Strategic inventories under limited commitment Supplementary Appendix Fabio Antoniou∗ and Raffaele Fiocco†
1
Introduction
This Supplementary Appendix complements the paper and proceeds as follows. Section 2 establishes the main formal results with a longer time horizon setting and collects the proofs of Remarks 1 and 2 (Section 7.2 of the paper). Section 3 formally investigates nonlinear pricing policies (Section 7.3 of the paper). Section 4 demonstrates that the producer’s strategic incentives to hold inventories persist when inventories are only partially observable (Section 7.4 of the paper). Section 5 provides an explicit characterization of the results for the case of linear demand. Section 6 considers a framework with linear demand and quadratic production costs.
2
Longer time horizon
Absence of inventories
In the absence of inventories, it follows from Dudine et al. (2006)
that the producer’s profits under limited commitment can be written as Πns (p1 ) = Πc (p1 ) − sb
T X
ns Dsτ (pτ ) ,
(A1)
τ =1
where Πc (p1 ) =
PT
τ =1 (pτ
− c) Dτ (pτ ) denotes the aggregate profits under full commitment
ns (p ) is positive in and pτ = p1 + (τ − 1) sb is the price in period τ . The buyer stockpiling Dsτ τ
any period τ ∈ {1, ..., T − 1} if the unit buyer stockpiling cost sb is not excessively large. The T equilibrium pricing policy {pns τ }τ =1 satisfies the following first-order condition
Πc|1
(p1 ) − sb
T X
ns Dsτ |τ (pτ ) = 0.
(A2)
τ =1 ns ns Given the buyer stockpiling Dsτ −1 inherited from period τ − 1, the buyer stockpiling Dsτ (pτ )
in period τ is the outcome of the following recursive maximization problem ns ns ns ns ns Zτns Dsτ −1 = max (pτ − c) Dτ (pτ ) − Dsτ −1 + Dsτ (pτ ) + Zτ +1 (Dsτ (pτ )) . pτ
∗
University of Ioannina, Department of Economics, and Humboldt University of Berlin, School of Business and Economics. Email address:
[email protected] † Universitat Rovira i Virgili, Department of Economics and CREIP, Avinguda de la Universitat 1, 43204 Reus, Spain. Email address:
[email protected]
1
It follows from the envelope theorem and the binding no-arbitrage constraint pτ +1 = pτ + sb that
∂Zτns +1 ns ∂Dsτ
= − (pτ + sb − c). Taking the first-order condition for pτ yields
ns ns ns Dsτ (pτ ) = Dsτ −1 − Dτ (pτ ) − (pτ − c) Dτ |τ (pτ ) + sb Dsτ |τ (pτ ) ,
(A3)
ns (·) over time. which describes the evolution of Dsτ
Proof of Remark 1. Under the supposition of the remark, the producer can accumulate in the first period the aggregate amount of inventories for the following T − 1 periods that maximizes the producer’s ex ante profits and makes the constraint of sequential optimality slack in equilibrium. The producer’s profits are given by Π∗ (p1 ) = Πc (p1 ) − sp
T X
(τ − 1) Dτ (pτ ) ,
(A4)
τ =2
where pτ = p1 + (τ − 1) sb . Using the following first-order condition Πc|1 (p1 ) − sp
T X
(τ − 1) Dτ |τ (pτ ) = 0,
(A5)
τ =2
we find that
p∗1
= c−
PT ∗ ∗ τ =2 (τ −1)Dτ |τ τ =2 φτ −sp , ∗ D1|1 (A4) that Π∗ (p∗1 ) = Πc (pc1 )
D1∗ +
It follows from (A1) and
PT
where φτ (·) ≡ Dτ (·) + (pτ − c) Dτ |τ (·). for sp = 0 and Π∗ (p∗1 ) decreases with sp ,
c c while Πns (pns eTp > 0 for the 1 ) < Π (p1 ) is independent of sp . Hence, there exists a threshold s
unit inventory holding cost such that Π∗ (p∗1 ) > Πns (pns eTp , where seTp is 1 ) if and only if sp < s defined by Π
c
(p∗1 )
−Π
c
(pns 1 )
+ sb
T X
ns Dsτ (pns τ )
−
seTp
τ =1
T X
(τ − 1) Dτ (p∗τ ) = 0.
