A Modified Partial Adjustment Model ofAggregate U.S. Agricultural Supply

Jeffrey T. LaFrance and Oscar R. Burt

Aggregate U.S. agricultural supply response is modeled through a modified partial adjust-

ment model, where the effects of weather and other temporal stochastic effects are structured

to be purely static, while the effects of price and technology, or trend, are dynamic. The model

is applied to a time series of aggregate U.S. farm output, aggregate U.S. crop production, and

aggregate U.S. livestock and livestock products production for several sample periods within

the period 1911-1958. The three aggregate output indexes are tested for irreversibilities in

supply response, and no evidence of a definitive irreversible supply function is found for any

of the dynamic supply models. The use of a nonstochastic difference equation to model the

aggregate farm output and crop production equations results in short-run elasticity estimates

that are somewhat smaller than previous studies suggest while the long-run elasticities are

somewhat larger.

Although there are serious limitations toan aggregate measure of agricultural out-put in a supply response framework, theconcept is frequently used in agriculturaleconomics, and a few attempts have beenmade to empirically estimate equationsfrom time series data (Griliches; Tweetenand Quance). This paper examines somealternative specifications from the basicNerlove model used by Griliches and theconsequences to empirical estimates ofsupply elasticities. The various specifica-tions focus on the way in which stochasticcomponents of a dynamic regressionequation are treated, and they have im-plications for time series estimation ofsupply response equations for individualfarm commodities as well as aggregate in-dices.

Nerlove (1956, 1958) presented the fol-lowing output adjustment model and ap-plied it to corn, cotton and wheat produc-tion in the United States. Consider an

Jeffrey LaFrance is Assistant Professor and Oscar Burtis Professor in the Department of Agricultural Eco-nomics and Economics at Montana State University.

Contribution of Montana Agricultural ExperimentStation, Journal Series No. 1362.

individual producer with output level Ytin period t and a "desired" or "long-runnormal" level of output Y*t which is afunction of price Pt and technology Tt,

Y*t = a + oP + P Tt. (1)

Because of adjustment costs and fixed as-sets, the output adjustment achieved in anyperiod is assumed to be a (constant) frac-tion of the difference between the desiredoutput and the previous period's output,

(2)Yt - Y = 6(Y*t - Yt,1)0< 6 < 1.

Direct substitution for Y*t in (2) and solv-ing for Yt gives the dynamic supply equa-tion

(3)Yt = ab + 1 6Pt + 06Tt + (1 - 5)Yt-1 .

The usual estimation procedure is to adda disturbance term to (3) and apply themodel to aggregate data.

The first attempt to empirically esti-mate a dynamic aggregate U.S. agricul-tural supply function was Griliches' studyin which he applied ordinary least squaresto a structural equation of the form

Western Journal of Agricultural Economics, 8(1), 1983, pp. 1-12

Western Journal of Agricultural Economics

In Y, = a + b In Pt + c In W, (4)+ d In Yt-_ + gt + ut,

where Yt is the index of farm output; Pt isthe March index of prices received for allfarm products deflated by the March in-dex of prices paid for production items,farm wages, taxes, and interest; Wt isStallings' index for the effects of weatheron farm output; ut is a random distur-bance, and a, b, c, d and g are constants.Griliches analyzed aggregate farm outputand two subaggregates: all crops, and live-stock and livestock products.'

This model can be interpreted as amodified partial adjustment model inwhich the adjustment is linear in loga-rithms. The use of a linear trend in a log-arithmic relationship to approximatetechnological change can be interpretedas a tacit assumption that the supply curvehas shifted to the right at a constant (com-pounded annually) percentage rate.

Two aspects of the approach of Grilich-es warrant closer attention. First, the im-plied adjustment equation is in actual out-put rather than planned or expectedoutput, i.e., the level of output that wouldprevail given average conditions on thevariables not subject to control by farmers.Griliches (p. 291) discussed the problemas follows:

Measured output is not necessarily equal toplanned output, due to 'weather' and otherrandom effects. This . . . factor would leadto a downward bias in the estimate of thecoefficient of lagged output since the ad-justment assumed by the model proceedsfrom the previously 'planned' output, ofwhich actual output is not an error-freemeasure.

