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A study of the demand relationship between
fixed-weight and random-weight citrus
Zhifeng Gao, jonq-Ying Lee and Mark G. Brown
Zhifeng Gao (corresponding author)
Assistant Research Scientist
Food and Resource Economics Department
University of Florida
Email: [email protected]
Jonq-Ying Lee
Economist
Florida Department of Citrus
Email: [email protected]
Mark G. Brown
Economist
Florida Department of Citrus
Email: [email protected]
Selected poster prepared for presentation at the Agricultural & Applied Economics Association
Annual Meeting, Milwaukee, WI, July 26-28, 2009
Copyright 2008 by [Zhifeng Gao, Jonq-Ying Lee and Mark G. Brown]. All rights reserved.
Readers may make verbatim copies of this document for non-commercial purposes by any means,
provided that this copyright notice appears on all such copies.
Fresh citrus are sold either by the piece (or by the pounds) or by the bag/case; and
these fresh citrus are displayed in the same produce section in retail stores. The two
types of citrus, i.e., random-weight (RW) and fixed-weight (FW) citrus; are usually
displayed side by side and priced differently. The two different types of packaging are
used to accommodate the different needs of consumers, i.e., some prefer 5-pound bags,
while others prefer to buy in pieces. The qualities of the fruit in different packaging may
different in size, variety, and external look. Whether these two types of packaging of the
same citrus compete against each other has important implication for promotional tactics
used by retailers and the profitability of Florida citrus growers. If the random-weight and
the fixed-weight citrus of the same kind are close substitutes; strategies need to be
developed to avoid the direct competition of the same citrus fruit for consumers’ dollars.
The purpose of this study is to examine the demand relationships among four RW and
FW citrus; i.e., grapefruit, oranges, tangelos, and tangerines. The Rotterdam demand
system will be used to analyze the demand relationship among these citrus.
Method and Data
The traditional consumer problem of choosing that bundle of goods which
maximizes utility, subject to a budget constraint, m; formally, this problem can be written
as follows:
(1)
Maximize
u = u(q, a)
Subject to
p’q = m
where u is utility; p and q are price and quantity vectors with pi and qi being the price and
quantity of good i, respectively; a is a vector of retail promotional tactics in terms of
percent of all commodity volume (%ACV), respectively; and m is total expenditure.
The first order conditions for problem (1) are
(2)
∂u/∂q = λp,
p’q = m,
where λ is the Lagrangean multiplier which is equal to ∂u/∂m.
The solution to (2) is the set of demand equations
(3)
q = q(p, m, a);
and the Langrangean multiplier equation
(4)
λ = λ(p, m, a).
Following Brown and Lee (1993), an approximation to demand (3) is the
Rotterdam model which can be written as
(5)
widlnqi = μiDQ + Σj πijdlnpj + ΣjΣk βijkdajk,
i = 1, 2, . . ., n.
Where dlnxi = log(xit – xit - 52), xi = pi, qi; daj = dajt – dajt - 52 (the time subscribe was
omitted for brevity); wi = piqi/m is the budget share for good i; μi = pi(∂qi/∂m) is the
marginal propensity to consume; DQ = Σi widlnqi is the Divisia volume index; πij =
(pipj/m)sij is the Slutsky coefficient, with sij = (∂qi/∂pj + qj∂qi/∂m) or the element in the ith
row and jth column of the substitution matrix; βijk = wi(∂lnqi/∂ajk) is a promotional tactic
coefficient indicating the impact of the kth tactic used in promoting product j on the
demand for product i. The general restrictions on demand are
(6)
adding up: Σi μi = 1 and Σi πij = 0; Σi βijk = 0
homogeneity: Σj πij = 0; and
symmetry: πij = πji.
The promotional (feature ad and display) coefficients can be written as (Brown and Lee
1993, 2002)
(7)
βijk = -Σj πih γhjk,
i, j = 1, 2, . . ., n,
where γhjk =∂ln(∂u/∂qh)/∂ajk for i, h = 1, . . ., n.
Expressions (7) can be used to impose restrictions on the effects of retail
promotional tactics on demand (Brown and Lee 1993, 2002; Duffy 1987, 1989; Theil
1980). Because of the limited observations available for the study, the parameter space is
reduced to a manageable size. Following Theil (1980), we assume that promotional
tactics only affect marginal utility of the brand in question, resulting in the restriction βijk
= - πij γjjk, and that tactic k is equally effective across brands, further resulting in γjjk = γk.
