NBER WORKING PAPER SERIES
HUMAN CAPITAL SPILLOVERS IN FAMILIES:
DO PARENTS LEARN FROM OR LEAN ON THEIR CHILDREN?
Ilyana Kuziemko
Working Paper 17235
https://www.doczj.com/doc/1616334891.html,/papers/w17235
NATIONAL BUREAU OF ECONOMIC RESEARCH
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July 2011
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Human Capital Spillovers in Families: Do Parents Learn from or Lean on their Children? Ilyana Kuziemko
NBER Working Paper No. 17235
July 2011
JEL No. H23,I2,I28,J12,J13,J24
ABSTRACT
I develop a model in which a child's acquisition of a given form of human capital incentivizes adults in his household to either learn from him (if children act as teachers then adults' cost of learning the skill falls) or lean on him (if children's human capital substitutes for that of adults in household production then adults' benefit of learning the skill falls). I exploit regional variation in two shocks to children's human capital and examine the effect on adults. The rapid introduction of primary education for black children in the South during Reconstruction not only increased literacy of children but also of adults living in the same household ("learning" outweighs "leaning"). Conversely, the 1998 introduction of English immersion in California public schools appears to have increased the English skills of children but discouraged adults living with them from acquiring the language ("leaning" outweighs "learning"). Whether family members learn from or lean on each other has implications for the externalities associated with education policies.
Ilyana Kuziemko
361 Wallace Hall
Princeton University
Princeton, NJ 08544
and NBER
kuziemko@https://www.doczj.com/doc/1616334891.html,
1Introduction
Parents are often a child’s?rst teachers,and economic models have long recognized the role parents play in passing on human capital to their children(see Becker and Tomes1979, Becker and Tomes1986as well as the response by Goldberger1989).In contrast,these models generally assume that children’s human capital has little contemporaneous e?ect on parents and other adults in their household;it generally does not enter into the household production function and is not transferred to adults by peer e?ects or some other form of learning.1The empirical treatment of intergenerational transmission of human capital has followed the theoretical literature in focusing chie?y on the transmission from parents to children(see,for example,Behrman and Rosenzweig2002,Sacerdote2002,Plug2004,Black et al.2005,and Oreopoulos et al.2006).
In this paper,I model the transfer of human capital from children to adults.In contrast to the classic models of intergenerational human capital transmission,which?nd that,all else equal,an increase in parents’human capital leads to an increase in that of their children, I show that children’s human capital investment can either increase or decrease that of the adult members of their household.The sign of the e?ect depends on the household production function and the learning technology.
Suppose a child exogenously acquires a new skill.On the one hand,an adult can learn from the child,as the cost to adults of learning the skill will fall if their children can teach it to them.This“learning e?ect”suggests positive human capital spillovers from children to adults.On the other hand,an adult can lean on the child,as the bene?t to adults of acquiring the skill will fall if children’s human capital can substitute for that of adults in the household production function.This“leaning e?ect”suggests negative spillovers.Moreover,the model I present o?ers a framework for predicting which types of human capital lend themselves to“leaning”versus“learning.”The higher the cost of alternative methods of acquiring the
1Ehrlich and Lui(1991)assume children’s human capital a?ects parents in their old-age and thus parents invest in their children’s human capital because they will one day depend on their children’s income.But the direction of the investment in this model is still from parents to children.
skill,the more adults will learn from their children.The more children’s human capital can directly increase adults’consumption,the more adults will lean on their children.
The empirical work focuses on two examples where children received a plausibly exoge-nous shock to their human capital and estimates its e?ect on the human capital investment of the adults living with them.During the Reconstruction era following the U.S.Civil War, the federal government created the Freedmen’s Bureau to administer thousands of schools for black children in the South.Whereas essentially no black children in the Confederacy had access to formal schooling before the Civil War,over ten percent of those between the ages of ten and twelve were enrolled in school during the1869-1870school year.2Using household-level variation based on children’s ages interacted with county-level variation in the educational levels of black children,I?nd that living with a literate child increased the probability a black adult would be literate himself.Thus,on net,Reconstruction-era Southern blacks appear to have learned from their literate children.
