PR 2 2 194§' University of California College of Agriculture Agricultural Experiment Station Berkeley, California A STATISTICi:^, iU^ALYSIS OF THE DOIaESTIC DEiAM) FOR LEJONS, 1921-1941 by G« M» Kuznets and Lavvrence R. Klein June, 1943 Giannini Foundation of Agricultural Economics Mimeographed Report I<fo» 84 UNIVERSITY OF CALIFORNIA UBRARY COLLEGE OF AGRICULTURE DAVIS TABLE OF CONTENTS Page Introduction ....... • 1 Statistical considerations 4 Demand for summer lemons 11 Demand for winter lemons ...... 34 Allocation of shipments «•• 47 Summary 55 Appendix A. Series used in the study 57 Appendix B. Price and quantity residuals and tests of randomness 83 Appendix C. Net regression charts 88 Appendix D» Auxiliary tables 101 Appendix E» The use of mean monthly maximum temperatures in the summer lemon regression analysis 107 ... . . VJ.^i;^,;.j: i'.-fe.t^.l' '■'■■f:'- -''i 'W-')S- . . • . v.-^. 1^ .i* ♦^1 xj;. ii LIST OF TABLES Page Table 1. Weekly average consiomption of lemons by native-white nonrelief families classified by income and region, selected cities, May-August, 1936 12 Table 2* Summer lemons: Basic series used in the demand analysis, 1922-1941 15 Table Summer lemons: Regression coefficients and auxiliary constants, 1922-1941 16 Table 4» Sumraer lemons: Coefficients of separate determination . « • • • 20 Table 5. Summer lemons; Cadculated coefficients of price flexibility and demand elasticity, 1922-1941 23 Table 6. Simmer lemons: Regression coefficients and auxiliary constants for additional equations, 1922-1941 . 32 Table 7. Tfinter lemons; Basic series used in the demand analysis, 1921-22 to 1940-41 38 Table 8, Winter lemons: Regression coefficients and auxiliary constants 1921-22 to 1940-41 (1936-37 excluded) 39 Table 9. Ti'inter lemons: Calculated coefficients of price flexibility'' and demand elasticitj^, 1921-22 to 1940-41 (1936-37 excluded). . 42 Table 10. Winter lemons: Coefficients of separate determination 44 Table 11. Allocation of Ho vember -April and May-October lemon shipments to maximize f .o.b. returns for an undetermined and for a fixed volume of annual shipments, 1921-22 to 1940-41 49 Table 12, Shipments of summer and winter lemons maximizing value on tree and minimizing average annual f .o.b. price for fixed value on tree • ••• 54 Table 13. Seasonal average f.o.b, prices of California lemons, 1921-22 to 1940-41 58 Table 14. Estimates of correlation betvreen California Fruit Grovrers Exchange and industry weighted average prices under various assumptions ......••..••«•....•••••••• ^ Table 15. Comparison of f.o.b. prices of lemons excluding and including exports, 1935-36 to 1940-41 62 Table 16. Average sizes of lemons shipped by the California Fruit Growers Exchange and proportion of shipments graded Sunkist, 1930-31 to 1940-41 63 Table 17. YiTinter lemons: Domestic shipments, per-capita domestic ship- ments. United States supply, and per-capita supply, 1921-22 to 1940-41 65 I iii Page Table 18, Smnmer lemons: Domestic shipments. United States supply, and per-capita United States supply, 1922 to 1941 67 Table 19, Industry shipments of winter and summer lemons for fresh consumption, 1921-22 to 1940-41 69 Table 20. YiTinter lemons: California Fruit Grovrers Exchange domestic shipments and outside domestic shipments, 1921-22 to 1940-41. . 71 Table 21. Summer lemons: California Fruit Growers Exchange domestic shipments and outside supply, 1922-1941 ..... 72 Table 22. Indexes of total and per capita nonagri cultural income pay- ments in the United States, 1921-22 to 1940-41 74 Table 23. Indexes of v/inter and summer temperatures, 1921-22 to 1940-41 • . 77 Table 24. Population weights employed in the construction of indexes of summer and virinter temperatures 79 Table 25. Index of the incidence of influenza, 1921-22 to 1940-41 82 Table 26. Summer lemons: Price and quantity residuals 84 Table 27. Summer lemons: Tests of randomness of residuals ........ 85 Table 28, Winter lemons: Price and quantity residuals 86 Table 29, TiTinter lemons: Tests of randomness of residuals 87 Table 30. Tfinter lemons: The effect of adjusting domestic shipments in 1936-37 on the net regression coefficients and on the co- efficients of multiple correlation ........ ...... 102 Table 31, Summer lemons: Inter correlations of the series employed in the regression analyses 103 Table 32, Summer lemons: Reciprocal correlation matrices 104 Table 33, TJinter lemons: Intercorrelations of the series employed in the regression analyses 105 Table 34, ViTinter lemons: Reciprocal correlation matrices ......... 106 Table 35. Mean maximum temperatures in 22 cities, May-September, 1922 to 1941 109 Table 36, Average rail and boat unloads of lemons in May, June, July, August, and September in 22 cities, 1931 to 1938 , , 110 Table 37, Summer lemons: Regression coefficients and auxiliary constants obtained v/hen mean maximum temperature in 22 cities is used as the temperature variable ........... Ill Table 38, Suiiiner lemons: Correlation matrix and its inverse obtained vrhen mean maximum temperature in 22 cities is used as the temperature variable ........... 112 1 I LIST OF FIGURES Page Fig. 1. California f.o.b. price and donestio shipments of (A) winter and (B) summer lemons, 1921-22 to 1940-41 3 Fig. 2, Domestic shipments of (A) winter and (B) summer lemons and shipments maximizing f.o.b. returns, 1921-22 to 1940-41 .... 50 Fig. 3. Summer lemons: Net regressions of price on (a) United States supply, (B) index of nonagricultural income payments, and (C) index of temperature. Arithmetic form ....... ... 89 Fig. 4. Summer lemons: Net regressions of United States supply on (a) price, (B) index of nonagricultural income payments, and (C) index of temperature, 1922-1941. Arithmetic form ... 90 Fig; 5. Summer lemons: Net regressions of price on (a) per capita United States supply, (B) index of per capita nonagricultural income payments, and (c) index of temperature. Arithmetic form 91 Fig. 6. Summer lemons: Net regressions of per capita United States supply on (a) price, (B) index of per capita nonagricultural income payments, and (c) index of temperature. Arithmetic form 92 Fig. 7. Summer lemons: Net regressions of price on (A) United States supply, (b) index of nonagricultural income payments, and (C) index of temperature, 1922-1941. Logarithmic form .... 93 Fig. 8. Summer lemons: Net regressions of United States supply on (a) price, (B) index of nonagricultural income payments, and (C) index of temperature. Logarithmic form ........ 94 Fig. 9. ITinter lemons: Net regressions of price on (A) domestic ship- ments, (B) index of nonagricultural income payments, (C) index of temperature, and (d) time* 1921-22 to 1940-41. Arithmetic fom 95 Fig. 10. Tlinter lemons: Net regressions of domestic shipments on (a) price, (B) index of nonagricultural income pajmients, (C) index of temperature, and (D) time, 1921-22 to 1940-41. Arithmetic form 93 Fig. 11. Winter lemons: Net regressions of price on (a) per capita domestic shipments, (b) index of per capita nonagricultural income payments, (c) index of temperature, and (D) time. Arithmetic form 97 Fig. 12, TJinter lemons: Net regressions of per capita domestic shipments on (a) price, (B) index of per capita nonagricultural income payments, (c) index of temperature, and (d) time, 1921-22 to 1940-41, Arithmetic form 98 • • « Page Fig, 13» Winter lemons: Net regressions of price on (A) domestic ship- ments, (B) index of nonagricultural income payments, (C) index of temperature, and (d) time, 1921-22 to 1940-41. Logarithmic form ...••*...• 99 Fig. 14. Winter lemons: Het regressions of domestic shipments on (a) price, (B) index of nonagricultural income payments, (B) index of temperature, and (c) time, 1921-22 to 1940-41 Logarithmic fox*m .•...•.*••.••... • 100 • » ♦ » » , i. ! » A STATISTICAL ANALYSIS OF THE DOMESTIC DEMAND FOR LEMONS, 1921-1941 G» M« Kuznets ^ and Lawrence R» Klein Introduction This report presents some of the results of a study of the lemon industry now in progress at the Giannini Foundation of Agricultural Economics of the Univer- sity of California, Specifically, this paper describes a statistical analysis of 3/ the demand relations facing handlers of California lemons. The period under review is the twenty-year period 1921 to 1941. The geo- graphic market for the purposes of this study is the continental United States. During the last half of the period, that is, since 1932, practically all of the lemons marketed in the United States were produced in California. This, however, is not true of the first part of the period. In some of the earlier years imports constituted a significant fraction of our total lemon supply. California lemons are marketed in' the United States throughout the year. The marketing year is generally taJcen to be the twelve-month period, November 1 to October 31. Numerous references in the trade literature point to the notion that the demand for lemons marketed during the winter months is affected by the prevalence of colds and influenza, \Yhile the demand during the summer months is greatly responsive to weather conditions. It would thus appear that greater homogeneity would be attained by a separate consideration of the demand situation prevailing during the winter and summer months. For this reason separate analyses are y Instructor in Agricultural Economics, Junior Agricultural Economist in the Experiment Station, and Junior Agricultural Economist on the Giannini Foundation. y Research Assistant on the Giannini Foundation, June to September, 1942. y A "handler" is defined by the California Agricultural Prorate Act as "any person receiving agricultural commodities from the producer for the purpose of marketing the same." Statutes of California, 1933. Chapter 754. Hp I 1..:. \, ,. • •• ^ ' ■• " 'jo ai TO' \ 1' 'it presented in this paper for lemons shipped during the six months November through April (winter lemons) and for those shipped in May through October (summer lemons) of each marketing year. The time sequence of seasonal average f«o*b« prices and domestic shipments of lemons is portrayed for the months November through April in panel A, and for 4/ May through October in panel B of figure !• It vdll be observed that both ship- ment series exhibit a marked upward trend. Accompanying the upjvard trend in ship- ments is a price trend vrhioh exhibits, at least for winter lemons^ an obvious discontinuity roughly near the middle of the twenty-year period. For both winter and summer lemons the price trend is up^vard for the nine years 1922 through 1930. For the remaining eleven years the trend in the f .o^b. prices of v/inter lemons is horissontal, while the trend of the prices of summer lemons is declining. The tendency for the differential between f .o.b, prices of winter and summer lemons to diminish is evident. Attention should also be drawn to the marked season-to-season fluctuations in the separate price series. The expected inverse relation between the year-to-year movements of prices and shipments is discernible for both winter and summer lemons, but exceptions to this tendency may be noted. Such is the course that prices and shipments have taken during the period under consideration. YJhat are the factors that have fashioned this sequence? How much and in what manner have these factors influenced the two time series? These are the questions considered in this paper. 4/ These series 6a*e given in appendix A. The price series consist of the average prices, f.o.b, packing house, of lemons marketed by the California Fruit Growers Exchange. ,4 rfoae '}o ■ ; • ■» ■ » 2.50 ! ■■■J 1 . , . , ■ , _ _ ■ i ■ i ■ ■ 1S21-22 1923-24 1925-26 1927-28 1929-30 1931-32 1933-34 1935-36 1937-38 1939-40 Fig. 1. — California f.o.b. price and doniestic shipnents of (A) winter and (B) suramer lemons, 1921-22 to 1940-41. The equations of the least sqviares price trend lines are (1) p^, = 3.7022 + 0.1283 t-j^ (origin at 1925-26), (2) p;; = 3.0027 - 0.0066 tg (origin at 1935-36), (3) p^ = 4.5433 + 0.1230 (origin at 1926), (4) p" = 3.7627 - .1255 tp (origin at 1936). (Data from tables 13, 17, and 18, appendix A.) ■:.v.:,.>>. «7tir«c. .rwf i.-^i -safiKasi, j-swoiw' -f^-T^'S '/SsfS:' '51 •H 1- 4. Statistical Considerations Formally, the observations of price and quantity recorded in the raarket may be viewed as the simultaneous solutions of the equations, ^'t (P* ^t* a!^, a^) » 0 ' , „ ^ t = 1, 2, M 6t (P» <1J \» \» \) =0 where p is price, q is quantity, f^ is the demand schedule at time t, is the supply schedule, and b^ are the parameters of the ixio schedules. It is probably psychologically more defensible and logically more convenient to viev/ the market observations as the simultaneous solutions of the above equations in a stochastic model, after a random component has been added to eachj that is, as the solutions of tt tt 0.^,0 V, t II „ i/ = x,t;,.».,i^ St (p» ^t' \* \* ^t^ " ° where s and s. are random variables, t t If one of the tv/o cxarves f^ or g^ remained fixed while the other shifted, the price-quantity scatter would trace out the stable curve. This situation, it has been argued, is approximated for some cominodities, such as agricultural staples. — Clearly, however, this condition is not met by our data. It is reasonable to assume that the supply curve for lemons has not remained fixed, but has shifted in response to varying crop conditions and to changes in other factors. But it cannot be assumed that the demand schedule has remained fixed. The demand for lemons is presumed to be affected by weather conditions during the summer months and by the prevalence of colds and influenza during the vd.nter. These are abruptly changing factors \7hioh should induce erratic shifts of the demand curve from one period to another. — ■ Schultz, H. The theory and measurement of demand. The University of Chicago Press. Chicago, 1938. p. 136. " '^h e. 7 I With both schedules shifting, the parameters of the theoretical demand schedule could still be estimated if (1) the form of the demand curve, (2) the complete set of shift variables, and (3) the distribution functions of the random components were known. Not all of these items of information, however, are equally important for the determination of statistical demand functions. It is argued here that useful demand equations can b© secured even if nothing is loiovm concerning the form of the demand function and the distribution functions of the rsindom components. Let us suppose that we are able to catalogue and measure the important factors which cause the demand to shift, and that these factors are, on the uriaole, distinct from those that affect supply. Let us then choose some function and some method of fit — being guided in this primarily by considerations of simplicity and convenience but taking care not to gainsay whatever information is supplied by economic theory and our acquaintance with the commodity and its markets. Finally, let us suppose this that with these shift variables and/function we have secured an excellent fit (the measure of goodness of fit being consistent with our assumptions concerning the random components). The resulting equation is not the market demand schedule of the theoretical economist, but it is something more than just an estimation equation for predicting price and quantity. It is clearly an equation pertaining to the demand side of the market. Such equations, it is felt, should give us an insight into the demand relations prevailing in the market. The foregoing argument constitutes in a sense the rationale of this study. The remainder of this section is devoted to a more detailed consideration of the procedure v/e have follovred, 1, Shift Variables ,— Among the shift variables we have employed are indexes measuring the level of temperatin-es prevailing in the important metropolitan districts of this country, income payments used as a measure of consumer purchasing power, and "time," There is little doubt that the temperature variables measure factors which act on the demand side of the market. The most obvious influence of temperature on supply is through the effect on crop conditions, but the geographical t * -■ ( : •: • r: :•: :.: n-j :t;:i-.i;.v-.' -,,; •;:!• - ; ■ . .■ • ^ n» Tjioaalo ai J- J. ©•Mi f? . ■ 6. coverage of the temperature variable excludes this poeolbility. "Time" should be adequate for "explaining" shifts in demand due to widespread advertising campaigns and gradual changes in consumption habits. Income payments, in so far as they represent consumer purchasing power, are a demand factorj but the role played by this variable is equivocal because this index could be used as a measure of general business conditions and thus be simultaneously (although with a smaller weight) a supply factor. These variables considered as a group, however, are seen to be important on the demand rather than the supply side of the market, 2» Form of Equations .— In this study, price or quantity was expressed as a linear function of the shift variables or of their logarithms. By this it vms implicitly assumed that the shifts of the demand schedule are parallel. At a time t the net relation betv/een price and quantity is p «» aq + d^ or log p = a 'log q + d^, where and d^ are the shift parameters of the demand schedule. It is evident that d^ is a weighted average of the values of the shift variables at time t,and that f d^ is a similar average of their logarithms — the weights in both oases consisting of the coefficients of the shift variables. The conventional character of this procedure must be emphasized. There is a temptation to a'ttaeh too much significance to the specific form of the net relations, especially if the fit of the total equation is good. However, other functions describing markedly different net relations might fit the data equally well and possibly better than the functions selected by the investigator. It is chiefly because of the uncertainty concerning the form of the net relations that tviTO different, yet a priori equally admissible functions, vrere fitted. But it is not intended that this choice imply that the true relations lie somewhere betv/een the net relations described by the two functions. 3. Method of Fit .— • If the statistical results of this paper are to be looked upon merely as estimation equations, then the following model may be adopted. era. » - .Jb'+ 'p« '•""q 'ei -^^'sJ-^V :rtt. . \i: ■• ••1^.' 1 : ri '■" 'eicff <" 9d -ft- (i = 1, 2, N) "i " * °2i^2 + + Osi^s where the X^'s are the N observations of the dependent variable, m^i^ = E(X^) is the expectation of X^, the e^^'s are independent random variables with standard errors (T^ =Cr(unknown), — ' the Oj^'s are known numbers, and the a^'s are iinknown para- meters. This specification makes possible the calculation of the best unbiased linear estimate of the dependent variable and of the unbiased estimate of its 7/ standard error. — ' Similarly, the unbiased estimate of the standard error of the coefficients of the above linear estimate (net regression coefficients) may also be calculated,. 4.. Tests of Significance . — In the course of the investigation questions usually arise concerning the satisf actoriness of the equation as a whole and con- cerning the significance of the coefficients of the equation. Criteria either in terms of the standard error of the regression function or in terms of the standard error of the regression coefficients may be formulated by the investigator to answer such questions.. For oxtimple, the equation may be deemed satisfactory if the ratio of the standard error of the equation to the estimated value is less than some nvmiber k.. Or a given independent variable may be eliminated if the ratio of its coefficient to the standard error is less than a certain fixed value. The advantage of such "absolute" criteria is that they do not involve knowledge of the distribution function of the residuals e^^. Such tests, however, are in the main unsatisfactory 6/ ^ — ^ More generally, (7^ = B^p' where B^^'s are knovm numbers.. Throughout we assme that » 1 (i =1, 2, ,.»,,N). Unbiased in the sense that the mathematical expectation of the estimate is equal to the true value and"best"in the sense that of all unbiased linear estimates it has the smallest standard error. A detailed derivation of the estimate and of its standard error is given in: David, F..H.. and J. Neyman. Extension of the Markoff theorem on least squares.., Statistical Research Memoirs 2:105-116. 1938. ■ . I * ai*i lo 1' ■,, ••-11^ '^P Cits i> ; osij x - : ■ - ffpas ri.'.riif. sa-?/' f?- 1 Pn^- ^."v!--'J'F'-5"'>^-'*' ''*c%^ •' • r;- e i 9ii 8. because of their extreme arbitrariness and because, vrLthout knov/ledge of the distri- bution functions* only very weak probability statements can be made. The assumption that the residuals follow the normal law removes the arbitrari- ness inherent in the use of "absolute" criteria in multiple regression. However, such an assumption as to the distribution of the residuals may be considered arbi- trary in that there are many other equally admissible assimptions about the pro- bability function of the residuals. It may be desirable to consider the hypothesis most frequently tested in this paper, namely, that one of the parameters, say ag, is zero in some detail so that the nature of the test may be properly appreciated. Let U be the set of hypotheses ascribing to aj^, ag,Cr any real values with the restriction that J"> 0, and let u be a subset of U consisting of hypotheses specifying = 0 but ascribing to the other parameters a]^, a^^^* CT any real values (Cr>0). According to the Neyman-Pearson principle of likelihood the test of the hypothesis ag * 0 should be based on the value of the ratio L Pu / Pu or a constantly increasing or decreasing function of L , where p^ is the maximum of the joint probability function of (Xj_, X2, Xjj) relative to hypotheses con- tained in u, and p^ a similar maximum probability relative to hypotheses contained in U. A low value of L implies that the set of observations (Xi, Xjj) is much more probable on the basis of some hypothesis included in (U-u) than on the basis of the hypothesis tested and would thus lead to a rejection of the latter. Direct calculations lead to the following expression for L , for the hypo- thesis tested a =0: s T M 2 n/2 where r is the partial correlation Qoefficient« It is somewhat more convenient xs « 8 / — I Neyman, J, and .E. S. Pearson. On the use and interpretation of certain test criteria for purposes of statistical inference. Part I. Biometrika 20A: 175-240. 