Because of the dependence of ultimatefarm output upon weather and other ran-dom factors, it appears that farmers canat best plan for an expected level of out-put, given an average season in terms of

1The price variable for the livestock model is theprevious year's average annual price index, ratherthan March of the current year.

2

these uncontrollable factors. Such an "ex-pected" output level is a conceptual,unobservable variable, and the relevantproblem is how to specify an empiricallyestimable adjustment equation in thisvariable.

The second characteristic of the Gril-iches model that must be considered is thefact that the introduction of the laggeddependent variable on the right hand sideof the regression equation, without fur-ther restrictions on the exogenous vari-ables, imposes the same geometric lag pat-tern on all exogenous variables. While thisis inconsequential if the only regressors areprice and linear trend, we would expectthat this period's weather would tend tohave only a contemporaneous effect. Giv-en this assumption the dynamic effects ofthe weather index should be eliminated.

The third problem arises when ordi-nary least-squares regression techniquesare applied to a time-series model withthe lagged dependent variable included asone of the predetermined explanatoryvariables. It is well-known that ordinaryleast-squares estimates obtained frommodels with lagged dependent variablesand serially correlated errors result inbiased and inconsistent estimates. In anytime-series model, we would expect theresiduals to be autocorrelated because ofleft out explanatory variables that are cor-related over time. An approach to han-dling these last two problems is developedbelow where it will be seen that they areboth part of the same overall problem ofdynamic specification.

It is our opinion that rigid behavioralhypotheses about producer behavior areat best crude approximations to aggregatebehavior, especially when posited in theusual linear form with aspirations for em-pirical testing. A priori reasoning pro-duces several equally plausible models,and the data base is incapable of discern-ing the correct model with a high proba-bility, even if it were within the set tested.Thus, the approach adopted here is to use

July 1983

A Model of U.S. Agricultural Supply

the rational distributed lag model as arather general additive approximation tothe unknown structure of the aggregatesupply function, without formally speci-fying a behavioral model and deriving asystem of product supply and demandequations.

Alternative Specifications

One question raised in the previous sec-tion is whether the adjustment process isin actual quantities produced or in whatwould be "average" quantities associatedwith the period. Lagged output under av-erage conditions for random effects onoutput would be preferable to Yt_1 in (2),implying that Yt-, in the right-hand sideof (3) is an independent variable subjectto measurement errors. Note that"planned" output would differ from"long-run normal" output as a result ofadjustment costs and fixed assets, and adynamic adjustment equation is still war-ranted.

A second issue raised above is the ques-tion of purging the effects of a weathervariable from the dynamic aspects of sup-ply response. Assuming that we have anindex for the effects of weather, Wt, whichenters additively with parameter y, thenby defining the variable Yt = Yt - yWt, wecan rewrite the adjustment equation (2) interms of output net of weather effects

Yt - Yt_- = 6(Y*t- Yt1 ). (2a)

The dynamic supply equation (3) then be-comes

Yt = abt + 6Pt, + yW, + 0bT, (3a)

+ (1 - 6)(Yt-l - TWt-).

The weather variable in (3a) enters with-out any distributed lag response; a simpleproof is by induction. Clearly, Yt/dWt =3y, and

(5)dYt/dWt_ = (1 - b)[OY,t_ 1/W,_- -]

= (1 - )[y - y] = 0.

Since dYt/OWt_1 = 0, it follows inductivelythat dYt/dWt_j = 0 for j > 1, so that elim-ination of weather effects from the partialadjustment equation simultaneously re-moves the weather variable from any dis-tributed lag response in the derivedregression equation (3a).

An operational statistical model is ob-tained by adding a disturbance term to(3a), but the equation is nonlinear in pa-rameters. 2 Practical estimation could bewith nonlinear least squares or a condi-tional linear least-squares search on 6.

