Does the growth in higher education mean a decline in the quality of degrees ? The role of economic incentives to increase college enrolment rates

In this paper we construct a theory about how the expansion of higher education could be associated with several factors that indicate a decrease in the quality of degrees. We assume that the expansion of tertiary education takes place through three channels, and show how they are likely to reduce average study time, academic requirements and average wages, and inflate grades. First, universities have an incentive to increase their student body through public and private funding schemes beyond a level where they can keep their academic requirements high. Second, due to skill-biased technological change, employers have an incentive to recruit staff with a higher education degree. Third, students have an incentive to acquire a college degree due to employers’ preferences for such qualifications, the university application procedures and through the growing social value placed on education. We develop a parsimonious dynamic model in which a student, a college and an employer repeatedly make decisions about requirement levels, performance and wage levels. Our model shows that if i) universities have the incentive to decrease entrance requirements, ii) employers are more likely to employ staff with a higher education degree and iii) all types of students enrol in colleges, the final grade will not necessarily induce weaker students to study more to catch up with more able students. In order to re-establish a quality-guarantee mechanism, entrance requirements should be set at a higher level. CEPS Working Documents are intended to give an indication of work being conducted within CEPS’ research programmes and to stimulate reactions from other experts in the field. The opinions expressed in this document are the sole responsibility of the authors and do not necessarily represent the official position of CEPS. ISBN 978-94-6138-453-9 Available for free downloading from the CEPS website (http://www.ceps.eu) © CEPS 2015


Introduction
In this paper we study how the expansion of higher education can result in a decline in the quality of degrees. We measure the quality of degrees through studying time, academic requirements, average wages and the signalling power of grades (which is reduced through grade in ‡ation).
The issue of studying time as an indicator of quality in degrees has recently gained attention. In the United States a decline in full-time college students' studying time has been observed empirically and has itself become a subject, both academically and politically ( Moreau and Leathwood (2006) …nd that students'engagement in paid work has increased in England. Bekhradnia (2012:7) …nds that in the UK degrees take less time to complete than those in continental Europe and that less time is spent studying per week; a characteristic that mainly seems to apply to post-1992 universities, according to Bekhradnia (2012).
Reduced studying time has been associated with improved learning technology ( Manthei and Gilmore 2005). Findings indicate that there was an increase in part-time over full-time students, especially in Europe. Hakkinen (2006) …nds that in Finland 50% of students were working in 2000, which increased the time-to-degree. Working students spent 19 hours per week studying whereas non-working students spent 35 hours per week studying. Eurostudent (2005) …nds that more than 50% of students in Europe are employed. In the Netherlands and Estonia the student employment rate is two-thirds. At the same time Eurostudent (2005) provides evidence that students who work for 11 to 15 hours a week spend less time studying. In Germany and Romania the impact is seven hours less and in Bulgaria, the Czech Republic and Lithuania one hour less. Babcock and Marks (2010 a,b) show that even when controlling for part-time students by studying only full-time students, studying time still declines. The increase in part-time students alone therefore does not seem to be a su¢ cient explanation for the decrease in studying time.
Most of the prominent work in the relevant US literature associates declining studying 1 The amount of time US secondary students spend on homework has changed very little in the last 20 years. At high school, students did not study more in the 1940s than they did in the 1960-80s (Gill and Schlossmann 2003). However, according to Larson (2001), American children and adolescents spend less time on homework than their Asian or European counterparts and have more free time.
2 time with a decrease in college requirements and standards. This decrease arises from a set of reasons related to a more vulnerable position of faculty sta¤ (Brint 2011;Clotfelter, 2011;Arum and Roksa 2011;Bok 2006;Johnson 20002; Babcock and Marks 20101;Fischer 2011;Clack 2011): the amount of temporary positions as opposed to permanent positions has increased, publication pressure has risen and student evaluations matter more in securing an academic job. Grade in ‡ation has also been documented and associated with decreasing studying time (Johnson 2003;Rojstaczer 2011;Rojstaczer and Healey 2010). The decline in studying time has also been associated with the fact that college enrolment rates increase and less prepared, less motivated students enter campus (Brint 2011, Bound, Lovenheim andTurner 2009;Clack 2011).
Notwithstanding the wide array of factors that were considered as determinants for the decline in studying time, the increase in college enrolment rates has been accorded less importance in the debate about quality of degrees, but -taken together with a decrease in college requirements -it may have a considerable impact on the time spent studying (Briggs 2010). Indeed, Briggs (2010) predicts a decline in the number of average hours spent studying as the share of students entering college increases, but he has not given a clear theory on how this could actually happen.
The contribution of this paper is to provide a theoretical reasoning of how the expansion of tertiary education -together with a decrease in college requirements -could decrease average time spent studying, average wages attributed to college graduates and contribute to grade in ‡ation. We propose a model in which a university, multiple employers and a student can play a dynamic game with incomplete information. We focus on the undergraduate tertiary level of education since universities are ranked according to teaching quality, entry and achievement requirements and future employability of students 2 . In our model, expansion of education reduces average studying time because expansion turns out to be a low-ability expansion (the number of low-ability students accepted to university, whose studying time is zero, increases while the number of high-ability students stays the same). Machin and McNally (2007:11) already raise this possibility. An expansion towards more low-ability students entering college could be taking place because 1) college requirements decrease because universities are setting requirements that will maximise the pool of students, 2) all students are better o¤ if they get accepted to a university with a high rank and 3) employers are more likely to employ highly skilled individuals 3 . Our model shows that under these three conditions, the grade will not necessarily act as a mechanism to make weaker students study to catch up with more able students. In order to re-establish a quality guarantee mechanism, entry requirements should be set at a higher level.
We choose the UK as an empirical example of our model since students there seem to spend little time studying compared to their continental European counterparts (Eurostudent 2011 andBekhradnia 2012).
The remainder of the paper is organised as follows: the following section provides empirical evidence of tertiary education expansion and decline in the quality of higher education in the UK. Section 3 delineates the theoretical background for our model. Section 4 presents In this section we provide empirical evidence on the expansion of tertiary education in the UK since 1950 compared to other European countries, using several indicators of quality of higher education (as mentioned above these are studying time, academic requirements, average wages and the signalling power of grades)in order to empirically support the key variables in our model. In this section we focus on the United Kingdom since students in the United Kingdom seem to be spending little time studying compared to their continental European counterparts (Eurostudent 2011 andBekhradnia 2012). Furthermore, universities in the United Kingdom might be more strongly a¤ected by the need for a large number of students than continental European universities. The reason for this is that the United Kingdom is among the countries with the largest share of private funding through student fees with 263 in 2009 (Estermann and Pruvot 2011).

