Propuestas Técnico- Económicas de control operacional

In this subsection, we study how the exploitation-induction tradeoff impacts the equi- librium market outcome in the PF model. As in the SC model, we first characterize the impact of the PF market size coefficient vectors, {β_tpf : T ≥ t ≥ 1}.

Theorem 3.5.3 For each period t, the following statements hold:

(a) For each i and j ̸= i, y_i,tpf∗ is continuously increasing in β_i,tpf₋₁ and independent of β_j,tpf₋₁.

(b) For each i and j ̸= i, πpf_i,t∗ is continuously increasing in β_i,tpf₋₁ and continuously decreasing in β_j,tpf₋₁.

(c) For each i, j, and γt , p pf∗

i,t (γt) is continuously decreasing in π pf∗ j,t .

(d) If the PF model is symmetric, γ_s,tpf∗ is continuously increasing in π_s,tpf∗. If, in ad- dition, the monotonicity condition (3.17) holds, β_s,tpf is continuously increasing in πpf_s,t∗ as well.

(e) If the PF model is symmetric and π_s,tpf∗ is increasing in β_s,tpf₋₁, γpf_s,t∗ is continuously increasing in β_s,tpf₋₁, whereas ppf_i,t∗(γt) is continuously decreasing in βs,tpf−1. If, in addition, the monotonicity condition (3.17) holds, β_s,tpf is continuously increasing in β_s,tpf₋₁ as well.

Theorem 3.5.3 demonstrates that the market size coefficients {β_i,tpf : 1 ≤ i ≤ N, T ≥ t ≥ 1} quantify the intensity of the exploitation-induction tradeoff in the PF model. More specifically, a larger β_i,tpf₋₁ implies more intensive exploitation-induction tradeoff of firm i in period t.

As in the SC model, we use “ ˜ ” to denote the benchmark case without the service effect and the network effect, where the market size evolution function ˜αi,t(·) ≡ 0 for each firm i and each period t. Thus, the exploitation-induction tradeoff is absent in this benchmark model, and it suffices for the firms to myopically maximize their current- period profits. The following theorem characterizes the impact of the service effect and the network effect in the PF model.

Theorem 3.5.4 (a) For each firm i and each period t, y_i,tpf∗ ≥ ˜y_i,tpf∗, zpf_i,t∗ ≥ ˜z_i,tpf∗, and πpf_i,t∗ ≥ ˜π_i,tpf∗.

(b) For each firm i and each period t, ppf_i,t∗(γt) ≤ ˜ppf_i,t∗(γt) for all γt. Moreover, if the PF model is symmetric and (3.17) holds, xpf_i,t∗(It, Λt, γt) ≥ ˜xpfi,t∗(It, Λt, γt) for all i, t, (It, Λt)∈ S, and γt∈ [0, ¯γs,t]N.

(c) Consider the symmetric PF model. For each period t, γ_s,tpf∗ ≥ ˜γ_s,tpf∗. Thus, γ_i,tpf∗(It, Λt)≥ ˜γi,tpf∗(It, Λt) for all i and all (It, Λt)∈ S.

Consistent with Theorem 3.4.4(a), Theorem 3.5.4(a) shows that, the service effect and the network effect drive the competing firms to maintain higher service levels in the PF model. Theorem 3.5.4(b) reveals the impact of the exploitation-induction tradeoff upon the competing firms’ price and inventory strategy in the PF model. Specifically, given any outcome of the first-stage promotion competition γt, in the second-stage price competition, each firm i should charge a lower sales price under the service effect and the network effect, so as to exploit the network effect and induce higher future demands. Moreover, in each period t, the equilibrium post-delivery inventory levels contingent on any realized promotional effort vector γtare also higher in the PF model under the service effect and the network effect. Theorem 3.5.4(c) sheds light on how the exploitation- induction tradeoff influences the equilibrium promotion strategies under the service effect and the network effect. In the symmetric PF model, the equilibrium promotional effort of each firm i in each period t is higher under the service effect and the network effect.

Note that, in the PF model, the equilibrium price and inventory outcomes under the service effect and the network effect, p_s,tpf∗(γ_ss,tpf∗) and x_i,tpf∗(It, Λt, γss,tpf∗), may be either higher or lower than those without market size dynamics, ˜ppf_ss,t∗(˜γ_s,tpf∗) and ˜x_i,tpf∗(It, Λt, ˜γss,tpf∗). This phenomenon contrasts with the equilibrium market outcomes in the SC model, where the equilibrium sales price [post-delivery inventory level] of each firm in each period is lower [higher] under the service effect and the network effect (i.e., Theorem 3.4.4(b-i,iii)). This discrepancy is driven by the fact that, in the PF model, each firm observes the promotion decisions of its competitors before making its pricing decision. Hence, under the service effect and the network effect, the competing firms may either decrease the sales prices to induce future demands or increase the sales prices to exploit the better market condition from the increased promotional efforts (recall that γ_s,tpf∗ ≥ ˜γ_s,tpf∗). In general, either effect may dominate, so we do not have a general monotonicity relationship between either the equilibrium price outcomes (i.e., ppf_s,t∗(γ_ss,tpf∗) and ˜ppf_s,t∗(˜γ_ss,tpf∗)) or the equilibrium inventory outcomes (i.e., xpf_i,t∗(It, Λt, γss,tpf∗) and ˜x

pf∗

i,t (It, Λt, ˜γss,tpf∗)). Therefore, the exploitation-induction tradeoff in the PF model is more involved than that in the SC model. The competing firms only need to trade off between generating current profits

and inducing future demands intertemporally in the SC model, whereas they need to balance this tradeoff both inter-temporally and intra-temporally in the PF model.

