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09-11-2021

# Asymmetric Information and Differentiated Durable Goods Monopoly: Intra-Period Versus Intertemporal Discrimination

Authors: Didier Laussel, Ngo Van Long, Joana Resende

Published in: Dynamic Games and Applications | Issue 2/2022

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## Abstract

A durable good monopolist faces a continuum of heterogeneous customers who make purchase decisions by comparing present and expected price-quality offers. The monopolist designs a sequence of price-quality menus to segment the market. We consider the Markov perfect equilibrium (MPE) of a game where the monopolist is unable to commit to future price-quality menus. We obtain the novel results that: (a) under certain conditions, the monopolist covers the whole market in the first period (even when a static Mussa–Rosen monopolist would not cover the whole market), because this is a strategic means to convince customers that lower prices would not be offered in future periods and that (b) this can happen only under the stage-wise Stackelberg leadership assumption (whereby consumers base their expectations on the value of the state variable at the end of the period). Conditions under which MPE necessarily involves sequentially trading are also derived.

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Appendix
Available only for authorised users
Footnotes
1
p.156.

2
Technically, as noticed by Inderst [9],“while the real time in which the market is served goes to zero as the time between periods shrinks, the number of periods it takes to clear the market also increases”.

3
More precisely, consumers may use the Tesla online platform (https://​www.​tesla.​com/​en_​GB/​models/​design#battery) to parametrize a wide range of characteristics of the car (including the personalized dashboard). At the end, different consumers may end up getting differentiated variants of their cars.

4
See Laussel et al. [18] for a model in which a monopolist firm producing a non-durable good gets full information on customers’ preferences after their first purchase, using such information to make personalized price-quality offers (first-degree price discrimination). Laussel et al. [19] also look at the case of non-durable goods. Their setup departs from the present model (and also from [18]) since there, the monopolist is unable to adjust the specification of the product (quality) when it gets information on customers’ tastes (after the customers’ first purchase).

5
We are grateful to an anonymous referee for drawing our attention to this point.

6
For example, McKinsey [24] argues that “Personalization is teetering on the edge of the buzzword precipice. But companies that can figure out what it really means and how to take advantage of it are already outstripping their competition.” For more information, visit https://​www.​mckinsey.​com/​business-functions/​marketing-and-sales/​our-insights/​perspectives-on-personalization-at-scale [Access date: 25 February 2020] .

7
In the Gap case the price tends instantaneously toward the lowest type’s valuation. The profit is substantially eroded but not entirely dissipated. Immediate full market-coverage still holds.

8
Laussel et al. [20] show that the standard results for Coasian dynamics must be modified when the durable-good monopolist participates in two distinct markets with consumption network effects (the primary market and the aftermarket).

9
In static settings, the issue of optimal product design has been widely studied, following the seminal work of Mussa and Rosen [26]. Some recent contributions include Deneckere and McAfee [3] and Johnson and Myatt [11]. Johnson and Myatt [10] look at the problem of optimal product design in a competitive setup.

10
The outside option may be, for instance, the possibility of buying another product.

11
The customer base is “super strong” if the lowest type customers’ maximum net surplus (gross surplus minus production cost) is strictly positive, i.e., if a benevolent social planer would always strictly prefer to supply them.

12
That does not mean that this consumption does not yield permanent per period benefits.

13
For instance, Kumar ([15], p. 900), wrote that “Each consumer is in the market for only one unit of the good and exits after making the purchase” . Inderst ([9], p. 174) stated that “Consumers want to buy at most a single good”. ([28], p. 6) wrote that “In the baseline setting, buyers have unit-demand for the product and exit the market upon purchasing either of the two varieties.” On the contrary, for a model where upgrading is possible, see Takeyama [30].

14
They remark that “after all, in these models, durability simply amounts to sales permanently depleting the demand for the good.

15
For instance, one may think of a cataract operation, or a removal of an internal organ such as “Appendix” or the tonsil.

