Discussion Paper No. 8

Why Are Concavity Conditions Not Satisfied in the Cost
Function?: The Case of Japanese Manufacturing Firms
during the Bubble Period

Kazuo Ogawa

September 2008

ＧCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

September 2008

Why Are Concavity Conditions Not Satisfied in the Cost Function?:
The Case of Japanese Manufacturing Firms during the Bubble Period*

Kazuo Ogawa
Institute of Social and Economic Research
Osaka University* This paper is a complete revision of one previously circulated under the title “How Important Is
Land as a Production Factor for Japanese Manufacturing Industries?” An earlier version of this paper
was presented at the informal seminar in honor of Professor Saito held at Kobe University. The
author is grateful to the participants at the seminar for their helpful comments and suggestions. This
research was partially supported by a Grant-in-Aid for Scientific Research (19330044) from the
Ministry of Education and the Japan Economic Research Foundation. All remaining errors are, ofcourse, my own.

Abstract

This paper examines empirically some of the reasons why Japanese manufacturingfirms frequently fail to satisfy concavity conditions of the cost function. We focus on
the “bubble period” in the 1980s when land was in great demand for reasons related toboth production and speculation, and land prices soared. By estimating the translog costfunction with land as one of production inputs for manufacturing firms, we find that
violation of concavity resulted from borrowing constraints and large adjustments of
employment. We also demonstrate that elasticities of substitution between land andother inputs and input demand with respect to land rental prices are both estimated withlarge biases if the firms violating concavity are not excluded from the analysis.

JEL Classification Number: D24, R14
Keywords: Land, Translog cost function, Concavity
Correspondence to:Kazuo Ogawa, Institute of Social and Economic Research, Osaka University,
6-1 Mihogaoka, Ibaraki, Osaka, 567 JAPAN
tel: (81)6-879-8570 fax: (81)6-878-2766 E-mail: ogawa@iser.osaka-u.ac.jp

11. Introduction
One of the anomalies of empirical economics is the failure of concavity conditions inestimating cost functions. From a theoretical perspective, concavity of the cost function
is a basic tenet because concavity ensures a firm’s rational behavior for cost
minimization. Therefore, many attempts have been made to incorporate concavity into
estimating cost functions. Jorgenson and Fraumeni (1981) and Diewert and Wales(1987) proposed methods for imposing concavity conditions globally in the context of
cost function estimation, while Terrell (1996) and Ryan and Wales (2000) devised
methods for incorporating local concavity conditions into the cost function, maintaining
flexibility of the functional forms. Imposing concavity either globally or locally isdesirable to obtain the parameter estimates used for inferring the production structure of
the firm and evaluating the effects of various policy measures on the firm’s behavior.1
However, it is important to recognize that such an approach is based on the implicit
assumption that all of the firms in the sample are indeed minimizing production costs. If
some of the firms are incapable of minimizing production costs due to extraneous
circumstances, then imposing concavity conditions on the cost function misspecifies themodel and therefore yields inconsistent estimates of the cost function parameters.
The purpose of this study was to empirically investigate firms violating concavity
conditions of the cost function and identify economic factors responsible for violating
concavity conditions. This research is based on Japanese manufacturing firm data fromthe period during the 1980s known as the “bubble period” when land and stock prices
soared as exemplified by the average rate increases of 25.3% and 26.2% for the land
prices in six large metropolitan areas and the stock price index of the Tokyo Stock
Exchange (TOPIX), respectively, during this period.
The bubble period is ideal for our analysis because its sharp rise in land prices
encouraged firms to retain and/or increase their land holdings for speculative purposes
unrelated to production. Additionally, land played a collateral role in loan contracts

1

For empirical studies that estimate the parameter estimates of the cost function by imposing

concavity conditions on the cost function, see Kumbhakar (1990, 1992) and Chua et al. (2005).
2

