Proposals to Expand the Employment of the Coefficient Eta and the Methods of Its Determination and Significance

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310+1 TO K L r'M W TEB E M rh O J ^ M T OF TdE COEFFICIENT ETA A m T m LETHODS OF IIS DETERiAXlMTlQtf AND SIGNIFICANCE

by Koxoaa H m rvey fULngatram Ir.

A thesis submitted In partial fulfillment of the requirements of the degree of Doetor of Philosophy, in the Department of Corneree in the Graduate College of the State University of Iowa August, 1050

ProQ uest N u m b e r: 10902190

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Cofe, 8. '

TABUS OF COHTJaiTS

Chapter

Page

I

Introduction* •,•«•*••,•..•••••••••••.........

1

XX

A Consideration of* Symbols.......

4

.....

XXX An Assumption Qonoerning Mills• Cycle Data* ••.... X?

V VI VII

7

Determination of The Regression Lin© and fhe Related Elasticities by The Direot Least Squares Method* *.............................. *........ Determination of the Regression Lin© and the Related Elasticities by The Method......

9 SO

Determination of the Regression Line and theRelated Elasticities by the "b— jr* Method* ........ .. 25 aqr A Substitute Method of Checking the Elasticities found In the First Three Methods*.... 31

VIII Replacing The Basic Data Figures With Their Logarithms *. ................... IX X XI XII

37'

A Demand Curve, and the Determination of Its .... Elasticities

45

A Curvilinear Regression and Pertinent Trend Point Elasticities* ....

57

Elasticities in Manufacturing - Product, Capital and Labor.

71

.......

Elasticity of Farm'Product Prices and Prioes Paid by the Farmer ....

83

XIII Elasticity of Crime and Unemployment,....... . •. •.

95

XIV

Elasticities of Demand for Electrical Energy.«••.. 108

Appendix X Elasticity * A Hypothetical Analogy..........

131

Appendix II Footnotes.

136

Bibliography.

.... ................ ,.•*•« .... .......

137

ill table

or

tables

Table

Page

1.4

The Corrected Supply Curve Figures for An Average Business Cycle.•...............................13

0*4

Determination of the Regression H a © by the Direct least Squares Method; and the Elasticities ♦.......15

3.5

Determination of the Regression lines. Its Elasticity and the Tread Point Elasticities ......................... .21 Using the wbv_rH Method.

4.5

Determination of the Regression, Its Elasticity and the Trend Point Elasticities Using the wbxyrw M e t h o d .

5.? 5.7 7.7 8.7 0.8

.

Derivation of the Equation for Elasticity Used In the Substitute Method. ......

.37 .50

The Substitute Method and the Eta Trend Point Elasticities. .......................

....54

The Substitute Method and the Mbv.„w Trend Point E l a s t i c i t i e s .

.........35

The Substitute Method and the Point Elasticities. .....

Trend

The "Corrected” Figures for the Supply Curve Using the Logs of the Basie Data. ....

10.8 The Three Regression Lines for a Supply Curve Using the Logs of the Basic Data...............

36 .38 .....39

11.9 The Link Relatives for Wheat Production Figures; 1893 to 1910, Inclusive. ..... .......46 15.9 The Logs of the Link Relative of the Wheat Production Figures....................................47 15.9 The Deflated Price Figures for the Wheat Market, From Average Market Prices for the Tears 1898-1910... ,49 14.9 The Link Relatives and Their Logs for the Deflated Wheat Prices ..............

.

.50

iv 15*9

The "Gorrected" Figures for the Wheat Market; Production and Price...... ..... .........

16.9

The Three RegressIon Lines for a Demand Curve Using the Logs of the Link Relatives.................53

17.10 Summation Figures for Mills* Cycle Data..............59 18.10 Corrected Figures for a Parabola Mills * Cycle Data............................ *.......60 19.10 Corrected Figures for a Parabola Mills* Cycle Data..........

.......61

00.10 Finding Values for the Constants in A Parabolic Regression............................... 63 01*10 Computation of Trend points for the Parabola.......,*64 ££*10 Trend Point Elasticities for the Parabola............ 66 £3*10 Linear and Curvilinear Correlation...................69 £4*10 Reliability of Measures of Correlation Found In the Case of Mills * Cycle Data*.................... 70 £5.11 Indexes of Manufacturing, United States (Adapted from Douglas and Cobb)......*...............73 £6*11 Summation Figures for Capital, Labor and Product....*74 £7.11 Corrected Figures for Capital and Product*..*........75 £8.11 Corrected Figures for Labor and Product*...... .... .76 89*11 Capital and Product Determination of the Three Regression Lines and Elasticities....................77 30.11 Labor and Product Determination of the Three Regression Lines and Elasticities.*«•**••*•••«•***.**78 31*18 Indexes of Farm Prices............................ .85 32.13 Summation Figures for Farm Product Prices and Prices Paid by the Farmer .......

