University Competition and Transnational Education: The Choice of Branch Campus

2.1 Universities

Following De Fraja and Iossa (2002), the objective function of a university, i,i=A,B, is written as:

where W captures the three elements that a university bureaucracy ^[4] cares about:

1. the number of enrolling students ni, where

   ni={ni1+ni2ifaBCisopenedni1ifnoBCisopened

1. the quality of the student body Θi (i. e. average ability of students attending i), and

2. the expenditure on research Ri,

while

are the fixed costs associated with opening a branch campus. The first partial derivatives of W in eq. [1] are all positive, and Wnn⋅,WRR⋅,WΘΘ⋅<0, and that the second cross derivatives are sufficiently small so that the relevant second-order conditions are satisfied. This implies that W is approximately separable.

Each university has a budget determined by the amount of tuition fees collected from the enrolled students, fni, where f>0 represents the fees per student. Tuition fees f are set exogenously: universities are not free to choose what students are charged in fees. ^[5] Therefore, each university:

1. decides whether or not to open a BC in Country 2 at a fixed cost F>0; and

2. chooses the required standard necessary to admit a student in each location/campus. We denote this by xij∈x_ij,1,i=A,B,j=1,2, where x_ij>0 is the lowest possible admission standard. This implies that only students who reach at least standard xij are accepted at institution i with campus in Country j.

Further, suppose that teaching ni students carries a cost of

with c'(nj)>0, c''(nj)⩾0,j∈1,2 and limn→1c'(nj)=+∞. Thus the teaching cost is considered separately for each university site. This assumption represents better a university technology in the real world: the costs are increasing and convex within each campus, due to the number of staff, classroom size, equipment, laboratories, and so on.

2.2 Students

Students differ in ability, denoted by θ∈0,1. In each country, students’ distribution by ability is Gθ, with G0=0,G1=1 and density g(θ)=G'(θ). The admission standard set by a university, xij, is in the same support as ability, so that xij∈0,1 and xij=θij where θij is the lowest ability student that can be accepted by university i in Country j. ^[6] If a student attends university, she graduates with certainty at the end of the university period. Still with certainty, in the labour market she will receive a wage surplus for being a graduate (“graduate wage”) Sxij, depending on the university admission threshold xij. Effectively the wage surplus captures the peer effects at university: the higher the admission standard, the smarter the people a student interacts with Del Rey (2001), De Fraja and Iossa (2002) and Cesi and Paolini (2014). To put it simply, being with smart people makes you smarter.

A student’s payoff function (if attending university) depends on her ability θ, the admission standard, xij, the tuition fee, f and the mobility cost T:

where Uxij,θ=Sxij+θ is a student’s utility with S'(xij)>0, and

For the sake of simplicity, we assume Sx_ij>f. In other words, the lowest possible graduate wage is higher than tuition fees. This ensures that every student is willing to attend university irrespective of f. A possible interpretation is that a government agency designs tuition fees in order to give incentives to the largest number of students to attend university. This assumption simplifies the analysis as f does not play any role in determining the demand function of students, but only determines a university’s budget.

To simplify the analysis, all students from Country 1 attend university in Country 1, even if at least one BC is present in Country 2. ^[7] On the other hand, in Country 2 there is an exogenous number of students β∈0,1, denoted as “privileged” who can borrow, either from their family or the banking system, the amount of money to cover the mobility costs t. β is independent of a student’s level of ability.

A student who does not attend university has a reservation utility of U0,0<Uxij,θ, for all xij∈x_ij,1, so that a student from the developed country would surely attend university if admitted. A student from the developing country would surely attend university if admitted and either

1. there is a BC, or

2. there is no BC but she belongs to the group of privileged students and Uxij,θ−f≥t.

The next definition is convenient.

2.3 Students’ Admission and University Attendance

In this section we establish university attendance which depends on the admission standard/threshold set by each university. Using [1] we rewrite the payoff functions of the two universities as

where nixAj,xBj and ΘixAj,xBj are the number of students admitted and the average quality of students at university i,i=A,B, respectively, given the admission standards xAj,xBj for locations j=1,2. Notice that we have replaced research expenditure Ri with Ri=fnixAj,xBj−cnixAj,xBj, i. e. the difference between total revenue from tuition fees and teaching costs.

