The Use and Potential of Optimal ControlModels in Agricultural Economics

David Zilberman

Scientific disciplines are constantly ex-amining and refining the set of analyticaltools they apply in their investigation. Thisprocess of assessment and modification hasimportant effects on the educations of newprofessionals, the requirements from prac-titioners, the selection of research topics, the"real world" applicability and relevance ofresearch topics, and the agenda of the profes-sion.

At the present, I believe that the minimalarsenal of a practicing agricultural economistincludes diagrammatic analysis of outcomesin competitive and monopolistic markets,basic macroeconomic models, linear pro-gramming, and simple regression. Nonlinearprogramming, simultaneous equationeconometric models, and maybe dynamicprogramming are frequently applied toolsand are part of the "mainstream" methods ofthe profession. Optimal control analysis,however, is a relatively new analytical tool;its usefulness is in the midst of a process ofevaluation. The verdict is not out yetwhether it becomes a mainstay of agriculturaleconomics analysis or whether it becomes amarginal tool without widespread use (likegame theory).

This presentation will argue that controltheory models are here to stay; that optimal

David Zilberman is an assistant professor in the Depart-ment of Agricultural and Resource Economics, Universi-ty of California, Berkeley.

Giannini Foundation Paper No. 654 (for identificationonly.) Special thanks to Gordon C. Rausser, MargrietCaswell, and Jeffery La France for their comments andsuggestions. The research leading to this report wassupported by the University of California, Water Re-sources Center, as part of Water Resources CenterProject UCAL-WRC-W-622.

control is a powerful tool which expands therange of issues dealt with by agriculturaleconomists and increases their effectiveness.Therefore, it should become part of themainstream tools of agricultural economics.

After a sketchy description of optimal con-trol models, some of their main applicationsand contributions to agricultural economicswill be summarized, and then extremely pro-mising avenues for future application will bepresented using examples and tentative re-sults.

Optimal Control Models andTheir Use in Economics

Optimal control models describe theevolvement of a system over a time horizonand determine optimal levels of decision var-iables over time. The state of the system atany point in time is characterized by statevariables. Time-dependent variables deter-mined by the decision-maker are control var-iables. The system may also be affected bysome unconrollable random variables.Changes over time in the state variables areaccording to equations of motion which areassumed to be functions of the state vari-ables, control variables, and random vari-ables at the moment of change. The decision-maker's objective in the general case is tomaximize the expected value of the sum overtime of temporal utility functions which arefunctions of time and levels of the control,state, and random variables at each momentof time. Thus, the optimal control problemdetermines the controls that maximize thevalue of the objective function subject to theequations of motion given the initial values ofthe state variables.

There are significant differences in meth-odology and focus of analysis between deter-

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ministic optimal control models (which as-sume that the system is not affected by ran-dom variables) and optimal control modelswith stochastic elements. Deterministic con-trol models are used frequently as analyticaltools applying the maximum principle ofPontryagin et al. that, under reasonable con-ditions, translates the intertemporal opti-mization problem to many temporal optimi-zation problems of the same form and, inessence, extends the comparative staticanalysis to a dynamic framework. It yieldsqualitative results, suggests testable hypoth-eses, and derives relationships that can besolved numerically (many times with dynam-ic programming). Optimal conrol modelswith elements of uncertainty are much morecomplex than the deterministic ones and arerarely used to derive analytical results. Theyare mostly applied to derive numerical solu-tions to empirical problems and are amen-able to various types of sensitivy analyses.Significant methodological research has beenconducted, however, to improve the solutionconcepts and techniques for these models.Particular attention has been given to situa-tions where some of the uncertainty is theresult of lack of information about the truevalues of the parameters of the system. Re-cent solution concepts for these cases involveconstant estimation of the system's parame-ters over time as more information is re-vealed, and the results of the process oflearning are incorporated to derive bettercontrols. The most advanced technique,namely, adaptive control, takes into accountthe expected gains from future learning indetermining optimal control levels. 1

While methodological developments andapplications of optimization models to dy-namic economic systems occurred during the1950s, the introduction of the "maximumprinciple" in the 1960s elevated optimal con-trol to prominence as a research tool ineconomics. It seemed especially promising

'A good exposition of optimal control methodologiesunder certainty and uncertainty with application toagriculture appears in Rausser and Hochman.

