Moss, Charles B.

Schmitz, Andrew

Zanzig, Blair

English

Research Papers in Economics

Natural Disasters and the Yield Curve of Florida
Valencias
by
Blair R. Zanzig
Charles B. Moss
and
Andrew Schmitz
*
Submitted for the 
Western Agricultural Economics Association's
1997 Meetings
July 13-16,  Reno, Nevada
                    
* Blair Zanzig is a masters student in the Food and Resource Economics Department at the
University of Florida.  Charles Moss is an Associate Professor in the Food and Resource
Economics Department at the University of Florida.  Andrew Schmitz is professor and
Eminent Scholar in the Department of Food and Resource Economics, University of
Florida, where he holds the Been Hill Griffin III Endowed Chair.Abstract
Natural Disasters and the Yield Curve of Florida Valencias
This study estimates a model of the age-yield relationship for Florida Valencia
oranges as a Sigmoid curve using a modified hyperbolic tangent function.  From this
model,  the absence of natural disaster risk in the lower interior region is shown to have
resulted in a higher yield function compared to other areas of the state.  It is also shown
that the rate of technology adoption influences the yield function.
Keywords: freeze, Valencia oranges, yield functions, technology adoption, hyperbolic
tangent and natural disasters.
JEL: 711, 717, 210Natural Disasters and the Yield Curve of Florida Valencias
I. Introduction
The Florida citrus industry has undergone a considerable transformation during the
past fifteen years.  As figure 1 shows, orange production has surged in the lower interior
counties while tailing off dramatically in the upper interior region that had been historically
a major orange producer.
1  A major reason for this shift south is thought to be the
increased exposure to freeze damage in the upper interior counties.  For example, the
upper interior counties were hit particularly had when back to back freezes in  1983 and
1984 destroyed virtually all of the citrus groves.   Additionally new technology has
allowed production to occur in the lower interior regions where environmental conditions
precluded production earlier.  Critical to being able to examine the cause and effect of
these production movements is an estimation of orange yield curves.  In the past this has
been hampered by the difficulty in identifying a supply function for Valencia oranges.  This
paper takes a first step towards identifying a reasonable supply function by estimating a
model of the age-yield relationship for Valencia oranges.  From this model, clear
observable effects of reduced freeze exposure in the lower interior region are found.  It is
also shown that the rate of technology adoption influences the yield function.
Knowledge of certain physiological characteristics of orange tree growth is
essential to understanding the model developed here.  Orange trees demonstrate a high
degree of juvenility, such that a young tree may require several years prior to bearing fruit.
                    
