation (58) can be rewritten as

bxx=---- - --=^ --j

Ex^ rx z^"
- 8o .

But "by equations (56) and (57) > the two right-hand terms are equal to

b-, X and "b^ x respectively. Hence,

^xx= Vix -^^ygx = 1- (5Ô.2)

The same demonstration can "be extended to cases in which x is the sum of

any number of variables.

2. In equation (58.1) above, it was assumed without proof that x = y,+ yp.

This follows from the fact the X = Y, + Yg for every observation. If the

observations are summed and then divided by their number to obtain arithmetic

means,

(59)

(60)

in which the bars denote arithmetic means. For any given observation

X = Yj^ + Y2. X may be subtracted from the left-hand term and (Y^ + Yp^ ^^^^ *^®

right-hand term without destroying the equality. Hence,

X - X = (Yi - Yj) + (Y2 - Y2) (61)

which, by definition, is (62)

X = y^ -^ 72-

3. From basic regression formulas,

^ix = Yl - VX (63) ,

EX = ^1 + ^Y2

n

or

n n

X = Yi + Yg,

a

a.„, = X - b^ X = 0, as t^ = 1 . (65)

Adding equations (63) and {6k) gives

But this equation also may be written as
. 81 .

Hence, the sume of the a values is equal to zero. This demonstration also

can te extended to any number of subdivisions of x.

Note 2. Demand Relationships in Utility Theory

A theoretical criterion for determining the nature of the demand relationship between two comnodities is given by è^ V ^ o, where Ü, total utility,

axj^àxg ^

is an increasing function of the quantities consumed (x and Xp) of each of the

two commodities. This statement implies that ^ U N,0 ( and also that Q^ U yO).

cl^l 0/X2

The expression d ^ shows the effect upon the rate of increase in utility per

unit of x^ of consuming additional units of X2. If an additional unit of Xg reduces

the per unit contribution of x, to consumer satisfaction, the two commodities

are competing, and o U • 0 ^ If an added unit of x^ has no effect on the per

unit contribution of Xn to consumer satisfaction, the commodities are independent

in demand, and <^ n = 0. Finally, if an added unit of Xg increases the

^ i 2

per unit contribution of x-,, the two commodities are completing, and ^ U yn

This criterion is in terms of the utility surface of an individual consumer

and is not subject to statistical measurement without severe restrictions.

Footnote 6, page 13, said that, according to utility theory, the

coefficients b ^ and b^. of equations (3) and (U) respectively should be equal

in all cases if consumers are rational. This result follows from the basic

criterion stated above. If utility is assumed to be measurable, we have

^ U = p and j U = Vo* As the order of differentiation is immaterial, we

have also that è ^ U = <j jj . But this is equivalent ( on the

measurability assumption) to <J 1 = ^2 . The left-hand term of this equality

<)X2 ^xi

is equivalent to b^^g and the right-hand term to be b^^. This is "Hotelling's

condition," presented by Schultz (U2).

Hote 3. Some Demand Interrelationships in

the Meat and Poultry Group 17/

The notation is symbolic. For example: 'LJ'^ stands for the price of all

meat at retail; ^ is quantity (per capita consuSption except in equations (6)

IJJ All variables expressed as first differences of logarithms of annual

observations, 1922-Ul, except for turkeys for which the period is 1929-**^1*
- 32 -

and (6w)); and ;¿ is disposable personal Income per capita. The subscripts are

m = all meat, £ = pork, b = beef, 1 = lamb, ç = chicken, and t = turkey. The

subscript pvl stands for pork, veal and lamb, and pbv stands for pork, beef, and

veal. Nuinbers In parentheses are standard errors of the regression coefficients.

Meats

P^^"*^ » -1.50 q« + 0.87 y (67)

"^ (0.08)^ (0.03)

E^ = 0.98

P„^^^ =-0.02 q^ , -1.16 q^+ O.9O y (68)

(0.13)^''^ (0.07) P (0.06)

R^ = 0.97

p.^ (^) = -1.08 q^ + 0.88 y -0.U2 q^^^ (69)

(0.11) (0.06) (0.07)

R^ = 0.95

p (^^ = -0.01 -0.50 q, + 0.78 y -0.65 q,^,^ Í70)

(O.lU) 1 (0.06) (O.lU)

E^ = O.9U

Chickens

vj""^ = -0.75 q. + 0.76 y -O.U2 q^ (71)

(0.18) (0.09) (0.16)

E^ = 0.86

P^(r) = -0.01 -0.87 q^ + 0.82 y -O.U3 q^ (71.1)

E = 1.00 (Forced result, assuming perfect correlation

of random errors in all variables.)

Turkeys

P^i^) = 0.02 - 1.21 q^ + 1.06 y - 0.97 q (72)

(0.25) (0.20) (O.lrô) °

E^ = 0.90

p^^^^ = 0.02 - 1.09 g^ + i.n y - 0.96 q^ - 0.23 Q, (72.1)

(0.19) (0.11) (0.28) ° (0.11)

E =0.97 Special analysis allowing for effects of estimated

errors In all variables^ but without forcing

perfect correlation.)
- 83-

Commente

According to these equations, retail prices of "beef and lamb were sul)stantially affected by supplies of other meats—particularly pork. But supplies of

"beef, veal, and lamb had no statistically significant influence on the retail

price of pork.

The retail price of chickens vas significantly affected by the supply of

red meats as well as by the supply of chicken.

No retail price series is available in the case of ttirkeys. The unadjusted

analysis (equation (72)) suggests that prices of turkeys are influenced by the

supply of chicken. Supplies of pork in October-December may also influence

turkey prices but this is not so clearly established. (See equation (72.1).)