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for studies of economic growth and mac-roeconomics.

While the performance of control theory inmacroeconomics has been disappointing[Wallish] and the theory of economic growthhas not fulfilled all the expectations it raised[Hahn and Matthews], these applicationshave produced knowledge and human capitalthat have generated surprisingly high yieldswhen used for other topics. By replacingtime with space as the domain of the statesand control variable, optimal control modelshave been used successfully in urbaneconomics [Mills]. Moreover, optimal con-trol models with human capital index as do-main have been essential in the evolvementof the promising new research on optimalincome tax schemes [Cooter]. Theevolvement of application of optimal controlmodels in economics should teach us a lessonregarding its potential contribution to ag-ricultural economics. It may be that, heretoo, some of the most important contribu-tions may be in analyzing systems that do notnecessarily vary along a time dimension butalong other dimensions.

Applications of Optimal Controlto Capture Dynamic Aspects ofAgricultural Economics

The higher tendency of agriculturaleconomics, relative to other fields of econom-ics to incorporate details of physicalprocesses in its models, may suggest that itoffers relatively more opportunities to applydynamic control models than other fields.Indeed, there are several particular areaswhere the use of dynamic optimal control canimprove (or has improved) significantly thequality of results obtained by agriculturaleconomists. One of these areas is farm andproduction management. The use of dynamiccontrol models in this area makes it possibleto present agricultural production as aprocess of growth, which it is. For example,Yaron et al. used a dynamic framework todetermine optimal irrigation policy for a fieldcrop (wheat). In this model, yield is a func-tion of two state varibles -soil salinity and

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moisture - over the life of the crop. Irriga-tion is a control variable that affects the statevariables and, through them, it affects yield.The model is applied using dynamic pro-gramming, and optimal (profit maximizing)irrigation strategy is selected from a feasibleset.

The usefulness of optimal control is in-creasing as one moves from considering ag-ricultural and resource management prob-lems, which are totally in the private sectordomain, to problems which involve publicpolicies.

A very significant and productive researchhas been conducted applying dynamic opti-mal control models for the management ofrenewable and exhaustible natural resources[Dasgupta and Heal]. Special attention hasbeen paid to optimal management of fisher-ies, forests [Clark], and pesticides [Howittand Rausser]. These applications of optimalcontrol have drastically altered the perspec-tive and improved the quality of answers andsolutions suggested by resource economiststhat have resulted in the revamping of long-held policies and practices of resource man-agement.

While deterministic optimal control mod-els have been more prominent in theeconomic research of natural resources man-agement policies, optimal control modelswith stochastic components have been pro-minent in more traditional agricultural policyapplications.

Stochastic and adaptive control are naturaltools of planning and management for mar-keting boards, producer cooperatives (orcartels), and government agencies adminis-trating commodity programs. The first appli-cation of these techniques for agriculturalpolicies was Gustafson's dynamic pro-gramming model for optimal management ofpublic stocks. Rausser and Hochman pre-sented stochastic dynamic models for optimalmanagement of marketing boards, Freebairnand Rausser used adaptive control for opti-mal determination of beef import quotas, andseveral authors recently developed optimalcontrol models for the management of com-

modity programs [for example, Burt, Koo,and Dudley]. While the use of optimal con-trol for agricultural policy analysis is increas-ing, it is still a limited tool in its applicationand impacts. This lackluster performancecannot be completely explained by the lack oftraining of many of the practitioners of ag-ricultural policy analysis that prevent themfrom using this tool. While this is one factor,there seem to be more substantial reasons.

