estimation of moments and production decisions under uncertainty

8/8/2019 Estimation of Moments and Production Decisions Under Uncertainty

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Estimation of Moments and Production Decisions Under UncertaintyAuthor(s): Elie Appelbaum and Aman UllahSource: The Review of Economics and Statistics, Vol. 79, No. 4 (Nov., 1997), pp. 631-637Published by: The MIT PressStable URL: http://www.jstor.org/stable/2951415

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ESTIMATION OF MOMENTS AND PRODUCTION DECISIONSUNDER UNCERTAINTYElie AppelbaumandAmanUllah*

Abstract-The purpose of this paper is to examine production decisionsunder output price uncertainty. Using a nonparametric estimation tech-nique to estimate the first four moments of the unknown price distributionand applying duality, we provide a simple empirical framework for theanalysis of supply and demand decisions under price uncertainty. Themodel is used to examine the importance of higher moments in the firm'sproduction decisions and to investigate underlying attitudes towardrisk.

I. Introductionr HEREexists a vast theoretical literatureon the effects of- uncertaintyon firm behavior.The most common assump-tion in the literature s thatfirms maximize a von Neumann-Morgenstern utility function.' Hence, for example, if theprice of output is unknown, the whole price distributionplays a role in the firm's decisions. Consequently, allmoments of the price distribution could be important inproduction decisions. There are also numerous studies thatuse a mean-variance framework, where only the first twomoments of the distribution play a role.2 Unfortunately,while the theoretical literatureon production under uncer-tainty is vast, there are very few empirical studies applyingthese models.3 In particular,given the difficulties in obtain-ing estimates of varying higher moments, the importanceofhigher moments has not been examined empirically.The purpose of this paper is to examine productiondecisions under output price uncertainty.Using a nonpara-metric estimation technique to estimate the first four mo-ments of the price distribution and applying duality, weprovide a simple empiricalframeworkfor the analysis of theeffects of uncertaintyon productiondecisions. The model isused to examine the importance of higher moments inproduction decisions and to study underlying attitudestowardrisk.

Applying the model to the U.S. printing and publishingand the stone, clay, and glass industries, we find that highermoments play a significant role in determining input andoutput decisions. Specifically, we test for and reject riskneutrality. We find that, for both industries, productionresponses indicate the existence of risk aversion and areconsistent with behavior under decreasing absolute riskaversion.

II. Theoretical FrameworkConsider a competitive firm whose technology is given bythe production function y = F(x), where y is output, x is avector of inputs, and F is a continuous, nondecreasingquasi-concave production function. We assume that whenthe firm makes its decisions, it knows the input price vectorw, but does not know the output price p. The price of y isdistributed according to the (cumulative) distribution func-tion G, with E(p) = p andVar((p) = U2. It is useful to definethe price as p = p + e-p +E, where E and e= oEarerandom variables whose distribution functions are Ge and

GE with E(e) = E(E) = 0 and Var(e) = ca2.We assume that the firm maximizes expected utility ofprofits, E[U(IT)] = E{U[py - wx - r]},where uTs profit, ris a fixed cost,4and Uis a von Neumann-Morgensternutilityfunction with U' > 0. The solution to the firm's problemdefines the (dual) indirect (expected) utility function Vas

max EjU[(p +? E)y- wx - r]: y ? F(x)} (1)-V(W, p-,r, , p)where p represents higher moments of G, and V is continu-ous and convex in the moments.5 The firm's demand andsupply functions can be easily obtained from the indirectexpected utility function V.Applying the envelop theoremtoequation (1) we get

av - -E[U'U())]x( (2)a

-= yE[U'(Tr)] (3)dvar - E[U'(r)] (4)

so that the firm's input demand and output supply functionsare given by6

av jayXi =- X 5)dwi dry = - _ / (6)dp5 r

Received for publication July 25, 1994. Revision accepted for publica-tion June 4, 1996.* York University, Toronto, and University of California at Riverside,respectively.Financial supportfrom SSHRCC is gratefullyacknowledged. We wish tothank the referees for their useful comments and suggestions.I See for example, Sandmo (1971), Batra and Ullah (1974), Appelbaumand Katz (1986), and Dalal (1990). For additional references see Hey(1979).2 More recently non-expected utility models have also been applied toproductionunderuncertainty.See Chew and Epstein (1992).3 For examples of empirical studies that consider the effects of uncer-tainty on firm behavior see Parkin (1970), Just (1974), Antonovitz and Roe(1986), Appelbaum (1991, 1993), andAppelbaumand Kohli (1993).

