Monopoly and Quality Omission

Abstract

This study delves into a market characterized by vertical product differentiation. Product qualities are represented on a one-dimensional interval scale. The research investigates the equilibrium within a monopoly scenario, considering a production cost that is strictly convex. The monopoly offers a strategy comprising various quality–price combinations, with consumer choices determining profits. The analysis involves a comparison between two analogous models: one with a continuous range of consumers and the other with a finite number of consumers. The study explores disparities in the potential for market failure between these two settings. Notably, numerical illustrations underscore these divergences in both market contexts.

1. Introduction

Market failure arises when goods are distributed inefficiently in an equilibrium, leading to a lack of optimal social welfare in terms of both quantity or quality. More precisely, a quantity-based (or quality-based) market failure occurs when the quantities (or qualities) that reach equilibrium differ from the socially optimal set of quantities (or qualities).

The impact of monopolies on social welfare has garnered significant attention from both researchers and practitioners, examining effects on both quantity (for instance, [1,2]) and quality (such as [3,4]). Regarding quantity considerations, the curvature of demand curves becomes crucial in determining the direction of the output effect (as observed by [1,5]). Furthermore, for welfare to increase alongside price discrimination, a prerequisite is the overall expansion of output (as pointed out by [2]). On the quality front, the composition of the equilibrium offer spectrum—whether it encompasses a single quality level or multiple quality levels—can exert an influence on social welfare (as exemplified by a numerical instance in [4], where a multi-product offer spectrum contributed to market failure).

This study examines a market characterized by vertical product differentiation1 and two comparable models: one with a continuous range of consumers and the other with a finite number of consumers, wherein product qualities are depicted along a one-dimensional interval. A monopoly firm operating with a strictly convex production cost is at the center of the analysis. The study delves into disparities in the occurrence of market failure between these two market setups.

2. Literature Review

In the realm of monopoly models encompassing a continuous range of consumers, there are several notable examples. Ref. [6] explored the optimal market strategy of a monopoly producer with a strictly convex production cost function who offers multiple product qualities, where the buyers acquire, at most, a single unit of the product. They noted that the equilibrium offer spectrum could appear either as a continuous curve or as a single point representing a specific quality–price combination. Building on this, ref. [3] delved into how the introduction of a new product could impact price structure and consumer welfare. Continuing this line of inquiry, ref. [7] analyzed scenarios in which a monopoly could segment the market entirely by providing a full range of qualities when no production costs were involved. Ref. [8] extended this approach to cases where products differ not only in a vertical attribute but also in a horizontal attribute and analyzed how this extension affects equilibrium. Ref. [9] presented two distinct assumptions concerning production costs: they can be classified as either fixed or variable in relation to quality improvement. Ref. [10] examined the potential for an innovative firm to enter a monopoly market, where incumbent firms employed “limit qualities” as strategies to deter entry. Ref. [11] revolved around a market for imperfectly durable goods, demonstrating that offering varied quality versions of a product becomes advantageous when inherent product durability is high. Ref. [4] conducted a comparison between equilibrium outcomes and social optimum, exploring how a monopoly could achieve a multi-product offer spectrum. In a related development, ref. [12] formulated a model wherein the quality of a product experienced a negative impact as more consumers purchased it. Their research demonstrated that when there is any level of heterogeneity among consumers, a monopolist will consistently find it more profitable to offer multiple quality levels of the product, each at different prices. Ref. [13] discussed the conditions leading to no market failure. More recently, ref. [14] characterized efficient market demands with fixed surplus levels.

Other monopoly models have a finite number of consumers. This analysis was originally developed by [15,16] and later by [17] who derived the conditions where a monopolistic firm only offers the highest quality. Ref. [18] analyzed a dynamic game and derived that the assumption of a continuum of consumers has proved misleading in the context of durable-goods monopoly.

