Competition for the international pool of talent

Appendices

Appendix: A

1.1 Proof of Lemma 1

We start by deriving the equilibrium tuition fees (8) and (9). Country 1 chooses t ₁ to maximise net revenues R ₁ according to Eq. 6, taking t ₂ and quality levels (q ₁,q ₂) as given. The corresponding first-order condition for given Δq>0 is, after some rearrangements,

$$ \frac{\partial R_{1}}{\partial t_{1}}=N\left\{ \frac{\partial\widehat{a} }{\partial t_{1}}\left[ p\tau\left( \underline{w}+\widehat{a}q_{1}\right) +t_{1}-\alpha q_{1}\right] +\widehat{a}\right\} =0\text{,} $$

(21)

from which the best-response function $t_{1}=t_{1}^{br}(t_{2};q_{1},q_{2})$ can be derived:

$$ t_{1}=\theta_{1}t_{2}+\frac{p\tau\underline{w}-c(q_{1})}{\frac{p\tau q_{1} }{\rho{\Delta} q}-2};\quad\theta_{1}:=\frac{\frac{p\tau q_{1}}{\rho{\Delta} q}-1}{\frac{p\tau q_{1}}{\rho{\Delta} q}-2}\text{.} $$

(22)

Using Eq. 7, we can analogously determine the first-order condition for the tuition fee chosen by country 2 and the best-response function $t_{2}=t_{2}^{br}(t_{1};q_{1},q_{2})$:

$$ \frac{\partial R_{2}}{\partial t_{2}}=N\left\{ -\frac{\partial\widehat{a} }{\partial t_{2}}\left[ p\tau\left( \underline{w}+\widehat{a}q_{2}\right) +t_{2}-\alpha q_{2}\right] +\left( 1-\widehat{a}\right) \right\} =0\text{,} $$

(23)

and

$$ t_{2}=\theta_{2}t_{1}+\frac{\rho{\Delta} q+c(q_{2})-p\tau\underline{w}} {\frac{p\tau q_{2}}{\rho{\Delta} q}+2};\quad\theta_{2}:=\frac{\frac{p\tau q_{2}}{\rho{\Delta} q}+1}{\frac{p\tau q_{2}}{\rho{\Delta} q}+2}\text{.} $$

(24)

Combining Eqs. 22 and 24 yields equilibrium tuition fees

$$\begin{array}{@{}rcl@{}} t_{1}^{\ast}(q_{1},q_{2}) & =&\frac{1}{1-\theta_{1}\theta_{2}}\left[ \frac{p\tau\underline{w}-c(q_{1})}{\frac{p\tau q_{1}}{\rho{\Delta} q} -2}+\theta_{1}\frac{\rho{\Delta} q+c(q_{2})-p\tau\underline{w}}{\frac{p\tau q_{2}}{\rho{\Delta} q}+2}\right], \end{array} $$

(25)

$$\begin{array}{@{}rcl@{}} t_{2}^{\ast}(q_{1},q_{2}) & =&\frac{1}{1-\theta_{1}\theta_{2}}\left[ \theta_{2}\frac{p\tau\underline{w}-c(q_{1})}{\frac{p\tau q_{1}}{\rho{\Delta} q}-2}+\frac{\rho{\Delta} q+c(q_{2})-p\tau\underline{w}}{\frac{p\tau q_{2} }{\rho{\Delta} q}+2}\right], \end{array} $$

(26)

which finally can be reduced to Eqs. 8 and 9. These tuition fees are uniquely determined: As the best responses (22) and (24) are continuous functions of the opponent’s tuition fee, and as the corresponding best-response curves have slopes of 𝜃 ₁≤ 1/2 and 𝜃 ₂∈(1/2,1), these curves intersect only once. The second-order condition $\partial ^{2}R_{2}/\partial {t_{2}^{2}}=-N\left (\partial \widehat {a}/\partial t_{2}\right ) \left [ p\tau \left (\partial \widehat {a}/\partial t_{2}\right ) q_{2}+2\right ] <0\Leftrightarrow -p\tau q_{2}-2\rho {\Delta } q<0$ is satisfied for all (q ₁,q ₂); the second-order conditions $\partial ^{2} R_{1}/\partial {t_{1}^{2}}=N\left (\partial \widehat {a}/\partial t_{1}\right ) \left [ p\tau \left (\partial \widehat {a}/\partial t_{1}\right ) q_{1} +2\right ] <0$ is fulfilled for $p\tau q_{1}-2\rho {\Delta } q<0\Leftrightarrow q_{2}>[(p\tau /2\rho )+1]q_{1}$. Under the latter condition, both second-order conditions are satisfied. Then, the second-stage equilibrium is unique and given by the tuition fees (8) and (9) and the allocation of students (11). In particular, this is the case for the quality levels $q_{2}^{\ast }>q_{1}^{\ast }=0$ that follow from conditions (12) and (13) and for quality levels in the larger neighbourhood of $q_{2}^{\ast }$ and $q_{1}^{\ast }$. (We assume p τ+2ρ−α>0 to guarantee $0<\hat {a}^{\ast }<1$ and to exclude boundary solutions, which were not characterised by Eqs. 8 and 9 for $q_{2}^{\ast }>q_{1}^{\ast }=0$. This assumption underlies our arguments above. Further details are provided on request.)

As the second-order condition $d^{2}R_{2}/d{q_{2}^{2}}=-\partial ^{2}F/\partial {q_{2}^{2}}<0$ is fulfilled (see Eq. 13), the solution $(t_{1}^{\ast }(q_{1}^{\ast },q_{2}^{\ast }),t_{2}^{\ast }(q_{1}^{\ast } ,q_{2}^{\ast }),q_{1}^{\ast },q_{2}^{\ast },\hat {a}^{\ast })$ constitutes a local equilibrium if we can additionally prove that both governments achieve positive net benefits, i.e. R _i >0 for $q_{2}^{\ast }>q_{1}^{\ast }=0$. First, we show that net variable rents r _i (q ₁,q ₂) = τ W _i + N _i [t _i −c(q _i )]>0 are strictly positive. Inserting equilibrium values $t_{1}^{\ast }$, $t_{2}^{\ast }$ and $\hat {a}^{\ast }$, which are defined by Eqs. 8, 9 and 11, in the net variable rents (see the corresponding terms in Eqs. 6 and 7) yields, after some rearrangements,

$$\begin{array}{@{}rcl@{}} r_{1}(q_{1},q_{2}) & =&N\left( \frac{\alpha+\rho}{p\tau+3\rho}\right) \left( \frac{\rho\left( 2\rho{\Delta} q-p\tau q_{1}\right) +\alpha\left( 4\rho{\Delta} q-p\tau q_{1}\right) }{2\left( p\tau+3\rho\right) }\right), \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} r_{2}(q_{1},q_{2}) & =&N\left( \frac{p\tau+2\rho-\alpha}{p\tau+3\rho }\right) \left( \frac{\left( p\tau+2\rho-\alpha\right) \left( 2\rho{\Delta} q+p\tau q_{2}\right) }{2\left( p\tau+3\rho\right) }\right) \text{,} \end{array} $$

(28)

which not only shows that net variable rents r ₁ and r ₂ decline with quality q ₁, but more importantly that

$$\begin{array}{@{}rcl@{}} r_{1}(q_{1},q_{2}) & >&0\quad\text{if}\quad p\tau q_{1}-2\rho{\Delta} q<0, \end{array} $$

(29)

$$\begin{array}{@{}rcl@{}} r_{2}(q_{1},q_{2}) &>&0\quad\text{if}\quad\left( \frac{p\tau}{2} q_{2}+\rho{\Delta} q\right) (p\tau+2\rho-\alpha)>0. \end{array} $$

(30)

Thus, net variable rent r ₁(q ₁,q ₂) is strictly positive if the second-order condition $\partial ^{2}R_{1}/\partial {t_{1}^{2}}<0\Leftrightarrow p\tau q_{1}-2\rho {\Delta } q<0$ is satisfied, which is the case for $q_{2}^{\ast }>q_{1}^{\ast }=0$. Similarly, net variable rent r ₂(q ₁,q ₂) is strictly positive, as we assume p τ+2ρ−α>0 (which guarantees that $\hat {a}<1$). With $q_{2}^{\ast }>q_{1}^{\ast }=0$, country 1 generates a strictly positive benefit R ₁>0 (see Eq. 6) because F(0)=0 and $r_{1}(q_{1}^{\ast },q_{2}^{\ast })>0$.

Country 2 also generates a strictly positive rent R ₂, as $\lim \limits _{q_{2}\rightarrow 0}R(0,q_{2})=0$ implies $q_{2}^{\ast }=\arg \max R_{2}(q_{2})>0\Leftrightarrow R_{2}(q_{2}^{\ast })>0$.

