DWC-8702
Investment Policies for Steel-Producing Countries
An Empirical Analysis of the Use of Iron Ore by
Major Steel-Producing Countries
Theophilos Priovolos
Division Working Paper No. 1987-2
March 1987
Commodity Studies and Projections Division
Economic Analysis and Projections Department
Economics and Research Staff
The World Bank
Division Working Papers report on work in progress and are
circulated to stimulate discussion and comment.
INVESTMENT POLICIES FOR STEEL-PRODUCING COUNTRIES
AN EMPIRICAL ANALYSIS OF THE USE OF IRON ORE BY
MAJOR STEEL-PRODUCING COUNTRIES
Theophilos Priovolos
March 1987
The World Bank does not accept responsibility for the views
expressed herein which are those of the author and should not be
attributed to the World Bank or its affiliated organizations. The
findings, interpretations, and conclusions are the results of
research supported by the Bank; they do not necessarily represent
official policy of the Bank. The designations employed and the
presentation of material used in this document are solely for the
convenience of the reader and do not imply the expression of any
opinion whatsoever on the part of the World Bank or its affiliates
concerning the legal status of any country, territory, city, area,
or of its authorities, or concerning the delimitation of its
boundaries, or national affiliations.
TABLE OF CONTENTS
Page
SUMMARYS ....... ..*0-.0.0........ ....................................... iv
I. INTRODUCTION ................................................................ 1
II. THE MODEL .... ........................................................ 3
III. EMPIRICAL ANALYSIS . ... .............................................. 11
A. Cobb-Douglas Production Function Estimates . . 12
B. Steady States . . ................. .... 19
IV. POLICY CONCLUSIONS ... ...... .............................. . 24
ANNEX 1: DATA BANK SOURCES ......... . . 33
ANNEX 2: NON-MONOTONIC EQUILIBRIUM PATHS .. ...... .. ................. 34
ANNEX 3: ESTIMATING THE ELASTICITY OF SUBSTITUTION BETWEEN
IRON ORE AND OTHER FACTORS OF PRODUCTION .................. 36
REFERENCES ......................o........ 45
!~~~~~~~~~~~ -iii -
LIST OF TABLES
Page
TABLE 1: ESTIMATES OF PARAMETERS OF COBB-DOUGLAS
PRODUCTION FUNCTIONS FOR STEEL WITH VARIABLE RETURNS TO SCALE .... 15
TABLE 2: ESTIMATES OF PARAMETERS OF COBB-DOUGLAS
PRODUCTION FUNCTIONS FOR STEEL WITH CONSTANT RETURNS TO SCALE .... 18
TABLE 3: STEADY STATES OF STEEL OUTPUT/CAPITAL STOCK RATIO (Z)
AND IRON ORE CONSUMPTION/RESERVES RATIO (d) ...................... 20
TABLE 4: ASSUMPTIONS FOR DIFFERENT PRICE PATHS ............................ 23
TABLE 5: GROWTH RATES FOR IRON ORE CONSUMPTION/RESERVES
RATIO: 1972-83 .. ............................ 24
TABLE 6: RELATIVE IMPORTANCE OF FACTORS OF PRODUCTION IN THE
STEEL PRODUCTION PROCESS: AN INTERNATIONAL COMPARISON ............ 26
ANNEX TABLE 3.1: ELASTICITIES OF SUBSTITUTION BETWEEN INPUTS
IN THE STEEL PRODUCTION PROCESS .......................... 39
LIST OF FIGURES
FIGURE 1: PHASE DIAGRAM OF d AND Z ....... 7
FIGURE 2: PRICE PATHS BASED ON DIFFERENT VALUES FOR Z* AND d* . . 22
FIGURE 3: WORLD DEPLETION RATE (DSWW) AND
INTERNATIONAL IRON ORE PRICE (IOP) ............................. 25
FIGURE 4: STEEL OUTPUT TO CAPITAL RATIO, SELECTED COUNTRIES .............. 28
FIGURE 5: IRON ORE CONSUMPTION TO WORLD RESERVES, SELECTED COUNTRIES ..... 30
- iv -
SUMMARY *
Based on the analysis by Stiglitz (1974) and Dixit (1976) on natural
resources dynamics this study shows that steel industries with high capital
stock shares relative to labor or other raw material inputs should seek to
increase their steel output/capital ratio and their consumption of iron ore to
world iron ore reserves ratio in order to approach efficiency. The opposite
policies should be adopted if the capital stock share is low relative to labor
and other raw materials. The analysis also notes that constant monitoring of
iron ore depletion plans is necessary in order to minimize error from choosing
inappropriate initial conditions.
The study analyses empirically the operations of nine major steel
producing countries namely: the United States, the Federal Republic of
Germany, France, the United Kingdom, Italy, Japan, India, Brazil and the
Republic of Korea. It shows that during the 1973-83 period most steel
producers tended to use iron ore efficiently in their production process. The
study shows that the investment policies of the steel industries in the United
States, the Federal Republic of Germany, France and India should aim at
further decreasing their output/capital ratio and their consumption of iron
ore to world reserves ratio to achieve efficiency. On the contrary, the
investment policies of the Republic of Korea should aim at increasing the
steel industry's output/capital ratio and the consumption of iron ore to world
reserves ratio to achieve efficiency. The efficient investment strategies of
Japan, Italy and Brazil are not as clear. Very likely, Japan should follow
policies similar to those in the United States or the Federal Republic of
Germany. Italy should try to increase its output/capital ratio while
decreasing its consumption of iron ore to world reserves ratio. Lastly,
drawing conclusions for Brazil's strategy to achieve efficiency was found to
be difficult due to data problems.
* This paper has greatly benefitted from discussions and suggestions from
R.C. Duncan, D. Tarr and M. Imran, who read and commented on earlier
drafts. Tamar Dunietz and Moshe Buchinsky provided excellent research
assistance.
I. INTRODUCTION
1. During the 1960-84 period, iron ore prices increased a mere 1.2% p.a.
while steel merchant bars and the manufacturing unit value (MUV) increased by
5% p.a. and 4.9% p.a. respectively.l/ In view of the decline of iron ore
prices in real terms, several questions are raised: Have the major steel
producing countries of the world adjusted to declining real iron ore prices?
Are their production methods still efficient? What should their policies be in
order to achieve efficiency? This paper aims at providing some answers to
these questions by an empirical analysis of the operations of nine major steel
producing countries namely: the United States, the Federal Republic of
Cermany, France, the United Kingdom, Italy, Japan, India, Brazil and the
Republic of Korea.
2. Iron ore is used almost exclusively for the production of pig iron
and sponge iron. In the steel making process per se iron ore is a minor input
compared with pig iron and scrap. Ninety-eight percent of iron ore goes into
pig iron and sponge iron production, 1% of iron ore is consumed directly in
the steel making process itself and 1% is consumed in other industries such as
the cement industry. Annual world iron ore production exceeds 450 million tons
in Fe content of which about 90% is produced by 12 countries.
3. According to the US Bureau of Mines, world iron ore resources were
estimated to contain some 235 billion tons of iron in 1984. Of these resources
about 89 billion tons were considered to be reserves and of these 63 billion
1/ See Commodity Trade and Price Trends, The World Bank, Economic Analysis
and Projections Department, 1986 Edition, Washington, D.C.
