Perennial Crop Supply Response: A Kalman Filter Approach

Perennial Crop Supply Response: A Kalman Filter Approach Keith C. Knapp and Kazim Konyar

A state-space model for perennial crop supply response is developed. New plantings and removals depend on the existing age structure of the crop and expected values for future prices and other exogenous variables. Acreage in individual age categories evolves depending upon existing acreage, new plantings, and removals. The Kalman filter and an iterative parameter search provide maximum-likelihood estimates of the unknown parameters and age group acreages from observed data on total acreage and production. An empirical application for alfalfa shows that existing acreage has differential impacts on new plantings and removals depending upon age.

Key words: Kalman filter, perennial crops, state space models, supply.

A significant literature has developed on the estimation of perennial crop supply response. A number of studies are based on single-equation regression models for either aggregate output, aggregate acreage, or changes in these variables. Akiyama and Trivedi provide a review. This approach is limited because decisions on new plantings and removals depend on existing acreage in the various age categories, as well as existing total acreage and expectations about future prices, yields, and other variables. In ad- dition, the structural models used to derive the reduced-form estimating equations typically assume desired levels of acreage or output a s a function of prices and a partial adjustment or stock adjustment model to determine investment or changes in acreage or output. With stable prices and other exogenous variables and an adjustment coefficient between zero and one, this implies convergence to a steady-state or long- run equilibrium, which need not be the case for perennial crops (Knapp).

Several studies have provided separate estimates of new planting and removal equations which account, at least in part, for age distribution effects. These include French and Bres- sler for lemons: Arak for coffee; Alston, Free-

Keith C. Knapp is an associate professor of resource economics, University of California, Riverside. Kazim Konyar is an agricultural economist, U.S. Department of Agriculture.

Phyllis Nash provided assistance with the graphics. The authors are also grateful to two anonymous referees for comments moti- vating an expansion of the original analysis.

bairn, and Quilkey for oranges; French, King, and Minami for cling peaches; Akiyama and Trivedi for tea; and Hartley, Nerlove, and Pe- ters for rubber. However, these studies utilize data on new plantings, removals, and age distribution which are not generally available for most crops in most regions. Bellman and Hart- ley propose an approach utilizing "neoclassical econometrics," but no empirical estimates are given. Knapp investigates a normative model for alfalfa.

For many perennial crops there is limited or no data on new plantings area, removals, and area in individual age groups. Thus, the problem remains of econometrically estimating perennial crop supply response with aggregate acreage and production data while still accounting for age-group dynamics in a theoretically correct manner. This paper proposes the use of state-space models and the Kalman filter to solve this problem. State-space models have system dynamics with (possibly) unobservable state variables and measurement equations relating the state variables to observable variables. The Kal- man fdter generates optimal estimates of the state variables and their variance-covariance matrix. The perennial crop problem is reformulated here a s a state-space model, and the Kalman filter is used in conjunction with an iterative search to generate parameter estimates as well as estimates of new plantings, removals, and existing acreage by age group. An empirical application to California alfalfa production is given.

Copyright 1991 American Agricultural Economics Association

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Theory

An individual producer is considered. The producer makes new plantings and removes existing acreage to maximize profits over an infinite horizon. The model, derived from Knapp, is also a simplified version of the model in Bellman and Hartley. The following variables are defined: A~, is new plantings, beginning of year t; Ai t is existing area of the perennial crop at the beginning of year t which will be in the ith year of production during year t if produced, i = 2, . . . , n; and Ri, is the area of the perennial crop in the ith year of growth which is removed from production at the beginning of year t, i = 2, . . . , n. Here n is the maximum number of age categories considered. A~~ and Ri t a r e control variables, while Ait , i = 2, . . . , n are the state variables.

The age group dynamics are given by

(1) Ai, t+ 1 = A i _ l , t - R i _ l , t ,

where i = 2, . . . , n and R~t = 0. Profits from the perennial crop depend on the cost of new plantings and removals, annual production costs, the opportunity cost of land planted to the perennial crop, and revenue received. Present value of profits is given by

(2) ~ oq . . . . . Ant, Rlt . . . . . R,,.t; Pt), t = l

where a is the discount factor, Pt is a vector of prices and other exogenous variables, and "rr denotes annual profits a s a function of acreage, removals, and prices. The producer's problem is to maximize (2) subject to (1) and nonnegativity conditions on the variables.