*86

33*18 Summation Figures for Farm Product Prices and Prices Paid by the Farmer* .... ...87

V

34.15

Centered Squares for Farm Product Prices and Prices Paid by the Farmer .................

88

35.15

Centered Cross Products for Farm Product Prices and Prices Paid by the Farmer*

89

36.IE

Values for the Constants of a Tarn Prices Parabola.......................................

90

OomputaLtidh of Trend ‘points for the Farm Prices Regression..............................

91

Computation of Trend Points for the Farm Prices Regression*,*........... ............. .

92

39.13

Farm Prices Trend Point Elasticities*.*,*.**.*,

93

40.13

The Link Relatives for Unemployment Figures, Hew York State* ........ ......

97

37.13 38*13

41*13

The Link Relatives for Admission to Hew York State PrIsons«,.*•*«.•**,.,.*...... •••**98

43.13

The Logs of the Link Relatives of the Unemployment Figures, .....

*99

43*13

The Logs of the Link Relatives of the Admissions to Prison Figures*................__ ...100

44*13

Crime and Unemployment - The "Corrected" Figures*.*101

45.13

Crime and Unemployment - Computing the Coefficient of Correlation** * 1

0

3

46.13

Crime and Unemployment - Determination of Three Regression Lines and Their Elasticities* * .........*103

47.14

The Link Relatives and Their Logs for Residential or Domestic Energy Sales and Average Revenue per Kwhrs. (1927*1936)***... *******....... ......*.112

48*14

The Corrected Figures and Mean Elasticity for Residential or Domestic Demand for Electrical Energy* .......

49*14

Measure of Significance of Mean Coefficient of Elasticity for Residential or Domestic Demand for Electrical Energy. .....

*113

114

Vi 50.14

The Link .Relatives and Their Logs for Residential or Domestic Energy Sales and Average Revenue per Ktorhr. ( 1 9 3 7 - 1 9 4 8 ) , . . , . *,,.„, •., .... *....... 115

51.14

The Corrected Figures and Mean Elasticity for Residential or Domestic Demand for Electrical Energy. * ...................,.,........ ..........115

52„14

Measure,of Significance,of Mean Coefficient of Elasticity for Residential or Domestic veimna for Electrical Energy..............................,117

53.14

The Link Relatives and Their Logs for Small Light and Power Energy Sales and Average Revenue per Ewhr. {1927-1936},..,...,...*.......,......... 118

54*14

The Corrected Figures and Mean Elasticity for Small Light and Power Demand for Electrical Energy {1927-1936) , .......... .. .119

55.14

Measure of Significance of Mean Coefficient of Elasticity for Small Light and Power Demand for Electrical Energy.... ............ ...120

56.14

The Link Relatives and Their Logs for Small Light and Power Energy Sales and Average Revenue per Km hr. {1937-1948),..,.......,....,........ ...,121

57.14

The Corrected Figures and Mean Elasticity for Small Light and Power Demand for Electrical Energy (1937-1948) ........

123

58.14

Measure of Significance of Mean Coefficient of Elasticity for Small Light and Power Demand .... .123 for Electrical Energy.

59.14

The Link Relatives and Their Logs for Large Light and Power Energy Bales and Average Revenue per Kwiar. {1927-1936) .... ^ .124

60.14

The Corrected Figures and Mean Elasticity for Large Light and Power Demand for Electrical Energy {1927-1936).......

61.14

Measure of Significance of Mean Coefficient Elasticity for Large Light and Power Demand for Electrical Energy........

126 of 127

vii ©£*14

The Link Relatives and Their Logs for Large Light and Power Energy Sales and Average Revenue per Kwhr. { 1 9 3 7 - 1 9 4 8 } * . . * * . 1 2 8

©5*14

The Corrected Figures and Mean Elasticity for Large Light and Power Demand for Electrical Energy { 1 9 5 7 - 1 9 4 8 ..... *.......

64,14

Measure of Significance of Bean Coefficient of Elasticity for Large Light and power Demand for Electrical E n e r g y *

129

.*150

vin TABL& OF GHART3 Charts

Page

1*

Regression of Price on quantity of an Average Business Cycle as Determined by the Direct Least Squares Method*•••*•**•*••*••***•••14

£*

Regression of Price on quantity of an Average Business Cycle as Determined by the wbyX” and ttbXyl,,!* Methods* ••••***•*«•*••****.••••£3

3#

Regression of Price on quantity Substituting the Logarithms In The Direct Least Squares Method*«•••*••.»•**••••«.•*•**.** *43

4.

