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The equity-productivity tradeoff:
European Economic Review 32 (1988) 1585-1
lan
Jere R. BEHRMAN a brad Bank, Washington, DC 20433, USA Received May 1986, final version received June 1987 A governmental welfare function is specified to explore the equity-productivity tradeoff in the allocation of governmental schooling resources. The constrained maximization ode1 is estimated for Brazil using expected incomes in each geographical area as the outcome of interest and governmental resources for each area as the input. The equity-productivity tradeoff in Brazilian governmental preferences is estimated to be considerable, with a welfare curvature close neither to the utilitarian nor to the Rawlsian extremes, but close to the intermediate CobbDouglas value. This implies neutral rather than reinforcing or compensating behavior in the allocation of governmental resources, with the result that the distribution of such resources across areas is more equal than is that of expected incomes.
In recent years distributional as well as growth goals have been increasingly emphasized in the development literature (e.g., Ahluwalia, Chenery et al., Cline). In this literature schooling is seen as a major policy instr&ment with which productivity and distributional goals can be simultaneously pursued [e.g., Colclough ( 1982), World Bank ( 1980,198 I)]. ut in the distribution of public inputs for schooling, there can be a considerable tradeoff between productivity and distributional goals. The geographical areas in which the returns to public spending on schooling, that e marginal productivity of schooling from society’s point of view, are highest are ot necessarily those in which a concern for equity spending be concentrated. An obvious exa le in most ehrman is William R. enan, Jr. Professor of Economics and Co-Director of the Center for Analysis of Developin the University consultant to the r&all was Chic Ith and Nutrition World Bank Research Project 672-21, ‘Studies o original draft prepared in 1983 e are gr?tefu ichael L _-Tart; 1v. and to an anonymous referee, Taubman for useful comments. None of these i any view expressed here. 2921/88/$3.50 0 1988, lsevier Science Publishers
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J.R. Behrman and N. Birdsall, The equity-productivity tradeoff
is the urban-rural dichotomy: &~ral areas tend to be poorer, but available estimates suggest that marginal returns to schooling are hrgher in urban areas, probably because of larger supplies of complementary in ilts. Typically, schooling investments also are higher in urban areas. n this paper we model and estimate the equity-productivity trade0 implicit in the geographical allocation of public resources to pr secondary schooling in a major developing country, Brazil. While an equ productivity tradeoff in the allocation over space of schooling res been recognized by some, we know of no previous efforts to quantify the nature of this tradeoff. We do so by estimating the critical parameters of the implicit welfare function which governs the allocation of public resources by razihan society among its citize . Our method, which we apply using data across 50 geographical areas in razil, provides us with an estimate of the amount of inequality aversion plicit in society’s al!ocation of public resources to schooling across these areas. ‘t his estimate SL’;gests that this welfare function is neither utilitarian with a pure productiv.ty concern nor awlsian with a pure distributional concern. Instead, there 1s an important productivity-equity tradeoff, with somewhat more emphasis on productivity than on equity.
2.I. Gweral formulation
ss governing the geographical allocation of resources to schoolmplex in Brazil. Schooling is funded by a combination cf tocal, state, government resources, with states providing as much as twoe prancing (in the early 1970s which is the time period that we ) and the federal government providing 20 to 25 percent. We assume that the resulting allocation by a number of different government licit societal welfare function budget constraints.’ We do not assume sly maximize such a welfare y maximize such a function. her complicated entities recent public finance Craig, Christenson and ‘3ne can conceive a number of different processes Iii which govern ir respective contributions to educational expenditures. For example, there ~i~ib~~n~ in which local governments and the federal government ‘both are reaction function of the other. Data are not available that erefore, we assu
J.R. Behrman and N. Birdsall, The equit
1587
razil comprise t e rural areas of two very small states due to sampling err
opolitan area of Rio
tween rural and urban1 irreas, so that some of the tradeoff between productivity and equity is actually reflected in the allocation within states of resources to urban versus rural areas. There is also, however, a tradeoff between ity and distributional objectives in the allocation of federal reven e states. At the regional level, the ost obvious distinction is between the highly industrialized southeast an the very poor agricultural northeast. Both local and federal direct expenditures on schooling are higher in the southeast. One analyst of schooling expenditures has estimated, for example, that state and local spending on secondary schools in the northeast in 1975 would have to have been more than three times as great as it was to bring secondary enrollment rates and quality to the southeast level (De Mello e Souza, p. 56). Our rough calculations based on available data on federal direct expenditures for education in the three IL. :or regions (northeast, southeast, and frontier) indicate however, that federal government spending could reduce that disparity substantially; if all federal expenditures on education were directed to the northeast, spending per pupil there, combining state and federal fun could be about the same as state sapending per pupil in the southeast.2 owever, what evidence there is indicates that per capita federal spending on education is if anything slightly higher in the southeast than in the northeast,3 so that federal spending does not compensate. Of course, it is possible that in the absence of distributional objectives, the federal government might direct even more of its spending toward the southeast, if the marginal returns to schooling in the southeast are sufficiently greater than for the other regions. The actual skewness of spending depends not only on the dispersion of marginal returns to schooling investments among areas, but ,nt of inequality aversion in society’s implicit welfare function. also on the ev I = Thus the productivity-equity tradeoff occurs at both the state and the federal government level. e do not attempt to distinguish explicitly between the two, but note that the federal government probably spends enough on schooling to equalize total spending among the three major regi and thus could probably equalize spending among equalize spen in states well mig ‘Federal exp enditures per capita for education ave 61, 55 and 65 (1975 cruzeiros) for t frontier, northeast and southeast regions. Total expenditures per student, inclu are 642, 419, and 1196 for the three regions, respectiv percent of the population, federal expenditures are about 3 per student in each ar reg!on, and if directed to the northeast could assure near e an llinger, Tables 12 and 30. har and Dillinger, op. cit.
J.R. Behrman and N. Birdsall, The eq
1588
center than is politically and administrative done. Thus in a sense it is the implici government that we are estimating. e assume that this welfare functi capita income of indivi eligible to areas? Differences in i among the in area are assumed to be distributed si for the areas? Therefore the welfare fu income of an average individual eligible to times, where ni is the number of such
w= W(Y, ,...,
Yl,...,
Y, ,..., Ym)=
y-producticitytradeof
feasible, it could in principle be elfare function of the federal efined over the led in each of m iduals within each geographical und the geographical means n merely repeats the expected schooled in the ith area (I$) ni in the ith area: if(K)9**-94nf(KJ)*
(1)
.