This is a dynamic programming problem. Un- der appropriate assumptions about rr, an optimal value function Jt exists for each year which gives the present value of profits in future years under optimal operation:

( 3 ) J t ( A 2 t , . . . , Ant ; P t , . . . , Po) oz

E a src(A*s, An*s, R*, R*" Ps), �9 � 9 �9 �9 �9 ~ n,s, S = t

where A* and R* denote optimal levels of the state and control variables; J~ is a function of the state variables and depends parametrically on prices and other exogenous variables.

Optimal values for the decision variables in each year are chosen to maximize

Amer . J. Agr . Econ.

(4) 7F(Al t , " ' ' , An t , R l t , . . . , Rn, t ; P t )

q- a J t + l ( A 2 , t + l , . . . , A,,,t+l; Pi+l, . . . , P~)

subject to (1) and the nonnegativity conditions. Optimal decision mies can then be written as

(5) A* = f t ( A 2 t , . . . , Ant'~ P t , . . . , P~o)

(6) R* ----" g i t (A2 t , . . . , Ant ; P t , . . . , P~)

for i = 2, . . . , n. Thus, the optimal values for Alt and Rit depend on existing acreage of the crop in each age group and future prices. A theoretical model for perennial crop supply response is given by (1), (5), and (6) after specification of the approp¡ functions. ~

Empirical Model

A linear model is specified for empirical estimation. Observations are assumed to be available for total acreage and production of the perennial crop and various exogenous variables believed to influence crop supply. However, new plantings, removals, and acreage by age category are not directly observable. This is the sit- uation for many perennial crops.

New plantings are given by

n m

(7) A~t = E ajAjt + E bjZit + et, j = 2 j = l

where Zjt are exogenous variables, m is the number of exogeneous va¡ and el is a white-noise error term; Zt = (Zlt . . . . . Zmt) includes expected values for future prices and other exogenous variables. Removals are specified as

n m

( 8 ) R i t - - Z c i j A j t -q- E d i j Z J t -q- eŸ j = 2 j = l

r where i = 2, . . . , n and eit is a white-noise error term. Total acreage during year t is then

1 The model in this section of the paper implicitly assumes that future values of prices (and other exogenous variables) are either known with certainty or that the conditions implying certainty equivalence are met. Neither of these assumptions is likely to be completely true in practice. The implication is that the optimal decision rules for new plantings and removals [eqs. (5), (6) in the text] depend paramet¡ on all moments of the probability distribution for future prices and other exogenous variables and not just the expected values as implied in the text. In the ensuing empirical analysis, we follow most studies in empirical agricultural supply response analysis and consider only the expected values of p¡ and other exogenous variables, while acknowledging that a more complete empirical analysis would include higher moments as well.

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(9) A,

ti

Z a = (Ai, - Rit ) q- e , , i = l

where e~ is again a white-noise error term. Equation (9) assumes that total acreage (At) is subject to measurement error. Total production is given by

n

(10) Qt = E Yi t (A i t - Rit) -}- eq, i 1

where y , is crop yield for age-category i in year t. The Yi, are assumed to be known coefficients; e q is also a white-noise error term which is in-

a n r dependent of e t , e l , and eit.

The complete empirical model is equations (1) and (7)-(10). The coefficients to be estimated are a v, b/, c~ 3, di3, and the covariance matrices

a r for the error term vectors (e,, e q) and (e~', e,). Given estimates of these coefficients and the expected values and variance-covariance matrix for A~o and Rio, then estimates of Ait and Rit, t = 1, . . . , T, can also be constructed. These estimations are carried out by recasting the model in state-space form and using a Kalman filter combined with an iterative search over the parameters. This procedure is described in the next section.

Estimation Procedure

values and variance-covariance matrices for the initial values of b 3 and d~ 3 must also be specified.