-2-
tons were located in industrial and developing countries. At 1984 steel and
iron ore consumption levels, iron ore reserves could satisfy steel demand for
194 years. The percentage of world iron ore produced that is traded inter-
nationally has risen from 30% in 1961 to 48% in 1984. Six countries, of which
one is industrial, accounted for more than 75% of the iron ore exports. Three
groups: Japan, the European Economic Community (EEC) and the United States,
accounted for 70% of total imports in 1984. Nine countries, the United States,
the Federal Republic of Germany, France, the United Kingdom, Italy, Japan, the
Republic of Korea, India and Brazil produced 42% of world crude steel
production (or 59% of industrial and developing countries world crude steel
production) in 1984. Iron ore price growth prospects remain bleak. In addition
to the declining demand for steel, the oversupply of iron ore persisting from
the last decade and the relative simplicity of replacing one source of supply
with another have made the iron ore market quite competitive. In the last four
years, international iron ore prices have been declining continuously in real
terms, and for 1987 price prospects are equally gloomy because of the abun-
dance of iron ore capacity and the slackness in steel demand.
4. The following three sections of this study will present: (a) the
model from which the equilibrium and efficiency conditions and relevant steady
state paths are derived; (b) the empirical results of the analysis; and (c)
policy conclusions. A Cobb-Douglas production technology is used to estimate
the parameters that determine steady states and equilibrium price and
depletion paths. The production technologies and iron ore needs of the nine
largest steel-producing nations are compared. Policy conclusions are derived
as to the optimal strategy for steel producers to reach equilibrium production
paths.
-3-
II. THE MODEL
5. The analysis focusses on the special but useful case of an economy in
which the steel industry is characterized by a Cobb-Douglas technology of the
form
Y = A eXt Ka L MY R6 (1)
where
Y = aggregate steel output
A = constant
x rate of output-augmenting technological progress
(assumed to be constant)
R = utilization of iron ore
L = supply of labor
K = capital stock
M = utilization of other raw materials and inputs
a, 8, y, 6 > 0
For the purposes of this section of the study nothing is gained by assuming
different production functions for different countries and different stages of
steel production. We may also write
Y = C +K
where
C = consumption
-4-
K = net investment
Y may be thought of as net output or a depreciation rate can explicitly be
assumed.l/ It is also assumed that labor grows at a constant rate n, i.e.,
L/L = n. The labor supply at time t being
L = L ent
0
Raw materials (M) other than iron ore are assumed to grow at a constant rate
as well, i.e., M/M = m. The supply of these raw materials at time t is
mt
M = Me e. In addition it is assumed that R is subject to the stock
0
constraint
f R(t) dt S
t=O0
where
SO is iron ore stock at time o
In this model it is also assumed that there is no deterioration of capital and
the equation for the accumulation of capital takes the form
K = sY or K/K = sY/K (2)
The propensity to save (s) is assumed constant. From (1), by taking growth
1/ For simplicity purposes, the analysis here deals with one sector model
only. Modification of this assumption has not been attempted successfully
due to the complexities involved in deriving steady state paths.
-5-
rates we have:
(Y/Y) = x + a(K/K) + Bn + ym + 6(R/R)
The condition for efficient asset allocation is: 1/
Cs(Y/K) = (Y/Y) - (R/R) (3)
If Z = Y/K, x= 1 - s and (4)
d = R/S (5)
From (1) to (5) the following differential equations are obtained: 2/
Y/Y = {(8n + ym + X - axZ)/(l - d)} + aZ (6)
R/R = {(Bn + ym + X - axZ)/(l - 0)} (7)
Z/Z = Y/Y - K/K
= {(8n + ym + X - axZ)/ (1 - d)} + (a - 1 + x) Z (8)
1/ See Dixit (1976) and Stiglitz (1974). Efficient asset use requires the
marginal products of capital and of the resource (R) to grow at equal
rates.
2/ These are the same equations as in Stiglitz (1974), Solow (1974b) and
Dixit (1976).
d/d = R/R + d
= {(Bn + ym + X - axZ)/(l - d)} + d (9)
The two lines with Z = 0 and d = 0 intersect at the point (Z , d ) given by
Z = {(Bn + ym + X)/(s (1-a) + 6(a-s))} (10)
d = Z (a - s) (11)
These two points are the efficient steady state. 1/
6. The phase diagram, Figure 1, shows the solutions of the differential
equations. The steady state is a saddle-point. As Z goes to Z it goes to 0
or - on all but the two stable paths.
Efficiency in use requires also that the relative price of iron ore
to steel is equal to the marginal product of iron ore, i.e.,
P = 6Y/R = 6 eXt Ka Li MY/R1-6 (12)
where P = relative iron ore price (with the denominator the steel price).
1/ For proof and for stability/Jacobian conditions see Stiglitz (1974) or
Dixit (1976). Identical notation as that of Dixit has been used here to
show that the problem and its solution are the same as that presented in
Dixit's (1976) discussion on natural resources dynamics.
-7-
FIGURE 1: Phase Diagram of d and Z
Z z
SOURCES: DIXIT (1976) AND WORLD BANK, ECONOMIC ANALYSIS
AND PROJECTIONS DEPARTMENT.
,~~~~~~~ ,
-8-
The rate of change of the price would then be
P = a-, = aZ (13)
which makes it endogenous to Z. Equation (12) shows that price is inversely
related to the depletion rate. The choice of too high a depletion rate is
equivalent to the choice of too low a price.
7. The phase diagram show that the depletion rate will continue to
remain high while the price will continue to remain low. As Z goes to Z and d
goes to -, d/d goes to 1 (see equation (11)). Nordhaus and Tobinl/ show that
by integrating equation (11) a finite amount of time is necessary for d to go
to -. If the initial choice of the depletion rate is too low, the price will
be high and the reserves of iron ore will never be used up. The solution of
the differential equation (9) will be a function of Z and thus of its initial
conditions from the solution of equation (8). Taking KO, Lo, Mo and Ro as
given, equation (1) gives the following relation:
Z( = {A La MY Ro }i do
This is a relation that can be represented in the diagram of (d,Z) space. Two
curves are presented, curve C1 is generated with a ZO smaller than curve C2.
Ko is large relative to Lo and Ro in Cl curve. The opposite is true in curve
C2. The appropriate initial conditions should then be those that correspond to
the curve that meets one of the stable paths into the saddle-point. The
important conclusion is that stable paths are the only desirable ones but the
ability of an economy to select them is very doubtful. Even with perfect
1/ For proof see Nordhaus and Tobin (1972).
-9-
foresight over a finite or an infinite horizon,l/ the error of choosing the
wrong initial conditions and thus, for example, setting the price too high or
too low and discouraging or encouraging iron ore utilization is difficult to
avoid.
8. If the steel industry's Ko is large relative to Lo, Mo and Ro the
initial values of Z (Z0) and d (do) should be made low relative to steady
state values (Z* and d*) and they should increase steadily through time. The
opposite should happen if the initial capital stock is relatively low. Undue
compliance with the observation that current iron ore reserves at current
rates of consumption will last 194 years is not desirable as the wrong initial
depletion plans or rates of consumption will lead to ever-increasing error if
followed relentlessly.
9. Dixit2/ shows that removal of some of the restrictions of the Cobb-
Douglas production'function is possible. However, he shows that increasing the
number of factors, resources and capital goods does not yield many general
qualitative results. He also shows that making endogenous the consumption
growth path of steel or allowing for an elasticity of substitution different
from one or a variable elasticity increases the possibilities for non-
monotonic paths; for example, the iron ore consumption per capita can fall
then rise and
1/ For proof, Stiglitz (1974).
2/ See Dixit (1976) for an analysis of equilibrium conditions and paths of a
two-factor production function with constant returns to scale, Y=F(K, R)
and a maximization problem of the following iso-elastic utility function:
rO u (c) e Ptdt
where p = interest rate in utility terms.
Annex 2 demonstrates some of the characteristics of non-monotonic
equilibrium paths, given the removal of some of the restrictions of the
Cobb-Douglas production function.
- 10 -
finally fall again along an optimum path. In the long run, if the elasticity
of substitution (a) between iron ore (R) and capital stock (K) is smaller or
equal to one, i.e., a 5 1 and the limit of the marginal product of capital is
zero then:
R = K = C _ p (14)
R K C E
where
p = interest rate in utility terms
E = elasticity of marginal utility, i.e., the relative rate at
which undiscounted marginal utility falls as consumption
per head increases.
