Indice de Egocentrismo

A

Proofs

Proof of Proposition 1: Let’s look for participatory RE equilibrium, where φ = 1 and q > 0. All we need to do is to check the four conditions specified in Definition 1.

First, we look at retailer’s problem: Given belief ˆφ, the retailer maximizes total profit π = pE min( ˆφD, q) − cq + rE( ˆφD)+wE((1 − ˆφ)D) + wE( ˆφD − q)+= (p−w)E min( ˆφD, q)−cq +( ˆφr + w)ED, which is a typical newsvendor problem (plus a constant ( ˆφr + w)ED), and therefore the optimal order quantity q◦ is given by ¯Fq_ˆ◦

φ



= _p−wc ∧ 1, where x ∧ y means min(x, y).

Since in equilibrium, the retailer’s belief is consistent with the outcome, we have ˆφ = φ = 1. Thus, q◦ = ¯F−1



c p−w ∧ 1



. Since q > 0 in the participatory equilibrium, we must have p − c > w and thus q◦= ¯F−1_p−wc .

Note in equilibrium, we also need consumer’s belief to be consistent with the outcome, i.e., ˆ

ξ◦ = A(q◦) = E min(φD, q◦)/E(φD) = E min(D, ¯F−1(_p−wc ))/ED.

Finally, we go back to consumer’s decision. To ensure φ = 1, we need us ≥ uo, i.e., hs≤ ˆξ◦ho.

Therefore, we need hs ≤

E min(D, ¯F−1₍ c p−w))

ED ho.

Based on the analysis above, we find the conditions for a participatory equilibrium are hs≤ E min(D, ¯F−1(_p−wc ))

ED hoand p−c > w. And the equilibrium outcome is φ = 1 and q

◦ _{= ¯F}−1 c p−w

 .

Proof of Proposition 2: If min(hs, hb) ≤ ho, then the profit function π = pE min (D, q) − cq + rE min (D, q)+

wE (D − q)+= (p + r − w)E min(D, q) − cq + wED, which is a typical newsvendor problem (plus a constant wED). Thus, the optimal order quantity q∗ is given by ¯F (q∗) = _p+r−wc ∧ 1. Then, if p − c > w − r, we have q∗= ¯F−1_p+r−wc ; otherwise, q∗ = 0.

If min(hs, hb) > ho, then no customer comes to store and thus the retailer will stock nothing

Proof of Proposition 3: If hs∈ E minhD, ¯F−1 c p−w∧1 i ED ho, ho # and p − c > w − r, then

π∗ = pE min (D, q∗) − cq∗+ rE min (D, q∗) + wE(D − q∗)+ = (p + r − w) E min (D, q∗) − cq∗+ wED

> wED = π◦

where the inequality is due to the fact that q∗> 0. This completes the proof. Proof of Proposition 4: If hb ≤ ho< hs and p − c > w − r, then q∗ = ¯F−1

 c p+r−w  > 0 because of Proposition 2. Then,

π∗ = pE min (D, q∗) − cq∗+ rE min (D, q∗) + wE(D − q∗)+ = (p + r − w) E min (D, q∗) − cq∗+ wED

> wED = π◦

where the inequality is due to the fact that q∗> 0. This completes the proof.

Proof of Proposition 5: If hs≤ E min  D, ¯F−1  c p−w  ED ho and p − c > w, we have q ◦ _{6= q}∗ _{when r > 0.} Then

π◦ = pE min (D, q◦) − cq◦+ rED + wE(D − q◦)+ = (p◦− w) E min (D, q◦_{) − cq}◦_{+ rED + wED}

> (p∗− w) E min (D, q∗) − cq∗+ rED + wED

> (p∗− w) E min (D, q∗_{) − cq}∗_{+ rE min (D, q}∗_{) + wED}

= p∗E min (D, q∗) − cq∗+ rE min (D, q∗) + wE(D − q∗)+ = π∗

where the first inequality is due to the strict concavity of the newsvendor profit function and q◦ is the unique maximizer of π◦, and the second inequality is because ED > E min(D, q) for any q. This completes the proof.

ists. Same as what we did in the base model, we are going to look for participatory equilibrium, where α_s◦ > 0, α◦_so > 0 and q◦ > 0. All we need to do is to check the four conditions specified in Definition 2.

