Cost–Benefit Analysis of Transport Policies: An Application to Subsidies in Air Transport Markets

Abstract

 In this chapter, we use the cost-benefit analysis (CBA) to evaluate subsidies in transport markets. The CBA of any public intervention requires the comparison of the situations with and without intervention, identifying winners and losers, and carefully analysing the key parameters that would allow the public policy to induce the desired effects in the market. In particular, we focus on subsidies for resident passengers in air transport markets. We prove that the effectiveness of the subsidy strongly depends on the particular characteristics of the route, such as the level of competition, the proportion of residents, or the shape of residents’ and non-residents’ demand functions. Thus, any evaluation of the effects of such subsidies has to be performed route by route, taking into account the particular characteristics of the route, the market share of residents and non-residents, and the degree of competition.

Appendices

Appendix 1: Mathematical Expressions of the Model

Equilibrium in the Absence of Public Subsidies: The Monopoly Case

The monopoly carrier chooses the ticket-price $p_{S}$ that solves the following maximization program:

$$\mathop {{\text{Max}}}\limits_{{P_{s} }} \alpha N\left( {p_{S} - c} \right)x^{R} + \left( {1 - \alpha } \right)N\left( {p_{S} - c} \right)x^{{{\text{NR}}}} ,$$

(5.23)

with $x^{R} = \frac{{a_{R} - P_{S} }}{{b_{R} }}$, ${\text{and}}\,x^{{{\text{NR}}}} = \frac{{a_{{{\text{NR}}}} - P_{S} }}{{b_{{{\text{NR}}}} }}$.

The first-order condition of the above maximization program is given by:

$$\frac{N}{{b_{R} b_{{{\text{NR}}}} }}\left( {b_{{{\text{NR}}}} \alpha \left( {a_{R} + c} \right) + b_{R} \left( {1 - \alpha } \right)\left( {a_{{{\text{NR}}}} + c} \right) - 2p_{S} \left( {\alpha b_{{{\text{NR}}}} + \left( {1 - \alpha } \right)b_{R} } \right)} \right) = 0.$$

(5.24)

Solving the first-order condition we obtain the following optimal ticket-price:

$$p_{S}^{0} = \frac{{b_{{{\text{NR}}}} \alpha \left( {a_{R} + c} \right) + b_{R} \left( {1 - \alpha } \right)\left( {a_{{{\text{NR}}}} + c} \right)}}{{2\left( {\alpha b_{{{\text{NR}}}} + \left( {1 - \alpha } \right)b_{R} } \right)}}.$$

(5.25)

By substituting the optimal ticket-price in the demand function of each passenger, we obtain the following demanded quantities per passenger (residents and non-residents, respectively):

$$\begin{aligned} x_{0}^{R} & = \frac{{a_{R} - p_{S}^{0} }}{{b_{R} }} = \frac{{\alpha a_{R} b_{{{\text{NR}}}} + 2a_{R} b_{R} \left( {1 - \alpha } \right) - \left( {1 - \alpha } \right)b_{R} a_{{{\text{NR}}}} - c\left( {\alpha b_{{{\text{NR}}}} + \left( {1 - \alpha } \right)b_{R} } \right)}}{{2b_{R} \left( {\alpha b_{{{\text{NR}}}} + \left( {1 - \alpha } \right)b_{R} } \right)}}, \\ x_{0}^{{{\text{NR}}}} & = \frac{{a_{{{\text{NR}}}} - p_{S}^{0} }}{{b_{{{\text{NR}}}} }} = \frac{{\left( {1 - \alpha } \right)b_{R} a_{{{\text{NR}}}} + 2\alpha a_{{{\text{NR}}}} b_{{{\text{NR}}}} - \alpha a_{R} b_{{{\text{NR}}}} - c\left( {\alpha b_{{{\text{NR}}}} + \left( {1 - \alpha } \right)b_{R} } \right)}}{{2b_{{{\text{NR}}}} \left( {\alpha b_{{{\text{NR}}}} + \left( {1 - \alpha } \right)b_{R} } \right)}}. \\ \end{aligned}$$

(5.26)

Equilibrium with Ad Valorem Subsidies for Residents: The Monopoly Case

When the route is operated by a monopolist and an ad valorem subsidy only for residents is introduced, the airline solves the following maximization program:

$$\mathop {{\text{Max}}}\limits_{{P_{S} }} \alpha N\left( {p_{S} - c} \right)x^{R} + \left( {1 - \alpha } \right)N\left( {p_{S} - c} \right)x^{{{\text{NR}}}} ,$$

(5.27)

where $x^{R}$ and $x^{{{\text{NR}}}}$ represent, given the price that they finally pay, the quantity demanded by residents and non-residents, respectively. Thus, $x^{R} = \frac{{a_{R} - \left( {1 - \sigma } \right)p_{S} }}{{b_{R} }}$ and $x^{{{\text{NR}}}} = \frac{{a_{{{\text{NR}}}} - p_{S} }}{{b_{{{\text{NR}}}} }}$.

