Publications Empirical Studies HOCKE, KYN, WAGENER  PRODUCTION FUNCTIONS ESTIMATES FOR SOVIET INDUSTRY

1.   Introduction

1.1 The main aim of this study is to estimate production functions for Soviet industry on the basis of the performance in the years 1951—1970. For this purpose different specifications and estimation procedures will be tried. First experiments indicated that the results diverge on a wide range. Therefore, we think it necessary to show possible implications of this fact. To evaluate the effects of different parameter estimates in production functions the growth potential of Soviet industry according to a very simple neo-classical model will be used as a paradigm. This does not necessarily imply that we think such a model is representative for a Soviet-type economy. However, the growth potential can very well serve to demonstrate.

 (1)                         Y(t) = F(K,L,t) (2)                          I(t) = aY(t) (3)                     dK/dt  =  I(t) (4)                     dL/dt  =  lL(t)

Here Y, K, L, I  represent output. capital stock, labor and investment respectively
a is the share of investment in total output and
l
is the rate of growth of labor force.

The growth path Y(t) apparently depends on  a, l and the explicit form of pro­duction function F(K, L, t). The main problem in accomplishing our task is to obtain good estimates of the parameters of F(K, L, t). It is known that estimation of macroeconomic production functions for Soviet-type economies creates specific problems, because the absence of market mechanisms eliminates the possibility of using the so called marginal conditions. According to the established practice we shall, therefore, estimate the production functions directly, by regressing output on the quantities of capital and labor. To simplify our task, we shall estimate only production functions with constant elasticity of substitution and with Hicks neutral technical change, that is the CES production function:

 (1a) Y = A(t) [dK-w + (1 - d)L-w ] -n/w

or its special form (for w = 0), the Cobb-Douglas production function

 (1b) Y = A(t)KbLg

Some alternative assumptions about the form of A (t) will be used.

1.2 Defining a so called equilibrium rate of growth  ge as the rate of growth which maintains the capital-output ratio constant. Keeping a constant, we know that the actual rates of growth will asymptotically converge to  ge. When we call  ge the growth potential of the economy this has to be understood in a paradigmatic way. For it is perfectly clear that in the context of Soviet industry a cannot be taken constant ex ante. However, we are not interested in demonstrating how the deliberate choice of a by the policymaker influences growth,1) but rather in assessing the effects of different production function parameter estimates.

In the model (1) to (4) with the production function

 (1c) Y(t) = A(t) F(K,L,t)

the equilibrium rate of growth ge is given by

 (5) ge = (hLl + r)/(1 - hK)

where    hL  = (Y / L )( L /Y  ) , hK  = (Y / K )( K /Y )   are respectively the labor elasticity and the  capital elasticity of output and r = dA/dt (1/A) is the rate of Hicks-neutral technical change.

1.3 In the case of a unitary elasticity of substitution (i.e. w = 0), and a constant rate of technological change r, the equilibrium rate of growth ge is constant. It follows from (1 b) and (5) that

 (5a) ge = (g l + r)/(1 - b)

which for constant returns to scale  (b + g = 1) can be simplified to

 (5b) ge = l + r/(1 - b)

The rate of technological change is very important for determining the long-run growth potential, however, the other parameters of the production function namely the elasticities b and g are important as well. This is very rarely acknowledged. Many economists tend to judge the growth potential of the economy only according to the ‘dynamic efficiency” which is frequently identified with the rate of technological change. The importance of the parameter b and g for the evaluation of the long run growth potential can be easily illustrated on the following example:  Suppose that we are comparing two economies A and B characterized by the Cobb-Douglas production function with the parameters:

A:                   bA  =  .2       gA  =  .9   rA  =  .045

B:                    bB  =  .7       gB  =  .1   rB  =  .025

Let us further assume that the rate of growth of labor force is for both countries the same

lA  =  lB  =  .01.

Using the formula (5a) we find the growth potential of the economy B is greater than the growth potential of the economy A

geB  =   .0866   >   .0675   =   geB

although both the rate of technological change and the degree of economies of scale were greater in the economy A.

1.4.      The case in which the equilibrium rate of growth is not constant, is even more relevant for Soviet-type economies. Changes in ge may result from two different causes:

(a)        The rate of technological progress may vary

 (6) r = r (t).

If we still assume that the elasticity of substitution is equal to 1 than the equilibrium rate of growth will be

 (7a) ge(t) = [gl +  r(t)]/(1 – b)

ge(t) would be declining if r(t)is declining and it would be increasing in the opposite case. In empirical studies it is usual to approximate  r(t) by the linear function:

 (6a) r(t) = r0  +  r1 t.

This approximation will be used also in the present paper. We should stress, however, that if the “true” function  r(t) is nonlinear then the linear approximation (6a) may not be suitable for the long run forecast of economic growth, even if it explains the rates of growth in the observed period quite well. Obviously the discrepancy between the “true” function r(t) and its linear approximation (6a) can become enormous at very high t’ s. Suppose, for example, that the. “true” function  r(t) is a monotonous decreasing function which asymptotically converges to a value r* >  0. If  the function  r(t) (6a) is estimated from, let us say, 20 observations, then the estimated value of the parameter r0 will be positive end estimated value of r1 will be negative. The linear approximation of r(t) extrapolated beyond the year  t =  - r 0  / r 1   would be negative and therefore inconsistent with the above assumption.

(b) The equilibrium rate of  growth ge will not be constant if the elasticity of substitution is not equal to 1 (except for the very special case v = 1 and r =  1 - hk     Suppose a production function (la) with w ¹  0 and  r =  (dA/dt)(1/A) > 0. In this case both output and capital stock would grow faster than labor if the economy is on the equilibrium path of growth. With w ¹  0 the capital elasticity of output is a function of the capital-labor ratio

 (8) hK = n [1 + (1 – d)/d (K/L)w ] –1/w

In the case of low elasticity of substitution (i.e. if  w > 0)  hK   is declining along the equilibrium path. If on the contrary the elasticity of substitution is high (w< 0) the capital elasti­city of output will increase. The labor elasticity of output hL   will change in the opposite direction. Assuming constant l and r the equilibrium rate of growth (5) must be declining if the elasticity of substitution is less than 1 and it must be increasing if the elasticity of substitution is greater than 1.

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