The state variables for the empirical model in state-space form are Ait, i = 1 . . . . , n; R , , i = 2 . . . . , n ; b j t , j = 1 , . . . , m ; a n d d o t , i = 2 . . . . , n, j = 1, . . . , m. The observed variables are A, and Q,, t = 1, . . . , T. The measurement equations are (9) and (10), while system dynamics are given by equations (11), (1), and (12)-(14). The unknown parameters to be estimated in the state-space model are av, cij , the variance- covariance matrices for the error terms, and the expectations and variance-covariance matrix for the state variables at t = 0. Let this set of unknown parameters and initial conditions be de- noted U.

With these equations and variable definitions, the model is in state-space form for given values of the parameters and initial conditions in U (Judge et al.). The Kalman filter can then be used to give optimal estimates of the state-space state variables and their variance/covariance matrix for given values in U, as well as the value of the log-likelihood function. Iterations over the values of the parameters and initial conditions in U are then conducted to find maximum-likelihood estimates for all unknown coefficients. Details for the particular empirical application considered here are available from the authors upon request.

The perennial crop empirical model defined by equations (1) and (7)-(10) is first recast in state- space form. Substituting (1) into (7) and (8) gives

(11) n m

Alt = E a3(A3-',t-' - R3-' , t- ' ) + E bjZ3t + et' 3=2 j=~

(12) n m

Rit = ~ ci j (Aj -1,t-1 -- RJ -1 , t - l ) -~ 2 dijZJ t + eŸ j - 2 j = l

where i = 2, . . . , n. In (11) and (12) the Z3t are exogenous variables with known values; however, the coefficients b 3 and dij are unknown parameters. In the state-space model, b 3 and di3 can therefore be treated as state variables with system dynamics given by

(13) b 3, = bj,,-1 and

(14) di3t = di3,,-1,

where i = 2, . . . , n a n d j = 1 . . . . . m. Mean

Data

The model is applied to alfalfa production in Califomia. Alfalfa is typically planted for three to four years and then removed from production. Therefore, the empirical model is estimated with four age groups. The estimations are carried out using data from 1945-85. Except where noted, data sources are described in Kon- yar and Knapp.

Age-specific yields are required to specify the quantity measurement equation (10). Based on Knapp, relative yields are defined by Yi, = aiy2t, i = 1, . . . , n and (al, . . . , a,) = (.66, 1., .92, .84). There has been a significant upward trend in California alfalfa yields. A time-series regression for the period under consideration gives

igt = 4.31 + .054"t R 2 = .92, (72.76) (21.12)

where 37t denotes the trend in average yield (tons/ acre) and t-statistics are given in parentheses. The average expected yield in each period is as-

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sumed to equal the trend of the average yield in that period:

n

E Yit /n= Yt, i=1

which implicitly defines Y2t and hence Yit for each period t using the definitions and coefficients described above.

From the theoretical model, new plantings and removals depend on existing acreage and expected profitability of growing alfalfa. Expected profitability depends in tum on expected revenue from alfalfa minus production costs and the opportunity cost of using land for alfalfa production. Prices received for competing crops are used here as a surrogate for land opportunity costs. The new plantings and removal functions are therefore specified as

(15)

(16)

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Empirical Results

Table 1 gives the estimated coefficients for the new plantings equation (15). The negative age- group coefficients in the new plantings equation imply that, everything else equal, an increase in existing acreage results in a decrease in new plantings. This is consistent with a priori expectations based on theoretical considerations. 2 Also, both the magnitude (absolute value) and statistical significance of the age-group coefficients aj decrease as j increases. Thus, existing acreage in older age categories has less influence on new plantings than does existing acreage in younger age categories.

The coefficient for TRALF* is positive, while the coefficients for PCINDX* and CCINDX* are negative. Thus, new plantings increase as expected gross revenues from alfalfa production

4

Alt = E a j ( A j - 1 , t - 1 - Rj-l,t-1) + bl -li- b2 TRALF* + b3 PCINDX* + b 4 CCINDX* + el and j = 2

4

Rit = E cij(Aj -l,t-1 - R J -1,t-l) -}- dil + di2 TRALF* + PCINDX* + di4 CCINDX* + eŸ j = 2

where TRALF* is the expected total revenue from alfalfa production (S/acre), PCINDX* is the expected value for a production cost index, and CCINDX* is the expected value of an index for the prices of competing crops; TRALF* is further defined as the expected price of alfalfa times expected yield, where expected yield is calculated from the above regression of alfalfa yields on time.

