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Licensed Unlicensed Requires Authentication Published by De Gruyter June 14, 2014

Dynamic Price Discrimination with Customer Recognition

  • Ching-Jen Sun EMAIL logo

Abstract

This paper studies a general two-period model of product line pricing with customer recognition. Specifically, we consider a monopolist who can sell vertically differentiated products over two periods to heterogeneous consumers. Each consumer demands one unit of the product in each period. In the second period, the monopolist can condition the price–quality offers on the observed purchasing behavior in the first period. In this setup, the monopolist can price discriminate consumers in two dimensions: by quality as well as by purchase history. We fully characterize the monopolist’s optimal pricing strategy when there are two types of consumers. When the type space is a continuum, we show that there is no fully separating equilibrium, and some properties of the optimal contracts (price–quality pairs) are characterized within the class of partitional perfect Bayesian equilibria.

JEL Classification: D42; L11

Appendix

Proof of Proposition 5

Suppose to the contrary that there exists a fully separating continuation equilibrium with N=[θ_,a] for some aθ_,θˉ). Let (p1(q1(θ)),q1(θ))C1 denote the contract taken by consumer θ in the first period, where θ(a,θˉ]. WLOG we assume aζ. Since each consumer θ(a,θˉ] fully reveals his type in the first period, the firm will extract all surplus from them in the second period. Hence in the second period, the firm will offer (u(ϕ(θ),θ),ϕ(θ)) to the consumer who took (p1(q1(θ)),q1(θ)) in the first period. The utility of type θ who takes the offer for type θˆ in the first period is V˜(θˆ,θ)=u(q1(θˆ),θ)p1(q1(θˆ))+δmax[0,u(ϕ(θˆ),θ)u(ϕ(θˆ),θˆ)], and the value function is V(θ)=maxθˆ[θ_,θˉ]V˜(θˆ,θ). The general envelope theorem (Milgrom and Segal 2002, Theorem 2) can be applied here, and V() is differentiable a.e. Pick any θ at which V() is differentiable. The right-hand derivative at θ is V+(θ*)=Dθ+V˜(θ^,θ)|θ^=θ=θ*=θ[u(q1(θ^),θ)p1(q1(θ^))+δ(u(ϕ(θ^),θ)u(ϕ(θ^),θ^))]|θ^=θ=θ*=uθ(q1(θ*),θ*)+δuθ(ϕ(θ*),θ*). On the other hand, the left-hand derivative at θ is V(θ)=DθV˜(θˆ,θ)|θˆ=θ=θ=θ[u(q1(θˆ),θ)p1(q1(θˆ))]|θˆ=θ=θ=uθ(q1(θ),θ), a contradiction. Q.E.D.

Proof of Proposition 6

Let {(pi,qi)} be the first-period contracts offered by the firm with qi<qi+1, and {θi} be the cutoff points induced by {(pi,qi)}. Obviously θ1ζ, for the firm cannot get any profit in either period if θ1<ζ. Define Ai=[θi,θi+1]. For each Ai the firm chooses in the second period a cutoff point θAi[θi,θi+1] and a nondecreasing function q(|Ai):[θAi,θi+1][0,1] to maximize

[11]θAiθi+1Sq(θ|Ai),θuθq(θ|Ai),θF(θi+1)F(θ)f(θ)f(θ)dθ.

Let {θAi,q(|Ai)} denote the optimal second-period strategy in Ai. We notice that q(θi+1|Ai)=ϕ(θi+1). The incentive constraints for θi+1 and γ[θAi,θi+1) require that

u(qi+1,θi+1)pi+1u(qi,θi+1)pi+δθAiθi+1uθ(q(θ|Ai),θ)dθ
u(qi,γ)pi+δθAiγuθ(q(θ|Ai),θ)dθu(qi+1,γ)pi+1

Therefore for any γ[θAi,θi+1)

[12]u(qi+1,θi+1)u(qi+1,γ)u(qi,θi+1)u(qi,γ)δγθi+1uθq(θ|Ai),θdθ0

Dividing both sides by (θi+1γ) and letting γθi+1 gives us

[13]uθ(qi+1,θi+1)uθ(qi,θi+1)δuθ(ϕ(θi+1),θi+1).