This welfare function is maximized su The first constraint is that total pub1 be greater or equal to the sum of such res in all of the geographical areas. If Ri is to eligible individuals in the ith area, this c
RzCniRi,
0 two major constraints. urces devoted to schooling (R) rces allocated to all individuals erage public resources devoted
(2)
m
In order to focus on the estimation of the government allocation of public schoo resources is assumed to be given? The second constraint (or set of co come-generation function for I$ whit average le th of schooling (Si) an schooling ( in the ith geographical area: &= Y”(Si,
uity-productivity tradeoff in the resources, the total of such aints) is the expected average s posited to depend upon the average resources devoted to
(3) ength of schooling is supported ates, of which Psacharopoulos
‘The overall welfare function may be con defense, national prestige), but we as schooling resources is separable from t ly because of its impact on exp ic analysis of the impa tradition).
many other outcomes (e.g., national art related to the allocation of public s. The assumption that schooling is of is a simplification that is common in (e.g., earnings functions in the inter rences in the distribution of individual etc. in different geographical areas, but to the sources of these
provides a recent review. nt in school, however, the quality of schooling that the quality of schooling is etermined basi resources per eligible student (RJ to schooling.* of schooling is posited to be a private decision de n of expected income levels associated with different schooling levels, given schooling costs (includi school emphasized by Mincer and associated with a given increment i the quality of schooling. That is, an extra year posited to increase the expected in lower-quality schooling. Therefore we posit that the length of schooling depends on the quality of schooling9 - which is again represented by the allocation of public resources - and by other factors related to family background, expected labor market outcomes, etc. as summarized in t vector Xi” Si = Si(Ri, Xi).
WV
Substitution of this relation into the expected income function in relation (3) givesl l & = x( Ri, Xi).
cw
‘We find empirical support for the importance of school quality in Brazil in Behrman and Birdsall (1983,1985). Also see Heyneman and Loxley. *For the primary and secondary levels of schooling under study, this is a reasonable approximation for most societies. There are some cases (e.g., the Republic of Korea, the Philippines) in which private resources also may be quite important in determining schooling quality. To analyze such casts, these private resources also would have to be included in relation (3). ‘We assume that migration decisions rmong areas are not made on the basis of schooling quality, so Ri is given. Post-schooling migration may occur and is incorporated into expected income outcomes. The assumption of no substantial migrati?n in response to perceived school quality differentials is much more palatable for our sample than it might be for many others for two reasons. (1) Our geographical areas are fairly large so usually fairly long-distance migration would be required to change areas in response to differential schoolin schooling levels are quite low (the sample mean is about three grades) so the tend to be young enough that it is unlikely they woul their own to obtain better primary echoohng, nor th for this reason. For smaller geographical distances and higher schooling levels, of course, in least of students, if not of their fa many societies migration ( tials. response to school quality di rminants of the length of schooling for current children of the “Birdsall investigates the adults in the sample which we use in Section quality on the length of their SC theoretical reasons why such an 1 ‘The asterisk in relation (3) is used t
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J.R. Behrman and N. Birdsall, The equity-productivity tradeofl
This is the form in which we represent the second set of constraints in what follows, Xi is assumed to be basically exogenous to the decision re the allocation of public schooling resources. e allocation of public resources for primary an secondary schooling is assumed to be the outcome of the maximization of the welfare function in relation (1) subject to the budget constraint in relation (2) and the expected income function in relation (3B). We assume that the welfare effect of increasing the ith income is positive (iYW/iJF> 0)and that the i of more public schooling resources on expected income is positive Under these assumptions, all public schooling resources are allocated and the equality in eq. (2) holds. We also assume that the relevant functions are smooth, with non-increasing returns to scale and diminishing returns to each nt, and that the maximization leads to an interior solution. In this first-order conditions for the maximization are
ere R is the Lagrangian multiplier for the budget constraint. n behind our estimation procedure can be illustrated by ratio of the first-order conditions for the ith and jth areas
slope of the expected income possibility frontier. a tangency, at which int the marginal rate of substitution
or each there is a relation like
only productivity matters. l3 There is indifference among points on such a curve with considerably different distributional implications. two extremes these are an infinite num in which there are equity-productivity t 2.2. Explicit functional forms and relative versus a solute inequality awersio To estimate the equity-productivity tradeoff from relation (4), ex assumptions must be made for the expected income function in relation and for the welfare function in relation (1). Expected
income function:
We consider a partial log-linear expected
income function In x=a+blnRi+g(Xi).
(3C)
The last right-side term can be any functional form which satisfies the nonincreasing returns to scale and diminishing returns to individual variable assumptions made above (e.g., if g(Xi) =p In Xi, relation (3C) is a CobbDouglas expected income production function). For substitution into relation (4) the partial derivative of 5 with respect to Ri is require
Since x depends on Xi, this partial derivative depends on the other average area-specific characteristics relating to family background and expected labor market outcomes and therefore is shifted by Xi as is required for the identification of the welfare surface (see subsection 2.1) even though Xi does not appear explicitly in relation (3D). Government werfare function: The we fare functio nown CES form
e thank a referee for pointing out that maximizing states of the world may lead to a Wawlsian preferen (in a Gini coeffkient sense) at the same time the lowe
‘benefits’ is ambiguous. If we interpret benefi
Behrman and N. Birdsall, The equity-productioity tradeo$
JR
159%
ith this welfare function c summarizes the equity-productiv A higher value of c implies greater inequalit awlsian and only the lowest ex tirely. If c = levant. For values of c ere is an equity-productivity tradeoff. his welfare function implies constant relative i equality aversion. pe of an isowelfare curve jth areas”
-- 3 -Y-j -c
dK
0
W=
VliF
’
w
is relation says that along an isowelfare curve in the ith-jt -weighted) relative expected into e between individuals in area i and area j is relevant.a6 aximization of the CES welfare function subject to the budget constraint and to the expected income function in relation (3C) and taking logarithms gives In i=(1
-C)Ill
q+k,
n (1 -c)%/E,. Note that Xi does not a ar explicitly in relation tifying the curvature of the welfare ecause all of the im act of Xi is represented by licit appearance 0 14As an alternative parameter repr
we also considered the Kohm-Pollak (K-P) welfare function (see and Sah and Behrman and Craig) which also has a oneaversion: W = -(l/h)In [&/@e-kY’], where n=Cni. A ality aversion. For LO, the K-P welfare function r h-, 00, the K-P welfare function
area:
where Z=g(Xi) -g(Xj). Relations (SA) a public resources (R) and of e depends on Z (and thus the inequality aversion (c), and the elasticity o government-allocated school resources (b).
that the distri across areas nt relative respect to
oefficients of 2 in relations Tables 1 and 2 summarize the values of (5A) and (5B), depending on different value nd b. As discussed above, c e Rawlsian case of 00, with may range from the utilitarian case of zero y. If the impact of pub the Cobb-Douglas value of one also not nd if the returns to scale m school resources o income is pos decreasing, constant or not ction function of expected incom erefore this range increasing, b is bounded by zer b is included in the tables. Before examining the entries in the tables, it is useful to note that positive entries imply a distributicn of R or Y that favors the area X, and negative entries tion relation (SC) is Co of one has a special sign& uction function cas
J.R. Behrman and N. Birdsall, The equity-productivity tradeof
Since [XJXj)P gives the inequality due to1 the inequality in the distribution of rest of the exponent (which is the same in SA’ as in SA and the same ’ as in SB) tells whether the impact of the inequality in X is reinforced, weakened or unaltered by the allocation of R, depending upon whether the rest of the exponent is greater than, less than or equal to one.‘* First consider the exponents for the relative RJR, relation in table 1. The ouglas extent of relative inequality aversion (c= 1) obviously is critical. On the Rawlsian side of the Cobb-Douglas case (c > l), school Table
1
Values of coefficient of Z in relation for distribution of public school resources (RJRj) as dependent on relative inequality aversion (c) and elasticity of expected income with respect to public school resources (b).
