Publications   Empirical StudiesHOCKE, KYN, WAGENER  PRODUCTION FUNCTIONS ESTIMATES FOR SOVIET INDUSTRY

4. Estimations of Production functions 

 4.1 In this section the estimates of several alternative forms of macroeconomic production functions will be reported. The variants of Cobb-Douglas production functions were esti­mated by linear regression from logarithms of the observed data. For each variant we shall compare the results obtained from aggregated time-series (LATS) and from pooled cross-section and time-series (CS-TS) data. The number of observations In the LATS approach was 20 (1951 - 1970). However, during the successive transformation to eliminate serial correlation one or two degrees of freedom may have been lost. For LATS approach we shall report only results ob­tained from the transformed data.  

The number of observations in the CS-TS approach was 200 (10 sectors and 20 years). After the transformations to eliminate autocorrelation, the number of degrees of freedom was reduced by 20. In the CS-TS case we report also the results obtained before the transformation of variables by the straight application of OLS without and with sectoral dummy variables. It should be mentioned here that the autoregressive, coefficients (AC) reported for the CS-TS estimates are the first order autoregressive coefficients calculated simultaneously from all the sectoral time series to indicate the degree of autocorrelation in the whole model. They differ from the second order autoregressive coefficients calculated separately for each sector, which were used for the transformation of variables. D-W stands for the Durbin-Watson statistic, and SER for the standard error of regression.      

4.2       Table 5 shows the estimates of the first and most simple variant of the Cobb-Douglas production function

(21)Y/L =A (K/L) b e e

    Table 5

Variant

b*

R2

DW

AC

SER

1a

LATS

with L

.742

(,064)

.893

2.1

-.05

 

1b

LATS

with L*

.775

(.045)

.948

2.1      

-.03

 

1c

CS-TS

OLS

.285

(.050)

.208

.1

.95

.474

1d

CS-TS

dum.

.752

(.057)

.961

.3

.84

.108

1e

CS-TS

trans.vars.

.704

(.123)

.93

2.1

-.05

.002

  Except for the first step in the CS-TS approach all the estimates of b are similar. The very high value of b* is explainable here by the fact that no technical change was included in the model. It is to be noticed that in the CS-TS approach the use of sectoral dummies improved R2 considerably and the transformation of variables eliminated autocorrelation. The transformation increased the standard error of the estimated coefficient b

Table 6 brings estimates of the model

(22)Y =A KbLgee  

     Table 6

Variant

b*

g*

n*

R2

DW

AC

SER

2a

LATS

with L

.722

(.094)

.304

(.282)

1.026

.976

1.9

-.04

 

2b

LATS

with L*

.766

(.054)

.275

(.167)

l.042

.967

2.1

-.03

 

2c

CS-TS

OLS

.365

(.063)

.774

(.058)

1.139

.584

.1

.96

.465

2d

CS-TS

dum.

.554

(.103)

.848

(.267)

l.402

.984

.3

.84

.095

2e

CS-TS

trans.

    vars.

.602

(.643)

541

(.226)

1.143

.999

1.2

-.17

.002

The LATS approach provides high estimates of the capital elasticity of output and relatively low estimates of the labor elasticity of output. On the other hand the CS-TS approach seems to indicate more normal. relations. LATS indicates almost constant returns to scale while the CS-TS approach shows economies of scale. ­The next model contains the Hicks-neutral technical change and assumes constant returns to scale.

(23)Y/L =A (K/L) b e rt + e  

  Table 7

Variant

b*

r*

R2

DW

AC

SER

3a

LATS

with L

.458

(.181)

.020

(.012)

.900

2.0

-.02

 

3b

LATS

with L*

.628

(.119)

.012

(.009)

.947

2.0

-.02

 

31c

CS-TS

OLS

.193

(.052)

.042

(.009)

.386

.01

.96

.418

3d

CS-TS

dum.

.142

(.195)

,045

(.013)

.978 

.3

.84

.081

3e

CS-TS

trans.

    vars.

.319

(.400)

.033

(.030)

.994

2.2

-.04

.002

  There is a visible discrepancy between LATS and CS-TS approaches. The LATS estimates of b are considerably higher and the estimates of r considerably smaller than the estimates of the respective coefficients obtained by the CS-TS method.

The following model allows for a .time trend and elasticities of scale:

(24)Y =A KbLg e rt + e  

         Table 8

Variant

b*

g*

n*

r*

R2

DW

AC

SER

4a

LATS

with L

.700

(.320)

.430

(.270)

1.120

-.014

(.003)

.990

2.0

-.04

 

4c

CS-TS

OLS

.070

(.087)

.888

(.054)

.958

.055

(.012)

.673

.1

.96

.413

4d

CS-TS

dum.

.135

(.181) 

1.069

(.255)

1.204

.038

(.014)

.989

.3

.84

.077

4e

CS-TS

trans. vars.