Regression of Price on quantity Substituting the Logarithms in the **by3ew and the wbXyr” Methods**********************.***#44

5.

The Three Regression Lines for a Given B e m u d Curve Doing the Logs of the Link Relatives.««.•.•*•••,•••••.»•••••••••*•*••••••••*.52k

6*

Three Regression Lines of Product on Capital Doing Logarithms.•«••*•.**.«*•.•••,...*.•.•.79

7*

Curvilinear Regression of Prices paid by the Parmer on Farm Produet Pri ees* •..,.*•*••.••**«.• 94

ix

ACKMOVflEiXIiMEM?

The author wishes to express his deep appreciation for the guidance and in­ struction in the preparation of this thesis so generously given to him by Professor George R* Davies*

1 Chapter X .3MTRODOCWOM Elasticity may waXl be considered aImply as the comparison of the relative rate© of change existing between any two given variable©, Traditionally, this has not been the concept, 3Frotm the day© of Alfred Marshall until quite recently, the usage of elasticity has been limited rather largely to the well-known field of demand curve analysis with an occasional extension to supply curve problems.

This is well illustrat­

ed, for example, in the numerical definition of elasticity as offered by Moulding*

"the percentage change In the quan­

tity (demanded or supplied) which would result from a one percent change in prlc©.*^ This definition is sufficiently accurate and suf­ ficiently representative of the thought in the field to serve as an illustration, although Moulding himself apolo­ gise© somewhat for this definition as being open to certain mathematical objections. This thesis has two general purposes* first, to make a plea for a broader interpretation of elasticity so as to include the consideration of any two given variables; second, to indicate, to describe, and, to some degree, to evaluate the various statistical methods that may be used to

deiemiu© the elasticity or elasticities in a given problem# In this connection certain new techniques have been devel­ oped, to Implement or better validate the me® of this ®tailsileal ■tool*' With reference to the first purpose, Moulding*®, 'definition might well be expanded and changed to read a® fol­ lows*

the percentage change in the dependent variable which accompanies a one percent change in the independent variable* Elasticity may well be considered a short step fur­ ther in statistical analysis than correlation, and it may be viewed in the same general application*

Correlation is wide­

ly accepted In almost any situation where the variables are plausibly considered interdependent«

It remains to the indi­

vidual employing the techniques to determine just where such should be used and how the results should be qualified*

Cor­

relation considers the relationship of average interdepend­ ence, or how closely variability in one series follows vari­ ability in another series, with the element of change implied in most economic analyses.

Elasticity merely goes on© step

further and measures the rate of change in one variable ac­ companying the rate of change in another variable. There seems to be no logical reason, therefor©, to keep this statistical concept confined within the narrow

» limit© of demand and supply curve analyses.

The correlation

concept has not bean so limited, and It appear© that In a dynamics economy a comparison of rates of change among a ■vari­ ety of type© of variables, not merely two types of variable©, 1© of increasing concern to the student of business and eco­ nomies • A rather simple analogy I© to be found In Appendix 1 illustrating further this author *s belief in the possibili­ ty of expanding the usage of the concept# furthermore, the various examples in the study have purposely employed data widely different in nature in order to indicate the possible wide rang© of application of elasticity techniques. Ooaeemin these example# in centered more upon the various aspects of the methodology involved in derivingelas­ ticities as well as extending the scope of application, rather than upon the economic significance of the specific finding© in any one illustration.

To expound upon such sig­

nificance is not within the province of this study and fur­ thermore, would extend the limits of the work more than is desirable#

4 Gbapber IX a aa^aiDBmfioH of symbols

Before proceeding to the first of the Illustrations Of elasticity and the methodology involved, some consider®# tion will be given to the subject of the symbol* which have been used and are toeing used In this area. Acthording to a statement by a. G. B. Allen,, there exists no singly accepted or established mathematical symbol which Is in common usage today for the expression of the term elasticity.2

It appear® that the symbol most commonly used

in the traditional type analysis, Is the Greek eta ()}}.

Ini­

tially, it was employed to indicate a relative change in (juantity as applied to demand curve analyses, and it has been logically extended to include supply curve analyse® as well. In his work2 Henry Schultz suggests, however, the employment of the small & Instead of the eta for supply curves, i'rederlck 0. Mills, on the other hand, utilized the £ to re­ place eta only when elasticity Is toeing measured in a tempo# rai framework,

furthermore, he employs an £ to indicate

price flexibility in the ©am© connection; price flexibility being her© in effect the elasticity of price in either a sup­ ply or demand curve situation.

