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Estimation and testing of alternative production function models
SYBRAND SCHIM VAN DER LOEFF RINS HARKEMA Erasmus
University,
Rotterdam
Estimation and Testing of Alternative Production Function Models * The paper deals with four different decision models for a firm operating with a constant elasticity of substitution production function. The assumptions underlying the different models and their consequences for estimation are carefully specified. Maximum likelihood estimates of the parameters are obtained of the full four-equation models, using data pertaining to the Dutch manufacturing sector. Finally likelihood ratio test procedures are developed in order to determine whether one or more of the decision models can be rejected on the basis of this data.
1. Introduction The developments in the past two decades in computer technology have made estimation of non-linear models a commonplace occurrence. Opening up new possibilities has, in a sense, made the choice for the applied researcher more difficult. Inasmuch as the model specification had consequences for estimation, the choice of the model was usually based on computational rather than theoretical grounds. Linear versions were chosen over non-linear ones, and single-equation methods over full information methods. Once the restriction with regard to computational effort is slackened the problem of choosing between model specifications becomes apparent. In the field of estimating the parameters of a production function this general development has also taken place. Once the restrictiveness of the log-linear Cobb-Douglas production function had been realized, the attention was focused on the more general (non-linear) constant elasticity of substitution (CES) production function. The parameters of the CES production function continued to be estimated by means of linear or linearized versions of side relationships or the function itself. Concurrently, the realization *This research was undertaken in part while the first author had a research fellowship at CORE, the help of which is gratefully acknowledged. The authors are indebted to an anonymous referee for many stimulating and valuable comments and suggestions on an earlier version of this paper. Journal of Macroeconomics, 0 Wayne State University
Winter 1981, Press, 1981.
Vol.
3, No.
1, pp. 33-53
33
Sybrand Schim uan der Loeff
and Rins Harkema
that single-equation methods to estimate the parameters of the production function could produce estimates afflicted with simultaneity bias, led to the specification of models which admitted of consistent single-equation estimation methods. Given the present state of computer technology, the full model (side relationships inclusive) can, however, easily be handled and, as could be expected, produces more efficient estimates of the production function [see, e.g., Zellner and Richard (1973) and Schim van der Loeff and Harkema (1976) 1. In view of the multiplicity of the model specifications, it seems interesting to develop and apply procedures from which a selection can be made. To this end we set out in this paper to specify four alternative models of firm behavior that have been developed in the existing literature [e.g. Hoch (1958) and Zellner, Kmenta, and Dreze (1966)] . The implications for estimating the parameters of the production function within the context of each of these models are reiterated. In Section 3 the estimation of these models is discussed and a brief description of the data is given. In Section 4 the testing procedure is described and applied, and finally in Section 5 some conclusions are drawn.
2. Four Models of Firm Behavior In this section we shall state the assumptions underlying the four models of firm behavior that are under review in this paper. It should be noted at the outset that it is the decision model of the firm that is the subject of this section, whereas in the following sections the problems with regard to estimation and testing will be discussed. Denoting’ by X,, the value of gross production in constant prices and by p, its price (index), we shall assume that the demand schedule for output is given by the following relation X0 = &pzeui,
(1)
where Q, is the constant price elasticity of demand for X,, 3, denotes factors influencing demand for X0 other than its own price, and the disturbance term u i represents purely random shifts in demand. Production of X0 takes place according to two relations:
‘Time 34
subscripts
will
be suppressed
in this
section.
Alternative
Production
Function
Models
V = eg’(81K-P + ~zL-p)-l’pe”o,
(2)
0 < a < 1,
(3)
and M = aX,e’O,
where V denotes value added in constant prices, K and L represent measures for inputs of capital and labor services, respectively, and M is the value of material inputs in constant prices. From (2) and (3), it is clear that value added is assumed to be generated according to a CES production function with constant returns to scale and a rate of (Hicks-neutral) technical progress of 100 g per cent, whereas the technology relating material inputs to gross production is assumed to be of the fixed coefficients type. The disturbance term, uO, is supposed to represent random shocks in production due to acts of nature, whereas l 0 allows for inefficiencies in the productive process and therefore is assumed to take on non-negative values only [see, e.g., Schmidt (1976)]. Gross production, value added and material inputs in constant prices, are linked by the identity V=X,,-M.
(4)
Note that the domain of E, should be restricted to the interval [0, -In(a)] to ensure that X0 takes on nonnegative values only. The implicit deflator p,, the “price” of value added, is obtained by the so-called double-deflation procedure. Therefore we have to add an identity relating gross production, value added, and material inputs in current prices p,v
= POX, - P,M
7
(5)
where p0 and p, denote the prices of X, and M, respectively. The four models of firm behavior that will be considered derive from different assumptions regarding the determination of the optimal levels of factor inputs by the firm, and we state them explicitly. Assumption 1 In order to determine the optimal each period, the firm maximizes profits 7F=pJ-TK-WL,
levels of factor inputs
in
(6) 35
Sybrand Schim van der Loeff
and Rins Harkema
using the levels of factor inputs as instruments and taking the prices of the services of capital and labor r and w as well as the price of material inputs p, as given. The bars over 6, and v serve to stress the fact that it is assumed that the stochastic nature of these variables is disregarded by the firm. This is the familiar model as formulated by Hoch (1958), and we shall henceforth refer to it as model 1. The model of Zellner, Kmenta, and D&e (1966), model 2, is summarized in Assumption 2. Assumption 2 The optimal levels of factor inputs are determined mizing the mathematical expectation of profits E(lT) = E(p,V)
- rK - WL,
by maxi-
(7)
again using the levels of factor inputs as instruments and with r, w and p, given. In addition, we consider two more models which state that the firm behaves as if production were deterministic and demand stochastic (model 3) and conversely (model 4). Especially in models 2,3, and 4, the straightforward substitution of equations (1) through (5) in order to obtain an expression for expected profits in terms of K and L yields a system that is very hard to deal with from a statistical point of view. These problems arise because of the linearity of the relationship between p,, p,, and p,. More specifically (3), (4) and (5) yield the exact relationship ( p,lp,)
= 1 - (Yeso + ae’O(p,lp,)
.
(8)
In order to obtain a log-linear relationship (with constant coefficients) we take the Taylor expansion of (8) around the (constant) values
(PO/P,)
= ( PJP,)”
3
(PAtlP,)
= (PM/P,)‘,
and
in terms of ln( p,/p,). 36
% =m[Osm<
-In(o)]
ln( p,/p,)
and l ,, yielding
Alternative
Production
Function
Models
ln(nJp,) = ln(p,/p.)* - 1 + [(p,lp,)“l -I (1 - aemP - (P,/P,)” + (P,/P,)” ln(rhlrh)“l~
+ aem[(T.~J~~)~I -‘(p,lrb)” ln(PM /PA + ae* [( P,/P,) ‘I -’ I( PM/~,)” - lI(% -m)+R,
(9)
where R is the residual error of approximation. Combining the error term in e,, and the residual error of approximation, R, into one disturbance term u:, we may rewrite (9) as
pO = e”pY,p~..T e”; ,
(10)
with
c = ln(p,/p,)* - 1 + [(P,/P,)‘~ -’ (1 - aem Y
[l - (PM/P,)’ + (p,l~,)*
ln(p,lp,)*lI
9
= aem[( P,/PJ’I -~(P,/PJ* ,
and ,t Ul =
ae’” [( P,/P,)‘~ -’ [(P,/P,)’
- 11k, - 4 + R .
We can now substitute for p, and X0 in (1) to obtain a relationship in terms of value added and its price with the help of (3), (4), and (10) to obtain
V = {p:eul
(11)
with q = (1 - Y)‘lo 2
and 37
Sybrand Schim van der Loeff
and Rins Harkema
= ui + qou:
%
+ In(1 - oe’o) .
Alternatively (11) may also be derived by reformulating the specification of the technology and employing the partial duality approach as explicated by Bruno (1978). To that purpose we delete (3) and replace (4) by using the Cobb-Douglas relationship between gross production, value added, and material inputs: x0 = MP v’-+ Maximizing an optimal
.
(44
(expected) profits subject to (1) and (4a), we obtain quantity of material inputs M M = [&lb,,,
exp
[
V(l--P)(l+lhO) pp;l
-
] 1/ tl--P(l+l/%,)l
(1 - PM + rl3uo l-p(1+7);l)
1
02)
’
where TV= 8(1 + q or) exp {a2/2}, a2 denoting the variance of E = (1 - 8)(1 + q,‘)u, - ~0’2~:. Substituting (12) into (4a), the associated value of gross production in constant prices r?, is found to be z.
