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What Can We Learn from Education Production Studies?

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Abstract

The results of a Becker–Peltzman–Stigler model of local school district decision-making yields biased or inconsistent efficiency measures when some school outputs are not measured. Empirical investigation of data for 95 Tennessee counties in the 1999–2000 academic year finds that Data Envelopment Analysis (DEA) efficiency measures, and efficiency rankings based on those measures, are highly sensitive to changes in the number of output measures used. An artifact of the DEA process causes increasing correlation of efficiency scores with the inverse of per pupil expenditures as outputs increase. Hence, high-stakes policy initiatives should not be based on such scores.

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Notes

  1. Hanushek [1979, pp. 361–362] noted that “… with information about only one output, estimation of the reduced form might be quite misleading. The estimated effects of the various inputs will reflect both the production technology (the effect of each input on the single output) and the choice between outputs, not simply the production technology”.

  2. More formally [Kumbhakar and Lovell 2000],

    where P(x) is the feasible output set : P(x)={y: x can produce y} and x is an input vector.

  3. An alternative, as in Peltzman [1993], is to consider a political utility function that is a weighted sum of the utilities of different interest groups. As the influence of competing interest groups is not the focus here, this specification is not pursued. These models are more general than median voter models [Peltzman 1993; Downes and Pogue 1994] as the proportion of eligible voters choosing to vote, and the identity of the median voter, may vary across districts or in response to the administrator's actions.

  4. Although the elimination of diseconomies of scope may seem reasonable on its face, this need not be so. Akerlof and Kranton [2002], for example, suggest that racial integration was detrimental to student performance by, in part, increasing the number of disenchanted students — those who felt as if they were excluded from the school culture. Over time, schools were able to partially reverse the slide in student performance by expending resources to make more students feel included in the school. Thus, increasing “diversity” in the form of racial integration may have increased the cost of student “performance” as measured by standardized tests.

  5. The effects when costs respond to Z, but the value function V does not, are similar to those derived in this case, if cost responds differentially across outputs, C1 Z >C2 Z for example. Without a differential cost effect, changes in Z will not affect the relative quantities of the outputs chosen when V is independent of Z.

  6. To see this, construct the efficiency index

    where Q 1 is the observed output for any district and max Q1 is the maximum feasible output of Q1 at cost level B.1

    In Figure 1, this yields

    The reader may confirm that the result in Figure 1 is exactly the same when changes in Z favor Q2 instead of Q1: ɛ1 Z < ɛ2 Z , dQ1/dZ < 0, dQ2/dZ>0, for Z1>Z2.

  7. In fact, Pritchett and Filmer [1999] derive superficially similar results, but, since they focus on the empirical finding that “inputs do not matter” in producing educational outcomes, they miss the implications for output measurements as opposed to input measurement.

  8. More formally, suppose that at the same level of cost district 1 is inefficient (λ1< 1) in producing vector y of n outputs, while district 2 is efficient (λ2=1) in producing vector y* of n outputs, but only j < n outputs are observed. The observed output of district 1 is then y1=λ1y j , where y j is the observed elements of y, and the observed output of district 2 is y2=λ2y j *=y j *, where y j * is the observed elements of y*. If y2 < y1, then district 2 will appear to be less efficient that district 1, while in reality district 2 is more efficient, λ2>λ1.

  9. No Tennessee school district crosses a county line, and most counties have exactly one school district. Several counties have small school districts restricted to early grades, so that students pass to the county-wide district upon entering middle or high school. In order to make all school districts comparable, it was necessary to aggregate data to the county-level.

  10. On the recommendation of a referee, we excluded measures of achievement on standardized tests; these measures are known to correlate highly with socioeconomic status and are therefore not necessarily good measures of school performance. Thus our emphasis on the value-added measures which, by calculating gains from each student's starting point, “implicitly control for socio-economic status and other background factors to the extent that their influence on the post-test is already reflected in the pre-test score” [Ballou et al. 2004, p. 38].

  11. The counties are Shelby (which includes the city of Memphis), Fayette (immediately to the east of Shelby), and Hardeman (immediately to the east of Fayette).

  12. As shown at the bottom of Table 2, SES is a composite variable, created as the first principal component of three variables, all for the year 2000: percentage of pupils not eligible for free or reduced price meals; percentage of county population 25 years or older who have at least 4 years of college; and county per capita income.

  13. These data are found at http://www.mtsu.edu/~eaeff/downloads/TennSchoolCosts2000.xls and documented in Eff [2008]. The teacher cost index is based on a hedonic regression over a set of 60,000 Tennessee public school teachers. Though the index, strictly speaking, measures only variation in the cost of a constant-quality teacher across counties, it seems probable that it would also capture much of the inter-county variation in other labor and operating costs.