τ =2
∗ c ns c Since p∗1 < pns 1 under relatively mild conditions and, in addition, p1 ≥ p1 and p1 > p1 , it holds P P 1 ns (pns ). Πc (p∗1 ) > Πc (pns eTp is such that seTp Tτ=2 (τ − 1) Dτ (p∗τ ) > sb Tτ=1 Dsτ τ 1 ). Then, s
Binding sequential optimality constraint small, i.e., c
0. τ =2
τ =1
τ =1
ns (·) > D is (·). Given the buyer stockpiling D is We derive the conditions under which Dsτ sτ sτ −1 is (p ) in period τ is the outcome of the inherited from period τ − 1, the buyer stockpiling Dsτ τ
following recursive maximization problem is is is is is is Zτis Dsτ −1 = max pτ Dτ (pτ ) − Dsτ −1 + Dsτ (pτ ) − sp Iτ (pτ ) + Zτ +1 Dsτ (pτ ) , pτ
where Iτis (pτ ) =
PT
is (p ) (for τ < T ) is the amount of inventories that the − Dsτ τ
t=τ +1 Dt (pt )
producer carries from period τ to period τ + 1. It follows from the envelope theorem and the binding no-arbitrage constraint pτ +1 = pτ + sb that
∂Zτis+1 is ∂Dsτ
= − (pτ + sb ). Taking the first-order
condition for pτ yields is is is is Dsτ (pτ ) = Dsτ −1 − Dτ (pτ ) − pτ Dτ |τ (pτ ) + sp Iτ |τ (pτ ) + sb Dsτ |τ (pτ ) ,
(A7)
is (·) over time. which describes the evolution of Dsτ ns (·) described by (A3) and D is (·) described by We are now in a position to compare Dsτ sτ ns (·) = D is (·) = 0. Using (A3) and (A7) yields (A7). Proceeding backwards, we find that DsT sT ns is ns is DsT −1 (·) = DsT −1 (·) − cDT |T (·), which implies DsT −1 (·) > DsT −1 (·) (for c > 0). It follows ns (·) > D is (·) for τ ∈ {1, ..., T − 2} if and only if from (A3) and (A7) that Dsτ sτ
−c
T X
Dt|t (pt ) + sp
t=τ +1
T X
is It|t (pt ) + sb
t=τ +1
T h i X is ns Dst|t (pt ) − Dst|t (pt ) > 0. t=τ +1
ns (·) > D is (·) if and only if Suppose T = 3. It holds Ds1 s1
−cD2|2 (p2 ) − cD3|3 (p3 ) − sp D3|3 (p3 ) + p3 D3|33 (p3 ) + sb cD3|33 (p3 ) > 0, where the inequality follows (for c > 0) as long as the curvature of D3 (·) is not too pronounced, such as with linear demand. High inventory turnover Suppose that the producer accumulates inventories only for the following period. This inventory strategy can be interpreted as “high inventory turnover”. Given that the constraint of sequential optimality is binding in equilibrium,2 the producer’s profits can be written as ish
Π
c
(p1 ) = Π (p1 ) − sb
T X τ =1
ish Dsτ (pτ )
− sp
" T X
# Dτ (pτ ) −
ish Ds1 (p1 )
,
(A8)
τ =2
ish (·) denotes the amount of buyer stockpiling in period τ under high inventory where Dsτ is pis if and only if turnover. Using (A6) and (A8), we find that Πish pis τ >Π τ 2
The ex ante optimal pricing policy cannot be sustained in equilibrium when inventories are accumulated only for the following period, since the producer succumbs to the temptation to produce the amount of inventories that maximizes the continuation profits.
3
" T # T h T X X i X is is ish is is is is ish is sb Dsτ pτ − Dsτ pτ + sp (τ − 2) Dτ pτ − Dsτ pτ + Ds1 p1 > 0, τ =1
τ =2
τ =1
where the second expression in square brackets is generally positive. We derive the conditions is (·) > D ish (·). Given the buyer stockpiling D ish inherited from period τ − 1, under which Dsτ sτ sτ −1 ish (p ) in period τ is the outcome of the following recursive maximization the buyer stockpiling Dsτ τ
problem h i ish ish ish ish ish ish p D (p ) − D + D (p ) −(c + s ) I (p )+Z D (p ) , Zτish Dsτ = max τ τ τ τ p τ τ sτ −1 sτ τ τ +1 sτ −1 pτ
ish (p )+D ish (p where Iτish (pτ ) = Dτ +1 (pτ +1 )−Dsτ τ τ +1 ) (for τ < T ) is the amount of inventories sτ +1
accumulated in period τ in order to cover the demand in period τ +1. It follows from the envelope theorem and the binding no-arbitrage constraint pτ +1 = pτ +sb that
∂Zτish +1 ish ∂Dsτ
= − (pτ + sb ). Taking
the first-order condition for pτ yields ish ish ish ish Dsτ (pτ ) = Dsτ −1 − Dτ (pτ ) − pτ Dτ |τ (pτ ) + (c + sp ) Iτ |τ (pτ ) + sb Dsτ |τ (pτ ) ,
(A9)
ish (·) over time. which describes the evolution of Dsτ is (·) = D ish (·) = 0. Using Proof of Remark 2. Proceeding backwards, we find that DsT sT is ish (A7) and (A9) yields DsT −1 (·) = DsT −1 (·) = DT (·) + pT DT |T (·). For T = 3, we have is (·) = D ish (·) and I is (·) = I ish (·). It follows from (A7) and (A9) that Ds2 s2 2 2 is ish ish Ds1 (·) = Ds1 (·) + cI2|2 (·) . is (·) > D ish (·) if and only if I ish (·) > 0 (for c > 0). Since I ish (·) = D (·) − D ish (·) It holds Ds1 3 s1 2 s2 2|2 ish (·) = D (·) + p D ish ish and Ds2 3 3 3|3 (·), we obtain I2|2 (·) = −D3|3 (·) − p3 D3|33 (·). Then, I2|2 (·) > 0
