295
The Linear Approximation of the
CES Function with n Input Variables
AYOE HOFF
Danish Research Institute of Food Economics
Abstract The Taylor approximation to the n-input constant elasticity of
substi-tution (CES) function is presented and compared to Kmenta’s well-known approximation for n = 2. The n-input approximation is, as for n = 2, a translog function, but with more complex restrictions on the translog parameters. Bias and consistency of the Taylor approximation to the n-input CES function are dis-cussed, and it is argued that the approximation will only give reliable results for a limited regime of CES parameters and input values. A test of CES structure of the n-input translog function is performed for the fleet of Danish trawlers oper-ating in the North Sea from 1987–99.
Key words CES production function, Taylor approximation, translog function,
multiple inputs.
JEL Classification Codes C12, C52, D24.
Introduction
The Constant Elasticity of Substitution (CES) production function together with the Cobb-Douglas (CD) and the translog function are widely applied functions in pro-duction theory. All three have been applied in connection with fishery economics, where increasing attention is currently being paid to establishing valid production relationships between landings (weight or value) and effort exerted, such as fishing time, labor, and fishing power. Proper understanding of such relationships is impor-tant when discussing management initiatives with the aim to sustain the fish resource.
Often in production theory only two inputs, e.g. capital and labor, are present. Thus, the theory of production functions has in many cases been developed in the two-input case. Within the fishery more than two inputs are often present, as fishing effort may include several different factors (fishing time, fishing power, labor, stock, etc.). It is thus often necessary to consider the n-input forms of the above-de-scribed production functions.1
The translog form and the logarithm of the CD are both linear, which makes these forms easy to estimate employing standard econometric packages. The two-in-put forms of these are easily extended to n intwo-in-puts, and the resulting forms, also
Ayoe Hoff is a research fellow at The Royal Veterinary and Agricultural University, Danish Research Institute of Food Economics, Fisheries Economics and Management Division, Rolighedsvej 25, 1958 Frederiksberg C, DK – Denmark, email: [email protected].
I wish to thank Peter Allerup, Danish University of Education; Hans Frost, Danish Research Institute of Food Economics; and Niels Vestergaard, University of Southern Denmark, for valuable discussions and support on the work presented herein.
1 This has been the case in the EU-project, ‘The Relationship between Fleet Capacity, Landings, and the
being linear, are easy to apply. However, the CES form is non-linear and cannot be linearized analytically. Estimating functional parameters for the CES function thus includes non-linear fitting techniques, which are generally complicated, in the two-as well two-as n-input ctwo-ase. It may therefore be advantageous, if possible, to employ a linear approximation of the CES function when estimating the parameters of this form. Kmenta (1967) presents such a linearization of the two-input CES function, employing a Taylor approximation, the result of which is a restricted form of the general translog function. Contrary to this, an extension to n inputs of Kmenta’s lin-ear approximation of the two-input CES function has not been performed to date. This paper presents this extension; i.e., the linear Taylor approximation to the n-in-put CES function, which is again a translog function, but with more general restrictions on the translog parameters than in the two-input case. This result is, as such, applicable for all productions, independent of the number of inputs employed.
It must be stressed that Kmenta’s linearization of the two-input CES function, as well as the general n-input result presented below, are both approximations of the exact CES function, based on Taylor expansions of the CES function around the elasticity of substitution equal to unity. As such, the linearizations are only appli-cable for elasticities of substitutions in the neighborhood of unity. In this connection, the paper presents a general discussion of bias and consistency of the Taylor approximation to the CES function, and suggests guidelines for the range of CES parameters (elasticity of substitution, returns to scale) and input values for which the approximation can be used. When the CES parameters and input values are outside these ranges, the exact form of the CES function must be employed, even though this includes more complicated estimation techniques.
