University Competition: Symmetric or Asymmetric Quality Choices? Eve Vanhaechty University of Antwerp, Faculty of Applied Economics, Prinsstraat 13, B-2000 Belgium,
[email protected] Wilfried Pauwels University of Antwerp, Faculty of Applied Economics, Prinsstraat 13, B-2000 Belgium,
[email protected] April 2006
We gratefully acknowledge …nancial support from the PAI project P5/26, funded by the Belgian government. We wish to thank Jan Bouckaert, Bruno De Borger, Elena Del Rey and Robert Gary-Bobo for helpful comments on earlier versions of this paper. y Corresponding author.
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Abstract: In this paper we model competition between two publicly …nanced and identical universities deciding on the quality of their teaching. The education o¤ered by the two universities is di¤erentiated horizontally and vertically. If horizontal di¤erentiation dominates, the Nash equilibrium is symmetric, and the two universities o¤er the same quality levels. If vertical di¤erentiation dominates, the Nash equilibrium is asymmetric, and the high quality university attracts the better students. Symmetric and asymmetric equilibria may also coexist. The three driving forces behind these results are: a single crossing condition for the utility of the students, the peer group e¤ect, and the students’ mobility costs. We also compare the monopoly and the duopoly case and …nd that a shift from monopoly to duopoly increases teaching quality. Key words: (higher) education, research, competition, horizontal and vertical dominance, asymmetric equilibria
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1
Introduction
Most empirical studies report a negative e¤ect for physical distance on the possibility that a student enrols in a certain university. Sá, Florax and Rietveld [21], for instance, investigate the determinants of university entrance for Dutch high school graduates. Dutch universities are publicly funded. At the time of the study tuition fees are centrally determined and uniform across institutions. Rationing of supply (i.e. setting admission criteria) is allowed but non-existent. The authors …nd that the choice of potential students is negatively a¤ected by the distance between a student’s home and the location of the university. Surprisingly, the quality of teaching does not seem to play a signi…cant role in the students’choice behavior. The authors suggest that this can be explained by the fact that the quality di¤erences between Dutch universities are relatively small. It seems that in the Netherlands a situation of uniform quality of universities goes hand in hand with immobile students. This makes one wonder whether student mobility would be accompanied by large quality di¤erences between universities. The aim of this paper is to study the interplay between vertical and horizontal di¤erentiation of publicly funded universities. In the game developed in this paper both universities decide on the quality of their teaching while having a …xed physical location. Their payo¤ is speci…ed as a weighted sum of teaching quality and available research funds. Each student is characterized by a geographical location, and by a level of innate ability. Given these two characteristics, students rank the two universities in order of their preference. This ranking depends on two critical considerations. First, there are mobility costs. Each student is located at a certain distance from each university, implying a mobility cost for each university. Students with di¤erent locations face di¤erent mobility costs. These costs give rise to horizontal di¤erentiation between the universities. Secondly, universities also o¤er study programs of di¤erent quality levels. This gives rise to vertical product di¤erentiation. For our results it is important to consider domination of one type of di¤erentiation by the other. If all students living su¢ ciently close to a particular university, prefer that university to the other, for all levels of ability, we say that horizontal di¤erentiation dominates vertical di¤erentiation (i.e. there is horizontal dominance). Conversely, if all students with a su¢ ciently high (low) ability
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level prefer the high (low) quality university, for any given location, then vertical di¤erentiation dominates (i.e. there is vertical dominance). We show that each of the two types of domination gives rise to a di¤erent type of equilibrium. If horizontal di¤erentiation dominates, the equilibrium quality levels o¤ered by the two universities will be the same. The Nash equilibrium is symmetric. (This resembles the situation in the Netherlands.) If, on the other hand, vertical product di¤erentiation dominates, equilibria occur in which the two universities o¤er di¤erent equilibrium quality levels. The high quality university attracts the better students. The Nash equilibrium is asymmetric. This is remarkable since the two universities are ex ante identical. This result is consistent with the literature dealing with horizontal and vertical di¤erentiation within the …eld of industrial organization theory (e.g. Anderson, de Palma and Thisse [1] and Irmen and Thisse [14]). A basic result in this literature can be formulated as follows: minimal di¤erentiation is possible in one dimension, only if di¤erentiation is su¢ ciently large in the other. Applied to our model, this means that minimal di¤erentiation in quality (symmetric quality levels) is only possible when mobility costs are su¢ ciently large (when there is horizontal dominance). Depending on the exact parameter values the following equilibrium con…gurations occur: one symmetric equilibrium, two asymmetric equilibria, one symmetric and two asymmetric equilibria, and no equilibrium. There are three basic characteristics that drive our results. First, preferences of the students have to satisfy a single crossing property. In particular, a student’s e¤ort required to obtain a degree of a certain quality decreases as the student’s ability increases, and, for a given level of ability, a student’s required e¤ort increases as quality increases. This seems to be a very reasonable property of any student’s preferences. In the absence of this property, there can be no vertical product di¤erentiation in equilibrium.1 Secondly, there is the peer e¤ect. The larger the average ability level of the student body, the smaller the teaching cost required to realize a degree of a given quality level. Without this e¤ect, there does not exist an equilibrium in which the universities are vertically di¤erentiated. Finally, as already noted, there are mobility costs. If students do not care about the geographical location of the universities, we do not …nd an equilibrium. 1 This
will not be explicitly proven in the paper, but we know it from previous work.
