Network Externalities and the Overprovision of Quality by a Monopolist.

Luca Lambertini [*]

Raimondello Orsini [+]

We investigate the behavior of a monopolist supplying a vertically differentiated good with network externalities. Assuming a convex unit cost of quality improvements, we show that the presence of network externalities may yield oversupply of quality compared with the social optimum, when partial market coverage emerges at equilibrium. Overall, the incentive to expand output increases in the extent of network externalities, thereby partially counterbalancing the social damage produced by the quality distortion.

1. Introduction

The increasing relevance of high-tech sectors, such as telecommunication, hardware--software, and audio--video industries, justifies the growing amount of attention devoted to the analysis of markets for goods that generate network externalities (Katz and Shapiro 1985, 1986, 1994; Farrell and Saloner 1985, 1986). An example taken by the audio--video industry is the current discussion concerning the possibility of adopting a new standard for hi-fi equipments, DVD (digital versatile disc) or SACD (super audio compact disc). The network effect consists of the fact that the utility from purchasing equipment characterized by a given standard is positively related to the number of consumers who purchase a compatible equipment, for two reasons. The first is that the larger the number of users of a given standard, the wider the range of software made available for that standard. The second is that a larger network of users is usually associated with more efficient customer service and technical assistance.

If consumers are sensitive to the size of the market, the producer's choice of the output level can be compared to the provision of product quality, in that allowing consumers to access a larger network is essentially like providing them with a good characterized by a superior quality. Therefore, the existence of network effects in markets where intrinsic quality is endogenous prompts a reassessment of the welfare implications of market power against either perfect competition or social planning.

To provide a theoretical framework capable of evaluating this issue, we build upon Spence's (1975) seminal paper on vertical differentiation, introducing a network effect into consumer preferences. In the considerable amount of literature stemming from Spence's paper (Sheshinski 1976; Mussa and Rosen 1978; Maskin and Riley 1984), the main question is whether a monopolist supplies the socially optimal quality or distorts it so as to induce self-selection on the part of consumers.

The earliest contributions (Spence 1975; Sheshinski 1976) deal with a single-product monopolist. Their main conclusions are that (i) for a given output level, quality is over- or under-supplied by the monopolist as compared with social planning, depending on whether the marginal valuation of quality is above or below the average valuation of quality (if they coincide, the monopolist supplies the same quality as the social planner); and (ii) the monopolist under-supplies quality if his output is close to the socially optimal one.

We adopt a model where a single-product monopolist supplies a good whose production entails a variable cost, which is assumed to be convex in the quality level, and consumers' utility function contains a network externality component. We proceed in two steps. First, we consider the general case where the forms of both the cost function and the consumer distribution are left unspecified, except for the overall features of technology. In this framework, we characterize the role of the interplay between the income distribution of consumers and the network externality in determining the quality choice of a profit-seeking monopolist vis-a-vis a social planner. We find that, for any given income distribution, there may exist a range of network effects such that Spence's conclusions are reversed. As a second step, we investigate in detail a more specific model where we adopt some standard assumptions concerning consumer distribution and technology. In particular, we consider the case of a uniform consumer distribut ion, where it is known that, if network effects were absent, a profit-maximizing monopolist would supply the same quality as a social planner, as long as partial market coverage obtains (Spence 1975). This allows us to describe the behavior of equilibrium prices, qualities, and output levels, as well as to establish the parameter regions where full and partial market coverage alternatively obtains, under both regimes. The main result is that, as long as partial market coverage is observed under both regimes, both equilibrium qualities are decreasing in the amount of network externalities, with the socially optimal quality decreasing faster than the monopoly quality, yielding overprovision of quality by the monopolist. This is driven by the planner's incentive to lower the price so as to increase market coverage as much as possible. As intuition suggests, the number of consumers able to purchase in equilibrium is always larger under social planning. Moreover, in the parameter region where this happens, the dea dweight loss increases in the extent of network externalities. However, this phenomenon tends to disappear as soon as full market coverage emerges at equilibrium under social planning. In the remainder of the parameter range, the deadweight loss is either decreasing or constant with respect to the network effect.

The paper is organized as follows. Section 2 presents the general model. A specific formulation is introduced in section 3, where we derive both the monopoly equilibrium and the social optimum, which are then compared in section 4. Section 5 concludes.

2. The Model

Consider a monopoly market for a good whose utility depends both on intrinsic characteristics, which are represented by quality q, and on total consumption x. Consumers are characterized by a parameter [theta], which represents the individual marginal willingness to pay for quality: [theta] is distributed with density f([theta]) over the interval [[theta] - 1,[theta]], with [theta][greater than or equal to]1. The population is normalized to 1. Each consumer buys at most one unit of the good, and the resulting net surplus is

U = max{[theta]q + [alpha]x - p, 0}

where p is the price charged by the monopolist, and [alpha] (the same for all the agents) is a positive coefficient representing the weight of the network externality in the utility function. Let [theta]([alpha], p. q, f([theta])) define the marginal willingness to pay of the consumer who is indifferent between buying and not buying:

[theta]([alpha], p, q, f ([theta])) [equivel] {[theta]: [theta]q + [alpha] [[[integral].sup.0].sub.0]f(z)dx - p = 0}.

