## Annotation 1 Estimating the Productivity Effect of Social Capital

## 1 Estimated Equation

The following Cobb–Douglas production function that allows for social capital stocks explicitly is presumed (one term earlier variables are used for private capital and social capital to deal with the endogeneity problem:

$$ \mathbf{Y}_{\mathrm{t}}=\mathrm{A}(\mathrm{H}_{\mathrm{t}}\cdot\mathrm{L}_{\mathrm{t}})^{\alpha}(\mathrm{CU}_{\mathrm{t}}\cdot\mathrm{K}_{\mathrm{t}-1})^{\beta}\mathbf{G}^{\gamma}_{\mathrm{t}-1} $$

Y = total production, A = technical level of economy, H = number of hours worked, L = number of employees, CU = operating rate, K = private capital stock, G = social capital stock

Subscript t denotes the term.

At estimation, a linear homogeneity of production function $ (\alpha+\beta=1) $ relating to the labor and private capital was postulated to estimate the following logarithmically converted estimated equation in the least square method:

$$ \ln\mathrm{Y}_{\mathrm{t}}=\mathrm{c}+(1-\beta)\ln\mathrm{H}_{\mathrm{t}}\cdot\mathrm{L}_{\mathrm{t}}+\beta\ln\mathrm{CU}_{\mathrm{t}}\cdot\mathrm{K}_{\mathrm{t}-1}+\gamma\ln\mathrm{G}_{\mathrm{t}-1}+\varepsilon_{\mathrm{t}}(\varepsilon\colon\mathrm{error}\mathrm{term}) $$

The results of such estimation are summarized in the table below. Case 2 presents the results of estimation for social capital broken down into seven fields under the jurisdiction of the Ministry of Land, Infrastructure, Transport and Tourism (MLIT) (roads, ports and harbors, airports, sewage systems, urban parks, flood control and seacoasts). Case 3 presents the results of estimation for social capital limited to the three fields of traffic (roads, ports and harbors and airports).

Social Capital Stocks	Case 1	Case 2	Case 3
Constant term (c)	-8.916	-8.917	-9.329
(-7.872) **	(-8.310) **	(-8.781) **
$ \beta $	0.315	0.309	0.267
(2.822) **	(2.855) **	(2.481) *
$ \gamma $	0.226	0.229	0.266
(2.700) *	(2.851) **	(3.263) **
Number of samples	35	35	35

(Notes) 1 The parentheses under the coefficient estimate enclose a t-value, which has been subjected to a robust Newy-Wet correction for the serial correlation.

2 * is significant with a significance level of 5%. ** is significant with a significance level of 1%.

## 2 Data

## (1) Total production

For 1975 to 1979, the rate of change in the total production from a year earlier was determined from the values of subtotals “Gross Domestic Product (Real) by Economic Activity” in the Cabinet Office’s Fiscal 1988 Annual Report on National Accounts (1990 Basis, 68SNA), so that the total production was retrospectively estimated from the values of subtotals in 1980 in “Gross Products by Economic Activity (Real)” in the Cabinet Office’s Fiscal 2009 Annual Report on National Accounts (2000 Basis, 93SNA). For 1980 to 2009, the values of subtotals in “Gross Products by Economic Activity (Real)” in the Cabinet Office’s Fiscal 2009 National Accounts (2000 Basis, 93SNA) are used.

## (2) Number of hours worked

For 1975 to 1979, yearly ratios were calculated from five-year data for 1980 to 1984 on surveyed industry totals (30 persons or more) and from the number of hours worked for the same period out of the data on “Number of Employees, Number of Employers and Number of Hours Worked by Economic Activity” in the Cabinet Office’s Fiscal 2009 Annual Report on National Accounts (2000 Basis, 93SNA). Then, data on surveyed industry totals (30 persons or more) for 1975 to 1979 out of the Ministry of Health, Labour and Welfare’s Monthly Labor Statistics Survey was multiplied by the five-year average of these yearly ratios as a conversion rate. Data for 1980 to 2009 was cited from “Number of Employees, Number of Employers and Number of Hours Worked by Economic Activity” in the Cabinet Office’s Fiscal 2009 Annual Report on National Accounts (2000 Basis, 93SNA).