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In the paper we will analyze the monopoly as well as the duopoly case. We show that an increase in competition – a move from monopoly to duopoly – always raises the average quality level of teaching. The literature on quality competition between universities is very limited. A …rst important contribution was made by Del Rey [8]. In her model universities compete in two stages: …rst they select a quality level and afterwards they set an admission standard2 . The model we will use in our paper is similar to Del Rey’s model. However, in Del Rey’s model there is no vertical product di¤erentiation in equilibrium. Only symmetric Nash equilibria occur. We extend her model by introducing the single crossing condition already mentioned, and by using a di¤erent speci…cation for a university’s teaching cost function. Moreover, for simplicity, we dropped the possibility for universities to select an admission standard, but we know from previous work that this does not matter for our main conclusions. A second important reference is De Fraja and Iossa [6]. One of their main results is that asymmetric equilibria in admission standards occur, provided mobility costs are not too high. In this paper we assume universities compete in quality, and this quality is clearly positively related with the average ability level of the students enrolling in a university. Similar to De Fraja and Iossa [6] we …nd a link between the asymmetry of the equilibrium and the height of the mobility costs. However, we generalize this result by linking the symmetry or asymmetry of the equilibria to the properties of horizontal and vertical dominance and not only to mobility costs. Moreover, as opposed to De Fraja and Iossa [6] we also show that a symmetric equilibrium and two asymmetric equilibria can coexist. Finally, our model has a much more explicit structure (cost function of teaching, objective function of the university, . . . ) giving more insight into the nature of the possible equilibria. Apart from papers on quality competition between publicly funded universities, other important papers have to be distinguished. Some papers are concerned with pro…t maximizing universities (or schools) competing in tuition fees (e.g. Rothshild and White [20] and Epple and Romano [12]), with universities who have to decide which new programs they launch (Del Rey and Wauthy [11]), with competition between non-pro…t and for-pro…t universities (Del Rey and Romero [10]), or with 2 This
is the minimal level of ability required for admission at the university.
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quality di¤erentiated universities having to decide on the workload of their bridging programs3 (Vanhaecht [24]). The structure of the paper is as follows. In section 2 we describe the behavior of the students and the universities. Section 3 analyzes the case of a monopolistic university. In Section 4 we solve the duopoly case. Section 5 concludes.
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The model
In this section we describe the basic ingredients of the model. We …rst specify the behavior of the students. We then analyze the decisions of the universities.
2.1
The students
Consider a unit mass of students. Students are characterized by their physical location x, and by their innate ability (or talent) level a. These two characteristics are assumed to be uniformly and independently distributed on [0; 1]
[0; 1]. We want to describe how a student with characteristics
(x; a) chooses between two universities. The two universities di¤er in their …xed physical location, and they can choose the quality of their degrees. University 1 is physically located at x = 0, and university 2 at x = 1. Moreover, university 1 o¤ers a degree of quality level q1 , while university 2 o¤ers a degree of quality level q2 . We assume that a student with ability a and physically located at a distance x from university 1 and a distance (1
x) from university 2 enjoys the following utility
levels from attending university 1 and 2, respectively, u1
=
+ q1
(1
a)q1
cx
u2
=
+ q2
(1
a)q2
c(1
(1) x):
(2)
First, simply attending a university augments the student’s utility with the constant . We assume that
is high enough, so that the student always prefers attending a university to not attending.
3 Bridging
programs are de…ned as the extra courses students have to attend when switching from one university’s
bachelor’s program to another university’s master’s program.
