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Journal of Economic Dynamics&Control28(2004)2485–2510
/locate/econba
The allocation of public funds in a hierarchical
educational system
Xuejuan Su
Department of Economics,Indiana University,Wylie Hall105,Bloomington,IN47405,USA
Received8October2002;accepted6January2004
汽车手刹Abstract光辉岁月粤语翻译谐音
This paper studies the dynamic e ects of allocating public funds between basic and advanced education.Holding the size of total public education fundsÿxed,I identify the e ects of their composition on aggregate e ciency and(in)equality.First,since basic education is a prerequisite for attending advanced education,there exists a lower bound on funding it:allocation policies below this l
ower bound are strictly Pareto dominated.Whether a corresponding lower bound on funding advanced education exists or not depends on the development stage of the economy and the size of total public funds.Secondly,while an allocation policy favoring basic education generates the usual redistribution from top to bottom,one favoring advanced education may result in rever redistribution from bottom to top.Last,through the inter-generational link,short-run allocation policies may have long-run e ects.A simple rule-of-thumb is that for an economy in its early development stage,focusing on basic education for su ciently long duration is the only way out of polarization and low aggregate e ciency,contrary to the actual policies pursued by many of the less developed economies.
?2004Elvier B.V.All rights rerved.
Keywords:Education policy;Hierarchical education;Public resource allocation;Redistributive e ect
1.Introduction
Any educational system is intrinsically hierarchical.Individuals have to learn basic knowledge before they can study more advanced knowledge.Conquently,within the commonly obrved three-stage system–primary,condary and tertiary education, individuals have toÿnish lower levels successfully
before they can enter higher lev-els,and their academic achievements in lower levels determine their preparations and qualiÿcations for the higher levels.上升星座天秤座
香菇炒芹菜E-mail address:xuesu@indiana.edu(X.Su).
0165-1889/$-e front matter?2004Elvier B.V.All rights rerved.
doi:10.1016/j.jedc.2004.01.003
2486X.Su/Journal of Economic Dynamics&Control28(2004)2485–2510 Table1
School participation rate and survival rate
CS Enrollment ratio(%)Survival rate(%)
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Years(1)Prim.(2)Second.(3)Tert.(4)(5)=(3)
(6)=
(4)
Bangladesh**589437488 Brazil**89771177318 Chad*65781142 Japan**91001004810048 Lesotho**778213274 Niger**83051173 United Kingdom*119994589559 United States*109587729276 Countries with1999/2000data are indicated with*and tho with2000/2001data are indicated with**.
No wonder,then,that when compulsory schooling legislation is prent,1it exclu-sively regulates school participation at lower levels,and leaves that at higher levels to private choice.Conquently,school participation is relatively uniform in lower levels, yet it exhibits pronounced di erence in higher levels across individuals.Furthermore, due to the hierarchical nature of education,drop-out is much more likely than drop-in at any intermediate stage,so the participation rate shrinks when a cohort of individ-uals moves up along the education ladder.Table1provides some cross-country data revealing this pattern.While the participation rate shrinks only moderately in devel-oped economies like the U.S.,the U.K.and Japan,it shrinks drastically in some less developed economies like Chad,Lesotho and Niger.
At any educational stage,the quality of education depends on the resource availabil-ity,and it both a ects and is a ected by individuals’participation decisions.Our focus is on the public education funds,the major component of total education expenditure for almost all countries.Adjusting the aggr
egate funding level by participation rate, expenditure per pupil is a strong indicator of schooling quality at a given stage.Data in Table2reveal that countries di er in their allocation policies concerning public education funds,and the relative schooling quality across educational stages exhibits prominent variations.While the relative schooling quality between higher stage and lower stage remains clo to1in developed economies,it ris beyond10in some less developed economies.This rais the main issue of the current paper.
Given the hierarchical nature of the educational system,di erent stages of educa-tion are not perfectly substitutable,so a change in the schooling quality at one stage may not be readily o t by an opposite change at another stage.Conquently,the resource allocation policy may have profound impact on theÿnal outcomes.This pa-per conducts positive analysis of the dynamic e ects of the allocation of public funds across educational stages.Given aÿxed size of public education funds,how does their
1UNESCO(1999)report shows that over95%of all countries have such legislations.