The main empirical results presented in the paper assume naive expectations for prices and production costs. Thus, expected alfalfa prices, PCINDXt* and CCINDX* just equal their actual values lagged one year. P C I N D X (1977 = 100) is based on U.S. Department of Agriculture (USDA) statistics, while CCINDX (1975 = 1.0) is constructed by calculating a weighted average of prices received for eight Califomia field crops and then using 1975 as the base year. A separate section also considers quasi-rational expectations in the sense of Nerlove, Grether, and Car- valho. In this case, expectations are calculated from ARIMA models for the individual time series.

Finally, the error tenns in the measurement and systems equations are assumed to be mu- tually independent. This implies that only the variances of the individual error terms need to be estimated; the covariances are assumed to be z e r o .

increase, as expected production costs decrease, or as the expected returns from other crops decrease. These results are also consistent with a priori considerations. The R e value for the new plantings equation is somewhat low (.293) sug- gesting that other functional forms or explana- tory variables could be tried in future work. The Durbin-Watson statistic does not suggest the presence of serially correlated error terms.

Coefficient values for the removal equations (16) are also given in table 1. The age-group coefficients cij are all positive for i = j and sig- nificantly larger in absolute value than the coefficients c• i ~ j , by an order of magnitude or more. These results are consistent with biological considerations in growing perennial crops. In general, some fraction of existing acreage is removed each year because of poor stands, dis- ease, or pest damage. Also, the length of time alfalfa stands are kept in the ground varies re- gionally due to climatic differences. For example, alfalfa stands may typically last three years in the Imperial Valley in California but four years in the Central Valley. Thus, holding prices constant, removals in age group j will be positively

2 Imagine the system is initially at a steady state. If aj > 0, then an increase in existing acreage over the steady-state level would increase new plantings and total acreage would continually increase over time. If aj < 0, then an increase in existing acreage decreases new plantings and the system can return to a steady state.

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Table 1. Est imated New Plantings and Removals Equations for the Alfalfa Supply Response Model (naive expeetations)

Independent Dependen t Variable

Variable A~ R2 R3 R4

A2 - . 5 8 4 .489 x 10 -~ - . 1 8 4 x 10 -3 - . 3 8 7 • 10 -2 ( .170) ***a ( .618 x 10-4) *** (.723 • 10-4) *** (.683 x 10-4) ***

A3 - . 3 0 8 .375 x 10 -2 .304 .146 x 10 -2 ( .156)** (.567 x 10-4) *** ( .664 • 10-4) *** (.628 x 10-4) ***

A 4 - . 1 7 1 - . 4 3 8 x 10 -2 - . 1 9 5 x 10 -2 .297 ( .158) ( .574 x 10-4) *** ( .672 X 10-4) * * * ( .635 X 10-4) * * *

Constant .843 - . 4 5 1 x 10 -2 - . 4 8 2 x 10 -2 - . 4 4 6 x 10 -3 ( .150)*** (.545 x 10-4) *** (.638 x 10-4) *** (.603 x 10-4) ***

TRALF* .729 X 10 -3 - . 9 4 6 x 10 -5 - . 1 1 8 X 10 -4 - . 5 3 2 x 10 -5 ( .487 x 10-3) * ( .177 • 10-6) *** (.207 x 10-6) *** ( .196 x 10-6) ***

PCINDX* - . 1 2 5 x 10 -2 .123 x 10 -4 .187 x 10 -4 .596 • 10 -5 ( .128 x 10 -2) ( .466 x 10-6) *** ( .546 • 10-6) *** ( .516 x 10-6) ***

CCINDX* - . 3 2 3 .297 x 10 -2 .372 x 10 -2 .155 x 10 -2 ( .159)** (.577 x 10-4) *** ( .676 x 10-4) *** (.638 x 10-4) ***

R: .293 .999 .999 .999 D - W b 1.78 2.15 2.65 1.72

Note: Results are from the OLS regressions in the last iteration of the iterative KF/OLS algorithm. a Standard deviations of the estimated coefficients are in parentheses; single, double, and triple asterisks denote significance at the 10%, 5%, and 1% levels, respectively. b D-W is the Durbin-Watson statistic.

related to, and most strongly affected by, existing acreage in the same age group.

