HUMAN CAPITAL INEQUALITY AND ECONOMIC
GROWTH:SOME NEW EVIDENCE*
Amparo Castello ?and Rafael Dome ?nech
This paper provides new measures of human capital inequality for a broad panel of countries.
Taking attainment levels from Barro and Lee (2001),we compute Gini coef?cients and the distribution of education by quintiles for 108countries over ?ve-year intervals from 1960to https://www.doczj.com/doc/a99561756.html,ing this new cross-country data on human capital inequality two main conclusions are obtained.First,most countries in the world have tended to reduce the inequality in human capital distribution.Second,human capital inequality measures provide more robust results than income inequality measures in the estimation of standard growth and investment equa-tions.
How is inequality generated?How does inequality evolve over time?How does inequality inˉuence other variables such as economic growth?Numerous re-searchers have tried to answer these questions over the years.Initially,economists paid attention to factors that determine income inequality as,for example,in the inˉuential work of Kuznets (1955),who analysed the effects of economic devel-opment upon the evolution of the distribution of income.In contrast,more recent literature addresses the question of how income or wealth distribution affects the growth of income,that is,it focuses on the potential effects of inequality on economic growth through different channels.1
In spite of the distribution of wealth being the relevant inequality source in theoretical models,the scarcity of available data on the distribution of wealth for a broad number of countries and for suf?ciently long periods leads most empirical studies to use income inequality data as a proxy for wealth inequality.2On other occasions,the distribution of wealth is proxied by the distribution of land.For example,Alesina and Rodrik (1994)and Deininger and Squire (1998)include land inequality along with income inequality to analyse the relationship between initial inequality in the asset distribution and long-term growth.
However,income and land inequality may be insuf?cient measures of wealth inequality since other variables such as human capital are also important determi-nants of wealth and growth.Thus,in some models that analyse the relationship between inequality and economic growth,the role played by human capital
*The authors wish to thank the editor,two anonymous referees,J.E.Bosca ?,D.Checchi,M.R.Sanmart??n and participants at the XV Annual Congress of the European Economic Association and at the 2001Annual Conference of the Royal Economic Society for very helpful comments.R.Dome ?nech acknowledges the ?nancial support of CICYT SEC99-0820.The data set is available at http://iei.uv.es/$rdomenec/human/h_ineq.html 1
We can distinguish three different mechanisms in this literature:the effects of wealth inequality on ?scal policy (Alesina and Rodrik,1994;Persson and Tabellini,1994),the effects on sociopolitical instability (Alesina and Perotti,1996),and ?nally,the effects on human capital accumulation (Galor and Zeira,1993).Excellent surveys of these theories are found in Perotti (1996)and Aghion et al.(1999).2
Measures of income inequality in growth regressions have been used by Persson and Tabellini (1994),Clarke (1995),Perotti (1996),Barro (2000),Banerjee and Duˉo (2000)or Forbes (2000).
The Economic Journal ,112(March ),C187±C200.óRoyal Economic Society 2002.Published by Blackwell Publishers,108Cowley Road,Oxford OX41JF,UK and 350Main Street,Malden,MA 02148,USA.
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endowment is very important if not crucial,since the distribution of income is mainly given by the distribution of human capital.For instance,Glomm and Ravikumar (1992),Saint-Paul and Verdier (1993)or Galor and Tsiddon (1997),among others,present models in which the source of inequality is mainly deter-mined by the distribution of human capital.But,at the same time,inequality affects human capital accumulation.In fact,some of the more interesting theories of how inequality affects growth (eg Galor and Zeira,1993)are based on the inter-action between imperfect credit markets,asset inequality and human capital accumulation.
The interest in these mechanisms at the theoretical level contrasts with the scarcity of empirical results.Due to the lack of available data on human capital inequality,little attention has been devoted to the inˉuence of human capital distribution on economic growth in empirical studies.Some exceptions are Bird-sall and London ?o (1997),and Lo
?pez et al.(1998).This ?rst study analyses a sample of 43countries and uses the standard deviation of years of education as the measure of human capital inequality.The problem with the standard deviation,however,is that it is an absolute measure of dispersion;thus it does not control for differences in the mean of the distribution.The second study uses a wider range of human capital inequality indicators but focuses on a reduced number of 12Asian and Latin American countries.
The objective of this paper is to provide new human capital inequality measures,that allow us to make a ?rst approximation of the relationship between these indicators and economic growth in a broad number of countries.In particular,using the recent information contained in Barro and Lee's (2001)data set about educational attainments,we calculate a human capital Gini coef?cient.Addition-ally,to improve the information provided by this aggregate measure of inequality,we also compute the distribution of education by quintiles,in line with a number of contributions to the analysis of the relationship between the distribution of income and economic growth.3
Apart from their intrinsic interest,an additional advantage of our indicators is that they conveniently complement the information provided by income in-equality measures.Clearly,income distribution data suffer from several problems.Besides the limitations related to their quality,the main problem arises with the different de?nitions of income used to measure inequality.Although the income distribution data set of Deininger and Squire (1996)has represented a signi?cant improvement in coverage and quality compared with previous data sets,there is room for improvement in other inequality indicators,especially in developing countries where income inequality data are more scarce.
Our new measures of human capital inequality allow us to have a closer look at the relationship between inequality and economic growth.The main ?ndings il-lustrate that human capital inequality measures provide more robust results than income inequality measures in the growth regressions.Moreover,the results
3
For instance,Persson and Tabellini (1994)use the third quintile as a measure of equality,Perotti (1996)combines the third and fourth quintile to capture the notion of `middle class',and Deininger and Squire (1996)calculate the top to the bottom quintile ratio as a measure of inequality.C188
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suggest that human capital inequality negatively inˉuences economic growth rates not only through the ef?ciency of resource allocation but also through a reduction in investment rates.
The structure of this paper is as follows.Section 1presents the procedure used to obtain the Gini index and the distribution of education by quintiles,and it compares the information provided by these different measures.Section 2analyses the distribution of the human capital inequality indicators across countries and their evolution from 1960to 2000.Section 3studies the relationship between human capital inequality and economic growth.Finally,Section 4contains the conclusions reached.
1.Measuring Human Capital Inequality
This section describes how we have obtained the measures of human capital in-equality for a broad cross section of countries.We take schooling ?gures from Barro and Lee (2001)to construct a standard representation of inequality such as the Gini coef?cient.The choice of this index to analyse inequality in the distri-bution of human capital is mainly due to the fact that it is the one normally used in international comparisons of income distribution.Nevertheless,as pointed out by Deininger and Squire (1996)among others,it is dif?cult to characterise in-equality by such a simple measure.For this reason,to extend the information provided by the Gini index we also report ?gures of the distribution of education by quintiles.