X.Su/Journal of Economic Dynamics&Control28(2004)2485–25102487 Table2
Allocation of public educational expenditure
%of Total education expenditure Ratio of exp.per pupil
Prim.(7)Second.(8)Tert.(9)(10)=(8)=(3)
(7)=(2)
(11)=
(9)=(4)
(7)=(2)
Bangladesh**38.143.010.1 2.3 3.4
Brazil**34.035.721.1 1.4 3.5
Chad*57.525.916.6 3.216.5
Japan**354015.2 1.10.9 Lesotho**48.627.716.7 2.18.9
Niger**49.328.619.9 3.512.1
United Kingdom*24.446.720.1 2.0 1.4
United States*31.435.626.2 1.2 1.1 Countries with1999/2000data are indicated with*and tho with2000/2001data are indicated with**.
allocation across di erent stages a ect individuals’participation decisions and hence human capital and income?What are the policy e ects on(in)equality?In particular, does the di erence in participation decisions at lower and higher stages lead to rever redistribution of resources from low-income to high-income families?And to what ex-tent can this rever redistribution be o t through income tax?What are the policy e ects on aggregate e ciency?Is there a Pareto-dominant allocation policy,or does a policy change necessarily beneÿt some individuals at the expen of others?Finally what are the long-run e ects of an allocation policy?
The issues cannot be analyzed without looking into the internal structure of the educational system.Instead of viewing the entire education process as a black-box, I introduce the hierarchical structure.Namely,in this model,the educational system consists of two stages.The human capital output produced at the lower stage acts as an input into the education process at the higher stage.This approach enables us to identify the composition e ect free from the size e ect of public ed
ucation funds. Economists have studied the size e ect of public education funds from various per-spectives;but very few have studied the composition e ect bad on the hierarchical structure of the educational system.One exception is Driskill and Horowitz(2002), who explicitly model the education hierarchy.They consider a social planner’s prob-lem in a standard growth framework and study the optimal investment in hierarchically produced human capital as well as in physical capital.They show that the optimal in-vestment program,part of which is the investment plan at di erent stages of education, depends on the relative scarcity of physical capital and the di erential levels of human capital,and hence on the development stage of an economy.However,that paper sheds no light on the policy e ects in a decentralized decision-making tup,and it cannot be ud to study the distributional e ects across individuals.As will be outlined below, individuals di er in their participation decisions,and the distributional e ects constitute an important aspect of the allocation policy,hence should not be ignored.The current paperÿlls the gaps.
2488X.Su/Journal of Economic Dynamics&Control28(2004)2485–2510
Within a dynamic framework of successive generations,this paper models education as a hierarchical two-stage system with two distinct technologies.The lower and higher stages are also referred to as the basic and advanced stages.The technology at the basic level is standard,where hu例假腰疼
man capital is produced with conventional inputs such as an individual’s pre-school preparation(also called initial qualiÿcation),schooling time and the relevant schooling quality.The technology at the advanced level us the hu-man capital output produced from basic education as an input.This factor determines individuals preparation for advanced education(also called augmented qualiÿcation). Poorly prepared individuals(with augmented qualiÿcations below some threshold level) simply cannot beneÿt from advanced education even if they do attend.2Only su -ciently prepared individuals(with augmented qualiÿcations above the threshold level) can beneÿt from advanced education,and they make their own participation decisions at the advanced stage.
Individuals’participation decisions at the advanced stage vary according to their augmented qualiÿcations,which in turn depend on the schooling quality at the basic stage.Conquently,their beneÿts from public education funds depend on the alloca-tion policy.There are two opposite forces at work.On the one hand,individuals with low qualiÿcations,not attending advanced education themlves,helpÿnance it through income tax,so there is rever redistribution of resources from the poor to the rich.3 On the other hand,individuals with high qualiÿcations earn more income and pay more taxes,so the rever redistribution may be partly or even fully o t.(See Bevia and Iturbe-Ormaetxe(2002).)Given aÿxed size of public education funds,di erent alloca-tion policies elicit quite d
i erent individual respons in school participation decisions, and therefore have di erent distributional e ects.