such that large firms, which had relied heavily on bank loans to finance investment prior
to the bubble period, began to raise funds directly from capital markets owing to overall
liberalization of financial market. This forced banks to search for new customers, andthey eventually lent money to smaller and/or nonmanufacturing firms, establishing newties that were nonexistent prior to the bubble period. Land assets held by the borrowers
played vital collateral roles in mitigating the informational asymmetry between
creditors and debtors during this period.2 When the purchase of land is driven primarily
by such speculative motivations, land acquisition does not necessarily contribute toproduction and it is highly unlikely that cost minimization is attained under thesecircumstances. Also, when factor prices change, the optimal combination of inputs for afirm will change accordingly and the firm may not be able to choose the optimal levelof inputs due to borrowing constraints, resulting in unattained cost minimization forsome firms. The above discussions suggest that a goal of identifying factors responsible
for the violation of concavity conditions is best achieved by examining the behavior ofJapanese manufacturing firms during the bubble period.
We estimate the translog cost function of value-added type using the factor inputs oflabor, capital, and land based on manufacturing firm data for five industries: chemicals,
iron and steel, machinery, electrical machinery, and transport equipment. After
identifying the firms that fail to satisfy concavity conditions, we determine the factorsthat cause violation of concavity conditions. Furthermore, we compare the elasticities ofsubstitution and the input demand elasticities with respect to the land rental pricebetween the unconstrained case including the firms that violate concavity conditions
and the case excluding the firms that violate concavity conditions.
The proportion of firms violating concavity conditions is approximately 50% of thetotal firms regardless of the industry. The probit analysis reveals that a large rate ofchange in labor and less dependence on bank loans increase the probability of violating

2

For empirical evidence to support the collateral role of land in Japan, see Ogawa et al. (1996) andOgawa and Suzuki (1998, 2000).

3

concavity conditions. We also find that the degree of substitution between land andother input factors as well as input elasticities with respect to land rental price are both
estimated with large biases unless the firms violating concavity are carefully excluded
from the analysis.
The paper is organized into five sections. Section 1 consists of this introductory
discussion. In Section 2, we specify the translog cost function together with the costshare equations to be estimated. Then we explain the procedure for identifying the firmsthat violate concavity conditions. Section 3 deals with econometric issues and an
explanation of the panel data set. Section 4 presents the estimation results and identifies
factors responsible for the violation of concavity conditions. We also gauge the extent
to which the violation of concavity conditions affects the estimates of the elasticities ofsubstitution as well as the input demand elasticities with respect to the land rental price.
Finally, our conclusions are summarized in Section 5.

2. Theoretical Framework: Characterization of Cost Function and Concavity
Conditions
The real value of a manufacturing firm is primarily the result of some combination ofcapital, land stock, and labor factors. I represent a firm's production technology using
the translog cost function so that the flexibility of production structure can beincorporated into the model. Any degree of substitutability or complementarity of
production factors can be attained under the translog cost function.3
The translog cost function is specified as follows:3

For an exposition of translog cost function, see Christensen, Jorgenson, and Lau (1971, 1973).

Jorgenson (1986) gives a comprehensive survey of modeling producer behavior.

43

ln C ( p, y ) = α 0 + ∑ α i ln pi + α y ln y +
i =1

3
1 3 3
α
ln
p
ln
p
+
α iy ln pi ln y
∑ ∑ ij i j ∑2 i =1 j =1
i =1

N
T
1
2
+ α yy (ln y ) + ∑ γ f DF f + ∑ λt DTt + ε
2
f =1
t =1

(1)

where C ( p, y ) : nominal value-added or total costpi : factor price of i-th input

y : real value-added
DF f :dummy variable for the f-th firm
DTi :dummy variable for year

ε : disturbance term
i = 1,2,3 for capital stock, labor and land stock, respectively

The cost function of Eq. (1) incorporates nonconstant returns to scale and time-varying
technical progress, represented by time dummies. The factor demand function is derived
by employing Shephard’s Lemma. We obtain the following cost-share equations:si ≡3pi X i= α i + ∑ α ij ln p j + α iy ln y
C
j =1

(2)

(i = 1,2,3)
We estimate the cost-share equations jointly with the cost function. In so doing, we
impose the following integrability conditions of the cost function on the system.
1) symmetry α ij = α ji2) cost exhaustion

3

3

3

i =1

i =1

i =1

∑ α i = 1 , ∑α iy = 0 and ∑ α ij = 0( j = 1,2,3)

33) linear homogeneity in factor prices∑α = 0 (i = 1,2,3)
j =1

ij

Note that one of the components in Eq. (2) is linearly redundant.
5Concavity of the cost function is data-dependent and we examine whether they aresatisfied for each observation based on the parameter estimates. Note that we are
examining local but not global concavity evaluated at each observation. Specifically,
the concavity of the translog cost function in the input prices implies that the following
matrix must be negative semidefinite.4
Φ = Α + ss ′ − Ω

(3)

[ ]

where A ≡ α̂ ij

α̂ ij :parameter estimates of the translog costfunctions ′ ≡ [s1 , s2 , s3 ]⎡ s1
Ω ≡ ⎢⎢ 0
⎢⎣ 0

0
s2
0

(i, j = 1,2,3)

si : cost share of the i-th input
0⎤
0 ⎥⎥
s3 ⎥⎦

Therefore we calculate the eigenvalues of the Φ matrix for each observation andidentify the firms satisfying concavity conditions if none of the eigenvalues takepositive values.