By the mean-value theorem, a[qi,qi+1] such that

[14]uθq(a,θi+1)(qi+1qi)=uθ(qi+1,θi+1)uθ(qi,θi+1)δuθ(ϕ(θi+1),θi+1).

Hence

[15]qi+1qiδuθ(ϕ(θi+1),θi+1)uθq(a,θi+1)δuθ(ϕ(θi+1),θi+1)uθq(κ(θi+1),θi+1)δminθ[ζ,θˉ]uθ(ϕ(θ),θ)uθq(κ(θ),θ).

Q.E.D.

Proof of Proposition 7

Let (p1,q1) be the first-period contract offered by the firm and θ1l be the first-period cutoff point induced by (p1,q1). Define A=[θ1l,θˉ] and N=[θ_,θ1l]. In the second period, the firm can offer different menus of contracts in A and N. Let θAl be the cutoff point in A,θNl the cutoff point in N,q2(|A):[θAl,θˉ]0,1] a nondecreasing quality offering in A, and q2(|N):[θNl,θˉ]0,1] a nondecreasing quality offering in N. Then the firm’s profit function is

Π=[1F(θ1l)][p1c(q1)]+δ{θAlθ¯[S(q2(θ|A),θ)uθ(q2(θ|A),θ)1F(θ)f(θ)]f(θ)dθ
+θNlθ1l[S(q2(θ|N),θ)uθ(q2(θ|N),θ)F(θ1l)F(θ)f(θ)]f(θ)dθ}

To solve this problem, first let us determine the first-period price p1 given (θ1l,θAl, θNl,q1,q2(|A),q2(|N)). Consider consumer θ1ls behavior. In the first period, θ1l can choose to be the lowest type in A and gain no surplus in the second period, or the highest type in N and acquire some information rent. To satisfy the incentive constraint, p1 should be set to make θ1l indifferent between these two alternatives. By taking the contract (p1,q1), consumer θ1ls utility is u(q1,θ1l)p1. If he rejects the contract, then his utility will be δθNlθ1luθ(q2(θ|N),θ)dθ by the envelope theorem. Hence in equilibrium it must be the case that u(q1,θ1l)p1=δθNlθ1luθ(q2(θ|N),θ)dθ, or equivalently p1=u(q1,θ1l)δθNlθ1luθ(q2(θ|N),θ)dθ. Then the profit function can be rewritten as

[16]Π=[1F(θ1l)][u(q1,θ1l)c(q1)δθNlθ1luθ(q2(θ|N),θ)dθ]+δ{θAlθ¯[S(q2(θ|A),θ)uθ(q2(θ|A),θ)1F(θ)f(θ)]f(θ)dθ+θNlθ1l[S(q2(θ|N),θ)uθ(q2(θ|N),θ)F(θ1l)F(θ)f(θ)]f(θ)dθ}

The optimal choice of q1 must solve maxq1[0,1]u(q1,θ1l)c(q1). Hence q1=ϕ(θ1l). On the other hand, we note that the optimal choice of q2(θ1l|N) is ϕ(θ1l) as well. Furthermore, we also need to check the incentive compatibility for any θA and θN. Given that q1=q2(θ1l|N)=ϕ(θ1l), it is straightforward to verify that the incentive compatibility holds for all types.

For a given θ1l, let {θAl,θNl,q2(|A),q2(|N)} denote the optimal second-period strategy. To determine the optimal value of θ1l, we analyze the first-order condition for θ1l. We claim that θ1l>θl by showing that Πθ1l>0θθl.26 There are two cases to be considered: θAl=θ1l and θAl>θ1l.