First, we look at retailer’s problem: Given belief ˆαo, ˆαs, ˆαso, the retailer maximizes total profit

π = pE min (( ˆαs+ ˆαso) D, q) − cq + rE ( ˆαs+ ˆαso) D + wE ˆαoD + wE_αˆsαˆ_{+ ˆ}so_αso(( ˆαs+ ˆαso) D − q)+=  p − w αˆso ˆ αs+ ˆαso 

E min(( ˆαs+ ˆαso)D, q) − cq + (( ˆαs+ ˆαso)r + ( ˆαo+ ˆαso)w)ED, which is a typical

newsvendor problem (plus a constant (( ˆαs+ ˆαso)r + ( ˆαo+ ˆαso)w)ED), and therefore the optimal

order quantity q◦ is given by ¯F  q◦ ˆ αs+ ˆαso  = c p−w_{αs+ ˆˆ}αsoˆ_αso ∧ 1.

Since in equilibrium, the retailer’s belief is consistent with the outcome, we have ˆαi = α◦_i,

i = o, s, so. Note, in any participatory equilibrium, ξ > 0. Then, α◦so

α◦ s+α◦so = v−p 2H−(v−p). Thus, q◦ = (α◦_s+ α◦_so) ¯F−1  c p−w_2H−(v−p)v−p ∧ 1 

. Since q◦ > 0 in the participatory equilibrium, we must have p − c > w_2H−(v−p)v−p and thus q◦ = (α◦_s+ α_so◦ ) ¯F−1



c p−w_2H−(v−p)v−p

 .

Note in equilibrium, we also need consumer’s belief to be consistent with the outcome, i.e., ˆ

ξ = A(q◦) = E min((α_s◦+ α◦_so)D, q◦)/E((α◦_s+ α◦_so)D) = E min(D, ¯F−1( c

p−w_2H−(v−p)v−p ))/ED, which

is indeed greater than 0. Thus, there are customers who are willing to come to store.

Based on the analysis above, we find the condition for a participatory equilibrium is p − c > w_2H−(v−p)v−p . And the equilibrium store inventory level q◦ = ¯F−1



c p−w_2H−(v−p)v−p

 . Proof of Proposition 7: Note the retailer’s profit can be expressed as

π =  p + r − w α ∗ so+ α∗bo α∗ s+ α∗b + α∗so+ α∗bo  E min ((α∗_s+ α∗_b + α∗_so+ α_bo∗ ) D, q)−cq+w(α∗_o+α∗_so+α∗_bo)ED

which is a typical newsvendor problem (plus a constant w(α_o∗+ α∗_so+ α∗_bo)ED). Thus, the optimal store inventory level is given by q∗= (α∗_s+ α∗_b + α∗_so+ α∗_bo) ¯F−1 c

p+r−w α∗so+α ∗ bo α∗_{s +α}∗ b+α∗so+α∗bo ∧ 1 !

Proof of Proposition 8: Denote A◦_l = {(hs, ho)| max(v − p − ho, −hs+ ξ◦(v − p)) < 0} and A∗_l =

{(h_s, ho)| max(v − p − ho, v − p − hs, v − p − βshs− βoho) < 0}. To prove part (i), we simply need to

show that A∗_l ⊂ A◦_l. Since ξ◦ < 1, for any (hs, ho) ∈ A_l∗, we will have (hs, ho) ∈ A◦_l. Thus, A∗_l ⊂ A◦_l.