It may be argued that as the linearization of the general CES function is only an approximation and is further only applicable for a certain range of CES parameters, the exact form of the CES function should, as a rule, be used instead of the translog approximation. This is a sound argument, which the author of this paper also sup-ports. However, there is one reason why the linear approximation to the CES function must be discussed for two as well as n inputs: Kmenta’s two-input result has often been cited in connection with the general translog function, but during the years it seems that two miscomprehensions of the original result have emerged. Firstly, it is often stated that the general, two-input translog function is equal to a CES function, given certain restrictions on the translog functional parameters. This is wrong, since the translog is only an approximation to the CES function and only applicable for certain ranges of the CES parameters, as discussed above. The second miscomprehension is when Kmenta’s two-input result is cited in connection with the n-input translog function; i.e., that the n-input translog function is an approximation to the n-input CES function, given the same restrictions on the translog parameters as in the two-input case (i.e., Kmenta’s original result). This is a wrong conclusion, as it is shown in this paper that Kmenta’s Taylor approximation for the two-input case is, in fact, not directly applicable in the n-input case, but that a more general extension of the results must be employed. Thus, as the Kmenta result is often cited in the literature, this cannot be ignored. The miscomprehensions of the result must be corrected, which is why it is important firstly to extend the result correctly to the n-input case, and secondly to discuss the range within which the result is applicable. This is, as mentioned above, the purpose of this paper.
The paper is introduced with a short presentation of the concepts of elasticity of substitution and the constant elasticity of substitution production function. Next, Kmenta’s linearization of the two-input CES function is presented, followed by a presentation of the more general n-input result. Then follows a discussion of the consistency of the approximation to the CES function; i.e., the range of CES param-eters and input values for which the approximation is valid. The paper is concluded
with an application of the CES approximation to Danish trawlers operating in the North Sea in the period 1987–99.
Elasticity of Substitution
The degree of substitutability between the input factors of a production is an essen-tial concept within production theory. Hicks (1932) was the first to introduce and discuss a dimensionless measure of substitutability of input factors, the so-called elasticity of substitution, for a two-factor production. The Hicks elasticity of substi-tution is defined as the relative change in the proportion of the two input-factors as a function of the relative change of the corresponding marginal rate of technical sub-stitution: σ = d (x2 / x1) x2 / x1 d (MP1 / MP2) MP1/ MP2 = d ln(x2 / x1) d ln(MP1 / MP2) , (1)
where x1 and x2 are the input values, and MP1 and MP2 are the marginal products of the technology. It is straightforward to show that when the technology is described by the production function y = f(x1,x2), equation (1) reduces to:
σ = (x1f1+ x2f2) x1x2
f1f2
(2 f12f1f2 − f11f22 − f22f12)
, (2)
where fi = ∂f/∂xi and fij = ∂2f/∂xi∂xj. When more than two inputs are present in the
technology, no ‘empirical’ formula exists for the elasticity of substitution. Chambers (1988) lists three alternatives, which are the most widely used and well known in the literature: (i) the Direct Elasticity of Substitution (DES), (ii) the Allen Partial Elasticity of substitution (AES), and (iii ) the Morishima Elasticity of Substitution (MES).
The DES between input factors xi and xj (i,j = 1,…,n) is defined by:
σijDES = (xifi+ xjfj) xixj fifj (2 fijfifj − fiifj 2 − f jjfi 2); i ≠ j (3) i.e., Hick’s two-input elasticity of substitution extended directly to the n-input case. As mentioned by Chambers (1988), this measure should be interpreted as a short-run elasticity, since it measures the degree of substitutability between factors i and j, keeping all other factors fixed, and not allowing for adjustment of the other factors when one factor is changed (see also Blackorby and Russel 1989).
