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Copyright 2005, Board of Trustees, University
of Florida
The Maret for
A RAt v, ona
Wit." er Fmotoes
txpectati os
ntert .e.oreta o s
The Market for Winter Tomatoes:
A Rational Expectations Interpretation
by
J. Scott Shonkwiler
Robert D. Emerson
J. Scott Shonkwiler is an Assistant Professor and Robert D. Emerson
1s an Associate Professor, Food and Resource Economics Department,
University of Florida, Gainesville, Florida 32611
TABLE OF CONTENTS
Page
List of Tables ........................... ...........1ii
Model Specification ..................................3
Modeling Expectations ................................ 5
Estimation Results .... ............................. 8
Model Implications .................................11
Conclusions ............................. ........... 13
Footnotes ...........................................18
References ..........................................20
LIST OF TABLES
Page
Table 1. Maximum Likelihood Estimates .............. 15
Table 2. Derived Reduced Form Results .............. 16
Table 3. Impact, Interim and Longrun
Acreage Elasticities ..................... 17
The Florida Agricultural Market Research Center
A Service of
the Food and Resource Economics Department
of the
Institute of Food and Agricultural Sciences
The purpose of this Center is to provide timely, applied research
on current and emerging marketing problems affecting Florida's agri
cultural and marine industries. The Center seeks to provide research
and information to production, marketing, and processing firms, groups
and organizations concerned with improving and expanding markets for
Florida agricultural and marine products.
The Center is staffed by a basic group of economists trained in
agriculture and marketing. In addition, cooperating personnel from
other IFAS units provide a wide range of expertise which can be applied
as determined by the requirements of individual projects.
The Market for Winter Tomatoes:
A Rational Expectations Interpretation
Florida produces about 90 percent of domestic fresh tomato supplies
during the November to midJune season. Since the Cuban trade embargo
of February 1962, Mexico has become the major foreign supplier of fresh
winter tomatoes. The fresh winter tomat market in the U.S. has been
almost equally shared by Florida and Mexico in recent years. The
rivalry for this tirket reached the point in 1979 that Florida growers
charged that Mexican producers were dumping their product on the U.S.
market. Many Florida growers suggest that unless Mexican tomato imports
are limited the Florida tomato industry will effectively disappear. /
Primary attention in this paper is focused on domestic tomato
acreage allocations in light of the increasing Mexican influence on the
winter tomato market. The paper does not attempt to evaluate the
welfare effects of Mexican imports on the Florida tomato industry. In
addition to the emphasis on acreage response, the domestic quantity and
price effects are also considered.
Most efforts to explain acreage devoted to agricultural crops
utilize some variant of a cobweb model since the ultimate price of the
crop cannot be ascertained until the completion of the production
cycle. Nevertheless, if our supply and demand framework is truly
representative of the way the market works, an alternative to lagged
price for representing the anticipated price for the crop is the price
prediction of the supply and demand model itself. Thus, acreage would
be argued to be dependent on the expected price of the crop; and
expected price would be quite simply the solution of the supply and
demand model for price on the basis of information available at planting
time. The underlying hypothesis is that producers are acting in
accordance with the information available to them at planting time and
in a way represented by the interaction in the supply and demand
model. Viewed in this way, planted acreage decisions would be based on
such factors as anticipated Mexican imports and their attendant effect
on the price of tomatoes. One distinction between this and the
traditional cobweb framework is that in the latter case lagged imports
may influence planted acreage via their effect on lagged prices, but any
impact expected imports might have on current price is ignored in the
acreage relationship. Additionally, by utilizing the expected price to
explain acreage, all information in the supply and demand model
including expected imports is brought to bear to assist in explaining
acreage. This concept is referred to in the economics literature as the
rational expectations hypothesis as introduced by Muth1/
An economic model of the winter tomato market incorporating the
rational expectations hypothesis is specified in the following
section. This is followed by a discussion of alternative price
expectation hypotheses. The parameters of the supply and demand model
are then estimated for the rational expectations and the cobweb
models. The final section of the paper presents an evaluation of the
results and assesses their implications.
Model Specification
We specify a supply and demand model for Florida tomatoes since
Florida produces nearly all of the domestic winter fresh tomatoes. The
demand equation is an aggregate U.S. demand function. The supply side
is decomposed into an acreage equation and a yield equation. The
rationale for the latter is that the decision to plant must be made
considerably before the market time. Nevertheless, a number of factors
can alter the yield once acreage is devoted to tomatoes.
