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Heterogeneous Production Functions, Growth Convergence and Regional Disparity ① Jia Nan [email protected] The Southwestern University of Finance and Economics Gan Li [email protected] SWUFE and Texas A&M University Jan 2009 Postal Address: No.55, Guang Hua Cun, The School Of Economics, The Southwestern University Of Finance And Economics, Qingyang District, Chengdu, China email: [email protected] Abstract: The literature about economic growth convergence and regional disparity assumes the same gross production functions across countries or regions. This paper allows the possibilities of different production functions for different regions. Such differences come from the observed differences in the factor shares of income. In steady states, a region with a higher capital share would have a higher per capita output. Thus, the output differences among regions will be persistent if they have different factor shares. Using the province-level data in China, I show that the capital share in income indeed has a very significant and positive effect on the level and the growth rate of per capita output. Key words: factor share, production function, growth convergence, regional disparity JEL classification : O180 ,O470,R110 ① For helpful comments I thank Li Gan, Shihe Fu, Wenbin Zang, Zhen Lei, Liwei Shan, Xiaochen Cai, Xiaoling Chen, Yi Li and seminar participants at South West University of Finance and Economics. For financial support I thank the Leading Academic Discipline Program , 211 Project for SWUFE(the 3rd phase) and the Humanities and Social Science project “Dynamic optimization and regulation for Monetary policy”(08JC790082)by the State Education Commission. Any opinions expressed are those of the author. 1 1. Introduction The neo-classic growth model suggests that economies with similar initial conditions would achieve similar growth rate under the assumptions of decreasing returns to factors, constant returns to scale, and similar aggregate production functions. In this type of models, the poor economy would grow faster than the rich economy. In the long-run, the growth rate of different economies would converge and the differences among economies would be gradually reduced. This is the so-called “convergence hypothesis”. On the other hand, in the new growth models, the accumulation of physical capital, human capital and knowledge capital would often have increasing returns to scale. Economies with higher levels of capitals would have growth faster. As a consequence, the growth of different economies would diverge and the disparities among different economies would increase. Empirically, a standard test of the two theories is to test the convergence of various economies. In their classic paper, Mankiw, Romer and Weil (1992) (MRW hereafter) introduce human capital into the standard Solow model to develop an Augmented Solow Model. They find evidence that countries converge at about the rate that the augmented Solow model predicts if holding population growth and capital accumulation constant. This finding has been considered as the evidence of “conditional convergence”. A large number of empirical studies has since used alternative econometric methods and data samples, such as Barro and Sala-i-Martin (1992), Bernard and Durlauf (1995), Islam (1995), Quah (1996, 1997), Dowrick and Ngugen (1989), Bernard and Jones (1996), Andrade (2003), Mathur (2005) and D’uva (2007). Most of them arrive at the same conclusion as MRW, i.e., economic growth across countries has conditional convergence. The topic of convergence across regions in China has also generated wide interests in the literature. Most studies agree that the regional disparity in China was reduced before 1990 and then was increased after 1990. Meanwhile there is conditional convergence among regions and club convergence in the east region, the middle region, and the west region. Those papers include Cai and Du (2000), Shen and Ma (2002), Lin and P. Liu(2003), Wang (2004), Xu and Shu (2005), Peng (2005, 2006), Wang and Zhang (2006), Zhang (2006), and Teng and Liang (2006). Figure 1 is the real per capita output of 30 provinces in China. From the figure, the difference 2 of real per capita output has been reduced before 1990 and then increased constantly among different provinces. Within each of the east region, the middle region, and the west region, the differences also increase. Comparing the three regions, the regional disparity is the largest in the east region, and the smallest in the west region. 