1928, 'Mcf bna aawf ^ one 9. to use the following constantly decreasing function of L , (iJ-s) ( L-2A - 1) (iJ-s) 1-r xs. It will be recognized that L ^ is the ratio of the net regression coefficient of X on s to the unbiased estimate of its standard error. The distribution of this ratio for the case a^ = 0 is knora, -i/ and thus the probability of rejecting a true hypothesis may be reduced to any desired level. It has been shown that a common best critical region for all admissible 10/ alternatives e;:ists if the hypotheses in (U-u) specify a > 0 or a < 0. ^ This s region is defined by the inequality L j_ for a^ > 0 or by the inequality L^_^- t^ for ag<0. In these inequalities t^ is obtained from tables of the integral of the Student-Fisher t distribution for (ll-s) degrees of freedom and for (probability of rejecting a true hypothesis) based on one tail of the distribution. A test based on the best critical region is to be preferred to other tests because for any set level of significance, its power (the probability of rejecting the hypothesis v/hen it is v/rong) is not less than the power of any other test determined at the same level of significance. The use of "standard errors" in connection with time-series data is frequently criticized by economists and statisticians. Aside from the question of the formal satisfaction by the data of the specific assumptions underlying the coEEiionly used tests, much of the strength of the criticism is derived from a general feeling concerning the lack of roalism in the basic specifications. The notion of a random sample when applied to describe a configuration of prices or quantities over time appears to be highly inappropriate, not only because the price or quantity 9/ — ' Fisher, R, A. Appli 1925. Application of "Student's" distribution, Metron 5:90-104. 10/ — Kolodziejczyk, S. On an important class of statistical hj'-potheses. Bio- metrika 27:161-190. 1935, ■'uJ&il'-i.iitih. tv:~ Is. 9.?t,f')L<:<; f-<^* ^^-t^^f^^Wwrf ■'• ' * I 9jf o ;r aj^if >■ , vo-- | 10. observations may be shoim to be serially related, but because the sequence of prices or quantities appears to be unique. When separated from a consideration of the specific conditions which must be met by the data in each problem, such atti- tudes become an expression of the convictions of the investigator and are difficult to evaluate objectively. The randomness of residuals with regard to time is a condition of importance in the application of classical regression methods to time -series data. The formal satisfaction of this condition in a given problem can be tested, at least approxi- mately, and such tests have been carried out ia this paper. The normality of residuals, however, was not actually tested since there were too few observations on which to base such a test. flf ■■■ ' ■ ■■■■■■ ■ . ,:';i'}vi^r](!(/k .pJ-jJft'ir V-h Hi Demand for Summer Lemons The shift variables considered explicitly in the analysis of demand for lemons during the summer months were consumer income, index of temperatures, time, and volume of production. Of these only the first two variables proved to be statistically significant, but there are indications that such trend factors as growth of population during the period under review, advertising, etc., did exert an influence on the demand for susfimer lemons in this country. Consumer Income . In table 1 is given the average weekly consumption of lemons by income class obtained in some of the regional samples of the 1935-36 Consumer Purchases Study, It is seen that in these samples the relation be- tv/een family income and consumption of lemons is positive and that f,or the range of income considered, the relation is approximately linear, ly It would have been of interest to combine such budget data with time series "market" data on prices and quantities as suggested recently by Marsohak.-^V In this manner additional Information provided by the budget data could be utilized, Hovrever, while this approach may be suitable in a study of demand for goods" at the retail level, it is not well adapted to analyses of demand at other stages of the marketing process. The existence, at a given time, of a relation between consumer income and lemon consumption is, therefore, used here only to suggest that consumer income may be an important factor affecting demand for lemons. For a description of the particular income index employed (index of nonagri- cultural income payments) and a discussion of its limitations, the reader is referred to appendix A, Here it should be pointed out that both prices and income are used throughout this study in current dollars — no attempt has been made to deflate these series. It is largely a matter of individual prefer- ence as to whether demand is to be defined in money or in real (deflated) prices. It may be deemed desirable to eliminate from prices changes assoc- iated with the movement of the general price level and to consider the effect on demand for a specific commodity of changes in real rather than money incomes. However, the crude adjustments commonly employed for this purpose are of ques- tionable validity; their use may actually distort the relations which the analysis seeks to describe. Weather » The effect of weather on the demand for summer lemons has long been observed by the trade. Since the principal use of lemons during the A similar relation is fovind in the September-November data for regions cited in table 1 and in the village and farm samples tabulated by the Bureau of Home Econ- omics, See; United States Bureau of Home Economics, Family food consumption and dietary levels. Farm series, Misc, Pub, 405, Urban and village series. Misc. Pub. 452, Washington, D.C., 1941, IH/ Marschak, J, Money illusion and demand analysis, Reviev; of Economic Stat- - istics 25;40-48. February, 1943, i^/ V/orking, E, J, Indications of changes in the demand for agricultural pro- ducts. Jour, Farm Econ, 14:239-256, 1932, ,.7~.^'" . i '='rv-' '•••'lafm^^'- "il' */:.-':'r--:tr •,p-''.-)Of' c ierr3;,i? (tfi J? .■ ©•re"''.' ■• ' '-^ zrc^Hri''- cr'.->.4^if)>,..'!.^'i':: ■■^:.\eij,l-X'j-, nfi ^^:-^ . . ^.^.oL-^Jrrijijf} :^<=-(f • ■nifU:ri>fi':iti*{\:{,h f^.Mt^ci:iifrf}\ ^^^^■^■^:.^P>^ [it^Ui.p^):)l-^.^:t^:^.'ri'^'.^ r . ;;-v' ^^i;;.:;i•.-V>c;^■ -0<i:^,:V-<,:i.m^,«^s^;r*O^}y|^^^;^^^ . f vX^Vf .ti^^ S^-V -i :>■■•■■ TABLE 1 Weekly Average Consumption of Lemons by Native-White Nonrelief Families Classified by Income and Region, Selected Cities, May-August, 1936 Average weekly family consumption of lemons Income class New York City and Chicago New England and East Central Southeast West Central and Rocky- Mountain Pacific North- west Dollars Dozens 500- 999 .07 .06 ,28 ,08 ,08 1,000-1,499 .12 .13 .45 .15 .14 1,500-1,599 .11 .17 .53 .21 .14 2,000-2,999 .21 .20 .64 .25 .26 3,000-3,999 .27 .38 .78 .33 .37 4,000-4,999 .21 .41 .94 .47 .39 Source of data: United States Bureau of Labor Statistics. Study of consumer purchases. Family expenditures in selected cities, 1935-36, Volume 11: Food. Bulletin No. 648, Govt. Print. Off. T/fashington, D. C. 1940. ■ ; 'T - ^5" 'jr* '>"^ vc V X :*•♦■*'; « .I'a * *i ■ .0 13. summer months is in cooling beverages, weather conditions affecting consump- tion of cold drinks may also be expected to affect the demand for lemons* It is unlikely that temperature is the only weather factor affecting the summer demand for lemons. Other factors such as relative humidity, amount of precipitation, wind velocity, hours of sunshine, etc., may be required to measure properly the effect of weather. Furthermore, each of these variables is in itself complex, and various characteristics of their temporal distribu- tion would have to be examined to determine the most appropriate measure or combination of measures to be employed, YJith the few observations at our dis- posal, it was obviously unwise to attempt a comprehensive coverage of these factors. In the analyses to be described, w© have used the average, for a number of cities, of daily positive departures of maximum temperatures from an arbitrary reference point to represent the vreather variable. A mention of th^ reasons leading to the choice of this measure will be found in appendix A.iS' Trend Factors . Under the rubric "trend factors" are included demand influ- ences whose aggregate effect is to change the demand for lemons slowly and smoothly. During the twenty years under consideration the population of the continental United States increased by some 25 millions of persons} extensive advertising of lemons was carried on yearly^ new outlets and better marketing facilities were developed. It is reasonable to assume that in the aggregate such factors have increased the demand for lemons slowly and smoothly. Because of the difficulty of measuring the separate net effects, of such variables, only their aggregate contribution can be considered.!^ It is usual in statistical demand studies based on time series data to account for the influence of trend factors by the introduction of a "time" variable. This practice has been followed in the present study. In the cor- relation of time series the variable "time" has also a more formalistio duty to perform, namely to randomize residuals. It is, of course, highly desirable that the contribution of "time" be minor relative to the contribution of the other more explicit independent variables; otherwise a large part of the variance of the dependent variable is allocated to factors not properly identi- fied. Vol ume of Domestic Production , The variables we have considered thus far are factors which might reasonably be expected to affect the consumer demand for lemons. The statistical analyses reported in this paper, however, are oriented not toward consumer demand, but toward the derived demand, at the f.o.b. level, facing handlers of lemons. Hence, factors unrelated to consumer demand, but directly affecting the derived demand must also be considered. Such factors are certainly more difficult to measure than those already dis- cussed. While it was not possible to undertake a thorough study of these variables, some attempts have been made to test their influence. Another temperature index is described in appendix E. 15/ The influence of the population factor was partly eliminated in some of the regression equations described below by the use of per-oapita data. Such an adjust- ment, however, obviously ignores the effect of changes in the age distribution of the population. £3;'' ^riW ;-v><^cf! =ly&tfc r^i^jii '^4ndV'-7 ' ,^.,;rr- 'i./v ■.■ t:■^ ;,i.v.:' Ki;;- ,. X;^...:^ :•: .• 14 Knowledge (expectations) by dealers of the volume of lemons available for disposal as fresh fruit may exert a net influence on the level of f»o»b» prices. In years of unusually large crops a given level of shipments may result in prices substantially lower than those called forth in years of short crops (factors affecting consumer demand remaining constant) by an identical level of shipments. A widespread expectation of an unusually large crop may create an unwillingness on the part of dealers to buy extensively in the present market because of the anticipation of a fall in prices. On the other hand, a wide- spread knowledge among dealers of the shortness of the crop may induce a will- ingness to pay "scarcity" prices even though the supply of lemons in the whole- sale markets may be ample. A striking illustration of the operation of this mechanism is described in the next section. The measurement of the influence of this variable is complicated by the fact that since 1924 marketing agreements containing provisions for limitation of shipments have been in effect in the lemon industry. Because of the presence of control over volume of shipments, it is possible that a greater expression of this factor is to be found in years characterized by short crops than in years of large crops. However, until 1941 marketing agreements were not in- dustry-vri.de. Fiirthermore, in years of large crops, grower pressure to increase shipments regardless of demand conditions may be expected. Y^hatever the reasons, the fact is that, for the period under review, the year-to-year fluctuations in quantity shipped. 16/ A large crop, then, has meant more often than not large shipments and hence anticipated Icvi prices. Volume of Im p orts . Mention has already been made of the fact that during the first half of tho period under review large quantities of lemons were imported into this country. This source of supply, obviously, cannot be left out of consideration in a demand study, even though the problems created by imports are nov/ only of historical interest. Little is known concerning the substitute ability relations between imported and domestic lemons; hence not much can be said concerning the expected direction and degree of influence of the import variable. The statistical analysis may throw some light on the above-mentioned relationships. The series used in the analysis of demand for summer lemons are given in table 2, A linear form expressing f.o.b. prices as a function of total United States supply of lemons, nonagri cultural income payments, and the temperature index was found to fit the data rather closely (see equation Is, table 3). An examination of the scattering of residuals about the various net regression lines, shown in figure 3, appendix C, did not reveal marked departures from linearity, ll/ The correlation between first differences in industry annual shipments and first differences in annual production is .55. 11/ The charts given in appendix C v/ere constructed by drawing a trace of the regression surface in a plane with the dependent variable measured on the vertical axis and each of the independent variables measured on the horizontal axis. The points represent the residuals (see table 26, appendix B) as calculated from the re^ gression equation, plotted as deviations from the net regression line corresponding to the values of the independent variable at which they were calculated. production were fluctuations in At i ■ ■^5-'" , . ■ "' •.- . ex.: . . < — t-r:— - I'- r. , ■ •■; '-f -, , . '• 'r • .. .. . -• ■. •v"Ui^' •'^r t ■ ■ ♦•.■•i;;';.:»ofl ;.v .;i . - • ■• .- ' ■' ■ ■ - 3 but TABLE 2 Summer Lemons: Basic Series Used in the Demand Analysis, 1922-1941 i'er capita Index of - — rer capita Year F .o.b. United United Domestic Imports C.F.G.E. Outside nonagri- nonagri- index of May- price States States shipments for con- domestic supply cultural cultural tempera- October supply supply sumption shipments income * i ncome tures "if 1 2 0 Xp 2d Xqt Xqo cc X-, 3 0 X, o 1 2 3 4 c D c o n 1 Q 0 Q y T A dollars 10,000 pounds 10,000 10,000 T A AAA 10 ,000 10,000 per cent per cent per cent per box boxes boxes boxes boxes boxes 1922 4.48 328 2.27 235 93 201 127 88 88 100 1923 5.23 312 2.13 202 110 193 119 100 97 92 1924 2.74 369 2.48 322 47 276 93 101 96 80 1925 4.93 399 2.64 289 110 267 132 110 103 120 1926 3.23 403 2.63 328 74 300 103 113 103 88 1927 4.88 351 2.26 289 62 259 92 116 104 75 1928 4.86 350 2.22 303 47 281 69 119 104 89 1929 5.36 355 2.22 322 33 292 63 123 106 105 1930 5.18 388 2.40 333 55 308 80 111 94 138 1931 4.50 399 2.44 381 18 348 51 96 81 150 1932 4.41 312 1.90 307 5 283 29 74 62 110 1933 3.60 354 2.14 347 7 311 43 71 59 136 1934 3.85 406 2.44 404 2 368 38 81 67 142 J. A'l fi HI. D •x J. U 36 87 71 106 431 2.56 4? 7 A 408 23 104 85 151 1937 4.86 387 2.28 384 3 369 18 108 87 127 1938 2,87 435 2.55 435 414 21 99 79 115 1939 3.17 470 2.73 470 445 25 107 65 140 1940 3.04 461 2.66 461 422 39 114 90 106 1941 3.19 550 3.14 550 507 43 135 105 145 X 4.114 393.8 2.429 360.2 33,4 331.6 62.2 102.9 88.3 115.8 0.860 56.5' 0.264 82.5 37.9 79.7 36*4 16.0 11.4 23.6 * 1935-39 = 100. t 1922 = 100. Sources of data: Col. 1: Table 13. Cols, 8 and 9: Table 22. Cols. 2, 3, 4, and 5: Table 18. Col. 10: Table 23. Cols. 6 and 7: Table 21, .1 c (1 |; TABLE 3 Summer Lemons: Regression Coefficients and Auxiliary Constants, 1922-1941 (Figures in parentheses are standard errors) Net reeression coefficients * Adjusted coefficient 01 multiple correlation Elasticity 1 1 Flexibility 01 price f t Equation Dependent variable (x) Constant term c F»o»b. price! ^1 Ui.ited j States j supply Nonagri- cultural income ^3 Index of tempera- tures ^4 Is 2s ^1 ^2 4 ,o4o2 185.6147 -41.2473 (4.4092) (0.0022) (0.0067) 2.1239 (0.2387) (0.0047) 1.3774 (0.1607) 0.9050 0.9569 -0.4309 -1.9620 (x) c X 1 Fer capita United State s supply ^2 Per capita nonagri cul- tural in- come x° ^3 ^4 R 3s 4s ^1 4.8712 (C.0250) -3.6617 { 0 .3^90^ 0.0565 f 0 .0075^ 0 .0146 (0.0016) 0.0272 ( G fcC'046 ) ( 0 .0073 (0.0010) 0.9089 0.9401 -0.3888 -2.1615 ^(x) C Log x^ Log Xg Log X3 R f 5s Log 3.5555 -2.0854 (0.2385) 1.0356 (0.1747) 0.0074 (0.013) 0.8922 -2.0854 6s ^Og Xg 1.6757 -0.3965 (0.0454) 0.4865 (0.0613) 0.0036 (0.0004) u.you^ j — u.o^joa (Footnotes on next pagej a. Table 3 continued: * "^2. ~ average f ,o.b. price in dollars per box. Xg = United States supply of lemons in units of 10,000 boxes. x° = Per capita United States supply of lemons in pounds, Xj = Index of nonagricultural income payments, 1935-39 = 100. x° = Per capita index of nonagricultural income -ayments in oer cent. 3 x^ = Index of daily maximum temperature departures from 80° in 39 metropolitan districts, May through October. At the centroid. In natural logarithms. -3 18, A linear equation expressing the United States supply of summer lemons as a function of the f,o,b, prices, nonagrioultural income, and the index of summer temp- eratures (see equation 2s, table 3) was found to give an excellent fit to the data. With supply as the dependent variable, the adjusted multiple correlation coefficient is 0«9569 v/hi«h is markedly higher than the corresponding coeffif>ient of 0.9050 with price a« the dependent variable* The scatters of supply residuals about the vajfiouB net regression lines are shown in the panels of figure 4,- appendix C. It is evident from these charts that the use of curved net regressions would not mark- edly improve the fit of the equation. The results of tests of randomness of residuals derived from the two equations are found in table 27, appendix B. Neither the magnitudes of the lagged correlatiq;i coefficients nor the probabilities derived from the Wallis-Moore test force the rejection of the hypothesis that the residuals constitute a random series. In both equations Is and 2s as, in fact, in all of the regression equations discussed in this paper — the signs of the net regression coefficients agree with the direction of influence expected on a priori grovinds. Nonagrioultural income pajipients and the temperature index are related directly to both the f «f>»b, prices and the United States supply of summer lemons, while the two latter variables are related inversely* The inverse net price-supply relationship is, of course, in agreement with the demand relationship discussed in theoretical economics. The direct variation of both prices and quantities with income and temperature is also entirely reasonable. An increase in purchasing pov/er should express itself either in higher prices of lemons, if the quantity of lemons purchased remains constant, or in larger quantities of lemons purchased if the price remains constant. A similar relation is evident for the seasonal demand factor, temperature. Of some interest is the determination of the relative contribution of the several independent variables. In almost all statistical demand studies the two variables of prim© importance are price and quantity} the investigator usually J '^^rva^^'k^ rl'^--i^t. sf^m^ B^S^vJ . ■ ., j^.. , ja ...rji t--'"' ■• ■■• rift* i-i>-;. ;/.v; , vlciri ti: sjc^ . ' ..„]?;. v;;; '• ^ ■■ • "..cv vi;; ;u -:v; -w;; o';-- ^i^jj:'.? :: •^ ■; rarr a;/ • ■■ ■" ' •' ••■ ■ — ■. * — • -.W i;/-: exv, r,.^. .-'V-':' al ■■ '■•■v;.;; •■ . •: ■,'iJ,;i-.; :;rvi/ / "ii-;./;, ■x-'. -i-., -.r'l.: \- .i'liai i- Or:. 5^10 : . ; .;■ :. : '■ .' , ;• rr:-;v;' ■ ■ "c •-. . . . ,-::':-r-ui'y'y --iiT'i-; - A:: 'r l-..-. • , ; a' J.- ;-: — •; -;- ?v;: .i;z<«t^-";^ ••.w.-:>;^.j,rii rJ o .;;;vi-:-.J : i.:. , ^ ■;■;::) v ■ '. :■ • ■ . Id, expects a significant part of the variation in price to be associated vdth the variation in the amount sold, and inversely. The supply-price relationship is also the most strategic from the point of view of policy, fcr these are ordinarily the only two variables in the formulation which are capable of being controlled. Of interest is also the contribution of the temperature factor to price and supply variance, since this non-eoonomio seasonal factor has been given great prominence in the discussions of the demand for siimmer lemons found in trade publications* The net regression coefficients in the two equations cited above cannot be taken as they stand to measure the contribution of the independent variables in "accounting" for the price or supply variance. In fact, the problem of measuring the relative importance of independent variables may not have a unique solution; certainly, none of the measures at present available can be deemed satisfactory. Of the several measures of relative importance commonly employed, the coefficient of separate determination, d. . ■ iff . . r^ j, is, probably, the least ambiguous because of m 2 its additive property^ d^j = R^, where is the multiple correlation coefficient. The ambiguity introduced by the occasional appearance of negative values of d^^ can be removed by a minor adjustment which does not disturb the additive property of this coefficient. 18/ The first two rows of table 4 give the coefficients of separate determination for equations Is and Zsi It will be observed that either f .o.b, price or the quan- tity of summer lemons accounts for the major part of the variance of the dependent variable. This is particularly evident in equation 16 in which approximately 73 per cent of the variance of f.o.b* prices is allocated to the United States supply > 9 per cent to nonagrioultural income, and only 2 per cent to the temperature variable. 18/ o i di . , , V — ' The adjustment is of the form d, » d. . ^"^^T < (ijjtj)* ij I 10 131 J^'a'ilii ibtfb'rVi iikilfi;^/ li" It. ii-'' -■■''■'sXiiir -j/w ■ ■ ■ _ 20. TABLE 4 Summer Lemons: Coefficients of Separate Determination Equation U6 peii 0.011 u variable F.o.b, price United States supply Per capita United States supply Index of nonaf,r i- cul bural inc ome Per capita nonagri- cul tural income inaex oi tempera- tures ^1 ^2 -I ^3 0 X 3 ^4 Per cent Is ^1 73.4 9.4 2,0 2e ^2 35.9 27,3 29,7 3s ^1 58.8 24.7 1,9 4s x° 2 41,1 26,5 22,6 58 Log 75.6 6,8 0,4 6s LOG 35.7 24,7 31,6 Source of data: Based on equations Is through 6s given in table 3, Coefficients based on equations Is and 3s adjusted to remove negative signs. I i 7, ■'.'•rr;, -in .