Because of the complexity of optimal con-trol models with elements of uncertainty,they determine policies for specificparametric values and situations that may betemporary and of limited interest. Thespecificity of the results shifts the emphasis ofmany of the policy-oriented control articlesto the technique they use. This emphasis ontechnique is a turn off to many people whofirst want to learn what they will gain from anew methodology before understanding itsdetails. It seems that current optimal controlmodels in agricultural policy lack a facility forqualitative sensitivity analysis over time thatis able to generate generalized hypothesesand theories the way comparative statics doesin economic modeling of static systems. Re-cent methodologies for comparative dynam-ics developed by Aoki may alleviate thisproblem somewhat.

Moreover, the perceived strength of opti-mal control in agricultural policy is in deter-mining the best policies given the objectivefunction and constraints of the economic sys-tem. The controversies in agriculturaleconomics are mostly with regard to the ex-act objectives of policymakers and the con-straint structure they face. Therefore, thereis a higher premium to models that betterclarify objectives and, in particular, con-straints of the system than to models thatobtain "better" numerical results for systemsbased on shaky ground. Thus, optimal con-trol models will become more prominent inagricultural policy analysis if and when theywill be applied to improve our understandingof the working of the agricultural economyand to overcome essential shortcomings ofstandard models such as badly specified

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aggregates, unsatisfactory modeling ofprocesses of technological change, etc.

Use of Optimal Control When theSystem's Domain is Not Time

As the experience of general economicssuggests, the introduction of optimal controlto analyze dynamic systems may have impor-tant methodological and educational exter-nalities. These models train the user to ana-lyze choice problems where the key variablesare not scalers but functions defined on aninterval. In the dynamic control models, thearray of the system is time; however, thesame technique can be applied when thearray is distance, product quality, or humanability. Thus, optimal control enables theeconomist to analyze problems where theatomistic units of the system have an essen-tial element of heterogeneity.

Agricultural and resource economists haveapplied optimal control models to systemswith domains different than time. For exam-ple, optimal control models were applied todesign policies for management of waterquality and allocate water quantities along ariver basin [Hochman, Pines, and Zilber-man] where distance replaced time as thearray of the system. Nevertheless, it seemsthat the potential of this line of research hashardly been tapped.

Explicit consideration of dimensions ofhetrogeneity among key factors in agricultur-al economic systems have many advantages.First, it expands the range of issues andpolicies addressed by agricultural econo-mists. Second, it overcomes some of thedeficiencies and drawbacks of existing mod-els, and, third, it expresses rigorously argu-ments that have been previously presentedonly heuristically. These points can be illus-trated by the following very simple regionalmodel of irrigation where land quality varia-tions are explicitly considered. 2

Currently, most models of agriculturalproduction consider inputs to be homogen-ous. However, the effectiveness of inputs,2 This model relies heavily on my joint work with Mar-griet Caswell.

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such as fertilizer and water, depends on somemeasures of quality of the lands in which theyare being applied. Some modern applicationtechnologies are introduced to improve theeffectiveness of these inputs on certain typesof lands, and they can be considered quality-augmenting technologies. For example, dripirrigation assists the soil in holding waterand, therefore, improves irrigation effec-tiveness (compared to traditional methods offlooding and farrows) [Caswell]. Differencesof land quality (measured for our purposes bywater-holding capacity) affect the benefits ofdrip irrigation and, thus, the tendency to useit. Thus, regional models of irrigation shouldexplicitly consider land quality differentials.

The Irrigation Example

Consider a region specializing in a cropwhich price is given by P. Let q=f(e) be a peracre production function of a certain cropunder a traditional irrigation technologywhen q denotes output per acre, e is effectivewater per acre (water attains the crop), andf(.) has the standard properties f'>0, f"<O,and also lim f'(e) = oo. Effective water is the

e->0product of applied water per acre x and thequality measure of the land e, e=Ex. Themeasurement of the quality of a certain pieceof land is the share of irrigated water it allowsto attain the crop, thus O0s<1. Let the totalland of the region be denoted by L, and thedensity function of the land-quality distribu-tion be g(s). Thus, for small Ae, the amountof land with quality in the interval ( - As/2,e + Ae/2) is approximated by L g(8)Ae. As-sume that g(E)>0 for O0e<1.