There are different ways of computing the Gini coef?cient.Since the Barro and Lee data set provides information on the average schooling years and attainment levels,the human capital Gini coef?cient (G h )can be computed as follows:4
G h
12H 3i 0 3j 0 x i à x j n i n j
1
where H are the average schooling years of the population aged 15years and over,i and j stand for the different levels of education,n i and n j are the shares of population with a given level of education,and x i and x j are the cumulative av-erage schooling years of each educational level.Following Barro and Lee (2001),we consider four levels of education:no schooling (0),primary (1),secondary (2)and higher education (3).De?ning x i as the average schooling years of each educational level i ,we observe that 5
x 0 x 0 0Y
x 1 x 1Y
x 2 x 1 x 2Y
x 3 x 1 x 2 x 3X
2
4
This expression has also been used in two recent papers by Thomas et al.(2000)and Checchi
(2000)to obtain a human capital Gini coef?cient.In contrast with our approach,Checchi (2000)uses the information of the education for the population aged 25years and over.For many developing countries a large portion of the labour force is younger than 25.Since most of our sample is composed of developing countries,we have used the educational information for the population aged 15and over.5
In terms of the variables of Barro and Lee's data set,we have that x 0 0,x 1 pyr 15a lp 15 ls 15 lh 15 Y x 2 syr 15a ls 15 lh 15 ,x 3 hyr 15a lh 15,n 0 lu 15,n 1 lp 15,n 2 ls 15,n 3 lh 15and H tyr 15X
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Expanding expression (1)and using (2),the Gini coef?cient can be computed as follows:
G h n 0
n 1x 2 n 2 n 3 n 3x 3 n 1 n 2
n 1x 1 n 2 x 1 x 2 n 3 x 1 x 2 x 3
X
3
Besides the Gini coef?cient,Barro and Lee's data set can also be used to obtain the distribution of education by quintiles.Table 1shows two examples that illus-trate how these measures have been obtained for two countries which are at op-posite extremes of the distribution.A good example of a large concentration of education,with a Gini coef?cient close to one,is Yemen in 1975.In this country 98.8%of the population had no schooling.This means that,in terms of quintiles,the ?rst four quintiles have no education,and all education is concentrated in the top quintile.On the contrary,a Gini coef?cient close to zero would represent the case where the attainment level in each quintile is similar.A good example is the United States in 1980where the share of total education attained by each quintile is around 0.2.
Although Barro and Lee's data set is the best available source on human capital stocks for a large sample of countries to date,as Krueger and Lindahl (2000)or De la Fuente and Dome ?nech (2001)have pointed out,the measurement errors for different education levels,due to the poor quality of the original sources,can be particularly substantial in some countries.To the extent that we use Barro and Lee's data set,we acknowledge the presence of some measurement errors which may distort the inequality variables we have computed.
Our data set includes 108countries from 1960to 2000,with a total of 935observations.All countries for which human capital variables are available have
Table 1
Two Examples of the Distribution of Education by Quintiles
Education levels Quintiles
i
x i n i s p s p s e x s u s Q s G h Yemen 000.98810.200.200000.990
(1975)
1 5.20.00520.200.40000
2 4.50.00530.200.60000
3 2.00.00140.200.8000050.20 1.000.0980.098 1.0United 000.00910.200.20 1.898 1.8980.1610.080
States 1 5.90.05120.200.40 2.279 4.1760.353(1980)
2 5.50.66230.200.60 2.279 6.4550.5463
2.9
0.281
40.200.80 2.5068.9610.7585
0.20
1.00
2.861
11.822
1.000
De?nitions:i is the education level (no schooling (0),primary (1),secondary (2)and tertiary (3)),x i is the average schooling years of each educational level,n i is the share of no schooling,primary,secondary and higher schooling attained by the population aged 15years and over,s is the quintile,p s is
the share of the population in each of the ?ve intervals, p s is the cumulative population share,e x s is the total schooling years attained by each of the quintiles,u s
s m 1e x m
and the cumulative share of education attained by each quintile is Q s u s a u 5.As an example,in the case of the United States in 1980,
since n 0 n 1`0X 2,the total schooling years attained by the ?rst quintile (e x
1 is given by e x 1 n 0x 0 n 1x 1 0X 20àn 1àn 0 x 1 x
2 1X 898X C190
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been classi?ed in seven different groups,and basic descriptive statistics of the Gini coef?cients and the distribution of education by quintiles are shown in Table 2.The second column of Table 2(G h )illustrates that South Asian countries are the group that,on average,have the largest human capital Gini coef?cient (0X 697).On the contrary,the countries with a more egalitarian distribution of human capital are the Advanced Countries with a mean of 0X 208.With regard to the quintiles,the third column shows that the ranking among countries in terms of human capital equality,measured by the third quintile (Q 3),is equal to the one obtained with the Gini coef?cient.However,the order differs with the ratio of the bottom quintile to the top one (column 4).Although South Asian countries show on average a larger Gini coef?cient than Sub-Saharan African countries,the lowest 20%of the population receives more education in the former group than in the latter.As a result,South Asia is the region with the greatest inequality,measured by the Gini coef?cient,whereas Sub-Saharan Africa is the region with the greatest inequality,measured by the ratio of the bottom to the top quintile.
The ?fth column (D ln G h )indicates that all groups of countries,with the exception of Advanced Countries,have decreased the inequality in the distribution of human capital.The average schooling years of the population aged 15years and over (H )is displayed in the sixth column.It shows that the economies with a higher stock of human capital are also the countries in which education is more evenly distributed.In fact,the correlation between G h and H is very high (à0X 90 .How-ever,Fig.1shows that the dispersion among both indicators increases substantially as the Gini index falls.We also ?nd many countries that,in spite of having the same average schooling years,signi?cantly differ in the distribution of education.According to recent World Bank ?gures,in the 1980s India and Indonesia,both highly populated countries,had similar levels of income inequality.They also had similar levels of average years of schooling:approximately 3.6years of formal edu-cation.However,as Fig.1shows,the distribution of education was quite different since the proportion of the population with no schooling and,at the other extreme,with a university education was much higher in India than in Indonesia.
Finally,there is a surprisingly low correlation between the human capital and the income Gini coef?https://www.doczj.com/doc/a99561756.html,paring the second and the seventh columns,we can appreciate that the countries with the lowest and the greatest inequality in the
Table 2
Average Human Capital Inequality Indicators by Groups of Countries
G h
Q 3Bot 20/Top 20
D ln G h H G y Middle East &North Africa 0.5830.1650.032)0.073 3.9310.403Sub-Saharan Africa
0.6370.1190.005)0.044 2.4300.481Latin America &the Caribbean 0.3670.3390.127)0.026 4.7840.494East Asia &the Paci?c 0.3770.3310.092)0.068 5.5580.405South Asia
0.6970.0800.010)0.041 2.4000.349Advanced Countries 0.2080.4550.3620.0037.9400.339Transitional Economies
0.223
0.447
0.299
)0.015
7.045
0.285
G h is the human capital Gini coef?cient,Q 3is the third quintile,Bot 20a Top 20is the ratio of the bottom to the top quintile,H is the stock of human capital and G y is the income Gini coef?cient.2002]
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distribution of education do not coincide with those in the distribution of income.These differences explain the low correlation between both inequality indicators (0X 27)shown in Fig.
2.
Fig.1.Human Capital Inequality and the Level of Human Capital
(average years of
schooling)
Fig.2.Income Inequality and Human Capital Inequality
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2.Variations Within and Across Countries
This section explores the variability of human capital inequality measures across and within countries.Li et al.(1998),using the Deininger and Squire (1996)data set,?nd that income inequality is relatively stable within countries but it varies signi?cantly among countries.This section tries to answer the following questions:How strong are the differences in human capital inequality among countries?Have these differences persisted over time or have they been reduced?
