PROCESAMIENTO Y ANALISIS DE DATOS

In equilibrium, drivers choose strategies in response to the regulator’s choice of price levels and the number of medallions. It is, however, more convenient to characterize the equilibrium in terms of the regulator’s choice of prices and vacancy rates, as it makes the equilibrium easier to solve and makes the analysis consistent with the previous taxicab literature. For exclusive cruising regulation and combined regula- tion, we show that there exists an equivalent representation where the variables of interest are represented in terms of price and vacancy rates. This is the most natural extension of the Douglas Cruising framework to the multiple location setup. The next result shows that exclusive cruising equilibria can be represented in terms of prices and vacancy levels that satisfy a series of conditions.

Proposition 1 (Stationary Markov Equilibrium - Exclusive Cruising Regulation) An exclusive cruising equilibrium can be characterized by {(pi, Vi)}i∈{1,2} such that

for i ∈ {1, 2},

1. Demand: The quantities demanded satisfy qi = qi(pi, Vi) and ˜qi = ˜qi(pi, Vi).

2. Flow Behavior: The number of transitioning taxis is Ti = ˜qj.

3. Individual Rationality: The discounted expected profit must be non-negative

pi Qi ≥ c (Qi+ Vi) if ˜qi = 0

Conversely, for any pair {(pi, Vi)}i∈{1,2} such that the above conditions are satisfied,

there exists a unique exclusive cruising equilibrium {(pi, NiLEC)}i∈{1,2} with the same

values of {(Vi, Qi, Ti)}i∈{1,2}.

Under exclusive cruising, incentive compatibility is satisfied trivially. The flow condition is satisfied because taxis have to transition back to their affiliated loca- tion to be able to pick up passengers; therefore, ˜qi taxis will have to transition back

to their original location. The individual rationality condition depends on whether drivers drive passengers to the other location. When drivers drive passengers to the other location, the drivers’ unaffiliated location will have the lower expected dis- counted profit. The expected discounted profit will be positive provided the condition in the proposition is satisfied. When drivers do not drive passengers to the other location, individual rationality only requires nonnegative discounted profit in their affiliated location. The drivers will receive a nonnegative discounted profit provided the expected revenue exceeds the cost, this will occur provided the condition in the proposition is satisfied. Given a price vacancy representation, the total number of medallions for affiliation i will satisfy:

N_iLEC = Vi+ Qi+ ˜qi

Since taxis pick up and drop off passengers with high frequency, the discount rate will be close to 1; therefore, we look at the limiting behavior of the individual rationality constraint. Taking the limit as δ → 1 implies:

The total revenue drivers receive must exceed the total cost they incur.

Looking at prices and vacancy levels instead of prices and the total number of taxis allows us to state the regulator’s problem in a simpler way. The regulator chooses a set of prices and vacancy levels to maximize total surplus subject to the zero discounted profit constraints:

W T SLEC = X i∈{1,2} 1 ωi  Z Qi 0 pi(Q, vi) dQ − c (Qi+ Vi+ ˜qi) − φi(Qi+ Vi+ ˜qj) + (ωi− 1) (Qipi− c (Qi+ Vi+ ˜qi))  subject to δpi Qi ≥ c (Qi+ δ ˜qi+ Vi) if ˜qi > 0 pi Qi ≥ c (Qi+ Vi) if ˜qi = 0

To develop a representation of combined equilibria in terms of price and vacancy, we need to characterize incentive compatible vacancy rates. The driver’s search behavior will be based on the expected revenue drivers receive in each of the locations and the transition probabilities. The expected revenue and transition probabilities gives conditions on the vacancy rates and transitioning that is observed. For a given price and vacancy rate, we can look at whether these prices and vacancy rates lead to an outcome that is incentive compatible.

Proposition 2 (Search Behavior - Combined Regulation) For each combined equi- librium with price and vacancy levels {(pi, Vi)}i∈{1,2}, the following conditions hold:

1. If Rν

2. If Rν

i > Rνj for some i, j ∈ {1, 2} where j 6= i, define

R∗_j = δ ρ ˜ ν jj 1 + δ(1 − ρv ii) Rν_i (2.10) Then Ti = 0 and Vj = ˜qi− ˜qj Tj = 0 if Rνj > R ∗ j Vj+ Tj = ˜qi− ˜qj if Rνj = R ∗ j Tj = ˜qi− ˜qj Vj = 0 if Rνj < R ∗ j

Alternatively, for prices and vacancy rates {(pi, Vi)}i∈{1,2}, when condition (1) or

(2) is satisfied there exists incentive compatible strategies that induces these price and vacancy levels such that the stationary flow condition is satisfied. Any such strategies induce the same {(Qi, Vi, Ti)}i∈{1,2}.