(Utilitarian)
O
c=l (CobbDouglas)
b+O
1
1 -c>o
0
O
1 ->l l-b
1 -b(l
00
--0
C-*0
b-,1.0
C--,00
1 < c < 00
(Rawlsian)
l-c<0
--oo
l-c
l-c -c)
>o
0 l-b(l-c)
l-c
0
l-c -l<-
c
-1 b -1
C
Table 2 Value of coetkient of Z in relation for distribution of expected incomes (YJq) as dependent on relative inequality aversion (c) and elasticity of expected income with respect to public school resources (b). c-0
c=l
(Utilitarian)
o
b-+0
1
1
O
--1
b-,1.
1
1
1
l-b
T-!?(I 4)
00
I ->I C
C-+00
(Cobb-Douglas)
1
(Rawlsian)
1
1
1 1
>I
1
<1
0
l-b(l-C) 1
1 --
0
J.R. Behman and N. Birdsall, The equity-productivity trade
resources are distributed so as to favor or compensate those areas with poorer X (i.e., the exponents are negative), though this tendency is wea the more effective are school resources in altering expected income (i.e., the greater is 6) and the closer relative inequality aversion is to the CobbDouglas case. On the utilitarian side of the CobbDouglas case (cc l), public school resources are distributed with the areas with more X also receiving more R, thus reinforcing differences in X. For the Cob&Douglas value of inequality aversion (c = l), the government is neutral in its allocation of public resources, neither compensating for nor reinforcing inequalities in the distribution of X. If the expected income production relation is CobbDouglas, an interesting question is whether the exponent is greater or less than one. The critical value at which the exponent in the RJR, relation equals one is c*=bJ(l
+b).
W)
c* is zero if b is zero and approaches 0.5 monotonically as b approaches one. If c is below c* and if the expected income production function is CobbDouglas, school resources are distributed more unequally than is the impact of the distribution of X alone on the distribution of expected incomes. This occurs at the utilitarian extreme (c+O) in which only total expected income counts, but also for values of c greater than zero if school resources are relatively effective (i.e., b is large). Now consider the exponents for the relative expected income relation (x/q) in table 2. The pattern of values of one again is interesting. The first row indicates that the coeffkient approaches one as b approaches zero because public school resources are ineffective in such a case - despite the much different degrees of relative inequality aversion in the first row of table 1. As b approaches zero the distribution of income reflects only the impact of the distribution of X, independent of the distribution of R. The third column also has coefficient values of one. This is the Cobb-Douglas case in which equity and productivity considerations are balanced off so that expected incomes are distributed identically to the impact of the expected X’s even though the R’s are equally distributed (see zero values in the third column of table 1) for all permissible values of the effectiveness of ubhc resources on expected incomes (i.e., all values of b). Note that in this case expected incomes are distributed as unequally as is the impact of rovision of equal public school resources to all. If c is less ouglas value of one and b is positive, even though the may be more equal t distribution of expecte of the X’s.
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J.R. Behrman and N. Birdsall, The equity-productivity tradeo$
inequalities in X (i.e., the relevant coefficients in table 1 are negative), this results in a more equal distribution of expected incomes than the impact of the distribution of X alone would imply. Ocly if c approaches infinity in the awlsian limi , however, are expected incomes actually equalized.
3.1. Data Our data are from the razilian l/100 Public Use sample of households of the 1970 census. We consider only males in order to avoid complications due to selectivity regarding female labor force participation. We limit the age range to 15-35 in order to have a better representation of public schooling resource aliocation (see below). We consider m= 50 geographical areas because the data set permits the matching of areas of schooling with the public resources devoted to schooling at this level of aggregation. The 50 eas are comprised of the rural areas of 24 states plus urban areas of 26 of razil’s 27 states in 1970. To represent the average expected income for those eligible to be schooled in each area (Q, we use the mean 1970 monthly income of those who were in that area at the time they were of school age.” This income may be earned in this area or elsewhere, if there was post-schooling migration. By using this measure, we assume that the actual migratory experience of the sample was that expected a priori and that the migratory costs were relatively small compared to permanent income levels. The value of this variable ranges fairly considerably (with a standard deviation of 188 as to a mean of 230 cruzeiros) despite the averaging out of individual for those with the same geographical area of origin. nt the average government provision of resources per child in ), we would like to have data on these resources in real terms at the time that the schooling was undertaken. Unfortunately, such data are not available. As a proxy, therefore, we use the mean education (years of area as calculated from the same sample, under ajor resource being allocated is teachers of e use the nominal values despite evidence of geographical price variations because price deflators are not available for the areas of interest. Given that the pries levels probably are correlated with income levels due to higher rents in higher-income, more-urban locations (see Thomas), the use of nominal instead of real income may bias our estimates, as the following nstrates. Let Y= Yl/Pi where Y; is the nominal exp ed income in the ith area and Pi price deflator for the ith area. Then relation (43) can written as lnRi=k+(L-c)ln YT(1 -c) In Pi. If the last term is excluded since Pi is not observed, the estimate of the coenicient omitted variable bias: E[ 1 -c] = 1 -c-( 1 -c)r, where E is the n Y;. If r is positive as utilitarian case) if the e true
J.R. Behrman and N. Birdsall, The equity-productivity
1597
differential training and quality.20 The mean value of teacher education in the sample is 8.2 grades, with a standard deviation across the geograp areas of 2.3 rades. Teacher e ucation ranged in 1970 from a low of about 3 years in the rural northeast to 11 years in the urban south. Tea&er education is a better proxy for average government resources more correlated it is with teacher salaries, the more corre salaries are with other public school expenditures, the m re uniform teacher training is across areas, and the more stable over e is the geographical distribution of these resources. In Behrman and sall (1983) we summarize the available (fragmented) evidence about the positive association of teachers’ salaries with other inputs. Generally such associations are fairly high, and salaries of teachers are a high proportion of current expenditure in schools. However, our proxy is far from ideal. A preferable measure of school inputs would be for the earlier period when males earning income in 1970 were actually of school age. The median year in which males in our sample were 10 years old was 1955, and expansion of the school system was considerable in the 1950s and 1960s. It is unlikely that there was any important change in the ranking among regions of teacher education, but some measurement error is inevitable due to change over time in the geographical distribution of resources. To limit this measurement error we limit our sample to young men (up through age 35). 3.2. Estimates and implications to estimate relation (4 We use ordinary-least-squares proceduresZ1 the assumption that measurement error and heterogeneity result in an additive disturbance term: In Ri = 0.32 In x + 0.42, (0.04)
R2 = 0.50,
SE = 0.22.