Case 1. θAl>θ1l.Πθ1l in this case is (where we use the envelope theorem when taking a derivative to the second-period profit function):

Πθ1l=f(θ1l)[S(q1,θ1l)δθNlθ1luθ(q2(θ|N),θ)dθ]+[1F(θ1l)][uθ(q1,θ1l)θ1lδθNlθ1luθ(q2(θ|N),θ)dθ]+δS(q2(θ1l|N),θ1l)f(θ1l)θNlθ1luθ(q2(θ|N),θ)dθf(θ1l)]=[(1F(θ1l))uθ(q1,θ1l)f(θ1l)S(q1,θ1l)]+δS(q2(θ1l|N),θ1l)f(θ1l)(1F(θ1l))θ1lθNlθ1luθ(q2(θ|N),θ)dθ]=[(1F(θ1l))uθ(q1,θ1l)f(θ1l)S(q1,θ1l)]+δS(q2(θ1l|N),θ1l)f(θ1l)(1F(θ1l))uθ(q2(θ1l|N),θ1l)]δ(1F(θ1l))[θNlθ1lθ1luθ(q2(θ|N),θ)dθuθ(q2(θNl|N),θNl)θNlθ1l]=(1δ)f(θ1l)[S(ϕ(θ1l),θ1l)1F(θ1l)f(θ1l)uθ(ϕ(θ1l),θ1l)]δ(1F(θ1l))[θNlθ1lθ1luθ(q2(θ|N),θ)dθuθ(q2(θNl|N),θNl)θNlθ1l]

Fact 1. θ1luθ(q2(θ|N),θ)0 and θNlθ1l>0. Invoking the envelope theorem on the second-period value function at some θ on N,

Ψ(θ1l,θ)=maxq2(θ|N)[0,1]Φ(q2(θ|N))=S(q2(θ|N),θ)uθ(q2(θ|N),θ)F(θ1l)F(θ)f(θ),

we get θ1lΨ(θ1l,θ)=uθ(q2(θ|N),θ)f(θ1l)/f(θ)<0; hence, θNl is increasing in θ1l.q2(θ|N) solves the first-order condition Φq=0 (or Φq>0 if q2(θ|N)=1). Applying the implicit function theorem, we can get θ1lq2(θ|N)0. Thus

[17]θ1luθ(q2(θ|N),θ)=uθq(q2(θ|N),θ)θ1lq2(θ|N)0.

Fact 2. If θ1lθl, then we have

[18]S(ϕ(θ1l),θ1l)1F(θ1l)f(θ1l)uθ(ϕ(θ1l),θ1l)0,

for S(q,θl)1F(θl)f(θl)uθ(q,θl)q[0,1] is zero at most and θ1lθl.

Combining these two facts, we conclude that Πθ1l>0 when θ1lθl. Hence, the optimal choice θ1l(θl,θˉ) in this case.

Case 2. Now suppose θAl=θ1l. Then Πθ1l has one more term:

[19]δS(q2(θ1l|A),θ1l)uθ(q2(θ1l|A),θ1l)1F(θ1l)f(θ1l)f(θ1l),

which is non-negative when θ1lθl. Therefore, the optimal choice θ1l(θl,θˉ).

Next we study the comparative statics. Since θ1l(θl,θˉ), it is sufficient for us to analyze Πθ1l for θ1l(θl,θˉ). In this range, θAl=θ1l, and Πθ1l can be expressed as Πθ1l=f(θ1l)χ(ϕ(θ1l),θ1l)+δγ(θ1l), where

[20]χ(ϕ(θ1l),θ1l)=S(ϕ(θ1l),θ1l)1F(θ1l)f(θ1l)uθ(ϕ(θ1l),θ1l)
[21]γ(θ1l)=f(θ1l)χ(ϕ(θ1l),θ1l)S(q2(θ1l|A),θ1l)uθ(q2(θ1l|A),θ1l)1F(θ1l)f(θ1l)
(1F(θ1l))θNlθ1lθ1luθ(q2(θ|N),θ)dθuθ(q2(θNl|N),θNl)θNlθ1l