3. Econometric Issues and Data Description

Since we employ a panel data set of firms in estimation, the firm-specific effects aretaken into consideration explicitly as firm dummies in the translog function. Note thatfirm-specific effects do not appear in the cost-share equation. A random disturbanceterm is also added to each cost-share equation and we assume that the resulting
disturbance vector together with the one in the translog cost function is multivariate
normally distributed with a mean vector zero and constant variance–covariance matrix.
Then we apply the maximum likelihood estimation to the system.
Equations (1) and (2) are estimated on the basis of panel data of the Japanese4

For an example, see p. 1887 in Jorgenson (1986).

6

manufacturing firms listed on the Tokyo Stock Exchange. The panel data set employed
for estimation is constructed from the database of the Japan Development Bank. Our
total sample consists of 342 firms, each chosen using the criterion that neither a change
in the term of account settlements nor any large mergers or acquisitions occurred during
the sample period from 1979 to 1993. The cost function and cost-share equations are
estimated separately for each of the five industrial categories: chemicals (80), iron andsteel (38), machinery (91), electrical machinery (71), and transport equipment (62), withthe total number of firms for each category indicated in parentheses. The firms for thecost analysis are categorized because firms in the same industries are more likely toshare common production technologies; thus, the parameters characterizing theirproduction technologies are expected to be estimated with greater precision. A moredetailed explanation of the data construction procedure is provided in the Data
Appendix.
Table 1 shows the sample average of the investment rate, defined as grossinvestment divided by the beginning-of-period capital stock, rate of change in real land
stock, and the rate of change in employment or number of employees for the fiveindustries. The average is calculated separately for the “pre-bubble period” (1979–1985)
and the “bubble period” (1986–1993) with the standard deviations indicated inparentheses. Among these industries, the average rate of change in land stock is higher
during the bubble period for chemicals, iron and steel, and machinery, and thecorresponding fourfold and threefold higher standard deviations for iron and steel andmachinery, respectively, imply that land purchasing behavior varied considerably acrossfirms during the bubble period.

4. Estimation of Results and Implications
To what extent are concavity conditions violated?

Table 2 shows the proportion of the firms that violated the concavity condition of thetranslog cost function for the entire period from 1979 to 1993. 5 On average,

5

The parameter estimates of the translog cost function are given in the Appendix.
7

approximately 50% of the observations do not satisfy concavity conditions. However,wide variations across time and industry are apparent. The average proportion of firms
violating concavity conditions is less than 50% for the chemicals and iron and steel
industries but more than 50% for the machinery, electrical machinery, and transport
equipment industries. These proportions range from 32.5% (chemicals in 1980) to
69.2% (machinery in 1993) and there seems to be no discernible trend or regularity inthe proportion of firms violating concavity conditions.

What factors are responsible for violation of concavity conditions?

Once we identify observations that violate the concavity conditions of the costfunction, we can empirically identify the factors that cause violations of concavity
conditions. Concavity conditions are violated owing primarily to two basic processes.
First, large changes in quasi-fixed inputs over a short time period require a firm toreadjust its organizational structure, which may be accompanied by the relocation of
employees and/or machinery. This process will result in additional expenses andallocations and prevent minimization of production costs.6 Furthermore, large volumetransactions of land stock might be motivated for speculative gains in which case thechange in land stock is unrelated to the change in manufacturing output. Therefore, cost
minimization is not attained.
The second process relates to optimization in that even if factor prices change, a firm
does not necessarily choose the optimal combination of inputs under the newly changed
circumstances due to borrowing constraints. In this case, the firm will be forced tochoose an input combination that is suboptimal, and therefore, production costs will not
be effectively minimized. This scenario seems quite likely for the case of purchasing
indivisible capital stock. When a firm has a close relationship with a bank, it is lesslikely that borrowing constraints will exist for that firm. We measure bank–firm

6This inference is along the same line as the adjustment cost story of demand for quasi-fixed inputs.

A more rigorous approach incorporating the adjustment cost needs formulation of cost minimization
is obtained from dynamic aspects.