There are many sources of randomvariation other than weather effects whichlead to a disturbance term in (3), includ-ing aggregation over individual farms,omitted variables, and other minor speci-fication errors. If the disturbance termadded to (3) or (3a) to get a statisticalmodel is dominated by factors that do notreflect changes in output capacity, thenthe lagged dependent variable would bebetter defined as the lagged expectationof the regression equation. This statisticalmodel for (3a) would be

Yt = ab + /36P + yW, + 08Tt

+ (1 - 6)(,t-_ - 7Wt-) + ut, (6)

where mt = E(Y I W,) and Y, = rt + u,. Thepartial adjustment model associated with(6) is

lit li -q- = (Y*t - it 1)' (2b)

where it = 7t - 'Wt.The three versions of partial adjust-

ment, (2), (2a), and (2b), represent differ-ent numbers of random componentspurged from the measure of output usedas the dependent variable in a dynamicregression equation. Nothing is removedin (2); weather effects are purged in (2a);while in (2b), both weather effects and theentire disturbance term of the regressionequation are purged.

A priori reasoning suggests going at least

2 There is, however, sufficient identification to esti-mate a, i, and 0 from a linear regression by com-bining estimates of 6, a' = a6, d' = /6, and 0' = 06.

3

LaFrance and Burt

Western Journal of Agricultural Economics

as far as (2a) by purging weather effectsfrom output, but moving all the way to(2b) might remove some components thatindirectly reflect the fixed assets and man-agement inertia which are important inspecification of the partial adjustment hy-pothesis. The essential question is just whatthe disturbance term comprises, and itmust be recognized that this term is a con-struction of the statistician with many va-garies as to its actual source.

In an aggregate supply equation, wewould expect much of the disturbanceterm to stem from aggregation problemsin the dependent variable and from thesingle aggregate price index used to sub-sume all individual commodity prices. Itmakes sense to purge the former com-ponents, but not the latter from the partialadjustment equation. In a sense, the lim-itations of using a single price index canbe viewed as specification error, given thedefinition of the dependent variable, inthat a more complete vector of priceswould better explain aggregate output.Some of our empirical results suggest thisto be the case.

If ut in (6) has the classical properties,then replacing rnt with Yt_ would pro-duce a disturbance following a first-ordermoving average process, u, - (1 - )ut_.Ordinary least squares estimates wouldthen produce biased and inconsistent es-timators.3 The disturbance term is likelyto be autocorrelated in any specification,but there is no a priori reason to supposethat it will be restricted to a first-orderMarkov process with parameter -(1 - 6).The structure of dynamic models must bediscovered from the data with the help ofeconomic theory in most applications, andin these situations there is a distinct ad-vantage in not having the lagged depen-dent variable serving as an independent

3 Of course, a search over 6 with the structure of theerror term accounted for, using a criterion of con-ditional OLS given 6 produces asymptotic maxi-mum likelihood estimators which are the same asnonlinear LS estimators from (6).

variable in the regression model. If thedisturbance term in (6) is misspecified, andthe remainder of the equation is correctlyspecified, least-squares estimates are stillconsistent, which is not the case when rnis replaced by Y,_.

Least-squares estimates of the unknownparameters in (6) can be calculated withnonlinear least-squares algorithms bytreating the {it} as unobservable variables,but nevertheless, as subject to least-squaresestimation because each is implicitly afunction of the parameters (ac, 3, ', 0, 6)plus the initial condition parameter 70. Thislatter parameter can be given an a prioriestimate of Y0.4 The computational algo-rithm used in this study is a modified Mar-quardt nonlinear least-squares routinemuch the same as that commonly used intime-series packages to deal with a mov-ing average error process. The estimationprocedure yields estimates that areasymptotically equivalent to maximumlikelihood under normality, and with oneindependent variable is equivalent to themethod of Maddala and Rao except forthe handling of initial conditions. Addi-tional details on the practical estimationof models of this type in (6) are given inBurt.

Another advantage in using (6) is thesimplicity with which general distributedlags on the independent variables can beapproximated by the rational lag modeland superimposed on the partial adjust-ment model of (2).5 Let L be the lag op-erator where LXt = X,1_ and

4 A Monte Carlo study by Schmidt found that usingY0 for ?0o was about as efficient as estimating r?0 si-multaneously with the other parameters.