Therefore we have 2Πθ1lδ=γ(θ1l). If γ(θ1l)>0, then Π is (locally) supermodular at (δ,θ1l), which implies that the optimal first-period cutoff point θ1l is increasing in δ. Similarly, if γ(θ1l)<0, then θ1l is decreasing in δ. So we have signθ1lδ=signγ(θ1l). The first-order condition Πθ1l|θ1l=θ1l=0 implies that signχ(ϕ(θ1l),θ1l)=signγ(θ1l). Thus we have signθ1lδ=signχ(ϕ(θ1l),θ1l). On the other hand, we observe that S(q,θ) is supermodular on (q,θ) by the single crossing property. Hence the optimal first-period quality q1=ϕ(θ), which maximizes the social surplus at θ, is increasing in θ. Therefore signq1δ=signθ1lδ. Q.E.D.

Proof of Proposition 8

Case 1. When δ is high. This is a direct consequence of proposition 6 and the fact that q[0,1].

Case 2. When S(q,θ) is log submodular. First, we show that any two-contract strategy is strictly dominated by a single-contract strategy when S(q,θ) is log submodular. To make the analysis clear, we divide the proof into three steps.

Step 1. Write down the profit function. Let C1={(pi,qi)}i=12 be any two-contract strategy with q1<q2, and {θi}i=12 with θ1<θ2 the cutoff points induced by C1. Define Ai=[θi,θi+1] (let θ3=θˉ) and N=[θ_,θ1]. Let {θAi,q(|Ai)} denote the optimal strategy in Ai, and {θN,q(|N)} the optimal strategy in N. As explained in the proof of Proposition 7, (p1,p2) must satisfy the following equations:

[22]u(q1,θ1)p1=δθNθ1uθ(q(θ|N),θ)dθ
[23]u(q2,θ2)p2=u(q1,θ2)p1+δθA1θ2uθ(q(θ|A1),θ)dθ

Thus the profit function can be written as:

[24]Π=[1F(θ2)][u(q2,θ2)u(q1,θ2)+u(q1,θ1)[δθNθ1uθ(q(θ|N),θ)dθδθA1θ2uθ(q(θ|A1),θ)dθc(q2)]+[F(θ2)F(θ1)][u(q1,θ1)δθNθ1uθ(q(θ|N),θ)dθc(q1)]+δ{θA2θ¯[S(q(θ|A2),θ)uθ(q(θ|A2),θ)1F(θ)f(θ)]f(θ)dθ+θA1θ2[S(q(θ|A1),θ)uθ(q(θ|A1),θ)F(θ2)F(θ)f(θ)]f(θ)dθ+θNθ1[S(q(θ|N),θ)uθ(q(θ|N),θ)F(θ1)F(θ)f(θ)]f(θ)dθ}

Step 2. We claim that θ1>θl. There are two cases to be considered: θA1=θ1 and θA1>θ1. First, let us assume θAl>θ1l. Using the envelope theorem, the first-order condition w.r.t. θ1 in this case is :

[25]Πθ1=[1F(θ2)][uθ(q1,θ1)δθ1θNθ1uθ(q(θ|N),θ)dθ]f(θ1)[u(q1,θ1)δθNθ1uθ(q(θ|N),θ)dθc(q1)]+[F(θ2)F(θ1)][uθ(q1,θ1)δθ1θNθ1uθ(q(θ|N),θ)dθ]+δS(q(θ1|N),θ1)f(θ1)f(θ1)θNθ1uθ(q(θ|N),θ)dθ]=[1F(θ1)][uθ(q1,θ1)δθ1θNθ1uθ(q(θ|N),θ)dθ]f(θ1)S(q1,θ1)+δS(q(θ1|N),θ1)f(θ1)=[(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)]+δf(θ1)S(q(θ1|N),θ1)(1F(θ1))uθ(q(θ1|N),θ1)]δ(1F(θ1))[θNθ1θ1uθ(q(θ|N),θ)dθuθ(q(θN|N),θN)θNθ1]

Fact 1. θNθ1θ1uθ(q(θ|N),θ)dθuθ(q(θN|N),θN)θNθ1<0, as demonstrated in the proof of Proposition 7.