In1998,California voters passed Proposition227,which replaced bilingual education with English immersion in public schools.When classes ended for the summer in1998,29percent of English-learners received core academic instruction in their native languages;when classes resumed that September,only11percent https://www.doczj.com/doc/1616334891.html,ing geographic variation in compliance with Proposition227across California,I?nd that although children living in highly compliant areas are more likely to speak English after the reform,the adults living with children in these areas are less likely to be English pro?cient.Thus,on net,immigrant parents appear to lean on their English-pro?cient children.In both this and the Reconstruction context,the results are driven by households with children of school-going age,suggesting that children’s human capital acquisition,and not some omitted variable,is driving the e?ect on adults.
The results in this paper may interest a variety of researchers and policy-makers.First, the model I present highlights the possibility of“negative”human capital spillovers,which has received little attention among economists studying peer e?ects.Of course,economists 2My calculations from the1870Integrated Public Use Microdata Series(IPUMS).
have studied free-riding in the context of public-goods games and team work e?ort,but,in the context of human capital,have generally assumed individuals learn from their peers.
Second,as most educational policies target children,determining the extent of human capital spillovers to older members of the household would allow policy-makers to better compare the marginal social bene?ts and costs of these policies.In the case of English acquisition,gains to children may be tempered by the negative spillovers on adults.My results suggest that if policy-makers wish to assimilate entire immigrant families,separate programs may need to target adults as teaching only children may in fact slow adults’progress.In contrast,the literacy patterns of Reconstruction-era Southern blacks suggest, as in Miguel and Kremer(2004),that program evaluations that consider only the e?ects on a policy’s prime targets may systematically underestimate its social bene?ts.
Third,the transmission of human capital from children to adults likely plays an especially important role in developing countries.Unlike many developed countries in which average educational attainment has plateaued,educational attainment in developing countries is still rising with each successive cohort,so children often have more total years of schooling than their parents and thus opportunities to teach them new information and skills.To my knowl-edge,few if any development economics papers have examined whether interventions that target children a?ect the adults with whom they live.Given the scarcity of resources of gov-ernments and NGOs in developing countries,promoting investments with positive spillovers to parents and addressing situations with negative spillovers could lead to important welfare gains.
Fourth,even in cases where parents and children have the same level of formal education, children often invest more in learning how to use new technologies(e.g.,computers).3Thus, in settings with rapid technology growth,child-to-adult spillovers may play an especially
3I have found very little academic research on the implications of age-speci?c technology adop-tion.However,marketing research suggests that one-half of U.S.children have helped their parents use the Internet to shop at online stores,plan vacations,get driving directions,or download tax forms.See Gardner(2007).
important role.
Finally,the results in this paper might interest researchers and policy-makers in the area of bilingual education and immigration reform.Although Proposition227remains contro-versial in California,Massachusetts and Arizona have since passed similar initiatives(The Economist,2008).Taking the opposite approach,districts in Georgia and Utah have hired teachers from Mexico to conduct classes in Spanish to their growing population of Hispanic students(Thompson,2009).Between1979and2006,the number of students K-12speaking a foreign language at home has tripled,and the trend shows no sign of reversing(United States Department of Education,2008).Understanding the e?ects of di?erent educational philosophies on immigrant students and their families is likely to remain essential to op-timally crafting public policy for many years to come;indeed,both Presidents Bush and Obama have stressed English pro?ciency requirements in their comprehensive immigration reform proposals.
The paper is organized as follows.Section2presents a simple model to illustrate the interactions between children’s and adults’human capital investments.Section3provides background on Reconstruction and the Freedmen’s Bureau,as well the data,empirical strat-egy and results for the literacy analysis.Section4is the analogue to Section3but focuses on Proposition227and English acquisition of immigrants in California.Section5concludes and o?ers directions for further research.