5 Technology is ignored momentarily to simplify thediscussion. A separate lag structure on Tt could beincluded in the same fashion as for price. If T, isapproximated by a linear trend as in most empiricalmodels, the dynamic specification is of no conse-quence except for the small effects of initial con-ditions. This assertion follows from the specialnature of the linear trend and the linear lag oper-ator L.

4

July 1983

A Model of U.S. Agricultural Supply

3(L) = / 0 + 31L + . + kLk

o(L)= 1 - oL -... L

Dropping the technology variable for sim-plicity, (1) is generalized to

Y*t = a + f(L)Pt/w(L) (7)

which on substitution into the partial ad-justment equation (2b) in "expected" out-put levels, net of weather effects, yields

t = ab. + i(L)6Pt /w(L) + -yWt (8)+ (1 -6)(t-1 - yWt-1):

Multiplying by o(L) and rearrangingyields

(9)t = ' + ±(L)6P, + y/Wt + . . + tnWt-

+ X1Tt- 1 + 1 _ _ + Xnnt-n,

where n = m + 1 and the {^y} and {X} arefunctions of 6, y, the {1j}, and the {aw}.Although it is not obvious in (9) that thereare no dynamic weather effects, an induc-tive proof like that given below (3a) isstraightforward.

The Data and Some Limitations

In view of the obvious limitations of us-ing aggregate output and price indices,the statistical models used here must berecognized as rather crude and subject toconsiderable specification error. Thisproblem is compounded by various gov-ernment programs designed to alter agri-cultural production and income which aretoo numerous and varied to handle withconcomitant variables in the regressionequation. Rapid technological changeduring the post World War II period alsogreatly complicates modeling supply re-sponse when a smooth trend is used tomeasure the effects of technology.

Output is measured by the USDA indexof farm output for the aggregate farmoutput model, the index of output of allcrops for the all crops model, and the in-dex of output of livestock and livestockproducts for the livestock model. The out-

put price variables used are the Marchprice index for all farm output, the Marchprice index for all crops, and the annualprice index for livestock and livestockproducts lagged one year. 6 The aggregateoutput price and the all-crops price weredeflated by the annual index of prices paidfor production items, farm wage rates, in-terest and taxes lagged one year. The live-stock price was deflated by the annual av-erage price paid by farmers for feed,lagged one year.7 The weather variable isStallings' index for the appropriate aggre-gate index, a series available for the years1900-57. The only serious attempt to up-date the deflator was by Kost, but the ex-tension was only through 1962.

Although the data are available back to1910, our results suggested that there wasprobably a serious deficiency in theweather index somewhere before 1913.This conclusion was reached on the basisof model sensitivity to inclusion of theseearliest three years in the sample with andwithout the weather index as an explan-atory variable. Specifications emanatingfrom (6) were particularly helpful in thisregard because no can be estimated as aparameter and compared to Y0 as an apriori approximation to ro. This is essen-tially a least squares "backcast" to checkfor specification error.

The importance of weather on agricul-tural output, the considerations in theabove paragraph, the problems of gov-ernment programs, and a surge in tech-nological change in the 1950s lead us tolean heavily on the sample period 1914-51, using 1913 as an initial condition ob-

6 The choice of periods for prices was nearly the sameas that used by Griliches; the only exception wasthat we used a lagged annual deflator on Marchprice instead of the current March deflator.

7 Data for 1910-56 are summarized in USDA Agri-culture Handbook No. 118, Vol. 1, 1957. Samplemeans of the variables for 1914-51 are: weatherindex .996, trend 32.5, aggregate price .957, cropprice .943, livestock/feed price 1.113, total output.568, and crop output .644.

5

LaFrance and Burt

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July 1983 Western Journal of Agricultural Economics

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A Model of U.S. Agricultural Supply

servation. We also estimated the sampleperiod 1921-51 and 1921-57 for compar-ison with Griliches' results and because ofsome question about the accuracy of out-put and price indices prior to 1920.