To study the possible existence of a general within-country time trend during the period 1960±2000,as well as the existence of country speci?c effects,we consider the following simple linear trend model:
HCI it a i b d t u it
4
where HCI is a human capital inequality indicator (G h and Bot 20a Top 20 ,i is the country,t the year and d a trend.As expected,the results indicate that the null hypothesis of equal country speci?c effects (a i a Y V i is rejected.Moreover,the coef?cient for a general trend is negative for the Gini index (à0X 021),positive for Bot 20a Top 20(0X 007)and statistically signi?cant in both cases.Therefore,the dif-ferences in the distribution of education across countries are substantial and,in general,countries have tended to reduce the inequality in human capital distri-bution over the period under study.
The preceding exercise shows how the mean of these indicators has evolved over time,but we also want to analyse how the dispersion and relative position of each country has changed from 1960to 2000.The easiest way to test the constancy in the dispersion of the Gini coef?cient is through a time plot of its standard devi-ation.Nevertheless,the standard deviation is not enough to characterise the dis-tribution when it is not a normal one.In particular,the Bera±Jarque test rejects the null hypothesis of normality of the distribution of the Gini coef?cient and the ratio of the bottom to the top quintile.For this reason,in Fig.3we have represented the non-parametric estimation of the density functions of G h using a truncated gaus-sian kernel for a distribution in the interval 0Y 1 .6As we can see,the distribution of G h in 1960clearly shows two modes,but slowly the density concentrates around a Gini coef?cient of 0X 25.Thus,although the distribution is clearly not normal,in 2000the standard deviation of the Gini coef?cient is well below that of any other preceding year.
This general reduction in the mean and the dispersion of human capital in-equality will be analysed thoroughly.In particular,with the purpose of analysing the relative position of each country and its evolution over time,we take into account the concept of Quah's probability transitional matrix.Quah (1993,1996)takes each country's per capita GDP relative to the world average and discretises the set of possible values into intervals at 1/4,1/2,1and 2.The probability transitional matrix indicates the probability that an economy in income group i
6
We thank F.J.Goerlich for providing the programme that implements such a procedure.The main result shown in Fig.3is robust to changes in bandwith parameters.2002]
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transits to income group j between two different years.The same qualitative information has been represented in Fig.4.In this ?gure,we plot the relative position of the Gini index across countries in 2000against their relative position in 1960,dividing the sample into the ?ve intervals considered by Quah.The cutting points between intervals are the logs of 1/4,1/2,1and 2times the (geometric)average of the Gini coef?cient (G h .These intervals allow us to con?rm the existence of convergence or polarisation of the Gini coef?cient.If countries had converged,they should have changed from their initial intervals in 1960to the one around the average in 2000.On the contrary,if there had been a polari-sation in human capital Gini coef?cients,countries should have changed from their initial intervals to the extremes.The diagonal simply reˉects the logs of the values of the Gini index in 1960multiplied by the sample average rate of growth between 1960and 2000.Thus,countries win or lose in relative positions according to their vertical distance to the diagonal.As we can see in Fig.4,some countries that showed a high inequality in 1960(that were in the ?fth interval)have reduced inequality,moving to the fourth interval by 2000.Likewise,some countries that were in the fourth interval in 1960have reduced inequality,moving to the third interval by 2000,as in the case of Korea,Taiwan and Hong Kong,which are good examples of an important reduction in human capital inequality.On the contrary,some OECD countries have worsened their relative positions.
Finally,Fig.4also shows the adjusted values of the Gini index in 2000after estimating an equation where the Gini index in 1960is the regressor,both in logs.The smaller the slope of the adjusted line,the bigger the convergence process is.As we observe in this ?gure,it seems that during the period 1960to 2000there has been a process of convergence in the values of the Gini coef?cients since the
slope
Fig.3.Density Functions of the Gini Coef?cient,1960±2000
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of the regression line is smaller than the slope of the diagonal line (the estimated coef?cient is equal to 0X 673with a t-ratio equal to 16X 49 .
3.Human Capital Inequality and Economic Growth
In this section,we focus on the effect that human capital inequality can exert on economic growth rates.To consider this issue,we add inequality variables to an equation where the average economic growth is explained by initial per capita income and the average accumulation rates of human and physical capital.Next,we increase the set of explanatory variables to prove the robustness of the initial results.Finally,we repeat the exercises with different samples.
The results of this section should be seen as a ?rst attempt to evaluate the effects of human capital inequality upon economic growth using these measures rather than as a de?nitive analysis of the determinants of growth or as a test of any particular model.Additionally,although we are aware of some recent developments in the econometric analysis of economic growth and convergence,such as the
contribu-Fig.4.Convergence of the Human Capital Gini Coef?cient (G h )in logs
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tions of Islam (1995),Caselli et al.(1996),and Lee,Pesaran and Smith (1997,1998),our approach relies deliberately on standard cross-section and pooling regressions in order to facilitate the comparison with previous results in related literature.7
The data used in our growth regressions comes from Barro and Lee (1994,2001)with the exception of the variables concerning inequality.With regard to income inequality,G y is proxied by Deininger and Squire's (1996)high quality income Gini coef?cient.Although we would like to include this variable at the beginning of the period (close to 1960),the problems with the availability of high quality income Gini data restrict us to including this variable as an average.8
In our initial regression,the average growth rate of per capita income from 1960to 1990(D ln y )is the dependent variable and income inequality is included as an explanatory variable along with the logs of human capital accumulation (ln s h ),de?ned as the total gross enrollment ratio for secondary education (taken from UNESCO),physical capital accumulation (ln s k ),de?ned as the ratio of real domestic investment to real GDP,the black market premium (BMP )to control for government distortions of markets,and initial per capita income (ln y ).The results (not shown here)are as expected:on the one hand,the coef?cients of the accumulation of factors are positive and statistically signi?cant and,on the other,initial income per capita and income inequality coef?cients are negatively and signi?cantly related to per capita income growth.9Nevertheless,this result is not robust to the inclusion of additional variables to the set of regressors.In particular,as obtained by Deininger and Squire (1998),the coef?cient of G y is not statistically signi?cant when we include regional dummies,which capture permanent and speci?c characteristics of the regions that otherwise could bias the coef?cients of some explanatory variables.Moreover,when we add human capital inequality in
1960,measured by the human capital Gini coef?cient (G h
60
)in this basic growth equation,the coef?cient of G y
even becomes positive,as shown in column (1)of
Table 3,whereas the coef?cient of G h
60is negative and signi?cant.These initial results suggest,therefore,that the negative relationship between income inequality and growth,obtained in previous studies,is not robust to the inclusion of other variables in the regression.
In column (2)of Table 3,income inequality has been excluded from the set of explanatory variables.Although our human capital inequality indicator may contain some measurement errors as we have mentioned earlier,in contrast to the results for income inequality,the coef?cient of human capital inequality is again negative and statistically signi?cant when regional dummies are present.A similar result is obtained in column (3)when we include other explanatory variables often con-sidered in the literature:the log of the average population growth (as in Mankiw et al.