Enrolment in higher education
The expansion of tertiary education needed to be preceded by the universalisation of upper secondary education (Beblavy et al 2012:2). The US was the leading country in the expansion and universalisation 4 of upper secondary education in the 20th century. US enrolment rates in the upper secondary sector rose from 18% in 1910 to 73% in 1940 (Goldin 1998:347). Europe followed this example around 30 years later in the form of government e¤orts to make upper secondary education available to the masses (Beblavy 2012:2). As a consequence, tertiary education could equally expand (for an excellent review of this expansion see Trow 2007 or Schofer and Meyer 2005) -a phenomenon observed across Europe at di¤erent rates since the 1970s or 1980s (OECD 2012, Eurostat 2006 andMachin andMcNally 2007). The OECD (2012:9) measures that in 2012 over 503 of university-age young adults are enrolled in higher education in OECD countries. Eurostat (2006) shows that between 1998 and 2006, the number of students enrolled in tertiary education rose from 15 million to 19 million in the EU-27 area. Bratti and de Blasio (2008) provide evidence for Italy. Beblavy et al (2012) provide evidence for increasing enrolment in tertiary education in Germany, Slovakia, Czech Republic, Spain, the United Kingdom and Sweden 5 . In Figure 1 we show enrolment rates in tertiary education in the UK. We can see that enrolment in higher education increased only slightly between 1953 and 1989 but then surged very quickly and grew by about 400% from 5 to 25%. Compared to certain other European countries, expansion in the UK happened rather abruptly as well as rather late. As Figures 2 to 5 show, the expansion of tertiary education occurred at di¤erent speeds and dynamics across the various countries. Whereas we can observe a rather steady increase in tertiary enrolment rates in Sweden, the expansion in Spain happened more rapidly and more steeply.

Quality of tertiary degrees
As mentioned at the beginning of this paper and of this section, we measure the quality of higher education degrees through four elements: studying time, colleges' academic requirements, average wages and the 'signalling power'of grades. In this section we provide empirical evidence on studying time, academic requirements and the signalling power of grades (though examining grade in ‡ation). We do not analyse wage development in the United Kingdom and leave this discussion to our theoretical model.
Studying time of full-time students decreased in the US from 40 hours a week in 1961 to 27 hours a week in 2007 as Babcock and Marks (2010) show. In Europe, there are no such clear-cut numbers. Europe shows a high heterogeneity across countries in the time students spend studying (Eurostudent 2005(Eurostudent , 2008(Eurostudent , 2011 6 . According to the Eurostudent …ndings, time devoted to studying (classroom time and personal study time) ranges, on the Bachelor National evidence from the UK attests tothe fact that standards are declining at universities there (University World News 2008, The Guardian 2010). Based mainly on qualitative research, House of Commons (2008-2009) …nds that stakeholders and of higher education institutional actors in the UK believe that standards at university have worsened: a university degree is seen as being worth less than before by employers, students and academics. In the report, employers speak of their observation that students appear to be less motivated and have a less than ideal learning approach. Employers are said to focus more now on previous work experience than on degrees when selecting job candidates. Students are quoted as saying that what they had been taught at school early on was now being taught at university (ibid: 112). Academics believe that certain degrees have lost value compared to …ve, ten or twenty years ago; essays have declined in quality and students appear to be less well selected (House of Commons 2008-2009: 111). In fact, as shown in Figure 6, requirements to enter college have decreased as acceptance rates have an increasing trend. One reason for these higher acceptance rates may be an issue tackled by the House of Commons (2008-2009), which provides evidence of academic sta¤'s interest to increase the number of students admitted 7 .
Evidence on grade in ‡ation in the US has been provided by Rojstaczer