The concluding section of this literature review undertakes a comparative examination of the aforementioned approaches—the continuum of consumers and finite number of consumers—within the context of an oligopoly. Ref. [19] investigates scenarios involving distinct production costs, identifying the circumstances conducive to the emergence of an endogenous monopoly equilibrium when dealing with a finite number of consumer cases. On the other hand, ref. [20] explored situations where an upper bound may exist to the number of firms which can coexist in a noncooperative price equilibrium within a continuum of consumers. Building on their work, refs. [21,22] identified the conditions under which an endogenous duopoly with positive profits can establish equilibrium. Notably, each firm in the duopoly offers an endogenous single quality–price pair. Conversely, when production costs are symmetric, ref. [23] delves into a sequential game involving a duopoly, while [19] explores a simultaneous game within an oligopoly. In these instances, both studies delve into equilibria characterized by zero profits. Based on the literature, this analysis undertakes a comparison of these dual approaches within the context of a monopoly scenario. Recently, ref. [24] analyzed the stability of cartels involving vertically differentiated products. They discovered that if market shares remain unchanged from pre-collusion levels, the firm with the lowest price-cost margin under competition has the greatest incentive to deviate from the collusive agreement. This research was extended by [25], whose findings suggest that firms are highly motivated to coordinate prices when dealing with vertically differentiated products. Additionally, ref. [26] explored price collusion in the context of a repeated game, providing a detailed characterization of the collusive pricing equilibrium and assessing its impact on market shares and overall welfare.

In summary, this paper conducts an examination of scenarios involving a strictly convex production cost function. The investigation seeks to uncover disparities in the occurrence of market failure between these two market structures. The principal finding illustrates that when dealing with a continuum of consumers and complete market segmentation (with no pooling), the attainment of welfare maximization is reliant on the full servicing of the market. However, it is important to note that the equivalent model based on a finite number of consumers does not necessarily yield the same result, highlighting a contrast in the outcomes of the two approaches.

3. The Model with a Continuum of Consumers

Consider a product quality model in which the range of qualities is given by the interval $D=(0,\overline{Q}]$, where $\overline{Q}$ represents the highest available quality. Buyers are continuously distributed along the interval $T=[a,b]$, which represents their preferences for product quality, and it is assumed that $a>0$. Let p be the price associated with quality Q, forming a quality–price pair $(Q,p)\in D\times R_{++}$. The utility function for a consumer t choosing the quality–price pair $(Q,p)$ is given by $U_t(Q,p):T\times D\times R_{+}⟶R_{+}$, $U_t(Q,p)=t·Q-p$. This represents the utility for consumer t, where $t\in T$. The baseline utility level, $U_t^{*}$, occurs when the consumer makes no purchase, which is defined as $U_t^{*}=U_t(0,0)=0$.

Additionally, each consumer $t\in T$ is characterized by their income $e_t$, where $e(·)$: $T⟶R_{++}$. The net income of consumer t after purchasing the quality–price pair $(Q,p)$ is denoted as $y_t=e_t-p$. This represents the remaining income after the purchase for each consumer $t\in T$.

There is a monopoly firm. Denote by $\overline{Q}$ the maximal quality that the monopoly can offer. Let $C(Q)$ denote the marginal cost function of the monopolist producing quality Q.

We make the following assumptions:

Assumption 1A 

(Non-negative net income for all consumers). For any consumer $t\in T$ who chooses the pair $h=(Q,p)$, it holds that $e_t\ge t·\overline{Q}$, for all $t\in T$2.

There is an offer spectrum of quality–price pairs that is offered by the monopolist, where the pairs are arranged by increasing order of qualities3. This offer spectrum, denoted by s, is defined as a nonempty and closed subset of $D\times R_{++}$, where s is bounded. That is, for all $h=(Q,p)\in s$, it holds that $Q\le \overline{Q}$ and $p\le b\overline{Q}$4. Moreover, for all $\widehat{h}=(\widehat{Q},\widehat{p})$, $\tilde{h}=(\tilde{Q},\tilde{p})\in s$, $\tilde{Q}\ge \widehat{Q}\iff \tilde{p}\ge \widehat{p}$5, and the offer spectrum is bounded by $(\overline{Q},b\overline{Q})$. Each consumer $t\in T$ selects one of the pairs in the offer spectrum s or chooses not to purchase at all.

Assumption 2A 

(Priority for higher quality). If a consumer is indifferent between two quality–price pairs, she will choose the one with the higher quality.

The offer spectrum s generates the partition ${(N_h)}_{h\in s}$, where $N_h$ denotes the consumers $i\in N$ for whom $e_i\ge p$, and prefer $h\in s$ over all other pairs in s.