Hence, equation (7) becomes
βijk = - πij γk.
Imposing the forgoing promotional restrictions, the demand model (5) can be
written as
(8)
widlnqi = μiDQ + Σj πij(dlnpj - Σk γkdajk),
i, j = 1, 2, . . ., n.
In this case, the demand elasticity of a retail promotional tactic is
(9)
(∂lnqi/∂lnajk) = -(πij γk)ajk/wi.
The marginal impact of a tactic on demand is estimated as (this result is an
approximation, see Barten for further discussion)
dqi = -(πij γk/wi)qidajk;
and the marginal impacts on retail revenue can be written as
(10)
pi dqi = -pi (πij γk/wi)(qidajk).
Note that
(11)
Σipidqi = -Σi (πij γk/wi)piqidajk
= -γk*m*dajkΣi (πij/wi)(piqi/m)
= -γk*m*dajkΣi πij
= 0,
because of the adding-up restriction, Σiπij = 0. Thus, in the Rotterdam model, although
any change in promotional activities would reallocate total expenditure to across goods,
total expenditure remains constant.
Demand model (1) was applied to weekly sales data for fresh citrus provided by
Freshlook Marketing Group for the period from 01/08/2006 through 11/23/2008, a total
of 151 weeks. Seven types of citrus fruits were used in the study: RW and FW
grapefruit, RW and FW oranges, tangelos, and RW and FW tangerines. Because of RW
tangelos were not sold year around, the RW and FW tangelos were aggregated into one
category, tangelos. Table 1 shows the sample statistics of these variables.
As shown in Table 1, more citrus were sold in RW packages per store than by
pieces. The average prices for RW citrus were lower than those sold in pieces. Oranges
accounted for more than 60% of the dollar sales in grocery stores; which is followed by
tangerines (20%); grapefruit (14%); and tangelos (3%). The quantity shares follow a
similar pattern like the dollar shares, i.e., orange accounted for 62% of the total quantity
sold in grocery stores; which is followed by tangerines (18%); grapefruit (17.5%), and
tangelos (2%). Generally, oranges had the most featuring; which is followed by
grapefruit, tangerines, and tangelos. Orange had most temporary price reduction; which
is followed by tangerines, tangelos, and grapefruit.
The average pounds sold per store and the dollar shares were used as the
dependent variables qi and wi, respectively; the average prices were used as explanatory
variables pj. Two types of retail tactics were included in the study; i.e., feature and
temporary price reduction. These two retail tactic variables were measured by the share
of fruit sold under feature and temporary price reduction, respectively. All variables
were 52nd differenced to eliminate the seasonal patterns. Iterative seemingly unrelated
regression method was used to estimate the parameters μ and π. Results are presented in
Table 2.
Results
Table 2 shows the iterative seemingly unrelated regression estimates of equation
(1) with homogeneity and symmetry imposed. The data for (1) add up by construction
and the equation for RW tangerines was deleted (Barten 1969). The estimates are
invariant to the equation deleted, and the parameters of the deleted equation can be
recovered by using the adding up demand restrictions in equation (2) or by simply rerunning the model deleting a different equation.
The marginal propensities to consume (MPC, μi) for all citrus fruit are positive
and statistically different from zero except the one for RW tangerines. All own-price
Slutsky coefficients are negative and statistically different from zero. All except three
cross-price Slutsky coefficients are positive. Of the 21 cross-price Slutsky coefficients,
11 are positive and statistically different zero; three are negative and statistically different
from zero (between RW oranges and RW tangerines; between RW oranges and tangelos,
and between RW tangerines and tangelos); and the remaining seven are not statistically
different from zero. The positive cross-price Slutsky coefficient estimates indicate the
substitution relationship between RW oranges and FW oranges; FW oranges and both
RW and FW grapefruit; FW tangerines and the two types of grapefruit and oranges; RW
tangerines and FW grapefruit and FW tangerines; and between tangelos and FW
grapefruit and FW oranges. The negative cross-price Slutsky coefficient estimates
indicate the complementary relationship between RW tangerines and RW oranges;
tangelos and RW oranges; and tangelos and FW tangerines.
The parameter estimates for featuring and TPR are positive and statistically
different from zero, indicating that both featuring and TPR had positive impact on the
demand for citrus fruits. Also note that the coefficient estimate for features is larger than
the one for TPR, an indication that features had a larger impact on citrus fruit sales than
TPR.