Then, market demand is

x = [[[integral].sup.0].sub.0]f([theta])d[theta]

where

[theta] = max{[theta] - 1, [theta]}.

When max{[theta] - 1, [theta]} = [theta], partial market coverage (pmc) obtains; when instead max{[theta] - 1 [theta]} = [theta] - 1, full market coverage (fmc) obtains, that is, x - 1. Consumer surplus is

CS = [[[integral].sup.0].sub.0] U(.)f([theta]) d[theta].

On the supply side, production involves total costs C = c(q)x, with c', c" [greater than] 0. This amounts to saying that there are constant returns to scale and unit production costs c(q) are convex in quality. The profit function is then

[[pi].sub.M] = [p - c(q)].[[[integral].sup.0].sub.0]f([theta]) d[theta].

Social welfare is SW [[pi].sub.M] + CS. Except for the presence of network effects, this setup replicates the model introduced by Spence (1975). In his proposition 1, Spence (1975, p. 419) proves that, for a given output level, the monopolist undersupplies quality compared with the social optimum, iff the marginal willingness to pay of the average consumer is higher than the marginal willingness to pay of the marginal consumer, that is, the poorest consumer who is able to buy. A situation where, for any f([theta]), the average consumer is surely richer than the marginal one is that where full market coverage obtains under both regimes. Therefore

REMARK 1. Under full market coverage, the monopolist undersupplies quality as compared with the social planner, independently of the consumer distribution and the size of network effects.

PROOF. See Appendix A.

Notice that, when the market is fully covered, the size of [alpha] is obviously irrelevant, since the size of the network is given and equal to one. Under partial market coverage, we know from Spence (1975) that the oversupply or undersupply of quality by the monopolist as compared with the social optimum is determined only by the shape of the consumers' distribution, f([theta]). We are going to show below that introducing network effects may drastically modify the acquired wisdom. This is summarized in the following:

PROPOSITION 1. Given f([theta]), in the monopoly optimum where [partial][[pi].sub.M]/[partial]q = 0, the sign of [partial]CS/[partial]q under partial market coverage depends upon {[alpha], [theta]( [alpha], p, q, f ([theta]))}.

PROOF. See Appendix B.

The provision of quality by the monopolist might follow a schizophrenic rule. Given f([theta]), the monopolist may oversupply quality as long as x [less than] 1, and surely undersupplies quality when x = 1. The intuition behind the latter result is that optimal quality supply, under both monopoly and social planning, becomes independent of network effects as soon as the whole extent of potential demand is exploited, that is, the size of the network coincides with the size of the market. The schizophrenic rule followed by the monopolist entails the possibility that, in the neighbourhood of x = 1, the optimal monopoly quality [q.sub.M] becomes discontinuous, with [[q.sup.pmc].sub.M] [neq] [[q.sup.fmc].sub.M].

3. Quadratic Unit Costs and Uniform Distribution

To show that the monopolist may indeed end up oversupplying product quality, we investigate a version of the model that is widely adopted in the existing literature. In particular, we assume that (i) c(q) = t[q.sup.2], t [greater than or equal to] 0, so that C(q, x) = t[q.sup.2]x; and (ii) the population of consumers is uniformly distributed over [[theta] - 1, [theta]]. In this case, the marginal consumer is characterized by a willingness to pay [theta] = (p - [alpha] [theta])/(q - [alpha]). Hence, under partial market coverage and full market coverage, respectively, market demand is

x = [theta] - p - [alpha][theta]/q - [alpha] = q[theta] - p/q - [alpha] for all {p, q, [alpha]} such [theta] [epsilon] ([theta] - 1, [theta]] (pmc);

x = 1 for all such that [theta] [less than or equal to] [theta] - 1 (fmc).

In either case, monopoly profits are[[pi].sub.M] = (p - t[q.sup.2])x.

Profit Maximization 

In this section, we first treat separately the alternative settings of partial and full market coverage. Then we proceed to establish the parameter ranges where the monopolist adopts, alternatively, one regime or the other.

Partial Market Coverage

Suppose [theta] [epsilon] ([theta] - 1, [theta]]. Then, partial market coverage obtains and the monopolist's profits are given by

[[[pi].sup.pmc].sub.M] = (p - t[q.sup.2])[[theta] - p - [alpha][theta]/q-[alpha]]. (1)

This expression has to be maximized with respect to the two choice variables: price and quality. The first-order condition with respect to price is

[partial][[[pi].sup.pmc].sub.M]/[partial]p = [[theta].sub.q] - 2p + t[q.sup.2]/q - [alpha] = 0

which leads to p = q([theta] + tq)/2. Plugging this expression for p in Equation 1 and taking the derivative with respect to q yields

[partial][[[pi].sup.pmc].sub.M]/[partial]q = q([theta] - tq)([theta]q - 2[alpha][theta] + 4[alpha]tq - 3t[q.sup.2])/4[(q - [alpha]).sup.2]

The quality level provided by the monopolist is therefore [1]

[[q.sup.pmc].sub.M] = [theta] + 4[alpha]t + k/6t, (2)

where k = [square root][[theta].sup.2] + 16[[alpha].sup.2][t.sup.2] - l6[alpha]t[theta]. Observe that [[q.sup.pmc].sub.M] [epsilon] R+ for all [alpha] [less than or equal to] [theta]/[4t(2 + [square root]3)]. Moreover, [[q.sup.pmc].sub.M] = [theta]/(3t) if [alpha] = 0 (see Lambertini 1997). Equilibrium price and output are:

[[p.sup.pmc].sub.M] = ([theta] + 4[alpha]t + k)(7[theta] + 4[alpha]t + k)/72t (3)

[[x.sup.pmc].sub.M] = 8[[alpha].sup.2][t.sup.2] - [[theta].sup.2] - 8[alpha][theta]t + (2[alpha]t - [theta])k/3(2[alpha]t - [theta] - k) (4)

From Equations 2, 3, and 4, it can be easily verified that, under pmc, quality and price are decreasing while output is increasing in [alpha].