8relationships using the ratio of bank loans to total assets denoted by BANK .
We can econometrically formulate the ideas described above by the following probit
model. The dichotomous dependent variable, denoted by Z i , takes the value 1 if the ith
observation satisfies concavity conditions of the cost function; otherwise, it takes thevalue 0. The explanatory variables are the absolute value of the rate of change in capitalstock, labor, and land stock for the ith observation, represented by g ij ( j = 1,2,3) , andthe bank–firm relationship (BANK ) . Then our probit model is written as

3

T

j =1

j =1

Z i = β 0 + ∑ β j g ij + θBANK i + ∑η j DT j + ν i(4)where ν i : disturbance termTime dummies (DT j ) are added in Eq. (4) to account for macroeconomic shocks. The
coefficients of β j are expected to be negative and that of θ to be positive. Equation (4)is estimated for each of the five industries (Table 3). The coefficient estimate of theabsolute value of the rate of change in employees is significantly negative formachinery and electrical machinery, implying that a large rate of change in employees
is likely to cause violation of concavity conditions. It hints that the firms incur extra
costs in adjusting a large volume of labor input factors. Unexpectedly, the coefficientestimates of the absolute value of the rate of change in capital stock and land stock arestatistically insignificant, irrespective of industry. Closer bank–firm relationships
significantly increase the probability that concavity conditions are satisfied forchemicals, machinery, and electrical machinery. This implies that borrowing constraints
do prevent some firms from attaining cost minimization. To summarize, failure ofconcavity conditions arises mainly from a large change in employees and borrowing
constraints but not from massive trading of land for speculative purposes.

Consequence of ignoring concavity conditions

Characteristics of the production technology are often expressed in terms ofelasticity of substitution between inputs and input elasticities with respect to input
9prices. They provide useful information in quantitatively evaluating the effects of policychange on demand for factor inputs. However, if the estimates of elasticity of
substitution and input elasticities with respect to input prices are calculated using thesample that includes the firms violating concavity conditions of the cost function, thenany inferences based on such estimates are highly suspect.
In this subsection, we quantitatively evaluate the severity of this problem bycomparing the estimates of elasticity of substitution among inputs and input elasticities
with respect to land rental price between the case in which the whole sample including
the firms that violate concavity is used and the case excluding those violating concavity.
As for the measure of elasticity of substitution, we calculate the Allen partial
elasticities of substitution. The Allen partial elasticity of substitution between the ith
input and the jth input, denoted by σ ij , is defined as

σ ij = 1 +

α ij

(5)

si s jTwo inputs are a substitute if σ ij is positive, while they are a complement if σ ij isnegative. Based on the coefficient estimates of the translog cost function, the elasticities
of substitution are computed for two different samples. One sample consists of all thefirms in the sample, regardless of whether concavity conditions are satisfied. The othersample consists of only the firms that satisfy concavity conditions. In both cases, theelasticities of substitution are computed for each observation of the sample and thenaveraged out. The elasticities of substitution thus calculated are given in Table 4. Thereis no discernible difference in the estimates of the elasticity of substitution between
capital and labor, irrespective of industry. However, the elasticity of substitution
between land and other inputs are estimated with large biases. When the firms violating
concavity conditions are included in calculation, complementarity between capital andland is overestimated for chemicals, iron and steel, and machinery, while substitutability
between capital and land is overestimated for electrical machinery and transport
equipment. Complementarity between land and labor is also overestimated for transport10

equipment and substitutability between land and labor is underestimated for machineryand electrical machinery.
I now discuss the extent to which the elasticities of factor inputs with respect to land
rental price are biased without paying proper attention to the firms violating concavity
conditions. The elasticity of the ith input with respect to land rental price (ε iL ) is
calculated as

ε iL = σ iL s L(6)

where σ iL : elasticity of substitution between land and i-th input

The computed values are shown in Table 5. Note that own price elasticities are
estimated with large biases except for iron and steel. Own price elasticities are all
negative and stable around –0.68 (transportation equipment) to –0.43 (machinery) when
the firms violating concavity conditions are excluded in the calculation. However, own
price elasticities are underestimated in the absolute value for chemicals, electricalmachinery, and transport equipment when the firms violating concavity conditions areincluded in the calculation. For machinery, own price elasticity is estimated to bepositive and the evidence can be interpreted as follows. An increase in land rental priceinduces the firms satisfying concavity conditions to reduce the land inputs as production
factors, so that own price elasticity takes a positive value. In contrast, the firms violating
concavity conditions might demand land not for production purposes, but for
speculative purposes. If this is the case, then an increase in the land price might generate
the expectation that the land price will rise further in the future, which might prompt thefirms to increase land purchases in pursuit of speculative gains. Thus, a negative
response of the land purchase to a land price hike by the firms satisfying concavity
conditions is offset by a positive response by the firms violating concavity conditions.
This situation was quite likely during the bubble period in Japan.
To sum up, unless proper attention is paid to the firms violating concavity conditionsof the cost function, we end up obtaining biased estimates of the elasticity of
substitution among inputs and input elasticities with respect to factor prices and we are11misled into making erroneous inferences regarding policy evaluations.