Theoretical Background
In this section we discuss various economic theories that form the backbone of the model presented in section 4. Our model is characterised by three distinguishing features that explain the expansion of tertiary education. We show how education expansion may lead to a potential decrease in studying e¤ort under certain circumstances: 1. The value of high skills on the labour market: employers employ only students who went to university -there is a quest for highly skilled individuals. It is important to note here that by "highly skilled" individuals we refer to those who went to university. They can be of low or high (innate) ability.
2. The role of universities and the value of expansion: universities maximise the number of students accepted since funding obtained by university increases with the number of students.
3. The rising importance of university degrees: all potential students are strictly better o¤ if they get accepted to university and if the ranking of this university is high enough to guarantee them a job.
Below we describe the theoretical basis for each of these features of the model.

The value of high skills on the labour market
The employer employs only students with high skills. He or she is modelled based on a screening model (Stiglitz 1975; Rothschild and Stiglitz 1976;Arrow 1973 andWiles 1974) an education economic model in which the employer is mainly interested in high educational attainment when making hiring decisions on a potential employee. In a screening model the 8 Figure 7: Percentage of upper class …rst class grades awarded by higher education institutes in the United Kingdom, Source: Higher Education Statistics Agency employer cannot observe the actual ability of potential employees and he or she therefore screens applicants according to their educational attainment to determine an applicant's ability of learning. Hiring applicants with university degrees reduces the employer's risk of hiring someone with a diminished capacity for learning. Implicit characteristics of this model, upon which results are dependent, are: (1) students with higher ability are better in absolute terms; the ranking of abilities is unambiguous (such screening is called hierarchical), (2) labour supply is inelastic, (3) individuals have perfect information on their own abilities, (4) on-the-job screening is not possible, (5) screening is exact, (6) the information acquired is general, in the sense that it is not …rm speci…c (Stiglitz 1975:286). In the screening model as in Stiglitz (1975) there can be a separating equilibrium or a pooling equilibrium 8 the latter characterises a situation in which all workers receive the same average wage whereas the former characterises a situation in which high ability types receive a high wage and low ability types receive a low wage. In both types of equilibria the existence of the low ability type decreases the net income of the high ability type and the existence of the high ability type will provide the low ability type with at least his or her marginal product (Stiglitz 1975:285). The screening model has several implications: …rstly, social returns di¤er from private returns and are not necessarily positive in the presence of screening. In fact, screening may tend to lead to higher inequality even if the screening mechanism increases the productivity of …rms (Stiglitz 1975:299). There is thus a trade-o¤ between distributive versus productivity concerns. Secondly, there is a tendency to over-invest in education, especially in publicly funded schools: education serves mainly as a signal and therefore education systems should choose the least costly method of transmitting this signal. In consequence, the screening model is often used as a theoretical basis to decrease public spending on education (Quiggin 1999). Huang and Capelli (2001), Bedard (2001) and Weiss (1995) o¤er some empirical evidence of the model and of the importance of higher education for employers.
In our model we add -compared to the Stiglitz and Rothschild (1976) model -a new decision-maker, the university, whose requirements a¤ect both the wage contract and studying time. However, given university requirements, the interaction between the students and the employers in each period t is similar to that in other standard models of screening and there is a unique symmetric perfect Bayesian equilibrium (SPBE). 11 We are interested in comparative statics, focusing on the e¤ects of university requirements on students'equilibrium studying time, grades and wages. As a further extension in comparison with Stiglitz (1975), we consider an interaction between the decision-makers that is repeated over time, with a new student taking decisions in every period. The equilibrium of this repeated interaction is then de…ned as a set of players'strategies and beliefs that constitute an SPBE in each period. The fact that university degrees determine employment and the wage contract can be bedded into a wider (sociological) theory about the expansion of higher education. Theories about the expansion of tertiary education being linked to economic development (re ‡ected by the employers'interests described above) are termed as socioeconomic functionalist explanations (Beblavy et al. 2012:3). These theories are based on the fact that national economic development creates demand for skills and supply is given through resources created through economic development (Collins 1971;Goldin 1998). This assumption is in line with recent …ndings and predictions that high skills matter and will matter most on the labour market (CEDEFOP 2010).

