Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors in Input Demand? *

Transcription

1 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors in Input Demand? * Mauricio Vaz Lobo Bittencourt ** Armando Vaz Sampaio *** Abstract This study investigates the primal and dual approaches for production in the presence of stochastic errors in output and input demands, and policy implications when such errors are not taken into account. A synthetic dataset is used to econometrically estimate the primal and dual functions associated with a given technology. Results show that both formulations are unbiased, consistent and efficient, even in the presence of a Cobb-Douglas technology. Not accounting for such errors can lead to wrong policy recommendations in a productive sector. Any kind of policy created to improve the total production of a particular sector should consider these issues before applying them to real data. Keywords: Stochastic Errors, Input Demands, Duality, Monte Carlo Simulation. JEL Codes: C13, C51, C81, D24. * Submitted in July Revised in March The authors thank Brian Roe and Ratapol Teratanavat for their contributions and suggestions, as well as the financial support from CAPES. ** Professor, Graduate Program in Economic Development (PPGDE-UFPR) and Department of Economics at UFPR; Visiting Professor at The Ohio State University (USA); and CNPq fellow. mbittencourt@ufpr.br or bittencourt.1@osu.edu *** Professor, Graduate Program in Economic Development (PPGDE-UFPR) and Department of Economics at UFPR. avsampaio@ufpr.br Brazilian Review of Econometrics v. 31, n o 2, pp November 2011

2 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio 1. Introduction Duality is very useful in economics because it is a straightforward and natural way to elaborate and analyze economic problems. However, dual representation does not exist without its primal counterpart. Although the primal approach allows for an immediate and intuitive interpretation, the dual method can be analytically more appropriate for some complex problems. Notwithstanding, the debate about which method is more suitable to the analysis of economic issues is still alive. Duality proved to be a very thriving approach in empirical research when it became popular 30 years ago. According to Just (2000), to capture the empirical benefits of duality, both primal and dual implications of its estimates should be compared against other empirical studies, regardless of whether the estimated relationships have been derived by primal or dual approaches, i.e., in case of output, the primal method would be the production function whereas the dual method would be the profit function (or cost function). It is widely known that primal and dual methods in the consumer and production theories are theoretically equivalent provided that there are not any complications. According to the available dataset, duality may provide some theoretical insight into the economic problems under study. Nevertheless, there are many studies that focus on empirical problems associated with a broad array of characteristics of a dual analysis, which are different from the regularity conditions presented in textbooks. For instance, Burgess (1975) and Appelbaum (1978) found different results for the primal and dual methods in the presence of risk and stochastic error. Pope (1980) and Pope (1982a) suggested that duality results do not hold for stochastic models. Thompson and langworthy (1989) and Pope and Just (2002) suggest that the functional form may affect elasticity estimates in the primal and dual methods. Lusk et al. (2002) underscored the need of high-quality databases to estimate dual relationships in the presence of measurement error and low variability of relative prices for different sample sizes. In this context, the present paper aims to assess duality properties in an empirical study by including the Hicks-neutral technical change, stochastic errors in production, and stochastic errors in input demand in the profit maximization problem using synthetic data from a Monte Carlo simulation. The method implies some assumptions about production, each of which has substantial precedents in the literature. Theoretically, one would expect the results of primal and dual methods to be similar, as the dual function is closely related to the primal function. Empirically, dual estimation is more appealing, since the necessary amount of information is less restrictive than that of the primal function, given that dual estimation requires only data on profits, input prices, and production levels, as also observed by Young (1982). According to Zellner et al. (1966), the ordinary least squares (OLS) method is consistent with the Cobb-Douglas technology by assuming that production shocks 296 Brazilian Review of Econometrics 31(2) November 2011