.330

(.426)

.636

(.555)

.966

.030

(.036)

.999

2.1

-.07

 .002

The discrepancy between LATS and CS-TS results is again apparent. While the LATS approach estimates a very high b and slightly negative rate of technical change, the CS-TS approach gives relatively low b and a rate of technical change between 3 and 5.5 per cent. No clear conclusions about the returns to scale can be made.

The last two models of the CD-production function were estimated with a linear trend in the rate of technical change. The first variant assumed constant returns to scale

(25)Y/L =A (K/L) b exp( rot +  ˝ r1t2 + e)  

   Table 9

Variant

b*

ro*

r1*

R2

DW

AC

SER

5a

LATS

with L

.521

(.181)

-.033

(.022)

.003

(.00l)

.969

2.0

-.04

 

5b

LATS

with L*

.727

(.133)

-.027

(.030)

.002

(.0015)

.965

2.0

-.04

 

5c

CS-TS

OLS

.183

(.034)

.059

(.003)

-.00l6

(.003l)

.425

.1

.96

.419

5d

CS-TS

dum.

.125

(.182)

.063

(.020)

-.00l6

(.00l3)

.980

.3

.83

.077

5e

CS-TS

trans.vars.

.338

(.430)

.039

(.049)

-.0009

(.0028)

.996

2.1

-.07 

.002

       This variant gives interesting results. Firstly, we see a similar pattern as before in the estimation of b. The LATS estimates are high, while the CS-TS estimates are rather low. Secondly, we see a quite clear although not always statistically significant trend in the rate of technological change. The sign of the trend is different in LATS and CS-TS approaches. In the LATS estimates the initial rate of technological change r0 is negative (about -3 percent) while the rate of technological change is increasing by about . 2 to .3 percent annually so that it would reach + 2 percent at the end of the observed period. The 1951-1970 average would be slightly below zero, as in the preceding model (4a).  

On the contrary the CS-TS approach shows a relatively high initial rate of technological change (4-6 percent) declining by .09 to .16 percent annually. This means that the rate of technical change would be down to approximately 2 percent at the end of the observed period (1970). The average rate of technical change in the whole period would be 3 - 4 per cent.  

Finally the similar model was repeated with relaxed assump­tions about the returns to scale.

(26)Y =A KbLg exp( rot +  ˝ r1t2 + e)  

          Table 10

Variant

b*

g*

n*

ro*

r1*

R2

DW

AC

SER

6c

CS-TS

OLS

.068

(.087)

.788

(.054)

.856

 

.074

(.035)

-0018

(.003l)

.208

.1

.95

.474

6d

CS-TS

dum.

.121

(.179)  

1.060

(.250)

1.180

 

.055

(.021)  

-.00l4

(.00l3)

.990

.3

.83

.074

6e

CS-TS

trans.

    vars.

.335

(.438)  

.630

(.540)

.965

 

.041

(.053)

-.0010

(.0027)

.999

2.1

-.05

.002

          The results shown in the Table l0 are consistent with the previous findings of the CS-TS approach although they are statistically not significant. The capital elasticity of output is rather low, re­turns to scale around 1, the rate of technical change was initially quite high but it has been declining by .1 - .2 per cent annually.  

For a general conclusion of the tables 5 through 10 we may state that  
1. the use of man-hours instead of man-years diminishes the standard errors of the factor elasticities of production. The dummies and transformed variables improve the R
2 statistic in the CS-TS approach but increase the standard errors of the parameters so drastically as to make them statistically insignificant;

2. the LATS approach produced relatively high values of the capital elasticity of production together with low estimates for the time trend, sometimes even negative ones. The CS-TS approach, on the other hand, favors in the trade off between the capital production elasticity and the time trend the latter;

3. where the time trend was allowed to vary, the LATS approach 1ead to a negative initial value and a rising rate of change while the CS-TS approach gave the opposite result. In both cases the value of the rate of change was about 2 per cent at the end of the period (1970). The consequences which these two different sets of results would have for our understanding of Soviet industrial growth during the 50s and the 60s should not be overlooked;

4. it may be interesting to compare the variants 3e, 4e, 5e and 6e.The estimated parameters of these variants are very close to each other and seem quite reasonable, although they all are statistically not significant. The capital elasticity of output is in all four cases between .32 and .34, the average rate of technical change is close to 3 per cent and shows in the last two models a declining trend of about .1 per cent annually.     

