A Consideration of the Decision to Reallocate Educational Resources on the Basis of a Comparison of Academic Area Test Scores across Schools

doi:10.4236/ce.2010.13025

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Creative Education

2010. Vol.1, No.3, 162-165

A Consideration of the Decision to Reallocate Educational

Resources on the Basis of a Comparison of Academic Area Test

Scores across Schools

Melvin V. Borland, Roy M. Howsen

Department of Economics, Western Kentucky University, Bowling Green, USA.

Email: roy.h [email protected]

Received July 30th, 2010; revised September 25th, 2010; accepted September 30th, 2010.

Although suggestions for the reallocation of educational resources for individual schools or school districts are

often made on the basis of comparative test scores by academic area across such individual schools or school

districts, such scores, it is shown, are neither necessarily nor sufficiently informative for the systematic determi-

nation of utility increasing reallocations of those resources. To the extent that different administrators hold dif-

ferent educational utility functions in terms of test scores, then, even under assumptions of concurrent, but indi-

vidual utility maximizations, different distributions of educational resources would be expected. Students in in-

dividual school or school districts, therefore, would be expected to have comparative test scores that are neces-

sarily low in at least one academic area and high in at least one other, mutatis mutandis. Reallocations made on

the basis of comparative test scores by academic area within individual schools or school districts, therefore,

cannot, it is shown, be expected to systematically increase educational utility.

Keywords: Reallocation Resources, Test Scores

Comparative test scores by academic area for individual

schools or school districts are extensively reported by the vari-

ous media and the subject of much discussion by policy makers,

educational authorities, and members of the general public.1 It

is on the basis of such comparative scores across schools or

districts that reforms in resource allocation are suggested. The

typical suggestion given the report of comparative test scores

by academic area is to use relatively more resources in the aca-

demic area(s) associated with the relatively low comparative

test score(s) and, therefore, necessarily fewer resources in the

academic area(s) associated with the relatively high compara-

tive test score(s), given that the availability of such resources is

constrained. Is such a change in resource allocation necessarily

educational utility-increasing (Leithwood & Stager, 1989; Si-

mon, 1976)?2 For the purpose of stimulating empirical work on

the relationship between comparative test scores and resource

allocation, consider the following model of optimal resource

allocation and the discussion of exogenous effects given optim-

al resource allocation.

Optimal Resource Allocation

The educational utility function considered in this paper is

limited to the scores, q and v, associated respectively with the

area s of quantitative reasoning and verbal proficiency, only.

This limitation on the number of areas for which test scores are

associated allows us with ease of exposition to concentrate on

the particular variables in question in this model of optimal

resource allocation, that is, on the amount of resources, rq, used

in the production of quantitative reasoning and the amount of

resources, rv, used in the production of verbal proficiency and

results in no loss of generality.

Rational school administrators under these conditions are

expected to maximize educational utility, u, as a function, u, of

q and v, i.e. , to maximize

( )

,,uu qv=

(1)

subject to

Rr r= +

(2)

In Equation (2) above, R is the sum of the number of units of

resources distributed to the areas of quantitative reasoning and

verbal proficiency, that is, of rq and rv, respectively.3 The budg-

et constraint for the school administrator is implied by the finite

number of units of resources available for distribution.

The absolute test scores associated with the areas of quantita-

tive reasoning and verbal proficiency are assumed to be related

to the quantities of resources, rq and rv, by production processes,

q and v,4 respectively. The proposition that time, at least, corre-

lates positively with measured achievement is supported by a

considerable literature.5 (Levin & Tang, 1987; Roberts et al,

1986; Karweit, 1984; Penham & Lieberman, 1980; Fisher &

Filby, 1929; Wiley & Harnischfeger, 1974; Carroll, 1963)

By substitution, where the production processes, q and v, in

which the school administrator has the finite number of units of

resources available for distribution state the levels of educa-

tional achievements by area as functions of rq and rv, respec-

tively, Equation (1) can be rewritten as

( )

u uqrvr=

(3)

Form the Lagrange function,

( )

qvf qv

u uqrvrRrr=− −−

(4)

where λ is as yet an undetermined multiplier, to solve for the

M. V. BORLAND ET AL.

163

optimal quantities of rq and rv. The 1st-order conditions for the

maximization of educational utility are determined from the

solution of the three partials of Equation (4) with respect to rq,

rv, and λ, each set equal to 0. As such, the utility-maximizing

conditions require the distribution of available resources so that

ur ur=

6 (5)

