Study versus television

Abstract

The great majority of studies on the effect of school quality on academicoutcomes do not take account of changes in student choices concerning effortif school quality, e.g. class size, changes. We show that empiricalestimates of the ‘total’ effect of changes in school qualitycould be quite different from the ‘partial’ effect holding otherinputs (including student effort) constant. The main parameters governingthis difference are the extent to which inputs in the education productionfunction are substitutes or complements and how kinked is the benefit from ahigher mark. The difference depends also on student ability, thestudent’s distaste for effort and the curvature of the educationproduction with respect to effort.

JEL classification

I21; I28

1 Introduction

The majority of studies on the effect of ‘school quality’ on academicoutcomes do not take account of changes in student choices concerning effort whenschool quality, e.g. class size or teacher quality, changes.¹ Inparticular, students might respond to changes in their school quality by adjustingthe time and effort devoted to study (with a consequent change in leisure or marketwork). Thus, the ‘partial’ effects of school quality on academicoutcomes corresponding to production function parameters may differ from empiricalestimates of the ‘total’ effects (which include effects via effortresponse). This is similar to the distinction between production function parametersand average policy effects in Todd and Wolpin (2003) wherefamily inputs are allowed to respond to changes in school quality.

From a policy point of view, both the total and partial effects of an increase inschool quality are of interest. The total effect on student achievement is of courseimportant for policy makers: An intervention aimed at improving student academicoutcomes will typically be considered a success if the goal is achieved, and afailure otherwise (irrespective of specific mechanisms, including possible effortresponse). However, the partial effect on achievement (holding effort constant) isalso important, since it provides knowledge of education production functionparameters and the mechanisms through which an intervention works (or why it doesnot work). Furthermore, if students respond to an intervention which increasesschool quality by reducing time and effort devoted to study, benefit-cost analysisthat only have total effects as benefit will underestimate ‘full’benefits. These include, inter alia, more leisure, increased earnings frommarket work and less need for intervention by parents. In the formal model of thispaper we focus on educational outcomes and student leisure, but benefits are muchwider than that.

Empirical estimates of class size effects are often rather small and insignificant.Reasons for this may be non-random sorting of students into schools and that schoolsreduce class size when students are more ‘disruptive’ (Lazear 2001). Another explanation may be that students typicallyreduce effort in response to a reduction in class size.

A few papers model and estimate the response of parental inputs to changes in schoolquality. Houtenville and Conway (2008) consider a theoreticalmodel in which student achievement depends on parental effort and school resources,and parents maximize utility, which is a function of student achievement, leisureand consumption, subject to time and budget constraints. In this model, an increasein school resources may induce parents to increase or reduce their effort dependingon the form of the utility and production functions. Their empirical analysisindicates that parental effort and per-student spending have positive effects onstudent achievement, and that some measures of parental effort are affectednegatively by per-student spending. However, the estimated effect of per-studentspending on achievement is not affected by whether or not parental effort isincluded in the model. A similar result is found in Datar and Mason (2008): controlling for parental involvement does not change estimatedeffects of class size on test scores for children in kindergarten and first grade.Bonesrønning (2004) finds zero or positive effects onparental effort of reducing class size. Das et al. (2011) consider a dynamic household optimization model where child testscores depend on school and household inputs. Assuming that households makedecisions regarding their own inputs before they know the amount of school inputs,they are only able to respond to anticipated changes in school inputs. Using datafrom Zambia and India the authors find that household school expenditure is reducedwhen anticipated school grants are increased, and that anticipated grants have noeffect on student test scores whereas unanticipated grants have significant positiveeffects.

Only very few papers consider models of student response to changes in schoolquality. In the theoretical model of De Fraja et al. (2010) student effort, parental effort and school effort aresimultaneously determined as a Nash equilibrium. In this very general model, achange in an exogenous variable, e.g. an increase in school resources, may increaseor reduce the equilibrium level of effort of students (and parents and schools)depending on the form of the utility and education production functions and thevalues of exogenous variables. In their empirical analysis, the measure of studenteffort is based on (a factor analysis of) general attitude variables such as whetherstudents like school, whether they think homework is boring and whether they want toleave school. The authors do not find any significant effect of class size on theirmeasure of student effort, but they do find that student effort is reduced when‘school effort’ is increased, where school effort is based on (a factoranalysis of mainly) whether streaming and disciplinary methods are used. They findthat student and parental effort are positively correlated (where parental effort isbased on a factor analysis of mainly the teacher’s opinion of parents’interest in their child’s education) and that class size has no effect onparental effort. The attitudinal variables used by De Fraja et al. (2010) may be poor proxies for effort or time spent on homework,and they may not be expected to be much affected by (marginal) changes in schoolresources. Furthermore, the authors ignore the important issue of the endogeneity ofclass size and assume observed class size variation to be exogenous. These problemsmay explain why they do not find any effect of class size on their measure ofstudent effort.