The significant estimate of c =0.68 suggests relative inequality zcrlsion somewhere between the extreme utilitarian case (c =0) and the Cob implies that the government indeed is making a Douglas case (c ore in the SC ooling of those in productivity-equity trade by investin 20Teachers salaries dominate public schooling expenditures in most developing countries. See World Bank (1980). Some readers might think that it would be desirab to adjust our meas by class size. We do not have the data required for such an adjust owever recent resea suggests that class size has very little impact on student achievement. See Glass and Smith and Jamison. 21 ay cause an upward bia 22 ouglas case, in turn, is extremes. In the statement in the text we are assumin s than 0.32 so that trre unbiased esti
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J.R. Behrman and N. Birdsall, The equity-productivity tradeofl
areas with poorer prospects (due to the factors in Xi) than it would do if it ut the concern with were attempting only to maximize total product. inequality probably does not result in suffkient devotion of schooling resources to those in areas with poorer prospects to compensate co hese poorer prospects, though the bias towards the utilitaria pure investment case discussed in the appendix precludes certainty in this regard. How does our estimate of c equal to 0.68 fit into the discussion of tables 1 2? If F is less than 0.32 so the bias in this and 2 at the end of 32 towards the utilitarian case of c equal to estimate of c is less zero,23 this value of relative inequality aversion is on the utilitarian side of the Cobb-Douglas value of one, but above c* for all possible expected income elasticities with respect to public school resources. In such a case, if the expected income generation relation is Cobb-Douglas, our estimate suggests that the Brazilian implicit concern with equity is suffkient that public school reson:rccs are distributed more equally across areas than is the impact of other ctors which determine expected incomes (X). But at the azilian implicit concern with produ tivity is sufficiently same time, the strong (and the stribution of school resources thus s4’kiently unequal) so that the resulting distribution of expected incomes is more unequal than the distribution of the impact of X alone. This is true even though, as noted above, the distribution of public school resources is less unequal than is the distribution of X. If 6 is greater than 0.32, however, the concern with uality is su ciently strong so that the public school resources are distributed to favor the areas with less satisfactory prospects (due to the so that relative expected incomes are greater in such areas e impact of the X alone. That is, the ublic school resources so as to compensate partially for the distribution of X.24
welfare function for the districharacterized neither
N. Birdsall, The equit
Stochastic
c nsiderationsand simultaneitybias
Direct estimation of relation (4 y ordinary-least-squares ethods wit an additive disturbance term involves simultaneity bias because when society chooses an allocation of Ri across areas it also is choosing a distribution of x across areas. Were data available on Xi, it would be possible to obtain unbiased estimates of the equity-pro ctivity tradeoff in the societal welfare function by directly estimating the forms of Ri and I$ as depe on Xi. Unfortunately, in the case under examination, data are not available wit which to estimate such reduced forms. As a second-best approach, therefore, we now consider the nature of the simultaneity bias if we estimate relation (4B) using ordinary-least-squares procedures with an additive disturbance term. Let wi be that disturbance term and ri and yi be the deviations fro the means for In Ri and In I$, respectively. Then relation (4 ) can be written as ri=( 1 -C)yi+
Wia
The expected value of the least-square estimate is
E[l-c)=
ECriYil EL;C l -c)Yi + wi)Yil= 1- c + ECwiYil E[yfl= 4Y’I ’ ECY’I
(6)
where E is the expectations operator. Now let wi= ui+ vi, where u1 arises due to heterogeneity in the expected income production possibility frontier across areas, and tli is due to differences in actual expected income. Such heterogeneity results in a tangency at a point like B in fig. 1, instead of at A as happens on the average. Under the standard assumption that yi and vi are uncorrelated, the numerator in the last term in relation (6) reduces to E[uiy,l. From relation
Locus of Average Tangencies
IFiF i . Aver:ge outcome and a d
etero
city
i
1600
J.R. Behrman and N. Birdsall, The equity-productivity tradeo$
(3C) each unit decrease in Yi caused by such heterogeneity results in a 6 reduction in yi, SO E[uiyi]=bE[y,Z] and E[l-c]=l-c+b. Therefore 1 -c is biased upwards since tS>O, and c is biased downwards, i.e., towards the utilitarian or pure-investment case.
Ahluwalia, MS., 1976, Inequality, poverty and development, Journal of Development Economics, 307-342. Behrman, Jere R. and Nancy Birdsall, 1983, The quality of schooling: Quantity alone is misleading, American Economic Review 73, no. $928-946. Behrman, Jere R. and Nancy Birdsall, 1985, The quality of schooling: Reply, American Economic Review 75, no. 5, 1202-l 205. Behrman, Jere R. and Stephen G. Craig, 1987, The distribution of public services: An exploration of local governmental preferences, American Ecollc.mic Review 77, no. 1, 37-49. Behrman, Jere R., Robert A. Pollak and Paul Taubman, 1982, Parental preferences and provision for progeny, Journal of Political Economy 90, no. 1,52-73. Behrman, Jere R. and Raaj Kumar Sah, 1984, What role does equity play in the international distribution of development aid?, in: Moises Syrquin, Lance Taylor and Larry E. Westphal, eds., Economic structure and performance (Academic Press, New York) 295-315. Birdsall, Nancy, 1985, Public inputs and child schooling in Brazil, Journal of Development Economics 18, no. 1, 67-86. Blackorby, C. and D. Donaldson, 1980, A theoretical treatment of indices of absolute inequality, International Economic Review 2 1, 107-l 36. Chenery, H.B., MS. Ahluwalia, C.L.G. Bell, J.H. Duloy and R. Jolly, 1974, Redistribution with growth (Oxford University Press, London). Christianson, V. and E.S. Jansen, 1978, Implicit social preferences in the Norwegian system of t taxations, Journal of Public Economics 10, 217-245. illiam R., 1975, Distribution and development: A survey of literature, Journal of opment Economics 1, no. 4,359-$00. Christopher, 1982, The impact of primary schooling on economic development: A view ‘of the evidence, World Development 10, 167-185. Souza, Alberto, 1979, Financiamento da educacao e acesso a escola no Brazil, Rio de etulio Vargas/Centro de
studos Fiscais, 1980, Regionalizacao das transacoes do
ary L. Smith, 1979, Meta-analysis of reasearch on class size and achievement, ion and Policy Analysis 1, 2-16. Loxley, 1983, The effects of school quality in academic ievement across 29 high and low incoae countries, n Journal of Sociology 88, no. my’s ‘as if’ proposition: s the median income voter really decisive?, 45-65. uced class size and other alt atives for improving schools: An ene V. Glass, Leonard S. Cahen, ary L. Smith and Ni ola N. Filby, esearch and policy (Sage rience and earnings (National
Bureau of Economic ate a
1971, A theory ~~~ust~ce( arcelo, 1979, Who benefits (Oxford University Press, Oxford).
ent report, 1980 (Oxfo World Bank, 1981, World development report, 1981 (Otiord University Press, New York).
lan
Jere R. BEHRMAN a brad Bank, Washington, DC 20433, USA Received May 1986, final version received June 1987 A governmental welfare function is specified to explore the equity-productivity tradeoff in the allocation of governmental schooling resources. The constrained maximization ode1 is estimated for Brazil using expected incomes in each geographical area as the outcome of interest and governmental resources for each area as the input. The equity-productivity tradeoff in Brazilian governmental preferences is estimated to be considerable, with a welfare curvature close neither to the utilitarian nor to the Rawlsian extremes, but close to the intermediate CobbDouglas value. This implies neutral rather than reinforcing or compensating behavior in the allocation of governmental resources, with the result that the distribution of such resources across areas is more equal than is that of expected incomes.