=
[p””
v’-PpPp~~]
exp
[
-
l/
tl--P(l+l/%)l
PO - P)U +c3uo l-p(l+l-t;l)
1
(13)
.
Hence, the associated value of gross production is
exp Substituting
[
-
PO - PNl + -f-l,‘)“uo 1-p(l+q;l)
-rlo
, Ul
I
*
(14) and (12) into (5), we once again obtain Y = P(l + q,‘)
38
-l
in current
3
prices
(14) (11) with
Alternative
Production
Function
Models
and Ul
=
[1
+
(1
-
Yhol uo - qo(l - Y)
It should be noted that this approach also requires approximating a linear relationship by a log-linear one, viz., (4) by (4a). This is necessary in view of the fact that the real value added series has been obtained by the method of double deflation and that consequently (4) and (5) are satisfied identically by the data employed. If forced to choose between the two technology representations, we slightly prefer the first one, the main reason being that the ratio between material inputs and gross production in constant prices shows a substantially smaller variation than the corresponding ratio in current prices. Nevertheless this argument can not be considered as very preponderant because of the poor quality of the data for gross production as well as material inputs, this indeed being the very reason why the model has been reformulated in terms of value added. We shall assume that u. and ur are distributed according to a bivariate normal2 distribution with zero mean and covariance matrix X0, where
(15) The optimal levels of factor inputs g and J? are obtained from the first-order conditions for a (expected) profit maximum and depend upon the model. Generally the first-order conditions may be written as exp [(1/2)a’Z,a] * exp(-pgi)B,
pX’+pIz--(l+p) [l + (l/q)]
= r exp(b’u’)
,
(164
and exp [(1/2)a’X,a} ?he
normality
assumption
pX”’ is made
L--(l+‘) here
for the sake
of convenience
only. 39
Sybrand Schim van der Loeff and Rins Harkemu TABLE Model
1.
Survey of Four Models of Firm Behavior p
X
P” .
P
2
Pv
V
3
Pv
t
i%
v
4
aI
a2
1+’
-’ -fl
0
rl 1
--
b,
1+l+p rl
-’
0
--
11
* exp{-pgi}fJ,[l
1+11
0 rl
+ (l/9)]
b,
rl 1 1
1+1+, 11
= w exp(b’u’),
0
Wb)
where u” = (uoul)’ and the elements of the two-dimensional vectors a and b as well as the specification of p and X depend on the model under consideration. A survey of the four alternative combinations that are considered is given in Table 1. The four equations (2), (ll), (16a) and (16b) determine the endogenous variables p,, V (or 6, and V, as the case may be) I? and L^ given the (exogenous) variables r, w, and p, and appropriate restrictions on the values that the parameters can take on, viz.
-l
(17)
-uJ
3. Estimation To enable us to write the likelihood functions associated with the models discussed earlier, we will need a further specification of the link between the optimal levels of factor inputs I?t and &, which are unobservable in practice, and the actual levels K, and L,. For the sake of simplicity, we shall assume that, apart from error terms, the adjustment process is instantaneous, i.e., 40
Alternative
Production
Function
Models
K, = dteUlr , and
L, = &eu3r (t = 1, . . . . T) .
08)
In the stage of estimation, value added as well as its price are treated as random variables regardless of the model under consideration so that (2) and (11) are treated as estimating equations and no distinction is made any longer between variables with and without bars. It will be assumed that the four-dimensional vectors of error terms u: = (u,, . . . vSt) (t = 1, . . . . I’) are identically and independently distributed according to a multivariate normal distribution with zero mean and variance-covariance matrix X. Taking logarithms the models can concisely be written as3
vt = Bu, , where the elements
of v, are given by
vor = ln(V,)
+ In [O,K;’
+ 02L;‘]
vlt = ln(V,)
- rl in(h)
- In(k)
v2t
09)
= ln(V,)/a
- ln(K,)/a
/p - gi ,
(20)
y
- pgi - ln(r,/p,
J
+ ln{e,(l+ l/q)exp [U/2)a’&all,
%t= ln(V,)/u- lnW,)lu - wi - ln(qlrQ + ln Ce,U- llrl)exp t(1/WI;,411
,
and the four by four matrix B is given by -1
0
0
0
0
1
0
0
h
b,
-o+l-d
0
_b 1
b,
0
-
B=
(21)
-he elasticity of substitution in order to conserve on space.
will
be denoted
-0
+ P)-
interchangeably
as o or (1 + p)-’
41
Sybrand Schim van der Loeff
and Rins Harkema
where the specification of the elements in the vectors a and b in (20) and (21) is given in Table 1 for each of the four models. The absolute value of the Jacobian of the transformation from the structural disturbances U, to the vector of observed quantities y, = [In (V,) In ( p,)ln(K,) In (L,)] is given by 1J,( = 1JI = u (t = 1, . . . . T) .
(22)
The Zellner-Kmenta-Dreze model (model 2) allows for consistent estimation of the production function parameters by means of single-equation (nonlinear) methods under the assumption that the disturbance term of the production function is uncorrelated with the disturbance terms of the factor demand equations. This wellknown result has been generalized by Hodges (1969) for the case of CES production functions. If in addition it is assumed that u10 in (15) is zero, single-equation methods yield consistent estimates of the production function parameters in the context of model 4 as well. Given the nonlinearities in these models amplified by the shape of the Jacobian and the restrictions on the parameter space (17), this seems an appreciable advantage. However, one is caught in the dilemma that the production function parameters appear elsewhere in the model and, as a consequence, the estimation of the production function by itself leads to inefficient estimates of these parameters. In Schim van der Loeff and Harkema (1976), it was indicated that incorporation of the extra information about the parameters of the production function contained in the factor demand equations leads to more precise estimates, in particular with regard to the elasticity of substitution. Here our purpose, however, is to compare the four models of firm behavior, and therefore the full models are estimated. Implementation In order to implement these models, we need one further assumption, usually implicit in all empirical work, namely, that the models presented earlier provide an adequate description of the behavior of the aggregate of economic agents to which our data pertain. In a strict sense this description involves making assumptions about the aggregative properties of the models and will usually involve conditions on the functional forms of the supposed underlying individual relationships, but it seems outside the scope of the present paper to pursue this line of thought at great length. Instead the overall assumption is made that the data 42
Alternative
Production
Function
Models
at hand can be viewed as having been generated by the models described. In order to estimate these models, we need data on V,, for which an index of gross value added against factor prices has been taken, on K,, for which we use an index of energy consumption as proxy for inputs of capital services, and on L,, for which an index on labor volume corrected for shortening work weeks, sickness and accident leave, etc., and the effects of sex, age, and schooling has been calculated. All data pertain to the Dutch manufacturing sector and cover the period from 1950 through 1972. As for the price variables, more or less standard indices were available, albeit for p,, an index had to be reconstructed with the implicit deflator of value added and an index measuring the price of gross production by means of the definitional relations (4) and (5). Since a large part of the production of the Dutch manufacturing sector is exported (approximately 30 per cent), an index of the non-price effects on the demand for output, i.e., &, [c.f. (l)] has been specified according to
where p,, is an index of the price of competitors on foreign markets and m,, is an index of world imports reweighted according to importance for Dutch products. More details about the data can be found in Schim van der Loeff (1974). Here we confine ourselves to presenting the data, that are actually used in the Appendix B at the end of the paper. Since some of the data are only available in index numbers, the models are transformed to wet = ln(k;)
+ In [S&L” + (1 - S)L,‘]
/p - gt = got,
wit = ln(Y,) - rl ln( pvtlpct) - 5 In b,,) - Kln(p,,lp,,) = ult, wzt
= (1 + fdln(YJ - (1 + dln(K,) - pgt - ln(r,/p,)
= 4% + wgt
b2Yu
-
(1 +
P)&
,
>
= (1 + dln(kT,) - (1 + dln(L,) - pgt - ln(wtIpVf) , = hfAJ,+
b2Vll
-
(1 +
P)IL3t
1
(24) 43
Sybrand Schim van der Loeff
and Rins Harkema
where the time index (and the trend variable) takes on the values -13 (corresponding to the year 1950) through 9 (corresponding to the year 1972), excluding the value 0 which corresponds to the base year 1963. All variables have a subscript bar to stress the fact that they are measured as index numbers, whereas the disturbance terms, utt, are equal to the original disturbance terms, u,,, in deviation from their base year values, i.e., att = u,, - Us,, (i = 0 , . . . . 3). The parameter 6 = O,K,P(OIK~P + OpLip)-l, and the demand for output relation has been reparametrized such that q +
K =
Q.