2Model
2.1Overview
This section provides a simple model of how adults’optimal level of human capital investment depends on the human capital of their children.As in the standard model of human capital investment(e.g.,Ben Porath1967;Becker1964),adults weigh the bene?t of the investment (the increase in consumption)against its price(the time,opportunity or psychic cost).
Children change the standard model in two ways.On the one hand,children can decrease the cost of human capital investment for their relatives.For example,suppose that in order to learn English immigrant parents can either study at home with their pro?cient child or attend an English as a Second Language(ESL)class.Not only can they save money and time if their child acts as their private tutor,they may also“save face”as they can avoid making potentially embarrassing mistakes in front of strangers.This decrease in the price of investment leads parents to invest more in human capital acquisition.I call this phenomenon the“learning e?ect.”
On the other hand,if children’s human capital can substitute for that of adults in house-hold production,then pro?cient children provide many of the bene?ts adults would enjoy from acquiring the human capital themselves.For example,a literate or English-pro?cient child can read contracts,bills or coupons,and confer with landlords,doctors and teachers on behalf of their family members;children’s human capital may even assist adults in?nding better jobs.4The ability of children’s human capital to directly increase adults’consumption decreases adults’incentive to invest in human capital themselves.I call this phenomenon the “leaning e?ect.”
2.2Mechanics
I modify the classic returns-to-education model with the above ideas in mind.Adults max-imize a separable utility function positive and concave in consumption and negative and convex in the cost of investment.5Adults’consumption y is a positive and concave function of both their own human capital k and their children’s human capital c,so y=y(k,c). Adults’human capital k is a positive and concave function of their investment in human capital,which I denote by e,as one can think of investment in this context as“e?ort”or
4Basu et al.(2001)use data from Bangladesh to show that having a literate member of the household is associated with higher wages for non-literate members.
5The convex,negative relationship between cost of investment and utility follows from the standard assumption that marginal utility diminishes with any“good;”in this case,the“good”would be lack of investment costs.
“education.”
The cost of investmentλis increasing and convex in e.Importantly,there is a comple-mentarity between adults’investment e and their children’s level of human capital c,so that λec<0.As described above,having a pro?cient child can reduce the per-unit psychic or monetary cost of investment e.
With the above assumptions in mind,I specify adults’utility as:
ψ(y(k(e),c))?λ(e,c).(1)
As described above,y k,y c,k e,λe,λee are positive andλec is negative.As utility is a positive, concave function of consumption,ψ >0andψ <0.
Adults choose e?so as to satisfy the following?rst-order condition:
ψ y k k e=λe(2)
Equation(2)yields the standard result that individuals set e?so that the utility gain due to the increase in consumption associated with a marginal increase in e(the left-hand side of the equation)equals the increase in disutility associated with higher investment costs (right-hand side).
The main comparative static addressed in the empirical work is the e?ect of children’s human capital on the human capital of adult household members,or?k(e?).On the one hand, e?,and thus k(e?),will increase with c because of the“learning e?ect.”Having a pro?cient child serve as a tutor decreases parents’per-unit cost of investment(more formally,recall thatλec<0).As the right-hand side of the equation falls with an increase in c,individuals must increase e to satisfy the?rst-order condition.
On the other hand,e?will decrease with c because of the“leaning e?ect.”An increase in children’s human capital directly increases adults’consumption by y c,thus lowering adults’marginal utility of consumptionψ .Therefore,adults will decrease investment so as to equal-ize the marginal utility of consumption and the marginal dis-utility of investments costs in
equation2.All else equal,if adults can rely on children’s human capital to increase household consumption they will invest less in human capital themselves.
The idea of competing incentives is expressed more formally below:
Proposition.The e?ect of children’s human capital on that of adults in the household,
?k(e?)
