Publications   Empirical StudiesOldrich & Ludmila Kyn: Production Functions for Eastern Europe



Earlier, we have described 40 models of production functions, which were estimated by three different approaches for six East European countries and sometimes even with alternative data specifications. This required to run thousands of regressions only fraction of which is reported in the Appendix.


Pure cross-section analysis

Cross-sectional estimates of the macroeconomic production functions were obtained for Czechoslovakia, Poland, Hungary, Bulgaria and Rumania. The estimates of capital elasticities of output b for Czechoslovakia, Poland and Hungary are very similar - around .25 - .3 - and quite close to the theoretically expected values. Almost all of these estimates are significantly different from 0 on the 2 per cent level of significance. The estimates of b for Bulgaria and Rumania are considerably smaller (.05-.15) and they are not significantly different from 0. It is not clear whether this difference in estimated parameters reflects true differences in capital elasticities of output or simply the fact that our capital data for Bulgaria and Rumania contained relatively much larger errors of measurement than the capital data for the other countries.


The estimated b fluctuated slightly in short run. In the long run we can see slightly increasing trend in Hungary and Bulgaria but no visible trend in Czechoslovakia, Poland and Rumania. Generally the assumption of constant b seems quite reasonable. It implies that we may represent estimates of b for individual years by their overall arithmetic mean.

Except for Rumania only small and statistically insignificant deviations from constant returns to scale were found. The assumption of constant returns to scale is apparently quite reasonable for these countries.

Table 1:  Mean Values of Factor elasticities
(obtained from cross-sectional estimates)


 Model CDCRS  








Czechoslovak economy  






Czechoslovak industry






Polish industry






Hungarian industry 






Bulgarian economy 






Rumanian economy







The estimated constants at = lgA(t) show in all cases the clear upward trend. The cross-section estimates will be, therefore, compatible with the time series estimates containing technical progress. Assuming (2.1) it is possible to get rough estimates of the rates of technical progress (total factor productivity) by regressing the time series of the constants at obtained from cross-sectional estimates on the time variable t.

Table 2: Average rates of technical change
(obtained from constants of cross-section regressions)


 Model CDCRS



stand. error 


Czechoslovak economy




Czechoslovak industry




Polish industry




Hungarian economy




Bulgarian economy




Rumanian industry




Table 2 shows very high rates of technical change in postwar Eastern Europe. With the exception of extremely high rates for Bulgaria these results roughly correspond to the estimates obtained in both pure time series and combination of time series and cross-section approaches.

Click bellow to display the tables:

Appendix Table 1 

Appendix Table 2 

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The cross-sectional estimates of the Kmenta's linear approximation of the CES production function for the Czechoslovak economy and industry are shown in table 3 in the Appendix. 

Click bellow to display the tables:

Table 3 Czechoslovakia (cross section CES)

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The coefficient d* was calculated from the estimates of d by standardizing the units of measurement so that the capital-output ratio in machine building (sector 4) would be equal to 1 in all years. The coefficient d* is thus slightly smaller - by less than f/2  - than the capital elasticity corresponding to the average capital-labor ratio for the whole economy. The comparison of d*  with estimates of b in the CD approach show that the assumption of the nonunitary elasticity of substitution does not make any difference for the estimated capital elasticity of output. Unlike the CD approach we see here quite clear trends in d*: increasing for the whole economy and declining for industry.

The addition of the CES term to the regression equation made almost no difference for the estimated returns to scale. The estimated elasticity of substitution was generally very high, nevertheless not significantly different from 1.

Pure Time-Series Analysis

 The pure time series approach was used to estimate sectoral production functions for Czechoslovakia, Poland and Bulgaria. Very many models were actually run for each country, however, space allows only for summarizing the results in the table 3abc.

Click bellow to display the tables:

Table 3a Czechoslovakia

Table 3b Poland

Table 3c Bulgaria

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Appendix Table 4 Czechoslovakia

Appendix Table 5 Czechoslovakia

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The most obvious conclusion from surveying all time series regressions was that very few of them gave statistically good and economically meaningful results. Either R2 were very low, or the parameters were not significant or they did not have meaningful values. Parameters were also extremely sensitive to the form of model and type of data used. The addition of new variable, quite frequently caused a drastic change in the estimates of coefficients at other variables. 

It is, however, interesting to note, that the whole set of all time series estimates gives much more reasonable picture than individual estimates alone. We can draw from them several interesting conclusion:

(1) The mean values of estimated coefficients calculated from sectoral coefficients8 of each model (Table 3) show surprising similarity and with only few exceptions are economically meaningful and consistent with results of cross-sectional analysis. The coefficients in Tables 1, 2 and 3 do not disprove the hypothesis that the cross-sectional estimates are mean values of sectoral parameters.

The mutual compatibility of the cross-sectional and the time series estimates is obvious at least for Czechoslovakia and production function with constant returns to scale.



The time series estimates of models with CRS for Poland give the mean value of b somewhat larger than cross-sectional estimates but the estimate of the average rate of technical change is almost exactly the same as the rate implied by the cross-sectional estimates. The results of models with NRS for Poland are bad partly because 13 observations do not provide enough degrees of free­dom for reliable estimation of 5 parameters.

The mean values of b obtained for Bulgaria were - with two exceptions - between .25 and .35. This would indicate a normal value of capital elasticity rather than low values suggested by the cross-sectional approach. The average rate of technical change is on the other hand somewhat lower - around 6 per cent instead of almost 8 per cent implied by the cross-sectional estimates.