The results in table 1 also show that removals decrease as expected alfalfa gross revenues increase and increase as expected production costs and returns from competing crops increase. All coefficient values in the removal equations are significant at the 1% level, the R 2 values ate quite high, and the Durbin-Watson statistics are not suggestive of serial correlation in the error terms.

Figure 1 shows estimated new plantings and removals of age categories 2 and 3 from the KF smoother recursions. Estimated new plantings exhibit considerable variability. Removals of second-year acreage are almost constant; removals of third-year acreage exhibit somewhat greater variability. Estimated values for year 4 removals (not reported) are somewhat lower on average than year 3 removals and exhibit a slightly lower range of variation. Time-series plots for acreage in age categories 2 -4 (also not reported) are qualitatively similar to the time- series plot for new plantings in figure 1 after a suitable number of lags. Because removals are positive, acreage in each age category is somewhat less than acreage in the previous age category, and this also tends to dampen the observed variability over time.

The observed variables in this analysis (other than exogenous variables) are assumed to be alfalfa production and total acreage in California.

Fitted values for Ait and Rit are calculated from equations (1) and (15)-(16) with the error terms set to zero and where estimated values from the Kalman filter smoother recursions are used for the unobserved acreage and removal variables on the right-hand side of these equations. The fitted values for Ait and Ri, are used to calculate fitted production and total acreage, and these are plotted in figure 2 along with actual values. This procedure is analogous to computing fitted values for ordinary least squares (OLS) regressions.

Fitted production tracks actual alfalfa production quite well, with significant deviations in only a few years. The model tracks total alfalfa acreage somewhat less well. Although the fitted results follow the overall observed trends in actual areas, there are significant deviations in several years. These results correspond with the estimated standard deviations of the error terms for the measurement equations. For production, the estimated standard deviation of el q is 9.8 • 1 0 - 6

tons and the standard deviation of et is 4 x 104

acres�9 (Note that both standard deviations are relatively low in comparison to an average total alfalfa area of 1 6 �9 1 x 10 acres and yield of 5.4 tons/acre over the period 1945-85.)

Long-run equilibrium or steady state occurs for values of Ait and Rit which, once achieved, are maintained indefinitely for given values of the exogenous variables. Long-run equilibrium

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0 . 5 5 -

0 . 5 0

0 . 4 5

0 .40

O. 35 u1

f_

~ 0 . 3 0

r

ID .,-4

0 . 2 5

~E

0.20

0.15

0.~.0

0.05

0.00

R2

i , , , , , , , , I , , , , , , , ' ' t , ", , , , ' ' ' ' I , ' ' ' ' ' ' ' ' i

~945 ~955 1965 1975 ~985

Year

Figure 1. Time-series plots for estimated new alfalfa plantings in California (Al), and re- movais in age-categories 2 (R2) and 3 (R3)

values for A, and Ri, can be calculated from equations (1) and (15)-(16) after dropping the time indices. This constitutes a system of seven equations in seven va¡ (noting that R1 -- 0), which was solved using M a t h e m a t i c a (Wolf- ram) to obtain steady-state values of Ai and R i as functions of the exogeneous variables. Total alfalfa acreage in long-run equilibrium is then given by

(17) A* = 1.32 + 1.17 • 10 -3 T R A L F * - 1.99 x 10 -3 P C I N D X * - .512 C C I N D X *

for the estimated parameter values in table 1. Using 1983-85 average values for the exogenous variables, long-run equilibrium total acreage is estimated as 1.08 million acres which is quite close to the average of 1.1 million acres over the period 1945-85.

Price elasticities are reported in table 2 for quantity supplied in the Califomia alfalfa market. Short-run (1 year) elasticities are calculated from equation (9) after substituting in the esti-

mated equations (15)-(16), dropping the error term, and using 1983-85 average estimated values for Ait from the KF smoother recursions. Long-run elasticities are calculated using equation (17). All calculations are based on 1983- 85 average values for p¡ yields, and acreage. The results suggest that alfalfa supply is inelastic in both the short and long run.