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In other words, we assume that there is no binding participation constraint for students. Second, a student incurs a mobility cost which is taken to be proportional to the distance between her own physical location and that of the university at which she enrolls. A student located at a distance x from university 1 faces a mobility cost of cx when attending university 1, and of c(1
x) when
attending university 2. Third, the quality of the degree o¤ered a¤ects a student’s utility in two di¤erent ways. On the one hand, a student’s future wage premium due to university education is increasing in the quality level of the university chosen.4 On the other hand, obtaining a degree at a higher quality university requires a higher investment of e¤ort from the student. The e¤ect of this e¤ort cost is given by (1
a)q1 , where
is a positive number. The required e¤ort cost decreases
with the ability a of the student.5 We neglect discounting. From now on we assume, however, that
= 2. Hence, (1) and (2) reduce to
u1
=
+ (2a
1)q1
cx
u2
=
+ (2a
1)q2
c(1
(3) x):
(4)
The reasons for doing this can be summarized as follows.6 First, we want to assure that the utility levels imply the following single crossing property @ 2 ui = 2 > 0: @qi @a
(5)
See Mirrlees [18] and Spence [22]. It means that the net gain from an increase in quality is always higher for a higher ability student. Or, equivalently, the marginal e¤ort cost for a degree of a given quality is decreasing in a student’s ability level. Second, we assume that a high ability student bene…ts from attending a high quality university, while it makes an excess demand on a low ability 4 Chevalier
and Conlon [4] …nd a wage premium of up to 6% for males graduating from the most prestigious
(highest quality based on di¤erent quality measures) universities in the UK. Moreover, they try to control for the fact that high-ability students tend to select a high-quality university. Similarly, Brewer et al. [3] conclude for the US that, even after correcting for selection into the type of university, prestigious private universities yield signi…cantly higher earnings compared to public universities. 5 Hence, e¤ort is not chosen by the student herself. The student chooses a university with a certain quality level and this implies the required e¤ort. 6 We do not comment on what happens if
6= 2. The discussion would be to lengthy and confusing. Moreover, a
wider range of di¤erent cases would have to be investigated.
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student, @ui = 2a @qi Third, setting
1>0,a>
1 : 2
(6)
= 2 implies that either vertical di¤erentiation or horizontal di¤erentiation between
the two universities dominates. This will become clear later on. Finally, assuming that both universities’ market shares to
1 2,
= 2; …xes
independent of the quality they provide to their students.
The average ability of the students attracted to the universities will still depend on the quality o¤ered. Consequently, universities can not compete in quantity (student numbers), and we can really focuss on quality competition. Again, this will be clari…ed later on. We now analyze the students’ choices between the two universities. The students who are indi¤erent between studying at university 1 and 2 mark the boundary between the two universities’ markets. Setting u1 equal to u2 and solving for x yields the market boundary, denoted x ^(a) x ^(a) =
(2a
1)(q1 2c
Students with characteristics (x; a) such that x x
q2 ) + c
:
(7)
x ^(a), prefer university 1. Students for whom
x ^(a) prefer university 2. For a given quality di¤erence q1
q2 , equation (7) de…nes a straight
line in the (x; a)-space. Students to the left of this line prefer university 1. The number of these students is denoted by d1 , the demand for university 1. Students to the right of this line prefer university 2. The number of these students is denoted d2 , the demand for university 2. Since we assume that students always prefer attending a university to not attending one, it follows that d1 + d2 = 1. Whenever the universities o¤er di¤erent quality levels, we will call university 1 the high and university 2 the low quality university, q1
q2
0. Consequently, the market boundary
given in (7) has a positive slope. The distance between a student’s physical location and the location of a university di¤erentiates the two universities horizontally. The quality di¤erence between the two universities di¤erentiates them vertically. Comparable to Anderson, De Palma and Thisse [1] and Degryse [7] we distinguish between two possible cases: horizontal and vertical dominance. On the one hand, if 0
q1
q2
c,
we say that there is horizontal dominance: both universities attract a positive market share for all ability levels. The slope of the market boundary is then smaller than one. See Figure 1. The demands for university 1 and 2, and the average ability level of the students attracted to university 8
a 1
xˆ(a ) d2
d1
0
x
1
Figure 1: Horizontal dominance
1 and 2 become d1 =
Z1
3 21 Z 1 4 a(^ x(a))da5 a1 = d1
x ^(a)da;
and d2 =
Z1
(1
(8)
0
0
21 Z 1 4 a2 = a(1 d2
x ^(a))da;
0
0
On the other hand, if the inequality q1
q2
3
x ^(a))da5 :
(9)
c holds, we say that there is vertical dominance: the
high quality university 1 obtains the entire market for high ability students, while the low quality university 2 attracts all low ability students. The slope of the market boundary is larger than one. See Figure 2. The demands for university 1 and 2, and the average ability level of the students attracted to university 1 and 2 are now given by d1 =
Za
x ^ (a) da +
d2 =
Za^ 0
da +
da;
a
a ^
and
Z1
Za
(1
Za Z1 1 a1 = [ a^ x (a) da + ada] d1
(10)
Za^ Za 1 a2 = [ ada + a(1 d2
(11)
a
a ^
x ^ (a))da;
0
a ^
x ^ (a))da]
a ^
with a ^=
1 2
c
a=
1 c + : 2 2(q1 q2 )
2(q1
q2 )
(12)
and
Using (7)-(13), we …nd that, for all values of q1
q2 9
0 and of c,
(13)
a 1
xˆ(a )
d1
a
d2
aˆ 0
1
x
Figure 2: Vertical dominance
d1 = d2 =
1 : 2
(14)
It follows that, independent of the quality di¤erence and of the mobility cost, both universities’ demands always equal one half of the total student population. Remember that this results from the fact that we set
equal to 2. Furthermore, although both universities attract exactly the same
number of students, it is clear from Figure 1 and Figure 2 that a higher quality university always attracts a student body with a higher average ability level. The case q1
q2
0 can be treated very similarly. The market boundary (7) will then have
a negative slope. Expressions for both universities’ demands can be obtained by changing the subindexes 1 and 2.
