quasi-fixed inputs in u.s. and japanese manufacturing: a generalized leontief restricted cost...

Quasi-fixed Inputs in U.S. and Japanese Manufacturing: a Generalized Leontief Restricted CostFunction ApproachAuthor(s): Catherine MorrisonSource: The Review of Economics and Statistics, Vol. 70, No. 2 (May, 1988), pp. 275-287Published by: The MIT PressStable URL: http://www.jstor.org/stable/1928312 .

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QUASI-FIXED INPUTS IN U.S. AND JAPANESE MANUFACTURING: A GENERALIZED LEONTIEF

RESTRICTED COST FUNCTION APPROACH

Catherine Morrison* Abstract-A Generalized Leontief variable cost function is developed which permits analytical derivation of steady state levels of quasi-fixed inputs and capacity output, and does not require specification of a numeraire input. This framework is then used to evaluate the factor demand patterns of Japanese and U.S. manufacturing, based on both one quasi-fixed (capital) input and two quasi-fixed (capital and labor) input specifications and both static and dynamic optimization. The findings provide evidence of considerable flexibility of demand responses in Japan, which may have contributed to Japan's relatively strong economic performance in the volatile 1970s.

I. Introduction

N the past two decades numerous empirical I studies based on flexible functional forms have been published characterizing the behavior and technology of firms. Although flexible functional forms in general are designed to capture complex input substitution patterns, the vast majority have been based on the translog (TL) functional form developed by Christensen-Jorgen- son-Lau (1973). Reasons for this preference to- ward the TL form might include the fact that its logarithmic form facilitates empirical imposition of homogeneity constraints, regularity conditions, and the calculation of elasticities. The Generalized Leontief (GL) functional form proposed by Diewert (1971) has been established as a useful alternative for long-run production studies but has not enjoyed the same degree of popularity.1

Most empirical studies of production behavior have also tended to assume instantaneous adjustment of all inputs to full equilibrium levels. This is due at least in part to the smooth time trends followed by numerous economic variables that helped rationalize full equilibrium assumptions. In

addition, although theoretically the short-run fixity of some inputs such as capital has been acknowledged at least since the time of Marshall, it is only recently that the distinction between short- and long-run functional forms has been well established. Events of the last decade-for example, unexpected sharp energy price increases and substantial cyclical variations in production -have drawn increased attention to the usefulness of distinguishing temporary or subequilibrium from full equilibrium. This has provided motivation for development of empirical research based on short-run or restricted cost functions.

The studies in the existing small empirical liter- ature based on restricted cost functions are based primarily on either a translog variable cost function or a normalized quadratic cost function.2 These functions are generally applied to time series data, and in this context have been denoted short- run as distinguished from long-run functions.3 A problem with the translog function for short-run studies is that due to its nonlinear logarithmic form one cannot analytically compute the full equilibrium level of the fixed inputs, but instead must rely on iterative numerical techniques. Some researchers have reported difficulties in obtaining numerical convergence with the translog variable cost function and thus with computing estimates of long-run elasticities.4 When the normalized quadratic form is used it is necessary to choose one variable input as the numeraire. As a conse- quence, the demand function for the numeraire

Received for publication August 27, 1986. Revision accepted for publication September 16, 1987.

* Tufts University and the National Bureau of Economic Research.

Support from the National Science Foundation, Grant # SES-8309352 is gratefully acknowledged. I am also indebted to Ernst R. Berndt and David 0. Wood for providing the U.S. data, to Takamitsu Sawa of Kyoto University for the Japanese data, and to an anonymous referee for helpful comments. 1 Interestingly, both the translog and Generalized Leontief

forms were considered already a decade earlier by the agricultural economists Heady-Dillon (1961).

2 Translog restricted cost or profit functions have been estimated by, for example, Atkinson-Halvorsen (1976), Brown- Christensen (1981), Bemdt-Wood (1984) and Berndt-Hesse (1986). Normalized quadratic variable cost functions have been employed in Berndt-Fuss-Waverman (1980) and Morrison- Berndt (1981).

3 There is really no explicit time dimension in the adjustment process, however, so the terms "partial static equilibrium" and "full equilibrium" used by Brown and Christensen (1981) are equally valid and more general.

See, for example, Berndt-Hesse (1986). Note also that additional problems with the standard approaches to using translog functional forms have recently been addressed by Kula- tilaka (1987).

Copyright ? 1988 [ 275 ]



276 THE REVIEW OF ECONOMICS AND STATISTICS

input differs from those for the other variable inputs. This asymmetry causes lack of invariance of empirical results to the choice of numeraire input.5 These difficulties suggest that the functional forms used to date in empirical production studies that distinguish between long- and short- run have drawbacks that may be dealt with by a suitable choice of functional form.

The first purpose of this paper is to present a short-run extension of the Diewert Generalized Leontief function that explicitly incorporates the effects of quasi-fixed inputs, yet does not suffer from the drawbacks mentioned above. Specifi- cally, the GL restricted cost function presented here, an extension of Parks (1971), Woodland (1975), and Diewert and Wales (1987), permits analytical, closed-form derivation of long-run equilibrium levels of the quasi-fixed inputs and does not require an arbitrary choice of numeraire input. Also, it can easily be employed in studies based either on static or dynamic optimization.