The structural model is specifid Ly the following four equation
system with all variables in logarithms:
(1) At = + aa t + a2Ct + a3Rt + a At+ It
(2) Yt 0 + Pt Lt + 2Wt + + 2t
(3) = Y + YiPt Y1Dt + Y2Mt + Y31t + 3t
(4) Qt = At + Yt
where
At = acreage of tomatoes planted;
Pt = expected price per carton of tomatoes;
Ct = expected cost of a carton of tomatoes;
Rt = prime interest rate prevailing during the time planting
decisions are made (July through January);
Yt = yield of 30 pound cartons per planted acre;
Pt = season average price per carton of tomatoes;
Lt = hourly wage of piecerate farm workers in Florida
(midseason);
Wt = weather index constructed from the detrended yields of
cucumbers and green peppers two vegetable crops grown in
many of the same areas as tomatoes during the winter season;
Xt = adoption of plastic mulch: 0 prior to 1973 season; 1
thereafter;
Qt = quantity of tomatoes shipped;
Dt = U.S. personal consumption expenditures price deflator (October
through June);
Mt = quantity of Mexican tomatoes imported (October through June);
It = U.S. total real disposable income (October through June).
Since product prices and production costs are uncertain at the time
the acreage decision is made, we argue that decisions are made on
expected price and expected cost. The specification of expectations is
treated in the following section. Much of the land used to grow
tomatoes is rented. Consequently, the tomato grower may be less
interested in the production of other crops than in the opportunity cost
of funds used on purchased inputs, as reflected by the prime rate
(Rt). Finally, the producer may not be able to fully adjust acreage to
an optimum level due to mistakes or increased costs, thus he can only
make a partial adjustment as reflected by the inclusion of AtI.2/
Tomato yields (Y ) are strongly influenced by the number of times
the crop is harvested. Because tomatoes do not mature uniformly, fields
of staked tomatoes may be picked five times and ground plants may be
picked twice depending on crop and market conditions (Zepp and
Simmons). The harvesting decision is argued to depend on current
product price (the price is known at harvest time in contest to the
acreage question) and the wage rate. Furthermore, the specification
assumes that it is the ratio of price to wage rate that is important.
They thus have the same coefficient except for sign. Tomato yields are
also highly affected by cold weather which can either destroy the plant
or inhibit the fruit from setting. The introduction of fullbed plastic
mulch during the 1970s represented aun laporLant change in technology as
reflected through an increase in yields.
The demand for tomatoes (Qt) is specified at the wholesale level
including the usual economic variables, deflated price and deflated
disposable income. In addition, Mexican imports are included as a
demand shifter for Florida tomatoes./ The final equation (4) closes
the system with the identity that quantity demanded is equal to yield
per acre times the number of acres planted.
Modeling Expectations
Two expectational variables are utilized in our supply response
model, both of which occur in the acreage equation. These are expected
price and expected cost of production, neither of which can be known at
the time the crop is planted. Typically ad hoc extrapolative models for
expectation variables of the form
(5) Zt ti
i ti.
are used. The number of lags and the types of restrictions to put on
the A1 are usually subjectively determined. Moreover, such procedures
ignore the information specified in the structural model for how prices
are determined in the market. Nerlove has recently argued that supply
response models have not become more "econometrically relevant" since
his work of over two decades ago. Shonkwiler similarly comments that:
A fundamental difficulty with such expectation formation
models concerns their arbitrariness and lack of theoretical
basis. These models are not necessarily accurate
representations of economic behavior implied by the underlying
economic structure.
The rational expectations model presented in this paper is offered as an
alternative to the traditional agricultural supply response approaches.
Despite Muth's original casting of the rational expectations model
in terms of market supply and demand relations, empirical applications
of the rational expectations hypothesis (REH) have almost exclusively
appeared in the macroeconomics province (vid. Shiller's review). In the
agricultural sector where fixed biological lags separate the production
decision and consequent output, supply response models have typically
employed extrapolative mechanisms to represent expected prices or
returns (c.f. Askari and Cummings, Nerlove). By contrast, the REH
maintains that participants in the market act as if they were solving
the supply and demand system in forming their price expectations. Thus,
the rational expectations model and models incorporating extrapolative
types of expectations are two competing frameworks for explaining
acreage variations.
There are several reasons to expect that Florida tomato growers
form rational expectations. Note first that production is found in a
small geographic area which implies that producers face similar economic
and climatic environments. The highly commercial and concentrated
nature of the Florida tomato industry may produce a situation more
conducive to the use of rational expectations by producers (DeCanio).