The equilibrium allocation of students is $\hat {a}^{\ast }$. From Eq. 3 follows that all individuals with ability $a\geq \hat {a}^{\ast }$ study in the high-quality country 2, while all students with $a<\hat {a}^{\ast }$ study in country 1.

1.2 Proof of non-existence of symmetric equilibrium

To show that a symmetric solution cannot exist, we first analyse tuition fee competition, assuming that the two countries had chosen identical educational qualities q ₁ = q ₂=:q in the first stage. Assume that students who are indifferent between the two countries study in each of the two countries with probability 0.5. For undifferentiated quality levels, the variable net rent then amounts to

$$ r_{i}|_{\Delta q=0}= \left\{\begin{array}{lll} \tau W+N(t_{i}-c(q)) & \text{if}\quad t_{i}<t_{j},\\ \frac{1}{2}[\tau W+N(t_{i}-c(q))] & \text{if}\quad t_{i}=t_{j},\\ 0 & \text{if}\quad t_{i}>t_{j}, \end{array}\right. $$

(31)

where $W=pN{{\int }_{0}^{1}}(\underline {w}+aq)da=pN(\underline {w}+q/2)$. The fixed costs of providing quality are already sunk and therefore irrelevant for tuition fee competition. Countries have an incentive to undercut their competitor in order to attract all foreign students as long as r _i is positive, thereby engaging in a race-to-the-bottom leading to tuition fees $t_{1}=t_{2}=\alpha q-p\tau (\underline {w}+q/2)$ and r _i =0.

This result of the second stage affects, in turn, the overall rent R _i after taking quality competition into account. If the variable net rent in Eqs. 6 and 7 is zero, only the fixed costs remain, i.e. R _i =−F(q), i=1,2. Then educational qualities q ₁ = q ₂>0 cannot constitute an equilibrium. One country could unilaterally deviate and choose, for instance, an educational quality of zero, thereby reducing fixed costs F(q) to zero, while the ensuing tuition fee competition would lead to positive net variable revenues according to Eq. 27. Thus, net benefits of this country would rise from −F(q) to a positive value.

Next, the solution q ₁ = q ₂=0, implying R ₁ = R ₂=0, cannot be an equilibrium either. The reason is that one country, say country 2, can then gain from unilaterally raising its quality to $q_{2}^{\ast }$ and realising positive net revenues R ₂>0, as explored in the proof of Lemma 1.

1.3 Proof of Proposition 1

First of all, note that

$$ \frac{\partial\rho(p)}{\partial p}=(1-\tau)-(1-\tau_{ROW})\gamma\geq0 $$

(32)

can be signed unambiguously by Assumption 1. This finding can be used to get, from Eqs. 1 and 11,

$$ \frac{d\hat{a}^{\ast}}{dp}=-\frac{\tau\left[(1-\tau_{\text{\tiny{ROW}}}) \gamma+\alpha\right]+3\alpha\frac{\partial\rho}{\partial p}}{(p\tau +3\rho)^{2}}<0\text{,} $$

(33)

and thus $\partial (1-\hat {a}^{\ast })/\partial p>0$. This proves part (iii) of Proposition 1.

Now, part (i) follows from

$$ \frac{d{\Delta} q^{\ast}}{dp}\gtreqqless0\quad\overset{(q_{1}^{\ast} =0)}{\Leftrightarrow}\quad\frac{dq_{2}^{\ast}}{dp}=-\frac{d^{2}R_{2}/\left( dq_{2}dp\right) }{d^{2}R_{2}/d{q_{2}^{2}}}\gtreqqless0 $$

(34)

$$ \overset{(13)}{\Leftrightarrow}\quad\frac{d^{2}R_{2}} {dq_{2}dp}=\frac{d\left[ \frac{N}{2}(p\tau+2\rho)(1-\hat{a}^{\ast} )^{2}\right] }{dp}\gtreqqless0. $$

(35)

Further, we can show that

$$ \frac{d^{2}R_{2}}{dq_{2}dp}=\frac{N(1-\hat{a}^{\ast})}{2}\left[ (\tau +2\frac{\partial\rho}{\partial p})(1-\hat{a}^{\ast})-2(p\tau+2\rho )\frac{\partial\hat{a}^{\ast}}{\partial p}\right] >0\text{,} $$

(36)

where the last inequality results from ∂ ρ/∂ p≥0 (see Eq. 32), $\partial \hat {a}^{\ast }/\partial p<0$ (see Eq. 33) and $1-\hat {a}^{\ast }>0$. That means that from Eq. 36 follows that the sign of dΔq ^∗/d p is positive. Thus, $dq_{2}^{\ast }/dp>0$ also results, as $q_{1}^{\ast }=0$ irrespective of p.

Considering the tuition fee differential (10), we get

$$\begin{array}{@{}rcl@{}} \frac{d{\Delta} t^{\ast}}{dp} & =&{\Omega}\frac{d{\Delta} q^{\ast}}{dp}+{\Delta} q^{\ast}\left\{ \frac{d{\Omega}}{dp}+\frac{\partial{\Omega}}{\partial\rho }\frac{\partial\rho}{\partial p}\right\} \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} & =&{\Omega}\frac{\Delta q^{\ast}}{p}\left\{ \underbrace{\frac{d{\Delta q^{\ast}}}{dp}\frac{p}{{\Delta q^{\ast}}}}_{:=\varepsilon_{\Delta q,p} >0}+\underbrace{\frac{\partial{\Omega}}{\partial p}\frac{p}{\Omega} }_{:=\varepsilon_{\Omega,p}<0}+\underbrace{\frac{\partial{\Omega}} {\partial\rho}\frac{\rho}{\Omega}}_{:=\varepsilon_{\Omega,\rho} >0}\cdot\underbrace{\frac{\partial\rho}{\partial p}\frac{p}{\rho} }_{:=\varepsilon_{\rho,p}\geq0}\right\} \text{,} \end{array} $$

(38)

where ε _Δq,p :=(dΔq/d p)(p/Δq)>0, ε _Ω,p :=(∂Ω/∂ p)(p/Ω)<0, and ε _ρ,p :=(∂ ρ/∂ p)(p/ρ)≥0 follow directly from Eqs. 34–36, 10, and 32. Furthermore, ε _Ω,ρ =(∂Ω/∂ ρ)(ρ/Ω)>0 results from

$$ \frac{\partial{\Omega}}{\partial\rho}=\frac{p\tau\left( \alpha +2\rho\right) +3\rho^{2}}{\left( p\tau+3\rho\right)^{2} }>0\text{.} $$

(39)

The derivative (38) directly implies (14), which is stated in part (ii) of Proposition 1. Furthermore, for q ₁=0, we get $t_{2}^{\ast }=2t_{1}^{\ast }+p\tau \underline {w}$ and $t_{1}^{\ast }=(1/2)(t_{2}^{\ast }-p\tau \underline {w})$ (see Eqs. 8 and 9), leading to ${\Delta } t^{\ast }=t_{2}^{\ast }-t_{1}^{\ast }=t_{1}^{\ast }+p\tau \underline {w}=(1/2)(t_{2}^{\ast } +p\tau \underline {w})$ and thus $d{\Delta } t^{\ast }/dp=dt_{1}^{\ast } /dp+\tau \underline {w}=(1/2)(dt_{2}^{\ast }/dp+\tau \underline {w})$. Then, dΔt ^∗/d p<0 implies $dt_{1}^{\ast }/dp<0$ and $dt_{2}^{\ast }/dp<0$. However, $dt_{1}^{\ast }/dp>0$ or $dt_{2}^{\ast }/dp>0$ is possible if dΔt ^∗/d p>0. To summarise, the impact of the stay rate p on the tuition fee differential and the tuition fees ${t_{1}^{P}}$ and ${t_{2}^{P}}$ is ambiguous.

1.4 Proof of Proposition 2

Part (i) of Proposition 2 follows from

$$ \frac{d{\Delta} q^{\ast}}{d\gamma}\gtreqqless0\quad\overset{(q_{1}^{\ast} =0)}{\Leftrightarrow}\quad\frac{dq_{2}^{\ast}}{d\gamma}=-\frac{d^{2} R_{2}/\left( dqd\gamma\right) }{d^{2}R_{2}/d{q_{2}^{2}}}\gtreqqless0 $$

(40)

$$ \overset{(13)}{\Leftrightarrow}\quad\frac{d^{2}R_{2} }{dqd\gamma}=\frac{d\left[ \frac{N}{2}(p\tau+2\rho)(1-\hat{a}^{\ast} )^{2}\right] }{d\rho}\frac{d\rho}{d\gamma}\gtreqqless0 $$

(41)

and, using Eq. 11 and the fact that $d\rho /d\gamma =\left (1-p\right ) \left (1-\tau _{ROW}\right ) >0$ (see Eq. 1),

$$ \frac{d^{2}R_{2}}{dqd\gamma}>0\quad\Leftrightarrow\quad3\rho p\tau +6\rho^{2}+2\alpha p\tau+3\alpha\rho>0\text{,} $$

(42)

where the last inequality always holds. Thus, dΔq ^∗/d p>0, implying $dq_{2}^{\ast }/dp>0$ because $q_{1}^{\ast }=0$ irrespective of p.