Formally, $i\in N_h$ is a consumer i that belongs to $N_h$, $h\equiv (Q,p)\in s$, if $e_i\ge p$, and

$$t_i·Q\ge p \mathrm{and} t_i·(Q-\tilde{Q})\ge (p-\tilde{p})$$

for all $\tilde{h}\in s$, with the last inequality holding strictly for all $\tilde{h}\in s$ where $\tilde{h}>h$.

Similarly, $T_0$ represents the set of consumers opt not to buy any product. This set is defined as:

$$T_0=\{t\in T:0>\underset{\{(Q,p)=h\in s\}}{Max}\{t·Q-p\}\}.$$

Assumption 3A 

(A strictly convex cost function). The production cost $C(Q)$ is twice continuously differentiable and strictly convex, i.e.,

$C(0)=0$, $C^{\prime }(.)>0$, and $C^{″}(.)>0$. In particular, we assume that $C^{\prime }(0)<a$ and $C^{\prime }(\overline{Q})\ge b$.

Denote by h $\equiv ($q,p) ($\overline{h}\equiv (\overline{\mathrm{q}},\overline{\mathrm{p}})$) the pair with the lowest (highest) quality that the monopoly offers and assume that for all $(Q,p)\in [s]$ we have q* ≤ q ≤ Q ≤ $\overline{\mathrm{q}}\le Min\{{\overline{q}}^{*},\overline{Q}\}$, where $a=$ $C^{\prime }({\underset{¯}{\mathrm{q}}}^{*})$, $b=$ $C^{\prime }({\overline{q}}^{*})$6. Finally, we assume that for all $Q>$ q* we have $C^{\prime }(Q)\ge a$.

The indifferent consumer t between purchasing pair h and not making a purchase at all can be expressed as t $={\displaystyle \frac{\underset{¯}{\mathrm{p}}}{\underset{¯}{\mathrm{q}}}}$, where if t $>a$ (t $\le a$), then the market is partially served (fully served). In addition, the market share $\Delta t$ of the monopolist can be calculated as $\Delta t=b-$t. Finally, we define the consumer surplus for consumer t, denoted by $C{S}_t$, as $C{S}_t=t·Q-p$. This represents the surplus consumer t gains from selecting the quality–price pair $(Q,p)$, for each $t\in T$.

3.1. Social Welfare

Let us suppose there exists a social planner whose goal is to maximize social welfare by selecting a range of qualities that the monopolist provides to consumers. The measure of social welfare, denoted as W, is defined as

$$W=\underset{t=\underset{¯}{\mathrm{t}}}{\stackrel{b}{\int }}[t·Q_t^W-C(Q_t^W)]dt,$$

(1)

where t $\ge a$.

3.2. Equilibrium

In a state of equilibrium, the monopolist attains profit maximization by presenting consumers with a selection of quality–price combinations.

3.3. Results

Before turning to the results, the main points of this model can be summarized as follows. We examine a product quality model with a continuous range of buyers, who are distributed along an interval representing an ordered set of taste preferences for different product qualities. There is a monopolistic firm with a strictly convex production cost where its objective is either to maximize profit under equilibrium by offering quality–price pairs to consumers, or, under social welfare, to maximize welfare by offering quality for these consumers.

This exploration leads to the subsequent results, where all proofs appear in Appendix A.

Result 1A 

(A Complete Segmentation under Welfare Maximization). Building upon Gayer’s findings [13], we can derive the following:

When the condition $b\le C^{\prime }(\overline{Q})$ holds, a scenario of complete market segmentation arises in the context of welfare maximization. In other words, if $t=C^{\prime }(Q_t^W)$ for all $t\in T$, then each consumer will opt for a distinct quality that aligns with the social optimum.

Note 1A. 

From Result 1, it becomes evident that a balance is achieved between the marginal utility and the corresponding marginal cost for each quality level. This implies that the social planner will present a range of qualities [q*,${\overline{q}}^{*}]$ encompassing the interval of consumers $[a,b]$. Specifically, consumer a will select quality q*, while consumer b will choose quality ${\overline{q}}^{*}$.

Note 2A. 

Building upon Mussa and Rosen’s findings [6], we can derive the following:

a. 

