Returns to Scale in Agriculture: A Suggested Theoretical Framework. Arianto A. Patunru 1 Department of Economics, University of Indonesia

Transcription

1 Returns to cale in Agriculture: A uggested Theoretical Framework Arianto A. Patunru 1 Department of Economics, University of Indonesia This version, ay 26, 2004 tandard economic theory suggests that increasing returns come from indivisible fixed assets. However, many analysts argue that the long run size distribution of machinery, land, structures, irrigation, herds, and flock are continuous. Our objective here is to provide a framework to claim that there are significant returns to scale in agriculture despite the contradictions in past studies. We consider the importance of management ability in production process (efford, 1986). The short-run rigidities or lumpiness 2 of management ability, we believe, would result in scale economies. Therefore, larger farms with higher ability to utilize more will exhibit lower unit costs. We start with behavioral explanation for returns to scale based on the lumpiness of management ability. Next, we develop an econometric model that will allow us to test the behavioral explanation. Behavioral explanation. Quasi-fixity of management ability means that this input cannot adjust instantaneously to exogenous shock (e.g. technology change) that may otherwise lower the costs. An introduction of new technology, for example, requires advancement in management ability so that it can be used to produce in more efficient manner. However, it is very likely that knowledge and skills are developed not fast enough to catch up with the technology. The rapid change in computer technology, for instance, does not instantaneously contribute to the production process 1 I thank Carl Nelson for guidance. 2 Throughout the analysis, we refer short-run rigidity, lumpiness and quasi-fixity to same meaning. The term quasi implies that the input in-question can adjust even in the short-run, but slower than variable inputs. 1

2 in average farms, since it takes significant amount of time for the management to learn. uch short-run constraint limits the farmers ability to be operating at the optimal long-run size. They may, however, be producing in the area of increasing returns to scale. Gradual increase in management ability will allow them to take advantage of this situation in the form of cost reduction. Therefore, as we mentioned above, farms that are larger or more highly utilized will be able to produce more efficiently, in terms of lower unit costs (Paul, 1999, 2000). To be more systematic, we use the concept of capacity utilization and total factor productivity, TFP (Paul, 1999; orrison, 1993) 3. In addition, dual approach (Diewert, 1984) is employed 4. That is, we assess this problem using a cost function. Finally, since we assume quasifixity of management ability, we need to examine a restricted cost function, instead of a total cost function. One study has found that using total cost in the case of quasi-fixed inputs would result in biased estimations (Caves et al, 1981) 5, since it implies the prices of all inputs together, including the quasi-fixed inputs. A variable cost function as a restricted function, on the other hand, explicitly represents quasi-fixity and therefore represents short-run costs. Assuming only management ability is lumpy, we then can present the variable cost function as VC( w,,,, t), where w is the vector of variable input prices, represents the management ability level, is output, and t is the shift variable (e.g. time as proxy to technology). We include in the function to account for the quasi-fixity nature of the management ability, i.e. allowing for partial adjustment (e.g. investment) in knowledge (Paul, 3 Although the concept of capacity utilization is closely linked to capital input, here we address it to management ability as the quasi-fixed input. 4 There are some caveats of using dual approach (see e.g. undlak, 1996). We choose dual approach since it more conveniently fits with capacity utilization measure. As orrison put it, it is directly amenable to specification and interpretation of relevant economic parameter (orrison, 1993). In addition, Park and won (1995) has reported that primal approach (production side) may incur biases in the case of estimating non-constant returns to scale and needs to be corrected. 5 The estimation procedure will be discussed in the next section. 2

3 1999). However, we assume no depreciation 6. By this construction, it follows that the short-run total cost function is therefore TC = VC( ) + w where w is the market price of management ability. Employing envelope theorem, we can define a shadow value for the management ability as Z =. This shows the value of extra unit addition to management ability in terms of VC the potential reduction in variable input costs in producing the given output. In addition, the shadow value for investment in knowledge is Z =. And corresponding shadow cost VC function is then TC* = VC( ) + Z. Farm s input demand decision that is constrained by quasi-fixed management ability will result in a sub-equilibrium that differs from long run equilibrium when full adjustment is completed. This leads to a deviation of shadow economic values of this input from its corresponding market price. Therefore, w Z in the short run but w = Z in the long run (assuming a constant long-run returns to scale). Following Park and won (1995), we can derive the TFP function from the cost side as follows: w v v Z Z i i i TFP = ε = ε (1) TC v TC TC where = ln TC ε t and ε = ln TC ln. 7 We define the latter as the short-run cost elasticity with respect to output. We use this measure of short-run cost curvature as the short-run returns to scale, since it is the inverse of short-run scale economies (orrison, 1993; Paul, 1999). The terms wi and v i refer to the price and quantity of i-th variable input. We can also calculate the primal TFP (see Park and won, 1995) and relate it to the dual as: ε = ε. t ε = 6 Therefore, 1 7 This generalized TFP is preferable than the traditional measure, since it decomposes the productivity into scale economies, technical change, and utilization rate. 3