Obviously the transformed disturbances, y,,, will be serially correlated even when the original disturbances are not, as we have assumed them to be. More specifically, from the assumption that the vectors of disturbances u: = (uot . . . uS1) are identically and independently distributed, according to a multivariate normal distribution with zero mean and variance covariance matrix X, it follows that the joint distribution of the vector of transformed disturbances g = (~1~~ . . . y’, yi . . . u;)‘, where yt = (uor . . . ~~0’ is also multivariate normal with zero mean and covariance matrix (I + LL’) 0 X, where the symbol 0 denotes the usual Kronecker product and L is the vector of order T (=22) with all elements equal to unity. Now we can write the likelihood function of the system (24) which, apart from an irrelevant constant, is seen to be l(e,X Idata) = - (T/2)
In [det(x)]
- (T/2)
tr(X-‘M)
+ T ln(lJl), (25)
where 8 = (6 p g 17 5 K)’ is the vector and M is the moment matrix c
of structural
parameters
(B-‘w,)(B-‘w,)’ t
- (T + I)-‘[
T(B-‘q)][
T
(B-‘-t)]}
y
(26)
where the summation runs from t = -13 to t = 9, excluding the value zero. The elements of the vector w i = (w,, . . . wSL)are explicated in (24) and the specification of the matrix B is given in (21) and Table 1 for each of the four models under consideration. 44
Alternative
Production
Function
Models
Lest the four models become observationally equivalent, we have to impose restrictions on the covariance matrix X4 In line with earlier specifications [see, e.g., Zellner, Kmenta, and D&e (1966)], we shall assume E [uotuL,,] = 0 (i = 2,3) for all t. Concentrating with respect to X-’ yields the concentrated likelihood function Z* and the maximum likelihood estimates for the structural parameters are obtained by maximizing Z*(B]data)
= (]JI)“/det(%)
(27)
with respect to 8 as usual. As shown in Appendix A, the elements of 2 are nontrivial functions of the elements of the moment matrix M because of the restrictions on the elements of Z. Maximizing (27), subject to the constraints from (17), yields a border optimum for -rl in view of the restriction q < -1. The reason for this may be the presumably poor quality of the price index for the price of gross production that was used to ‘reconstruct’ a price index for material inputs. In any case, it was decided to proceed conditionally on a fixed value for q0 of -2.5 which lies within the range of values that have been found for the Netherlands. As a consequence, no standard error for -rl has been computed. Asymptotic standard errors of the estimates for the remaining structural parameters have been calculated by means of the Hessian matrix of the concentrated log-likelihood function evaluated at its optimum. The resulting estimates and asymptotic standard errors are exhibited in Table 2. On the basis of these more or less identical estimates, albeit the models where production is assumed stochastic in the decision stage (models 2 and 4) yield somewhat lower estimates for u, nothing can be said as to which model is most appropriate. In the next section model selection will be discussed.
4. Testing the Model Specification In order to test whether the data do support one or more of the models of firm behavior that have been considered, a comprehensive model is specified that contains any of them as specific cases. To that purpose specification (24) is adopted without ‘Maximizing (25) with respect to the unrestricted P-’ the estimated I matrix 2 = M, it can be seen that the function is independent of b, and b,.
matrix and concentrated
substituting likelihood
45
Sybrand Schim van der Loeff 2.
TABLE
Behavior
Conditional Estimates of the Four E [uotu,,] = 0 (i = 2,3)”
1
Model
2
Model
3
Model
4
Models
of Firm
Assuming
s Model
and Rins Harkema
*Asymptotic
u
0.575 (0.042) 0.560 (0.043) 0.569 (0.043) 0.567 (0.040) standard
0.778 (0.060) 0.742 (0.069) 0.774 (0.065) 0.744 (0.065) errors
5
0.0249 (0.0021) 0.0262 (0.0024) 0.0254 (0.0023) 0.0257 (0.0020)
are shown
on b, and
the restrictions model
g
between
b,. More
1.25 (0.12) 1.31 (0.13) 1.25 (0.13) 1.29 (0.12)
K
-1.24 (0.37) -1.04 (0.38) -1.15 (0.39) -1.12 (0.38)
parentheses.
specifically,
we
w, = Bu,
consider
the
(28)
with -1
0
0
0
0
1
0
0
4
Jr
.+
+
B= -0
+ P) 0
0 -(I+
P
)I
(29)
where two parameters 9 and + have been added that are to be estimated. The maximum value of the concentrated likelihood function associated with the comprehensive model (28) can be compared with the maximum values of the concentrated likelihood functions associated with the restricted models. The latter values are obtained by substituting the maximum likelihood estimates in Table 2 into the concentrated likelihood function associated with the comprehensive model and taking account of the restrictions of + and + as explicated in terms of b, and b, in Table 1. Twice the reduction in the natural logarithm of the likelihood function is asymptotically distributed as x2 with degrees of freedom equal to the number of restrictions imposed on the parameters. A test statistic can thus be constructed to ascertain which of the four 46
Alternative
Production
Function
Models
restricted models may be rejected on the basis of the data. Despite the relatively small number of observations, this test could at least for illustrative purposes be applied to the choice problem at hand viz., which of the four models does best describe the Dutch manufacturing sector in the 1950’s and 1960’s. First, however, it should be noticed, as is clear when writing out the covariance matrix of w,, that without further restrictions on the covariance matrix of u, only a linear combination of + and JI can be estimated. In order to estimate + and JI separately, we impose the further restriction that E [QU,,] = 0 (i = 2,3) for all t which makes X block diagonal and guarantees identifiability of + and JI as well as the other structural parameters. Maximizing the concentrated likelihood function yields once again a border optimum for q. Proceeding as before conditionally on a value of -2.5 for q,,, we obtain the conditional maximum likelihood estimates for the remaining parameters as exhibited in Table 3. Of course, in view of the additional restrictions on the covariance matrix, these estimates cannot be compared with those in Table 2. It is clear that the estimates of + and JI are not in line with any of the four models under consideration (viz. + = 0 in models 1 and 3, and +=1+q-l+p = 0.4 in models 2 and 4, and + = 0 in models 1 and 4, and JI = -q-l = 0.9 in models 2 and 3). In order to perform a formal likelihood ratio test, the four models of firm behavior have been reestimated with the additional constraint on the covariance matrix that E [ul,u,,] = 0 (i = 2,3). The resulting estimates are shown in Table 4. They seem in line with the estimates in Table 2, though in both models 2 and 3 the elasticity of substitution is estimated substantially higher, whereas the price elasticity q is estimated substantially lower. The numerical values of the test statistics and the outcomes of the tests at a five per cent level of significance are summarized in Table 5. From Table 5, the conclusion can be drawn that on the basis TABLE Assuming
6 0.489 (0.060) *Asymptotic
3.
Conditional Estimates of the Comprehensive E [u,,u,,] = E [u,,u,,] = 0 (i = 2,3)”
u
R
5
0.761
0.0307 (0.0036)
(0.12)
(0.084) standard
errors
are shown
1.12 between
K
-1.41 (0.40)
4
2.02 (0.54)
Model
*
-0.0518 (0.16)
parentheses. 47
Sybrand Schim tlan der Loeff
and Rins Harkema
TABLE 4. Conditional Estimates of the Four Models Behavior Assuming E [uotu,,] = E [u,,u,,] = 0 (i = 2,3)”
Model
1
Model
2
Model
3
Model
4
*Asymptotic
6
u
g
5
0.573 (0.045) 0.529 (0.044) 0.532 (0.046) 0.565 (0.043)
0.782 (0.060) 0.896 (0.073) 0.904 (0.073) 0.749 (0.065)
0.0250 (0.0023) 0.0274 (0.0022) 0.0269 (0.0022) 0.0258 (0.0022)
1.17 (0.12) 1.39 (0.13) 1.38 (0.13) 1.21 (0.12)
standard
errors
are shown
between
of Firm K
- 1.30 (0.43) -0.752 (0.41) -0.785 (0.41) -1.16 (0.43)
parentheses.
of this data the models in which the firm is assumed to take into account uncertainty with regard to the demand for its product are rejected. As to the choice whether the firm takes into account uncertainty with regard to production, the tests yield no conclusive evidence. A procedure, which could be followed in order to obtain some more insight as to which one of the models, 1 or 4, receives support from the data is: let the data choose between models 1 and 4 given that it is one of them. To be more specific, the maximum value of the concentrated likelihood function in which JI = 0 and + is unrestricted is compared with those where + is restricted to equal 0 (model 1) or 1 + q-l + p (model 4), respectively. This test can, of course, be carried out for both specifications of the covariance matrix. If this test is carried out for the original specification of the covariance matrix viz. E [uotu,,] = 0, E [ul,u,,] # 0 (i = 2,3), model 1 is rejected whereas model 4 is not, at the five per cent level of significance. If the test is carried out in the second (block diagonal) specification of the covariance matrix, both models 1 and 4 are rejected. TABLE
5.