Using Eq. 10, part (ii) of Proposition 2 follows from

$$ \frac{d{\Delta} t^{\ast}}{d\gamma}=\frac{\partial{\Delta} t^{\ast}}{\partial \rho}\cdot\frac{\partial\rho}{\partial\gamma}+\frac{\partial{\Delta} t^{\ast}}{\partial{\Delta} q^{\ast}}\cdot\frac{d{\Delta} q^{\ast}}{d\gamma}>0, $$

(43)

where ∂Δt ^∗/∂Δq ^∗=Ω(p,ρ)>0 (see Eq. 10), dΔq ^∗/d γ>0 (see Eqs. 40–42), ∂ ρ/∂ γ=(1−p)(1−τ _{R O W} )>0 (see Eq. 1), and

$$ \frac{\partial{\Delta} t^{\ast}}{\partial\rho}=\frac{p\tau(\alpha +2\rho)+3\rho^{2}}{(p\tau+3\rho)^{2}}{\Delta} q^{\ast}>0 $$

(44)

(see Eqs. 10 and 39). Moreover, dΔt ^∗/d γ>0 implies $dt_{1}^{\ast }/d\gamma >0$ and $dt_{2}^{\ast }/d\gamma >0$, since ${\Delta } t^{\ast }=t_{2}^{\ast }-t_{1}^{\ast }=t_{1}^{\ast }+p\tau \underline {w}=(1/2)(t_{2}^{\ast }+p\tau \underline {w})$ (see proof of Proposition 1) and thus $d{\Delta } t^{\ast }/d\gamma =dt_{1}^{\ast }/d\gamma =(1/2)dt_{2}^{\ast }/d\gamma$.

Also, using Eqs. 1 and 11 yields

$$ \frac{d(1-\hat{a}^{\ast})}{d\gamma}\gtreqqless0\quad\Leftrightarrow\quad -\frac{\partial\hat{a}^{\ast}}{d\rho}\frac{\partial\rho}{\partial\gamma }\gtreqqless0\quad\Leftrightarrow\quad3\alpha-p\tau\gtreqqless0\text{,} $$

(45)

which proves part (iii) of Proposition 2.

1.5 Proof Proposition 3

Applying comparative statics again yields

$$ \frac{\partial{\Delta} q^{\ast}}{\partial N}\gtreqqless0\quad\overset {(q_{1}^{\ast}=0)}{\Leftrightarrow}\quad\frac{\partial q_{2}^{\ast}}{\partial N}\gtreqqless0\quad\overset{(13)}{\Leftrightarrow}\quad \frac{\partial}{\partial N}\left[ \frac{N}{2}(p\tau+2\rho)(1-\hat{a} ^{\ast})^{2}\right] \gtreqqless0\text{,} $$

(46)

with

$$ \frac{\partial}{\partial N}\left[ \frac{N}{2}(p\tau+2\rho)(1-\hat{a}^{\ast})^{2}\right] =\frac{1}{2}(p\tau+2\rho)(1-\hat{a}^{\ast} )^{2}>0\text{.} $$

(47)

Thus, dΔq ^∗/d N>0 and, as $q_{1}^{\ast }=0$ irrespective of p, $dq_{2}^{\ast }/dN>0$, as stated in part (i) of Proposition 3.

Part (ii) follows from

$$ \frac{d{\Delta} t^{\ast}}{dN}=\frac{\partial{\Delta} t^{\ast}}{\partial{\Delta} q^{\ast}}\cdot\frac{\partial{\Delta} q^{\ast}}{\partial N}>0\text{,} $$

(48)

where ∂Δt ^∗/∂Δq ^∗=Ω(p,ρ)>0 (see Eq. 10) and ∂Δq ^∗/∂ N>0 (see Eqs. 46 and 47). In line with our previous reasoning, dΔt ^∗/d N>0 implies $dt_{i}^{\ast }/dN>0$, since ${\Delta } t^{\ast }=t_{2}^{\ast }-t_{1}^{\ast }=t_{1}^{\ast }+p\tau \underline {w}=(1/2)(t_{2}^{\ast }+p\tau \underline {w})$ and thus $d{\Delta } t^{\ast }/dN=dt_{1}^{\ast }/dN=(1/2)dt_{2}^{\ast }/dN$.

Part (iii) follows directly from Eq. 11, as a ^∗ is independent of N.

Appendix: B

1.1 Competition between private universities (Section 4.4)

Let us start by analysing the competition between two private universities for the international pool of talent. As indicated in the paper, we explore a slightly more general model in this appendix than the one informally discussed in Section 4.4 of the paper. As in this section of the paper, the private universities are not interested in how their decisions affect future tax payments, but in contrast to this section, they now receive a prestige benefit from well-educated alumni. This additional benefit enters their objective function

$$ R_{i}^{P}=\lambda A_{i}+N_{i}[t_{i}-c(q_{i})]-F(q_{i})\text{,} $$

(49)

where λ≥0 captures the weight assigned to the prestige benefit, A _i stands for the aggregate human capital generated by university i, and the superscript P indicates the scenario with private competition. In the paper, we discuss the special case of λ=0. As we will see, considering the more general case of λ≥0 does not alter the results informally discussed in the paper.

1.1.1 Quality and tuition fee competition revisited 1

The threshold value (3) still characterises the demand of the international students for the two universities. Thus, the two universities generate human capital $A_{1}=N{\int }_{0}^{\hat {a}}aq_{1} da=\frac {1}{2}\hat {a}^{2}q_{1}N$ and $A_{2}=N{\int }_{\hat {a}}^{1}aq_{2} da=\frac {1}{2}\left (1-\hat {a}^{2}\right ) q_{1}N$, respectively. Using these terms to rearrange objective function (49) yields

$$ R_{1}^{P}=\hat{a}N\left\{ \frac{1}{2}\lambda\hat{a}q_{1}+t_{1}-c(q_{1} )\right\} -F(q_{1})\text{,} $$

(50)

$$ R_{2}^{P}=\left( 1-\hat{a}\right) N\left\{ \frac{1}{2}\lambda\left( 1+\hat{a}\right) q_{2}+t_{2}-c(q_{2})\right\} -F(q_{2})\text{.} $$

(51)

The differences between these objective functions and their counterparts (6) and (7) are that the tax revenue term τ p is replaced with the prestige term λ, and that the wage component $\underline {w}$ is not included (as future tax revenues are now not taken into account). These differences will become important when we discuss the impact of changes in the socioeconomic environment on education policy and student allocation.

First, we analyse the tuition fee competition in the second stage under the new circumstances. The universities choose their tuition fees t ₁ and t ₂, respectively, to maximise their objectives $R_{1}^{P}$ and $R_{2}^{P}$, taking the quality levels (q ₁,q ₂) and their rival’s tuition fee as given. Using threshold value (3) and differentiating objective functions (50) and (51), we get

$$ \frac{dR_{1}^{P}}{dt_{1}}=N\left[ \frac{d\hat{a}}{dt_{1}}\left( \lambda \hat{a}q_{1}+t_{1}-\alpha q_{1}\right) +\hat{a}\right] =0\Leftrightarrow t_{1}^{P}=\hat{a}\rho(q_{2}-q_{1})-\lambda\hat{a}q_{1}+\alpha q_{1} $$

(52)

and

$$\begin{array}{@{}rcl@{}} \frac{dR_{2}^{P}}{dt_{2}} &=&N\left[ -\frac{d\hat{a}}{dt_{2}}\left( \lambda\hat{a}q_{2}+t_{2}-\alpha q_{2}\right) +\left( 1-\hat{a}\right) \right] =0\\ &\Leftrightarrow& t_{2}^{P}=(1-\hat{a})\rho(q_{2}-q_{1})-\lambda\hat {a}q_{2}+\alpha q_{2}\text{.} \end{array} $$

(53)

Then, using Eqs. 52 and 53 gives us the threshold level

$$ \hat{a}^{P}=\frac{\alpha+\rho}{\lambda+3\rho} $$

(54)

and the tuition fee differential

$$ {\Delta} t^{P}\left( {\Delta} q\right) :=t_{2}^{P}-t_{1}^{P}={\Omega}^{P} (p,\rho){\Delta} q\text{,\quad where\quad}{\Omega}^{P}(p,\rho)=\frac {\rho\left( \alpha+\rho\right)}{\lambda+3\rho}\text{.} $$

(55)