7
For example,using panel techniques to control for time invariant country-speci?c effects,Forbes (2000)gets very different results of the effects of income inequality on economic growth to the ones obtained by Alesina and Rodrik (1994),Perotti (1996),Persson and Tabellini (1994),or Deininger and Squire (1998).8
The inclusion of the income Gini at the beginning of the period signi?cantly reduces the size of the sample.As the variability of this coef?cient is very low throughout the whole period (see Li et al.,1998)the inclusion of this variable as an average does not change the main results.9
A negative relationship between income inequality and growth can be found in Alesina and Rodrik (1994),Persson and Tabellini (1994),Clarke (1995)or Perotti (1996).C196
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(1992),augmented by the rate of technical progress and the depreciation rate,assumed to be 0.02and 0.03,respectively)and public consumption over GDP.The preceding regressions consider the direct effect of inequality on growth,but it is also possible that inequality variables are related indirectly to economic growth through the accumulation of factors.Column (4)shows that once the rates of human and physical capital accumulation are ruled out from the set of explanatory variables,initial human capital inequality has a bigger negative effect on the economic growth rates,suggesting that the indirect effects through the accumu-lation rates are also very important.However,given the high correlation between
human capital inequality and its stock,it is possible that G h
60
may be picking up the effect of the level of human capital on growth.In column (5)we test this
Table 3
Cross-section Regressions
D ln y
ln s k
(1)
(2)(3)(4)
(5)(6)(7)(8)(9)Constant 0.1220.1420.1470.1330.1290.118)1.208 1.969 1.653(5.19)(7.20)(4.65)(7.96)(7.05)(5.82)(1.89)(1.75)(1.33)ln y 60)0.011)0.011)0.013)0.011)0.011)0.010)0.015)0.077)0.082(4.14)(5.17)(6.01)(5.95)
(6.01)
(4.82)
(0.21)(1.06)(1.15)ln s h 0.0060.0060.0060.2010.2850.279(2.01)(2.46)(2.53)(2.08)(3.17)
(2.94)
ln s k 0.0050.0070.005(1.35)(2.65)(1.74)G y
0.0380.0330.846(2.12)(1.95)(1.61)G h 60
)0.021)0.017)0.016)0.028)0.022)0.030)0.717)0.849)0.604(2.76)(3.08)(2.70)(6.37)(2.21)(5.89)(2.93)(3.86)(1.67)BMP )0.004)0.006)0.006)0.005)0.006)0.003)0.022)0.266)0.268(1.74)
(2.81)
(3.36)(3.07)(3.11)(1.37)(0.20)
(1.60)(1.60)ln n 0X 05 )0.0040.7270.672(0.39)(2.52)(2.35)g )0.051)0.063)0.061)0.063)0.593)0.518(2.54)(3.38)(3.25)(3.01)(2.17)(2.08)ln H 60
0.0020.076(0.71)(0.83)Latin America )0.019)0.014)0.016)0.020)0.020)0.022)0.298)0.346)0.347(4.67)(5.78)(5.07)(8.34)(8.34)(6.24)(2.53)(4.15)(4.17)Sub-Saharan Africa )0.022)0.018)0.018)0.026)0.026)0.028)0.243)0.093)0.103(4.18)(5.01)(5.05)(8.32)(8.38)(6.52)(1.12)(0.49)(0.55)East Asia 0.0190.0190.0170.0180.0170.0180.011)0.091)0.119(3.90)(4.35)(3.75)(3.56)(3.23)(3.17)(0.10)(0.81)(1.07)"R
20.7440.7710.7890.7580.7560.7560.5780.7060.704N.obs.
67
83
83
83
83
67
67
83
83
Note:Cross-country regressions.White's heteroscedasticity-consistent t ratios in parentheses.Explanatory variables:per capita income in 1960(y ),total gross enrollment ratio in secondary education (s h ),the investment rate (s k ),the income Gini coef?cient (G y ),the human capital Gini
coef?cient in 1960(G h
60),the black market premium (BMP )in 1960,average population growth (n ),total years of schooling of the population aged 15and over in 1960(H 60)and public consumption (g ,from PWT 5.5).Latin -America ,Sub -Saharan Africa and East Asia are regional dummies.Dummies for Botswana and Philippines in cols.(1)to (6),Bangladesh in col.(7),and Mozambique in cols.(8)to (9)are included,since their residuals exceed more than three times the standard error of the estimated residuals.
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hypothesis.Although the coef?cient is now smaller than in column (4),the results suggest that the effects of the distribution of human capital are important and signi?cant.Column (6)shows the results of adding income inequality to the speci?cation estimated in column (4):the coef?cient of G y is again positive
whereas that of G h
60
is negative and statistically signi?cant.Taking these results together,the coef?cient of human capital inequality ranges approximately be-tween à0X 015and à0X 03.As G h fell by 0X 10points on average from 1960(G h 1960
0X 41)to 2000(G h 2000 0X 31),the effects on the average annual growth rates range between 0X 15and 0X 30%.
An alternative way of analysing the indirect effects of human capital inequality on growth is pursued in columns (7)to (9),using now the log of accumulation of physical capital (s k )as the dependent variable.In column (7)we observe that initial human capital inequality has a signi?cant and negative effect on physical capital accumulation.On the contrary,the results concerning income inequality are quite different since the effect of the income Gini coef?cient on physical capital accumulation seems to be positive.Columns (8)and (9)show that the negative effects of human capital inequality on the investment rate survive the inclusion of other variables such as population growth,public consumption and the initial stock of human capital.
Finally,we have analysed the robustness of these results.10First,all the results hold when we examine their sensitivity to the exclusion of atypical observations in our sample.Second,we test if results in Table 3hold with pooled data.Extending the data in the temporal dimension allows us to use lagged variables as instruments to control for the existence of endogenous explanatory variables,such as the accumulation rates.The regressions with pooled data con?rm the results obtained in the previous exercises:human capital inequality has a negative and statistically signi?cant effect upon growth mainly through the investment rate.Third,we have also analysed the robustness using different measures of inequality.In particular,the results with the third quintile as a measure of equality are quite similar to the ones obtained with the Gini coef?cient.Finally,these negative effects of human capital inequality are also supported when we do the same regressions for a sample of developing countries only.
4.Conclusions
The main objective of this paper has been to provide indicators of human capital inequality for a large sample of countries and years,and to analyse their inˉuence on the economic growth process.To construct the indicators of human capital inequality,we have distributed school attainment levels by quintiles and we have calculated a human capital Gini coef?cient.One of the main advantages of these indicators is that they may conveniently complement the information provided by income inequality measures.
Using these new indicators two main ?ndings are obtained.First,the variability of human capital inequality indicators is greater across countries than within each
10
The results are available upon request.
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country.Nevertheless,as a result of a general reduction in human capital in-equality,a process of convergence in human capital equality has taken place.Second,whereas the negative effect of income inequality on economic growth rates is not robust to the inclusion of regional dummies to the set of regressors,the cross-country and pool regressions suggest that there is a negative effect of human capital inequality on economic growth rates.This result is robust to changes in the explanatory variables,the exclusion of atypical observations,the use of instru-mental variables to control for endogeneity problems and the utilisation of dif-ferent measures of human capital inequality.