Finally, the intense competition with Mexican imports and the
information collection and dissemination service of the Florida Tomato
Committee suggest that growers take important supply and demand forces
into account when making production decisions.
In its most general form, the rational expectations interpretation
of expected price, Pt is the mathematical expectation of Pt given all
information available when the expectation is formed. In a structural
economic model this information consists of the predetermined variables
and the model's reduced form parameters (Wallis). Specifically, the
econometric model presented in equations (1)(4) yields the following
reduced form equation for the price variable4 /
(6) aP + BPt + YPt a a2C 3Rt At+ 0 Lt
2 Wt 3Xt + YO Dt+ Y2Mt + Y3 t
Taking the expectation of the above equation, the expected price, Pt, is
given by
S 1* *
(7) P t ( + 1 Y1) ( a2Ct 33Rt 4 At1 0 1 t
W X + Y YD + YM + YI
82 t t 0 1 t 2Mt 3 t
where asterisks on the righthandside of (7) denote the expectations of
the current exogenous variables.
The consequences of expression (7) are immediate since it
explicitly shows that expected price depends not only upon expected
levels of imports but also upon the other predetermined variables or
forecasted exogenous variables in the system. And, this dependence is
given exactly as a function of the structural model's parameters.
The values of the exogenous variables are forecasted by the
following relations:
(8) Zit = 6io + 611 itI + it
since there is no other structural information concerning their
generation (Wallis). The exception to (8) occurs for the weather index
which is created with zero mean. Because this variable is not expected
to have any systematic component, we assume that its forecasted value is
identical to its mean value.
The alternative cobweb model is specified by substituting Pt1 for
P in the acreage equation (1). The model is dynamic since lagged
acreage is included as an explanatory variable. This particular form
for acreage response models has been found to explain tomato acreage
fairly well (Morris).
Estimation Results
The data used are the nineteen winter seasons from 196162 through
197980.! Under the REH the variables exogenous to the system, Ct, L ,
A it t
Dt, Mt and It, were predicted using (8) and had squared correlations
with their actual values of .80, .98, .998, .86, and .994
respectively. Substituting these predicted series in (7) gives an
expression for P which is linear in the variables but nonlinear in
terms of the structural parameters. Estimation of the four equation
system (1)(4) is accomplished by replacing Pt in (1) with expression
(7). The resulting acreage equation under the REH is then:
(9) At 0 + a R Y aA
80 + t 8 t + Y t 2 t + t
+ 2Ct + 3R + 4 At1 It
Under the alternative of a dynamic cobweb model the acreage equation
appears as
(10) At = a0 + t + a2Ct + a3Rt + a At1 + t
Under the REH the system is highly nonlinear in the parameters in
addition to having parameter restrictions across equations. Although
limited information methods such as two stage least squares could be
utilized to estimate the parameters equation by equation, the occurrence
of crossequation parameter restrictions make this much less attractive
than usual. A particularly appropriate method for this simultaneous
equations model is full information maximum likelihood (although still
conditional on the forecasts of C L D M and It)/ The
disturbances pit' 2t' p3t as well as ult', 2t' 3t are assumed to
follow joint normal distributions.
The maximum likelihood parameter estimates for both expectations
models are presented in Table 1. Note that the parameters may also be
interpreted as elasticities since all variables are in logarithmic
form. It is apparent that the REH "fits" the data better than the
cobweb hypothesis since its calculated likelihood is greater. In fact
using a test suggested by Revankar, the variable Pt_ was included in
the REH acreage model and the full system was reestimated. The
rationale for inclusion of Pt, was that the REH model was now nested
within a model with an extrapolative component. The likelihood ratio
test of the restricted rational expectations model versus the more
general expectations model yielded a calculated chisquare value of
.0134 with one degree of freedom. The critical value for rejecting the
rational expectations model at the .05 level would require a chisquare
value greater than 3.84.
The REH and the cobweb model parameter estimates, however, show
many close similarities. The results for the acreage equation appear
reasonable in terms of the coefficients' signs and magnitudes. The
rational expectations variable Pt_ enters the equation quite
significantly and with a magnitude twice that of the naive
expectation. As expected, the opportunity costs associated with the
rate of interest and the expected production costs have a negative
effect on the acreage planted.