3 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? are unknown at the decision-making stage, such that input demand ought to be based on expected production. As a matter of fact, the causality relationship of errors in production will not be taken into consideration for errors in input demand, as they are generated by a Monte Carlo simulation and are totally independent from this transmission problem, as pointed out by Mundlak and Hoch (1965). This paper raises the possibility that both dual and primal methods may yield good results when some empirical complications are introduced into the model. A dataset based on the behavior of some representative agent is built from the Monte Carlo simulation and used to econometrically estimate the primal and dual functions associated with a given technology. The goals of this paper are: first, to empirically assess the properties of the OLS estimator for the primal and dual methods, in view of a Hicks-neutral technical change, in the presence of stochastic errors in production and in input demand in the optimization process, which are not observed by an econometrician. Second, to show the implications of economic policies when the presence of these errors is not taken into account by policymakers. The paper is organized as follows. Section 2 presents a literature review on duality and the major problems with the transmission of shocks throughout the productive process. Section 3 deals with the source of data and describes the main characteristics of the model to be assessed. Section 4 discusses the major results obtained in the three preceding sections. Section 5 contains the final remarks. 2. Literature Review According to Pope (1982b) and Taylor (1989), some advantages of the dual method include: easy applicability, flexibility in measurement, fewer data requirements, and its appropriateness for assessing more problems than the primal method. The downside is that not all problems can be dealt with by the dual method (e.g., production of multiple products, models under risk aversion and uncertainty, nonlinear and dynamic models). A profit-maximizing (dual) formulation in the presence of errors or distortions may lead to incorrect choices for resource allocation. Taylor (1989) states that a critical assumption for this event is that Hotelling s lemma does not hold. Thus, it is not possible to obtain Hotelling s results from the standard profit function in the presence of stochastic errors in output and input demands. The only exception is when the production function is quadratic in relation to inputs. 1 As remarked by Pope and Just (2001, 2003), problems with identification of errors in the supply/demand system may include: a) stochastic representation of the production function, b) correlation of regressors with errors in input demand, 1 This is a substantial problem when there is transmission of optimization errors from the choice of inputs. See Pope and Just (2001) for further details and for the mathematical proposition. Brazilian Review of Econometrics 31(2) November

4 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio c) correction and assessment of simultaneous equation bias, and d) consistent representation of stochastic elements with the dual method. Mundlak and Hoch (1965) say that the estimation of the Cobb-Douglas production function can be inconsistent if input demands are not independent from the error term specified in the production function. Depending on the degree of transmission of production shocks to the input demand functions, the estimates are not consistent. In this paper, a multiplicative production shock is used, not affecting input demand. For example, it can be assumed that the final output level relies on weather conditions, which is common in the agricultural setting. It is not hard to fancy a situation in which weather conditions affect only the final output, with decisions about inputs being made at the outset of the growing season. Therefore, error in production is explained by weather events during the growing season. Nonetheless, decision makers can quickly respond to seasonal shocks by changing their decision on inputs, compromising the production error with the decision on inputs. Depending on the farmer s capacity to react, the effect of these shocks may affect the researcher s capacity to identify the basic economic structure of the problem in a consistent fashion. According to Pope and Just (2001, 2003), if decision makers do not know about the stochastic variations when the decisions on inputs are taken, then disturbances in production cannot affect the decision on inputs. But errors in the decision on inputs may or may not affect output, depending upon whether they affect the effective amount of inputs used or whether they only represent measurement errors. This implies that the transmission of errors between production, factor, and supply is not symmetric. It is reasonable to think that these influences are totally or partially transferred to input demand. So, when that occurs, standard econometric estimates of production and of dual functions are inconsistent, as pointed out, for instance, in Mundlak and Hoch (1965), McElroy (1987), Pope (1996), Moschini (2001), and Pope and Just (2001, 2002, 2003). Pope (1996) and Moschini (2001) showed that the presence of stochastic errors in the decision about inputs along with stochastic shocks to production can generate nonlinear errors-in-variables that would yield inconsistent estimates in conventional econometric procedures. 2 Kumbhakar and Tsionas (2011) also underline the importance of error specification (optimization or measurement) in primal and dual frameworks that were originally constructed in a deterministic way. Those authors contribute to the literature for they discuss, both theoretically and empirically, about the consequences of how stochastic errors are specified in primal and dual formulations using a flexible functional form of the translog type and assuming a production 2 McElroy (1987) drew attention to the increase in the complexity of input demand structures in the presence of stochastic errors specified in an additive fashion. 298 Brazilian Review of Econometrics 31(2) November 2011