•.xrna i 21. The variance of quantity is partitioned more evenly among the independent variables in equation 2s j the allocations to f»o»b« price, income, and temperature variables are respectively 36 per cent, 27 per cent, and 30 per cent. Not much confidence can be felt in the reversal of the relative importsince of the income and tempera- ture factors in the two equations, for such results are not obtained when other 19/ measures of relative importance are employed. — ' In view of the great prominence given weather factors in trade descriptions of the behavior of prices of summer lemons, it is somewhat surprising to find that the foregoing rough measures assign relatively little importance to the temperature . 20/ variable. — < However, it is doubtful whether the total effect on demand for lemons of changes in weather conditions has been or could be ascertained by means of the temperature index we have employed. As was stated previously, temperature is not the only condition of relevance; furthermore, only one measure of the temper- ature variable was used. On the other hand, the possibility that the effect of weather has been given undue emphasis in trade discussions of interseasonal varia- ^. ^ , . , 21/ tion or lemon prioee and shipments cannot be ruled out of consideration. — ' The values of the coefficient of flexibility of price calculated from equa- tion Is at the centroid and of the coefficient of elasticiiy of demand calculated 22/ from equation 2s, similarly at the centroid, are given in table 3. — • Demand for 19/ The squares of the pertinent partial correlation coefficients are: r?o = 0.8454, r^, = 0.7350, r^ „ = 0,6910, = 0.8319, and r^ 12.34 13.24 14.23 23.14 24.13 0.8212, Y\rith regard to goodness of fit of the equations, temperature is an important variable, for its omission results in a marked reduction of fit. For example, with temperature omitted the adjusted coefficient of multiple correlation (price depen- dent) is only 0.6699 — a considerable reduction from its former value of 0,9050, 21/ __/ The statistical results pertain to interseasonal behavior of prices and quantities; temperature may v/ell be the major factor affecting the intraseasonal variation of prices and shipments. 22/ cp = 3- " p ^2 and H = ^""2 % = b^, ""l , where the ^ ^^2 ^ ^ % 21 ^ x's are means and b^^ is the net regression coefficient (i denoting the dependent (Continued on next page) ... • . .•.r-n^-.U' . . -. TV: 22« (Footnote 22/ continued) variable). For significance tests in connection with f and rj. see: Girshick, M»A« The application of the theory of linear hypotheses to the coefficient of elasticity of demand. Jour. Amer, Stat. Assn. 37:233-237. 1942, summer lemons at the f.o.b. level, based on these equations and measured at the centroid, is seen to be inelastic. Since in the linear formulation the values of flexibility and elasticity change from point to point on the regression hyperplane, it is of interest to consider the magnitudes of the coefficients of flexibility and 23 / elasticity in each of the years included in the analysis. — As can be seen from the enclosed tabulation (table 5), the absolute values of the seasonal coefficients of flexibility are all greater than 1 and correspondingly the absolute values of the seasonal coefficients of elasticity are all less than 1. While the reciprocals of the coefficients of flexibility do not agree exactly with the values of the elasticity coefficients, differences are to be expected from the very nature of the two estimation equations. It will be noted further that there is a trend in the calculated values of . This trend is due to the upward trend in supply, for it is the property of negatively sloped linear demand laws that flexibility increases 24 / with an increase in quantity. — ' Obviously, the trend in the calculated values of 1^ is opposite to that for (|3 • ' The findings with regard to coefficients of elasticity and flexibility accord with a priori expectations concerning the nature of the demand for lemons. Lemons are used in various drinks, in cooking and baking, for flavoring and seasoning, as garnish, as salad dressing, and as hair rinse. In all but a few uses lemons are 23/ The values of ^ and for year t were calculated from the expressions <f =» b,o and h = b„, ^"^^ where x' is the estimated f.o.b, price 1 t l** I ''t '^^ it Y ^ X' It 2t » for the year t calculated from equations Is and x is the estimated supply calcu- lated from equation 2s ^ It is possible, however, v/ith a multidimensional linear equation that <^ is not a monotonically increasing function of quantity. X bst. TABLE 5 Summer Lemons: Calculated Coefficients of Price Flexibiliby and Demand Elasticity, 1922-1941 Year Hay-October Price flexibility Demand elasticity \x Per capita formulation Price 1 xexi D 1 -11 uy Demand e last ic ity K 1 2 A •± 1922 -1.4668 -0.5678 -1,9558 -0.4611 1923 -1.2803 -0,6981 -1,5444 -0,5635 1924 -2.1408 -0.2845 -2.6788 -0,2418 1925 -1.8397 -0.5335 -2.2533 -0,4544 1926 -2.2974 -0.3221 -2.7914 -0.2808 1927 -1.6245 -0.6027 -1.8349 -0.5114 1928 -1.4414 -0,5561 -1.6129 -0,4871 1929 -1.3222 -0,5969 -1.4542 -0,5302 1930 -1.5228 -0,5371 -1.7097 -0.4892 1931 -1.7522 -0.4521 -1.9465 -0.4165 1932 -1.4719 -0.5823 -1.5776 -0.5244 1933 -1.7772 -0.3957 -1.9300 -0.3656 1934 -2.2918 -0,4026 -2.4957 -0.3728 1935 -3.1826 -0,3713 -3.3884 -0,3427 1936 -2.0098 -0.4342 -2.1304 -0.4059 1937 -1.6527 -0,5147 -1.7073 -0,4808 1938 -2.8976 -0,2716 -2.9927 -0.2563 1939 -2.8169 -0.2753 -2.8725 -0.2626 1940 -3.1942 -0.2797 -3.1420 -0.2653 1941 -3.5595 -0.2435 -3.5378 -0.2362 Source of data: Calculated from equations Is, 2s, 3s, and 4s respectively for the values of the independent variables which prevailed in the various years. 1 i ^, - ; ■ . ' • . . : : p " " 7" • ■ ■ ' ' ' i . ■■ ' f r •■■ . ' * ; : .. .. . •i : • ' ■ ' '! . ' .". ■ • *■ " ' • i V • . ■ . i ■ ■ . . i i I \.. i : -.-•.Trsr'.-.-;;rr.7r-.— vjr.iiii ;-- 24. an important ingredient but at the same time a minor element of the cost of the total product.. This vrould lead one to expect that consumption of lemons is not highly responsive to changes in their prices, that is, that the demajid for lemons is inelastic. The use of equations Is and 2s for estimation purposes should be accompanied by some measure of the uncertainty of the estimate. The unbiased estimates of the 2&/ variance of the best linear unbiased estimate of and Xg are given herewith,. — 2 2 2 2 M = 0.0070 + 0,0153 z + 0.0115 z + 0.0124 z. - 0.0162 zoZ, - /^f-|_ 2 3 4 ^ -0.0179 ZgZ^ + 0.0113 ZgZ^ and 2 f. 2 /J^ " = 14.18 + 14.39 2^ + 14.62 + 14.43 z" - 3.36 z^z, + 2 '3 2 ^4 '1''3 +0.335 z,z^ + 3.72 z z, 14 3 4 where the z^'s are the "noxTnalized" values of the independent variables for which t x-^ or is estimated. The statistical uncertainty accompanying the estimate x is t f then expressed in the form of the confidence interval x - j^f^h $ ^ * ^ac/^i" where t^ is the Student-Fisher t at the level of significance r)C»and 1 - oC is 2 the confidence coefficient. The expressions for yt>t^ will not be stated explicitly for the other estimation equations to be considered in this section. However, / 2 = ^D: Ifi.lJPl ivhere IR| is the determinant of the matrix of inter- ' H (N-n)|Rii|2 correlations between the n variables, jR^ij is the minor of in iRj , |d! is the determinant of the matrix D -1 ^2 ^3 • ^n 1 ''23 • . » *"2n H ^23 1 . .. ^Zn '^2n '^3n - . • 1 1 — . 2 and z. = i_ (x^ - x^^). The formula for used here is equivalent to the expres- ^ 0\ / ^1 sioa derived in: David, F. N.. and J» Neyman. Extension of the Markoff theorem on least squares. Statistical Research Memoirs 2:105-116. 1938. -. : / ;:.y; r..V^-'Cf!i**.l?C'; a-Si.iln.'IV'. • 5; : si. . L ^ 25*- sufficient information is given in this paper to enable the calculation of the standard error of the regression equation for any of the estimation equations given , 2 2 in tables 3 and 8, Attention should also be drawn to the fact that ( + S^), each i v/here S. is the unbiased estimate of the variance of/x^, is the appropriate variance 26 / for determining the confidence limits of the future observation. — i An attempt was made to introduce a measure of the factor discussed in previous pages under the rubric "volume of production." First differences in annual production of lemons and first differences in California Fruit Growers Ex- change lemon picks and the Exchange May 1 storage were not found to be related to either price or quantity residuals secured from equations Is and 2s respectively. These measures, then, if introduced as additional independent variables in the equations vrould not reduce significantly the "unexplained" variance of the dependent 27 / • variable. — ■ Hovrever, first differences in Exchange May-October lemon picks appeared to be related to price and quantity residuals. It is difficult to see why first differences in picks alone should bear a significant net relation to prices or shipments, since May 1 storage has been an important constituent of the supply of lemons available for shipments during the summer months. Because complete data on — ^ See Eisenhart, C. The interpretation of certain regression methods and their use in biological and industrial research. Annals of Math. Stat. 10:162-186. 1939. Also, Ezekiel, Mordecai, Methods of correlation analysis. Second edition. Chapter 19, John YiTiley & Sons, Inc., New York, 1941. "^iJ Let j be the additional independent variable under consideration and i the dependent variable. Then where r . is the correlation between residuals and the additional variable, Rj is the coefficient of multiple correlation between j and the other variables including i, N is the nvmiber of observations, and n the number of constants in the regression equation. The magnitude of t may be deduced from this expression knowing Tgj and assigning some plausible value to R.. — •■■ .•c'-". : - - , • ,\- -.i>- . .... mI.TTV -- ..^ tof: y -^^ ^, 26. lemon piftks foSr the pdi^iod under review were not available at the time these cal- culations vrere made, further consideration of this factor was not attempted, An attempt was also made to account for the influence of the trend factors by- introducing "time" as a separate variable in the estimation equations. In the equations given below, is f,o,bo price, Xg United States supply of summer lemons, Xg index of nonagricultural income payments, x^ the index of summer temperatures, and Xg "time" measured in units of six months from an origin midway between 1931 and 1932, The figxires in parentheses are standard errors of the net regression coefficients. It is seen that the added variable does not have a significant in- fluence. With f,o,b, price as the dependent variable, the standard error of the coefficient of Xg is greater than the coefficient. In the other equation^ the ratio of the coefficient of x to its standard error (t = 1,586) falls short of 5 the value of t corresponding to the 5 per cent level for 15 degrees of freedom x' = 4,5214 - 0.0210 x„ + 0,0450 x^ + 0,0279 x. + 0.0032 x^ (0,0030) (0,0072) (0,0050) (0,0129) t X = 206,4872 - 36,3209 x. + 1,9942 x, + 1,1373 x^ + 0,7911 x^ 2 13 4 5 (5,2356) (0.2424) (0.2157) (0.4988) As vj-ill be indicated in some detail below, "time" was found to be an important factor in the analysis of the data on winter lemons. The question naturally arises as to v/hy trend factors fail to play a role in the analysis of the demand for lemons during the summer months. Should it be concluded that the factors which "time" is presumed to represent did not affect the demand for summer lemons during the period under consideration? 28/ — ' The scatter diagrams of residuals against first differences in picks were based on only eighteen observations. 7. . 27. A partial answer to this question is found in the happenstance that the index of temperatures exhibits an upward trend during the period under review. This trend may have, to some extent, played the part of the catoh-all "time" variable, A further circumstance to be considered in this connection is that the quantity variable used in the summer equations consisted of domestic shipments of lemons plus imports for consumption; whereas in the winter equations the quantity variable consisted only of domestic shipments. The use of United States supply, rather than domestic shipments, as the quantity variable makes the trend factors less important, because the trend in imports in the period under consideration runs exactly counter 29/ to the strong upward trend in domestic shipments, ' The results secured when domestic shipments of summer lemons are employed together with "time" constitute an empirical confirmation of this point. These equations, both vdth price and with domestic shipments (x^^) as the dependent variables, are given below. The figures in parentheses are standard errors, x' = 5.492 - 0,0209 x , + 0,0320 x^ + 0,0247 x. + 0,0691 x^ I 2d 3 4 o (0,0030) (0,0062) (0.0049) (0,0207) x' = 253.95 - 36,3242 x, + 1.3769 x„ + 0.9857 x. + 3.9632 x^. 2d J. o t o (5.2951) (0.2451) (0,2182) (0.5045) "Time" (x^.) is novr significant in both equations and the adjusted multiple correla- tion coefficients, 0.8969 for price dependent and 0.9813 with domestic shipments as the dependent variable, are of about the same magnitude as those previously secured. Thus it is questionable whether economic meaning should bo road into the failure of "time" to contribute significantly to the fit of the sumr.ier equations. 29/ The following equations represent the linear trend in domestic shipments of summer lemons and United States supply of sxammer lemons, respectively. Both quan- tity series are in units of 10,000 boxes, x' = 360.23 + 6.5921 t (Domestic shipments) 2d x' = 393.80 + 3.6865 t (United States supply) 2 i. its (r- 28. It appears rather that the explanation lies in the charaoteristics of the variables used. . The reality of the upward trend of summer temperatures is of no particular relevance to the problem of eliminating the effect of trend factors. It is of primary importance, however, in connection with the proper measurement of the effect of the temperature factor, ■ A detailed study of the methods employed in the construction of the temperature index failed to reveal how the trend in the index could be an artifact of the procedure employed in its construction. Nevertheless, the possibility that it is an artifact cannot be completely ruled out of considera- 30/ tion. — ' Thus the question of the separate contributions of the smooth and erratic components of the temperature index must be considered. The case for temperature as a significant factor affecting demand of lemons during the summer months would be strengthened if it could be shown that each component of the temp- erature index makes a significant contribution. The answer to this question is already given indirectly by the equations just cited, for in those equations both the temperature index and "time" are statistically significant. More direct evidence is obtained by separating the smooth and erratic components of the temperature variable and introducing each into the regression equation in conjunction with other variables previously employed. The smooth component of the temperature index was defined by the least squares equation x' = 115,75 + 1,2402 x^ where x^ represents "time," and the erratic component by 4 5 5 the algebraic residuals of the temperature observations from this trend line t 31/ (x^ - x^),'~^n the regression equations given below, x^ is f.o.b.. price of summer lemons, Xp is the United States supply of summer lemons, x is the index of In this connection see appendix E in which a different temperature index is used as an independent variable. ^ k parabolic regression of tompcrature on time r.-ould have probably described the smooth component of the temperature index more fully, Hov;ever, the linear trend is an excellent approximation to the smooth component of the index. as ..7 Tti C 3i 29. nonagricultural income payments, Xr- = — i (x - 115.75) is the adjusted linear "^ 5 1.2402 4 ^ component of the temperature index and z its erratic component. A comparison of the coefficients of Xg and z with their standard errors (given in parentheses) reveals that both coefficients are significant. x^ = - 0,0209 Xg + 0.0451 x^ + 0,0374 Xg + 0.0278 z j (0,0030) (0,0073) (0.0131) (0,0050) I t X = C„ - 36.3057 X, + 2.0012 x_ + 2,2011 Xc + 1.1358 z 2 2 1 3 5 (5.2945) (0,2455) (0.3687) (0.2190) It is likely that groi/rbh of population is a major trend influence; hence by using per-oapita data the effect of at least this trend factor may to some extent be given implicit consideration. Two new variables, per-capita United States supply of summer lemons and per-capita nonagricultural income, were used, replacing United States supply and nonagricultural income, respectively. The net regression co- efficients and other pertinent statistics for the analysis of the per-capita data are given in the third and fourth row of table 3 (equations 3s and 4s). The net regression lines and scatters of residuals derived from these equations are shown in the various panels of figures 5 and 6. TThere comparison is possible the results are seen to be remarkably similar to those secured in the original analysis. The adjusted coefficients- of multiple correlation differ but slightly from those pre- viously secured. The estimates of flexibility and elasticity at the centroid are consistent with the values derived from equations Is and 2s. The sajiie statement applies to the annual flexibility and elasticities shown in columns 3 and 4 of table 5, The use of per-capita income has apparently given income greater relative importance in equation 3s than it had in equation Is (see table 4). The significance of such a difference, hovrever, is difficult to establish* The per-capita formulation also allows us to test the hypothesis that the population factor is the only trend factor of significance in connection with summer 30. lemons* If this hypothesis were true, a trend factor, such as the trend component of the temperature variable, would fail to exert a significant influence in the per- oapita formulation. The introduction of the trend and erratic components of the temperature variable (x^) into the per-capita estimation equations replacing x^ leads to the following equations (in normalized form): I 1 z = -1.0647 z° + 0.9059 z° + 0.3650 z^. + 0.6122 z 2 3 5 (0.1464) (0,1426) (0.1461) (0.1070) and 0 o Zg = -0,7318 z^ + 0.8025 + 0.4206 z^ + 0.4985 z . ^,.^..„ . v..^ (0.1006) (0.0930) (0.0949) (0*0921) where = (x.^ - x^)/(r^, Zg is the trend component, and z the erratic component of the temperature variable. The figiores in parentheses are standard errors. Since for 15 degrees of freedom t at the 5 per cent level is 2,131, it is seen that the coefficient of z_ in both equations is statistically significant. This result suggests that growth of population is not the only trend factor that may have shifted the demand for summer lemons upward during the period under reviev/. Thus far all of the equations were based on a formulation in which the dependent variable was treated as a linear function of the independent variables. A linear form in the logarithms of the variables has also been frequently employed in statistical demand studies, mainly, one supposes, beoause it leads to constant coefficients of elasticity and flexibility. The net regression coefficients and auxiliary constants secured when the logarithmic form was fitted to the data are given in the last two rows of table 3. Expressed on the arithmetic scale, equations 5s and 6s are as follows i 359.3 -2.0854 1.0356 0.0074 x^ "3 Cg = 4.739 x^ -0.3965 0.4685 0,0036 X4 (5s) (6s) oris- s-?JrT{-^' ■ ('..« h.Ti"?cJ to rrri-ir-.r-'-oic'-rii' r"*'T nXr'.Trar''i s/ut -iaiaiLj 4i0Oi3 :.• i. e -ijjc iayvai ,<■•■, ■.■rH') ••x.s*^j ci t. _ , .;ra 'to w-w-'ivea •?xQ{jny .boi-xsu a'c-jf' •ij'ii»r!v-i> 0-ti*&<.jt/ ^nrBm ^tija^vw;-. -ao'i . ^ii •.v«^'i-x ;0 31. The logarithmic form fits the data about as well as the linear form (see the adjusted multiple correlation coefficients in table 3), The demand for summer lemons at the f,o»b« level is again shown to be inelastic* This is apparent from the values of the coefficients of price flexibility and demand elasticity shown for these equations in table 3, It will be noted that the values of these coefficients, differ but little from the values of price flexibility and demand elasticity at the oentroid, derived from the previous equations. To test the possibility that trend factors may have a measurable influence in the logarithmic form, "time" was introduced linearly as an independent variable. The results secured were very similar to those obtained when "time" was introduced in equations Is and 2s. The ratios of the net regression coefficients of "time" to their standard errors in both equations of the logarithmic form fell short of the 32/ t's corresponding to -tiie 5 per cent level for 15 degrees of freedom. — ' It thus appears that also in the logarithmic form trend factors are of little significance, once the effects attributable to the other independent variables are eliminated. Tv/o other sets of regression equations require a brief comment,. In the first set» imports of lemons are treated as a separate variable; in the second, the supply series is broken dovm into the quantity of lemons marketed by the California Fruit Growers Exchange, and the quantity sold by other marketing agencies. The shipment data used in these formulations are cited in table 2, The equations, vJhen domestic shipments of summer lemons and imports for consumption during the months May through October are used as separate variables (see equations 7s and 8s of table 6), show little improvement over the original formulations. The values of the adjusted multiple correlation coefficients are of I'irith the logarithm of f.o.b. price as the dependent variable, the ratio of the net regression coefficient to its standard error was less than 1. In the other equation, the t-ratio vms 1.653. The t-ratio corresponding to the 5 per cent level for 15 degrees of freedom is 2.131. •. . • ... ■■'.. -■■i-. ' : ■ ■ ■ ■■ . : ■■ . • ■ • •• ■ a.!-'i>'.fcj. i-j.'.j- .. • -,. . .yiisj. '. ^.'■'.v- ii/^' i.^ •i-.— vr.- .*-<srA" V ■ .v^i «. ■.t-.'-.v^wu .■--r.-^'- l.-'WjAl.JaVr J-nIJ-/,. ■ ., v ., "V - .■ •itfwfii'^^v oL^ift^t. f^x\f'--'^ TABLE 6 r Lemons: Regression Coefficients and Auxiliary Constants for Additional Equations, 1922-1941 (Figures in parentheses are standard errors ) Net repression coefficients * Adjusted Equation variable 1 Constant term F.o.b. price Domestic shipments Imports for consumption N onagri - cultural income Index of temper- atures coefficient of multiple correlation Elastic- ity of demand Flexi- bility of price (x) c ^1 2d x„. 