The fact that the monopolist expands production as the weight/of network externalities becomes more relevant can be explained as follows. The location of the marginal consumer, which determines the extent of market coverage, is directly related/to the price-quality ratio. As the network effect increases, the monopolist reduces quality and therefore marginal cost. Because the latter is convex, the reduction in the price level outweighs the reduction in quality. As a result, the price-quality ratio decreases and output gets larger. [2]

The welfare implications of these distortions can be traced out calculating the level of consumer surplus and monopoly profits:

[[[pi].sup.pmc].sub.M] = [[8[[alpha].sup.2][t.sup.2] - [[theta].sup.2] - 8[alpha][theta]t + (2[alpha]t - [theta])k].sup.2]/54t([theta] - 2[alpha]t + k)

[[CS.sup.pmc].sub.M] = ([theta] + 4[alpha]t + k)[[8[[alpha].sup.2][t.sup.2] - [[theta].sup.2] - 8[alpha][theta]t + (2[alpha]t - [theta])k].sup.2]/108t[([theta] - 2[alpha]t + k).sup.2]

The resulting level of social welfare, [[SW.sup.pmc].sub.M] = [[CS.sup.pmc].sub.M] + [[[pi].sup.pmc].sub.M], is

[[SW.sup.pmc].sub.M] = ([alpha]t - [theta])[8[[alpha].sup.2][t.sup.2] - [[theta].sup.2] - 8[alpha][theta]t + (2[alpha]t - [theta])k][[[theta].sup.2] - 4[alpha][theta]t + 16[[alpha].sup.2][t.sup.2] + (4[alpha]t + [theta])k]/18t[([theta] - 2[alpha]t + k).sup.2]

A priori, the effect of a variation of [alpha] on profits, consumer surplus, and welfare is ambiguous, since [partial][[q.sup.pmc].sub.M]/[partial][alpha] [less than] 0, [partial][[p.sup.pmc].sub.M]/[partial][alpha] [less than] 0, and [partial][x.sup.pmc].sub.M]/[partial][alpha] [greater than] 0. However, as long as k [epsilon] R, it can be shown that

[partial][[[pi].sup.pmc].sub.M]/[partial][alpha] [greater than] 0 [forall] [alpha], [theta]; [partial][[CS.sup.pmc].sub.M]/[partial][alpha] [greater than] 0 [forall] [alpha] [epsilon] [0, [theta]/16t); [partial][[SW.sup.pmc].sub.M]/[partial][alpha] [greater than] 0 [forall] [alpha] [epsilon] [0, (4 - [sqrt]13)[theta]/6t).

In the remainder of the paper, we will show that [0, (4 - [sqrt]13)[theta]/(6t)) is wider than the range where partial market coverage obtains in equilibrium. The effect on profits of an increase in the weight of network externalities is positive because the marginal cost decreases more rapidly than price, while output increases. Consumer surplus benefits from an increase in [alpha] because the reduction in the quality level is more than compensated by the lower price, the larger market coverage, and the associated supplementary network effect.

Full Market Coverage

In this case, profits write [[[pi].sup.fmc].sub.M] = p - t[q.sup.2]. Since [[[pi].sup.fmc].sub.M] is always increasing in price, the monopolist chooses the highest price compatible with full market coverage:

p(q) = ([theta] - 1)q + [alpha]

Therefore, the monopolist chooses the quality maximizing [[[pi].sup.fmc].sub.M] = ([theta] - 1)q + [alpha] - t[q.sup.2]. The first-order condition is

[[[[partial][pi].sup.fmc].sub.M]/[partial]q = [theta] - 1 - 2tq = 0.

yielding = [[q.sup.fmc].sub.M] = ([theta] - 1)/2t as the optimal quality. The price set by the monopolist is [[p.sup.fmc].sub.M] = p([[q.sup.fmc].sub.M]) = [([theta] - 1).sup.2]/2t + [alpha]. This proves immediately that the optimal monopoly price under full market coverage is everywhere increasing in [alpha].

The corresponding profits and social welfare are [[[pi].sup.fmc].sub.M] = [alpha] + [([theta] - 1).sup.2]/(4t) and [[SW.sup.fmc].sub.M] = [alpha] + [theta]([theta} - 1)/4t).