5 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? cost minimization problem. The results show that stochastic errors (nonlinear and additive) specified in the dual cost system are accurate error functions both in the primal method and in the production function. However, as a consequence, error structures in primal and dual methods are different, bringing important implications for their econometric estimations. 3 The present paper differs from that of Kumbhakar and Tsionas because it uses a dual profit maximization problem as starting point and focuses specifically on the importance to verify the properties of the OLS estimators and of the instrumental variable (IV) when there is transmission of stochastic errors in input demand to production, using a Cobb-Douglas production function, in addition to not assuming the transmission of stochastic errors in production to input demand. 4 Pope and Just (2002) demonstrated that an ad hoc inclusion of disturbances for specification of supply and demand derived under certainty may compromise integrability. The main problem is that econometric methods assume independent errors, but optimization errors impose dependence. Hence, if effective production depends on effectively used inputs, which differ from those initially predicted (expected), as noted by Mundlak and Hoch (1965), input errors are transmitted to final output, causing econometric estimation problems. According to Mundlak and Hoch (1965), in the case of total transmission of production shocks to input allocation, the OLS estimator is inconsistent and may be upward biased in cases of decreasing returns to scale. A simple IV method yields consistent estimates. Under partial transmission, OLS and IV estimators are inconsistent and may be biased upward or downward. According to Mundlak (1996), the case of no transmission estimated by OLS yields unbiased, consistent and efficient estimates, as also concluded by Zellner et al. (1966) for a Cobb-Douglas production function. 3. Data and Theoretical Model 3.1 Standard profit maximization Initial assumptions are related to the agent s problem and to the technology to be adopted. The problem here is based on a profit-maximizing agent. For that purpose, a representative farmer that maximizes his profit could be considered. Using a Cobb-Douglas 5 technology with decreasing return to scale, the agent maximizes 3 Kumbhakar and Tsionas use two alternative ways to model the theoretical problem of introducing stochastic errors into the dual cost minimization method, justifying what the authors referred to as primal or dual systems. 4 OLS and IV estimates are expected to be biased and inconsistent due to the transmission of stochastic errors in input demand to production (but not vice versa), even in the case of a Cobb-Douglas technology, as outlined by Mundlak and Hoch (1965), Zellner et al. (1966) and Mundlak (1996). 5 A survey and a comparative theoretical discussion between the use of CES and Cobb- Douglas functional forms in macroeconomic forecasting models are shown in Miller (2008). For other studies on the theoretical or empirical characteristics of these functional forms, see Berndt Brazilian Review of Econometrics 31(2) November

6 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio his profit conditional on the available Cobb-Douglas technology. This technology basically consists of two inputs: capital and labor. Therefore, the agent s profit maximization problem is represented by: max Π (w 1, w 2, p) = py w 1 x 1 w 2 x 2 x 1, x 2 s.t.f(x 1, x 2 ) = x a 1x b 2 y where w i is the price of the i-th input; x i is the quantity of the i-th input used; y is the level of production; p is the output price; a and b are positive parameters of labor (x 1 ) and capital (x 2 ) inputs, respectively. Since this profit function satisfies all the theoretical properties of a profit function, such as: being nonnegative, not decreasing in p, not increasing in w i, being convex and continuous, and having positive linear homogeneity, it is possible to recover the underlying production function from this specific profit function. This means that one can apply the dual theory to recover the technology used. Once again, the main question posed by this paper is: Which method (primal or dual, or both) can best estimate the technology used? To answer this question, it is necessary to make some assumptions about the initially known technology to be used. First, input prices are considered to be exogenous. Second, it is necessary to fix parameters a and b such that it is possible to guarantee a technology with decreasing return to scale. To meet this criterion, we arbitrarily use a = 0.7 and b = Once the primal function is estimated, the production function can be recovered easily. The same applies to the dual function when it is possible to use Hotelling s lemma in the estimated profit function to obtain input demand, 7 which can also be used to recover the production function. By means of a Monte Carlo simulation, a dataset with 60 observations was generated (monthly data for 5 years) for input prices, input quantity, expected production, and also production shock (due, for example, to weather changes) that may affect final output in each period. Wages vary uniformly at the average of 0.6 monetary units; price of capital also varies uniformly at the average of 0.5 monetary units; likewise, the expected output price varies uniformly at the (1976), Griliches and Mairesse (1995), and Fraser (2002). 6 Since a Monte Carlo simulation is used for the data generating process, the selection of magnitudes of the Cobb-Douglas function coefficients is only restrictive if the sum is less than 1; so, the magnitudes can take any combinations that fulfill this requirement, and then, specifically, the values 0.7 and 0.2 are only one of the possibilities. 7 However, in the presence of some errors in input demand, the application of Hotelling s lemma for the profit function does not lead to input demand and output supply in the conventional manner (Pope and Just, 2001). The main consequence for the econometric estimation would be an inconsistent OLS estimator. The proofs for these important comments are not within the scope of this paper. 300 Brazilian Review of Econometrics 31(2) November 2011