Suppose the region has one source of waterfor irrigation (no rain) and annual supply isgiven by Z. Suppose also that the regionalauthority wants to allocate water to maximizeaggregate profit. The regional optimizationproblem in this case is

1(1) Max P Lf f(£-x) g(g) de

x(e) 0

subject to

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1(2) L f g() x() d = Z.

0

This problem is reformulated to the formof a "textbook" deterministic control problemby introducing a state variable, Z() - theamount of water available to lands that arenot more inferior than e. The equation ofmotion and initial condition associated withZ(e) are

(3)

(4)

Z= -g(e) x(e)L

Z(O)= Z.

(Dotted variables denote derivative with re-spect to the domain variable, in this case,land quality).

Thus, the reformulated control problemconsists of solving (1) subject to (3) and (4).Using the maximum principle, define theHamiltonian

H() = L g(E) [Pf(x- )-X(E) x(E)]

where X(E) is the costate variable of the equa-tion of motion (3). It can be interpreted as theshadow price of irrigated water for land ofquality e. Assume that all the water will beutilized, i.e.,

of water. (Unlike the case of allocation ofresources over time, profits obtained at dif-ferent land qualities are treated equally. The"discount rate" is zero.) Thus, let the shadowprice of irrigated water be denoted by k.Using X(E)= , condition (7) suggests alloca-tion of water for irrigation such that themarginal productivity of irrigation isequalized along land qualities. Since e= x- ,it implies that the ratio of marginal productsof effective water of two land qualities is theinverse of the land quality ratio,

af (X l)de - E2

e (x2'F 2)0e

Equation (8) and the concavity of the pro-duction function imply that lands of higherquality utilize more effective water than low-er quality lands and thus produce more yield.Note, however, that actual irrigation maydecline with higher quality land and the ad-vantage of better land is reflected in lowerwater use as well as higher yield.

Differentiating (7) with respect to landquality yields:

(8)

X(e)(9)X() x(E) -[1 + 'qfe le- 1

(5) Z(1)=0.

The maximum principle suggests that anoptimal solution consists of values of x(E),

X(E), and Z(E), 0<E<1 satisfying equations (3)to (5) and

OZ(6) ~= H )=az 08 <1

and

OH Of(7) - (E) = P f (e).E - X() = 0ax ae

0<E<1.

Condition (6) suggests that differences inland quality does not affect the shadow price

where lf, = f' e/f' is the elasticity of the mar-ginal productivity of effective water, and it isnegative. Thus, higher land qualities requiremore (less) irrigation when the elasticity ofmarginal productivity of effective irrigation(rqfe)is greater (smaller) than -1.

Assuming that farmers in the region areprofit maximizers, the solution to the optimalcontrol problem in (1), (3), and (4) suggests aPareto efficient water-pricing rule and, alter-natively, a Pareto efficient rule for directallocation of water according to land quality.Obviously, water pricing or allocationschemes that are obtained ignoring landquality may be suboptimal or infeasible inthe long run. For example, if decision-makers treat land as an homogenous input

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with quality that equals £, the average quali-ty in the region, then they impose a waterprice of f'(E Z/L). If this price is smaller(greater) than the optimal price derived fromthe control problem, there will be excessdemand (supply) for water. If quantities (notprices) are the allocation tool and homogenei-ty of land is assumed, each acre will receiveZ/L units of water, and there will be a loss ofprofit to the region. The cost of this errordepends on the degree of variability of landquality with the region. An application of asimilar model to evaluate alternative policiesto control waste disposal practices of dairiesfound that a waste disposal rule consideringheterogeneity among dairies can attain theregional environmental standard at a third ofthe cost of a direct regulation based on a falsehomogeneity assumption [Zilberman].