4 The inclusion of fixed costs is, essentially, for convenience. The sameresults can be obtained (slightly differently) when r = 0.5 Proofs are given in Appelbaum (1993) and can be obtained from theauthors upon request.6 See, for example, Pope (1980), Chavas (1985), Chavas and Pope(1985), and Dalal (1990).

? 1997 by the PresidentandFellows of HarvardCollege andthe Massachusetts nstituteof Technology [ 631 ]


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632 THE REVIEW OF ECONOMICSAND STATISTICSwhere xi and wi are the ith elements of x and w, respectively.Equations(5) and (6) are the equivalents of Roy's identity inconsumertheory.Without uncertainty, additional properties of dual func-tions can be easily obtained, since either the objectivefunction (in the theory of the firm) or the constraint (in thetheory of the consumer) is linear.Specifically, dual functionsusually also satisfy homogeneity and monotonicity restric-tions (see Diewert (1982) and Epstein (1981)). Theserestrictions are used to obtain qualitative results to test theunderlying theory and to reduce the number of parametersthat need to be estimatedin empirical applications.Since profits are transformed nonlinearly by the utilityfunction, similar properties do not necessarily hold for theindirect utility function V. Specifically, under output priceuncertainty,demand and supply functions are not necessar-ily homogeneous, symmetric, anddownward(upward) slop-ing. Appelbaum (1993) shows that the indirect utilityfunction V satisfies the following properties:

1. V is nonincreasing in input prices and fixed cost, butnondecreasingin expected output price.2. V is decreasing, increasing, or constant in a if the firmis risk averse, risk loving, or risk neutral,respectively.3. If changes in a or r affect demand and supply, the firmcannot be risk neutral.4. Vis not homogeneous of degree 1 inp5,w, r, or inp, w,r, a. Risk neutrality is a necessary and sufficientconditionfor linearhomogeneity of V.Linearhomoge-neity of V inp, w, r (riskneutrality)is a necessary andsufficient condition for demand and supply functionsto be homogeneous of degree 0 inp3,w, r.5. The Slutsky equation (equivalent to the one in con-sumertheory) is given by

axi axi axiw1 WjdV -x (7)

where 3xiawj) dV=O - Si is thecompensatedubstitu-tion effect (holding V constant). The [Si] matrix issymmetric negative semidefinite. However, as can beseen from equation (7), this does not imply that the[axilawj] matrix is also negative semidefinite. Thusinput demand functions are not necessarily downwardsloping.7Similarly,we define

ay ay ay_ _ dv=o ar Y (8)PdV=O ar

where thecompensated upplyeffectSP= (&YIP) dV=O ispositive. But again, this does not imply that the supplyfunction is upwardsloping.

6. (a) Input demand functions are symmetric (axilawj = axjlawi) (for any utility or density function) ifand only if V is weakly separablein input prices.(b) Demand and supply are reciprocally symmetric(aylawj = axj?a)) if and only if V is weakly separablein input prices and expected output price.8

