A study of the demand relationship between fixedweight and randomweight citrus Zhifeng Gao, jonqYing Lee and Mark G. Brown
Zhifeng Gao (corresponding author) Assistant Research Scientist Food and Resource Economics Department University of Florida Email: zfgao@ufl.edu JonqYing Lee Economist Florida Department of Citrus Email: jonqying@ufl.edu Mark G. Brown Economist Florida Department of Citrus Email: mgbrown@ufl.edu Selected poster prepared for presentation at the Agricultural & Applied Economics Association Annual Meeting, Milwaukee, WI, July 2628, 2009
Copyright 2008 by [Zhifeng Gao, JonqYing Lee and Mark G. Brown]. All rights reserved. Readers may make verbatim copies of this document for noncommercial 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., randomweight (RW) and fixedweight (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 5pound 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 randomweight and the fixedweight 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 p
i
and q
i
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 =
λ
p, 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) w
i
dlnq
i
=
μ
i
DQ +
Σ
j
π
ij
dlnp
j
+
Σ
j
Σ
k
β
ijk
da
jk
, i = 1, 2, . . ., n. Where dlnx
i
= log(x
it
– x
it  52
), x
i
= p
i
, q
i
; da
j
= da
jt
– da
jt  52
(the time subscribe was omitted for brevity); w
i
= p
i
q
i
/m is the budget share for good i;
μ
i
= p
i
(
∂
q
i
/
∂
m) is the marginal propensity to consume; DQ =
Σ
i
w
i
dlnq
i
is the Divisia volume index;
π
ij
= (p
i
p
j
/m)s
ij
is the Slutsky coefficient, with s
ij
= (
∂
q
i
/
∂
p
j
+ q
j
∂
q
i
/
∂
m) or the element in the ith row and jth column of the substitution matrix;
β
ijk
= w
i
(
∂
lnq
i
/
∂
a
jk
) is a promotional tactic coefficient indicating the impact of the
k
th tactic used in promoting product
j
on the demand for product
i
. The general restrictions on demand are (6) adding up:
Σ
i
μ
i
= 1 and
Σ
i
π
ij
= 0;
Σ
i
β
ijk
= 0 homogeneity:
Σ
j
π
ij
= 0; and symmetry:
π
ij
=
π
ji
.
The promotional (feature ad and display) coefficients can be written as (Brown and Lee 1993, 2002) (7)
β
ijk
= 
Σ
j
π
ih
γ
hjk
,
i
,
j
= 1, 2, . . ., n, where
γ
hjk
=
∂
ln(
∂
u/
∂
q
h
)/
∂
a
jk
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
β
ijk
= 
π
ij
γ
jjk
, and that tactic
k
is equally effective across brands, further resulting in
γ
jjk
=
γ
k
. Hence, equation (7) becomes
β
ijk
= 
π
ij
γ
k
. Imposing the forgoing promotional restrictions, the demand model (5) can be written as (8) w
i
dlnq
i
=
μ
i
DQ +
Σ
j
π
ij
(dlnp
j

Σ
k
γ
k
da
jk
),
i
, j = 1, 2, . . ., n. In this case, the demand elasticity of a retail promotional tactic is (9) (
∂
lnq
i
/
∂
lna
jk
) = (
π
ij
γ
k
)a
jk
/w
i
. The marginal impact of a tactic on demand is estimated as (this result is an approximation, see Barten for further discussion) dq
i
= (
π
ij
γ
k
/w
i
)q
i
da
jk
; and the marginal impacts on retail revenue can be written as (10) p
i
dq
i
= p
i
(
π
ij
γ
k
/w
i
)(q
i
da
jk
). Note that
(11)
Σ
i
p
i
dq
i
= 
Σ
i
(
π
ij
γ
k
/w
i
)p
i
q
i
da
jk
= 
γ
k
*m*da
jk
Σ
i
(
π
ij
/w
i
)(p
i
q
i
/m) = 
γ
k
*m*da
jk
Σ
i
π
ij
= 0, because of the addingup restriction,
Σ
i
π
ij
= 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.