In recent years distributional as well as growth goals have been increasingly emphasized in the development literature (e.g., Ahluwalia, Chenery et al., Cline). In this literature schooling is seen as a major policy instr&ment with which productivity and distributional goals can be simultaneously pursued [e.g., Colclough ( 1982), World Bank ( 1980,198 I)]. ut in the distribution of public inputs for schooling, there can be a considerable tradeoff between productivity and distributional goals. The geographical areas in which the returns to public spending on schooling, that e marginal productivity of schooling from society’s point of view, are highest are ot necessarily those in which a concern for equity spending be concentrated. An obvious exa le in most ehrman is William R. enan, Jr. Professor of Economics and Co-Director of the Center for Analysis of Developin the University consultant to the r&all was Chic Ith and Nutrition World Bank Research Project 672-21, ‘Studies o original draft prepared in 1983 e are gr?tefu ichael L _-Tart; 1v. and to an anonymous referee, Taubman for useful comments. None of these i any view expressed here. 2921/88/$3.50 0 1988, lsevier Science Publishers
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J.R. Behrman and N. Birdsall, The equity-productivity tradeoff
is the urban-rural dichotomy: &~ral areas tend to be poorer, but available estimates suggest that marginal returns to schooling are hrgher in urban areas, probably because of larger supplies of complementary in ilts. Typically, schooling investments also are higher in urban areas. n this paper we model and estimate the equity-productivity trade0 implicit in the geographical allocation of public resources to pr secondary schooling in a major developing country, Brazil. While an equ productivity tradeoff in the allocation over space of schooling res been recognized by some, we know of no previous efforts to quantify the nature of this tradeoff. We do so by estimating the critical parameters of the implicit welfare function which governs the allocation of public resources by razihan society among its citize . Our method, which we apply using data across 50 geographical areas in razil, provides us with an estimate of the amount of inequality aversion plicit in society’s al!ocation of public resources to schooling across these areas. ‘t his estimate SL’;gests that this welfare function is neither utilitarian with a pure productiv.ty concern nor awlsian with a pure distributional concern. Instead, there 1s an important productivity-equity tradeoff, with somewhat more emphasis on productivity than on equity.
2.I. Gweral formulation
ss governing the geographical allocation of resources to schoolmplex in Brazil. Schooling is funded by a combination cf tocal, state, government resources, with states providing as much as twoe prancing (in the early 1970s which is the time period that we ) and the federal government providing 20 to 25 percent. We assume that the resulting allocation by a number of different government licit societal welfare function budget constraints.’ We do not assume sly maximize such a welfare y maximize such a function. her complicated entities recent public finance Craig, Christenson and ‘3ne can conceive a number of different processes Iii which govern ir respective contributions to educational expenditures. For example, there ~i~ib~~n~ in which local governments and the federal government ‘both are reaction function of the other. Data are not available that erefore, we assu
J.R. Behrman and N. Birdsall, The equit
1587
razil comprise t e rural areas of two very small states due to sampling err
opolitan area of Rio
tween rural and urban1 irreas, so that some of the tradeoff between productivity and equity is actually reflected in the allocation within states of resources to urban versus rural areas. There is also, however, a tradeoff between ity and distributional objectives in the allocation of federal reven e states. At the regional level, the ost obvious distinction is between the highly industrialized southeast an the very poor agricultural northeast. Both local and federal direct expenditures on schooling are higher in the southeast. One analyst of schooling expenditures has estimated, for example, that state and local spending on secondary schools in the northeast in 1975 would have to have been more than three times as great as it was to bring secondary enrollment rates and quality to the southeast level (De Mello e Souza, p. 56). Our rough calculations based on available data on federal direct expenditures for education in the three IL. :or regions (northeast, southeast, and frontier) indicate however, that federal government spending could reduce that disparity substantially; if all federal expenditures on education were directed to the northeast, spending per pupil there, combining state and federal fun could be about the same as state sapending per pupil in the southeast.2 owever, what evidence there is indicates that per capita federal spending on education is if anything slightly higher in the southeast than in the northeast,3 so that federal spending does not compensate. Of course, it is possible that in the absence of distributional objectives, the federal government might direct even more of its spending toward the southeast, if the marginal returns to schooling in the southeast are sufficiently greater than for the other regions. The actual skewness of spending depends not only on the dispersion of marginal returns to schooling investments among areas, but ,nt of inequality aversion in society’s implicit welfare function. also on the ev I = Thus the productivity-equity tradeoff occurs at both the state and the federal government level. e do not attempt to distinguish explicitly between the two, but note that the federal government probably spends enough on schooling to equalize total spending among the three major regi and thus could probably equalize spending among equalize spen in states well mig ‘Federal exp enditures per capita for education ave 61, 55 and 65 (1975 cruzeiros) for t frontier, northeast and southeast regions. Total expenditures per student, inclu are 642, 419, and 1196 for the three regions, respectiv percent of the population, federal expenditures are about 3 per student in each ar reg!on, and if directed to the northeast could assure near e an llinger, Tables 12 and 30. har and Dillinger, op. cit.
J.R. Behrman and N. Birdsall, The eq
1588
center than is politically and administrative done. Thus in a sense it is the implici government that we are estimating. e assume that this welfare functi capita income of indivi eligible to areas? Differences in i among the in area are assumed to be distributed si for the areas? Therefore the welfare fu income of an average individual eligible to times, where ni is the number of such
w= W(Y, ,...,
Yl,...,
Y, ,..., Ym)=
y-producticitytradeof
feasible, it could in principle be elfare function of the federal efined over the led in each of m iduals within each geographical und the geographical means n merely repeats the expected schooled in the ith area (I$) ni in the ith area: if(K)9**-94nf(KJ)*
(1)
.