It is clear that ∂Θi/∂xij>0: an increase in a university’s admission standard increases the average ability of that university’s students. According to the timing of the game, universities set their admission standards following their decision on the opening (or not) of a branch campus. Therefore there are four distinct possibilities, depending on the first-stage subgames:

1. Neither university opens a branch campus, hence the relevant standards to set are xA1 and xB1.

2. University A opens a branch campus; the standards to be set are xA1, xA2 and xB1.

3. University B opens a branch campus; the standards to be set are xA1, xB1 and xB2.

4. Both universities open a branch campus; the relevant standards are xA1, xA2, xB1 and xB2.

A natural question arising here is whether the standards chosen will be symmetric, meaning that the universities set the same admission standards, or asymmetric, where universities choose different admission standards. A preliminary result is the following.

Lemma 1 establishes that an equilibrium requires setting different admission standards.

In relation to subgame 4 (see above) we clarify that one can distinguish the following possibilities: (a) University A sets a higher standard in both countries, (b) University B sets a higher standard in both countries, (c) University A sets a higher standard in country 1, (d) University A sets a higher standard in Country 2. Cases (a) and (b) and cases (c) and (d) are clearly specular. In what follows, we focus on equilibria where university A sets the higher standard in every country in which it operates, and we call university A “elitist”. ^[8] We therefore state,

The following propositions describe university attendance of students from Country 1 and 2, respectively.

The next proposition establishes university attendance of students from Country 2. For these students, university attendance depends on (i) whether or not one or two branch campuses are opened, and (ii), in the case where only university B opens a branch campus, whether t is greater or not than t∗ (see Definition 1). Indeed, a student with high ability may be admitted to university A, but the mobility costs are high so that the student may prefer to attend the branch campus of university B. The equivalent of Proposition 1 for students of Country 2 follows.

Propositions 1 and 2 allow us to simplify the interaction between universities and students so that, in the first stage, the strategy space is binary, and consists of the decision of whether to open (or not) a branch campus in country 2 and in the second stage, the strategy space is given by the admission standards, xAj,xBj∈[0,1]. The equilibrium concept is subgame perfect equilibrium by backward induction. Figure 1 depicts the timing of the game.

Figure 1:

The simplified game.

To make further progress and obtain explicit closed-form solutions for the remainder of the paper we specify the functional relationships as follows:

where Sxij=wxij,w>0 is a graduate’s wage. The specifications chosen (eqs [3], [4], [5] and [6]) respect the properties of the associated general functions. In particular, note that the average ability of students increases with xij. Finally, assuming a uniform distribution for ability is standard in the literature of university competition (Del Rey, 2001; De Fraja and Iossa, 2002; Cesi and Paolini, 2013 and Carroni, Cesi and Paolini 2015).

2.4 Admission Standards

The following Lemma 2 shows the number of admitted students and is a direct consequence of Propositions 1 and 2.

Based on these results for the case where University B opens a BC and University A does not, Lemma 3 below provides the critical value for mobility cost t∗ that determines whether a student from country 2 will attend country 1 or not (see Definition 1).

Notice that university A (B) prefers wxA−t≤>wxB, as shown by:

and

2.5 Investment in BC

In the first stage, each university decides whether to open a BC according to the competitor’s strategy. For brevity, we focus on the case where t≤t∗. ^[9] The following table shows the payoff matrix according to whether university A and B decide to invest in a BC.

The full derivation of πANN; πBNN,πAFN; πBFN,πANF; πBNF and πANN; πBNN is given by eqs [13], [15], [19] and [17], respectively, in Appendix A.1. This allows us to compare profits in every scenario and to find the equilibria in the decision of opening a BC. The subgame perfect equilibrium is therefore determined as the Nash equilibrium of the 2×2 payoff matrix above, according to values of the establishment (fixed) costs of opening a BC, F.