In equilibrium, the offer spectrum takes the form of a connected curve.

b. 

No distortion occurs in the highest quality. In other words: $\overline{q}={\overline{q}}^E={\overline{q}}^W$, where $b=C^{\prime }(\overline{q})$7.

Hence, Note 2 part b underscores that altering the highest quality lacks motivation. Such a distortion would decrease the marginal profit linked to that specific quality, and there would be no compensating benefit from adjustments to other qualities.

Derived from Note 2 part a, we can represent all pertinent prices as functions of p8 and formulate the monopoly’s profit function as follows:

$$\pi =[\underset{¯}{\mathrm{p}}-C(\underset{¯}{\mathrm{q}})]·(b-\underset{¯}{\mathrm{t}})·K=[\underset{¯}{\mathrm{p}}-C(\underset{¯}{\mathrm{q}})]·\left(b-{\displaystyle \frac{\underset{¯}{\mathrm{p}}}{\underset{¯}{\mathrm{q}}}}\right)·K,$$

(2)

where $K\ge 1$9, and ${\displaystyle \frac{\partial K}{\partial \underset{¯}{\mathrm{p}}}}={\displaystyle \frac{\partial K}{\partial \underset{¯}{\mathrm{q}}}}=0$.

The first-order condition with respect to p yield the following price function:

$$\underset{¯}{\mathrm{p}}={\displaystyle \frac{b\underset{¯}{\mathrm{q}}+C(\underset{¯}{\mathrm{q}})}{2}}={\displaystyle \frac{b+C(\underset{¯}{\mathrm{q}})/\underset{¯}{\mathrm{q}}}{2}}·\underset{¯}{\mathrm{q}}=\underset{¯}{\mathrm{t}}·\underset{¯}{\mathrm{q}}$$

(3)

From (3) we have

$$\underset{¯}{\mathrm{t}}={\displaystyle \frac{b+C(\underset{¯}{\mathrm{q}})/\underset{¯}{\mathrm{q}}}{2}}.$$

(4)

Inserting (3) to (2) yields

$$\pi ={\displaystyle \frac{{\left(b-C(\underset{¯}{\mathrm{q}})/\underset{¯}{\mathrm{q}}\right)}^2}{4}}·\underset{¯}{\mathrm{q}}·K,$$

(5)

where $K\ge 1$ and ${\displaystyle \frac{\partial K}{\partial \underset{¯}{\mathrm{p}}}}={\displaystyle \frac{\partial K}{\partial \underset{¯}{\mathrm{q}}}}=0$.

From (4), the first-order condition with respect to q can yield the optimal value of quality q.

Corollary 1A 

(Emergence of an Endogenous Multi Product Quality Spectrum). Drawing from the work of [4], it can be inferred that under equilibrium conditions where $b\le C^{\prime }(\overline{Q})$, an endogenous multi-product offer spectrum materializes10.

Prior to delving into the principal outcome of this model, we establish the concept of market failure within the context of quality: A market failure occurs when the collection of qualities in equilibrium differs from the social optimum set of qualities.

Result 2A. 

a. 

In scenarios where $b\le Min\{2a-C({\underset{¯}{q}}^{*})/{\underset{¯}{q}}^{*},C^{\prime }(\overline{Q})\}$, the occurrence of market failure can be averted.

b. 

When the condition $C^{\prime }(\overline{Q})\ge b>2a-C({\underset{¯}{q}}^{*})/{\underset{¯}{q}}^{*}$ is met, market failure is observed. Specifically, this leads to the omission of qualities in relation to the case of welfare maximization11.

Note 3A. 

i. 

Consequently, in alignment with the conditions described in part a of Result 2, a state of fully served equilibrium emerges, characterized by complete market segmentation. This implies an identical range of qualities for both welfare maximization and the fully served equilibrium, effectively eliminating market failure.

ii. 

Conversely, as detailed in part b of Result 2, a partially served market arises, resulting in market failure.

iii. 

Specifically, in the context of part b of Result 2, quality omission signifies the incentive to exclude lower quality options. This results in the monopoly presenting a range of qualities [q^E,${\overline{q}}^{*}]$ for consumers within the interval [t, b], respectively. Notably, consumer t (where t $>a$) selects quality q, with q^E > q*, while consumer b will continue to opt for quality ${\overline{q}}^{*}$.