The second purpose of this paper is to illustrate the empirical usefulness of the GL cost function by employing data from the manufacturing sectors of Japan and the United States to characterize short- and long-run price elasticities of demand. Although capital has generally been specified to be the only fixed input in most previous studies of U.S. firms, for Japan the existence of "lifetime" contracts and substantial bonus payments implies that both labor and capital inputs might be better considered as fixed in the short run. The large fluctuations in input prices and output demand experienced at least since the 1970s also suggests that future expectations may be an issue. The multiple fixed input framework developed here allows both labor fixity and dynamic optimization to be specified as alternatives for empirical implementation.

II. The Theoretical Structure

Incorporation of fixed inputs into a model for analysis of firm behavior first requires specification of a restricted or partial static equilibrium function, generally a cost function, as a basis for estimation. The advantages and disadvantages of using different functional forms as approximations to the general restricted cost function differ from

those for full equilibrium analysis. Interrelation- ships among the variable inputs, produced output and fixed input use are more complex than in an instantaneous adjustment model. Further, both short-run and long-run behavior must be calcula- ble from the restricted cost function, and when the number of quasi-fixed inputs is increased empirical tractability must be ensured.

Many researchers6 have simply adapted the long-run translog (TL) cost function for short-run analysis by using the stock(s) of fixed inputs(s) as argument(s) of the cost function instead of their prices. However, numerically imputing long-run behavior from the short-run TL cost function has sometimes caused practical difficulties. Bemdt and Hesse (1986), for example, report considerable difficulty in obtaining numerical convergence for their long-run elasticity calculations based on the TL restricted cost function. Although this difficulty is a numerical issue, it clearly is 'exacerbated if curvature conditions are close to being violated.

Other studies have used cost of adjustment dynamic models based on the work of Lucas (1967) and Treadway (1974) to characterize investment behavior.7 With this approach, problems arise not only because the adjustment matrix is not tracta- ble in general for multiple quasi-fixed inputs,8 but also because it is difficult in this already complex framework to incorporate the effects of nonstatic expectations. In addition, to ensure tractability of estimating the endogenous adjustment process, the models have been based on a quadratic cost function, which, to satisfy linear homogeneity restrictions, must be normalized by an arbitrary choice of a numeraire input price. This results in asymmetry of demand equations, and invariance to the choice of numeraire.

Although these difficulties are not as severe as many faced by empirical researchers, they imply that alternative specifications may be useful. One functional form which may be used as the basis for a theoretical and empirical model of the firm's short-run behavior is a suitable extension of the Generalized Leontief (GL).

S This was discussed by Mahmud et al. (1986) among others.

6 Including Brown and Christensen (1981), Bemdt and Wood (1984), Bemdt and Hesse (1986), Pindyck and Rotemberg (1983) and Morrison (1985), among others.

7 These studies include Berndt, Fuss and Waverman (1980), as well as Epstein and Denny (1980) and Morrison and Bemdt (1981).

8 Some restrictions on interactions must usually be imposed. See Morrison and Bemdt (1981).



QUASI-FIXED INPUTS IN MANUFACTURING 277

The traditional GL cost function is a functional form in the square root of prices. Beyond this point, a variety of generalizations have been appended to account for, for example, technical change and returns to scale (Parks (1971), Woodland (1975), Bemdt and Khaled (1979), Diewert and Wales (1987)) and for fixed inputs (Mork (1978) and Mahmud et al. (1986)). These extensions vary in their emphases and therefore on the methods used for "generalizing" the GL.

In this study, the standard GL form is extended to allow for fixed inputs in a manner more com- parable to Parks-Woodland-Diewert-Wales than Berndt-Khaled-Mork. The additive structure of Parks-Woodland is retained, as are the interaction and price-sum terms of Diewert-Wales. However, the interaction and intercept terms in Diewert- Wales are simplified to allow for a symmetric representation of additional arguments of the cost function, thereby facilitating incorporation of multiple quasi-fixed inputs. Such a GL restricted (or variable) cost function can be written as

y- CijPi, PJ+ E E8imPiSmn

?EPiY E Ymn n i m n + yS E8EikPiXk

i k

+ EAiE E Ymk Sm5xk ] i m k

+ EPi E E -YlkXk Xi ( ik I

where pi is the price of variable input i and i, j are subscripts denoting variable inputs; Xk is the stock of quasi-fixed input k and k, 1 denote fixed inputs; and m, n enumerate output (Y) and other exogenous arguments of G not included in the returns to scale specification, such as the state of technology t or investment in quasi-fixed inputs AXk. Note that this functional form varies in its treatment of the different types of arguments, particularly with respect to output. This construction allows nesting of different scale specifications within the general functional form.9

Demand equations for empirical implementation may be derived from (1) based on Shephard's Lemma. Alternatively, to reduce possible hetero- skedasticity, input-output equations can be estimated. These equations may be written as

vi dG 1 5 - 2?1 = ai1(p1/pi)5 y apjy =

? jESijS 5 + E Y-YmnS2isn5 m m n

+ 2Y2Ymk SmXk ] m k

? Y * E Y2kXIXk5Xi5. (2) k I

It is clear from (1) that if the variable input prices all increase by the proportion X, G must increase by X, so linear homogeneity in prices is satisfied. Equivalently, all the demand equations including the "demand" for fixed inputs-the desired xk* levels-remain constant if all prices increase equiproportionally, as can be seen by in- spection of (2) and also (14) below. In addition, as with most functional forms,10 global convexity (concavity) in the Xk (pi) variables, such that 32G/dx2 > 0 and d2G/dpi < 0 is not ensured by this form. It also has problems in being imposed because this would require constraining all the 3ik and Ylk terms to be nonpositive (convexity) and all 8ij terms to be nonnegative (concavity), implying substitutability of all inputs.'1 Curvature at each sample point can be tested, however, by computing the second derivatives of G and check- ing signs. Finally, the GL functional form represented by equations (1) anJf (2) allows for interactions among all exogenous Y, t and Xk variables, or more formally, is flexible enough to ensure a local second order approximation to a general restricted cost function.'2