5. Concluding Remarks

In this paper, we examined empirically why some manufacturing firms failed tosatisfy concavity conditions of the cost function. We focused on the bubble period
during the 1980s in Japan when land prices soared and land was in great demand not
only for production purposes but also for speculation. By estimating the translog costfunction with land as one of the production factors, violation of concavity was found toresult primarily from borrowing constraints and large adjustments in employment.
Furthermore, elasticities of substitution between land and other inputs and input
elasticities with respect to land rental prices were found to be estimated with large
biases if the firms violating concavity conditions were not excluded from the analysis.

12References[1] Christensen,L.R., Jorgenson,D.W., and L.J.Lau(1971).” Conjugate Duality and theTranscendental Logarithmic Production Function,” Econometrica 39, pp.255-256.

[2] Christensen,L.R., Jorgenson,D.W., and L.J.Lau(1973).” Transcendental Logarithmic
Production Frontiers,” Review of Economics and Statistics 55, pp.28-45.

[3] Chua, C. L., Kew, H., and J. Yong(2005). “Airline Code-share Alliances and Costs:
Imposing Concavity on Translog Cost Function Estimation,” Review of Industrial
Organization 26, pp.461-487.

[4] Diewert,W.E. and T.J. Wales(1987).”Flexible Functional Forms and Global
Curvature Conditions,” Econometrica 55, pp.43-68.

[5] Hayashi,F. and T.Inoue(1991)." The Relation between Firm Growth and Q with
Multiple Capital Goods: Theory and Evidence from Panel Data on Japanese Firms,"
Econometrica 59, 731-753.

[6] Hoshi,T. and A.K.Kashyap(1990).” Evidence on Q and Investment for JapaneseFirms,” Journal of the Japanese and International Economies 4, pp.371-400.

[7] Jorgenson,D.W.(1986).” Econometric Methods for Modeling Producer Behavior,” inGriliches, Z.and M.D.Intriligator, (eds.) Handbook of Econometrics, Vol.3 (Amsterdam:
North- Holland), pp.1841-1915.

[8] Jorgenson, D.W. and B.M. Fraumeni(1981).” Relative Prices and Technical
Change,” in Berndt, E.R. and B.C. Field, (eds.), Modeling and Measuring Natural
Resource Substitution, Cambridge : M.I.T. Press.

13[9] Kumbhakar, S.C.(1990).” A Reexamination of Returns to Scale, Density andTechnical Progress in U.S. Airlines,” Southern Economic Journal 57, pp.428-442.

[10] Kumbhakar, S.C.(1992).” Allocative Distortions, Technical Progress and Input
Demand in U.S. Airlines: 1970-1984,” International Economic Review 33, pp.723-737.

[11] Ogawa,K.,Kitasaka,S.,Watanabe,T.,Maruyama,T.Yamaoka,H. and Y.Iwata(1994).”
Asset Markets and Business Fluctuations in Japan,” Keizai Bunseki, No.136, Economic
Research Institute, Economic Planning Agency.

[12] Ogawa,K., Kitasaka,S., Yamaoka,H. and Y.Iwata(1996). “Borrowing Constraints
and Role of Land Asset in Japanese Corporate Investment Decision,” Journal of theJapanese and International Economies 10, pp.122-149.

[13] Ogawa,K. and K.Suzuki(1998). “Land Value and Corporate Investment: Evidence
from Japanese Panel Data,” Journal of the Japanese and International Economies 12,
pp.232-249.

[14] Ogawa,K. and K.Suzuki(2000). “Demand for Bank Loans under Borrowing
Constraints: A Panel Study Japanese Firm Data,” Journal of the Japanese andInternational Economies 14, pp.1-21.

[15] Ryan, D.L. and T. J. Wales(2000).” Imposing Local Concavity in the Translog andGeneralized Leontief Cost Functions,” Economics Letters 67, pp.253-260.

[16] Terrell, D.(1996).” Incorporating Monotonicity and Concavity Conditions in
Flexible Functional Forms,” Journal of Econometrics 11, pp.179-194.

14

Data Appendix

We give brief explanations how the data series are constructed with special
emphasis on stock variables such as capital stock and land stock.