Forecasted alfalfa area was calculated under alternate initial conditions assuming 1983-85 average values for yield and the exogenous variables. One simulation uses estimated values from the KF smoother recursions to specify initial (1985) values for Ait and Ri,. Since 1985, total alfalfa acreage is quite close to the long-run equilibrium acreage calculated earlier, the calculated time path is relatively constant; there are some minor fluctuations in the early years because individual age groups are not at long-run equilibrium levels. To further explore the dynamic behavior of the estimated system, a lower initial total acreage equal to .4 • 106 acres and a higher initial total acreage of 2 • 106 acres

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8 .0 -

7 . 51

7 .0 -

2 6 .5 -

B .O - '~

5 . 5 - ==

~ 5 .0-

4.51

4.01

t 945

v q -,- , V *

' ' ' ' ' ' ~ ~ , i , , , , , , , , , i , ' ' ' ' ' ' ' ' I , ' ' ' ' ' ' ' ' 1

1955 t 965 1975 i 9B5

YEAR

i . 25 .

i . 20 -

t . 15 -

i . 05 "

~.oo-

0.95"

O.gO.

945

A ~

i955 t965 i975 t 985

YEAR

Figure 2. Actual (*) and fitted ( - ) production (a) and acreage (b) for the California alfalfa market

are also considered. In both cases the system converges to a long-run equilibrium in approx- imately four to five years after some initial fluctuations about that level.

Quasi-Rational Expectations

This section considers quasi-rational expectations in the sense of Nerlove, Grether, and Car- valho. The authors show (pp. 302-8) that under the rational expectations hypothesis, price expectations can be represented as an ARIMA model with coefficients depending on the coefficients in the structural model. Quasi-rational expectations then consist of estimating ARIMA

Table 2. Price-Elasticities of California Al- falfa Quantity Supplied (naive expectations)

Alfalfa Price* PCINDX* CCINDX*

One-year .41 - . 2 0 - . 3 6 Long-run .61 - .29 - .54

Elasticities calculated at 1983-85 average price, yield and acreage levels.

Perennial Crop Responses 847

models for the price expectations but ignoring the cross-equation parameter restrictions and then using predicted values from the ARIMA models for expected price.

Quasi-rational expectations are implemented in this study as follows. First, standard time- series methods (Shumway) are used to identify the following models:

log alfalfa price: ARIMA (2, 2, 0) log PCINDX: ARIMA (0, 2, 2) log CCINDX: ARIMA (0, 1, 1)

where the entire data series 1945-85 is used in the identification process. Second, for each year t from 1955 to 1985, the above ARIMA models are estimated using data from 1945 to year t - 1, and forecasts generated for years s, s = t, . . . , t + 3. The four-year average is then used to specify TRALF*, PCINDX*, and CCINDX* in equations (15) and (16). Finally, the unknown parameters and acreage and removal variables are estimated as before, except that the estimation is carried out for the years 1955-85 instead of 1945-85 under naive expectations.

Table 3 gives the estimated coefficients for the new plantings and removal equations under quasi-rational expectations. The signs of the estimated coefficients in table 3 are roughly comparable to those under naive expectations (table 1). Exceptions are the signs for some age group coefficients in the removal equations and the signs for CCINDX* in the new plantings equation and the removal equations for age groups 2 and 3. In particular, an increase in the price of competing crops under the quasi-rational expectations hypothesis implies an increase in new plantings area in contrast to what might be expected a priori. The estimated coefficients under quasi-rational expectations are roughly comparable in order of magnitude to the estimated coefficients under naive expectations. Other than that, the most notable differences are higher age- group coefficients and lower TRALF* coefficients (in absolute value) under quasi-rational expectations.

Table 4 gives the price elasticities for Cali- fornia alfalfa supply under quasi-rational expectations. Here the own-price elasticity is implau- sibly low (3% in both the short and long run). The production cost elasticities are quite close to those under naive expectations. The elasticity of competing crop prices is positive in contrast to both the results under naive expectations and a priori reasoning. Long-run equilibrium acreage is calculated as 1.04 million acres, which is slightly lower than estimated under naive ex-

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Table 3. Estimated New Plantings and Removal Equations for the Alfalfa Supply Response Model (quasi-rational expectations)