In the following, we assume α<λ+2ρ to guarantee $\hat {a}^{P}<1$. The corresponding tuition fees are given by

$$\begin{array}{@{}rcl@{}} t_{1}^{P}\left( q_{1},q_{2}\right) & =&\frac{\rho\lbrack\rho{\Delta} q-\lambda q_{1}+\alpha(q_{2}+2q_{1})]}{\lambda+3\rho}\text{,} \end{array} $$

(56)

$$\begin{array}{@{}rcl@{}} t_{2}^{P}\left( q_{1},q_{2}\right) & =&\frac{\rho\lbrack2\rho{\Delta} q-\lambda q_{1}+\alpha(q_{1}+2q_{2})]}{\lambda+3\rho}\text{.} \end{array} $$

(57)

Second, we explore quality competition in the first stage. Using the solutions (54), (56) and (57), we get the derivative

$$ \frac{dR_{1}}{dq_{1}}=\hat{a}^{P}N\left[ \frac{\lambda}{2}\hat{a}^{P} +\frac{\partial t_{1}^{P}}{\partial q_{1}}-\alpha\right] -\frac{\partial F}{\partial q_{1}}=-\frac{N}{2}(\lambda+2\rho)(\hat{a}^{P})^{2} -\frac{\partial F}{\partial q_{1}}<0\text{,} $$

(58)

which implies $q_{1}^{P}=0$, and the first-order condition

$$\begin{array}{@{}rcl@{}} \frac{dR_{2}}{dq_{2}} & =&(1-\hat{a}^{P})N\left[ \frac{\lambda}{2}(1+\hat {a}^{P})+\frac{\partial t_{2}^{P}}{\partial q_{2}}-\alpha\right] -\frac{\partial F}{\partial q_{1}}\\ & =&\frac{N}{2}(\lambda+2\rho)(1-\hat{a}^{P})^{2}-\frac{\partial F}{\partial q_{2}}=0\text{,} \end{array} $$

(59)

which implicitly determines the optimal quality level $q_{2}^{P}>0$.

In line with the solution in the case of the basic model, the threshold level (54), the tuition fees (56) and (57), the first-order condition (59) and the quality level $q_{1}^{P}=0$ jointly characterise a local equilibrium with $q_{2}^{P}>q_{1}^{P}=0$ and $t_{2}^{P}>t_{1}^{P}$. In this local equilibrium, the high-quality university attracts the brightest students, i.e. those with $a\in \left [ \hat {a}^{P},1\right ]$, whereas the low-quality university takes the other students (see Lemma 1 in the paper for comparison). Note that when competition is between private universities, the outcome is independent of whether the universities are located in two different countries or the same country.

Comparing the corresponding local equilibria in the cases of private and public competition reveals similarities and differences. Firstly, the low-quality university sets its quality level equal to zero in both cases for the very same reason. It differentiates its quality level as much as possible from the quality level of its competitor to soften tuition fee competition.

Secondly, the high educational quality is higher (lower) under private competition than under public competition if and only if the prestige parameter λ is greater (smaller) than the average tax term p τ. Also, a higher (lower) quality q ₂ goes hand in hand with a higher (lower) share of students who are enrolled at the high-quality university. These conclusions follow from comparing the first-order conditions (59) and (13) and the threshold levels (54) and (11), which reveals that $q_{2}^{P}\gtreqless q_{2}^{\ast }\Leftrightarrow \lambda \gtreqless p\tau \Leftrightarrow 1-a_{2}^{P}\gtreqless 1-a_{2}^{\ast }$.

Thirdly, tuition fees and the tuition fee differential can be greater or smaller under private competition. However, if the high educational quality is the same under private and public competition (i.e. if λ = p τ), then the tuition fees of both universities are higher under private competition than under public competition while the tuition fee differential is the same under the two forms of competition. This conclusion follows from comparing tuition fees (8), (9), (56) and (57) and tuition fee differentials (10) and (55), which reveals that, for λ = p τ, $t_{i}^{P}=t_{i}^{\ast }+p\tau \underline {w}>t_{i}^{\ast }$ and thus ${\Delta } t_{i}^{P}={\Delta } t_{i}^{\ast }$.

In the special case of λ=0, which is discussed in the paper, the universities assign no weight to the prestige benefit and consider international students only as an additional source of net revenues. Then, as private universities ignore the fiscal externalities of international students, the education quality q ₂ and the quality differential Δq are definitely lower under private competition than under public competition. The tuition fees of the two universities and the tuition fee differential might be higher or lower. In any case, fewer students are then enrolled at the high-quality university.

1.1.2 Competition in a changing environment revisited 1

First, let us analyse how a marginal change in the stay rate p affects the local equilibrium. Differentiating equilibrium threshold (54) gives

$$ \frac{d\hat{a}^{P}}{dp}=\frac{\partial\hat{a}}{\partial\rho}\frac {\partial\rho}{\partial p}=\frac{\lambda-3\alpha}{\left( \lambda +3\rho\right) ^{2}}\underbrace{\frac{\partial\rho}{\partial p}}_{\geq 0}\gtreqless0\quad\Leftrightarrow\quad\lambda-3\alpha\gtreqless0\text{ if }\frac{\partial\rho}{\partial p}>0 $$

(60)

and thus $\partial (1-\hat {a}^{P})/\partial p\gtreqless 0\Leftrightarrow 3\alpha -\lambda \gtreqless 0$ if ∂ ρ/∂ p>0, where d ρ/d p≥0 follows from (32). In contrast to the case of λ=0, which is discussed in the paper, and in contrast to the case of public competition, the share $(1-\hat {a}^{P})$ of foreign students who study at the high-quality university can even decline with the stay rate p if the prestige parameter λ is sufficiently large.

Next, we show that quality $q_{2}^{P}$ and the quality differential Δq ^P increase with the stay rate p. This conclusion follows from

$$ \frac{d{\Delta} q^{P}}{dp}\geq0\quad\overset{(q_{1}^{P}=0)}{\Leftrightarrow }\quad\frac{dq_{2}^{P}}{dp}=-\frac{d^{2}R_{2}^{P}/dq_{2}^{P}dp}{d^{2}R_{2}^{2}/d(q_{2}^{P})^{2}}\geq0\quad\Leftrightarrow\quad $$

(61)

$$\begin{array}{@{}rcl@{}} \frac{d^{2}R_{2}^{P}}{dq_{2}^{P}dp} &=&(1-\hat{a}^{P})N\left[ \left( 1-\hat{a}^{P}\right) \frac{\partial\rho}{\partial p}-\left( \lambda+2\rho\right) \frac{d\hat{a}^{P}}{dp}\right] \\ &=&(1-\hat{a}^{P})N\left[ \frac{3\rho\left( \lambda+2\rho+\alpha\right) +2\alpha\rho}{\left( \lambda+3\rho\right)^{2}}\right] \frac{\partial \rho}{\partial p}\geq0\text{,} \end{array} $$

(62)

where we used (59), (60) and ∂ ρ/∂ p≥0 (see Eq. 32). For τ = τ _{R O W} and γ=1, ∂ ρ/∂ p=0 results and the quality q ₂ does not alter in response to changes in the stay rate q ₂. Obviously, quality $q_{1}^{P}=0$ is unaffected by any socioeconomic changes.

Differentiating the tuition fee differential (55) with respect to the stay rate yields

$$\begin{array}{@{}rcl@{}} \frac{d{\Delta} t^{P}}{dp} &=&{\Omega}^{P}\frac{dq_{2}^{P}}{dp}+{\Delta} q^{P}\frac{\partial{\Omega}^{P}}{\partial\rho}\frac{\partial\rho}{\partial p}\\ &=&{\Omega}^{P}\frac{q_{2}^{P}}{p}\left\{ \underbrace{\frac{d{\Delta} q^{P}} {dp}\frac{p}{\Delta q^{P}}}_{=:\varepsilon_{\Delta q,p}\geq0}+\underbrace {\frac{\partial{\Omega}^{P}}{\partial\rho}\frac{\rho}{{\Omega}^{P}}}_{=:\varepsilon_{{\Omega}^{P},p}>0}\cdot\underbrace{\frac{\partial\rho }{\partial p}\frac{p}{\rho}}_{=:\varepsilon_{\rho,p}\geq0}\right\} \geq0. \end{array} $$

(63)

where the last inequality follows from $dq_{2}^{P}/dp\geq 0$ (see Eq. 61), ∂ ρ/∂ p≥0 (see Eq. 32) and

$$ \frac{\partial{\Omega}^{P}}{\partial\rho}=\frac{\lambda\left( \alpha +2\rho\right) +3\rho^{2}}{\left( \lambda+3\rho\right)^{2} }>0\text{.} $$

(64)

Furthermore, for q ₁=0, we get $t_{2}^{P}=2t_{1}^{P}$, leading to ${\Delta } t^{P}=t_{2}^{P}-t_{1}^{P}=t_{1}^{P}$. Then, $dt_{1}^{P}/dp=d{\Delta } t^{P}/dp\geq 0$ and, as $t_{2}^{P}=2t_{1}^{P}$, $dt_{2}^{P}/dp=2d{\Delta } t^{P}/dp\geq 0$. That is, the tuition fees $t_{1}^{P}$ and $t_{2}^{P}$ and the tuition fee differential increase with the stay rate p.