These 1st-order conditions require that the ratios of the mar-

ginal utilities of the last unit of resources spent on quantitative

reasoning and the last unit of resources spent on verbal profi-

ciency be equal to the ratio of their associated prices.7 In this

case, the associated prices are in terms of units of resources and

are, therefore, each equal to 1. These conditions imply that, if

the increase in utility that would result from spending an addi-

tional unit of educational resources on either quantitative rea-

soning or verbal proficiency was greater than that which would

result from spending an additional unit on the other, utility

could be increased if and only if resources are reallocated to

that area with the larger marginal utility and, since R is a finite

number of units, necessarily from that area with the smaller

marginal utility. The multiplier, λ, is interpreted, in this case, as

the marginal educational utility of resources, that is, as the

change in educational utility with respect to a change in the

finite number of units of resources available for distribution.8

Exogenous Effects Given Optimal Resource

Allocations

If the allocation of available resources for an individual

school or school district satisfies Equation (5), levels of achieve-

ment associated with the various areas of testing, q and v, are

implied by production processes, q and v, and educational utili-

ty is maximized. Comparative area scores for that individual

school or school district associated with such utility-maximiz-

ing levels of academic achievement, however, are dependent

not only on levels of the absolute achievement of students in

that individual school or school district, but on levels of the

absolute achievement of students in other individual schools or

school districts to which comparisons are made. And such other

levels of absolute achievement are implied by other, although

nevertheless optimal, allocations of educational resources with

respect to the associated utility functions of the other individual

schools or school districts.

There may be several explanations, under such a comparison,

for the existence of different allocations of educational re-

sources across individual schools or school districts, even under

assumptions of concurrent, but individual utility maximization.

One such explanation is the existence of different educational

utility functions across such individual schools or school dis-

tricts to which different allocations are made. Given the educa-

tional utility function for an individual school or school district,

suppose that the allocation of resources satisfies Equation (5).

Note that, even if the school administrator of the individual

school or school district is the only school administrator to do

so with respect to the educational utility function attached to

that individual school or school district, that is, to allocate re-

sources in a utility-max imizing way, it may, nevertheless, have

a comparative score in at least one academic area that is low

and a comparative score in at least one other academic area that

is high relative to that for other individual schools or school

districts to which comparisons are made, mutatis mutandis.

Suppose that the educational utility functions for two differ-

ent school groups, i and j, are given by different functions, ui

and uj,

( )

uu qv=

(6)

and

( )

uu qv=

(7)

each of which are assumed to have the typical properties asso-

ciated with such functions. By substitution,

( )

i iqivi

uu qrvr=

(8)

and

() ()

( )

j jqjvj

uuqrvr=

(9)

The expected allocations that would satisfy the associated

conditions for utility maximization for individual school groups,

i and j, imply the existence of a relatively low comparative

score in at least one academic area and a relatively high com-

parative score in at least one other academic area for each indi-

vidual school or school district, to the extent that different

school administrators of different school groups will have acted

differently to maximize different educational utility functions.

Note that such individual school or school districts are, never-

theless, each maximizing educational utility, but with respect to

different educational utility functions, ui and uj, respectively.

Even though ui is maximized with rqi and rvi that yield values of

qi and vi, respectively, the relative positions of qi and vi in a

report of comparative test scores is dependent not only on qi

and vi, but by the values of qj and vj resulting from the choice of

other individual school or school district administrator, j,

beyond the control of the administrator for the individual

school or school district, i, in question.

Let the values for rqi and rvi result in area test scores, for ex-

ample, of 41 and 59, respectively, for individual school group i.

Let the values for rqj and rvj result in test scores, for example, of

53 and 47, respectively, for individual school group j. The

comparative scores for group i are relatively low in quantitative

reasoning and high in verbal proficiency and for group j are

relatively high in quantitative reasoning and low in verbal pro-

ficiency.9 If such values for rqi and rvi and for rqj and rvj exist

where utility-maximizing conditions hold, that is, if qirq = virq

and qjrq = vjrq, for qroups, i and j, and if more resources, then,

are spent on quantitative reasoning and necessarily less re-

sources are spent on verbal proficiency in i and if less resources,

then, are spent on quantitative reasoning and necessarily more

resources are spent on verbal proficiency in j, the reallocations,

in response to the comparative scores across school groups in

this case, would have necessarily negative effects on educa-

tional utility for both groups.