Likelihood
Ratio Tests of Four Models of Firm
Behavior
48
Model
-2 In h
1 2 3 4
5.90 9.16 14.1 4.02
Outcome
of test
do not reject reject reject do not reject
Alternative
Production
Function
Models
This last point demonstrates an obvious shortcoming of the testing procedure, which it has in common with, for example, the test of separate families of hypotheses due to Cox (1961). In this case, it is hard to find conclusive evidence in favor of any of the four models of firm behavior. If put to the chore of choosing one anyway, we would, on the basis of this data, have a slight preference for model 4 in which it is assumed that the firm takes into account random fluctuations on the production side whereas the stochastic nature of the demand for its product is disregarded in deciding on the optimal quantities of factor inputs. Given the state of aggregation of the data, the oversimplified specification of our models and the small amount of data on which the large sample tests are based, the conclusion, needless to say, is tentative. 5. Conclusion In this paper four models of firm behavior are set out which incorporate varying degrees of awareness on the part of the decision maker of the stochastic nature of his production and demand schedules, or more specifically, which incorporate different assumptions about the degree to which this stochastic nature is taken into account in making decisions as to which quantities of factor inputs are optimal. The implications for estimating the parameters of the production function are well known [see e.g. Hodges (1969)], but in order to obtain efficient estimates, the full models should be estimated in the sense that the production function should be estimated jointly with the factor demand equations as well as with the demand schedule for output. The question, then arises which model of firm behavior is to be considered or, in other words, what behavior on the part of the decision maker has given rise to the observed data. A testing procedure has been developed on which inferences about this question can be based. The results and shortcomings of this testing procedure are discussed at the end of the preceding section. Appendix
A
On Concentrating
the Log-Likelihood
Functions
In order to concentrate the log-likelihood respect to the elements of X where
function
(25) with
49
Sybrand Schim uan der Loeff
and Ains Harkema
Z=
(A-1)
or, equivalently, the elements of T = X’ where the elements of VI (as well as those of M) are indexed accordingly, we consider the following Lagrangian expression
+
2F2NJo3
-
(A-2)
)
4Jl3*dJJrll)
where the restrictions uor = utO = 0 (i = 2,3) are expressed in the alternative but equivalent form Jloj = $,,I&, IJJ;,’ = 0. Differentiating with respect to the elements of q, and tar and k2 and setting the resulting expressions equal to zero yields
(1) 6,
= m,, (i,j = 2,3) ,
and
h =oo=moo, (2)
GOi = csio= 0 (i = 2,3) )
(3)
GoI = m,, + t9,,mo2
(4)
ell = ml1 - 2(mo2~,,
(5)
61i
= 9,
(A.3)
+ h3m03) +Y: y + mo3h3) +,,42
)
+ moiCo13~~ 0 = 2,3) ,
as an implicit solution for the elements of 2. An explicit solution can be obtained by means of the identity %@ = I. From this identity it is clear that . I (1) *,,*A’ = - ~o,mG9 (2) 50
$,,$Gr
= -(m3,e12
- m32G.13)(m22m33 - mz2)-l
,
64.4)
Alternative (3)
913K
= -(m,,~,,
Production
Function
- m32612Nm22m33
- msl
Models *
Finally an explicit expression for GOI in terms of the elements of the matrix M is obtained by substituting the expressions (A.4) (1) through (3) into equations (A.3) (3) and (5) which yields ,. =01
=
%o
bo,
(%2%3
-
m
:,,
-
ml2
(mo2m33
- ml3 (mo3 m2, - mo2m32)1[m,h2m33
-
mo3m3*)
- mt,)
- mo2h02m33 - mmm3,)- mo3h,.e22 - mozm32)l -‘. Substituting 8 for X in (25), we obtain the concentrated function (27) since tr(e-‘M) reduces to a constant: t&l
M) = $,m,
+ 2$,,m,,
+ +33m33 + 55ii,h2 + 2~,,(m,,
+ $,,m,,
(A.5)
likelihood
+ $2pm22 + 2+3zm32
+ moz+02GY:)
+ mo3+ol+L,l)
= Jl,m, + -NJ,, [m,, + (k2m02+ 4i3m03)+Y,1 1 +
91,
h
- 2($,,mo2+ 4i3m03 )9,, $2 1 + Lm22
+ 2G,,m,, + $,,m,, +
Ni,
+ 2k3h3 + mo3~ol~~,l),
(ml2+ mo24,,GYi) (A.@
where use has been made of the restrictions $,, = $,,$oI $ rll (i = 2,3), and the same quantity has been added and subtracted on the right-hand side of the equality sign. Upon comparing with A.3(1) to (5), (A.@ is seen to be the trace of the product of 2 and its inverse. Hence tr@-‘M) is equal to the order of 2 (=4).
51
Sybrand Schim van der Loeff and Rins Harkema Appendix B TABLE 1. Time-Series
1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972
47 49 49 54 59 64 68 71 71 78 87 90 95 100 110 116 122 126 139 153 166 172 181
72.0 78.5 78.7 80.5 84.3 87.9 88.7 93.8 91.9 92.5 96.2 97.4 98.4 100.0 107.2 114.3 117.8 122.4 124.1 133.1 134.6 141.2 152.5
54.5 57.5 58.7 61.2 65.6 69.5 74.9 75.3 77.7 81.1 87.6 89.9 94.9 100.0 107.7 114.0 120.9 132.2 147.7 165.8 179.5 193.1 212.3
Data on the Dutch Manufacturinn
62.6 70.4 76.1 71.5 70.9 75.8 90.1 107.2 101.6 91.9 95.2 89.1 95.2 100.0 116.7 125.5 138.5 137.0 134.9 149.0 166.1 163.9 147.3
80.6 82.2 80.8 82.8 85.9 88.1 89.6 90.4 89.9 92.0 95.4 97.6 98.4 100.0 100.6 102.8 102.5 101.5 101.9 94.8 95.5 95.3 94.3
34.1 37.7 39.8 41.0 46.4 51.6 58.0 65.0 66.8 68.3 75.8 83.7 90.9 100.0 115.8 127.7 142.3 154.7 168.0 195.4 225.9 253.7 280.7
87.5 103.2 107.0 101.8 99.0 100.5 194.2 106.8 103.4 101.0 103.2 99.8 99.2 100.0 101.9 103.4 105.6 105.6 104.5 108.1 114.7 114.7 116.7
38.7 42.6 43.5 45.0 49.3 55.4 59.5 63.5 64.4 70.6 80.5 84.9 92.4 100.0 110.7 112.1 129.0 133.0 151.2 173.9 192.1 192.1 207.5
Sector
86.0 112.5 103.2 98.8 98.9 100.0 99.4 100.9 99.9 99.6 97.7 97.1 97.7 100.0 104.8 105.4 112.0 111.7 114.0 106.6 112.8 116.9 117.7
References Bruno, M. “Duality, Intermediate Inputs and Value-Added.” In Production Economics: A Dual Approach to Theory and Applications, Volume 2. M. Fuss and D. McFadden, eds. Amsterdam: North-Holland, 1978. Cox, D.R. “Tests of Separate Families of Hypotheses.” In Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability. J. Neyman, ed. Berkeley: University of California Press, 1961. Hoch, I. “Simultaneous Equation Bias in the Context of the Cobb-Douglas Production Function.” Econometrica 26 (October 1958): 566-78. Hodges, D.J. “A Note on Estimation of Cobb-Douglas and CES 52
Alternative
Production
Function
Models
Production Function Models.” Econometrica 37 (October 1969): 721-25. Schim van der Loeff, S. “Estimation of a CES Production Function for the Dutch Manufacturing Sector.” Econometric Institute, Rotterdam, Working Paper, 1974. -. and R. Harkema. “Three Models of Firm Behavior; Theory and Estimation with an Application to the Dutch Manufacturing Sector.” Review of Economics and Statistics 58 (February 1976): 13-21. Schmidt, P. “On the Statistical Estimation of Parametric Frontier Production Functions.” Review of Economics and Statistics 58 (May 1976): 238-39. Zellner, A., J. Kmenta, and J. Dreze. “Specification and Estimation of Cobb-Douglas Production Function Models.” Econometrica 34 (October 1966): 784-95. -. and J.F. Richard. “Use of Prior Information in the Analysis and Estimation of Cobb-Douglas Production Function Models.” International Economic Review 14 (February 1973): 107-9.