If p is positive the growth rate of consumption for iron ore should decline
with time. If a > 1 then
R.= p + n . (1-E) _ (a - l)n = P + n - can (15)
where
n = limit of marginal product of capital (if a > 1 then n > 0)
Relation (15) indicates that growth of iron ore consumption may be sustained
if n is large enough. Thus the elasticity of substitution between iron ore and
the other inputs into the production technology govern the possibility of
sustaining growth of iron ore consumption and depletion growth rates.1/
1/ The model presented here does not consider uncertainty. This is quite
difficult to handle. In the literature of natural resources, Dasgupta and
Heal (1974) have studied a simple kind of uncertainty, namely about the
date at which a new technology will enable us to do without the resource
altogether. It was found that uncertainty has the effect of raising the
discount rate.
- 11 -
III. EMPIRICAL ANALYSIS
10. This section estimates and carries out an analysis of the production
functions of the nine most important steel-producing countries of the western
world namely, the United States, Japan, the Federal Republic of Germany,
France, the United Kingdom, Italy, Brazil, India and the Republic of Korea.
The following results are presented: (a) estimates of the coefficients of
Cobb-Douglas production functions under constant and variable returns to
scale; (b) steady states of steel output capital ratio (Z) and of the iron ore
depletion rate (d); and (c) the iron ore equilibrium prices and consumption
under alternative initial conditions. The estimates elasticities of
substitution between iron ore and the other factors of production (based on a
CES production function) are presented in Annex 3. These estimates have been
made to test the elasticity of substitution implicit in assuming a Cobb-
Douglas production function.
11. The primary sources for the data are World Steel Dynamics (WSD),1/
International Iron and Steel Institute (IISI)2/ and UNCTAD.3/ The World Bank's
data banks have been used to double check the consistency of the primary data.
All assumptions regarding future employment growth and output growth rates are
consistent with World Bank forecasts as of January 1986. Annex 1 gives a list
of all variables used in the empirical analysis, their source, units and
sample period.
1/ See World Steel Dynamics (1985) for company steel data.
2/ See IISI (1985) for country steel data.
3/ See UNCTAD (1984) for country iron ore data.
- 12 -
A. Cobb-Douglas Production Function Estimates
12. The coefficients of equation (1)
Y = A Ka La MY R6 ext
are estimated with variable returns to scale (v * 1) and constant returns to
scale (v = 1). The analysis of Klein (1953) for railroad operations is the
basis of analysis with variable returns to scale. Equation (1) is rearranged
so that endogenous variables are on the left hand side (lhs) and predetermined
or exogenous variables are on the right hand side (rhs) of the equation, i.e.,
Ka L8 MY R6 = YA e xt eu = C (16)
where u is a disturbance term
The production unit is assumed to minimize costs by varying the four endogen-
ous inputs K, L, M and R subject to the condition that it must meet given
steel output demand. The first order conditions from the cost minimization
problem using production function (16) are:
a C/K = (P /9) e
K
w
3 C/L = (P /L1) e
y C/M = (P I9) e
M
6 C/R = (P It) e
R
- 13 -
Ka La MY R6 - C = 0
Where PX' PL' PM, and PR are the prices of capital, labor, raw materials, and
iron ore and Q is a Langrangean multiplier and V, W, X and Z are disturbance
terms. The first four equations can be rearranged to yield
K = a (X-V)
PRR 6
L a 8 (X-W)
--e
PRR 6
MM=y e(X~Z
PRR 6
Upon further rearrangement and taking logarithms this is equivalent to:
ln (a/6) = ln(PK K/PRR) + (V-X)
ln(0/6) - ln(P LL/PRR) + (W-X)
ln(Y16) ln(P MM/PR R) + (Z-X)
The unknown parameters on the lhs of these expressions can simply be estimated
by taking the sample means of the rhs by equating a/6, 8/6 and y/d, respect-
ively, to the (geometric) sample average of the ratio of the corresponding
production factor rewards or
In( 1)= 1 Zln( R
- 14 -
and similarly for the other expressions. The main advantage of this method is
that the dependent variables and the disturbance term enter additionally in
the estimators. The estimates are unbiased and consistent even though the
terms involved are not independent. To obtain the remaining parameters of the
production function (16), the equation can be rewritten upon taking logarithms
as:
lnR + a lnK + lnL + Yt lnM = - lnA + tlnY - Xt + u
The lhs can be constructed now in the manner of two stage least squares by
substituting the estimates of a/6, 8/6, and y/t for the corresponding para-
meters. The lhs variable is then used as the dependent variable in a least-
squares regression on the predetermined variables Y and t, yielding estimates
of parameter ratios from which the original parameters are readily identified.
The sum of coefficients a, B, y, and 6 provides an estimate of the returns to
scale. The results of this estimation method are presented in Table 1.
13. A second method has been used to estimate the parameters of equation
(1), i.e., by imposing the assumption of constant returns to scale. From the
equations to be estimated the marginal productivity conditions are derived
from the profit maximization problem under perfect competition as follows:l/
a Y/K = (P /P) e
K
1/ Wallis (1973) shows that the marginal productivity conditions are
identified.
- 15 -
TABLE 1: ESTIMATES OF PARAMETERS OF COBB-DOUGLAS PRODUCTION
FUNCTIONS FOR STEEL WITH VARIABLE RETURNS TO SCALE
PARAMETERS/A/B WW U J D F C I B N K
A 0.01 0.01 0.001 0.01 0.00 *** 0.00 **i 0.002 0.00
aL 0.22 0.13 0.26 0.18 0.53 *** 0.53 *** 0.23 0.37
8 0.54 0.90 0.21 0.49 0.91 *** 0.32 * 0.53 0.10
y 0.31 0.19 0.68 0.38 1.12 *** 0.62 - 0.67 1.13
6 0.16 0.15 0.14 0.14 0.35 *** 0.11 *** 0.22 0.12
X -0.001* 0.02 -0.02 0.001* 0.02* *** 0.03 *** -0.03 -0.20
v 1.23 1.37 1.29 1.19 2.91 *** 1.58 *** 1.65 1.72
R2 0.82 0.92 0.42 0.72 0.24 0.80 0.83 0.22 0.56 0.98
DW 1.18 1.64 2.39 1.54 2.22 0.49 1.59 1.23 0.92 1.40
/A COUNTRY CODES:
WW INDUSTRIAL AND DEVELOPING COUNTRIES
U UNITED STATES
J JAPAN
D GERMANY, F.R.
F FRANCE
G UNITED KINGDOM
I ITALY
B BRAZIL
N INDIA
K KOREA, R. OF
/B * T TEST INSIGNIFICANT.
*** WRONG SIGN.
SOURCE: WORLD BANK, ECONOMIC ANALYSIS AND PROJECTIONS DEPARTMENT.
- 16 -
3 Y/L = (P /P) e
L
x
y Y/M = (P m/P) e
6 Y/R = (P /P) e
R
where P is the price of steel output. These equations can be alternatively
written in the form:
Ina = ln(P K/PY) + V
K
and similarly for the other equation. If E(V) = 0 then the estimates of
ln a = n Zln( py ) etc.
are unbiased and consistent. The disturbance term also enters additionally
here. The unbiasedness property disappears but consistency remains when we
unscramble an estimate of a as exp (lna). The resulting estimate is the sample
geometric mean of the input's (capital) share in total output.
n | -
K
a =p etc.
Py
The estimates a, 3, y, 6 will tend to have a sum close to 1 as long as the
data obey the identityl!/
1/ Some errors due to aggregation may occur as the country data are derived
from company balance sheets in current US$; subsequently they are deflated
into real US$.