    
1 Upper interior counties include: Lake, Marion, Orange, Osceola, Putnam, Seminole,
Sumter, and Volusia counties.  Lower interior counties include: Charlotte, Collier, DeSoto,
Glades, Hardee, Hendry, Highlands, Lee, Okechobee, and Polk counties.2
 To minimize such unproductive stages, growers graft a scion (the top part of the tree)
onto a rootstock system.  Choosing the rootstock and scion independently allows the
grower to take advantage of specific horticultural characteristics of various cultivors.
(Jackson et al., 1989)  As a result, orange trees can become productive in as little as 3
years compared to the alternative 6-7 years.  Also the surface area of the orange tree
canopy is crucial to the yield of the tree.  Unlike other crops like pecans, where fruit may
develop anywhere in the canopy, oranges are found only on the outside surface area. 
Thus as a tree develops, it gains increased productive capacity.  Overall the age of a
particular tree plays a crucial role in determining how much fruit the tree will yield. 
Several types of oranges are produced in Florida including: early, midseason and
late (Valencia) varieties.  Valencia oranges are the focus of this study, because the are the
most suitable for juicing, and occupy the largest share of the Florida orange crop. 
Valencia oranges are particularly susceptible to freeze damage because they are not ready
to harvest until the late winter months. Originally the major orange producing region of
the state was the St. Johns River basin in the Northeast.  After the freeze of 1894-95,
production moved to an area known as the Ridge or the upper interior located roughly in
the central part of Florida.  After major freezes in 1957 and 1962, production edged south
but was prevented from completely relocating in the South by the inclement growing
conditions of the Florida wetlands.  As new agronomic advances were made production
became possible in the lower interior regions.  After a series of record freezes during the
1980's in the upper interior counties, production made a final dramatic shift south.  As a
final illustration of the impact of this latest push south, figure 2 compares two
representative counties, Lake county in the upper interior and Hendry county in the lower3
interior.  After the back to back freezes of 1983 and 1984, orange production in Lake
county virtually vanished, while production in Hendry county boomed.  While overall
production has changed only nominally the impact at the local levels has been substantial.
II. A Model of Regional Valencia Orange Yields
Given the importance of the age of an orange tree to its productive capacity, an
understanding of the precise age-yield relationship becomes crucial to explaining shifts in
orange production.  Mathematically, the yield of an orange tree can be expressed as
U=å a   age ii
i
             (1)
where "i  is the technical coefficient of production and is unique to each age cohort.  A
reasonable method of estimating "i  might be to employ a standard OLS regression.  There
are however several limitations to this approach.  The most critical is the assumption
implicitly made when using OLS that Y is a linear function of age.  Such an assumption
would mean that in each additional year a tree yields an equal increase in production. 
Evidence, however, suggests gains in yield level off at some maturity point and that in
general the relationship between yield and time is nonlinear.  Additionally, we have no a
priori reason to believe that each year's change in yield is equivalent to the prior year’s
change.
Given such limitations it is not surprising that using OLS to model the age-yield
curve produces undesirable results.   In regressions not reported here, the OLS estimation
produced several negative coefficients.  These results clearly indict the appropriateness of
the OLS model.  The negative coefficients would suggest as the number of trees with the4
negative coefficient increases the lower the overall grove production declines.  Such
absurd results motivate the search for alternative functional forms.
Given the physiological characteristics of orange tree production a more
reasonable hypothesis is that the functional form of the age-yield relationship follows a
Sigmoid 'S' shape.  That is, over the early years of a tree’s life, the change in yield is
relatively flat.  As the tree grows older the change in yield increases more sharply until
maturity where the growth rate flattens out once again.  The hyperbolic tangent function
produces such a functional form.  A formal representation can then be expressed as:
a
b
bb  =    ii age
max
{ tanh( )}
2
1 01 ++                    (2)
where $0 is an intercept term, $1 is a slope parameter and $max represents the maximum
yield.  Such a specification requires the curve to exist in a positive domain, with a
symmetrical Sigmoid 'S' curve centered at $max/2.  $0, as the y-intercept term, corresponds
to the minimum level of production since the range of ages is strictly positive and thus the
portion of the function to the left of the y-axis is meaningless.  $1 allows for an
unrestricted slope, thus allowing the function to take a variety of appearances including
almost linear or a combination of convexities. 
In order to estimate the model, we can substitute (2) into (1) to give an estimate of
expected yield () $ U.  The $ parameters are then estimated by nonlinear least squares. 
Formally, estimating the age-yield relationship can be expressed as the following
minimization problem: 
Min
 , 
  
01 bb b max,
( $ ) UU ii
i
å - 25
Given the sensitivity of nonlinear estimation to the initial values, the above
minimization problem can be modified by substituting a linear approximation for the
hyperbolic tangent function.  The results of the linear function can be used as a "best
guess" starting point for the original nonlinear model.  After solving the nonlinear model
for $max, fixing $0 and $1, successive iterations can be performed alternating the
parameters held fixed while substituting the previous results as the new "best guess" initial
value.  This iterative process is carried out until the marginal value of changing any
parameter is close to zero. 
III.  Results
The above model was estimated for four separate citrus producing regions as
defined by the Florida Department of Citrus.  Panel data ranging from 1967-1993 across
Florida counties provided the sample for estimation.  Each observation consisted of the
number of trees i years old, for i=3 to 24 for each county.  This tree age profile was then
matched to each county’s level of production for the appropriate year.
2  Only trees at least
3 years old were included since younger trees are largely considered nonbearing.  Age
profile inventory data was constructed from various editions of the Florida Agricultural
Statistics Service's (FASS) Tree Survey.  Production data was taken from various issues
of Citrus Summary also prepared by FASS.  
Table 1 presents the estimated $ parameters by region.   Substituting those values
into (2) the technical production coefficients can be calculated and are reported in table 2.
                    