Using quasi-experimental variation in class size, a recent study estimates studentand parental response to class size in grades 4-6 (Fredriksson P, Öckert N,Oosterbeek H: Inside the black box of class size effects: Behavioral responses toclass size variation, unpublished). Their measures of student effort are time spenton homework and reading outside school. In their theoretical model (based onAlbornoz F, Berlinski S, Cabrales A: Motivation, resources and the organization ofthe school system, unpublished), the education production function determinesstudent skills as the product of student ability and student effort, so class sizeonly affects student skills through student effort. They assume that student utilitydepends on student effort and ‘rewards to effort’ which are determinedby parents’ and teachers’ utility maximizing behaviour. In their model,class size always has a negative effect on student effort, but a non-negative effecton parental effort. In this paper we consider a theoretical parametric model whichincludes a more general education production function and a more flexible studentutility function, which depends on an educational outcome and effort, in order toanalyse how student response to a change in school quality may depend on the valuesof important parameters of the production and utility functions. We ignore parentalresponse to a change in school quality. This simplification is more appropriate whenconsidering older students (e.g. 15-year-olds).

Some papers estimating the effects of course-specific (or subject-specific) schoolquality inputs on student academic outcomes provide indirectly an indication thatstudents’ change of effort in response to changes in school quality may beimportant. Aaronson et al. (2007) estimate the effectof teacher quality and find, e.g., statistically significant effects of mathematicsteachers on mathematics test scores, but also significant effects of Englishteachers on mathematics test scores (and of mathematics teachers on English testscores). Heinesen (2010) estimates significant negativeeffects of subject-specific class size on examination marks in the same subject, butthe results also indicate negative effects on marks in other subjects. Oneinterpretation of the effects of subject-specific school inputs on student outcomesin other subjects is that they are due to spill-over effects between subjectsinduced by student reallocation of effort between subjects.

In this paper we consider simple parametric models with endogenous student effort andshow that students’ responses to changes in their school quality could implyquite large differences between total and partial effects on academic outcomes ofchanges in school quality and could lead to empirical estimates of total effectswhich are either larger or smaller than corresponding partial effects. The mainparameters governing the sign of the difference between total and partial effectsare shown to be the extent of substitution in the education production function andhow kinked is the benefit from a higher mark. The absolute value of the differencealso depends on the student’s distaste for effort, the curvature of the‘production’ of the final mark with respect to effort and thestudent’s ability in the course of study. Our main conclusion is that reliableestimates of the partial effect of school quality on academic outcomes requiresinformation on academic effort, including time use. This suggests a new round ofdata collection.

2 The basic idea

We begin with an illustration of the basic idea. Assume that an educational outcome(e.g. test scores or examination marks) is a concave function of student effort fora fixed level of school quality, and that student utility is increasing in theeducational outcome and decreasing in effort. For a given level of school quality,students choose effort, and thereby (ignoring uncertainty) the educational outcome,to maximize utility. If school quality increases, education production possibilitiesincrease for each level of effort. This is illustrated in Figure 1 for four different cases of production and utility functions. In eachpanel of Figure 1 the concave curves represent theeducation production functions before and after the increase in school quality(solid and dashed lines, respectively), and the convex curves are studentindifference curves. The points marked by A are the initial utility maximizingchoices of effort and outcome, and points marked by B are the corresponding choicesafter the increase in school quality. The ‘partial’ effect of anincrease in school quality is given by the vertical shift of the production functionat point A (i.e., holding effort fixed at its initial optimal level). However, theupward shift in the production function implies that students may obtain a higheroutcome with less effort. This ‘income effect’ tends to reduce effort.The substitution effect may enhance the negative effort response if the marginalproduct of effort decreases when school quality increases, or it may work in theother direction if effort and school quality are complements in production. Ofcourse, the optimal response of students also depends on the form of the utilityfunction. The two left panels of Figure 1 illustratecases where effort is reduced (more so in the upper panel) implying that the totaleffect on the educational outcome is smaller than the partial effect. In the upperright panel, effort is unchanged (no difference between total and partial effect),and in the lower right panel effort is increased (the total effect exceeds thepartial effect).

Figure 1

Education production relations between outcome and student effort for twodifferent levels of school quality, and student indifference curves -four different cases of production and utility functions.

3 A parametric model

To fix ideas we consider a parametric model. We begin with the simplest case in whicha student takes only one course.² The outcome is a mark y.This mark is the result of school quality, s, student ability,μ, and student effort, h. ³ To make ourmain points as cleanly as possible we ignore uncertainty and take a simpleparameterisation for the production function:

$$y=s^{\lambda }+\mu h^{\eta }+\varrho s^{\lambda }\mu h^{\eta }$$

(1)