4. The Model with a Finite Number of Consumers

Consider a product quality model in which the range of qualities is given by the interval $D=(0,\overline{Q}]$, where $\overline{Q}$ represents the highest quality. A finite number of n consumers indexed by i in the set $N=\{1,\cdots ,n\}$, encompasses the order of their taste for qualities. Let p denote the price associated with quality Q, forming a quality price pair $(Q,p)\in D\times R_{++}$. We define the utility function for consumer i, denoted as $U_i(Q,p)$, where $U_i(Q,p):N\times D\times R_{+}⟶R_{+}$ is given by $U_i(Q,p)=t_i·Q-p$, for every $i\in N$, where $0<t_1<\dots <t_H$. We denote by $U_i^{*}=U_i(0,0)=0$ the utility level derived by consumer i if she does not make a purchase. Additionally, each consumer $i\in N$ is characterized by her income $e_i$, where $e(·):N⟶R_{++}$. The net income of a consumer i after purchasing the quality–price pair $(Q,p)$ is denoted as $y_i=e_i-p$, for every $i\in N$.

There is a monopoly firm. Denote by $\overline{Q}$ the maximal quality that the monopoly can offer. Let $C(Q)$ denote the marginal cost function of the monopolist producing quality Q.

We make the following assumptions:12

Assumption 1B 

(A non-negative net income for all consumers). For each consumer $i\in N$ who chooses the pair $h=(Q,p)$, we have $e_i\ge t_i·\overline{Q}$.

There is an offer spectrum of H quality–price pairs: $s=$$((Q_1,p_1),\dots ,(Q_H,p_H))$ that is offered by the monopolist, where the pairs are arranged by their increasing order of qualities $Q_1<\dots <Q_H$. It can easily be seen that for all $\widehat{h}=(\widehat{Q},\widehat{p})$, $\tilde{h}=(\tilde{Q},\tilde{p})\in s$, we have $\tilde{Q}\ge \widehat{Q}\iff \tilde{p}\ge \widehat{p}$, which yields $p_1<p_2<\dots <p_H$. In addition, we have $0\le H\le n$, and the offer spectrum is bounded by $(\overline{Q},t_n\overline{Q})$.

Each consumer $i\in N$ selects one of the pairs in the offer spectrum s or chooses not to purchase at all.

Assumption 2B 

(Priority for higher quality). If a consumer is indifferent between two pairs, she will buy the one with the higher quality.

The offer spectrum s generates the partition ${(N_h)}_{h\in s}$, where $N_h$ denotes the consumers $i\in N$ for whom $e_i\ge p$, and prefer $h\in s$ over all other pairs in s.

Formally, $i\in N_h$ is a consumer i that belongs to $N_h$, $h\equiv (Q,p)\in s$, if $e_i\ge p$, and

$$t_i·Q\ge p\mathrm{and}{t}_i·(Q-\tilde{Q})\ge (p-\tilde{p})$$

for all $\tilde{h}\in s$, where the latter inequality is strict for all $\tilde{h}\in s$ where $\tilde{h}>h$.

Similarly, $N_0$ denotes the set of consumers who choose not to buy any brand:

$$N_0=\{i\in N:0>\underset{\{(Q,p)=h\in s\}}{Max}\{t_i·Q-p\}\}.$$

Assumption 3B 

(A strictly convex cost function). The production cost $C(Q)$ is twice continuously differentiable and convex, i.e.,:

$C(0)=0$, $C^{\prime }(.)>0$, and $C^{″}(.)>0$. In particular, we assume that $C^{\prime }(0)<t_1$ and $C^{\prime }(\overline{Q})\ge t_n$13.

4.1. Social Welfare

Now, we assume there is a social planner whose objective is to maximize social welfare by choosing a (not necessarily unique) set of qualities that the monopolist offers consumers. The social welfare W is given by

$$W=\underset{i=1}{\sum ^n}[t_i·Q_i^W-C(Q_i^W)],$$

(6)

where $0\le Q_1^W\le \dots \le Q_n^W\le \overline{Q}$.

4.2. Equilibrium

In a state of equilibrium, the monopolist attains profit maximization by presenting consumers with a selection of quality–price combinations.