4 orrison (1993) has developed a cost-based measure of capacity utilization. he constructed a capacity utilization index for the case of fixed input and constant long-run returns to scale as CU c = 1 ε, where the last term here is the elasticity of total costs with respect to TC management ability. ince this index is the ratio of shadow cost function TC* and short-run total cost function T, we know that if the market price for management ability is less (greater) than its shadow value, i.e. w < Z ( Z w > ), then the management ability is over- (under-) utilized. Overutilization implies that the costs in the short-run increase more than proportionately with output, although they increase proportionately in the long run. We now can derive the long-run cost elasticity, ε = d ln TC as: d ln lntc lntc ε = + = ε + ε ln ln Therefore, for constant long-run returns to scale, we obtain TC c (2) ε = 1 ε = CU. In this case the capacity utilization index of management ability equals the short-run returns to scale. If the index is greater than one, there are potential gains from cost reduction due to increase in management ability. Here, farm is operating in the area of increasing returns to scale. If this is next supported by empirical estimation, we say: there are increasing returns to scale in agriculture, due to the lumpiness of management ability. Now we would relax the assumption of constant long-run returns to scale (orrison, 1993). If, instead of one, the long-run cost-output elasticity is equal to some η, we modify the short-run elasticity into ε = η( 1 ε ) = CU ε. Therefore, the short-run elasticity is a TC c 4

5 combination of long-run returns to scale 8 and capacity utilization. If, in addition, homotheticity is ε ε TC not satisfied, we have ε η( ) = 1. 9 η We can see now the interrelationship between these measures. If the long-run cost-ouput elasticity is less than one (i.e. economies of scale) and so is the short-run cost-output elasticity, then both index greater- and less than one indicate that the farm is operating in increasing returns to scale region. That is, on the decreasing part of the short-run and long-run average cost curve. Overutilization, however, is closer to the minimum point of these curves. In addition, the shortrun cost-output elasticity that is less than one implies a falling TFP. Econometric model. We now turn to developing an econometric model for the behavioral explanation above. First of all, we need a functional form approximation of VC( w,,,, t) that captures the cross-effects among all the arguments (i.e. no a priori restrictions on the shapes of the curves representing the production technology). At this point, it is tempting to use a translog cost function as has been done e.g. by Brown and Christensen (1981). This flexible function allows straightforward calculations of elasticities and returns to scale since they depend only on the parameter estimates. However, it is appropriate for long-run or instantaneous adjustment only, while its logarithmic nature creates difficulties for short-run analysis. That is, the derivation of shadow values requires a highly complicated algorithm (Paul, 1999). In our case, we are interested in the gradual movement of the lumpy management ability. Berndt, Fuss, and Waverman (1980) use a normalized quadratic function to allow for input fixity. In this case η = = 8 That is, C C AC TC* where C and AC are the marginal- and average costs, respectively. Thus, the longrun returns to scale is the inverse of the long-run cost elasticity. Note that we abstract from unnecessary confusion between economies of scale and economies of size, since we assume cost-minimizing farm. (see e.g. Chambers, 1988). 9 It can be shown that lower returns to scale will be obtained if we mistakenly use the formula for homothetic case. ee Oum et al (1991). 5

