An Imperfectly Competitive Market Model of the U.S...

An Imperfectly Competitive Market Modelof the U.S. Lettuce Industry

Michael D. Hammig and Ron C. Mittelhammer

An econometric model was specified to represent the U.S. lettuce industry.Cursory examination of the industry structure suggests that imperfect competition mayprevail in the lettuce market. Therefore, relations were specified that allowed for thepossibility of imperfectly competitive behavior to affect market equilibrium outcomes.Specifically, a supply price equation was specified to account for the influence of marketpower of large growers, particularly during seasons of geographically concentratedproduction. Results do not contradict the hypothesis that imperfect competition exists inthe lettuce market.

The perfectly competitive model providesthe basic analytical framework for the vastmajority of research directed at agriculturalcommodity markets although analysts oftenrecognize many deviations from the competi-tive norm within those markets. The assump-tions of the competitive model are wellknown and, even though they may not bestrictly representative of a given market,they often serve as useful approximations.However, in some agricultural commoditymarkets the competitive model is not appro-priate. The lettuce market exemplifies a casewhere the tenets of perfect competition donot comfortably apply.

This paper presents part of a largerU.S.D.A. study to develop quarterly marketmodels for a selected set of fresh saladvegetables. Though the market structures formost fresh vegetables are in general termssimilar, the lettuce market displays someunique characteristics. The bulk of the cropis produced in California; Monterey Countyin the spring and summer and Imperial

Michael D. Hammig is Agricultural Economist, NationalEconomics Division, Economics, Statistics, andCooperatives Service, USDA, and Ron C. Mittelham-mer is Assistant Professor, Washington State University.

The authors gratefully acknowledge the helpful com-ments of the editor and the anonymous reviewers.

County in the fall and winter [U.S. Depart-ment of Agriculture]. A relatively small num-ber of large producers control a significantportion of the total supply of lettuce. InCalifornia, eight producers have at timessupplied nearly one-half of that State's prod-uction. 1

Given a situation where a small number ofproducers control a large proportion of totalproduction, the opportunity for the extrac-tion of above-normal profits may exist. Thispaper presents an attempt to construct asimulation model of the U.S. lettuce marketwith notions of an imperfectly competitivemarket structure as its base. The model isthen used to assess the impact on the lettucemarket of a relative increase in wages paid tohired farm labor.

Model Structure

The market for lettuce is completely forfresh use. Neither long-term storage norsignificant processing of lettuce takes place.Imports of lettuce are negligible; however,significant quantities are exported - mostlyto Canada - each year. The econometricmodel developed for this study consists of

'These figures were obtained from 4/1/74 to 3/31/75based on unpublished data from the State of CaliforniaIceberg Lettuce Research Board.

1

Western Journal of Agricultural Economics

five estimated relationships and two iden-tities. Acreage planted, acreage harvested,supply price, domestic demand and exportdemand were estimated directly. Total do-mestic supply and yield were determined byidentities.

Acreage Planted

In the process of producing lettuce, thefirst decision a producer must make relates tothe acreage that he plants in a given season.Generally a minimum of ten weeks passbetween planting and a decision concerningthe harvest. Therefore the acreage planteddecision must be based on information availa-ble well in advance of the time when actualquantities produced and consumed, andprices, are known to producers.

The key factors affecting acreage plantedare hypothesized to include the expectedprice, the risk of error in determining thatprice expectation, the variable costs of pro-ducing the crop, and existing investment infixed assets used in the production of lettuce.The acreage planted equation was specifiedas:

(1) APt = ao + alPEt + a2Rt +

a3CPt + a4APt-4 + a5Dsp +

a6Dsum + a7 Dfall + vt,

where APt is the number of acres planted forproduction of lettuce in quarter t; PEt is theexpected price of lettuce per cwt.; Rt is therisk associated with price expectations; CPt isthe index of prices paid by farmers for itemsused in production, 1967 = 100; Dsp, Dsum,and Dfall are dummy variable intercept shif-ters for the spring, summer, and fall quar-ters, respectively; and vt is the disturbance.