The profit of the monopolist is given by

$$\pi =\underset{i=1}{\sum ^n}[p_i^E-C(Q_i^E)],$$

(7)

where $0\le Q_1^E\le \cdots \le Q_n^E\le \overline{Q}$.

In addition, we have the following n price constraints:

$$p_1^E=t_1·Q_1^E\mathrm{and}$$

(8)

$$p_i^E=p_{i-1}^E+t_i·(Q_i^E-Q_{i-1}^E),$$

(9)

for all $i=2,3,\cdots ,n$.

Inserting (8) into (9) yields

$$p_i^E=t_1·Q_1^E+t_2·(Q_2^E-Q_1^E)+\cdots +t_i·(Q_i^E-Q_{i-1}^E),$$

(10)

for all $i=2,3,\cdots ,n$.

Summing all of the n price equations yields

$$\underset{i=1}{\sum ^n}{p}_i^E=\underset{i=1}{\sum ^n}[(n-i+1)·t_i-(n-i)·t_{i+1}]·Q_i^E.$$

(11)

Inserting (11) to (7) yields

$$\pi =\underset{i=1}{\sum ^n}[(n-i+1)·t_i-(n-i)·t_{i+1}]·Q_i^E-\underset{i=1}{\sum ^n}C(Q_i^E).$$

(12)

4.3. Results

Before turning to the results, the main points of this model can be summarized as follows. We analyze a product quality model involving a finite number of consumers, each distributed along an interval that represents an ordered ranking of their preferences for different product qualities. There is a monopolistic firm with a strictly convex production cost where its objective is either to maximize profit under equilibrium by offering quality–price pairs to consumers or, under social welfare, to maximize welfare by offering qualities for these consumers.

Result 1B 

(A complete segmentation under welfare maximization). Building upon Gayer’s findings [13], we can derive the following: In the case where $t_n\le C^{\prime }(\overline{Q})$, we have a complete market segmentation under welfare maximization. That is, when $t_i=C^{\prime }(Q_i^W)$ for all $i\in N$, each consumer will choose a different socially optimum quality14.

Note 1B. 

Thus, Result 3 indicates that the marginal utility is equal to the equivalent marginal cost for each of the n qualities.

Result 2B 

(No quality distortion of the highest quality). In the case where $t_n\le C^{\prime }(\overline{Q})$, there is no quality distortion of the highest quality. That is, $Q_n=Q_n^E=Q_n^W$, where $t_n=C^{\prime }(Q_n)$15.

Note 2B. 

Thus, Result 3 shows that there is no incentive to distort the highest quality since it decreases the marginal profit from this quality without any compensation from any of the other qualities.

Result 3B 

(Market failure). In the case where $t_n\le C^{\prime }(\overline{Q})$, there will be market failure. Specifically,

a. 

When $t_i>$$(n-i)(t_{i+1}-t_i)$ for all $i\in N$, in equilibrium, the market is fully served and $0<Q_i^E<Q_i^W$ for all $i=1,\dots ,(n-1)$); i.e., in equilibrium the monopoly will distort all of its qualities relative to the social optimum, except its highest quality.

b. 

Let k, where $1\le k<n$, be the highest consumer index for which $t_k\le$ $(n-k)(t_{k+1}-t_k)$. In this case, the market is partially served, where $Q_i^E<Q_i^W$ for all $i=(k+1),\dots ,(n-1)$; i.e., in equilibrium, the monopoly will omit its lowest k qualities and will distort the middle $(n-k-1)$ qualities relative to the social optimum but will not change its highest quality.

Note 3B. 

Thus, Result 5 suggests that there is an incentive to distort/omit the lower qualities because the monopoly will increase the price of higher qualities (due to the increased gap between the levels of qualities—$\Delta Q$), and this may increase its total profit.

4.4. Special Cases

Now we invert the assumption that $t_n\le C^{\prime }(\overline{Q})$ and deal briefly with the following complementary cases:

I.

$t_1<C^{\prime }(\overline{Q})<t_n$.

Corollary 1B. 

In the case where $t_1<C^{\prime }(\overline{Q})<t_n$,

a. 

There will be market failure with no quality distortion of the highest quality (equivalent to the former case).

b. 