53
University,
Rotterdam
Estimation and Testing of Alternative Production Function Models * The paper deals with four different decision models for a firm operating with a constant elasticity of substitution production function. The assumptions underlying the different models and their consequences for estimation are carefully specified. Maximum likelihood estimates of the parameters are obtained of the full four-equation models, using data pertaining to the Dutch manufacturing sector. Finally likelihood ratio test procedures are developed in order to determine whether one or more of the decision models can be rejected on the basis of this data.
1. Introduction The developments in the past two decades in computer technology have made estimation of non-linear models a commonplace occurrence. Opening up new possibilities has, in a sense, made the choice for the applied researcher more difficult. Inasmuch as the model specification had consequences for estimation, the choice of the model was usually based on computational rather than theoretical grounds. Linear versions were chosen over non-linear ones, and single-equation methods over full information methods. Once the restriction with regard to computational effort is slackened the problem of choosing between model specifications becomes apparent. In the field of estimating the parameters of a production function this general development has also taken place. Once the restrictiveness of the log-linear Cobb-Douglas production function had been realized, the attention was focused on the more general (non-linear) constant elasticity of substitution (CES) production function. The parameters of the CES production function continued to be estimated by means of linear or linearized versions of side relationships or the function itself. Concurrently, the realization *This research was undertaken in part while the first author had a research fellowship at CORE, the help of which is gratefully acknowledged. The authors are indebted to an anonymous referee for many stimulating and valuable comments and suggestions on an earlier version of this paper. Journal of Macroeconomics, 0 Wayne State University
Winter 1981, Press, 1981.
Vol.
3, No.
1, pp. 33-53
33
Sybrand Schim uan der Loeff
and Rins Harkema
that single-equation methods to estimate the parameters of the production function could produce estimates afflicted with simultaneity bias, led to the specification of models which admitted of consistent single-equation estimation methods. Given the present state of computer technology, the full model (side relationships inclusive) can, however, easily be handled and, as could be expected, produces more efficient estimates of the production function [see, e.g., Zellner and Richard (1973) and Schim van der Loeff and Harkema (1976) 1. In view of the multiplicity of the model specifications, it seems interesting to develop and apply procedures from which a selection can be made. To this end we set out in this paper to specify four alternative models of firm behavior that have been developed in the existing literature [e.g. Hoch (1958) and Zellner, Kmenta, and Dreze (1966)] . The implications for estimating the parameters of the production function within the context of each of these models are reiterated. In Section 3 the estimation of these models is discussed and a brief description of the data is given. In Section 4 the testing procedure is described and applied, and finally in Section 5 some conclusions are drawn.
2. Four Models of Firm Behavior In this section we shall state the assumptions underlying the four models of firm behavior that are under review in this paper. It should be noted at the outset that it is the decision model of the firm that is the subject of this section, whereas in the following sections the problems with regard to estimation and testing will be discussed. Denoting’ by X,, the value of gross production in constant prices and by p, its price (index), we shall assume that the demand schedule for output is given by the following relation X0 = &pzeui,
(1)
where Q, is the constant price elasticity of demand for X,, 3, denotes factors influencing demand for X0 other than its own price, and the disturbance term u i represents purely random shifts in demand. Production of X0 takes place according to two relations:
‘Time 34
subscripts
will
be suppressed
in this
section.
Alternative
Production
Function
Models
V = eg’(81K-P + ~zL-p)-l’pe”o,
(2)
0 < a < 1,
(3)
and M = aX,e’O,
where V denotes value added in constant prices, K and L represent measures for inputs of capital and labor services, respectively, and M is the value of material inputs in constant prices. From (2) and (3), it is clear that value added is assumed to be generated according to a CES production function with constant returns to scale and a rate of (Hicks-neutral) technical progress of 100 g per cent, whereas the technology relating material inputs to gross production is assumed to be of the fixed coefficients type. The disturbance term, uO, is supposed to represent random shocks in production due to acts of nature, whereas l 0 allows for inefficiencies in the productive process and therefore is assumed to take on non-negative values only [see, e.g., Schmidt (1976)]. Gross production, value added and material inputs in constant prices, are linked by the identity V=X,,-M.
(4)
Note that the domain of E, should be restricted to the interval [0, -In(a)] to ensure that X0 takes on nonnegative values only. The implicit deflator p,, the “price” of value added, is obtained by the so-called double-deflation procedure. Therefore we have to add an identity relating gross production, value added, and material inputs in current prices p,v
= POX, - P,M
7
(5)
where p0 and p, denote the prices of X, and M, respectively. The four models of firm behavior that will be considered derive from different assumptions regarding the determination of the optimal levels of factor inputs by the firm, and we state them explicitly. Assumption 1 In order to determine the optimal each period, the firm maximizes profits 7F=pJ-TK-WL,
levels of factor inputs
in
(6) 35
Sybrand Schim van der Loeff
and Rins Harkema
using the levels of factor inputs as instruments and taking the prices of the services of capital and labor r and w as well as the price of material inputs p, as given. The bars over 6, and v serve to stress the fact that it is assumed that the stochastic nature of these variables is disregarded by the firm. This is the familiar model as formulated by Hoch (1958), and we shall henceforth refer to it as model 1. The model of Zellner, Kmenta, and D&e (1966), model 2, is summarized in Assumption 2. Assumption 2 The optimal levels of factor inputs are determined mizing the mathematical expectation of profits E(lT) = E(p,V)
- rK - WL,
by maxi-
(7)
again using the levels of factor inputs as instruments and with r, w and p, given. In addition, we consider two more models which state that the firm behaves as if production were deterministic and demand stochastic (model 3) and conversely (model 4). Especially in models 2,3, and 4, the straightforward substitution of equations (1) through (5) in order to obtain an expression for expected profits in terms of K and L yields a system that is very hard to deal with from a statistical point of view. These problems arise because of the linearity of the relationship between p,, p,, and p,. More specifically (3), (4) and (5) yield the exact relationship ( p,lp,)
= 1 - (Yeso + ae’O(p,lp,)
.
(8)
In order to obtain a log-linear relationship (with constant coefficients) we take the Taylor expansion of (8) around the (constant) values
(PO/P,)
= ( PJP,)”
3
(PAtlP,)
= (PM/P,)‘,
and
in terms of ln( p,/p,). 36
% =m[Osm<
-In(o)]
ln( p,/p,)
and l ,, yielding
Alternative
Production
Function
Models
ln(nJp,) = ln(p,/p.)* - 1 + [(p,lp,)“l -I (1 - aemP - (P,/P,)” + (P,/P,)” ln(rhlrh)“l~
+ aem[(T.~J~~)~I -‘(p,lrb)” ln(PM /PA + ae* [( P,/P,) ‘I -’ I( PM/~,)” - lI(% -m)+R,
(9)
where R is the residual error of approximation. Combining the error term in e,, and the residual error of approximation, R, into one disturbance term u:, we may rewrite (9) as
pO = e”pY,p~..T e”; ,
(10)
with
c = ln(p,/p,)* - 1 + [(P,/P,)‘~ -’ (1 - aem Y
[l - (PM/P,)’ + (p,l~,)*
ln(p,lp,)*lI
9
= aem[( P,/PJ’I -~(P,/PJ* ,
and ,t Ul =
ae’” [( P,/P,)‘~ -’ [(P,/P,)’
- 11k, - 4 + R .
We can now substitute for p, and X0 in (1) to obtain a relationship in terms of value added and its price with the help of (3), (4), and (10) to obtain
V = {p:eul
(11)
with q = (1 - Y)‘lo 2
and 37
Sybrand Schim van der Loeff
and Rins Harkema
= ui + qou:
%
+ In(1 - oe’o) .
Alternatively (11) may also be derived by reformulating the specification of the technology and employing the partial duality approach as explicated by Bruno (1978). To that purpose we delete (3) and replace (4) by using the Cobb-Douglas relationship between gross production, value added, and material inputs: x0 = MP v’-+ Maximizing an optimal
.
(44
(expected) profits subject to (1) and (4a), we obtain quantity of material inputs M M = [&lb,,,
exp
[
V(l--P)(l+lhO) pp;l
-
] 1/ tl--P(l+l/%,)l
(1 - PM + rl3uo l-p(1+7);l)
1
02)
’
where TV= 8(1 + q or) exp {a2/2}, a2 denoting the variance of E = (1 - 8)(1 + q,‘)u, - ~0’2~:. Substituting (12) into (4a), the associated value of gross production in constant prices r?, is found to be z.
