Table Of ContentLectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. KUnzi
Mathematical Economics
201
Paul M.C. de Boer
Price Effects
in Input-Output Relations:
A Theoretical
and Empirical Study
for the Netherlands 1949-1967
Springer-Verlag
Berlin Heidelberg New York 1982
Editorial Board
H. Albach A. V. Balakrishnan M. Beckmann (Managing Editor)
P. Dhrymes J. Green W. Hildenbrand W. Krelle
H. P. KOnzi (Managing Editor) K. Ritter R. Sato U. Schittko
P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. H.P. KOnzi
Universitat ZOrich
CH-8090 ZOrich, Schweiz
Author
Dr. Paul M.C. de Boer
Econometric Institute, Erasmus University Rotterdam
P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
ISBN-13: 978-3-540-11550-2 e-ISBN-13: 978-3-642-46460-7
DOl: 10.1007/978-3-642-46460-7
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© by Springer-Verlag Berlin Heidelberg 1982
2142/3140-543210
TABLE OF CONTENTS
Page
List of main symbols
1. Introduction
1.1. Preliminary remarks
1.2. Outline of the study 4
2. Theory of production and costs 5
2.1. Introduction 5
2.2. Assumptions underlying neo-classical theory of production and
costs 6
2.3. Elasticities of substitution 9
2.4. On the existence of a unique cost minimum 11
2.5. Neo-classical theory of costs 14
2.5.1. General theory 14
2.5.2. Homogeneous production functions 20
Appendix 2.A. Proof of theorem 1 22
Appendix 2.B. Proof of corrolary 24
Appendix 2.C. A generalization of lemma 2 of Barten, Kloek and Lempers 25
Appendix 2.D. A modified version of proposition 7 of Shephard 26
3. Constant elasticities of substitution class of production functions 27
3.1. Introduction 27
3.2. Neo-classical properties of the CES production function 29
3.3. Application of the theory of costs to the h-homogeneous CES
production function 33
3.4. The two-level CES production function 36
Appendix 3.A. Theoretical restrictions on the parameters of the two-
level CES production function 39
Appendix 3.B. Derivation of input demand relations, the cost function,
the Allen partial elasticities of substitution for the two-level
CES production function and a reformulation for time series
analysis 44
Page
4. Theory of price effects in input-output relations: some models 49
4.1. Introduction 49
4.1.1. Traditional input-output analys'is 49
4.1.2. Generalized input-output analysis: an overview 53
4.2. Generalization of input-output analysis based on one-level CES
production functions 56
4.3. Introduction of technical change into the generalized input-output
model 62
4.4. The two-level CES productio.n function in input-output analysis 64
4.5. Aggregation of the demand relations for a firm to a demand relation
of an industry 68
4.6. Two other studies dealing with substitutability in input-output
analysis 70
4.6.1. The approach by Theil and Tilanus (1964) 70
4.6.2. The approach by Kreyger (1978) 71
Appendix 4.A. The CBS classification of sectors 74
Appendix 4.B. Constancy of input-output ratios 75
5. Methods of estimating price effects in input-output relations 77
5.1. Introduction 77
5.2. Conceptualization 78
5.3. Estimation of the model 80
5.4. Applying methods of estimation to generalized input-output models 85
Appendix 5.A. Maximum likelihood estimation in case of autocorrelation
within an input group 87
6. Estimated price effects in input-output relations: the Netherlands,
1949-1967 91
6.1. Introduction 91
6.2. Assembling the data 93
6.3. Performance of one-level CES models, 1949-1958, compared with tra
ditional models 99
6.3.1. Parameter estimates 99
6.3.2. Predictive performance 100
IV
Page
6.4. Application of one-level CES models to 1949-1966 103
6.4.1. General 103
6.4.2. Comparison between methods 103
6.4.3. Comparison between models 106
6.5. Comparison of the estimates between the two periods of observation 108
6.6. Application of the two-level CES model 110
6.6.1. Performance of the two-level CES model, 1949-1958 110
6.6.2. Extension of the period of observation to 1949-1966 112
6.6.3. Comparison of the estimates between the two periods of
observation 114
6.7. Summary of the conclusions 115
Appendix 6.A. Tables relating to the one-level CES models 118
Appendix 6.B. The statistics of Spearman, Theil and Somermeyer 121
Appendix 6.C. Tables relating to the two-level CES model 123
7. Summary and conclusions 125
Notes 130
References 132
Author's index 136
Subject index 137
v
LIST OF MAIN SYMBOLS
Chapter 1
y output
x input of factor of production m (= 1, ... , M)
m
Chapter 2
JRM M-dimensional Euclidean space
D non-negative orthant ofJRM
Dl set of interior points of D
D2 set of boundary points of D
L(y) production input set
E(y) efficient subset of L(y)
f(x) production function
Pm price of input m (= 1, ... , M)
H Hessian matrix with typical element fmm'
(m, m' = 1, ••• , M)
f vector of first-order partial derivatives with typical
x
element f = ~ (m = 1, ... , M)
m dX
m
E(v,w) elasticity of v with respect to w
IFijl cofactor of element (i,j) of H bordered with fx (the
bordered Hessian)
IFI determinant of the bordered Hessian
C total cost