2.2
The universities
In this section we …rst specify a university’s cost function of teaching. Then we describe the budget constraint. Finally, we introduce the payo¤ function of a university. The teaching cost Ti of university i is modelled as follows Ti = (1 with 0
0 to have an asymmetric equilibrium in quality
Furthermore, a university’s teaching cost increases with the the quality qi provided. Remark that we assume that marginal costs are increasing in quality and that dropping this assumption leads to the disappearance of the asymmetric equilibrium. The budget constraint of a university is kept very simple. A university receives a lump sum budget F from the regulator. These funds can be used by a university to …nance its teaching activities or to spend on research. Research funds are denoted by Ri . The budget constraint is given by F = Ti + Ri : Remember that above we derived that di =
1 2
(16)
independent of the quality provided by the university.
Hence, by making the funding dependent on a university’s number of students the regulator will still not be able to in‡uence the behavior of the universities. Studying the e¤ects of di¤erent funding mechansims lies outside the scope of this paper. The speci…cation of a public university’s objective function is not straightforward. In line with De Fraja and Iossa [6], Del Rey [8] and Kemnitz [15] we assume that universities are interested in the “prestige”of their institution. This prestige depends on the number of students, on the average ability of the student body (as an indicator of quality), and on the expenditures on research. Since in our model the number of students enrolling in a university is …xed, we use the following speci…cation Ui = qi + Ri :
(17)
The term qi measures the teaching quality of the university, and Ri represents the funds available 7 Theoretical
considerations on peer group e¤ects can be found in e.g. Epple and Romano [12] and Winston
[25]. Empirical investigation on peer group e¤ects in tertiary education is done by e.g. Betts and Morell [2] and Zimmerman [26].
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for research. The weight attached to the latter equals . From (15) and (16) it follows that Ri = F
(1
ai )qi2 :
(18)
The university’s payo¤ function can …nally be written as Ui (q1 ; q2 ) = qi +
F
(1
ai )qi2 :
(19)
A clear weakness of this speci…cation is that the research output of a university is measured by the size of the research budget. This neglects the quality aspect of the research. Moreover, there may be economies of scope between teaching and research. A high quality of teaching will also bene…t the quality of research, and vice versa.8 We can introduce economies of scope in our ~ i , can be de…ned such that a model as follows. The true size of the research funds, denoted by R fraction
ai )qi2 . The “net”teaching
of it is covered by the “gross”teaching expenditures Ti = (1
expenditures are T~i = (1
ai )qi2
~i = the true research expenditures R
~ i . From the budget constraint F = T~i + R ~ i , we then obtain R 1 1
F
(1
ai )qi2 =
1 1
Ri : If
~ i will exceed Ri . < 1, R
The payo¤ function (19) can then be written as ~ i = qi + ~ R i Ui (q1 ; q2 ) = qi + R with ~ =
1
: As 0
: An increase in the economies of scope
between teaching and research is then equivalent to an increase in weight involving
(20)
, and to an increase of the
given to research funds in (19). In what follows, however, we will always use the notation ~i. and Ri . When interpreting the results one can easily shift to ~ and R
The complete game can now be speci…ed as follows. First, the universities simultaneously decide on their quality levels q1 and q2 . Students observe these levels, and decide to enrol in one of the two universities. This was shown in Figure 1 and Figure 2. Knowing qi and ai , each university calculates its teaching cost Ti . Subtracting this cost from the total government subsidy F yields the available research funds Ri . 8 In
the current context, economies of scope exist if there are cost e¢ ciencies to be gained by jointly producing
teaching and research output, rather than producing each of them separately. Empirical con…rmation of economies of scope between undergraduate teaching, graduate teaching and research can be found in Hashimoto and Cohn [13] and Koshal and Koshal [16].