9 In earlier versions of this study a more symmetric form was specified where all xm variables were treated as Xk variables except that the entire function was multiplied by Y to facilitate specification of input-output equations. Although this form is

equally as useful and more elegant than the specification in the current paper, the desire to test constant returns to scale restrictions suggested that the current form is more useful.

10 See Diewert and Wales (1987) for further discussion. h This is consistent with long-run concavity in all prices,

once the long-run desired values of Xk and xi are substituted into the cost function.

12 This can be checked using the rules outlined in Diewert and Wales (1987), adjusted slightly to take into account the short-run nature of the function. This form in (1) has just enough parameters to be flexible.




Although the generality of the GL form in (1) is appealing, empirical researchers have found it difficult to identify independently the impacts of technology, quasi-fixed inputs and returns to scale. It is therefore often useful for the GL function to be constrained a priori to long-run constant returns to scale (CRTS) where all long-run output elasticities equal one; i.e., d ln vl/d ln Y = d ln xl/d ln Y = 1.13 Equation (1) has been constructed so that these restrictions can be satisfied simply by setting YYk = YmY = iY= 0. Incorpo- rating these restrictions and making the common assumption that the Sm vector only includes t and Y results in the form:

G = Y. r* E a1ijap5 Pi5 iti [ji

+ Epiyttt1 i I

+ >EE8ikPi Xk i k

+2 -EPi Ytkt Xk] i k k

+ Epi Y 2 2YklXk5' Xi5 (3) i k I

The corresponding input-output equations are v dG 1l

_ a d y-Eaij(Pj1Pi)' y 9piY ,,i

?i6. t5 Yttt

+ Y-.5[Y .X5 + E8ikXk k

+2 * EYtkt Xk ] k

+ y- . TW yk-5Xk5i (4)

k I

Equations (4) therefore comprise a system of estimable equations representing the short-run de-

13 Here

d ln v/d In Y = Y/v,

V(I v/aY + 9 Vi/9Xk * dxk/dY),

k$ Y,

and

d In Xk/d In Y = Y/x,* (8Xk/axl *dxk/dY)-

mand behavior of the firm facing a CRTS technology.

In addition to the system of input-output equations, information about the quasi-fixity of inputs can also be included for estimation purposes. Consider the definition of the shadow value of quasi-fixed input k (the potential reduction in variable costs from having an additional unit of Xk, -aG/aXk = Zk):

Zk = - .5* [Y2. i*EYk1XkPXi ? Y 5xk

i l .~~~~~~~~~~. ,ik -pi+ 2. 2PiYtk t

(5) This expression explicitly captures the dependence of the shadow value on the relationship between xk and all input prices, all quasi-fixed inputs, Y, and t.

With only one quasi-fixed input, perfect competition and long-run CRTS, after payments to all variable inputs have been made returns to the firm can be attributed to the fixed input. In this case, (5) can be appended to the system of estimating equations (4) by calculating the ex post return to Xk as Zk = (P. Y - G)/xk for the left hand side of the equation, where P is the output price. In the multiple quasi-fixed input case the left hand side shadow values are not individually well- defined. This occurs because the nonparametric residual calculation of the ex post value of the quasi-fixed inputs is not readily decomposed into the individual returns to each fixed input. It is possible, however, to estimate the model using the sum of the shadow values as a single dependent variable. In this case one can calculate (P Y- G) = (Revenue-G)-R-net = EkZk *Xk as the dependent variable, and add all equations represented by (5) multiplied by their input levels for the right hand side of the expression:

net EPi . EyklXk Xi ki l

?Y5X,5* (Eik Pi

+ 2 - EPiytk t )}].

(6) An additional possible estimating equation rep-

resenting profit maximizing behavior is the output




price equation P = MC (or P - Y/G = MC- Y/G = (dG/dY)- Y/G) proposed by Mork, which is, however, equivalent to the shadow value equations (5) and (6) in the short run.14 More specifically, with output price equal to marginal cost and perfect competition, an estimating equation based on the revenue relationship can be constructed by calculating

d = E aijp;5pf + DeitPi p t 5 + EPiYttt ay

i i +.-5 * Y5[ 2E ik Pi.Xk

i k

+2 * EAi Fytkt Xk.5

(7)

and estimating this equation, substituting output price P on the left hand side.

Using both (6) and (7) as estimating equations with CRTS is, however, redundant because they contain equivalent information. I.e., recall from (4) that with CRTS it must be the case that

1= EGY + Gk' k

or E d G Xk E -ZkXk

Gk k aXk G k G -(PY- G)

G PY

=l1l- - y= 1 - ,GY (8) G

where a ln G/ ln Y= EGY and a InG/d ln xk = EGk by definition. Thus all of the information in (7) is contained in (6) as long as the ex post rate of return to the quasi-fixed inputs is calculated resid- ually as YkZk * xkG = (PY - G)/G in the data construction process.

