Anexo 1 Informe de Supervisión del Contrato de Obra 464 y 574 del 2014

The previous section investigates volume-based loyalty programs, under which customers receive the same number of points from each purchase, no matter how much they pay. In practice, some companies use expense-based programs, under which customers receive points based on the amount of money they spend. This section extends our results to expense-based programs.

Consider American Airlines that provides an expense-based loyalty program: it issues 5 miles for each dollar spent and requires 25,000 miles for a free flight. Then, the total number of points issued per unit depends on the price paid. For instance, a customer gets 4,000 miles for a ticket priced at $800, but only gets 1,500 miles for a ticket discounted to $300 on the same flight. Let n be the number of points issued per dollar spent; here, n= 5. As before, we use M to denote the number of points required for a free unit; in our example,M = 25,000, so customers are entitled to a free flight after spending $5,000 (e.g., 7 full-fare flights or 17 discounted flights). This results in Ni = npi and Mi =M, where i=L, H.

With expense-based loyalty programs, most of our results remain unchanged. We begin with the following proposition.

Proposition 2. In the equilibrium,

(i) Low types buy q∗_L, then redeem q_A∗; high types buy q_H∗.

(ii) p∗_i = vi

1−nw∗;

(iii) There exists K¯ such that

(a) if K≤K¯, then q_L∗ = 0 andq∗_A, q_H∗ satisfy

q∗_A+q_H∗ =K, q∗_A= σ ∗_np∗ H M E[q ∗ H ∧X].

(b) If K >K¯, then q∗_H =q and q_A∗,q∗_L satisfy q_A∗ +q_L∗ + ¯q =K, q∗_A= σ ∗_np∗ L M q ∗ L+ σ∗np∗_H M E[¯q∧X]. Here, q¯= ¯F−1(vL vH).

(iv) The equilibrium profit is vL·q_L∗ +vH·q∗_A+vH ·E[q∗_H ∧X];

(v) R∗ =vH, σ∗ = _vv_HL, w∗ = v_ML.

A quick glance at Proposition 2 reveals several similarities to Proposition 1. First, Proposi- tion 2(i) shows that with expense-based programs, it remains optimal to induce low-types to buy before redeeming, resulting in awards being valued at the high valuation, i.e.,R∗=vH, as in volume-based programs. Second, the profit function in Proposition 2(iv) remains un- changed and shows that, by extracting a price premium for loyalty points, the firm again receivesvH from each of theqAredeemed units. In other words, whether they are volume- based or expense-based, loyalty programs enable the firm to extract the high valuation from each unit redeemed by a low-type. Finally, Proposition 2(v) shows that the redemp- tion probability σ∗= vL

vH and the value of loyalty pointsw

∗ ₌ vL

M match our earlier results for volume-based programs.

However, there are a couple of differences between Proposition 1 and Proposition 2. First, to extract all consumer surplus, the firm chooses p∗_i =vi+np∗iw, which gives p∗i =

vi

1−nw∗. As a result, the price premium for loyalty points enters as a multiplicative factor ₁₋1_nw∗ in expense-based programs, instead of an additive term N w∗ in volume-based programs. Second, the exact form of revenue management decisions differ from volume-based programs. Nevertheless, the optimal capacity allocation rule for expense-based programs follows the same logic as before. We start with a protection level ¯q for high-types, `a la Littlewood. If this protection level and the associated award liability exceeds capacity, then all units are priced high and the protection level is adjusted downward to meet capacity. Otherwise,

the firm protectsq∗_H = ¯q units for high-types and additionally allocatesq_L∗ units for sale at the low price so that all capacity is used up, after considering all associated award liability. Consequently, the rule of thumb described in the previous section still applies: the optimal award capacity can be achieved by matching the redemption probability toσ∗.

One noteworthy result in Proposition 2 is that the protection level ¯q is identical to that in the classic model. This is because in expense-based loyalty programs, both the price premium and the cost of issuing points are proportional to the valuation of the customer who made the purchase. As a result, the critical fractile ¯q determining the protection level is simply the ratio of customer valuations, as in the classic model. This finding leads us to our next result. The result suggests that Expense-based programs not only retains the property of inducing low types to pay the high price, but also simplifies the calculation of revenue management. An expense-based program protects the same number of units for high-types as in the classic Littlewood model, suggesting that capacity allocation decisions can be made without considering loyalty programs. The firm only needs to collect historical information about prices to make decisions on the protection levels.