2i x^ 3 R a ^ r t 7s ^1 4.0437 -0.0196 ^0 .002o; -0.0172 \ u •uufiy } 0.0423 \ U .UU 1 ^ J 0.0289 (0.0048) 0.9022 -1,5076 8s ^2d 185.1003 -40.5952 [ 5 .3425; -1.0280 2.1283 1.3611 (0.2011) 0.9788 -0.4636 Exchange domestic Outside ^(x) c ^1 shipments X 2e shipments ^2o X 3 ^4 R ^e ' e 9s 1 4.1525 -0.0201 (0.0024) -0.0188 (0.0046) 0.0435 (0.0072) 0.0287 to. 0050) 0.8996 -1.6205 10s ^2e 188.3385 -40.8252 (4.9562) -1.0308 ! (0.1385) 2.1274 (0.2466) 1.3523 (0.2004) 0.9772 1 -0.5065 * X = Seasonal average f.o.b. price in dollars per box. X = Domestic shipments of lemons in units of 10,000 boxes. 2d ^2i ~ Imports of lemons for consumption in units of 10,000 boxes. Xgg = Domestic shipments of lemons by the California Fruit Growers Exchange in units of 10,000 boxes. XgQ = Domestic shipments of lemons outside California Fruit Growers Exchange plus imports for consumption, in units of 10,000 boxes. X = Index of nonagricultural income payments, 1935-39 = 100. 3 x^ = Index of daily maximum temperature departures from 80° in 39 metropolitan districts. May through October, At the centroid. ' r ~:..- — , 1 ..... -.^^2;^ ■ ... .^1 I I i. :.' 1 ■ 1 *• * - K - - ; ^ - 3 ^ 3 ■r' ■ -.-.ilio^'iJ^^:..- . 1 ■ X .. ; ;. ... ; J. , V 33. about the same magnitude as those previously secured, and the residuals in the two different price equations (equations Is and 7s) agree precisely in sign and closely 33/ in magnitude, — i The coefficients of domestio shipments and imports for consumption agree closely in magnitude. The difference between them in relation to the magni- tudes of their standard errors is so small as to obviate the necessity of testing its significance. Thus, for the period May through October, domestic shipments and imports of lemons for consmnption are not distinguishable in their contributions toward the estimation of f ,o,b, prices. The second formulation attempts to test a similar hypothesis with regard to lemons marketed by shippers other than the California Fruit Growers Exchange, The multiple regression equations relating to this formulation are given in the last two rows of table 6 (equation 9s and 10s), In these equations, domestio shipments of lemons by the California Fruit Growers Exchange (x„ ) and shipments of lemons 2e for fresh consumption by other marketing agencies plus imports for consumption (x„ ) u* O replace the United States supply of stmmer lemons as the quantity variable. Since the difference between the coefficients of Exchange domestio shipments and outside shipments is small and not significant (see equation 9s), the hypothesis that lemons marketed by the California Fruit Growers Exchange and other lemons si ipped for fresh consumption during the summer months are statistically differentiable components of the total United States supply of lemons must be rejected. The correlation betiireen price residuals of equations Is and 7s is ,98« According to the "special theory of related demands" (see: Schultz, H, Theory and measurement of demand, 569-604p, University of Chicago Press, 1938), the negative sign of the coefficient X2o indicates that lemons marketed by the Exchange and lemons marketed by other shippers are competing. The equality of the coefficients suggests that lemons marketed by the two groups of shippers are per- fectly competing. The other conditions on the parameters of the linear demand function discussed by Schultz cannot be tested because of lack of data on prices received by shippers outside the Exchange, 34. Demand for YiTinter Lemons Regression equations in which price of winter lemons was expressed as a linear function of doraeatic shipments of lemons, consumer income, and time were found to give a poor fit to the data. Equally unsatisfactory were the equations in which the logarithm of price was expressed as a linear function of the logarithms of shipments, income, and time. A study of the residuals derived from these equa- tions revealed that the fit could not be improved by any reasonable modification of the form of the net regression functions. Apparently oystematic factors, in addition to those considered above, have affected the demand for lemons during the winter months* References to the prevalence of colds and influenza as a variable in account- ing for the up-vmrd shift of demand for winter lemons are not infrequently found in citrus trade publications* Hot lemonade is probably one of the common home remedies for colds, a use of lemons which has received extensive mention in the advertising campaigns of the California Fruit Gro^rers Exchange. Accordingly, it appeared worth while to introduce a measure of this factor into the regression equations. Since statistics on the incidence of colds are not available, only an index of the incidence of influenza could be constructed (see table 25, appendix /). This index was introduced linearly into the regression equations, and also some non-linear transformations of this variable to both algebraic and transcendental functions were used. It appeared, however, that no reasonable function of this index v;ould be statistically significant. First differences in annual production of lemons and first differences in Exchange lemon picks and Exchange November 1 storage were considered as possible independent variables, but their contribution did not prove significant. Other variables used in the regression equations, along v/ith price and supply, were storage rates of lemons and various combinations of data pertaining to storage, picks, and shipments during the winter months. However, these variables also proved ■ : ■' ■'•'r-'f'v-'y'k'S:'!:?!'' .(.'i.f.:l''.r.'- OT t>5' '^'/;v, ^ • ■ : . ■ ■ ■ ■ ■ -.■j'i'ti/Oo?).« ".'Ti oiJM f .. ■■- ♦i'l- *, ■<}.-?v-».-- C'5i ■ S! f ■ .'9 1 -Ift 35. to be of little significance in the analysis* These unsatisfactory results may be due to the fact that the unit of time employed is too long a period in which to con- sider relations among picks, average storage rates, and shipments because of too extensive double counting. In a long enough period, the same lemon may be counted in picks, storage, and shipments as it passes through each of these stages on the vra.y to final consumption. The failure of the influenza index to improve the fit of the regression equa- tions may be due to poor reporting of influenza case rates, but the possibility must be considered that even adequate data on incidence of influenza may not provide a measure v/hich would prove to be statistically significant. Apparently the attack 35/ rate of influenza is not related to that of common colds. Thus, v^hile the incidence rate of influenza may itself be related to consumption of lemons, because of the relatively small frequencies involved the net effect of this factor may not be statistically measurable. In searching about for a variable v/hich could take the place of an index of incidence of colds, it occurred to the wTiters that severity of winter climate may be related to incidence of colds and consequently to consumption of lemons during the winter months. Some evidence may be cited in support of this conjecture, particularly with reference to temperature. To test this conjecture statis- tically, an index of average temperatures for the months December, January, and 35/ Frost, Tf. H. and M. Gover. Incidence and time distribution of common colds in several groups kept under continuous observation. Pub. Health Rep. 47:1815. September 2, 1932. These virriters found no relation betvireen incidence of colds and influenza in the groups \inder observation. 36/ Gover, M., L. J. Reed, and S. D. Collins. Time distribution of common colds and its relation to corresponding weather conditions. Pub. Health Rep. 49:811-824. July 13, 1934. In this study deviations of the weekly incidence of respiratory attack during an eighteen-month period among students in various universities in the United States vrere correlated with deviations of v/eather conditions from mean or normal. A weak but consistent inverse relation vms observed betiveen weekly de- partures from the norm of the respiratory attack rate and deviations from mean temperature. Other vj-eather conditions such as hours of sunshine, wind velocity, precipitation, etc., did not show a consistent association with incidence rate of colds. i tit ti&CSJ^.O i-.^Vj6 J... 36, February was constructed and introduced as an added variable into the regression equations. This variable proved to bo statistically significant. Before going on to the discussion of the regression equations, a point of procedure must be emphasized. The period considered is that beginning with the season 1921-22 and ending with the season 1940-41, but one season is omitted, namely 1936-37, This omission requires somewhat extensive discussion, for v;ith only twenty observations at our disposal, the rejection of one season represents a loss of 5 per cent of the total information* In January 1937 severe frosts occurred in southern California causing pre- sumably extensive damage to the lemon crop. Successive forecasts of the 1936-37 lemon crop by the California Cooperative Crop Reporting Service provide a quantita- tive expression of current judgment concerning the extensiveness of the damage. The forecast of the crop as of January 1, 1937, just prior to the freeze, was 8,316,000 boxes. The forecast as of February 1, just after the freeze, was about 2,500,000 boxes less, namely 5,832,000 boxes. The forecasts of the volume of crop 37/ on March 1 and April 1 were of about the same magnitude aa on February 1. V/ith the prospects of an extremely short crop facing the trade, the movement of lemons was brisk and the prices received, relative to the quantity moved, were quite high, In point of fact, however, the frost damage to the lemon crop vra.s not as extensive as was believed. The final production figures for 1936-37 were 7,579,000 boxes or almost 2,000,000 boxes above the February, March, and April forecasts. ^ 37 / — i California Cooperative Crop Reporting Service. California Citrus Fruit Report as of January 1, 1937; February 1, 1937; March 1, 1937; April 1, 1937. Sacramento, California. (Mimeo.) 38/ See: California Fruit Growers Exchange. Annual report of the general manager for the year ended October 31, 1937. 39/ California Cooperative Crop Reporting Service. California citrus crops production and utilization estimates, 1909-10 to 1937-38. Sacramento, California, September 15, 1939. 1 'u •iv »8i. ii* fan 37. It is believed that this combination of factors ^ms unique in the twenty-year period under review. As an alternative to omitting 1936-37, the domestic shipment figure for that season v;as reduced to the level of expectations based on the just quoted fore- casts of the lemon crop by the California Crop Reporting Service, The use of the adjusted shipment figure gave results almost identical with those secured when the season was omitted. Because of the some¥/hat arbitrary character of the adjustment. stants corresponding to six equations, each based on nineteen observations. The series used are given in table 7, The equations are of the same form as those 4-1 / calculated for summer lemons, — i It will be noted that the adjusted multiple cor- relation coefficients are of about the same magnitude as those secured for analogous summer regression equations; that the linear form, per-capita formulation, and the logarithmic form fit the data equally ^toII; that the estimation equations with shipments as the dependent variable fit the data better than the regression equation with f,o,b, price as the dependent variable; that all of the net regression co- efficients are apparently significant; and that the demand for winter lemons — The net regression coefficients and auxiliary constants v/hen all of the observations v/ere used and v/hen the 1936-37 shipment observation was adjusted are given in table 30, appendix D, It should be pointed out that in calculating per-capita domestic shipments and per-capita nonagricultural income, population figures as of July 1 were employed. In this analysis it would have been formally more correct to have used population figures adjusted to January 1 of each year. In view of the fallibility of the annual population estimates, especially of the estimates of farm population, such an adjustment v>ras not deemed v;orth v/hile. Later calculations indicated that the use of population figures interpolated to January 1 of each year would have had the following effects on the two per-capita series 9 per-capita doraestio shipment figures would each have changed by 0.01 and 5 values of the index of per-capita nonagricultural income v/ould have each ohai:ged by 1 point. Such changes could have but an entirely negligible effect on the magnitudes of the various statistical con- stants calculated in the winter lemon analysis • however, it was deemed preferable to omit the season. In table 8 are given the net regression coefficients and auxiliary o on- 40/ 'ia at-, TABLE 7 ■winter Lemons: Basic Series Used in the Demand Analysis, 1921-22 to 1940-41 ! - ■■ j TnHpv of P p y CA-D 1 1 a Year F.o.b. Domestic Per capita C.F>G.E. Outside nonagri- nonagri- Index of November- price shipments domestic domestic domestic cultural cultural tempera- April shipments shipments shipment" income * income tures f" "^1 ^2 ^2 ^2e ^2o ^3 x° ■ ' -^4 1 2 3 4 5 6 7 8 dollars 1.000 pounds 1.000 1,000 per cent per cent per cent boxes boxes boxes 1921-22 3.48 1,641 1.14 1,423 218 71 71 81 1922-23 4.20 1,308 0.89 1,226 82 83 81 78 1923-24 2.38 1,911 1.28 1,754 157 92 88 85 1924-25 3,87 1,648 1.09 1,491 157 93 87 82 1925-26 3.33 1,750 1 ,14 1,623 127 100 91 80 1926-27 2.73 1,954 1.26 1,570 384 102 91 82 1927-28 4.81 1,517 0.96 1,393 124 102 89 84 1928-29 3.58 2,146 1.34 1,881 265 106 91 81 1929-30 4.94 1, 558 0.96 1,410 148 105 89 86 1930-31 3.09 2,008 1,23 1,713 295 91 76 87 1931-32 2.49 1,984 1.21 1,615 369 75 63 100 1932-33 3.09 1,761 1,07 1,556 205 61 51 91 ±yo»^— o*± 1 .25 1,820 253 69 57 76 1934-35 2.18 2,609 1.56 2,077 532 74 61 79 1935-36 3.72 2,339 1.39 2,174 165 82 67 67 1936-37 3.79 2,559 1.30 2,393 166 93 75 90 1937-38 3.50 2,283 1,34 2,078 205 88 70 84 1938-39 2.56 2,491 1.45 2,143 348 90 71 87 1939-40 2.99 2,746 1.58 2,345 401 97 76 77 1940-41 2.42 3.013 1.72 2.459 554 109 85 83 « f 3.293 2,038.9 1.256 1,776.4 262.6 88,9 76.6 82.6 tr ^ 0.762 440.0 0.216 341.2 132.8 13.6 12,4 6.5 * 1925-1929 = 100. f 1931-32 = 100. Excluding 1936-37. Sources of data: Col. 1: Table 13. Cols, 2 and 3: Table 17, Cols. 4 and 5: Table 20, Cols. 6 and 7: Table 22, Col, 8: Table 23. '» . , • ■ • V.'i ; i T - ■ .1 i * is 1 ..'.\ • • - i ' i • - i ■ r ' 1 ! ,■■.-*■■• 1 1 ... .. - ( . 1 -JS-i. G''i.-X--- ThBLE 8 Winter Lemons: Regression Coefficients and Auxiliary Constants 1921-22 to 1940-41 (1936-37 excluded) (Figures in parentheses are standard errors) 1 Equation Net regression coefficients * 1 1 -Adjusted ! Flexibility r Dependent variable ^x) Constant term c F.o .b. price Domestic shi pments ^Jonagri- cultural income x^ J Ir>dpx of temper- atures Time ^5 of multiple correlation R of demand Iw 2vv ^1 ^2 10.9472 3,690.0778 -294.9264 (37.9659) -0.0028 (0.0004) 0.0225 (0.0059) 7.3125 (1.9295) -0.0484 (0.0124) -15.9275 (4.0358) 0.0636 (0.0132) 24.8053 (2.3908) 0.9022 0.9697 -0.4764 -1.7044 ^(x) c "l Far capita domestic shipments yj ^2 Per capita nonagri- cultural income 0 ^3 ^4 X 5 R 'L r T t 3w 4w X 1 ^0 10.9687 2.2768 1 . -0.1825 (0.0232) -4.4674 (C.5679) 0.0245 (Q.0071) 0.0050 (0.0014) -0.0474 (0.C124) -0.0096 (0.0025) 0.0479 (0.0113) 0.0115 10.0016) 0.9021 0.9514 -0 .4786 -1.7038 1 i I Log x^ Log Xg Log X, 1 ^ Log R i 5w Log x-j^ 5.4798 -1.5816 (0.2191) (0.4510 (0.1544) ! -1.1632 1 0.0169 (0.3182)j(C.0040) 0.8698 -1,5816 6w Log Xg 3.2845 -0.4984 ' (0.0590) 0.2556 (0.0872) -0.6412 1 0.0117 (0.1816)i(0.0013) 0.9623 -0.4984 (Footnotes on next page) • : - ■ - * ■ , . * - : ■ : .... V..''..i.- Z.'. 5 i I 5'- .1 - ■ > ;■ " ■ ■ ' : , .:.7. f " " j * i :l i ■ - ■■■ ■ .. .... » - ■ f. - ■■ . • f (Table 8 continued) * = Seasonal average f.o.b. prices in dollars per box. = Domestic shipments of California and Arizona lemons in units of 1,000 boxes. x° = P.r capita domestic shipments of California and Arizona lemons in pounds. Xj = Index of nonagri cultural income payments, 1924-25 to 1928-29 = 100. x° = Per capita index of nonagricultural income payments in per cent. x^ = Index Of mean temperatures in December, January, and February in 32 metropolitan districts, 1931-32 = 100. x^ = Time in units of half-years, origin half-way between 1930-31 and 1931-32, ■f At the centroid. ^ In natural logarithms. O 41. measured at the centroid is consistently inelastic in all formulations. Attention is also drawn to the annual values of price flexibility and demand elasticity based on equations Iv;, 2w, 3w, and 4w (see table 9). For tliree of the nineteen seasons under consideration — namely, 1922-23, 1927-28, and 1929-30 ~ the values of the coefficient of price flexibility indicate that the demand for winter lemons v/as elastic. Otherwise, the remaining entries confirm the presumption that an inelastic demand for winter lemons faced handlers of lemons during the period under review. The net regression lines and scatters of residuals derived from the equations cited in table 8 are shown in the various panels of figures 9, 10, 11, 12, 13, and 14, appendix C. In no case is it apparent that a modification of the form of net regressions vrould result in a significant Lmprovement of fit. The results of the tests of randomness of residuals are summarized in table 29, appendix B. It is evident that the hypothesis of randomness of residuals with respect to time cannot be rejected on the basis of these tests. An examination of the scatters of residuals about the net regression of price and of shipments on temperature (see panel C of figs. 9 to 14, appendix C) might lead to the conclusion that the temperature variable would not contribute significantly, were it not for the close fit of the observations for 1931-32 and 1935-36, To test this suggestion, the seasons 1931-32 and 1935-36 were omitted and two equations, one with f.o.b, price and the other with per-capita shipments as the dependent variable, vrere fitted to the remaining seventeen observations, 42/ —J The lanbiased estimates of the variance of the best linear xuibiased estimates jf XI and X2 (equations Iw and 2w) are, /^\^ = 0.0060 + 0.0243z^ + 0,0064z| + 0.0065z^ + 0,0233Zg - O.OOSSzgZg + +O.OO62Z2Z4 - O.O409Z2Z5 + O.OOOSZgZ^ + 0.0093ZgZ5 - 0»D048z4Z5 ^■oA z 2 2 ? 2 M, ^ = 642,74 + 836. 74z^ + 688,13Z2 684.43z^ + 761.76Zg - 37O.I2Z1Z3 + 2 +347.73zj^z^ + 627.9IZ1Z5 + Z^,bdz^z^ - 149.24z2Zg + 184.91z^Z5, where z. = J_ (x. - x. ). !>■;*■■ TABLE 9 T/inter Lemons: Calculated Coefficients of Price Flexibility and Demand Elasticity, 1921-22 to 1940-41. (1936-37 excluded) Per capita formulation Year November— April Price flexibility Demand elasticity Price flexibility T Demand elasticity M 1921-22 -1.5619 -0.7220 -1.7760 -0.637C ) 1922-23 -0.3276 -0,8684 -0.8900 -0.7922 ■ 1923-24 -1.9620 -0.3627 -2.1491 -0,336'- r 1924-25 -1.2260 -0.7132 -1.3079 -0.663J > 1925-26 -1.2675 -0.5190 -1.3444 -0.49 5E 1926-27 -1.6230 -0.3831 -1.7303 -0.372] L 1927-28 -0.9183 -0.9415 -0.9435 -0.9205 ) 1928-29 -1.8584 -0.5292 -1.9095 -0.527] L 1929-30 -0.9204 -0.9359 -0.9238 -0.942' 1930-31 -1.7353 -0.4481 -1.7352 -0.453£ i 1931-32 -2.2855 -0.3791 -2.2358 -0.383J 5 1932-33 -1.4905 -0.4926 -1.4611 -0.500J 1933-34 -1.6654 -0.4360 -1.6325 -0.448; 5 1934-35 -3.5114 -0.2568 -3.3389 -0,264' 7 1935-36 -1.7513 -0.4671 -1.6976 -0.488^ I ± C tj U— ij 1 1937-38 -1.8502 -0.4516 -1.7916 -0.481( 1938-39 -2.4053 -0.2927 -2,2904 -0.311 7 1939-40 -2.6908 -0.3251 -2.4013 -0,348 3 1940-41 - 3.6239 -0.2442 -3.2754 -0.263 1 Z Source of data : Calculated from equations Iw, 2vir, ,3w, and 4w respectively for the values of the independent variables which prevailed in the various years. 43. The two equations (in standard form) ai'e given below (the figures in paren- theses are standard errors of the |S* - coefficients). The adjusted coefficients of multiple correlation are fairly high, 0.8902 for the first equation and 0.9494 for the second equation. The t-ratio for the coefficient p is 1.979. Since the set of admissible alternatives specifies negative values of ^ we need to consider only one tail of the t-distribution. For 12 degrees freedom t Qg is 1.782 and thus the null hypothesis can be rejected. The t-ratio for the coefficient is 2,455. These results tend to strengthen the presumption that the temperature factor is a significant variable affecting the demand for lemons during the winter months. ' t z = -l,3023z„ + 0,4029z - 0.2412z. + 0.7415z, 1 2345 (0.1844) (0.1282) (0.1219) (0.1960) I z' = -0.6189Z, + 0,2794z„ - 0,l939z. + 0.6177z<, 2 i o 4 5 (0.0876) (0.0879) (0.0790) (0.0907) I The coefficients of separate detennination for all of the regression equa- tions summarized in table 8 are given in table 10. It will be noted that the con- tribution of winter temperatures to the variance of the dependent variable is of about the same magnitude as that of income in all of the formulations thus far considered. Each of these variables contributes less than 10 per cent to the variance of the dependent variable. Vfith price as the dependent variable, the contribution of the trend factors is somewhat larger, betv/een 15 and 20 per cent, but still considerably smaller than the contribution of the quantity variable. Hovrever, in the equations with domestic shipments as the dependent variable, the trend factors are seen to "account" for a large proportion of the variance of the dependent variable. Thus in equation 2'w, 55 per cent of the variance of the dependent variable is allocated to the trend factors and only 33 per cent of the variance to the price factor. It is for this reason that the per-capita formulation is of particular significance in the winter analysis, for obviously it is necessary 44. TABLE 10 Tlinter Lemons: Coefficients of Separate Determination Eqvation Dependent variable F.o.b, price Domestic shipments Per capita domestic shipments Index of nonagri- oultural income Per capita nonagri- oultural income Inde: temps IC jr- 3 S Time ^1 ^2 ^3 0 ^3 ^4 ^5 Per cent Iw ^1 56,4 5,2 4.{ i 19.3 2w X 2 33.0 3,0 4.] L 55,3 3w \ 55,6 8,4 5. J ) 16,3 4w 2 41,5 4,4- 5,^ t 41,3 5w Log Xj^ 56.7 3,8 5,4 t 18,0 6w Log Xg 35,3 2,1- 3.£ 53,1 Source of data: Based on equations Iw through 6w given in table 8, Coefficients based on equations Iw, 3w, 4w, and 5w adjusted to remove negative signs. 1 1 '( ■ -A V ., .. .