In: Proceedings of the 1991 symposium on systems analysis inforest resources; 1991 March 3-6; Charleston, SC. Asheville, NC:U.S. Department of Agriculture, Forest Service, SoutheasternForest Experiment Station; 1991: 225-240.

COST FUNCTION APPROACH FOR ESTIMATING DERIVED

DEMAND FOR COMPOSITE WOOD PRODUCTS1

Thomas C. Marcin2

Abstract.-A cost function approach was examined for using the con-cept of duality between production and input factor demands. Atranslog cost function was used to represent residential constructioncosts and derived conditional factor demand equations. Alternativemodels were derived from the translog cost function by imposing pa-rameter restrictions.

Keywords: Production economics, economic duality, factor demand

INTRODUCTION

Composite wood products are the fastest growing seg-ment of the forest products industry. The demand forthese products is largely a derived demand that arisesprincipally in the construction industry. This paper dis-cusses potential problems in model construction and de-velops a modeling approach to estimate conditional fac-tor demand based on the translog cost function. A costfunction relates the total cost of production for a level ofoutput to input factor prices and technical change. Gen-erally, a translog cost function is paired with its factor-share equations to obtain accurate estimates of para-meters. Thus, the factor shares are functions of the samevariables and parameters as those of the cost function.

The principle of duality in the theory of productioncan be used to specify cost functions from which the de-mand for factor inputs can be derived under certain as-sumptions of cost minimization and perfect competition(McFadden 1978). The cost function approach has sev-eral advantages. First, the cost function can be speci-fied from factor input prices for a given level of output.This leads to computational simplicity in computing fac-tor demands. Second, the conditional factor demandequations can be obtained simply by partial differentia-tion of the corresponding indirect cost function. Third,the cost function provides a convenient way for speci-fying empirical functional forms for application to pricedata where the quantity of factor demands is not known(Fuss 1977a).

1 Presented at the 1991 Systems Analysis in Forest Re-sources Symposium, Charleston, SC, March 3-7, 1991.2 Economist, USDA Forest Service, Forest Products Lab-oratory, Madison, WI 53705-2398. The Forest ProductsLaboratory is maintained in cooperation with the Uni-versity of Wisconsin. This article was written and pre-pared by U.S. Government employees on official time, andit is therefore in the public domain and not subject tocopyright.

This paper first considers the translog cost functionas developed by Christensen and others (1973) for rep-resenting construction costs and the derived conditionalfactor demand equations. Previous work by Rockel andBuongiorno (1982) is then reviewed, using the translogcost function to model construction costs. The availabil-ity of data for different levels of aggregation is discussed.Finally, alternative model specifications for empirical ap-plications are examined.

COST FUNCTION

The general specification of any cost function beginswith the general form

Then, using Shephard lemma (Shephard 1970) andthe envelope theorem (Beattie and Taylor 1985), the re-sultant factor demand equation for the ith factor con-ditional on the output level Q is the conditional factordemand. The conditional factor demand is the partialderivative of pi on the cost function

where xiis ith input factor demand.

Translog Cost Function

The translog cost function has the advantage of flexi-bility of specification and can be applied to multiproduct,multifactor production. Unfortunately, it is impossibleto provide an explicit solution to the underlying produc-tion function mathematically. The translog productionfunction and the translog cost function do not generallycorrespond to the same technology, although they can

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be regarded as close approximations for the same tech-nology. Thus, the translog cost function is most useful instudies of factor demand and product supply. However,the translog production and cost functions provide a localsecond-order approximation to any price frontier. i.e., anyarbitrary cost function (Christensen and others 1973).Additionally. for many production and cost functionsused in econometric studies. the translog functions doesprovide a global approximation (Denny and Fuss 1977).

The production structure can be represented bytranslog cost function and a time trend to represent tech-nology as follows:

The factor share equations for each input of thetranslog cost function are

and

where S is the factor share and Xi is the quantity de-manded for the ith factor input.

Because the shares must add to 1, the restrictions onthe parameters are

In addition, rij = rji (i # j) must be true; if it is nottrue, the cross partial derivatives will not be equal andsymmetry conditions will be violated.

