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A production function for the Swedish pig iron industry
i?ngineenhg Costs ~nd~r~~cti~~
Economics, 7 (1984) : 13-317
313
Elsevier Sdeme Publishers B.V., Amsterdam - Print& ir The Nethedands
Department of Economics, Univenity College of North Wales, Bangor, Gwynedd LL57 200
(Gnsat Britain)
Brn lnstitutitrnen for National&onomi,
University of lJme& 301-87 Ume8 (Sweden)
From the 1960s onwards production sf pig iron has been steadily going up in Japan, and rn the 19 70s South Korea undertook a major programme of pig iron production. In the Western countries, production rose somewhat in the 196Os, but declined or remained constant in the 19 70s. Against this background, we have looked at data for the Swedish industry for the period 1960-1975
searching for some clu,ns. Duting this perio the majority of plants in Sweden enjoyed rate of return on capital of 12% on averag For most .firms this rate has declined ove time, and in three of the plants the margin ctivity of capital went below zero. The nal productl’vity of variable facytrs has held its own, perhaps reflecting better work practices.
During the 198Os, production of pig iron and ferro all~~s steadily went up in Japan, from 1.028 to 5.809 million metric tons
For example, production of pig iron and ferro alloys in that country was only 3.4 thousand metric tonnes in 1970. the figure had risen to 43 1 thou uring this period pro
465% in ten years. was only IO%, from 1.335 to metric tonnes, in the same pend was 37%, startin tic tonnes in 196
7 thousand in 1970 an
events it wou “Production data dis
into different countries appear in Table 37 of of Statistics [S]. We have consulted a number of issues. July 1967, January 1974 and September 1979 and 1980. 0167-188X/84/$03.00
0 1984 Elsevier Science Publishers B.V.
314 In this note we confine ourselves to looking at the Swedish pig iron industry. We employ a variable elasticity of substitution production function, which allows for productivity to explained by the input ratio. A full scale study should repeat this exercise for a number of countries and compare the results. We had ready access only to Swedish data but it is our hope that others will describe the performance of this industry elsewhere, There would then be a br+ for proper comparison. In the Swtdish pig iron industr, r results indicatr that the marginal product& of capital ha:j fallen over time. The rate ..I’ return on capital lies in the main between 19.2% and 5.3%. For the majority of the plants it is about 12% on average.
MPI = ( Y/X,) (A
l
+
A3 In X,) (2.2)
al elast~~t~es ei=aln
In Y=A,,+A,In (2.1j n the
above e
ar~~al
tion In stands for natural
products
deemed
as:
Y/a lnX,=At+AjTnXz
e2 = a In Y/a In X2 = Az + A3 In X1 The second derivatives of X, and X2 are, respectively,
(2.3)
with respect to
aMP,IaXI = (El - 1) (MPJX,) aMP,/aX, = (Ez ..- 1) (M&IX,)
(2.4)
The cross derivative is, aMP,/aXz
= aMP,/aX, = (w1
+ A,) x Y/(X,X,)
(2.5)
TLe return
to scale is the sum of the above marginal elasticities: e=E]+E*
In the Bell Journal, two formulations of the production function were suggested. The first is the multiplicative function employed by Vinod [I], and the second is the additive function suggested by Sudit [2]. Ths additive function does not seem appropriate for our data. Vinod’s formulation is an extension of the Cobb.-Douglas function, but allowing for variable elasticilies. Suppose t:here are two inputs, X1 and Xz, and one output, Y. The non-homogeneous production function utilised in this note is a log quadratic formulation. By taking logalithms of both the right and left hand sides, it ian be written as:
a
(2.6)
The elasticity of substitution, u, can be reduced to the expression below [ I] : u = E/(E + 2A,j
(2.7)
We note that ifA:, = 0, u is equal to 1 and the production function, eqn. 2.1, becomes the Cobb--Douglas function.
