I show that Swan’s (1970) independence result requires a multiplicative interaction between durability and all other quality attributes. Because there is no compelling argument for a multiplicativity in quality, monopolists tend to distort durability, even with constant marginal costs. Distortions in durability and other quality aspects are aligned exactly when the marginal cost of quality do not increase too much with durability.Durability; quality; monopoly

Strausz, Roland

English

Research Papers in Economics

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September 2009 
 
 
 
Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. 
Discussion Paper No. 271 
Monopoly Distortions in 
Durability and 
Multi-Dimensional Quality 
 
Roland Strausz* 
 
 Monopoly Distortions in Durability and
Multi–Dimensional Quality
Roland Strausz∗
Humboldt–Universit¨ at zu Berlin
September 3, 2009
Abstract
I show that Swan’s (1970) independence result requires a mul-
tiplicative interaction between durability and all other quality at-
tributes. Because there is no compelling argument for a multiplica-
tivity in quality, monopolists tend to distort durability, even with
constant marginal costs. Distortions in durability and other quality
aspects are aligned exactly when the marginal cost of quality do not
increase too much with durability.
Keywords: Durability, quality, monopoly
JEL Classiﬁcation No.: L15
∗Humboldt-Universit¨ at zu Berlin, Institute for Micro Economic Theory, Spandauer
Str. 1, D-10178 Berlin (Germany), strauszr@wiwi.hu-berlin.de, tel:+49.30.2093.5676;
fax:+49.30.2093.5938. I thank an anonymous referee for helpful comments and sugges-
tions. This research was supported by the Deutsche Forschungsgemeinschaft through
SFB/TR-15 ”Governance and the Eﬃciency of Economic Systems”.
11 Introduction
The seminal paper Swan (1970) demonstrates the surprising result that a
standard monopolist has no incentive to distort durability. The result implies
that durability choice is independent of market structure. Consequently,
there is no reason for governmental intervention concerning durability choice
even if markets are not competitive.
The result has found its way in most standard text books of industrial
organization (e.g., Tirole, 1988, p.102ﬀ; Carlton and Perloﬀ, 1989, p.628ﬀ;
Scherer and Ross, 1990, p.608ﬀ). It is well known however that Swan’s inde-
pendence result fails if marginal costs are not constant (e.g. Sieper and Swan
1973; Levhari and Peles 1973). More precisely, the independence result holds
when the cost function displays a multiplicative structure between durability
and quantity; constant marginal cost is but one example. Despite its speci-
ﬁcity, a constant return to scale technology is nevertheless the appropriate
benchmark for comparing the competitive outcome to a monopolistic one in
the long run, because, in the long run, the industry’s marginal cost equals
the minimum average cost at a production plant and is therefore constant.
This note shows, however, that, in the presence of additional quality at-
tributes, Swan’s independence result requires that durability and these other
attributes interact also multiplicatively. Yet, in contrast to the argument that
marginal costs tend to be constant in the long run, there is no compelling
reason why the interaction between durability and other quality characteris-
tics should be multiplicative. The exact interaction depends on technological
considerations alone.
To understand the intuition why additional quality attributes invalidate
Swan’s independence result, it is helpful to understand ﬁrst why, without
additional quality attributes, Swan’s result holds. The ﬁrst ingredient that
drives Swan’s result is that durability choice can be rephrased as a question
about cost minimization only. Because the monopolist’s incentive to reduce
2costs are undistorted, the monopolist picks, for a given output level, the ef-
ﬁcient durability level. This by itself is, however, insuﬃcient to guarantee
an undistorted durability level, because the cost–minimizing durability level
will, in general, depend on the ﬁrm’s output level and a monopolist typically
produces an ineﬃciently low output. For this reason, the second ingredient
that drives Swan’s independence result is the constant return to scale tech-
nology. It implies a multiplicative interaction between durability and output
in the ﬁrm’s cost function. This ensures that the cost–minimizing durability
level is independent of the ﬁrm’s output.
Now, for Swan’s independence result to hold also in the presence of ad-
ditional quality attributes, the cost–minimizing durability level must be in-
dependent of these quality attributes as well. Hence, just like in the case
of the ﬁrm’s output, durability must interact multiplicatively with all these
attributes for durability to remain undistorted. Without multiplicativity,
the direction of the distortion in durability follows the distortion in quality
when a higher durability level lowers or does not raise the marginal cost of
quality too much. In this case, it is optimal for the monopolist to choose an
ineﬃciently high durability exactly when it is optimal for him to choose an
ineﬃciently high quality level.
Durability in connection with additional quality attributes have been in-
vestigated in other economic contexts. Focusing on Coase’s durable–good
monopolist, Chi (1999) studies how quality choice alleviates the monopo-
list’s problem and Inderst (2008) shows how the durable–good monopolist
may oﬀer diﬀerent qualities to discriminate between diﬀerent type of con-
sumers. Strausz (2009) studies how, in a repeated games model, reduced
durability acts as an incentive device to choose credible quality levels in
other dimensions.
32 The Setup
I illustrate my arguments in a straightforward extension of Kleiman and
Ophir (1966) to goods with two–dimensional product characteristics as used