Student effort and school quality are normalized so that 0 < h< 1 and 0 < s ≤ 1. The parametersη and λ capture the curvature in theoutput with respect to student effort and school quality, respectively, andϱ measures the degree of substitution or complementarity ofthe two inputs. To ensure that production is increasing and concave in both weassume that 0 < λ < 1,0 < η < 1, ϱ> - 1 and $\varrho \ge -{\left(\bar{\mu }\right)}^{-1},$ where $\bar{\mu }$ is the maximum value of μ. If ϱ< 0 then s and h are substitutes inproduction, and if ϱ > 0 they are complements. To motivatethis function, note that education production may be considered to consist of twolearning processes, learning at school (represented by the term s^λ) and learning at home doing homework (represented by μ h^η), and an interaction effect between the two processes represented by the lastterm in (1). The marginal effect of school resources may be smaller forwell-prepared/high-ability students (ϱ < 0) if the primarygoal of teaching is to ensure that all students obtain a basic level of skills.Also, the marginal effect of effort (and ability) may be smaller when schoolresources are high: when the learning process at school is more effective there maybe smaller returns to effort at home to learning the curriculum. In conventionalproduction functions including the Cobb-Douglas and CES functions, inputs arecomplements, i.e. the marginal product of one input (e.g. h) increases whenthe amount of the other input (s) is increased. However, as argued above,h and s may be substitutes in production, so thatthe marginal product of h is reduced when s increases,and vice versa.⁴

In empirical studies a parameter of interest is the partialelasticity of the outcome with respect to school quality, holding effortconstant:

$$\epsilon =\frac{\text{∂}\ln y}{\text{∂}\ln s}{\mid }_h$$

(2)

Note that in the parameterisation (1) the partial elasticity is not independent ofeffort and student ability. The same is true for the corresponding derivative∂ y/∂ s, unless ϱ = 0. If effort is fixed then we canrecover the partial elasticity from observing variation in y due toexperimental variation in s.

Typically neither experimental nor non-experimental studies of effects of schoolquality have access to data on student effort or time use, and therefore theinterpretation of estimated effects as education production function parameterspresumes that effort is fixed. However, it is perfectly reasonable to assume thatsome students might respond to the change in constraints with adjustment in effortexpended. To capture this, let preferences be represented by the utility function:

$$u=\frac{y^{1-\sigma }}{1-\sigma }+\delta \ln \left(1-h\right)$$

(3)

where the maximum time available for study is normalised to unity (as notedabove).⁵ The parameter δ captures the taste forleisure and the parameter σ (> 0) captures thecurvature in the concern about the outcome; higher values ofσ give a more kinked benefit function. That is, for highervalues of σ the student will have a high return below athreshold value of y (‘passing’) and a low returnabove.

Denoting the optimal choices by $\left(ĥ,ŷ\right)$the total elasticity is given by:

$$\widehat{\epsilon }=\frac{d\ln ŷ}{d\ln s}$$

(4)

Just as the partial elasticity is a parameter of interest in empirical studies, so isthe total elasticity, which may be recovered by observing variation in the outcome(including variation via effort response) due to experimental variation ins. However, in general, the total elasticity will not be equal to thepartial elasticity ε. We define the elasticitydifference as

$$\Delta =\widehat{\epsilon }-\epsilon$$

(5)

The sign of the elasticity difference will depend on the sign of the effortelasticity,$\text{∂}\ln ĥ/\text{∂}\ln s$. Clearly the elasticity difference Δ will bepositive ($\widehat{\epsilon }>\epsilon$) if the student responds to the increase in school quality byputting in more effort, and it will be negative if the effort elasticity isnegative. If s and h are substitutes in production(ϱ ≤0) the effort elasticity and the elasticity differenceare negative. But if they are complements (ϱ > 0) the marginalproduct of effort is increased when school quality increases, and therefore it maybe optimal for the student to increase effort. Whether it is in fact optimal toincrease effort depends on the size of ϱ and the otherparameters of the model, especially the curvature of the benefit function withrespect to the outcome (σ). Thus, even if s andh are complements it may still be optimal to reduce effortbecause of the ‘income effect’: an increase in s enablesstudents to obtain a larger y with less effort (more leisure).

First, we show that the effort elasticity (and therefore the elasticity difference)is negative if σ > 1 or ϱ ≤ 0. Themarginal effect of school quality on effort is found by inserting (1) into (3) anddifferentiating the first-order condition:

$$\begin{array}{ll}\frac{\partial ĥ}{\partial s}& =-\frac{{\text{∂}}^{2}u/\partial ĥ\partial s}{{\text{∂}}^{2}u/\text{∂}{ĥ}^2}\\ \frac{{\text{∂}}^{2}u}{\partial ĥ\partial s}& =\lambda s^{\lambda -1}\mu \eta h^{\eta -1}{y}^{-\sigma }[\varrho (1-\sigma )-\sigma y^{-1}]\\ \frac{{\text{∂}}^{2}u}{\text{∂}{ĥ}^2}& =\mu \eta h^{\eta -2}(1+\varrho s^{\lambda })y^{-\sigma }\left[\right(\eta -1)-\sigma y^{-1}\mu \eta h^{\eta }(1+\varrho s^{\lambda }\left)\right]-\frac{\delta }{{(1-h)}^2}<0\end{array}$$