100000 50000 19 78 19 81 19 84 19 87 19 90 19 93 19 96 19 99 20 02 20 05 0 west Figure 1:Real per capita output cross-provinces The literature has provided several explanations on increasing regional disparity since1990. One explanation is the scarcity of the human capital endowment, the market distortion and the lack of openness (Cai and Du, 2000; Shen and Ma, 2002; Wang and Fan, 2004; Wang and Zhang, 2006). Other people suggest that the government development strategies are inconsistent to the regional comparative advantage (Lin and P. Liu, 2003; Lin and M. Liu, 2003). A third alternative explanation is that the labor force in the east region grows faster than the capital accumulation. As a consequence, the marginal return to capital remains higher in the east than in the west (Liu, 2001; Shen and Tang, 2006). A fourth one is that there are technology spillover obstacles (Fu and Wu, 2006). However, all empirical studies so far are built upon the assumption that all countries or regions have identical aggregate production functions. This paper relaxes the assumption, allowing the production function to be different across regions. Therefore, it provides a new alternative interpretation about expanded regional disparity cross-provinces in China. This paper argues that the differences in factor 3 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 120000 100000 80000 60000 40000 20000 0 1978 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 120000 100000 80000 60000 40000 20000 0 150000 1982 middle east 1980 real per capita GDP Real per capita GDP 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 120000 100000 80000 60000 40000 20000 0 1978 Real per capita GDP Real per capita GDP Whole country shares in output in different regions lead differences in aggregate production functions, which in turn would generates persistent differences in output. This hypothesis is tested by using province-level data from 1990 to 2005 of China. In this paper, I find that that the capital share in output affects real per capita output significantly. The higher capital share, the higher the real per capita output. As a result, regional disparity may increases. I also find that regions with larger differences in factor shares have larger differences in their per capita output. For example, provinces in the east region has the largest variation in their capital shares, they also have the largest variation in the per capita output. Further, the growth rates still satisfy the conditional convergence and the club convergence when controlling factor shares. Yet the output elasticity of factors again has very significant impacts on growth rate. The higher the output elasticity of capital, the higher the growth rates the region. The effect leads to regional disparity become larger. The rest of this paper is organized as follows: Section 2 provides the interpretation about expanded regional disparity of this paper. Section 3 details the empirical model as well as variables and data. Section 4 shows the empirical results. Section 5 concludes. 2. A New Explanation The current literature on economic growth across regions or countries imposes very strong homogeneity assumptions on aggregate production function. The assumption implies that the capital share in the national income (or the output elasticity of the capital), and the labor share in the national income (or the output elasticity of the labor) across countries or regions are identical. This assumption may be true within a country since the capital share in national appears to be fairly constant over a long period of time. Figure 2 show the time series for the United States dating back to 1935 and Figure 3 shows the United Kingdom dating back to 1855 (Mitchell, 1988). These graphs suggest that the labor share moves very little over time within a country. 4 Labor share 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1935 1945 1955 1965 1975 1985 Figure 2: labor share and GNP:United States 1935-1985 0.8 Labor share 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1855 1875 1895 1915 1935 1955 1975 Figure 3: labor share and GDP: United Kingdom 1855-1980 Across countries, however, there appear to be substantial differences in income shares. Collin (2002) plots labor shares of 81 countries in most recent years available (1987–92), against levels of real per capita output (Figure 4). Clearly there are wide disparities in labor shares across countries, ranging from 0 to 0.7. It also appears a positive relationship between labor shares and real per capita GDP. There are several possible explanations on why factor shares might differ across countries. One explanation is that countries do not operate the same aggregate technology. The second explanation is that countries may share a common aggregate technology but have different institutional arrangements. A third alternative is that some countries might face imperfect factor markets, so that wages are not equated to marginal products (Gollin, 2002). 