Any change in the allocation of resources of the type typi-

cally suggested on the basis of comparative test scores across

school groups would result in less than the maximum educa-

tional utilities with respect to such different educational utility

functions. A claim to the contrary would violate the original

assumption that the initial allocations of resources, from the

associated solutions of the three partials of Equation (4), were

utility-maximizing. As such, comparative scores for the various

areas of testing to which resources are distributed are neither

M. V. BORLAND ET AL.

164

necessarily or sufficiently informative for the systematic deter-

mination of utility-increasing reallocations of educational re-

sources. The relative position of an individual school or school

district is determined not only by their decision to allocate re-

sources to the various academic areas for which comparisons

are made, but the decision to allocate resources by other

schools or school districts over which they have no control .

Summary and Conclusion

Is the typical suggestion of change in resource allocation

given the report of comparative test scores by academic area

necessarily utility-increasing? If different school administrators

allocate resources under utility-maximizing conditions, but with

respect to different educational utility functions, then even in

this case, the students in such schools or school districts would

be expected to have comparative test scores that are necessarily

low in at least one academic area and high in at least one other

academic area, mutatis mutandis. To reallocate educational

resources, therefore, on the basis of the existence of compara-

tive test scores can not be expected to be utility-increasing for

any individual school or school district, despite the existence of

low and high comparative scores in such individual schools or

school districts. A claim to the contrary would violate the orig-

inal assumption that the initial allocations of resources are util-

ity-maximizing. Unless empirical work on the relationship be-

tween comparative test scores and resource allocation is under-

taken to provide knowledge of comparative marginal utilities

by academic area, reallocations of available resources sug-

gested by policy makers, educational authorities, and members

of the general public will have unknown consequences on edu-

cational utility.

Re ferences

Carroll, J. (1963). A model of school learning. Teachers College

Record, 64, 723-733.

Denham, C., & Lieberman, A. (Eds. ) (1980). Time to Learn. Washing-

ton, D.C.: National Institute of Education.

Fisher, C., Marliave, R. & Filby, N. (1979). Improving teaching by

increasing ‘academic learning time’. Educational Leadership, 37,

52-54.

Karweit, N. (1984). Time-on-task reconsidered: Synthesis of research

on time and learning. Educational Leadership, 41, 32-35.

Leithwood, K., & Stager, M. (1989). Expertise in principals’ problem

solving. Educational Administration Quarterly, 25, 126-161.

doi:10.1177/0013161X89025002003

Levin, H., & Tsang, M. (1987). Economics of student time. Ec onomics

of Education Review, 6, 357-364. doi:10.1016/0272-7757(87)90

019-7

Roberts, R., Schrader, R., & Harryman, M. (1986). Productive use of

time: an attack on declining achievement. Journal of Human Beha-

vior and Learning, 3, 32-40.

Simon, H. (1976). Administrative Behavior: A Study of Decision- Mak-

ing Processes in Administrative Organization, 3rd ed., New York:

The Free Press.

Wiley, D., & Harnischfeger, A. (1974). Explosion of a myth: Quantity

of schooling and exposure to in instruction, major educational ve-

hicles. Educational Researcher, 3, 7-12.

Endnotes

1Such comparative test scores most commonly take the form of per-

centile scores by academic area derived from absolute scores for com-

parisons to other individual schools or school districts. Sometimes,

however, absolute test scores are transformed into stanines, despite the

condition that neither parents nor counselors are likely to be able to

provide a definition of a stanine.

2Such a question with respect to the results of administrative deci-

sions has long existed and is currently relevant. See K. Leithwood and

M. Stager, “Expertise in Principals’ Problem Solving,” Educational

Administration Quarterly, v25, n2 (May 1989), and H. Simon, Admini-

strative Behavior: A Study of Decision-Making Processes in Adminis-

trative Organization

3Equation (1) is assumed to be continuous, with first and second par-

tial derivatives that are continuous, and strictly quasi-concave in its

arguments. The first partials of Equation (1) are assumed to be strictly

positive. Attempts, in general, to affect levels of achievement, in terms

of scores associated with the various areas of testing, are not con-

strained to those that redistribute a fixed quantity of educational re-

sources, but include those that increase educational resources by trans-

fers from non-educational areas and by increases in the educational day

and/or year. Nevertheless, given the determination for R in a more

general equilibrium framework, distributions of educational resources

are assumed to be constrained to R.