=
[p””
v’-PpPp~~]
exp
[
-
l/
tl--P(l+l/%)l
PO - P)U +c3uo l-p(l+l-t;l)
1
(13)
.
Hence, the associated value of gross production is
exp Substituting
[
-
PO - PNl + -f-l,‘)“uo 1-p(l+q;l)
-rlo
, Ul
I
*
(14) and (12) into (5), we once again obtain Y = P(l + q,‘)
38
-l
in current
3
prices
(14) (11) with
Alternative
Production
Function
Models
and Ul
=
[1
+
(1
-
Yhol uo - qo(l - Y)
It should be noted that this approach also requires approximating a linear relationship by a log-linear one, viz., (4) by (4a). This is necessary in view of the fact that the real value added series has been obtained by the method of double deflation and that consequently (4) and (5) are satisfied identically by the data employed. If forced to choose between the two technology representations, we slightly prefer the first one, the main reason being that the ratio between material inputs and gross production in constant prices shows a substantially smaller variation than the corresponding ratio in current prices. Nevertheless this argument can not be considered as very preponderant because of the poor quality of the data for gross production as well as material inputs, this indeed being the very reason why the model has been reformulated in terms of value added. We shall assume that u. and ur are distributed according to a bivariate normal2 distribution with zero mean and covariance matrix X0, where
(15) The optimal levels of factor inputs g and J? are obtained from the first-order conditions for a (expected) profit maximum and depend upon the model. Generally the first-order conditions may be written as exp [(1/2)a’Z,a] * exp(-pgi)B,
pX’+pIz--(l+p) [l + (l/q)]
= r exp(b’u’)
,
(164
and exp [(1/2)a’X,a} ?he
normality
assumption
pX”’ is made
L--(l+‘) here
for the sake
of convenience
only. 39
Sybrand Schim van der Loeff and Rins Harkemu TABLE Model
1.
Survey of Four Models of Firm Behavior p
X
P” .
P
2
Pv
V
3
Pv
t
i%
v
4
aI
a2
1+’
-’ -fl
0
rl 1
--
b,
1+l+p rl
-’
0
--
11
* exp{-pgi}fJ,[l
1+11
0 rl
+ (l/9)]
b,
rl 1 1
1+1+, 11
= w exp(b’u’),
0
Wb)
where u” = (uoul)’ and the elements of the two-dimensional vectors a and b as well as the specification of p and X depend on the model under consideration. A survey of the four alternative combinations that are considered is given in Table 1. The four equations (2), (ll), (16a) and (16b) determine the endogenous variables p,, V (or 6, and V, as the case may be) I? and L^ given the (exogenous) variables r, w, and p, and appropriate restrictions on the values that the parameters can take on, viz.
-l
(17)
-uJ
3. Estimation To enable us to write the likelihood functions associated with the models discussed earlier, we will need a further specification of the link between the optimal levels of factor inputs I?t and &, which are unobservable in practice, and the actual levels K, and L,. For the sake of simplicity, we shall assume that, apart from error terms, the adjustment process is instantaneous, i.e., 40
Alternative
Production
Function
Models
K, = dteUlr , and
L, = &eu3r (t = 1, . . . . T) .
08)
In the stage of estimation, value added as well as its price are treated as random variables regardless of the model under consideration so that (2) and (11) are treated as estimating equations and no distinction is made any longer between variables with and without bars. It will be assumed that the four-dimensional vectors of error terms u: = (u,, . . . vSt) (t = 1, . . . . I’) are identically and independently distributed according to a multivariate normal distribution with zero mean and variance-covariance matrix X. Taking logarithms the models can concisely be written as3
vt = Bu, , where the elements
of v, are given by
vor = ln(V,)
+ In [O,K;’
+ 02L;‘]
vlt = ln(V,)
- rl in(h)
- In(k)
v2t
09)
= ln(V,)/a
- ln(K,)/a
/p - gi ,
(20)
y
- pgi - ln(r,/p,
J
+ ln{e,(l+ l/q)exp [U/2)a’&all,
%t= ln(V,)/u- lnW,)lu - wi - ln(qlrQ + ln Ce,U- llrl)exp t(1/WI;,411
,
and the four by four matrix B is given by -1
0
0
0
0
1
0
0
h
b,
-o+l-d
0
_b 1
b,
0
-
B=
(21)
-he elasticity of substitution in order to conserve on space.
will
be denoted
-0
+ P)-
interchangeably
as o or (1 + p)-’
41
Sybrand Schim van der Loeff
and Rins Harkema
where the specification of the elements in the vectors a and b in (20) and (21) is given in Table 1 for each of the four models. The absolute value of the Jacobian of the transformation from the structural disturbances U, to the vector of observed quantities y, = [In (V,) In ( p,)ln(K,) In (L,)] is given by 1J,( = 1JI = u (t = 1, . . . . T) .
(22)
The Zellner-Kmenta-Dreze model (model 2) allows for consistent estimation of the production function parameters by means of single-equation (nonlinear) methods under the assumption that the disturbance term of the production function is uncorrelated with the disturbance terms of the factor demand equations. This wellknown result has been generalized by Hodges (1969) for the case of CES production functions. If in addition it is assumed that u10 in (15) is zero, single-equation methods yield consistent estimates of the production function parameters in the context of model 4 as well. Given the nonlinearities in these models amplified by the shape of the Jacobian and the restrictions on the parameter space (17), this seems an appreciable advantage. However, one is caught in the dilemma that the production function parameters appear elsewhere in the model and, as a consequence, the estimation of the production function by itself leads to inefficient estimates of these parameters. In Schim van der Loeff and Harkema (1976), it was indicated that incorporation of the extra information about the parameters of the production function contained in the factor demand equations leads to more precise estimates, in particular with regard to the elasticity of substitution. Here our purpose, however, is to compare the four models of firm behavior, and therefore the full models are estimated. Implementation In order to implement these models, we need one further assumption, usually implicit in all empirical work, namely, that the models presented earlier provide an adequate description of the behavior of the aggregate of economic agents to which our data pertain. In a strict sense this description involves making assumptions about the aggregative properties of the models and will usually involve conditions on the functional forms of the supposed underlying individual relationships, but it seems outside the scope of the present paper to pursue this line of thought at great length. Instead the overall assumption is made that the data 42
Alternative
Production
Function
Models
at hand can be viewed as having been generated by the models described. In order to estimate these models, we need data on V,, for which an index of gross value added against factor prices has been taken, on K,, for which we use an index of energy consumption as proxy for inputs of capital services, and on L,, for which an index on labor volume corrected for shortening work weeks, sickness and accident leave, etc., and the effects of sex, age, and schooling has been calculated. All data pertain to the Dutch manufacturing sector and cover the period from 1950 through 1972. As for the price variables, more or less standard indices were available, albeit for p,, an index had to be reconstructed with the implicit deflator of value added and an index measuring the price of gross production by means of the definitional relations (4) and (5). Since a large part of the production of the Dutch manufacturing sector is exported (approximately 30 per cent), an index of the non-price effects on the demand for output, i.e., &, [c.f. (l)] has been specified according to
where p,, is an index of the price of competitors on foreign markets and m,, is an index of world imports reweighted according to importance for Dutch products. More details about the data can be found in Schim van der Loeff (1974). Here we confine ourselves to presenting the data, that are actually used in the Appendix B at the end of the paper. Since some of the data are only available in index numbers, the models are transformed to wet = ln(k;)
+ In [S&L” + (1 - S)L,‘]
/p - gt = got,
wit = ln(Y,) - rl ln( pvtlpct) - 5 In b,,) - Kln(p,,lp,,) = ult, wzt
= (1 + fdln(YJ - (1 + dln(K,) - pgt - ln(r,/p,)
= 4% + wgt
b2Yu
-
(1 +
P)&
,
>
= (1 + dln(kT,) - (1 + dln(L,) - pgt - ln(wtIpVf) , = hfAJ,+
b2Vll
-
(1 +
P)IL3t
1
(24) 43
Sybrand Schim van der Loeff
and Rins Harkema
where the time index (and the trend variable) takes on the values -13 (corresponding to the year 1950) through 9 (corresponding to the year 1972), excluding the value 0 which corresponds to the base year 1963. All variables have a subscript bar to stress the fact that they are measured as index numbers, whereas the disturbance terms, utt, are equal to the original disturbance terms, u,,, in deviation from their base year values, i.e., att = u,, - Us,, (i = 0 , . . . . 3). The parameter 6 = O,K,P(OIK~P + OpLip)-l, and the demand for output relation has been reparametrized such that q +
K =
Q.