Second, we analyse how a marginal rise in the income in the rest of the world, i.e. a marginal rise in γ, affects the local equilibrium. A marginal increase in the ROW parameter γ has an ambiguous effect on the share $(1-\hat {a}^{P})$ of students who study at the high-quality university:

$$ \frac{d\hat{a}^{P}}{d\gamma}=\frac{\partial\hat{a}^{P}}{\partial\rho} \frac{\partial\rho}{\partial\gamma}=\frac{\lambda-3\alpha}{\left( \lambda+3\rho\right)^{2}}\left( 1-p\right) \left( 1-\tau_{ROW}\right) \gtreqless0\quad\Leftrightarrow\quad\lambda-3\alpha\gtreqless0 $$

(65)

and thus $\partial (1-\hat {a}^{P})/\partial p\gtreqless 0\Leftrightarrow 3\alpha -\lambda \gtreqless 0$, where we used ∂ ρ/∂ γ=(1−p)(1−τ _{R O W} )>0 (see Eq. 1). However, in the special case of λ=0, a higher income in ROW leads to a (weakly) higher share of students who are enrolled at the high-quality university.

Next, following the same line reasoning as before, we can show that quality q ₂ and the quality differential Δq ^P increase with the ROW income parameter γ. This conclusion follows from

$$ \frac{d{\Delta} q^{P}}{d\gamma}>0\quad\overset{(q_{1}^{P}=0)}{\Leftrightarrow }\quad\frac{dq_{2}^{P}}{d\gamma}=-\frac{d^{2}R_{2}^{P}/dq_{2}^{P}d\gamma }{d^{2}R_{2}^{2}/d(q_{2}^{P})^{2}}>0\quad\Leftrightarrow\quad $$

(66)

$$\begin{array}{@{}rcl@{}} \frac{d^{2}R_{2}^{P}}{dq_{2}^{P}d\gamma} &=&(1-\hat{a}^{P})N\left[ \left( 1-\hat{a}^{P}\right) \frac{\partial\rho}{\partial\gamma}-\left( \lambda+2\rho\right) \frac{d\hat{a}^{P}}{d\gamma}\right] \\ &=&(1-\hat{a}^{P})N\left[ \frac{3\rho\left( \lambda+2\rho+\alpha\right) +2\alpha\rho}{\left( \lambda+3\rho\right)^{2}}\right] \frac{\partial \rho}{\partial\gamma}>0\text{,} \end{array} $$

(67)

where we used Eqs. 59, 65 and again ∂ ρ/∂ γ=(1−p)(1−τ _{R O W} )>0 (see Eq. 1).

Furthermore, the tuition fee differential Δt ^P increases with the ROW income parameter γ, since

$$ \frac{d{\Delta} t^{P}}{d\gamma}={\Omega}^{P}\frac{dq_{2}^{P}}{d\gamma}+q_{2}^{P}\frac{\partial{\Omega}^{P}}{\partial\rho}\frac{\partial\rho} {\partial\gamma}>0\text{,} $$

(68)

where the last inequality follows from $dq_{2}^{P}/d\gamma >0$ (see Eq. 66), ∂ ρ/∂ γ=(1−p)(1−τ _{R O W} )>0 (see Eq. 1) and ∂Ω^P/∂ ρ>0 (see Eq. 64). Using again $t_{2}^{P}=2t_{1}^{P}$ and thus ${\Delta } t^{P}=t_{1}^{P}$, we can conclude that $dt_{2}^{P}/d\gamma =2dt_{1}^{P}/d\gamma =2d{\Delta } t^{P}/d\gamma >0$. That is, the tuition fees $t_{1}^{P}$ and $t_{2}^{P}$ also increase with the ROW income parameter γ.

Finally, we analyse how an increase in the size of the international pool of talent affects the local equilibrium. Inspection of Eq. 59 reveals that

$$ \frac{dq_{2}^{P}}{dN}\!>\!0\!\Leftrightarrow\!\frac{d^{2}R_{2}^{P}}{dq_{2}^{P} dN}=\frac{\partial}{\partial N}\left[ \frac{N}{2}\left( \lambda +2\rho\right) \left( \!1-\hat{a}^{P}\!\right)^{2}\right] =\frac{1} {2}\left( \lambda+2\rho\right) \left( \!1-\hat{a}^{P}\!\right) ^{2}\!>\!0\text{,} $$

(69)

where the last inequality is obviously satisfied.

Furthermore, $dq_{2}^{P}/dN>0$ (see Eq. 69), ∂Δt ^P/∂Δq ^P=Ω^P>0 (see Eq. 55) and dΩ^P/d N=0 (see again Eq. 55) imply that $d{\Delta } t^{P}/dN=(\partial {\Delta } t^{P}/\partial {\Delta } q^{P})(d{\Delta } q^{P}/dN)={\Omega }^{P}(dq_{2}^{P}/dN)>0$. Using again $t_{2}^{P}=2t_{1}^{P}$ and thus ${\Delta } t^{P}=t_{1}^{P}$, we get $dt_{2}^{P}/dN=2dt_{1}^{P}/dN=2d{\Delta } t^{P}/dN>0$. Thus, the tuition fees $t_{1}^{P}$ and $t_{2}^{P}$, the tuition fee differential Δt ^P, the education quality $q_{2}^{P}$ and the quality differential Δq ^P all increase with the cohort size parameter N. As Eq. 54 is independent of N, the allocation of students, i.e. $\hat {a}^{P}$, is unaffected by changes in the size of the pool of talent.

1.2 Domestic competition (Section 4.4)

As discussed in Section 4.4 of the paper, many international students might have regional preferences and prefer studying in one country over doing so in another for a variety of reasons, including cultural preferences, language skills and personal relationships. In this case, universities within a country are in fiercer competition with each other than universities in different countries. Also, public universities compete with private universities in many countries, and governments find it easier to influence decisions of public universities. We take up these issues by analysing now domestic competition between a public university and a private one, assuming that N international students intend to study in the country where the two universities are located. While the government can control the education quality and tuition fee of the public university, it has no sway over the decisions of the private university. Otherwise, the universities compete in education quality and tuition fees as they did in the previous sections. This analysis complements our preceding analysis of international competition.

In principle, there are two possible equilibria, one in which the public university provides a higher education quality and one in which the private university does so (unless the objective functions of the two different universities are too ‘asymmetric’, in which case only one of the two equilibria might exist). We focus now on the case in which the quality of education is higher at the public university than at the private rival. Adjusting the objective functions of the public and the private university to this new setting, we obtain

$$ R_{1}^{P}=\hat{a}N\left[ t_{1}-c(q_{1})\right] -F(q_{1})\text{,} $$

(70)

$$ R_{2}^{S}=\left( \!1-\hat{a}\!\right) N\left\{ \tau p\left[ \underline{\!w}+\frac{1}{2}\left( \!1+\hat{a}\!\right) q_{2}\!\right] \!+t_{2}-c(q_{2} )\!\right\} +\hat{a}N\tau p\left( \underline{\!w}+\frac{1}{2}\hat{a} q_{1}\!\right) -F(q_{2})\text{,} $$

(71)

where the superscripts P and S stand for ‘private’ and ‘state’ (i.e. public). For simplicity, we set λ=0, which reduces the objective function of the private university to Eq. 70. Objective function (71) contains all expected tax revenues that will arise in this country, irrespective of whether the tax payer graduated at the public or private university. However, it only includes the tuition fee revenues and education costs of the public university. Thus, the government aims at maximising the contribution of international students to the public budget (including the budget of the public university).