III. Empirical ImplementationA. EconometricSpecification

Having discussed the theoretical framework, we nowprovide examples of empirical applications. To implementthe model empirically, we first have to specify a functionalform for the indirect expected utility function. Given thisfunctional form, if the moments of G were known, we couldsimply estimate the system of equations (5) and (6).Unfortunatelythe moments of the distribution G are gener-ally not known and will, therefore,have to be estimated.Forexample, assuming rationalexpectations,the firm forms itsexpectationsof the moments of G by estimatingthe demandfunction,using market nformationon variables hat determinedemand.Given estimatesof the momentsof the pricedistribu-tion, we can estimate the firm's demand and supplyfunctions.The model can then be used to test regularityconditions andhypothesesregarding ttitudesowardrisk. It can alsobe usedtoestimate he effects of uncertainty nd the importance f variousmoments in determiningthe firm's productiondecisions.The examples we provide apply the model to the U.S.printing and publishing (PP) and the stone, clay, and glass(SCG) industries. These industries were chosen simply asexamples of possible applications of our model, but alsosince they are considered competitive.9We also applied themodel to the U.S. paper and allied products, furniture andfixtures, and the textile industries. We do not reportresultsfor these industries since they were similar.We assume thatthere are threecompetitivelypriced nputs nthe productionprocess-labor xl, capitalXk, and intermediategoods (materials) m-whose prices arewl, Wk,andwm,respec-tively.The dataare ortheperiod 1948-1989. This is theupdateddata set fromJorgenson t al. (1987), where a discussionof thedataconstruction an be found. To ensure that the results areindependent f unitsof measurement, llprices nthepaperwerenormalized nd measurednreal,rather hannominal, erms. 0To conform with the analysis above, where a fixed costwas used, we break problem (1) into two steps. First, wedefine the restricted indirect expected utility function J asthe solution to the problem,

maxy xl x E{U[(p + UE)y - WIXI- WMX - r](9):y F(xl,Xm, Xk)}=J(w1, Wm, , r,Xk,Y,p)

7 This corresponds to the standard result in consumer theory, whereMarshallian demand functions are not necessarily downward sloping, butcompensated demands always are.

8Which implies that technology must satisfy constant returnsto scale.9See Appelbaum (1982).10As an alternative, we also used the aggregate price index (fromJorgenson et al. (1987)) to calculate real prices. This yielded similar resultsas far as the elasticities and test results are concerned.


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UCTIONDECISIONSUNDER UNCERTAINTY 633where r = wkxk.The solution to problem (1), V, is then givenby

maxxk {J(w1, WMi, p, r, Xk, a, p): r = WkxkI(10)

--V(w, p, o, P).In the empiricalanalysis we work with the restrictedfunction,11ndwe use the firstfourmomentsof the pricedistribution, jsl, ..., p4. From a theoretical viewpoint itseems ratherunlikely hathigherordermomentswill playarole n thefirm'sdecisions. nfact,theredonotseem toexisttheoreticalresults in the literatureon production heoryunderuncertainty, hichdependon higherordermoments.Thuswe assume hatthe restrictedndirectexpectedutilityfunction s given bythetranslogunction

ln J = aoo+ oi ln wi + 0.5 I a'oihln wi ln Whi i h+ j11nj + 0.5 z j,Bgn ,lj n gI I g+ a ikln wi In k + ' yYijnwi ln Sl1i iij+ f31k ln I lnxk + ,kln xk (11)

+ 0.513kk(lnXk)2+ Oairln wi ln r + Bjrln j ln ri J+ Pr ln r + 3rkn Xkn r + 0.5,13rr(ln)2,i,h = ,m, j,g = 1,... .,4

withthesymmetryestrictions ih = aXhi, jg = fg3,andyij =yji.ApplyingRoy's identity oequation 11)we gettheinputdemandandoutputsupply"share" quations correspond-ingto equations5) and(6)) as

aci+ tih ln Wh + 'Yij n pjh i+ aik ln Xk+ ai, ln r

Si =S r +? aOirlnwi + jr ln jl (12)i I+ rk n Xk+ Irrln rh, i = 1,m, j = 1, ... , 4

p+ -yilnwi +E lj3ln gjl J+ Ilk ln xk + ,lrln r

-Sy - Pr + Otirlnwi + 'jrIn ,j (13)+ frk Inxk + rr In r

where si = wixilr for i = 1, m and sy = pylr -tlyIr are theinputand output"shares."Since the systemof equations(12) and (13) is homogeneousof degree zero in theparameters e use the normalization13r - 1. (14)

For empiricalmplementationhemodelhas tobe imbed-ded withina stochastic ramework.To do this we assumethatequations12) and(13) arestochasticdueto "errorsnoptimization."Wedefine heoptimizationrrorsn theshareequations at time t as vy(t), vl(t), and vm(t).We denote thecolumn vector of disturbances at time t as v(t) {vy(t),vj (t), vm(t)l nd assumethatthe vectorof disturbancessidentically and independently,oint normallydistributedwith meanzeroandnonsingularovariancematrix l,

