Competition Before Sunset: The Case of the Finnish ATM Market

Appendix

Proof of Proposition 2. For the moment, let us assume that the non-negativity constraints on network sizes do not bind. Then the first-order condition for the problem of choosing the socially optimal f reads as

dWdf=dXBdf[Δc−ΔM+t(1−Δs−2XB)]+dsBdf[∂XB∂sB[Δc−ΔM+t(1−Δs−2XB)]+t(XB−sB)−ksB]=0.

Using equations (3) and (33) and simplifying gives

(35)dWdf=Δc−ft−1k[12[Δc−ΔM+t(1−sI−sB)]−ksB]=0. (35)

The first-order condition for the socially optimal a is analogously given by

dWda=dsBda[∂XB∂sB[Δc−ΔM+t(1−Δs−2XB)]+t(XB−sB)−ksB]+dsIda[∂XB∂sI[Δc−ΔM+t(1−Δs−2XB)]+t(1−sI−XB)−ksI]=0.

After using equations (3) and (34), this can be simplified to

(36)dWda=t(1−sI−sB)−ksB−ksI=0. (36)

Plugging equation (36) into equation (35) yields

dWdf=Δc−f−t2k[Δc−ΔM+k(sI−sB)]=0.

Next, substituting equations (34) and (33) for s_I and s_B gives

dWdf=Δc−f−t2(Δc−ΔMk+f−Δc2k)=0.

Solving this equation for f yields after some manipulation

(37)f=f∗∗≡Δc+2t(ΔM−Δc)4k+t. (37)

Inserting equations (33) and (34) into equation (36) yields

dWda=tt+k−(a−cI2k+a−f−cB2k)=0.

Solving this equation for a gives

(38)a(f)=ktt+k+f+cI+cB2. (38)

Substituting equation (37) for f in equation (38) then gives

(39)a(f∗∗)≡a∗∗=ktt+k+cI+t4k+t(ΔM−Δc). (39)

Finally, substitution of equations (37) and (39) for f and a in equations (33) and (34) results in the equilibrium network sizes of

(40)sB∗∗=t2k(kt+k−ΔM−Δc4k+t) (40)

and

(41)sI∗∗=t2k(kt+k+ΔM−Δc4k+t). (41)

Comparing equations (40) and (41) with equations (31) and (32) produces the following observations: i) If ΔM≥Δc, then sI∗≥sI∗∗>0 and sB∗∗≥sB∗. ii) If Δc≥ΔM, then sB∗≥sB∗∗>0 and sI∗∗≥sI∗. Thus, because we work in the range of parameter values where sB∗>0 and sI∗>0 we also have that sB∗∗>0 and sI∗∗>0, satisfying our initial assumption of positive network sizes.      ▀

Proof of Proposition 3. Note first from equation (5) that since 0≤X_B(f, s_B, s_I)≤1 (we later verify that this holds in equilibrium) and since c_B≥0, a necessary condition for π_B(f, s_B, a, s_I)≥0 is that f>a. However, if f>a and c_B≥0, then a–f–c_B<0. Equation (33) then implies that the constraint s_B≥0 is violated, leading to sBR=0.

Let us for the moment assume that a≥c_I implying that π_I(f, a, s_I, s_B)≥0 and Δs=s_I=(a–c_I)/2k≥0. Then, after substituting equation (3) with Δs=s_I=(a–c_I)/2k for equation (5) and some straightforward calculations, we can re-express the constraint π_B(f, a)≥0 as

−2kf2+f[2k(a+t−cB+ΔM)+t(a−cI)]−2k[ΔM(a−cB)+t(a+cB)]−t(a−cI)(a−cB)≥0.

The left-hand side is a quadratic polynomial in f. Its two roots are given by

(42)fR(a)=f(a)±f(a)2−[ΔM(a−cB)+t(a+cB)]−t(a−cB)(a−cI)2k (42)

where

f(a)=2k(t+a+ΔM−cB)+t(a−cI)4k=t+a+ΔM−cB2+t(a−cI)4k.

can be found after substitution of Δs=s_I=(a–c_I)/2k for equation (7).