The expected price variable is defined as athree-year geometrically declining weightedaverage of past observed prices as

(2)3

PEt = I Wi Pt-4ii=l

where w=.54369.2 This specification forprice expectation is motivated, in part, byNerlove's adaptive expectations hypothesis,where expectations are adjusted by weighteddifferences between actual and expected out-comes on price. When this type of expecta-tions hypothesis is properly transformed toeliminate unobservable variables, an infiniteseries of geometrically declining weightedpast prices results. Expectations variablesused in the model presented here are alsobased on geometrically declining weightedlagged prices. However, the lags are finite. Itis hypothesized that producers cease to relyon past experience in forming current expec-tations after an appropriate time span. Ex-perimentation with various lag lengths hasshown that the three-year lag provides thebest empirical results.

The risk associated with price expectationsis represented by the square root of a weight-ed average of squared deviations betweenactual and expected prices relative to theexpected price as

(3)3

Rt = ( I Wi(Pt-4i- PEt_4i) 2 )2/PEt.i=l

This specification of risk provides a measureof the accuracy of expectation formationrelative to the level of expectations. Thus,the same degree of accuracy, as measured bythe numerator, would imply less perceivedrisk if the price expectation is high than itwould if the price expectation were relativelylow.

Using information from previous studies,economic theory, and USDA commodityspecialists, prior stochastic constraints wereincorporated in the model through the mixedestimation technique [Theil and Goldberg-er]. Studies by Nerlove and Addison, Ham-mig (1978), Lin, and Traill give insights intothe expected range of values for elasticities

2This value was obtained by solving the equation w3+

w2 + w = 1. The subscript, t, refers to a quarterly timeperiod, thus t - 4i refers to an annual lag.

2

July 1980

Hammig and Mittelhammer

with respect to expected price, risk, andproduction costs. All sources of prior infor-mation agree that the elasticities of acreagewith respect to expected price and variableproduction costs will be in the inelasticrange. It has also been observed that acreageelasticities with respect to risk variablessimilar to the one used here fall between zeroand -. 1. Thus, the restrictions appliedthrough mixed estimation, as used in thispaper, are determined by first establishingthe appropriate interval of acceptable valuesfor the relevant parameter to be estimated.The midpoint of the interval is taken as theprior point estimate, and 95 percent proba-bility limits, assuming normally distributedprior information, are established by assum-ing that two standard deviations span theinterval on either side of the point estimate.Thus, the coefficients were stochasticallyconstrained such that the mean level elas-ticities of acreage planted with respect toexpected price, risk, and costs of productionwere 5. ± .5, -. 05 ± .05, and -. 5 ± .5,respectively, with .95 probability, and mixedestimation was applied to the hypothesizedrelation. All covariances among these restric-tions were assumed to equal zero. (For amore detailed discussion of the prior infor-mation applied to the acreage planted equa-tion see Hammig (1979).)

Acreage Harvested, Yield, and Pricing

The assumed goal of firms engaged in theproduction and sale of lettuce is to obtain thelargest possible profit given certain opera-tional constraints. Market demand forcesconstrain the amount of product that may besold. Physical production requirements con-strain the level of output that may be at-tained. The availability and accuracy of infor-mation regarding the general market situa-tion further constrains the actions a firm maytake.

Under perfect competition producers max-imize their profits by providing supplies upto the point where marginal cost is equal tothe market price, and the aggregate supply

curve for the industry is the horizontal sum-mation of the portions of individual firmmarginal cost curves lying above averagevariable costs. Market price is established atthe point where the aggregate supply curveintersects the aggregate demand curve.

In the case of the lettuce market, since alarge portion of production is controlled by asmall number of growers, the competitivesupply price may not be attainable. That is,large growers, recognizing their power tosignificantly control supplies, may insist on alarger than normal profit margin in thelettuce they sell, and thus put upward pres-sure on the market price. The vigor withwhich such a course would be pursued woulddepend on the elasticity of competing sup-plies, and the elasticity of market demand forlettuce. If both elasticities are sufficientlyinelastic, grower profit can be enhancedthrough relatively small reductions in thequantity of lettuce offered for sale.

Lettuce quantity supplied is, by definition,the product of the number of acres harvestedand the yield per harvested acre. Yields varydirectly with the frequency and intensitywith which the field is harvested. Greater(within biological limits) and lesser yields peracre can be had through increased or de-creased application of harvesting inputs(primarily labor). Acreage planted providesthe upper bound on the potential number ofacres harvestable. On the average, four tofive percent of planted lettuce acreage isunharvested due to crop damage and due toeconomic decisions not to harvest.