He also refer© to H. L.

Moore*® use of the Greek Phi 1$) for the same general purpose.^ Another divergence in the use of symbols is found

9 in Eeanebb 1* Boulting*s Booaomlo Analysis ♦

Boulding uses

the g tor til© elasticity of .quantity la a given supply curve analysis., and the Q r m k

4 tor 'P i

a demand curve analysis#

the elasticity of quantity in v

Still other symbols may be found

occasionally# Because of the differences in the usage of symbols, a plea must be mad© for the employment of a simplified, uni­ form terminology for tin© indication of elasticity#

It is ap­

parent that the variety of symbols which could be used to ex­ press elasticity might be most numerous, if one wished to bring into the symbolism specific references as to its indi­ vidual employment♦

Since we plead for simplicity and the

avoidance of ambiguity, let the capital j| be the general ex­ pression for the coefficient of elasticity ip all cases with but one stipulation.

In order to indicate which of the two

types of elasticities possible in a given problem is under discussion, use will be made of the subscripts gx and xy: that is, iSyg will be used to Indicate the rate of change of £ com­ pared with that of g and of x compared with that of £#

to indicate the rate of change It is not believed necessary

to use further qualification within these symbols themselves. A study of the particular setting of any given problem should readily indicate any further information which may b© desira­ ble and necessary concerning tc.e variable© and their relation­ ship#

&

The symbols' suggested above wlll.be utiliss®#. throughout this particular thesis,

The Bta might well^hege

been used la place of th® B except for. the fact that .It;!;.,■■■• presently carries an Implication of rigidity in Its applica* tion due to it historical usage and employment*

9 Chapter III ah m m m v x m , a o K m x i a miixs* o m i m m Froderiok 0* Mill.® of GoXmbia University M s wfi® considerable effort to egcpand the concept of elasticity, and the first general are® of diaouaalon la given over t0:a consideration of one of his Illustrations. fher© may be some littl© reluctance, at first, in the reader’s mind to accept a basic assumption made herein* namely, that the data of an average business cycle may- be included within the category of a supply curve * Historical­ ly, It is true that a supply curve has generally been consid­ ered to be the interrelationship of price and quantity for a single commodity or commodity group.

Ckwmlll and Blodgett de­

fine a supply curve as "the schedule of a series of quanti­ ties of an economic good which, In a given market at a given time* would be offered at sale at a corresponding set of prices* As the associated movements of prices and quantities are followed, in Mills1 study, It becomes obvious that rela­ tions in time are also being dealt with,

1‘hese temporal rela­

tionships are not among those assumed in many studies of a sup­ ply nature, but they are common enough so that the author ha® no great concern in including them in this thesis, theoretically, a supply curve may be considered as a measure of the result® in a market when all things are held

constant except demand.'

fhi® definition, it Is 'believed,can

be applied equally to single commodities, commodity groups, and to the total market behavior,

The extension of the his­

torical concepts relating to'coordinated prices and.quantity movements makes possible a much wider application of su&h tool© of analysis.

The author'’agrees that price and quantity

factors In am average business cycle do not const!tut© a ;sup­ ply curve In the

simplest sense; however,

they do requlf®

the same form of m m lysis* If on© is able to accept the validity of an average business cycle, It Is considerably easier to view its "aver­ age dw components

in the roles of supply or demand curve analy­

ses*

whose work has been drawnupon, has also used

Mr. Mills,

this same assumption in hi® various studies of the determina­ tion of the elasticities for the interstag© periods*

Bather

than dealing with the elasticities from one stag© to another, the present author has considered elasticity on the basis of a smoothed line*

It is, however, on the same general assump­

tion that this author begins hi® treatise.

9 Chapter IV of t o K&Gs&aaxoH u m

and thb

esuvssd

M jm iQ V S tm It TOE MKSCT 1BASV Wm&MM W

D

the basio figures for these first analyses are drawn from a study by Frederick 0# Mills; a book entitled Price"quantity Interactions in Business Cycles.” 'Specifically, the figures used come from table 18, "Average Movements of Aggregate Values, Average Chit Prices, and Physical Volume In Business Cycle®, All Coiimditles*,^ which table includes the following: Stage Measures Reference Cycle Stages Value Price Quantity

I

IX

8b 94 90

9? 99 98

XIX 105 104 101

IV

118 no

108

V

135 113 118

VI

VII

VIII

116 110 107

99 99 99

as 91 94

XX 85

90 94

these composite figures, averages for all oommodlties, were derived from stage averages which defined price, quantity, and value patterns for ©4 individual commodities. The stage averages for the individual commodities were given equal weight In the derivation of the general averages in the above table*

Mr, Mills• sample group of commodities includ­

ed, for the period of his study, about one-third by value of the total goods exchanged in the United States.