The yield equation illustrates the sensitivity of supply to current
prices and labor costs. While the price elasticity of yield may appear
unduly large, recall that this is essentially a structural supply
equation. Thus, a sharp increase in yields acts to depress price
through the resulting increase in supply.7/
The demand equation shows an own price elasticity of about .8 for
Florida tomatoes which appears reasonable in terms of other studies
(Nuckton). The effect of Mexican imports on the demand for Florida
tomatoes is extremely significant and the .8 parameter suggests they are
very close substitutes. The high income elasticity of demand is not
surprising given that fresh winter tomatoes may be considered somewhat
of a luxury item.
Model Implications
The structural models were solved for their reduced forms so that
each endogenous variable could be expressed as a function of only the
predetermined variables. The calculated reduced form parameters are
presented in Table 2 along with their associated validation measures.
Again, recall that all coefficients may also be interpreted as
elasticities.
As discussed above, the REH variable, P may be expressed as a
linear combination of all predetermined and forecasted exogenous
variables (except Wt). Since Pt is consistent with the reduced form
forecast of Pt from the original model, their reduced forms should be
related. This relationship is immediate if we assume that each
forecasted exogenous variable is an unbiased estimate. Then, for
example, we see that the coefficient on M in the expectations equation
equals the sum of the coefficients on Mt and Mt in the price equation.
The reduced form results in Table 2 underscore the importance of
imports on expected prices and quantities in the Florida tomato
industry. With regard to Florida tomato prices and quantities sold, a
10 percent increase in observed imports reduced price by 2.68% (1.66%)
and quantity by 5.91% (6.53%) under the REH (dynamic cobweb). In other
words, a 10 percent increase in Mexican tomato imports, ceteris paribus,
has the effect of reducing total Florida tomato revenues by 8.59%
(8.19%). This relationship highlights the competitive nature of the
winter tomato market.
The reduced form thus suggests that the dominant effect of Mexican
tomato imports is on the reduction of domestic supply, not domestic
price. As Mexican imports increase, not only does this reduce domestic
supply, but correctly anticipated increases in imports (or lagged
imports under the cobweb model) reduce domestic acreage. This strongly
suggests that Mexican imports have had a significant influence on the
contraction of the Florida industry in terms of acres planted.
Moreover, the reduced form equation for acreage is the one equation that
reveals a difference in predictive ability between the REH and the
cobweb models. The R2 is somewhat over twice as high as for the REH and
the mean absolute error is about onethird less with the REH model as
compared to the cobweb. Thus, although the REH does not offer
significant improvements in predicting price or quantity over the
simpler cobweb, the predictive ability for acreage is greatly improved
with the additional information brought to bear through the rational
expectations interpretation.
The long runeffects of Mexican imports on Florida tomato acreage
are illustrated with a dynamic analysis of the reduced forms. (Table 3).
The long run response of acreage to imports is nearly four times as
great under the REH model as compared to the cobweb model (.447 vs.
.123). The results thus suggest considerably more acreage adjustment
to anticipated imports, for example, than would be reflected by the
simpler cobweb model. Moreover, the REH estimates suggest that 42
percent of the acreage adjustment due to changes in anticipated imports
occurs in the current time period. By contrast the cobweb model cannot
reflect any adjustment in the current period.
Conclusions
The primary focus of this paper is interpreting the effect of
Mexican tomato imports on the highly structured and centralized Florida
winter tomato market. The unique feature of the approach taken in this
paper is the specification of a rational. expectations framework on the
producers' expected price in the acreage decision. Previous efforts at
estimating acreage and supply equations have typically assumed ad hoc
price expectation specifications in order to compensate for the
production time lag in the acreage decision equation. The most common
of these is the cobweb model. By contrast, the rational expectations
specification allows for the possibility that producers utilize all
information available to them at the time planting decisions are made.
They subsequently adjust their plantings on the basis of this
information and in accordance with its implications as reflected through
the supply and demand model.
We find that the rational expectations specification is consistent
with the data for the winter tomato market. Moreover, the results
suggest superior performance in interpreting acreage decisions than for
the more typical cobweb model. As might be anticipated, the differences
in yield and price predictions are not greatly different in the two
cases. The reason is simply that the essential difference in the two
approaches is in the specification of the acreage equation. There is
very little difference between the two models in either the composition
or the coefficients of the reduced form equations other than acreage.