Qf, foralls, tifs =tE[v(s)v(t)] = O, if t 0 s (15)

where l is a 3 X 3 positivedefinitematrix.B. EmpiricalResults

First we estimate he momentsof the pricedistribution.We assume hat he demandunctions givenby

p = D(z) + e (16)wherethe variables n the vectorz are takenas aggregateoutputof the industry,he U.S. aggregateoutputandpriceindexes alsotaken romJorgensont al. (1987)),anda timetrend.Weestimate he conditionalon z) first our momentsof p nonparametricallys follows. Considerthe scalarrandomvariablep and the 1 X q randomvector z =[Zi, . . ., Zq]. Then the rth-order conditional moment of pgiven z is mr(Z)= E(prIz) for r = 1, 2, .... For r = 1,ml(z) E(plz) is the conditionalmeanof p; for r = 2,m2(Z) = E(p21z) is the second conditional moment of pgiven z; and so on. The problemwe considerhere is thenonparametricernelestimationof mr(z)basedon the data{pi, zi},i = 1, . . ., n. For r = 1, the well-known Nadaraya(1964)andWatson1964)kernelestimators

nm. (z) = piwi(z) (17)i=l111But,of course, his doesnot mean hatwe takexkas fixed.All it meansis that hesolution s conditional n the changingvaluesof Xk.


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634 THE REVIEWOF ECONOMICSAND STATISTICSwhere

IZi Z\ I Ii Zwi(z) = K h )/ K h (18)and K(.) is the kernel function and h the window width orsmoothing parameter.The kernel K(-) has the propertiesthatit is nonzero, integratesto unity, and is symmetric, such as amultivariate normal density with a zero mean and an identitycovariance matrix.Intuitively, the estimator m4(z) is the weighted average ofthe pi values correspondingto those zi's which are aroundz,the point at which mlhs calculated. The weights are given bythe kernel function K(.), which is usually chosen to be asymmetricdensity aroundzero and is such that it gives a lowweight to these observations zi that lie far from z. It is wellknown that the choice of kernel does not seem to matteragreat deal (see, for example, Ullah (1988, p. 643)). Here weuse the productof normal kernels K(z) = HlflIK(zj), whereK(zj) is a standardnormaldensity.The window width h is animportantparameter,andits choice determines the "size" orthe interval around z over which the observations areaveraged. Usually the largerthe h is, the less is the varianceand the smootherthe curve, but the largerthe bias. The h thatminimizes the asymptotic integratedmean-squarederror ofm is of nl/(4q) In our calculations we take h = n-l (4+q) anduse the [zil,... , Ziq]data scaled with their standard devia-tions. For details on the choice of kernel K and windowwidth h see Marron(1988) and Ullah (1988).We note that mi (z) is the sample estimate of the popula-tion average of p-values conditional on z, E(p z). Thereforeone can write the nonparametric estimators of mr(z), theaverage ofpr conditional on z, as

nMrI(Z) = Epwi (z) (19)i=l

where wi(z) is as in equation (18). Using this result we canwrite the nonparametricestimator of the rth conditionalmoments around the mean asP2(PIz) m2(z)-m 2(z) (20)P3(p|Z) = m3(Z)- 3 h2(Z)4 z(z) + 2m13(Z) (21)

(PIZ) = i4(z) - 4 z3(Z) h1(Z) (22)+ 6fi2(z)m2l(z)- 3 4(z).

The asymptotic properties of Mir(z) are well established inthe literature.(See, for example, Singh andTracy (1977)).Given the nonparametric estimates of the first fourmoments of p, we estimate equations (12) and (13) for thetwo industries using maximum likelihood with the normal-

ization and symmetry restrictions imposed. The parameterestimates are given in table 1. Given these parameterestimates, we check for local regularityrestrictions (at thepoint of approximation). Local monotonicity requires thatoti< 0, otr< 0, and PI > 0. The parameterestimates in table1 show that these condition are satisfied. We calculated Siand SP at the point of approximation,and found them to benegative definite and positive definite, respectively, for bothindustries. Checking for convexity of V in the first momentof the distribution, we found it to be satisfied locally.12 Wetested and rejected linearhomogeneity of Vinpj,r, wi,wmforboth industries, which implies that we must reject riskneutrality.Finally,we test for the symmetryof inputdemandfunctions (the weak separability restrictions). The localrestrictions for weak separability of V in input prices aregiven by the restrictions13