In short,these ?ndings indicate that education inequality is associated with lower investment rates and,consequently,lower income growth.Countries that in 1960showed greater inequality in the distribution of education have experienced lower investment rates than countries which showed less inequality.These lower investment rates have in turn meant lower income growth rates.Policies,therefore,conducted to promote growth should not only take into account the level but also the distribution of education,generalising the access to formal education at dif-ferent stages to a wider section of the population.Universitat Jaume I,Castello ?n,Spain Universidad de Valencia,Spain
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双代号网络图计算方法是每年建造师考试中的必考题，小到选择题、大到案例分析题，笔者在此总结2种计算方法，并附实例，供大家参考学习，互相交流，考出好成绩。 双代号网络图计算方法一 一、要点： 任何一个工作总时差≥自由时差 自由时差等于各时间间隔的最小值（这点对六时参数的计算非常用用） 关键线路上相邻工作的时间间隔为零，且自由时差=总时差 最迟开始时间—最早开始时间（最小） 关键工作：总时差最小的工作 最迟完成时间—最早完成时间（最小） 在网络计划中，计算工期是根据终点节点的最早完成时间的最大值 二、双代号网络图六时参数我总结的计算步骤（比书上简单得多） ①② t过程 做题次序： 1 4 5 ES LS TF 2 3 6 FS LF FF
步骤一： 1、A 上再做A 下 2 3、起点的A 上=0，下一个的A 上 A 上 4、A 下=A 上+t 过程（时间） 步骤二： 1、 B 下再做B 上 2、 做的方向从结束点往开始点 3、 结束点B 下=T （需要的总时间=结束工作节点中最大的A 下） 结束点B 上=T-t 过程（时间） 4、B 下=前一个的B 上（这里的前一个是从终点起算的） 遇到多指出去的时，取数值小的B 上 B 上=B 下—t 过程（时间） 步骤三： 总时差=B 上—A 上=B 下—A 下 如果不相等，你就是算错了 步骤四： 自由时差=紧后工作A 上（取最小的）—本工作A 下 =紧后工作的最早开始时间—本工作的最迟开始时间 （有多个紧后工作的取最小值） 例：
双代号网络图计算方法二 一、双代号网络图6个时间参数的计算方法（图上计算法） 从左向右累加，多个紧前取大，计算最早开始结束； 从右到左累减，多个紧后取小，计算最迟结束开始。 紧后左上-自己右下=自由时差。 上方之差或下方之差是总时差。 计算某工作总时差的简单方法：①找出关键线路，计算总工期； ②找出经过该工作的所有线路，求出最长的时间 ③该工作总时差=总工期-② 二、双代号时标网络图 双代号时标网络计划是以时间坐标为尺度编制的网络计划，以实
一、双代号网络图6个时间参数的计算方法（图上计算法） 从左向右累加，多个紧前取大，计算最早开始结束； 从右到左累减，多个紧后取小，计算最迟结束开始。 紧后左上-自己右下=自由时差。 上方之差或下方之差是总时差。 计算某工作总时差的简单方法：①找出关键线路，计算总工期； ②找出经过该工作的所有线路，求出最长的时间 ③该工作总时差=总工期-② 二、双代号时标网络图 双代号时标网络计划是以时间坐标为尺度编制的网络计划，以实箭线表示工作，以虚箭线 表示虚工作，以波形线表示工作的自由时差。 双代号时标网络图 1、关键线路 在时标双代号网络图上逆方向看，没有出现波形线的线路为关键线路（包括虚工作）。如图中①→②→⑥→⑧ 2、时差计算 1）自由时差 双代号时标网络图自由时差的计算很简单，就是该工作箭线上波形线的长度。 如A工作的FF=0，B工作的FF=1 但是有一种特殊情况，很容易忽略。
如上图，E工作的箭线上没有波形线，但是E工作与其紧后工作之间都有时间间隔，此时E工作 的自由时差=E与其紧后工作时间间隔的最小值，即E的自由时差为1。 2）总时差。 总时差的简单计算方法： 计算哪个工作的总时差，就以哪个工作为起点工作（一定要注意，即不是从头算，也不是 从该工作的紧后算，而是从该工作开始算），寻找通过该工作的所有线路，然后计算各条线路的 波形线的长度和，该工作的总时差=波形线长度和的最小值。 还是以上面的网络图为例，计算E工作的总时差： 以E工作为起点工作，通过E工作的线路有EH和EJ，两条线路的波形线的和都是2，所以此时E 的总时差就是2。 再比如，计算C工作的总时差：通过C工作的线路有三条，CEH，波形线的和为4；CEJ，波 形线的和为4；CGJ，波形线的和为1，那么C的总时差就是1。
双代号网络图时间参数的计算 二、工作计算法 【例题】：根据表中逻辑关系，绘制双代号网络图，并采用工作计算法计算各工作的时间参数。
紧前- A A B B、C C D、E E、F H、G 时间 3 3 3 8 5 4 4 2 2 （一）工作的最早开始时间ES i-j --各紧前工作全部完成后，本工作可能开始的最早时刻。
3 6 14 （二）工作的最早完成时间EF i-j EF i-j= ES i-j + D i-j 1 ?计算工期T c等于一个网络计划关键线路所花的时间，即网络计划结束工作最早完成时间的最大值，即T c = max {EF i-n} 2 .当网络计划未规定要求工期T r时，T p= T c 3 .当规定了要求工期T r时，T c 2. 其他工作的最迟完成时间按逆箭头相减，箭尾相碰取小值”计算。--在不影响计划工期的前提下，该工作最迟必须完成的时刻。 （四）工作最迟开始时间LS i-j LS i-j = LF i-j —D i-j --在不影响计划工期的前提下，该工作最迟必须开始的时刻。 （五）工作的总时差TF i-j TF i-j = LS i-j —ES i-j 或TF i-j = LF i-j —EF i-j --在不影响计划工期的前提下，该工作存在的机动时间。 FF i-j = ES j-k — EF i-j 作业1 :根据表中逻辑关系，绘制双代号网络图。 