Fact 2. q(θ1|N)=1, since S(q,θ) is log submodular.

Fact 3. δuθ(q1,θ1)uθ(q(θ1|N),θ1) from the incentive constraint.

Fact 4. f(θ1)S(q(θ1|N),θ1)(1F(θ1))uθ(q(θ1|N),θ1)0 if θ1θl, since S(q,θl)1F(θl)f(θl)uθ(q,θl) is zero at most and θ1θl.

Fact 5. (1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)0 if θ1θl, since S(q,θl)1F(θl)f(θl)uθ(q,θl) is zero at most and θ1θl.

Fact 6. We show that if θ1θl, then

[(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)]+δf(θ1)S(q(θ1|N),θ1)(1F(θ1))uθ(q(θ1|N),θ1)]>0.

From Facts 2–5, we can get

[26][(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)]+δf(θ1)S(q(θ1|N),θ1)(1F(θ1))uθ(q(θ1|N),θ1)]=[(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)]+δf(θ1)S(1,θ1)(1F(θ1))uθ(1,θ1)][(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)]+uθ(q1,θ1)uθ(1,θ1)[f(θ1)S(1,θ1)(1F(θ1))uθ(1,θ1)]=f(θ1)uθ(1,θ1)[uθ(q1,θ1)S(1,θ1)uθ(1,θ1)S(q1,θ1)]=f(θ1)S(1,θ1)S(q1,θ1)uθ(1,θ1)[uθ(q1,θ1)S(q1,θ1)uθ(1,θ1)S(1,θ1)]>0.

The last inequality comes from the log submodularity of S(q,θ).

Combining Facts 1 and 6, we conclude that Πθ1>0 for any θ1θl. Hence, the optimal choice of θ1 is strictly greater than θl.

Now consider the second case θA1=θ1. In this case, the change of θ1 has two additional effects. The first-order condition w.r.t. θ1 becomes:

[27]Πθ1=[(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)]+δf(θ1)S(q(θ1|N),θ1)(1F(θ1))uθ(q(θ1|N),θ1)]+δ(1F(θ2))uθ(q(θ1|A1),θ1)δf(θ1)S(q(θ1|A1),θ1)(F(θ2)F(θ1))uθ(q(θ1|A1),θ1)]δ(1F(θ1))[θNθ1θ1uθ(q(θ|N),θ)dθuθ(q(θN|N),θN)θNθ1]=[(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)]+δf(θ1)S(q(θ1|N),θ1)(1F(θ1))uθ(q(θ1|N),θ1)]δf(θ1)S(q(θ1|A1),θ1)(1F(θ1))uθ(q(θ1|A1),θ1)]δ(1F(θ1))[θNθ1θ1uθ(q(θ|N),θ)dθuθ(q(θN|N),θN)θNθ1]

The log submodularity of S(q,θ) implies that q(θ1|A1)=q(θ1|N)=1, therefore

[28]Πθ1=(1F(θ1))uθ(q1,θ1)f(θ1)S(q1,θ1)δ(1F(θ1))θNθ1θ1uθ(q(θ|N),θ)dθuθ(q(θN|N),θN)θNθ1.

As shown in the first case, this term is positive if θ1θl. Hence θ1>θl.