The refer­

ence cycle stages are nine in number; the first four are the

expansion stages of the average business cycle, the fifth stage is centered at the peak or reversal point of the cycle, and the last four are the contraction stages* This thesis is interested in only the quantity and the price averages, for each of the Reference Cycle stages, in the determination of regression lines and elasticities for a supply curve*

The quantity averages and the price averages

will hereafter bear the usual labels of X and J respectively* The first method of analysis used in the determina­ tion of the regression line and the elasticities of the sup­ ply curve is one known as the Direct least Squares Method*8 The mathematical derivation of the regression or trend line is based on certain statistically acceptable equations which will be Indicated below*

A short explanation of the logical

derivation of the final equation used in this method will assist the reader in understanding the application of such method in any further analysis* The determination of the Tread line, or rather "points of trend"f Is based on the following: (1)

T * a * bX

The factor £ is a value that defines the height of the trend line at the place selected as the origin or beginning point where X is taken as 'he.ro*

The factor b Is the measure of the

slop© which is brought about by the introduction of th©

11 additional quantity interval®, —

in this case denoted by J.

In this way the trend at any given measure or quantity is the height at th© point of origin a, plus th© product of the meas­ ure of th© slop© £ and the quantity intervals between th© point ,;©f origin and th© present given quantity measure for which th© trend is sought* This may b© more readily seen by glancing at th© example, in Table 8*

The quantity measure or JE# *OT ^he first

of th© Average Cycle Stages is 90.

Assuming the validity of

a and b for the moment, the trend £ or rather J£L$ since we are measuring th© price factor trend, for the quantity average J (.90) is a

or -14.6199b ♦ 1*15439 {times90); or

which isthetrend value of T at that

69.0V515;

particular quantity

measure of £• The above equation, however, is in turn dependent upon the discovery of th© values of th© constants, a and b • The acceptable equation used for at is* (2j

a - ^

* bBj,.

Kj and My are the means of the X ana Y measures, or the quan­ tity and the price In this analysis. The equation used for b, 1st (3)

b H w 8 ♦ 1)V8 - 1 W ........

Since th© equation for b is the solution of a quadratic, it has alternate values, so that th© required value of b takes th© same sign as the Zxy

involved*

The w, in its turn, is determined from the follow­ ing?

Before going on to see Bow these last three ©quetioas are used In th© present analysis* the reader should' examine Table 1- In order to see hew the quantities %x^,,*::g2^£ dud %y^. the "corrected" figures measured as deviations from their means, have been derived from the X and Y measures* through the us© of ©enterlag equations# ed the numerical values for X%*» ftxv and

Having once determin­ it should be

comparatively easy to follow their?substitution in the first equation#

The solution of the remaining equations is accom­

plished in order to arrive at a trend figure or a trend point * for eaoh of the independent values of X*

The location of

each of these points in any given graphic analysis will, if the computations have been correct, show them all capable of interconnection by a single straight line* known as the trend line*

If one or more of the computed trend items does not

fail on the single straight line denoted above, it Is manda­ tory to examine th© data and the computations for errors# Chart 1 gives a graphic presentation of the regres­ sion line found by the Direct Least Squares method, for the price and quantity measures of an average business cycle * Here has been established a trend line for the Y, or price measures, while the Jg factors have remained the independent# The small centered squares on this chart denote th® position

ta b lh ;

1,4

Corrected 3upply Curve figure a for An ■Average Bueiaeee Oyol©

L

r

XT

x3

.

yg

90 98 101 108 118 107 99 94 94

94 99 104 110 118 110 99 91 90

8460 9708 10604 11880 18644 11770 9801 6534 8460

8100 9604 10801 11664 12544 11449 9801 8856 0336

8836 9801 10616 12100 12544 12100 9801 @281 8100

903

909

91675

91035

92379

2X3

tt

91035 » 0 *

«o

m

9Q6G1

Xac2

«*

434

IXI

•

91675

• ■815-

ZXZT

@30827

1r~ -0

«

-91205

i.acy

«

472

zrz —

■

■m i n im .... inijPjWWiitM

log Xo/log

-

3 3 B

■

^

Therefore, as might be expected, the spread of the trend lines is considerably greater*

The regression line

found by the Direct least Square® method is far las® oember©d in its relative position to the other two regression lines than was the former one©* the b

this time it lies much nearer to

r line than'to the

regression line*

The elastioitles for the various trend lines are computed by the same method as was used previously, and are found in Table 16 * Mention has been made previously In this study that regression lines need not necessarily b® straight lines , but might well be curvilinear in nature*