, 3rd. ed., New York: The Free Press, 1976.

4Equation (4) is also assumed to be continuous, with first and second

partial derivatives that are continuous, and strictly quasi-concave in its

arguments. The first partials of Equation (4) are assumed to be strictly

positive, as well. An analysis over time can be simplified by invoking a

composite-resource theorem that states that, if prices of a group of

resources change in the same proportion over time, the optimal choice

of resources is consistent with the solution implied by the consideration

of resources as a single resource.

5See H. Levin and M. Tsang, “Economics of Student Time,” Eco-

nomics of Education Review, v6, n4 (1987), pp. 357-364; R. Roberts, R.

Schrader, and M. Harryman, “Productive Use of Time: An Attack on

Declining Achievement,” Journal of Human Behavior and Learning, v3,

n3 (1986), pp.32-40; N. Karweit, “Time-on-Task Reconsidered: Syn-

thesis of Research on Time and Learning,” Educational Leadership, v41,

n8 (May 1984), pp. 32-35; C. Denham and A. Lieberman,

co-editors, Time to Learn, Washington, D.C.: National Institute of

Education (1980); C. Fisher, R., and N. Filby, “Improving Teaaching

by Increasing ‘Academic Learning Time’,” Educational Leadership,

v37, n1 (October 1979), pp. 52-54; D. Wiley and A. Harnischfeger,

“Explosion of a Myth: Quantity of Schooling and Exposure to Instruc-

tion, Major Educational Vehicles,” Educational Researcher, v3, n4

(April 1974), pp. 7-12; J. Carroll, “A Model of School Learn-

ing,” Teachers College Record

6The terms, urq = urv, represent the 1st partials of u with respect to rq

and rv, res pecti ve ly.

, v64, n8 (May 1963), pp. 723-733.

7The 2nd-order conditions require that the score transformation func-

tion is increasing at the quantities at which the 1st-order conditions are

satisfied. If the educational utility function in terms of rq and rv is a

regular strictly quasi-concave function over a domain, the quantities at

which the 1st-order conditions are satisfied are unique educational

M. V. BORLAND ET AL.

165

utility-maximizing quantities over that domain.

8Since urq = urv for the rq and rv that would result from the solution of

the three partials of Equation (4), uq is less than uv for the rq + Δr and rv

- Δr that would result with the reallocation of a unit of resources, Δr,

implied by the typical suggestion of change in resource allocation to

quantitative reasoning and from verbal proficiency. The condition that

uq would be less than uv is implied by the condition that urq = urv are

both less than 0. The condition that urq = urv are both less than 0 is con-

sistent with the 2nd-order condition for utility maximization that results

from the strict quasi-concavity assumption, previously stated. The

terms, urq and urv represent the 2nd partials of u with respect to rq and rv,

respectively.

It is not necessary, however, that the values for rq and rv satisfy the

utility-maximizing conditions for the reallocation typically suggested

by policy makers, educational authorities, and members of the general

public to have a negative effect on educational utility. Indeed, even if

the initial allocation of resources is not consistent with educational

utility maximization, the typical suggestion of change may not be utili-

ty-increasing. In particular, if uq is less than uv, the reallocation typical-

ly suggested by policy makers, educational authorities, and members of

the general public would also have a negative effect on educational

utility. And such a condition may exist, despite the existence of the

concurrent condition that q is less than v. It is only such, if uq is greater

than uv, that the suggested reallocation may be utility-increasing. How-

ever, even if uq is greater than uv, the suggested reallocation is not nec-

essarily utility-increasing. The sign of the change in educational utility

for the typical suggestion of change in resource allocation would de-

pend on the magnitude of the suggested reallocation. As such, test

scores for the various areas of testing to which resources are distributed

are neither necessarily or sufficiently informative for the systematic

determination of utility-increasing reallocations of educational re-

sou rces.

9Such an assignment of scores is consistent with the existence of the

conditions referenced above.