The AES between two input factors xi and xj of the n-input production is defined
by: σijA = xkfk k=1 n
∑
xixj Hij H ; i ≠ j; H = 0 f1 K fn f1 f11 L f1n M M M fn fn1 K fnn , (4)where H is the bordered Hessian matrix, |H| is the determinant of this, and Hij is the
(i,j)th cofactor.2 Comparison with equation (2) shows that when n = 2, the AES re-duces to the Hicks definition for two inputs. Blackorby and Russel (1989) criticise this measure for being inadequate even though it tries to remedy the drawbacks of the DES measure. They show that the AES measure does not preserve the properties of the original Hicks measure (equation 1), such as providing information on rela-tive factor shares, etc. They prefer the MES, which is defined as:
σij MES = fj xi Hij H − fj xj Hij H ; i ≠ j. (5)
Blackorby and Russel (1989) argue that this measure does preserve the properties of the original Hicks measure and should thus be preferred to the DES and the AES measures.
The Constant Elasticity of Substitution Production Function
Arrow et al. (1961) have shown that the elasticity of substitution is constant in the two-input case, given the Hicks elasticity of substitution σ (equation 1), if the pro-duction function has the form:
y = f2(x1, x2) ≡ γ(β ⋅x1 −ρ + (1 − β)x2−ρ)−1/ρ; σ ≠1 γ ⋅ x1β ⋅x2(1−β); σ =1 . (6)
For this function, the (constant) Hicks elasticity of substitution between the input factors is given by σ = 1/(1 + ρ), when σ≠ 1 (ρ≠ 0). It is easy to show that the two-input CES function reduces to the Cobb-Douglas function given in the lower half of equation (6) in the limit σ→1, ( i.e. ρ→0).
Uzawa (1962), McFadden (1963), and Blackorby and Russel (1989) have de-rived the CES forms for the three different measures of n-input elasticity of substitution mentioned above. The Blackorby-Russel form [constant MES, cf. equa-tion (5)] is a direct generalizaequa-tion to n inputs of the Arrow et al. two-input form [equation (6)], while the Uzawa (constant AES, cf. equation 4) and McFadden [con-stant DES, cf. equation (3)] forms are more complicated but may be reduced to the Blackorby-Russel form as a special case. Blackorby and Russel (1989) show that the MES measure of substitution [equation (5)] is constant for the given production if and only if the production function has the form:
y = f ( x1,K, xk) = γ βkxk −ρ k=1 n
∑
−1/ρ ; ρ ≠ 0 γ ⋅ xk βk k=1 n∏
; ρ = 0 ; βk = 1 k=1 n∑
. (7)2 The ijth cofactor of a matrix is defined as the determinant of the matrix where the ith row and jth
As for the two-input case, it is observed that this form reduces to the Cobb-Dou-glas form in the limit ρ = 0. The constant elasticity of substitution between any two inputs under MES is σ
ij
MES = 1/(1 + ρ). Blackorby and Russel (1989) argue that the
reason why the Uzawa and the McFadden forms are not directly comparable with the ‘basic’ CES structure given by Arrow et al. (equation 6) is that neither the DES nor the AES measures are natural generalizations of the two-variable elasticity of substitu-tion (equasubstitu-tion 1). They continue by stating that since the MES measure gives a CES structure, which is directly comparable with the two-input case, the MES measure must be the ‘true’ measure of elasticity of substitution for multiple inputs.
As noted by Kmenta (1967), the Blackorby-Russel form (equation [7]) has re-turns to scale equal to unity, which is quite restrictive. A more general form, which allows the returns to scale to be different from unity, is given by Kmenta (1967):
y = γ βkxk −ρ k=1 n
∑
−ν/ρ ; βk = 1 k=1 n∑
, (8)where v is the return to scale.
The Translog Approximation of the Two-Input CES Function
As mentioned in the Introduction, the CES function cannot be linearized analyti-cally. Kmenta (1967), however, notes the two-input CES function (equation 6) may, when the parameter ρ is in the neighborhood of zero (i.e., when the elasticity of sub-stitution σ is in the neighborhood of unity), be approximated by a linear Taylor series expansion, which has the form:
ln(y) = ln(γ)+ ν ⋅ β ⋅ln(x1)+ ν ⋅(1− β)⋅ln(x2) (9) − ρν 2 ⋅ β ⋅(1− β)ln( x1) 2 − ρν 2 ⋅ β ⋅(1− β)ln( x2) 2 + ρν ⋅ β(1− β)ln( x 1)⋅ln(x2).