- 17 -
PY = PKK + P LL + PMm + PRR
In that case the arithmetic mean of factor shares has a + 8 + y + & = 1
(variations in prices might give slight departures from 1). Estimates
of a, 8, y, and 6 using this method are presented in Table 2.
14. Tables 1 and 2 indicate the following: (i) Returns to scale are not
constant but increasing. The results of Table 1 are generally statistically
superior to those of Table 2. The R2 of the equations of Table 1 are higher
than those of Table 2 (with the exception of the R2 for Japan and France);l/
(ii) The results of Table 1 show that in most cases 6 > y > a > S. That is
the elasticities of labor with respect to steel output are higher than those
of raw materials (other than iron ore), capital and iron ore. (With constant
returns to scale, the elasticities of raw materials other than iron ore become
more important than those of labor.) In Japan and in the Republic of Korea the
elasticities of raw material (other than iron ore) and of capital become more
important than those of labor. The labor elasticity of steel production in the
United States (and in France2/) is the highest among all the countries
considered in Table 1. India seems to have a relatively high labor
elasticity; India also has the highest iron ore elasticity among the
I/ The Durbin-Watson test for India's and the western world's equations
indicate some serial correlation.
2/ Statistically, France's elasticity coefficient is less reliable.
- 18 -
TABLE 2: ESTIMATES OF PARAMETERS OF COBB-DOUGLAS PRODUCTION
FUNCTIONS FOR STEEL WITH CONSTANT RETURNS TO SCALE
PARAMETERS/A/B WW U J D F c I B N K
A 0.03 0.04 0.02 0.02 0.01 0.01 0.004 0.003 0.02 0.01
a 0.11 0.06 0.15 0.08 0.12 0.09 0.25 0.13 0.00 0.21
a 0.26 0.39 0.12 0.23 0.21 0.32 0.22 0.12 0.19 0.05
Y 0.51 0.48 0.60 0.60 1.68 0.60 0.60 0.74 0.58 0.57
6 0.08 0.07 0.08 0.07 0.08 0.07 0.05 0.05 0.08 0.06
x -0.014 -0.019 -0.02 -0.003* -0.022 -0.009* -0.008 0.043 -0.022 0.091
v 0.96 0.99 0.96 0.98 1.10 1.08 1.11 1.04 0.93 0.89
R2 0.61 0.40 0.66 0.08 0.36 0.07 0.09 0.24 0.37 0.75
DW 1.82 2.46 2.59 1.31 2.19 0.96 2.12 0.83 1.00 1.61
/A COUNTRY CODES:
WW INDUSTRIAL AND DEVELOPING COUNTRIES
U UNITED STATES
J JAPAN
D GERMANY, F.R.
F FRANCE
C UNITED KINGDOM
I ITALY
B BRAZIL
N INDIA
K KOREA, R. OF
/B * T TEST INSIGNIFICANT.
*** WRONG SIGN.
SOURCE: WORLD BANK, ECONOMIC ANALYSIS AND PROJECTIONS DEPARTMENT.
- 19 -
countries considered.l/ Italy has the highest capital elasticity; and, (iii)
The parameter X (the rate of output-augmenting technical progress) takes
values that in most cases are either insignificant statistically or negative.
Only for the United States and Italy is X significant and positive,2/ and
varies between 0.02 and 0.03 for the 1972-1983 period.
B. Steady States
15. Equations (10) and (11) of Section II identify the steady state
values of Z (the steel output to capital ratio) and d (the iron ore
consumption to iron ore reserves ratio). Based on values of Z a set of price
paths may be determined as follows:
*
p p e (17)
where p is the iron ore/steel price ratio.
16. The equilibrium price path may also be estimated by equation (17) if
Z values are replaced by Z equilibrium values, i.e., the solution of equation
(8). Table 3 presents the Z and d values for four different cases. From the
foregoing results, the most reliable results from the statistical and economic
point of view are those of Case 1, i.e., for the Cobb-Douglas production
function with variable returns to scale and no technical progress. For the
United States, the Federal Republic of Cermany, France and Italy, however, the
results of Case 3 would be preferred to those of Case 1 as the Cobb-Douglas
production function includes technical progress coefficients.
1/ France's iron elasticity seems excessively high and the R2 indicates that
it should be viewed with caution. Table 2 gives a very different picture
than that of Table 1 for France.
2/ In Table 2 X is positive and significant only for Brazil.
- 20 -
TABLE 3: STEADY STATES OF STEEL OUTPUT/CAPITAL STOCK RATIO (Z)
AND IRON ORE CONSUMPTION/RESERVES RATIO (d)
PARAMETERS/A WW U J D F G I B N K
CASE 1 /B
Z 0.11 0.07 0.12 0.05 0.09 0.12 0.17 0.31
d 0.02 0.10 0.03 0.01 0.05 0.06 0.03 0.10
CASE 2 IC
Z 0.18 0.21 0.15 0.12 0.19 0.13 0.18 0.46 0.28 0.23
d 0.01 0.01 0.02 0.00 0.02 0.01 0.04 0.03 0.01 0.03
CASE 3 /D
Z 0.10 0.21 0.00 0.05 0.14 0.36 0.04
d 0.02 0.03 0.00 0.01 0.01 0.17 0.01
CASE 4 /E
Z 0.07 0.09 0.03 0.09 0.89 0.09
d 0.00 0.00 0.00 0.02 0.06 0.00
/ IF 6.1 0.0 5.1 4.4 1.1 2.7 4.8 6.1 4.2 6.1
n IF 2.0 0.7 0.4 -0.1 0.4 0.1 0.1 2.0 1.9 1.4
m IF 3.4 2.5 2.9 2.2 2.4 1.9 2.5 5.9 5.0 4.0
/A COUNTRY CODES:
VW INDUSTRIAL AND DEVELOPING COUNTRIES
U UNITED STATES
J JAPAN
D GERMANY, F.R.
P FRANCE
G UNITED KINGDOM
I ITALY
B BRAZIL
N INDIA
K KOREA, R. OF
/B CASE 1 USES ESTIMATES OF PARAMETERS OF THE COBB-DOUGLAS PRODUCTION FUNC-
TION WITH VARIABLE RETURNS TO SCALE AND WITHOUT TECHNOLOCICAL PROGRESS,
i.e., x - 0 .
IC CASE 2 USES ESTIMATES OF PARAMETERS OF THE COBB-DOUGLAS PRODUCTION FUNC-
TION WITH CONSTANT RETURNS TO SCALE AND WITHOUT TECHNOLOGICAL PROGRESS,
i.e., )x - 0
/D CASE 3 IS AS CASE 1 BUT WITH TECHNOLOGICAL PROGRESS, i.e., x * 0.
/e CASE 4 IS AS CASE 2 BUT WITH TECHNOLOCICAL PROGRESS, i.e., x . 0.
IF . - SAVINCS RATIO (IN Z); THIS RATIO HAS BEEN CALCULATED BY REGRESSINC THE
CHANCE OF NET WORTH IN REAL TERMS OF STEEL PRODUCING COMPANIES IN THE
COUNTRIES CONSIDERED HERE TO THE STEEL OUTPUT OF THESE COMPANIES.
n * GROWTH OF LABOR EMPLOYED (IN 1); IT HAS BEEN ASSUMED THAT THIS GROWTH
RATE IS EQUAL TO THAT OF THE POPULATION OF THE COUNTRY IN QUESTION.
WORLD BANK POPULATION FORECASTS HAVE BEEN USED; THE CROWTH RATES ARE
THOSE OF THE 1985-90 PERIOD.
m s CROWTH RATE OF RAW MATERIAL SUPPLY OTHER THAN THAT OF IRON ORE (IN Z);
IT IS ASSUMED THAT THIS CROWTH RATE WILL BE EQUAL TO THAT OF CDP;
WORLD 8ANK JANUARY 1986 FORECASTS HAVE BEEN USED; THE GROWTH RATES ARE
THOSE OF THE 1985-90 PERIOD.