This welfare function is maximized su The first constraint is that total pub1 be greater or equal to the sum of such res in all of the geographical areas. If Ri is to eligible individuals in the ith area, this c
RzCniRi,
0 two major constraints. urces devoted to schooling (R) rces allocated to all individuals erage public resources devoted
(2)
m
In order to focus on the estimation of the government allocation of public schoo resources is assumed to be given? The second constraint (or set of co come-generation function for I$ whit average le th of schooling (Si) an schooling ( in the ith geographical area: &= Y”(Si,
uity-productivity tradeoff in the resources, the total of such aints) is the expected average s posited to depend upon the average resources devoted to
(3) ength of schooling is supported ates, of which Psacharopoulos
‘The overall welfare function may be con defense, national prestige), but we as schooling resources is separable from t ly because of its impact on exp ic analysis of the impa tradition).
many other outcomes (e.g., national art related to the allocation of public s. The assumption that schooling is of is a simplification that is common in (e.g., earnings functions in the inter rences in the distribution of individual etc. in different geographical areas, but to the sources of these
provides a recent review. nt in school, however, the quality of schooling that the quality of schooling is etermined basi resources per eligible student (RJ to schooling.* of schooling is posited to be a private decision de n of expected income levels associated with different schooling levels, given schooling costs (includi school emphasized by Mincer and associated with a given increment i the quality of schooling. That is, an extra year posited to increase the expected in lower-quality schooling. Therefore we posit that the length of schooling depends on the quality of schooling9 - which is again represented by the allocation of public resources - and by other factors related to family background, expected labor market outcomes, etc. as summarized in t vector Xi” Si = Si(Ri, Xi).
WV
Substitution of this relation into the expected income function in relation (3) givesl l & = x( Ri, Xi).
cw
‘We find empirical support for the importance of school quality in Brazil in Behrman and Birdsall (1983,1985). Also see Heyneman and Loxley. *For the primary and secondary levels of schooling under study, this is a reasonable approximation for most societies. There are some cases (e.g., the Republic of Korea, the Philippines) in which private resources also may be quite important in determining schooling quality. To analyze such casts, these private resources also would have to be included in relation (3). ‘We assume that migration decisions rmong areas are not made on the basis of schooling quality, so Ri is given. Post-schooling migration may occur and is incorporated into expected income outcomes. The assumption of no substantial migrati?n in response to perceived school quality differentials is much more palatable for our sample than it might be for many others for two reasons. (1) Our geographical areas are fairly large so usually fairly long-distance migration would be required to change areas in response to differential schoolin schooling levels are quite low (the sample mean is about three grades) so the tend to be young enough that it is unlikely they woul their own to obtain better primary echoohng, nor th for this reason. For smaller geographical distances and higher schooling levels, of course, in least of students, if not of their fa many societies migration ( tials. response to school quality di rminants of the length of schooling for current children of the “Birdsall investigates the adults in the sample which we use in Section quality on the length of their SC theoretical reasons why such an 1 ‘The asterisk in relation (3) is used t
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J.R. Behrman and N. Birdsall, The equity-productivity tradeofl
This is the form in which we represent the second set of constraints in what follows, Xi is assumed to be basically exogenous to the decision re the allocation of public schooling resources. e allocation of public resources for primary an secondary schooling is assumed to be the outcome of the maximization of the welfare function in relation (1) subject to the budget constraint in relation (2) and the expected income function in relation (3B). We assume that the welfare effect of increasing the ith income is positive (iYW/iJF> 0)and that the i of more public schooling resources on expected income is positive Under these assumptions, all public schooling resources are allocated and the equality in eq. (2) holds. We also assume that the relevant functions are smooth, with non-increasing returns to scale and diminishing returns to each nt, and that the maximization leads to an interior solution. In this first-order conditions for the maximization are
ere R is the Lagrangian multiplier for the budget constraint. n behind our estimation procedure can be illustrated by ratio of the first-order conditions for the ith and jth areas
slope of the expected income possibility frontier. a tangency, at which int the marginal rate of substitution
or each there is a relation like
only productivity matters. l3 There is indifference among points on such a curve with considerably different distributional implications. two extremes these are an infinite num in which there are equity-productivity t 2.2. Explicit functional forms and relative versus a solute inequality awersio To estimate the equity-productivity tradeoff from relation (4), ex assumptions must be made for the expected income function in relation and for the welfare function in relation (1). Expected
income function:
We consider a partial log-linear expected
income function In x=a+blnRi+g(Xi).
(3C)
The last right-side term can be any functional form which satisfies the nonincreasing returns to scale and diminishing returns to individual variable assumptions made above (e.g., if g(Xi) =p In Xi, relation (3C) is a CobbDouglas expected income production function). For substitution into relation (4) the partial derivative of 5 with respect to Ri is require
Since x depends on Xi, this partial derivative depends on the other average area-specific characteristics relating to family background and expected labor market outcomes and therefore is shifted by Xi as is required for the identification of the welfare surface (see subsection 2.1) even though Xi does not appear explicitly in relation (3D). Government werfare function: The we fare functio nown CES form
e thank a referee for pointing out that maximizing states of the world may lead to a Wawlsian preferen (in a Gini coeffkient sense) at the same time the lowe
‘benefits’ is ambiguous. If we interpret benefi
Behrman and N. Birdsall, The equity-productioity tradeo$
JR
159%
ith this welfare function c summarizes the equity-productiv A higher value of c implies greater inequalit awlsian and only the lowest ex tirely. If c = levant. For values of c ere is an equity-productivity tradeoff. his welfare function implies constant relative i equality aversion. pe of an isowelfare curve jth areas”
-- 3 -Y-j -c
dK
0
W=
VliF
’
w
is relation says that along an isowelfare curve in the ith-jt -weighted) relative expected into e between individuals in area i and area j is relevant.a6 aximization of the CES welfare function subject to the budget constraint and to the expected income function in relation (3C) and taking logarithms gives In i=(1
-C)Ill
q+k,
n (1 -c)%/E,. Note that Xi does not a ar explicitly in relation tifying the curvature of the welfare ecause all of the im act of Xi is represented by licit appearance 0 14As an alternative parameter repr
we also considered the Kohm-Pollak (K-P) welfare function (see and Sah and Behrman and Craig) which also has a oneaversion: W = -(l/h)In [&/@e-kY’], where n=Cni. A ality aversion. For LO, the K-P welfare function r h-, 00, the K-P welfare function
area:
where Z=g(Xi) -g(Xj). Relations (SA) a public resources (R) and of e depends on Z (and thus the inequality aversion (c), and the elasticity o government-allocated school resources (b).
that the distri across areas nt relative respect to
oefficients of 2 in relations Tables 1 and 2 summarize the values of (5A) and (5B), depending on different value nd b. As discussed above, c e Rawlsian case of 00, with may range from the utilitarian case of zero y. If the impact of pub the Cobb-Douglas value of one also not nd if the returns to scale m school resources o income is pos decreasing, constant or not ction function of expected incom erefore this range increasing, b is bounded by zer b is included in the tables. Before examining the entries in the tables, it is useful to note that positive entries imply a distributicn of R or Y that favors the area X, and negative entries tion relation (SC) is Co of one has a special sign& uction function cas
J.R. Behrman and N. Birdsall, The equity-productivity tradeof
Since [XJXj)P gives the inequality due to1 the inequality in the distribution of rest of the exponent (which is the same in SA’ as in SA and the same ’ as in SB) tells whether the impact of the inequality in X is reinforced, weakened or unaltered by the allocation of R, depending upon whether the rest of the exponent is greater than, less than or equal to one.‘* First consider the exponents for the relative RJR, relation in table 1. The ouglas extent of relative inequality aversion (c= 1) obviously is critical. On the Rawlsian side of the Cobb-Douglas case (c > l), school Table
1
Values of coefficient of Z in relation for distribution of public school resources (RJRj) as dependent on relative inequality aversion (c) and elasticity of expected income with respect to public school resources (b).