In the absence of market power to affectprice, the supply curve for lettuce could bespecified in a standard manner where thesupply price at harvest time would dependon the quantity supplied, and the prices ofinputs used in harvesting and selling thecrop. However, since a large portion of totalproduction is concentrated in the hands of afew large growers, there is reason to believethat the large growers can influence theselling price. Thus quantity and input pricesdo not exhaust the list of variables influenc-ing supply and pricing behavior. It is there-

3

Lettuce Market Model


fore hypothesized that the behavior of thelarge growers may be influenced by theirperceptions of market conditions, especiallythe behavior of competing suppliers. Giventhe data available, this latter phenomenonwas proxied in the supply price equation byincluding variables that represent the expect-ed level of quantity demanded, the anticipat-ed potential quantity of lettuce supplied, andvariables identifying seasons of the yearwhen lettuce production is more geographi-cally dispersed and less concentrated inCalifornia (spring and summer).

The supply-price equation was specified as

(4) Pt = At(Qs,t) a1(WGt/PDYt)T2

(IWPIt)"3 eut,

where

(5) At = K exp(bo + bi(Qd*,t/Qs*,t)

+b 2 Dsp + b3 Dsum),

K is a constant; Qd*,t/Qs*,t is the ratio ofexpected quantity demanded to anticipatedpotential lettuce quantity supplied; Dsp andDsum are dummy variables having the value1 in the spring and summer quarters, respec-tively, and the value zero elsewhere; Qs,t islettuce quantity supplied; WGt is an annualindex of migratory worker wage rates,1967= 100; PDYt is an annual index of laborproductivity in vegetable production,1967 = 1.0; IWPIt is the quarterly industrialwholesale price index used to proxy for costsother than labor in the harvesting and sale oflettuce; and ut is the disturbance term.

The variables Qd*,t, Qs*,t, Dsp, andDsum are used to proxy perceptions ofmarket power available to major lettuceproducers. Spring and summer dummy vari-ables are included as indicators of the timeperiods when total production is least re-stricted to the major producers and whencompetition for regional markets is signifi-cantly increased due to active lettuce produc-tion by producers throughout the nation. Theratio Qd*,t/Qs*,t is used to proxy percep-

4

tions of the relative demand/supply situationthat could exist given demand expectationsand expectations of potential maximum sup-plies forthcoming from the known plantedacreage. In the same manner that the priceexpectation is formed by equation (2), theexpected demand variable is defined as ageometrically declining weighted average ofpast demands,

(6)3

Qd*,t = wi

Qdt-i,i=l

where w is as defined in (2). More informa-tion is available regarding potential supply,since the number of acres planted is known atharvest time. Expected potential quantitysupplied is defined as the product of currentacreage planted and expected yields; wherethe yield expectation is also formed in thesame manner as (2)- a geometrically declin-ing weighted average of past yields. It ishypothesized that as expected demand in-creases relative to expected potential supplyin a harvest period, large producers perceivean increase in their ability to affect marketoutcomes, and the supply price - andultimately the market price - will tend toincrease. That is, large producers will actbased on their expectations of the power theywill have to control the market. In periodswhere they expect short supplies relative todemand they will exert more upward pres-sure on the price. However, in periods ofmore significant competition for markets(spring and summer), it is hypothesized thatthe market power of large producers isundercut, and supply-price and market pricemore nearly approach perfectly competitivelevels.

The migratory worker wage index, de-flated by an index of labor productivity, isincluded to represent the labor cost compo-nent of lettuce production. The IWPI vari-able is included to account for all other costs.Under either perfect or imperfect competi-tion, market equilibrium prices will increasewhen costs are increased. The magnitude of

July 1980


the price increase will depend on the marketpower exercised by producers and/or theelasticities of supply and demand for thegiven product. Johnson and Zahara haveshown that labor constitutes up to 40 percentof total lettuce harvesting costs. Therefore,using mixed estimation, the coefficient onmigratory worker wages was constrained toreflect an elasticity of .4 + .3, .95 probabili-ty, in anticipation of the effect of labor costson supply price.