The role of universities and the value of expansion
We consider universities as public institutions providing education and making choices according to public choice theory: in public choice models, it is usually assumed that heads of public institutions aim to maximise their own utility. They will promote the bene…ts of the institutions only in a way that it is bene…cial to them personally. Niskanen (1968) argues that heads of institutions will bene…t from expansion of their institutions beyond the social optimum. A reason why managers of educational institutions will bene…t from an expansion of their institution stems from not-for-pro…t (NFP) theory (Holland and Ritvo 2008). NFP theory concerns institutions that do not distribute surplus revenues as dividends or pro…t, such as universities. Surplus revenues should be kept in the institution to serve its goals. Not-for-pro…t theory states that in the absence of a pro…t motive, NFPs tend to expand since that is the way for managers to increase their prestige and personal perquisites. It is therefore not necessary for the income of universities to expand linearly with number of students -it may if students pay tuition or if state pays per student -but even if there are more complicated funding mechanisms, if they lead to immediate or eventual expansion of university budget following an increase in number of students, our condition is satis…ed so that a linear assumption about the correlation of university funds with the number of students is an acceptable approximation.

The growing importance of university degrees
The assumption that all potential students are strictly better o¤ if they get accepted to university is based on the features of the labour market, the fact that universities do have the incentive to take up as many students as the government quota -or university capacity -allows and the fact that the social reputation of a university degree has risen with the opening of universities to a larger population. Rubinson and Fuller (1992) show that educational expansion has also occurred as a result of education having become a status symbol. This theoretical explanation for the expansion of tertiary education can be termed sociological or con ‡ict, competition and organisation theory (Beblavy et al 2012:3). These theories put less emphasis on market powers as an explanatory factor but rather emphasise social status. Due to competition between di¤erent status groups for success in education, the expansion of education rises beyond purely functional requirements (Bourdieu and Passeron 1977;Collins 1971 and1979). In times of high competition among social status groups, as can be the case with a rising population, expansion should be more rapid than in times with less competition (Rubinson and Fuller 1992).

The Model
In this section we construct a parsimonious model to explore theoretically how the expansion of tertiary education can decrease studying time.
Consider the following dynamic model. There are three essential entities that make decisions: a student with ability a that is distributed according to a probability distribution function (p.d.f.) f with support [a; a], a university and employers. Implicitly, there is also a fourth party -a government or an education authority -that imposes restrictions on the university by setting (a) the maximal increment in the number of students per period (the quota of students) and (b) a rank condition according to which an expansion of the student pool at the university is only allowed if the rank of the university exceeds a certain threshold. In the model, we take the quota of incoming students and the threshold for the rank as given and …xed over time, so that there is no active role for the government.
Time is discrete, t 2 f1; 2; :::g, and the three decision makers interact repeatedly over time. Each time period a new cohort of students substitutes the previous, so that a new representative student makes decisions at each time t. The timing of events is the following: 1. the university sets the level of requirements, r t , that determines the entry conditions and the grades; 2. M employers simultaneously select a wage contract, given the university requirements and the grades; 3. students with ability high enough to be accepted to the university choose the studying e¤ort.
Below we describe each of these steps and the interaction between the decisions of the university, students and the employers in more detail. First, the university sets the level of requirements, r t , that determines the entry conditions and the grades. Only students with ability a > r t pass the entry requirements and get accepted to the university. Upon acceptance they can guarantee themselves a positive level of utility, provided that the rank of the university is higher than a certain threshold (below which the wages are zero). Other potential students, with abilities below r t , are not accepted and make no study-related decisions; their utility is set to be zero. Thus, all potential students are strictly better o¤ if they get accepted to the university when the rank of the university is high enough to guarantee them a job, and hence the student pool in this case is represented by all students with abilities in (r t ; a]. Suppose that at t = 1, a r 1 < a. This initial value of requirements is set arbitrarily within the speci…ed range and determines the initial pool of students, N 1 . To make things interesting though we assume that r 1 is large enough so that the rank of the university at t = 1 is above the threshold R set by the authorities; this condition is formally stated and discussed below. 9 At any t 2 the requirements, r t , are chosen by the university so as to maximize the pool of students, N t , within the quota assigned by the authorities. Implicit in this is the idea that the larger the number of students, the larger the funding that the university obtains, either on a public or on a private basis. For this reason the university has an incentive to reduce its requirements and accept all students within the quota. Yet, it can only do so as long as with the new reduced standards its rank, R t (r t ), is above the threshold R set by the authorities. One can think of this as a situation where the authorities care about prestige of the university and in order to promote high standards, they only allow the university to expand when its rank is higher than a certain level. 10 When the time t requirements are set, the grade of a student with ability a is determined by where s t is the studying time.
Next, M independent employers simultaneously choose the wage contract. They only employ people who went to the university (students with abilities above r t ), conditional on the fact that the university rank is above the publicly known threshold, that is, R t R. If R t < R, the employers do not employ anyone. The employers are willing to employ students with abilities above a certain level A at wage w Ht , and those with abilities below A -at wage w Lt , where w Ht > w Lt . 11 However, the employers do not observe the abilities of students and can only judge them from the grades. They select the wage contracts that may pool or separate high-and low-ability students, depending on the employers'beliefs about which types of students will choose which grade. The wage levels, w Ht and w Lt are determined via a standard Bertrand competition 12 between employers. Moreover, only two di¤erent wages may exist in equilibrium since as we will show, only two di¤erent grades -signals of ability, may be observed.
Finally, given the university requirements and the wage contracts of the employers, a student of any ability a decides how much time to allocate to studying. Studying more improves the grade and this may increase the wage if the wage contracts of the employers are designed to separate high-and low-ability students; if the wage contracts are pooling, studying time does not a¤ect the wage. Also, studying reduces the leisure time. The student's 9 See (1). 10 Notice that even without the prescription from the authorities, the university can only guarantee itself a positive number of students if its rank is above R. For instance, when students are obliged to pay an amount " > 0 for their education, no one would apply to the university if the rank is below R. This is clear from the speci…cation of students preferences and wages described below. 11 We assume the most interesting scenario, when r 1 < A < a. 12 In a Bertrand competition, sellers set prices simultaneously (in our case the employers set wages). Buyers take these prices as given and set quantities (in our case students set e¤ort). utility is increasing in wage and in leisure; therefore, the choice of the amount of studying time depends crucially on the type of the wage contract -pooling or separating. If the contract is pooling, all students tend to choose the same studying time, minimal that is required to receive the grade that grants the employment. If the contract is separating, then high-ability students study more because for them the returns to studying are higher.
More formally, the dilemmas faced at each time period by the university, employers and students, in their chronological order, are described below.