-. • • .v - • •Sv^-'- ■■ ■■ tf ■ 45. to account more explicitly for the multitude of influences that are represented by- time. However, the introduction of the population factor»while reducing the con- tribution of "time" still leaves it of considerable importance. Thus in equation 4w, roughly 41 per' cent of the shipment variance is allocated to the trend factors. The major importance of the time variable in the shipment equations is an undesirable feature of these equations and one which should induce considerable caution in their use. In all of the equations thus far considered, the quantity variable was re- presented by dOTiestio shipments of lemons during the months November through April of each year. In the summer analysis imports were shown to be a homogeneous part of the total supply of lemons during the months May through October. Such, appar- ently, is not the case during the winter months. The use of United States supply of lemons during the winter months — that is, of domestic shipments plus imports — as the quantity variable reduces the adjusted multiple correlation coefficient from 0,9022 to 0.5144 with price as the dependent variable and from 0.9697 to 0,8666 with United States supply as the dependent variable. It is not satisfactory, either , to include imports separately as an independent variable, for then imports add little to the fit of either the price or the shipment equations. This analysis has found no empirical grounds for including imports explicitly or implicitly in the equations for vdnter lemons. The lack of relevsinoe of the import variable during the v/inter months is difficult to explain, especially in viev/ of the fact that the relative magnitudes of imports for consumption during the two seaccns were quite similar. It is possi ble that lemons imported during the viinter months were intended primrily for consumption during the summer months. The fact that heavy imports of lemons have usually occurred in Februaiy, March, and April lends some support to this conjecture. 43/ Hovrever, domestic shipments of lemons also exhibit practically the same pattern. For the ten-year period 1921-22 through 1930-31, imports of lemons for con- sumption during the months November, December, and January constituted 19.4 per cent of the California Fruit Grovrers Exchange movement of lemons into fresh consumption. For the months February, March, and April this proportion rose only to 21,3 per cent. ; ■!■ ■4 f ni Tg-^f^8. •I ^v»-^ 4 *0' V'?f t •■. 46. It is also possible that due to sudden and large shifts of demand induced by \ireather factors, the product differentiation whioh obtains during the winter is much weakened during the summer months. While the fifst possibility appears, off-hand, to be more plausible than the second, neither has direct evidence which could be cited in its support. As was the case in the analysis of data on summer lemons, the replacement of domestic shipments by two quantity variables, one consisting of domestic shipments by the California Fruit Growers Exchange and the other of domestic shipments by other marketing agencies, leads to the conclusion that such a breakdovm of the quantity variable has little effect. Below is given the price equation in which Exchange domestic shipments (x„ ) and domestic shipments of shippers outside of the Exchange (x„ ) are used ae two independent variables. The other variables in the equation are f,o,b, prices (xj^), index of nonagricultural income (xg), index of temperatures in December, January, and February (x^), and "time" (xg). The figures in parentheses are standard errors of the partial regression coefficients. While the adjusted multiple correlation coefficient is somewhat lower for this equation than for equation Iw (0,8502 as against 0,9032), the coefficients of the two quantity series are almost identical in magnitude, f x^ = 3,2926 - 0.0028X2^ - 0.0027x2q + 0,0226x2 - 0,0492x_^ + 0,0642X5 (0,0006) (0,0010) (0,0062) (0,0155) (0,015l) '■ '•. "ill' 08^0 -oiT:*^ ■ '33*7 -Bvi f.r. 'r "J-- c;. Allocation of Shipments In this section we shall confine ourselves to a brief consideration of two types of allocation of shipments over time. In the first class are included allo- cations designed to maximize returns to the industry; the second is comprised of allocations designed to maximize shipnents or minimize prices for fixed "net" returns to the industry. The basic interval of time for which most marketing decisions are made in the lemon industry is much shorter than the six-month period on which the analyses described in the preceding section were based, Hovraver, if the results of these analyses are to be at all useful in connection with the problem of allocation, a time interval shorter than six months cannot be employed. Thus the allocation of shipments that v/ill be considered is that betvTeen the two periods November through April and May through October of each marketing year. The discussion to follow is pertinent to decisions concerning shipments for other time intervals only in so far as such decisions affect the total quantity shipped during each of the two six- month periods, ViTe shall begin by determining the allocation required to maximize f ,o,b. returns. In this simplified case all costs to growers are disregarded. Further- more, neither returns from diversion of lemons to processing nor the long-run effects of the allocation are considered. Subsequently we shall take into account some of the costs and the additional revenue from processed fruit received by the grovrers. The long-run effects of the allocations discussed in this section, however cannot be determined on the basis of the information at present available. For a given marketing year let the price-quantity relation in each of the two subperiods be, I xi - a + a^xg yj^ = b + b2y2 where X], is the f,o,b, price per box and Xg domestic shipments for the six-month ... ... . ■'•.■<: ::•>;■■ ■' ■ ■ ,:..U-v. .... - . Mr •■■ 48. period November through April, and where yj^ and yg denote respectively f.o •b. prices and domestic shipments for the period May through October. The f»o»b« returns for a given year are given by: 2 2 R - axg + byg + agXg + b^yg The values of Xg and yg maximizing R ("optimum shipments") are easily shown to be 2 ' 2b If the constraint x + y^ ■ q, where q is a fixed value, is introduced, 2 shipments maximizing f.o.b, returns are given by the expressions: ^1 ^ b - a + 2b9q 2 ^^^2 * ^2^ (B) Yg = q - The shipments of winter and summer lemons maximizing f .o.b, returns with and without constraint for the years 1921-22 to 1940-41 are found in table 11. One set of these figures is based on equations 7s and Iw ~ the other, on equal bions 8s and 2w solved for f .©.b, price, To determine the constrained optimum shipments. f.o.b. returns were maximized under the condition that the sum of optimum shipments of winter and summer lemons for each marketing year equal the actual quantity 'shipped during the year. In figure 2 the series of optimum shipments are i -epresented graphically together with the actual domestic shipments of winter and summer lemons. The shading of the area betvreen estimates of optimum shipments, derived foi - v/inter lemons from equations Iw and 2w and for simmer lemons frcaa equations 7s ant i 8s, is intended to suggest the existence of a zone of uncertainty for these estimates in ^^-/ These equations were chosen because they contain explicitly the relevant variable to which allocation can be applied, namely, domestic shipments. TABLE 11 Allocation of November- April and May-October Lecion •Shipments to Maximize F.O.B. Returns for an Undetermined and for a Fixed Volume of Annual Shipments, 1921-22 to 1940-41 ■ Unrestricted opt imum allocation Restricted ortimum allocation Domestic s hipments Equations Iw and 7s Squs-tions 2w and 8s E'^uations Iw and 7s E^Ui^tions 2w and 8s November- fjiay- Novembe r- Nove;r.ber- iVi&y NovsiT^ber- May- Nove^r/oer- May- April October April October April October April October April October ^2 ^2 4 0 ^2 4 0 ^2 ^2 1 ^2 t ^2 1 2 3 4 5 6 7 8 9 10 10.000 boxes 1921-22 164 1 235 135 231 122 206 148 251 152 247 1922-23 131 202 144 231 132 205 127 206 130 203 1923-24 191 322 144 251 132 231 193 320 195 318 1924-25 165 289 150 263 137 235 167 287 171 282 1925-26 175 328 157 258 144 235 194 310 196 307 1926-27 195 289 158 257 145 235 187 298 189 295 1927-28 15? 303 158 277 146 256 166 289 168 286 1928-29 215 322 165 299 153 278 195 342 197 340 1929-30 156 333 163 301 151 276 173 316 177 312 1930-31 201 381 158 310 147 288 205 377 209 373 1931-32 198 307 143 263 134 244 184 321 188 318 1932-33 176 347 147 277 138 257 188 335 192 331 1933-34 207 404 166 295 155 275 228 383 232 380 1934-35 261 416 168 276 157 257 265 412 268 409 1935-36 234 427 184 326 172 304 247 414 250 411 1936-37 1937-38 228 435 176 295 166 276 256 407 259 404 1938-39 249 470 176 322 167 302 268 451 272 447 1939-40 275 461 190 305 180 285 290 445 293 442 1940-41 301 550 192 356 182 • 335 318 533 323 529 Sources of data: Col. l: Col. 2 of table 7, Col. 2: Col. 4 of table 2. Cols. 3 and 4: Based on expressions (A), see text, applied to equations Iw of table 8 and 7s of table 6. Cols. 5 and 6: Based on expressions (A), see text, applied to equations 2w of table 8 and 8s of table 6. Cols. 7 and 8: Based on expressions (B), see text, with the constraint ^ ^2 ~ ^^2 * ^2» applied to equations Iw of table 8 and 7s of table 6, , Cols. 9 and 10: Based on equations (B), see text, with the constraint ^tg * ~ ^2 * ^2* ^'PP^^®^ ^° equations 2w of table 8 and 8s of table 6. * : I i ' . i :. , t V ■ r ; - ■ -■ • - • -■ : ■- ■ ■. - ; -■ V? I .• .', ; 1 ■. -J ):_-■*. '•1 i ^-c- ! ■-■?tT '! B : Mi., : ... i .; :y:. T : ■ . ; ■ u ^ .. . ' '■ ..... "Tj j' TT--.'-"- T J i i J :■: - 4 .V 1 - .. -4 ; • ■■ I. *' * . • . . . ^ . to « M O o ta •a G a « 3 o -p V EH «9 <I> X o o » c OS 10 o J3 a 500 400 300 200 300 ■ 200- 100 November-April Constrained optimun /Shipments omestic shipments cm shipments 1923-24 -26 -28 -30 -32 -34 -36 -38 -40 Fig 1921-22 , 2. — Domestic shipments of (A) winter and (B) simmer lemons and shipm.ents maximizing f.o.b. returns, to 1940-41. (Data from table 11.) 51, addition to that implied by the standard errors of the estimates. Since equations with price as the dependent variable and equations vri.th quantity as the dependent variable are a priori equally aopeptable, the differences bebv/een the results secured on the basis of these equations should in some way become a part cf the measure of the "true" optimum shipments necessarily lie v/ithin these zones. Inspection of figure 2 leads to the following observations: (l) The optiminn shipments of vdnter and summer lemons in the case of the unconstrained maximum arg considerably smaller than the actual shipments. For stunraer lemons in only one season and for winter lemons in only three seasons have the actual shipments been of the same order of magnitude as the shipments maximizing f ,o.b, returns. The gap betv/een the actual and optimum shipments is markedly greater during the latter part of the period than during the earlier years. Furthermore, during the latter half of the period the differences between the actual and optimum shipments of summer lemons are larger than the differences for v/inter lemons. (2) The optimum shipments of vrinter and summer lemons in the case of the constrained maximum do not differ greatly from the actual shipments. Hov/ever, beginning vdth 1933 the optimum shipments of summer lemons are slightly but consistently smaller than the actual shipments. For mnter lemons this tendency is, obviously, in the opposite direction. Thus both sets of calculations, those for the constrained maximum and those for the unconstrained maximum, suggest that during the latter half of the period, relative to prevailing demand conditions, an overshipment of summer lemons has taken place. This conclusion is in harmony v:ith the observation made earlier (page 2 ) concerning the downvrard trend of prices of simmer lemons during roughly — » The studies of Frisch and Koopmans are relevant in this connection: Frisoh, R. Statistical confluence analysis by means of complete regression systems. Oslo. 1934; and Koopmans, T. Linear regression analysis of economic time series. Haarlem. 1937. uncertainty to accompany . • t\- ...... 1. .9 etB - ■ ■ . c - • - . --^ ■■ ; :'■ r- ■■ ^■ _ j; :. ■' ■■> mil ■ '■ » ■:;-v ■;:■:)•= «:.•::•" ■ ■ ■- v •.;..•.>•■• - ■ -; • • -■■ I ■' . . ... ■>■• ■;. ■ ; , , 1 ^iid tot MOila ■- • ■•■ t> ■ . ^ ; , w.-..:..' ... . '.■Cr. I i ■ "" . ' ■ .:v -t! "<•!■" ;-r. •;!: • i,. J;^':Xo:i' i- • . ' , • i 1 52. the same part of the period* It may appear from the figures cited in connection with the unconstrained maximum that a drastic curtailment of shipments of summer and winter lemons in many of the years considered would have increased markedly f ♦©•b. returns to the industry for the period. It should be emphasized that such a oonolusion is un^mrranted♦ The formulation employed herein deep not permit aasepament of the long-run effects of an allocation policy* An obvious oonsequenoe of a drastic curijailment of lemon shipments is a shift pn the part of consumers to substitutes* It is probable that within the range of lempn prices which have ijhus far prevailed, the level of prices of technically substitutable oommodities has not had an important influence on the demand for lemons. However, in higher price ranges — effectuated by markedly our- tailed shipments -t^ substitution may beocraie a factor of major importance. Introduction of oosts and addition^-l revenues arising fvcsa, the sale of lemons for manufacture has the effect of further decreasing optimum shipments derived for the unconstrained maximum. These nevj- elements are introduced by maxi- mizing for ?L given year "value on tree" defined as y « R ^ (g + h) (xg + yg) + (Q - yg) z, where g represents picking and hauling costs per box, h represents the packing- house charge and selling (including advertising costs), Q the total lemon crop, and, finally, z the price "on tree" of lemons used for prooessingi R, as before, denotes f ,o.b, returns for packed and loose fruit. The reader will note the simplifying assumption of constant average "variable" costs and the further assumption of a fixed "on tree" price of lemons diverted to manufaotwe. Cultural costs, interest on investment, and depreciation alloT.mn9e are not included in this formulation. The shipments maximizing V are given by . x° ^ e,%^ r ^ 2 2 (C) * 0 e + h + 8 -.t;.; '■. ^ . . \ b-i':.— i. c i: V- J; ^ ^v.;;■■ M:v-.- ::■ •., ojii; »>bvj^f;^'4^i;•■^v^> J vcnjt^u C'ini;j.; -.j.f'i- !:■■ •^Z -vg-is^^fUi^c^^ia sr ■ v't-gX " ' ■ ' ■ ■ - ■ \ ■ fUv«- 53, It is apparent from expressions (C) that shipments maximizing 7 will be smaller than shipments maximizing R, that is, that the demand at the "on tree" level is more inelastic than the demand for lemons at the f.o.b, level. The introduction of costs and additional revenue from processing has no effect on the magnitudes of optimum shipments derived for the constrained maximum. The application of the remaining criterion involves the maximization of U = + y-g under the condition V = or the minimization of W » R (xg + 72)"^ under the same constraint. The values of x^ and maximizing U or minimizing V/ are given by: (D) (^c -F) ^2 (ag •.bg) (^c -F) + ^2) where P = Qz - agCxg')^ - h^iv^* f • In table 12 are indicated the values of optimum shipments for the three years 1937-38 through 1939-40 calculated from (c) and (D). The differences between the optimum shipments maximizing "value on tree" and the shipments minimizing the annual v/eighted average prioo for a fixed "on tree" value are very large. Since the magnitude of such differences is a function of the value selected for V no significance attaches to the particular magnitudes of optimum shipments given in table 12, These are to be viev/ed as numerical illustrations of the expressions (C) and (D) designed to make their meaning more concrete. Of some interest, however, is the following observation. Let d be the difference betiveen the optimum shipments calculated from expressions (C) and (D), that is (x*" - x") or (y'" - y**), and let d' be the value of d for = 0, It is easily seen that d where is the maximum value of V^., Thus relatively large differences between optimum shipments derived from expressions (C) and (D) are to be expected for a large range of values of V» I'.xjLSiiiiii ■ - . . v^'V r - -'"^ • !. ery- ^0.1 ... « r ' ■- ' ' ' ■ ■ ■ ' ■ ' ■ ■ ■ T i. f ■ ■7—' f;Si--fi,h7 tie, 0'.'jr it ""i »i f » » >" . s. .j ' • . 3«f (ffX itfUT (P) 8«ox8ae'?',xd aoi^ fetfvl.te; • .. « * TABLE 12 Shipments of Summer and Winter Lemons Maximizing Value on Tree and Minimizing Average Annual F.O.B. Price for Fixed Value on Tree Year Annual pro- duction 0 Costs f.o.b. packing house less cultural costs e + h Price on tree for manufa c- ture jg Shipments maxi- mizing value on tree Ve lue on tree V Shipments minimizing prices for fixed value on tree November- April »? 3C2 May- October yz November- April ^2 May- October II ♦ ^2 1 2 3 4 5 6 7 8 10,000 dollars dollars 10,000 10,000 1,000 10,000 10,000 boxes per box per box boxes boxes dollars boxes boxes 1937-38 930 1.32 0.11 150 259 12,518 258 411 1938-39 1,111 1.32 0.05 152 287 13,876 269 452 1939-40 1,198 1.28 0.22 1 163 267 13,040 291 447 Sources of data: Cols. 1 and 2: California Fruit Growers Exchange. Marketing Research Department. Statistical information on the lemon industry. February, 1942. Col. 3; United States Bureau of Agricultural Economics. Agricultural prices. Vi/ashington, D. C. August 28, 1942. (Mimeo. ) Cols. 4 and 5: Based on expressions (C), see text, equations Iw and 7s and cols. 1, 2, and 3« Col. 6: Estimated on the basis of equations Iw and 7s and data in cols. 1. 2. and 3. cn Cols. 7 and 8: Based on expressions (D), see text, eqiiations Iw and 78, and cols. 1, 2, 3, and 6. '■'■L'y.-t^'B- •i^ t: J? ■ 'a9 1 ■J} i s. • . V vrr : f * * ' ^1 • ,» - ! ■ . .i • a * 55. Summary Most of the variation of the average f .o.b, prices of California suramer lemons during the twenty-year period 1922 to 1941 can be accounted for by con- comitant variation in three factors: namely, (l) total quantity of lemons mar- keted as fresh fruit in the United States, (2) nonagricultural income payments in the United States, and (3) maximum temperatures prevailing during the summer months. Most of the variation of the average f .o.b. prices of winter lemons can be accounted for by concomitant changes in (1) domestic shipments for fresh consumption of California winter lemons, (2) nonagricultural income payments in the United States, (3) average lev^l of temperatures prevailing in December, January, and February, and (4) trend factors not individually identified but whose aggregate effect is represented by the variable "time." This variable failed to make a significant contribution in the analysis of summer lemons largely because of the presence of a marked trend in the temperature index. Additional evidence, however, makes it probable that during the period the demand for summer lemons relative to annual changes in the purchasing povrer of consumers and to annual changes in the level of maximum temperatures shifted upvmrd. In order to obtain a more comprehensive picture of the characteristics of demand for summer and winter lemons, both price and quantity were employed as dependent variables in the regression analysis. In each case the dependent variable (and its logarithm) was expressed as a linear function of the independent variables (and of their logarithms). In addition, equations based on per-capita quantity and per-capita income were fitted. The results secured with these different treatments were found to be in substantial agreement. On the basis of these analyses several major conclusions appear to be warranted: (1) The demand for summer lemons on an f.o.b. basis was inelastic during the period; the demand for v/inter lemons on an f.o.b. basis vms also in- elastic during most of the period. There appears to be little or no difference in the elasticities of demand for sviramer and winter lemons, (g) Changes in income 56. have affected the demand for smmner and winter lemons. Increases in nonagricultural income payments have been associated with upward shifts of demand for both summer and winter lemons, (3) Changes in weather conditions, more specifically temperature, have affected the demand for both summer and winter lemons. An upward shift of demand for lemons during the summer months is associated with increase in tempera- tures; an upv/ard shift of demand for lemons during the winter months is associated with decrease in temperatures, (4) During the tv/enty-year period the demand for both summer and winter lemons, relative to the other independent variables, has shifted upvards. It is likely that factors other than growth of population are responsible for a large part of this upv/ard shift. The problan of allocation of lemon shipments betv/een the tvro periods of the marketing year, November to April and May to October, v/as given brief consideration. Shipments maximizing f ,o,b, returns or "value on tree" — the latter determined under a set of simplifying assumptions — were found to be, for most of the years of the period, considerably smaller than the actual domestic shipments of winter and summer lemons for fresh consumption. The conclusion that revenue to growers could be increased in the long rim if drastic limitation of shipments were instituted is not implied by the foregoing since long-run effects are not treated in the formula- tions underlying these calculations. The allocation of a fixed annual volume of shipments to maximize returns to growers was also considered. For the twenty-year period, the allocation of the actual annual shipments so as to maximize returns to growers was shown to result in shipments of winter and summer lemons not greatly different from the actual shipments. However, since 1932-33 there has been apparently a slight but consistent tendency to vmdership during the months Hovember to April and to overship during the months May to October, This and other evidence suggest that in recent years the shipments of summer lemons were somev/hat large relative to the demand conditions prevailing during the year. : ..70 -trfd- od' evlrr^. U J , ooiaeJ to . ,.-r)oXr/' ■ ' • '- ■■■'>■»;' ....... . • - ■ • — . ' .' - .4 ;.j o .; ;"i:,;;f'i ■..fr-Yvs:^%Nr , ..... . C9i :i3:. . ii^.. -i./T;:- .