(6)

Thus, the sign of $\partial ĥ/\mathrm{\partial s}$ is equal to the sign of ${\text{∂}}^{2}u/\partial ĥ\partial s$. It is obvious that $\partial ĥ/\mathrm{\partial s}<0$ if (ϱ ≤ 0 and σ < 1) or if(ϱ ≥ 0 and σ > 1). However, $\partial ĥ/\mathrm{\partial s}$ is also negative when ϱ < 0 and σ> 1. Thus, when ϱ < 0 and σ > 1 we have

$$\frac{{\text{∂}}^{2}u}{\partial ĥ\partial s}<0\iff \varrho (1-\sigma )-\sigma {(s^{\lambda }+\mu h^{\eta }+\varrho s^{\lambda }\mu h^{\eta })}^{-1}<0$$

(7)

$$\iff s^{\lambda }+\mu h^{\eta }+\varrho s^{\lambda }\mu h^{\eta }<\frac{\sigma }{1-\sigma }\frac{1}{\varrho }$$

(8)

This inequality always holds given the assumed restrictions on the parameters. TheRHS of the inequality tends towards its lower limit $\text{max}(1,\overline{\mu })$ from above whenσ → ∞ and $\varrho =\text{max}(-1,-1/\overline{\mu })$. It is easy to show that: (a) for $\varrho =\text{max}(-1,-1/\overline{\mu })$ the LHS of (8) is always less than or equal to $\text{max}(1,\overline{\mu });$ (b) when $\text{max}(-1,-1/\overline{\mu })<\varrho <0$ the inequality (8) also holds (the derivative of the LHS withrespect to ϱ is smaller than the derivative of the RHS). Thus, $\partial ĥ/\partial s<0$ if ϱ ≤ 0 or σ > 1.

We now examine further the determinants of the direction and size of the elasticitydifference Δ. Although simple, the parametric model does not yieldclosed form expressions for the elasticities of interest. We therefore have toresort to simulations to illustrate how they vary with the parameters. Without lossof generality we can take λ = 0.4.⁶

We take a grid over the values given in Table 1 (andcalculate the elasticities at s = 1).⁷ These values are,of course, wholly arbitrary and serve only to illustrate the variation in theelasticity difference. Since the maximum value of μ is hereequal to 10, concavity is ensured whenϱ ≥ - 0.1. For each set of parameter values,the partial elasticity is calculated holding h fixed at the optimallevel given the parameters and the initial level of school quality, whereas thetotal elasticity is calculated letting h adjust to its new optimallevel induced by the increase in s.

Table 1 Simulation parameter values

With this range of parameter values the minimum and maximum of the elasticitydifference Δ are -0.12 and 0.00, respectively. Thus, even whenϱ attains its maximum value of the grid (0.075)Δ is not positive for any combination of the otherparameters within the grid of Table 1. Table<2 shows that the parameter values that induce the extreme valuesof Δ are very different. As expected, the largest negative valueof Δ is obtained when s and h arestrong substitutes in production (ϱ attains its minimum). Also,Δ is more negative if the benefit function is more curved(high σ), if the student has a higher taste for leisure (highδ), if the elasticity of output with respect to effort is high(high η), and/or if student ability (μ) is relativelylow (although in this case not at its minimum).

Table 2 Extreme values of the difference between total and partialelasticities

When s and h are strong complements in productionΔ may be substantially positive. This is illustrated inTable 3 where ϱ is fixed at 2.Here the largest negative value of Δ is obtained for about thesame values of (μ,δ,σ,η) asin Table 2 (except that μ is 2instead of 3), whereas the largest positive value of Δ (0.055)is obtained for the same parameter values except that σ (thecurvature of the benefit function) is at its minimum instead of itsmaximum.⁸

Table 3 Extreme values of the difference between total and partial elasticitieswhen school quality and effort are strong complements (rho=2)

Within any school, we would expect that the parameters(μ,δ,σ,η,ϱ)are heterogeneous. For example, how important it is to attain more than a simple‘passing’ grade will vary from student to student, implyingheterogeneity in the parameter σ. Moreover, the distributions of theseparameters may not be independent. For example, high ability students (highμ) who aspire to further education may have a lower concern forsimply passing.

4 Differential effects

The results of Summers and Wolfe (1977), Krueger (1999), Angrist and Lavy (1999), Browningand Heinesen (2007) and Heinesen (2010)indicate that reducing class size has larger positive effects for students fromdisadvantaged backgrounds, and Heinesen (2010) also findsthat low-ability students benefit significantly more than high-ability students.Aaronson et al. (2007) find that teacher-qualityeffects are relatively larger for lower-ability students.

