Ajuste de parámetros del modelo

Since the willingness-to-pay of the customers in each period is increasing in the size of the associated network, the firm may launch network expanding promotion campaigns to enlarge the network size and, hence, increase its profitability. The network expanding promotion strategy is commonly used in practice for products with network externali-ties. For example, in February 2015, Microsoft discounted the 12-month Xbox Live Gold membership by 33 percent to both expand the size of Xbox Live and promote the sales of Xbox One (see, e.g., [153]). In the case where the associated network is an online communication network (i.e., r_n(·) ≡ 0), network expanding promotion is the effort and investment the firm makes in social media marketing to attract customers to create and share the messages about the product in the network (i.e., through the electronic word-of-mouth). As an example, in October 2014, Apple bought Twitter’s Promoted Trend at a daily cost of$200,000 to engage Twitter users for the new iPad Air 2 launch (see, e.g., [93]).

To model the network expanding promotion of the firm, let n_t be the number of customers who join the associated service network in period t in addition to the social customers who purchase the product. The total cost of attracting n_t customers into the network is c_n(n_t), where c_n(·) is a continuously differentiable and convexly increasing function of n_t with c_n(0) = 0. Note that the network expanding promotion do not change

the inventory dynamics of the firm, but they do have some impacts on the network size dynamics. More specifically, with network expanding promotion, the network size at the beginning of period t− 1 is given by: Nt−1 = θD_t(p_t, N_t) + ηN_t+ n_t+ ϵ_t.

We define

v^p_t(It, Nt) := the maximum expected discounted profits with network expanding promotion in periods t, t− 1, · · · , 1, when starting period t with an inventory level I_t and network size N_t;

and (x^p_t^∗(I_t, N_t), p^p_t^∗(I_t, N_t), n^∗_t(I_t, N_t)) as the optimal pricing and inventory policy. As in the base model, we assume that, in the last period (period 1), the excess inventory is salvaged with unit value c, and the backlogged demand is filled with ordering cost c, i.e., v₀^p(I0, N0) = cI0 for any (I0, N0). Employing similar dynamic programming and sample path analysis methods, we characterize the optimal policy in the model with network expanding promotion in the following lemma.

Lemma 6 Define a sequence of functions {π^pt(N_t) : t = T, T − 1, · · · , 1} and a sequence of pricing and inventory policies{(x^pt(N_t), p^p_t(N_t), n_t(N_t)) : t = T, T−1, · · · , 1} as follows:

π_t^p(N_t) = max

(xt,pt,nt)∈F^pJ_t^p(x_t, p_t, n_t, N_t), (2.11) where J_t^p(x_t, p_t, n_t, N_t) = R_t(p_t, N_t) + βx_t+ Λ(x_t− ¯Vt+ p_t− γ(Nt))− cn(n_t)

+G^p_t(θ( ¯V_t− pt+ γ(N_t)) + ηN_t+ n_t),

with G^p_t(y) := E{rn(y + θξ_t+ ϵ_t) + απ^p_t−1(y + θξ_t)}, π^p0(·) ≡ 0, and (x^p_t(N_t), p^p_t(N_t), n_t(N_t)) := argmax_{(xt,pt,nt)∈Fp}J_t^p(x_t, p_t, n_t, N_t).

(a) π^p_t(·) is concave, continuously differentiable, and increasing in Nt. J_t^p(·, ·, ·, ·) is jointly concave and continuously differentiable in (x_t, p_t, n_t).

(b) If I_t ≤ x^pt(N_t), (x^p_t^∗(I_t, N_t), p^p_t^∗(I_t, N_t), n^∗_t(I_t, N_t)) = (x^p_t(N_t), p^p_t(N_t), n_t(N_t)) and v_t^p(I_t, N_t) = cI_t+ π_t^p(N_t); otherwise, x^p_t^∗(I_t, N_t) = I_t. If I_T ≤ x^pT(N_T),

(x^p_t^∗(I_t, N_t), p^p_t^∗(I_t, N_t), n^∗_t(I_t, N_t)) = (x^p_t(N_t), p^p_t(N_t), n_t(N_t)) for all t and (I_t, N_t) with probability 1.

Lemma 6 demonstrates that a network-size-dependent base-stock/list-price/promotion policy is optimal in the model with price discrimination. By normalizing the value of cur-rent inventory, we can reduce the state space dimension of the dynamic program to 1.