5 Labor share/real per capita GDP 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000 Real per capita GDP Figure 4 labor share and real per capita GDP in 81 countries 1987-1992 Given the possible differences in factor shares, it is natural to allow that aggregate production functions to be different across regions. Such differences in aggregate production functions may contribute to the disparity in output. As a consequence, economic disparity may continue to increase across countries or regions. 3. Model and Data 3.1 Model MRW (1992) demonstrate that the Solow model has impressive empirical explanatory power. MRW approach the issue of non-steady-state behavior by estimating rates of convergence and relating these to the parameters of the model. I take a more direct approach: Based on the Solow model, the steady-state per capita output depends on the capital share. In addition, I find that the growth rates may also depend on the capital share. Therefore, regions with different capital shares may have increasing disparity. Based on the Solow model, the motion equation per capita is k sy (t ) (n g )k (t ) , where y is the level of output per effective unit of labor, y capital per effective unit of labor, k Y ; k is the stock of EL K , E is defined as the labor augmenting EL technological progress; s is the saving rate; n is the labor growth rate; δ is depreciate 6 rate; and finally, g is exogenous technological progress rate. Output per capita is: y (t ) k (t ) In the steady state: sk (t ) (n g )k (t ) Or s k n g 1 y (1) In (1), since k is defined as the stock of capital per effective unit of labor, so, capital stock per capita k l , is: k l K / L . Rearrange (1): s k l n g 1 E y (2) From (2), output per capita is affected by the saving rate s, depreciate rate δ, labor growth rate n, exogenous technological progress rate g, labor augmenting technological progress E and capital share α. If these variables are the same in different countries or regions, output per capita will ultimately converge. However, if one of these variables is different, output per capita may not converge, and regional disparity may persist. Current literature has studied the heterogeneity of some of these variables. This paper focuses on the capital share parameter α, the key parameter in the production function. If the capital shares are different in two regions, the capital stock per capita may not be the same even though all the other variables are the same. Figure 5 shows that any two economies with different capital shares have different steady state, so their income disparity may not disappear. 7 b y” a y’ 0 2 1 Figure 5:Output Elasticity of Capital and the Steady States Taking logarithm of (2), we have: s ln y ln 1 n g ln E (3) For any two regions in a country, however, it is often plausible to assume they have the same depreciation rate δ and exogenous technological progress rate g. Because E denotes labor-augmented technological progress, it can be regarded as the human capital. Two regions may have different levels of human capital, different saving rates s, and different labor growth rates n. More importantly, they are allowed to have the different capital share . Taking partial derivative with respect to , s, n, E of (3): ln y 1 s ln E 0 ln 2 1 n g ln y 1 0 1 s s ln y 1 0 1 n g n ln y 0 ln E (4) Accordingly, we can establish the following regression model: ln y i a 0 a1ˆ i a 2 tˆi a3 s i a 4 ni a5 E i 8 (5) Where ̂ is the average output elasticity of capital in the same region. In (5), we also include the total factor productivity tˆ since it is obvious that this variable may affect the output level. It is expected that a1>0, a2>0, a3>0, a4<0, and a5>0. Further, to examine if the regional growth rate converges, I use the growth rate to test if there is absolutely convergence under heterogeneous production function by using the following model: ln y it a 0 a1 ln y i 0 * ˆ it i (6) Where ln y i 0 denotes real per capita output in the initial period, ln y it denotes the growth rate of per capita output in the end of the period. Equation (7) includes some control variables: ln y it a 0 a1 ln y i 0 * ˆ it a 2ˆ it a 3 tˆit a 4 s it a 5 nit a 6 E it i (7) It is easy to show that: Where, ln y it a1 * ̂ it ln y i 0 ln y it a1 * ln y i 0 a 2 ˆ it If (8) (9) ln y it 0 , then there is conditional convergence in regional economic ln y i 0 growth; If ln y it 0 and is significant, then capital share has positive impacts ˆ it on economic growth, and the differences in α will result in different output in different regions. 