Publications 
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 CobbDouglas production functions were estimated by linear regression from logarithms of the observed data. For each variant we shall compare the results obtained from aggregated timeseries (LATS) and from pooled crosssection and timeseries (CSTS) 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 CSTS 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 CSTS 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 CSTS 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. DW stands for the DurbinWatson 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 CobbDouglas production function
(21)  Y/L  =  A (K/L) ^{b} e ^{e} 
Table 5
Variant  b*  R^{2}  DW  AC  SER  
1a  LATS  with L  .742 (,064)  .893  2.1  .05  
1b  LATS  with L*  .775 (.045)  .948  2.1  .03  
1c  CSTS  OLS  .285 (.050)  .208  .1  .95  .474 
1d  CSTS  dum.  .752 (.057)  .961  .3  .84  .108 
1e  CSTS  trans.vars.  .704 (.123)  .93  2.1  .05  .002 
Table 6 brings estimates of the model
(22)  Y  =  A K^{b}L^{g}e^{e} 
Table 6
Variant  b*  g*  n*  R^{2}  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  CSTS  OLS  .365 (.063)  .774 (.058)  1.139  .584  .1  .96  .465 
2d  CSTS  dum.  .554 (.103)  .848 (.267)  l.402  .984  .3  .84  .095 
2e  CSTS  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 CSTS approach seems to indicate more normal. relations. LATS indicates almost constant returns to scale while the CSTS approach shows economies of scale. The next model contains the Hicksneutral technical change and assumes constant returns to scale.
(23)  Y/L  =  A (K/L) ^{b} e ^{rt +} ^{e } 
Table 7
Variant  b*  r*  R^{2}  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  CSTS  OLS  .193 (.052)  .042 (.009)  .386  .01  .96  .418 
3d  CSTS  dum.  .142 (.195)  ,045 (.013)  .978  .3  .84  .081 
3e  CSTS  trans. vars.  .319 (.400)  .033 (.030)  .994  2.2  .04  .002 
The following model allows for a .time trend and elasticities of scale:
(24)  Y  =  A K^{b}L^{g} e ^{rt +} ^{e } 
Table 8
Variant  b*  g*  n*  r*  R^{2}  DW  AC  SER  
4a  LATS  with L  .700 (.320)  .430 (.270)  1.120  .014 (.003)  .990  2.0  .04  
4c  CSTS  OLS  .070 (.087)  .888 (.054)  .958  .055 (.012)  .673  .1  .96  .413 
4d  CSTS  dum.  .135 (.181)  1.069 (.255)  1.204  .038 (.014)  .989  .3  .84  .077 
4e  CSTS  trans.  .330 (.426)  .636 (.555)  .966  .030 (.036)  .999  2.1  .07  .002 
The discrepancy between LATS and CSTS results is again apparent. While the LATS approach estimates a very high b and slightly negative rate of technical change, the CSTS 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 CDproduction 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( r_{o}t + ˝ r_{1}t^{2} + e)^{ } 
Table 9
Variant  b*  r_{o}*  r_{1}*  R^{2}  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  CSTS  OLS  .183 (.034)  .059 (.003)  .00l6 (.003l)  .425  .1  .96  .419 
5d  CSTS  dum.  .125 (.182)  .063 (.020)  .00l6 (.00l3)  .980  .3  .83  .077 
5e  CSTS  trans.  .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 CSTS 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 CSTS approaches. In the LATS estimates the initial rate of technological change r_{0} 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 19511970 average would be slightly below zero, as in the preceding model (4a).
On the contrary the CSTS approach shows a relatively high initial rate of technological change (46 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 K^{b}L^{g} exp( r_{o}t + ˝ r_{1}t^{2} + e)^{ } 
Table 10
Variant  b*  g*  n*  r_{o}*  r_{1}*  R^{2}  DW  AC  SER  
6c  CSTS  OLS  .068 (.087)  .788 (.054)  .856  .074 (.035)  0018 (.003l)  .208  .1  .95  .474 
6d  CSTS  dum.  .121 (.179)  1.060 (.250)  1.180  .055 (.021)  .00l4 (.00l3)  .990  .3  .83  .074 
6e  CSTS  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 CSTS 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 manhours instead of manyears diminishes the standard errors of the factor elasticities of production. The dummies and transformed variables improve the R
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 CSTS 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 CSTS 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 tradeoff 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 “tradeoff” 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 CSTS 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 CSTS 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 timeseries 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 manhours 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*  R^{2}  
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
 7a  8a  9a  10a  
Year  h_{K}  h_{L}  h_{K}  h_{L}  h_{K}  _{ }h_{L}  _{ }h_{K}  h_{L} 
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  h_{K}  h_{L}  h_{K}  h_{L}  h_{K}  _{ }h_{L}  _{ }h_{K}  h_{L} 
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.