With probability 1, the optimal policy is independent of the starting inventory level in each period, as long as the initial inventory level I_T is below the optimal period-T base-stock level in the first period x^p_T(N_T).

As in the model with price discrimination, Theorems 2.4.3, 2.4.5, 2.4.6, and 2.4.7 can be generalized to the model with network expanding promotion. We now demonstrate the effectiveness [ineffectiveness] of network expanding promotion in the model with [without]

network externalities.

Theorem 2.5.2 (a) Let 0 < ι < 1, and ¯S(N ) := sup{∆ : P(Nt−1 ≥ ∆|Nt = N )≥ ι}.

If

(1− ι)[r_n^′( ¯S(N )) + α(p− c)γ^′( ¯S(N ))] > c^′_n(0), (2.12) then n^∗_t(I_t, N ) > 0 for all I_t. Moreover, ¯S(N ) is continuously increasing in N and, for each 0 < ι < 1, there exists an N^∗(ι) ≥ 0, such that (2.12) holds for all N < N^∗(ι).

(b) If γ(·) ≡ γ0 and (∑t−1

τ =0(αη)^τ)r^′_n(0) ≤ c^′n(0), n^∗_t(It, Nt)≡ 0 for all It and Nt≥ 0.

Theorem 2.5.2 characterizes the dichotomy on when the firm should offer network expanding promotion. More specifically, Theorem 2.5.2(a) shows that, when either (i) the intensity of network externalities is sufficiently strong or (ii) the associated service network is sufficiently profitable (as characterized by inequality (2.12)), it is optimal for the firm to offer network expanding promotion to customers as long as the current network size is sufficiently low (i.e., n^∗_t(It, Nt) > 0 if Nt ≤ N^∗(ι)). The intuition behind Theorem 2.5.2(a) is that, if a lower bound of the marginal value of offering network expanding promotion, (1− ι)[r^′n( ¯S(N )) + α(p− c)γ^′( ¯S(N ))], dominates its marginal cost c^′_n(0), the firm should offer network expanding promotion to customers. Here, ¯S(N ) can be interpreted as the threshold such that, conditioned on Nt = N , the probability that the network size in period t− 1 exceeds ¯S(N ) is smaller than ι, regardless of the pricing strategy the firm employs. Hence, network expanding promotion are effective in exploiting network externalities, especially when N_t and, thus, the potential demand is low. On the other hand, Theorem 2.5.2(b) shows that if network externalities do not exist (i.e., γ(·) ≡ 0) and the associated service network is not sufficiently profitable (i.e., (∑t−1

τ =0(αη)^τ)r^′_n(0)≤ c^′n(0)), it is optimal for the firm not to offer any network expanding promotion.

Next, we study the impact of network expanding promotion upon the firm’s optimal policy.

Theorem 2.5.3 Assume that two inventory systems are identical except that one with network expanding promotion and the other without. For each period t and each network size N_t≥ 0, the following statements hold: (a) p^pt(N_t)≥ pt(N_t); (b) y_t^p(N_t)≤ yt(N_t); (c) x^p_t(N_t)≤ xt(N_t); and (d) π^p_t(N_t)≥ πt(N_t), where the inequality is strict if n_t(N_t) > 0.

Theorem 2.5.3 highlights how the firm should adjust its price and inventory policy with network expanding promotion. More specifically, we show in Theorem 2.5.3(a) that, with the same network size (and, hence, the same potential market size), the firm should charge a higher sales price with network expanding promotion. Since both the sales price and the network expanding promotion helps induce future demands via network externalities, the adoption of network expanding promotion allows the firm to increase the sales price to generate higher profit in the current period. As a result, the optimal expected demand and the optimal base-stock level are lower with market expanding promotion. In Theorem 2.5.3(d), we show that network expanding promotion can improve the profitability of the firm.

To summarize, network expanding promotion helps the firm exploit network exter-nalities by boosting the network size in each period. In particular, network expanding promotion facilitates the firm to induce future demands with network expanding promo-tion, while generating higher current profits with a higher sales price. The firm should offer network expanding promotion when the intensity of network externalities is suffi-ciently strong or the associated service network is suffisuffi-ciently profitable.