The simple model described above is consistent with these findings since the totalelasticity, $\widehat{\epsilon }$, is decreasing in student ability, μ. The absolutevalue of the total effect of school resources on marks ($dŷ/\mathit{\text{ds}}$) is also decreasing in μ for many combinations ofvalues for the other parameters. When ϱ = 0 (i.e., s andh are neither substitutes nor complements in production), thederivative ∂ y/∂ s in the production function (1) holding h fixed doesnot depend on μ. However, the total effect $\partial ŷ/\mathrm{\partial s}$ depends on μ because students respond tochanges in school quality by changing effort. This is illustrated inFigure 2 which shows how $\widehat{\epsilon }$ and $dŷ/\mathit{\text{ds}}$ vary with μ in the model consisting of (3) and(1) where(δ,σ,η) = (1.0,1.05,0.4). Inthe lower part of the figure ϱ = 0, and the lower right panelshows that $dŷ/\mathit{\text{ds}}$ is decreasing in μ. The lower left panel shows thatthe total elasticity decreases much more in μ. This is not surprisingsince y determined by the production function (1) holding h fixedand also the optimal value $ŷ$ are strongly increasing in μ. In the upper part ofthe figure ϱ = -0.05 (i.e., s and h aresubstitutes in production). Whereas the total elasticity varies with μin much the same way in the two parts of the figure, $dŷ/\mathit{\text{ds}}$ decreases much more in the upper part of the figure, wheres and h are substitutes, than in the lower part.

Figure 2

Total elasticity and absolute change in outcome when school resources areincreased by 10%. The two upper panels are for rho=-0.05, thetwo lower panels are for rho = 0.

Figure 3 illustrates differential effects with respect tothe parameters of the utility function, i.e. the curvature with respect to theeducational outcome (σ) and the taste for leisure (δ),given constant values of student ability (μ = 5) and thecurvature of the production function with respect to effort(η = 0.4). In the special case whereϱ = 0, school quality (s) and effort(h) are neither substitutes nor complements in production, and thederivative with respect to s in the production function, holdingh constant, is equal to λ s^λ-1, i.e. independent of the initial levels of hand y. Assuming λ = 0.4 and s = 1 asabove and considering a 10% increase of s, the partial effect ony is 0.04. This is represented by the horizontal line in the upperleft panel of Figure 3, whereas the two downward slopingcurves in this panel show the total effects for δ = 0.5 (solidline) and δ = 2 (dashed line), respectively. Whereas the partialeffect is constant, the total effect is decreasing in both σ andδ. When both σ and δ aresmall (about 0.5) the total effect is about 6% smaller than the partial effect; thedifference is about 12% if instead δ is large (2); and it isabout 40% if σ is large (1.95). The upper right panel ofFigure 3 shows the corresponding relationship betweenthe elasticity difference Δ and σ for thetwo extreme values of δ (and again for ϱ= 0). The numerical value of the elasticity difference increases in bothσ and δ, and it is about -0.07 when bothparameters are large (about 2) and in this case the partial elasticity is 0.16. Therelation between the elasticity difference and the utility function parameters isalmost the same when ϱ = -0.05, i.e. when s andh are substitutes in production; see the lower left panel. Thelower right panel shows the relation when s and h arestrong complements (ϱ = 2): when σ is smallthe elasticity difference is positive (h increases in response to theincrease in s), and more so when δ is large; but whenσ is large the elasticity difference is negative and largenumerically (of about the same size as when ϱ = 0 or ϱ= -0.05).

Figure 3

Differential effects with respect to parameters of the utility function:Total and partial effects and elasticity differences.

5 A pass/fail mark

If in this simple model the curvature parameter in the utility function is above athreshold (σ > 1), students respond to an increase in schoolquality by decreasing their effort. Conversely, if σ< 1 (and ϱ > 0), students may in some casesrespond by increasing their effort. To investigate in more detail the effect of thecurvature of the benefit function we consider an extreme case in which:

$$u=\mathbf{I}\left(y\ge y^{\ast }\right)+\delta \ln \left(1-h\right)$$

(9)

where I (y ≥y^∗) is an indicator function that takes value unity if y≥ y^∗ and zero otherwise (with y^∗ > s^λ). This corresponds to a pass/fail mark. In this case students will eitherset:

$$\begin{array}{l}h=0\Rightarrow u=0\\ h={\left(\frac{y^{\ast }-s^{\lambda }}{\mu (1+\varrho s^{\lambda })}\right)}^{\frac{1}{\eta }}\Rightarrow u=1+\delta \ln \left(1-{\left(\frac{y^{\ast }-s^{\lambda }}{\mu (1+\varrho s^{\lambda })}\right)}^{\frac{1}{\eta }}\right)\end{array}$$

(10)

where we assume that the passing grade is attainable for some feasible level ofeffort: y^∗ < s^λ+ μ (1 + ϱ s^λ). The student chooses to exert effort if:

$$\delta \le -{\left(\ln \left(1-{\left(\frac{y^{\ast }-s^{\lambda }}{\mu (1+\varrho s^{\lambda })}\right)}^{\frac{1}{\eta }}\right)\right)}^{-1}$$

(11)

This illustrates that, ceteris paribus, a pass grade is more likely ifcomplementarity in production, student ability or school quality are high or if thestudent has a low taste for leisure.⁹ For a given level of schoolquality we have three groups of students: bad fails; marginal fails (students whofailed but were close to choosing to pass) and passes. If we increase school qualitythen the bad fails continue to exert no effort and fail, the marginal fails increasetheir effort (the level given in (10) with the new level of s) and passstudents reduce their effort and still pass. Thus we have three different responsesto the policy change: negative, zero and positive.