Demand elasticity estimates based on the demand parameter estimates (μi and πij)
and sample means are shown in Table 3. The estimated income elasticities (εim = μi/wi)
show the impacts of a one percent increase in the total expenditure of these citrus fruits
on the sales of the individual fruit. Results show that if the total expenditure on citrus
fruits is increased by one percent, the sales of FW oranges and FW tangerines would
increase by more than one percent and the rest of the citrus fruit would increase by less
than one percent while the demand for RW tangerines would decrease by 0.32%.
The uncompensated own-price elasticity estimates (εii = πii/wi – wiεim) show the
demands for all FW citrus fruits are price elastic and RW citrus are price inelastic. Note
that tangelos are the aggregate of RW and RW tangelos; hence we cannot compare the
magnitudes of their own-price elasticity estimates. FW oranges had the highest ownprice elasticity, followed by FW grapefruit, FW tangerines, tangelos; and RW tangerines
had the lowest own-price elasticity among the seven types of citrus studied. The
uncompensated cross-price elasticity estimates (εij = πij/wi – wjεim) are generally small
relative to their own-price elasticities except the ones between FW oranges and tangelos
(0.8656); FW oranges and FW grapefruit (0.5595); FW tangerines and tangelos (0.5267); and FW grapefruit and tangelos (0.5004). These positive cross-price elasticity
estimates show that they are close substitutes. In addition, the cross-price elasticity
estimates between FW and RW grapefruit; FW and RW oranges are statistically not
different from zero, indicating that they are not substitutes; however, the cross-price
elasticity estimate indicates that FW tangerines is a substitute of RW tangerines.
Elasticity estimates for feature ads and temporary price reduction are presented in
Tables 4 and 5; respectively. The demand elasticity estimates for feature ads show that
feature ads for FW oranges had the highest demand elasticity, followed by those for FW
grapefruit, FW tangerines, RW oranges, tangelos, RW grapefruit, and RW tangerines had
the smallest feature ads demand elasticity. The cross-feature-ads demand elasticity
estimates are relatively small as compare to the own-feature-ads demand elasticities.
Similar pattern was found for the demand elasticities for temporary price reductions.
Concluding Remarks
The Rotterdam demand system was used to analyze the demand relationships
among four citrus fruits – grapefruit, oranges, tangelos, and tangerines. The emphasis of
this study was placed on the demand relationship between RW and fixed-weight citrus
fruits. Results indicated that FW and RW grapefruit (and FW and RW oranges) are not
substitutes; however, RW tangerine is a substitute of FW tangerines.
Results also show that promoting FW grapefruit would not influence the demand
for RW grapefruit and vice versa. Results also show that promoting FW oranges (or FW
tangerines) would decrease the demand for RW oranges (or RW tangerines) and vice
versa.
References
Barten, A. P. (1969). “Maximum Likelihood Estimation of a Complete System of
Demand Equations,” European Economic Review, 1: 7-73.
Brown, M. G. and J. Lee (1993). “Alternative Specifications of Advertising in the
Rotterdam Model,” European Review of Agricultural Economics, 20: 419-36.
Brown, M. G. and J. Lee (2002). “Restrictions on the Effects of Preference Variables in
the Rotterdam Model,” Journal of Agricultural and Applied Economics, 34: 1726.
Duffy, M. H. (1987). “Advertising and the Inter-Product Distribution of Demand,”
European Economic Review, 31: 1051-70.
Duffy, M. H. (1989). “Measuring the Contribution of Advertising to Growth in Demand:
An Econometric Accounting Framework,” Internal Journal of Advertising, 8: 95110.
Theil, H. (1980). System-wide Explorations in International Economics, Input-Output
Analysis, and Marketing Research, New York: North-Holland.
