Measuring productivity dynamics in Japan: a quantile approach

Abstract

This paper presents an approach for estimating changes in firms’ productivity. We apply the quantile approach, which estimates the changes in the productivity distribution of surviving firms. Using this method, the paper clarifies productivity dynamics in terms of both average change and dispersion in Japan from 1987 to 2014. The main results of the analysis are as follows: During a boom or normal period, the productivity distribution shifts to the right and the productivity dispersion decreases. Conversely, during a recession, the productivity distribution shifts to the left and the productivity dispersion expands. The analysis also gives quantitatively significant results. During the historically rare global financial crisis of 2008, the weighted (simple) average of manufacturing productivity in Japan fell by only 0.2% (5.4%). We identified that this counter-intuitive result was due to a significant change in the shape of the productivity distribution and found that the crisis would reduce productivity by more than 22% if the effects of changing the shape of the distribution were adjusted.

Appendices

Appendices

1.1 Appendix A: Derivation of (1)

Denote $\tilde{\phi }$ as the random productivity which is unobservable. The cumulative distribution function (CDF) and probability density function (PDF) of the random variable are, respectively, denoted by $\tilde{F}(.)$ and $\tilde{f}(.)$. What we observe is $\sigma$, which is related to $\tilde{\phi }$ by three factors in QA; (i) Shift (S), (ii) Truncation (T), and (iii) Dilation (D).

(i) Shift (S) is an effect that affects all establishments equally regardless of the level of productivity. The effect can be shown by $\phi =\tilde{\phi }+S$. The CDF is, therefore, given by $F(\phi )=\tilde{F}(\tilde{\phi })=\tilde{F}(\phi +S)$. In the probability distribution, the effect of the shift (S) is represented by the parallel movement of the distribution, as shown in Fig. 1.

(ii) Truncation (T) captures changes in the proportion of establishments that are below the minimum level of productivity required to survive in the market. Suppose that the lowest level of productivity of the establishment operating in the market is given by $\underline{\phi }$. In this case, an establishment with a productivity lower than $\underline{\phi }$ is not observed in the data; $F(\underline{\phi })=T$. The TFP of such establishments with lower productivity than $\underline{\phi }$ is zero: $\phi =\tilde{\phi }$ if $\tilde{\phi }\ge \underline{\phi }$ and $\phi =0$ if $\tilde{\phi }< \underline{\phi }$. The sum of the probabilities must be 1, but at this stage the following equations hold:

$$\begin{aligned} \int _{\underline{\phi }}^{\infty } f(\phi )d\phi =1 \ \ \mathrm{and} \ \ \int _{-\infty }^{\infty } f(\phi )d\phi =1-T \end{aligned}$$

In order to satisfy the law of probabilities, by dividing both sides of the equation by $1-T$, we get:

$$\begin{aligned} \int _{-\infty }^{\phi } f(\phi )d\phi =\frac{\int _{-\infty }^{\phi }\tilde{f}(\tilde{\phi })d\tilde{\phi }-\int _{-\infty }^{\underline{\phi }}\tilde{f}(\tilde{\phi })d\tilde{\phi }}{1-T} \ \mathrm{and} \ F(\phi )=\frac{\tilde{F}(\tilde{\phi })-T}{1-T}, \ \mathrm{where} \ \phi \ge \underline{\phi }. \end{aligned}$$

Since the productivity of the establishment is zero if its productivity is less than $\underline{\phi }$, we have

$$\begin{aligned} F(\phi )=\max \{0,\frac{\tilde{F}(\tilde{\phi })-T}{1-T}\} \ \ \ \mathrm{where} \ T\in [0,1]. \end{aligned}$$

(13)

Figure 1(b) visually shows the difference between the Shift effect and the Truncation effect. In the figure, although the distribution moves in parallel to the right due to the Shift effect, there is no Truncation effect because the proportion of establishments with low productivity that cannot be activated in the market does not change.

(iii) Dilation (D) captures the effect that changes in productivity differ among establishments according to the original level of their productivities. The relationship between $\tilde{\phi }$ and $\phi$ can be expressed as $\phi =D\tilde{\phi }$ when the productivity of the establishment increases by a factor of D. In this case, the CDF is given by $F(\phi )=\tilde{F}(\tilde{\phi })=\tilde{F}(\phi /D)$. If there is the Shift effect in addition to the Dilation effect, $\phi =D\tilde{\phi }+S$ holds, and the CDF is as follows.

$$\begin{aligned} F(\phi )=\tilde{F}(\tilde{\phi })=\tilde{F}\left( \frac{\phi -S}{D}\right) \end{aligned}$$

(14)

Substituting (14) into (13), we obtain (1).