in Strausz (2009). In particular, I assume that the good is characterized by
a durability level d and a second quality characteristic which I capture by
a one–dimensional parameter q > 0. The durability level d describes the
length of time that the product works. The quality q measures the utility
which consumers derive from a functioning unit. For simplicity, I assume
that durability is deterministic: it yields the consumer a constant utility for
a time d and no utility thereafter.
Consumers have a total mass of one. Each consumer needs at most one
functioning unit of the good. Consumers diﬀer in their appreciation of quality
q. In particular, a consumer of type α ∈ [0,1] receives an instantaneous utility
of αq from a good with quality q. Hence, a good with characteristics (q,d)
yields consumer α a discounted utility of
v(q,d|α) ≡
Z d
0
αqe
−rtdt =
(1 − e−dr)qα
r
,
where r represents the common discount rate. The function v(q,d|α) repre-
sents consumer α’s willingness to pay for a good (q,d); a consumer α buys
the good (q,d) exactly when
p ≤ v(q,d|α). (1)
Assuming that consumers are distributed according to the c.d.f. F(α), the
buying decision (1) yields the inverse demand function
P(x|q,d) = F
−1(1 − x)(1 − e
−dr)q/r,
where F −1(.) is the inverse of the c.d.f. F(.).
Let c(q,d,x) denote the cost of producing a quantity x of the good (q,d).
In this case, proﬁts equal
Πt(q,d,x) = xP(x|q,d) − c(q,d,x).
4For simplicity, I consider only stationary choices of q, d, and x. If the ﬁrm
chooses durability d, consumers must repurchase the product every d periods.
Hence, the ﬁrm’s overall discounted proﬁt from producing an inﬁnite stream
of x products (q,d) every d periods is
Π(q,d,x) =
∞ X
i=0
￿
e
−dr
￿i
Πt(q,d,x) =
q
r
xF
−1(1 − x) −
c(q,d,x)
1 − e−dr . (2)
The main question is how the monopolistic solution compares to the eﬃ-
cient one. Because consumer surplus is the integral under the inverse demand
function, an output x of goods (q,d) generates social welfare
Wt(q,d,x) =
Z x
0
P(˜ x|q,d)d˜ x − c(q,d,x).
An inﬁnite production of x units of goods (q,d) every d periods yields a social
welfare of
W(q,d,x) =
∞ X
i=0
￿
e
−dr
￿i
Wt(q,d,x) =
q
r
Z x
0
F
−1(1 − ˜ x)d˜ x −
c(q,d,x)
1 − e−dr . (3)
3 Results
Let us ﬁrst assume Swan’s framework where quality q is ﬁxed and the ﬁrm
only chooses the output and durability level. A diﬀerentiation of (2) and (3)
with respect to durability d yields the identical ﬁrst order condition
c(q,d,x)e
−drr = cd(q,d,x)(1 − e
−dr). (4)
It illustrates that, when quality q is ﬁxed, then for a ﬁxed quantity level x,
the monopolist’s durability decision is undistorted. A closer inspection of
(2) and (3) reveals the intuition behind this result: Durability choice only
aﬀects the cost of production and not the ﬁrm’s revenue or the consumer’s
surplus. The optimal choice of d, therefore, follows from cost minimization
considerations alone. With respect to cost minimization, the monopolist’s
incentives are aligned with eﬃciency.
5Yet, the identical ﬁrst order conditions do not guarantee that the mo-
nopolist’s durability choice is undistorted. This is because they still depend
on output x. But if we can express the cost function c(q,d,x) as a product
ˆ c(q,x)˜ c(d), the term ˆ c(q,x) cancels out and the ﬁrst order condition simpliﬁes
to
˜ c(d)e
−drr = ˜ cd(d)(1 − e
−dr). (5)
In this case, optimal durability d is independent of quantity x.
From a long run perspective, the case of constant marginal cost is a
compelling benchmark and it underscores the relevance of Swan (1970)’s
independence result.
Let us now consider that the case where quality q is no longer ﬁxed, but
is optimally set by the ﬁrm. In this case, Swan’s independence result would
still hold if durability δ interacts multiplicatively with quality q. Yet, we
cannot make a compelling argument why this should be the case. Indeed,
monopoly distortions in quality provision are well–studied in industrial or-
ganization. The direction of the distortion is, in general, ambiguous. Two
eﬀects distort the monopolist’s quality choice. First, the monopolist’s choice
depends on the behavior of the marginal consumer, whereas, from a welfare
perspective, the average eﬀect over all consumers is the appropriate mea-
sure. Second, because the optimal quality choice depends on output x and
the monopolist’s output decision is distorted downwards, the quality choices
are diﬀerent due to diﬀerences in output choices. The direction of these two
eﬀects are indeterminate and, ultimately, depend on the ﬁrm’s production
technology. To make these observations more precise, let us compare the ﬁrst
order conditions
∂Π
∂q
= 0 ⇒ cq(q,d,x) = F
−1(1 − x)(1 − e
−dr)/r; (6)
and
∂W
∂q
= 0 ⇒ cq(q,d,x) =
Z x
0
F
−1(1 − ˜ x)d˜ x(1 − e
−dr)/r. (7)
6reveals the two types of distortions. The two ﬁrst order conditions diﬀer
structurally and depend on the output level x.
From these ﬁrst order conditions, we can investigate how the feedback
eﬀect distorts durability choice. Considering durability d as a function of
quality q and diﬀerentiating (4) with respect to quality q yields, after a
rearrangement of terms,
∂d
∂q
=
rcq(q,d,x)e−dr − (1 − e−dr)cqd(q,d,x)
r2c(q,d,x)e−dr + (1 − e−dr)cdd(q,d,x)
.
The sign of ∂d/∂q, which determines the direction of the feedback eﬀect,
depends on the cross partial derivative cqd. If cqd is negative, the sign is
unambiguously positive, because at an optimal solution cdd > 0 and cq > 0.
In this case, the sign of ∂d/∂q is positive and, as a result, the distortion
in durability follows the distortion in quality: ineﬃciently high quality goes
hand in hand with ineﬃciently high durability. In contrast, if the cross partial
derivative cqd is positive and relatively large in comparison to the derivative
cq, the distortions in quality and durability counteract each other.
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Monopoly Distortions in Durability and Multi-Dimensional Quality

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