a/l Yij (lr=-Yin1= c m j= 1, ...,4. (23)OLM 'Ymj ?tmrWe test for and reject these restrictions (which implies thatalso global weak separability is rejected). Since weakseparabilityin inputprices is a weaker restrictionthanweakseparability in input and expected output prices, the rejec-tion of conditions (23) implies that reciprocal symmetry ofsupply and demandfunctions is also rejected.To examine the effects of the exogenous variables on thefirm'sproductiondecisions, we use the parameterestimatesto calculate the correspondingelasticities. First,demandandsupply elasticities with respect to the moments of the pricedistributionaregiven by

1iYijlA PJr,a for i = , m, j = 1, . . . ' 4 (24)and

3lj/81 Jirg j = 2,3, 4oyj P1/81 3lIrb - 1, j = 1 (25)

wherealnJ alnJ alnJ_ ,i =lI,m, 81 , d-a Inwi' ',8 a Inpl r a In r'

Supply and demand are unaffected by the higher moments,I2, P3, p4, if andonly if the correspondingelasticities satisfy:Oij= Oyj= 0 for all i = 1, m andj = 2, 3, 4. The local

12 Since all of the parameters (in equation (11)) which do not involveinput prices or VIu o not appear in the estimated equations (12) and (13),we cannot check for convexity in the higher moments.13 Note that since VilVj = Cl/Cj (as can be seen in footnote 15), inputprices also have to be weakly separable in all moments. They do not haveto be weakly separable, however, in Xk, that is, we can have C(w, y, Xk) =C(h(w, Xk), y, Xk). Thus the local restrictions include all parameters thatinvolve r and the moments, but not necessarily those that involve xk.


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PRODUCTIONDECISIONSUNDER UNCERTAINTY 635TABLE 1.-PARAMETER ESTIMATES AND CORRESPONDING t-VALUES OF V

Parameter PP SCG Parameter PP SCGPrk 1.95964 0.086472 atm -1.02026 -1.12624.43135 0.254489 -65.0617 -28.2099Pir 0.22142 0.2443 Oamk 0.622997 0.385732.5563 3.48776 1.93902 0.914664Pmr 0.031689 -0.0203097 a-MM -0.972658 -0.877410.319331 -2.02871 -4.66885 -3.54634Ilr -0.050309 0.066191 'Yml -0.917606 -0.612233-0.531951 0.860386 -6.28237 -3.70809P2r -3.34453 -1.49213 Ym2 -1.68526 -0.318282-2.83301 -3.39262 -1.69856 -0.625071P3r 4.98317 2.18391 Ym3 2.82585 0.9337333.01428 3.47576 2.05143 1.293484r -2.07623 -0.878857 Ym4 -1.25581 -0.506554-3.16477 -3.46395 -2.31612 -1.73893

Prr -0.761543 -0.640011 Pi 1.01414 1.09604-6.33353 -8.37801 81.6327 39.585al -1.02571 -1.08343 lk -0.74533 -0.130297-98.2394 -45.8247 -2.2525 -0.372145atlk 0.273067 -3.21572 rii 7.97461 4.87276

0.796953 -0.920726 1.93964 2.25312aO11 -0.333291 -0.387216 12 1.25326 0.525427-2.31578 -2.55631 0.98912 0.097219atim 0.344917 -0.12676 13 -2.80924 -0.75622.9052 0.712326 -1.78075 -1.1406'Yni -0.067936 0.05436 114 1.32417 0.436595-0.583739 0.434707 2.18741 1.6799'Y12 -2.68923 -0.785898-2.41714 -1.83519'Y13 3.99511 1.301852.57121 2.14026'Y14 -1.65951 -0.574926-2.70326 -2.34684