The market price for lettuce is establishedwhen quantity demanded at a given price isequal to the quantity that will be supplied atthat price. If the market price is to beestablished above the perfectly competitiveprice, a portion of production must be with-held from the market - presumably by thelarge growers who exercise their marketcontrol to achieve higher profit through high-er prices. As stated previously, quantitysupplied can be varied by changing thenumber of acres harvested and/or by varyingthe intensity with which harvesting is pur-sued. When the market is in equilibrium, thequantity demanded at the market price es-tablished must be equal to the number ofacres harvested times average yield per acreharvested.

The model achieves the equality in thefollowing way. Together with the establish-ment of supply offer curves associating quan-tities supplied with supply prices for indi-vidual growers, the profit maximizing mix ofnumber of acres harvested and yield is estab-lished. It is hypothesized that for the aggre-gate supply-price curve, the higher the sup-ply price and the associated quantity sup-plied, the greater will be the number of acresharvested and the higher will be the intensityof the harvest (yield). Since the number ofacres available for harvest is fixed, whileintensity of harvest may vary depending onprevailing conditions, it is further hy-pothesized that yield is more responsive toprice than is acreage harvested. In addition,as real wages paid to labor increases, ceterisparibus, it would be expected that feweracres would be harvested in the aggregate,

and the intensity of the harvest (yield) wouldbe decreased. Also, the aggregate level ofacreage harvested relative to acreage plantedmay be affected by varying yield and weatherpossibilities in the spring and summer whenthe growing of lettuce is more geographicallydispersed.

Of course, acreage harvested cannot ex-ceed acreage planted, and thus the acreageharvested relationship was modeled in thefollowing general form:

(7) AHt = Pt APt,

where AHt is acreage harvested in acres; APtis acreage planted in acres; and o < at P 1. Aparticular functional form that exhibits therequired bounds on Pt, and that performedwell in empirical testing, was the logisticfunction.

(8) Pt= 1/(1 + exp(Zt))

where Zt is a set of factors affecting theacreage harvested decision. In this study

(9) Zt = Vo + V1Pt + V2(WGt/PDYt)

+V 3Dsp + V4Dsum + Et

where all variables are as defined previously,with Et being the disturbance. By substitu-tion of (9) into (8), and then (8) into (7), andwith further algebraic manipulations andlinearization in logarithms, the estimatingequation was

(10) In APt 1) = Yo + -/1PtAHt

+ y2(WGt/PDYt) + y3 Dsp +

y4Dsum + Et.

In equilibrium, the average yield is deter-mined as the value that when multiplied byacreage harvested results in a quantity sup-plied that is equal to quantity demanded, allat the established market price. Quantitydemanded, acreage harvested, average yield,quantity supplied, and final market price are

5



all affected endogenously by consumers andproducers in the lettuce market; where someproducers may follow policies that restrictacreage harvested and/or yield in order torestrict quantity supplied and thus increaseprices above perfectly competitive prices.

Derived Domestic Demand

The domestic demand for lettuce isspecified at the grower level, following theconventional tenets of demand theory, as

(11) Qdt/POPt = go + gl Pt/CPIt + g2

It/(CPIt POP) + g3Tt + Ut

where Qdt is domestic consumption of lettucein million cwts. in quarter t; Pt is the farmlevel price of lettuce in dollars per cwt.; It isdisposable income in billions of dollars; CPItis the consumer price index with 1967 =1.0; POPt is population in millions; and Tt is ataste and habit shifter of demand. It isassumed that the marketing margin consistsof fixed per unit and/or percentage markups,and thus no explicit reference to marketingmargin components is made (see George andKing, p. 56-59).

3(12) Tt = 80 + E bi(Qdt-4i/POPt-4i)

i=l

3with E bi = 1. That is, the taste and habit

i=lshifter is a linear function of the weightedaverage of per capita consumptions for threeprevious corresponding quarters.3

Shiller's method [Shiller] was usedto impose the hypothesis of smoothly declin-ing weights on 81, 82, and 83 usingthe mixed estimation technique. The deriva-tion of the implied stochastic constraint onthe parameters of the demand equation ap-pears in abbreviated form in the Appendix.In addition, through examination of the re-search of George and King, and through

3 Empirical testing of various lag lengths revealed thatthe three-year lag performed best.

6

consultations with U.S.D.A. commodityspecialists, prior stochastic constraints on themean level elasticities with respect to priceand income of -. 10 ± .10 and .20 +.20, respectively, with probabilty .95, wereimposed using mixed estimation. The inter-vals for these constraints were generated inan independent manner. Thus the restric-tions are assumed to be subjectively inde-pendent and the covariance between the twois zero.