- 21 -
17. Based on Case 1 of Table 3 the following points may be made: (i) the
steady state consumption level of iron ore of the United States and the
Republic of Korea are the highest; they are followed by the steady state
values of Italy and France. The lowest steady state levels of consumption are
those of the Federal Republic of Germany, Japan and India.l/ The only country,
among the nine examined here, that has a steady state consumption level below
that of the western world as a whole is that of the Federal Republic of
Germany; and, (ii) the countries that have a steady state output/capital value
higher than that of the western world are the Republic of Korea, Japan, Italy
and India.2/ The countries that have the lowest output/capital value are the
Federal Republic of Germany, the United States and France.
18. Equation (17) for the western world, based on the assumptions of
Table 3, could take the form:
* 0.024t
p = pOe
The paths of the iron ore/steel price ratio are presented in Figure 2. This
price ratio is dependent on the initial price (po)--1983 has been used as the
initial year--and the value of the steady state Z^ (which depends inter alia
on the long-term growth rates of employment and the supplies of iron ore raw
materials, n and m). By changing the assumptions about the long-term growth of
1/ The results regarding India have to be regarded with caution as the
Durbin-Watson of the production function in Table 2 indicates the
existence of serial correlation.
2/ See footnote 1/.
- 22 -
Figure 2: Price Paths Based on Different Values For Z' and d*: 1972-2000
(Iron Ore Price Deflated by Steel Price)
Ratio
0.096
i P
J-.
JJ
0.056 (3
I~~~~~~P
0.036
1972 1992 2012
SOURCE: SEE TABLES 3 AND 4.
- 23 -
employment and other raw materials, different Z steady states and price paths
p are derived. Four such paths are shown in Figure 2. The assumptions behind
these four paths p , P(1)' p (2) and p(3) are presented in Table 4. As
pointed out
TABLE 4: ASSUMPTIONS FOR DIFFERENT PRICE PATHS
PRICE PATH
GROWTH RATES /A p P(l) P(2) P(3)
n (Z) 2.0 -3.53 -3.53 0
m (Z) 3.4 3.40 -14.96 0
/A FOR DEFINITIONS OF n AND m SEE TABLE 3.
in Section II, the steady state Z is a saddle point and only the solutions of
equations (8) and (9) may lead to it. At any other point of Figure 1 policy
should be induced to reach the steady state or the path of the solution of
equations (8) and (9). As the solution of equations (8) and (9) is unknownl/
it cannot be asserted with absolute certainty that the actual iron ore/steel
price or the actual steel output/capital ratio or the actual iron ore
depletion rate is on the efficient path or not. The movement of the Z and d
variables over time towards their steady state values or away from them may
provide an indication as to whether steel producers are moving on their
efficient path or whether some policy changes are needed.2/
1/ The solution of these differential equations is difficult.
2/ As p, Z and d depend on "exogenous" long-term values of m and n, any
policy conclusions depend as well on the values of these parameters.
-24-
IV. POLICY CONCLUSIONS
19. As expected, the depletion of iron ore, i.e., the ratio of iron ore
consumption to world iron ore reserves is negatively related to iron ore
prices (see Figure 3).1/ The growth rates of the iron ore consumption to
reserves ratio for the nine countries examined in this study and the western
world are presented in Table 5. Only Italy and the Republic of Korea showed
positive average growth rates for the 1972-1983 period. Iron ore prices
increased by an annual average rate of 5.4% during the same period. For the
United States, France, United Kingdom and Brazil their consumption to reserves
ratio declined on average far more than the western world as a whole. The
western world's iron ore consumption to reserves ratio declined by an average
of 3.4% during the 1972-1983 period.
TABLE 5: GROWTH RATES FOR IRON ORE CONSUMPTION/RESERVES RATIO: 1972-83
COUNTRY (Z)
WESTERN WORLD -3.40
UNITED STATES -7.77
JAPAN -1.38
GERMANY, F.R. -0.23
FRANCE -4.72
UNITED KINGDOM -4.71
ITALY 0.92
BRAZIL -3.88
INDIA -1.41
KOREA, R. OF 36.77
1/ The Rotterdam CIF price of Brazilian iron ore sinter shipment is used as
the international iron ore price.
- 25 -
FIGURE 3: WORLD DEPLETION RATE (DSW) AND INTERNATIONAL
IRON ORE PRICE (IOP)
(DSWW)
0.004-
0.0039
0.0038
0.0037
0.0036
0.0035
0.0034
0.0033
0.0032
0.0031
0.003
0.0029
0.0028
0.0027
0.0026
0.0025
0.0024
0.0023
0.0022
0.0021 -
0.002-
1972 1975 1978 1981
(lOP)
29-
28 -
27 -
25-
25 -
24-
23 -
22 -
21-
(L 20-
19 1
i8
17
16
1s
14
13
12
11
1972 1975 1978 1981 1984
YEARS
SOURCE: WORLD BANK, ECONOMIC ANALYSIS AND PROJECTIONS DEPARTMENT
- 26 -
20. Table 6 shows the relative importance of different factors of
production in the steel production process. In nearly all countries the share
of capital stock in the production process is small relative to the share of
all other factors of production, with the exception of the share of iron ore.
In the case of the Republic of Korea, the share of labor is smaller than that
of capital. The country with the smallest share of capital in its production
process is the United States followed by the Federal Republic of Germany,
France and India. Japan, the Republic of Korea and Italy have larger shares of
capital in their production process than the other countries.
TABLE 6: RELATIVE IMPORTANCE OF FACTORS OF PRODUCTION IN THE
STEEL PRODUCTION PROCESS: AN INTERNATIONAL COMPARISON
(Z)
FACTORS OF
PRODUCTION U J D F G I B N K
CAPITAL (K) 0.7 2.0 1.3 1.3 - 3.3 - 1.3 2.3
LABOR (L) 11.6 4.0 8.5 7.1 - 4.9 - 7.0 1.5
RAW MATERIALS (M) 1.4 7.5 3.8 5.0 - 5.5 - 5.1 9.9
IRON ORE (R) 0.6 0.8 0.7 0.8 - 0.5 - 0.9 0.5
/A COUNTRY CODES:
WW INDUSTRIAL AND DEVELOPING COUNTRIES
U UNITED STATES
J JAPAN
D GERMANY, F.R.
F FRANCE
G UNITED KINGDOM
I ITALY
B BRAZIL
N INDIA
K KOREA, R. OF
SOURCE: SEE TABLE 1.
- 27 -
21. Based on the analysis of Section II, countries with a small initial
capital share relative to other factors of production should choose as high as
possible initial steel output-to-capital ratio and as high as possible iron
ore consumption-to-reserves share relative to their steady state values and
aim at decreasing them over time in order to reach either their efficient path
or their.steady state. The opposite should be true for countries with large
initial capital shares. The results of Table 6 indicate that the United
States, the Federal Republic of Germany, France and India should follow the
first pattern in order to reach their steady state. The Republic of Korea
should follow the second pattern. Japan and Italy are less clear cases and
they could follow either pattern.
22. Figures 4 and 5 plot the historical path of steel output to capital
ratio divided by the appropriate steady state and the historical path of iron
ore consumption-to-reserves ratio divided by the appropriate steady state,
respectively.