The function given in equation (9) is recognized as a translog function:
ln(y)= α0+ α1⋅ln(x1)+ α2⋅ln(x2)+ α11ln(x1)2+ α22ln(x2)2+ α12⋅ln(x1)⋅ln(x2), (10) for which the parameters fulfil the conditions:
α1 + α2 = ν;ρ ⋅ ν ⋅ β(1− β) = α12 = −2α11 = −2α22. (11)
The above presentation shows that if a two-input technology is believed to have: (i) constant elasticity of substitution, and (ii) this elasticity of substitution is in the neighborhood of unity, then the input and output values observed for the technology may be fitted to the translog form given in equation (10). The restrictions stated in equation (11) must then be utilized to (i) test whether the estimated translog form does, in fact, approximate a CES function, and (ii) estimate the CES parameters γ, v,
ρ, and β. It must be emphasized that a given two-input technology may in fact be CES, even though the estimated translog form does not obey the restrictions given in equation (11). The reason may be that the necessary condition that σ must be close to unity is not fulfilled.
The Translog Approximation of the N-Input CES Function
Kmenta’s (1967) result presented in the preceding section is often cited in the two-input and the n-two-input case. However, the result has, to my knowledge, not been extended to the n-input case (to date), and should not be cited when more than two inputs are present. A simple first order Taylor series expansion of the general n-input CES function (equation 8) shows that Kmenta’s result (equations 9–11) is not di-rectly applicable when more than two inputs are present. On the contrary, when the CES function (equation 8) with n inputs is approximated by a first-order Taylor ap-proximation around ρ = 0, the result is still a translog function as in the two-input case, but the restrictions on the translog parameters are not equal to the two-input restrictions, but an extension of these. Hence, it may be shown that the general CES function (equation 8) may for σ in the neighborhood of unity be approximated by the translog form: 3
ln(y) = ln(γ)+ αiln(xi)+ αijln(xi)ln( xj) j=i n
∑
i=1 n∑
i=1 n∑
, (12)for which the parameters obey:
(I) αk = νβk (13) (II) αk k=1 n
∑
= ν (III) αij αiαj αk k=1 n∑
i≠j = 2αii αi 2 αk k=1 n∑
− αi = 2αjj αj 2 αk k=1 n∑
− αj = ρ.Condition (III) may be rewritten to the following form:
αiji≠j = −2αii 1+ αk αj k≠(i, j )
∑
= −2αjj 1+ αk αi k≠(i, j )∑
, (14)which can be compared directly with the two-input case. Comparison of equation (14) and equation (11) shows that the n-input approximation is reduced to Kmenta’s result when n = 2.
Thus, if a n-input technology is believed to have constant elasticity of substitu-tion, and if it is believed that this elasticity of substitution is in the neighborhood of unity, then the input and output values observed for the technology may be fitted to the n-dimensional translog form given by equation (12), and the restrictions (III) stated in equation (13) must then be utilized to test whether the estimated translog
form does, in fact, approximate a CES function. If this is the case, the three condi-tions (I), (II), and (III) can then be employed to estimate the CES parameters γ, v, ρ, and β. It must be stressed that the technology considered may, in fact, be CES, even though the estimated linear form does not comply with the restrictions given in equation (13).
Consistency of the Translog Approximation to the CES Function
Thursby and Knox Lovell (1978) discuss the consistency of the CES parameters es-timated with Kmenta’s translog approximation (equation 9) to the two-input CES function (equation 6). They point out that the translog approximation is a truncated Taylor series and that the hereby-estimated CES parameters must thus be biased by truncation errors. They moreover stress that the estimated CES parameters may not even be asymptotically consistent when the examined sample size increases, as the Taylor series underlying the approximation is a power series in ln(x1/x2), which, as such, only converge to the true CES function when ln(x1/x2) is within the conver-gence circle having radius |1/(ρβ)|. Thursby and Knox Lovell perform a series of Monte Carlo simulations to test how well the two-input approximation estimates the CES parameters and generally conclude that ‘the CES parameters are estimated con-sistently only under the most favourable circumstances’ (Thursby and Knox Lovell 1978).