The significance of the rational expectations statement is that
producers can and do adjust not only to historic price information, but
more importantly to current information not necessarily reflected in
past prices. If anticipated imports are higher than previously,
domestic producers adjust in part by reducing plantings. As trade
restrictions on tomatoes are reevaluated with Mexico, it is significant
to note that the results of this paper suggest that Florida producers
quickly and correctly adjust to the anticipated level of Mexican
imports. Moreover, the impact is primarily on quantities rather than
prices, and furthermore, a significant part of the quantity impact is
attributed to acreage rather than the complete impact being absorbed by
economic abandonment and the ensuing waste of resources once the crop
has been planted.
Table 1. Maximum Likelihood Estimates
Equation Parameters Variable REH Model Cobweb Model
Acreage
Yield
*b
Pt
C
Rt
At1
Pt/Lt
Wt
Xt
Pt/Dt
Mt
It
Demand
Loglikelihood
1.312
.917
.731
.472
.756
.588
2.208
.209
.215
6.04
.786
.801
2.087
(1.26)a
(.332)
(.251)
(.146)
(.322)
(.490)
(1.22)
(.073)
(.119)
(1.14)
(.203)
(.118)
(.204)
95.3664
2.116
.460
.484
.256
.544
.050
3.939
.319
.0143
6.42
.837
.792
2.112
(.715)
(.152)
(.158)
(.082)
(.183)
(1.18)
(3.05)
(.150)
(.304)
(1.43)
(.239)
(.132)
(.232)
93.9708
aEstimated asymptotic standard errors are in parentheses.
bThis is PtI under the cobweb model.
Table 2. Derived Reduced Form Results
Dependrat Predetermined Variables
Variables Model Intercept PV C 1 A 1 L1 U I DL Mt 1 1 R2a tAIb
t RER .549 .560 .361
Cobweb 2.116 .460 .484 .256
Yt R 3.89 .413 .266 .427 .580 .055
Cobweb 7.05 .380 .400 .211 .449 .691 .056
t RE 4.44 .147 .095
Cobweb 4.93 .081 .085 .045
.152 .580 .055
.095 .691 .056
P RH 2.03 .187 .121 .193 .737 .070
Cobweb 1.78 .096 .101 .054 .114 .825 .067
.050
.184 .188 .489 .518 .624 1.862
.268 2.88Z
.093 .580 .591 1.54 .136 .138 .361 .382
.003 .691 .653 1.74
.043 .580 .591 1.54 .048 .050 .128 .136
.003 .691 .653 1.74
.055 .263 .268 .697 .062 .063 .163 .173
.003 .175 .166 .442
.897 4.362
.924 3.532
.839 1.10Z
.834 1.102
.906 6.26Z
.924 6.08Z
REX 2.03
Cobweb
.187 .121 .193
.055
.201 .205 .534 .564
aSquared correlation between predicted reduced form and actual values.
"Mean absolute percentage error of predicted series.
Table 3. Impact, Interim and Longrun Acreage Elasticities
Period REH Cobweb
s %AAt/%AM* %AAt/%AMt__s
Impact Interim Impact Interim
0 .188 .188 .0671 .0671
1 .109 .297 .0300 .9071
2 .063 .360 .0139 .111
3 .036 .396 .0064 .1174
4 .021 .417 .0030 .1204
0 .447 0 .1230
17
FOOTNOTES
1/ Nelson has mentioned that "'rational' has deteriorated in current
usage to little more than a synonym for 'unbiased' or 'optimal
extrapolative'," (p. 331). Our usage conforms with Muth's
definition that rational expectations "are essentially the same as
predictions of the relevant theory," (p. 316).
2/ Kennan has shown that when a decisionmaker is faced with a
quadratic loss function containing both a disequilibrium cost and an
adjustment cost the partial adjustment model can serve as a
description of optimal behavior.
3/ One can logically argue that the demand equation should be specified
as the demand for domestic and imported tomatoes. However, this
introduces additional severe nonlinearities into the system as well
as the requirement for data to explain the level of exports from
Mexico. This is beyond the scope of this paper and is left for
future work.
4/ The disturbance terms have been deleted here since they will
disappear upon taking expectations.
5/ Data on P and A were from Florida Vegetable Summary, C was from
Costs and Returns from Vegetable Crops in Florida, R, D, and I were
from Department of Commerce, L was from Walker and Florida Farm
Labor, Q was from Annual Report of the Florida Tomato Committee, and
M was from Preview of Mexico's Vegetable Production for Export.
6/ See, for example, Intriligator, p. 412.
7/ In fact, if the underlying harvesting production function is assumed
to be of the form
Y = aH
where H represents the labor input, thb elasticities from the
estimated supply equation
Y p2.208 L2.208
imply that the elasticity of production for the labor input be 8 =
.688.
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