3.2 Data and measurement The empirical analysis of this paper uses provincial level data in China. The sample period is 1990-2005 because the purpose of this paper is to discuss why the economy gap becomes larger among provinces in China after 1990, and because one of the key variables, labor compensation, is available only after 1990. The city of Chongqing is included in Sichuan because of missing data in early years. The variables used in the analysis include real output, capital stock, labor force, 9 labor compensation, saving rate, labor growth rate, human capital and price index. The official data of capital stock is not available in China. Following the literature, I use the capital stock data of Zhang, Ying, Wu and Zhang (2004). Because their capital stock data is deflated by constant price of 1952, all the nominal variables are deflated using the constant price of 1952 to be consistent with the capital stock data of Zhang et al (2004). Following MRW (1992), the percentage of the labor force with the secondary school education is used as a proxy for the rate of human-capital accumulation. All other data are come from various years of China Statistical Yearbook. To estimate the value of capital share α, we use the percentage of labor compensation in GDP. Then we can calculate the implied capital share by assuming constant return to scale: ̂ 1 labor compensation GDP (10) To reduce measurement errors, we consider two types of labor shares: the average labor shares between1990 and 2004, and the average labor shares 1990-2000. Figure 6 shows the ratio of labor compensation to GDP in thirty provinces. Labor share varies from 0.3 to 0.8. Table 1 has the descriptive statistics by three regions within China for labor shares and for per capita real output. The difference of the labor share within the east China② area is the largest, and within the west China is the smallest. 1. The area with the largest differences of labor share has the largest difference in per capita output. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Figure 6:Labor Share in Different Provinces ② I partition the three areas conventionally. The east area includes Beijing, Tianjing, Hebei, Liaoning, Shanghai, Jiangshu, Zhejiang, Fujian, Shandong, Guangdong, Guangxi and Hainan. The middle area includes Shanxi, Neimeng gu, Jilin, Helong jiang, Anhui, Jiangxi, Henan, Hubei, Hunan. The west area includes Chongqing, Sichuan, Guizhou, Yunan, Xizang, Shanxi, Gansu, Nixia, Qinghai, Xinjiang. 10 Variable Real per capita output Labor share Table 1:Descriptive Statistics of Variables Obs Mean Std.Dev. Area ③ Mix Max East 12 13269.7 7381.966 4651.364 32156.64 Middle 9 7136.358 2071.75 4782.829 10568.54 West 9 5595.108 1995.092 2902.795 10190.84 East 12 .4763484 .0695394 .3489196 .616737 Middle 9 .5543737 .0612278 .4527116 .611435 .4663748 .7369851 West 9 .5587092 ④ .0764226 Finally, we can calculate the total factors productivity by estimating Solow residual. Consider a Cobb-Douglas production function: Y (t ) A(t ) K (t ) L(t )1 (11) Where, Y is real output in period t, K is capital stock, L is labor force, A is total factors productivity, is output elasticity of capital. Taking logs of both sides of (11), and then take a first difference, we have: ln Yt ln At ln K t 1 ln Lt The Solow residual is basically ln At , calculated by the difference: ln At ln Yt ln K t 1 ln Lt Therefore, tˆ ln Y ˆ ln K (1 ˆ ) ln L (12) We can see from (11), the total factor productivity is also correlated to capital share. Different output elasticity of capital will bring different TFP. 4. Empirical Results 4.1 The effect of heterogeneous production function on the level of the output The regression equation in (5) is used to estimate the impacts of capital share on the per capita output level. The key parameter of interest is the coefficient of the ③ In the east area, the differences of real per capita output among other regions and Beijing as well as Shanghai are very large. If taking the two observations out, the variance of real per capita output in the east is 2867.94, which is still bigger than middle and west. ④ Because of the special geography environment, the labor share in Xizang is remarkably higher than the average level of the west, so as to its variance. If taking Xizang’s data out, the standard variance of labor share in the west is smaller than the east and middle. 