To estimate the production function, we used plant level data over ;he period 196 l-1975. Output is measured in kilotonnes and labour input is measured in kilohours. able cost are measured on crowns. The data are described in with labour and capital stock e conclude that these
with a “process is only a minor co
315 proportional to output and affords IN substitutian possibilities. The main elements of variable cost co sist of energy and repair bills. e arguments of the specification finally accepted are variable cost (VAR) and capital stock (CAP). The variable cost is the sum of the costs of energy, labour and depreciation value of the capital stock. The depreciation value of the capital stock is included to reflect capital usage in the production prf cess. The BLS estimate of the multiplicative function
half of the estimates of CJobtained by some investigators appears to lie above unity 13 I. ‘She elasticity of substitution between cal ital and variable cost being not very much TABLE 1 Elasticities andmarginal productivities MPVAR
MF’CAP
15.03 15.25 15.23 14.87
0.00 -0.12 -0.11 -0.02
1975
0.99 1.14 14.51
-0.33
1960 1965 1970 1975
1.31 1.31 1.29 1.28
1.10 1.10 1.10 1.10
12.13 11.77 11.91 12.18
1.16 1.02 0.62 0.64
1960 1965 1969 1975
1.32 1.24 1.23 1.22
1.10 1.11 1.11 1.11
11.74 13.96 13.92 14.21
1.07 0.93 0.78 0.72
1960
1.31 1.10 11.85
1.09
1970 1974
1.23 1.11 14.07 1.25 1.11 13.56
0.84 0.83
1960 1965 1970 1975
1.21 1.19 1.20 1.20
1.11 1.11 1.11 1.11
14.59 15.07 14.72 14.84
0.62 0.51 0.58 0.57
1960 1965 1970 1975
1.31 1.31 1.27 1.26
l.liE 1.10 1.10 1.10
12.92 11.92 12.44 12.66
1.78 1.01 0.67 0.69
8960 1965 1970 6975
1.23 1.20 1.12 1.12
1.11 1.11 1.12 1.12
13.91 14.53 15.48 15.31
0.76 0.58 -0.02 0.03
YeaI
F
1960 1965 1970 1975
1.10 1.12 11.08 1.12 1.08 1.12 1.08 1.13
(3.1)
is shown below. The estimates are indicated with the conventional symbol A) and the t--statistic are in brackets.
In (OUTPW)
= A0 + A&( VAR) + Azln(CAP) + AJn( VAR) ln(CAP)
Ah,= 1.645 (8.99)
R, =
A^, = 1.3 1 (18.66)
b, = -0.060 (2.22)
0.191 (2.22)
R BAR SQUARE = 0.98 This equation is well determined, with acceptable significance levels for the estimated coefficients. tLASTICBTIES AND
Sample values of the marginal productivities of capital and variable cost appear in Table 1. The elasticity of substitution, o, and the scale elasticity, E, are also tabulated. Only a sample of values are shown in the table, there being 336 observations. The data span the period 1960-1975 for most plants, but some were commissioned in the 70s. of substitution We that variation of the scale elasti wider. The range of values are
1961 1965 8970 1975 1.960 1965 B97@ 1975
0
0.10 -0.11 -0.22 -0.16 0.48 0.47 0.49 0.27
J t I
4
3 16 greater than unity would indicate that the scope for energy conservation through investment is perhaps somewhat more exited t%lah one would have b&hed. This conslusion ghoutd be qualified. A complete chmge in e technology would render invalid the prcrduction function estimated here. The substitution possibilities for, say, energy input might then be increased. productivity of capital has In three groups, the marginal uctivity went below zero. Two of the lants involved an part of the nationaltar which incurred severe losses over sample peri.cd. Presumably they nal productivity of To calculate the m e price (1960) of pig capital, we note that iron (less cod of ore) is 160 X IO3 crowns per kilotonne. From the values of MPCAP tabulated in Table 1, we therefore note that the rate %3freturn on capital lies in the main between 19.2% and 5.3%. For the majority of the plants it is about 12% on average. For most firms, the rate of return on capital had declined over time. In one of the plants, this rcte h:n gone down from 12.2% to 0.48% between 1960 and 1975. However, the decline has been more moderate in most piants. On rh e whole, the decline in the martial productivity of capital has been 3.3% per annum*. The decline in the marginal prodluctivity of variable factors is not as renounced. In som:: of the pi fact gone u
vith
to
capital
is negative,
(2) ‘The cross acceleration terms and the direct acceleration of the variable cost :ue positive but small. The first observation indicates that the marginal productivity of capital would not grow with the introduction of additional capital.. The positive acceleration terms observed perhaps indicate technological progress, for which there is no explicit proxy in our equation. Pn any event, these terms are very small. An explicit time trend in the equation was rejected. The steady decline in the marginal productivity of capital and the lack of sens;tivity of this parameter to the introduction of new capital would suggest that organisstion of work has much to do with th.e changing fortunes of the industry.