University decision making
The university's funding is proportional to the number of students. In fact, assuming that the tuition fee per student or a government subsidy per student is …xed at a certain level p, the total tuition fee or subsidy that the university obtains at time t is p N t , where N t is the number of students. Therefore, the goal of the university at any period t is to maximize the size of its student pool 13 . However, two restrictions are imposed externally. First, at any requirements level set by the university, its rank, R t , should not fall below R, the …xed given constant. We will refer to R t R as the rank condition. Secondly, at each time t the student pool cannot be increased by more than a a , the quota extension for the total number of students. This second restriction can arise as a result of regulation by the government or the education authority or it can simply re ‡ect facility constraints at the university, that is unable to extend its studying facilities (such as computers, libraries, study materials and space) faster than at a certain rate. Naturally, a a should not exceed 1 N 1 .
Thus, the university chooses its requirements r t at time t with the objective to expand its student pool subject to two constraints: The de…nition of the rank accounts for the teaching quality as captured by T Nt -the ratio of the number of teachers to the number of students (T < 1 and reasonably T < N 1 , the initial number of students), the level of entry and grade requirements, r t , the employment prospects of time t student cohort as captured by the average wage, Avw t , and other factors -included in constant term . The weights , and are all positive constants and + + < 1. The value of R is arbitrary; however, the interesting case is when it is neither too high, nor too low. Namely, assume that R is such that a) initially, at t = 1 the rank condition is satis…ed, that is R 1 (r 1 ) > R (in fact, let R 1 (r 1 ) R), and b) if requirements decline to the lowest possible level, r t = a, when all students are accepted to the university, the rank condition does not hold, R t (a) < R. Given the de…nition of the rank above and using the expression for the average wage that will be derived later, this means that T + a+ w Ht a w Lt a A(w Ht w Lt ) a a + < R T a a a r 1 + r 1 + w Ht a w Lt r 1 A(w Ht w Lt ) a r 1 + (1)

Employers'decision making
There are M independent employers in the market. For simplicity assume that M = 2; the analysis for M > 2 can be easily extended. The employers have identical technology that is characterized by the feature that all workers with ability above A (high-ability workers) can perform the work that entitles them to the same wage w Ht . Also, all workers with ability below A (low-ability workers) can do the work that entitles them to the same wage w Lt . Assume that r 1 < A < a. In equilibrium, each employer seeks to maximize his expected pro…t, equal to the expected work productivity of the employee net of the wage payment. We assume that the work productivity of a high-ability individual is equal to his ability times constant factor x H and the work productivity of a low-ability individual is equal to his ability times constant factor x L . We consider x L x H to make sure that the work productivity (and the equilibrium wage) of a high-ability individual is higher than the work productivity (and the equilibrium wage) of a low-ability individual. The wages w Ht and w Lt are determined endogenously, via standard Bertrand competition between the employers.
Since the ability is not observable by the employers, they form beliefs about the ability based on the grade that the job applicant receives at the university. That is, having observed the grade g t , the employers assign some probability to the fact that the worker is of high-ability type (even for o¤-the-equilibrium path choices of g t ). Given these beliefs, the employers design the wage contract that either "pools" or "separates" di¤erent types of workers, so that wages are allocated to workers consistently with the beliefs.
Consider …rst the following beliefs of the employers: g t g t corresponds to a A g t < g t corresponds to a < A for some threshold grade g t . These beliefs lead to a separating contract that o¤ers the high wage w Ht to anyone with the grade g t g t and the low wage w Lt to anyone with the grade g t < g t . Later, in section 5, we show that the separating beliefs of the type described above, with a uniquely de…ned level of g t , are actually the only equilibrium beliefs of employers.
Moreover, the employment takes place and the wages are paid only if the rank condition is satis…ed, R t R. If it is not satis…ed, the employers are not interested in employing anyone (wages are zero).

