Publications |
1 | This paper^{1 }presents estimates of macroeconomic production functions for Poland, Czechoslovakia, East Germany, Hungary, Rumania and Bulgaria. The estimated production functions were variants of the general type | ||
(1 | Y =A(t)F(K, L)U(e, t) | ||
where | Y, K, L and t are output, capital, labor and time respectively, U is random disturbance, F is either Cobb-Douglas or CES production function and the term A(t) represents impact of the output augmenting technical change. | ||
All the variants with Cobb-Douglas production function were linearized either by logarithmic or by the "rates of growth" transformation. The CES functions were approximated by Kmenta's formula, so that only ordinary least squares method was used for estimation. | |||
One of the purposes of this paper was to compare results obtained by three different approaches, namely (a) the pure cross-section analysis, (b) the pure time series analysis and (c) the combination of cross-section and time series analyses. The sensitivity of the estimated parameters to the variations in the form of production function and to the changes in data was also tested. | |||
Although the main aim of our research was to estimate trends in A(t), it turned out to be necessary to pay also a great attention to the reliable estimation of the shape of the function F(K,L), because both estimates are mutually dependent and both parts of the production function are crucial for understanding the dynamic behavior of the economic system. | |||
3 |
Four particular questions were examined: | ||
(1) What is the rate of technical progress in the Soviet-type economies? (2) Are the factor elasticities in Soviet-type economies different then those in other countries (3) Does the Soviet-type economic system cause significant economies or diseconomies of scale? (4) Is the elasticity of substitution in Soviet-type economies close to one, very low or very high? | |||
Models of Production Functions | |||
In this section several variants of the production function (1.1) will be formulated and assigned labels. These labels will be used later when the estimated parameters of individual models are reported. We shall consider five forms of the term A(t), which differ in assumptions about the rate of technical change | |||
(1) No technical progress (NT P) | |||
(2.0) | A(t) = A_{0} | ||
(2) Constant rate of technical progress (CRTP) | |||
(2.1) | A(t)=A_{0} exp(rt) | ||
(3) Trend in the rate of technical progress (TRTP) | |||
(2.2) | A(t) =A_{0}exp(rt + lt^{2}/2) | ||
(4) Recessions in the rate of technical progress (RRTP) | |||
(2.3) | A(t)= A_{0} exp(rt + mSs_{t}) | ||
where | s_{t} is a dummy variable for recession years. | ||
(5) Recessions and trend in the rates of technical change (RTRTP) | |||
(2.4) | A(t)= A_{0} exp(rt + mSs_{t} + lt^{2}/2) |
Further we shall consider four forms of the term F(K, L) which differ in the assumption about both the elasticity of substitution and the returns to scale. | |||
(1) Cobb-Douglas production function with constant returns to scale (CDCRS) | |||
(2.6) | F(K, L) = K^{b}L^{1-b} | ||
(2) Cobb-Douglas function with nonconstant returns to scale (CDNRS) | |||
(2.7) | F(K,L) = K^{b}L^{g} | ||
(3) CES function with constant returns to scale (CESCRS) | |||
(2.8) | F(K, L) = [dK^{-w} + (1 - d)L^{-w}]^{-1/w} | ||
(4) CES function with nonconstant returnes to scale | |||
(2.9) | F(K, L) = [dK^{-w} + (1 - d)L^{-w}]^{-n/w} | ||
Let us denote the factor elasticities by h_{K}, and h_{L } | |||
h_{K }=_{ }(¶Y/¶K)(K/Y)_{ }h_{L }=^{ }(¶Y/¶L)(L/Y)_{ } | |||
In the Cobb-Douglas case h_{K }=_{ } b and h_{L }= g. In the CES case factor elasticities are generally not constant. They depend on the capital-labor ratio | |||
(2.10) | h_{K }= n{ 1 + [(1 - d)/ d ] (K/L)^{ w }}^{-1} | ||
(2.11) | h_{L }= n{ 1 + [d /(1 - d)] (K/L)^{ -w }}^{-1} | ||