Obviously the transformed disturbances, y,,, will be serially correlated even when the original disturbances are not, as we have assumed them to be. More specifically, from the assumption that the vectors of disturbances u: = (uot . . . uS1) are identically and independently distributed, according to a multivariate normal distribution with zero mean and variance covariance matrix X, it follows that the joint distribution of the vector of transformed disturbances g = (~1~~ . . . y’, yi . . . u;)‘, where yt = (uor . . . ~~0’ is also multivariate normal with zero mean and covariance matrix (I + LL’) 0 X, where the symbol 0 denotes the usual Kronecker product and L is the vector of order T (=22) with all elements equal to unity. Now we can write the likelihood function of the system (24) which, apart from an irrelevant constant, is seen to be l(e,X Idata) = - (T/2)
In [det(x)]
- (T/2)
tr(X-‘M)
+ T ln(lJl), (25)
where 8 = (6 p g 17 5 K)’ is the vector and M is the moment matrix c
of structural
parameters
(B-‘w,)(B-‘w,)’ t
- (T + I)-‘[
T(B-‘q)][
T
(B-‘-t)]}
y
(26)
where the summation runs from t = -13 to t = 9, excluding the value zero. The elements of the vector w i = (w,, . . . wSL)are explicated in (24) and the specification of the matrix B is given in (21) and Table 1 for each of the four models under consideration. 44
Alternative
Production
Function
Models
Lest the four models become observationally equivalent, we have to impose restrictions on the covariance matrix X4 In line with earlier specifications [see, e.g., Zellner, Kmenta, and D&e (1966)], we shall assume E [uotuL,,] = 0 (i = 2,3) for all t. Concentrating with respect to X-’ yields the concentrated likelihood function Z* and the maximum likelihood estimates for the structural parameters are obtained by maximizing Z*(B]data)
= (]JI)“/det(%)
(27)
with respect to 8 as usual. As shown in Appendix A, the elements of 2 are nontrivial functions of the elements of the moment matrix M because of the restrictions on the elements of Z. Maximizing (27), subject to the constraints from (17), yields a border optimum for -rl in view of the restriction q < -1. The reason for this may be the presumably poor quality of the price index for the price of gross production that was used to ‘reconstruct’ a price index for material inputs. In any case, it was decided to proceed conditionally on a fixed value for q0 of -2.5 which lies within the range of values that have been found for the Netherlands. As a consequence, no standard error for -rl has been computed. Asymptotic standard errors of the estimates for the remaining structural parameters have been calculated by means of the Hessian matrix of the concentrated log-likelihood function evaluated at its optimum. The resulting estimates and asymptotic standard errors are exhibited in Table 2. On the basis of these more or less identical estimates, albeit the models where production is assumed stochastic in the decision stage (models 2 and 4) yield somewhat lower estimates for u, nothing can be said as to which model is most appropriate. In the next section model selection will be discussed.
4. Testing the Model Specification In order to test whether the data do support one or more of the models of firm behavior that have been considered, a comprehensive model is specified that contains any of them as specific cases. To that purpose specification (24) is adopted without ‘Maximizing (25) with respect to the unrestricted P-’ the estimated I matrix 2 = M, it can be seen that the function is independent of b, and b,.
matrix and concentrated
substituting likelihood
45
Sybrand Schim van der Loeff 2.
TABLE
Behavior
Conditional Estimates of the Four E [uotu,,] = 0 (i = 2,3)”
1
Model
2
Model
3
Model
4
Models
of Firm
Assuming
s Model
and Rins Harkema
*Asymptotic
u
0.575 (0.042) 0.560 (0.043) 0.569 (0.043) 0.567 (0.040) standard
0.778 (0.060) 0.742 (0.069) 0.774 (0.065) 0.744 (0.065) errors
5
0.0249 (0.0021) 0.0262 (0.0024) 0.0254 (0.0023) 0.0257 (0.0020)
are shown
on b, and
the restrictions model
g
between
b,. More
1.25 (0.12) 1.31 (0.13) 1.25 (0.13) 1.29 (0.12)
K
-1.24 (0.37) -1.04 (0.38) -1.15 (0.39) -1.12 (0.38)
parentheses.
specifically,
we
w, = Bu,
consider
the
(28)
with -1
0
0
0
0
1
0
0
4
Jr
.+
+
B= -0
+ P) 0
0 -(I+
P
)I
(29)
where two parameters 9 and + have been added that are to be estimated. The maximum value of the concentrated likelihood function associated with the comprehensive model (28) can be compared with the maximum values of the concentrated likelihood functions associated with the restricted models. The latter values are obtained by substituting the maximum likelihood estimates in Table 2 into the concentrated likelihood function associated with the comprehensive model and taking account of the restrictions of + and + as explicated in terms of b, and b, in Table 1. Twice the reduction in the natural logarithm of the likelihood function is asymptotically distributed as x2 with degrees of freedom equal to the number of restrictions imposed on the parameters. A test statistic can thus be constructed to ascertain which of the four 46
Alternative
Production
Function
Models
restricted models may be rejected on the basis of the data. Despite the relatively small number of observations, this test could at least for illustrative purposes be applied to the choice problem at hand viz., which of the four models does best describe the Dutch manufacturing sector in the 1950’s and 1960’s. First, however, it should be noticed, as is clear when writing out the covariance matrix of w,, that without further restrictions on the covariance matrix of u, only a linear combination of + and JI can be estimated. In order to estimate + and JI separately, we impose the further restriction that E [QU,,] = 0 (i = 2,3) for all t which makes X block diagonal and guarantees identifiability of + and JI as well as the other structural parameters. Maximizing the concentrated likelihood function yields once again a border optimum for q. Proceeding as before conditionally on a value of -2.5 for q,,, we obtain the conditional maximum likelihood estimates for the remaining parameters as exhibited in Table 3. Of course, in view of the additional restrictions on the covariance matrix, these estimates cannot be compared with those in Table 2. It is clear that the estimates of + and JI are not in line with any of the four models under consideration (viz. + = 0 in models 1 and 3, and +=1+q-l+p = 0.4 in models 2 and 4, and + = 0 in models 1 and 4, and JI = -q-l = 0.9 in models 2 and 3). In order to perform a formal likelihood ratio test, the four models of firm behavior have been reestimated with the additional constraint on the covariance matrix that E [ul,u,,] = 0 (i = 2,3). The resulting estimates are shown in Table 4. They seem in line with the estimates in Table 2, though in both models 2 and 3 the elasticity of substitution is estimated substantially higher, whereas the price elasticity q is estimated substantially lower. The numerical values of the test statistics and the outcomes of the tests at a five per cent level of significance are summarized in Table 5. From Table 5, the conclusion can be drawn that on the basis TABLE Assuming
6 0.489 (0.060) *Asymptotic
3.
Conditional Estimates of the Comprehensive E [u,,u,,] = E [u,,u,,] = 0 (i = 2,3)”
u
R
5
0.761
0.0307 (0.0036)
(0.12)
(0.084) standard
errors
are shown
1.12 between
K
-1.41 (0.40)
4
2.02 (0.54)
Model
*
-0.0518 (0.16)
parentheses. 47
Sybrand Schim tlan der Loeff
and Rins Harkema
TABLE 4. Conditional Estimates of the Four Models Behavior Assuming E [uotu,,] = E [u,,u,,] = 0 (i = 2,3)”
Model
1
Model
2
Model
3
Model
4
*Asymptotic
6
u
g
5
0.573 (0.045) 0.529 (0.044) 0.532 (0.046) 0.565 (0.043)
0.782 (0.060) 0.896 (0.073) 0.904 (0.073) 0.749 (0.065)
0.0250 (0.0023) 0.0274 (0.0022) 0.0269 (0.0022) 0.0258 (0.0022)
1.17 (0.12) 1.39 (0.13) 1.38 (0.13) 1.21 (0.12)
standard
errors
are shown
between
of Firm K
- 1.30 (0.43) -0.752 (0.41) -0.785 (0.41) -1.16 (0.43)
parentheses.
of this data the models in which the firm is assumed to take into account uncertainty with regard to the demand for its product are rejected. As to the choice whether the firm takes into account uncertainty with regard to production, the tests yield no conclusive evidence. A procedure, which could be followed in order to obtain some more insight as to which one of the models, 1 or 4, receives support from the data is: let the data choose between models 1 and 4 given that it is one of them. To be more specific, the maximum value of the concentrated likelihood function in which JI = 0 and + is unrestricted is compared with those where + is restricted to equal 0 (model 1) or 1 + q-l + p (model 4), respectively. This test can, of course, be carried out for both specifications of the covariance matrix. If this test is carried out for the original specification of the covariance matrix viz. E [uotu,,] = 0, E [ul,u,,] # 0 (i = 2,3), model 1 is rejected whereas model 4 is not, at the five per cent level of significance. If the test is carried out in the second (block diagonal) specification of the covariance matrix, both models 1 and 4 are rejected. TABLE
5.