Student decision making
Given the wage contract of the employers, a representative student with ability a > r t who is accepted to college at time t, chooses the time for study and for leisure with the objective to maximize his or her utility at time t. If the wage contract is separating, of the type described above, the optimization problem of a student is As it was mentioned earlier, l t and s t denote leisure and studying time, respectively, g t is the grade, and w t is the wage. Given the university requirements, r t , a grade g t is obtained if a student with ability a studies for s t units of time. The student is employed only if the rank condition holds and then the wage is either w Ht if the student's grade is above the threshold g t set by the employer, or w Lt if the grade is below this threshold.

Equilibrium
Equilibrium in period t is a standard symmetric perfect Bayesian equilibrium, where all decision makers achieve their objectives subject to the constraints they face and the beliefs of the employers are a) the same, b) "correct" on the equilibrium path, that is, con…rmed by the rational choices of high-and low-ability students. Formally, equilibrium is the combination of requirements r t set by the university, the beliefs and the wage contract of the employers and studying times s t chosen by students with all abilities a 2 (r t ; a] such that: 1. the size of the student pool, N t , is maximized subject to N t N t 1 (r t 1 ) + a a and R t (r t ) R; 2. the employer's beliefs on the equilibrium path are correct, i.e. consistent with the choices of students. If the corresponding wage contract is separating, workers with ability above A receive the wage w Ht , workers with ability below A receive the wage w Lt and both wages are determined via Bertrand competition between the employers; if the wage contract is pooling, all workers receive the same …xed wage, determined via Bertrand competition; 3. the utility of any student with ability a 2 (r t ; a] is maximized given the employers' beliefs (on and o¤ the equilibrium path), wage contract, the university requirements and the de…nition of a grade, g t = maxfs t a r t ; 0g; 4. the beliefs of the employers o¤ the equilibrium path are arbitrary.
Below we characterize the equilibrium in any period t 2. 14 As we will demonstrate, this equilibrium is unique.The equilibrium of the entire model is then de…ned as a situation where in each t 2 the SPBE of is played.