7 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? average of 2 monetary units. A random lognormal shock is also generated, which determines the actual or observed output. To generate the labor and capital input demands, it is necessary to solve the agent s profit maximization problem in order to obtain the input demand expressions that maximize the profit conditional on a given technology. After substituting the production function into the objective function, the firstorder conditions for the problem above are given by: w 1 = apx a 1 1 x b 2 (1) w 2 = bpx a 1x b 1 2 (2) ( ) b ( ) 1 b b x (1 a b) (1 a b) a 1 1 (w 1, w 2, p) = p (1 a b) (3) x 2 (w 1, w 2, p) = w 2 ( b w 2 w 1 ) 1 a ( ) a (1 a b) (1 a b) a 1 p (1 a b) (4) w 1 These expressions are the optimal input demands for a profit-maximizing agent. Maximum profit is given by the following expression: Π (w 1, w 2, p) = p [( x a 1 ( w 1, w 2, p )) ( x b 2 (w 1, w 2, p) )] w 1 x 1 w 2 x 2 (5) Based on the preceding assumptions and using the expression above, input demands and expected output are calculated from the generated data for output price, input prices and random shocks. Random shocks were included as multiplicative random production shock to represent changes in productive (weather) conditions. It is thus possible to affirm that ỹ = y exp(ε), where ε is a random normal shock and ỹ is the observed production. Five hundred simulations are generated using the Monte Carlo procedure, providing a set of 500 data on 60 observations used to estimate the primal and dual coefficients through the mean of regression coefficients. In order to capture technological changes over time, it is also assumed that the model contains a Hicks-neutral technical change component. This technical change is regarded as disembodied, or as an investment-neutral technical change, implying that output levels increase without new capital investments. According to Chambers (1988), a production function is Hicks-neutral if, and only if, it can be written as: y = f (φ (x 1, x 2 ), t) (6) As the Cobb-Douglas technology is linearly homogeneous and homothetic relative to labor and capital, expression (6) implies that: Brazilian Review of Econometrics 31(2) November

8 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio f(x 1, x 2, t)/ x 1 = 0 (7) t f(x 1, x 2, t)/ x 2 Analyzing expression (7), one notes that technical change does not alter the ratio of marginal products used in the production process. Therefore, technological change will not alter the marginal rate of technical substitution. Due to the homotheticity of the underlying Cobb-Douglas technology used in this paper and to the assumption of Hicks neutrality, technology will simultaneously be cost and profit neutral. This implies that technical change does not modify the optimal input ratios or the profit-maximizing input ratios. 3.2 Modified (Expected) profit maximization The complication to be included in this analysis concerns the possible existence of random errors not only in the production function, due to production factors that cannot be controlled by the agent, but also in capital and labor input demand due to errors during the agent s decision-making process. Hence, the contribution of the present paper goes beyond that of Pope (1996), as they considered input demand to have a deterministic behavior. Discussions and details about the implications and consequences of stochastic errors in the model can be found in Moschini (2001). Both papers deal with the problems associated with the use of the ex-post and ex-ante cost function, but the expected level of production, which is relevant to cost minimization, is not observed. This paper contains two sources of errors: primal error due to the stochastic production function and the errors in input demand. According to McElroy (1987) and Moschini (2001), the major consequence of these errors is that if the equations are estimated by the dual approach, the model will belong to the class of nonlinear errors-in-variables models, yielding inconsistent estimates. Although agents know about the presence of errors in input demand, they cannot avoid them. This means that entrepreneurs choose inputs such as x = f( x, e), where x stands for an unobserved vector of inputs, and e are the random uniform errors in input. Moschini (2001) developed and suggested a method based on the expected profit maximization problem, which yields consistent estimates of basic technology parameters because it removes the errors-in-variables problem. According to Mundlak and Hoch (1965) and Pope and Just (2001, 2003), it is assumed that stochastic errors in production are not transmitted to input, but that an inverse transmission occurs, i.e., from inputs to production. Therefore, OLS estimators must be inconsistent and biased. In the Appendix, there is a simple case where the OLS is not consistent, just because errors are not properly captured by the economic agent, as a result of differences in input quality or in managerial capacity, as suggested by Brown and Walker (1995). In other words, as outlined by Pope and Just (2001, 2002, 2003), since the errors in the selection of inputs are transmitted to production, Hotelling s lemma does not hold, and the 302 Brazilian Review of Econometrics 31(2) November 2011

