Table 5 Notations in the objective function.

x^r

\({x}^{r}={\bar{x}}^{r}\cdot cV{A}^{r}\), where x^r is the total output of region r, and \({\bar{x}}^{r}\) is the initial value of x^r. cVA^r is the ratio of the production-approach GRP taken from the calibrated national account data over that of the initial MRIOs (\(cV{A}^{r}=vaOb{j}_{prod}^{r}/{\overline{va}}_{prod}^{r}\)). Specifically, the total output of Tibet for years 1997, 2002 and 2007 are estimated from the x^Tibet of year 2012, since there are no IO tables of Tibet for these years. For example, \({x}^{Tibet,2007}={x}^{Tibet,2012}\cdot crV{A}^{Tibet,07-12}\),where \(crV{A}^{Tibet,07-12}\) is the rate of change of Tibet’s value added between years 2007 and 2012;

\({h}_{ij}^{r}\) and \({\bar{h}}_{ij}^{r}\)

\({h}_{ij}^{r}\) is the column structure of the provincial tables, and \({\bar{h}}_{ij}^{r}\) is the initial value of \({h}_{ij}^{r}\); Specifically, for the “scrap and waste” sector of some provinces, the rate of value added over the total input (VA rate) in the official input-output table of 1997, 2002 and 2007 are equal to 1, which goes against the economic common sense. To deal with this issue, we take the column structure of the province with the highest VA rate as the initial value of the column structure of the few provinces with VA rate of “1” in 2007. For the year 1997 and 2002, since the VA rates of the “scrap and waste” sector in most provinces equal to 1, we take the column structure of the corresponding provinces in 2007 as the initial value.

\(inv{t}_{{i}^{\ast }}^{r}\)

\(inv{t}_{{i}^{* }}^{r}\) is the changes in inventory of sector i*, region r. sector i* refer to those sectors with zero changes in inventory in the corresponding national IO tables, which are mostly service sectors;

\(strin{c}_{ii}^{r}\) and \(strincOb{j}_{ii}^{r}\)

\(strin{c}_{ii}^{r}=\frac{v{a}_{ii}^{r}}{{\sum }_{ii=1}^{4}\,v{a}_{ii}^{r}}\), where \(v{a}_{ii}^{r}\) represents the income-approach GRP term ii, region r. \(strincOb{j}_{ii}^{r}\) is the objective value of \(strin{c}_{ii}^{r}\), which is calculated from the calibrated national account data;

\(strpr{d}_{jp}^{r}\) and \(strprdOb{j}_{jp}^{r}\)

\(strpr{d}_{jp}^{r}=\frac{v{a}_{jp}^{r}}{\mathop{\sum }\limits_{jp=1}^{9}\,v{a}_{jp}^{r}}\), where \(v{a}_{jp}^{r}\) represents the production-approach GRP sector jp (with 9 aggregated sectors), region r. \(strprdOb{j}_{jp}^{r}\) is the objective value of \(strpr{d}_{jp}^{r}\), which is calculated from the calibrated national account data;

\(strprdMa{x}_{jp}^{r}\) and \(strprdMi{n}_{jp}^{r}\)

\(strprdMa{x}_{jp}^{r}\) is the adjusted structure of production-approach GRP, whose elements equal to \(strprd-ad{j}_{max}\), where \(ad{j}_{max}\) is the adjustment value to make \(strprd\) less than the larger value between the \(strprdOb{j}_{jp}^{r}\) and the official ones (uncalibrated values);Similarly, \(strprdMi{n}_{jp}^{r}\) is the adjusted structure of production-approach GRP, whose elements equal to \(strprd+ad{j}_{min}\), where \(ad{j}_{min}\) is the adjustment value to make strprd greater than the smaller value between the \(strprdOb{j}_{jp}^{r}\) and the official ones (uncalibrated values).

pex_i and pim_i

pex_i and pim_i represent the sum of provincial outflow and inflow by provinces of sector i, respectively.