One of the authors (S.P.C.) is pJatefu1 for a travel grant from the SSRC (U.K.).
H.D. Vinod, 1972. Nonhomogeneous production timetions and applications to telecommunications. Bell J. Econ. Manag. Sci., 3(2): 531-543. Ephraim F. Sudit, 1973. Additive nonhomogeneous production functions in telecommunications. Bell J. Econ. Manag. SC&,4(2): 499-514. U. Kazi, 1980. Tbe variable elasticity of substitution production function* a case study for Indian manufacMing industries. Oxford Econ. P., 32(l): 163-175. Sores Wibe, 1980. Tekn& och aggregering i produktonsteorin (Engineering and Aggregation in Production Theory), PhD Thesis, University ofUme& Sweden, 1980. Monthly Bulletin +f Statistics, New York: United Nations.
d~~va~ves of the output with capital indicate e
respect
diXrect accelera
a
I5 blast furnaces were inciuded i-1 the survey:
317 Some plants had more th,an one identical {i.e. same site and vintage) furnace. Since they had the same input figures, we included data for only one furnace, for each group of identical furnaces. This was done to avoid bias in statistical estimation. Thus, only 10 furnaces were taken into account in this exercise. Our data are annual and span the period 1960- 1975, but not for all furnaces. Some of them were commissioned in the 1970s. Special features of the data are noted beloxr. (i) Capital input. In most cases direct observations wem available for only the latter part of the sample period, usually between 1970 and 1975. Earlier data were calculated
from indirect information published in Forsakringsbolagens Forvarderings Kommittee [4Q. (ii) Depreciation and repair are in 1960 prices and am included in the variable cost. (iii) The capital stock is measured in replacement value in 1960 prices. (iv) Fuel input consists of the sum of coke and oil. (v) Production is the yearly production capacity measured in kilotonnes. Actual production data are corrected for deliberate stops due to e.g. vacation. Both labour and fuel inputs are also similarly corrected. (Received Match tember 7. 1983)
17, 1983: accepted in revised form Sep-
Economics, 7 (1984) : 13-317
313
Elsevier Sdeme Publishers B.V., Amsterdam - Print& ir The Nethedands
Department of Economics, Univenity College of North Wales, Bangor, Gwynedd LL57 200
(Gnsat Britain)
Brn lnstitutitrnen for National&onomi,
University of lJme& 301-87 Ume8 (Sweden)
From the 1960s onwards production sf pig iron has been steadily going up in Japan, and rn the 19 70s South Korea undertook a major programme of pig iron production. In the Western countries, production rose somewhat in the 196Os, but declined or remained constant in the 19 70s. Against this background, we have looked at data for the Swedish industry for the period 1960-1975
searching for some clu,ns. Duting this perio the majority of plants in Sweden enjoyed rate of return on capital of 12% on averag For most .firms this rate has declined ove time, and in three of the plants the margin ctivity of capital went below zero. The nal productl’vity of variable facytrs has held its own, perhaps reflecting better work practices.
During the 198Os, production of pig iron and ferro all~~s steadily went up in Japan, from 1.028 to 5.809 million metric tons
For example, production of pig iron and ferro alloys in that country was only 3.4 thousand metric tonnes in 1970. the figure had risen to 43 1 thou uring this period pro
465% in ten years. was only IO%, from 1.335 to metric tonnes, in the same pend was 37%, startin tic tonnes in 196
7 thousand in 1970 an
events it wou “Production data dis
into different countries appear in Table 37 of of Statistics [S]. We have consulted a number of issues. July 1967, January 1974 and September 1979 and 1980. 0167-188X/84/$03.00
0 1984 Elsevier Science Publishers B.V.
314 In this note we confine ourselves to looking at the Swedish pig iron industry. We employ a variable elasticity of substitution production function, which allows for productivity to explained by the input ratio. A full scale study should repeat this exercise for a number of countries and compare the results. We had ready access only to Swedish data but it is our hope that others will describe the performance of this industry elsewhere, There would then be a br+ for proper comparison. In the Swtdish pig iron industr, r results indicatr that the marginal product& of capital ha:j fallen over time. The rate ..I’ return on capital lies in the main between 19.2% and 5.3%. For the majority of the plants it is about 12% on average.