,can be positive or negative.It is a positive function of(?λec).This term represents ?c
the extent to which learning from pro?cient children can lower the per-unit cost of adults’human capital investment(the“learning e?ect”).It is a negative function of y c,the direct contribution of children’s human capital to adults’consumption(the“leaning e?ect”).
Proof.See appendix.
While the model does not specify the sign of the e?ect,it does suggest when the sign is likely to be positive or negative.The learning e?ect is especially strong when the alternative to learning from one’s children is especially costly.For example,as I discuss in the next section,very few black adults in the former Confederacy had access to schooling themselves. So even if they incurred some psychic cost related to the embarrassment of learning from a child,there did not exist a viable alternative.
Conversely,the leaning e?ect is likely to be especially important if theψterm of utility were a function only of simple items such as food or clothing,as their consumption value should be independent of an individual’s human capital.However,the marginal utility of other consumption items may depend on one’s own human capital stock.For example,the consumption value of movies or newspapers depends on having not only the resources to purchase the ticket or paper but also pro?ciency in the local language.Individuals might also derive utility directly from the sense of personal accomplishment gained from learning a new skill.In such cases,having a pro?cient child is a poor substitute for acquiring human capital oneself.
2.3Discussion
The model obviously makes many simplifying assumptions and is meant mostly for illustra-tive purposes.For example,I make no real distinction between household production and adults’consumption and implicitly assume that parents’consumption increases even when the increase in household production is due entirely to their children’s e?orts.Instead,chil-dren may refuse to contribute to household production if they want their parents to learn the skill themselves.Similarly,children’s human capital acquisition may change the bargaining power within the household.These e?ects would act to dampen any“leaning”incentive.
Moreover,the model assumes children’s human capital is determined outside the model. Instead,children may simply refuse to invest in human capital if they know their parents will free-ride o?of them,thus making children’s human capital endogenous to parents’expected behavior.If children only learn when they believe their parents will learn as well,then the leaning mechanism is e?ectively shut o?.
Obviously,identifying plausibly exogenous sources of variation is essential for estimating the key comparative statics in the model and is the focus of the remainder of the paper. The variation I exploit arises from educational interventions that target children.Children exposed to the intervention acquire higher levels of human capital and I use this variation to estimate the e?ects on adults’human capital investment.Moreover,the children most a?ected are often quite young and thus may be less likely to act strategically in deciding how much e?ort to invest at school.
3Did former slaves learn from their literate children?
The empirical analysis begins with an investigation of literacy spillovers within Southern black households during Reconstruction.I start by providing some historical context,?rst on the incentives for black adults to learn to read and write,and then on the educational opportunities provided by the Freedmen’s Bureau.I then describe the individuals I sample
from the1870Census as well as my empirical strategy and results.I conclude the section with a series of robustness checks.
3.1Background
Reconstruction and the Freedmen’s Bureau
One of the goals of Reconstruction—which generally refers to the policies implemented in the South by the federal government after the Civil War—was to address the economic needs of former slaves.This process actually began before the war ended:any time Union soldiers captured Southern territory they had to decide how to treat individuals held as slaves, especially after the Emancipation Proclamation of1863o?cially declared such individuals free.As captured territory grew,the Department of War o?cially established the Bureau for Refugees,Freedmen and Abandoned Lands on March3,1865,more than a month before the Confederate surrender.The Freedmen’s Bureau,as it became known,was one of the key institutions of the Reconstruction era and provided former slaves in the Confederacy with basic food rations,medical care,job placement,and,most famously,education.