Construction of Capital Stock
Our basic strategy to construct a series of the physical depreciable capital stock is tofollow the perpetual inventory method, as discussed in Hayashi and Inoue(1991) . Our
benchmark capital stock is that in the fiscal year of 1970. It is assumed that the bookvalued benchmark is equal to the capital stock in terms of replacement cost basis. The
physical depreciation rates (δ ) are based on those reported in Hayashi and Inoue(1991).
Given the benchmark value of the depreciable stock, real investment series, anddepreciation rate, we obtain the series of capital stock from the following formula. For
detailed explanations see Hayashi and Inoue(1991).
K t = (1 − δ )K t −1 + I t(A-1)

where K t : capital stock at the end of t-th period
I t : investment

Construction of Land StockWe also follow the perpetual inventory method in calculating the series of land stock,
along the lines suggested by Hoshi and Kashyap(1990) and Hayashi and Inoue(1991).
We choose the fiscal year of 1970 as our benchmark period. The benchmark stock ofland at the market price is obtained by multiplying the book value of land stock in thebalance sheet by the market-book ratio, 5.27, which is borrowed from Ogawa etal.(1994). The net investment in land at the market price (NILAND ) is calculated as theincrement of land, which is evaluated at current price, minus the decrement of land atcurrent price. The decrement of land in the balance sheet is originally book-valued, sothat it is converted into market-value under the LIFO-type assumption that the land sold15

in period t was purchased in the most recent period, period t-1. Then the land stock at
the market price is constructed by the following formula.

LANDYt = LANDYt −1(

ptL+ NILANDt
ptL−1

(A-2)

)

where LANDYt = ptL Lt : stock of land at market priceptL :land priceFinally the real land stock series is obtained by dividing the land stock at market priceby land price index.

Labor Input
Number of employees at the beginning of period

Rental Price of Capital Stock
The rental price of capital stock is computed as follows:ptI (Rt + δ )(A-3)

where: ptI investment goods deflator
Rt : borrowing interest rate of the firm which is computed as the sum ofinterest and discount paid and bond interest expenses
divided by the sum of short-term and long-term loans
payable, bonds payable and notes receivable discounted.

δ : depreciation rate, which is assumed to be constant (0.08 per annum)

Rental Price of Land Stock
The rental price of land stock is computed as follows:
ptL Rt(A-4)

Wage Rate
The wage rate is computed as the average monthly wages times 12 to convert into
16

annual basis.

Nominal Value-added
Nominal value-added is defined as the sum of payment to capital stock, land stockand labor.

Land PriceIn the first place the rate of change in land price is calculated as a weighted averageof the rate of change in land price of six large city areas and that of other areas. The
weight is the proportion of land at market price held by the corporations in six large city
areas. Then a time series of land price is constructed using the rate of change in land
price calculated above so that the land price in 1990 can be unity.

17Appendix
Estimation Results of the Translog Cost Function
Chemicals
Parameter

Coefficient estimates

α0
α1
α2
α3
α11
α12
α 13
α 22
α 23
α 33

2.0340(5.54)***
0.2964(4.97)***0.4147(6.00)***0.2889(6.67)***0.0648(6.27)***-0.0233(-2.82)***-0.0415(-8.61)***0.0235(2.84)***-0.0002(-0.04)
0.0417(7.64)***
0.6389(15.86)***

αy
α yy

0.0121(3.21)***

α1 y0.0154(6.54)***α2y-0.0051(-2.26)**

α3y-0.0102(-6.07)***

DT 1980-0.1331(-15.38)***-0.2034(-20.53)***-0.2610(-23.47)***-0.2790(-23.01)***-0.3136(-24.12)***-0.3724(-24.93)***-0.3711(-23.66)***-0.3677(-23.16)***-0.4161(-24.17)***-0.4426(-24.45)***-0.4847(-23.89)***-0.5029(-24.21)***-0.5417(-25.77)***-0.5595(-25.71)***4073.291183

DT 1981
DT 1982
DT 1983
DT 1984
DT 1985
DT 1986
DT 1987
DT 1988
DT 1989
DT 1990
DT 1991
DT 1992
DT 1993

Log of likelihood
Number of observations
18

Iron and SteelParameter

Coefficient estimates

α0
α1
α2
α3
α11
α12
α 13
α 22
α 23
α 33

1.6175(2.09)**
0.2118(1.87)*
0.7927(7.04)***-0.0044(-0.06)
0.0668(3.33)***-0.0213(-1.31)-0.0455(-5.30)***0.0094(0.61)
0.0119(1.37)
0.0336(3.50)***0.5174(5.58)***

αy
α yy

0.0494(5.04)***

α1 y0.0264(8.37)***α2y-0.0358(-9.71)***α3y

0.0094(3.89)***DT 1980-0.1456(-11.97)***-0.1229(-10.38)***-0.1543(-11.45)***-0.2338(-16.01)***-0.2056(-13.89)***-0.1936(-12.69)***-0.1739(-10.12)***-0.1521(-9.45)***-0.1163(-7.93)***-0.0984(-6.18)***-0.1203(-5.67)***-0.1235(-5.94)***-0.1290(-8.22)***-0.1725(-11.81)***
2018.81542