Even if the shadow value or output price equations are not used for estimation, the expressions (5) and (7) allow the shadow values and marginal cost to be imputed in a parametric form. This facilitates assessment and interpretation of the

extent and direction of deviations from equilibrium in terms of costs, reflected by the difference between the shadow value of Xk, Zk' and the market price, Pk. The analytical representation in (5) also allows direct derivation of the steady state or desired levels of fixed arguments X4* in closed form. Thus representation of the gap between actual xk and Xk* that measures the extent of subequilibrium in terms of input levels is possible.

More specifically, the shadow value expression in (5) lends itself directly to analytical calculation of Xk* from the steady state relationship Pk = Zk; in full equilibrium the market rental and shadow value of all fixed inputs must be equal.15 Imposing this equality and solving for X4* results in

=* . V Y *S ? 2. - E 0)1 k ( k 'Pi PYkt)

+ EPi7Yk1(k*,)Xi5 (Pk + JPi Ykk'fl.

(9) Note that x * is homogeneous of degree zero in

prices, as is required by theory. Also, Ykk > 0 implies that the long-run elasticity of capital with respect to Pk is negative and thus that the curvature conditions on Xk with respect to Pk are satisfied."6 Finally, although curvature conditions of x4* with respect to pi are not defined, the convexity conditions for xk, d2G/dx2 > 0, imply that the 8ik terms should not al be positive (capital may not be complementary with all variable inputs). This would be expected both from standard theoretical results and also because if an input is fixed in the short run some other input, by definition a substitute, must overshoot its long-run value to compensate for the fixity in producing the given amount of output.

One difficulty remains with multiple fixed input specifications. For full equilibrium X4* can only be

"1In the long run price must be set equal to long-run marginal costs, which also takes aXk*/dY into account. More specifically, total costs are G + 2k Pk Xk, so short-run marginal costs are simply aG/I Y since Xk does not change, but long-run marginal costs must take the change from Xk to Xk*, condi- tional on Y, into account.

15 This is a standard result which is conceptually obvious; the market and implicit values of the quasi-fixed inputs must be equal in full equilibrium or further adjustment would be desired. It is also analytically simple to motivate. In full equilibrium it must be the case that the SRAC and LRAC curves are tangent. This tangency condition is dTC/dK = 0, where TC = G + PK * K. Straightforward differentiation then implies that - aG/dK = PK, where - aG/dK is by definition the shadow value Zk. 16Although xk* may either increase or decrease (or stay

constant) with changes in pi, curvature requirements imply that this tendency should decrease over time. The conditions for this to hold are, however, complex and difficult to impose.




calculated correctly if it is computed at the x, levels, implying that computations of these xk and x, values must be accomplished simulta- neously. Since the GL form allows for analytical specification of the x,, these expressions may be substituted directly into (9), implying that numerical techniques are not necessary. Clearly, difficulties associated with this computation increase as the number of quasi-fixed inputs increases.

The next step in constructing the GL variable cost framework is to specify the form of the substitution elasticities. The short-run or subequilibrium elasticities are defined as

SR a av pj(10) -a In pj apj Xk=Xk(1

Similarly, an elasticity of variable input demand with respect to output changes can be specified as

ESR a lnv= av Y (11) 'y l InY aY vi' Clearly these short-run elasticities are straightforward to compute using the parameters from the system of estimating equations (4).

The long-run elasticities are nearly as easy to specify empirically as the short-run elasticities, especially for single x models, given the analytical expression of x4 in (9). Once x4 has been calculated, G and the demand equations are evaluated at x at each observation to compute long-run costs and demand.'7 The corresponding long-run elasticities can then be derived first by computing the short-run adjustment of the dependent variable (cost or input demand) with respect to a change in the exogenous variable, and then adding the associated long-run change.18 More specifically, for the long-run elasticity of variable input j with respect to a change in the price of variable input i,

LR Pi [__ dk dXk1 E1 - i dXk , ~ (1 2)

V1L~ Xk=Xk k dJ

where Xk . Own-price, fixed input-price and output elasticities are computed similarly. Note again that with multiple quasi-fixed inputs which are interdependent (Ykl = 0, k * 1) the dx4/dpi terms must take adjustment of all quasi-fixed inputs to full equilibrium levels into account. This can be accomplished by simulation of these elasticities or by using the derivatives of (9) with the x* values substituted.

The theoretical framework as developed so far allows representation of either a single or multiple fixed input cost function. Thus short-run firm optimization with only capital fixed, or with including quasi-fixed labor to take labor rigidities into account, may be modeled. One possible extension of these results allows partial adjustment processes and thus incorporates dynamic optimization. Although such a framework is not amena- ble to explicit consideration of investment decisions which would add a true time-dimension to the problem of current factor demand behavior, more restrictive dynamic structures may usefully be incorporated.

More specifically, one possible extension`9 is to adapt the approach of Berndt, Fuss and Waverman (1980) (BFW), which includes explicit incorporation of an investment equation based on an endogenous adjustment matrix implying geo- metrically declining adjustment over time. How- ever, since the BFW model is based on a quadratic cost function, the second derivatives of G are simply parameters, allowing for straightforward calculation of the adjustment matrix. With the GL form the second derivatives are very complex and an analytical solution for the investment equation is intractable, causing the BFW procedure to be inoperable.