Examination* however,

of the charts so far in this thesis reveal regression lines as nothing else than straight'lines*

Some explanation of

this fact is felt to be necessary*, If the data case® and the regression line® 'had been plotted on an ordinary chart and had not been essenti­ ally reduced to a log chart, these regression lineamight well have been best fitted by a curve of some type, such as a rectangular hyperbola*

However, because of the mathematical

relationships brought about by a reduction of both the meas­ ure® and the data to logarithmetlo measurements, th@ curves became straight lines; this same result is also forthcoming If quasi-log reductions, such as nth root or nth reciprocals

m ar© employed*

This may b© deduced by examination of the

equation© below, whloh are first shown as expressions of curved lines, but which, when reduced to logs, become straight liness (1 8 )

T

logY (1 3 )

«* a X -b

*

Y

LogY

log. a t b log X

• a®i b x

•

log a 1 bX log ®

Oompaxiaon can be made between equation 13 above and equation 1, where the trend line point was based on the expression X « a+feX* same®

The basic relationship is the

In the first method all the trend line points were

capable of interconnection by a single straight line*

The

expression T » a+bx is also a function of the regression line in question, which is a straight line*

In a demand

curve analysis, since the slope is a negative one, the ex­ pression is log J * log a - b log J|#

Equation 15 is an

expression of another curve which may by the same method be reduced to a straight line through the use of logs.

This

replacement of a curved line by a straight one is common practice In mathematics. If, however, the above methods were not employed, and it appeared that a parabolic regression line would give 13 the best fit to the data , the procedures concerning elasticities would be somewhat different*

The next chapter

Illustrates such procedures although fitting Ilea largely outside the prayino© of tills studyj and it Is aonstdered to be preliminary and rather subsidiary in nature .to th© deter* miaation of ©laetiolttes.

In determining whether.to us© a

Straight or curved regression line, resort may be had;.to the Chi~Squar© test, or linear and curvilinear oorr©labIons may be utilized for the same purpose•

Chapter X

a

wsmxhimm hsohx^xok

i w m w ?

w ©

and

p o m iiiAmaiTOS'

Mention has been mad© previously that an effort would b© mad# to tit a curved regression line and to find til© pertinent trend point elasticities in order to illustrate til© techniques involved in such problem. therefore* a ©lapi© parabola will be fitted to the data of Mills* average business cycle, as indicated at page 9 of this thesis#

Much

of Chapter 4 was devoted to the fitting of a straight line trend to this same data, on the assumption that such'a straight line would give an adequate fit#

The following ex­

ercise will, therefore, tend to substantiate or disprove this original assumption* as well as serve in the capacity of illustrative technique# Th© second degree parabola may be described by the equation;

(14)

I

- a ♦ ME + OX8

There exists a number of possibilities in the fit­ ting of such a parabola, the most direct general method would be to solve algebraically the normal equations after calculating the required summations and substituting them la the proper manner#

$uch a procedure, however, is capable of

simplification by centering the data* the X ’s# the M squares and T*s#

By so doing and by considering the Xs series as Xg

m and the X series as

these ©Rations then becomes

(15) (16)

♦

0Zxg2

*

2'xl3r

«

tx^f

The problem then becomes on© of calculating and cen­ tering the summations.

The summations may be foun& in Table

IfI the centering procedure as well as the resultant "correct­ ed** figures are to be seen in Tables la and 1®, Of next concern is the finding of the values for the weights or coefficients "a", ”bw , and we %

Through the

use of the corrected figures already determined, the equa­ tions above (15) and (16) are capable of solution for the de­ sired coefficients & and & by substitution and the processes of algebra.

This work has been done and may be found at

Table SO. The value of the coefficient "a" may be found by substituting the known values of b and & in the first uncent­ ered normal equation! (17)

Ha

*

bXX ♦ cZX^ « Zt

Tile resulting value for & is $4*3075345 and its determination may be found also In Table SO, while the values for b and o are 0.4433347 and 0,0031845, respectively. Having determined th© values of the constants, It is now possible to establish trend line points for each of the data oases, based upon the trend equation indicated earlier, I.e. i

39 TA0U3 17.10 ■ f m S P M S V * . (Be® Table 1*4) *i 90 98

2]2

IT X

£

107 99 94 94

6100 9604 10201 11664 12544 11449 9801 8856 8636

94 99 104 110 112 110 99 91 90

8640 ©702 10504 11680 12544 11770 9801 8554 6460

903

91035

909

©1678

101 108

112

V 761400 900796 1060904 1286040 1404928 1269390 970299 804076 795240 9390073

t

*?• 729000 941192 1030301 1259712 1404928 1223043 970299 830584 830584 9221643

2

*a_ 65610000 92236816 104060401 136048896 157351936 131079601 96039601 78074896 78074896 938697043

m

SAB18 18,10 003?reoted Figures For A Parabola Mills* Gyol© 0afca ( B m fabl© 17,10) -

aSx^y

m

KtWV' W

ZjtjT

m

Jt^Y

m

XX T - Exxzy / H A ©1678 1 0 - SXjZY -

-

EV

T

38082?