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Before solving the game between the two universities we …rst study the case in which one university has a monopoly. This is an important benchmark case.
3
The monopoly case
In this section we consider the case where there is only one university. As we assumed that all students always want to attend a university, the demand for the single university equals the total student population, dm = 1, independent of the quality of its teaching. The average ability level am equals 12 . The monopolistic university’s payo¤ function can be written as Um (qm ) = qm +
h
F
(1
2
i 2 )qm :
(21)
The university maximizes this function with respect to qm . The …rst order condition requires that 1
(1
2 )qm
= 0. Hence, the quality choice which satis…es this becomes qm =
order condition,
2 (1
2)
1 (2
):
The second
< 0, is satis…ed. The following theorem easily follows.
Theorem 1 A monopolistic university’s optimal quality level is given by qm =
1 (2
)
:
Its optimal teaching expenditures and research funds are Tm =
1 2
2 (2
)
1
and Rm = F
2
2 (2
)
:
The university’s maximal payo¤ is equal to Um = F +
1 2 (2
)
:
The following comparative statics results easily follow from Theorem 1. First, the larger the peer group e¤ect , the higher the monopolist’s quality choice, the more it spends on teaching, and
13
hence the lower the research funds. In absence of the peer group e¤ect ( = 0) the monopolist’s quality choice equals
1 2
. Moreover, the higher the monopolist’s preference for research , the lower
its quality choice, the less it spends on teaching and the higher the size of the research funds. In the limiting case, a pure research oriented university devotes all funds to research and o¤ers education of zero quality, lim qm = 0 so that teaching costs are zero. !1
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The duopoly case
In this section we solve the duopoly game. In this game both universities simultaneously decide on quality levels q1 and q2 . Comparing these quality levels the students sort themselves over the two universities. Depending on the parameter values, the following Nash equilibrium con…gurations can appear: (I) one symmetric equilibrium in which the quality levels of the two universities are equal, (II) two asymmetric equilibria in which one university provides higher quality education than the other, (III) one symmetric and two asymmetric equilibria, and (IV) no equilibrium. We analyze each of these possibilities. Before doing so, we have to make an important technical remark. When solving this game, we always have to restrict ourselves to one of the two cases de…ned above: horizontal (q1
q2
c) or vertical dominance (q1
q2
c). In other words, we restrict ourselves to
a certain area in the (q1 ; q2 ) space. See Figure 3. From the previous section we know that for each case there is a speci…c way in which ai depends on q1 and q2 : Using this dependence, we investigate whether there is an equilibrium in quality levels. Ex post we have to check whether this equilibrium actually lies within the area considered However, even if this is the case, it is not impossible that, given the rival’s strategy, it pays for a university to choose a quality level in a di¤erent area.
4.1
Horizontal dominance: symmetric quality choices
In this section we investigate the case of horizontal dominance. This means that we assume that the quality di¤erence between the two universities is smaller than the mobility cost faced by the students: q1
q2
c. We know that the average abilities of both universities’student bodies are
14
q1
q1 = q2 + c q1 = q2
A1
q1 = q2 − c
A2 A2 A3
q2
0
Figure 3: Important areas in the (q1 ; q2 ) space
then given by
21 3 21 Z Z 4 5 4 a1 = 2 a(^ x(a))da and a2 = 2 a(1 0
with
0
x ^(a) =
(2a
1)(q1 2c
q2 ) + c
3
x ^(a))da5
:
(22)
(23)
We insert these expressions into the payo¤ functions of the two universities U1
= q1 +
F
(1
a1 )q12
(24)
U2
= q2 +
F
(1
a2 )q22
(25)
, and we maximize each payo¤ function with respect to the quality level of that university. The resulting system of …rst order conditions has four solutions. However, only the following symmetric solution q1S
=
q2S
=
3c (2
p p 3 c (3c ( 2 + )2
)
2 )
(26)
satis…es all conditions for a Nash equilibrium for a reasonable set of parameter values. First, the solution given in (26) is feasible if and only if c>
2 : 3 ( 2 + )2
(27)
Second, second order conditions are satis…ed if and only if c>
8 : 9 ( 2 + )2 15
(28)
q1
q1 = q2 + c q1 = q2
A1
q1 = q2 − c
A2 A2 A3
q2
0
Figure 4: Deviations from the symmetric equilibrium
Third, let us denote the reaction function of university i by Ri . Local stability of the equilibrium then requires that 1
@R1 (q2S ) @q2
@R2 (q1S ) > 0: @q1
(29)
This condition is satis…ed if and only if c>
6 : 5 ( 2 + )2
(30)
Remark that satisfaction of condition (30) implies satisfaction of conditions (27) and (28). Finally, at the beginning of this subsection we assumed that both universities’quality choices are situated within area A2 in the (q1 ; q2 ) space. See Figure 3. We are sure that, within this area, (26) gives both universities’ best strategy. Ex post we have to check whether none of the universities has an incentive to use a strategy outside area A2 . First, given that q2 = q2S , we have to investigate whether it pays for university 1 to increase its quality to a quality level q2S + c + " in area A1 . This will not be interesting for university 1 as long as U1 (q1S ; q2S )
U1 (q2S + c + "; q2S ). Second, given that
q2 = q2S , we have to investigate whether it pays for university 1 to decrease its quality to a quality level q2S
c
" in area A3 . This will not be the case if and only if U1 (q1S ; q2S )
U1 (q2S
c
"; q2S ).