6 More than one course of study

As we discussed in the introduction, results in papers using course- orsubject-specific variation in school quality indicate that students’ effortresponse may be important. Thus, Aaronson et al. (2007) find significant effects of, e.g., both math and English teacherson math test scores, and Heinesen (2010) finds significantnegative effects of course-specific class size on marks in the same course, but alsoindication of negative effects on marks in other courses. One possible mechanismwhich may explain these cross-course effects (or spill-over effects between courses)is student reallocation of effort between subjects. This may be illustrated by anextension of the above model framework to the more general case with more than onecourse of study. For simplicity, consider the case with two courses. We assume thatthe utility function is additive in the two outcomes (with the same curvatureparameter) and leisure:

$$v=\frac{y_1^{1-\sigma }}{1-\sigma }+\frac{y_2^{1-\sigma }}{1-\sigma }+\delta \ln \left(1-h_1-h_2\right)$$

(12)

where y _i and h _i are the outcome (a mark) and student effort in subject i,respectively.

We allow student ability (μ _i ) to differ between subjects, but for simplicity we assume that the otherparameters in the two subject-specific production functions are identical, and weassume their form to be similar to (1):

$$y_i=s_i^{\lambda }+{\mu }_i{h}_i^{\eta }+\varrho s_i^{\lambda }{\mu }_i{h}_i^{\eta },i=1,2$$

(13)

Denoting the optimal choices by (${ĥ}_1,{ĥ}_2,{ŷ}_1,{ŷ}_2$) we may consider four total elasticities, namely elasticities ofthe two outcomes with respect to each of the two school quality inputs:

$$\begin{array}{ll}{\widehat{\epsilon }}_{\mathit{\text{ii}}}& =\frac{d\ln {ŷ}_i}{d\ln s_i},i=1,2\end{array}$$

(14)

$$\begin{array}{ll}{\widehat{\epsilon }}_{\mathit{\text{ij}}}& =\frac{d\ln {ŷ}_i}{d\ln s_j},i,j=1,2,i\ne j\end{array}$$

(15)

The two total ‘own resource’ elasticities in (14) consist of a directeffect of increased school resources in subject i on academic outcomein the same subject and an indirect effect through changed effort in subjecti. The two total cross-elasticities (15) are different from zero if anincrease in school resources in one subject induces students to change effort in theother subject. The partial own-resource elasticities, ε _ii = ∂ lny _i /∂  lns _i , consist of only the direct effect, holding effort fixed. The partialcross-elasticities, ε _ij = ∂ lny _i /∂ lns _j ,i ≠ j, are zero. The sign of each of the fourelasticity differences ${\Delta }_{\mathit{\text{ij}}}={\widehat{\epsilon }}_{\mathit{\text{ij}}}-{\epsilon }_{\mathit{\text{ij}}}$ (i,j = 1,2) is equal to the sign of thecorresponding effort elasticity $\text{∂}\ln {\widehat{h}}_i/\text{∂}\ln s_j$.

As in the model with only one course of study, the signs of the elasticitydifferences are mainly determined by the signs of σ andϱ. Thus, we now show that $d{ĥ}_i/{\mathit{\text{ds}}}_i<0$ and $d{ĥ}_i/{\mathit{\text{ds}}}_j>0$ (i,j = 1,2; i≠ j) if σ > 1 or ϱ≤ 0. To see this, we insert the production functions (13) into theutility function (12) and differentiate the first-order conditions:

$$\begin{array}{ll}\frac{\text{∂}{ĥ}_i}{\text{∂}{s}_i}& =-\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_i\text{∂}{s}_i}\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_j^2}/D\\ \frac{\text{∂}{ĥ}_i}{\text{∂}{s}_j}& =\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_j\text{∂}{s}_j}\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_i\text{∂}{ĥ}_j}/D\\ D& =\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_1^2}\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_2^2}-{\left(\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_1\text{∂}{ĥ}_2}\right)}^2\end{array}$$

(16)

where

$$\begin{array}{ll}\frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_i\text{∂}{s}_i}& =\lambda s_i^{\lambda -1}{\mu }_i\eta h_i^{\eta -1}{y}_i^{-\sigma }[\varrho (1-\sigma )-\sigma y_i^{-1}]\\ \frac{{\text{∂}}^{2}v}{\text{∂}{ĥ}_i^2}& =A_i-\frac{\delta }{{(1-h_1-h_2)}^2}\text{,}\frac{{\text{∂}}^{2}v}{\text{∂}{\widehat{h}}_1\text{∂}{ĥ}_2}=-\frac{\delta }{{(1-h_1-h_2)}^2}\\ A_i& ={\mu }_i\eta h_i^{\eta -2}(1+\varrho s_i^{\lambda })y_i^{-\sigma }\left[\right(\eta -1)-\sigma y_i^{-1}{\mu }_i\eta h_i^{\eta }(1+\varrho s_i^{\lambda }\left)\right]\end{array}$$