PP SCGEquation R2 Durbin-Watson R2 Durbin-WatsonOutput 0.950401 1.21012 0.914576 1.3508Labor 0.976429 1.05238 0.958801 1.19269Materials 0.934851 1.27183 0.899135 1.39501

restrictionsor these elasticities o be zeroaregivenby'Yj + oir = ? j = 2, 3, 4, i = , m. (26)

131i+ PA,3y =?We testandreject heserestrictionsor both ndustries,husrejecting hat productiondecisionsare (locally)unaffectedby highermoments.14gain,thisimpliesthatriskneutralitymustberejected.Having rejectedrisk neutrality, an we go furtheranddeterminehatwe actuallyhave riskaversion?Oneway tocheck if there is risk aversionis to see whether V is

decreasingn variance.Unfortunatelyllparametershatdonot involvew, V1I, r r arelost in the differentiationf theindirectutilityfunction.Consequentlyome of the param-eters involvingthe secondmomentsdo not appear n ourestimatedsystem, so that we cannotcheck whetherV isincreasingor decreasing n the second moment.We can,however,checkthis indirectly.As is well known,withriskaversion,outputwill be lowerunderuncertaintyomparedwiththe certaintyorriskneutrality)ase.'5Theconverse s

14 Given hat helocalrestrictions rerejected, t is clear hat he globalrestrictions ill berejected swell.

15See Sandmo1971).Toseethis,note hat ince nputpricesareknown,we can obtain the optimal level of output from the problemmaxyE{U[(p + rE)y - C(w,y) - r]}, where C(w,y) is the usual cost.function(which implies that VilVj= CVlCj).This yields the first-orderconditionp5 aClay+ f, wherea = - [cov(U', p)]IE(U') is themarginalcostof uncertainty.Withrisk aversion he "full marginal osts"are given


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636 THE REVIEW OF ECONOMICSAND STATISTICSTABLE 2.-ELASTICITIES EVALUATED AT THE POINT OF APPROXIMATION

PP SCGElasticity Estimate t-Statistic Estimate t-Statistic

Oil 0.0159 0.1438 0.0160 0.1603012 -0.7227 -2.962 -0.7667 -3.851013 1.0881 3.087 0.9823 3.401014 -0.45831 -3.183 -0.34820 -2.9540mi 0.8490 5.014 0.6098 4.5820m2 -1.6927 -4.020 -1.2095 -3.8460m3 2.2134 3.671 1.3548 2.9640m4 -0.8453 -3.479 -0.4290 -2.292Oyl 6.8130 1.680 3.5119 1.797oy2 -2.1087 -2.838 -1.4441 -3.665oy3 2.2130 3.255 1.4939 3.6610y4 -0.7705 -3.287 -0.4805 -3.175Oil -0.4536 -3.347 -0.3983 -2.975Oin -0.3045 -2.366 -0.0861 -6.018Oml -0.1166 -0.991 0.3568 2.538omm - 0.0149 -0.076 -0.4240 -1.831Oyl 0.1544 1.471 0.2939 2.920oym -0.6830 -4.293 -0.3142 -2.345

also true, that is, if output is lower under uncertainty, wemust have risk aversion. It is thereforepossible to check forrisk aversion by comparing output levels with and withoutuncertainty (or with risk neutrality). To obtain the values foroutput under certainty we note that without uncertainty, Vbecomes the usual profit function, which is linear homoge-neous in w, V1, r.16 This is obtained by imposing linearhomogeneity (in w, V1, r) and the requirement that aVhar=-1, I3jg= 0 for all j = 2, 3, 4. Thus we estimate the modelwith these restrictions imposed and use the parameterestimates to calculate the predictedvalues of output,for bothindustries. Comparing the estimated values of output withthe actual ones, we find that outputin the restricted case (nouncertainty) s higher than output under the unrestrictedcase(with uncertainty) for all observations. Thus we concludethatwe do indeed have risk aversion.The estimated elasticities with respect to the moments ofthe price distribution, calculated at the point of approxima-tion, and their standard errors are reported in table 2. Astable 2 shows, the elasticities of labor with respect to allmoments have the same sign as the those of materials withrespect to these moments. Specifically, input elasticities withrespect to the first moment (expected output price) aresignificantly positive, whereas input elasticities with respectto the second moments are significantly negative for the twoindustries. Hence a higher expected output price willincrease demand for both inputs, but a higher variance willdecrease it. As for input elasticities with respect to the thirdand fourth moments, we find that they are significantlypositive andnegative, respectively, for both industries.