The result of Thursby and Knox Lovell logically extends to the n-input case, meaning that: (i) the CES parameters estimated by the translog approximation are expected to be biased by truncation errors and (ii) the Taylor series underlying the approximation is a power series in the (n – 1) terms ln(xi/x1), whose convergence
ra-dii will depend on the (unknown) CES parameters, again resulting in an inconsistency of the estimated CES parameters if the input variables are outside the convergence radii.
Hoff (2002) performs a series of simulations of the translog approximation to a three-input and five-input CES function, respectively. It is shown that: (i) the ability of the translog approximation to predict true CES structure quickly decreases with increasing ρ, and that ρ should generally not exceed 0.1–0.2 for the approximation to be valid; (ii) the translog approximation quickly fails to predict CES structure if the returns to scale exceeds unity (increasing returns); and (iii) the translog approxi-mation generally fails to predict true CES structure if the range of the input values exceeds more than one order of magnitude. In contrast, it is also shown that the translog approximation will predict true CES structure in most cases if the output variable has some stochastic variation, s0, but that the consistency of the hereby esti-mated CES parameters will, in this case, depend strongly on s0 and the number of observations. Finally, it is shown that when s0 decreases, the chance for neglecting the hypothesis of CES structure increases due to the truncation error of the Taylor function, even though the hypothesis is true.
Thus, great care should be exerted when employing the translog approximation to the CES function, as it may, in many cases, fail to predict true CES structure of data. If the test (equation 13) fails, it is recommended to fit the exact CES function to data. And even if equation (13) predicts CES structure, the result should still be viewed with some caution, as this may be due to inconsistency of the approximation (outside the convergence circle) or to large variation in the observed data. It is gen-erally recommended also to fit data to the exact form.
Application: Danish Trawlers in the North Sea
The test of whether the translog production function is an approximation of the CES production function has been applied in a case study of the fleet of Danish trawlers operating in the North Sea during the period 1987–99. The employed data set is based on official catch and vessel information provided by the Danish Directorate of Fisheries. Only trawlers for which the monthly aggregated catch value of cod and flatfish in the North Sea constitutes more than 50% of the total monthly aggregated catch value in the North Sea are included.4 Moreover, only trawlers operated by full-time skippers are considered. Analysis of the production structure has been performed on a monthly level, and the data set includes monthly aggregated values of: (i) total catch-value obtained by the fleet (measured in mean prices over the pe-riod considered) in the given month; (ii) total crew working on the fleet in the given month; (iii ) total number of days at sea (North Sea) employed by the fleet in the given month; and (iv) mean vessel horsepower, weighted by vessel fishing time, ob-served in the fleet during the given month.
Since stock information is not available for all species caught by the selected fleet, only stock data for cod, haddock, whiting, saithe, plaice, and sole are included. The value of these species caught by the fleet constitutes 66% of the total catch value during the period. This is considered to be a reasonable amount. Since it has not been possible to obtain information on stock levels of the remaining species caught by the fleet, it has been chosen to employ the stocks of these six species as a measure of the total stock available. A single stock index has been estimated by summing the six stocks each weighted by the mean price of the species over the pe-riod 1987–99. This stock index has been included in the data set, and is employed as input.
The data set contains five variables (one output and four inputs) each compris-ing 156 observations (one for each month in the period 1987–99). All variables have been normalised by dividing them with their averages over the period. This reduces possible collinearity and confines the input and output variables to the same order of magnitude (around unity), which is important for the test of the CES approximation; cf. the discussion of the previous section. Moreover, each variable has been season-ally adjusted to avoid spurious regression due to seasonal trends.5 One outlying observation has been removed to obtain normally distributed residuals in the regres-sion.6 In all, 155 observations have been used to fit the translog function.