1.2 Appendix B: Derivation of (3) and (4)

If $T_1>T_0$, the change in variable $\phi \rightarrow (\phi -S)/D$ turns $F_0$ into

$$\begin{aligned} F_0 \left( \frac{\phi -S}{D} \right) = \max \left\{ 0, \dfrac{\tilde{F} \left( \dfrac{\phi - S_1}{D_1} \right) -T_0}{1-T_0} \right\} . \end{aligned}$$

Dividing by $1-T$ and adding $-T/(1-T)$ to all terms leads to

$$\begin{aligned} \dfrac{F_0 \left( \dfrac{\phi -S}{D} \right) -T}{1-T} = \max \left\{ \dfrac{-T}{1-T}, \dfrac{\tilde{F}\left( \dfrac{\phi - S_1}{D_1} \right) -T_1}{1-T_1} \right\} . \end{aligned}$$

We obtain

$$\begin{aligned} \max \left\{ 0,\dfrac{F_0\left( \dfrac{\phi -S}{D} \right) -T}{1-T}\right\} = \max \left\{ \dfrac{-T}{1-T}, \dfrac{\tilde{F}\left( \dfrac{\phi - S_1}{D_1} \right) -T_1}{1-T_1} \right\} = F_1(\phi ). \end{aligned}$$

Next, if $T_1<T_0$, the change in variable $\phi \rightarrow D\phi + S$ turns $F_0$ into

$$\begin{aligned} F_0 \left( D\phi + S \right) = \max \left\{ 0, \dfrac{\tilde{F} \left( \dfrac{\phi - S_1}{D_1} \right) -T_0}{1-T_0} \right\} . \end{aligned}$$

Dividing by $1-T$ and adding $-T/(1-T)$ to all terms leads to

$$\begin{aligned} \dfrac{F_0 \left( D\phi + S \right) -\dfrac{-T}{1-T}}{1 -\dfrac{-T}{1-T}} = \max \left\{ \dfrac{-T}{1-T}, \dfrac{\tilde{F}\left( \dfrac{\phi - S_1}{D_1} \right) -T_1}{1-T_1} \right\} . \end{aligned}$$

We obtain

$$\begin{aligned} \max \left\{ 0,\dfrac{F_0\left( D\phi + S \right) - \dfrac{-T}{1-T}}{1-\dfrac{-T}{1-T}}\right\} = \max \left\{ \dfrac{-T}{1-T}, \dfrac{\tilde{F}\left( \dfrac{\phi - S_1}{D_1} \right) -T_1}{1-T_1} \right\} = F_1(\phi ). \end{aligned}$$

1.3 Appendix C: Derivation of (10)

Following the computation method of the minimization criterion suggested by Combes et al. (2012), this appendix provides the detailed computation methodology in Section 3.1.3. In this approach, the parameter $\theta$ is estimated by minimizing $M(\theta )$ from $\hat{m}_\theta (u)$ and $\hat{\tilde{m}}_\theta (u)$ for any rank of $u \in [0, 1]$. To evaluate $\hat{m}_\theta (u)$ and $\hat{\tilde{m}}_\theta (u)$, we approximate true quantiles $\lambda _t$ by $\hat{\lambda }_t(u)$. First, we list productivity in ascending order for each period. The set of log productivities in period t is

$$\begin{aligned} \varvec{\phi _t} =[\phi _t(0), \phi _t(1), \cdots , \phi _t(N_t -1)], \end{aligned}$$

where $N_t$ is the number of firms in period t and $\phi _t(0)< \phi _t(1)< \cdots <\phi _t(N_t -1)$. For $k \in \{ 0, \cdots , N_t -1\}$, the sample quantiles at the observed rank are $\hat{\lambda }_t(k/N_t) = \phi _t(k)$. According to these quantiles, we approximate the uth quantile of $N_t$ using the following equation for any $u \in ( 0, 1)$:

$$\begin{aligned} \hat{\lambda }_t(u) =(k_t^* + 1 - u N_t ) \hat{\lambda }_t \bigg ( \frac{k_t^*}{N_t} \bigg ) + (u N_t - k_t^*) \hat{\lambda }_t \bigg ( \frac{k_t^* +1}{N_t} \bigg ), \end{aligned}$$

where $k_t^* = \lfloor u N_t \rfloor$ and $\lfloor \cdot \rfloor$ indicates the integer part. The empirical counterparts of $m_\theta (u)$ and $\tilde{m}_\theta (u)$ are

$$\begin{aligned} \hat{m}_\theta (u)= & {} \hat{\lambda }_1(r_T(u)) - D \hat{\lambda }_0 (T + (1-T) r_T(u)) - S, \\ \hat{\tilde{m}}_\theta (u)= & {} \hat{\lambda }_0 (\tilde{r}_T(u)) - \frac{1}{D} \hat{\lambda }_1 \bigg (\frac{\tilde{r}_T(u) - T}{1-T}\bigg ) + \frac{S}{D}, \end{aligned}$$