23. From these graphs, the following points are noteworthy:l/ (i) The
United States started to operate "optimally" sometime after 1979 as far as the
steel output-to-capital share (Z) is concerned, and sometime after 1975 as far
as the iron ore consumption-to-reserves ratio (d) is concerned. In order to
reach its steady state, the United States will have to continue decreasing its
steel output-to-capital ratio and its iron ore consumption-to-reserves ratio;
(ii) The Federal Republic of Germany did not operate "optimally" for the 1975-
81 period, as far as the steel output-to-capital share (Z) is concerned. Since
1981, its policies regarding Z have been "optimal". Germany's policies
l/ The word "optimal" is used here to describe desirable behavior, consistent
with the previously described expectations for reaching steady states.
FIGURE 4: STEEL OUTPUT TO CAPITAL RATIO, SELECTED COUNTRIES
GERMANY U.S.A.
(DELATED BY 15 STEADY STATE VALUE_DEFLATED eV ITS STEADY STAIE VAE)
0.069 0.047
O.OU9 - 0.047 -
0.058-8 0.044-
0.057- 0.045-\
0.046- 0.044 -
0.05- 0.043 -
0.054- 0.042-
0.053 - 0.041 -
0.052 -0.04
0 0.081 0.039
0.08 F 0.036
0.049- 0.0/37-
0.042- 0.036 -
0.047- 0.032-/
0.048- 0.034 -
0.041 - 0.0133
0.04.4 -0.032
0.043 - 0.031 -
0.042 -0.03
0.041 0.029
0.04 - T
1972 1975 1978 1981 1972 1975 1971 1961
YEARS YEARS
FRANCE JAPAN
DEFLA.MD sy rrs SEAuy sTATE vAwE) (DEFLATED BY ITS STEADrY STATE VALUE)
0.026 .1
0.026 - ~~~~~~~~~~~~~~~~~~~~~~0.017
0.024-
/ ~~~~~~~~~~~~~~~~~~~~~0.018
0.022-
0.02-
0.018 0.014-
0 0
F- 0.016 F: 0.013-
0.014-001-
0.012 -~~~~~~~~~~~~~~~~~~~~~~~.1
0.011
0.01
0.01
0.008
0.006 0.000
0.004 - - 0.006 -l . I*
1972 1975 1976 1961 1972 1975 1767 1961
YEARS YEARS
FIGURE 4 (CONTINUED): STEEL OUTPUT TO CAPITAL RATIO, SELECTED COUNTRIES
ITALY KOREA
(DELATED BY FM S1EADY STATE VALUE) _ _( AUt Of SI EADY UTAIE VMJE)
0.011 - _ _ _ _ .0_ _0 ,___-_-_-.-- -
0.014
0.013 0.002
0.012-
0.011 W GAN
~~~~~~~~~~~o.om __ __
0.01 -
O.000 -
O.Ot~~~~~~~~~~~~~.0 -
0.008 -
0.007-
0.006 ~~~~~~~~~~~~~~~~~~~~~~~~0.004
0o.. 000
0~~~~~~~~~~~.0oo - s . 003 --r---_--r----
0.003
0.002 -0.002-
0.001
1672 1171 1173 l"I 1372 177 173 ins '0
INDIA
SOURACE W MB EIO I A START VALIET
0.011 -
0.013
0.017-
0.016
0.013
80.014-
0.013-
0.012-
0.011
0.01
1172 1375 1371 lilt
SOURCE: WORLD BANK, ECONOMIC ANALYSIS AND PROJECTIONS DEPARTMENT
FIGURE 5: IRON ORE CONSUMPTION TO WORLD RESERVES, SELECTED COUNTRIES
GERMANY U.S.A.
0.0 62 - -0 .__ - _ _ _ _ _- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
0.06
0.1
0.0C X
0.066- 0.06
0.064-
0.062- 0.06
0.06 007i
0.046~~~~~~~~~~~~~~~~~~~~~~~~~00
OA"4 0.06
0.044
0.042 -.oj
0.04~~~~~~~~~~~~~~~~~~~~~~~~~00
0.0m~~~~~~~~~~~~~~~~~~~~~~~~~~00
0.036.0
0.002 . .6 . . , . 0---. . . . .
1072 1675 1676 151 1672 1671 157 1461 w
FRANCE JAPAN
O .004 - - _ _ _ _ _ _ . 4
0.0042 0.042
0.004 0.041-
0.04-
0.0m 6
0.0036-
0.036
0.0034 -0
F- 0.037-
0.0032-
0.0m 6
0.003-
0.0m 6
0.0025 0.034-
0.0026 0.033-
0.0024 0.032-
0.0022 0.031
0.002 0.03
1672 1175 1176 61" 1672 1076 l56 66
FIGURE 5 (CONTINUED): IRON ORE CONSUMPTION TO WORLD RESERVES, SELECTED COUNTRIES
ITALY KOREA
0.0014 - _ _ __0.0009 -_ _ __ _ _ __
0.0006
0.0007-
0.0012 -
0.0000
0.0011 -000
0.001 0.0004-
0.0003-
0.00090
0.0002-
0.0006 -000
0.0007 - _ , , , , , , . , W , 0-,,,,
172 1976 176 1971 1972 1976 1971 1361
YEAS YFAMWA
INDIA
0.00116
0.0011-
0.00106
0.001-
0.00066-
0.
0.0006
0.00076
0.00076
1972 1976 1978 1901
ECRS
SOURCE: WORLD BANK, ECONOMIC ANALYSIS AND PROJECTIONS DEPARTMENT
- 32 -
regarding its iron ore consumption-to-reserves ratio have been "optimal" for
the whole period with the exception of 1974 and 1980. Germany should continue
to aim at decreasing Z and d; (iii) France was following "optimal" policies
regarding Z up to 1978 (with the exception of 1974 and 1976). Since then (with
the exception of 1982) policies regarding Z have tended to be opposite to
those expected in order to reach the steady state or an "optimal" production
path. France's policies regarding d have been "optimal" for the whole period
under investigation. The 1974 and 1980 years were the only exceptional ones.
France's policies should continue aiming at decreasing Z and d; (iv) India has
been following policies that were not "optimal" as far as Z is concerned up to
1977 (with the exception of 1975). Since then Z has been declining as
"optimal" policies would have demanded. India's policies regarding d seem to
be less clear. The iron ore consumption-to-reserves ratio has been declining
for the 1972-73, 1975-76, 1977-80 and 1981-82 periods. In order to reach its
"optimal" steady state, India should aim at reducing Z and d; (v) The Republic
of Korea is characterized by an increasing Z and an increasing d. The years
for which this pattern is reversed are very few (1976, 1977, 1978 and 1982 for
Z and 1982 and 1983 for d). This is the expected "optimal" path for Korean
steel producers who will have to continue aiming at increasing Z and d until
the steady state is reached; (vi) Japan's graphs seem to indicate that
"optimal" policies are similar to those for the United States, the Federal
Republic of Germany, France and India, i.e., declining Z and d. In 1973, 1974
and 1979 Z increased, and in 1973 and 1979 d increased. Japan may be on an
"optimal" path tending to reach its steady state; and, (vii) Italy's steel
industry is characterized by a constantly increasing Z and a decreasing d. As
was earlier noted, Italy's policies may not be contrary to "optimal" behavior.
- 33 - ANNEX I
DATA BANK SOURCES
Variable/
Name Description Period Unit Source
YC Steel Revenues 1972-1983 Mill US$ WSD
LC Labor Cost 1972-1983 Mill US$ WSD
IC Interest and Depreciation Cost 1972-1983 Mill US$ WSD
CC Iron Ore Consumption Cost 1972-1983 Mill US$ WB
MC Other Raw Material Cost 1972-1983 Mill US$ WB
PROFIT Profit/Loss Steel Industry 1972-1983 Mill US$ WSD
Y Steel Production 1972-1983 Mill T WSD, IISI
L Labor Employed 1972-1983 Thousand WSD
ARI Capital Stock 1972-1983 Mill 1980 US$ WSD, WB
C Iron Ore Consumption 1972-1984 Mill T(Fe cont) WB
MR Other Material Cost 1972-1983 Mill 1980 US$ WB
P80 Wholesale Price Index Steel 1972-1984 1980=100 WSD
PMUV MUV Index 1972-1984 1980=100 WB
PC Import Unit Value 1972-1984 $/T (Fe cont) WB
PY Steel Deflator 1972-1983 $/T WB
IOP Brazilian North Sea CIF
Iron Ore Price 1972-1985 $/T WB
RES Iron Ore Reserves 1972-1985 Mill T B. of M.