Export Demand

Virtually all of the lettuce exported by theU.S. is shipped to Canada. In addition, in thewinter, spring, and fall quarters, virtually allof the lettuce consumed by Canadians isimported from the U.S. Canadian produc-tion of lettuce is commercially significantonly in the summer.

The demand for U.S. exports of lettuce istherefore essentially Canadian consumer de-mand in winter, spring, and fall, and Cana-dian excess demand in the summer. Theexport demand function utilized was

(13) EXPt = ho + hi Pt/(ERt *

CPIcan,t) + h2 Ican,t/CPIcan,t + h3

POPcan,t -Qscant + et

where EXPt is quantity of lettuce exported inquarter t in million cwts; Pt in U.S. farmprice in dollars per cwt; ERt is the exchangerate expressed in dollars U.S. to dollarsCanadian; POPcan,t is Canadian population inmillions; and Qscan,t is Canadian production oflettuce in million cwts. Stochastic prior con-straints on the mean level price and incomeelasticities were applied in the same manneras for estimation of domestic demand, andmixed estimation was applied to (13).

Estimation Results

The structural equations requiring statisti-cal estimation in the lettuce market modelare presented in Table 1. Prior as well assample information was available for all the

July 1980


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relations except acreage harvested, and themixed estimation technique was applied inthose cases. The prior information used inthese relations, along with correspondingposterior estimates are given in Table 2. Theacreage planted equation did not containendogenous explanatory variables, so OLSwas an appropriate estimation procedure. Allother relations required a simultaneousequation estimation technique and 2SLS wasused. Sample data for the years 1954-1977were used to estimate the system.

Results of the acreage planted relationwere as anticipated; all signs were as expect-ed and the X2 test that the prior informationis not contradicted by the sample data indi-cates that sample and prior information arecompatible at conventional levels of type Ierror [see Theil]. Mean level elasticities withrespect to expected price, risk, and costs ofproduction were .4170, -. 0561, and-. 2011, respectively.

The 2SLS mixed estimation results of thesupply price equation are shown in equation2 of Table 1. Signs of all coefficients conformto expectations. The X2 test of compatibilityindicates compatible prior and sample infor-mation. Elasticities of supply price withrespect to production, wages, and other costsall fall in the inelastic range at .5187, .2934,and .3611 respectively. These results tend toconfirm the hypothesis that imperfectly com-petitive behavior exists in the lettuce market.The significant negative effects of the springand summer dummy variables imply that, infact, prices are induced upward duringperiods of restricted competition. Totalquantities produced and consumed remain

remarkably constant across all seasons, lend-ing some strength to the argument that thedummy variables are not mere proxies forseasonal demand shifts.

Estimated results of the acreage harvestedequation appear in equation 3 of Table 1.Conventional theory suggests that the coeffi-cient on price should be negative and thecoefficient on wages should be positive. Giv-en these signs, as price increases the propor-tion of acres harvested to acres planted willalso increase, and as wages increase relativeto productivity this proportion will fall. Theresults obtained in estimation conform to theexpected pattern. Mean level elasticities ofacreage harvested with respect to price andwages are .006 and - .009, respectively. Thelow elasticities suggest that economic vari-ables have relatively minor impact on theharvesting decision.

The domestic demand and export demandequations incorporated the same prior no-tions regarding elasticities with respect toprice and income. In the case of domesticdemand, results were as anticipated. Themean level price and income elasticities ofdemand were estimated to be -. 1223 and.1877, respectively.

In the export demand equation, the priorincome elasticity was found to be incompat-ible with the sample information by a X2 testat the .05 level of type I error [see Theil], andthe prior input was discarded. Prior know-ledge of the price elasticity was quite com-patible with the sample, and the resultsobtained using only the one constraint gaveelasticities of export demand with respect to

TABLE 2. Comparison of Prior and Posterior Estimates in the U.S. Lettuce Market Model

Elasticity Prior Estimate Posterior Estimate

AP,PE .5 ± .5 .417AP,R -. 05 ± .05 -. 056AP,CP -. 5 + .5 -. 201P,WG .4 ± .3 .293Qd,P -. 1 ± .1 -. 122Qd,l .2 ± .2 .183EXP,P -. 1 ± .1 -.102

8

July 1980


price and income of -. 1016 and .6750,respectively.