The above optimal control model of irriga-tion can be expanded to incorporate adoptionof land quality augmenting technologies likedrip irrigation. Assume that the annual costof drip is k dollars per acre (independent ofquality)3 . This may be the sum of annualizedinvestment cost and annual setup cost. Theuse of drip irrigation improves water effec-tiveness of lands from e to h(e) where h(O)0,h(8)>e, h'(e)>0, h"(E)O0, and h(1)=1. Lety7() be the extent of adoption of drip irriga-tion for land quality £. Its range is given by

(10) 07y(e) 1.

The Hamiltonian for the new optimizationproblem is

(12) H=g(8)L {y[Pf(h-x)-k]

+ (1 - y) Pf(e.x) - x+ l(1 - y)}

where rq(e) is the shadow price of the con-straint (10). Applying the maximum principleand the Kuhn-Tucker conditions for (12)yields necessary conditions for optimal solu-tions (assuming all the water is utilized):

(13) _H A() =-)=0OZ

Hence, X(e)= WE.

(14) - O (e)= g(e) L-{yPf'(h-x)-hax

+(1 -- ) Pf'(x-e).e- -}= 0

(15) H (e)= g(e) L*{Pf(h x)a'Y

-Pf(e'x) - k - -}<0

(16) aH ()=l--y0Af )

aH 'Y=0,y

(1 - y) = 0.

Using these definitions, the objective func-tion for the expanded irrigation problem is

1(11) Max Lf {y[Pf(xh)- k]

X(E),Y7() 0

+ (1 -y) Pf(x -)} g(E)dE

(For convenience, E was omitted from somevariables that are functions of e.) And theconstraints are equations (3), (4), and (10).

3Fixed cost, k, could have been assumed a function ofland quality. This would have complicated the analysiswithout adding a lot of insights.

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Conditions (14) to (16) suggest that dripirrigation will be adopted fully for land quali-ty E if the gains from adoptions will cover thefixed cost of drip. It will not be adopted at allif the opposite is true, i.e.,

(E) = 1

(17) O<y(e) < 1 if max {Pf(hx)- x}x

-max {Pf(ex)-x-k} = 0x <

7() = O.

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The analysis of the region's equilibrium inCaswell has proven that, given k, P, and Z,drip irrigation is adopted for a medium rangeof land qualities (e, e), while the gains fromthe new technology will not warrant adoptionon the better or worse land qualities.Moreover, increases in output price or re-duction in the fixed cost of the new technolo-gy increases the range of land qualities wheredrip is adopted. In addition, an increase inaggregate water supply increases (decreases)the range of land qualities where drip isadopted if the elasticity of the marginal pro-ductivity or effective water is smaller (larger)than 1 in absolute value.

The effects of changes in k, P, and Z onaggregate profits, its distribution, and aggre-gate adoption (measured in acres) can becomputed using the solution to the regionalcontrol problem. Since these parameters areaffected by actual policy tools (for example,the fixed cost of drip k can be affected by acredit subsidy, extension effort reducing set-up cost, research and development policyimproving effectiveness of drip, etc.), thistype of model extends our ability to analyzesituations of technological change and to de-velop policies to control adoption of newtechnologies.

Two-Stage Optimal Control Problems

The previous sections encourage applyingoptimal control to temporal decision prob-lems with heterogeneous activity units.Many real-life situations involve intertem-poral decisions for systems with heterogene-ous activity units. In many cases, these prob-lems can be solved using optimal control intwo stages. First, temporal optimizationproblems are solved for each point in timeand determine the optimal behavior of activi-ty units and the resulting temporal benefitsas functions of variables that depend on time.In the second stage, the benefits functionsand the dynamics of the time-dependent var-iables are incorporated into a dynamic opti-mal control problem that determines the op-timal path of controllable time-dependentvariables.