Results
In the following assume for the sake of simplicity that abilities of students are distributed uniformly on [a; a], so that f is the p.d. f. of the uniform distribution, f (a) = 1  a a on [a; a]. Then the c.d.f. is F (a) = a a a a for a 2 (a; a] (0 below a and 1 above a) and the truncated p.d.f. e f t -with the support on the interval (r t ; a] for a r t < a -is de…ned as Our …rst proposition states that as soon as the threshold A, separating high-and lowability students is high enough, the beliefs of the employers and the corresponding separating wage contract of the type described above are in fact equilibrium beliefs and contracts in any period t.
Proposition 1 If the threshold of abilities A is su¢ ciently high, then the set of strategies and employers'beliefs characterized below constitutes an equilibrium at any t 2: 1. The university requirements are de…ned recursively by r t = maxfr t 1 ; r g, where r is the unique solution of R t (r t ) = R. Given the initial requirements r 1 , r t is de…ned uniquely in every t 2: if r 1 (t 1) > r r t = r otherwise 2. The beliefs of the employers are such that all students with the grade g t g t are of high ability (with a A) and all the other students are of low ability (with a < A), where the threshold grade, g t , is equal to The corresponding separating wage contract o¤ers the high wage w Ht to students with g t g t and the low wage w Lt to students with g t < g t . The wages, w Ht and w Lt , are equal to 3. The studying time of the student of any ability a < A is 0, while the studying time of the student of any ability a A is These studying times lead to g t = 0 and g t = g t , respectively. Both types of students accept the o¤ered wage contract and earn wages w Ht (if of high type) and w Lt (if of low type).
The next proposition states the "uniqueness" result. Namely, it claims that the separating equilibrium described in Proposition 3.1 is unique and no pooling equilibrium exists.
Proposition 2 At any t 2, there exists no equilibrium in which the wage contract of the employers is pooling.
The proofs of Proposition 3.1 and 3.2 are provided in the Appendix. Thus, the unique equilibrium outcome at any time t 2 is such that (a) the university sets the requirements that are by lower than the ones in the previous period, as soon as this level does not fall below the lowest acceptable threshold, r ; (b) the employers o¤er the separating wage contract that allocates high wage to students of high ability and low wage to students of low ability, and the wages are determined a la Bertrand; (c) the students either do not study or study just enough to obtain the threshold grade g t and accept the wage contract of the employers.
It is easy to see that the long-run equilibrium value of requirements is r , the value at which the rank condition holds as equality, R t (r ) = R. This induces the long-run value of the threshold grade The long-run studying time of any low-ability student is 0, while long-run studying time of any high-ability student is We are interested in the e¤ects that the university requirements have on equilibrium studying time, grades and wages. Notice that the equilibrium wage of high-ability students (with a A) is constant, una¤ected by r t , while the wage of low-ability students (with a < A) is increasing in r t , that is, it declines when r t declines. This is so because Bertrand competition between the employers who have separating beliefs induces them to set w Ht equal to the expected productivity of the high-type students, x H E(a j a A), and w Lt equal to the expected productivity of the low-type students, x L E(a j r t a < A). The former conditional expectation is constant since the pool of high-ability graduates, with a A, remains the same as requirements decline. On the other hand, the latter conditional expectation moves together with r t : smaller r t leads to a larger pool of low-ability graduates, hence their expected ability and wage become smaller.
The equilibrium grade of low-ability students is constant and equal to 0, while the equilibrium grade of high-ability students, g t , increases as requirements become lower.
Finally, the equilibrium studying time of low-ability students remains …xed at 0, while the equilibrium studying time of high-ability students, s t , increases as r t declines. The reason why the studying time of high-ability students increases has to do with the fact that as r t declines, declining low-to-high wage ratio, w Lt w Ht , creates incentives for low-ability students to study and receive high wage, i.e. imitate high-ability students. For this not to happen, the studying time required to earn high wage increases, by just enough to discourage imitation by low types.
These e¤ects of the requirements on equilibrium studying time, grades and on wages induce prediction for the e¤ects on the average/expected equilibrium studying time, Avs t , the average/expected equilibrium grade, Avg t , and the average/expected equilibrium wage, Avw t , of time t cohort of students. Clearly, the average wage declines because the wage and the number of high-ability students are …xed, while the wage of low-ability students decreases and the number of low-ability students becomes larger.
On the other hand, the e¤ects of declining university requirements on the average grade and average studying time are not so straightforward. Comparative statics results (derived in the Appendix) suggest that as the requirements decline, the average grade increases and the average studying time decreases, with the latter being true at least as soon as (i) the ratio of productivity factors, x L x H , is su¢ ciently low and/or (ii) the ratio of students'preferences for leisure or wages is su¢ ciently small and high types constitute only a relatively small proportion of the total student pool. Intuitively, as requirements become weaker, the average grade increases because the threshold grade, g t , increases and hence, even though the number of high-ability students remains the same, their grades rise and this turns out to overcompensate an increase in the pool of low-ability students whose grade is zero. Similarly, the decline in average studying time is a result of an increase in the number of low-ability students, whose studying time is zero. This e¤ect turns out to outweigh an increase in the studying time of high-ability students, at least as soon as the impact of the declining low-tohigh wage ratio (that increases s t ) is muted by low productivity factors ratio, x L x H , and/or if students'preferences allocate su¢ ciently higher weight to leisure than to wages (so that s t of high-ability students does not increase much) and high types constitute only a relatively small proportion of the total student pool. 15 The proposition below summarizes these conclusions: The equilibrium of the model is characterized by the following comparative statics: 1. The equilibrium wage of high-ability students, w Ht , is constant, independent of r t , while the equilibrium wage of low-ability students, w Lt , decreases when r t declines.
2. The equilibrium studying time of low-ability students is …xed at 0, while the equilibrium studying time of high-ability students increases when r t declines. As more low-ability students enter university, the increased studying time of the high ability students will not compensate the growing number of low ability students and average studying time will decline.
3. The equilibrium grade of low-ability students is …xed at 0, while the equilibrium grade of high-ability students increases when r t declines.
4. The average equilibrium wage, Avw t , decreases and the average equilibrium grade, Avg t , increases when r t declines. The average equilibrium studying time, Avs t , decreases at least as soon as (i) x L x H is su¢ ciently small and/or (ii) is su¢ ciently small and A is su¢ ciently large.