Results

Aggregate Output Response

Results of fitting several first order dif-ference equations to aggregate farm out-put are given in Table 1 (numbers in pa-rentheses are asymptotic standard errors).The two sample periods for each equationare paired, one above the other, with ana-b designation on the equation numberto denote the periods beginning in 1914and 1921. The model designation Yt_, ver-sus E(Yt_,) refers to the assumptions asso-ciated with partial adjustment equation(2a) and (2b), respectively. Reported elas-ticities are at sample means, and adjustedR-square includes the explanatory valueof the autoregressive error structure.

Equations using E(Yt_,) as the laggedoutput measure were estimated withE(Yo) = mo treated as an additional param-eter, but the estimate is not reported. Theestimate of r0 was always very close to Y0,and the first observation was saved in first-order autoregressive error specificationsbecause u0 = Yo - 0 provides an initialcondition for the error process.

In equations 1.3 and 1.4 in Table 1, weuse an index of total farm output for thedependent variable and separate price in-dices for the crop and livestock sector re-sponses. The improved fit, as measured byadjusted R-square, and the reduced stan-dard errors on the regression coefficientsimply that separate price indices reducethe specification error associated with theweakness of a single aggregate price in-dex. The better fit of equations 1.4 com-pared to 1.3 suggests that the partial ad-justment equation (2b) is superior to (2a)when the regression equation is well spec-ified.

Coefficients estimated for each separateprice index must be interpreted as partialeffects on the total aggregate output in-dex, holding the other price index con-stant. Reported elasticities for equations1.3 and 1.4 were computed from an ag-gregation formula as follows. Two mainassumptions were used to derive the for-mula: (1) the index of prices paid forfeed used to deflate the livestock price in-dex can be interpreted as an exogenousinput price, and (2) the net effect on totaloutput of each separate price index is re-stricted to its own component of output.8

By definition of the index of prices re-ceived for all farm products,

Pt= wiPct + W2PLt,WI + W2 = 1,

where Pt is the index of prices receivedfor all farm products in year t, Pct is theindex of prices received for all crops, PLtis the index of prices received for livestockand livestock products, w1 is the relativeimportance of crops, and w2 is the relativeimportance of livestock and livestockproducts during a given weight-base pe-riod for the constructon of the aggregateprices received index. Feed consumed iscalculated as a fixed proportion of live-stock production (USDA, Handbook No.118, Vol. 2, p. 34), and seed crops are cal-culated as a percentage of the average val-ue of all crops during a given weight-baseperiod. Therefore, total output is definedas

Qt = VQct + V2QLt,O < V1, v2 < 1,

where Qt is total farm output in year t,QCt is all crop production, Qt is total live-stock production, vI is the percentage of

8 The logical basis for calculating a single price elas-ticity is compromised somewhat by the prices paidfor feed index appearing in the livestock price vari-able, creating a negative correlation between thetwo price variables. This negative correlation willtend to bias the elasticity upward when using theformula derived here.

7

LaFrance and Burt

Western Journal of Agricultural Economics

all crop production not used for seed, andv2 is the percentage of livestock produc-tion net of feed consumed.

For aggregated indices not usually muchdifferent in magnitude, it seems reason-able to assume that for small changes inthe price indices,

anddQt/Pct - aQct/OPct

aQt/aPLt - QLt/OPLt,

allowing the approximation

aQt/Pt - V13 1 + V2f 2,

where i31 and /2 are coefficients on the cropand livestock price indices, respectively.This leads to an estimated aggregate priceelasticity of

e - (V1 31 + V2 t 2 )(WlPct + w2 PLt)/Qt.

The weights used to calculate the elastic-ities in Table 1 are based on 1937-41 and1935-39 for prices and quantities, respec-tively. The actual weights are w, = .422,w2 = .578, v, = .9935, and v 2 = .4018.

In an attempt to delineate an upperbound on the long-run price elasticity,equation 1.5 in Table 1 was estimated withtrend omitted, which imputes the effectsof technological change to price effects.Since this is a two-price equation, the elas-ticity has an additional upward bias fromthe aggregation procedure (see footnote8).