Overall the results of estimation of the fivestructural equations support the hy-pothesized relationships inherent in thestructure of the lettuce market. All signsagree with the expected direction of forcesoperating in the market, and the compatibili-ty tests uniformly confirm that sample andprior information are, in fact, compatible. Asnoted above, the model is completed byidentities to determine quantity supplied(the sum of domestic and export demands)and yield (quantity supplied divided by har-vest acreage). The complete model can thenbe used to simulate activities in the lettucemarket.

Model Simulation

Since the reduced form of the lettucemarket model in nonlinear, the Gauss-Seideltechnique was used to simulate the system.Model solutions were obtained for the years1960 through 1977, and these results werecompared to actual market outcomes to mea-sure the validity of the complete model.Measures of goodness of fit are presented inTable 3.

Recent union activity on behalf of lettuceharvest laborers in California indicates thatthe possibility exists for harvest labor costs tosignificantly increase in coming years. Theeffects of increased wages on the lettucemarket can be examined by the simulationmodel. The United Farm Workers (UFW)has announced a goal of obtaining wageincreases of approximately 40 percent abovecurrent levels [Washington Post, Washing-ton Star]. Many lettuce producers are resist-ing UFW pressure to obtain such increases.

The impact on the market of large wageincreases was evaluated by comparing simu-lation results obtained under two sets ofassumptions. Baseline projections weremade assuming normal trend adjustments inwages (seven percent per year) and otherfactors as an average of changes over the pastfive years. A second set of projections wasdeveloped for the market under the assump-

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tion of similar normal trend adjustments forall factors except wages. Wages were in-creased in the first year by 40 percent andsubsequent annual wage increases were heldat seven percent. The comparison of the twosimulations is, therefore, of an effective wagedifference of 33 percent in 1979 increasingmoderately in following years. Both simula-tions were extrapolated over the period 1979-83 to allow lagged effects to be resolved overtime. Simulation results for domestic quanti-ty demanded, quantity supplied, and thesupply price are presented in Table 4.

The effect of the 33 percent immediatewage increase over normal levels is to reduceslightly quantities demanded and supplied,and to increase prices. However, the mag-

nitudes of the differences are relativelysmall. Quantities differ by less than twopercent for all seasons over the five yearperiod. Price differences average slightlyover seven percent for the same period.

Some implications of these results can beinterpreted by examining the cost structureof lettuce production. Using the work ofJohnson and Zahara it can be shown that, for1975, a 33 percent increase in wages wouldtranslate to a 6.5 to 14.0 percent increase inlettuce harvesting costs. Since harvestingcosts constitute about two-thirds of totalvariable costs of production, the costs ofproducing lettuce would increase by 4.3 to9.3 percent due exclusively to the change inwages paid. Based on the results of this

TABLE 4. Lettuce Market Simulation Results under Baseline Assumptions and with In-creased Wages

Baseline Projections Projections with Increased Wages

Year/ Quantity Quantity Quantity Quantityseason Supplied Demanded Price Supplied Demanded Price

1,000 cwt. $/cwt. 1,000 cwt. $/cwt.

1979Winter 14.64 13.50 8.44 14.53 13.40 9.11Spring 15.25 14.07 8.85 15.15 13.97 9.52Summer 14.61 14.32 8.34 14.52 14.23 8.98Fall 14.24 13.08 9.25 14.14 12.98 9.93





10

July 1980


study, the seven percent increase in prices,projected by the simulation including higherwages, would translate into revenues accru-ing to growers between 6.5 and 7.2 percentabove those that would be obtained underthe baseline scenario. Revenue increasesbetween 6.5 and 7.2 percent concurrent withcost increases between 4.3 and 9.3 percentimply that the bulk of the cost increases willbe passed forward by producers to otherparticipants in the lettuce marketing system.

Conclusion

This study has endeavored to present aneconometric model representing a marketwhere elements of imperfect competitionpotentially exist. Through the modeling ofthe supply price relationship, imperfectlycompetitive behavior is allowed to enter thedetermination of market activities. Thoughthe results of estimation and simulation of thecomplete model are not conclusive as to theexistence of imperfectly competitive be-havior in the U.S lettuce market, the struc-ture of the production side of the marketsuggests that imperfect competition warrantsconcern, and empirical results do not ruleout the possibility that lettuce growers exer-cise some degree of market power.