as long as D3 (·) is not too convex. It is rather straightforward to show that inventory accumulation is sequentially optimal. Once the second period has been reached, the producer must decide whether to accumulate inventories for the third period or not. It follows from our analysis that inventory accumulation mitigates the buyer stockpiling incentives in the second period. Therefore, the producer prefers to hold inventories in the second period as well (for sp small enough). Slack no-arbitrage constraint
If the no-arbitrage constraint is not binding between period
τ and period τ + 1, it follows from Dudine et al. (2006) that pτ + sb < pτ +1 never occurs in equilibrium and we must have pτ + sb > pτ +1 , which implies that buyer stockpiling is strictly dominated in period τ . A natural reason is that the demand falls in period τ + 1. When the ex ante optimal pricing policy {p∗τ }Tτ=1 is also sequentially optimal, inventory accumulation in the first period allows the producer to approach the full commitment solution, irrespective of whether the no-arbitrage constraint is binding or not. Note that the producer may not resort to any inventories in some periods when the no-arbitrage constraint is not binding. For instance, consider the full commitment case where the no-arbitrage constraint is not binding between periods T − 1 and T , i.e., pcT −1 + sb > pcT , and the price in period T is set at the static monopoly
4
level, i.e., pcT = pm T . In this case, the producer can replicate the full commitment price without using any inventories in period T . Suppose now that the constraint of sequential optimality is binding in equilibrium. In line with Dudine et al. (2006), let (T1 , ..., Tm ) be a sequence of dates, with 1 ≤ T1 ≤ ... ≤ Tm ≤ T such that the no-arbitrage constraint is not binding between periods Ti and Ti +1, i.e., pTi +sb > pTi +1 , for any i ∈ {1, ..., m}. Consider one specific time interval {Ti + 1, ..., Ti+1 }, such that the ns (·) > 0 under limited commitment. Since no-arbitrage constraint is binding and therefore Dsτ
Ti+1 is effectively a terminal period, we can proceed backwards from period Ti+1 to period Ti + 1, ignoring the rest of the game. If the producer accumulates inventories in period Ti + 1 to cover the aggregate demand until period Ti+1 and replicates the pricing policy in the absence of is (·) < D ns (·), inventories, it follows from our analysis that buyer stockpiling can decline, i.e., Dsτ sτ
which makes the producer better off.
3
Nonlinear pricing
Following Hendel et al. (2014), we assume that the producer offers the buyers a nonlinear pricing policy of the form {Qτ , Pτ } that specifies a quantity Qτ and a payment Pτ in each period τ ∈ {1, 2}. Let Vτ (·) be the buyer willingness to pay for consumption in period τ , where the (continuously differentiable) function Vτ (·) is increasing and concave, i.e., Vτ |τ (·) > 0 and Vτ |τ τ (·) < 0, with Vτ (0) = 0. The following remark characterizes the equilibrium under full commitment. Remark 3 Suppose that the producer offers the buyers a nonlinear pricing policy that consists of a menu of bundles {Qτ , Pτ } in each period τ ∈ {1, 2}. When the unit buyer stockpiling cost sb is sufficiently large, the full commitment equilibrium exhibits the following features: (i) no buyer stockpiling, i.e., Dscnl = 0; fb cnl (ii) first period consumption below the static efficient level, i.e., Qcnl 1 ≤ Q1 , where Q1
0; fb (iii) second period consumption at the static efficient level, i.e., Qcnl 2 = Q2 ; (iv) first period payment P1cnl = V1 Qcnl ; 1 cnl cnl 0 0 0 (v) second period payment P2 = V1 Q1 +V2 Qcnl − V1 Qcnl 2 1 − Ds + V2 (Ds ) − sb Ds .
Proof of Remark 3. The producer faces the following maximization problem max
Q1 ,Q2 ,P1 ,P2
P1 + P2 − c (Q1 + Q2 )
subject to the following buyer participation constraints max {V1 (Q1 − Ds ) + V2 (Q2 + Ds ) − P1 − P2 − sb Ds } ≥ 0 Ds
(A10)
max {V1 (Q1 − Ds ) + V2 (Q2 + Ds ) − P1 − P2 − sb Ds } ≥ Ds
max {V1 (Q1 − Ds ) + V2 (Ds ) − P1 − sb Ds } Ds
5
(A11)
max {V1 (Q1 − Ds ) + V2 (Q2 + Ds ) − P1 − P2 − sb Ds } ≥ V2 (Q2 ) − P2 . Ds
(A12)
Condition (A10) is the standard participation constraint, which ensures that buyers prefer to purchase the good in each period rather than stay out of the market. Conditions (A11) and (A12) are the participation constraints that guarantee the buyers’ unwillingness to skip a purchase event in any period. The first-order condition for Ds associated with the buyer surplus maximization problem on the left-hand side of constraints (A10), (A11) and (A12) is given by −V1|1 (Q1 − Ds ) + V2|2 (Q2 + Ds ) − sb ≤ 0.