  14. In a box and whisker plot, the median is indicated by a heavy line, the box encloses the second and third quartile, and the whiskers extend out to a distance of 1.5 times the box, unless the minimum or maximum are less than that amount, in which case they extend out to the minimum or maximum. The points beyond the whiskers are considered outliers.

  15. The R2+ is created through a backward selection process when regressing preexisting outputs on a newly entered output. At each step, only those preexisting outputs with positive coefficients are retained (regardless of significance); the process stops when all coefficients are positive, and the R2 from this final regression is designated the R2+. Requiring all coefficients to be positive is necessary since the weights μ r in equation (7) are restricted to be positive: a DEA model is unlike a regression model, since adding an output highly negatively correlated to preexisting outputs would in fact add variation not already captured by preexisting outputs.

  16. Since Δθ will be larger for models with small numbers of outputs, and R2+ will be larger for models with larger number of outputs, a negative correlation could be an artifact of number of outputs. For this reason, we calculated the correlation between Δθ and R2+ separately for each of the different numbers of outputs in the initial model. Table 5 reports the weighted mean of these 11 different correlations, where the weights are the number of different models within each category.

  17. This is a simplification, since each county k in fact has unique weights μ rk , and not the same weights μ r as shown here.

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Acknowledgements

For helpful comments on earlier versions of the paper, we thank Reuben Kyle and Charles Baum; seminar participants at Middle Tennessee State University; session participants at the 2002 Southern Economic Association meeting; and three anonymous referees. We retain responsibility for any remaining errors or omissions.

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Authors and Affiliations

  1. Department of Economics and Finance, Middle Tennessee State University, Box 27, Murfreesboro, 37132, TN, USA

    Ellis Anthon Eff & Christopher C Klein

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Appendices

APPENDIX A

Detailed results for Case II: B Z =C Z =0

Imposing the condition B Z =C Z =0 on equations (5) and (6) gives

where [D]>0 and the signs depend on those of the bracketed terms in the numerator. These can be rewritten by solving the first-order condition in equation (4) for C1 and substituting this expression into the equations above:

where ɛ iZ is the elasticity of the marginal value of output i with respect to Z. These equations have opposite signs, except for case of equal elasticities (ɛ1X=ɛ2X ) when each equation is equal to zero. Both dQ i /dZ=0 corresponds to the case in which household socioeconomic status does not affect the demand for schooling. Under the assumptions in the text, however, increases in Z favor output Q1, such that

That is, as socioeconomic factors change to favor output Q1, output of Q1 will increase and Q2 will decline, holding the educational budget (B) constant.

The elasticities are equal if V is separable in the Q i and Z, such that V(Q1, Q2, Z) ≡ U(Q1, Q2)F(Z). In this case a change in Z causes no change in the first-order conditions [∂(V1/V2)/∂Z=0]. That is, changes in school district demographics cause no change in the district's choice of outputs.

APPENDIX B

Detailed results for Case III: B Z >0, C Z < 0

It is a straightforward, if tedious, exercise to show (see below) that

where I=Q1/maxQ1 is the efficiency index, e QZ is the elasticity of Q1 with respect to Z, and e MZ is the elasticity of max Q1 with respect to Z. Under the assumptions of the model, both elasticities will be positive, as both the quantity of Q1 produced and the budget will increase as Z increases. The direction of change in the efficiency index as Z increases depends on the relative magnitudes of these two elasticities.

The term e QZ is just equation (5) expressed as an elasticity. The term e MZ shows the relative proportional change in the maximum Q1 as the budget (cost) changes with Z and is determined by the scale properties of the single-output cost function, C(Q1, 0; Z). For example, if C(Q1,0; Z) displays constant returns to scale, then a proportional change in cost is associated with an equal proportional change in max Q1, such that e MX =C1/C=B Z /B. Similarly, economies of scale imply e MZ >B Z /B and diseconomies of scale imply e MZ < B Z /B.

The scale properties of a cost function are related to the degree of homogeneity of the cost function at any point. If costs are linearly homogeneous, C(kQ1,0)=kC(Q1,0) where k is a positive constant, there are constant returns to scale. For scale economies, C(kQ1,0) < kC(Q1,0), and for diseconomies, C(kQ1,0)>kC(Q1,0).

To see that dI/dZ=(e QZ –e MZ )(I/Z), consider the general form dI/dX:

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Eff, E., Klein, C. What Can We Learn from Education Production Studies?. Eastern Econ J 36, 450–479 (2010). https://doi.org/10.1057/eej.2009.39

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