MPI = ( Y/X,) (A
l
+
A3 In X,) (2.2)
al elast~~t~es ei=aln
In Y=A,,+A,In (2.1j n the
above e
ar~~al
tion In stands for natural
products
deemed
as:
Y/a lnX,=At+AjTnXz
e2 = a In Y/a In X2 = Az + A3 In X1 The second derivatives of X, and X2 are, respectively,
(2.3)
with respect to
aMP,IaXI = (El - 1) (MPJX,) aMP,/aX, = (Ez ..- 1) (M&IX,)
(2.4)
The cross derivative is, aMP,/aXz
= aMP,/aX, = (w1
+ A,) x Y/(X,X,)
(2.5)
TLe return
to scale is the sum of the above marginal elasticities: e=E]+E*
In the Bell Journal, two formulations of the production function were suggested. The first is the multiplicative function employed by Vinod [I], and the second is the additive function suggested by Sudit [2]. Ths additive function does not seem appropriate for our data. Vinod’s formulation is an extension of the Cobb.-Douglas function, but allowing for variable elasticilies. Suppose t:here are two inputs, X1 and Xz, and one output, Y. The non-homogeneous production function utilised in this note is a log quadratic formulation. By taking logalithms of both the right and left hand sides, it ian be written as:
a
(2.6)
The elasticity of substitution, u, can be reduced to the expression below [ I] : u = E/(E + 2A,j
(2.7)
We note that ifA:, = 0, u is equal to 1 and the production function, eqn. 2.1, becomes the Cobb--Douglas function.
To estimate the production function, we used plant level data over ;he period 196 l-1975. Output is measured in kilotonnes and labour input is measured in kilohours. able cost are measured on crowns. The data are described in with labour and capital stock e conclude that these
with a “process is only a minor co
315 proportional to output and affords IN substitutian possibilities. The main elements of variable cost co sist of energy and repair bills. e arguments of the specification finally accepted are variable cost (VAR) and capital stock (CAP). The variable cost is the sum of the costs of energy, labour and depreciation value of the capital stock. The depreciation value of the capital stock is included to reflect capital usage in the production prf cess. The BLS estimate of the multiplicative function
half of the estimates of CJobtained by some investigators appears to lie above unity 13 I. ‘She elasticity of substitution between cal ital and variable cost being not very much TABLE 1 Elasticities andmarginal productivities MPVAR
MF’CAP
15.03 15.25 15.23 14.87
0.00 -0.12 -0.11 -0.02
1975
0.99 1.14 14.51
-0.33
1960 1965 1970 1975
1.31 1.31 1.29 1.28
1.10 1.10 1.10 1.10
12.13 11.77 11.91 12.18
1.16 1.02 0.62 0.64
1960 1965 1969 1975
1.32 1.24 1.23 1.22
1.10 1.11 1.11 1.11
11.74 13.96 13.92 14.21
1.07 0.93 0.78 0.72
1960
1.31 1.10 11.85
1.09
1970 1974
1.23 1.11 14.07 1.25 1.11 13.56
0.84 0.83
1960 1965 1970 1975
1.21 1.19 1.20 1.20
1.11 1.11 1.11 1.11
14.59 15.07 14.72 14.84
0.62 0.51 0.58 0.57
1960 1965 1970 1975
1.31 1.31 1.27 1.26
l.liE 1.10 1.10 1.10
12.92 11.92 12.44 12.66
1.78 1.01 0.67 0.69
8960 1965 1970 6975
1.23 1.20 1.12 1.12
1.11 1.11 1.12 1.12
13.91 14.53 15.48 15.31
0.76 0.58 -0.02 0.03
YeaI
F
1960 1965 1970 1975
1.10 1.12 11.08 1.12 1.08 1.12 1.08 1.13
(3.1)
is shown below. The estimates are indicated with the conventional symbol A) and the t--statistic are in brackets.