1.2.1 Quality and tuition fee competition revisited 2

Let us again first consider the tuition fee competition in the second stage, taking into account that the threshold value (3) still determines the demand of the international pool of talent for the two universities. The universities set their tuition fees simultaneously and non-cooperatively, leading to the first-order conditions

$$ \frac{\partial R_{1}^{P}}{\partial t_{1}}=N\left[ \frac{\partial\hat{a}}{\partial t_{1}}(t_{1}-\alpha q_{1})+\hat{a}\right] =0\quad\Leftrightarrow \quad t_{1}^{PS}=\rho(q_{2}-q_{1})\hat{a}+\alpha q_{1}, $$

(72)

and

$$\begin{array}{@{}rcl@{}} \frac{\partial R_{2}^{S}}{\partial t_{2}} & =&N\left\{ -\frac{\partial \hat{a}}{\partial t_{2}}\left[ p\tau\hat{a}(q_{2}-q_{1})+t_{2}-\alpha q_{2}\right] +(1-\hat{a})\right\} =0\\ &\Leftrightarrow&\quad t_{2}^{PS}=\left[ \rho(1-\hat{a})-p\tau\hat {a}\right] (q_{2}-q_{1})+\alpha q_{2}\text{,} \end{array} $$

(73)

where the superscript PS stands for variables in the case of domestic competition between a private and a public (state) university. Inserting threshold value (3) into Eqs. 72 and 73 and rearranging the resulting terms yield the threshold level

$$ \hat{a}^{PS}=\frac{\alpha+\rho}{p\tau+3\rho}\text{,} $$

(74)

and the tuition fee differential

$$ {\Delta} t^{PS}:=t_{2}^{PS}-t_{1}^{PS}={\Omega}^{PS}(p,\rho){\Delta} q\text{,\quad where\quad}{\Omega}^{PS}(p,\rho)=\frac{\rho(\alpha+\rho)}{p\tau+3\rho }\text{.} $$

(75)

The corresponding tuition fees are given by

$$\begin{array}{@{}rcl@{}} t_{1}^{PS}\left( q_{1},q_{2}\right) & =&\frac{\rho\lbrack\rho{\Delta} q+\alpha(q_{2}+2q_{1})]+\alpha p\tau q_{1}}{p\tau+3\rho}\text{,} \end{array} $$

(76)

$$\begin{array}{@{}rcl@{}} t_{2}^{PS}\left( q_{1},q_{2}\right) & =&\frac{\rho\lbrack2\rho{\Delta} q+\alpha(q_{1}+2q_{2})]+\alpha p\tau q_{1}}{p\tau+3\rho}\text{.} \end{array} $$

(77)

Second, we explore the quality competition in the first stage. Using the solutions (74), (76), and (77), we can calculate the derivatives

$$ \frac{\partial R_{1}^{P}}{\partial q_{1}}=\hat{a}^{PS}N\left( \frac{\partial t_{1}^{PS}}{\partial q_{1}}-\alpha\right) -\frac{\partial F}{\partial q_{1} }=-N\rho(\hat{a}^{PS})^{2}-\frac{\partial F}{\partial q_{1}}<0\text{,} $$

(78)

implying the optimal quality level $q_{1}^{PS}=0$, and

$$\begin{array}{@{}rcl@{}} \frac{\partial R_{2}^{S}}{\partial q_{2}} & =&(1-\hat{a}^{PS})N\left( \frac{p\tau}{2}(1+\hat{a}^{PS})+\frac{\partial t_{2}^{PS}}{\partial q_{2} }-\alpha\right) -\frac{\partial F}{\partial q_{2}}\\ & =&\frac{N}{2}(p\tau+2\rho)(1-\hat{a}^{PS})^{2}-\frac{\partial F}{\partial q_{2}}=0, \end{array} $$

(79)

implicitly determining the quality level $q_{2}^{PS}>0$.

Jointly, the threshold level (74), the tuition fees (76) and (77), the first-order condition (79) and the quality level $q_{1}^{PS}=0$ determine a local equilibrium with $q_{2}^{PS}>q_{1}^{PS}=0$ and $t_{2}^{PS}>t_{1}^{PS}$. In the scenario in which the public university provides the high-quality education, it attracts the brightest students, i.e. those with $a\in \left [ \hat {a}^{PS},1\right ]$, whereas the private university takes the other students.

Comparing these equilibrium values with those in the case of international public competition reveals astonishing similarities and a few differences (see Eqs. 8–13). The levels of education quality, the quality and tuition fee differentials and the threshold level $\hat {a}^{PS}$ are the same as the ones in the case of international public competition. That is, $q_{1}^{PS}=q_{1}^{\ast }$, $q_{2}^{PS}=q_{2}^{\ast }$, Δq ^PS=Δq ^∗, Δt ^PS=Δt ^∗ and $\hat {a} ^{PS}=\hat {a}^{\ast }$. Both tuition fees $t_{1}^{PS}$ and $t_{2}^{PS}$ exceed their counterparts $t_{1}^{\ast }$ and $t_{2}^{\ast }$ by $p\tau \underline {w}$. That is, $t_{1}^{PS}=\left [ \rho (\alpha +\rho )q_{2}^{PS}\right ] /\left (p\tau +3\rho \right ) =t_{1}^{\ast }+p\tau \underline {w}$ and $t_{2}^{PS}=\left [ 2\rho (\alpha +\rho )q_{2}^{PS}\right ] /\left (p\tau +3\rho \right ) =t_{2}^{\ast }+p\tau \underline {w}$.

The intuition for this outcome is as follows: Compared to the situation under international public competition, the negative effects of raising the tuition fee of the public university are cushioned. Now, students who switch from the public to the private university to avoid such higher fees still stay in the country and end up as domestic tax payers with probability p. Their tax payments will be lower if the private university provides a lower education quality, but they will not vanish altogether. As a result, the public university charges a higher tuition fee, and the private university follows suit, since the private university (ignoring the fiscal benefits generated by international students) anyway tends to charge higher tuition fees, and since tuition fees are strategic complements. This evenly increases the tuition fees without affecting the tuition fee differential and the allocation of the students across the two universities.

How does domestic competition change the incentive for the public university to invest in education quality? On the one hand, attracting another student means that this student pays $p\tau \underline {w}$ more in tuition fees under domestic competition than under international public competition. This strengthens the incentive to raise education quality. On the other hand, the gain in terms of additional tax payments is $p\tau \underline {w}$ lower under domestic competition, as this is the expected amount the students pay in taxes even if they graduate at the private university with quality level q1P S=0. As the two opposing effects cancel each other out exactly, the net incentives to invest in education quality are the same under domestic and international competition, and so are thus the equilibrium quality levels $q_{2}^{PS}$ and $q_{2}^{\ast }$.

1.2.2 Competition in a changing environment revisited 2

As a result of the strong similarities between the equilibrium in the case of international public competition and the one in the case of domestic competition between a public and a private university, it is no surprise that the effects of socioeconomic changes on the education quality levels, the quality and tuition fee differentials and the allocation of students are the same under these two forms of competition. Formally, this conclusion follows from the fact that $q_{2}^{PS}=q_{2}^{\ast }$ implies $\partial q_{2}^{PS}/\partial p=\partial q_{2}^{\ast }/\partial p$, $\partial q_{2}^{PS}/\partial \gamma =\partial q_{2}^{\ast }/\partial \gamma$, $\partial q_{2}^{PS}/\partial N=\partial q_{2}^{\ast }/\partial N$, and that Δt ^PS=Δt ^∗ implies ∂Δt ^PS/∂ p = ∂Δt ^∗/∂ p, ∂Δt ^PS/∂ γ = ∂Δt ^∗/∂ γ, ∂Δt ^PS/∂ N = ∂Δt ^∗/∂ N, etc.

Similarly, the effects of a change in the income in ROW and the size of the pool of talent on the tuition fees are the same in the current extension as in the basic model in the paper. Formally, $t_{i}^{PS}=t_{i}^{\ast } +p\tau \underline {w}$ implies $\partial t_{i}^{PS}/\partial \gamma =\partial (t_{i}^{\ast }+p\tau \underline {w})/\partial \gamma =\partial t_{i}^{\ast }/\partial \gamma$ and $\partial t_{i}^{PS}/\partial N=\partial (t_{i}^{\ast }+p\tau \underline {w})/\partial N=\partial t_{i}^{\ast }/\partial N$. Moreover, inserting q ₁=0 into Eqs. 76 and 77 gives the relationship $t_{2}^{PS}=2t_{1}^{PS}$, leading to ${\Delta } t^{PS}=t_{2}^{PS}-t_{1}^{PS}=t_{1}^{PS}$. Then, $dt_{2}^{PS}/dk=2dt_{1}^{PS}/dk=2d{\Delta } t^{PS}/dk$, k = p,γ,N. That is, both tuition fees increase or decrease along with the tuition fee differential, as the socioeconomic environment changes. Thus, an increase in the ROW income parameter or the size of the pool of talent raises not only the tuition fee differential but also both tuition fees. Also, both tuition fees respond as ambiguous to changes in the stay rate as the tuition fee differential does. Consequently, all conclusions stated in Propositions 1 to 3 and Corollary 1 carry over to the case of domestic competition without qualifications. There are no qualitative changes of the comparative statics results.

The situation is slightly more complicated in the corresponding equilibrium in which the private university is the high-quality provider. Such an equilibrium can exist under certain circumstances even if λ=0. However, even in this equilibrium, if it exists, the tuition fee differential Δt and threshold value $\hat {a}$ are given by Eqs. 10 and 11, which characterise the corresponding variables in the case of international public competition. Also, the comparative statics conclusions in Propositions 1 to 3 with respect to the levels of education quality, the quality and tuition fee differential and the allocation of students across the two university are qualitatively the same. In this sense, our comparative statics conclusions are fairly robust across different forms of competition.