Table 1. Sample statistics – 01/08/06 through 11/23/08 Grapefruit FW
Grapefruit RW
Price 0.83
1.15
(0.16) (0.11)
Lbs/store 352
294
(211) (140)
$ Share 0.0464
0.0981
(0.0097) (0.0177)
Lbs share 0.0942
0.0808
(0.0289) (0.0182)
% Featuring 0.1251
0.1051
(0.1971) (0.1436)
% Price Reduction 0.1594
0.1547
(0.1668) (0.1081)
Numbers in parentheses are standard errors. Orange FW
Orange RW
0.93
(0.20)
1,409
(830)
0.1768
(0.2196)
0.0291
(0.0091)
0.1714
(0.2282)
0.0221
(0.0170)
0.0834
(0.1164)
0.1927
(0.1621)
0.0347
(0.0305)
0.1952
(0.1505)
0.0285
(0.0174)
0.1512
(0.0910)
0.1944
(0.1990)
75
(49)
1.72
(0.52)
113
(70)
0.1729
(0.1159)
0.2657
(0.0671)
Tangelos
0.1938
(0.1676)
676
(704)
0.4183
(0.0923)
0.3570
(0.0804)
1.56
(0.36)
929
(370)
0.2012
(0.0537)
Tangerines RW
1.41
(0.24)
Tangerines FW
1.25
(0.26)
0.0794
(0.1351)
0.1608
(0.1151)
0.1697
(0.1214)
Table 2. Demand parameter estimates
μi
Grapefruit FW
0.0173
(0.0062)
Grapefruit RW
0.0246*
(0.0081)
Orange FW
0.2741*
(0.0253)
Orange RW
0.3957*
(0.0352)
Tangerines FW
0.2513*
(0.0387)
Tangerines RW
-0.0091*
(0.0039)
Tangelos
0.0460*
(0.0088)
γ1 (Feature)
0.2610*
(0.0775)
πij
Grapefruit FW
Grapefruit RW
Orange FW
Orange RW
Tangerines FW
Tangerines RW
Tangelos
-0.0638*
(0.0076)
0.0026
(0.0071)
0.0294*
(0.0058)
0.0031
(0.0073)
0.0190*
(0.0063)
0.0099*
(0.0029)
0.0195*
(0.0062)
-0.0653*
(0.0094)
0.0312*
(0.0069)
0.0108
(0.0094)
0.0137*
(0.0079)
0.0021
(0.0034)
0.0090
(0.0074)
-0.2573*
(0.0289)
0.1101*
(0.0275)
0.0425*
(0.0226)
0.0047
(0.0037)
0.0392*
(0.0081)
-0.1817*
(0.0465)
0.0982*
(0.0326)
-0.0163*
(0.0049)
-0.0243*
(0.0113)
-0.1739*
(0.0380)
0.0107*
(0.0042)
-0.0103
(0.0083)
-0.0122*
(0.0029)
0.0011
(0.0021)
-0.0342*
(0.0081)
γ2 (Price Discount)
0.1727*
(0.0754)
*Statistically different from zero at α = 0.05 level.
Table 3. Income and uncompensated elasticity estimates
Income
Elasticity
Grapefruit FW
Grapefruit RW
Compensated Price Elasticity
Orange FW
Orange RW
Tangerines FW
Tangerines RW
Tangelos
Grapefruit FW
0.3739*
(0.1342)
-1.3926*
(0.1647)
0.0203
(0.1561)
0.5595*
(0.1216)
-0.0890
(0.1443)
0.3454*
(0.1441)
0.2032*
(0.0634)
0.4070*
(0.1343)
Grapefruit RW
0.2508*
(0.0822)
0.0153
(0.0734)
-0.6902*
(0.0975)
0.2674*
(0.0675)
0.0055
(0.0849)
0.0965
(0.0868)
0.0139
(0.0350)
0.0829
(0.0753)
Orange FW
1.3624*
(0.1256)
0.0831*
(0.0308)
0.0214
(0.0397)
-1.5527*
(0.1375)
-0.0225
(0.1287)
-0.0241
(0.1276)
-0.0154
(0.0195)
0.1478*
(0.0403)
Orange RW
0.9462*
(0.0840)
-0.0364*
(0.0186)
-0.0670*
(0.0266)
0.0729
(0.0653)
-0.8302*
(0.0965)
0.0712
(0.0885)
-0.0659*
(0.0127)
-0.0908*
(0.0273)
Tangerines FW
1.4536*
(0.2239)
0.0426
(0.0388)
-0.0632
(0.0561)
-0.0464
(0.1147)
-0.0400
(0.1542)
-1.2571*
(0.2484)
0.0206
(0.0269)
-0.1100*
(0.0479)
Tangerines RW
-0.3186*
(0.1376)
0.3631*
(0.1035)
0.1039
(0.1226)
0.2296*
(0.1255)
-0.4387*
(0.1583)
0.4316*
(0.1564)
-0.4193*
(0.1014)
0.0483
(0.0726)
1.3259*
0.5004*
(0.2534)
(0.1809)
*Statistically different from zero at α = 0.05 level.