(Utilitarian)
O
c=l (CobbDouglas)
b+O
1
1 -c>o
0
O
1 ->l l-b
1 -b(l
00
--0
C-*0
b-,1.0
C--,00
1 < c < 00
(Rawlsian)
l-c<0
--oo
l-c
l-c -c)
>o
0 l-b(l-c)
l-c
0
l-c -l<-
c
-1 b -1
C
Table 2 Value of coetkient of Z in relation for distribution of expected incomes (YJq) as dependent on relative inequality aversion (c) and elasticity of expected income with respect to public school resources (b). c-0
c=l
(Utilitarian)
o
b-+0
1
1
O
--1
b-,1.
1
1
1
l-b
T-!?(I 4)
00
I ->I C
C-+00
(Cobb-Douglas)
1
(Rawlsian)
1
1
1 1
>I
1
<1
0
l-b(l-C) 1
1 --
0
J.R. Behman and N. Birdsall, The equity-productivity trade
resources are distributed so as to favor or compensate those areas with poorer X (i.e., the exponents are negative), though this tendency is wea the more effective are school resources in altering expected income (i.e., the greater is 6) and the closer relative inequality aversion is to the CobbDouglas case. On the utilitarian side of the CobbDouglas case (cc l), public school resources are distributed with the areas with more X also receiving more R, thus reinforcing differences in X. For the Cob&Douglas value of inequality aversion (c = l), the government is neutral in its allocation of public resources, neither compensating for nor reinforcing inequalities in the distribution of X. If the expected income production relation is CobbDouglas, an interesting question is whether the exponent is greater or less than one. The critical value at which the exponent in the RJR, relation equals one is c*=bJ(l
+b).
W)
c* is zero if b is zero and approaches 0.5 monotonically as b approaches one. If c is below c* and if the expected income production function is CobbDouglas, school resources are distributed more unequally than is the impact of the distribution of X alone on the distribution of expected incomes. This occurs at the utilitarian extreme (c+O) in which only total expected income counts, but also for values of c greater than zero if school resources are relatively effective (i.e., b is large). Now consider the exponents for the relative expected income relation (x/q) in table 2. The pattern of values of one again is interesting. The first row indicates that the coeffkient approaches one as b approaches zero because public school resources are ineffective in such a case - despite the much different degrees of relative inequality aversion in the first row of table 1. As b approaches zero the distribution of income reflects only the impact of the distribution of X, independent of the distribution of R. The third column also has coefficient values of one. This is the Cobb-Douglas case in which equity and productivity considerations are balanced off so that expected incomes are distributed identically to the impact of the expected X’s even though the R’s are equally distributed (see zero values in the third column of table 1) for all permissible values of the effectiveness of ubhc resources on expected incomes (i.e., all values of b). Note that in this case expected incomes are distributed as unequally as is the impact of rovision of equal public school resources to all. If c is less ouglas value of one and b is positive, even though the may be more equal t distribution of expecte of the X’s.
1596
J.R. Behrman and N. Birdsall, The equity-productivity tradeo$
inequalities in X (i.e., the relevant coefficients in table 1 are negative), this results in a more equal distribution of expected incomes than the impact of the distribution of X alone would imply. Ocly if c approaches infinity in the awlsian limi , however, are expected incomes actually equalized.
3.1. Data Our data are from the razilian l/100 Public Use sample of households of the 1970 census. We consider only males in order to avoid complications due to selectivity regarding female labor force participation. We limit the age range to 15-35 in order to have a better representation of public schooling resource aliocation (see below). We consider m= 50 geographical areas because the data set permits the matching of areas of schooling with the public resources devoted to schooling at this level of aggregation. The 50 eas are comprised of the rural areas of 24 states plus urban areas of 26 of razil’s 27 states in 1970. To represent the average expected income for those eligible to be schooled in each area (Q, we use the mean 1970 monthly income of those who were in that area at the time they were of school age.” This income may be earned in this area or elsewhere, if there was post-schooling migration. By using this measure, we assume that the actual migratory experience of the sample was that expected a priori and that the migratory costs were relatively small compared to permanent income levels. The value of this variable ranges fairly considerably (with a standard deviation of 188 as to a mean of 230 cruzeiros) despite the averaging out of individual for those with the same geographical area of origin. nt the average government provision of resources per child in ), we would like to have data on these resources in real terms at the time that the schooling was undertaken. Unfortunately, such data are not available. As a proxy, therefore, we use the mean education (years of area as calculated from the same sample, under ajor resource being allocated is teachers of e use the nominal values despite evidence of geographical price variations because price deflators are not available for the areas of interest. Given that the pries levels probably are correlated with income levels due to higher rents in higher-income, more-urban locations (see Thomas), the use of nominal instead of real income may bias our estimates, as the following nstrates. Let Y= Yl/Pi where Y; is the nominal exp ed income in the ith area and Pi price deflator for the ith area. Then relation (43) can written as lnRi=k+(L-c)ln YT(1 -c) In Pi. If the last term is excluded since Pi is not observed, the estimate of the coenicient omitted variable bias: E[ 1 -c] = 1 -c-( 1 -c)r, where E is the n Y;. If r is positive as utilitarian case) if the e true
J.R. Behrman and N. Birdsall, The equity-productivity
1597
differential training and quality.20 The mean value of teacher education in the sample is 8.2 grades, with a standard deviation across the geograp areas of 2.3 rades. Teacher e ucation ranged in 1970 from a low of about 3 years in the rural northeast to 11 years in the urban south. Tea&er education is a better proxy for average government resources more correlated it is with teacher salaries, the more corre salaries are with other public school expenditures, the m re uniform teacher training is across areas, and the more stable over e is the geographical distribution of these resources. In Behrman and sall (1983) we summarize the available (fragmented) evidence about the positive association of teachers’ salaries with other inputs. Generally such associations are fairly high, and salaries of teachers are a high proportion of current expenditure in schools. However, our proxy is far from ideal. A preferable measure of school inputs would be for the earlier period when males earning income in 1970 were actually of school age. The median year in which males in our sample were 10 years old was 1955, and expansion of the school system was considerable in the 1950s and 1960s. It is unlikely that there was any important change in the ranking among regions of teacher education, but some measurement error is inevitable due to change over time in the geographical distribution of resources. To limit this measurement error we limit our sample to young men (up through age 35). 3.2. Estimates and implications to estimate relation (4 We use ordinary-least-squares proceduresZ1 the assumption that measurement error and heterogeneity result in an additive disturbance term: In Ri = 0.32 In x + 0.42, (0.04)
R2 = 0.50,
SE = 0.22.