(A13)
It can be easily verified ex post that condition (A13) holds with the (strict) inequality in equilibrium. This implies that buyers choose Dscnl = 0 in equilibrium. The first-order condition for Ds associated with the buyer surplus maximization problem on the right-hand side of constraint (A11) writes as −V1|1 (Q1 − Ds ) + V2|2 (Ds ) − sb ≤ 0.
(A14)
Let Ds0 (Q1 ) ≥ 0 be the value for Ds that satisfies (A14). Adding constraints (A11) and (A12) yields after some manipulation V1 (Q1 ) + V2 (Q2 ) − P1 − P2 ≥ V1 Q1 − Ds0 (·) + V2 Ds0 (·) − sb Ds0 (·) − V1 (Q1 ) ≥ 0, where the second inequality follows from the definition of Ds0 (·). Therefore, constraint (A10) is redundant. Since the producer’s profits increase with P1 and P2 , constraints (A11) and (A12) are binding in equilibrium. Substituting these constraints into the producer’s maximization problem yields max 2V1 (Q1 ) + V2 (Q2 ) − V1 Q1 − Ds0 (·) + V2 Ds0 (·) − sb Ds0 (·) − c (Q1 + Q2 ) .
Q1 ,Q2
Using (A14), it follows from the envelope theorem that the derivative of the expression in square brackets with respect to Q1 reduces to V1|1 (Q1 − Ds0 (·)). The first-order conditions for Q1 and Q2 are respectively given by 2V1|1 (Q1 ) − V1|1 Q1 − Ds0 (Q1 ) − c = 0 V2|2 (Q2 ) − c = 0. Let Qfτ b be the static first-best (efficient) quantity, where Vτ |τ
Qfτ b − c = 0, τ ∈ {1, 2}. We
find that under full commitment the first period equilibrium quantity (and consumption) Qcnl 1 fb is such that Qcnl 1 ≤ Q1 (given Vτ |τ τ (·) < 0 and the concavity of the producer’s profit function), fb 0 where Qcnl 1 < Q1 for Ds (·) > 0. The second period equilibrium quantity (and consumption)
Qcnl is such that Qcnl = Qf2 b . It follows from the binding participation constraints (A11) 2 2 and (A12) that the equilibrium buyer payments are P1cnl = V1 Qcnl in the first period and 1 cnl cnl cnl cnl 0 0 0 P2 = V1 Q1 + V2 Q2 − V1 Q1 − Ds + V2 (Ds ) − sb Ds in the second period. As the producer’s aggregate profits Πcnl increase with sb , i.e.,
6
∂Πcnl ∂sb
= Ds0 (by the envelope theorem),
a larger sb makes it more desirable for the producer to deter buyer stockpiling. We refer to the quadratic example below for the derivation of the threshold value for sb above which the solution formalized in the remark is optimal. We now turn to the characterization of the equilibrium under limited commitment. Remark 4 Suppose that the producer offers the buyers a nonlinear pricing policy that consists of a menu of bundles {Qτ , Pτ } in each period τ ∈ {1, 2}. When the unit buyer stockpiling cost sb is sufficiently small, the limited commitment equilibrium exhibits the following features: lcnl (Q ) < 1; (i) buyer stockpiling, i.e., Dslcnl > 0 such that 0 < Ds|1 1
(ii) first period consumption below the static efficient level, i.e., Qlcnl − Dslcnl < Qf1 b ; 1 (iii) second period consumption at the static efficient level, i.e., Qlcnl + Dslcnl = Qf2 b ; 2 (iv) first period payment P1lcnl = V1 Qlcnl − Dslcnl + V2 Dslcnl − sb Dslcnl ; 1 (v) second period payment P2lcnl = V2 Qlcnl + Dslcnl − V2 Dslcnl . 2 Proof of Remark 4. Proceeding backwards, the producer’s second period maximization problem is given by max P2 − cQ2
Q2 ,P2
subject to the following buyer participation constraint V2 Q2 + Dslcnl (Q1 ) − P2 ≥ V2 Dslcnl (Q1 ) , where Dslcnl (·) denotes the buyer stockpiling inherited from the first period (see below) and V2 Dslcnl (·) is the second period buyer surplus if the second period policy is rejected. Since the producer’s second period profits increase with P2 , the buyer participation constraint is binding in equilibrium. Substituting this constraint into the producer’s second period maximization problem and taking the first-order condition for Q2 yields V2|2 Q2 + Dslcnl (·) − c = 0.