0.1292
(0.2183)
0.8656*
(0.2266)
-1.2545*
(0.3019)
-0.5267*
(0.2643)
-0.0072
(0.0617)
-1.0327*
(0.2334)
Tangelos
Table 4. Elasticity estimates for featuring Featuring in Grapefruit FW
Grapefruit RW
Orange FW
Orange RW
Tangerines FW
Tangerines RW
Tangelos
Grapefruit FW
0.0449*
(0.0119)
-0.0016
(0.0041)
-0.0293*
(0.0096)
-0.0034
(0.0080)
-0.0183*
(0.0074)
-0.0047*
(0.0019)
-0.0087*
(0.0039)
Grapefruit RW
-0.0009
(0.0023)
0.0182*
(0.0050)
-0.0147*
(0.0053)
-0.0056
(0.0050)
-0.0063*
(0.0037)
-0.0005
(0.0008)
-0.0019
(0.0018)
Orange FW
-0.0048*
(0.0016)
-0.0043*
(0.0015)
0.0590*
(0.0147)
-0.0278*
(0.0081)
-0.0095*
(0.0058)
-0.0005
(0.0004)
-0.0040*
(0.0012)
Orange RW
-0.0002
(0.0006)
-0.0007
(0.0006)
-0.0121*
(0.0035)
0.0220*
(0.0072)
-0.0105*
(0.0043)
0.0008*
(0.0003)
0.0012*
(0.0006)
Tangerines FW
-0.0036*
(0.0014)
-0.0022*
(0.0013)
-0.0113*
(0.0070)
-0.0288*
(0.0118)
0.0450*
(0.0146)
-0.0014*
(0.0005)
0.0012
(0.0011)
Tangerines RW
-0.0114*
(0.0047)
-0.0020
(0.0034)
-0.0076
(0.0057)
0.0290*
(0.0102)
-0.0168*
(0.0068)
0.0093*
(0.0032)
-0.0008
(0.0015)
Tangelos
-0.0183*
(0.0082)
-0.0071
(0.0067)
-0.0522*
(0.0160)
0.0355*
(0.0179)
0.0133
(0.0115)
-0.0007
(0.0013)
0.0205*
(0.0083)
*Statistically different from zero at α = 0.05 level. **Statistically different from zero at α = 0.10 level. Table 5. Elasticity estimates for temporary price reduction Temporary Price Reduction in Grapefruit FW
Grapefruit RW
Orange FW
Orange RW
Tangerines FW
Tangerines RW
Tangelos
Grapefruit FW
0.0379*
(0.0152)
-0.0015
(0.0041)
-0.0212*
(0.0092)
-0.0023
(0.0054)
-0.0136*
(0.0067)
-0.0059*
(0.0031)
-0.0123*
(0.0067)
Grapefruit RW
-0.0007
(0.0020)
0.0178*
(0.0074)
-0.0106*
(0.0050)
-0.0037
(0.0034)
-0.0047**
(0.0031)
-0.0006
(0.0010)
-0.0027
(0.0027)
Orange FW
-0.0040*
(0.0017)
-0.0041*
(0.0019)
0.0428*
(0.0169)
-0.0185*
(0.0082)
-0.0070**
(0.0047)
-0.0007
(0.0005)
-0.0057*
(0.0024)
Orange RW
-0.0002
(0.0005)
-0.0007
(0.0006)
-0.0088*
(0.0039)
0.0146*
(0.0069)
-0.0078*
(0.0042)
0.0011*
(0.0005)
0.0017*
(0.0010)
Tangerines FW
-0.0030*
(0.0015)
-0.0021*
(0.0014)
-0.0082**
(0.0055)
-0.0191*
(0.0104)
0.0335*
(0.0154)
-0.0017*
(0.0009)
0.0017
(0.0016)
Tangerines RW
-0.0096*
(0.0049)
-0.0019
(0.0034)
-0.0055
(0.0043)
0.0193*
(0.0092)
-0.0125*
(0.0068)
0.0119*
(0.0056)
-0.0013
(0.0022)
Tangelos
-0.0155*
(0.0085)
-0.0069
(0.0069)
-0.0379*
(0.0162)
0.0236*
(0.0144)
0.0099
(0.0089)
-0.0008
(0.0017)
0.0289*
(0.0147)
*Statistically different from zero at α = 0.05 level. **Statistically different from zero at α = 0.10 level.