If the expected revenue is the same in each location, drivers that transition forgo revenue in the period they transition but do not receive a higher payoff from being in the other location; therefore, drivers will not want to transition. The net flow of occupied taxis from location j to i, ˜qj− ˜qi, must be made up by vacant taxis moving

from location i to j. If

Vi ≥ ˜qj − ˜qi (2.11)

for i ∈ {1, 2}, there are enough vacant taxis at location i to make up the difference in the number of occupied taxis traveling between locations. When equation (3.3) holds, drivers can direct their search such that the net flow of vacant taxis from

location i to j equals the flow of occupied taxis from location j to location i.9

When the expected revenue differs, drivers will prefer to be in the location with the higher expected revenue. Depending on the relative payoffs of the two locations and the transition probabilities, the drivers at the low-value location will either prefer to search for passengers while relocating to the high-value location, would prefer to transition to the high-value location, or would be indifferent between the two actions. There will be some critical value for revenue in the low-value location such that the driver will be indifferent between searching for passengers and transitioning. For expected revenue levels above that, drivers would prefer to search. For expected revenue levels below the critical value drivers would prefer to transition.

Proposition 2 established the the set of the prices and vacancy levels that lead to incentive compatible outcomes that have a stationary flow of taxis. When individu- ality rationality is satisfied for the prices and vacancy rates, the outcome will be an equilibrium. Formally:

Proposition 3 (Metro-Level Equilibrium) A combined equilibrium can be charac- terized by {(pi, Vi)}i∈{1,2} and {(qi, ˜qi, Ti)}i∈{1,2} such that

1. Demand: The quantities demanded satisfy qi = qi(pi, Vi) and ˜qi = ˜qi(pi, Vi).

2. Flow Behavior: Either condition (1) or (2) in Proposition 2 is satisfied. 3. Individual Rationality: The discounted expected profit must be non-negative:

If Ti+ Vi > 0 for i ∈ {1, 2}:

R_iν ≥ c if Rν₁ = Rν₂ (2.12)

9_{When the expected revenue is equal at the two locations, multiple sets of strategies can induce}

the same equilibrium prices and vacancy rates. All of these equilibria are payoff equivalent; there- fore, we do not need to distinguish between equilibria by looking at which location the vacant taxis end up.

 1 − δ qi+ Vi Qi+ Vi   Qj Qj + Vj + Tj pj − c  + δ ˜qj+ Vj + Tj Qj+ Vj + Tj   Qi Qi+ Vi pi− c  ≥ 0 if Rν_i > Rν_j (2.13) Otherwise, Rν

i ≥ c for all i such that Ti+ Vi > 0

Conversely, for any {pi, Vi}i∈{1,2}such that the above conditions are satisfied, there

exists a combined equilibrium with the same {Vi, Qi, Ti}i∈{1,2}.

When the expected revenue at each location the same, individual rationality will be satisfied whenever the expected revenue is greater than or equal to the cost. When the expected revenue is different, the discounted expected revenue must exceed the cost. This means that the expected revenue could be below the cost in the low- revenue location, provided there is sufficiently high revenue in the high-revenue loca- tion. The individual rationality condition are easier to interpret when δ approaches 1. The limiting nonnegative discounted profit condition is

X i∈{1,2} pi Qi ≥ X i∈{1,2} c (Qi+ Ti+ vi)

The total revenue that drivers receive must exceed the total cost they incur. All of the drivers choose strategies with the same expected payoff; therefore, in the limit their expected discounted profit will be the same regardless of the individual driver’s current location. Since the limiting profit is the same, the individual driver’s profit will be positive if the total revenue exceeds the total cost.

Using the price-vacancy representation of the equilibrium, the regulator’s maxi- mization problem is to maximize

W T SM L = X

i∈{1,2}

 Z Qi

0

− φi(Qi+ Vi+ Ti) + (ωi− 1) (Qipi− c (Qi+ Vi+ Ti))



(2.14)

subject to the constraints implied by Proposition 2. Given a preferred set of prices and vacancy rates, the regulator will need to assign

NM L = X

i∈{1,2}

(Vi+ Qi+ Ti)

medallions. The medallions cover the taxis that are occupied, vacant, or transitioning in each location.