Figure 1 shows the yearly averages of the five variables after normalisation and seasonal adjustment. It is seen that catch value, crew, and fishing time (days at sea) are approximately correlated, indicating that the variable inputs crew and fishing time are adjusted in the short run according to the economical situation of the fish-ery. Horsepower shows a steadier pattern, as this variable is more difficult to vary in the short run. Finally, the stock index shows a declining trend.
In order to estimate the connection between landed value (Y), average horse-power (HP), vessel fishing time (days at sea, DAS), labor (crew, C), and stock (S) for the fleet of trawlers operating in the North Sea, the following translog function has been fitted to the above described data set:
4 For a detailed description of the data set refer to Rodgers et al . 2003. 5 By the X11 seasonal adjustment method.
log(Y ) = fTL(HP, DAS, C, S)≡ α0 + α1⋅log(HP)+ α2⋅log(DAS) (15)
+ α3⋅log(C) + α4 ⋅log(S )+ α11⋅log(HP)⋅log(HP)+ α22 ⋅log(DAS)⋅log(DAS)
+ α33⋅log(C )⋅log(C )+ α44 ⋅log(S)⋅log(S)+ α12 ⋅log(HP)⋅log(DAS)
+ α13⋅log(HP)⋅log(C )+ α14 ⋅log(HP)⋅log(S)+ α23 ⋅log(DAS)⋅log(C )
+ α24⋅log(DAS)⋅log(S)+ α34⋅log(C )⋅log(S).
The regression parameters resulting from the estimation are shown in table 1. Some autocorrelation is detected in the fit; therefore, one autocorrelated term has been in-cluded, of lag 1. Two tests have been performed for the estimated model, firstly whether the translog function is an approximation to the CES function, employing the condition given in equation (13), and secondly whether the translog can be re-duced to a Cobb-Douglas function; i.e., the joint hypothesis that none of the cross parameters, αij, are significantly different from zero. Three tests, Wald, Likelihood
Ratio, and Lagrange Multiplier, are performed for each hypothesis.
The results of the tests are presented in table 2. The table shows that the hypoth-esis that the translog is an approximation of the CES function is rejected at the 5% level in one out of the three tests, but at the 10% level it is rejected in two out of the
Figure 1. Yearly Average Values of Total Catch Value, Horsepower, Crew, Days at
Sea, and Stock for the Danish Trawler Fleet Operating in the North Sea (1987–99)
Note: Each variable has been normalised by its overall average value over all years. All variables are seasonally adjusted.
three tests. If the hypothesis of CES structure is believed to be true, the substitution factor ρ can be evaluated as the average of the factors given by the test (equation 13). The average of these is 1.12 (giving an elasticity of substitution of 0.47), while the standard deviation is 3.37, indicating a large variation of the ρ values estimated by the test. The latter points to some uncertainty whether the estimated translog is, in fact, an approximation to a CES function. This is further supported by the fact that the average substitution factor is far from zero.
Table 2 further shows that the hypothesis that the translog function can be re-duced to a Cobb-Douglas function is accepted in two out of the three tests at the 5% level, indicating that data may follow a Cobb-Douglas form, having constant elastic-ity of substitution, σ, equal to unity and substitution parameters, r, ρ, equal to zero. In this connection, the exact CES function:
Table 1
Estimated Parameters for the Translog Function (equation 15) Fitted to Yearly Aggregate Observations for Danish Trawlers in the North Sea (1987–99)
Translog Parameter Value Translog Parameter Value
Intercept –0.01 (±0.02) log(S) log(S) 1.09 (±0.96)
Log( HP(a)) 0.52 (±0.12)*** (b) log(HP)log(DAS) –0.79 (±0.61)
Log( DAS(a)) 1.20 (±0.08)*** Log(HP)log(C) 0.67 (±0.47)
Log( C(a)) –0.30 (±0.11)*** log(HP)log(S) 0.59 (±1.00)
Log( S(a)) –0.48 (±0.16)*** Log(DAS)log(C) –0.74 (±0.61)
Log( HP)log(HP) –0.01 (±0.42) Log(DAS) log(S) –1.45 (±0.61)**
Log( DAS)log(DAS) 0.69 (±0.30)** log(C)log(S) 1.19 (±0.80)
Log( C)log(C) –0.07 (±0.29) Lag1 in AR model 0.35 (±0.08)***
Adj R2 0.88
Notes: Numbers in parentheses are standard deviations of the regression parameters. (a) HP = Horsepower, DAS = Days at Sea, C = Crew, S = Stock.