Instead of directly running schools for Southern blacks,the Bureau funded religious and philanthropic organizations to do so.For example,the American Missionary Association alone was responsible for the instruction of over40,000students by1866(Butchart,1980). By the time of its?rst full school year in1865-1866,the Bureau supported964schools and 90,000pupils.Those numbers increased to2,677and150,000,respectively,by the1869-1870school year(Jones,1980,p.224).However,enrollment fell after1870as the federal government disbanded the Freedmen’s Bureau and began to bring Reconstruction to a close.6
6By the terms of the Supplementary Freedmen’s Bureau Act of1866(passed to continue and extend the authority created in the original act of1865),the Freedmen’s Bureau was to be disbanded during the summer of1868.A last-minute bill was passed to extend its general authority until January1869and to continue its educational mandate inde?nitely.However,educational funding was severely cut and the federal role in the education of Southern blacks e?ectively ended by the close of1870(Morris,1981,p.243).By the early1870s Northern support for reconstruction had waned and the1876“compromise”that granted Republican Rutherford Hayes the presidency in exchange for the withdrawal of federal troops from the South o?cially ended Reconstruction
Historical evidence of leaning and learning
Southern black adults during Reconstruction had many incentives to learn to read and write from their children.Collins and Margo(2006)estimate an eleven percent labor-market return to literacy for blacks in the South in1870,using occupational status as a proxy for income. There may have been an even higher return with respect to wealth and consumption,condi-tional on occupation,as literacy protected former slaves from signing exploitative contracts.“[T]he ability to read was also crucial for the freed people as they became involved in labor contracts and as they tried to acquire property.Former slaves recognized the importance of being able to read contracts,as one recalled,‘so they would know how to keep some of them white folks from gittin’land’way from’em if they did buy it’”(Cornelius,1991,p.143).
Moreover,children often acted as teachers to both their parents and other members of the community.Historian Heather Williams writes of the black schoolchild:“as soon as he learned a lesson,he became responsible for teaching it to someone else”(Williams,2006, p.139).Freedmen’s Bureau o?cials noted this phenomenon in each of their semi-annual reports.By the third report,the Commissioner tries to estimate the extent of these spillover e?ects:
Adults have acquired con?dence that they also can learn;even the aged are peer-
ing into these printed pages with some hope that knowledge is for them.Thou-
sands of children who have become advanced are teaching parents and older
members of the family;so that nearly every freedman’s home in the land is
a school-house...[and]whole families have become pupils...We scarcely dare es-
timate the number who are at the present time in some process of elementary
learning.To say that half a million of these poor people are now studying...would
be a very low estimate(Alvord,Third Semi-annual Report,1867,p.5).
Conversely,former slaves might have relied on their children not to teach but to perform tasks that required literacy.Learning to read and write likely exacted a lower pecuniary, opportunity and psychological cost for children,and adults may have found it more e?cient to have their children specialize in literacy-intensive tasks.A former slave testifying before (Woodward,1991,pp.197-198).
a Congressional committee in1871stated:“I have a son I sent to school when he was small.
I make him read all my letters and do all my writing.I keep him with me all the time”(Williams,2006,p.103).
In short,the rapid education of black children provided adults incentives to both learn and lean,and the rest of this section empirically assesses which incentive outweighed the other.
3.2Data
I use data from the Integrated Public Use Microdata Series(IPUMS)sample of the1870 Census.I focus on black individuals ages25through60,as younger individuals might them-selves have been school children between1865and1870and I wish to isolate as much as possible the learning-through-children e?ect from any e?ect of adults themselves attending formal classes.I further restrict my sample to those born in any state of the slave-holding South and residing at the time of the Census in the former Confederacy,re?ecting the fact that Reconstruction did not apply to the slave-holding border states that remained in the Union.7Although not all Southern blacks were slaves at the time of the Civil War,the vast majority of the individuals I sample would have been,though I return to the issue of free blacks in the pre-bellum period in robustness checks.8Finally,I restrict attention to those counties in which at least100black individuals are sampled in the1870IPUMS,as some of the empirical work uses county-level averages of black literacy and this restriction reduces the noise in this measure.
Table1reports summary statistics for both the sample of adults described above as well as children between the ages of ten and eighteen who otherwise meet the sample re-
7Results are robust to restricting the sample to those actually born in the Confederacy or including those currently residing in slave-holding non-Confederate states.
