Because tuition increases to a type reduce attendance if the college's preferred tuition is below the posted price, the optimal admission policy is simple. All students are admitted to the college if and only if 16 for satisfying All students of the type are admitted if 11 does not bind. Hence, the admission threshold for minorities is lower. Our assumption that private colleges maximize a quality index is arguably well motivated, given colleges' preoccupation with rankings. It facilitates testing, and we find significance of the key underlying parameters.
However, the proper college objective and the role of diversity in it is an open question.
Market power and price discrimination in the US market for higher education
Colleges might, for example, discount to minorities because their utilities directly enter the college objective, rather than because diversity increases quality. Research focused on identifying what college objective best fits the data is of much interest. See the next point in the text. In this vein, it is interesting to compare the tuition results to the case where colleges maximize profits.
The main objective of the article is to determine the empirical content of the pricing equation in Distinguishing quality and profit maximization empirically would require distinguishing relatively subtle differences between equilibria under the two alternatives. Likewise, the quality maximizer has stronger incentives to attract minorities. To test the implications of equation 14 , we need to close the demand model and derive the conditional market shares for each private college. For that, we need to derive the admission policies of state schools. Next, we summarize a model of state colleges that generates state college policies that have these properties with details in an Appendix from the authors upon request.
This can lead their admission threshold to be higher or lower, depending on the relative weight in the quality function on the peer effect to the resource effect. Note, too, these admission thresholds will each be lower for minority status. In the empirical implementation, consistent with the data, we allow multiple state colleges that vary by quality and then let tuitions and admission thresholds vary among them.
With these results in hand, we are now in a position to turn to empirical analysis and determine whether the predictions of this model regarding pricing and demand are consistent with the observed data. To estimate the model, we need to invoke some additional parametric assumptions. Assumption 1. The assumptions above then imply that the conditional choice probability for type is given by, for j : 20 The pricing equation 12 for private colleges can then be written: 21 Effective marginal costs at private colleges are given by: 22 The pricing function is then: In addition, we simplify notation by writing the marginal resource costs as: 24 We treat the as additional parameters to be estimated.
The model implies an appealing decomposition of tuition. From 21 , observe that tuition to student is a convex combination of the student's effective marginal cost and cost adjusted income. The absence of market power would imply tuition equal to EMC , which arises only in the limit as , implying idiosyncratic preferences are irrelevant. The linkage to income reflects market power over the student type. The weight on income increases with the student type's market share at the college, indicating increased market power over the student.
This indicates that market power declines as idiosyncratic preferences become less important. The information set of the econometrician can be characterized as follows. Assumption 2.
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We observe a sample. Let denote the state of student i , the minority status, ability, income, and the tuition at college j. Note that we observe the tuition at the college, attended in equilibrium, but not at colleges that are not attended in equilibrium. Let denote an indicator which is equal to one if student i attends college j and zero otherwise. L is known.
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Prices for all students i at private colleges that are not paying the posted price are measured with classical error: 25 where is i. Consider the subsample of students that attend private colleges and are not paying the full posted price, that is, the subsample of students at private schools that obtain some institutional aid. Using this subsample, we can identify and estimate most of the parameters of the model using the predictions of the model about price discrimination. In particular, we can implement the following semiparametric estimator.
We estimate the conditional market shares for all students for the private college that is attended in the data. We use a simple flexible logit estimator using a quadratic approximation in b and y , where the coefficients depend on m and s. We then use the estimated logit model to predict the conditional choice probability denoted by. Alternatively, we could use nonparametric techniques such as kernel or sieve estimators.
Substituting the estimator of the conditional market share into the pricing equation, we obtain: 26 where is the measurement error term. We use a bootstrap algorithm to estimate the standard errors to account for the sequential nature of the estimation procedure. All the empirical results reported in this article are based on this estimator.
One nice property of this estimator is that it is consistent for large N , but small J. This scenario is relevant for most practical applications. We do not allow for random coefficients in our utility specification because of data limitations.
source url We also do not observe large changes in the choice set over time or the full portfolio of colleges that a students chooses at the application stage. In principle, it is not difficult to extend our semiparametric estimator to account for random effects in the demand side of the model. This section discusses how to identify and estimate the levels of these parameters using a modified version of the estimator suggested by Berry Two additional challenges to estimation are present that are typically not encountered in standard demand analysis.
First, the potential choice set of a student is unobserved by the econometrician. This allows us to characterize the relevant choice set for each student in the sample. Hence, institutional aid and net tuition policies of all private colleges are functions of income and ability as long as the price maximum is not binding.
A key challenge encountered in estimation is that the institutional aid is observed only at the college that is attended in equilibrium. The econometrician does not observe the financial aid packages and, hence the net tuition, that were offered by the colleges that also admitted the student, but were ultimately rejected by the student.
As a consequence, we cannot directly evaluate the conditional choice probabilities for each student. However, we can consistently estimate the institutional aid functions of each college type, using nonparametric techniques such as kernel or sieve estimators. Given these consistent estimators, we then can compute the conditional choice probabilities of each student. Consider the full sample of all students, including those students that attend private colleges and that pay the posted price, as well as students attending public colleges and universities.
Let our estimator be denoted by. The third set denotes the set of all private colleges to which the student is admitted, and the last set is the outside option. We then nonparametrically estimate the prices for each student at each college to which the student was admitted based on the observed tuition levels, using a local quadratic polynomial.
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Let us denote these estimates by. For private colleges, we use a local polynomial smoothing estimator LOESS to estimate the tuition function. The polynomial constructs a nonparametric estimate of tuition based on ability and income, and interpolates only where the observed data span.
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That is, if we observe an individual with a similar ability and income attending the college with a given tuition, the LOESS estimator calculates a polynomial relationship among tuition, ability, and income within the relevant bandwidth and predicts tuition locally. However, the resulting admission set—where a tuition can be predicted—is really a combination of admission and matriculation.
Hence, we assume that if a college accepts an individual with ability , it accepts all individuals where. Substituting the nonparametric estimates of the tuitions into the conditional choice probabilities, we obtain: 28 The quality levels for each school are determined by the fixed point of the following mapping: 29 where: is initial guess of the quality, is the average empirical market share of college j observed in the data, and is the predicted average market share using the initial guess about the vector of qualities: 30 We can identify 's for each college, subject to a normalization such as.
The normalization of quality is necessary as market shares add up to one.
Using the fact that , we obtain the the following regression model: 31 Define 32 and note that is known at this point. This estimator is consistent despite the fact that and may not be independent because the regression above does not have an intercept. Note that the last step of the estimator requires a large number of colleges or preferably multiple markets. Finally, consider identification of college cost functions.
Note that the fixed costs of operating a private college F are not identified from our analysis.