(17)

Since A _i < 0, we have ${\text{∂}}^{2}v/\text{∂}{ĥ}_i^2<0.$ Furthermore, D=A₁A₂ - (A₁ + A₂) δ /(1 - h₁ - h₂)² > 0. Thus, $\text{∂}{ĥ}_i/\text{∂}{s}_i<0$ and $\text{∂}{ĥ}_i/\text{∂}{s}_j>0$ iff $\varrho (1-\sigma )-\sigma y_i^{-1}<0,$ and this inequality holds if σ > 1 or ϱ≤ 0 by arguments similar to the one-course case.

If we take a grid over the same values of the parameters as given in Table 1, where now both μ₁ and μ₂ vary between 1 and 10, we obtain the extreme values of theelasticity differences ${\Delta }_{11}={\widehat{\epsilon }}_{11}-{\epsilon }_{11}$ and ${\Delta }_{12}={\widehat{\epsilon }}_{12}$ and the associated parameters shown in Table 4.¹⁰ The ‘own resource’ elasticity differenceΔ₁₁ has extreme values -0.12 and -0.00, and the same is true forΔ₂₂ (not shown in the table since the model is symmetric in the twocourses). The elasticity difference Δ₁₁ (and Δ₂₂) vary with the parameters in basically the same way asΔ in the one-course case (except for the dependence on abilityin the other subject): The extreme negative value is obtained when σ,δ and η are at their maximum, whenϱ is at its minimum, and when ability in the two subjects is low(although not at the minimum); the maximum value (zero) is obtained when whenσ, δ and η are at theirminimum, when ϱ is at its maximum, and when ability is at itsmaximum in the same subject and at its minimum in the other subject. The two crosselasticities ${\widehat{\epsilon }}_{12}$ and ${\widehat{\epsilon }}_{21}$ have extreme values -0.00 and 0.05. The extreme positive value of ${\widehat{\epsilon }}_{12}$ is obtained when σ, η andμ₁ are at their maximum values, and when ϱ,δ and μ₂ are at their minimum values. Thus, the elasticity of outcome in onesubject with respect to school resources in the other subject is high whenϱ is negative and numerically large,σ is high, ability in the first subject is high and ability inthe other subject is low. The minimum of ${\widehat{\epsilon }}_{12}$ is obtained when ϱ = 0 and when σ,δ, μ₁ and μ₂ are high, and η is low.

Table 4 Extreme values of differences between total and partial elasticities inmodel with two courses of study

Table 5 illustrates that when school quality and studenteffort are strong complements in production (ϱ = 2 in thetable) then the elasticity difference of Δ₁₁ may be substantially positive (as for Δ in themodel with a single subject) and ${\widehat{\epsilon }}_{12}$ may be substantially negative. The maximum of Δ₁₁ is obtained when σ and δ aresmall and η and μ₁ and μ₂ are high. The minimum of ${\widehat{\epsilon }}_{12}$ is obtained for the same parameter values, except thatμ₁ is small. However, Table 5 also shows thatwhen σ is large, Δ₁₁ may be substantially negative and ${\widehat{\epsilon }}_{12}$ may be substantially positive even when s andh are strong complements in production.

Table 5 Extreme values of differences between total and partial elasticities inmodel with two courses of study when school quality and effort arestrong complements (rho = 2)

Figure 4 shows how the total elasticities ${\widehat{\epsilon }}_{\mathit{\text{ij}}}$ and the absolute change in the outcomes D${ŷ}_{\mathit{\text{ij}}}$, which is the increase in ${ŷ}_i$ when s _j is increased by 10%, vary with ability in subject 1 when (μ₂,δ,σ,η,ϱ) =(5,0.5,1.95,0.6,-0.05). Thus, the values of δ,σ and η are chosen so that the crosselasticities are relatively large. The own-subject elasticity ${\widehat{\epsilon }}_{11}$ and the absolute change D${\widehat{y}}_{11}$ are decreasing in μ₁ corresponding to the results in the one-subject case. The otherown-subject elasticity ${\widehat{\epsilon }}_{22}$ (and D${ŷ}_{22}$) do not depend much on μ₁, while the cross elasticity ${\widehat{\epsilon }}_{21}$ (and D${ŷ}_{21}$) are decreasing in μ₁, and ${\widehat{\epsilon }}_{12}$ (and D${ŷ}_{12}$) are increasing in μ₁. The figure illustrates that the cross elasticities may be rather largecompared to the own-subject elasticities. For instance, when μ₁ = 5 the cross elasticities are about one third of the own-subjectelasticities. Thus, when school resources in one course is changed, this may havesubstantial effects on outcomes also in other courses, and the mechanism behindthese cross effects in this simple model is students’ reallocation of effortbetween subjects: When school resources in one course increase it may be optimal forstudents to reduce effort in this course and increase leisure and effort in theother course. In Figure 4ϱ = -0.05 so that s _i and h _i are assumed to be substitutes in production. Settingϱ equal to zero produces a rather similar figure although thecross elasticities are a little smaller compared to the own-course elasticities (forμ₁ = 5 the ratio is 0.27). Assuming inputs to be strong complementsin production implies that the ratios of cross elasticities to own-courseelasticities are smaller.