16
The quality level being selected according to equation (2) so as to maximize the social surplus of each consumer’s type.

17
See below for more details.

18
Note that $$h(\theta )\frac{1}{r}q(\theta )$$ is not the information rent $$U(\theta )$$ of type $$\theta$$.

19
With the uniform distribution, $$h(\theta )={\overline{\theta }}-\theta$$, and $$\theta -h(\theta )=2\theta -{\overline{\theta }}$$, which is positive iff $$\theta \ge {\overline{\theta }}/2$$.

20
To compute the optimal cutoff, we use the condition $$\frac{1}{r}(2\theta ^{*opt}-{\overline{\theta }})\left[ q^{m}(\theta ^{*})\right] -B-\frac{1}{2}\left[ q^{m}(\theta ^{*opt})\right] ^{2}=0=>\frac{1}{r^{2}}(2\theta ^{*opt}-{\overline{\theta }})^{2}-B-\frac{1 }{2r^{2}}(2\theta ^{*opt}-{\overline{\theta }})^{2}=0,$$implying: $$\theta ^{*opt}=\frac{{\overline{\theta }}+r\sqrt{2B}}{2}=\frac{2 \underline{\theta }+{\widehat{\theta }}}{2}>\underline{\theta }.$$

21
Function $${\widetilde{v}}(.)$$ is defined in Lemma 1 by Eq. (11).

22
See Laussel et al. [18], Claim 1, for a proof of this result (in “Appendix”) in a different context (a model with non-durable goods, where the monopolist gets full information on consumers’ type $$\theta$$ after their first purchase).

23
A similar condition has been obtained in Laussel et al. [18] for the case of a monopolist that sells non-durable goods (and gets full information on consumers’ preferences after their first purchase).

24
See, e.g., Long [21] for an exposition of the concept of stage-wise Stackelberg leadership and for a review of that literature.

25
This follows from Eq. (28), where $$\theta _{n+1}$$ is replaced with $$\underline{\theta }$$, and where $$U(\underline{\theta })=0$$.

26
The first case (Case A) arises when $$\Theta (n+1)$$ is such that $$q^{m}(\Theta (n+1)|\Theta (n))>\beta q^{se}(\Theta (n+1)),$$ whereas the second case (Case B) arises when $$\Theta (n+1)$$ is such that $$q^{m}(\Theta (n+1)|\Theta (n))<\beta q^{se}(\Theta (n+1)).$$

27
It should be noticed in addition that this occurs here in one period (the initial one) while in the standard durable model it takes (in the limit, as the duration of each period becomes infinitesimal) an infinite number of periods.

28
The proof is analogous to the proof of Lemma 1.

29
Notice that the numerator is positive since $$\underline{\theta }\ge \frac{ {\overline{\theta }}+r\sqrt{2B}}{2}$$ (full market coverage by the static MR monopolist) and the denominator is positive as well since the market is strong.

30
Note that by Assumption A4 and $$h({\overline{\theta }})=0$$, we have $$v\left( {\overline{\theta }}\right) >0$$.

31
Notice that $$q^{se}(\theta )=\arg \underset{q}{\max }\frac{\theta }{r} q-c(q).$$

32
Notice that the strict monotonicity of $$q^{m}(\theta |\Theta (n))$$ with respect to $$\theta$$ (which follows from our assumptions) ensures that the solution is unique.

33
Notice that this condition is weaker than the condition $${\overline{\theta }} \le \underline{\theta }\left[ (2-\beta )-\sqrt{(4-\beta )({\overline{\beta }} -\beta )}\right] ;$$.

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Metadata
Title
Asymmetric Information and Differentiated Durable Goods Monopoly: Intra-Period Versus Intertemporal Discrimination
Authors
Didier Laussel
Ngo Van Long
Joana Resende
Publication date
09-11-2021
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 2/2022
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00405-z

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