Two similar deviations exist for university 2. See Figure 4. Unfortunately, we can not give the exact restrictions on the parameter values which prevent both universities from deviating from the equilibrium in (26). Intuitively, however, it is clear that for large values of the mobility cost c deviating to qiS + c + " or qiS
c
" is too costly. In the next subsection we show numerically that 16
for a reasonable set of parameter values (26) holds, and that no university deviates from it. Now consider Theorem 3.
Theorem 2 Assume that (30) holds. Then, there exists a symmetric, stable and local subgame perfect Nash equilibrium given by q1S
=
q2S
=
3c (2
)
p p 3 c (3c ( 2 + )2
2 )
:
In the next subsection we compute that for a reasonable set of parameter values this equilibrium is global. From theorem 3 we conclude that a su¢ ciently high value of the mobility cost faced by students induces the universities to o¤er the same quality to their students. Two interesting properties of this equilibrium are worth mentioning. First, within area A2 both universities’reaction functions are downward sloping. See Figure 5. This follows from @ 2 U1 = @q1 @q2
q1 6c
< 0 and
@ 2 U2 = @q2 @q1
q2 6c
< 0:
(31)
Hence, in this equilibrium the quality choices of the two universities are strategic substitutes: when university 1 raises its quality choice, university 2 will react by lowering its quality choice, and vice versa. Second, the equilibrium is symmetric, not only in the sense that both universities select the same quality level q1S = q2S , but also in the sense that they attract a student body with the same average ability aS1 = aS2 = 12 ; spend the same amount on teaching T1S = T2S , spend the same amount on research R1S = R2S and hence have the same payo¤ U1S = U2S . See Figure 6. Let us summarize some comparative statics. First, we …nd that an increase in the peer group e¤ect
raises the equilibrium quality level. From (26) it is not immediately clear what happens
when we neglect this peer group e¤ect ( = 0). It is easy to check, however, that in that case both universities provide education with a quality level equal to marginal utility of research
1 2
: Second, an increase in the
decreases the equilibrium quality level. Moreover, if both universities
are entirely focused on maximizing research funds, both devote all funds to research and o¤er education of zero quality, lim qm = 0; so that teaching costs are zero. !1
17
q2 R1 R2
q1
Figure 5: Reaction functions: symmetric equilibrium
a 1
n2*S
n1*S
0
1 2
1
x
Figure 6: Symmetric division of students
From theorem 2 it follows that an increase in the mobility cost leads to a reduction in the symmetric quality level @qiS 0: @q1
(38)
the mobility costs go to in…nity, students simply attend the university closest to their own physical location,
lim (^ x(a)) =
c!1 1 0 We
1 . 2
dropped an asymmetric solution with negative quality choices.
19
q1 q1 = q2 + c q1 = q2
A1
q1 = q2 − c A2
A3
A2
q2
0
Figure 7: Deviations from the asymmetric equilibrium
Unfortunately, we can not translate this requirement into an explicit condition on the parameter values. In the numerical examples of the next section we do check for stability of the asymmetric solution. Third, initially we assumed that q1
q2
c. This requires satisfaction of the following
condition c
0 and
@ 2 U2 = @q2 @q1
c2 q2 (q2 + 2q1 ) 12(q1 q2 )4
< 0:
(41)
This means that an increase in q2 induces university 1 to increase q1 , while an increase in q1 induces university 2 to decrease q2 . See Figure 8.11
Second, at the equilibrium described in
Theorem 4 university 1 o¤ers a higher quality level, and hence it attracts a student body with a higher average ability compared to university 2. See Figure 9. Despite the higher average ability level of university 1’s student body, it spends more on teaching than university 2, T1A > T2A : Hence, university 1 has less research funds R1A < R2A : Remark that under conditions (37) and (39), the equilibrium payo¤ of university 1 always exceeds the equilibrium payo¤ of university 2: U1A = q1A + R1A > U2A = q2A + R2A . Next we summarize some comparative statics. In line with previous results, an increase in decreases the asymmetric quality levels, while an increase in 1 1 In
increases the asymmetric quality
this Figure we see that, for certain parameter values, the reaction functions are not continuous. As q2 starts
to increase from 0, university 1 reacts by increasing q1 : However, when q2 reaches a certain level it is no longer a best reply for university 1 to stay on increasing q1 (probably because the marginal cost of quality is increasing). Its best reply is now situated in another area, i.e. that in which it becomes the lowest quality university. Of course, the same reasoning applies to the reaction function of university 2.
