Appendix A Concavity of the airline carrier’s programme
Conditions sufficient to obtain a maximum, if one exists, depend on the concavity of £. The Lagrangian can be decomposed into various functions, of which concavity ensures an optimal solution is obtained. £ is expressed as follows:
The first expression corresponds to profit made by the airline carrier ¿?2n
p(lV)lV — C(1V) — aN. This profit is concave in TV > 0 iff ",Y < 0, which gives the sufficient condition (CSLCC):
We will ensure that IV 0 at equilibrium. This condition is typically satisfied if passenger demand p(TV) is linear in TV. Otherwise, the condition requires that if the effects of the price on the second-order number of passengers are d2»
increasing — that is, ——- > 0 — then the latter remain limited in relation to first-order demand effects, in addition to convexity effects of airline carrier costs, which is a working hypothesis of the model. The second expression corresponds to profit made by the airport, multiplied by the multiplier p > 0 : (1 + /z)(rS(r,TV) - Cff(TV) + aiv).
As is the case for the airline carrier, the concavity of this programme is desirable in the interests of analysis. It is guaranteed provided that
We will ensure that at equilibrium r 0 (CS’irport) is guaranteed, in that the airport’s variable costs are convex and in that the impact of the externality on the demand for rental space is decreasing in line with the number of passengers in the airport. Returns from the positive externality are assumed as decreasing for the sake of practicality. (CS^lrporJ is typically checked concerning demand for space S(r,N), which is linear in r. Enfm, (c.S''j,por j. Lastly, the condition (CS3) requires checking. It is valid provided that the effect of the externality on the demand for rental space is not too high in relation to the effects on costs and the direct effect on the price S(r,N). Finally, the expression + (1 + — CF^ is linear in TV and therefore does not change
the concavity of the general programme.
To conclude, if conditions (CSLCC), (cS’irport) and (csflrport) are satisfied, a local maximum can be obtained. If conditions are strictly met, a global maximum can be obtained. The conditions for our specifications are as follows:
Appendix B Proof of proposition 2
First-order conditions give
A certain number of constraints need to be checked to correcdy calibrate the model. More specifically, we must also ensure that N > 0, N < N, r > 0. Both cases should be considered, depending on whether the constraint is saturated. Let’s assume that /i = 0:
All constraints require a minimum a, and thus, they are all compatible with each other. We now need to determine the relevant lower boundary for a. We will compare a, and ct0. The sign of their difference a, — a0 is
dependent on the sign — 2S/3 + N/32 + + 9)p which can be decom-
/
posed into two elements, — S (3 + N [32 + 2(7 + 9)p) > 0 since ct, > 0 and — S ¡3 + 2(7 + 9^p) > 0 if ct, > 0. Thus, the sign of the difference is positive -that is, ct, > ct0. Equally, the sign of the difference between ct, and ct0 depends on the same condition and is therefore true for all admissible ct parameters. Comparison of thresholds ct, and ct, as for ct, and ct, which gives the following result: ct, > ct3 and ct, > ct, on the condition that ct, > 0 and ct, > 0 -therefore imposing an additional condition to categorise ct, and ct,. We define Ct = ct,.
The constraint on aeronautical charges is not saturated if the price cap set by the regulator is sufficiently high. This question cannot be fully answered without setting further parameters. However, a nonzero set of parameters exists, for which <7 < a. These parameters satisfy the sufficient constraint at optimum:
If the parameters satisfy this constraint, then it is not necessary to consider the case p. > 0.
The solution for aeronautical charges is therefore a decreasing function of A, as shown in the sign of the first derivative <7 in relation to A. After calculation, the sign of the derivative depends on the sign of the following expression S/3 + 2p(Na — 7 — 0). This sign is negative for all a>a.
The derivative of N with respect to the externality parameter (3 is posi-S(2S/3 + W2-4(7 + 0)/9
tive provided that ct > ct, =-------=—7=--------—. This does not add
4 4Np(s+N/3)
further constraints, because the sign of the difference of ct4 and fl depends on (—2S/3 + N/32 + 4(7 + 9)p^, the negativity of which was shown earlier. For all ct > ct, ------> 0 .
“ d(3