With aÿxed size of public funds,di erent allocation policies also di er in their e ects on aggregate e ciency.Some policies lead to equilibrium outcomes that are strictly Pareto dominated.Since basic education is esntial for producing human capital output in its own right and providing the necessary input,the augmented qualiÿcation, in advanced education,there exists a lower bound on its funding level.Allocation policies that fund basic education below this lower bound are strictly Pareto dominated by the one meeting this lower bound.On the other hand,whether a corresponding lower bound on funding advanced education exists or not depends on the initial distribution and the size of total public funds.When a fraction of the population has extremely low initial qualiÿcations,and/or when the size of the total public funds is not big enough, any allocation of funding to advanced education always hurts tho individuals who are unprepared for the advanced stage,and hence the allocation policies may not be Pareto ranked.Thus in this model,policy e ects depend crucially on the development stage of an economy.
2Think that students who can barely do algebra have nothing to gain to sit in an advanced analysis cour. 3In the United States data,Hann(1970),Radner and Miller(1970),Peltzman(1973)and Bish
op(1977)ÿnd some evidence that students from high income families are more likely to attend higher education. Le Grand(1982)documents similar phenomenon for the United Kingdom.For a recent political economy analysis on this issue,e Fernandez and Rogerson(1995).
X.Su/Journal of Economic Dynamics&Control28(2004)2485–25102489 Outside the Pareto improvable range,where there is no con ict of interests across individuals,there exists a range of allocation policies that beneÿt some individuals at the expen of others.Conquently there may be a trade-o between aggregate e -ciency and equality.This trade-o ,again,depends crucially on the development stage of an ,the distribution of individuals’initial qualiÿcations.Since basic education is universally beneÿcial and advanced education may be exclusive,the policy e ects are esntially determined by how exclusive advanced education is in an econ-omy.In a less developed economy,the majority of the population has extremely low initial qualiÿcations,so advanced education is extremely exclusive.Then an allocation policy favoring basic education is most likely to improve both the aggregate e ciency and equality.In a developed economy,a substantial fraction of the population has high initial qualiÿcations and can beneÿt more from advanced education.Therefore an alloca-tion policy favoring basic education is likely to improve equality but hurt the aggregate e ciency.This intra-generational trade-o is an important aspect of the policy e ects.
Through the inter-generational link,higher human capital levels of parents imply better pre-school preparations and hence higher initial qualiÿcations of children,so an allocation policy that a ects the current generation’s decisions on human capital accumulation has long-lasting impacts over all future generations.Conquently when there are multiple steady states,convergence to which steady state is path-dependent. Numerical examples are ud to illustrate the simple rule-of-thumb.For an economy in its early stage of development,allocating most of the public funds on basic education for a su ciently long-time span is the only way out of the‘underdevelopment trap’and polarization in the long run.
The rest of the paper is structured as follows.Section2lays out the model. Section3characterizes the equilibrium and analyzes the policy e ects on equality and aggregate e ciency.When analytical results cannot be obtained,some numerical examples are provided in Section4.Section5analyzes the long-term policy e ects. Section6concludes and discuss directions for future work.All proofs are relegated to Appendix A,as is the algorithm ud in the numerical analysis.In Appendix B, the original model is extended to include both physical capital and private education expenditure.It is shown that the extended model shares the same qualitative features of the original model.
2.Two-stage education model
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This is a successive generations model with heterogeneous individuals.4Within a generation,individuals di er in their pre-school preparations(initial qualiÿcations), which is assumed to be an increasing function of the human capital levels of their parents.5
4The successive generations modeling approach to the study of distributional dynamics follows Banerjee and Newman(1991)and Perotti(1993).
5That children’s pre-school preparations depend on the human capital levels of parents is a commonly adopted speciÿcation in the literature.Just to mention a few,e Glomm and Ravikumar(1992),Eckstein and Zilcha(1994),Benabou(1996)and Kaganovich and Zilcha(1999).