Step 3. We show that any two-contract strategy is dominated by a one-contract strategy. From step 2, we know that θ1>θl. It is then easy to see that θA2=θ2 and θA1=θ1. We also know that q(θ|N)=q(θ|A1)=q(θ|A2)=1; hence, the profit of this two-contract strategy is

[29]Π=[1F(θ2)][u(q2,θ2)u(q1,θ2)+u(q1,θ1)δθNθ2uθ(1,θ)dθc(q2)]+[F(θ2)F(θ1)][S(q1,θ1)δθNθ1uθ(1,θ)dθ]+δ{θ1θ¯S(1,θ)f(θ)dθθ2θ¯uθ(1,θ)(1F(θ))dθθ1θ2uθ(1,θ)(F(θ2)F(θ))dθ+θNθ1[S(1,θ)uθ(1,θ)F(θ1)F(θ)f(θ)]f(θ)dθ}

Now if we just offer one quality q2 and set the cutoff point at θ1, then the profit is:

[30]Π˜=(1F(θ1))[u(q2,θ1)δθNθ1uθ(1,θ)dθc(q2)]+δ{θ1θ¯[S(1,θ)uθ(1,θ)1F(θ)f(θ)]f(θ)dθ+θNθ1[S(1,θ)uθ(1,θ)F(θ1)F(θ)f(θ)]f(θ)dθ}

After rearranging terms, the difference is:

[31]Π˜Π=[1F(θ2)]S(q1,θ2)[1F(θ1)]S(q1,θ1)+[1F(θ1)]S(q2,θ1)[1F(θ2)]S(q2,θ2)=[1F(θ1)]q1q2Sq(q,θ1)dq[1F(θ2)]q1q2Sq(q,θ2)dq=q1q2θ1θ2Sq(q,θ)f(θ)(1F(θ))Sqθ(q,θ)dθdq>0

by the log submodularity of S(q,θ). Therefore, the firm will not offer two contracts in the first period.

The N-contract case has a recursive structure, and we can duplicate the procedure of the proof above to show that there is a (N−1)-contract that dominates this N-contract strategy. Repeating this argument, we conclude that the firm’s optimal strategy is to offer a single contract in the first period. Q.E.D.

Acknowledgments

I would like to acknowledge James Peck, Howard P. Marvel, Huanxing Yang, Lixin Ye, Nejat Anbarci, Jan Bouckaert, an anonymous referee, and an associate editor for their helpful comments. All remaining errors are my own.

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  1. 1

    Amazon promised this practice would not happen again after a public uproar.

  2. 2

    The Total Rewards card collects gamblers’ characteristics like age and gender and playing habits like how much they spend per trip, what their favorite games are, and even how fast they pull a slot-machine lever.

  3. 3
  4. 4

    A function f(x,y)C2 is said to be log supermodular if fxyffxfy>0.

  5. 5

    The examples include Mussa and Rosen (1978), Johnson and Myatt (2003) and Villas-Boas (1998) on product design decisions, Stokey (1979) on intertemporal price discrimination with commitment, Deneckere and McAfee (1996) and McAfee (2006) on damaged goods, Courty and Li (2000) and Gale and Holmes (1993) on advance purchase discounts, Bhargava and Choudhary (2001) and Varian (2001) on versioning information goods, and Anderson and Song (2004) and Nevo and Wolfram (2002) on coupons.

  6. 6

    By integrating the models in Mussa and Rosen (1978) and Stokey (1979) into a common framework, Salant (1989) provides an explanation for these two inconsistent findings.

  7. 7
  8. 8

    See Fudenberg and Villas-Boas (2007), Armstrong (2006) and Stole (2005) for the recent developments on this topic.

  9. 9
  10. 10

    Given the firm’s optimal first-period strategy, θH’s incentive constraint is binding, and multiple continuation equilibria could arise depending on the choice made by θH. Following the literature, here we implicitly assume θH always chooses the “right” bundle qˉ.Bester and Strausz (2001) and Stole (2001) show that the firm may obtain higher profits when θH plays mixed strategies. However, as there is no suitable equilibrium refinement for us to select a unique equilibrium, we feel that there is no reason why θH has an incentive to commit himself to a complicated mixed strategy that is favorable to the firm.