- 57, APPEiroiX A Series Used in the Study The Price Series *— The price series employed in this paper consist of the average prices, f»o.b» basis, received for lemons marketed by the California Fruit Growers Exchange during the six-month periods November through April and May through October, for each of the twenty years 1921-22 through 1940-41 (see table 13 )• The average price for each six-month period was obtained by dividing the total amount of money (reduced to an f .o»b, basis) received by the Exchange for lemons shipped during the period, by the number of boxes shipped* It is impossible to determine directly the degree to which the Exchange prices are representative of prices received for all California lemons since data on sales by other shippers of California lemons are not available to us. Such factors as differentiation of product, variation in time and place of shipments, and variation in composition of .shipments \vould make for differences between prices received by the Exchange and those received by other marketing agencies. However, Exchange prices would exert a predominant influence on the weighted average price for the industry as a whole. For the period under reviev/ the California Fruit Grovrers Exchange has consistently marketed the major proportion of California lemons* For winter lemons, on the average, the Exchange handled about 88 per cent of the total domestic shipments; for summer lemons the Exchange percentage was even higher, approximately 92 per cent. In no year did the Exchange percentage fall below 80 46/ for winter lemons and 86 for summer lemons, ■— ' Some notion as to the magnitude of the correlation to be expected between the weightedraverage price of lemons marketed by the California Fruit Growers Exchange and the industry weighted- average price may be obtained by the follovdng calculation. These percentages are based on figures given in tables 20 and 21, vl-j...'.>-3 ni fcsai^ sc. > or ; f prt': ■• r. ''.t 'ij.^;:! ' yr^ jo:-: f'-'r. f\rr. f •■'lo'rxrjB') '^■-i -f'-''' ' rfp, 'i^ff-^r '.rT ■''^j.'- 10 R^pV tocf ;> erf od- a . xo 3 orfcf- lo ei -f 9di bjb noi.-^on 58. TABLE 13 Seasonal Average F.O.B. Prices of California Lemons 1921-22 to 1940-41 Period Period Year November- April May- October Year November- April May- Octob 3r dollars per box dollars per box dollars per box dollars per box 1921-22 3.48 4.48 1931-32 2.49 4.4: L 1922-23 4.20 5,23 1932-33 3.09 3.6( 1923-24 2.38 2.74 1933-34 3.20 3.8! 5 1924-25 3.87 4.93 1934-35 2.18 3,3J ) T Q O C OP. ly c 0— CD OmOO o.2o 1935-36 3.72 4.5: L 1926-27 2.73 4.88 1936-37 3,79 4.8e 1927-28 4.81 4.86 1937-38 3.50 2.8' 7 1928-29 3.58 5.36 1938-39 2.56 3.1' J 1929-30 4.94 5.18 1939-40 2.99 3.0^ 1930-31 3.09 4.50 , 1940-41 2AZ 3.1£ Source of data: Based on prices received by the California Fruit Growers Exchange for lemons shipped during these periods. Beginning May, 1926, prices of fruit sold loose are included. Beginning November, 1935, prices of exported lemons are excluded. 59. On the assumption that the proportion of the total shipments of domestic lemons controlled by the Exchange is constant from year to year and equal to pi, the cor- relation between the Exchange weighted average price and the industry weighted average price is given by the expression, Pl + k P2 where k is the ratio of the standard deviation of the weighted average prices re- ceived by shippers outside the Exchange to the standard deviation the Exchange weighted average prices, r^^g the correlation between the two sets of weighted average prices, and pg = l-p^^* Considering r^^^ as a function of r^g, the minimm value of the former is given for r^g = - k PgP]_"^» namely. (kp^)' It Pl The minimum value of r^^ may thus be calculated for various values of p^ and k« The values of r^^g and r°^ obtained when to p-j^ is assigned the value of the ratio of Exchange domestic shipments to industry domestic shipments and to k plausible limits, are given in table 14» Values of r^^^ obtained when P]^ is assigned the lowest observed value during the period under review are also indicated in the table. Under all assvimptions, and these by no means favored the appearance of high corre- lations, the correlation coefficients secured are very high. These results suggest that a satisfactory use in regression analysis may be made of Exchange prices as approximations to the weighted average industry prices. Several observations must be made concerning the year-to-year comparability of the series. Beginning mth May 1926, prices of lemons sold loose are included in our series. Since prices of packed lemons are available for each of the twenty years under consideration, greater homogeneity might have been achieved by exclud- ing data on loose lemons. However, sales of the latter have increased significantly during the period under review. Rather than to disregsird a factor of growing jj. ..c>x ja '.-.Tii-ij,'!!: ■ -too aifi, ^ •9 ^btt« ii3;^,v .vj^'^*j&:\;..?'''<3L-,.. v fli be' . ■o .. <■• ■ ^'.i" a©iiv.. 4?f'5i~-^,cf«i. ■4.,-"' - ■ .J J 2.' i /if-.. TABL!i 14 Estimates of Correlation Between California Fruit Growers Exchange and Industry V^eighted Average Prices Under Various Assumptions Ratio of C.F.G.E. shipments to industry shipments Pi P2 = 1 - Pi Correlation between C.F.G.i. and out- side weighted average prices 'l2 Correlation between C.F.G.E. and in- dustry weighted average prices ^It Minimum value of ^t r° 1 1 0.8752 * 0.1248 1 —0 .1426 n QROfi u .yoyo 0.8752 0.1248 2 -0.2852 0.9584 0.7961 t 0.2031 1 0 0.9690 0.9202 + 0.0798 1 -0.0867 0.9962 0 .9202 0.0798 2 -0.1734 0.9924 0.8560 4 0.1440 1 0 0,9862 * Ratio of Exchange domestic shipments of winter lemons 1921-22 through 1940-41 to industry domestic shipments. Smallest observed ratio of Exchange domestic shipments of winter lemons to industry domestic shipments for the period 1921-22 through 1940-41. :^ Ratio of Exchange domestic shipments of summer lemons, 1922 through 1941, to industry domestic shipments, ^ smallest observed ratio of Exchange domestic shipments of summer lemons to industry domestic shipments for the period 19? 2 through 1941. i i •; \ i t. i ; 'f • - ■ * . '• ■ : - J..-- ..... . i ■ i ! i i ; T ,■ ■ r ; \ J i 4 ■.■ ■;■ i. . ; . -^r '., '.. " ; ...... . J.. -J., 61. Mpol'tance, it was thought preferable to attbept the risk of a slight upward bias in the fii»st tour or five observations of -the series. Of possibly greater importance as far' as the homogeneity of the price series is concerned is the fact that prior to 1935-36 the series include prices of ex- ported lemons. For the six years for which export data are available, the prices based on domestic shipments differ by as much as 10 cents a box in only one season from prices based on total shipnents (see table 15 )• It is unlikely that differences of this magnitude would be found for the earlier years of the period. Total exports during the last six years were greater relatively than during the 47 / previous fourteen years. — « Furthermore, in the more recent years a somewhat larger proportion of lemons than previously was exported to destinations other than Canada; the f ,o.b. prices for the latter country are ordinarily closely in line with prices of like grades and sizes of domestic lemons marketed in continental United States. 48/ Finally, it must be noted that the grade and size composition of shipments varied from year to year (see table 16). The lemons to which the prices refer have not constituted a strictly homogeneous commodity. Had monthly data on grades and sizes been available for the whole of the period, these sources of heterogeneity could have been removed. In the absence of this information we are unable to 49/ assess the contribution of these factors to the variance of the f.o.b. prices. 47/ For the period 1921-22 through 1934-35, exports of domestic winter lemons were 5.8 per cent of total shipments of domestic lemons for fresh consumption; for the period 1935-36 through 1940-41, the percentage of exports v/as 8.1 per cent. Analogous figures for summer lemons are 4.1 per cent and 7.0 per cent. See data on exports given in tables 17 and 18. 48/ — I For the period 1925-26 through 1934-35 exports of lemons to Canada comprised approximately 74 per cent of total lemon exports. The corresponding figure for the period 1935-36 through 1939-40 is somewhat less than 60 per cent. 49/ ._y One other possible source of inoomparability should be mentioned. In November 1929 the California Fruit Growers Exchange extended its operations to Arizona and from that time on prices of Arizona lemons marketed by the Exchange are included in the price series. The volume of Arizona lemons marketed by the Exchange, however, is negligibly small. W W 1 3-* • . ft ©OA ;. ■.■••■J:; -.-..i ; ;;-;.>u;;or> ■iix^J.-; ./ .-,r^,•^ ;r ;.■ A '■ •■ • .-i;:i:::'v ;^v;.^^.: r;// .^C'-"-* iS^;-?:)^ -^^rw -v:?- ^i:.^: t: .■:.:ib vi:.;; -KwA «i-:; fti:-;:?=tr^r •■v-.o.; ,-j:j- ;;v:!;; :.v ■ . ; ■■i.Tsb 99Q -iv, O.V i::.; ;;-::^!n -jrr;-!^:!^ •t;^^ 'i^^c*!- 62, TABLE 15 Comparison of F.O.B, Prices of Lemons Excluding and Including Exports, 1935-36 to 1940-41 November- April May -October Year F.o.b. price excluding exports F.o.b. price Xiiv> XUU XXig exports F.o.b. price XUU XZig exports F.o.b, price j.nc xuu xiig exports 1 2 3 4 dollars dollars dollars dollars 1935-36 3.72 3.75 4,51 4.49 1936-37 3.79 3.81 4.86 4.87 1937-38 3.50 3.49 2.87 2.82 1938-39 2.56 2.51 3.17 3.07 1939-40 2.99 2.99 3.04 3.04 1940-41 2.42 2.42 3.19 3.19 Source of data: Prom records of the California Fruit Growers Exchange. I I r jr v-:- I . . - 1 ^ J! 63. TABLE 16 Average Sizes of Lemons Shipped by the California Fruit Growers Exchange and Proportion of Shipments Graded Sunkist, 1930-31 to 1940-41 Aver a le size shipments prraded Sunkist November-Octobe I Oc** April — October r J. 3 Number per box Per cent 1930-31 334 357 57,9 1931-32 341 363 57.3 1932-33 355 368 58.8 1933-34 363 378 57.5 1934-35 361 355 59.8 1935-36 367 374 55.0 1936-37 O 1 0 ^ 'iOt 1 1937-38 355 359 61.3 1938-39 364 367 54.7 1939-40 363 359 62.0 1940-41 ! 354 i. 349 65.3 Sources of data: Cols. 1 and 2; Based on packed lemon shipments of the Calif- ornia Fruit Growers Exchange. Average size for each season was computed by weighting the monthly average sizes by the corresponding monthly shipments. Col. 3: Based on reports furnished to the California Fruit Growers Exchange by district exchanges. Includes both packed and loose fruit. 4- t i t-.. - V ;,^ — , \ • „■ ■ y ■ J j • j. ^ ■ • i i - i ' ■ » ■ : : ■ ! id . ■ •* ■ ' C -. J i i i : 1 6 t • 1* i [ ■ -.v ] 1 ■ s • - j J, •-' ! ^* f . 1 .'■ i 1 I '** ' i f i ■ ■.V '6 • ! i i t .> C" f ( 1 1 [ 1 • ;. _ : ! 1 1 eiirto 64. The Quantity Series ^— » The quantity series employed in the regression analyiOB are given for winter lemons in table 17 and for summer lemons in table 18, In column 1 of these tables are given estimates of shipments of California and Arizona winter and summer lemons for fresh consumption. Since these are the basic components of the final series, their construction will be described in detail* The sources of the other entries are indicated fully in the notes accompanying the tables. Estimates of the industry shipments of domestic lemons for fresh consumption were obtained by adjusting the estimates of packed shipments of domestic lemons by the ratio of packed and loose shipments of lemons by the Exchange to their packed domestic shipments. This step in the construction of the series is based on the assumption that the ratio of sales of loose lemons by the industry to total industry sales of lemons is identical with the analogous ratio for the Exchange. The validity of this plausible assumption cannot be directly examined since reliable estimates of sales of loose lemons by shippers outside of the Exchange are not available for any part of the period under review. However, since the Exchange has marketed the major proportion of the California lemon crop in each of the twenty years, the ratio of loose to packed fruit for the industry as a whole could not be very different from the ratio for the Exchange. In turn, estimates of packed shipments of domestic lemons wore obtained by adjusting the Exchange packed shipments by the ratio of industry rail and boat ship- ments of domestic lemons to the Exchange rail and boat shipments. Industry rail and boat shipments cannot be used directly to represent total industry shipments of lemons, because movement of lemons by truck is not included in these figures, and because of the difficulty of determining the proportion of lemons in mixed citrus car shipments. On the other hand, it is probable that the proportion of lemons in mixed citrus car shipments is not greatly different for shippers outside the Exchange from what it is for the Exchange. This is a plausible assumption and one v/hich affects only a relatively minor subtotal. ■■r n:-.;.; ■■ . . .. ' .■ ©Xiv vio'i oij.-^-; «s.-i't j-;..^- ■'• -• ■.• ■ -.■ . t€» ^ .i*.-/ v.. , ,'V. •11. •■^ --';.;-W.'V.>..i. ; .f." rfj;; r r. -v.; v?r^;..;'.-r5.. .V. ....... ^. ; i--'Oy..~.^!^''ir'-'^..-":L, _ "^w ■ " '.J Grid 'io .. ,r?:fo. '.i3 n<- '/x*.sion f> TABLS 17 'A'inter Lemons: Domestic Shipments, Per Capita Domestic Shipments, United States Supply, and Per Capita Supply, 1921-22 to 1940-41 Year Per capita November- Total Domestic Domestic United States domestic Imports for United States April shipments exports shipments population shioments consumption supply 1 2 3 4 5 6 7 thousand thousand thousand thousand pounds thousand thousand boxes boxes boxes persons boxes boxes 1921-22 1,756 115 1,641 109,873 1.14 384 2,025 1922-23 1,388 80 1,308 111,537 0.89 690 1,998 1923-24 2,034 123 1,911 113,302 1.28 137 2,048 1924-25 1,726 78 1,648 114,867 l.OS 384 2,032 1925-26 1,860 110 1,750 116,532 1.14 309 2,059 1926-27 2,139 185 1,954 118,197 1.26 96 2,050 1927-28 1,610 93 1,517 119,862 0.96 505 2,022 1928-29 2,291 145 2,146 121,526 1.34 156 2,302 1929-30 1,632 74 1,558 123,077 0.96 611 2,169 1930-31 2,147 139 2,008 124,039 1.23 69 2,077 1931-32 2,122 138 1,984 124,840 1.21 36 2,020 1932-33 1,823 62 1,761 125,578 1.07 60 1,821 1933-34 2,163 90 2,073 126,373 1,25 8 2,081 1934-35 2,779 170 2,609 127,249 1.56 3 2,612 1935-36 2,671 332 2,339 128,052 1.39 47 2,386 1936-37 2,691 132 2,559 128,823 1.30 8 2,5€7 1937-38 2,433 150 2,283 129,823 1.34 46 2,329 1938-39 2,790 299 2,491 130,878 1.45 2,491 1939-40 2,964 218 2,746 131,954 1.5S 2,746 1940-41 3.247 234 3.013 133.030 * 1.72 3.013 * Estimated. Sources of data: Col. 1: Col. 4, table 19. Col. 2: United States Department of Commerce. Monthly summary of foreign commerce of the United States, monthly issues. Col. 3 ; Col. 1 minus col. 2. Col. 4: United States Department of Commerce Bureau of the Census. Statistical A^strac^ ol^tTie United States, 1941. Population. Estimated as of July 1 for continental United States. Col. 5: Col. 3 divided by col. 4 and multiplied by 76 for conversion to pounds. w (Continued on next page) I 1 J -^iv^-i i t . ■ \ ■- > ■ 1 . . .; • » /- * ".:.' ' ' ■■ ■ ■ 1 \ ^ - . * f » ■ " ♦ '■■ , ; J ' F i ■ ^ . . . - .J • > *_ . t . ■ - ■ : -.»-■. ', ■ i -I .'i^-.ir - ' V. . . J . ■ ! - » 1 - .. * * .. • - " • •; » ■ * ; "' ■ i : i ■ f ■ ' , ■ I- ■ K ... i ' , " ) i J ' ^-e^v . ' . - - 4 , . . . . ■ " - T ^ ■■ ■ ' ., .»•■ ^, i ■ \r/: r ♦ » ■ i - ■ . ( "- " 1 ; : ■'.'5 ! ; . . / i f -.: -'■ ' I .... ^ ■ ■-•V. • i'^- r ^ .. - *■ '. ■. . . ■ ■ : I..- y . , .. • i . * ' * Table 17 continued. Sources of data: Col. 6: United States Department of Commerce. Monthly summary of foreign commerce of the Unite States, monthly issues. 1921-22 through 1932-33: "General imports" minus "exports of foreign lemons 1933-34 to date: "Imports for consumption." Exports of foreign lemons for November, 1921, through December, 1925, and all import figures converted to feoxes on basis cr 76 pounds per box. Col. 7: Col, 3 plus col.. 6. TABLE 18 Summer Lemons: Domestic Shipments, United States Supply, and Per Capita United States Supply, 1922 to 1941 Year ■ United United Per capita ivlay- Total Domestic Domestic Imports for States States United States October shipments exports shipments consumption supply population supply 1 2 3 4 5 6 7 thousand thousand thousand thousand thousand thousand pounds boxe s boxes boxes boxes boxes persons 1922 2,425 78 2,347 931 3,278 109,873 2.27 1923 2,104 85 2,019 1,096 3,115 111,537 2,13 1924 3,335 118 3,217 468 3,685 113,202 2.48 1925 2,964 77 2,887 1,101 3,988 114,867 2.64 1926 3,442 160 3,282 743 4,025 116,532 2.63 1927 3,043 155 2,888 619 3,507 118,197 2.26 1928 3,16£ 131 3,031 464 3,495 119,862 2.22 1929 3,377 155 3,222 332 3,554 121,526 2.22 1930 3,437 104 3,333 549 3,882 123,077 2,40 1931 3,941 129 3,812 176 3,988 124,039 2.44 1932 3,167 95 3,072 47 3,119 124,840 1.90 1933 3,567 96 3,471 69 3,540 125,578 2.14 1934 4,180 140 4,040 15 4,055 126,373 2.44 1935 4,535 376 4,159 4,159 127,249 2.48 1936 4,576 302 4,274 39 4,313 128,052 2.56 1937 3,964 124 3,840 26 3,866 128,823 2.28 1938 4,915 570 4,345 4,345 129,823 2.55 1939 5,234 532 4,702 4,702 130,878 2.73 1940 4,857 252 4,605 4,605 131,954 2.66 1941 5.775 276 5.499 5.499 133.030* 3.14 * Estimated. Sources of data: Col. 1: Col. 8, table 19. Col. 2: United States Department of Commerce. Monthly summary of foreign commerce of the United States, monthly issues. Cel., 3: Col, 1 minus col, 2.^ Col. 4: United States Department of Commerce. Monthly summary of foreign commerce of the United States, monthly issues. For 1922 through 1933: General imports minus exports of foreign lemons; 1934 to date: Imports for consumption. Exports of foreign lemons for 1922 through 1925 and all import figures converted to boxes on basis of 76 pounds per box. / ,\ *^ ° I' r (continued; . ■■ • i ;' ■ * ■ . f » . . \ 1 1* i t • 1 i • ■ • ■ ,; ; 1 -. I { ^ » I r .• f ; 1 * 1 ; ■ » . ■ , ■ • ■ . . » . . M V. .. i i i ' 1 ■ ■ '■ ■ \ \:"r ... f = . • ». ..■ ,• ; / Table 18 continued Sources of data: Col» 5: Col, 3 plus col, 4. Col, 6: Col, 4 of table 17. Col, 7: Col, 5 divided by col. 6 and multiplied by 76 for conversion to pounds. TABLE 19 Industry Shipments of '"anter and Summer Lemons for Fresh Consumption, 1921-22 to 1940-41 Year l&£l-22 1922- 23 1923- 24 1924- 25 1925- 26 1926- 27 1927- 28 1928- 29 1929- 30 1930- 31 1931- 32 1932- 33 1933- 34 1934- 35 1935- 36 1936- 37 1937- 38 1938- 39 1939- 40 1940- 41 1941- 42 November- Industry rail and boat shipments carlots 3,881 3,439 4,987 4,285 4,623 5,573 4,067 5,894 4,018 5,493 5,402 4,593 5,143 6,649 6,128 5,347 5,169 5,908 6,271 6,885 6.369 C.F.G.iii. rail and boat shipments April carlots 3,368 3,224 4,579 3,879 4,287 4,479 3,735 5,169 3,636 4,685 4,397 4,060 4,513 5,294 5,242 4,803 4,534 4,784 5,000 5,260 5.544 C.F.G.E. shipments 1.000 boxes 1,523 1,301 1,867 1,562 1,725 1,719 1»479 2,009 1,477 1,831 1,727 1,610 1,898 2,213 2,285 2,417 2,133 2,260 2,363 2,481 2.621 May-October Sources of data: Cols. 1, Exchange and industry Col. 3: packed and loose fruit estimated to be 2»86 p Industry C.F.G.E. Total rail and rail and C.F.G.E. Total industry boat boat shipments industry shipments shipments shipments shipments 4 5 6 7 8 1.000 carlots carlots 1,000 1,000 boxes boxes boxes 1,756 5,751 4,922 2,076 2,425 1,388 5,215 4,972 2,006 ? 104. 2,034 7,954 6,819 2,859 1,726 7,306 6,750 2,738 1,860 8,771 8,007 3,143 2,139 7,641 6,860 2,732 1,610 8,224 7,617 2,928 w y X u c> 2,291 8,460 7,675 3,064 "^77 1,632 9,010 8,319 3,174 2,147 10,072 9,195 3,598 2,122 7,910 7,291 2,919 3,167 1,823 8,943 8,004 3,193 3,567 2,163 10,515 9,586 3,810 4,180 2,779 10,877 9,928 4,140 4,535 2,671 11,003 10,123 4,209 4,576 2,691 8,526 7,994 3,717 3, 964 2,433 10,753 9,793 4,476 4,915 2,790 11,173 10,107 4,734 5,234 2,964 10,215 8,907 4,235 4,857 3,247 12,554 11,067 5,091 5,775 3.011 10.546 9.494 4.376 4,861 2, 5, and 6: Reported by the railroads to the California Fruit Growers Exchange, truck shipments are excluded. From the records of the California Fruit Growers Exchange. Consists of shipments of For 1921-22 through 1925-26 shipments of loose fruit not available, but were er cent of packed shipments during those years. (Continued on next page) CO a-ble 19 continued: Sources of data: Col. 4: Col. 3 multiplied by col. 1 divided by col, 2. V ^ ^ ?°"^* ''j. l^^ records of the California Fruit Growers Exchange. Consists of shipments of packed and loose fruit. For 1922 through 1925, shipments of loose fruit not available, but were estimated to be 3.02 per cent of packed shipments during those years. Col. 8: Col. 7 multiplied by col, 5 divided by col. 6. --3 O :■■ -f. TABLE 20 71 Winter Lemons: California Fruit Growers Exchange Domestic Shipments and Outside Domestic Shipments, 1921-22 to 1940-41 I Year Domestic C.F.G.E. Estimated C • F. G. E. Outside November- shipments shipments C.F.G.E. domestic domesti» April exports shipments shipments 1 2 3 4 5 1,000 1,000 1,000 1,000 1,000 boxes boxes boxes boxes boxes 1921-22 1,641 1,523 100 1,423 215 i 1922-23 1,308 1,301 75 1,226 82 1923-24 1,911 1,867 113 1,754 157 1924-25 1,648 1,562 71 1,491 157 1925-26 1,750 1,725 102 1,623 127 1926-27 1,954 1,719 149 1,570 384 1927-28 1,517 1,479 86 1,393 124 1928-29 2,146 2,009 128 1,881 265 1929-30 1,558 1,477 67 1,410 148 1930-31 2,008 1,831 118 1,713 295 1931-32 1,984 1,727 112 1,615 369 1932-33 1,761 1,610 54 1,556 205 1933-34 2,073 1,898 78 1,820 253 1934-35 2,609 2,213 136 2,077 53J > 1935-36 2,339 2,174 16 J 1936-37 2,559 2,393 16£ 1937-38 2,283 2,078 20{ 1938-39 2,491 2,143 34£ i 1939-40 2,746 2,345 40] [ ■ 1940-41 3,013 2,459 554 Sources of data: Col. 1: Col. 3, table 17. Col. 2: Col. 3, table 19. Col. 3: Col. 2 of table 17 multiplied by the ratio of California Fruit Growers Exchange rail and boat shipments of lemons to industry rail and boat shipments as reported by the railroads to California Fruit Growers Exchange • ^ Col, 4: Through 1934-35, col. 2 minus col. 3; beginning 1935-36, reported by the California Fruit Growers Exchange. Col. 5: Col. 1 minus col« 4. 351 3l i.o . TABLE 21 Summer Lemons: California Fruit Growers Exchange Domestic Shipments and Outside Supply, 1922-1941 Year Estimated C.F.G.E. Hay- Domestic C.F.G.E. C.F.G.E. domes t io U w lUC a U X U Outside October shipments shipments exports shicmsnta SUDPlv 1 2 3 4 c 6 1,000 1,000 1,000 1,000 1. 000 boxes boxes boxes V V "V \^ 1^ L/U AC? O boxes boxes 1922 ? ^4.7 C , \J*t t 9 n7fi d7 2,009 338 269 1923 <c,UUd 81 1,925 94 1, 190 19?4 O f CI 1 101 2,758 459 927 1925 9 RH 7 C , / OO 71 2,667 220 1. 321 1926 % 9R9 O p CO c 146 2,997 285 028 1927 •L ^ C 1 9 RRft 139 2,593 295 914 1928 o , wo J. 9 Q 9(3 TOT 121 2,807 224 688 1929 3 999 O , UDrr 2,923 299 631 1930 3 33'^ 1 74. ytj 3,078 255 804 1931 3 81? 1 J. / 3,481 331 507 1932 3 07? 9 Ql Q Q 7 2,832 240 287 1933 3 471 3 1 Q"^ 0 , J.