Output, capital, and labor in the short and long run.

• Author: Levy, Daniel

1. Introduction

   In traditional growth accounting calculations that originated from Solow's |32; 33~ seminal work, macroeconomists usually assign output elasticities of 0.30 and 0.70 to capital and labor inputs respectively. These values are based on the assumption that producers are operating in a competitive, profit maximizing, constant returns to scale environment in which factors of production are paid their marginal product. Brown |6~, Douglas |12~, and Intriligator |20~ have provided empirical support for these assumptions for the pre-war and inter-war periods. Their estimated capital and labor elasticities of the output were around 0.25 and 0.75 respectively.

   However, a recent study by Paul Romer |30~ concludes that the contribution of capital accumulation to long-run growth is substantially underestimated in the conventional growth accounting analysis and that the true capital elasticity of output may actually be greater than its share in total income because of positive externalities associated with investment. On the other hand, Romer suggests that the contribution of labor is considerably overestimated and that the true labor elasticity may actually be smaller than its share in income because of negative externalities associated with labor. In particular, according to Romer's estimation, the long-run capital and labor elasticities of output probably lie in the range of 0.7-1.0 and 0.1-0.3 respectively. But his estimates come from historical data of output, capital, and labor averaged over 10- and 20-year intervals for the periods 1890-1980 and 1839-1979, respectively. Therefore, as Romer himself suggests, the signal-to-noise ratio may be too small for a sensible interpretation of these figures, since the above time series contain only 7-9 observations. In addition, a long-run averaging of the data may not have completely eliminated the effect of business cycle fluctuations.

   Bernanke |3, 204~ expresses doubts about the correctness of Romer's estimates since "it cannot literally be true that output is independent of labor input, |and therefore~ this result must be caused by an estimation bias."

   The importance of capital accumulation in the growth of the U.S. economy is emphasized in other studies that analyze the sources of long-term growth in the U.S. economy |4; 5; 8; 10; 22; 23~. For example, Jorgenson |23, 25~ argues that "comparing the contribution of capital input with other sources of output growth for the period 1948-1979 as a whole makes clear that capital input is the most significant source of growth." Denison |11, 220~ makes a similar argument: "I do not share the other extreme view, sometimes encountered, that capital can be ignored because its significance is hard to establish if one fits a production function by correlation analysis. I stress again: capital is an important growth source. It has sometimes contributed importantly to differences in growth rates between periods and places. More capital formation would raise the growth rate."(1)

   From a theoretical point of view, this argument is not really new. The usual assumption used in standard microeconomic models that the stock of capital is fixed in the short run but variable in the long run implies that variations in the stock of capital will affect the output in the long run.(2) More importantly, dynamic models that involve some kind of transaction costs usually make similar predictions.(3)

   The purpose of this paper is to provide new empirical evidence on the relative importance of capital and labor in the determination of output in the short and long run. Unlike the studies cited above, the methodology applied here uses frequency domain analysis. The advantage of using the frequency domain framework is that it allows us to conduct the analysis on a frequency-by-frequency basis for describing empirical cyclical regularities in the data and examining the dynamic relationship between time series without the intervention of an econometric model. The frequency domain methodology used here is nonparametric and therefore requires no behavioral or distributional assumption about the time series of output, capital, and labor. The only requirement is that the series analysed be stationary. The quarterly time series of capital stock used here was constructed recently and thus differs from the data used by Romer |30~ and others. Despite these differences, the findings reported in this paper indicate that capital indeed is a far more important factor than labor in the determination of output at the zero frequency band. Furthermore, I show that the zero-frequency labor elasticity of output may well be close to zero, or even zero. An additional finding of this paper is related to the accelerator model of investment: it turns out that output leads capital at the zero frequency band which suggests that the traditional accelerator model may be a good description of the long-run investment process.

   Statistical evidence supporting these ideas are derived below by examining the capital-output and the labor-output relationships using cross-spectral analysis. Spectral analysis provides a useful framework for studying the issues raised here because in the frequency domain short-run and long-run relationships between time series can be characterized and analyzed by looking at the behavior of the series in the high and low frequencies, respectively. The main disadvantage of ordinary time domain regression analysis in the context discussed here is the fact that it implicitly treats all frequencies equally. In addition, as Chow |9~, Harvey |17~, and many others argue, although the information gained from frequency domain analysis is theoretically a transformation of its time domain analog, some dynamic and cyclical features of the data are easier to identify and interpret in the frequency domain.

   The paper is organized as follows. In the next section, I briefly review the statistical methodology used in this study. In section III, I describe the data set. Next, in section IV, I present and discuss the empirical results of the study. In section V, I discuss the implications of the findings for growth accounting in the context of U.S. business cycles. The paper ends with a brief summary of the main results and some concluding remarks.

2. The Methodology

   Spectral analysis makes it possible to conduct time series analysis in the frequency domain, where we think of a stationary series as being made up of sine and cosine waves of different frequencies and amplitudes. In a univariate case, we are interested in determining how much of the total variance ("power") of the series is determined by each frequency component. In a bivariate setup, spectral analysis provides a description of a linear relationship between time series at different frequencies.(4)

   For a covariance stationary univariate process |y.sub.t~, the autocovariance function is given by the expression

   |Gamma~(s) = E|(|y.sub.t+s~ - |Mu~) (|y.sub.t~ - |Mu~)~ (1)

   where |Mu~ is the mean of the process. It is usually assumed that both |Gamma~(s) and |Mu~ are time independent which is essential for past observations to be useful in describing the present or the future. It follows that |Gamma~(s) = |Gamma~(-s). The spectrum of the series |y.sub.t~ is defined as the Fourier transform of its autocovariance function, and is given by

   |f.sub.y~|Omega~ = (1/2|Pi~) |Summation of~|Gamma~(s)|e.sup.-isw~ where s is = - |infinity~ to |infinity~ 0 |is less than or equal to~ |Omega~ |is less than or equal to~ |Pi~ (2)

   where |Omega~ is the frequency and is measured in cycles per period (in radians).

   For a bivariate covariance stationary process (|y.sub.t~, |x.sub.t~), the cross covariance function given by

   ||Gamma~.sub.yx~ (s) = E|(|y.sub.t+s~ - ||Mu~.sub.y~)(|x.sub.t~ - ||Mu~.sub.x~)~, (3)

   measures the degree of linear association between the two stochastic processes for different time lags and is independent of time. The cross spectrum is the Fourier transform of the cross covariance function and is given by

   |f.sub.yx~(Omega) = (1/2|Pi~) |Summation of~ ||Gamma~sub.yx~(s)|e.sup.-isw~ where s = -|infinity~ to |infinity~ (4)

   which is a complex-valued function of |Omega~. Since the cross spectrum as given above cannot be examined directly, the usual practice is to compute and plot "squared coherence," "phase," and "gain." The squared coherence which is given by

   |C.sub.yx~(|Omega~) = |parallel to~|f.sub.yx~|(|Omega~).sup.2~/||f.sub.y~(|Omega~)|f.sub.x~(|Omega~)~ 0 |is less than or equal to~ |C.sub.yx~(|Omega~) |is less than or equal to~ 1 (5)

   is analogous to the square of the correlation coefficient between the series at each frequency. That is, it represents the degree to which one time series can be represented as a linear function of the other. The higher the |C.sub.yx~(|Omega~), the more closely related are the two series at frequency |Omega~. By its construction the coherence says nothing about the sign of the relation between the two series, nor anything about the timing of any lead/lag in the...