Independent Dependen t Var iable

Var iable A x R2 R3 R4

A2 - . 6 3 7 .58 x 10 -1 .39 x 10 -2 .35 x 10 -2 ( .107) ***a (.1 • 10-3) *** ( .12 x 10-3) *** (.1 • 10-3) ***

A3 - . 7 3 8 .56 x 10 -2 .54 x 10 -1 .39 x 10 -2 ( .102)*** ( .96 x 10-4) *** (.1 • 10-3) *** (.1 x 10-3) ***

A4 - . 6 6 .28 x 10 -2 .53 • 10 -3 .30 ( .097)*** (.9 • 10-4) *** (.1 • 10-3) *** ( .98 • 10-4) ***

Constant .994 - . 1 5 2 • 10 -1 - . 7 2 x 10 -2 - . 5 3 x 10 -2 ( .086)*** (.8 x 10-4) *** (.98 x 10-4) *** (.9 x 10-4) ***

TRALF* .40 x 10 -4 - . 3 4 x 10 -5 - . 3 2 • 10 -5 - . 3 6 • 10 -5 (.64 x 10 -4) ( .6 x 10-7) *** (.7 • 10-7) *** (.6 • 10-7) ***

PCINDX* - . 1 6 x 10 -2 .1 • 10 -4 .11 x 10 -4 .11 x 10 -4 ( .256 • 10-3) *** ( .24 • 10-6) *** (.3 • 10-6) *** ( .26 • 10-6) ***

CCINDX* .865 x 10 -1 - . 1 1 5 x 10 -2 - . 5 3 x 10 -3 .99 • 10 -3 ( .0453)** (.43 x 10-4) *** (.5 • 10-4) *** ( .46 x 10-4) ***

R 2 .77 .99 .99 .99 D - W b 1.61 2.15 3.1 2.5

Note: Results are from the OLS regressions in the last iteration of the iterative KF/OLS algorithm. a Standard deviations of the estimated coefficients are in parentheses; single, double, and triple asterisks denote significance at the 10%, 5%, and 1% levels, respectively. b D-W is the Durbin-Watson statistic.

Table 4. Price-Elasticities of California Al- falfa Quantity Supplied (quasi-rational expectations)

Alfa l fa Price* PCINDX* CCINDX*

One-yea r .03 - .25 .09 Long- run .03 - . 2 9 .11

Note: Elasticities calculated at 1983-85 average price, yield, and acreage levels.

pectations. Forecasted acreage was also estimated for the period 1986-2025 under the same initial conditions and values as before. As with naive expectations, total alfalfa acreage under quasi-rational expectations exhibits a damped cycle with eventual convergence to a long-run equilibrium. However, convergence to long-run equilib¡ takes noticeably longer (10-20 years) with supply response estimated under quasi- rational expectations, even assuming constant expected prices and costs during the forecast period.

Conclusions

Previous econometric studies of perennial crop supply response have been based on OLS estimates of single-equation, reduced-form regressions for aggregate output or acreage, or OLS

estimates of individual new planting/removal equations. The first approach does not ade- quately account for age-distribution impacts on new plantings and removals, while the second approach requires detailed data which are not always available. A state-space approach using the Kalman filter is proposed here. The structural model includes age-distribution effects and separate new planting, removal, and age-group dynamics. Both parameter and age-group acreage estimates are obtained from data on total acreage, production, and prices.

The model is applied to alfalfa production in California. Under naive price and cost expectations, the estimated model provides a reason- able fit to the data. The correct signs were obtained and the price coefficients are generally statistically significant. Estimated long-run equilibrium is close to average acreage over the sample period. The results suggest that alfalfa supply is inelastic in both the short- and long- run. Total acreage converges to long-run equilibrium under constant prices, but with some overshooting. The model was also estimated assuming quasi-rational expectations. The estimated coefficients are generally comparable in sign and order of magnitude to those under naive expectations. However, the own-price supply elasticity is much less than that under naive expectations, and the elasticity with respect to competing crop prices is positive. In general, the results here under naive expectations seem

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more consistent with a priori considerations than those under quasi-rational expectations.