«70 OO 3, 107 364 433 1934 4,040 3,810 128 3,682 358 373 1935 4,159 4, 140 349 7 7Qfl o , /yo 361 361 1936 4,274 4,078 196 235 1937 3,840 3,691 149 175 1938 4,345 4,133 212 212 1939 4,702 4,453 249 249 1940 4,605 4,217 388 388 1941 5,499 5,070 429 429 Sources of data: Col. 1: Col. 3, table 18. Col. 2: Col. 7, table 19, Col. 3: Col. 2 of table 18 multiplied by the ratio of California Fruit Growers Exchange rail and boat shipments of lemons to industry rail and boat shipments as reported by the railroads to the California Fruit Growers Exchange, Col. 4: Through 1935, Col. 2 minus col. 3; beginning 1936^ reported by the California Fruit Growers Exchange. Col. 5: Col. 1 minus col. 4. Col. 6: Col. 5 plus col. 4 of table 18. 73. The domestic shipments of ^idnter lemons by the California Fruit Growers Exchange and by all other handlers are given in table 20. Similar data for summer lemons are recorded in table 21, For years prior to 1935-36, shipment figures entered in the permanent records of the Exchange include exports. Exchange exports for the period 1921-22 to 1934-35 were estimated by adjusting industry export fig- ures by the ratio of Exchange rail and boat shipments to industry rail and boat shipments. This adjustment is based on the assumption that the ratio of Exohange exports to Exohange shipments is the same as the ratio of exports by other handlers of California and Arizona lemons to their total shipments. The Income Series .-- In table 22 are given indexes of nonagri cultural income payments based on data published by the United States Department of Agriculture, The entries in this table are simple averages of the monthly indexes for the six- month periods November through April and May through October. ^2/ It should, perhaps, be noted here that for the years prior to 1929 the index is based on W, I, King's and the Brookings Institution estimates, and since 1929 on United States Department of Commerce estimates of income payments. j At least two criticisms may be offered of the use of the index of nonagricul- tural income payments as a measure of consumer purchasing pov/er. Since lemons are produced by a small proportion of the total number of farmers, total income pay- ments would be a more suitable measure of consumer purchasing power than non- agricultural income payments. However, a still more suitable index would be one based on total income payments from which necessitous expenditures have been — ' A description of this index is given in: Bean, L, H,, P, H. Bollinger, and 0, V. Wells. Nonagri cultural income as a measure of dcsnestic demand. 45p, U.S. Agr. Adjust, Aimin. June 1937. 51/ — ■/ The entire series prior to 1929 is published in Leven, Maurice, Harold G. Moulton, and Clark ViTarburton, America's capacity to consume, p, 152. The Brook- ings Institution, V/ashington, D.C., 1934. "soflimws, tot # - '■ ■' '-^l o^rr - ■ ■ TABLE 22 Indexes of Total and Per Capita Nonagricultural Income Payments in the United States, 1921-22 to 1940-41 74. Year 1921- 22 1922- 23 1923- 24 1924- 25 1925- 26 1926- 27 1927- 28 1928- 29 1929- 30 1930- 31 1931- 32 1932- 33 1933- 34 1934- 35 1935- 36 1936- 37 1937- 38 1938- 39 1939- 40 1940- 41 Nonagricultural income payments November- April 1925-29 = 100 1 71 83 92 93 100 102 102 106 105 91 75 61 69 74 82 93 88 90 97 109 May-October 1935-39 = 100 88 100 101 110 113 116 119 123 111 96 74 71 81 87 104 108 99 107 114 135 Index of nonfarm population 1922 = 100 Per capita nonagricultural income payments 100 103 105 107 110 112 114 116 118 119 119 120 121 122 123 124 125 126 127* 128* November- April 71 81 88 87 91 91 89 91 89 76 63 51 57 61 67 75 70 71 76 85 May- October 88 97 96 103 103 104 104 106 94 81 62 69 Vi 85 87 79 85 90 105 ♦Estimated. Sources of data: i Col. 1: 1921 to May, 1941: United States Bureau of Agricultural Economics. Nonagricultural income payments. United States, 1919 to date. 2p* T/ashington, D. C, July 21, 1941. (Mimeo.) June, 1941 on: United States Department of Commerce. Survey of Current Business, monthly issues, adjusted by 88.105 per cent. Figures are simple averages of seasonally corrected monthly indexes. Loans and payments to World V/ar veterans on their adjusted service certificates in 1931 and 1936 are excluded. In the other years veteran bonus payments were of little significance. Col. 2: United States Bureau of Agricultural Economics. Nonagricul- tural income payments. United States, 1909-41. 2p. Y^ashington, D. C. Feb- ruary 2, 1942. Figures are simple averages of seasonally corrected monthly indexes. Loans and payments to Y.'orld lYar veterans on their adjusted service certificates in 1931 and 1936 are excluded. Col. 3: Col. 4 of table 17 minus estimates of farm population given in: United States Bureau of Agricultural Economics. Farm population estimates. Released July, 1940. (Mimeo.) Adjusted as of July 1 of each year. Col. 4: Col. 1 divided by col. 3, Col. 5: Col. 2 divided by col. 3 ' " / - < ■. ■ . : ■■■■■■■■ - '■ - ■ ' i ; ■ ■ . ■ 75. subtracted. The demand for most consumer goods, one would suppose, is affected not so much by total income payments as by the amount of freely disposable income avail- able to the consumers. At the time the calculations reported in this paper were made, estimates of total income payments on other than calendar year basis were not available for years prior to 1929, Later, total income payments for November-April and May- October of each year were estimated by the senior author for use in another demand study. Estimates of disposable income were also prepared by subtracting, from total income payments to individuals, necessitous living expenditures, medical expenditures, and selected federal taxes, Ylhile the three indexes ~ namely, nonagrioultural income payments, total income payments to individuals, and disposable income — exhibited some differences, their year-to-year fluctuations were strikingly similar. The correlations betvreen the three November-April indexes for the period 1922-23 through 1939-40 were r^^ = ,992, r^^ = ,984, and r^^ = ,996; the correlations for May-October were r^^ = ,991, r^^ = .984, and r^^ = ,998 (a » nonagrioultural income payments, b = total income payments, o = residual income). In view of the magnitude of these correlations it is doubtful that extensive recalculations were warranted. Indexes of Summer and Winter Temperatures ,— The construction of a tempera- ture index raises a number of troublesome questions. What should be the basic unit of time in which temperatures are to be collected? Should mean, maximum or minimum temperatures be used? For what localities should temperatures be recorded? Should actual temperatures, departures from normal in a given area, or departures from some arbitrary level constant for all localities be calculated? Eavr should temperatures be combined? Extensive trial may supply appropriate answers to all of these ques- tions, but extensive trial is time-consuming. Practical considerations make it impossible to avoid arbitrariness in the construction of the index. The index of summer temperatures constructed for this study consists, for each of the twenty six-month periods, of the sum of weighted daily positive depart- ures of maximum temperatures from 80° Fahrenheit in 39 metropolitan areas (see ■;J\r \'C \fjf ^r; fV^j. 76, table 23), Population weights were used in calculating the daily weighted means. In order to insure a satisfactory degree of sensitiveness to the index, it appeared advisable to employ as its basic component daily maximum temperatures. Considering the intended use of this variable, it is clear that a geographically representative measure of temperature would not be particularly suitable. Rather an index which summarized temperature conditions in the densely populated areas of this country was required. This led to a consideration of metropolitan districts as areas for which temperature data should be secured. The 39 metropolitan districts selected were the urban centers largest in population. The exclusion of pre- dominantly rural areas is open to criticism, since the available evidence suggests that the per-oapita consumption of lemons in rural areas is nearly on par with the per-capita oonsumptiom in urban districts, ^ The geographical extensiveness of rural regions, however, has precluded their inclusion in the index. It is likely that the relation between temperature and consumption of cold drinks has a lower bound within the range of observed temperatures; below a certain point changes in temperature fail to have a measvirable effect on the consumption of cold drinks. This consideration suggests the use of positi-ve departures of maximum temperatures from some level. For this purpose a constant reference point, namely 80° Fahrenheit, was employed for all districts and for all days. The use of a changing reference level, such as the "normal" maximum temperature in a certain district for a given day, would have partialled out the effect of differences between the general levels of temperature in the various metropolitan areas. It appeared desirable, however, that the index in part reflect such differences. The selection of 80° Fahrenheit as the reference level is, of course, arbitrary, but the use of any other reference point, provided that it was not extreme, would have, in all likelihood, given approximately the same results, 52/ — ' Annual report of the general manager of the California Fruit Growers Exchange for 1937, 36p, "i''-'^:^'"" ^-i^^-' •'ir'^jw: c civ ■ fit ::-iv;4:o ■, 'rt: •> 'Tf^ ;• jfV.^!- ii-; t;,.'!. T. '.1 I; .i ' C V'' '■ I: - .0 Ci TABLE 23 Indexes of ViTinter and Summer Temperatures 1921-22 to 1940-41 77. Year Index of temperatures Year Index of temperature 5S December- February * May- October f December- February * I'iay- October f 1 2 ■ — — — ~— — 3 4 per cent per cent per cent per "cent 81 100 1931-32 100 lie ) 78 92 1932-33 91 136 T 85 80 1933-34 76 142 X ^ c c 0 82 120 1934-35 79 106 80 88 1935-36 67 151 1926-27 82 75 1936-37 90 127 1927-28 84 89 1937-38 84 115 1928-29 81 105 1938-39 87 140 1929-30 86 138 1939-40 77 106 1930-31 87 150 1840-41 83 145 * 1931-32 =100. t 1922 = 100, Sources of data: Cols. 1 and 5: United States Weather Bureau, Climatic Summary of the United States, United States Meteorological Yearbook and Climatological Data for the United States by sections. Monthly mean temperatures December through February in 32 cities vreighted by the population in the correspond- ing metropolitan districts. The weights employed are given in table 24, Cols, 2 and 4: United States Weather Bureau, Climatological Data for the United States by sections. Positive departures of daily maximum tempera- tures in 39 cities from 80°, May through October, vreighted by the population in the corresponding metropolitan districts. The vreights employed are given in table 24. ^ :. . ■ ■ ( •.: : ■■ • ... .... — I ; • ; i . . • \ ■ 1 i : ; 1 . : „-,.. '■■■>.;** ' " ; ! I '. i \ . i ■ '■■u i ... ■; . • - .- ■Hii. 1 t .. . i :..f 1 JLc-; . - .r;iJCE v . V.,- i,i !f ji-JXl 78. On© other item requires mention in connection with the construction of the index, namely the weights employed in the calculation of the daily means. As was indicated previously, the positive departures of maximum temperatures from 80° Fahrenheit in each of the 39 metropolitan districts were weighted by the population in the district. For this purpose the population figures from the censuses of 1920, 1930, and 1940 were employed as weights for periods 1922 through 1925, 1926 through 1934, and 1935 through 1941 respectively (see table 24), The use of the three sets of weights was deemed sufficient to account for the relative shifts of populations in -ttiese areas. Other weights, such as average unloads of lemons in each of the metropolitan 55/ districts — ' or even income figures for each area, might have been employed in place of the weights we have used, or in conjunction with them. The weighting employed is based on the simple hypothesis that, at each level of temperature, consumption of cold drinks is a single-valued monotonically increasing function of the number of persons exposed to the temperature. The weighting disregards possible regional differences in lemon consumption, but it is doubtful that there is marked regional variation in the use of lemons in cold drinks, ^ If simple rather than composite weighting is to be employed, weights based on population appeared to be superior to other feasible types of weights, | The index of temperatures employed in the analysis of demand for winter lemons is based on mean temperatures during December, January, and February in 32 metropolitan districts. The weighted mean temperature for each of the three months 5^ 5y See appendix E for the use of average unloads as weights. The data in table 1 suggest the existence of at least one regional difference in consumption of lemons, namely that between the southeastern region of the United States and other regions considered in the Study of Consumer Purchases, The data reported in table 1 were secured during the months May through August of 1936, Analogous data secured during the months September through November, certainly a cooler period, reveal about the same difference between lemon consumption in the southeast and in the other regions. Thus it is not clear that this regional differ- tial is related to the use of lemons in thirst-quenching beverages. ; ©or .■^rl> yj .as! iioqoad-sm erirf lo liom 't'.; lie. i'tfc'.'^'' Ji) 8.;. V '! i ' ■ >,..j%- n.i. nt: TABLE 24 Population Weights Employed in the Construction of Indexes of Summer and Winter Temperatures Metropolitan districts Mi ay-October ii idex Decernber-Ffthrnqrv inrio^ 1922 through 1925 1926 through 1934' 1935 through 1941 1921-22 through 1924-25 1925-26 through 1933-34 1935-36 through 1940-41 Albany - Schenectady - Troy, N. Y. Atlanta, Georgia Baltimore, Maryland Birmingham, Alabama Boston, Massachusetts Buffalo, New York Chicago, Illinois Cincinnati, Ohio Cleveland, Ohio Columbus, Ohio Dallas, Texas i Denver, Colorado Detroit, Michigan Hartford, Connecticut Houston, Texas Indianapolis, Indiana Kansas City (Kansas, Missouri) Los Angeles, California Louisville, Kentucky Memphis, Tennessee Milwaukee, 'Aisconsin Minneapolis - St, Paul, Minnesota New Orleans, Louisiana New York - Northeastern New Jersey Oklahoma City, Oklah.oma Omaha, Neb, - Council Bluffs, Iowa Philadelphia, Pennsylvania 1 2 3 4 5 6 .011 .008 ,024 .008 .058 .019 .095 .018 ,027 .008 ,006 .008 .036 .011 .005 .010 .014 .031 .010 .005 .016 .020 .012 .247 .003 .007 .010 .008 .021 .009 .052 .018 .098 .017 .027 .008 .007 .007 .047 .011 .008 .009 .014 .052 .009 .006 7 .019 .011 .245 .005 .006 .009 .009 .022 .009 .049 .018 .094 .016 .025 .008 .008 .008 .048 .010 .011 .009 .013 .061 .009 .007 • UXO .019 .011 .244 .005 .006 .012 .026 .064 .022 .105 .020 .030 .009 .009 .040 .012 .011 .015 .011 .006 .018 .022 .273 .003 .008 .011 .024 .059 .021 .112 .019 .031 .009 .008 .054 .012 .011 .016 .010 .007 ,019 ,021 ,279 .005 .007 .011 .025 .057 .021 .109 .019 .029 .009 .009 .056 .012 .011 .015 .011 .008 .019 .022 .283 .005 .007 Pittsburgh, Pennsylvania Portland, Oregon fOTl ,049 .009 v664 .044 .009 .060 .042 .008 .079 .055 .010 .073 .050 .010 .070 .048 .010 Providence. Rhode Island .025 1 .022 .021 .028 .025 .024 (Continued on next page) I V- ■ ':;!■: ■ . V i i ' ■■ T^: •" " 1 ; ;'" .■ - "' ' .1 ' ■ \r:: i. ■ : ! Table 24 continued. May- October index December-February index 1922 1926 1935 1921-22 1925-26 Metropolitan districts through through through throuffh V 1 11 LI E^l 1 + Vi y*A 1 1 crVi 1925 1934 1941 1924-25 1 2 3 4 5 Richmond, Virginia ,006 .005 .005 .006 .006 .006 Rochester, New York .010 .009 .009 -0] 1 ♦ \JJ. -1. m n St, Louis, Missouri .031 .029 .029 .034 .033 .033 Salt Lake City, Utah .004 .004 .004 .005 .005 .005 San Francisco - Oakland, California .028 .029 .030 Scranton - Wilkes Barre, Pa, .017 .015 .013 .019 .017 ;oi5 Seattle, Washington .010 .009 .009 .011 .011 .011 Toledo, Ohio .008 .008 .007 .009 .009 .008 Washington, D,C, .015 .014 .019 .017 .016 .022 Total population 34,489,191 44,519,745 47,950,608 31,144,016 39,009,662 41,340,367 Sources of data: Cols, 1 and 4: Population figures for 1920 from United States Bureau of the Census, Fifteenth Census of the United States, 1930. Metropolitan Districts, Population and Area, Govt. Print. Off., '.'ashington, 1932, expressed as ratios of the total population in the cited metropolitan districts. Cols, 2, 3, 5, and 6: Population figures for 1930 and 1940 from United States Bureau of the Census, Sixteenth Census of the United States, 1940, Population, first series. Number of Inhabitants, United States Suratnary. Govt. Print. Off,, Washington, 1941, expressed as ratios of the total population in the cited metro- politan districts. CO o 1 h{ 4 '■■ . I \ ' r" i 1 . ■ ; i : -'t » ■ i.- . '. i i I / ♦ i • ■ • ■ , ■' * • ■- i 1 ' : i- .• J - , ..' 1 .1- ' ■■ ! ; » ■ . I - ' . ' : t. 1 • , r-v— ' i - . t ! t i I' ; — - i — j t 81. was obtained by weighting the mean monthly temperatxires in the 82 districts by the population in each district. As before, three different sets of population figures were employed, based on the 1920, 1930, and 1940 population counts and applied respectively to the periods 1921-22 through 1924-25, 1925-26 tlirough 1933-34, and 1934-35 through 1940-41 (see table 24). The final figure for each year is the sum of the weighted mean temperatures in December, January, and February adjusted for the difference in the number of days in these months. j Index of Influenza .— The index of influenza given in table 25 is based on the incidence of influenza statistics reported by the various state health boards to the United States Public Health Service. Considerable caution must be exercised in the use of this index. Incidence of influenza figures are far from complete for even the most recent years. Not all of the states have or are reporting nonfatal cases of influenza and, for states which do report, varous degrees of underreport- ing are to be expected. | The index was constructed by calculating the link relatives of the incidences for states reporting influenza in two successive years. The nvnnber of states on which the relatives are based varied, of course, from year to year. An alternative procedure would have been to base the index on a constant sample of states, that is, on states that have reported for each year of the period under reviev^. It is felt that the first procedure yields a more reliable index since it utilizes a larger number of reports. However, an index based on data secured from states •vrfiioh have reported influenza cases throughout the period showed excellent agree- ment with the index reported in table 25. The major difference between the two indexes was the greater magnitude for epidemic years of the values of the iniex based on a constant number of states. i .. . ■ • ' ; ; • \ ■ ■ ■' - ■ ' ■. -t ,,tbf.'' TABLE 25 82 Index of the Incidence of Influenza, 1921-22 to 1940-41 Year, November- April Index of incidence of influenza Year, November- April Index of incidencfi of influenza per cent per cent 1921-22 64 1931-32 21 1922-23 100 1932-33 90 1923-24 15 1933-34 13 1924-25 44 1934-35 28 1925-26 64 1935-36 38 1926-27 21 1936-37 1927-28 24 1937-38 15 1928-29 176 1938-39 33 1929-30 12 1939-40 44 1930-31 26 1940-41 162 Source of data: United States Public Health Service, Public Health Reports. The figures are link relatives reduced to 1922-23 base* 83, APPENDIX B Price and Quantity Residuals and Tests of Randomness i TABLE 26 Summer Lemons: Price and Quantity Residuals 84* Year Equation Mav- October Is 2s 3s 4s 5s 6s 7s 8s 9s 10s 10 000 in onn Uu J. XUFB 10,000 boxes boxes boxes 0,12 3 0,23 0,04 0,005 -0,002 0,03 4 0,06 4 0,24 3 0,16 0,00 -0,004 *0,012 0,15 4 0.20 4 -0,79 -28 -0,65 -0.12 -0,099 »-0»034 -0,78 -27 -0,81 -28 T Q o r 0,48 18 0,64 0,15 0,059 0.022 0,32 20 0,40 20 -0,37 -11 -0,22 -0,01 -0,021 -0,002 -0.44 -10 -0,40 -10 TOOT 0.45 17 0.37 0.07 0,061 0,022 0.49 17 0,46 17 -0,12 -10 -0.18 -0,07 -0,001 -0,007 -0.05 -11 -0,08 -11 -0,14 -15 -0,23 -0,10 -0.012 -0.015 -0.05 -16 -0.10 -16 lyou -0.04 -10 -0.04 -0.03 -0.007 -0.012 -0.08 -9 -0.07 -9 iyol -0.17 -12 -0.09 -0.04 -0.016 -0.012 -0.17 -11 -0,18 -11 iyo<: 0,07 0 0.00 -0.03 -0.001 -0.005 0.15 -1 0,12 -2 1933 -0,48 -21 -0.46 -0.12 -0.040 -0.017 -0.47 -21 -0,48 -21 19o4 0,22 12 0.27 0.07 0.034 0.017 0.22 12 0,21 12 1 Q C 0,71 39 0.71 0.21 0,085 0,046 0.74 39 0.73 39 i90D 0.11 3 0,11 0.01 0,016 0.004 0,13 2 0,13 2 1937 0.06 -2 -0.03 -0.04 0.011 -0.001 0,15 -3 0.12 -4 1938 -0.21 -1 -0,25 -0.02 -0.034 -0.004 -0.17 -1 -0.17 -2 1939 -0.25 -5 -0.31 -0.04 -0,036 -0.001 -0.25 -5 -0.24 -5 1940 0.08 13 -0.06 0.03 0.010 0.015 0.13 12 0.10 12 1941 0.02 10 -0.06 0.04 -0.012 0.004 0.01 10 0.00 10 Sourfce of data: Derived from equations Is through 10s of tables 3 and 6 applied to data given in table 2, *f • -i — "T " i * ... — j,. ■ » 1 i i 1 .i .0 i 1 • .' f ' ' f ■ ■• « i ; - '}•■'■'■ 1 • i; • ^';>*' ;, ■ • ■■■ J • - : ^ *J ■ ■ ■-' *; r r ; 1 ■ ■ ■• ; . I: ,■ i n ■ ?.-<SX-', ; . - , ; . » ■ - ! ■' 4 * ' / - ; , ! ' ( i • * 1 it'. : ' f '. ■ ■ *.,Vf. C i ' ■>■ ^» ' ■ i • ; ■* N;. ^. • i' .•lv.;o • \'i ■ (■;«■;■ ( t ' T t f . ■ 'N i * -■ ■ ■ (• i •• ' vi-i-i .•".)( ! , • .1^.0. ■ I i < TABLE 27 Summer Lemons: Tests of Randomness of Residuals Eauation Lagged oo- * W J» cat V.f>^Ai coefficient Wallis and Moore test 2 1 2 3 Is -0.3563 0.816 0,70 2s -0.0706 3,149 0.26 3s -0.2349 0,298 0.88 4s 0,3848 1.502 0.53 5s -0.1399 0.416 0.84 6s 0,1304 2.395 0.36 7s -0.2982 0.816 0,70 8s -0.0802 3,149 0.26 9s -0.3280 0.816 0.70 10s -0.0785 3,149 0.26 Sources of data: Col. 1: Correlations of residuals lagged one year, based on data given in table 26. The 1 per cent significance point for n = 20 at the negative tail is -0.524. (Anderson, R. L, Distribution of the serial correlation coefficient. Annals of Mathematical Statistics 13:1-13, 1942.) Cols. 2 and 3: Based on data given in table 26 and calcu- lated by combining all phases of more than two years. A given entry in col. 3 gives the probability that a Xp of the magni- tude of the corresponding entry in col, 2 or larger would be obtained in a random time series. For details of the test, see Vifallis, W, A, and G. H. Moore, A significance test for time series and other ordered observations. National Bureau of Economic Research Tech, Paper 1, September, 1941. :r.<- » i i; i t •' ! 1 ii asV- ■ !; i a^: i ! 80X 1 —■»;■.--■■ :?;.-.-&■ - " if." J.'t' «rft>©T xio 86. TABLE 28 Winter Lemons: Price and Quantity Residuals Year Equat ion November- April Iw 2w 3w 4w \J w dollars 1,000 boxes dollars pounds 10|S_ 1 1921-22 0.59 219 0.61 0.14 0.070 0 .045 1922-23 -0.15 -86 -0.27 -0.08 -0.067 -0 1923-24 -0.30 -24 -0.28 -0.01 -0.047 -0 .VJUD 1924-25 0.17 48 0.15 0.02 0.029 0 1926-26 -0.47 -142 -0.46 -0.09 -0.038 -0 1926-27 -0.58 -148 -0.62 -0.08 -0.056 -0 1927-28 0.26 11 0.26 0.01 0.017 -0 1928-29 0.40 151 0.45 0.10 0.086 0 1929-30 0.28 1 0.30 0.00 0.023 -0 .OOQ 1930-31 -0.10 -26 -0.08 -0.01 0.013 0 .009 1931-32 0.10 48 0,07 0.02 0.003 0 1932-33 -0.16 -89 -0,18 -0.06 -0.005 -0 .012 1933-34 -0.23 -91 -0,22 -0.05 -0.008 -0 UUcS 1934-35 0.13 105 0.09 0.06 -0.025 0, OAT 1935-36 0,04 -10 0.06 0.00 0.014 0. 