(2) The use of GNP instead of GVO as a left hand variable does not give radi­cally different estimates of capital and labor elasticities of output, however, it gives lower - although not much lower - estimates of the rate of technical change. The rates of technical change are respectable even if measured in terms of Western estimates of GNP.


(3) The estimates of the models with TRTP clearly indicate the existence of declining trends in the rate of technical change in all three countries. Not only the mean values of l as shown in Table 3 but also the individual values of those parameters for almost all sectors are negative. 

The magnitude of the declining trend varied, but most frequently it was between .1 and .5 per cent annually which implies that the rate of technical change in 20 post war years may have slowed down by approximately 2 to 5 per cent. It should be noted, that the declining trend is much less apparent if the output is measured by GNP.

(4) Models with the "recession parameter" m estimated for Czechoslovakia, show very clearly the cyclical fluctuations in the rates of technical change. On the basis of preliminary information the years 1953-1954, and 1962-1965 were selected as recession years. The estimates of models with dummies for recession years show that practically in all sectors the rate of technical change was in recession years lower than in the nonrecession years by about 4 per cent.


(5) Values of estimated parameters obtained from rates of growth are approxi­mately the same as those obtained from logarithms of observed data.

 In RG models the R2 s, were much smaller but the Durbin-Watson statistics were better and standard errors of estimated parameters were almost the same as in the LG models. Models with variable rate of technical change give generally better results than those with constant  r. On the other hand the results were more economically meaningful when the returns to scale were constrained to unity.


Combination of Cross-Section and Time-Series Analyses

The combination approach was applied in two steps:


 In the first step the ordinary least squares method was applied to the pooled cross-section and time-series data


In the second step the regressions were repeated with sectoral dummy variables.


Several common features are visible in the estimates from the pooled data9  

Appendix Table6 

Appendix Table 7 

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(1) The first step estimates (without sectoral dummy variables) may give low R2's, very low Durbin -Watson statistics10 but they give usually highly significant estimates of the regression coefficients. The introduction of sectoral dummy variables increased considerably R2's and usually also diminished standard errors of estimated coefficients, but the serial correlation has not been removed. The check on sectoral sums of squared residuals indicated also presence of heteroscedasticity.

(2)The most striking feature is the relative stability of estimates from the pooled cross-section and time-series data. Particularly it is evident, that the values of estimated coefficients are not sensitive to the form of the technical change term A(t).

(3)The third interesting feature is the fact that almost all the estimated coefficients have economically meaningful values.  Let us take for example the estimates of b. Out of 95 separate estimates of  b, only two were negative, two larger than 1 and 85 were in the interval from .1 to .5. (57 of them in the narrower range .15-. 3). Similarly we can see that the average rate of technical change is almost never unrealistically high or low, the occurrence quite frequent in the pure time series analysis. Out of 45 separate estimates of r  no one was negative, only two were smaller than 1 per cent, one was larger than 8 per cent but 42 of them were in the interval from 3 to 7 per cent.

The estimates of production functions obtained from the pooled cross-section and time-series data seem to confirm most of the conclusions which were made previously on the basis of pure time-series and cross-section estimates.


The capital elasticity of output in all six East European countries is most likely to be somewhere between .15 and .3. Even though some models gave different estimates of   b, there is no unambiguous evidence that the capital elasticity of any East European country would be out of this range.

The estimates of models where the returns to scale parameter n was not constrained to unity give very ambiguous results. Although the deviation of n from unity was in many cases statistically significant it was frequently not very large. Only some models with sectoral dummy variables resulted in more substantial deviations from constant returns to scale. But taking all the estimations together, no clear evidence for either increasing or decreasing returns to scale can be found for any country.

The estimates of the Kmenta's approximation of the CES production function give also ambiguous results. The estimated f for Poland was not significant and in some cases positive and in other negative. The estimates of f for Czechoslovakia were significant but also alternating signs. Estimates for Bulgaria indicate the elasticity of substitution to be significantly larger than one and those for East Germany significantly smaller than one. Our estimates therefore do not confirm the hypothesis that the Soviet-type economic system creates a consistent deviation of the elasticity of substitution from unity.

There are noticeable similarities in the estimated rates of technical change in the East European countries. Almost all the estimates of the average rate of technical change r for Czechoslovakia (GVO), Bulgaria and Romania are between 5 and 6 per cent. East Germany gives somewhat broader range 3 to 7 per cent. The average rate of technical change for Poland and Hungary was estimated in the range 3 to 5 per cent. The estimates of r from GNP data for Czechoslovakia gave - as expected - a slower rate of technical change, only around 3 per cent. This difference is caused partly by the fact that GNP does not grow as fast as GVO and partly by the very high capital elasticity b in the GNP estimations. It is also likely that some portion of the discrepancy between GVO and GNP estimates of r may have been due to the "inflation of constant prices" as was suggested earlier.

The models with variable rate of technical change gave radically different results for two groups of countries. The first group, which consists of Czechoslovakia (GVO), East Germany, Poland and Bulgaria, exhibits a clear and sizable declining trend of the rates of technical change. With few exceptions the estimates of the parameter l indicated a .2 to .4 per cent annual decline of the rate of technical change in the mentioned countries. Such a trend was not found in Hungary and Romania. The estimates of l on the basis of GNP data are more ambiguous. Sometimes they show the declining trend, but of much smaller size than the estimates from GVO data suggest.

Finally the estimations of the production functions with the 'recession parameter" m confirm our previous finding, i.e. the sizable and statistically significant fluctuations of the rate of technical change in Czechoslovakia.



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