The CES parameter d is dependent on units of measurement, and it is clear from (2.10) that it cannot be compared directly with the Cobb-Douglas b, except if the units of measurement of labor and capital are standardized in such a way that for some specific observation the capital-labor ratio is equal to one. It follows from (2.10) and (2.11) that in such a case | |||
(2.12) | h_{K }=_{ } nb and h_{L }= ng. | ||
| The proper choice of units of measurements can make nd to represent the capital elasticity of output and n(1 - d) the labor elasticity. This can be true, however, only for one particular observation. | ||
Finally, two alternative ways of randomization will be assumed | |||
(1) For estimates from logarithms (LG) | |||
(2.13) | U(e,t)=exp(e) | ||
(2) For estimates from rates of growth (RG) | |||
(2.14) | U(e,t)=exp(et) | ||
The combination of five forms of A(t) with four forms of F(K, L) and two ways of randomization gives 40 possible models, not all of which were actually estimated. The label for the whole model will be obtained by combining labels for its parts. For example the label CDCRS-CRTP-RG will mean Cobb-Douglas function with constant returns to scale and constant rate of technical progress, which is estimated in the rates of growth transformation. | |||
The regression equations for the Cobb-Douglas functions are quite obvious. The variants with constant returns to scale were estimated by regressing the labor productivity Y/L on the capital labor ratio K/L and alternative time variables. | |||
The variants with nonconstant returns to scale (CDNRS) were estimated mostly by regressing output (Y) on quantities of factors (K, L) and only in few cases by regressing labor productivity (Y/L) on capital output ratio (K/L) and labor (L). Both approaches give identical estimates of parameters, but may give different R^{2}. | |||
Kmenta's linear approximation of the CES production function were estimated in the logarithmic form only. The regression equation for the model CESCRSRTRTP-LG was | |||
(2.18) | lgY/L = a + dlgK/L + f(lgK/L)^{2 }+ rt + (1/2)l t^{2} + mS_{1}^{t}s_{t} + e | ||
and for the model CESNRS-RTRTP-LG | |||
(2.19) | lgY/L = a+ndlgK/L+ f(lgK/L)^{2 }+(n -1)lgL+rt+(1/2)lt^{2}+mS_{1}^{t}s_{t}+e | ||
or | |||
(2.20) | lgY/L = a+ndlgK+f(lgK/L)^{2}+(n-1)lgL+rt+(1/2)lt^{2}+mS^{ }_{t}^{t} s_{t}+ e | ||
It is known that (2.18) - (2.20) are approximations, which correspond to some other forms of the production function as well. Kmenta's formula^{2 }gives good approximation of the CES production function only in the close neighborhood of the Cobb-Douglas function. Then it follows that | |||
(2.21) | f = - (1/2) wnd(1-d) | ||
Having estimated d, v and f from (2.18), (2.19) or (2.20) we can calculate w according to | |||
(2.22) | w = - 2 f/[ nd(1-d) ] | ||
The elasticity of substitution can be then obtained as s = 1/(1 + w). The problem, however, is that w obtained in this way depends on units of measurement, while the true w of the CES function does not. The effect of the arbitrary choice of units of measurement can be demonstrated quite easily.^{3} | |||
There are two ways, how the problem of units of measurement can be at least partially overcome: | |||
(i) | Assuming 0 < d < 1 it follows, that whatever the units of measurement are d (1 - d) £ .25. With regard to (2.22) it must therefore hold | ||
(2.23) | w = -b(f/n) b ³ 8 | ||
This relation can help in estimating the minimum deviation of the elasticity of substitution from 1 | |||
(ii) The second way rests on the "standardization" of units of measurement by choosing k_{K} and k_{L} in such a way that K*/L* = 1 for certain selected observation. This does not really eliminate the problem, but it provides at least certain common ground for comparison of w's obtained from different regressions. |
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