Conclusion
In this paper we constructed a theory on how the expansion of higher education could be associated with several factors that indicate a decline in the quality of degrees. We focused on how the expansion of higher education -together with a decrease in university requirements resulting from budgetary considerations -might be associated with a decline in average studying time (student e¤ort), grade in ‡ation and a decrease in average wages provided to university graduates. We constructed a model in which a university, multiple employers and a student play a dynamic game with incomplete information. This model can be interpreted as an extension of the Rothschild and Stiglitz (1976) and Stiglitz (1975) model of screening in which the employer needs to employ a mechanism to sort employees into high-performing or low-performing workers. Other previous theoretical models also form the basis of our model: universities are modelled as non-for-pro…t organisations in which managers are interested in expanding their institutions.
By the means of our model we show that students'average studying time, average wages as well as the signalling power of grades decline because college enrolment rates increase beyond a level, at which academic requirements can be kept at a high level. In our model the expansion of education reduces average studying time because expansion is modelled as lowability expansion (the number of low-ability students accepted to university, whose optimal studying time is shown to be zero, is shown to increasewhile the number of high ability students stays the same). Machin and McNally (2007:11) already raise this possibility. This is the case because(1) college academic requirements decrease as a consequence of universities setting requirements so as to maximise the pool of students, (2) all students are better o¤ if they get accepted to a university with a high rank and (3) employers award a high wage to those with a high grade.
Our model shows that under these circumstances, the grade will not necessarily act as a mechanism to induce weaker students to study and catch up with more able students. In order to re-establish a quality guarantee mechanism, entrance requirements should be set at a higher level. 19 Now, the proof of the equilibrium still requires (1) to show that as long as A is su¢ ciently high, no pooling contract chosen by the competitive employer can break the candidate separating equilibrium, (2) to …nd the equilibrium wages, (3) to specify the equilibrium strategy of the university. Consider each of these points in turn.
(1) Notice that in equilibrium Bertrand competition over wages results in zero pro…ts for each employer. Consider the deviation by one of the employers to the pooling wage contract that is the best (among all pooling contracts) for the students and allows the employer to make positive pro…ts. This best pooling contract o¤ers the single wage that is equal to the average/expected wage in the whole population of students (minus small " > 0), and this wage is paid to a student with any grade g t 0. The average wage is equal to We show that even this best pooling contract cannot be accepted by both types of students, high-and low-ability, as soon as A is su¢ ciently high. Indeed, at least some high-ability students will not accept this contract if It is easy to see that this condition holds for any w Ht > w Lt if A = a and a > A. So, for A large enough, high-ability students do not accept the pooling contract and prefer the separating contract instead. Then the deviation to the pooling wage contract is not pro…table, and the candidate separating wage contract is an equilibrium.
(2) With separating wage contract, the expected pro…t of each employer from employing a high-ability student is equal to Similarly, the expected pro…t from employing a low-ability student is equal to Both employers have the same technology, hence, the standard Bertrand argument leads to zero pro…ts and where 1 a A and 1 A rt are the truncated probability density functions with the support on [A; a] and (r t ; A), respectively.
(3) Finally, consider the optimal choice of requirements by the university at time t. Given the objective of the university to increase its funds by maximizing the pool of students, the university reduces the requirements by "as much as possible" since this maximizes N t = R a rt f (a)da = a rt a a . Therefore, as soon as requirements are above r , the level at which the rank condition just holds, the university cuts them back.
Notice that the level of requirements r exists and is unique. Indeed, consider that using the expression for the average wage in (2) and the expression for N t , the rank of the university at time t can be written as: R t (r t ) = T a rt a a + r t + w Ht a w Lt r t A(w Ht w Lt ) a r t + = = T (a a) (a r t ) + r t + w Ht a w Lt r t A(w Ht w Lt ) a r t + All terms in the sum (apart from ) are increasing in r t , that is, the rank of the university moves together with the requirements. Therefore, the lowest level of requirements for which the rank condition holds is such that R t (r t ) = R. Due to the fact that R t (r t ) is monotonically increasing in r t and R satis…es (1), there exists a unique solution of R t (r t ) = R in (a; r 1 ), which we denote by r .
So, for any t 2 as soon as r t 1 < r (r 1 < r due to the second inequality in (1)) the optimal strategy of the university is to set r t = maxfr t 1 ; r g r t = r t 1 guarantees that N t = N t 1 (r t 1 ) + a a , i.e. the quota of students at time t is used fully and the number of students at t is maximized: a r t a a = a r t 1 a a + a a r t = r t 1 On the other hand, if r t 1 = r , then the requirements are not adjusted anymore. 17 Proof of Proposition 3.2 Consider the candidate pooling equilibrium. The best (for the students) pooling wage contract o¤ers the wage that is equal to the average/expected wage in the whole population of students (see (2)), paid irrespectively of a student's grade. The pro…ts of both employers are zero.
Below we show that there exists another contract such that the deviation by any of the employers to this contract is pro…table. Consider the contract that o¤ers w t = w Ht (minus small " > 0) if g t b g t and w t = w Lt if g t < b g t for some b g t > 0. We prove that there exists b g t such that this contract is accepted by high-ability students but rejected by low-ability students, so that the deviating employer raises positive pro…ts.
The contract is accepted by high-ability students, who choose the studying time b s t = b gt+rt a (just enough to receive the grade b g t ) as soon as u(1; Avw t ) < u(1 b s t ; w Ht ) for any a A

Equivalently,
w Ht a w Lt r t A(w Ht w Lt ) a r t < 1 b g t + r t a w Ht for any a A On the other hand, the contract is rejected by low-ability students, who choose the pooling wage contract with s t = 0 (and g t = 0), as soon as u(1; Avw t ) > u(1 b s t ; w Ht ) for any a < A

Equivalently,
w Ht a w Lt r t A(w Ht w Lt ) a r t < 1 b g t + r t a w Ht for any a < A Both conditions, (4) and (5) are satis…ed if and only if they hold as equality at a = A. Solving this equality for b g t produces b g t = A 1 w Ht a w Lt r t A(w Ht w Lt ) (a r t )w Ht ! r t Hence, the deviating contract breaks the candidate pooling equilibrium.

Proof of Proposition 3.3
We only need to prove part 4 of the proposition. Namely, we need to study the e¤ect of r t on Avg t and Avs t . First, notice that Consider the partial derivatives of Avg t and Avs t with respect to r t and evaluate their sign. Simple algebra leads to The sign of @Avst @rt is positive, at least as soon as (i) x L x H is su¢ ciently small and/or (ii) is su¢ ciently small and A is su¢ ciently large.