9 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? errors are regarded as optimization errors, because equation (8) is affected not only by production errors (ε), but also by errors in input demand (e). Thus, the primal and dual models to be estimated in this paper, given the presence of technical change and of errors in production and in input demands, are: ln y (x 1, s 2, t, ε) = k + a ln x 1(w 1, w 2, p, e) + b ln x 2(w 1, w 2, p, e) + γt + ε (8) ln Π (w 1, w 2, p, t, u) = α + ϕ ln p + µ ln w 1 + φ ln w 2 + βt + u (9) where u in (9) includes both e and ε from equation (8), and k and α are constant terms. Both e and ε have constant variance, are uncorrelated, and are independent by construction. 8 The estimates of a and b can be recovered by multiplying φ and µ by the negative of the inverse of ϕ. Equation (8) is estimated by two different procedures: OLS and IV, 9 and the results are then compared. 4. Results and Discussion 4.1 The standard profit maximization problem The econometric procedure is carried out in two stages. First, the primal problem is estimated by the production function using OLS and the observed production in natural log (ỹ) as dependent variable against the natural log of labor (x 1), natural log of capital (x 2) and shocks. Second, the dual problem is estimated by the natural log of the profit function (5) as dependent variable against the natural log of wage (w 1 ), natural log of price of capital (w 2 ), and input demands, given by (3) and (4). The results of the regressions estimated by the primal and dual methods are shown in Table 1. Note that the estimated coefficients a and b are very similar to those assumed in the Monte Carlo simulation. In both estimates, only a was significantly different from zero at 1%. To obtain the coefficients from the dual estimation, it was necessary to multiply the labor and capital coefficients by: 1/coefficient of p. The estimates did not show any significant change when the number of draws was reduced from 500 to 200, 100, 80, 50 and 20. So, the estimated coefficients were consistent with the known parameters, as expected. 8 i.e., in the simulation, the data were obtained in a way that the errors in production and in input demand were independent. 9 Output and input prices will be used as instruments in this case. The justification for that lies in the fact that these prices were constructed in an exogenous fashion in the simulation, and that they were theoretically important for the agents decision-making process, in addition to being independent from the residuals in the production function and in the input demand equations. Kumbhakar and Tsionas (2011) also used similar instrumental variables. Brazilian Review of Econometrics 31(2) November

10 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio Table 1 Estimation of primal and dual functions in the standard profit maximization problem with Cobb-Douglas technology with decreasing returns to scale Coefficients Primal t-value Dual t-value True values a b Using 500 draws. The t value is used to test whether the parameters are statistically different from zero. It was also tested whether the estimated coefficients were statistically equal to the known coefficients a and b. In Table 2, the results show that it is not possible to reject the null hypothesis that estimate a of the primal and dual functions is equal to 0.7, since the calculated t value was 0.03 and 0.003, respectively. The same hypothesis tests demonstrated that the estimates of b are statistically equal to 0.2, as the calculated t values were 0.04 and 0.05, respectively, for the primal and dual estimations. Table 2 Estimation of primal and dual functions in the standard profit maximization problem with Cobb-Douglas technology with decreasing returns to scale Coefficients Primal t-value Dual t-value True values a b Using 500 draws. The t value is used to test whether the parameters are statistically different from a = 0.7 and b = 0.2. The performance of both methods was similar if we compare the numbers obtained in Table 1 and the described t-tests. For that reason, the estimates were reassessed using a 95% confidence interval, as shown in Table 3. Table 3 95% confidence interval for primal and dual estimations Coefficients Primal Dual Lower CI Higher CI Lower CI Higher CI a b Bootstrapping resampling method using 2,000 replications, according to Davison and Hinkley (1997) and Efron and Tibshirani (1997). Even though the true (known) parameters are within the confidence interval 304 Brazilian Review of Econometrics 31(2) November 2011

11 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? in the primal and dual estimations, we can see in Table 3 that dual estimations performed better in terms of efficiency, as the distribution of the estimates is concentrated within a very narrow interval. Therefore, the interval of estimates for the primal formulation was larger than for the dual one. 4.2 Modified (Expected) profit maximization problem The results for primal and dual formulations with stochastic errors in input demand are shown in Tables 4 and 5, which display the average bias of the estimated parameters and the sample size with the respective true parameters a = 0.7 and b = 0.2. The average bias of the estimated parameters consists of the simple difference between each true parameter and the average value for the different estimations (one for each sample size) of the respective estimates of a and b. The tables show that the average bias decreases as sample size increases, though confidence intervals are larger with additional draws. This suggests that the estimator is consistent, but inefficient. The IV estimator seems to produce a smaller bias, but it is less efficient than the OLS estimator. Table 4 Average bias for primal and dual formulations Average bias (OLS) Coefficients T = 50 T = 100 T = 200 T = 500 T = 550 a b Average bias (IV) Coefficients T = 50 T = 100 T = 200 T = 500 T = 550 a b Average bias (Dual) Coefficients T = 50 T = 100 T = 200 T = 500 T = 550 a b Dual estimation appears to converge to smaller biases more quickly than does the primal method, although the absolute level of bias is larger for smaller sample sizes. The dual estimate also has a smaller confidence interval than those observed in the primal formulation, suggesting that the dual estimates have a smaller bias and are more efficient. From the theoretical standpoint, these results bring up an interesting issue, since several authors, such as Mundlak and Hoch (1965), Moschini (2001), Pope and Just (2001, 2002, 2003) suggest that the OLS and IV estimations of primal and dual equations in the presence of errors in the selection of inputs, which are Brazilian Review of Econometrics 31(2) November