Lin, W. "Measuring Aggregate Supply Response UnderInstability." American Journal of AgriculturalEconomics, 59 (1977): 903-907.

Nerlove, M. "Estimates of the Elasticities of Supply ofSelected Agricultural Commodities," Journal of FarmEconomics, 38 (1956): 496-509.

Nerlove, M. and W. Addison. "Statistical Estimation ofLong-Run Elasticities of Supply and Demand," Jour-nal of Farm Economics, 40 (1958): 861-880.

Shiller, Robert J. "A Distributed Lag Estimator DerivedFrom Smoothness Priors." Econometrica, 41 (1973):775-788.

Theil, H. "On the Use of Incomplete Prior Informationin Regression Analysis," Journal of American Statisti-cal Association, 58 (1963): 401-414.

Theil, H., and A. S. Goldberger. "On Pure and MixedStatistical Estimation in Economics." InternationalEconomic Review, 2 (1961): 65-78.

Traill, W. B. "Risk Variables in Econometric SupplyResponse Models." Paper presented at AAEA annualmeeting, State College, PA, 1976.

U.S. Department of Agriculture, "Vegetables - FreshMarket Annual Summary." Washington, D.C.: SRS,1978.

Washington Post. "Chavez's Lettuce Walkout is Wilt-ing." Washington, D.C., April 29, 1979.

Washington Star. "Chavez on the Lettuce Strikers."Washington, D.C., April 27, 1979.

References

George, P. S. and G. A. King. Consumer Demand forFood Commodities in the United States with Projec-tions for 1980. Giannini Foundation MonographNumber 26, March 1971.

Hammig, M. D. Supply Response and Simulation ofSupply and Demand for the U.S. Fresh VegetableIndustry. Unpublished Ph.D. dissertation, Washing-ton State University, 1978.

Hammig, M. D. "Regional Acreage Response by Quar-ter for Fresh Tomatoes: An Example of the Use ofMixed Estimation." Southern Journal of AgriculturalEconomics, 11 (1979): 69-74.

Johnson, S. S. and M. Zahara. "Lettuce Harvesting andPacking: Hand and Machine Harvest Alternative."Vegetable Situation. ERS, USDA, May 1976.

Appendix

The Shiller method of imposing smooth-ness priors on model parameters incorpo-rates prior information on finite differencesinvolving the function 8i = f(i) which repre-sents the relationship between the weightand the lag length. In the case at hand,

3Tt = 80 + E 8i(Qdt-4i/POPt-4i)

i=1

3with 8i= 1. It is further assumed that

i=181`82'83, i.e., weights decline over time, or

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at most remain unchanged, which, thereforeimplies that .333 s 81 < 1, 0 < 82 < .5, and 0<836..333. The first difference of the weights

are given as A812 = 61 - 2 and AS2 3 = 82 -

83. The second difference of weights is givenas A2 8123 = 81 - 282 + 83, and it is thislinear combination of parameters that will beconcentrated on. Specifically, we wish toestablish maximum and minimum values forA2 6123that are possible given the constraintsimposed on 81, 82, 83. This problem can bestated in terms of a simple linear program as

MAX or MIN A26 12 3 = 81-282 + 83

s.t. 8 i=l81>82

82>8381, 82, 83 - 0

The problem can be solved to obtain theresults that MIN A2 812 3 = -. 5 when 81 = 82

= .5, and 83 = 0; and MAX A2123 = 1 when

81=1, and 82 = 83 = 0. Thus, -. 5 < A26 1 2 3

<1.0.However, the parameters 81, 82, and 83 are

multiplied by the constant g3 as they appearin the demand equation, and thus priorbounds on g3A2 8123 = g381 - 2g3 82 + g3 83are required. It is suggested that 0 < g3 < 1,which allows for the effects of habit to decayover time, and also implies strict stability ofthe difference equation that is implied by thedemand equation. The stochastic constraintimposed in estimation was specified as

Pr (g38 - 2g3 82 + g383 = .25+.75) = .95

where the prior was assumed to be normallydistributed with point estimate = .25 andvariance = (.375)2. The prior is not con-sidered unduly restrictive, since the intervalreduces as g3 -> 0.

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July 1980

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