The impact of the error specification isillustrated by a log-linear equation for1921-57 (basically Griliches' equation 1.5) 9

which are equations 1.6 and 1.7 in Table1 with and without an autoregressive error.The long-run elasticity estimate increasedfrom .20 to .35 by introducing the auto-regressive error.

Many of the same equations were esti-mated with the variables in logarithms ex-

9 Griliches' equation estimated by OLS did not ex-clude the weather index from the distributed lagand resulted in an even smaller estimate of the coef-ficient on Yt_-, i.e., 0.298, and a smaller long-runelasticity of 0.14.

8

cept for trend, but the linear models con-sistently gave a better fit, and theparameter estimates were more stableacross sample periods. We also tried anexponential trend in the linear models, butit did not improve the fit or change theparameter estimates much except forsamples going through 1957.

Several rational lag specifications weretried on the price variable jointly with thepartial adjustment equation. For example,a partial adjustment/adaptive-expecta-tions model was considered quite plausi-ble on a priori grounds. A second-orderdifference equation resulted in a margin-ally significant coefficient on E(yt-2), butthe significance was traced to observationsfor 1934 and 1936, which are the worstdrought years in the series; consequently,the second order model was not given se-rious credence. Aggregation over all pricesprobably removes the opportunity tomodel any distributed lag price expecta-tions, even if they are important for someindividual commodities.

The primary differences in the specifi-cations presented here and those of Gril-iches are: (1) the deflator for March price,(2) exclusion of weather from a distrib-uted lag response, (3) a first order auto-regressive error structure, and (4) linearinstead of log-linear equations.

Aggregate Crop Response

Results for the crops index are quitesimilar to aggregate output with approx-imately the same long-run price elastici-ties as the single price models in Table 1.Several first order difference equations forthe 1914-51 period are given in Table 2.

Equations 2.1 and 2.2 illustrate the sen-sitivity of the lag structure to specificationof the error term when the lagged depen-dent variable is a regressor. Equations 2.1and 2.3 yield almost identical long-runelasticities, and there is little basis forchoosing between the stochastic versusnonstochastic difference equation models.

July 1983

A Model of U.S. Agricultural Supply

TABLE 2. Estimated Equations for Aggregate Crops Response (1914-51).

Short- Long-Equa- Auto- Run Runtion Crop Linear Lagged regressive Elas- Elas-No. Model Weathera Price Trend Output Error R2 ticity ticity

2.1 Yt_1 .214 .066 .0023 .669 -. 583 .872 .10 .29(.031) (.017) (.0006) (.098) (.157)

2.2 Yt,1 .234 .097 .0043 .337 .845 .14 .22(.043) (.023) (.0009) (.130)

2.3 E(Yt-1) .231 .055 .0019 .739 -. 049 .862 .08 .31(.044) (.016) (.0005) (.095) (.162)

2.4 E(Yt-1) .371 .077 .0027 .730 -.049 .842 .08 .28and logsb (.069) (.024) (.0008) (.102) (.162)

Note: Numbers in parentheses are asymptotic standard errors.a The weather variable is excluded from the distributed lag in all equations.b All variables in natural logarithms except a linear trend.

As in Table 1 for aggregate output, theformer tends to have a smaller coefficienton the measure of lagged response. Equa-tion 2.4, which is linear in logarithms ofall variables except trend, has nearly thesame dynamic structure as the linearmodel in 2.3, but the fit is not as good.

Aggregate Livestock Response

Since the weather index is unimportantin livestock output, it might be expectedthat our results would be close to those inGriliches' study because much of the dis-crepancy found in the aggregate outputand crops models seems to be due to dy-namic specification on the weather index.But another important aspect is specifi-cation of the structure of the disturbanceterm; recall the estimated negative first-order autoregressive structure for thelagged dependent variable equations inTables 1 and 2.