12 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio Table 5 Confidence interval for primal and dual formulations 95% confidence interval (OLS) Coefficients T = 50 T = 100 T = 200 T = 500 T = 550 a b % confidence interval (IV) Coefficients T = 50 T = 100 T = 200 T = 500 T = 550 a b % confidence interval (Dual) Coefficients T = 50 T = 100 T = 200 T = 500 T = 550 a b Brazilian Review of Econometrics 31(2) November 2011

13 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? found in the observed production function, should yield biased and inconsistent estimates, except for the quadratic functional equation (Pope and Just, 2001, 2003). 4.3 Policy implications What would happen if an economic agent (or an econometrician) did not take into account possible errors during the selection of inputs to be used? What would the main implications be for policymakers when they try to boost the productive sector and these errors are overlooked? To answer these questions, let us suppose that the government is willing to increase production at the sectoral level by way of subsidies on the price of capital used in a given sector. Thus, expected production and profit should increase in this sector. But that it is not an easy task because the problems that could affect decisions about production, as discussed earlier, could also generate some error in resource allocation and/or measurement errors in the major inputs used by policymakers in the productive process, or by the econometrician who is modeling the productive sector in order to assess the impact of the subsidy policy on the economy. As an example of the implications of not taking into account the presence of errors in input demand, let us suppose a 10% subsidy on the price of capital (w2) in the Cobb-Douglas production function, which implies that the random price of capital generated before the subsidy is reduced by 10%. Table 6 shows production and the profit value for each sample size, as well as the current values for production and profit when errors in input demand are taken into account. Table 6 Production and average profit with 10% subsidy on the price of capital used (in 1,000 units) Average quantity and sample size without errors in input Production T = 50 T = 100 T = 200 T = 500 T = 550 Levels 26,633 27,122 27,064 26,170 26,123 Profit 50,742 51,517 50,989 48,994 48,909 Average quantity and sample size with errors in input Production T = 50 T = 100 T = 200 T = 500 T = 550 Levels 53,360 55,085 55,291 53,382 53,281 Profit 96,427 99,359 98,780 94,801 94,652 Table 7 shows the magnitude of the average bias towards production and profit with subsidy on the price of capital when the error in input demand is not taken into account by policymakers or by the entrepreneur during the productive process. It is also possible to note that the estimation bias does not decrease as sample Brazilian Review of Econometrics 31(2) November

14 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio size increases. The Appendix shows that in a formal way because the OLS does not yield a good estimate when errors in input are not accounted for in case of measurement error. Table 7 Bias towards production and profit with a 10% subsidy on the price of capital used (in 1,000 units) Bias and sample size Production T = 50 T = 100 T = 200 T = 500 T = 550 Levels 26,728 27,962 28,227 27,211 27,158 Profit 45,685 47,841 47,790 45,806 45,743 In the presence of stochastic errors in input demand, not taking these errors into account may lead to smaller production and lower profit, as illustrated by Taylor (1989), where the optimal level of inputs to be used was not properly utilized, with a solution that was far away from the profit-maximizing levels of production inputs. Therefore, any type of policy that can increase the total production of a given sector should take into consideration the issues discussed here, thereby avoiding biases and inefficiency before it is applied to the real world. So, unsurprisingly, profit and production are higher with errors in input allocation than without them. This occurs due to how the model was developed, equations (8) and (9), with additive error terms in the log specification of the model, 10 and also due to the strong monotonicity of the production function ( y e > 0). 5. Final Remarks This paper raised the possibility that both dual and primal formulations could yield good results when some empirical complications are included in the model. A dataset based on the behavior of some representative agent is constructed using the Monte Carlo simulation and used to econometrically estimate the primal and dual functions associated with a given technology. One of the major contributions of this paper is the analysis of the econometric properties of primal and dual estimations for a profit-maximizing agent using data with possible stochastic errors in production and in input, which are not directly observed by the econometrician. Additionally, this paper sought to determine the possible influence of stochastic errors in input demand and in production on the solution to the profit maximization problem. The goal was to check the validity of the OLS estimator to obtain the 10 Negative errors in input demand were used in the log specification and the results for production and profit were exactly the opposite of those shown in Table 6. For instance, for T = 500, the average profit in the presence of errors was 43.6 and the average profit without the errors was Brazilian Review of Econometrics 31(2) November 2011