To illustrate this two-stage procedure, theirrigation problem mentioned above is ex-tended to be a dynamic optimization prob-lem. Assume that a region has an initial stockof water. This water stock is augmented overtime by rain (in the wet season) and depletedby irrigation (in the dry season). Using twoirrigation technologies (drip-flood) in man-ners described previously, the decision-maker's objective is to maximize the dis-counted aggregate profit of the region overan infinite time horizon. Thus, the regionaloptimization can be formulated as

(18)GO 1

max f e-r t Lf {Jy[P(t)x(e,t),y(e,t) 0 0

f(x h) - k(t)]

+ ( -y) P(t) f(x -)}g(e) dE dt

subject to

1(19) ;(t) = G(t)- x(t, ) g(e) dE

0

0 < /(t, e)~ 1 0<s(t) s(0) =

where r is the discount rate, s(t) is the stockof water at time t, and G(t) is the per periodcontribution of rain to the water stock. Here,the extent of adoption of drip and water useper acre depend on both time and land quali-ty.

The problem in (18) can be solved in twostages. First, solving temporary optimal con-trol problems defined by relations (11), (3),and (4), one derives rr[Z(t), P(t), k(t)] whichdenotes aggregate net profit as a function ofwater consumed, prices of output, and thefixed cost at time t. This aggregate relation-ship replaces the temporary maximand in(18), and Z(t) is introduced to (19) to generatethe second-stage optimal control problem

o00

(20) max f e- rt '[Z(t), P(t), k(t)] dtZ(t) 0

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subject to

;(t) = G(t)- Z(t)

s(O)=s.

The solution to this problem yields theoptimal plan of water allocation as well as theoptimal pricing scheme for water over time.Solving the optimal temporary equililbriumof (10), (3), and (4) for each time period giventhe optimal Z(t), one can derive the optimalwater use pattern and the extent of adoptionof drip irrigation over land quality and overtime. These last results can project the dy-namic path of the diffusion of drip irrigationunder the optimal water-pricing policy.

While the dynamic model introduced inrelations (18) and (19) assumes water quotas(or in a competitive economy assumes waterprices) to be the only decision variables andk(t) and P(t) are given parameters, one canexpand the analysis to make the fixed cost ofdrip irrigation a function of a governmentpolicy, say, government research and exten-sion effort, and use the model to determinethe optimal path of these policies as well aswater pricing over time.

The use of the two-stage optimal controlmodels to first determine temporal optimalbehavior of an heterogeneous population andthen to determine optimal dynamic behaviorcan overcome two shortcomings of main-stream economic models. First, it suggests anew approach to model processes of techno-logical innovation, and, second, it derivesmeaningful aggregates for use in dynamiceconomic models. The irrigation example hasillustrated both points somewhat. Discussingspecific models of diffusion and growth canilluminate them further.

Up to the present, the dominant approachfor modeling diffusion processes of new tech-nologies is the one suggested by Mansfieldwhich considers these processes to be mainlyprocesses of imitation. Recently, however,Mansfield's model has been criticized for itsweak microeconomic foundations, lack of em-phasis on the role of profit maximization be-havior, heterogeneity of the population of

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potential users and dynamic processes likelearning by doing, and cheapening of capitalrelative to labor over time. New models ofdiffusion that try to overcome Mansfield'sshortcomings [Feder and O'Mara; Davies]are similar in essence to the two-stage opti-mal control problem presented above. Thesemodels obtained S-shaped diffusion curvesand other patterns of diffusion processes con-sistent with empirical observation. Moreov-er, by incorporating the assumption of nega-tively sloped demand to these new models oftechnological adoption, one can obtain aquantitative formulation of Cochrane's modelof the "technological treadmill" [Kislev andShchori-Bachrach have made some progressin this direction]. This heuristic model hasbeen essential in depicting the distributionalimpacts of frequent technological changes inU.S. agriculture and in motivating and jus-tifying government policies. Thus, two stagesof optimal control have a lot of potential inmodeling better processes of technologicalchange and generating improved quantita-tive frameworks for policy analysis.

The irrigation example illustrates theusefulness optimal control procedures havein deriving well-defined temporary aggre-gates. One of the justified criticisms of theneoclassical theory of economic growth, in-troduced as part of the "Cambridge Con-troversy" [Harcourt] is the use of an ill-specified aggregate production function andcapital. The "putty clay" approach and vin-tage models are examples of how the use ofoptimal control for populations withhetergenous activity units can remedy thisproblem.