An alternative approach which is more feasible in this framework and incorporates certain dynamic aspects is that proposed by Pindyck and Rotemberg (1983) (PR). Their basic idea can be decomposed into two parts: (i) estimation of the Euler equation rather than an investment equation

17 Note that the capital elasticities-those elasticities which reflect the long-run impact on K of a change in an exogenous variable-can be considered "desired capital" elasticities, which capture the change in K* in response to a change in the economic conditions facing the firm.

18 See Berndt, Fuss and Waverman (1980) for a more complete discussion of this procedure.

19 One approach not discussed here requires adaptation of the variable cost function G into a value function and use of this to impute adjustment behavior, as in Epstein (1979). This is, however, not only a very complex procedure but requires a complete respecification of the model rather than an adaptation of it. It also has had limited empirical success to date. It may, however, be a fruitful approach to use in future research.




so that an analytical solution for investment de- termination is not required, and (ii) incorporation of anticipatory expectations by use of instrumental variables, thus "instrumenting out" the errors incorporated in the observed exogenous variables such as current prices that are not relevant for investment decisions. Instrumenting the current variables using past values of the variables causes the resulting estimates to be consistent with ra- tional expectations, provided that the instruments chosen adequately represent the information set available to the firm.

There are two ways to incorporate a PR-type procedure into the current model. The first, as in the original PR study, is to assume a simple additive external costs of adjustment model, where additional investment becomes increasingly costly due perhaps to pressures on the capital supplying market. This method simply requires appending one more equation to the variable input demand equations (4), a Euler equation incorporating the impact of quadratic external adjustment costs on capital accumulation decisions.

Such an approach is, however, unlikely to be empirically effective if external costs of adjustment are small and insignificant, as was suggested by Morrison (1983) and was also found to be the case in my preliminary empirical investigation. Alternatively, using the basic ideas underlying the PR study to simplify the estimation of a BFW-type model incorporating internal costs of adjustment appears more promising. Specifically, optimization is assumed to be based on a variable cost function with the Axk (AK with one quasi-fixed input and k in continuous time) as additional arguments incorporated as sm variables. Internal costs of adjustment are then represented by G(Ax) =

dG/dIA.xkl . Ax, with dG/dIAXkl > 0, d2G/ d IAxk 12> 0 and G(0) = 0. Note that although the curvature conditions cannot be imposed without excessively restricting the function, similarly to convexity of the function with respect to Xk, they can be checked. In addition, from the expression for dG/dAx, similar to (5) but adapted for the difference between the Sm and Xk variables, it is clear that G(O) = 0.

The optimization problem facing the firm is now to choose the path of Xk, or Axk each year, in addition to the vector of variable inputs at each time period, to minimize the present value of the stream of costs from producing output Y at each

future period.20 Minimization of the implied cost expression yields the Euler first order conditions (in terms of continuous time):

-G- rG,- p + G + GXXX = O, (13) as in BFW, where * denotes the second derivative of x with respect to time. Estimation can therefore be carried out for the system of equations including (4) and (13), using instrumental variables to adjust for unobserved expectations as in Pindyck- Rotemberg.

III. Empirical Implementation and Illustration

Empirical implementation of the GL model was carried out on data for U.S. manufacturing 1952-81 from Berndt and Wood (1984) and for Japanese manufacturing 1955-81 from Sawa (1986). All reported estimates are based on the Generalized Leontief CRTS restricted cost function represented by (3). Preliminary results using the NCRTS function given by (1) indicated some difficulties identifying separately the effects of technical change, quasi-fixity of inputs and returns to scale, so the potential extension to NCRTS was not pursued.

The original estimated model to be discussed is the static one-quasi-fixed-input (capital, K) (1QFI) model, with three variable inputs, labor (L), energy (E), and non-energy intermediate materials (M). In addition to the relevant system of variable input demand equations (4), the output-price equation (7) was appended for estimation. This equation was used instead of the shadow value equation because of the difficulties of identifying the ex ante and ex post values for the quasi-fixed inputs for the Japanese data.21

20 See BFW or Morrison-Berndt for clarification of the underlying steps.

21 Results were quite robust to the alternative shadow value or output price specifications. However, if only the variable input input-output equations were estimated, YKK was not well identified. In these cases YKK-a crucial parameter for imputing ZK-often had a large standard error and sometimes was insignificantly different from zero. One qualification to this is that in those runs where YYKK was insignificant or small, imposing higher values of YKK within two standard deviations of the estimated value generally provided results very compara- ble to those including the shadow value or output price equations. Although crude, such tests suggest that a more "legiti- mate" value for YKK consistent with other models cannot be rejected for those models estimated without the extra equations.




One extension to this static model is a two quasi-fixed input model (2QFI) including labor as fixed instead of variable, which may more effec- tively capture the institutional environment in Japan. Again, incorporating the output price equation was the preferred approach for estimation of that model.

An alternative version of the original model incorporates dynamic optimization into the one quasi-fixed input framework. Both external and internal adjustment cost approaches were investi- gated. In either case the system of equations used for estimation included the base equations (4) plus a Euler equation as outlined in the previous section. The internal costs of adjustment model is employed for the results discussed below since in preliminary analysis it appeared empirically more justifiable.22

With respect to econometric issues, any of these systems of equations can be estimated using iterative Zellner techniques, since simultaneity is not a problem. Even the dynamic model has a recursive ,structure and thus this type of systems estimation is valid. Additional complexities arise, however, once the possibility of nonstatic expectations or the potential for endogenous output choice is acknowledged. An iterative Three Stage Least Squares estimation procedure was therefore used as a basis for empirical implementation of both the static and dynamic frameworks.