Partial versus Full Market Coverage

The monopolist's optimal market coverage policy can be summarized as follows. Whenever both partial and full market coverage are admissible, that is, [theta] [epsilon] [[theta] - 1, [theta]], that is, [[x.sup.pmc].sub.M] [less than] 1, then pmc is optimal. If either condition is violated, then fmc obtains. The detailed partition of the parameter space {[alpha], [theta]} is described by:

REMARK 2. For all [theta] [epsilon] [1, [sqrt]3), [[[pi].sup.pmc].sub.M] [greater than] [[[pi].sup.fmc].sub.M] and pmc obtains for all [alpha] [epsilon] [0, (1 + 2[theta] - 2[sqrt]1 + [theta].sup.2])/(4t)). For all [theta] [epsilon] [[sqrt]3, 3), [[[pi].sup.pmc].sub.M] [greater than] [[[pi].sup.fmc].sub.M] and pmc obtains for all [alpha] [epsilon] [0, (-[[theta].sup.2] + 4[theta] - 3)/(8t)). In all other regions of {[alpha], [theta]), fmc obtains.

PROOF. See Appendix C.

The parabolic function [alpha] = (-[[theta].sup.2] + 4[theta] - 3)/(8t) jointly describes the loci [[[pi].sup.pmc].sub.M] - [[[pi].sup.fmc].sub.M] = 0 and [[x.sup.pmc].sub.M] -- 1 = 0 if and only if [theta] [epsilon] [[square root]3, 3]. To the left of [theta] = [square root]3, k becomes negative, and the two loci no longer coincide either with each other or with [alpha] = (-[[theta].sup.2] + 4[theta] - 3)/(8t). [3]

The above proposition also entails that (i) for all [alpha] [epsilon] ([3 - 2[square root]2]/[4t], 1/[8t]), the monopolist's optimal output is nonmonotone in [theta]; and (ii) for all [theta] [epsilon] [1, [square root]3), the monopolist's optimal output is discontinuous in [alpha]. Point (i) follows from the observation that the monopolist may find it profitable to serve all consumers even if the marginal willingness to pay for quality is relatively low, provided that a is above (1 + 2[theta] - 2[square root]1 + [[theta].sup.2])/(4t). To prove (ii) observe that [[x.sup.pmc].sub.M] [less than] 1 along [alpha] = (1 + 2[theta] - 2[square root]1 + [[theta].sup.2])/(4t), for all [theta] [epsilon] [1, [square root]). Conversely, [[x.sup.pmc].sub.M] = 1 along [alpha] = (-[[theta].sup.2] + 4[theta] - 3)/(8t), which is the relevant border between partial and full market coverage over [theta] [epsilon] [[square root]3, 3]. The discontinuity in the output level obviously goes along with a discontinuity in price and qua lity for all [theta] [epsilon] [1, [square root]3). The market coverage policy chosen by the monopolist in the space {[theta], [alpha]} is described in Figure 1.

The effects of both a and [theta] on the optimal quality can now be described. The following holds:

REMARK 3. Optimal quality is monotonically increasing in [theta] and nonincreasing in [alpha].

PROOF. See Appendix D.

The intuition behind the discontinuity observed in quantity, quality, and price when the market is relatively poor, that is, in the range [theta] [epsilon] [1, [square root]3), is as follows. Consider first the well-known situation without network externalities, where [[x.sup.pmc].sub.M] = 1 and [[[pi].sup.pmc].sub.M] = [[[pi].sup.fmc].sub.M] at [theta] = 3. As soon as some network effect operates, the two equations do not yield the same solution any longer, and the monopolist must evaluate the sign of [[[pi].sup.pmc].sub.M] - [[[pi].sup.fmc].sub.M] to choose between pmc and fmc. This is due to the fact that [alpha] is a substitute for [theta] in terms of willingness to pay. Therefore, the market can be fully covered also for [theta] [less than] 3, provided a is sufficiently large. If not, partial coverage obtains. This is reinforced by the fact that [[q.sup.pmc].sub.M] is decreasing in [alpha], whereas [[q.sup.fmc].sub.M] is independent of [alpha]. By choosing pmc the monopolist operates at lower unit costs as a increases.

Welfare Maximization

Consider now the behavior of a benevolent social planner maximizing social welfare with respect to price and quality, under partial and full market coverage, alternatively.

Partial Market Coverage

As is well known, network externalities being absent, the planner would price at marginal cost, serving twice as many consumers as the profit-maximizing monopolist and providing a product of the same quality (see Mussa and Rosen 1978). The effect of network externalities on optimal pricing and production decisions is described by the following first-order condition:

[partial][[SW.sup.pmc].sub.SP]/[partial]p = p(2[alpha] - q) - [alpha][theta]q + (q - [alpha])t[q.sup.2]/[(q - [alpha]).sup.2]

yielding [[p.sup.pmc].sub.SP] = q[[alpha][theta] + tq([alpha] - q)]/(2[alpha] - q). Then, optimal quality is

[[q.sup.pmc].sub.SP] = [theta] + 8[alpha]t + z/6t.

where z = [square root][[theta].sup.2] - 32[alpha][theta]t + 64[[alpha].sup.2][t.sup.2], which implies that [[q.sup.pmc].sub.SP] [epsilon] [R.sup.+] for all [alpha] [less than or equal] [theta]/[8t(2 + [square root]3)]. Plugging the optimal quality into [[p.sup.pmc].sub.SP] we get the planner's price as a function of parameters only:

[[p.sup.pmc].sub.SP] = ([theta] + 8[alpha]t + z)[[[theta].sup.2] + 40[[alpha].sup.2][t.sup.2] - 29[alpha][theta]t + ([theta] + 5[alpha]t)z]/18t([theta] - 4[alpha]t + z)

We prove the following:

REMARK 4. The social planner chooses (i) partial market coverage for all [alpha] [epsilon] [0, (8[theta] - 4[[theta].sup.2] - 3)/(32t)); (ii) full market coverage for all [alpha] [greater than or equal to] (8[theta] - 4[[theta].sup.2] - 3)/(32t).