In equilibrium, there may be a partial segmentation with regard to the highest quality; i.e., more than one consumer may choose the highest quality–price pair, which indicates that there may be fewer than n quality–price pairs in the offer spectrum.

c. 

In equilibrium, the highest pair includes the highest quality, meaning that $Q_H=\overline{Q}$.

II.

$t_1\ge C^{\prime }(\overline{Q})$.

Corollary 2B. 

In the case that $t_1\ge C^{\prime }(\overline{Q})$,

a. 

There will be no quality distortion of the highest quality (equivalent to both former cases).

b. 

In equilibrium, the social planner will offer the highest quality $\overline{Q}$ as a single quality.

c. 

Following Shitovitz et al. [17], in the special linear case of their model where $C(Q)=cQ$, and $t_1>c\ge 0$16, there may be no market failure (in contrast to both former cases) when the market is fully served since both the social planner and the monopoly will offer the highest quality $\overline{Q}$ as a single quality.

5. Numerical Examples

In the subsequent pair of parallel numerical illustrations, we undertake a comparison between both models.

In Example 1A, we establish that when dealing with a continuous spectrum of consumers and a state of fully served equilibrium, market failure is not assured. Conversely, in the corresponding Example 1B, which involves a finite number of consumers within a fully served equilibrium, market failure transpires, accompanied by a distortion in the lower quality offering.

Moving on to Examples 2A and 2B, we observe that market failure emerges in both scenarios when the equilibrium leads to partial market servicing. Specifically, in the context of the continuous consumer spectrum, a range of qualities is omitted, while in the finite consumer scenario, the omission is concentrated on the lower quality.

Example 1A. 

There is an interval $T=\left[2,3\right]$, which represents a continuum of consumers, each with preferences defined by their utility function: $U_t=tQ-p$, where $t\in T$. The range of product qualities is given by the interval $\left[0,4\right]$. The income of each consumer $t\in T$ is fixed at 15, where $15>b·\overline{Q}=3·4=12$. The monopoly’s cost function for producing a product of quality Q is $C(Q)=0.5Q^2$. Since $3=b\le 2a-C({\underset{¯}{q}}^{*})/$q* = $2·2-{\displaystyle \frac{0.5{({\underset{¯}{q}}^{*})}^2}{{\underset{¯}{q}}^{*}}}=4-0.5·2=3$, where $2=a=$$C^{\prime }({\underset{¯}{q}}^{*})=$q*, this signifies that the market is fully served. Moreover, the fact that there exists an identical range of qualities, namely $[2,3]$, for both the social optimum and equilibrium scenarios, indicates the absence of market failure.

Example 1B. 

There are two consumers, and their utilities are $U_1=2Q-p$ and $U_2=3Q-p$. The range of qualities is given by the interval $\left[0,4\right]$. The income of both consumers is $e_1=$$e_2=15>{\rho }_2·\overline{Q}=3·4=12$. The cost function of the monopoly is $C(Q)=0.5Q^2$, where $3=t_2<$$C^{\prime }(\overline{Q})=\overline{Q}=4$. Since $C^{\prime }(Q)=Q$, the socially optimum qualities are: $Q_1^W=t_1=2$ and $Q_2^W=t_2=3$. Since $p_1^E=t_1{Q}_1^E=2Q_1^E$ and $p_2^E=t_2{Q}_2^E-(t_2-t_1)Q_1^E=3Q_2^E-Q_1^E$, we have $\pi =p_1^E-C(Q_1^E)+p_2^E-C(Q_2^E)=Q_1^E$$-C(Q_1^E)+3Q_2^E-C(Q_2^E)$. Since ${\displaystyle \frac{\partial \pi }{\partial Q_1^E}}=0=1-Q_1^E$ and ${\displaystyle \frac{\partial \pi }{\partial Q_2^E}}=0=3-Q_2^E$, the equilibrium qualities are $Q_1^E=1<2=Q_1^W$ and $Q_2^E=3=Q_2^W$. Because $Q_1^E<Q_1^W$, market failure is observed, even though the market is operating under full servicing.

Example 2A. 