In the livestock equation the autore-gressive parameter estimate is positive, andwith the lagged dependent variable in theequation, it is not significant for samplestruncated before about 1960. The equa-tions estimated with the lagged expecta-tion of the dependent variable suggesteda stochastic difference equation with asmall positive autoregressive error param-

eter for samples within the period 1911-58. A recent Monte Carlo study of modelswith a "trended" variable and lagged de-pendent variable as regressors suggests thatordinary least squares is superior to gen-eralized least squares when the autore-gressive error parameter is between zeroand 0.5 [Maeshiro].

We concluded that little improvementcould be made over Griliches' estimatesof aggregate livestock response during1911-58. Attempts to estimate a relation-ship for the post World War II period werediscouraging. Regression coefficients ontrend and the lagged dependent variablewere too confounded to draw conclusionsabout long-run price response in recentyears. Apparently, technological changehas been too sporadic to capture with asmooth trend. Although we do not reportany equations here, the most consistent es-timates in our analysis across variousmodels were for the period 1921-51 witha short-run elasticity of 0.3 and with 0.7to 0.8 for the long-run. 10

Irreversibility Tests

The Wolffram technique for estimatingirreversible functions in time series, which

10 Detailed results including the fitted equations aregiven in LaFrance.

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LaFrance and Burt

Western Journal of Agricultural Economics

TABLE 3. Comparison of Griliches' Price Elasticities with the New Approach.

Griliches LaFrance and Burt

Short-Run Long-Run Short-Run Long-RunOutput Aggregate Elasticity Elasticity Elasticity Elasticity

Total Output 0.10 0.15 0.08 0.3 ± 0.10aAll Crops 0.16 0.23 0.08 0.28 + 0.13bLivestock and Products 0.20 0.70 0.30 0.70-0.80

a Taking 0.30 as a basic "unbiased" estimate of the long-run elasticity, this is an approximate 95% confidenceinterval based on asymptotic distribution theory and the two-price model.

b This is an approximate 95% confidence interval based on asymptotic distribution theory and equation 2.4.

was later simplified by Houck for its op-erational application, was applied to thethree output indices. In all cases, the trendvariable was confounded with the twoprice variables, one each for increasing anddecreasing prices. A positive trend in thereversible specification changed to a neg-ative trend when the single price was re-placed by the two separate price seriesto allow for irreversibilities. Adjusted R-square was essentially the same in theirreversibility specification with trend de-leted as in the reversible equation withtrend included. These results suggest thatthe two variables for rising and fallingprices are almost a perfect substitute fora single price variable jointly with lineartrend.

The model developed by Traill, Cole-man and Young for irreversible supply re-sponse was also tested, but their methodturned out to be rather infeasible becausethe absolute maximum price appears veryearly in the data for the total output andcrops indices. This confounded their"maximum price" variable with the in-tercept. We tried to remedy this by sub-jecting the previously experienced maxi-mum price to an exponential decay,

PtXj, 0 <X < 1,

where Pt is price in period t and t + j isthe period in which a maximum with re-spect to t is sought for the variable PtXj. Asearch was made on X using a conditionalleast squares criterion, but the revised

10

method did not improve the earlier re-sults.

We concluded that there is little evi-dence of irreversibilities in aggregate sup-ply response insofar as current economet-ric methodology can detect. Althoughthese methods did not reveal a definitiveirreversible supply function, we suspectthe outcome is a weakness in the methods.The simple deductive economic argumentfor irreversibilities in agricultural supply,especially in the aggregate, is most con-vincing (Johnson and Pasour).

Conclusions

A comparison of Griliches' elasticity es-timates with those obtained in this studyis presented in Table 3. The refinementsin specification of partial-adjustmentequations for supply response tend to pro-duce higher long-run and lower short-runprice elasticities than a straightforward useof the lagged output variable, as in Gril-iches' model. However, the differences inelasticities are small when an autoregres-sive error is specified in the equations con-taining the lagged dependent variable andthe weather variable is excluded from thedistributed lag. Differences between thelinear and log-linear specifications ac-count for some of the discrepancy in es-timated elasticities; our logarithmic equa-tions gave long-run elasticities about 20percent below the same linear equations.