15 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? technological parameters under the hypothesis of no transmission of production shocks to inputs, but also the transmission in the opposite direction, in addition to the consequences when these errors are not accounted for by policymakers at a more aggregate level. The empirical estimations showed that primal and dual formulations efficiently recover the true parameters of the Cobb-Douglas technology used. Dual estimation was better in terms of the range of the confidence interval and bias. Some possible empirical implications were also discussed using the hypothetical situation of subsidy on the price of capital in a given sector. Entrepreneurs or policymakers, by not taking into account the presence of stochastic errors in input demand, may make wrong decisions, with inefficient resource allocation and substantial bias due to misperception of these errors. An important theoretical and empirical contribution was the demonstration that, even in the presence of transmission of optimization errors in input demand to production, in an unobservable fashion, OLS and IV estimations of primal and dual formulations based on a Cobb-Douglas function were consistent and unbiased, unlike those expected by Zellner et al. (1966), Mundlak (1996) and Pope and Just (2002). In fact, the latter concluded that this could only happen with a technology in quadratic form. But, in practice, transmission of input to production implies that other classes of estimators may be used to obtain unbiased, consistent and efficient estimates. As suggested by Pope and Just (2001, 2003), depending on the source of error, and on how this error is specified (Kumbhakar and Tsionas, 2011), a different estimation method is necessary, and given the source of potential multiplicative error, a new method for modeling the producer s behavior is suggested. Hence, the logical extension of this paper would be to assess the reasons for the good performance of the OLS method in primal and dual estimations of a Cobb-Douglas technology. A possible future investigation may also include the potential transmission of errors in production to input demand, or also suggest a combination of the principles discussed in Pope and Just (2001, 2003) and Mundlak and Hoch (1965). Another possibility would be the inclusion of non-hicks-neutral technical change, since some empirical studies reject Hicks-neutrality in technical changes in production, as pointed out by Shumway (1995). References Appelbaum, E. (1978). Testing neoclassical production theory. Journal of Econometrics, 7: Berndt, E. (1976). Reconciling alternative estimates of the elasticity of substitution. Review of Economics and Statistics, 58: Brazilian Review of Econometrics 31(2) November

16 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio Brown, B. & Walker, M. B. (1995). Stochastic specification in random production models of cost-minimizing firms. Journal of Econometrics, 66: Burgess, D. F. (1975). Duality theory and pitfalls in the specification of technologies. Journal of Econometrics, 3: Chambers, R. (1988). Applied Production Analysis: The Dual Approach. Cambridge University Press. Davison, A. C. & Hinkley, D. V. (1997). Bootstrap Methods and their Application. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. Efron, B. & Tibshirani, R. (1997). Improvements on cross-validation: The bootstrap method. Journal of the American Statistical Association, 92: Fraser, I. (2002). The Cobb-Douglas production function: An antipodean defence? Economic Issues, 7: Griliches, Z. & Mairesse, J. (1995). Production functions: The search for identification. In Strom, S., editor, Econometrics and Economic Theory in the 20th Century: The 27th Ragnar Frisch Centennial Symposium. Cambridge University Press, Cambridge. Just, R. (2000). Econometric modeling of agricultural production: Are we in the right track? In Annual Joseph Havlicek Jr. Memorial Lecture. The Ohio State University, Columbus, Ohio. Kumbhakar, S. C. & Tsionas, E. G. (2011). Stochastic error specification in primal and dual production systems. Journal of Applied Econometrics, 26: Lusk, J., Featherstone, A., Marsh, T., & Abdulkadri, A. (2002). Empirical properties of duality theory. The Australian Journal of Agricultural and Resource Economics, 46: McElroy, M. (1987). Additive general error models for production, cost and derived demand or share systems. Journal of Political Economy, 95: Miller, E. (2008). An assessment of CES and Cobb-Douglas production functions. Working Paper 5, Congressional Budget Office. Moschini, G. (2001). Production risk and the estimation of ex-ante cost functions. Journal of Econometrics, 100: Mundlak, Y. (1996). Production function estimation: Reviving the primal. Econometrica, 64: Brazilian Review of Econometrics 31(2) November 2011