The significant estimate of c =0.68 suggests relative inequality zcrlsion somewhere between the extreme utilitarian case (c =0) and the Cob implies that the government indeed is making a Douglas case (c ore in the SC ooling of those in productivity-equity trade by investin 20Teachers salaries dominate public schooling expenditures in most developing countries. See World Bank (1980). Some readers might think that it would be desirab to adjust our meas by class size. We do not have the data required for such an adjust owever recent resea suggests that class size has very little impact on student achievement. See Glass and Smith and Jamison. 21 ay cause an upward bia 22 ouglas case, in turn, is extremes. In the statement in the text we are assumin s than 0.32 so that trre unbiased esti
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J.R. Behrman and N. Birdsall, The equity-productivity tradeofl
areas with poorer prospects (due to the factors in Xi) than it would do if it ut the concern with were attempting only to maximize total product. inequality probably does not result in suffkient devotion of schooling resources to those in areas with poorer prospects to compensate co hese poorer prospects, though the bias towards the utilitaria pure investment case discussed in the appendix precludes certainty in this regard. How does our estimate of c equal to 0.68 fit into the discussion of tables 1 2? If F is less than 0.32 so the bias in this and 2 at the end of 32 towards the utilitarian case of c equal to estimate of c is less zero,23 this value of relative inequality aversion is on the utilitarian side of the Cobb-Douglas value of one, but above c* for all possible expected income elasticities with respect to public school resources. In such a case, if the expected income generation relation is Cobb-Douglas, our estimate suggests that the Brazilian implicit concern with equity is suffkient that public school reson:rccs are distributed more equally across areas than is the impact of other ctors which determine expected incomes (X). But at the azilian implicit concern with produ tivity is sufficiently same time, the strong (and the stribution of school resources thus s4’kiently unequal) so that the resulting distribution of expected incomes is more unequal than the distribution of the impact of X alone. This is true even though, as noted above, the distribution of public school resources is less unequal than is the distribution of X. If 6 is greater than 0.32, however, the concern with uality is su ciently strong so that the public school resources are distributed to favor the areas with less satisfactory prospects (due to the so that relative expected incomes are greater in such areas e impact of the X alone. That is, the ublic school resources so as to compensate partially for the distribution of X.24
welfare function for the districharacterized neither
N. Birdsall, The equit
Stochastic
c nsiderationsand simultaneitybias
Direct estimation of relation (4 y ordinary-least-squares ethods wit an additive disturbance term involves simultaneity bias because when society chooses an allocation of Ri across areas it also is choosing a distribution of x across areas. Were data available on Xi, it would be possible to obtain unbiased estimates of the equity-pro ctivity tradeoff in the societal welfare function by directly estimating the forms of Ri and I$ as depe on Xi. Unfortunately, in the case under examination, data are not available wit which to estimate such reduced forms. As a second-best approach, therefore, we now consider the nature of the simultaneity bias if we estimate relation (4B) using ordinary-least-squares procedures with an additive disturbance term. Let wi be that disturbance term and ri and yi be the deviations fro the means for In Ri and In I$, respectively. Then relation (4 ) can be written as ri=( 1 -C)yi+
Wia
The expected value of the least-square estimate is
E[l-c)=
ECriYil EL;C l -c)Yi + wi)Yil= 1- c + ECwiYil E[yfl= 4Y’I ’ ECY’I
(6)
where E is the expectations operator. Now let wi= ui+ vi, where u1 arises due to heterogeneity in the expected income production possibility frontier across areas, and tli is due to differences in actual expected income. Such heterogeneity results in a tangency at a point like B in fig. 1, instead of at A as happens on the average. Under the standard assumption that yi and vi are uncorrelated, the numerator in the last term in relation (6) reduces to E[uiy,l. From relation
Locus of Average Tangencies
IFiF i . Aver:ge outcome and a d
etero
city
i
1600
J.R. Behrman and N. Birdsall, The equity-productivity tradeo$
(3C) each unit decrease in Yi caused by such heterogeneity results in a 6 reduction in yi, SO E[uiyi]=bE[y,Z] and E[l-c]=l-c+b. Therefore 1 -c is biased upwards since tS>O, and c is biased downwards, i.e., towards the utilitarian or pure-investment case.
Ahluwalia, MS., 1976, Inequality, poverty and development, Journal of Development Economics, 307-342. Behrman, Jere R. and Nancy Birdsall, 1983, The quality of schooling: Quantity alone is misleading, American Economic Review 73, no. $928-946. Behrman, Jere R. and Nancy Birdsall, 1985, The quality of schooling: Reply, American Economic Review 75, no. 5, 1202-l 205. Behrman, Jere R. and Stephen G. Craig, 1987, The distribution of public services: An exploration of local governmental preferences, American Ecollc.mic Review 77, no. 1, 37-49. Behrman, Jere R., Robert A. Pollak and Paul Taubman, 1982, Parental preferences and provision for progeny, Journal of Political Economy 90, no. 1,52-73. Behrman, Jere R. and Raaj Kumar Sah, 1984, What role does equity play in the international distribution of development aid?, in: Moises Syrquin, Lance Taylor and Larry E. Westphal, eds., Economic structure and performance (Academic Press, New York) 295-315. Birdsall, Nancy, 1985, Public inputs and child schooling in Brazil, Journal of Development Economics 18, no. 1, 67-86. Blackorby, C. and D. Donaldson, 1980, A theoretical treatment of indices of absolute inequality, International Economic Review 2 1, 107-l 36. Chenery, H.B., MS. Ahluwalia, C.L.G. Bell, J.H. Duloy and R. Jolly, 1974, Redistribution with growth (Oxford University Press, London). Christianson, V. and E.S. Jansen, 1978, Implicit social preferences in the Norwegian system of t taxations, Journal of Public Economics 10, 217-245. illiam R., 1975, Distribution and development: A survey of literature, Journal of opment Economics 1, no. 4,359-$00. Christopher, 1982, The impact of primary schooling on economic development: A view ‘of the evidence, World Development 10, 167-185. Souza, Alberto, 1979, Financiamento da educacao e acesso a escola no Brazil, Rio de etulio Vargas/Centro de
studos Fiscais, 1980, Regionalizacao das transacoes do
ary L. Smith, 1979, Meta-analysis of reasearch on class size and achievement, ion and Policy Analysis 1, 2-16. Loxley, 1983, The effects of school quality in academic ievement across 29 high and low incoae countries, n Journal of Sociology 88, no. my’s ‘as if’ proposition: s the median income voter really decisive?, 45-65. uced class size and other alt atives for improving schools: An ene V. Glass, Leonard S. Cahen, ary L. Smith and Ni ola N. Filby, esearch and policy (Sage rience and earnings (National
Bureau of Economic ate a
1971, A theory ~~~ust~ce( arcelo, 1979, Who benefits (Oxford University Press, Oxford).
ent report, 1980 (Oxfo World Bank, 1981, World development report, 1981 (Otiord University Press, New York).