(A15)
lcnl + D lcnl (·) = Qf b . Moving to the first This gives the second period quantity Qlcnl s 2 , where Q2 2
period, the producer’s maximization problem becomes max P1 − cQ1 + V2 Qlcnl + Dslcnl (·) − V2 Dslcnl (·) − cQlcnl 2 2 Q1 ,P1
subject to the following constraints V1 Q1 − Dslcnl (·) + V2 Dslcnl (·) − P1 − sb Dslcnl (·) ≥ 0 Dslcnl (Q1 ) = arg max V1 (Q1 − Ds ) + V2 (Ds ) − P1 − sb Ds . Ds
Recalling that P2 arises from the binding second period buyer participation constraint, the first constraint is a standard participation constraint that ensures the buyers’ willingness to accept the first period menu as well. The second constraint indicates that buyer stockpiling follows from the buyer surplus maximization problem. This yields the same first-order condition as in
7
(A14). Suppose that sb < selcnl ≡ V2|2 (0) − V1|1 b
Qlcnl . This implies that buyers choose 1 Ds =0
Dslcnl > 0 in equilibrium and therefore the first-order condition (A14) holds with equality. Note that Ds0 (Q1 ) = Dslcnl (Q1 ) for a given Q1 . Using (A14) and (A15), the first-order condition for profit maximization with respect to Q1 gives after some manipulation ih h i lcnl lcnl (Q1 ) V1|1 Q1 − Dslcnl (Q1 ) − c − sb Ds|1 (Q1 ) = 0. 1 − Ds|1 Applying the implicit function theorem to the first-order condition in (A14) (that holds with lcnl (Q ) < 1. Under limited commitequality) evaluated at Dslcnl (Q1 ), we obtain that 0 < Ds|1 1
ment, the first period equilibrium consumption Qlcnl − Dslcnl is such that Qlcnl − Dslcnl < Qf1 b 1 1 and the second period equilibrium consumption Qlcnl + Dslcnl is such that Qlcnl + Dslcnl = Qf2 b . 2 2 It follows from the binding buyer participation constraints that the equilibrium buyer pay ments are P1lcnl = V1 Qlcnl − Dslcnl + V2 Dslcnl − sb Dslcnl in the first period and P2lcnl = 1 V2 Qlcnl + Dslcnl − V2 Dslcnl in the second period. 2 Quadratic example For illustrative purposes, we now provide an example where buyers exhibit quadratic preferences. Suppose that the buyer surplus arising from the quantity consumed qτ is Vτ (qτ ) = ατ qτ − 21 qτ2 , which implies Vτ |τ (qτ ) = ατ −qτ , τ ∈ {1, 2}. It follows from Remarks 3 and 4 that Ds0 (Q1 ) = Dslcnl (Q1 ) =
1 2
(α2 − α1 + Q1 − sb ) (for Ds0 > 0). Using the results in
Remark 3, we find that the full commitment equilibrium quantities (sold) are Qcnl 1 =
1 (3α1 − α2 − 2c + sb ) and Qcnl 2 = α2 − c, 3
with associated payments P1cnl =
1 (3α1 − α2 − 2c + sb ) (3α1 + α2 + 2c − sb ) 18
P2cnl =
1 2 7α2 + 4α2 (c + sb ) − 11c2 − 4sb c − 2s2b . 18
and
The producer’s profits are Πcnl =
1 2 3α1 + 2α22 − 6α1 c + 5c2 − 2sb c − s2b − 2α2 (2c − sb ) . 6
(A16)
Using the results in Remark 4, the limited commitment equilibrium quantities (sold) are Qlcnl = α1 + α2 − 2c − 3sb and Qlcnl = 2sb , 1 2 with associated payments P1lcnl =
1 2 α1 + α22 − 2α2 sb − 2c2 − 4sb c − s2b and P2lcnl = 2sb (c + sb ) . 2
The producer’s profits are Πlcnl =
1 2 α1 + α22 − 2α1 c + 2c2 + 2sb c + 3s2b − 2α2 (c + sb ) . 2
8
(A17)
Using (A16) and (A17), it holds Πcnl ≥ Πlcnl if and only if sb ≥ senl enl b , where s b =
√ 4− 6 10
(α2 − c).
Since the limited commitment solution can be always replicated under full commitment, we find that for sb < senl b the solution in Remark 3 is no longer valid. A producer with full commitment powers prefers to induce buyer stockpiling. The solution in Remark 4 maximizes the producer’s ex ante profits in the presence of buyer stockpiling since it allows the producer to fully extract buyer surplus and to minimize the amount of buyer stockpiling. This implies that for sb < senl b the solution under full commitment coincides with the one under limited commitment.