Looking at the supply function, we find that supplyelasticities are significantly positive with respect to the firstmoment (expected output price), but significantly negativewith respect to the second moment, for the two industries.Hence a higher expected output price will increase thesupply of output,but a highervariance will decrease it. Thus(assumingthat U does not change sign), we conclude that forboth industrieswe have statistically significantriskaversion.As for supply elasticities with respect to the third and fourthmoments, we find that they are significantly positive andnegative, respectively, for both industries. Our findings thatdemand and supply functions are upward sloping withrespect to expected output price, but downwardsloping withrespect to the variance, are consistent with the standardresults, for the case of a risk-averse firm with decreasingabsolute risk aversion.17Next we calculate demand and supply input price elastici-ties, which are given by

Otij Otrj

rOXii OtriOii =-- -1, i 1,m (27)6i 6r

oyj -^ , j =1,m.6 r'These elasticities are given in table 2. Table 2 shows that allthe own price elasticities of demand have the right sign,indicating that the demand functions are negatively sloped.Crossprice elasticities of demand areusually not symmet-ric. Under uncertainty, however, the cross price elasticitiesof demand, Oij and Oji, o not even have to have the samesign.18 Indeed, as table 2 shows, demand elasticities are"sign symmetric" in the PP industry (laborand materials aremutuallysubstitutes),but not in the SCG industry.Our empirical results indicate that uncertainty had astatistically significant effect on production decisions. Thiswas due to the fact that risk neutralitywas rejected in bothindustries. The rejection of risk neutralityand the estimatedeffects of uncertainty on supply and demand suggest thatuncertaintyresulted in lower output produced (inputs used).What does this imply from a welfare point of view? First,letus consider the welfare implication for a given expectedoutput price. With risk aversion, the "full" marginalcosts19are higher due to the presence of marginal costs of uncer-tainty. The marginal costs of uncertainty are absent in thecase of risk neutrality (or no uncertainty), which is thereason why output is higher in that case. Clearly, a lower

by aClay+ 1,where1 > 0. Thusoutputwill be lower han n thecaseofriskneutrality, r no uncertaintywhen1 = 0). Note that t is the standardpractice n suchcomparisonso takethe price n the certainty ase as theexpectedprice see referencesn footnotes1 and2).16 In this case the problem s simply to maximizeprofits, rr= ply-C(w, y) - r, where C(w, y) is the usual cost function.

17 See Sandmo 1971) andAppelbaum ndKatz 1986).18 To see this, assume that 0,i > 0. Then we must have OLij > OLrLji18,r =

(oLri,jI8r)(OLrj8ijILrij8j) But given this, it is still possible to have oLij< ,i8jl8nwhich is required for Oji< 0.19See footnote 15.


8/8

PRODUCTIONDECISIONSUNDER UNCERTAINTY 637output level indicates a reduction in both producer andconsumersurplus,comparedto the risk-neutralcase, for anygiven expected (and actual) price.20The expected price ofoutput,however, will not be the same with and without riskneutrality (with or without uncertainty). Specifically, withrisk neutrality(or no uncertainty) we move to a lower pointon the same expecteddemandfunction, which will result in alower expected price and a higher output. Since outputwillbe higher and the expected (equilibrium)outputprice will belower with risk neutrality (or no uncertainty),it must be thecase that the sum of producerand consumer surplus will behigher. Furthermore,with a higher output and lower ex-pected price, consumer welfare will be higher. What aboutproducersurplus?Since the expected price is lower, but so isthe full marginal cost curve, we cannot determine whetherproducersurpluswill be higheror lower.Additional informa-tion on aggregate demand and supply elasticities is requiredto be able to make such a comparison.