11 capital share variable. Further, to investigate whether the income gaps within each of the east region, the middle region, and the west region areas are affected by different production function, two dummy variables are introduced: ln y i a 0 a1ˆ i a 2 tˆi a3 s i a 4 ni a5 D1i *ˆ a 6 D2 i *ˆ 1,west D1 0,otherwise Where, 1,east D2 0,otherwise I consider two alternative measures in output levels. The dependent variable in columns (1) and (3) use the per capita output in 2005, and the columns in (2) and (4) use the average per capita output during the period of 2001 and 2005. Further, when the dependent variable is the 2005 per capita output, the the average saving rate, the average labor growth rate, the average human capital stock and the average total factor productivity from 1990 to 2004. The regression results are reported in columns (1) and (2). Table 2: Output Elasticity of Capital and Real Per Capita Output Dependent variable: real per capita GDP (1) (2) (3) (4) 3.312** (1.44) 2.816** (1.28) 3.626** (1.49) 3.151** (1.44) Average total factor productivity 18.50** (7.46) 14.93** (5.72) 15.42** (7.03) 11.61** (5.57) Average saving rate 2.158* (1.16) 1.876* (1.03) 0.0692 (1.39) 0.0229 (1.29) Average labor growth rate -0.714 (7.64) -7.662 (5.50) 0.369 (7.22) -5.413 (5.32) Average human capital -0.632 (3.71) -0.672 (3.41) 3.538 (3.94) 2.614 (3.68) west* -0.428 (0.41) -0.412 (0.41) east* 0.611 (0.37) 0.586 (0.37) Average output elasticity of capital Constant 7.147*** (0.73) 7.547*** (0.58) 7.439*** (0.69) 7.838*** (0.56) 30 0.67 30 0.72 30 0.74 30 0.78 Observations R-squared Column (1) in table 2 is the regression of real per capita GDP in 2005 on the 12 average values of independent variables from 1990-2004. The results show that the capital share has a strong and significant positive impact on real output. In particular, if the capital share increase 1 percentage point, real per capita output increases about 3.312%。The results indicate that because the capital share is different among provinces or because of the different production function, the provincial disparity of the real per capita output becomes larger and larger. As a robustness check, the second column in table 2 uses the 2001-2005 average per capita output as the dependent variable, while independent variables are the average values from 1990-2000. The result shows that the output elasticity of capital still has a significant positive impact on real output.. Column (3) introduces regional dummy variables, using the 2005 per capita GDP as the dependent variable. The regression result indicates that capital share still has a very significant impact on the real output within the east region, the middle region, and the west region. That is to say, the differences of output per capita may be larger within three areas. For example, capital share increase per unit in the middle region of China, real per capita output increases about 3.626%; capital share increase per unit in the west part of China, real per capita output increases about 3.198% (t value is 2.35); capital share increase per unit in the east China, real per capita output increases about 4.237% (t value is 2.95). The impacts that factors share act on real output are all positive in three areas. Moreover, the impacts in the east are the biggest of three areas, which is consistent with the biggest differences of labor share in the east China. And also because of the biggest differences of labor share, the provincial gap is the biggest in the east. Further, the disparity of labor share in the west China is the smallest, so the provincial gap is the smallest in the west. The disparity of labor share in the middle China is a little bigger than the west, its provincial gap is a little bigger than the west. Therefore, the differences of output per capita among different regions have very close correlation with the differences of factors share. Column (4) is the robust test of result in column (3), which is introduced region dummy variables based on model 2. Compared with model 3, the effect degree of output elasticity of capital acting on real per capita output is lower. capital share increase per unit in the middle China, real per capita output increases about 3.151%; capital share increase per unit in the west China, real per capita output increases about 2.739% (t value is 2.16);capital share increase per unit in the east China, real per capita output increases about 3.737% (t value is 2.74). However, the output elasticity 13 of capital still has very significant positive impact on per capita output within three