There is an interval $T=\left[1.5,3\right]$, which represents a continuum of consumers, each with preferences defined by their utility function: $U_t=tQ-p$, where $t\in T$. The range of product qualities is given by the interval $\left[0,4\right]$. The income of each consumer $t\in T$ is fixed at 15, where $15>b·\overline{Q}=3·4=12$. The monopoly’s cost function for producing a product of quality Q is $C(Q)=0.5Q^2$. Since $3=b>2a-C({\underset{¯}{q}}^{*})/$q* = $2·1.5-{\displaystyle \frac{0.5{({\underset{¯}{q}}^{*})}^2}{{\underset{¯}{q}}^{*}}}=3-0.5·1.5=2.25$, where $1.5=a=$$C^{\prime }({\underset{¯}{q}}^{*})=$q*, this signifies that the market is partially served, where consumers $[1.5,2)$ will not buy at all. Furthermore, when comparing the quality range of $[1.5,3]$ under the social optimum with the reduced range of $[2,3]$ under equilibrium, it becomes clear that market failure is evident.

Example 2B. 

There are two consumers, and their utilities are $U_1=1.5Q-p$ and $U_2=3Q-p$. The range of qualities is given by the interval $\left[0,4\right]$. The income of both consumers is $e_1=$$e_2=15>{\rho }_2·\overline{Q}=3·4=12$. The cost function of the monopoly is $C(Q)=0.5Q^2$, where $3=t_2<$$C^{\prime }(\overline{Q})=\overline{Q}=4$. Since $C^{\prime }(Q)=Q$, the socially optimum qualities are $Q_1^W=t_1=1.5$ and $Q_2^W=t_2=3$. Since $p_1^E=t_1{Q}_1^E=1.5Q_1^E$ and $p_2^E=t_2{Q}_2^E-(t_2-t_1)Q_1^E=3Q_2^E-1.5Q_1^E$, we have $\pi =p_1^E-C(Q_1^E)+p_2^E-C(Q_2^E)=$$-C(Q_1^E)+4Q_2^E-C(Q_2^E)$. Since ${\displaystyle \frac{\partial \pi }{\partial Q_1^E}}=0=-Q_1^E$ and ${\displaystyle \frac{\partial \pi }{\partial Q_2^E}}=0=3-Q_2^E$, the equilibrium qualities are $Q_1^E=0<1.5=Q_1^W$ and $Q_2^E=3=Q_2^W$, where the market is partially served. Because $Q_1^E<Q_1^W$, market failure is observed.

6. Discussion and Conclusions

This paper explores a vertical product differentiation model involving a monopolist characterized by a strictly convex production cost function and a continuous spectrum of consumers. Additionally, a parallel analysis is carried out using a similar model that encompasses a finite number of consumers. The research aims to investigate variations in the occurrence of market failure between these two distinct market configurations.

This study offers several contributions. Firstly, it concentrates on equilibrium outcomes and the social optimum in a monopolistic scenario under two distinct market settings. Secondly, the paper undertakes a comparison of two approaches within the monopoly context, thereby differing from the works of Gayer [19,21,22], which compared these approaches within an oligopoly framework. Lastly, the study explores discrepancies in the potential for market failure between these two distinct market settings.

The central finding highlights that within the scope of complete market segmentation under welfare maximization, no occurrence of market failure emerges in a fully served equilibrium (as demonstrated in Example 1A). This outcome stands in contrast to the corresponding situation of complete market segmentation in a finite consumer scenario, where market failure persists even within a fully served equilibrium (as evidenced in Example 1B). Furthermore, when dealing with a partially served equilibrium, market failure arises alongside the omission of a range of qualities on the left side, in comparison to the corresponding set of qualities achieved under the premise of welfare maximization (as demonstrated in Examples 2A and 2B). The reasoning behind this contradiction lies in the fact that in both market scenarios, a monopolist is motivated to distort quality to maximize profits. However, in the scenario with a continuous spectrum of consumers, there is a constraint requiring the offered spectrum to be connected. This constraint inhibits any quality distortion, leading to outcomes where the range of equilibrium qualities either matches the upper range of socially optimal qualities or both ranges are precisely the same.

In the realm of future research, there is potential for extending this model to encompass the broader context of a (weakly) convex production cost function. This expansion would serve the purpose of identifying conditions that, within an equilibrium framework, facilitate an expansion of the range of product qualities, relative to the corresponding quality set attainable under the social optimum.17