8Based on tabulated data from the1860Census,which reported both slave and free black populations by state,I estimate that96percent of my sample would have been slaves in the pre-bellum period.My estimates coincide with those of Cramer(1997),who calculates that3.74percent of blacks in the Confederacy were free in1860.
strictions.Note that children’s literacy is almost twice that of adults,though is still only about19percent.Most adults have at least one child in their household,but less than half live with a child of“school-going age”(which I de?ne as between ten and fourteen,as in my sample children of those ages have the highest enrollment rates).Over nine percent of children“attended or were enrolled in school”at some point in the last twelve months.The corresponding number for adults is0.2percent,a fact to which I will return at the end of this section.9
One of the key explanatory variables used in this section is the county-level literacy rate among black children.(Note that I will often just use the term“child literacy rate”in the interest of brevity but,unless otherwise noted,this term refers speci?cally to the county-level literacy rate of black children.The same convention applies to the use of“adult literacy rate.”)I estimate this measure directly from the IPUMS by calculating for each county the literate share of all black children between the ages of10-18(the1870Census only asks the literacy question of children over nine years of age).
3.3Empirical strategy
Although the model relates adults’literacy to that of their children,regressing an indicator for whether an adult is literate on an indicator for whether he lives with a literate child is likely to produce a positively biased coe?cient on the latter variable,via any number of endogeneity scenarios.For example,an intrinsic aptitude or desire for learning to read may“run in the family.”Similarly,adults can teach children,which is after all the more traditional route of human capital transmission.
Instead,I proxy the probability of living with a literate child with the interaction between the literacy rate of black children in the respondent’s county and an indicator variable for whether the respondent is living with a child,which suggests the following di?erences-in-
9The1870Census has no other education measure,such as highest grade completed or total years of schooling.
di?erences estimation:
Literate ic=βChild-in-house i×Black-child-lit-rate c+μic+γX i+εic,(3)
where i indexes individuals and c indexes counties;Literate ic is an indicator variable for whether person i in county c reports being able to read;Black-child-lit-rate c is the estimated literacy rate among black children in county c;Child-in-house i is coded as one if a child lives in individual i’s household;μic is a vector of the two main e?ects of the interaction term; X i is a vector of individual covariates;andεic is the error term.10Loosely speaking,living in a county with high levels of child literacy is the treatment,and adults living with and without children are,respectively,the treatment and control groups.In other regressions, adults living with school-age children and all other adults living with children serve as the respective treatment and control groups.
Note that I do not have any variation across time as no literacy information exists for black slaves or their children in the1860Census.Although teaching slaves to read was explicitly outlawed in all confederate states except Tennessee(Frasier,2002,p.99),anecdotal evidence suggests that some slaves managed to acquire literacy in the pre-bellum South and literacy rates among free Southern blacks in the1860IPUMS appear to be about thirty percent (though recall this group accounted for only four percent of all Southern blacks).11 Though I cannot rely on variation across time,the cross-sectional variation I use arises from pre-determined characteristics at the county level interacted with pre-determined char-acteristics at the household level.In fact,there is likely some random component to geo-graphic variation in black children’s literacy.Bureau o?cials and aid societies often estab-lished schools in areas where union soldiers were located at the time of the Confederate
10I use a linear probability model instead of a probit model because I eventually estimate a county-?xed-e?ects model and want to guard against the incidental parameters problem(even when I eliminate counties with few observations in the IPUMS,I am left with over two hundred counties).However,in practice,using a probit model and including the?xed e?ects does not change any of the results.