工作 A B C D E F 紧前 工作 - A A B B 、 C D 、E 3 6 6 0 6 — 1 \i 3 F G(4) I 上卩1 0 0 0 3 3 6 9 3 4 14 T L8 0 z o T 5： ：1116 5 6 12 6 16 T Lfl J6 5 n N 0 0 0 3 3 0 6 9 3 4 14 5 6卩2 戶 - G(4) ：1114 L8 0 1 !0 4 :n 眇s Lfl 1(2) 11(2) 11 ■ Hl N r T 7 B(3) D(8) 6 E(5) X （六）自由时差 FF i-j --在不影响紧后工作最早开始时间的前提下, 该工作存在的机动时间。 6 k> K) ■1114 J E(5) 6 5： S F(4) D(8) 3 6 7 6 9 寄语：不管一建、二建，双代号是必考点，再复杂的网络图也能简单化， 本工作室整理了 三页纸供大家快速掌握，希望大家多学多练，掌握该知识 点，至少十分收入囊中。 双代号网络图六个参数计算的简易方法 一、非常有用的要点： 任何一个工作总时差≥自由时差 自由时差等于各时间间隔的最小值（这点对六时参数的计算非常用用） 关键线路上相邻工作的时间间隔为零，且自由时差=总时差 最迟开始时间—最早开始时间（最小） 关键工作：总时差最小的工作 最迟完成时间—最早完成时间（最小） 在网络计划中，计算工期是根据终点节点的最早完成时间的最大值 二、双代号网络图六时参数我总结的计算步骤（比书上简单得多） ① ② t 过程 做题次序： 1 4 5 ES LS TF 2 3 6 FS LF FF 步骤一： 1、A 上再做 A 下 2、 做的方向从起始工作往结束工作方向； 3、 起点的 A 上=0，下一个的 A 上=前一个的 A 下当遇到多指向时，要取数值大的 A 下 A 上 4、 A 下=A 上+t 过程（时间） 步骤二： 1、 B 下再做 B 上 2、 做的方向从结束点往开始点 3、 结束点 B 下=T （需要的总时间结束点 B 上=T-t 过程（时间） 4、 B 下=前一个的 B 上（这里的前一个是从终点起算的） 遇到多指出去的时，取数值小的 B 上 B 上=B 下—t 过程（时间） 步骤三： 总时差=B 上—A 上=B 下—A 下 如果不相等，你就是算错了 步骤四： 自由时差=紧后工作 A 上（取最小的）—本工作 A 下 =紧后工作的最早开始时间—本工作的最迟开始时间 （有多个紧后工作的取最小值） 例： 建筑工程双代号网络图是应用较为普遍的一种网络计划形式。它是以箭线及其两端节点的编号表示工作的网络图。 双代号网络图中的计算主要有六个时间参数： ES：最早开始时间，指各项工作紧前工作全部完成后，本工作最有可能开始的时刻； EF：最早完成时间，指各项紧前工作全部完成后，本工作有可能完成的最早时刻 LF：最迟完成时间，不影响整个网络计划工期完成的前提下，本工作的最迟完成时间； LS：最迟开始时间，指不影响整个网络计划工期完成的前提下，本工作最迟开始时间； TF：总时差，指不影响计划工期的前提下，本工作可以利用的机动时间； FF：自由时差，不影响紧后工作最早开始的前提下，本工作可以利用的机动时间。 双代号网络图时间参数的计算一般采用图上计算法。下面用例题进行讲解。 例题：试计算下面双代号网络图中，求工作C的总时差？ 早时间计算：ES，如果该工作与开始节点相连，最早开始时间为0，即A的最早开始时间ES=0； EF，最早结束时间等于该工作的最早开始+持续时间，即A的最早结束EF为0+5=5； 如果工作有紧前工作的时候，最早开始等于紧前工作的最早结束取大值，即B的最早开始FS=5，同理最早结束EF为5+6=11，而E 工作的最早开始ES为B、C工作最早结束（11、8）取大值为11。 最迟完成时间计算：LF，从最后节点开始算起也就是自右向左。 如果该工作与结束节点相连，最迟完成时间为计算工期23，即F的最迟结束时间LF=23； 中间工作最迟完成时间等于紧后工作的最迟完成时间减去紧后工作的持续时间。如果工作有紧后工作，最迟完成时间等于紧后工作最迟开始时间取小值。 LS，最迟开始时间等于最迟结束时间减去持续时间，即LS=LF-D； 时差计算： FF，自由时差=（紧后工作的ES-本工作的EF）； TF，总时差=（紧后工作的LS-本工作的ES）或者=（紧后工作的LF-本工作的EF）。 该题解析： 则C工作的总时差为3. 双代号网络图的绘制技巧 双代号网络图又称网络计划技术或箭条图，简称网络图。在我国随着建筑领域投资包干和招标承包制的深入贯彻执行，在施工过程中对进度管理、工期管理和成本监督方面要求愈益严格，网络计划技术在这方面将成为有效的工具。借助电子计算机，从计划的编制、优化、到执行过程中调整和控制，网络计划技术突现出它的优势，越来越被人们广泛认识、了解和使用。 1 绘图中普遍存在的问题 常听说大家对网络图的绘制比较头疼。因为在绘图时，工序与工序之间的逻辑关系难以把握、什么地方需要架设虚工序看不出来、前边工序什么时候相交、如何为后行工序做准备、网络图开始如何绘制、结尾如何收口等一系列问题都是我们绘制网络图必须遇到的问题和步骤。 如果掌握绘制技巧就能快速准确地完成绘图要求。下面我把这几年自己总结出来一套有效的方法介绍给大家。 ２网络图的绘制技巧 2.１网络图的三大要素网络图是由节点、工序和线路三大要素构成的。 ２.1.１节点 节点是用圆圈表示箭线之间的分离与交会的连接点。它由不同的代号来区，表示工序的结束与工序的开始的瞬间，具有承上启下的连接作用；它不占用时间，也不消耗资源。在网络图中结点分为开始结点、结束结点和中间结点三种。２.１.2 工序（工作） 工序是指把计划任务按实际需要的粗细程度划分成若干要消耗时间、资源、人力和材料的子项目。在网络图中用两个节点和一条箭线表示。箭线上方表示工序代号，下方表示工序作业时间。 2.１.３线路 线路是指在双代号网络图中从起点节点沿着箭线方向顺序通过一系列箭线和节点而达到终点节点的通道。一个完整的网路图有若干条线路组成，在诸多线路中作业时间相加最长的一条称为关键线路，宜用粗箭线、双箭线表示，使其一目了然。 2.２网络图的绘制技巧 要想快速准确地绘制双代号网路图，应先把工程项目的“工作明细表”分四步认真仔细的进行分析与研究。 ２.２.１网络图开头绘制技巧先从“工作明细表”中找出开始的工序。寻找的方法是：只要在“先行工序”一列中没有先行工序的工序，必定是开始的工序。这时候只需画一个 双代号网络图的绘制方法 一、根据题目要求画出工作逻辑关系矩阵表，格式如下： 二、根据工作逻辑矩阵表计算工作位置代号表，为了使双代号网络图的条理清楚，各工作的布局合理，可以先按照下列原则确定各工作的开始节点位置号和结束节点位置号，然后按各自的节点位置号绘制网络图。 位置代号计算规则： ①无紧前工作的工作（即双代号网络图开始的第一项工作），其开始节点位置号为零； ②有紧前工作的工作，其开始节点位置号等于其紧前工作的开始节点位置号的最大值加1； ③有紧后工作的工作，其结束节点位置号等于其紧后工作的开始节点位置号的最小值； ④无紧后工作的工作（即双代号网络图开始的最后一项工作），其结束节点位置号等于网络图中各工作的结束节点位置号的最大值加1。 三、绘制双代号网络进度计划表，按照下列绘图原则： 1、绘制没有紧前工作的工作箭线，使他们具有相同的开始节点，以保证网络图只有一个起点节点。 2、依次绘制其他工作箭线。这些工作箭线的绘制条件是其所有紧前工作箭线都已经绘制出来。在绘制这些工作箭线时，应按下列原则进行： ①当所要绘制的工作只有一项紧前工作时，则将该工作箭线直接绘制在其紧前工作之后即可。 ②当所要绘制的工作只有多项紧前工作时，应按以下四种情况分别予以考虑： 第一种情况：对于所要绘制的工作而言，如果在其多项紧前工作中存 在一项（且只存在一项）只作为本工作紧前工作的工作（即在紧前工作栏中，该紧前工作只出现一次），则应将本工作箭线直接画在该紧前工作箭 线之后，然后用虚箭线将其他紧前工作箭线的箭头节点与本工作的箭尾节 点分别相连，以表达它们之间的逻辑关系。 第二种情况：对于所要绘制的工作而言，如果在其紧前工作中存在多项只作为本工作紧前工作的工作，应将这些紧前工作的箭线的箭头节点合并，再从合并之后节点开始，画出本工作箭线，然后用虚箭线将其他紧前工作箭线的箭头节点与本工作的箭尾节点分别相连，以表达它们之间的逻辑关系。 第三种情况：对于所要绘制的工作而言，如果不存在第一和第二种情况时，应判断本工作的所有紧前工作是否都同时是其他工作的紧前工作（即在紧前工作栏中，这几项紧前工作是否均同时出现若干次）。如果上述条件成立，应将这些紧前工作的箭线的箭头节点合并，再从合并之后节点开始，画出本工作箭线。 第四种情况：对于所要绘制的工作而言，如果不存在第一和第二种情况，也不存在第三种情况时，则应将本工作箭线单独划在其紧前工作箭线之后的中部，然后用虚箭线将其他紧前工作箭线的箭头节点与本工作的箭尾节点分别相连，以表达它们之间的逻辑关系。 3、当各项工作箭线都绘制出来以后，应合并那些没有紧后工作的工作箭线的箭头节点，以保证网络图只有一个终点节点。 概念部分 双代号网络图是应用较为普遍的一种网络计划形式。它是以箭线及其两端节点的编号表示工作的网络图，如图12-l所示。 图12-1 双代号网络图 双代号网络图中，每一条箭线应表示一项工作。箭线的箭尾节点表示该工作的开始，箭线的箭头节点表示该工作的结束。 工作是指计划任务按需要粗细程度划分而成的、消耗时间或同时也消耗资源的一个子项目或子任务。根据计划编制的粗细不同，工作既可以是一个建设项目、一个单项工程，也可以是一个分项工程乃至一个工序。 一般情况下，工作需要消耗时间和资源（如支模板、浇筑混凝土等），有的则仅是消耗时间而不消耗资源（如混凝土养护、抹灰干燥等技术间歇）。在双代号网络图中，有一种既不消耗时间也不消耗资源的工作——虚工作，它用虚箭线来表示，用以反映一些工作与另外一些工作之间的逻辑关系，如图12-2所示，其中2-3工作即为虚工作。 图12-2 虚工作表示法 节点是指表示工作的开始、结束或连接关系的圆圈（或其他形状的封密图形）、箭线的出发节点叫作工作的起点节点，箭头指向的节点叫作工作的终点节点。任何工作都可以用其箭线前、后的两个节点的编码来表示，起点节点编码在前，终点节点编码在后。 网络图中从起点节点开始，沿箭头方向顺序通过一系列箭线与节点，最后达到终点节点的通路称为线路。一条线路上的各项工作所持续时间的累加之和称为该线路之长，它表示完成该线路上的所有工作需花费的时间。理论部分： 一节点的时间参数 1．节点最早时间 节点最早时间计算一般从起始节点开始，顺着箭线方向依次逐项进行。 （1）起始节点 起始节点i如未规定最早时间ET i时，其值应等于零，即 （12-1） 式中——节点i的最早时间； （2）其他节点 双代号网络图参数计算简易方法 一、非常有用的要点： 任何一个工作的总时差≥自由时差； 自由时差等于各时间间隔的最小值（这点对六时参数的计算非常用用）； 关键线路上相邻工作的时间间隔为零，且自由时差=总时差； 最迟开始时间—最早开始时间（最小） 关键工作：总时差最小的工作 最迟完成时间—最早完成时间（最小） 在网络计划中，计算工期是根据终点节点的最早完成时间的最大值。 二、双代号网络图六时参数的计算步骤（比书上简单得多） 最早开始ES 最迟开始LS 总时差TF 最早完成EF 最迟完成LF 自由时差FF 做题次序： 1 4 5 2 3 6 先求最早开始，再求最早完成，然后求最迟完成，第4步求最迟开始，第5步求总时差，第6步求自由时差。 步骤一： 1、先求最早开始，然后求最早完成； 2、做题方向：从起始工作往结束工作方向； 3、起点的最早开始= 0，下一个的最早开始=前一个的最早完成；当遇到多指向时，取数值大的最早完成。 最早完成=最早开始+持续时间 步骤二： 1、先求最迟完成，然后求最迟开始； 2、做题方向：从结束工作往开始工作方向； 3、结束点的最迟完成=工期T，（需要的总时间=结束工作节点中最大的最迟完成）， 结束点的最迟开始=工期T－持续时间； 4、最迟完成=前一个的最迟开始（这里的前一个是从终点起算的）；遇到多指向的时候，取数值小的最迟开始； 最迟开始=最迟完成－持续时间 步骤三： 总时差=最迟开始－最早开始=最迟完成－最早完成；如果不相等，你就是算错了； 步骤四： 自由时差=紧后工作最早开始(取最小的)－最早完成。 例： 总结起来四句话： 1、最早开始时间从起点开始，最早开始=紧前最早结束的max值； 2、最迟完成时间从终点开始，最迟完成=紧后最迟开始的min值； 3、总时差=最迟－最早； 4、自由时差=紧后最早开始的min值－最早完成。 注：总时差=自由时差＋紧后总时差的min值。 一、一般双代号网络图（没有时标）6个时间参数的计算方法（图上计算法） 6时间参数示意图： （左上）最早开始时间 | （右上）最迟开始时间 | 总时差 （左下）最早完成时间 | （右下）最迟完成时间 | 自由时差 计算步骤： 1、先计算“最早开始时间”和“最早完成时间”（口诀：早开加持续）： 计算方法：起始工作默认“0”为“最早开始时间”，然后从左向右累加工作持续时间，有多个紧前工作的取大值。 2、再计算“最迟开始时间”和“最迟完成时间”（口诀：迟完减持续）： 计算方法：结束工作默认“总工期”为“最迟完成时间”，然后从右到左累减工作持续时间，有多个紧后工作取小值。（一定要注意紧前工作和紧后工作的个数） 3、计算自由时差（口诀：后工作早开减本工作早完）： 计算方法：紧后工作左上（多个取小）-自己左下=自由时差。 4、计算总时差（口诀：迟开减早开或迟完减早完）： 计算方法：右上-左上=右下-左下=总时差。 计算某工作总时差的简单方法：①找出关键线路，计算总工期； ②找出经过该工作的所有线路，求出最长的时间 ③该工作总时差=总工期-② 二、双代号时标网络图（有时标，计算简便） 双代号时标网络计划是以时间坐标为尺度编制的网络计划，以实箭线表示工作，以虚箭线表示虚工作（虚工作没有持续时间，只表示工作之间的逻辑关系，即前一个工作完成后一个工作才能开始），以波形线表示该工作的自由时差。（图中所有时标单位均表示相应的持续时间，另外虚线和波形线要区分） 示例：双代号时标网络图 1、关键线路 在时标双代号网络图上逆方向看，没有出现波形线的线路为关键线路（包括虚工作）。如图中①→②→⑥→⑧ 般双代号网络图（没有时标）6个时间参数的计算方法（图上计算法） 6时间参数示意图: （左上）最早开始时间| （右上）最迟开始时间| 总时差 （左下）最早完成时间| （右下）最迟完成时间| 自由时差 计算步骤： 1、先计算“最早开始时间”和“最早完成时间” （口诀：早开加持续）： 计算方法：起始工作默认“ 0”为“最早开始时间”，然后从左向右累加工作持续时间，有多个紧前工作的取大值。 