6 adjustment matrixes are much easier to compute. However, the normalization yields results invariant to the chosen normalizing input (Paul, 1999). Given these considerations, we follow orrison (1993, 2000) and Park and won (1995) to use a generalized eontief function. As an alternative, one may use the method proposed by Rask (1995), that is, a generalized cfadden function 10. For the first approach, we choose a non-constant returns to scale short-run generalized eontief cost function, with two variable inputs (capital and labor) 11 as follows 12 : VC( w, w,, t,, I ) = {( α w + 2γ w w + α w ) + ( γ w + γ w I + γ w t + γ w + γ w I + γ w t ) + I t ( α + 2γ I + 2γ t + α I + 2γ I t + α t)( w + w )} + I t I {( γ w + γ w ) + ( γ + γ I + γ t )( w + w + α w + w ) (3) ( Using hephard s emma, we obtain the factor demand functions: It I I t t t )} VC 1 = w. 5 = { α + γ w w + γ + γ I + γ t + α + α I + α t + I t I t 2γ I + 2γ t + 2γ I t } + { γ + γ I t It. 5 + γ I. 5 + γ t } + γ (4) I t and 10 Rask (1995) estimates his model of Brazilian sugarcane production using generalized cfadden function, modified to allow for fixed inputs. He, however, did not consider the dynamic adjustment of those inputs. We need to adjust his model to account for quasi-fixity. 11 It is somewhat unusual to treat capital this way (i.e. without investment flow). However, our purpose here is to examine the management ability as the lumpy input. Otherwise, we can have two quasi-fixed inputs, and and another variable input (e.g. material). An example for multi (quasi-) fixed inputs, see orrison (1988). 12 Note, for convenience in notation, we replace to I. 6

7 VC 1 = w = { α + γ w w + γ + γ I I + γ t t + α + α I + α t + I t 2γ I + 2γ t + 2γ I t } + { γ + γ I t It. 5 γ I. 5 + γ t } + γ (5) I t To take market condition into account, suppose we have an inverse output demand function as follows (Park and won, 1995): P = β 0 + β + β P + β U + β p + β t (6) u p t where the variables are output price, output quantity, price of substitute, unemployment rate, interest rate, population, and preferences, respectively. Therefore, in addition to our equation-ofmotion for management ability and the profit maximizing condition R=C, we have six equations. It is common to use seemingly unrelated regression (URE) to estimate such six-equation system above. However, equation (6) and the market equilibrium condition give rise to endogeneity problem. This will lead to block-recursive model. Park and won (1996) suggest the use of Bayesian inference 13. The advantage of this technique is that the concavity conditions required for the cost function to be well-behaved is satisfied, besides the elimination of endogeneity problem 14. However, this Bayesian technique, although is easily calculated, it is conceptually difficult (Davidson and acinnon, 1993, p.676-9). Alternatively, as in orrison (1988), we can use iterative 3 estimation procedure without having to estimate the whole six equations 15. In this case, we only need to include factor + 13 If however, the complete system is estimated using three-stage least square --3 (at the cost of slightly upward bias over Bayesian estimation), the exogenous variables will serve as instrumental variables. 14 In fact the main purpose of using Bayesian inference is to ensure the concavity conditions while at the same time allowing for non-constant returns to scale. 15 Iterated 3 is preferred to iterated Zellner URE to anticipate the possibility of non-static expectation or endogenous output choice. 7

8 demand functions (/ and /) and add to this system an output-price equation, which is derived by taking the derivative of VC in (1) with respect to. This last equation implies an assumption of perfect competition. Another way is, if the dynamic optimization of the quasifixed input is of concern we can also estimate the model using /, /, and an Euler equation obtained from dynamically minimizing the total costs (orrison, 1993) 16. For the implementation of this generalized eontief model, a stochastic framework must be specified (Berndt, 1991). That is, we need to append disturbance term to the input-output equations. The resulting disturbance vector is assumed to be identically, independently distributed with mean vector zero and constant, nonsingular covariance matrix. These disturbances reflect the optimization errors by farm. Alternatively, the error term in the cost function is specified as υ = µ + ν, where the first component is the errors known by the farm but not by econometrician, and the second component is the noise observed by econometrician. The two components are independent, and they, as well as the error term associated with input-output equations, are independent with explanatory variables. The parameters obtained from the estimation procedure above are then used to calculate the significant economic parameters for analysis purposes. The short-run returns to scale parameter is computed from the estimated function. Next, we can investigate the long-run returns to scale as well as the capacity utilization using this parameter based on the behavioral model above. Prior to this, we can test the hypothesis of constant returns to scale by testing that all the coefficients associated with interaction terms between and the inputs are zero. That is, γ = γ = γ = γ I = 0. The symmetry of the input-output equations can also be tested, by investigating the cross-equation symmetry of the coefficients and then employing a Wald test 16 With additional complication of the problem, i.e. serially-correlated errors, orrison (2000) estimates similar system by using G (Generalized ethod of oments). 8