Age-structured models arise naturally in many areas of natural resource economics (Wilen). Examples include fisheries, forests, and wildlife herds. In many instances births and/or deaths may depend on the age-structure of the population; thus, accounting for age-structure is nec- essary for obtaining accurate theoretical and empirical models. Age-structured models also arise in other areas of economics, such as capital theory. In a given industry, for example, existing capital may be quite heterogenous with respect to age. Investment in new equipment therefore depends, in part, on the age-distribution of the existing capital stock. Economic models which explicitly account for the age distribution of physical and biological capital stocks are relatively scarce. This is due in part to difficulties in analyzing such models theoretically (e.g., Cushing) and in part to difficulties in obtaining age-specific data to conduct empirical analysis. Thus, although the focus of this paper is perennial crops, the method can also be applied to estimate age-structured models in other areas of economics when only age-aggregated data is available. 3 Current limitations on the use of this method include lack of generally applicable al- gorithms for state estimation in nonlinear models and parameter estimation in both linear and nonlinear models.

[Received April 1989; final revision received October 1990.]

References

Akiyama, T., and P. K. Trivedi. "Vintage Production Ap- proach to Perennial Crop Supply: An Application to

3 Dixon and Howitt (1979, 1980) use a Kalman filter to update estimated volumes and basal areas by age class in a normative study of forestry management. However, they assume that each of these variables is being sampled in each period. This contrasts with the approach here where there is assumed to be no age-specific data on ah annual basis and where annual production and acreage data is used to infer age-specific data.

Tea in Major Producing Countries." J. Econometrics 36( 1987): 133-61.

Alston, J. M., J. W. Freebairn, and J. J. Quilkey. "A Model of Supply Response in the Australian Orange Growing Industry." Aust. J. Agr. Econ. 24(1980):248-67.

Arak, M. "The Price-Responsiveness of Sao Paulo Coffee Growers." Food Res. Inst. Stud. 8(1968):211-23.

Bellman, R. E., and M. J. Hartley. "The Tree-Crop Prob- lem." Graduate School of Management Work. Pap. Ser. No. 87-25, University of California, Riverside, June 1987.

Cushing, J. M. "Nonlinear Mat¡ Models and Population Dynamics." Nat. Resour. Model. 2(1988):539-581.

Dixon, B. L., and R. E. Howitt. "Resource Production Un- der Uncertainty: A Stochastic Control Approach to Timber Harvest Scheduling." Amer. J. Agr. Econ. 62(1980):499-507.

"Uncertainty and the Intertemporal Management of Natural Resources: An Empirical Application to the Stanislaus National Forest." Giannini Foundation Monograph No. 38, University of California, Berke- ley, Sep. 1979.

French, B. C., and R. G. Bressler. "The Lemon Cycle." J. Farm Econ. 44(1962):1021-36.

French, B. C., G. A. King, and D. D. Minami. "Planting and Removal Relationships for Perennial Crops: An Application to Cling Peaches." Amer. J. Agr. Econ. 67(1985):215-23.

Hartley, M. J., M. Nerlove, and R. K. Peters, Jr. "An Analysis of Rubber Supply in Sri Lanka." Amer. J. Agr. Econ. 69(1987):755-61.

Judge, G., W. Griffiths, R. Hill, H. Lª and T.-C. Lee. The Theory and Practice of Econometrics. New York: John Wiley & Sons, 1985.

Knapp, K. "Dynamic Equilibrium in Markets for Perennial Crops." Amer. J. Agr. Econ. 69(1987):97-105.

Konyar, K., and K. Knapp. "Market Analysis of Alfalfa Hay: California Case." Agribus. 4(1988):271-84.

Nerlove, M., D. M. Grether, and J. L. Carvalho. Analysis of Economic Time Series: A Synthesis. New York: Ac- ademic Press, 1979.

Shumway, R. H. Applied StatŸ Time Series Analysis. Englewood Cliffs N J: Prentice-Hall, 1988.

Wilen, J. E. "Bioeconomics of Renewable Resource Use." Handbook of Natural Resource and Energy Econom- ics, ed. A. V. Kneese and J. L. Sweeney, vol. 1, chap. 2. Amsterdam: North-Holland Publishing Co., 1985.

Wolfram, S. Mathematica: A System for Doing Mathe- matics by Computer. Redwood City CA: Addison- Wesley Publishing Co., 1988.

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Perennial Crop Supply Response: A Kalman Filter Approach