003 1936-37 1937-38 0.10 -3 0.16 0.01 0.040 0. 014 1938-39 -0.29 -89 -0.27 -0.05 -0.036 -0. 019 1939-40 0.07 33 0,05 0.02 0.008 0. 006 1940-41 0.13 90 0.07 0.04 -0.022 -0. 003 Source of data: Derived from equations Iw through 6w of table 8 applied to data given in table 7. : -<3 x : ■■■•f 1 l i ■■ ! ■* ■ . 1 . .. « ' ■ ■" ■' ~. » * i f t 1 i > ■ i i i f • . ' .' * ■ '. ' • ! t t t t i ■ ". * - S 1 j i ii ■ 8 7, TABLE 29 Winter Lemons: Tests of Randomness of Residuals Equation Lagged cor- relation coefficient Wallls and Moore test 1 2 3 Iw 0.0583 3,399 0.23 2w -0.0888 6.862 0.05 Sw 0,0668 3.399 0.23 4w -0,1399 1.468 0,53 5w -0,0471 5.668 0,09 6w -0,1040 4.399 0.15 Sources of data; Col, 1: Correlations of residuals lagged one year, based on data given in table 28. The 1 per cent significance point for N = 15 at the negative tail is -0.597. (Anderson, R, L, Distribution of the serial correlation coefficient, Aimals ■ of Mathematical Statistics 13:1-13, 1942.) Cols, 2 and 3: Based on data given in table 28 for 1921- 22 through 1935-36 and calculated by combining all phases of more than two years, A given entry in col, 3 gives the probability that a of the magnitude of the corresponding entry in col, 2 or larger v;ould be obtained in a random time series. For details of the test, see V/allis, YJ, A. and G. H. Moore, A significance test for time series and other ordered observations. National Bureau of Economic Research Tech, Paper 1, September, 1941, • 4 • •— -t ■v.- - 1 .,1 ■.,r APPENDIX C Net Regression Charts 6.00 5.50 5.00 4.50 o I 4.00 ^ 3.50 0) 3.00 ^ 2.50 o 2.00 1.50 1.00 310 350 390 430 470 510 550 United States supply (Xp) 10,000 boxes 70 30 100 110 120 150 Index of nonagricultural income (x^) 70 80 90 100 110 120 130 140 150 Index of temperature (x^) Fig. 5. — Summer lemons: Net regressions of price on (A) United States supply (xip = 12.1858 - 0.0205 Xg), (B) index of nonagricultural income payments (x^5 = -0.4630 + 0.0445 X3), and (C) index of temperature (x^^ = 0.8536 + 0.0282 X4), 1922-1941. Arithmetic form. CQ g o o o o rH I I >f400 450 - 3.00 3.50 4.00 4.50 5.00 F.o.b. price (x^) — dollars a, §• to CO o -p cd +3 CO -d 0) -p •H c 350 300 70 80 90 100 110 120 130 Index of nonagricultural income (x^ ) 140 150 160 Index of temperature (x^) Fig. 4. — Summer lemons: Net regressions of United States supply on (a) price (xgi = 563.4914 - 41,2473 x^), (B) index of nonagricultural inco:ne payments (xgs = 175.3518 + 2.1239 X3), and (C) index of temperature (X24 - 234.5614 + 1.3774 X4), 1922-1941. Arithmetic form. 91. 70 80 90 100 110 120 130 140 150 Index of temperature (x^) Fig. 5. — Siunmer lemons: Net regressions of price on (A) per capita United States supply (x^po = 13.0064 - 3.6517 xg), (3) index of per capita nonagricul- tural income payments (xi^o = -0.8778 + 0.565 x§), and (C) index of temperature (^14 " + 0.0272 X4), 1922-1941. Arithmetic form. 1^- 92. 2.9 2.8 n 5 2.7 (X \ 2.6 2.5- rH & 2.4 P (0 ^ 2.3 •H & Q 2.2 u 0) ^ 2.1 2.0 2 .35 A .40 \.41 - 24 3'^\ -34 - \' 36 32 .\ . 31' X . 25 27 37" • 28* ox t 1 1,1 3o\ 29* o^<^'2.4 ,50 3. F. 00 3.50 4.00 4.50 5.00 5.50 o.b. price (x-j^) — dollars 1 I I 1 L 50 70 80 90 100 110 120 Index of per capita non- agriculUiral income (x°) 2.8 - w x> 2.7 - o a, 1 2.6 - ^ 2.4 §• 2.3 cS ■P •H 2.2 o 2.1 2.0 c 34 36 y .25 4lX / -31 .35 /<39 ^0* ^° ^38 '33 22 'y^ 32 .27 y ^23 •^26 '^8 •29 ' .24 ' . . 1 I ... 1 . ( 1 ! 1 70 80 90 100 110 120 130 140 150 160 Index of temperature (x^) Fig. 6. — Rummer lemons: Net regressions of per capita United States supply on (A) price (x^i =,3.3727 - 0,2295 x^), (B) index of per capita nonagricultujral income payments (xlso = 1.1373 + 0,0146 x§), and (C) index of temperature (x^4 = 1.5834 + 0.0073 X4), 1922-1941. Arithmetic form. J. 95. (D O •H o o O O O u o o o (-1 •^2° -15 2.50 ,00 .70 to o .60 .50 .40 .33 2.55 2.60 2.65 Log of supply (log xg) .30 2.70 2.75 X.70 to o O t£0 Q4 O HO O .50 40 C 35- 25 • 37. . 54' *53 36. y"b\ ) 41 59 .27 40./^ ?2 /'^^ 3 •38 16 - .24 • • 1 1.85 1.95 2.05 Log of income (log x^) 2.15 70 80 90 100 110 120 130 140 150 160 Index of temperature (x.) Fig. 7. — Summer lemons: Net regressions of price on (A) United States supply (log x^g = 6.0075 - 2.0854 log x^), (B) index of nonagricultural income payments (log x-j^^ = -1.4737 + 1.0356 log Xg), and (C) index of temperature (log x^^ = 0.2300 + 0.0032 x^), 1922-1941. Logarithmic form. i 94. 2.70- to o V) o f-3 2.70 1.85 1.95 2,05 2.15 Log of income (log x™) 90 100 110 120 130 140 150 160 Index of temperature (x^). Fig. 8. — Summer lemons: Net regressions of United States supply on (a) price (log x^^ =» 2.8305 - 0.3965 log x^), (B) index of nonagricultural income payments (log Xg^ - 1.6148 + 0.4865 log Xj), and (C) index of temperature (log x^4 = 2.4125 + 0.0015 x^), 1922-1941. Logaritlimic form. 4.5C- u H iH 4.0C- O j 3»5C~ '"In 3.0C- 0) o •H 2.5C- !h • • 2.0C- o o « m u ns (H iH O I I A 3.50 0) o •H U P. 80 90 Index of temperature (x^ 100 ) 14 16 18 20 22 24 26 28 50 Domestic shipments (xg) — 100,000 boxes J__, .. , 1 L_t- 1951 Seasons 1935 90 100 110 Index of no nagri cultural income (xj) Fig. 9. — Winter lemons: Net regressions of price on (A) domestic shipments (x^g = 8.9058 - 0.0028 (B) index of nonagricultural income payments (x{^ = 1.2950 + 0,0225 X3), (C) index of temperature (xf4 = 0.0484 X4), and (D) time (x^5 = 3.5295 + 0.0656 X5), 1921-22 to 1940-41. Arithmetic form. ■ 7.2948 CD cn 2.00 3.00 4.00 5.00 F.o.b. price (x^) — dollars Index of temperature (x^) (0 -p a ro a. o Id-" to o •H -P to O O o o o o 23 22 21 20 19 - 18 B 29 . 41 . - 22 35 „^ 40 28^^-^^ .32 '56 • 38 31 _ .27 26 39 ^^'^ - - 34 •33 1 70 80 90 100 Index of nonagricultural income (x^) 1922 1925 1930 Seasons 1935 1940 Fig . 10. — — Winter lemons: Net regressions of domestic shipments on (A) price (xg-^ = 3,010,03 - 294,93 Xj|)y^ (B) index of nonagricultural income payments (xgj = 1,388.5 + 7.3125 Xj), (G) index of temperature (X24 = 3,355.1 - 15.927 X4), and (D) time (X25 = 2,053.3 + 24.805 X5), 1921-22 to 1940-41. Arithmetic form. CO Index of temperature (x^) Seasons (xg) Fig. 11. — Winter lemons: Net regressions of price on (A) per capita domestic shipments (x^^po = 8.9028 - 4.4674 Xg), (B) index of per capita nonagricultural income payments (x^^^o = 1.4173 + 0.0245 Xg), (C) index of tempera- ture (x^^ = 7.2061 - 0.0474 x^), and (D) time (x£g = 3.3204 + 0,0479 x^), 1921-22 to 1940-41. Arithmetic form. 4 ft '^1 ■Ji- ll n 4 2.50 m 3.00 3.50 F.o.b. price (x^) 4.00 dollars 4.50 50 60 70 80 90 Index of per capita nonagricultural income (x°) to -p 2 p- •H CO 1.50- 1.40- -P CO i 1.30- 1.20- •H ^ 1.10 1.00 g p-i 1.00 1921-22 -25 -30 -35 -40 Index of temperature (x^) Seasons (xp 0.1825 ture Fig. 12. — Winter lemons: Net regressions of per capita domestic^ shipments on (A) price (x^^ = 1.8569 5 xn ), (B) index of per capita nonagricultural income payments (xg^o = 0.8746 + 0 (xg^ = 2.0503 - 0.0096 X4), and (D) time (xgg = 1.2625 + 0,0115 X5), 1921-22 to 1 0050 X5), (C) index of tempera- 1940-41. Arithmetic form. CO CO id V r i,.V J' .' I V J i T 1 'y': ; \ ft ?-5 '00' Ojyt.-?.."; o."0 i V,:... 0) o •H S-, o o to o 1-5 bO O H '2:10 2.20 2.30 2.40 2.50 Log of domestic shipments (log x^) o •H U a, O <+H q o tiD O 1-5 .60 .50 .40 .36 C •22 ^\ . 40 34^\^^ 25- .38 - ^\ 28 \^ * .30 35 • 26- \:31 27 24. • 33\ 39 J . ... ^\32 1.85 1.90 1.95 Log of temperature (log x^) 2.00 1.75 1.85 1.95 2.05 Log of nonagricultural income (log x^) 1921-22 -25 -30 Seasons -35 (X5) -40 Fig. 13— Winter lemons: Net regressions of price on (A) domestic shipments (log x{z " 4.1432 - 1.5816 log xp), (B) index of nonagricultural income payments (log xis = -0.3898 + 0.4610 log X5), (C) index of tempera- ture (log xi'4 = 2.7348 - 1.1632 log X4), and (D) time (log xfg = 0.5105 + 0.0073 x^), 1921-22 to 1940-41. Logarithmic form. 2.40 CO -p n a, •H w o --^ •H CS2 -P H m m bO S O O rH tJ ^ «H O w o 1-5 0) -p to -P « w 23 ^ fi o O rH O 2.30 2.20 .30 2.40 — 2.40 2.30 2.2CI .40 .50 .60 Log of f .o.b, price (log x^) 1.75 1.85 1.95 2.05 Log of nonagricultural income (log x^) m -p a CO o ^ -P X to e o O H O 2.30 2.20 3 1.75 1.85 1.95 2 JDS Log of temperature (log x^) 2.40 CO t •rt to •H CV •P X 03 O iH 2.30 o 3 2.2C- 1921-22 -25 -30 Seasons -35 -40 (X5) Fig. 14^-^ Winter lemons: Net regressions of domestic shipments on (A) price (log Xgi—*— 2.5518 - 0.4984 i Xg. J--r. ^^.u^ii<j . ii^^j i ^j^i ^„.v^w -..^x — — . V--/ I ^ ° <,!. log Xt), (B) index of nonagricultural income payments (log Xgg = 1.8008 + 0.2566 log xj), (0) index of temperature log x^4 = 3.5279 - 0.6412 log X4), and (D) time (log Xg5 Logarithmic form. 2.3024 + 0.0051 X5), 1921-22 to 1940-41. APPEiroiX D Auxiliary Tables TABLE 30 Winter Lemons: The Effect of Adjusting Domestic Shipments in 1936-37 on the Net Regression Coefficients and on the Coefficients of Multiple Correlation Net regressi on coefficients * Adjusted Equation Dependent variable F.o.b. price Domestic shipments Nonagri- cultural income Index of tempera- tures Time coefficient of multiple correlation ""l "^2 ^3 ^4 ^5 R Including 1936-37, not adjusted ^1 -0 .0027 (0.0005) 0.0232 (0.0082) -0.0352 (0.0176 0.0652 (0.0191) 0.7867 ^2 -240.5343 (46.5887) 6.8369 (2.5069) -12.9784 (4.9377) 27.3906 (2.9472) 0..9494 Adjusting 1936-37 shipment observa- Uon to 2,204,000 boxes ^1 -285.5297 (36.2236) -0.0028 (0.0004) 0.0233 (0.0060) 7.3266 (1.9156) -0.0451 (0.0124) -14.8370 (3.8206) 0.0677 (0.0131) 25,4577 (2.2611) 0,8941 0.9686 X = Seasonal average f.o.b. price in dollars per box. Xg = Domestic shipments of California and Arizona lemons in units of 1,000 boxes. X, = Index of nonagricultural income payments, 1924-25 to 1928-29 = 100. ^ and x^ = Index of mean temperatures in December, Januaryj/February in 32 metropolitan districts, 1931-32 =100. X5 = Time in units of ha lf years, origin halfway between 1930-31 and 1931-32. TABLE 31 Summer Lemons: Intercorrelations of the Series Employed in the Regression Analyses ""l "^3 0 V ^4 ^2d 2i V 2e 20 1 CiP IT 1 iOg Xg ^2 -0.572288 1 0 ^2 -0.548529 1 0.118360 a.452897 1 0 U vc / cOUO 1 A. \ -0 .026852 0.514790 0.347067 -0.130167 -0 •405177 1 ^2d -0,606340 0 .236590 0.592982 1 ^2i 0.469018 0.160461 -0,519741 -0.807169 1 ^2e -0.562579 0.248953 0.613403 1 ^2o 0.343487 0.157795 -0.543965 -0.772969 1 log Xj^ 0.005085 1 log 0.520499 -0.570926 1 log X3 ^0.144313 0*088095 0,430650 x, = x„ = Seasonal average f.o,b. price of summer lemons. United States supply of summer lemons. Per capita United States supply of summer lemons, Index of nonagricultural income. Index x>f per capi^ nonagri cultural income. ^ Index of temperature. "2d X2i ^26 - Domestic shipments of summer lemons, = Imports of summer lemons for consumption, = Domestic shipments of summer lemons by California Fruit Growers Exchange, -= Domestic shipments of summer lemons outside California Fruit Growers Exchange plus imports of summer lemons for consumption. o . 1 ! » ■ ■ \ ■ .... , pi . ~i TABLE 32 Summer Lemons: Reciprocal Correlation tlatrices Eauations Is and 2s Equations 6s and 7s ^3 ""l ''2d 21 ''3 ''4 X 6.562543 X 6.808934 2 8.833359 14.063709 Xo J 2d 12,769370 30.168585 ^3 -5 .438465 -8.469953 6,132164 ^21 5.165902 14.248319 8.648916 4 -5.079015 -8,105168 5,012417 5.688528 x. -5,359076 -12.466534 -5.967977 6.185965 Equations 3s and 4s -5 .401821 -11.763851 -4.591926 4.951939 6.088672 x° ''4 1 ^2 3 X4 Eauations 8s and 9s 1 0 X 2 6.829170 ^1 ^2e ^20 ^3 ^4 7,659574 10.223064 ^1 6,643819 o ""3 -6.467930 -8,176593 7.843269 ^2e 12.375084 28.082392 Y -5.095665 -6.655381 5.842061 5.540103 ^2e 5,282716 13.224280 7,912748 Equations 5s and 6s -5.382782 -12.008727 -5.740498 5.170648 log x^ log log x^ ^4 ^3 Log 5.823581 ""4 -5,239562 -11.263123 -4.412927 4.902235 6.005772 log Xg 7,710820 12.347127 log Xj -4.511113 -7.077073 5.085974 \ -4.694097 -7.487188 4.440520 5.561766 Tor the meaning of the various x's, see table 51« r TABLS 33 Winter Lemons: Intercorrelations of the Series Employed in the Regression Analyses 1 X 2 2 1 3 0 log X X log X log X ^2 — U •O'iOODVJ X x° 2 ^3 -0*710609 U »JL ooooo 1 X 1 ±. 0 ^3 0.340976 -0.169203 i • ^4 -0.209951 -0.174066 -0.207 373 -0.082620 -0.095509 1 0 .fi49398 0.740809 0.011540 -0 .427189 -0.042935 1 log \ -0.367142 1 log log log ^2 ^3 ^4 0,840508 0.003982 -0.059772 -0.666677 0.201064 -0,234293 1 0,106258 -0.163422 1 -0.071114 -5 Seasonal average f.o.b. price of winter lemons. Xg = Domestic shipments of winter lemons. x° = Per capita domestic shipments of winter lemons. Xj = Index of nonagricultural income. x° = Index of per capita nonagricultural income. c Index of temperature, x- = Time. t-- TABLE 34 '/Vinter Lemons: Reciprocal Correlation Matrices Equations Iw and 2w Equations 3w and 4w ^2 ''3 ^4 ^5 ^1 xo iZ ^4 ^1 6,913313 ^1 6.905547 2 10.986835 21.511390 r\ ^2 8.734010 13,545363 ^3 -2*770949 -4,861574 2.169333 0 ^3 -2,753762 -3.878092 d ,403504 ^4 2,848864 5,048229 -1.117880 2,249574 2.782599 3.913811 -1,030239 2.201530 ^5 -6.663260 -14,002538 3.048479 -3.142227 10,299944 -5.015852 -8.346937 2.854046 -2,233046 6,482752 Equations 5w and 6w log X^ log Kg log x^ log X 5 log x^i^ 6.171160 log Xg 9,195382 17.382883 log X3 -2.019832 -3.360642 1 .699643 log X4 2.487777 4.100113 -0,780249 2.053562 ^5 -5,306360 -10,975990 2.029677 -2.406959 8.125271 For the meaning of the various x's, see table 33, O 107* APPENDIX E The Use of Mean Monthly Maximum Temperatures in the Summer Lemons Regression Analysis The chief drav/back of the index of summer temperatures described in appendix A is that it is laborious to calculate. In this appendix are reported the results secured with the use of another temperature variable (see table 35 )# the extension of which to future seasons is relatively easy. This series consists of monthly mean maximum temperatures. May through September, in 22 cities weighted 55 / by 1931-38 average unloads of lemons in these cities for corresponding months. — ' The weights used are given in table 36. YiThile less sensitive than the index described in appendix A, this temperature variable yields quite similar results in the seasonal analysis* In table 37 are given the regression coefficients and auxiliary constants obtained when mean maximvim teiaperatures were used in place of the index of summer temperatures described previously. The other variables in the two equations pre- sented in table 37 are those used in equations Is and 2s of table 3. It is seen that with f.o.b, price as dependent variable, the adjusted coefficient of multiple correlation is of the same magnitude as that corresponding to equation Is (0.9031 as against 0.9050 secured with equation Is). With the United States supply as dependent variable, the adjusted coefficient of multiple correlation is somewhat lower than the coefficient of equation 2s (0.9482 as against 0.9569), but the difference between them is not large. The magnitudes of the regression coefficients of these equations differ somewhat from those of comparable coefficients of equations Is and 2s, but here, too, the discrepancies are not large. It is clear from the foregoing that the equations described in table 37 are acceptable substitutes 55 / — ' Wellman, H. R. Statistics pertaining to lemons, California Agr. Exp* Sta« October, 1940. (Processed). i'■^rJ.i•;.■. ■:••■! ,'\:r' .■•■<^::yy,i '(.r-:,^::. >.i'j:i'' c-'i-. ••c:.:;:-i.^.. • ' ii'V/^. M-f;- ■-i,ivo.-;:^- ;■(; » tc'.f^.; Ij:> ' , -;• .;• ■ • .■ . ; .' ' • . ■ Ih,":--.-' ( :v!i';.i;u.7r;-. *r:; -;-.'.;f--.u :J- acn-?' £"-:..-"i:-v:; 'f * « rrxn-.. '..•■i .:■ I- .-i.;; ■ r. .. t ■;;<. >;i>i-,\.l v;;.J- 'io Or?;:^' a.'. !:.!;-;; ;r!>«i.!f:; ,;*i';:7 f K': . i. ; i;;; :v ■ ;. .! •;. 1 i->.!::;0 ■■■.■i:}- f r.-' i:l ^ird^i^ri a-T , vi".;i.; vfi^, bu-'ru v,^;: '1:'\H,:-:: i'ii ■•.•■'>,r.-.l;/Or> !>^:^:^:j\l-n o-'J ^ -ii-v ;K.-;..f:j. :: ,:i>: •; v v.! ' ' V'-.?'/:'..' -7!;; :k:].\: ,{':r ji- ; rl-/;- ; ; :■. ■•;;•;>•,>•; ,fr:rf.' • Rv-i::!? cr>;:".;'i.-.;i-:^ ;,!'::■ ': o "'-.i-^c^i-ir.:. •■ 5 »!■ .!. ' r>v-f vi' ur^/y;- ■-•■'I't/'j •t:.^.!.,:- j ;! r'j't.-;/' om, vi t'.'.-:- ; 'j'siJiT.: ■■> m';- ^-^ : «-^-: i-n,.; Z'-i 108, for equations Is and 2s presented in the text* In table 37 are also indicated the coefficients of separate determination pertaining to the two equations. It is seen that the bulk of the variation of f.o.b. price is attributable to variation in the United States supply, and that income, and especially temperature, are relatively unimportant. These results are similar to those secured vn.th equation Is, and it is thus evident that the minor role played by temperature, as indicated by coefficients of separate determination, is not a consequence of the idiosyncracies of the index of summer temperatures described in appendix A. But here also it should be observed that from the point of view of goodness of fit, temperature is an important and, in fact, an indispens- able independent variable, since its omission results in a marked lowering of the 56/ fit of both equations, — ' One other point should be mentioned, namely, that the introduction of "time" as an additional independent variable does not improve significantly the fit of these equations. The ratios of the net regression coefficients of the added variable "time" to their standard errors are ,329 and 1,846 respectively with f«o,b, price and United States supply as the dependent variable. The null hypothesis here cannot be rejected since for 15 degrees of freedom the t-ratio at the 5 per cent level is 2,131, The explanation of the failure of "time" to contribute significantly, advanced in the text in connection viith. equations Is and 2s, is applicable here, for mean maximum temperatures also exhibit an upward trend during the period under review, 5§/ The coefficients of multiple correlation with f.o.b, price as the dependent variable obtained when United States supply, index of nonagri cultural income, and the temperature variable are omitted in turn are ^1(340) " 0,1416, ^'1(240) = 0.5627, RiC9'z\ " 0.6699, The first multiple correlation coefficient was not adjusted be- cause the adjustment resulted in an imaginary value of the coefficient. 67/ The equation of the trend in mean maximum temperatures is: x°' = 79,5950 + 0,0683 x_ 4 5 where x^ is time measured in half-years with the origin half-way between 1931 and □ 1932, The standard error of the coefficient of Xg is 0,0276 yielding t = 2,478, For 18 degrees of freedom the t-ratio at the 5 per cent level is 2,101, .•3 i/l bX 31. a t5 ■ ; -Zip .$,4.., , : ,. . ■■ ..-r.^bfl't . ■.: ».--'-'--■*.■- ■ vff l^.r«?^ -^:4yx;iJ3if:7. . .. . . f.r. 109. TABLE 35 Mean Maximum Temperatures in 22 Cities May-September , 1922 to 1941 Year May- September Mean maximum temperatures Year May- September Mean maximum temperatures degrees degrees 1922 79.5 1932 80.2 1923 79.0 1933 81.1 1924 76.6 1934 82.3 1925 79,9 1935 78,7 1926 77.8 1936 81,4 1927 76.6 1937 80.1 1928 77.6 1938 79.8 1929 78.7 1939 80.9 1930 31.2 1940 78.8 1931 80.7 1941 81.0 Source of data: Wellman, H. R. Statistics pertaining to lemons. California Agr* Exp. Sta. October, 1940, (Processed). Extended through 1941. The entries are monthly mean maximum temperatures May through September in 22 cities weighted by 1931-38 average unloads of lemons for correspond- ing months. For cities included see table 36, I X ; ■ r ' '** I ••• • ■ ! 1 • '. > ^ . '' ^' . . ■ -. 1 ^ ! „ ■• " . TABLE 36 110. Average Kail emd Boat Unloads of Lomons iu May« Juiio« July* August and September in 22 Cities, 1931 to 1938 Average unloads Cities May June July August September Carlots Atlanta, Ga» 25.1 34.4 25.6 24.4 Baltimore, Md. 49.1 95.1 100.4 69.9 Boston, Mass. 65.9 88.9 99.1 74.7 Buffalo, N. y. 19.4 26.4 19.4 18.5 Q c; Chicago, 111. 114.9 164.5 191.8 129.5 Cincinnati, Ohio 44.4 81.2 86.7 55.9 Cleveland, Ohio 47,4 65.4 58.6 40.1 Dallas, Texas 20.5 21.4 19.7 20.4 Denver, Colo. 15.4 23.0 22.8 14.9 7.0 Detroit, Mich. 59.9 93.1 79.7 52.0 28.0 Houston, Texas 18.1 25.3 19.2 18.4 14. R Kansas City, Mo. 24.5 38.7 38.0 26.5 in R Memphis, Tenn. 18.9 26.6 24.0 18.4 is.l Minneapolis-St. Paul, Minn. 29.0 39.1 38.1 23.7 New Orleans, La. 34.4 46.6 42.9 34.5 New York, N. Y. 259.9 387.5 387.2 310,4 190,1 Omaha, Neb» 10.2 17.1 19.5 10.8 4.9 Philadelphia, Pa. 91.0 153.4 152.5 113.6 52.5 Pittsburgh, Pa. 49.8 76.7 78,4 56.1 29.0 Portland, Ore» 17.2 15.9 12.4 12.7 9.5 St. Louis, Mo. 44.0 95.9 99.9 59,6 27.9 Seattle, T/ash. 19.0 20.3 13.1 16,8 11.2 Source of data: United States Department of Agriculture. Agricultural Marketing Service. Carlot unloads of certain fruits and vegetables in 66 cities and im- ports in 4 cities for Canada. Annual issues. VJ^ashington, D. C. (Processed) 1 « - . •,"8' ~ : ■ • ' ■ -•.••i-j-9G-f-- '. , « r '• ■ .f^^?'■i: ■ t - ^ « ~ ■ '• i - r , ■ i: j« : ■ . 1 : . . ,. ' 1 ■ i -J.. .0 -jaTi ' .- TABLE 37 Summer Lemons: Regression Coefficients and Auxiliary Constants Obtained When Mean Maximum Temperature in 22 Cities is Used as the Temperature Variable (Figures in parentheses are standard errors) Dependent variable Net recression coefficients * Adjusted coeffic- ient of multiple correla- tion R flexi- bility of price Elas- ticity of demand Percentage of variance of dependent variable ascribable to Constant term F.o.b. price ^1 United States supply ^2 Nonagri- cultural income ^3 Mean maximum tempera- tures 0 ^4 ^2 ^3 o \ X -26.1255 -0.0189 0.0490 0.4100 0.9031 -1.8067 70.9 10.8 2.7 1 (0.0020) (0.0071) (0.0695) ^2 -1,381,9485 -44.5889 2.5302 21,3450 0.9482 -0.4658 38.9 32.5 20.1 - (4.8363) (0.2732) (2.7835) I X, = Seasonal average f.o.b. price in dollars per box. Xg = United States supply of lemons in units of 10,000 boxes. Xj = Index of nonagri cultural income payments, 1935-39 = 100. x° = Mean monthly maximum temperature in 22 cities. May through September. 'f' At the centroid. ■ '\-'. .. - 1 [ - . » ■ • ! '. " 1 * . ■ 1 i < !■-'•* . '■ ~ :. ! ■- , .■ ■ • • t ■ ' . ■ i ■ • --> 1 f 4- i i-...v:-.;| j ; i i • 1""-.! — t t • ' . ■ ' ^"'iVf -J •_, i ' ' - ■ f. - . ' .- • - t ■f .■ 112, TABLE 38 Sumner Lemons: Correlation Matrix and its Inverse Obtained When Mean Maximum Temperature in 22 Cities is Used as the Temperature Variable Correlation matrix Reciprocal matrix ^1 ^2 ^3 0 ^4 ^1 ^^2 ^3 X° ""4 ^1 ^2 ^3 0 =^4 1 -0.572288 0.118360 0.036886 1 0,452897 0.341278 1 -0.312107 1 6.441653 7.984304 -5,875439 -4.796239 11.759235 -S.437056 -6.940941 7.182498 5.337810 c 5.211672 For the meaning of the x*s# see table 37. c - ■ ... ■■ " ■' • ■ ■ • i f-r • f 1. . \ > . 1 I