PROOF. See Appendix E.

In the space {[theta], [alpha]}, the boundary between pmc and fmc is described by the curve [alpha] = (8[theta] - 4[[theta].sup.2] - 3)/(32t). The above proposition also implies that the condition for [[q.sup.pmc].sub.SP] [epsilon] [R.sup.+] is never binding, in that [theta]/[8t(2 + [square root]3)] [greater than] (8[theta] - 4[[theta].sup.2] - 3)/(32t) for all [theta] [greater than or equal to] 1.

When [alpha] = 0, [[q.sup.pmc].sub.SP] = [theta]/(3t) = [[q.sup.pmc].sub.M], besides, [partial][[q.sup.pmc].sub.SP]/[partial][alpha] [less than] 0 over the whole admissible range. Hence, the socially optimal quality under partial market coverage is everywhere decreasing in [alpha]. Moreover, it can be verified that [[p.sup.pmc].sub.SP] = t[([[q.sup.pmc].sub.SP]).sup.2] - [alpha][[x.sup.pmc].sub.SP], which implies that, under social planning, consumption is subsidized to the full extent of the externality.

Full Market Coverage

Above [alpha] = (8[theta] - 4[[theta].sup.2] - 3)/(32t), full market coverage (x = [theta] - [theta] = 1) obtains. In such a case, social welfare is [[SW.sup.fmc].sub.SP] = [theta]q + [alpha] - q/2 - t[q.sup.2]. The first-order condition is

[partial][[SW.sup.fmc].sub.SP]/[partial]q = [theta] - 2tq - 1/2 = 0.

Therefore, [[q.sup.fmc].sub.SP] = (2[theta] - 1)/(4t). Social welfare in equilibrium is [[SW.sup.fmc].sub.SP] = [alpha] + [[theta]([theta] - 1)]/(4t) + 1/(16t).

4. Monopoly versus Social Planning

In this section we compare the choices of the social planner with those of the monopolist under both market coverage regimes. This is done by comparing the two optima where the planner and the monopolist choose both price and quality. Consider first the case where both the monopolist and the social planner cover the market only partially. In this region, it is straightforward to verify that \[partial][[q.sup.pmc].sub.SP]/[partial][alpha]\[greater than]\[partial][[q.sup.pmc].sub.M]/[partial][alpha]\. Since the monopolist and the social planner provide the same quality when [alpha] = 0, the above inequality proves the following:

LEMMA 1. In the parameter region where the social planner covers the whole market, the monopolist oversupplies quality as compared with the social optimum.

The interpretation of the above result [4] is that, when a network effect is operating, a reduction in quality is welfare improving because, in doing so, the planner increases the extent of market coverage and consumer surplus. From Spence (1975, proposition 1, p. 419), we know that, network effects being absent and given monopoly price p = q(0 + tq)/2, the following holds:

[partial]p/[partial]q = 1/x [[[integral].sup.0].sub.[theta]] [partial]p/[partial]q d[theta], (5)

as long as f([theta]) is uniform. Then, for any [alpha] [greater than] 0, the overprovision of product quality by the monopolist implies

[partial]p/[partial]q [greater than] 1/x[alpha] [[[integral].sup.0].sub.[theta]([alpha])] [parital]p/[partial]q d[theta] (6)

Since p is independent of [alpha], [partial]p/[partial]q is the same in Equations 5 and 6, and therefore we have that quality is oversupplied iff

1/x([alpha])[[[integral].sup.0].sub.[theta]([alpha])][partial]p/[part ial]q]d[theta][less than] 1/x [[[integral].sup.0].sub.[theta]][partialp/[partial]q d[theta],

which is true for all [alpha] [greater than] 0.

Consider now the region where both agents fully cover the market. It is easy to immediately [[q.sup.fmc].sub.SP] = [[q.sup.fmc].sub.M] + 1/(4t) and [[SW.sup.fmc].sub.SP] = [[SW.sup.fmc].sub.M] + 1/(16t). Hence, as from remark 1, then presence of network externalities does not modify Spence's (1975) finding concerning the incentive to undersupply quality, provided that the market is fully covered.

The comparison between [[q.sup.pmc].sub.M] and [[q.sup.fmc].sub.SP] remains to be carried out in the region where the social planner chooses full market coverage while the monopolist excludes some consumers from purchase:

LEMMA 2. When only the social planner fully covers the market, the monopolist oversupplies quality as compared with the social optimum in the following regions of the parameter space:

[theta] [epsilon] [1, [square root]5/2) and [alpha] [epsilon] [(8[theta] - 4[[theta].sup.2] - 3)/(32t), (1 + 2[theta] - 2 [square root]1 + [[theta].sup.2])/(4t));

[theta] [epsilon] [[square root]5/2, 3/2] and [alpha] [epsilon] [(8[theta] - 4[[theta].sup.2] - 3)/(32t), (8[theta] - 4[[theta].sup.2] - 3)/(16t)).