11Collins and Margo(2006)estimate that up to ten percent of slaves may have been literate in the late ante-bellum period.Also see Cornelius(1991).
surrender,an allocation that appears plausibly exogenous with respect to literacy patterns (Berlin et al.,1998,pp.40,154,161).Moreover,that variation is interacted with an additional source of variation(age structure of the individual’s household).Ideally,these interactions would not only be positively correlated with having a literate child,but also unrelated to adult literacy outside of my proposed mechanism,and potential endogeneity scenarios are discussed in detail later in the section.
3.4Results
Raw trends
Before turning to the regression analysis,I examine the relationship between adult and child literacy rates graphically.On the x-axis of Figure1,I plot the literacy rate of black children for each county.Figure1suggests that this measure has considerable variation across counties,though many appear clustered at zero.
Against this county-level child literacy rate I plot county-level literacy rates for three groups of black adults:those without children in their household,those with children in their household but none of whom are of school-going age(ages ten to fourteen),and those with school-age children in their household.All three series show a strong,positive correlation with the county-level literacy rates of black children,which is not surprising given the omitted-variables scenarios discussed in the previous subsection.The?rst series highlights this point: even though these adults do not live with children and thus are very unlikely to be subject to my proposed mechanism,their literacy rates are still strongly correlated with those of the children in their county.
Evidence in support of learning-from-children hypothesis cannot be found by looking at the overall correlation of county-level adult and child literacy rates,but by comparing that correlation for di?erent sets of adults.The hypothesis predicts that the correlation should be strongest for adults who have signi?cant contact with children and especially those who have contact with children of school-going age.Indeed,Figure1shows that compared to
adults who do not live with children,the literacy rates of adults who live with children appear more closely linked to the county-level child literacy rate.Moreover,the strongest correlation exists between child literacy rates and the literacy rates of adults who live with children of school-going age.
Regression results
Table2shows the results from estimating equation(3).As additional controls,I include dummy variables for age,gender and urban-versus-rural in all regressions.The estimated coe?cient on the child-literacy-rate variable suggests that living in a county where all black children are literate increases the probability that a black adult is literate by45.8percentage points,relative to his counterpart in a county where no children are literate.Equivalently, moving from a county with a?ve percent child literacy rate(the25th percentile for this variable)to one with a twenty-?ve percent rate(the75th percentile)is associated with a9.2 percentage point(0.458*(0.25-0.05)=0.092)increase.
Importantly,the interaction between the indicator for having a child in the household and the child literacy rate is positive and signi?cant.The point estimate suggests that for an adult living with a child,moving from the25th-percentile to the75th-percentile county increases the likelihood he is literate by an additional3.7percentage points(0.187*0.20=0.0374),or 36.3percent,given a baseline probability of0.103.Col.(2)adds county-level?xed e?ects, which reduces the coe?cient on Child-in-house×Black-child-lit-rate,though it remains positive and statistically signi?cant.
The results in the?rst two columns are consistent with adults learning from children. If learning is actually driving the results,then the e?ect should be driven by children of school-going age,as,say,a two-year-old child is unlikely to serve as a teacher to her parents. Col.(3)is the analogue to col.(1)except that adults living with school-age children(ages10 to14)serve as the treatment group and all other adults living with children as the control group.The coe?cient on the interaction term suggests that an adult living with a school-age
child in a county where all children are literate is roughly14percentage points more likely to be literate than is his neighbor who lives with children not of school-going age.In col.
(4),adding county?xed e?ects slightly reduces the coe?cient on the interaction term,but it remains positive and signi?cant.
I take the regression represented in col.(4)as my preferred speci?cation.Whether some-one lives with a child may obviously be correlated with many observable and unobservable traits that could be correlated with literacy,but the variation in the exact ages of the children in a household has at least some plausibly exogenous component.Moreover,I interact this information with the county-level child literacy rate,which has an independent,geographic dimension of variation.The validity of the results from my preferred speci?cation depends on whether this interaction term is orthogonal to other factors that could determine adult’s literacy,which I explore in the subsection below.