In particular, for the static model the expectations problem is theoretically unimportant since the firm's adjustment process is not incorporated; only short-term behavior with respect to variable inputs is represented. Even in this model, however, the possibility of endogenous output has prompted some empirical researchers to instrument the output level. Similar concerns about the firm's forward-looking behavior not modeled but perhaps captured in the data suggest that perhaps the "exogenous" prices should be instrumented to accommodate any errors in variables. Thus, a possibly preferable econometric approach is to estimate the static model by instrumental variables.

For the dynamic model, the necessity of estimating the model with instrumental variables is implied by construction, since once the adjust-

ment process is included the firm's forward-looking decisions become a crucial part of the behavioral model. To implement this model various instrument sets could be used, but Pindyck and Rotemberg generated their most satisfactory results using lagged values of the exogenous variables as instruments. This is an intuitively appealing approach if one believes some extension of adaptive expectations-using past values of the variables to represent current expectations-is valid.

An additional econometric issue is the pooling of the time series for these two countries. Pooled cross-section time-series methods are well developed and are quite straightforward to implement as long as only intercept terms are assumed to vary. The pooling method used in this paper is a structural approach in which different first order terms are assumed for each country in the variable input demand equations. These new terms are then also added to the variable cost function to retain integrability. Essentially this requires appending an additional country-effects dummy for Japan (denoted DJP)- to each individual input estimating equation by substituting (Ykk + YkJ

DJP) for Ykk and (/i3 + /3ij. DJP) for /3ii in all equations (where k = K and i = E, L, M for the 1QFI models and k = K, L and i = E, M for the 2QFI model).23

For the purposes of this study, such an approach appears preferable to the more common econometric-based techniques, since differential industry structure with respect to the quasi-fixed factors can be modeled using the structural method even if a shadow value equation is not estimated. More specifically, if one does not estimate an equation reflecting capital shadow valuation or accumulation no distinction between countries' behavior can be captured by the fixed-effect approach since a dummy is not included for capital. With the structural approach, differences are also captured in the demand equations.

Parameter estimates for the alternative models are presented in table 1.24 The results are quite

22 Incorporation of the PR procedure changed the results negligibly from the static models, suggesting that external costs of adjustment may not be very important for the manufacturing sector in each country.

23 This pooling adds enough extra information to stabilize the results as compared to individual country runs. For the empirical results below, this was particularly important for the Japanese data which cover a smaller time period than the U.S. data, although the overall patterns of results were quite robust ,across pooled and individual country specifications.

24 aKt iS estimated for the static specifications, but is omitted

in the dynamic model because of conceptual problems result-




TABLE 1.-THREE STAGE LEAST SQUARES PARAMETER ESTIMATES OF ALTERNATIVE MODELS (absolute value of t-statistics in parentheses)

Static Dynamic Static Dynamic 1QFI 2QFI 1QFI 1QFI 2QFI 1QFI

aLE 0.0200 -0.0116 8MK -1.6850 -2.0491 -1.5084 (5.233) (3.135) (31.446) (14.659) (36.892)

aLM 0.2128 0.2123 YLL 0.4402 (25.69) (22.019) (2.182)

aEM 0.0039 0.0190 0.0298 YLJ -0.1275 (0.831) (6.366) (7.448) (0.700)

aLL 0.2854 0.2310 8EL -0.5881 (7.054) (4.815) (11.480)

aEE 0.1069 0.2635 0.0177 8ML - 1.1272 (6.680) (9.224) (1.635) (17.575)

aMM 0.9008 1.7542 0.8776 YKL - 0.2325 (95.421) (46.091) (76.249) (0.612)

8Lt --0.0380 -0.0388 YKLJ 0.3743 (13.549) (14.636) (1.080)

8Et - 0.0005 -0.0141 - 0.0008 -0.0006 (0.469) (5.384) (1.318) (4.429)

8Mt -0.0336 -0.0567 -0.0316 8EK 0.0004 (14.923) (11.671) (10.018) (3.411)

aLJ -0.0438 -0.1018 8LK -0.0015 (3.240) (11.478) (4.815)

aEJ 0.0846 0.0076 0.0401 8MK 0.0013 (12.403) (1.595) (8.920) (4.325)

aMJ 0.1958 0.1526 0.1816 YtK -0.0044 -0.0356 (36.855) (15.597) (31.949) (2.664) (1.033)

YKK 1.3948 1.8429 0.5745 YtL 0.0385 (9.120) (2.808) (6.222) (5.397)

YKJ -0.6455 -1.1152 -0.3790 (12.087) (1.722) (11.752) R2 'S

8EK - 0.6861 -0.4058 -0.1642 L/} 0.9712 0.9889 (7.237) (3.206) (2.864) E/Y 0.6389 0.5716 0.9738

M/Y 0.6096 0.6757 0.9926 8LK -0.6370 -0.0677 PY- G 0.9212 0.4117

(4.404) (0.772) PK 0.2229

similar considering the important differences in model construction; the results for the 2QFI case deviate most from the overall pattern.

It is difficult to determine from the parameter estimates whether labor fixity is important for representing the technology and firm behavior of Japan and the United States in the 2QFI model.