PROOF. See Appendix F.

On the basis of lemmata 1 and 2, we characterize the provision of quality by the monopolist as follows:

PROPOSITION 2. When the monopolist chooses partial market coverage, there exists a parameter region where he oversupplies quality as compared with the social optimum, irrespective of the extent of market coverage chosen by the planner.

This region is defined by

[theta] [epsilon] [1, [sqrt]5/2) and [alpha] [epsilon] [0, (1 + 2[theta] - 2[sqrt]1 + [[theta].sup.2])/(rt));

[theta] [epsilon] [[sqrt]5/2, 3/2] and [alpha] [epsilon] [0, (8[theta] - 4[[theta].sup.2] - 3)/(16t)).

The partition of the parameter space {[theta], [alpha]} as from lemmata 1 and 2 and proposition 2 is in Figure 2. The monopolist chooses pmc for all [alpha] below the concave curve composed by [alpha] = (1 + 2[theta] - 2[sqrt]1 + [[theta].sup.2])/(4t) for [theta] [epsilon] [1, [sqrt]3] and [alpha] = (4[theta] - [[theta].sup.2] - 3)/(8t) for [theta] [epsilon] [[sqrt]3, 3]. The planner chooses pmc (respectively, fmc) below (respectively, above) curve A, that is, [alpha] = (8[theta] - 4[[theta].sup.2] - 3)/(32t). Moreover, [[q.sup.pmc].sub.M] = [[q.sup.pmc].sub.SP] along curve B, that is, [alpha] = (8[theta] - 4[[theta].sup.2] - 3)/(16t). Below [alpha] = (1 + 2[theta] - 2[sqrt]1 + [[theta].sup.2])/(4t) and curve B, the monopolist oversupplies quality. [5] This result is due to the planner's incentive to decrease price to increase market coverage as much as possible. Intuitively, the number of consumers able to purchase in equilibrium is always larger under social planning.

Figure 3 illustrates the behavior of equilibrium qualities under social planning and monopoly as [alpha] changes. In Figures 3a and b, the monopolist oversupplies quality compared with the social optimum. Observe that, although quality is usually thought of as an index of welfare, in this setting there is no such presumption, since welfare can be enhanced either by rising quality, or by expanding output. For a given [theta], the welfare-improving effect of expanding output is increasing in [alpha]. Combined with the behavior of qualities described by Figure 3, this entails that the ranking of social welfare levels is the same as the ranking of output levels. As long as pmc prevails, both [[x.sup.pmc].sub.M] and [[x.sup.pmc].sub.SP] are increasing and convex in [alpha], with [partial][[x.sup.pmc].sub.SP]/[partial][alpha] [greater than] [partial][[x.sup.pmc].sub.M]/[partial][alpha]. In the range where fmc prevails under social planning, while pmc obtains under monopoly, [[x.sup.fmc].sub.SP] - [[x.sup.pmc].sub.M] is decreasing and concave in [alpha]. The same applies to social welfare levels.

The deadweight loss due to monopoly power is nonmonotone in [alpha], for all [theta] [epsilon] [1, 3/2). For any [theta] [greater than or equal to] 3/2, the deadweight loss is monotonically nonincreasing in [alpha]. This implies that, whenever the market is sufficiently rich, the inefficiency usually associated with monopoly power is a lesser evil, the more so the more the externality component in consumer preferences becomes relevant. [6]

5. Concluding Remarks

We have reexamined the problem of the provision of quality by a single-product profit-maximizing monopolist, by introducing network effects in the model proposed by Spence (1975). On the one hand, our findings qualitatively confirm those of Spence in the case of full market coverage, where the monopolist under-supplies product quality as compared with the first best, independently of the shape of consumer distribution. On the other hand, the introduction of network externalities may drastically modify the monopolist's incentive towards the provision of quality under partial market coverage. First, we have analyzed a general setting where consumer distribution and technology remain unspecified. Then, we have confined our attention to a specific model with uniform consumer distribution and quadratic unit costs, proving that, under partial market coverage, equilibrium qualities decrease as network externalities become more relevant, with the socially optimal quality decreasing faster than the monopoly quality. Overprovision of quality by the monopolist thereby obtains, because of the planner's incentive to reduce the price and enlarge the network as much as possible.

An associated result is that, when the monopolist overprovides quality, the deadweight loss increases in the extent of network externalities. The welfare loss starts to decrease as soon as the social planner covers the market entirely.

The foregoing analysis has some relevant policy implications. When the market is sufficiently poor to induce even a benevolent planner to exclude some individuals from consumption, both the welfare distortion and the related incentive to regulate the monopolist are nonmonotone in the extent of network effects. This nonmonotonicity disappears as soon as full market coverage emerges at equilibrium under social planning, as in this situation the welfare loss is constant. Since network effects enhance the incentive to serve all consumers, the above argument holds that a large network effect tends to eliminate output distortions and leave in operation quality distortions only, the latter being less harmful than the former in most cases.

(*.) Dipartimento di Scienze Economiche, University of Bologna, Strada Maggiore 45, 40125 Bologna, Italy; E-mail [e-mail]; corresponding author.

(+.) Dipartimento di Scienze Economiche, University of Bologna, Strada Maggiore 45, 40125 Bologna, Italy; E-mail [e-mail].