9 with one-step Zellner-efficient estimation. Other particular measurement can also be undertaken using the estimated elasticity, depending on our interest. For example, the impact of management ability level on scale economies can be investigated by the derivative of cost-output elasticity with respect to management ability level. The TFP can also be calculated using the estimated elasticities. We can use the TFP formula above to see the contribution of different sources besides technology to productivity growth. Otherwise, given that our purpose is to contrast between returns to scale and technical change in their contribution to productivity growth, we can adjust our cost-time elasticity with returns to scale to disentangle them. The cost-side R productivity growth, adjusted for returns to scale is ε (1 )( ln = ε + ε ). Where the t second term in the left hand side reflects the extent to which the standard measure of technical change is biased, in the presence of scale economies. If this term is big enough, we have evidence to support the hypothesis that productivity growth is more likely due to scale economies rather than technical change 17. On the other hand, if we like to consider the direct effect of fixity of management ability, we adjust the productivity measure to be ε F = ε + εtc ( ln t ln ). Again, the last t term reflects bias of measurement, this time if the management ability input is fixed. Combining these adjusted measures together (see orrison, 1993) would give more convincing assessment on the hypothesis. Data. In order to perform the estimation above we recommend the data to be constructed as a panel of sample farms. This data set would include total costs of production as well as the 17 This is not sufficient, though, for such claim, since other sources may also contribute or the state is simply in subequilibrium. 9

10 variable costs, the quantity of output and quasi-fixed input, as well as the prices of output and variable inputs. Gross output is used if we allow for input substitutions 18. Time (year) will be taken as a proxy to technology. If we allow the farms to exit and entry in the industry, we may have an unbalanced panel data. As a proxy to management ability, we will follow efford (1985) to use performance ranking of each farm relative to other farms every year. This measure could be constructed based on output, goal attainment, cost, quality level of the output, etc. Alternatively, the level of skill, education, and training can be another proxy. Allowing fixed specific nature of farmers, we will employ a fixed effect model. If we are interested in geographical or size differences between farms, we may use dummy variables to control for varying returns. Finally, if Bayesian inference is chosen, we need to supply additional data associated with equation (6). 18 In this case, we need to calculate substitution elasticities by taking the derivative of log input with respect to log price of another input. 10

11 Reference Berdnt, E.R The Practice of Econometrics: Classis and Contemporary. Addison Wesley Publishing Co. Inc. Berndt, E.R.,.A. Fuss, and. Waverman Dynamic Adjustment odels of Industrial Energy Demand: Empirical Analysis of U.. anufacturing, EPRI Research Project. Electric Power Research Institute. Brown, R.. and.r. Christensen Estimating Elasticities of ubstitution in a odel of Partial tatic Equilibrium: An Application to U.. Agriculture, 1947 to in E.R. Berndt and B.C. Field (eds), odeling and easuring Natural Resource ubstitution. IT Press. Caves, D.W.,.R. Christensen and J.A. wanson Productivity Growth, cale Economies and Capacity Utilization in U.. Railroads, American Economic Review. 71, Chambers, R.G Applied Production Analysis. Cambridge University Press. Davidson, R. and J.G. acinnon Estimation and Inference in Econometrics. Oxford University Press. Diewert, W.E Duality Approaches to icroeconomic Theory. in.j. Arrow and.d. Intriligator (eds), Handbook of athematical Economics II. North Holland Press. efford, R.N Introducing anagement into the Production Function. Review of Economics and tatistics. 68(1), Feb, orrison, C.J Quasi-Fixed Inputs in U.. and Japanese anufacturing: A Generalized eontief Restricted Cost Function Approach. Review of Economics and tatistics. 70(2), ay, orrison, C.J A icroeconomic Approach to the easurement of Economic Performance. pringer Verlag. undlak, Production Function Estimation: Reviving the Primal. Econometrica. 64, Oum T.H.,.W. Tretheway, and. Zhang. A Note on Capacity Utilization and easurement of cale Economies. Journal of Business tatistics 9(1), Jan, Park, -R. and J.. won Rapid Economic Growth with Increasing Returns to cale and ittle or No Productivity Growth. Review of Economics and tatistics. 77(2), ay, Paul, C.J. orrison Cost tructure and the easurement of Economic Approach. luwer Academic Publisher. 11

12 Paul, C.J. orrison Cost Economies and arket Power in U.. Beef Packing. Giannini Foundation of Agricultural Economics. Rask, The tructure of Technology in Brazilian ugarcane Production, : An Application of a odified Generalized cfadden Cost Function. Journal of Applied Econometrics. 10(3), Jul-ep,