DT 1981
DT 1982
DT 1983
DT 1984
DT 1985
DT 1986
DT 1987
DT 1988
DT 1989
DT 1990
DT 1991
DT 1992
DT 1993

Log of likelihood
Number of observations

19

Machinery
Parameter

Coefficient estimates

α0
α1
α2
α3
α11
α12
α 13
α 22
α 23
α 33

2.1019(6.23)***0.0812(2.21)**0.4472(10.27)***0.4716(14.45)***0.0251(3.81)***-0.0035(-0.74)-0.0216(-6.64)***0.0247(5.75)***-0.0211(-6.48)***0.0427(18.81)***0.4572(8.05)***

αy
α yy

0.0331(5.37)***

α1 y0.0150(7.67)***α2y0.0025(1.10)

α3y-0.0175(-11.56)***

DT 1980-0.1064(-11.24)***-0.0994(-9.47)***-0.1028(-8.69)***-0.1151(-8.93)***-0.1351(-10.94)***-0.1627(-11.36)***-0.1643(-12.59)***-0.1653(-12.98)***-0.2343(-15.74)***-0.2520(-19.99)***-0.2452(-19.08)***-0.2454(-19.34)***-0.2081(-16.84)***-0.1851(-15.65)***4394.851356

DT 1981
DT 1982
DT 1983
DT 1984
DT 1985
DT 1986
DT 1987
DT 1988
DT 1989
DT 1990
DT 1991
DT 1992
DT 1993

Log of likelihood
Number of observations

20

Electrical Machinery
Parameter

Coefficient estimates

α0
α1
α2
α3
α11
α12
α 13
α 22
α 23
α 33

1.7576(6.16)***
0.6758(12.72)***-0.0389(-0.65)
0.3631(11.49)***0.0722(7.40)***-0.0822(-10.21)***0.0099(2.51)**
0.1171(14.75)***-0.0350(-9.05)***0.0250(7.60)***0.9278(35.75)***

αy
α yy

0.0131(6.01)***

α1 y0.0232(11.43)***α2y-0.0233(-11.19)**α3y0.0001(0.08)

DT 1980-0.2420(-24.51)***-0.2486(-23.99)***-0.3192(-27.35)***-0.3919(-29.76)***-0.4297(-29.15)***-0.5803(-34.82)***-0.6207(-34.79)***-0.6932(-35.61)***-0.8277(-40.42)***-0.9161(-42.63)***-1.0002(-44.05)***-1.1022(-46.10)***-1.1234(-46.35)***-1.1526(-47.09)***4634.25
1065

DT 1981
DT 1982
DT 1983
DT 1984
DT 1985
DT 1986
DT 1987
DT 1988
DT 1989
DT 1990
DT 1991
DT 1992
DT 1993

Log of likelihood
Number of observations

21

Transport Equipment
Parameter

Coefficient estimates

α0
α1
α2
α3
α11
α12
α 13
α 22
α 23
α 33

4.9098(11.85)***0.5828(8.48)***-0.1491(-1.88)*
0.5663(15.71)***0.0188(1.81)*-0.0536(-6.02)***0.0348(7.00)***0.1029(10.31)***-0.0493(-11.07)***0.0145(4.44)***
0.5211(8.75)***

αy
α yy

0.0300(4.69)***α1 y0.0210(8.49)***α2y-0.0103(-4.58)***α3y-0.0107(-7.34)***

DT 1980-0.1656(-16.45)***-0.1360(-12.23)***-0.0958(-8.22)***-0.1013(-8.82)***-0.1109(-8.94)***-0.1229(-10.22)***-0.1359(-10.45)***-0.1187(-8.89)***-0.1590(-11.50)***-0.1738(-11.98)***-0.1862(-11.66)***-0.2035(-12.56)***-0.1704(-11.03)***-0.1709(-11.90)***3564.45913

DT 1981
DT 1982
DT 1983
DT 1984
DT 1985
DT 1986
DT 1987
DT 1988
DT 1989
DT 1990
DT 1991
DT 1992
DT 1993

Log of likelihood
Number of observations

Note: The number in parenthesis is t-value. The coefficient estimates of firm
dummies are suppressed.
22

Table 1
Descriptive Statistics of the Rate of Change in Quasi-fixed Inputs(%)
Investment rate

Chemicals
Iron and steel
Machinery
Electrical machineryTransport equipment

1979-85

1986-93

13.8(10.9)
12.0(13.5)
15.2(16.4)
20.9(12.4)
17.4
(5.6)

14.9
(11.5)
15.1(27.8)
14.4(21.4)
16.2(28.4)
15.6(8.9)

Note: Number in parenthesis is standard deviation.