To deliver sharper insights on the managerial implications of the exploitation-induction tradeoff, we confine ourselves to the symmetric PF model for the rest of this section.

Theorem 3.5.5 Let two symmetric PF models be identical except that one with market

size evolution functions {ˆαs,t(·)}T_≥t≥1 and the other with {αs,t(·)}T_≥t≥1. Assume that, for each period t, (i) the monotonicity condition (3.17) holds, and (ii) κsb,t(·) ≡ κ0_sb,t for some constant κ0

sb,t.

(a) If ˆαs,t(zt)≥ αs,t(zt) for each period t and all zt, we have, for each period t, ˆβs,tpf ≥ β_s,tpf, ˆppf_i,t∗(γt) ≤ pi,tpf∗(γt) for all i and γt ∈ [0, ¯γs,t]N, and ˆγs,tpf∗ ≥ γ

pf∗

s,t . Thus, for each period t, ˆppf_i,t∗(It, Λt, γt)≤ ppfi,t∗(It, Λt, γt) and ˆγi,tpf∗(It, Λt) ≥ γi,tpf∗(It, Λt) for all i, (It, Λt)∈ S, and γt∈ [0, ¯γs,t]N.

(b) If, for each period t, ˆαs,t(zt) ≥ αs,t(zt) for all zt and ˆκ′sa,t(zi,t) ≥ κ′sa,t(zi,t) ≥ 0 for all zi,t, we have, for each period t, ˆβ_s,tpf ≥ β_s,tpf, ˆy_s,tpf∗ ≥ y_s,tpf∗, ˆppf_i,t∗(γt) ≤ ppf_i,t∗(γt), and ˆγs,tpf∗ ≥ γ

pf∗

s,t . Thus, for each period t, ˆp pf∗

i,t (It, Λt, γt)≤ pi,tpf∗(It, Λt, γt), ˆ

xpf_i,t∗(It, Λt, γt)≥ xpf_i,t∗(It, Λt, γt), ˆγ_i,tpf∗(It, Λt)≥ γ_i,tpf∗(It, Λt) for all i, (It, Λt)∈ S, and γt∈ [0, ¯γs,t]N.

Theorem 3.5.5(a) shows that, in the symmetric PF model, higher intensity of the network effect (i.e., larger αs,t(·)) drives all the competing firms to make more promotional efforts and charge lower sales prices for each observed promotion vector. Moreover, if the intensities of both the network effect and the service effect (i.e., the magnitudes of αs,t(·) and κ′_sa,t(·)) are higher, Theorem 3.5.5(b) demonstrates that all the competing firms are prompted to maintain higher service levels as well. Therefore, in the PF model, the exploitation-induction tradeoff is stronger with more intensive service effect and network effect.

Theorem 3.5.6 Consider the stationary symmetric PF model. Assume that, for each

period t, (i) the monotonicity condition (3.17) holds, and (ii) πpf_s,t∗ is increasing in β_s,tpf₋₁. For each period t, the following statements hold:

(a) β_s,tpf ≥ β_s,tpf₋₁, y_s,tpf∗ ≥ ypf_s,t∗₋₁, p_s,tpf∗(γ) ≤ ppf_s,t∗₋₁(γ) for each γ ∈ [0, ¯γs]N, and γ_s,tpf∗ ≥ γ_s,tpf₋₁∗ .

(b) ppf_i,t∗(I, Λ, γ)≤ ppf_i,t−1∗ (I, Λ, γ), xpf_i,t∗(I, Λ, γ)≥ xpf_i,t−1∗ (I, Λ, γ), and γ_i,tpf∗(I, Λ)≥ γ_i,tpf₋₁∗ (I, Λ) for each i, (I, Λ)∈ S, and γ ∈ [0, ¯γs,t]N_.

Analogous to Theorem 3.4.6, Theorem 3.5.6 justifies the widely used introductory price and promotion strategy. More specifically, this result shows that if the market is stationary and symmetric in the PF model, the competing firms should decrease the promotional efforts (i.e., γ_s,tpf∗) and service levels (i.e., y_s,tpf∗), and increase the sales prices contingent on any realized promotional efforts (i.e., ppf_s,t∗(γt)), over the planning horizon. Hence, Theorem 3.5.6 suggests that, in the PF model, the exploitation-induction tradeoff is more intensive at the early stage of the sales season than at later stages.

To conclude this section, we remark that, because of the aforementioned intra-temporal exploitation-induction tradeoff under the promotion-first competition, Theorems 3.5.5- 3.5.6 cannot give the monotone relationships on the equilibrium outcomes of each firm i’s sales price (i.e., ppf_i,t∗(It, Λt, γss,tpf∗)) and post-deliver inventory level (i.e., x

pf∗

i,t (It, Λt, γss,tpf∗)).