4
Inventory observability
To relax the assumption that the producer’s inventories are perfectly observable and to preserve at the same time the simplicity of our framework, we introduce a reduced form game with discrete choices that incorporates the main elements of the baseline model. The game involves two players, a producer and a representative buyer. The timing of the game unfolds as follows. (I) The producer decides whether to hold inventories (I) or not (O). (II) Nature determines that, in the presence of the producer’s inventories, the buyer observes inventories with probability σ ∈ (0, 1) and does not observe any inventories with the complementary probability 1 − σ. In the absence of the producer’s inventories, the buyer does not observe any inventories. (III) The buyer decides whether to stockpile the good (S) or not (N ). (IV) The producer’s and the buyer’s payoffs materialize. This framework reflects in a simple and natural manner the imperfect observability of inventories. When the producer holds inventories, the buyer can discover this decision only with probability σ. Therefore, σ captures the accuracy of the signal s ∈ {I, O} that the buyer receives about the producer’s choice between holding inventories (I) or not (O). The idea is that the producer attempts to inform the buyer about its inventories but some noise may prevent communication from being successful. When the producer does not hold any inventories, the buyer receives the signal that correctly identifies the producer’s choice, i.e., s = O. This assumption is not necessary and is only imposed for the sake of convenience. What matters for our purposes is that the buyer cannot perfectly observe the producer’s inventories. Given the signal s = O, the buyer does not know the producer’s inventory choice when deciding whether to stockpile the good (S) or not (N ). Figure 2 illustrates the extensive form of this game. Let Π be the producer’s profits and V the buyer surplus. The subscripts of the players’ payoffs indicate the players’ actions that lead to these payoffs. The producer’s profits are such that (i) ΠIN > ΠIS , (ii) ΠIN > ΠOS , (iii) ΠOS > ΠIS and (iv) ΠON > ΠIN . The rationale for the inequalities in points (i)-(iv) follows from the baseline model. Points (i) and (ii) state that the profits of a producer that accumulates inventories and removes buyer stockpiling are higher than the profits in the presence of buyer stockpiling, regardless of whether the producer holds inventories or not. Points (iii) and (iv) indicate that, for a given strategy of the buyer (stockpiling or not), the producer prefers to abstain from (costly) inventory activities. As regards buyer surplus, we have (v) VIN > VIS and
9
producer ◦ O
I N ature σ
1−σ buyer S
ΠOS VOS
buyer N
S
ΠON VON
N
ΠIS VIS
S
ΠIN VIN
ΠIS VIS
N
ΠIN VIN
Figure 2: Inventory observability
(vi) VOS > VON . Points (v) and (vi) show that the producer’s inventories mitigate the buyer stockpiling incentives. There exists a threshold σ e ≡
ΠOS −ΠIS ΠIN −ΠIS
∈ (0, 1) such that for σ ≤ σ e the producer never
accumulates inventories. Intuitively, when the signal about inventories is sufficiently noisy, the producer cannot benefit from holding inventories. Hereafter, we focus on the more relevant case σ > σ e. The game does not exhibit any perfect Bayesian equilibrium in pure strategies. The rationale is the following. Suppose that the producer chooses strategy I with probability 1. If the non-singleton information set in Figure 2 is reached, the buyer attaches a belief 1 to strategy I and therefore to being at the right decision node. This gives the producer σΠIN + (1 − σ) ΠIN = ΠIN . However, the producer has an incentive to deviate by selecting strategy O, which yields ΠON > ΠIN (see the inequalities in points (i)-(vi) above). Similarly, the producer cannot choose strategy O with probability 1 in equilibrium, either. Suppose that, by contradiction, this is an equilibrium. The buyer assigns a belief 1 to strategy O and therefore to being at the left decision node of the non-singleton information set in Figure 2. The producer obtains ΠOS . However, the producer prefers to switch to strategy I, which yields σΠIN + (1 − σ) ΠIS > ΠOS , where the inequality follows from σ > σ e. The game admits an equilibrium in mixed strategies. We define (ν, 1 − ν) as the probability distribution according to which the producer randomizes over the pure strategies I and O. The non-singleton information set in Figure 2 is reached with probability 1 − ν + ν (1 − σ) = 1 − νσ. Applying Bayesian updating, the probability that the buyer is at the right decision node of the non-singleton information set given that it has been reached is λ =
ν(1−σ) 1−νσ .
Put differently, λ denotes the
belief that the buyer assigns to the producer’s choice of the strategy I upon receiving the signal s = O. The buyer is indifferent between the pure strategies S and N in the nonsingleton information set if and only if λVIS + (1 − λ) VOS = λVIN + (1 − λ) VON . This yields λ=
VOS −VON VOS −VON +VIN −VIS
∈ (0, 1) (see the inequalities in points (v)-(vi) above). As λ =
ν(1−σ) 1−νσ ,
we
find after some manipulation that the equilibrium probability that the producer holds inventories
10
is ν ∗ =
VOS −VON VOS −VON +(1−σ)(VIN −VIS )
∈ (0, 1). Intuitively, an increase in σ leads to a higher ν ∗ , since
more accurate communication about inventories makes it more likely that the producer selects inventories in equilibrium. In the limit where σ → 1 and therefore communication tends to be perfect, the producer prefers to accumulate inventories, i.e., ν ∗ → 1, consistently with the baseline model. Turning to the buyer, we define (µ, 1 − µ) as the probability distribution according to which the buyer randomizes over the pure strategies S and N when being at the non-singleton information set in Figure 2. Since in equilibrium the producer must be indifferent between the pure strategies I and O, we have σΠIN + (1 − σ) [µΠIS + (1 − µ) ΠIN ] = µΠOS + (1 − µ) ΠON . The equilibrium probability of buyer stockpiling is µ∗ =
ΠON −ΠIN ΠON −ΠOS −(1−σ)(ΠIN −ΠIS )
∈ (0, 1) (recall
the inequalities in points (i)-(iv) above and σ > σ e). Therefore, the perfect Bayesian equilibrium of the game prescribes that the producer holds inventories with probability ν ∗ ∈ (0, 1) and the buyer stockpiles the good with probability µ∗ ∈ (0, 1).