  11. 11

    IRH is implied by ICH and IRL and hence ignored.

  12. 12

    Under the condition S(qˉ,θL)S(q_,θL)S(qˉ,θH)S(q_,θH)αS(q_,θL)S(q_,θH),δc[0,1] only if αS(qˉ,θH)S(qˉ,θL)0. Hence T2 is the optimal strategy directly if αS(qˉ,θH)S(qˉ,θL)<0. Moreover, δc=1 when αS(q_,θH)=S(q_,θL), and δc=δ when α=(1δ)S(qˉ,θL)+δS(q_,θL)(1δ)S(qˉ,θH)+δS(q_,θH).

  13. 13

    The optimal second-period strategy can be readily seen once we know the first-period strategy.

  14. 14
  15. 15

    If the firm cannot commit to its future actions, they show that in equilibrium consumers must play mixed strategies which do not appear in our model. The reason is that in their model δ=1, and here we focus our analysis on the case where 0<δ<1.

  16. 16

    Without loss of generality here we assume that the exit option is in all menus C1 and C2A. Consumers can always choose to not take any contract.

  17. 17

    It can be readily seen in the proof that Assumption 2 plays no role in the non-existence of fully separating equilibrium.

  18. 18
  19. 19

    Caillaud and Mezzetti (2004) study equilibrium reserve prices in sequential ascending auctions. They analyze the equilibrium reserve prices in the set of “equilibria with separation under participation.” An equilibrium with separation under participation is a perfect Bayesian equilibrium in which in the first-period bidders with valuations above some threshold v follow a symmetric, strictly increasing bidding strategy, and bidders with valuations below v do not participate. It is a special case of partitional PBE.

  20. 20

    Recall that ζ is determined by S(ϕ(ζ),ζ)=0.

  21. 21

    This result is not as straightforward as it seems. A more rigorous explanation is as follows. If the firm does business with consumers only once, then it is going to serve all consumers down to the type that gives it zero virtual profit, i.e., θl is determined by equating the social surplus to information rent. On the other hand, in the long-term seller−buyer relationship with customer recognition, the change of θ1l affects not only the first-period profit but also the second-period profit in segments A and N. Starting at θ1l=θl, let us look at the effect of a change in θ1l. Increasing the cutoff point from θl slightly has no effect on the profit in A since the maximization problem in A is the same as the static one, and the virtual profit is zero at θl. The effects on the first-period profit and the profit in N are intertwined and somewhat more subtle. By increasing the cutoff point from θl slightly, the firm chooses to postpone serving consumer θl to the second period, sacrificing the virtual profit (in the sense as in the static model) at θl in the first period but recouping it in the second period. Since the virtual profit at θl is non-positive, the net effect is non-negative. Moreover, postponing serving θl saves the firm information rent paying to all consumers θθl. Combining these effects and the fact that increasing θ1l from θl slightly has no effect on the profit in A, the firm finds it optimal to serve fewer consumers in the first period than it does in the static relationship.

  22. 22

    This paper focuses its analysis on nondurable goods. Fudenberg and Tirole (1998) study upgrades and buy-backs for a durable good. Since they assume that only a high-quality generation arrives in the second period, there is no possibility for consumers to downgrade the product. One possible and interesting extension of their basic model is to allow the firm to offer not only a high-quality version but also an inferior or damaged version in the second period.

  23. 23
  24. 24

    The trade-in options to purchase iPhone 5c/5s offered by Walmart and Best Buy could be viewed as a form of BBPD.

  25. 25

    See Armstrong (2006) for a discussion of this direction.

  26. 26

    Recall that θl is the optimal cutoff point in the static model.

Published Online: 2014-6-14
Published in Print: 2014-1-1

©2014 by De Gruyter

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