Next, let’s look at part (ii). To simplify notation, we denote ∆◦ = _2H−(v−p)v−p and ∆∗ =

α∗so+α∗bo

α∗

s+α∗b+α∗so+α∗bo. Note if βs+ βo> 1, we have ∆

◦ _{< ∆}∗_{. Thus,} p−c

If there are customers visiting store when there is no BOPS, we need to have w < p−c_∆◦. The

following analysis assumes this condition holds.

Suppose r = 0. If w ≥ p−c_∆∗ (this is possible since p−c_∆◦ > p−c_∆∗), then no one comes to store when

there is BOPS . In this case, π∗ = wE (α∗_so+ α∗_bo+ α∗_o) D = wE (α◦_so+ α◦_o) D < pE min ((α◦_s+ α◦_so) D, q) − cq + rE (α_s◦+ α◦_so) D + wEα◦_oD + wE α ◦ so α◦ s+ α◦so ((α◦_s+ α◦_so) D − q)+ = π◦

where the inequality is because q = 0 is also a feasible but not the optimal solution to the case where there is no BOPS. If w < p−c_∆∗, then there are consumers visiting store when there is BOPS.

Note (1) π◦− π∗ is continuous in w, and (2) π◦ > π∗ when w = p−c_∆∗. Thus, the analysis above

implies that there exists ¯w < p−c_∆∗ such that π◦ > π∗ if w > ¯w.

Proof of Proposition 9: Note ˜πs is a typical newsvendor profit function. Then, the optimal order

quantity ˜q∗ is given by ¯F (˜q∗) = _θp+rc ∧ 1. Thus, if θp + r − c > 0, we have ˜q∗ = ¯F−1_θp+rc , otherwise, ˜q∗ = 0.

Proof of Proposition 10: In the decentralized system, for any θ, denote optimal stock levels in the store as ˜q∗(θ). Suppose we have the same stock level in the centralized system. Then clearly we will achieve exactly the same total profit as ˜π∗(θ), since the BOPS revenue is just shared between channels in the decentralized case. Thus, we must have ˜π∗(θ) ≤ π∗. And due to the fact that π is strictly concave in q and q∗ > 0 (because of the assumptions that hb < min(hs, ho) and

p − c > w − r), we have ˜π∗(θ) = π∗ if only if ˜q∗ = q∗.

Next, let’s compare the optimal store inventory levels in both the centralized and decentralized systems:

• If θ < p−w_p , then _θp+rc > _p+r−wc . Thus, ˜q∗= ¯F−1_θp+rc < ¯F−1_p+r−wc = q∗. Since ˜q∗6= q∗_,

˜

π∗< π∗.

• If θ = p−w_p , then _θp+rc = _p+r−wc . Thus, ˜q∗= ¯F−1_θp+rc = ¯F−1_p+r−wc = q∗. Since ˜q∗= q∗, ˜

• If θ > p−w_p , then _θp+rc < _p+r−wc . Thus, ˜q∗= ¯F−1_θp+rc > ¯F−1_p+r−wc = q∗. Since ˜q∗6= q∗, ˜

π∗< π∗.

Finally, let’s prove ˜π∗ is quasiconcave in θ.

Let’s first show that ˜π∗ is nondecreasing in θ < p−w_p . If θ ≤ c−r_p , then ˜q∗ = 0 and thus ˜

π∗ = wED, which is independent of θ. If θ ∈ (c−r_p ,p−w_p ), then ∂ ˜_∂θπ∗ = ∂ ˜_∂qπ∗∂ ˜_∂θq∗ > 0, because ∂ ˜_∂qπ∗ > 0 if q < ˜q∗ and ∂ ˜_∂θq∗ > 0.

Then, let’s show that ˜π∗ is nonincreasing in θ > p−w_p . Note ∂ ˜_∂θπ∗ = ∂ ˜_∂qπ∗∂ ˜_∂θq∗ < 0, because

∂ ˜π∗

∂q < 0 if q > ˜q

∗ _and ∂ ˜q∗

∂θ > 0.

Thus, we can conclude that ˜π∗ is quasiconcave in θ.