    
2 Some observations were excluded due to missing data.  In particular, upper interior
counties severely damaged by the 1985 freeze were excluded.  Data was further limited since
tree inventories were only conducted biennially.6
 Figure 3 graphically depicts the estimated yield-age functions.  Quite clearly there are
stark contrasts between regional yield functions.  For all ages, citrus in the lower region
yields considerably more fruit.  Two factors together can explain this phenomenon.  First
production in most lower interior counties has only become possible recently because of
new technology.  The implication is that groves in the lower interior have been planted
using the latest high yield horticultural innovations.  Thus the contrasting yield functions in
figure 3 might more appropriately be viewed as a contrast between old and new
technologies rather than regional differences.  However technology cannot be the sole
factor at work. Assuming producers are rational, given such  a clear advantage of new
technology, all regions would adopt it.  The result would be a single age-yield relationship
instead of the four distinct curves found here.
Accounting for the regional differences in risk is the second factor at work. 
Natural disasters have to be considered.  The lower interior region generally avoids
freezes given its more tropical climate.  In contrast, the upper interior region, as discussed
earlier, has suffered extensive freeze damage at a variety of times.  Even if a freeze does
not destroy a tree, the damage to the fruit will result in lower production.  Since the model
of the yield function presented here is based on production data, lower production will
translate into lower yields.  If experimental data had been used the curve estimated would
have described a simple biological yield function.  Instead the model here implicitly
incorporates intragegional economic relationship influenced by such factors as frost
damage and differing rates of technology adoption. 7
Given the method of estimation, statistical testing of the results is nontrivial. 
Therefore only a qualitative evaluation of the model is offered presently.
3   Despite
allowing the intercept term ($0) to vary without restriction, the estimate of "i  for all
regions closely matches the coefficient for 3 year old trees offered by Muraro et al.
(1996a, 1996b, 1996c).  For example, in the lower interior region Muraro reports an
estimate of 0.7 while an estimate of 0.871 is reported here.
4  Additionally the age-yield
curves are consistent with the a priori expectations of a smooth 'S' curve reinforced by the
physiology of orange production.  Finally, as noted above, the model produces results that
are also consistent with known regional differences.
To further explore the economic implications of the specific yield functions, it is
useful to consider the annualized gross margin of an acre of Valencia orange trees.  This
approach is superior to simply comparing yields since cost differentials across counties
may more than offset higher yields.  First we assume that all startup costs are contained in
the first three years and that all revenue is generated in the third year and beyond.   Further
we assume a tree density of 120 trees per acre and that no tree loss occurs.  The density
assumption is reasonable in light of Muraro (1996) suggesting modern groves employ tree
densities of up to 150 trees per acre.  Given then the above assumptions an estimated yield
can be calculated for the acre over each year of its life, for simplicity we assume the only
relevant revenues are those occurring in the years 3-24.  An estimated revenue can be
                    
    
3 Future extensions of this paper will employ the method of "bootstrapping" the standard
errors to create a distribution that can then be used to perform hypothesis testing.
    
4 For the upper interior Muraro 1996 reports .6 versus the result estimated here of .497. 
Similarly for the east Muraro 1996 finds a coefficient of 1 compared to .709 found using the
model developed in section 2.8
calculated by assuming a constant price of $5.32 per box that corresponds to the actual
on-tree price of Valencia oranges in 1995-96.
5  Now the revenue calculated for the
hypothetical acre for each age cohort can be viewed as a series of payments made annually
over the course of 24 years, where the 1st three payments are zero.  By discounting that
stream of payments (assuming a 7% discount rate) and summing across years a total
present value of the revenue produced per acre is calculated. 
To facilitate comparison as well as allowing for the accounting of variable costs,
the present value of revenue is more easily interpreted as an annualized value.  The
annualized revenue for each region is reported in Table 3.  In order to account for
differences in production costs among regions, the annual total per acre variable cost 
(also in table 3) is deducted from the annualized revenue.
6  The resulting values are the
annualized gross margin or the profit to the owner not accounting for any startup costs. 
Obviously the lower interior is much more profitable than any of the other regions.
7
IV. Implications and Conclusions   
In light of the hypothesis regarding the cause of the yield differentials given above,
these financial statistics hold important economic implications.  For example, assume that
all differences are caused by differential exposure to freeze damage.  Under such a
scenario, any new freeze protection technology must be implemented at a cost less than or
                    