Under reasonable assumptions about thedistributions of the production parametersand properties of the micro production func-tions, Houthakker and Sato have obtainedaggregate relationships of the C.E.S. andCobb-Douglas forms; thus, suggesting asound theoretical explanation to the goodempirical fit of these functional forms andsupplying insights regarding their use andmisuse. Moreover, the putty-clay vintagemodels, which extend the range of issues

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dealth with by models of economic growth,suggest new measures of technologicalchange and supply new policy insights in thestatic [Hochman and Zilberman] and dynam-ic (using two-stage optimal control) settings[Zilberman].

Empirical Considerationsand Conclusions

While one may agree in principle thatoptimal control has immense potential whenapplied to temporary decision problems of aheterogeneous population and when solu-tions of such problems are introduced todynamic optimization problems, he may con-sider wide-scale application of this approachimpractical because of data limitations.

Empirical application of the temporaryanalysis requires estimation of the rules guid-ing the performance of activity units and thecapacity distribution of activity units. Exten-sion to dynamic analysis require estimationor prediction of these parameters over time.The dynamic extension is a lot simpler whenthe distribution of the activity units stay con-stant over time. For example, application ofthe temporal model of irrigation defined byrelations (10), (3), and (4) involves estimatingthe production function of the crop as a func-tion of effective water, estimation of watereffectiveness of drip irrigation as a function ofland quality, and estimation of land distribu-tion according to correctly defined qualitymeasures.

The example indicates the enormity of thedata requirement of the approach advocatedhere. There is evidence, however, that iden-tification of discriminating variables, the pre-diction of their effects on agents' behavior,and the estimation of the distribution ofpopulation according to these variables isfeasible and profitable. Marketing researchstudies, public opinion polls, etc., have datarequirements that crudely resemble the onesrequired by the proposed framework. Obvi-ously such studies obtain the data they need,and the mushrooming of public opinion re-search institutes is a "revealed preference"proof to the profitability of their activities.

The giant production linear programmingmodels, such as the ones developed byHeady at Iowa State University apply (seeHeady and Srivastava), in essence, an ap-proach closely relaed to the one suggestedhere, and the increase in detail and size ofdata bases used for these models should be asource for optimism.

Nevertheless, lack of data is still the maindeterrent to the widespread application oftemporal control models for heterogeneouspopulations and the continued reliance onneoclassical models assuming homogeneity ofpopulation. It seems, however, that thesedata problems reflect constraints of the pastthat will be removed in the future, as data-handling technologies are improving. Thisassessment follows the casual observationthat specification of models in economics anddata availability are determined more or lesssimultaneously, given data-processing tech-nology. Most of the empirical economic mod-els and data basis we use reflect the data-processing technology of the 1960s and1970s. Namely, they reflect the availability oflarge and fast computers with strong comput-ing capabilities, relatively expensive datacollection and storage cost, and almost nointercomputer communication. The currentproliferation of microcomputers, establish-ment of communication networks, and thecheapening of storage costs suggest that theapplication of more data-intensive analyticalmethods will become easier and cheaper; andtemporary optimal control models will bene-fit from this perceived reality.

This presentation has demonstrated thatoptimal control has made many inroads inagricultural economics. In some areas (not-ably, resource economics), it became a main-stream tool of analysis. In other areas, it isstill peripheral in use. In the areas whereoptimal control has risen to prominence, itsapplication has made a quantal difference byallowing analysis of new topics and by drasti-cally altering basic theories and beliefs. It hasbeen suggested that optimal control hasenormous potential in other areas, if it will beapplied to explicitly address the impacts of

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heterogeneity of activity units. This will al-low better understanding (and prediction) ofreality, addressing issues of equity and dis-tribution, and generate meaningful ag-greagtes over time. While in the past datarestrictions have prevented such applica-tions, they seem to become less of a problemas time goes on, making this application ofoptimal control promising and feasible.

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