— 3ST— w o

«• **©1203

Zx^T

at

47®

HS*28

-

JSEX2a

iXg2

«£XgS

£X22

*

-

-

--2— 2

iilgJSXg

938597043 *,

/ s a

- {EXg ) S

-0 Ix „3

- -S20819025 -

177780X8

8287371885

61 t

m

m

1

9

*

1

0

Oerraoted Figures For A ParabolaMilla*s_Gycle Bata (See table 17,10) *e*a7

it HSXgT - ZXggY

2*8?

*

IlgT

'* 9280073 J 0 * ZXgSY

ZXgY

-

ZXgpt

B

/

83700816 r —

T T * -O

it —919453$

ZXqF

0

HZac^Xg

m

2i 1x2 ZX1*E

m

9SS38

lEX^Xg *

ZXj&X&

2XxXg - ZXj.ZXg /

N

9221643 j e - 2Xx2Xa “ ""I--

m

1*-'

■

z B - *

» 1 - f2

y

*

0.90087■

m -1 «

0*09918 y

89.88)

e 8

« 27.2882908

{tables

0.1$ levsl

27.00)

Chapte rXI , ELASTICITIES IN m m F A O f m i m - m o r n m , capital

mo

labor

In a continuing effort to broaden the ooncept of til© coefficient of elasticity, attention Is now given to a time series of Indexes of manufacturing in the United States, Including product* capital, and labor for the years 1899 to 1922, inclusive,

From those index numbers as data, a number

of coefficients of elasticity have been derived. methods are used;

Tvrd general

on®, which has been Illustrated previously;

another, which has not as yet been mentioned# As has been stated elsewhere, concern in this thesis is centered rather more upon th© various aspects of methodology involved in deriving elasticities, as well as ex­ tending the application of this concept, than upon the eco­ nomic significance of the specific findings in any illustra­ tion#

However, if in the measurement of the differential re­

sponsiveness of the various factors involved in any given problem, certain, regularities are found to persist, it Is be­ lieved that such elasticities can be extremely useful in any given specific economic analysis.

Conclusions In such In­

stances should be qualified by the relative degree of their apparent statistical validity# The various Indexes U8*.~d have been adapted from

certain of those of Paul R* Douglas and Charles W# Cobb# 13 Hare were calculated index numbers of deflated value of capi­ tal used in manufacturing; the number of worhers employed in manufacturing; and an index of the physical product of manu­ factures in the United States for the years 1809 - 1922, These indexes with their appropriate logarithms are indicated In Table 85# Through preliminary experimentation, it appears that the relationship Is of a logarithmic nature and that a straight line log trend well fits the data*

Under these as­

sumptions, the elasticities of |>foduot against capital and product against labor have been sought, utilizing th© same methods found above in Chapter IX*

Once again the three dif­

ferent elasticities are found for each series, in Tables 29 and 30*

In both areas the three elasticities and trend lines

are found to center quite closely*

Such seems to be the case

when there exists a high correlation between th© basic data* The correlations in question prove to bo 0*9430 and 0*9345 re­ spectively, for th© data of product and capital, and product and labor.

As a matter of fact, only th© trend lines for th©

regression of product and capital have been drawn, and these are to be seen in Ghart 3*

Attempts were made to graph th©

trend lines for product and labor, but such regression lines were found to lie so close together as to make the necessary photographic reduction process completely impracticable, and their charting has been omitted for this reason*

*3?AEL$ 85.11 Indexes of Jfenufaetui*lag, United States (Adapted from Douglas end Cobb)

CAPITAL Mfl. 1899 1900 1901 1903 1903 1904 1905 1906 1907 1908 1909 1910 1911 1913 1915 1914 1915 1916 1917 1918 1919 1920 1931 1922

100 107 114 122 151 136 149 163 176 189 196 208 316 236 356 244 266 398 535 366 387 407 417 431

loa* .0000 #0394 •0569 .0864 .1173 .1399 .1733 .2132 •3455 .3672 .2967 .3181 .3345 .3541 .3729 .3874 .4249 .4742 .5250 .5635 .5877 .6093 .6201 .6345

LABOR Index 100 105 110 118 123 116 125 133 138 121 140 144 145 152 -tm . 154 ■182 196 200 193 193 147 161