Likelihood
Ratio Tests of Four Models of Firm
Behavior
48
Model
-2 In h
1 2 3 4
5.90 9.16 14.1 4.02
Outcome
of test
do not reject reject reject do not reject
Alternative
Production
Function
Models
This last point demonstrates an obvious shortcoming of the testing procedure, which it has in common with, for example, the test of separate families of hypotheses due to Cox (1961). In this case, it is hard to find conclusive evidence in favor of any of the four models of firm behavior. If put to the chore of choosing one anyway, we would, on the basis of this data, have a slight preference for model 4 in which it is assumed that the firm takes into account random fluctuations on the production side whereas the stochastic nature of the demand for its product is disregarded in deciding on the optimal quantities of factor inputs. Given the state of aggregation of the data, the oversimplified specification of our models and the small amount of data on which the large sample tests are based, the conclusion, needless to say, is tentative. 5. Conclusion In this paper four models of firm behavior are set out which incorporate varying degrees of awareness on the part of the decision maker of the stochastic nature of his production and demand schedules, or more specifically, which incorporate different assumptions about the degree to which this stochastic nature is taken into account in making decisions as to which quantities of factor inputs are optimal. The implications for estimating the parameters of the production function are well known [see e.g. Hodges (1969)], but in order to obtain efficient estimates, the full models should be estimated in the sense that the production function should be estimated jointly with the factor demand equations as well as with the demand schedule for output. The question, then arises which model of firm behavior is to be considered or, in other words, what behavior on the part of the decision maker has given rise to the observed data. A testing procedure has been developed on which inferences about this question can be based. The results and shortcomings of this testing procedure are discussed at the end of the preceding section. Appendix
A
On Concentrating
the Log-Likelihood
Functions
In order to concentrate the log-likelihood respect to the elements of X where
function
(25) with
49
Sybrand Schim uan der Loeff
and Ains Harkema
Z=
(A-1)
or, equivalently, the elements of T = X’ where the elements of VI (as well as those of M) are indexed accordingly, we consider the following Lagrangian expression
+
2F2NJo3
-
(A-2)
)
4Jl3*dJJrll)
where the restrictions uor = utO = 0 (i = 2,3) are expressed in the alternative but equivalent form Jloj = $,,I&, IJJ;,’ = 0. Differentiating with respect to the elements of q, and tar and k2 and setting the resulting expressions equal to zero yields
(1) 6,
= m,, (i,j = 2,3) ,
and
h =oo=moo, (2)
GOi = csio= 0 (i = 2,3) )
(3)
GoI = m,, + t9,,mo2
(4)
ell = ml1 - 2(mo2~,,
(5)
61i
= 9,
(A.3)
+ h3m03) +Y: y + mo3h3) +,,42
)
+ moiCo13~~ 0 = 2,3) ,
as an implicit solution for the elements of 2. An explicit solution can be obtained by means of the identity %@ = I. From this identity it is clear that . I (1) *,,*A’ = - ~o,mG9 (2) 50
$,,$Gr
= -(m3,e12
- m32G.13)(m22m33 - mz2)-l
,
64.4)
Alternative (3)
913K
= -(m,,~,,
Production
Function
- m32612Nm22m33
- msl
Models *
Finally an explicit expression for GOI in terms of the elements of the matrix M is obtained by substituting the expressions (A.4) (1) through (3) into equations (A.3) (3) and (5) which yields ,. =01
=
%o
bo,
(%2%3
-
m
:,,
-
ml2
(mo2m33
- ml3 (mo3 m2, - mo2m32)1[m,h2m33
-
mo3m3*)
- mt,)
- mo2h02m33 - mmm3,)- mo3h,.e22 - mozm32)l -‘. Substituting 8 for X in (25), we obtain the concentrated function (27) since tr(e-‘M) reduces to a constant: t&l
M) = $,m,
+ 2$,,m,,
+ +33m33 + 55ii,h2 + 2~,,(m,,
+ $,,m,,
(A.5)
likelihood
+ $2pm22 + 2+3zm32
+ moz+02GY:)
+ mo3+ol+L,l)
= Jl,m, + -NJ,, [m,, + (k2m02+ 4i3m03)+Y,1 1 +
91,
h
- 2($,,mo2+ 4i3m03 )9,, $2 1 + Lm22
+ 2G,,m,, + $,,m,, +
Ni,
+ 2k3h3 + mo3~ol~~,l),
(ml2+ mo24,,GYi) (A.@
where use has been made of the restrictions $,, = $,,$oI $ rll (i = 2,3), and the same quantity has been added and subtracted on the right-hand side of the equality sign. Upon comparing with A.3(1) to (5), (A.@ is seen to be the trace of the product of 2 and its inverse. Hence tr@-‘M) is equal to the order of 2 (=4).
51
Sybrand Schim van der Loeff and Rins Harkema Appendix B TABLE 1. Time-Series
1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972
47 49 49 54 59 64 68 71 71 78 87 90 95 100 110 116 122 126 139 153 166 172 181
72.0 78.5 78.7 80.5 84.3 87.9 88.7 93.8 91.9 92.5 96.2 97.4 98.4 100.0 107.2 114.3 117.8 122.4 124.1 133.1 134.6 141.2 152.5
54.5 57.5 58.7 61.2 65.6 69.5 74.9 75.3 77.7 81.1 87.6 89.9 94.9 100.0 107.7 114.0 120.9 132.2 147.7 165.8 179.5 193.1 212.3
Data on the Dutch Manufacturinn
62.6 70.4 76.1 71.5 70.9 75.8 90.1 107.2 101.6 91.9 95.2 89.1 95.2 100.0 116.7 125.5 138.5 137.0 134.9 149.0 166.1 163.9 147.3
80.6 82.2 80.8 82.8 85.9 88.1 89.6 90.4 89.9 92.0 95.4 97.6 98.4 100.0 100.6 102.8 102.5 101.5 101.9 94.8 95.5 95.3 94.3
34.1 37.7 39.8 41.0 46.4 51.6 58.0 65.0 66.8 68.3 75.8 83.7 90.9 100.0 115.8 127.7 142.3 154.7 168.0 195.4 225.9 253.7 280.7
87.5 103.2 107.0 101.8 99.0 100.5 194.2 106.8 103.4 101.0 103.2 99.8 99.2 100.0 101.9 103.4 105.6 105.6 104.5 108.1 114.7 114.7 116.7
38.7 42.6 43.5 45.0 49.3 55.4 59.5 63.5 64.4 70.6 80.5 84.9 92.4 100.0 110.7 112.1 129.0 133.0 151.2 173.9 192.1 192.1 207.5
Sector
86.0 112.5 103.2 98.8 98.9 100.0 99.4 100.9 99.9 99.6 97.7 97.1 97.7 100.0 104.8 105.4 112.0 111.7 114.0 106.6 112.8 116.9 117.7
References Bruno, M. “Duality, Intermediate Inputs and Value-Added.” In Production Economics: A Dual Approach to Theory and Applications, Volume 2. M. Fuss and D. McFadden, eds. Amsterdam: North-Holland, 1978. Cox, D.R. “Tests of Separate Families of Hypotheses.” In Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability. J. Neyman, ed. Berkeley: University of California Press, 1961. Hoch, I. “Simultaneous Equation Bias in the Context of the Cobb-Douglas Production Function.” Econometrica 26 (October 1958): 566-78. Hodges, D.J. “A Note on Estimation of Cobb-Douglas and CES 52
Alternative
Production
Function
Models
Production Function Models.” Econometrica 37 (October 1969): 721-25. Schim van der Loeff, S. “Estimation of a CES Production Function for the Dutch Manufacturing Sector.” Econometric Institute, Rotterdam, Working Paper, 1974. -. and R. Harkema. “Three Models of Firm Behavior; Theory and Estimation with an Application to the Dutch Manufacturing Sector.” Review of Economics and Statistics 58 (February 1976): 13-21. Schmidt, P. “On the Statistical Estimation of Parametric Frontier Production Functions.” Review of Economics and Statistics 58 (May 1976): 238-39. Zellner, A., J. Kmenta, and J. Dreze. “Specification and Estimation of Cobb-Douglas Production Function Models.” Econometrica 34 (October 1966): 784-95. -. and J.F. Richard. “Use of Prior Information in the Analysis and Estimation of Cobb-Douglas Production Function Models.” International Economic Review 14 (February 1973): 107-9.
53