1.3 Domestic students (Section 6)

In Section 6 of the paper, we argue that including domestic students into the analysis provides a more comprehensive picture of the competition for the international pool of talent, especially as universities often invest in ‘general’ education quality, i.e. in quality that benefits both domestic and foreign students in the same way. We now give a more detailed derivation of the results presented in Section 6 of the paper without, however, repeating all the basic assumptions of this variant of the model, as they are already stated in the paper.

Recall from the paper that there is a given pool of domestic talent of size D in each country, and that this pool is characterised by the same uniform ability distribution which also describes the pool of international students. However, these domestic students are internationally immobile. Only general (and not specific) education quality, denoted by $q_{i}^{D}$, is provided to both domestic and foreign students. As a preliminary step, let us consider the optimal education policy of each country in the case in which only domestic students exist. We look at this case first because we want to discover under which conditions all potential domestic students choose to attend the local university.

The wage of non-graduates is normalised to zero, implying that natives enrol at their domestic university if graduate wage $\underline {w}+aq_{i}$ exceeds the tuition fee for domestic students. Then, with a domestic pool of talent only, each government’s objective function, denoted by the superscript D o n l y, contains the wage sum of domestic graduates and the education costs, and is given by

$$ R^{Donly}=(1-\hat{a}^{D})D\left\{\left[\underline{w}+\frac{1}{2}(1+\hat {a}^{D})q\right] -c(q)\right\} -F(q)\text{,} $$

(80)

where $\hat {a}^{D}$ stands for the threshold ability level of the domestic students (we suppress the country index i for the time being). As already stated in the paper, tuition fees for domestic students and taxes paid by natives are welfare-neutral and thus do not enter the objective function. As a result, the tuition fees for domestic students are simply designed such that the optimal share of the domestic pool of talent receives higher education. Note that $\hat {a}^{D}=0$ implies that all domestic students attend university; thus, the market for domestic students is fully covered. As stated in the paper, we are interested in exactly this case, where all potential domestic students are enrolled, and where the two countries compete for the international pool of students only. This simple setting already allows us to sketch some implications of the interplay between the competition for the international pool of talent and a domestic student body. We first discuss the condition under which the case $\hat {a}^{D}=0$ emerges as optimal outcome. (Obviously, there always exists a set of tuition fee levels that guarantees $\hat {a}^{D}=0$.)

Taking the derivative of R ^Donly with respect to $\hat {a}^{D}$ yields

$$ \frac{\partial R^{Donly}}{\partial\hat{a}^{D}}=-D\left( \underline{w}+\hat {a}^{D}q-\alpha q\right) \text{,}\quad\text{implying}\quad\frac{\partial R^{Donly}}{\partial\hat{a}^{D}}\left. {}\right\vert_{\hat{a}^{D}=0} \leq0\Leftrightarrow\underline{w}\geq\alpha q\text{.} $$

(81)

Thus, each governments will implement a tuition fee such that all potential domestic students will enrol (i.e. $\hat {a}^{D}=0$) if the condition $\underline {w}\geq \alpha q$, the condition for a fully covered market for domestic students, is fulfilled. (Note that $\partial ^{2}R^{Donly} /\partial (\hat {a}^{D})^{2}=-Dq\leq 0$ for q≥0, which guarantees that $\hat {a}^{D}=0$ is indeed optimal if $\underline {w}\geq \alpha q$.)

Next, let us discuss the optimal level of education quality in the absence of international students. The derivative of R ^Donly with respect to the general quality level q,

$$ \frac{\partial R^{Donly}}{\partial q}=(1-\hat{a}^{D})D\left[\frac{1} {2}(1+\hat{a}^{D})-\alpha\right] -\frac{\partial F}{\partial q}\text{,} $$

(82)

can only be positive at q=0 if $\alpha <\frac {1}{2}(1+\hat {a}^{D})$ or, for $\hat {a}^{D}=0$, α<1/2. Only then can a positive education quality result, and we assume that α<1/2 is indeed satisfied. In fact, for a sufficiently large $\underline {w}$ and a sufficiently small α, the market for domestic students is fully covered in optimum, which can be achieved by setting the tuition fee for domestic students such that it falls short of the net graduate base salary $\left (1-\tau \right ) \underline {w}$, and the optimal quality level q ^Donly is positive and characterised by ∂ R ^Donly/∂ q=0 (assuming, as we have implicitly done before, that ∂ F(0)/∂ q is also sufficiently small; note that the second-order condition ∂ R ^Donly/∂ q=−∂ ² F/∂ q ²<0 is fulfilled).

1.3.1 Quality and tuition fee competition revisited 3

We now add international students to the model and, as mentioned above, consider general education quality, which affects both foreign and domestic students in the same way. We assume that the condition $\underline {w} \geq \alpha q$ is satisfied, meaning that all domestic students attend university when each government optimally chooses the tuition fees for its domestic students (adding international students does not affect this condition).

The governments of the two countries again enter tuition fee and quality competition for foreign students, whose demand is still characterised by threshold value (3). Rewriting the governments’ objective functions (18) in an extended form gives

$$ R_{1}^{D}=D\left[ (\underline{w}+\frac{1}{2}q_{1}^{D})-c(q_{1}^{D})\right] +\hat{a}N\left\{ \tau p\left[ \underline{w}+\frac{1}{2}\hat{a}q_{1}^{D}\right] +t_{1}-c(q_{1}^{D})\right\} -F(q_{1}^{D})\text{,} $$

(83)

$$\begin{array}{@{}rcl@{}} R_{2}^{D} &=&D\left[ (\underline{w}+\frac{1}{2}q_{2}^{D})-c(q_{2}^{D})\right] \\ &&+\left( 1-\hat{a}\right) N\left\{ \tau p\left[ \underline{w}+\frac {1}{2}\left( 1+\hat{a}\right) q_{2}^{D}\right] +t_{2}-c(q_{2}^{D})\right\} -F(q_{2}^{D})\text{.} \end{array} $$

(84)

Comparing these objective function with Eqs. 6 and 7 yields $R_{i}^{D}=D\left [ (\underline {w}+(1/2)q_{i}^{D})-\alpha q_{i}^{D}\right ] +R_{i}$. Therefore, as $\partial D\left [(\underline {w}+(1/2)q_{i}^{D})-\alpha q_{i}^{D}\right ] /\partial t_{i}=0$ holds, $\partial R_{i}^{D}/\partial t_{i}=\partial R_{i}/\partial t_{i}$ results. (Recall that t _i still stands for the tuition fees for international students, which can differ from those for domestic students.) Thus, the first-order conditions in the second stage (tuition fee competition) are identical in the cases with and without domestic students, and so are the tuition fees, the tuition fee differential and the allocation of students. Hence, the outcome of the tuition fee competition is still fully described by Eqs. 8, 9, 10 and 11.

Next, let us consider the quality competition in the first stage. The equilibrium quality levels are characterised by the conditions (19) and (20) in the paper. Comparing the first-order condition (20) with Eq. 13 reveals that $\partial R_{2}^{D}/\partial q_{2}=D(1/2-\alpha )+\partial R_{2}/\partial q_{2}>\partial R_{2}/\partial q_{2}$ for α<1/2, which implies $q_{2}^{D}>q_{2}^{\ast }$ for α<1/2. Similarly, $\partial R_{2}^{D}/\partial q_{2}=(N/2)(p\tau +2\rho )(1-\hat {a})^{2}+\partial R_{2}^{Donly}/\partial q_{2}>\partial R_{2}^{Donly}/\partial q_{2}$ yields $q_{2}^{D}>q_{2}^{Donly}$.

Furthermore, inspection of condition (19) shows that, for α<1/2, $\left . \partial R_{1}^{D}/\partial q_{1}\right \vert _{q_{1} =0}\gtreqless 0\Leftrightarrow D\gtreqless \lbrack \hat {a}^{2}(p\tau +2\rho )N-2(\partial F_{1}(0)/\partial q_{1})]/(1-2\alpha )$, leading to

$$ q_{1}^{D} \left\{\begin{array}{ll} =0 & \text{if}\quad D\leq\frac{\hat{a}^{2}(p\tau+2\rho)N-2(\partial F_{1}(0)/\partial q_{1})}{1-2\alpha},\\ >0 & \text{if}\quad D>\frac{\hat{a}^{2}(p\tau+2\rho)N-2(\partial F_{1}(0)/\partial q_{1})}{1-2\alpha}. \end{array}\right. $$

(85)

The role of the sizes of domestic and foreign student cohorts can immediately be inferred from Eq. 85. For α<1/2, educational quality in the low-quality country will be positive and thus higher than in the case without domestic students if D is sufficiently large or N is sufficiently small. Otherwise, the educational quality in this country will be zero, as it will also be if α≥1/2.