Some evidence from the R2's which indicate a significantly-higher fit for the 1QFI models, and particularly the dynamic specification, than for the 2QFI model, suggests that labor may not be as fixed as is often postulated. Evidence from the pooling parameters of the 2QFI model also suggests that the difference between the countries may be unimportant. For this model the parameter that captures the labor-effect directly, YLL, is positive and marginally significant, but the param-

ing from difficulties involved in solving the differential equations from the first order conditions with a cross K - t term.




eter reflecting the difference in labor structure between the two countries, YLJ' is small and insignificantly different from zero. Note, however, that for the 1QFI models aLJ is significantly different from zero, suggesting that the structure varies across countries. This difference between specifications is highlighted by estimates of the other parameters that distinguish between the two countries' responses; aLJ? aEJ, and YKJ are each sig-

nificantly different from zero for the 1QFI models but tend to be insignificant for the 2QFI model.

Comparisons and assessments of the alternative models may be carried out more readily using elasticity estimates. Labor responsiveness, for example, can be assessed by looking at the labor elasticities reported in table 2. For all models the Japanese own-elasticity for labor is substantially larger than that for the United States, and for the

TABLE 2.-SELECTED ELASTICITIES, REPORTED FOR 1972 (standard errors in parentheses)

IQFI-Static 2QFI-Static IQFI-Dynamic

United States Japan United States Japan United States Japan

SR LR SR LR SR LR SR LR SR LR SR LR

(EE -0.290 -0.323 -0.268 -0.269 -0.131 -0.554 -0.128 -0.394 -0.199 -0.365 -0.182 -0.200 (0.026) (0.032) (0.022) (0.023) (0.020) (0.049) (0.020) (0.040) (0.023) (0.044) (0.022) (0.026)

EEM 0.043 0.292 0.041 0.063 0.131 0.567 0.128 1.046 0.327 1.258 0.310 0.762 (0.054) (0.104) (0.050) (0.096) (0.020) (0.176) (0.020) (0.107) (0.044) (0.175) (0.042) (0.165)

EEL 0.243 0.196 0.227 0.225 0.435 0.242 -0.127 -0.372 -0.128 -0.187 (0.046) (0.056) (0.043) (0.045) (0.074) (0.074) (0.041) (0.076) (0.041) (0.049)

EEK -0.165 -0.021 -0.448 -0.894 -0.522 -0.374 (0.031) (0.056) (0.113) (0.053) (0.071) (0.118)

EEY 0.572 1.000 0.963 1.000 0.384 1.000 0.340 1.000 0.307 1.000 0.694 1.000 (0.074) (0.017) (0.082) (0.026) (0.051) (0.173) (0.082) (0.046) (0.048) (0.031) (0.085) (0.028)

EME 0.003 0.021 0.004 0.006 0.017 0.073 0.020 0.167 0.027 0.105 0.039 0.095 (0.004) (0.008) (0.004) (0.009) (0.003) (0.044) (0.003) (0.028) (0.004) (0.017) (0.005) (0.022)

cMM -0.188 -0.317 -0.220 -0.507 -0.017 -0.606 -0.020 -0.952 -0.221 -0.657 -0.331 -1.706 (0.009) (0.037) (0.011) (0.074) (0.003) (0.042) (0.003) (0.092) (0.010) (0.089) (0.015) (0.324)

EML 0.184 0.209 0.217 0.235 0.361 0.192 0.194 0.308 0.292 0.470 (0.007) (0.018) (0.008) (0.027) (0.098) (0.070) (0.009) (0.037) (0.013) (0.079)

EMK 0.087 0.266 0.172 0.593 0.224 1.140 (0.015) (0.048) (0.046) (0.093) (0.040) (0.250)

EMY 1.226 1.000 1.456 1.000 1.637 1.000 1.780 1.000 1.333 1.000 1.931 1.000 (0.016) (0.007) (0.025) (0.018) (0.013) (0.027) (0.018) (0.036) (0.013) (0.017) (0.026) (0.073)

ELE 0.037 0.030 0.070 0.069 0.158 0.171 -0.020 -0.059 -0.035 -0.052 (0.007) (0.008) (0.013) (0.014) (0.056) (0.025) (0.006) (0.012) (0.011) (0.014

CLM 0.398 0.451 0.741 0.804 1.012 0.848 0.370 0.589 0.645 1.038 (0.016) (0.037) (0.029) (0.086) (0.178) (0.193) (0.017) (0.068) (0.029) (0.161)

ELL -0.435 -0.445 -0.811 -0.815 -1.089 -1.256 -0.350 -0.407 -0.609 -0.660 (0.016) (0.021) (0.029) (0.034) (0.057) (0.118) (0.017) (0.033) (0.030) (0.053)

ELK -0.036 -0.058 -0.080 0.238 -0.122 -0.326 (0.015) (0.057) (0.098) (0.104) (0.028) (0.111)

ELY 0.907 1.000 0.901 1.000 1.000 1.000 0.833 1.000 0.733 1.000 (0.031) (0.009) (0.072) (0.028) (0.162) (0.148) (0.021) (0.011) (0.037) (0.044)

EKE -0.078 -0.005 -0.288 -0.235 -0.233 -0.061 (0.015) (0.012) (0.113) (0.020) (0.034) (0.019)

EKM 0.573 0.628 0.852 0.978 1.312 1.477 (0.089) (0.102) (0.117) (0.107) (0.192) (0.275)

EKL -0.110 -0.040 -0.142 0.088 -0.344 -0.192 (0.046) (0.042) (0.174) (0.039) (0.079) (0.067)

EKK -0.386 -0.584 -0.422 -0.831 -0.734 -1.224 (0.039) (0.067) (0.157) (0.111) (0.094) (0.219)

EKY 1.000 1.000 1.000 1.000 1.000 1.000 (0.022) (0.021) (0.160) (0.062) (0.036) (0.059)




1QFI models the long-run elasticity differs little from the short-run elasticity. This provides some evidence for labor flexibility rather than fixity, especially for Japan, even though labor fixity would seem a priori important in Japan.