In (OUTPW)
= A0 + A&( VAR) + Azln(CAP) + AJn( VAR) ln(CAP)
Ah,= 1.645 (8.99)
R, =
A^, = 1.3 1 (18.66)
b, = -0.060 (2.22)
0.191 (2.22)
R BAR SQUARE = 0.98 This equation is well determined, with acceptable significance levels for the estimated coefficients. tLASTICBTIES AND
Sample values of the marginal productivities of capital and variable cost appear in Table 1. The elasticity of substitution, o, and the scale elasticity, E, are also tabulated. Only a sample of values are shown in the table, there being 336 observations. The data span the period 1960-1975 for most plants, but some were commissioned in the 70s. of substitution We that variation of the scale elasti wider. The range of values are
1961 1965 8970 1975 1.960 1965 B97@ 1975
0
0.10 -0.11 -0.22 -0.16 0.48 0.47 0.49 0.27
J t I
4
3 16 greater than unity would indicate that the scope for energy conservation through investment is perhaps somewhat more exited t%lah one would have b&hed. This conslusion ghoutd be qualified. A complete chmge in e technology would render invalid the prcrduction function estimated here. The substitution possibilities for, say, energy input might then be increased. productivity of capital has In three groups, the marginal uctivity went below zero. Two of the lants involved an part of the nationaltar which incurred severe losses over sample peri.cd. Presumably they nal productivity of To calculate the m e price (1960) of pig capital, we note that iron (less cod of ore) is 160 X IO3 crowns per kilotonne. From the values of MPCAP tabulated in Table 1, we therefore note that the rate %3freturn on capital lies in the main between 19.2% and 5.3%. For the majority of the plants it is about 12% on average. For most firms, the rate of return on capital had declined over time. In one of the plants, this rcte h:n gone down from 12.2% to 0.48% between 1960 and 1975. However, the decline has been more moderate in most piants. On rh e whole, the decline in the martial productivity of capital has been 3.3% per annum*. The decline in the marginal prodluctivity of variable factors is not as renounced. In som:: of the pi fact gone u
vith
to
capital
is negative,
(2) ‘The cross acceleration terms and the direct acceleration of the variable cost :ue positive but small. The first observation indicates that the marginal productivity of capital would not grow with the introduction of additional capital.. The positive acceleration terms observed perhaps indicate technological progress, for which there is no explicit proxy in our equation. Pn any event, these terms are very small. An explicit time trend in the equation was rejected. The steady decline in the marginal productivity of capital and the lack of sens;tivity of this parameter to the introduction of new capital would suggest that organisstion of work has much to do with th.e changing fortunes of the industry.
One of the authors (S.P.C.) is pJatefu1 for a travel grant from the SSRC (U.K.).
H.D. Vinod, 1972. Nonhomogeneous production timetions and applications to telecommunications. Bell J. Econ. Manag. Sci., 3(2): 531-543. Ephraim F. Sudit, 1973. Additive nonhomogeneous production functions in telecommunications. Bell J. Econ. Manag. SC&,4(2): 499-514. U. Kazi, 1980. Tbe variable elasticity of substitution production function* a case study for Indian manufacMing industries. Oxford Econ. P., 32(l): 163-175. Sores Wibe, 1980. Tekn& och aggregering i produktonsteorin (Engineering and Aggregation in Production Theory), PhD Thesis, University ofUme& Sweden, 1980. Monthly Bulletin +f Statistics, New York: United Nations.
d~~va~ves of the output with capital indicate e
respect
diXrect accelera
a
I5 blast furnaces were inciuded i-1 the survey:
317 Some plants had more th,an one identical {i.e. same site and vintage) furnace. Since they had the same input figures, we included data for only one furnace, for each group of identical furnaces. This was done to avoid bias in statistical estimation. Thus, only 10 furnaces were taken into account in this exercise. Our data are annual and span the period 1960- 1975, but not for all furnaces. Some of them were commissioned in the 1970s. Special features of the data are noted beloxr. (i) Capital input. In most cases direct observations wem available for only the latter part of the sample period, usually between 1970 and 1975. Earlier data were calculated
from indirect information published in Forsakringsbolagens Forvarderings Kommittee [4Q. (ii) Depreciation and repair are in 1960 prices and am included in the variable cost. (iii) The capital stock is measured in replacement value in 1960 prices. (iv) Fuel input consists of the sum of coke and oil. (v) Production is the yearly production capacity measured in kilotonnes. Actual production data are corrected for deliberate stops due to e.g. vacation. Both labour and fuel inputs are also similarly corrected. (Received Match tember 7. 1983)
17, 1983: accepted in revised form Sep-