5
Linear demand
5.1
Discussion
We now explicitly characterize the results of the baseline model in a setting with a linear demand of the form Dτ (pτ ) = ατ − pτ , τ ∈ {1, 2}. The static monopoly price in period τ is pm τ =
ατ +c 2 .
Assumption 2 becomes the following. m Assumption 200 pm 1 + sb < p2 ⇐⇒ sb
0, Assumption 200 requires that α2 > α1 . Therefore, the demand grows in the second period. The formal results are provided in Section 5.2. The condition about the unit cost of production stated in Proposition 1 is c ≥ e c≡
α2 −α1 −2sb −sp . 2
This ensures that the ex
ante optimal inventory level is not excessive in the second period. Note that the threshold e c decreases with sb . This is because more expensive buyer stockpiling induces a higher second period price, which alleviates the producer’s temptation to discard a portion of inventories in the second period and to increase the price above the level at which inventories coincide with the quantity sold. Figure 3 illustrates the case c ≥ e c. Suppose first that the demand growth is sufficiently pronounced or the unit buyer stockpiling cost is relatively small, i.e., sb
sp ≥ 0, which allows us to focus on the most plausible parameter constellations and to limit the number of case distinctions. The following remark characterizes the results for the case c ≥ e c, where e c≡
α2 −α1 −2sb −sp . 2
The threshold values for sp are derived in the proof.
Remark 5 Suppose c ≥ e c. Then, the limited commitment equilibrium exhibits the following features. A. For sb
sp1 , then I ns = 0, Dsns = α2 +c−2sb −2pns 1 , and B. For sb ≥
α1 +α2 +2(c−sb )+sp , p∗2 = p∗1 + sb ; 4 α1 +α2 +2c ns pns , pns 1 = 2 = p1 +sb . 4
α2 −α1 , 4
(B.1) if sp ≤ sp2 , then the outcome in (A.1) applies; nn nn m (B.2) if sp > sp2 , then I nn = Dsnn = 0, and pnn 1 = p2 − sb , p2 = p2 =
α2 +c 2 .
Proof of Remark 5. In the absence of the producer’s inventories, it follows from the proof of Lemma 2 that the producer’s maximization problem is given by max (p1 − c) [α1 − p1 + Ds (p1 )] + (p1 + sb − c) [α2 − p1 − sb − Ds (p1 )] . p1
Using the first-order condition p2 =
α2 −Ds (p1 )+c 2
for the second period profit maximization
(i.e., the sequential optimality constraint) and the no-arbitrage constraint p2 = p1 + sb yields Ds (p1 ) = max {α2 + c − 2sb − 2p1 , 0}. Suppose first Ds (p1 ) > 0. Then, substituting Ds (p1 ) into the producer’s maximization problem and taking the first-order condition for p1 yields pns 1 =
α1 +α2 +2c 4
ns ns = 0 and D ns = α +c−2s −2pns = and pns 2 b s 2 = p1 +sb . Moreover, I 1
13
α2 −α1 −4sb . 2
The producer’s profits are Πns =
(α1 + α2 − 2c)2 + s2b . 8
(A18)
nn nn m If α2 − α1 − 4sb ≤ 0, then Dsnn = I nn = 0, with prices pnn 1 = p2 − sb and p2 = p2 =
α2 +c 2 .