L* Wage Rate Steel Industry 1972-1983 $/Year WSD
JC Interest Rate 1972-1983 % IMF/IFS
SOURCES: WSD - WORLD STEEL DYNAMICS; WB - WORLD BANK; IISI - INTERNATIONAL IRON AND
STEEL INSTITUTE; B. OF M. - U.S. BUREAU OF MINES; IMF/IFS - INTERNATIONAL
MONETARY FUND, INTERNATIONAL FINANCIAL STATISTICS.
-34- ANNEX 2
NON-MONOTONIC EQUILIBRIUM PATHS
1. Dixit (1976) shows that removal of some of the restrictions of the
Cobb-Douglas production function is possible. He shows that increasing the
number of factors, as in this case, or endogenizing steel consumption
increases the possibilities for non-monotonic paths. The resource use per
capita can then fall, rise and fall again along an optimal path.
2. Ingham and Simmons (1975) have also shown that the elasticity of
substitution can govern the possibility of sustaining growth in the
long run. Using a two-factor production function with constant returns to
scale (F (K, R) where K is capital stock and R resource use) and a
maximization problem of an iso-elastic utility function (IF u(c) e ptdt where
0
c = steel consumption and p = interest rate in utility terms) proves that the
equilibrium path of the depletion rate may be represented as:
d = d + {f(k) - h} - a f(k) (2.1)
d ~~k k
where
h = consumption of steel to capital stock ratio (= C/K)
k = capital stock to iron ore consumption ratio (=K/R)
f(k) = F(K, 1)
a - elasticity of substitution between K and R
3. The growth pattern for h is shown to be:
*
h t f (k) -p } _ {f(k) - h} (2.2)
- 35 - ANNEX 2
where
p = interest rate in utility terms ("rate of impatience").
£ = elasticity of marginal utility, i.e., the relative rate at
which undiscounted marginal utility falls as consumption per
head increases.
In the long-run h must ultimately equal to the following expression:
h = {p - n . (l-E)}/£ (2.3)
where n = lim f (k) = lim f(k) (2.4)
k
It is noteworthy that if a-S 1 then n = 0 while, if a > 1 then n > 0. The
asymptotic value for d would then be:
d = h + (a - 1) * n (2.5)
and the following expressions can be derived
k = n -h = C = (n - p)/e (2.6)
k
= a . n (2.7)
R = -d (2.8)
Expressions (2.3) to (2.8) indicate that as capital accumulates relative to
resource flow (k) the resource flow must decline (see (2.5), (2.7) and (2.8).
Ultimately, whether this will enable sustained growth of consumption (2.6)
depends on whether n will exceed p.
-36 - ANNEX 3
ESTIMATING THE ELASTICITY OF SUBSTITUTION BETNEEN IRON ORE
AND OTHER FACTORS OF PRODUCTION
4. As noted in Section II, the elasticity of substitution between iron
ore and capital (a) in a two-factor production technology governs the
possibility of sustaining growth of iron ore consumption. It was found there
that such growth may be sustained if the elasticity of substitution is greater
than one and if the limit of the marginal product of capital is large enough
to make the numerator of the rhs of equation (15) positive. Otherwise, and in
particular if a < 1, the growth of iron ore consumption may not be sustained.
5. In the Cobb-Douglas production function the elasticity of substi-
tution between inputs is constant and equal to one, which implies that optimal
growth of iron ore consumption may not imply sustained growth over time. It is
thus important to test the implied assumptions of the Cobb-Douglas production
function, i.e., to find out whether or not the elasticity of substitution
between iron ore and the other inputs in the steel production process is
greater than one.
6. From the pure profit maximization problem of the CES production
function of the form:
Y = y{6 N-0 + (1 - 6) N-0}-v/P
where
Y = output
N1, N2 = two inputs
1-a
0 o
a
a = elasticity of substitution
v - returns to scale
-37- ANNEX 3
we get the input marginal productivity conditions
P1a V (L.~ PPV= -(3.1)
ey = -p/v( Y )l+p Y-p+p/v = 31
-N = v (1-6) Y p/v Y Yl+ =~+/v (3.2)
ON2 N2 y 32
where
n1, n2 = input prices
y = output price
From any of these two equations the following equation is derived and used to
estimate the elasticity of substitution between N1 and N2 and the returns to
scale v.
Y ~~~~N2 1 m 33
ln- N = constant + oln-T-- + p a(l- V ) lnY (3.3)
The second term of the rhs part of the equation provides an estimate of
(a) and the third term provides an estimate for v. If we also allow neutral
technical progress to occur via changes in the parameters y, i.e.,
pxt
y = y where X is some constant rate, equation (3.3) may be rewritten as
follows:
y 1 1
In- constant + X pa - t + pa (1- -) ln Y + aln - (3.4)
N2 v v y
Equation (3.4) allows for the estimation of X. Alternatively, equation (3.1)
or (3.2) may be rewritten as follows:
-38- ANNEX 3
n2N2 2 (- ~ +(
ln y = ln v(l - 6) y - p(l- v) ln Y + (-a)ln- (3.5)
yY o v v Y
Estimates for a, v and X for the nine most important steel-producing countries
and the world (excluding the Eastern European countries) are presented in
Annex Table 3.1. Data are available for four different inputs (capital, labor,
iron ore and other raw materials), therefore elasticities of substitution have
been estimated between each of these inputs and all the other inputs as a
whole.