To pin down a, we maximize social welfare from equation (28) with respect to a when s_B=0. The first-order condition reads as

dWda=dsIda[∂XB∂sI[Δc−ΔM+t(1−Δs−2XB)]+t(1−sI−XB)−ksI]=0.

After using equations (3) and (34), and simplifying we get

(43)sI=sIR≡t+ΔM−Δc2k+t. (43)

To determine the optimal interchange fee, we substitute equation (34) for sIR in equation (43), and solve for a. This results in

(44)a=aR≡2k(t+ΔM−Δc)2k+t+cI. (44)

Substituting equation (44) for a in equation (42) gives

fR(aR)≡fR=t+ΔM±(t2k+t)2(t+ΔM−Δc)2−2cBt.

Note that both roots are larger than Δc, which would constitute the optimal foreign fee in the absence of zero profit constraint in the case where s_B=0. By the concavity of the welfare function, it is therefore optimal to implement as small f as possible compatible with the constraint π_B(f)≥0. This is given by the lower root of the above equation. As a result, we have πB(fR, aR, sIR, sBR)=0

where

(45)fR=t+ΔM−(t2k+t)2(t+ΔM−Δc)2−2cBt. (45)

From equations (43) and (44) we see that under restriction (19), sIR≥0 and a^R≥c_I, as we initially assumed. We should also assume that SR=sIR<1, which is equivalent to 2k>ΔM–Δc. Then, after substitution of sIR and f^R from equations (43) and (45) for Δs and f in equation (3), it is also straightforward to show that XB(fR, sIR, sBR)∈[0, 1] as we initially assumed.

Given (19) and Proposition 3, it is immediate that f^R≥f^*=Δc, aR≥cI≥as∗, and sB∗≥sBR=sBc=0. Claim sIR⋛sIc if k⋛t/3 follows from the comparison of equations (43) and (17).      ▀

Proof of Proposition 4. We show in the main text that the cap on the foreign fee binds. Then, solving the deployer I’s problem (8) under the assumption that the foreign fee is given by f^FCA(a)=a–c_B yields that

(46)a=aFCA(sB, sI)≡[t(1+Δs)+ΔM+cB+cI]2=a(sB, sI), (46)

which where the last equality follows from the comparison of a^FCA(s_B, s_I) with equation (10). This implies that the regulated foreign fee becomes

fFCA(sB, sI)=aFCA(sB, sI)−cB=t(1+Δs)+ΔM+Δc2.

It is immediate from the deployer B’s objective function (5) that ∂π_B/∂s_I<0 if f(a)=a–c_B. Thus sBFCA=sBc=0.

After substituting f^FCA(s_B, s_I) for equation (4), we may write the deployer I’s market share as

(47)XIFCA(sB, sI)=1+Δs4+ΔM−Δc4t=2XI(sB, sI) (47)

where the last equality comes from the comparison of XIFCA(sB, sI) with equation (13). Noting equations (46) and (47) and inspecting the deployer I’s objective functions from (8) and (16) shows that we may write the deployer I’s problem as

max sI≥0πI(sI, sB)=(t(1+Δs)+ΔM−Δc)28t−ksI22.

Using sBFCA=sBc=0 we quickly get that the deployer I’s optimal network size is given by

sIFCA=t+ΔM−Δc4k−t.

It is easy to verify that the second-order condition for the deployer I’s maximization problem is satisfied when

k>t4.

[This is also the second-order condition for a monopoly’s problem in Alexandrov (2008).] Under this assumption, we have sIFCA≥sIc=(t+ΔM−Δc)/(8k−t) [see equation (17)]. As a result, the total size of the ATM network increases since SFCA=sIFCA≥Sc=sIc.

Next, we substitute ΔsFCA≡sIFCA−sBFCA=sIFCA=(t+ΔM−Δc)/(4k−t) for equation (46). This yields

aFCA≡aFCA(sBFCA, sIFCA)=2k(t+ΔM+cB+cI)−tcI4k−t.

As a result, we have

fFCA=aFCA−cB=2k(t+ΔM+Δc)−tΔc4k−t.

Comparing f^FCA and a^FCA with f^c and a^c of equations (22) and (21), respectively, shows that under the restriction (19), a^FCA≥a^c and f^c≥f^FCA.      ▀