We thank Jonathan Hamilton, two anonymous referees, Giampietro Fabbri, and Carlo Scarpa for useful comments and suggestions. The usual disclaimer applies.

(1.) Second-order conditions are met throughout the calculations performed in the paper, although not shown for the sake of brevity.

(2.) As t tends to zero, the optimal monopoly quality would tend to infinity. Actually, the monopolist would produce the maximum quality that is technologically feasible, [q.sub.max]. The resulting output is lower than 1, and pmc obtains, for all [alpha] [epsilon] [0, ([q.sub.max])/2). It is easy to verify that, when production costs are nil, the social planner supplies all consumers with [q.sub.max], for all [alpha] [greater than or equal to] 0.

(3.) Observe that min [[pi].sub.M] = 1/(27t) at ([alpha] = 0, [theta] = 1). Therefore, if we considered a fixed cost F, we should assume F [less than or equal to] 1/(27t) to fully preserve the validity of the analysis and avoid considering the break-even condition.

(4.) Notice that this result can be sensitive to the specification of the cost function. If quality improvements hinge upon research and development activity whose costs are unrelated to output, the incentive to oversupply quality may disappear.

(5.) To the northeast of curve B, quality is always undersupplied at the monopoly equilibrium.

(6.) In the range of [theta] where the deadweight loss is nonmonotone with respect to [alpha], the interval wherein [partial]([SW.sub.SP] - [SW.sub.M])/ [partial][alpha] [greater than] 0 shrinks as t, that is, marginal production cost, increases. This is due to the incentive, for both the planner and the monopolist, to induce more consumers to buy, by exploiting the network effect rather than the hedonic effect. This is accomplished by raising the output level rather than the quality level.

References

Farrell, Joseph, and Garth Saloner. 1985. Standardization, compatibility and innovation. RAND Journal of Economics 16:70-83.

Farrell, Joseph, and Garth Saloner. 1986. Standardization and variety. Economics Letters 20:71-4.

Katz, Michael, and Carl Shapiro. 1985. Network externalities, competition, and compatibility. American Economic Review 75:424-40.

Katz, Michael, and Carl Shapiro. 1986. Technology adoption in the presence of network effects. Journal of Political Economy 94:822-41.

Katz, Michael, and Carl Shapiro. 1994. Systems competition and network effects, Journal of Economic Perspectives 8:93-115.

Lambertini, Luca. 1997. The multiproduct monopolist under vertical differentiation: An inductive approach. Recherches Economiques de Louvain 63:109-22.

Maskin, Eric, and John Riley. 1984. Monopoly with incomplete information. RAND Journal of Economics 15:171-96.

Mussa, Michael, and Sherwin Rosen. 1978. Monopoly and product quality. Journal of Economic Theory 18:301-17.

Sheshinski, Eytan. 1976. Price, quality and quantity regulation in monopoly situations. Economica 43:127-37.

Spence, A. Michael. 1975. Monopoly, quality and regulation. Bell Journal of Economics 6:417-29.

Appendix A

PROOF OF REMARK 1. Examine the situation where max {[theta] - 1, [theta]} = [theta] - 1, that is, full market coverage obtains. Here, x = 1 and the monopoly price must be p = ([theta] - l)q + [alpha]. The corresponding profit function is [[pi].sub.M] = ([theta] - l)q + a - c(q), and consumer surplus is CS = [[[integral].sup.[theta]].sub.[theta]-1]U(.)f([theta]) d[theta], where U = [theta]q - p + [alpha]. In the monopoly optimum, where [partial][[pi].sub.M]/[partial]q = [theta] - 1 - c = 0, the derivative of social welfare is

[partial]SW/[partial]q = [partial]CS/[partial]q = [[[integral].sup.[theta]].sub.[theta]-1][U.sub.q]f([theta]) d[theta] = [[[integral].sup.[theta]].sub.[theta]-1] [theta].f([theta])d[theta][greater than]0.

Appendix B

PROOF OF PROPOSITION 1. To prove the above claim, consider first the monopoly optimum when max{[theta] - 1, [theta]} = [theta]. The relevant first-order condition is

[partial][[pi].sub.M]/[partial]q = -[p - c(q)].f([theta]).[[theta].sub.q] + ([[p.sub.q] - c). [[[integral].sup.[theta]].sub.[theta]]f([theta])d[theta] = 0,

where [theta].sub.q] = [partial][theta][[alpha], p, q, f ([theta])]/[partial]q and [p.sub.q] = [partial]p/[partial]q. As quality increases, the location of the marginal consumer shifts in a way determined by:

[[theta].sub.q] = ([p.sub.q] -c)x/[p - c(q)]f([theta])

where sign {[[theta].sub.q]} = sign {[P.sub.q] - c}. Differentiating social welfare with respect to quality yields.

[partial]SW/[partial]q = [partial][[pi].sub.M]/[partial]q + [partial]CS/[partial]q = [partial][[pi].sub.M]/[partial]q + [[[integral].sup.[theta]].sub.[theta]]([theta]-[alpha].f([theta]).[[t heta].sub.q]-[p.sub.q])f([theta])d[theta] - \[theta]q - p + [alpha].[[[integral].sup.[theta]].sub.[theta]]f([theta])d[theta]\.f([ theta]).[[theta].sub.q].