The first-order condition of the above maximization program is given by:

$$\frac{N}{{b_{R} b_{{{\text{NR}}}} }}\left( {\alpha b_{{{\text{NR}}}} \left( { - 2p_{S} \left( {1 - \sigma } \right) + c\left( {1 - \sigma } \right) + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( { - 2p_{S} + c + a_{{{\text{NR}}}} } \right)} \right) = 0.$$

(5.28)

Solving the first-order condition we obtain the following optimal ticket-price:

$$p_{S}^{1} = \frac{{\alpha b_{{{\text{NR}}}} \left( {c\left( {1 - \sigma } \right) + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( {c + a_{{{\text{NR}}}} } \right)}}{{2\left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} \left( {1 - \sigma } \right)} \right)}}.$$

(5.29)

The ticket-price finally paid by residents with an ad valorem subsidy only for residents is given by:

$$p_{d}^{R} = p_{S}^{1} \left( {1 - \sigma } \right) = \frac{{\alpha b_{{{\text{NR}}}} \left( {c\left( {1 - \sigma } \right) + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( {c + a_{{{\text{NR}}}} } \right)}}{{2\left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} \left( {1 - \sigma } \right)} \right)}}\left( {1 - \sigma } \right).$$

(5.30)

By substituting the optimal ticket-price in the demand function of each passenger, we obtain the following demanded quantities per passenger (residents and non-residents, respectively):

$$\begin{aligned} x_{1}^{R} & = \frac{{a_{R} - p_{S}^{1} \left( {1 - \sigma } \right)}}{{b_{R} }} \\ & = \frac{{\alpha b_{{{\text{NR}}}} \left( {1 - \sigma } \right)\left( { - c\left( {1 - \sigma } \right) + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( { - c\left( {1 - \sigma } \right) + 2a_{R} - a_{{{\text{NR}}}} \left( {1 - \sigma } \right)} \right)}}{{2b_{R} \left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} \left( {1 - \sigma } \right)} \right)}}, \\ x_{1}^{{{\text{NR}}}} & = \frac{{a_{{{\text{NR}}}} - p_{S}^{1} }}{{b_{{{\text{NR}}}} }}\\ & = \frac{{\alpha b_{{{\text{NR}}}} \left( { - c\left( {1 - \sigma } \right) + 2a_{{{\text{NR}}}} \left( {1 - \sigma } \right) - a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} (a_{{{\text{NR}}}} - c)}}{{2b_{{{\text{NR}}}} \left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} \left( {1 - \sigma } \right)} \right)}}. \\ \end{aligned}$$

(5.31)

Equilibrium with Specific Subsidies for Residents: The Monopoly Case

When the route is operated by a monopolist and a specific subsidy only for residents is introduced, the airline solves the following maximization program:

$$\mathop {{\text{Max}}}\limits_{{P_{S} }} \alpha N\left( {p_{S} - c} \right)x^{R} + \left( {1 - \alpha } \right)N\left( {p_{S} - c} \right)x^{{{\text{NR}}}} ,$$

(5.32)

where $x^{R}$ and $x^{{{\text{NR}}}}$ represents, given the price that they finally pay, the quantity demanded by residents and non-residents, respectively. Thus, $x^{R} = \frac{{a_{R} - p_{S} + s}}{{b_{R} }}$ and $x^{{{\text{NR}}}} = \frac{{a_{{{\text{NR}}}} - p_{S} }}{{b_{{{\text{NR}}}} }}$.

The first-order condition of the above maximization program is given by:

$$\frac{N}{{b_{R} b_{{{\text{NR}}}} }}\left( {\alpha b_{{{\text{NR}}}} \left( { - 2p_{S} + s + c + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( { - 2p_{S} + c + a_{{{\text{NR}}}} } \right)} \right) = 0.$$

(5.33)

Solving the first-order condition we obtain the following optimal ticket-price:

$$p_{S}^{1} = \frac{{\alpha b_{{{\text{NR}}}} \left( {s + c + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( {c + a_{{{\text{NR}}}} } \right)}}{{2\left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} } \right)}}.$$

(5.34)

The ticket-price finally paid by residents with a specific subsidy only for residents is given by:

$$p_{d}^{R} = p_{S}^{1} - s = \frac{{\alpha b_{{{\text{NR}}}} \left( {s + c + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( {c + a_{{{\text{NR}}}} } \right)}}{{2\left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} } \right)}} - s.$$

(5.35)

By substituting the optimal ticket-price in the demand function of each passenger, we obtain the following demanded quantities per passenger (residents and non-residents, respectively):

$$\begin{aligned} x_{1}^{R} & = \frac{{a_{R} - p_{S}^{1} + s}}{{b_{R} }} = \frac{{\alpha b_{{{\text{NR}}}} \left( {s - c + a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( { - c + 2s + 2a_{R} - a_{{{\text{NR}}}} } \right)}}{{2b_{R} \left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} } \right)}}, \\ x_{1}^{\text{NR}} & = \frac{{a_{{{\text{NR}}}} - p_{S}^{1} }}{{b_{{{\text{NR}}}} }} = \frac{{\alpha b_{{{\text{NR}}}} \left( { - s - c + 2a_{{{\text{NR}}}} - a_{R} } \right) + \left( {1 - \alpha } \right)b_{R} \left( {a_{{{\text{NR}}}} - c} \right)}}{{2b_{{{\text{NR}}}} \left( {b_{R} \left( {1 - \alpha } \right) + \alpha b_{{{\text{NR}}}} } \right)}}. \\ \end{aligned}$$

(5.36)

Appendix 2: Some Empirical Evidence: The Case of Spain

Table 5.6 Ad valorem subsidy on flights from/to the Canary Islands from July 2018 to June 2019: interislands routes

Table 5.7 Ad valorem subsidy on flights from/to the Balearic Islands from July 2018 to June 2019: Interislands routes

Table 5.8 Ad valorem subsidy on flights from/to the Canary Islands from July 2018 to June 2019: domestic non-interislands routes

Table 5.9 Ad valorem subsidy on flights from/to the Balearic Islands from July 2018 to June 2019: domestic non-interislands routes