4.3       The results of our estimates clearly illustrate the trade-off between the estimates of the rate of technical change r and the estimates of the capital elasticity of output b. Whenever b* was small the rate of technical change r* was relatively high, whenever b* was high the estimated rate of technical change was small. This conforms with our expectations. The “trade-off” curve between b* and r* is clearly visible, in the Diagramm 3:

 

 

It depicts all the relevant pairs (b*,r*) from Tables 7,8,9 and 10. The LATS approach favors high values of b* and low values of r*, the CS-TS approach with the variables not corrected for serial correlation in the errors tends to high values of r* and very low values of b*. The CS-TS approach with the b* values close to .33 and the r* values close to .03, seems to be economically most meaningful. It is hard, if not impossible to say which pair represents the “true values of b* and r*  

4.4 The CES production functions were estimated only from the aggregate time-series using the nonlinear regression (NLATS). Four variants (numbered 7,8,9, and 10) of the function (la) were estimated (see Table 11). The variants 7 and 8 assumed constant returns to scale (n = 1) and the variants 7 and 9 assumed no technical change (r = 0). The most interesting for us are therefore variants 8 and l0. Each variant was estimated in two subvariants using either the number of employees (L) or the number of man-hours worked (L* ).  Unlike the preceding section, where the estimates of the constant A are not  reported, we do report them here. Although A is dependent on the units of measurement and difficult to interpret economically, it can be compared for the different models presented. Instead of parameter w we report directly the estimated value of the elasticity of substitution s = 1/(1+w).

Table 11

Variant

A*

d*

s*

r*

n*

R2

7a

with L

1.07

(.033)

.84

(.061)

.60

(.63)

 

 

.9996

7b

with L*

1.06

(.033)

.84

(.061)

.60

(.63)

 

 

.9995

8a

with L

.95

(.035)

.67

(.16)

.17

(.10)

.043

(.028)

 

.9988

8b

with L*

1.06

(.055)

.87

(.11)

.56

(.43)

.013

(.015)

 

.9971

9a

with L

.08

(.20)

.40

(.29)

.65

(.66)

 

1.53

(.50)

.9976

9b

with L*

1.76

(1.45)

.97

(.17)

.50

(1.25)

 

.89

(.17)

.9972

10a

with L

11.76

(12.51)

.98

(.073)

.11

(.11)

.059

(.010)

.46

(.23)

.9977

10b

with L*

20.87

(12.36)

.99

(.013)

.10

(.13)

.065

(.007)

.46

(.12)

.9977

                      Comparing these results with those of similar estimations (Weitzman, Kellman and Perez) we do not find any important discrepancies for the corresponding models with the only exception of the standard errors. These are much higher in our esti­mates than in those using the Weitzman data. Given the almost perfect correlation of the two sets of data, this is somewhat puzzling.

Table 12 Output Elasticities of Capital and Labor
(Calculated from the Estimated Parameters of the CES Production Function)

 

 7a

 8a

 9a

10a

  Year

hK

 hL

hK

hL

 hK

  hL

 hK

hL 

 1951

 .8353

 .1647

 .6115

 .3885

 .6017

 .9233

 .4461

 .0139

 1955

 .8093

 .1907

 .2992

 .7008

  5500

 .9799

 .3621

 .0979

 1960

 .7683

 .2317

 .0656

 .9344

 .4821

1.0479

 .0720

 .3880

 1965

 .7267

 .2733

 .0137

 .9863

 .4252

1.1018

 .0058

 .4542

 1970

 .6891

 .3109

 .0037

 .9963

 .3816

1.1484

 .0006

 .4594

average

 .7683

 .2317

 .1626

 .8174

 .4899

 1.040l

 .1699

 .2901

 

 

 7b

 8b

 9b

10b

  Year

hK

 hL

hK

hL

 hK

  hL

 hK

hL 

 1951

.8350

.1650

.8650

.1350

.8618

.0282

.3246

.0054

 1955

 .8081

.1919

.8377

 .1623

.8533

.0367

.2756

.0544

 1960

.7473

.2527

.7729

 .2271

.8293

 .0607

.0135

.3165

 1965

.7029

 .2971

 .7235

.2765

.8075

.0825

 .0007

.3293

 1970

.6642

.3358

 .6793

.3207

.7850

.1050

.0001

.3299

average.7540.2454 .7793.2207 .8295.0605 .1168.2132

The results of variant 10 should be left out of consideration. For if it is nearly impossible to separate the influence of two variables this will be all the more true for three if no further information is supplied. Also it was difficult to make this model converge. Regarding the remaining six estimates we find that the elasti­city of substitution is with one exception around .6. The standard deviation, however, is so high that neither the Cobb-Douglas nor the Leontief cases can be excluded. As a consequence of the low s and of the increasing capital labor ratio the capital elasticity of production hK is declining and the labor elasticity of production  hL is increasing in all cases (see Table 12). Like in the LATS approach the average value of  hK is very high in the NLATS approach except for the two cases 8a and 9a where high parameters for the time trend or for returns to scale have been estimated. The trade-off between the rate of change of efficiency and the capital elasti­city of production evidently holds true also in the CES model.

 

 

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