The producer’s profits are Πnn =
(α1 − c) (α2 − c) + 2 (α2 − α1 ) sb − 2s2b . 2
(A19)
In the presence of the producer’s inventories, it follows from the proof of Proposition 1 that the producer’s maximization problem is given by max (p1 − c) (α1 − p1 ) + (p1 + sb − c − sp ) (α2 − p1 − sb ) , p1
α1 +α2 +2(c−sb )+sp and p∗2 = p∗1 + 4 and Ds∗ = 0. The producer’s profits
which yields p∗1 =
sb . Moreover, I ∗ = D2∗ = α2 − p∗2 =
3α2 −α1 −2(c+sb )−sp 4
are
(α1 + α2 )2 − 4α2 (c − sb ) − 4α1 (c + sb ) + 4 c2 − s2b 2α1 − 6α2 + 4 (c + sb ) + sp + sp . Π = 8 8 (A20) ∗
Comparing the producer’s profits yields the following results. A. For sb
0 and
∂sp2 ∂c
=
√ 3α2 −α1 −2c−2sb (α2 −c)(2α2 −α1 −c−2sb )
− 2 > 0, where the inequalities follow from the assump-
tions on the parameters of the model. The following remark formalizes the results for the case c < e c. The threshold values for sp are derived in the proof. Remark 6 Suppose c < e c. Then, the limited commitment equilibrium exhibits the following features. A. For sb ≤
α2 −α1 −2c , 4
is is is (A.1) if sp ≤ sp3 , then I is = D2is − Dsis = α2 − pis 2 − Ds , Ds = α2 − 2sb − 2p1 , and
pis 1 =
α1 +α2 +2c−sp , 4
is pis 2 = p1 + sb ;
ns (A.2) if sp > sp3 , then I ns = 0, Dsns = α2 +c−2sb −2pns 1 , and p1 =
B. For
α2 −α1 −2c 4
< sb
0. Then, substituting Ds (p1 ) into the producer’s maximization problem and taking the first-order condition for p1 yields pis 1 = is is is and pis 2 = p1 + sb . Moreover, Ds = α2 − 2sb − 2p1 = is α2 − pis 2 − Ds =
Πis =
α1 +α2 +2c+4sb −sp . 4
α2 −α1 −4sb −2c+sp 2
and I is =
The producer’s profits are
(α1 + α2 )2 − 4c (α1 + α2 ) + 8sb (c + sb ) + 4c2 2α1 + 2α2 + 4c + 8sb − sp − sp . (A21) 8 8
If α2 − α1 − 4sb − 2c + sp ≤ 0, then Dsin = 0 and I in = D2in = α2 − pin 2 = pin 1
=
pin 2
α1 +α2 +2c−sp 4 is D2 − Dsis =
− sb and
Πin =
pin 2
=
pm 2 |c=0
=
α2 2 .
α2 2 ,
with prices
The producer’s profits are
α1 α2 − 2α1 (c + sb ) + 2sb (α2 − c − sb ) − α2 sp . 2
(A22)
Comparing the producer’s profits yields the following results. A. For sb ≤
α2 −α1 −2c , 4
the producer’s equilibrium profits are given by max Πns , Πis . Using
(A18) q and (A21), we obtain that Πns ≤ Πis if and only sp ≤ sp3 , where sp3 ≡ α1 + α2 + 2c +
4sb −
B. For
(α1 + α2 + 2c + 4sb )2 − 8sb c.
α2 −α1 −2c 4
< sb
sp4 , the producer’s equilibrium profits are given by max Πns , Πis . Using (A18) and (A21), we obtain that Πns ≤ Πis if and only if sp4 C. For sb ≥
6
α2 −α1 , 4
α1 +2c+4sb +2
the producer’s equilibrium profits are
(α1 +2c)2 +2(4α1 +5c)sb +16s2b . 3 nn given by Π in (A19).
Linear-quadratic framework
We briefly present the results in a framework with linear demand and quadratic production costs. Suppose that the demand function in period τ ∈ {1, 2} is Dτ (pτ ) = ατ − pτ and the producer’s cost function is C (Qτ ) = γQ2τ , where Qτ is the quantity produced in period τ and γ > 0 is a parameter. It follows from the proof of Lemma 4 that the producer’s maximization problem under full commitment reduces to max p1 (α1 − p1 ) − γ (α1 − p1 + I)2 + (p1 + sb ) (α2 − p1 − sb ) − γ (α2 − p1 − sb − I)2 − sp I, p1 ,I
15
which yields pcc 1 =
(1+2γ)(α1 +α2 )−2(1+γ)sb , 4(1+γ)
cc cc pcc 2 = p1 + sb and I =
2γ(α2 −α1 −sb )−sp . 4γ
It follows from the proof of Lemma 5 that the producer’s maximization problem under limited commitment can be written as max p1 (α1 − p1 ) − γ [α1 − p1 + Ds (p1 , I) + I]2 + (p1 + sb ) (α2 − p1 − sb ) p1 ,I
− γ [α2 − p1 − sb − Ds (p1 , I) − I]2 − sb Ds (p1 , I) − sp I, where Ds (·) follows from the constraint of sequential optimality, which yields Ds (·) = α2 − 2[(p1 +sb )(1+γ)+γI] . The equilibrium prices and 1+2γ 8γ(1+γ)(sb −sp )−sp isc isc p2 = p1 + sb and I isc = . 4γ
inventories are pisc = 1
The corresponding values in the absence of
the strategic effect (i.e., imposing Ds|I (·) = 0) are pnisc = 1 pnisc = pnisc +sb and I nisc = 2 1
(1+2γ)(α1 +α2 )−2(1+γ)sp , 4(1+γ)
2γ(3+4γ)sb −[1+8γ(1+γ)]sp . 4γ(1+2γ)2
Comparing
α1 +α2 −2(1+γ)[sp −2γ(α1 +α2 −sb )] , 4(1+γ)(1+2γ) isc nisc cc prices yields pτ > pτ > pτ ,
τ ∈ {1, 2}. The comparisons between the equilibrium levels of inventories are more cumbersome. For interior values of inventories and buyer stockpiling, it holds I isc > I nisc and I cc > I nisc . Moreover, we find that I isc R I cc .
16