areas. The larger the differences of factors share the region, the higher the impact of capital share acting on real per capita output, and the larger the economic gap within a area. 4.2 The impact of heterogeneous production function on the growth rate To test whether the difference of production function affects the convergence of growth rate, I use model (6) to examine the absolutely convergence and model (7) to examine the conditional convergence. After introducing the area dummy variables, I can examine the club convergence by using the following regression model: ln y it a 0 a1 ln y i 0 * ˆ it a 2ˆ it a 3 tˆit a 4 s it a 5 nit a 6 E it a 7 D1i a8 D2 i i The estimated results are displayed in table 3: Table 3:The capital share and the economic growth rate Dependent variable: economic growth rate (1) (2) (3) (4) (5) (6) Initial output per capita * 0.00959 (0.0074) -0.0632*** (0.019) -0.0678*** (0.020) 0.0111 (0.0067) -0.0455** (0.017) -0.0501*** (0.018) Output Elasticity of Capital 0.570*** (0.18) 0.561*** (0.18) 0.409** (0.16) 0.397** (0.16) Total factor productivity -0.117 (0.14) -0.142 (0.14) -0.222* (0.13) -0.247* (0.13) Saving rate 0.166*** (0.052) 0.148** (0.056) 0.133*** (0.046) 0.119** (0.050) Labor growth rate 0.461** (0.20) 0.414* (0.21) 0.439** (0.18) 0.390** (0.19) Human capital 0.00346 (0.0088) -0.0602 (0.082) 0.00818 (0.0079) -0.0577 (0.073) west* -0.0213 (0.017) -0.0190 (0.015) east* 0.107 (0.14) 0.111 (0.13) Constant 0.213*** (0.035) 0.142*** (0.039) 0.183*** (0.057) 0.173*** (0.031) 0.139*** (0.035) 0.180*** (0.050) Observation R-squared 30 0.06 30 0.53 30 0.56 30 0.09 30 0.56 30 0.60 14 Column (1) is a test of absolute convergence. The initial real output is the real per capita GDP in 1990. All other variables are at 2005 level. Based on the estimated coefficients and equation (8), the initial per capita output has positive but insignificant effect on the economic growth rate when the heterogeneity in production functions are included. Since the absolute convergence requires a negative coefficient, the result here clearly indicates no absolutely convergence among provinces. Column (2) is a test of conditional convergence. Since the coefficient of the interaction term between the initial per capita output and the capital share is significant and negative, there is conditional convergence among provincial growth rates. Further, capita share has a very significant and positive impact on the growth rate. Based on (9), the overall effect of capital share is positive, indicating that the provinces with higher capital shares have higher growth rates. Therefore, the difference in production functions in one of the important reasons causing regional differences. Column (3) is a test of the club convergence. It includes two regional dummy variables. In this model, the initial per capita output still has a significant and negative impact on the growth rate within the east region, the middle region, and the west region. This result implies that there is club convergence among the three areas, which is the same conclusion as current literature. Furthermore, capital share still has a very significant and positive impact on the growth rate. Again, the overall effect of the capital share is positive. That is, the higher the capital share, the higher the growth rate within three regions. The difference in production functions may increase the regional gaps. In order to alleviate the effect of business cycle, I put up robust tests by using the average growth rate covering 2001 to 2005 as dependent variables in Columns (4)-(6). The signs of coefficients are the same with the columns (1) and (3), and magnitude is slightly smaller. 5. Conclusion This paper proposes a new explanation on the increasing disparity among provinces in China after 1990. It argues that aggregate production functions may be different from different countries or regions. The difference is reflected by the capital share in total income. The higher the capital share, the higher the real per capita output in the region. Real output per capita is divergent in different countries 15 or regions. The empirical research using provincial level data in China indicates that the impact of capital share on real per capita output is strong and very significant. Provinces with higher capital shares have higher outputs. After controlling the heterogeneous capital shares, there is still conditional convergence in the provincial growth rate. And there is still club convergence within three areas. This conclusion is consistent with current literatures. 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