Rate of change inreal land stock
1979-85
1986-93
1.0
(3.5)
0.6
(5.0)
1.8
(9.8)
2.2
(9.4)
2.5
(10.7)

1.3
(4.3)
2.6
(19.9)
2.6(30.8)
1.9
(6.9)
1.4
(4.1)

Rate of change in
employment
1979-85
1986-93
-0.8
(5.8)
-2.4
(9.0)
-0.4
(8.5)
1.5
(7.9)
0.7
(6.2)

1.1
(4.7)
-1.1
(6.2)
0.0
(6.8)
1.7
(8.5)
0.8
(5.2)

Table 2 Proportion of Firms That Do Not Satisfy Convavity Conditions of the Cost Function(%)
Iron andElectricalTransport
Year
Chemicals
Machinerysteelmachinery equipment
1979
50.0
55.3
63.7
54.9
58.1
1980
32.5
44.7
59.3
60.6
56.5
1981
46.3
44.7
48.4
59.2
45.2
1982
51.2
57.9
64.8
59.2
48.4
1983
50.0
47.4
60.4
52.1
51.6
1984
43.8
42.1
59.3
53.5
41.9
1985
46.3
44.7
59.3
52.1
51.6
1986
45.0
55.3
58.2
62.0
41.9
1987
47.5
63.2
54.9
67.6
54.8
1988
52.5
42.1
49.5
59.2
45.2
1989
46.3
50.0
56.0
49.3
53.2
1990
38.7
42.1
50.5
46.5
51.6
1991
47.5
36.8
47.3
36.6
46.8
1992
52.5
47.4
57.1
64.8
51.6
1993
63.7
36.8
69.2
56.3
59.7
Mean

47.6

47.4

57.2

55.6

50.5

Table 3
Probit Analysis on the Determinants of Concavity Conditions of the Translog Cost Function

Industry

absolute value ofinvestment rateabsolute value ofrate of change inreal land stock

Explanatory variables
absolute value of
dependence onrate of change inbank loans
employees

log oflikelihood

fraction ofcorrect
prediction

numberofobservations

Chemicals-0.0415(-0.12)0.1420(0.14)

0.2989(0.36)

0.3690**(2.11)-804.101

0.57141183

Iron and steel

0.1544(0.38)

0.1342(0.21)-1.2519(-1.32)0.1768(0.61)-364.706

0.5923542

Machinery

0.0372(0.20)-0.2604(-0.85)-1.3429**(-2.08)0.6824***(4.02)-905.114

0.5841

1356

Electricalmachinery-0.2057(-0.87)0.1012(0.21)-1.0814*(-1.75)0.8975***(4.05)-708.825

0.5991

1065

Transport
equipment

0.2712(0.52)-0.7224(-1.26)0.1848(0.19)

0.0697(0.27)-626.584

0.5487913

Note: The coefficient estimates of time dummies are suppressed.

Table 4
Estimates of the Allen Partial Elasticities of Substitution
Sample satisfying
concavity conditions

Whole
sampleChemicals
Capital and labor
Capital and land
Land and labor

0.8449-0.04100.99710.8471-0.70240.9955Iron and steel
Capital and labor
Capital and land
Land and labor

0.8692-0.26481.24090.8703-0.7116
1.2622

Capital and labor
Capital and land
Land and labor

0.9686-0.12390.6821

0.9637-2.43770.0447

Electrical machinery
Capital and labor
Capital and land
Land and labor

0.47551.77060.08670.45812.4511-0.4706

Transport equipment
Capital and labor
Capital and land
Land and labor

0.68212.7814-0.2866

0.6847
4.1721-1.0174

Machinery

Note: Whole sample includes the firms violating concavity conditions as well as
those satisfying concavity conditions.

Table 5
Estimates of Input Demand Elasticities with Respect to Land Rental Price
Sample satisfying
concavity conditions

Whole
sampleChemicals
land
capital stocklabor-0.5136
0.03100.1639-0.3204
0.01240.1485Iron and steel
land
capital stocklabor-0.54180.00800.1659-0.4973-0.0037
0.1576land
capital stocklabor-0.4297
0.02730.11170.51990.00690.0842

Electrical machinery
land
capital stocklabor-0.48310.11820.0213-0.1792
0.10980.0115

Transport equipment
land
capital stocklabor-0.6752
0.21640.0094-0.5300
0.21050.0032

Machinery

See the note in Table 4.