17 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? Mundlak, Y. & Hoch, I. (1965). Consequences of alternative specifications in estimation of Cobb-Douglas production functions. Econometrica, 33: Pope, R. (1980). The generalized envelope theorem and price uncertainty. International Economic Review, 21: Pope, R. (1982a). Expected profit, price change, and risk aversion. American Journal of Agricultural Economics, 64: Pope, R. (1982b). To dual or not to dual? Western Journal of Agricultural Economics, 7: Pope, R. (1996). Empirical implementation of ex ante cost functions. Journal of Econometrics, 72: Pope, R. & Just, R. (2001). Distinguishing errors in measurement from errors in optimization. Mimeo (not published), 32 pages. Pope, R. & Just, R. (2002). Random profits and duality. American Journal of Agricultural Economics, 84:1 7. Pope, R. & Just, R. (2003). Distinguishing errors in measurement from errors in optimization. American Journal of Agricultural Economics, 85: Shumway, C. R. (1995). Duality contributions in production. Journal of Agricultural and Resource Economics, 20: Taylor, C. R. (1989). Duality, optimization, and microeconomic theory: Pitfalls for the applied researcher. Western Journal of Agricultural Economics, 14: Thompson, G. D. & langworthy, M. (1989). Profit function approximations and duality applications to agriculture. American Journal of Agricultural Economics, 71: Young, D. L. (1982). Relevance of duality theory to the practicing agricultural economist: Discussion. Western Journal of Agricultural Economics, 7: Zellner, A., Kmenta, J., & Dreze, J. (1966). Specification and estimation of the Cobb-Douglas production function. Econometrica, 34: Brazilian Review of Econometrics 31(2) November

18 Mauricio Vaz Lobo Bittencourt and Armando Vaz Sampaio Appendix: Inconsistency of OLS Estimation of the Primal Function with Stochastic Errors in Input Let us consider that it is possible to observe production and the inputs used in the production function as: y i = y i + u i and x i = x i + v i The lower-case letters represent the logarithmic form of variables Y (level of production) and X (input level). We may assume that: 1) E(u i ) = E(v i ) = 0; 2) V ar(u i ) = σ 2 u and V ar(v i ) = σ 2 v; 3) Cov(u i, v j ) = Cov(v i, v j ) = 0 for i j; 4) Cov(u i, vj ) = 0 for i, j, implying that errors in production are independent from errors in input. The problem arises when we try to estimate the production function using only observed variables y and x. Then, if the true production function is given by y i = β 1 + β 2 x i, we have: (y i u i ) = β 1 + β 2 (x i v i ) implying that y i = β 1 + β 2 x i + e i where e i = u i β 2 v i The error term e i has zero mean and constant variance, but its covariance with input level (x) is nonzero. Cov(x i, e i ) = E [(x i E(x i ))(e i E(e i ))] E(v i e i ) = E [v i (u i β 2 v i )] = E(v i u i ) E(β 2 v 2 i ) = β 2 σ 2 v Thus, if the OLS is used to estimate the primal function, we will obtain a biased and inconsistent estimate, as shown below: p lim b 2 = p lim [T 1 T i=1 (y i ȳ )(x i x )] [T 1 T i=1 (x i x )] = p lim β 2 + p lim[t 1 T i=1 (x i x )e i ] p lim[t 1 T i=1 (x i x ) 2 ] = β 2 β 2σ 2 v σ 2 x = β 2 ( 1 σ2 v σ 2 x 312 Brazilian Review of Econometrics 31(2) November 2011 )

19 Are Dual and Primal Estimations Equivalent in the Presence of Stochastic Errors? The OLS estimator is not only biased, as shown in the expression above, but also inconsistent, since the bias does not converge in probability to zero. It is interesting to note that the bias of estimator b 2 is proportional to the ratio between the variances of input errors (v i ) and the observed input level (x ). It is clear that if the errors occur only in final production, the OLS estimators are consistent and unbiased because p lim b 2 = β 2. But that holds if we consider that the error in production is not transmitted to input demand, i.e., the case of no transmission dealt with in Mundlak and Hoch (1965) and in the present paper. Brazilian Review of Econometrics 31(2) November