Estimating factor shares from nonstationary panel data

Abstract

The measurement of the sources of economic growth, essential for understanding the long-term perspective of any economy, depends on both the functional form that summarizes technology and factor share values. We estimate the share of physical capital in output, implied by a Cobb–Douglas production function, using fully modified and dynamic ordinary least squares estimators within a panel cointegration framework for 109 countries over the period 1960–2014. For several measures of labor, our estimates range between 0.46 and 0.56 for the worldwide set and vary significantly across regions. The panel estimates are: (i) similar when human capital is taken as a separate factor of production; and (ii) more robust than both the average of the individual cross-country estimates and those obtained in the model in differences. We also find that the CES production function estimates favor the Cobb–Douglas specification. Finally, we provide some evidence of a decline in the share of labor, in line with recent empirical studies.

Appendices

A: Regional groups

(see Table 8)

Table 8 Regional groups

B: Human capital as a separate factor of production

In our setup, human capital is labor augmenting, and so these two factors are “perfect substitutes” in production (i.e., human capital is only a measure of labor quality). Another approach including human capital (H) in a production function considers this variable as a separate factor of production as in Mankiw et al. (1992). For this case, Eq. (3) can instead be written as

$$\begin{aligned} Y_{it}=K_{it}^\alpha H_{it}^\beta L_{it}^{1-\alpha -\beta } \exp (a_i+\rho _i t+u_{it}), \end{aligned}$$

(6)

where $\alpha$ and $\beta$ denote the shares of physical and human capital in output, respectively. Once again, assuming constant returns to scale and perfect competition in factor markets, (6) can be written as

$$\begin{aligned} \frac{Y_{it}}{L_{it}}=\left( \frac{K_{it}}{L_{it}}\right) ^\alpha \left( \frac{H_{it}}{L_{it}}\right) ^\beta \exp (a_i+\rho _i t+u_{it}), \end{aligned}$$

(7)

or, equivalently,

$$\begin{aligned} \log (Y_{it}/L_{it})=a_i+\rho _i t+\alpha \log (K_{it}/L_{it})+\beta \log (H_{it}/L_{it})+u_{it}. \end{aligned}$$

(8)

Table 9 reports the estimation results for the worldwide and regional sets of countries, for the FMOLS and DOLS estimation methods and for population and employment used labor measures. The results indicate that the share of physical capital ranges from 0.35 to 0.66. For the worldwide sample, the share of physical capital ranges from 0.48 to 0.52. These results are similar to the estimates in Table 4. In most of the cases, the share of human capital estimates for the regional sets is not significant, whereas for the worldwide sample those estimates are significant and range from 0.05 to 0.14.

Table 9 Cobb–Douglas production function with human capital as a factor of production

C: The CES production function model

As is well known, the Cobb–Douglas production function constitutes a particular case of the CES production function. For this more general specification, the implied factor shares are not constant but depend on the factor/output ratio. In order to test for local misspecification, we consider a CES production function with two inputs given by

$$\begin{aligned} Y_{it}=\{\theta K_{it}^{-\delta }+(1-\theta ) L_{it}^{-\delta }\}^{-\frac{1}{\delta }} A_{it}, \end{aligned}$$

(9)

where $\theta$ and $1-\theta$ determine the optimal distribution of inputs, and $\delta$ determines the (constant) elasticity of substitution (which equals $\sigma =(1+\delta )^{-1}$). Under the assumption of competitive markets, the share of capital is given by $\theta (K/Y)^{-\delta }$ and the share of labor is given by $(1-\theta )(L/Y)^{-\delta }$. Kmenta (1967) proposed an approximation to the classical two-input CES production function

$$\begin{aligned} \log Y_{it}=\theta \log K_{it} +(1-\theta ) \log L_{it} -\frac{\delta }{2}\theta (1-\theta )(\log K_{it}-\log L_{it})^2+ \log A_{it}, \end{aligned}$$

(10)

by taking natural logs to both sides of (9) and computing a second-order Taylor expansion to $\log \{\theta K_{it}^{-\delta } + (1-\theta ) L_{it}^{-\delta }\}$ around $\delta = 0$. Alternatively, the same expression can be obtained by computing a first-order Taylor expansion to the transformed CES function around $\delta = 0$ (Henningsen and Henningsen 2011). Substituting (2) into (10) and defining $\beta =-\frac{\delta }{2}\theta (1-\theta )$ leads to

$$\begin{aligned} \log Y_{it}=a_i +\rho _i t+\theta \log K_{it} +(1-\theta ) \log L_{it} +\beta (\log K_{it}-\log L_{it})^2+u_{it}, \end{aligned}$$

(11)

or, equivalently,

$$\begin{aligned} \log (Y_{it}/L_{it})=a_i +\rho _i t+\theta \log (K_{it}/L_{it}) + \beta \{\log (K_{it}/L_{it})\}^2 + u_{it}. \end{aligned}$$

(12)

Therefore, by estimating (12) and testing $H_0:\beta =0$ we can evaluate whether the Cobb–Douglas function constitutes a valid description of the production process. Table 10 reports estimates of $\beta$ for the worldwide sample, which indicates that $\{\log (K_{it}/L_{it})\}^2$ is not statistically significant and favors the Cobb–Douglas specification. It should be noted that when we consider the worldwide sample, the estimates are also quite robust since there are enough observations both across countries and years.

Among regional estimates considering different methods and measures of labor, in general, estimates for the industrial and East Asia, South Asia and Pacific regions indicate that we cannot reject the null of $\beta =0$ in favor of the Cobb–Douglas specification; in the other three regions, most of the estimates indicate that the null is rejected; nonetheless, most of the coefficient estimates are close to zero.

Table 10 CES production function estimates