III. ConclusionUsing a nonparametricestimation technique to estimate

the firstfour moments of the unknown price distributionandapplying dualitytheory,this paperprovides a simple empiri-cal framework for the analysis of the decisions of firmsunder uncertainty.This frameworkis used to study underly-ing attitudes toward risk and to test for the significance ofhigher moments in productionchoices.Applying the model to the U.S. printing and publishingand the stone, clay, and glass industries,we find that highermoments play a significant role in determining input andoutput decisions. Specifically, we test for and reject riskneutrality n both industries.Moreover, we find thatproduc-tion responses indicate risk aversion and are consistent withbehaviorunder decreasing absolute risk aversion.REFERENCES

Antonovitz,F.,andT. Roe, "ATheoretical ndEmpiricalApproacho theValueof Informationn RiskyMarkets,"his REvIEw8 (1986),105-114.Appelbaum,E., "TheEstimationof the Degree of OligopolyPower,"Journal of Econometrics (1982), 287-299.

"Uncertaintyndthe Measurementf Productivity,"ournalofProductivityAnalysis 2 (1991), 157-170."AnApplication f DualityunderUncertainty," orkUniversityDiscussionPaper, resented tthe European conomicAssociationMeetings,Helsinki 1993).Appelbaum,E., and E. Katz, "Measuresof Risk Aversionand theComparativetaticsof Industry quilibrium,"mericanEconomicReview 76 (1986).Appelbaum,E., and U. Kohli, "ImportPrice Uncertaintyand theDistributionf Income,"hisREvIEw1993,forthcoming).Batra, R., and A. Ullah, "CompetitiveFirm and the Theoryof InputDemand under Price Uncertainty,"Journal of Political Economy 82(1974), 537-548.Chavas,J. P., "On heTheoryof theCompetitive irmunderUncertaintywhen Initial Wealth Is Random," Southern Economic Journal 51(1985),818-827.Chavas,J. P., and R. Pope, "Price Uncertainty nd CompetitiveFirmBehaviour:TestableHypothesesromExpectedUtilityMaximiza-tion," Journal of Economics and Business 37 (1985), 223-235.Chew, S. H., and L. G. Epstein,"A Unifying Approach o AxiomaticNon-ExpectedUtilityTheories:Corrigenda,"ournalofEconomicTheory (1992).Dalal,A. J., "SymmetryRestrictionsn theAnalysisof the CompetitiveFirm under Price Uncertainty," International Economic Review(1990).Diewert,W.E., "DualityApproachesn Microeconomics,"n K. J. Arrowand M. D. Intrilligator (eds.), Handbook of Mathematical Econom-ics, vol. 2 (Amsterdam: orth-Holland,982).Epstein,L., "GeneralizedDualityand Integrability," conometrica 9(1981), 655-678.

Hey, J. D., Uncertainty in Microeconomics (New York: New YorkUniversityPress,1979).Jorgenson,D. W.,F. M. Gollop,andB. Fraumeni, roductivity nd U.S.EconomicGrowthAmsterdam: orth-Holland,987).Just, R., "An Investigationof the Importanceof Risk in Farmers'Decisions," American Journal ofAgricultural Economics 56 (1974),14-25.Marron, . S., "AutomaticSmoothingParameter election:A Survey,"Empirical Economics 13 (1988), 187-208.Nadaraya,E. A., "On EstimatingRegression,"Theoryof ProbabilityApplications 9 (1964), 141-142.Parkin,M., "DiscountHouse Portfolioand Debt Selection,"Review ofEconomic Studies 37 (1970), 469-497.Pope,R.D., "TheGeneralized nvelopeTheorem ndPriceUncertainty,"SouthernEconomic Journal 21 (1980), 75-86.Sandmo, A., "On the Theory of the CompetitiveFirm under PriceUncertainty,"American Economic Review 61 (1971), 65-73.Singh,R. S., andD. S. Tracy,"StronglyConsistentEstimators f k-thOrderRegressionCurvesandRatesof Convergence," . Wahrsch.Verw.Gebiete 40 (1970), 339-348.Ullah, A., "Non-Parametric stimationof EconometricFunctionals,"CanadianJournal of Economics 21:3 (1988).Watson,G. S., "SmootherRegressionAnalysis,"Sankhya26 (1964),359-372.20Producersurpluss givenby the areabetween heexpectedpriceand"fullmarginal osts."

estimation of moments and production decisions under uncertainty

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