Begin by examining the strategy of university A according to university B decisions. If university B plays BC, university A would do the same for πA1FF>πA1NF, whereas if university B plays N, university A would play BC for πA1FN>πA1NN. Both these inequalities hold for all:

so that A has a dominant strategy: for F<FA it plays BC and for F>FA it plays N. Invoking dominance, we now turn to the behaviour of university B. For F<FA university A plays BC so that university B would also play BC when πB1FF>πB1FN, which occurs for all:

and would play N if F>FˆB. For F>FA university A plays N so that university B would play BC for πB1NF>πB1NN, which occurs for:

and would play N for F>F‾B.

Notice that the chain of inequalities for the threshold levels is FA>F‾B>FˆB. Therefore, in conjunction with the preceding discussion, we establish the following proposition, which is one of the main results of this paper. ^[10]

Figure 2 illustrates these equilibria for combination between the investment cost F and the graduate wage w, for given β and c. Also, notice that

university A gains more from the students’ qualification x than university B. Hence, the number of students n has relatively more importance in determining B’s profits. Students prefer to attend university A, so that the demand for higher education is filled from that institution, whereas university B serves only the remainder of the demand for higher education. Therefore university A has more incentives in investing in BC, given the same cost F.

Consider next how a variation in the share of privileged students may affect the equilibrium. Differentiating FA−FˆB with respect to β yields:

The result described in Corollary 1 can be explained as follows. A raise in the number of privileged students reduces the demand for university from students in the developing country. The non-elitist university is more affected by the fall in the demand relatively more than the elitist university, given the higher benefit from the latter from opening a branch campus.

Figure 2:

Illustration of proposition 3.

2.6 “Top” University

The analysis carried out so far aimed at stressing the importance of mobility costs in university interaction. This approach is relevant in many situations in which students’ decisions strongly depend on these costs. However, to highlight the role of mobility costs necessarily sets aside the role played by the prestige of attending the home campus of a top university. Indeed, there is substantial evidence suggesting that many students prefer to attend a top university even if the same institution opened a branch campus (for example, consider the proportion of Asian students in top American universities). In our framework, this corresponds to setting t<0 when attending university A. In words, the gain (in terms of reputation, extra salary independent of ability, and the like) for attending the home campus of a top university more than offsets the mobility costs for doing so. This section is devoted to explore this case.

Let us re-label University A as “top university”. As in the previous analysis, University A sets higher admission standards than University B. Moreover, denote tA<0 as the difference between mobility costs and the benefit of attending a top university. Nothing changes for University B, so that a student from the developing country who attends the home campus of University B still bears a cost of t>0. Lemma 2 changes as follows.

We solve the game following the previous structure. Equations [13], [20], [22] and [21] in Appendix A.2 are the equilibrium payoffs according to the decisions of opening a BC in the first stage. Using these results, we examine the investment decisions in BC in the first stage. We start by examining the strategy of university A according to university B decisions. If university B plays BC, university A would do the same for πA1FF>πA1NF, whereas if university B plays N, university A would play BC for πA1FN>πA1NN. Both inequalities hold for all:

Consider next the strategy of university B. If university A plays BC, university B would do the same for πB1FF>πB1FN, which occurs for all:

If university A plays N, university B would play BC the same for πB1NF>πB1NN, which takes place for:

For the sake of tractability, we set w=c. The results depend on the proportion of privileged students, as summarised in the following lemma.

Case [1] of Lemma 1 is qualitatively similar to the baseline model. When the proportion of privileged students is low, the top university has the same strategic behaviour of the elitist university. This is intuitive. Since a low proportion of students can afford to move to the developed country in order to attend the top university, the top university has an incentive in opening a BC.

The result changes for Cases [2] and [3] as follows.

Proposition 4 identifies a case ([2]) where it is the non-elitist University B that opens a BC while the top university does not. This is in contrast to the baseline case of Proposition 3. This result is due to the large proportion of privileged students, who can afford to move to the developed country. In this case University A has little incentive in opening a BC.