ANNEX TABLE 3.1: ELASTICITIES OF SUBSTITUTION BETWEEN INPUTS
IN THE STEEL PRODUCTION PROCESS /A/B
PARAMETERS WW U J D F c I B N K
1. IRON ORE AND ALL OTHER INPUTS
METHOD 1 /C
s 0.48 0.31 0.06 0.076
p *** 1.08 *** 2.23 *** *** *** 15.67 *** 12.160
v *** 0.59 0.75 *** *** *** 1.15 *** 0.790
R2 0.19 0.54 0.02 0.71 0.04 0.04 0.13 0.54 0.73 0.600
DW 1.53 1.56 1.70 1.69 2.05 3.00 1.69 2.36 3.00 2.610
METHOD 2 /C
s *** 0.42* *** 0.33 *** 0.19* 0.16* 0.31* *** 0.082*
p *** 1.38* *** 2.03 *** 8.09* 5.25* 2.23* *** 11.200
v *** 0.67* *** 0.76 *** 0.71* 0.73* 1.46* *** 0.690*
g *** 0.009* *** 0.001* * -0.013* 0.01* 0.27* *** 0.040*
R2 0.16 0.49 0.13 0.68 0.16 0.03 0.13 0.60 0.74 0.570
DW 1.19 1.46 1.69 1.78 2.00 2.52 2.26 2.62 3.16 2.830
METHOD 3 /C
s 0.43* 0.34 0.11 0.16 0.31* *** 0.08
R2 0.76 0.08 0.89 0.87 0.73 0.93 0.65 0.34 0.96 0.87
DW 1.19 1.46 1.69 1.78 2.00 2.52 2.26 2.62 3.16 2.83
*. . Il
PARAMETERS WW U J D F C I B N K
2. LABOR AND ALL OTHER INPUTS
METHOD 1 /C
s 0.65* 0.87 0.39 0.40 1.21 1.40 *** 0.01* 0.90 0.13*
p 0.53* 0.16 1.59 1.53 -0.17 -0.29 *** 90.03* 0.11 6.81*
v 0.75* 4.47* 12.78 28.59 0.29 0.81* *** 1.05* *** 2.01*
R2 0.08 0.38 0.79 0.80 0.54 0.94 0.49 0.14 0.89 0.95
DW 1.72 2.13 1.20 1.66 1.41 2.35 1.56 1.39 1.88 1.62
METHOD 2 /C
s 0.68 0.82 *** 0.26 0.60* 1.19 *** *** 0.98 0.12*
p 0.47 0.22 *** 2.85 0.07* -0.16 *** * 0.02 7.33*
v 4.57 *** *** 9.77 *** *** *** *** *** 2.02
g 0.24 *** *** 0.09 *** *** *** *** *** -0.003
R2 0.53 0.60 0.93 0.87 0.58 0.96 0.66 0.79 0.89 0.94
DW 1.38 1.21 0.67 2.66 0.81 2.58 1.30 3.18 2.09 1.60
3. CAPITAL STOCK AND ALL OTHER INPUTS
METHOD 1 /C
s 5.20* 0.27* 1.41 0.61 0.94 1.61 0.84* 0.43 0.75
p -0.81* 2.76* -0.29 0.64 0.06 *** -0.38 0.19 1.31 0.34
v 4.17* 1.44 0.28 2.40* *** *** *** *** 3.90
R2 0.72 0.00 0.74 0.14 0.80 0.10 0.78 0.85 0.50 0.88
DW 1.60 1.33 1.01 0.22 1.35 0.84 1.56 0.34 1.43 0.88
PARAMETERS WW U J D F C I B N K
METHOD 2A /C
s 0.64 0.03* 1.07 0.33 0.86 0.99 0.01 0.53 0.65
p 0.56 32.33* -0.07 2.03 0.16 0.11 99.00 *** 0.89 0.54
v *** 11.02* 0.10 3.09 *-* *** 5.69* ** 2.53* 1.67*
g 0.009* 0.59* -0.09* 0.03 0.08* -0.11* 0.30* -^* 0.02* -0.13*
R2 0.74 0.85 0.76 0.88 0.80 0.22 0.89 0.92 0.100 0.81
DW 1.42 1.73 1.21 2.29 1.88 0.65 1.94 2.54 0.56 1.01
METHOD 2b /C
s 0.97 0.23 0.89 0.54 0.80 0.61 0.83 0.59 0.32 0.72 a
p 0.03 3.35 0.12 0.85 0.25 0.64 0.21 0.70 2.13 0.39 .
v 2.04 0.26* *** 0.96 *** 3.97 ***
g *** *** *** -0.04 -0.05 0.02 *** 0.21 ***
R2 0.99 0.91 0.95 0.93 0.96 0.46 0.90 0.98 0.78 0.95
DW 1.41 1.56 1.10 1.31 1.57 1.51 2.17 1.02 1.23 1.18
4. "OTHER MATERIALS" AND ALL OTHER INPUTS (i.e., CAPITAL STOCK, LABOR AND IRON ORE)
METHOD 1 /C
s 1.90 1.22 0.94 1.29 0.91 1.04 *** 1.25 1.23 0.98
p -0.47 -0.18 0.06 -0.22 0.10 -0.04 *** -0.20 -0.19 0.02
v 0.74 0.32 0.59* 1.51 *i- 0.07 *** 2.04* *** **
R2 0.90 0.82 0.89 0.96 0.77 0.93 0.00 0.65 0.98 0.81
DW 1.42 1.08 1.31 1.86 1.01 1.36 0.55 1.82 2.28 2.51
;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
* . .
PARAMETERS WW U J D F C I B N K
METHOD 2 /C
s 1.30 1.30 1.04 1.26 0.60 1.20 0.77 1.62 1.32 0.73
p -0.23 -0.23 -0.04 -0.21 0.49 -0.17 0.30 -0.38 -0.24 0.37
v 0.82* 0.68 0.44* 2.50 *** 0.93* 0.53 *-k* *** ***
g 0.05* 0.06 -0.p6* 0.04 0.05* 0.11
R2 0.96 0.94 0.88 0.98 0.87 0.92 0.85 0.70 0.99 0.83
DW 1.67 1.87 1.05 3.16 1.88 1.94 2.07 2.28 2.69 1.74
/A COUNTRY CODES:
WW INDUSTRIAL AND DEVELOPING COUNTRIES
U UNITED STATES
J JAPAN
D GERMANY, F.R.
F FRANCE
G UNITED KINGDOM
I ITALY
B BRAZIL
N INDIA
K KOREA, R. OF
lB * T TEST INSIGNIFICANT.
*** WRONG SIGN.
/C THE PARAMETERS OF METHOD 1 ARE DERIVED FROM EQUATION (3.4) BUT WITHOUT THE TREND VARIABLE. THE
PARAMETERS OF METHOD 2 ARE DERIVED FROM EQUATION (3.4). THE PARAMETERS OF METHOD 3 ARE DERIVED
FROM EQUATION (3.5). TWO DIFFERENT SOURCES FOR CAPITAL HAVE BEEN USED TO ESTIMATE THE
"ELASTICITY" OF SUBSTITUTION OF CAPITAL AND ALL OTHER INPUTS. METHODS 1 AND 2a USE EQUATION
(3.4) AND CAPITAL DATA THAT RELATE TO REAL ASSETS. METHOD 2b USES EQUATION (3.4) AND CAPITAL
DATA THAT RELATE REAL NET WORTH OF STEEL COMPANIES IN THE COUNTRIES UNDER CONSIDERATION. ALL
EQUATIONS HAVE BEEN RUN FOR THE 1972-1983 PERIOD.
SOURCE: WORLD BANK, ECONOMIC ANALYSIS AND PROJECTIONS DEPARTMENT.
-43- ANNEX 3
7. The empirical results presented in Annex Table 3.1 indicate the
following: (i) The elasticity of substitution of iron ore is less than one
whenever the coefficients of the estimated equations are statistically
significant. In the majority of cases, the elasticity of substitution of iron
ore is not statistically significant. It is significant and less than one for
the United States, the Federal Republic of Germany, Brazil and the Republic of
Korea. For these countries, their optimal steady state will be reached by
increasing consumption of iron ore at a decreasing rate. During the 1972-83
period, substitution between iron ore and other raw materials, labor and
capital was not significant for all the other countries; (ii) In three of the
four cases when the elasticity of substitution of iron ore was significant,
returns to scale were found to be decreasing (v < 1). This is a surprising
result given the results of Section III-B. The technological progress
parameter X was statistically insignificant in almost all cases; (iii) The
elasticity of substitution between labor and all other inputs was found to be
less than one in most cases (with the exception of France and the United
Kingdom). For several countries, the statistics R2 and DW lent no credence to
the estimates; (iv) The elasticity of substitution between capital stock and
all other inputs is also estimated to be less than one (in all cases but that
of Japan and India) independently of the data basis of the capital stock. (v)
The elasticity of substitution between other materials and all other inputs
was greater than one in all cases but France and the Republic of Korea.
8. To summarize, using a CES production function approach, the
elasticity of substitution between iron ore and otherinputs in the steel
production process was found to be either insignificant statistically or, if
significant, less than one. Data were not available for all countries prior to
-44- ANNEX 3
1972. Although the results presented here (i.e., for some countries the
elasticity of substitution is less than one) do not alter the results of the
previous analysis, much important and useful empirical work remains to be done
when longer historical data series become available. Such empirical work
could, for example, trace the possible non-monotonic equilibrium path of iron
ore consumption and prices.l/
1/ See Annex 2 for a brief discussion of the possible removal of some
restrictions of the Cobb-Douglas production function.
- 45 -
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