     
5 The on-tree price represents the price received by the grove owner for a box net the
cost of picking and hauling.
    
6 The annual costs used here are taken from Muraro 1996a, 1996b, and 1996c.  They
represent the average variable cost for the 5 year period preceding 1995.
    
7 The strongly negative value for the east region may in part be explained by the fact that
oranges grown in the east are generally sold in the fresh market.  Thus results for the east
region may not be comparable to the juice oranges of the other regions.9
equal to the differential in annualized gross margin.  Relative to the upper and lower
interior this differential is $346.35 an acre per year.  Further these results have the
potential to explain the failure of grove owners to replant in the upper interior region
following the devastating freezes of the early 80’s.  To this day a great deal of the former
grove land sits idle.  Given the low gross margin found for the region of only $155.20 an
acre, it is more than reasonable that after accounting for start up costs the economic rents
to owners are close to zero.  Thus rational owners appear to be those who replanted
further south thus reinforcing the shift in orange production.10
References:
Jackson, L.K., R. Davis, J.M. Bulger, G.F. Fairchild, R. Muraro and D.P.H. Tucker.
Introduction  to the Florida Citrus Industry -- Its Production, Harvesting and Marketing
Practices.   Florida Cooperative Extension Service, IFAS, University of Florida.  1989
Muraro, Ronald and John Hebb.  Budgeting Costs and Returns for Indian River Citrus 
Production, 1995-96.  Economic Information Report 96-4.  University of Florida
Food  and Resource Economics Department.  August 1996a.
Muraro, Ronald and Thomas Oswalt.  Budgeting Costs and Returns for Central Florida
Citrus Production,  1995-96.  Economic Information Report 96-2.  University of Florida
Food  and Resource Economics Department.  August 1996b.
Muraro, Ronald, Fritz Roka, Robert Rouse and Robert Turley.  Budgeting Costs and
Returns for  Southwest Florida Citrus Production, 1995-96.  Economic Information
Report 96-3.   University of Florida Food and Resource Economics Department.  August
1996c.1112
Table 1
$ Parameters by Region
Parameter Lower Upper West East
$max  5.294  4.033  3.606  3.073
$0 -1.115 -1.329 -2.161 -0.938
$1  0.101  0.116  0.19  0.11213
Table 2
Technical Production Coefficients by Regions
Valencia Orange Trees
Age Of Tree Lower Upper West East
24 4.93 3.824 3.577 2.983
23 4.855 3.773 3.563 2.961
22 4.767 3.711 3.544 2.934
21 4.663 3.635 3.516 2.901
20 4.543 3.544 3.475 2.861
19 4.403 3.436 3.418 2.813
18 4.244 3.308 3.37 2.754
17 4.064 3.159 3.226 2.684
16 3.864 2.99 3.076 2.602
15 3.644 2.801 2.88 2.505
14 3.407 2.594 2.635 2.394
13 3.156 2.373 2.343 2.268
12 2.895 2.143 2.017 2.128
11 2.629 1.91 1.675 1.976
10 2.363 1.679 1.343 1.813
9 2.103 1.457 1.041 1.644
8 1.854 1.249 0.783 1.472
7 1.619 1.058 0.575 1.302
6 1.402 0.887 0.414 1.137
5 1.204 0.737 0.294 0.982
4 1.027 0.607 0.206 0.839
3 0.871 0.497 0.144 0.709
Table 3:
Financial Implications of Age-Yield Relationship
Annualized at 7% over 24 years
Revenue Cost Gross Margin
Lower $1,371.86 $870.23 $501.63
Upper $1,002.17 $846.89 $155.28
West $866.18 $870.23 ($4.05)
East $971.84 $1,085.39 ($113.55)

Natural Disasters and the Yield Curve of Florida Valencias

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