Thus, the factor shares are functions of the samevariables and parameters as those of the cost function.Therefore, if the factor shares are observable, then theequations corresponding to the share equations can bepooled with the cost equation when estimating parame-ters. In practice, however, it is often difficult to observethe factor shares so that parameters are estimated basedupon the cost function alone.

Alternative Models DerivedFrom Translog Function

The translog function is an approximation to the gen-eral nonhomothetic cost function, which is assumed tobe homogeneous in prices. Since the translog model iscomplicated, it is useful to look at other possible mod-els that can be derived from it by imposing further re-strictions on its parameters. For purposes of identifica-tion. the translog cost function is referred to as modelI and the subsequent models are called model II, model

III, etc. Model II, which is developed by imposing Hicks-neutral technology changes in the cost function over time,requires

The cost function can be further restricted to be ho-mothetic and homogeneous by requiring the cost functionto a separable function of prices. This model, model III,requires

Further constraints may be added to make thesecond-order price terms equal to zero, which results inunitary elasticity of substitution between factor inputs.This is model IV, which implies that

This, in fact, is the cost function corresponding to ageneralized Cobb-Douglas production function. Finally,the cost function can be reduced to that, of an ordinaryCobb-Douglas production function by adding constantreturns to scale. This requires CKQ = 0 and is model V.

Estimation of Cost Function

Data

The data available for estimating cost functions maybe classified as either macro- or microdata. Macrodata oraggregate data are available from the U.S. Departmentof Commerce. Previous work by Rockel and Buongiorno(1982) used monthly observations from the ConstructionReview published by the U.S. Department of Commerceto estimate the demand for wood and other inputs in res-idential construction. These authors used the Boeckhindex of building costs for residences as an index of theaverage cost of residence in the United States. This in-dex is based upon a survey of local costs for materialsand labor for 20 selected cities. These costs are then usedto calculate the cost of building a typical wood-framedhouse and a typical brick house (Levy 1977). The indexexcludes the cost of land and financing, and it is weightedbased upon actual labor and material costs observed from1926 to 1929. The Boeckh index has at least two majordrawbacks. First, the weights for the various shares ofconstruction costs have surely been much different in re-cent years than they were in the period 1926-1929. Sec-ond, the index does not incorporate any adjustments fortechnical change, which reduce the cost per unit of out-put. The Boeckh index overstates increases in the costper housing unit to the extent that technological progresshas occurred in the industry.

In a sense, Rockel and Buongiorno estimated theweights of the Boeckh index. The output variable usedin their model was the number of housing starts. A majorproblem with this measure of output is the heterogeneityof housing starts-the type, size, and quantity of housingstarts vary. A better measure of the aggregate output ofthe construction sector is the value of new constructionin constant dollars. Similarly, the current dollar valueof new construction is a reasonably accurate measure oftotal construction costs.

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The producer price index is used to provide measuresfor material input prices. These measures are reasonablygood estimates for existing products. The price of laborused by Rockel and Buongiorno is the average weeklyearnings of special trade construction workers. A betteralternative is to use an hourly wage rate because it pro-vides a constant unit of measure.

Additional data on factor shares are provided byinput-output tables for the United States, in which fac-tor shares can be calculated from the I/O tables. Thesetables provide a checkpoint for other data sources. Un-fortunately, these tables are not available in a frequent ortimely manner.

The use of monthly data entails statistical problemsrelated to autocorrelation, seasonality, collinearity, anddata aggregation. Autocorrelation can be tested by sev-eral procedures. The common test is the Durbin-Watsonstatistic. Values <1 or >2 indicate strong autocorrela-tion. Methods are also available for correcting autocorre-lation (for example, see Maddala (1977), pp. 257-291).Monthly data can exhibit strong autocoorelation ten-dencies, which need to be tested and corrected. The useof dummy variables in the model can be used to test forseasonality. Construction material prices may exhibitstrong seasonal patterns because the construction indus-try has strong seasonal patterns of activity. Problems ofcollinearity also exist for prices of many materials usedin construction. Finally, there are problems with dataaggregation. For example, there is no clear data set forcomposite wood products. Denny and May (1978) sug-gest the use of a two-stage estimation procedure: a mi-crofunction of a particular set of disaggregated inputs isestimated in the first stage and aggregate inputs are usedin the second stage. Puss (1977b) applied such a proce-dure to his work on energy.

