## Abstract

Over the past decades, the (private) college sectors in the higher education systems of several European countries have expanded their capacities massively. This happened even though colleges have been at a competitive disadvantage with universities which are publicly subsidized, while colleges must self-finance through tuition fees. The question arises how, in equilibrium, a diverse student population is allocated between these institutions and which factors may account for the college expansion over time. Moreover, the efficiency properties of the resulting human capital accumulation process are of special interest. Our paper explores these questions within an information-based theoretical framework. Individuals are screened for their (unobservable) innate abilities, and the precision of the screening mechanism, which is endogenous, balances demand and supply of educational services. We find that in the short term, when the college capacity is fixed, the introduction of college subsidies is not desirable in most cases. In the long term, the college sector may expand excessively, thereby establishing inefficiently low screening standards in the admission process to higher education.

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## Notes

- 1.
This applies to most countries in Europe (with the notable exception of the UK) and to Israel. The USA differs from our model, mainly because some elite US colleges possess strong scientific profiles and ambitious curricula that can easily compete with those of US state universities.

- 2.
In Israel, private colleges charge tuition fees which are, on average, three times higher than those charged by public universities. In France, Spain, and Italy, college tuition exceeds university tuition by a factor between 5 and 10. In Germany and in most East European countries, university education is (almost) free.

- 3.
Source: The Central Bureau of Statistics, Statistical Yearbook of Israel 2016, Chapter 8.13.

- 4.
Funding disparity between HE institutions has analogue in the American HE system as well. It has been demonstrated by Hoxby (2009) that students in more selective US universities are more heavily subsidized, as they pay a smaller fraction of the full cost of their education than do their counterparts in less selective schools.

- 5.
For more general dynamic models, with variable screening, see Eckwert and Zilcha (2004).

- 6.
Operating costs differ, because the university has better equipment and infrastructure, and because it hires more qualified and research-oriented faculty. Higher quality of staff and equipment allows the university to transform individual ability into human capital more efficiently than the college.

- 7.
This procedure may suggest that students are making sequential moves when choosing schools. Note, however, that the educational decisions can also be viewed as a simultaneous process where individual choice sets depend on the received signals.

- 8.
In reality, school decisions are based on an array of school characteristics. Costs as well as quality, in particular, may vary across schools within each sector. These variations are not captured by our model. Yet, as long as higher costs adequately reflect higher quality, our analysis is fairly robust. Suppose, for instance, that two colleges

*i*and*j*differ with regard to costs and quality, \(k^i_c\not =k^j_c\), \(\alpha ^i\not =\alpha ^j\), but are identical otherwise. It is easily verified that the incentive constraint in Eq. (7) is the same for both colleges, if and only if costs and qualities are aligned according to \((1+r)(k^j_c-k^i_c)=(\alpha ^j-\alpha ^i)wE_\theta [{{\tilde{a}}}|y_c]\). Else, some agents might be willing to attend college*i*but not college*j*, or vice versa. - 9.
For \(\kappa \)-values outside this interval, the admissible region in \((y_{c},y_u)\)-space is empty.

- 10.
This simplifying capacity assumption is justified by the fact that the capacities of existing universities tend to be sticky—partly because the public budget is fixed and partly because the infrastructure of research universities cannot be adjusted as easily as that of colleges. Empirically, over long time intervals the number of colleges exhibits much more variation than the number of universities. In Israel, for instance, just one single university has been established in the last five decades, while 60 new private colleges were set up in the last three decades alone (Statistical Yearbook of Israel 2016, Chapter 8.13). The situation in Germany has been similar. Here, the last three decades saw the establishment of 87 private colleges but only 13 publicly subsidized universities (Buschle and Haider 2016).

- 11.
Note that this result contrasts with some empirical findings in Bassanini and Scarpetta (2002).

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## Additional information

We wish to acknowledge receiving helpful comments and suggestions from two anonymous referees.

## Appendix

### Appendix

In this “Appendix,” we prove lemmas 1–4 and Proposition 5.

### Proof of Lemma 1

Differentiating Eq. (11) yields

from which the claim in the lemma follows immediately. \(\square \)

### Proof of Lemma 2

Differentiating Eq. (19), we obtain

\(\varphi :[0,1]\rightarrow [0,\frac{1}{4}]\) has been defined in Eq. (17). It is a strictly concave function that satisfies

Equations (36) and (37) in combination with Fig. 7 imply the claim in the lemma. \(\square \)

### Proof of Lemma 3

Define \(y^*_{c}(y_{u}):={\text {inf }}\{y_{c}\in [0,\frac{1}{2})|\gamma (y_{c},y_{u})\ge 1 \}\). Note that \(y_{c}^{*}(y_{u})\) is well defined since \(\underset{y_{c}\rightarrow \frac{1}{2}}{\lim }\gamma (y_{c},y_{u})=\infty ,y_{u}\in (\frac{1}{2},1]\). Since \(\gamma (y_{c},y_u)\) is strictly increasing in \(y_c\) and strictly decreasing in \(y_{u}\) for \((y_{c},y_{u})\in [0,\frac{1}{2}) \times (\frac{1}{2},1] \), the properties claimed in the lemma follow immediately. \(\square \)

### Proof of Lemma 4

Using Eqs. (12) and (19) in Eq. (28) we find

Substituting Eqs. (38) and (39) into Eq. (35) yields after some rearrangements

The RHS of Eq. (40) is strictly increasing in \(\kappa \) for \(\kappa =-\frac{1}{6}\) and strictly concave for \(\kappa \in [-\frac{1}{6},0)\). Moreover, direct substitution into Eq. (40) yields \(\frac{6}{\alpha w}\Phi (-\frac{1}{6})=0\) and \(\frac{6}{\alpha w}\Phi (0)<0.\) The claim in the lemma now follows immediately from the curvature in the graph of Fig. 8. \(\square \)

### Proof of Proposition 5

We show that under the restriction of the proposition the net value of human capital production in higher education, \(\eta =\eta _{c}+\eta _{u}\), is not maximal in the long-run equilibrium. For \(\kappa \in [-\frac{1}{6},\kappa ^{*})\), the college sector chooses \(y_{c}=0\) according to Proposition 4. We prove the proposition by showing that \(\partial \eta /\partial y_{c}|_{y_{c}=0}>0\).

Using Eqs. (12), (16), (19), (28), (31), (32), we calculate \(\eta \) as

Differentiating \(\eta \) with respect to \(y_{c}\), we obtain after some rearrangements

Since \(\kappa <0\) and \(\alpha <1\), the expression in (41) is positive. \(\square \)

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### Cite this article

Eckwert, B., Zilcha, I. The role of colleges within the higher education sector.
*Econ Theory* **69, **315–336 (2020). https://doi.org/10.1007/s00199-018-1163-3

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### Keywords

- Higher education
- College expansion
- Equilibrium screening mechanism
- Efficiency

### JEL Classification

- D80
- I21
- I23
- I25