PRODUCT

Log* .0000 . 0212 .0414 .0719 .0899 .0645 i0969 il239 .1399 ;0828 i1461 11584 *1614 a®is .1675 i1732 .1375 .2601 .2923 .3010 .2856 .2856 .1678 .2068

*CharaGteristic has been omitted

100 101 112 122 '124 122 143 152 ■151 126 155 159 153 177 184 189 ‘189 225 ‘227 ■223 218 231 179 240

.0000 .0043 .0492 .0364 .0934 .0864 .1563 ■.1818 .1790 .1004 .1903 .2614 .1847 .2480 V&84S; .'12279 .2765 ■18588 ;3560 ' .3483 .3385 13636 .2389 13802

74 TABLft »e*XX SuxaaaatioG Flyuraa

For Capital i Labor a&d Product

Capital » ZXX

m

i Labor * Xg ; product •» X

7*831®

XX^ *

3.48990074

(XX-)8

-

>

81,32789344

3.7870

XX2a -

0.78417420

(XX2)8

•

ZXj^Xg XT

13.89082900

- 1.58040459

- 4.9218

XT8 - 1.32367833 (XT)8

-

XX-jT

- 2.11318236

2XgT

-

XX.XT

- 38.84128080

XXgXT -

24.22116228

0.99078508

18.34243080

n SAS&B Z7.ll ® m reeted figures For Capital and Product i B m fable St. 11) E X j2

•

3 .4 6 9 9 0 0 7 4 j

- 0

»

8 .5 5 5 3 8 0 3 6

E*xa

-

0.91458018

z x xr

» 2,11313236 5 * ~1•60588343

m

e

o

{ZXj^®

-

N

0

tv

•

Q.50726691

ZT2

*

1.32367833 ; c

- e

• *1.00921509

Xya

»

0,31446324

-

*

ZXXZ T

ST

«

K

«

(2Y)2 N

8 1 ,38 769 344

38.54125080 ' 24

•

’

24,22116323 24

n nmm

ss.u

Correct#4 Figures For Labor and X^rodubt (See Table 26.11)

ZXjf

-

0.75417420 }

- O

» -0.57877204

0

•

Uig)2

-

H

13,89052900 24

.

Zx%Z

-

0.17540216

23CgY • 0.90076568 ; 0 • o

* ~0.76426794

Zx2y

•

0.22649774

2Y2

»

1.32367833 ; e

- o

« -1.00921509

ZyZ

m

* £%1Y

5

-

(£Y)2 N

0 431446324

* 18.34243050 24

»

24.22116225 24

n

tmrn @ 9.11 Capital and Pvodaot Determination of Th® Throe Regression line® and Klaeticltles (See Tables 27*11 and 28.11}

She Plgeot lea.at aaaae*. .tjg.tt.aflt w

-

8l3Exy

-

1.01453382

■£xx* - Ejr®

®jx •

" &

1,69086081

0.60011696

W2 - 8.85799382 b

- (w8 ♦ l ) ^ 2 - 1

-

£1.98417764 - 1

- ♦ 0.8708308S7

.......................................................

w b

* ♦ 0.S70330257

since the sign of the £xy is positive

The by^ Method:

bJ X . *

Q J *

1*69006021

*

* 2xx2

She bjy* Methodi

0,

0.50726691 0.91498018

3_

-

0.61991

v • i - eSffi ■

isofcw

£y2

b_ * *iy

»

0.61991678 — —

2—

» 1,705370000

ttASLR 30.XI Labor and Product Determination of The Three Regression Lines and Elasticities (See Tables 27.11 and 26.11) The Direct Least aqmtres Method; w

•

S2x^

IXg8

* Ey8

0* 4529904a

I

*

«* -3.25752865

-0.13906108

w£ -

10.61149421

b

-

(w8 ♦ l)1^2 -1 ■■-■■■!.M. w

b

» *1.853039218 since the sign of the Ixy is positive

-

13.40756427 - 1 ~ — * ..»*«.-» -3.25752885

Th. b ^ Mathods

iy*

b™.

0.28649774

» Zx2y

*

ixg2" b

1.35304 1

-

-

+1.353039212 or -0.739076883

1.89131

0.17540216

« 1.89130530

2 H»b

1 Method: xy ......... 1

3

y*

""

b_

3T m

x27 *

b_ 2

*

•

1.38637 1 1 l

2x Tv ~* »

6.K2649774 »

j-f—

O.ilMlil

1.38837883

1 o.?si>2< W d

79

o

o

CM

O «» *»•«•

4 .

«».*».■. an-**«.•*'•*•«*>*■•-•»*>•