In the monopoly optimum, [partial][[pi].sub.M]/[partial]q = 0, and the above expression simplifies to

[partial]SW/[partial]q = [partial]CS/[partial]q = [[[integral].sup.[theta]].sub.[theta]]([theta]-[alpha].f([theta]).[[t heta].sub.q]-[p.sub.q])f([theta] d([theta])d[theta]-[[theta]q-p + [alpha]x].f([theta]).[[theta].sub.q]

which can be rearranged as

[partial]CS/[partial]q = [[[integral].sup.[theta]].sub.[theta]]([theta] - [p.sub.q])f([theta]d[theta] - ([theta]q - p).f([theta]).([[theta].sub.q] - 2[alpha]x.f([theta]).[[theta].sub.q]. (A1)

Observe that, when [alpha] = 0, Equation A1 coincides with the well-known condition in Spence (1975). The presence of network externalities adds the last term where [alpha] appears explicitly, and modifies the other terms as well.

Appendix C

PROOF OF REMARK 2. As a first step, observe that [[x.sup.pmc].sub.M] - 1 = 0 along [alpha] - (-[[theta].sup.2] + 4[theta] - 3)/(8t), for all [theta] [epsilon] [[square root]3, 3]. To the left of [theta] = [square root]3, the locus [[x.sup.pmc].sub.M] - 1 = 0 does not belong to R, in that it lies above the reality condition for [[q.sup.pmc].sub.M], that is, [alpha] [less than or equal to] [theta]/[4t(2 + [square root]3)].

The decision concerning the extent of market coverage is taken according to the sign of [[[pi].sup.pmc].sub.M] - [[[pi].sup.fmc].sub.M]. The roots of [[[pi].sup.pmc].sub.M] = [[[pi].sup.fmc].sub.M] are

[[alpha].sub.1] = 1 + 2[theta] - 2[square root]1 + [[theta].sup.2]/4t for all [theta] [epsilon] [1, [square root]3]; [[alpha].sub.2] = -[[theta].sup.2] + 4[theta] -3/8t for all [theta] [epsilon] [[square root]3, 3].

When [theta] = [square root]3, [[alpha].sub.1] and [[alpha].sub.2] are tangent. To the right of [theta] = [square root]3, the locus [[x.sup.pmc].sub.M] - 1 = 0 coincides with the locus [[[pi].sup.pmc].sub.M] - [[[pi].sup.fmc].sub.M] = 0; to the left of [theta] = [square root]3, the locus [[x.sup.pmc].sub.M] - 1 = 0 [R], and the only relevant condition is [[[pi].sup.pmc].sub.M] - [[[pi].sup.fmc].sub.M] = 0.

Appendix D

PROOF OF REMARK 3. Consider first [[q.sup.fmc].sub.M]. This is increasing in [theta] and invariant in [alpha]. As to [[q.sup.pmc].sub.M], we know from lemma 1 that [partial][[q.sup.pmc].sub.M]/[partial][alpha] [less than or equal to] 0 for all [alpha] [epsilon] [0, [theta]/(4t[2 + [square root]3])]. Moreover, we have

[partial][[q.sup.pmc].sub.M]/[partial][theta] = 1 + [([[theta].sup.2] + 16[[alpha].sup.2][t.sup.2] - 16[alpha]t[theta]).sup.-1/2] ([theta] - 8[alpha]t)/6t. (A2)

A sufficient condition for Equation A2 to be positive is that [alpha] [epsilon] [0, [theta]/(8t)). This is surely true, in that [alpha] [epsilon] [0, 1/(8t)), and [theta] [greater than or equal to] 1.

Appendix E

PROOF OF REMARK 4. To assess the planner's incentive to entirely cover the market, the sign of the following expression has to be evaluated:

[lim.sub.[theta][right arrow][theta]-1]([partial][[SW.sup.pmc].sub.SP]/[partial]x) = ([theta] - 1)q + 2[alpha] - t[q.sup.2] = 4[[theta].sup.2] - 8[theta] + 3 + 32[alpha]t/16t

Notice that the above expression coincides with ([[x.sup.pmc].sub.SP] - 1). Its sign is positive for all [theta] [greater than] 1 + ([square root]1 - 32[alpha]t)/2, that is, [alpha] [greater than] (8[theta] - 4[[theta].sup.2] - 3)/(32t).

Appendix F

PROOF OF LEMMA 2. The region where only the social planner chooses full market coverage, is defined as follows:

[theta] [epsilon] [1, [square root]3] and [alpha] [epsilon] [max{0, (8[theta] - 4[[theta].sup.2] - 3)/(32t)}, (1 + 2[theta] - 2[square root]1 + [[theta].sup.2])/4t)];

[theta] [epsilon] [[square root]3, 3] and [alpha] [epsilon] [0, 4[theta] - [[theta].sup.2] - 3)/(8t)].

Comparing [[q.sup.pmc].sub.M] and [[q.sup.fmc].sub.SP], we find that [[q.sup.pmc].sub.M] = [[q.sup.fmc].sub.SP] along [alpha] = (8[theta] - 4[[theta].sup.2] - 3)/(16t). This curve crosses the locus [[[pi].sup.pmc].sub.M] = [[[pi].sup.fmc].sub.M] when [theta] = [square root]5/2.