21
q2 R1 R2
q1
Figure 8: Reaction functions: asymmetric equilibria
levels. Remark that when we neglect the peer group e¤ect (
= 0), there does not exist an
asymmetric equilibrium. Consider the following two interesting notes on the role of the parameter c in the asymmetric equilibrium. First, as opposed to (32), an increase in the mobility cost c now raises the equilibrium quality levels @qiA >0 @c
i = 1; 2:
(42)
Again, a change in the parameter c only in‡uences the average ability of the student body attracted to each university. As opposed to (33), we now …nd that @ai c = >0 @qi @c 3(q1 q2 )3
i = 1; 2:
(43)
This means that as the mobility cost increases, the positive e¤ ect of an increase in quality on the i average ability of a university’s student body ( @a @qi > 0) becomes larger and larger. It follows that a
university will increase the quality provided as the mobility cost increases. Second, we …nd that the a 1 n1* A
a n2* A
aˆ 0
1
x
Figure 9: Asymmetric division of students
22
degree of quality di¤erentiation in the asymmetric equilibrium tends to increase with the mobility cost @(q1A q2A ) > 0: @c
(44)
In fact, this states that as the horizontal di¤erentiation (measured by the size of c) between the universities increases, the vertical di¤erentiation (measured by q1A
q2A ) increases as well. This may
sound counterintuitive, but, remember that we need a low level of c for the asymmetric equilibrium to hold (see (37) and (39)), while the symmetric equilibrium requires a high level of c (see (30)). Hence, as c increases we move from the asymmetric to the symmetric equilibrium in quality, meaning that an increase in horizontal di¤erentiation leads to a decrease in vertical di¤erentiation between the universities. Finally, under conditions (37) and (39), the quality provided by university 1 (2) is higher (lower) than the quality choice of a monopolistic university, q2A < qm < q1A . Moreover, the average quality in this duopoly case exceeds the quality produced by a monopolistic university,
4.3
q1A +q2A 2
> qm :
Symmetric, asymmetric or no equilibrium?
In the previous two subsections we derived the su¢ cient conditions for local symmetric and asymmetric equilibria in quality. See Theorem 2 and 3. In this subsection we give numerical examples from which we derive the following relationship between the mobility cost and the type of equilibria. Theorem 4 For very low levels of the mobility cost there is no global equilibrium in quality. For intermediate levels of the mobility cost there are two global asymmetric equilibria in quality. For high levels of the mobility cost there is a global symmetric equilibrium in quality. In general Theorem 4 is comparable to the results of De Fraja and Iossa [6]. However, we …nd reasonable parameter values for which the symmetric and the two asymmetric equilibria coexist. Consider the following tables as an illustration.
23
Symmetric equilibrium c
q1
q2
(27)
soc
stab
global
0.4
0.9
2
0.562
0.562
sat
sat
not sat
not sat
0.5
0.9
2
0.531
0.531
sat
sat
sat
sat
0.6
0.9
2
0.515
0.515
sat
sat
sat
sat
(45)
Asymmetric equilibrium c
q1
q2
(28)
soc
stab
global
0.2
0.9
2
0.794
0.333
sat
sat
sat
not sat
0.3
0.9
2
0.823
0.345
sat
sat
sat
sat
0.4
0.9
2
0.861
0.361
sat
sat
sat
sat
0.5
0.9
2
0.906
0.380
sat
sat
sat
sat
0.6
0.9
2
0.955
0.401
not sat
sat
sat
sat
(46)
Figure 10 gives both universities’reaction functions when the symmetric and the asymmetric equilibria coexist. Figure 11 gives both universities’reaction functions when there is no equilibrium.