s {x € JRMlx ~ k, m = 1, ... ,M}
m
L Lagrangean function
A = A(y, PM): Lagrangean multiplier
Pl'
(0)
xm(y, Pl' PM): optimal demand relation for input m (= 1, ... , M)
Q(y,p) minimal cost function
dXm
x vector with typical element --- (m 1, M)
y dy
X matrix with typical element -d-X-m- (m, m' = 1, ••• , M)
P dPm'
A-1 adAy-1
Y
n-1
Ap- 1 vector with typical element ~ (m = 1, ... , M)
m
(0)
w. share of input j in total cost in the optimal situation
J
h degree of homogeneity of the production function
Chapter 3
p substitution parameter
1
0=-- ditto
l+p
c distribution parameter
m
1 m=m'
Kronecker delta
°mm' o m#m'
d distribution parameters of the two-level CES function
s
(s = 1,2)
distribution parameter of factor of production m in group
s (= 1,2)
substitution parameter of group s (= 1,2)
ditto
share of group s in total cost (s = 1,2)
share of factor of production m in total cost of group
s (= 1,2)
Chapter 4
x input into sector j of products of m
mj
Yj output of sector j
f. final demand of sector j
J
bkj contribution of primary factor k to the production of j
A matrix of input-output ratios in period (t-1) with typi
t-1
xij
cal element a .. =
l.J Yj
11. percentage change in sum of intermediate inputs of
J
sector j
Yj ditto, of gross volume of production of sector j
X ditto, of the demand for input m by sector j
mj
ditto, of the output price of sector m
Pm
h. degree of homogeneity of the production function of
J
sector j
elasticity of substitution between the intermediate in
°1j
puts of sector j
nj Hicksian generalized neutral technical progress of sector
j
{:,.tY vector of changes in total output
{:,.tf ditto, in final demand
VIII
diagonal matrix with hj on the main diagonal
vector of price changes
diagonal matrix with changes in prices on the main
diagonal
vector with typical element 0lj
diagonal matrix with 0ij on the main diagonal
matrix of intermediate deliveries in period (t-l) with
'typical element x .
m)
matrix of shares of inputs of sector m in total (inter-
mediate) cost of sector j in period (t-l) with typical
~
element w . = x ./( x ,.)
m) m) m'=l m )
E mj error in the demand relation of input m of sector j
0. elasticity of substitution between the group of interme
)
diate and of primary inputs
Bt_1 matrix of primary input ratios in period (t-l) with
typical element S . = ~j
~) j -1
1:r diagonal matrix W1. th o). + w1 ,Q, ). (01 ). -0.) ) on its main dia
gonal
~II ditto, with 0. on its main diagonal
)
lltPr vector of price changes of intermediate inputs
l\PII ditto, of primary inputs
Chapter 5
i subscript relating to inppt group (i = 1, NG)
j subscript relating to sector (j = 1, ... , NS)
variable to be explained
Yj
X. sub-matrix of the matrix X of explanatory variables
-1.,j
autocorrelation parameter
£.ij
2
o .. variance of the errors
-1.)
vector of parameters to be estimated
~j
-E1 •.•) vector of errors
(i, i ') th element of covariance matrix n ® r
~ii'
(i,i' = 1, ••• , NG)
V. covariance matrix of input group i (= 1, ••. , NG)
1.,j
IX
Chapter 6
k
log p (Po,Pl'Um) cost of living index
RM
log Q01 teal income index
kijt predicted error for period t of input i in sector j
(i,j=l, ••• ,NG)
y relative root mean square error
R Spearman's coefficient of rank order correlation
U Theil's inequality measure
V Somermeyer's inequality measure
x
1. INTRODUCTION
1.1. Pre Ziminary remarks
Input-output analysis is one of the most extensively used tools of economic
science. It has been introduced by Leontief (1941) who assumed that inputs into
a production process of a particular sector of economic activity is a constant
fraction of the output of that process in physicaZ terms.
National account statisticians, however, record the inputs and outputs of
sectors of economic activity in money flows. If those flows were voZumes (evalu
ated at constant prices, pertaining to a certain base year) they could represent
the physical amounts Leontief dealt with. Then, the Leontief assumption turns
into constancy of ratios of volumes of inputs to volumes of output. For an over
view of (traditional) input-output analysis we refer to section 4.1.1.
In practice, however, input-output tables in volumes are seldom available;
since as a rule they are expressed in monetary vaZues (i.e. evaluated at current
prices). In that case one generally assumes that the ratios between inputs (in
value terms) and outputs (in value terms) are constant.
In appendix B to chapter 4 we prove that the two variants described above
can be couched in terms of the (neo-classical) theory of costs subject to a
production function.
When input-output ratios are assumed to be constant in volume terms, the
production function, supposed to represent the technology of the sector of eco-
nomic activity under consideration, is of the Leontief type:
y min{X1, ••• , XM}, ( 1.1)
a1 aM
in which y: output, and
xm: input of factor of production m (= 1, .,., M), and
ai : (i = 1, ... , M) positive constants.
If, however, the ratios are supposed to be constant in value terms, the
production function is of the Cobb-Douglas type:
M Il
y C II x m, (1.2)
m=1 m