Supplies of poultry probably have some reverse effect upon prices of meat

but no attençt was made to measure it. This effect probably cannot be measured

from historical data, mainly because the absolute Importance of poultry meat,

in terms of pounds per capita, was only a fifth or a sixth that of meat. The

effect of supplies of poultry upon prices of meat is not likely to stand out

over the effects of errors in the data and variables omitted from the analysis.

Nevertheless, it is clear that poultry meat can and does serve to some extent

as a substitute for red meat when supplies of the latter are short.

Note U. An Approach To The Measurement of Competition

Between Apples From Different Producing Areas 18/

The variables are season average farm prices (2.), total production {g) and

total disposable Income (2;) •

p^ = price in Washington State

q^ = production in 11 western States

p = price in Hew York State

q = production in all other than the 11 western States.

Price and production variables do not refer to strictly comparable areas,

so the equations should be taúcen as Illustrative only.

Pt

(^) = -0.02 - 0.79 q^ -^l.OUy (73)

(O.OU) (0.12)

E^ = 0.96

18/ All variables are in terms of first differences of logarithms of crop-year

data, 1921-Ul.
- &k -

Pw^^^ = - 0-03 • 1-19 q^ + l.l6y (7^^)

(0.31) (OM)

E^ = 0.55

p^(f ) ^ . 0.02 - O.SIfq^ - o.38q^ ^ l-33y (T^^-l)

(0.20) (0.07) (0.30)

I? = 0.83

vj^^ = -0.01 . 0.68n + 1.23y (T5)

(0.08) (0.33)

E^ = 0.81

P^^^^ = - 0.02 - 0.21q^ - 0.65q^ + l-23y (75.1)

(0.2lf) (0.09) (0.36)

R^ = 0.82

Equation (73) implies a slightly elastic demand for apples during the

1921-Ul period.

Equations (7U) and (7^.1) indicate that the price of apples in Washington

State is strongly influenced "by the production of apples east of the Rocky

Mountains as well as in Washington and other western States. Production in the

east averaged larger than production in the west. If apples from the two areas

were perfect substitutes "bushel for bushel, q^ (in the logarithmic form of

equation (7•^.1))should have a larger coefficient than q^. The smaller coefficient obtained reflects significant differences in market demand.

Equations (75) and (75*1) imply that production of western apples had little

effect on the price of apples in New York State. This may be due in considerable

part to the fact that western production, much of which is on irrigated land,

was more stable than eastern production. The variance of q during I921-UI was

almost 8 times as large as that of 0. *^

A more refined analysis would consider (1) a larger number of producing areas,

with price and production variables properly associated; (2) the seasonal pattern

of marketings from each area; (3) competition between varieties in specific

markets; and (k) the fact that relative prices even of identical commodities in

widely separated areas would be Influenced by changes in relative production in

the two areas even though prices in consuming centers wore the same.

Note 5. SujG^^estions for Exploratory Analyses of Competition

between Grades of Beef Cattle and between

Grades and Cuts of Beef

Correlations between retail prices of different cuts of beef for 1922-Ul

appear to indicate significantly different elasticities of demand. Some simple

approaches to determine the extent to which this is true follow:
- 85-

1. Assume the same percentage variation from year to year in quantities

sold of two retail cuts of beef, that is, that they represent constant percentages of the carcass. Calculate the standard deviation of logarithmic nrice

changes in each cut.

2. As a first approximation, the elasticity of demand for each cut may he

written as

^l^$i (76)

1

/vi

<rp

where the q's and p's are the quantity sold and price at retail, respectively.

Even if ^ is unknown, the ratio between "^ and'7 could he estimated from

^ 1 2

the relationship:

'%-fl. (T8,

3* In the case of round steak and rib roast, the following regression

equation (based on logarithmic first differences) is obtained:

4ll roast = ^ |J:Jg)4oL steak. (^9)

E^ = 0.988.

As h differs significantOjr from 1, it appears likely that the elasticities

of demand also differ significantly.

h. However, the different price "behavior of the two cuts may he partly

due to different income elasticities, or to different competitive relationships

with pork, veal, and lamb. An equation explaining the average retail price

of all cuts of beef is as follows:

P^(r) = a -1.06 q^ + 0.88y -O.52 1^^^^ (Ö0)

The price of each individual cut of beef might be expressed as a function

of the same three independent variables and a comparison made of the various

coefficients as evidences of differential price flexibilities,

5, Conçetition between grades of beef (wholesale carcass prices) and

grades of cattle could be approached statistically by various means• One
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difi'lculty is lack of adequate data for the total supply of slaughter cattle

and "beef "by grade s

One approach is simply to compare relative prices of different grades of

(say) teef steers at Chicago with relative numbers slaughtered. A more formal

approach would "be to express the price of each grade as a function of slaughter

of the same and other grades and of a general demand factor such as disposable

Income. A preliminary inspection of data at Chicago on prices and numbers of

steers by grades suggests that the degree of competition between grades is limited

on a year-to-year basis. This would be consistent with a fairly rigid

stratification of the consumer market on a grade basis. Leading hotels and

restaurants may use Prlïïîe grade as a matter of policy; some major retail chain

stores are said to sell only Choice grade beef ; and so on. Some price competition doubtless occurs between adjacent grades. But if sufficiently accurate data

on quantities sold were available, even the adjacent grades might appear to be

imperfect substitutes and competition between (say) Prime and Medium grades

might be negligible.

(6) The available data may not be adequate to give solidity to a complete

demand structure for beef differentiated by grades and cuts, although they might

provide some useful Insights. To the extent that price analyses based on time

series were differentiated by cuts and grades, comparisons could be made with

the family budget data for individual cuts. Also, Inferences might be made as

to the demand for Individual grades on the basis of differences in average prices

paid for given cuts by different family Income groups.

U. S. GOVERNMENT PRINTING OFFICE : O—1954