q2 R1 R2
q1
Figure 10: Reaction functions: a symmetric and two asymmetric equilibria
24
q2 R2
R1 q1
Figure 11: Reaction functions: no equilibrium
5
Conclusion
We developed a duopoly model in which publicly funded universities compete in the quality of their teaching. A university’s teaching cost is decreasing in the average ability of its student body, and strictly convex in its quality choice. Universities care for research funds as well as teaching quality. Students are characterized by an ability level and a geographical location. The model captures two dimensions of product di¤erentiation. First, the mobility costs students incur when travelling to a university di¤erentiates the universities horizontally. Second, the quality di¤erence between the universities di¤erentiates them vertically. The two main results of this paper can be summarized as follows. First, we found that both universities will provide the same quality when horizontal di¤erentiation dominantes vertical di¤erentiation. This requires a mobility cost which is su¢ ciently high. Second, if vertical di¤erentiation dominantes horizontal di¤erentiation, the universities o¤er di¤erent quality levels. This requires a mobility cost which is su¢ ciently low. The two results imply that as the mobility cost increases we move from the asymmetric to the symmetric equilibrium in quality. In other words, the degree of vertical di¤erentiation tends to decrease when the degree of horizontal di¤erentiation increases. From our model it also follows that the average teaching quality produced in a duopoly exceeds the quality produced by a monopolist. Moreover, we show that universities who entirely focus on research and hence neglect their teaching provide zero quality, have the maximum of research funds and zero teaching costs. Finally, we …nd that if we would drop the assumption that more talented students allow a university to save on teaching costs (i.e. the peer group e¤ect), the resulting equilibrium would always be symmetric. 25
Our paper is closely related to the work of Del Rey [8] and De Fraja and Iossa [6]. There are however important di¤erences. First, as opposed to Del Rey [8], we included the assumption that a student’s preference for quality depends on her ability (i.e. the single crossing condition), and we used a di¤erent teaching cost function. Consequently, we found equilibria characterized by quality di¤erentiation. Del Rey [8] only found symmetric equilibria. Second, as opposed to De Fraja and Iossa [6] we assume that the universities compete in quality in stead of admission standards and we do not assume that all students necessarily prefer the highest quality university. This stems from the fact that in our model a student’s e¤ort cost is increasing in quality and decreasing in her ability so that some students prefer to enrol in a low quality university. Although the setting clearly di¤ers, we found the same correspondence between the size of the mobility cost and the type of equilibrium as De Fraja and Iossa [6] did. However, we generalized this result to a correspondence between horizontal and vertical dominance and the type of equilibrium. Moreover, De Fraja and Iossa [6] do not de…ne a region of the parameter values in which the symmetric and asymmetric equilibria coexist. Within the context of our model it might be interesting to look at the following two extensions. First, we can ask ourselves which quality levels maximize social welfare. Social welfare could e.g. be de…ned as the sum of the students’utilities minus the costs of education (Del Rey and Romero [10]). Second, we could include a participation constraint which could be binding for some types of students. Depending on their characteristics (x; a) some students may not …nd it worthwhile to participate in university education. In this case, the demand for a university will depend on the quality o¤ered and the mobility cost. This will surely in‡uence the competition between universities.
26
Appendix: No mobility costs (c = 0) We will see that, assuming q1
q2 and c = 0; the resulting equilibrium is asymmetric in quality.
But, we will prove that the low quality university 2 always has an incentive to deviate to q1A + ": If mobility costs are zero, all students with an ability a > 1 to the low quality university 2. Hence, using a1 =
3 4
1 2
prefer the high quality university
and a2 =
1 4
we maximize each university’s
payo¤ function with respect to its quality choice. We …nd the following asymmetric solution q1A (c = 0) =
2 (4
3 )
2
; q2A (c = 0) =
(4
)
:
Second order conditions are always satis…ed and the equilibrium obeys the assumption q1
(a30) q2 .We
claim however, that this equilibrium (a30) is only local since the low quality university 2 always has an incentive to deviate to q1A + ". Proof : Remark that when using solution (a30) we …nd that the payo¤ of university 1 always exceeds the payo¤ of university 2, U1A > U2A . Hence, if university 2 deviates to q1A + ", it attracts all the high ability students (those with a > 21 ) who used to attend university 1 and consequently improves its payo¤. See Figure 14. In this Appendix we assume that c = 0; but this argument can be extended for su¢ ciently small values of c > 0. To summarize, for very small values of the mobility cost c the asymmetric equilibrium only holds locally, because the low quality university has an incentive to surpass the high quality university. But, the higher the value of c; the higher the costs for the low quality university of surpassing the high quality university. See …gure 15. For c = 0:2; university 2 still …nds it optimal to jump to another area, i.e. to choose q1A + ": Hence, q2A U2 3.2 3.1
0.5 2.9
q2* A
1
1.5
2
q2
q1* A
2.8 2.7
Figure 12: Best reply of university 2 when c = 0
27
U2
c=0 c = 0.2 c = 0.4
3.2 3.15 3.1 3.05 0.5
1
1.5
2
q2
2.95 2.9 2.85
Figure 13: Best reply of university 2 as c increases
as de…ned in Theorem 3 is only a local best reply. For c = 0:4; however, q2A becomes university 2’s global best reply. Notice the analogy with De Fraja and Iossa [6].
28
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