Other studies to estimate production technologies andfactor demand equations have used annual data. For ex-ample, Christensen and others (1973) used annual datafrom the United States private domestic economy to fitparameters to the translog production and price possibil-ity frontiers. Denny and May (1978) used price and quan-tity input from Statistics Canada to measure prices andquantities of inputs and output of the Canadian manufac-turing sector to fit and test a translog cost function.

Monthly data were used in this study to examine thecost function approach. The output measure was residen-tial construction expenditures in constant dollars. Thecurrent dollar value of residential construction was usedas a measure of costs. Producer price indexes were usedas a measure of input prices. Factor prices were used forlumber and plywood, and combined indexes were used forhardboard, particleboard, and fiberboard and for otherconstruction materials. Hourly wage rates for construc-tion workers were used for measures of labor input price.Monthly data were compiled for 1960 to 1990 (U. S.Department of Commerce 1960-1990).

Parameters

Parameters for the various restricted forms of thetranslog cost functions were estimated with generalizednonlinear least square estimation procedures using ver-sion 4.1C of TSP (TSP International 1987). Single equa-tion methods were used to directly estimate the familyof restricted cost function forms. Because monthly data

were used, it was not possible to estimate factor inputcost share. Therefore, the cost function was estimated byitself.

Results of the regression estimates of the parametersof the translog cost function and its various restrictedforms are presented in Table 1. These preliminary re-sults indicate that the use of monthly data for this typeof analysis is complicated by several problems. Autocor-relation, seasonality, and collinearity of input variableslead to problems in obtaining expected signs and consis-tent parameter estimates.

The models presented in the Appendix provide a use-ful starting point for additional analysis of the cost func-tion approach. The Durbin-Watson statistic indicatessignificant autocorrelation (Maddala 1977). The param-eters for the proxy for composite products was positiveand significant in the generalized Cobb-Douglas costfunction and in all unitary elasticity models.

CONCLUSION

The cost function approach has several advantages.First, it provides a convenient way to obtain supply anddemand equations, which is consistent with traditional(primal) economic theory. Second, the dual approach isuseful in generating a functional specification for supplyand demand equations for econometric estimation. Fi-nally, the cost function approach provides a sound theo-retic approach for using price and cost data to estimate aconsistent set of factor demand equations.

Preliminary results using the duality approach werenot completely satisfactory using monthly data. Thevariable for composite products, which included par-ticleboard, fiberboard and hardboard, was significantand positive for all five models examined. This seemsto indicate a significant variation in price trends for thisvariable. A promising direction for future research isto develop a more refined data base using microdata orregional data and to specify a simultaneous-equationsmodel that would include factor share equations relatedto alternative costs of factor inputs and end-use markets.

Future directions for use of the duality theory for esti-mating factor demand can be divided into two categories:(1) improvements in the data base for use in model esti-mation and (2) refinements in the application of theoryfor model specification. Problems associated with the useof monthly data could be avoided by developing an alter-native data set based upon microdata from the construc-tion industry or other end-use industries. Regional datamight also be useful because much contemporary demandanalysis focuses on panel data sets from individual firmsor industrial surveys. An example of this type of work isthe work of Caves and others (1981) for the transporta-tion sector. In addition, it may de difficult to separatethe effects of changes in construction activity from trendsin factor prices in a monthly model because prices tend tomove with trends in activity. Thus, explanatory variablesrue probably not truly exogenous when national data areused. Therefore, simultaneous estimation methods maybe worthwhile. It would be useful to add a variable forconstruction financial costs and to examine the possi-ble effects of changes in real estate value on constructionexpenditure data. Theoretical considerations in modelspecifications include using a profit function approach be-cause the output is not a fixed quantity as specified in

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the cost function. The use of factor share equations ina simultaneous-equations model might eliminate someanomalies present in monthly data analysis. For exam-ple. it may be more useful to use annual data or cross-sectional data with factor share estimates even with asfew as 30 observations.

LITERATURE CITED

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