and we compute $\hat{m}_\theta (u)$ and $\hat{\tilde{m}}_\theta (u)$ at any rank u and $\theta$. Assuming $u_0 = 0$ and $u_K = 1$, we approximate

$$\begin{aligned} \int _0^1[\hat{m}_\theta (u)]^2 du \approx \frac{1}{2} \sum _{k=1}^K \{ [\hat{m}_\theta (u_k)]^2 + [\hat{m}_\theta (u_{k-1})]^2 \} (u_k - u_{k-1}), \\ \int _0^1[\hat{\tilde{m}}_\theta (u)]^2 du \approx \frac{1}{2} \sum _{k=1}^K \{ [\hat{\tilde{m}}_\theta (u_k)]^2 + [\hat{\tilde{m}}_\theta (u_{k-1})]^2 \} (u_k - u_{k-1}). \end{aligned}$$

We estimate the parameters $\theta$ by minimizing the sum of two quantiles,

$$\begin{aligned} \hat{\theta } = \arg \min _{\theta } M(\theta ) = \int _0^1[\hat{m}_\theta (u)]^2 du +\int _0^1[\hat{\tilde{m}}_\theta (u)]^2 du . \end{aligned}$$

1.4 Appendix D: Levinsohn and Petrin (2003) approach

We here present the estimation methodology proposed by Levinsohn and Petrin (2003). This method takes account of the endogeneity problem that capital inputs and productivity are correlated. According to Levinsohn and Petrin (2003), the production function is represented as $v_{it} = \beta _k k_{it} + \beta _l l_{it} + \varepsilon _{it}$, where $\varepsilon _{it} = \phi _{it} + e_{it}$, and $e_{it}$ is the error term that cannot be observed. $\phi _{it}$ is likely to be correlated with the capital inputs. In addition, $\phi _{it}$ has a positive relationship with intermediate inputs $m_{it}$, $m_{it} = g(k_{it},\phi _{it})$. This can be inverted $\phi _{it} = h(m_{it}, k_{it})$. Substituting this into production function leads to $v_{it} = \beta _l l_{it} + \Phi (k_{it}, m_{it}) + e_{it}$, where $\Phi (k_{it},m_{it}) = \beta _k k_{it} +h(m_{it}, k_{it})$. $\Phi (k_{it},m_{it})$ is replaced with a multi-order polynomial in $m_{it}$, $k_{it}$, and $\hat{\Phi }$. Then, this equation can be estimated by OLS. In addition, $\phi _{i,t-1}$ can be approximated by $\Phi (k_{i,t-1}, m_{i,t-1}) -\beta _k k_{i,t-1}$. From the above, the production function is rewritten as $v_{it} = \beta _l l_{it} + \gamma (\hat{\Phi }(k_{i,t-1},m_{i,t-1}) - \beta _k k_{i,t-1}) + e_{it}$, where $\gamma (\cdot )$ is an approximated term. This equation is estimated by nonlinear least squares, and $k_{it}$ does not have a correlation with the error term. Hence, we obtain consistent estimates of the production function coefficients.

1.5 Appendix E: TFP

In the analysis, we used the estimation results mainly based on Levinsohn and Petrin (2003). Another representative method is to use the method proposed by Wooldridge (2009). Using the generalized method of moments approach, Levinsohn and Petrin (2003) build the identification production function following Olley and Pakes (1996), whereas Wooldridge (2009) proposes a single-equation instrumental variable approach to control for the correlation between the inputs and unobserved productivity of establishments. Table 8 shows the estimation results of the production function derived using the Levinsohn and Petrin (2003) and Wooldridge (2009) approaches as well as OLS and OLS with fixed effects. The coefficients of $\ln k$ and $\ln l$ of both the Levinsohn and Petrin (2003) and the Wooldridge (2009) approaches are almost the same. They also have the expected signs and are statistically significant at the 1% level.

Using the data on the TFP of each establishment $i=1,2,...$, simple and weighted average $\ln$ TFPs in the industry at year t are calculated by $TFP_t=\sum _i \ln TFP_{ti}$ and $TFP_t=\sum _i \theta _{it} \ln TFP_{ti}$, where $\theta _{it}$ is the added-value share of establishment i in year t.

Table 8 Estimation of the production function

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1.6 Appendix F: Descriptive statistics of the productivity distribution

Table 9 presents the descriptive statistics of the productivity distribution. Between 1986 and 2014, productivity increased by 36% from 1.160 to 1.576 at the 10th percentile. On the contrary, the productivity rise at the 90th percentile remained at 11.7% from 3.42 to 3.82. This implies that the increase in productivity in firms with low productivity was relatively large and that the productivity gap between firms narrowed. This can also be confirmed by reading the table from left to right. In 1986, productivity at the 90th percentile was 2.95 times greater than that at the 10th percentile. That figure was 2.42 in 2014.

Table 9 Descriptive statistics of the productivity distribution