1.3.2 Competition in a changing environment revisited 3

As a final step, we will now show how including domestic students into the analysis affects the way changes in the socioeconomic environment shape international competition. We focus on the scenario with $q_{1}^{D}>0$ because our Propositions 1 to 3 and Corollary 1 would trivially remain valid if $q_{1}^{D}=0$ resulted. Importantly, as the allocation of students is still characterised by Eq. 11, all comparative statics results of Propositions 1 to 3 with respect to this threshold level remain valid without qualification.

Regarding the other equilibrium values, we begin by investigating the effects of the stay rate on the educational quality in the two countries. Consider first the effect on quality $q_{1}^{D}$. Since the income effect in Eq. 19 is independent of p, the impact of the stay rate on educational quality runs through the competitive effect only, and comparative statics leads to

$$ \frac{dq_{1}^{D}}{dp}\gtreqless0\Leftrightarrow\frac{d^{2}R_{1}^{D}} {dq_{1}^{D}dp}=-\frac{1}{2}\hat{a}N\left[ \left( \tau+2\frac{\partial \rho}{\partial p}\right) \hat{a}+2\left( p\tau+2\rho\right) \frac{d\hat{a}}{d p}\right] \gtreqqless0\text{.} $$

(86)

As ∂ ρ/∂ p≥0 (see Eq. 32) and $d\hat {a}/dp<0$ (see Eq. 33) work into opposite directions, the sign of the term $dq_{1}^{D}/dp$ is unclear. We can show that d q ₁/d p<0 results in the scenario with γ=0 (implying ∂ ρ/∂ p=1−τ) and α=0, while d q ₁/d p>0 holds in the special case with τ _{R O W} = τ and γ=1 (implying ∂ ρ/∂ p=0).

Regarding country 2, from Eq. 20 follows

$$ \frac{dq_{2}^{D}}{dp}>0\Leftrightarrow\frac{d^{2}R_{2}^{D}}{dq_{2}^{D} dp}=\frac{1}{2}(1-\hat{a})N\left[ (1-\hat{a})\left( \tau+2\frac {\partial\rho}{\partial p}\right) -2\left( p\tau+2\rho\right) \frac{d\hat{a}}{d p}\right] >0\text{,} $$

(87)

where ∂ ρ/∂ p≥0 (see again Eq. 32) and $d\hat {a}/dp<0$ (see again Eq. 33) yield the last inequality sign (see Eq. 36, which is completely in line with Eq. 87). That is, the educational quality in country 2 unambiguously increases with the stay rate, as it does in the basic model without domestic students. Since, however, the effect of the stay rate on $q_{1}^{D}$ is ambiguous, its impact on the quality differential Δq ^D cannot be determined without further specification. Calculations with the quadratic fixed cost function $F(q_{i})=(1/2)q_{i}^{2}$ indicate that dΔq ^D/d p<0 is only possible if the cost parameter α is rather large, which implies a threshold value $\hat {a}$ that is fairly close to one. Conversely, the quality differential Δq ^D increases with the stay rate p for a small cost parameter α, at least in the case of this quadratic fixed cost function. Then, Propositions 1 to 3 and Corollary 1 remain valid.

The impact of the stay rate on the tuition fee differential is already ambiguous in the basic model of the paper, and the ambiguity of the sign of dΔq ^D/d p further adds to the ambiguity of the sign of the term dΔt/d p, as comparing Eq. 38 with the following derivative shows:

$$\begin{array}{@{}rcl@{}} \frac{d{\Delta} t}{dp} &=&{\Omega}\frac{d{\Delta} q^{D}}{dp}+{\Delta} q^{D}\left\{ \frac{d{\Omega}}{dp}+\frac{\partial{\Omega}}{\partial\rho}\frac{\partial \rho}{\partial p}\right\} \end{array} $$

(88)

$$\begin{array}{@{}rcl@{}} &=&{\Omega}\frac{\Delta q^{D}}{p}\left\{\underbrace{\frac{d{\Delta q^{D}} }{dp}\frac{p}{{\Delta q^{D}}}}_{:=\varepsilon_{\Delta q^{D},p}\gtreqless 0}+\underbrace{\frac{\partial{\Omega}}{\partial p}\frac{p}{\Omega} }_{:=\varepsilon_{\Omega,p}<0}+\underbrace{\frac{\partial{\Omega}} {\partial\rho}\frac{\rho}{\Omega}}_{:=\varepsilon_{\Omega,\rho} >0}\cdot\underbrace{\frac{\partial\rho}{\partial p}\frac{p}{\rho} }_{:=\varepsilon_{\rho,p}\geq0}\right\}\gtreqless0\text{.} \end{array} $$

(89)

Next, we analyse the impact of an increase in the income in ROW. The effect of a higher income in the rest of the world on the educational quality in country 1 is ambiguous, as shown by

$$ \frac{dq_{1}^{D}}{d\gamma}\gtreqqless0\Leftrightarrow\frac{d^{2}R_{1}^{D}}{dq_{1}^{D}d\gamma}\gtreqqless0\Leftrightarrow-3\rho(p\tau+\rho -\alpha)-p\tau(p\tau-2\alpha)\gtreqqless0\text{.} $$

(90)

Inspection of Eq. 90 reveals that the term $dq_{1}^{D}/d\gamma$ becomes negative for small values of the cost parameter α (e.g. for α=0), while $dq_{1}^{D}/d\gamma$ is positive for large values of α (e.g. $\alpha \rightarrow p\tau +2\rho$, implying $\hat {a}\rightarrow 1$). That is, the education quality in country 1 decreases (increases) with the ROW income parameter γ if the education cost α are small (large). By contrast, the impact of the income parameter γ on education quality $q_{2}^{D}$ is clear-cut:

$$ \frac{dq_{2}^{D}}{d\gamma}>0\Leftrightarrow\frac{d^{2}R_{2}^{D}}{dq_{2}^{D}d\gamma}>0\Leftrightarrow3\rho p\tau+6\rho^{2}+2\alpha p\tau +3\alpha\rho>0\text{,} $$

(91)

where the last inequality, which is the same as the one in the case without domestic students (see Eq. 42), always holds. Thus, the education quality in country 2 always increases with the ROW income parameter γ. Nevertheless, the ambiguity of the sign of the term $dq_{1}^{D}/d\gamma$ carries over to the sign of the term dΔq ^D/d γ. The quality differential Δq ^D increases with the parameter γ if the education cost α is sufficiently small (as quality $q_{2}^{D}$ increases and quality $q_{1}^{D}$ decreases in this case), while the differential Δq ^D can decline if the the parameter α is sufficiently large (as we can show for the case of a quadratic fixed cost function, the increase in quality $q_{1}^{D}$ can exceed the rise in quality $q_{1}^{D}$).

Again, the ambiguity of dΔq ^D/d γ makes the effect of γ on the tuition fee differential Δt ^D ambiguous too, since

$$ \frac{d{\Delta} t}{d\gamma}=\frac{\partial{\Omega}^{D}}{\partial\rho} \frac{\partial\rho}{\partial\gamma}{\Delta} q+{\Omega}^{D}\frac{d{\Delta} q^{D} }{d\gamma}\gtreqqless0\text{,} $$

(92)

where the first-term of the right-hand side of the equation sign is positive, which follows from Eq. 39 and ∂ ρ/∂ γ=(1−p)(1−τ _{R O W} )≥0 (see Eq. 1). If the term dΔq ^D/d γ was sufficiently negative, then dΔt/d γ could be negative as well. Without further calculations, we cannot exclude this case. However, for sufficiently small education costs α, the quality differential Δq ^D increases with the ROW income parameter γ for sure, and so does the tuition fee differential Δt, all in line with Proposition 2.

Finally, consider the effect of the size of the talent pool on educational quality. Comparative statics reveals

$$ \frac{dq_{1}^{D}}{dN}<0\Leftrightarrow\frac{d^{2}R_{1}^{D}}{dq_{1}^{D} dN}<0\Leftrightarrow-\frac{1}{2}\hat{a}^{2}(p\tau+2\rho)<0\text{,} $$

(93)

$$ \frac{dq_{2}^{D}}{dN}>0\Leftrightarrow\frac{d^{2}R_{2}^{D}}{dq_{2}^{D} dN}>0\Leftrightarrow\frac{1}{2}(1-\hat{a})^{2}(p\tau+2\rho)>0\text{,} $$

(94)

where Eq. 94 is completely in line with Eq. 47. As $\partial q_{1}^{D}/\partial N<0$ and $\partial q_{2}^{D}/\partial N>0$, dΔq ^D/d N>0 results, which in turn implies

$$ \frac{\partial{\Delta} t^{D}}{\partial N}={\Omega}^{D}\frac{d{\Delta} q^{D}}{dN}>0 $$

(95)

(compare with Eq. 48). The conclusions about the quality and tuition fee differentials are identical to those reached in Proposition 3.