3.5Alternative hypotheses
Di?erential reactions to county characteristics
Obviously,there are important di?erences,both observable and unobservable,between coun-ties with high and low black child literacy rates.For example,the return to education might be higher in the former,and thus parents and other adults living with children might decide to invest more in education on their own,without any in?uence or assistance from their children.Similarly,counties with educational opportunities for black children might attract relatively educated or education-minded black parents.Of course,the speci?cations in Table 2start to address that possibility,by comparing adults with and without children and with and without children of school-going age,but it could still be the cases that adults with children are simply more responsive to human capital returns and educational opportunities than other adults.
One check on the likelihood of these scenarios is to explore whether literacy of white adults who otherwise meet the selection criteria exhibit the same patterns with respect to
black child literacy rates.Of course,whites are a highly imperfect comparison group,as even in the same county they inhabited di?erent economic and social environments.For example, the overall white literacy rates in my sample are80percent for adults and65percent for children,compared with11and19percent,respectively,for blacks.However,Reconstruction, which was at its height at the time of the Census,created a temporary moment when the economic and social opportunities for Southern blacks and whites were more similar than they had ever been before and would be for decades,and thus some of the same county-level factors that might have motivated blacks to invest in education or attracted educated blacks would have had the same e?ect on whites.As such,examining white literacy patterns can at a minimum serve as basic falsi?cation tests.
The regression reported in col.(5)of Table2is identical to col.(4)except that white adults with children in their household serve as the sample group.The coe?cient on School-age-child-in-house×Black-child-lit-rate is insigni?cant and in fact negative(though very close to zero).Thus,the literacy di?erences between white adults with school-age children and other whites with children have no apparent relationship to the black child literacy rate, in stark contrast to black adults.
Finally,I explore the relationship between black adult literacy rates and white child literacy rates.The speci?cation in col.(6)is identical to that in col.(4)except for an added W hite-child-literacy-rate×School-age-child-in-house interaction term.The coe?cient on this term is insigni?cant,close to zero and in fact negative,and,moreover,the coe?cient on the School-age-child-in-house×Black-child-lit-rate is unchanged from that in col.(4). Thus,black adults’literacy rates respond to factors highly correlated to black child literacy rates,not child literacy rates more generally.
Adults taught children
Another alternative explanation is that adults were teaching children,instead of children teaching adults.12Some Southern blacks had indeed attained literacy by the end of the Civil War,obviously independently of any e?ect the Freedmen’s Bureau would later have on their children.I thus try to eliminate from the regression sample adults who were potentially literate by the end of the war,in order to shut o?this potential source of reserve causality.
Many Southern black men learned to read while?ghting in the Union Army(Berlin et al., 1998).Therefore,col.(2)of Table3includes all women,but only men too old or young to have served.The point-estimate is essentially identical to that in Table2(which I reprint in col.(1)for ease of comparison).
Similarly,another group of Southern blacks who gained some literacy before the end of the War were freemen(and women).While it was illegal to teach a slave how to read in all states but Tennessee,no such restrictions existed for free Southern blacks.Assuming free blacks were indeed more likely to have been literate by the end of the Civil War(see subsection3.3for a discussion of literacy rates of free blacks),in col.(3)I exclude the three states with the highest free share of blacks as of the1860census(Louisiana,North Carolina, and Virginia).Again,the point-estimate appears largely invariant to this sampling choice, and actually increases several percentage points.
Adults attended school themselves
While the analysis has generally assumed that adults learned at home from children who passed on what they had learned in school,many adults attended classes themselves during Reconstruction.Indeed,a common image from the period is that of adults and students sitting side-by-side in classrooms.If adults with school-age children are more likely to attend
12Comparing the children of former slaves with the children of free northern blacks,Sacerdote (2005)?nds signi?cant intergenerational correlation with respect to literacy,though does not claim a causal interpretation.He shows evidence that by the second generation born after the Civil War the outcomes of the grandchildren of former slaves have almost fully converged with those of pre-bellum black freemen.
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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