Figure 4

Total elasticities and absolute change in outcomes when school resourcesare increased by 10%.

7 Conclusion

Typically, studies on the effect of school quality on academic outcomes do not takeaccount of students’ responses regarding academic effort or time use. Applyingsimple parametric models we have shown that students’ effort or time-useresponses to changes in school quality may cause large differences between the totalelasticity of changes in school quality and the partial education productionfunction elasticity (holding student effort constant). The main parametersdetermining the sign of the elasticity difference are the extent of substitutionbetween effort and school quality in the production function and how kinked is thebenefit from a higher mark in the student utility function. If effort and schoolquality are substitutes in production and/or if there is a marked kink in thebenefit function, students will tend to reduce effort when school quality isincreased implying that the total elasticity is smaller than the partial elasticity.The value of the elasticity difference also depends on the student’s distastefor effort, the curvature of the production function with respect to effort and thestudent’s ability in the course of study. In a model with two courses of studywe have shown that an increase in school resources in one course may reduce effortin that course, implying that the total ‘own-resource elasticity’ issmaller than the partial own-resource elasticity, but increase effort in the othercourse implying a positive (total) ‘cross elasticity’.

Our main conclusion is that reliable estimates of partial effects (based on educationproduction function parameters holding student effort constant) of school quality onacademic outcomes require - in addition to exogenous variation in school quality -information on academic effort and/or time use. This suggests a new round of datacollection.

It is important to note that our models are very simple in several respects. Forinstance, we focus on student effort response to a change of school quality andignore parental responses. Parental response is presumably very important foryounger students. Thus, our model is mostly relevant for older students (e.g.15-year-olds). Another limitation of our model is the assumption that school qualityaffects only the production function, but not the student utility function. Animportant aspect of school (and teacher) quality is to motivate and monitor studentsbetter, inducing them to put more effort in their school work. Thus, school qualitymay affect the parameters of the student utility function, but our model ignoresthis mechanism. In our model with two courses of study we focus on mandatory coursesand spill-over effects between these. It would be interesting to extent the model toenable analysis of effects of changes in school quality and other interventions onthe choice between optional courses, and academic outcomes given this choice (forinstance the major choice and GPA at university, see e.g. Arcidiacono etal.2012).

8 Endnotes

¹ Studies on the effect of class size include, inter alia, Hanushek(1996), Krueger (1999,2003), Angrist and Lavy (1999), Case andDeaton (1999), Hoxby (2000), Kruegerand Whitmore (2001), Heinesen (2010)and Fredriksson et al. (2013). Studies on the effect ofteacher quality include Rockoff (2004), Rivkin et al. (2005), Aaronson et al. (2007) andClotfelter et al. (2007a,2007b, 2010).Other measures of school quality used in the literature include expenditure perstudent and the teacher-student ratio (see e.g. the surveys in Hanushek 1996 Card and Krueger 1996 and Betts1996) and the number of teacher hours per student(Browning and Heinesen 2007).

² In the vastly simplified framework below we consider only aschool that has one class. A more general model would distinguish between classquality and school quality and allow for student selection based on within schoolquality differences.

³ We use Greek letters to denote preference and prodcutionparameters and Latin letters to denote choice variables. Thus s is thechoice of the school funding authorities. As discussed in the Introduction, weignore parental effort as an input in the production function which means that themodel is more relevant for older students.

⁴ Houtenville and Conway (2008) note thatschool resources and parental inputs may be substitutes in education production.

⁵ It is straightforward to allow for alternative uses of time suchas market work. This complicates the notation and analysis without adding much ofsignificance to the main points.

⁶ Results are qualitatively the same for other values ofλ.

⁷ Choosing other values of s produces qualitativelysimilar results.

⁸ The production functions and indifference curves ofFigure 1 are based on the parametric model above andthe four sets of parameters of Tables 2 and 3 (the upper panels of Figure 1correspond to Table 2 and the lower panels to table 3).

⁹ It is easily shown that $\frac{\text{∂}}{\partial s}\left(\frac{y^{\ast }-s^{\lambda }}{\mu (1+\varrho s^{\lambda })}\right)<0\iff 1+\varrho y^{\ast }>0,$ and that this inequality holds because of the restriction

$$\varrho \ge -1/\overline{\mu }.$$

¹⁰ The elasticities are calculated at λ = 0.4 ands₁ = s₂ = 0.5.