2、再计算“最迟开始时间”和“最迟完成时间” （口诀：迟完减持续）： 计算方法：结束工作默认“总工期”为“最迟完成时间”，然后从右到左累减工作 持续时间，有多个紧后工作取小值。（一定要注意紧前工作和紧后工作的个数） 3、计算自由时差（口诀：后工作早开减本工作早完）： 计算方法：紧后工作左上（多个取小）-自己左下=自由时差。 4、计算总时差（口诀：迟开减早开或迟完减早完）： 计算方法：右上-左上二右下-左下二总时差。 计算某工作总时差的简单方法：①找出关键线路，计算总工期; ②找出经过该工作的所有线路，求出最长的时间 ③该工作总时差=总工期-② 二、双代号时标网络图（有时标，计算简便） 双代号时标网络计划是以时间坐标为尺度编制的网络计划，以实箭线表示工作，以虚箭线表示虚工作（虚工作没有持续时间，只表示工作之间的逻辑关系，即前一个工作完成后一个工作才能开始），以波形线表示该工作的自由时差。（图中所有时标单位均表示相应的持续时间，另外虚线和波形线要区分） 示例：双代号时标网络图 双代号吋标网络图 1、关键线路 在时标双代号网络图上逆方向看，没有出现波形线的线路为关键线路（包括虚工作）如图中①一②一⑥一⑧ 2时差计算（这里只说自由时差和总时差，其余4个时差参见前面的累加和累减）1）自由 最新文件---------------- 仅供参考--------------------已改成-----------word文本 --------------------- 方便更改 赠人玫瑰，手留余香。 双代号网络图参数计算的简易方法 一、非常有用的要点： 任何一个工作总时差≥自由时差 自由时差等于各时间间隔的最小值（这点对六时参数的计算非常用用） 关键线路上相邻工作的时间间隔为零，且自由时差=总时差 最迟开始时间—最早开始时间（最小） 关键工作：总时差最小的工作 最迟完成时间—最早完成时间（最小） 在网络计划中，计算工期是根据终点节点的最早完成时间的最大值。 二、双代号网络图六时参数总结的计算步骤（比书上简单得多） 最早开始时间ES 最迟开始时间LS 总时差 最早完成时间EF 最迟完成时间LF 自由时差 简记为： A 上 B 上 总时差 A下 B下自由时差 ①② t过程 做题次序： 1 4 5 2 3 6 步骤一： 1、A上再做A下 2、的方向从起始工作往结束工作方向； 3、起点的A上=0，下一个的A上=前一个的A下； 当遇到多指向时，要取数值大的A 下 A上 4、A下=A上+t过程（时间） 步骤二： 1、B下再做B上 2、做的方向从结束点往开始点 3、结束点B下=T（需要的总时间=结束工作节点中最大的A下） 结束点B 上= T-t过程（时间） 4、B下=前一个的B上（这里的前一个是从终点起算的） 遇到多指出去的时，取数值小的B上 B下 t过程（时间） B上=B下—t过程（时间） 步骤三： 总时差=B 上—A 上 =B 下 —A 下 如果不相等，你就是算错了步骤四： 自由时差=紧后工作A 上（取最小的）—本工作A 下 例： 6 8 2 * 9 11 2 双代号网络图作为工程项目进度管理中，是最常用的工作进度安排方法，也是工程注册类执业考试中必考内容，对它的掌握程度，决定了实务考试的通过概率大小。 双代号网络图时间参数主要为6个时间参数（最早开始时间、最早完成时间、最迟开始时间、最迟完成时间、总时差和自由时差）的计算，按计算方法可以分为： 1、节点计算法 2、工作计算法 3、表格计算法 节点计算法最适合初学者，其计算方法简单、快速。 计算案例： 某工程项目的双代号网络见下图。（时间单位：月） [问题] 计算时间参数和判断关键线路。 [解答] 1、计算时间参数 （1）计算节点最早时间，计算方法：最早时间：从左向右累加，取最大值。 （2）计算最迟时间, 最迟时间计算方法：从右向左递减，取小值。 2、计算工作的六个时间参数 自由时差：该工作在不影响其紧后工作最早开始时间的情况下所具有的机动时间。 总时差：该工作在不影响总工期情况下所具有的机动时间。 通过前面计算节点的最早和最迟时间，可以先确定工作的最早开始时间和最迟完成时间，根据工作持续时间，计算出最早完成时间和最迟开始时间，以F工作为例，计算F工作的4个参数（以工作计算法标示）如下： 注:EF=ES+工作持续时间 LF=LS+工作持续时间 接下来计算F工作的总时差TF，在工作计算法中，总时差TF=LS-ES或LF-EF，在节点计算法，总时差TF可以紧后工作的最迟时间-本工作的最早完成时间，或者是紧后工作最迟时间-最早时间，以F工作为例计算它的TF： 接下来计算F工作的自由时差FF，根据定义：该工作在不影响其紧后工作最早开始时间的情况下所具有的机动时间，自由时差FF=紧后工作最早（或最小）开始时间-本工作最早完成时间ES，以F工作为例，F的紧后工作为G和H，G工作的最早开始时间为10（即4节点的最早时间），H工作的最早开始时间为11（即5节点的最早时间），G工作的时间最小，所以F的自由时差FF=G工作的最早开始时间ES-F工作的最早完成时间EF： 一、双代号网络图6个时间参数的计算方法（图上计算法）从左向右累加，多个紧前取大，计 算最早开始结束；从右到左累减，多个紧后取小，计算最迟结束开始。 紧后左上-自己右下=自由时差。上方之差或下方之差是总时差。 计算某工作总时差的简单方法：①找出关键线路，计算总工期； ②找出经过该工作的所有线路，求出最长的时间 ③该工作总时差=总工期-② 二、双代号时标网络图双代号时标网络计划是以时间坐标为尺度 编制的网络计划，以实箭线表示工作，以虚箭线 表示虚工作，以波形线表示工作的自由时差。 双代号时标网络图 1、关键线路 在时标双代号网络图上逆方向看，没有出现波形线的线路为关键线路（包括虚工作）如图中①一②一⑥一⑧ 2、时差计算1）自由时差 双代号时标网络图自由时差的计算很简单，就是该工作箭线上波形线的长度。 如A工作的FF=O, B工作的FF=1 但是有一种特殊情况，很容易忽略。 如上图，E工作的箭线上没有波形线，但是E工作与其紧后工作之间都有时间间隔，此时E X作的自由时差=E与其紧后工作时间间隔的最小值，即E的自由时差为1。 2）总时差。 总时差的简单计算方法： 计算哪个工作的总时差，就以哪个工作为起点工作（一定要注意，即不是从头算，也不 是 从该工作的紧后算，而是从该工作开始算），寻找通过该工作的所有线路，然后计算各 条线路的 波形线的长度和，该工作的总时差=波形线长度和的最小值。 还是以上面的网络图为例，计算E工作的总时差： 以E工作为起点工作，通过E工作的线路有EH ffi EJ,两条线路的波形线的和都是2,所以此时E 的总时差就是2。 再比如，计算C工作的总时差：通过C工作的线路有三条，CEH波形线的和为4; CEJ 波形线的和为4；CGJ波形线的和为1,那么C的总时差就是1双代号网络图六个参数计算方法(各实务专业通用)
双代号网络图最简单的计算方法
双代号网络图的绘制技巧
双代号网络图的绘制方法
双代号网络图计算(新)
双代号网络图计算最简便方法
双代号网络计划图计算方法简述
双代号网络计划图计算方法口诀简述
双代号网络图参数计算的简易方法(优选.)
双代号网络图时间参数计算技巧
双代号网络图解析实例
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