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A Model of U.S. Agricultural Supply

With the caveat that an average priceelasticity over the entire domain of a sup-ply equation is at best a crude indicatorof supply response behavior, our resultssuggest point estimates of .1 and .3 in theshort-run and long-run, respectively, foreach total output and crops indices overthe period 1914-51. We could not im-prove upon Griliches' estimates of priceelasticities for livestock and livestockproducts which were .2 and .7 in the short-run and long-run, respectively.

The lagged expectation measure of out-put yields an error term which appears toobey the classic assumptions for the cropindex, while actual lagged output givessuch an error term for the livestock andlivestock products index. These results areconsistent with lagged output being an in-direct estimate of the investment in fixedassets specialized to crop production whileit is more of a direct estimate of such as-sets used in livestock production. Thedominant role of breeding stock in live-stock production (particularly beef) sug-gests that lagged output would be a moredirect measure of fixed assets in livestockproduction than in crops production. Wenote that the livestock output index is forproduction, not marketings, and reflectschanges in inventories. These results sub-stantiate Griliches' assertion that the par-tial adjustment model he used employeda lagged output variable for the partialadjustment equation which was subject tomeasurement error, at least in the aggre-gate output and all crops models.

Partitioning various terms in the dy-namic regression equation out of the mea-sure used for lagged output should be use-ful in modeling supply response ofindividual commodities. The specificationof (2b) where the regression disturbanceterm is purged from the lagged depen-dent variable appears to be most appro-priate in well specified models, which aremore likely for individual commoditiesthan for an aggregate index of output.

References

Burt, Oscar R. "Estimation of Distributed Lags asNonstochastic Difference Equations." Staff Paper801, Economics Department, Montana State Uni-versity, 1980.

Griliches, Zvi. "Estimates of the Aggregate U.S. FarmSupply Function." Journal of Farm Economics,42(1960): 282-93.

Houck, James P. "An Approach to Specifying andEstimating Nonreversible Functions." AmericanJournal of Agricultural Economics, 59(1977): 570-72.

Johnson, Marc, A. and E. C. Pasour, Jr. "An Oppor-tunity Cost View of Fixed Asset Theory and theOverproduction Trap." American Journal of Ag-ricultural Economics, 63(1981): 1-7.

Kost, William E. "Weather Indexes, 1950-64:"Quarterly Bulletin of Michigan Agricultural Ex-periment Station, 47(Aug. 1964): 18-42.

LaFrance, Jeffrey T. "Estimating the Aggregate U.S.Agricultural Supply Function." M.S. Thesis, Mon-tana University, 1979.

Maddala, G. S. and A. S. Rao. "Maximum LikelihoodEstimation of Solow's and Jorgenson's DistributedLag Models." Review of Economics and Statistics,53(1971): 80-88.

Maeshiro, Asatoshi. "Small Sample Properties of Es-timators of Distributed Lag Models." Internation-al Economic Review, 21(1980): 721-33.

Nerlove, Marc. "Estimates of the Elasticities of Sup-ply of Selected Agricultural Commodities." Jour-nal of Farm Economics, 38(1956): 496-509.

Nerlove, Marc. The Dynamics of Supply: Estima-tion of Farmers' Response to Price. Baltimore:Johns Hopkins Press, 1958.

Stallings, James L. "Weather Indexes." Journal ofFarm Economics, 42(1960): 180-87.

Schmidt, Peter. "The Small Sample Effects of Var-ious Treatments of Truncated Remainders in theEstimation of Distributed Lag Models." Review bfEconomics and Statistics, 57(1975): 387-89.

Traill, Bruce, David Coleman, and Trevor Young."Estimating Irreversible Supply Functions."American Journal of Agricultural Economics,60(1978): 528-31.

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LaFrance and Burt

Western Journal of Agricultural Economics

Tweeten, Luther G. and C. Leroy Quance. "Positiv-istic Measures of Aggregate Supply Elasticities."American Journal of Agricultural Economics,51(1969): 342-52.

United States Department of Agriculture, Major Sta-tistical Series of USDA, Agricultural Handbook No.118, Vol. 1 and 2, 1957.

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July 1983