# (p.196) Appendix 6 Calculating Productivity

# (p.196) Appendix 6 Calculating Productivity

We measure firm productivity using the multilateral index developed by Caves, Christensen, and Diewert (1982) and later modified by Aw, Chung, and Roberts (2003). This productivity index offers several advantages over conventional parametric measures, e.g., the residuals from the Cobb–Douglas production function and its variants. First, the multilateral index is straightforward in computation and flexible enough to allow for heterogeneous production technology. Given that firms with varying degrees of technological sophistication compete with each other in China, this flexibility makes it particularly relevant to our setting. According to Van Biesebroeck, who examines the robustness of various productivity measures, “When measurement error is small (or outliers are properly controlled *ex post*), index numbers are among the best for estimating productivity growth and are among the best for estimating productivity levels” (2007: 529). Another advantage of the productivity index is that it allows for a consistent comparison of firm-level productivity across years. To compare any two firm-year observations that are transitive, this indicator expresses a firm’s output and inputs as deviations from a single reference point. This reference point is a hypothetical firm that operates for each 3-digit SIC industry during the base time period, i.e., 1998, the first year of the annual database, using the industry average for input shares, inputs, and outputs. We perform
(p.197)
robustness tests with alternative ways of measuring productivity. The productivity index is defined as follows:

where *i* denotes firm, *t* year, and *j* type of input (*j* = 1, ..., *m*). *Y* _{it} denotes output, and *X* _{ijt} denotes inputs including labor input, material input, and capital stock. *S* _{ijt} denotes input shares, defined as the ratio of labor costs to output for labor input, the ratio of material costs to output for material input, and one minus labor share and material share for capital input. The first term in Equation (1) captures the deviation of a firm’s output in year *t* from the average industry output in that year. The second term reflects the change in industry average output across all years in the study. The third and fourth terms repeat the same for each input *j*, which are summed using the input share for each firm (*S* _{ijt}) and the average input share for each 3-digit industry in each year as weights. The productivity index measures the proportional difference between the productivity of firm *i* in year *t* relative to the hypothetical firm in the base year.

We make the following three adjustments to correctly capture inputs and outputs in calculating productivity index. First, following Brandt, Van Biesebroeck, and Zhang (2012), we adjust output and material input using input price deflators and industry-level output. The 1998 to 2003 annual industrial surveys contain information on the value of a firm’s output in both nominal and real prices. We use the ratio of nominal output to real output to generate a firm-level price index, using 1998 as the base year. We then calculate the industry-level output price deflator by taking the weighted average of firm-level price indexes for each three-digit SIC industry using a firm’s output as weights.^{8} Because there is no information on firm-level nominal and real output
(p.198)
information available for the years between 2004 and 2009, we use the ex-factory price index at the two-digit SIC industry level from the *China Statistics Yearbook* for these years. We calculate the input deflator by taking the weighted average of the output deflators using the input coefficients from the 2002 Input-Output table as a weighing factor for each industry.

Second, the NBS database provides accounting-based information on firms’ nominal capital stock for each year, which allows us to trace capital stock increases from the time a firm appears in our annual data set. In order to estimate productivity, we also need to calculate the real capital stock for each year. For this purpose, we use the perpetual inventory method. For firms that appear in the 1994 NBS economy-wide census, we can observe their capital stocks as of 1994. For firms that are not in the 1994 economy-wide census, we can observe their capital stock when they first appear in the NBS annual industrial survey database. We can then estimate backward to determine firms’ nominal capital stock at the time of their establishment. To do so, we need three additional pieces of information: average capital growth rate, capital depreciation rate, and the investment deflator (to control for price changes in capital goods). Because some firms appeared in both the 1994 economy-wide census and the 1998 annual industrial survey, we can calculate the capital stock increase between 1994 and 1998 for these firms and consider it to be the average rate of growth of nominal capital stock between 1994 and 1998. Given that the capital growth rate may vary by industry and region, we make this calculation for each 2-digit SIC industry for each region. Using this average capital stock growth rate, we calculate backwards to determine a firm’s nominal capital stock at the time of establishment by discounting a firm’s nominal capital stock in the first year the firm appears in the data set. The real capital stock at the time of establishment is obtained by deflating the nominal capital stock with the Brandt and Rawski investment deflator (Brandt, Van Biesebroeck, and Zhang 2012). Closely following Brandt, Van Biesebroeck, and Zhang (2012), we calculate the real capital stock at time *t* by applying a 9% depreciation rate and deflating the annual nominal investment using the Brandt and Rawski
(p.199)
investment deflator. We can observe a firm’s actual investment from 1998 forward with the annual industrial survey and, therefore, use the observed change in a firm’s nominal capital stock for the nominal fixed investment for each year. We use the same depreciation and investment deflators to roll the real capital stock estimates forward. For firms that appear only in the 2004 economy-wide census, however, we cannot roll forward since we are unable to observe their actual investment history.

The third adjustment addresses labor costs. Because firms’ official financial reports of wages, employee benefits, and unemployment benefits may underestimate labor costs, we follow the method employed by Hsieh and Klenow (2009), which inflates total labor compensation by a constant factor, such that the ratio of sector-level labor compensation to total inputs is equal to the same ratio in the 2002 Input-Output table in that sector. In other words, we assume that all firms in an industry pay their workers extra compensation in the same proportion paid to workers’ official wages and benefits. Lastly, the 2008 and 2009 annual industrial survey lacks wage and material cost data. We extrapolate these factors by using previous years’ information for the firms, if they existed prior to 2008. For firms that newly entered in 2008 or 2009, we use the industry average weighted by employment.

## Notes:

(8)
. Following Brandt, Van Biesebroeck, and Zhang (2012), we do not use the firm-level price deflator, as it tends to be very noisy due to extreme outliers. We only generate a 3-digit industry-level price deflator, which is less aggregated than the 2-digit industry-level deflator available in the *China Statistics Yearbook*.