Labor and materials appear substitutable throughout, also suggesting potential flexibility that is greater in Japan. The other cross effects are, however, more ambiguous. The labor-energy relationship in the static models is consistent with some substitutability of labor resulting from energy price changes in both countries but more responsiveness in Japan. However, results for the dynamic model suggest complementarity to about the same degree for both countries. Interactions between labor and capital and labor responsiveness to output fluctuations also depend on the model used. For the 1QFI models labor and capital are complements for both countries, although with labor fixity incorporated labor and capital appear to be long-run substitutes in Japan, suggesting long-run flexibility. The labor-output elasticities suggest slightly less short-run labor flexibility for Japan than the United States, since in the short run for the United States both 1QFI models show a close to proportionate change in labor in the United States in response to output fluctuations (83% to 91%), whereas in Japan the adjustment is slightly lower (73% to 90%). Labor fixity implies that the corresponding short-run output elasticity for the 2QFI model is zero.

The labor-output results are consistent with the evidence of labor-capital complementarity that is larger in Japan. The complementarity would cause short-run responses to output changes to be at- tenuated more in Japan than the United States as a result of capital fixity. This difference may not be as important as it appears, however, if capital adjusts more quickly in Japan, as is suggested by some of the other estimates. In particular, the capital elasticities suggest less complementarity of capital with respect to labor price changes, and therefore more flexibility, in Japan than in the United States. The same is true for the capital- energy relationship. Similarly, capital is more substitutable with materials and has more own price responsiveness in Japan as compared to the United States.

The comparative flexibility of the countries' production structures in response to energy price shocks can also be assessed using these elasticities.

Surprisingly, for all models larger potential responsiveness to energy prices is evident in the United States than in Japan. However, substitutability of other inputs to accommodate energy price shocks appears stronger in Japan than the United States. For example, intermediate materials are substituted more in Japan to save high cost energy, although the standard error on this relationship is high. Also, as mentioned above more substitution of labor with energy price changes also appears possible, and the complementary relationship of capital with respect to energy is stronger in the United States than in Japan. The energy-output elasticities vary somewhat over models, generally indicating substantial short-run increasing returns to energy, especially in the United States. This suggests greater fixity of energy demand in the short run-a lack of flexibility to shocks that would affect energy use-in the United States.

Thus the results reported here are broadly com- patible with the conjecture offered by Sato and Suzawa (1983) concerning relatively substantial flexibility in Japanese production in response to energy price shocks, which was said to be due to relatively substitutable capital and energy in Japan. It also corroborates a similar hypothesis of vulner- ability of U.S. production to energy shocks suggested by Norsworthy and Malmquist (1983), because of the complementarity of energy and capital in the U.S. production process resulting from past cheap and plentiful energy supplies. The lower energy own price responsiveness for Japan, however, generates conflicting interpretations because the interactions are more complex than may ini- tially be recognized.

IV. Concluding Remarks

In this paper the standard instantaneous adjustment Generalized Leontief function has been extended to accommodate multiple quasi-fixed factors, and has been shown to have the desirable property that long-run equilibrium values of the fixed inputs, computed using shadow value rela- tionships, can be derived in closed form. The usefulness of this GL restricted cost function has then been illustrated using recent annual data from the manufacturing sectors of Japan and the United States. The altemative behavioral specifications, including static and dynamic optimization




and both capital and labor fixity, both illustrate the various types of models usefully estimated within this framework, and highlight certain differences in model assumptions that may be crucial for consideration of comparative productive structures in the two countries.

Among the empirical findings, the overall pattern is that more flexibility of labor is evident in Japan than would be a priori expected, possibly contributing to the Japanese ability to recover more quickly from problems such as the energy crisis in the 1970s and potentially the current reverse shock of downward adjusting energy prices. The evidence is also, generally, that more responsiveness is possible in Japan with respect to the effects of fluctuating energy prices on other input demands. However, energy own-price elasticities tend to be smaller in Japan than in the United States, both in the short and long run.

These findings about the comparative flexibility of the two production technologies are provoca- tive; if Japan is more flexible this could have been an important advantage during the volatile 1970s, contributing to its relatively strong productivity growth and other economic performance. Further research on the adjustment possibilities in the two countries in response to exogenous shocks will be very useful to further refine and interpret these results.

The GL restricted cost function framework proposed here provides a rich basis for extensions of current research. Clearly there are numerous forces generating varying economic trends in the manufacturing sectors of Japan and the United States. While the GL framework here has been focused on implications of allowing for short-run fixity of labor and capital inputs, it is of course also potentially useful for the analysis of numerous other issues involving, for example, further disaggrega- tion of inputs or by industry; assessment of tech- nological aspects such as the shadow costs of regulatory constraints, returns to scale, or imper- fect competition; or implications of alternative behavioral specifications such as variable profit maximization rather than cost minimization. These are all useful areas to direct future research.

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