(b) ** and *** indicate if the parameter is significantly different from zero at the 5% and 1% levels, respectively.
Table 2
Results of Tests for Whether the Translog Function (equation 15) describing the Production for Danish Trawlers in the North Sea (1987–99) can be
Reduced to a Cobb-Douglas and a CES Production Function
Reduction to: Test Statistic Pr > ChiSq
Cobb-Douglas Wald 10.95 0.0899 LR 63.73 < 0.0001 LM 10.84 0.0936 CES Wald 16.46 0.0579 LR 130.48 < 0.0001 LM 14.03 0.1212 Notes:
(a) Wald = Wald test, LR = Likelihood Ratio test, LM = Lagrange Multiplier test.
(b) The Statistic column denotes the test statistic. The Pr > ChiSq column denotes the χ2 test for whether
Y = β0
[
β1⋅HP−ρ + β2⋅DAS−ρ + β3⋅C−ρ + β4⋅S−ρ]
−ν/ρ, (16)
has been fitted to data, giving the results shown in table 3. It is observed that the substitution parameter, ρ, is not significantly different from zero, again indicating that a Cobb-Douglas function may be the correct choice of production function in the given example.
The conclusion of the above discussion is that the general translog form in the considered example is not an approximation to the CES form. However, the Cobb-Douglas function gives an acceptable approximation of the production, and it can be concluded that the elasticity of substitution is constant and equal to 1.
Conclusion and Discussion
This paper presents a generalization to n inputs of Kmenta’s (1967) translog linear-ization of the two-input CES function. It is shown that the n-input CES function may generally be approximated by a translog function employing a first-order trun-cated Taylor series, when the elasticity of substitution is in the neighborhood of unity. Moreover, it is shown that while simple restrictions hold for the translog pa-rameters in the two-input case, these restrictions become more complicated in the n-input case, but that the n-input result reduces to Kmenta’s result for n = 2 (Kmenta 1967).
The truncation error, bias, and consistency of employing the translog approxi-mation are discussed. It is stated that the translog approxiapproxi-mation is only valid for a limited range of the elasticity of substitution. Further, the translog approximation should be employed only if decreasing returns to scale are expected and if the input values of the problem can be scaled to have a variation of one order of magnitude at the most (Hoff 2002).
The results are applied to the fleet of Danish trawlers operating in the North Sea from 1987–99. A translog function has been fitted to the monthly landed catch value of the fleet, using fishing time, fishing power, labor, and stock as inputs. The translog function is tested for CES structure, and it is concluded that it is not an ap-proximation to the CES function in the given case. However, CES is present for the fleet, as the appropriate production form is believed to be Cobb-Douglas, having elasticity of substitution equal to unity.
Table 3
Estimated Parameters for the CES Function (equation 16) Fitted to Yearly Aggregate Observations for Danish Trawlers in the North Sea (1987–99)
Parameter Value β0 1.01*** β1 0.39*** β2 0.61*** β3 0 β4 0 υ 1.28*** ρ 0.28 Adj. R2 0.86
As a concluding remark, it must be emphasized that when the aim is to test whether a given technology has constant elasticity of substitution, in most cases it is not sufficient to fit data to a translog form and test whether this function predicts CES properties of data. If the test is neglected, the exact CES form should be fitted to the data.
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