Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach

Dynamic pricing of new products has been extensively studied in monopolistic and oligopolistic markets. But, the optimal control and differential game tools used to investigate pricing behavior on markets with a number of firms are not well-suited to model competitive markets with a large number of firms. Using a mean-field game approach, this article develops a setting where numerous firms optimize prices for a new product. We analyze a framework à la Bass with product diffusion and experience effects. The analytical contribution of the paper is to prove the existence and uniqueness of a mean-field game equilibrium, further characterized in terms of mean tendencies and market heterogeneity. We also demonstrate the possible emergence of one or more groups of firms with regards to their pricing strategy. Numerical simulations illustrate how differences in firm experience translate into market heterogeneity in sales and profits. We show that, on a market where the absolute price effect is stronger than the relative price effect, we observe the emergence of two groups of firms, characterized by different prices, sales, and profits. Heterogeneity in firms’ prices and profits is thus compatible with competitive markets.


X t
The variable capturing the distribution of cumulative demands of all firms at time t P t The variable capturing the distribution of prices of all firms at time t X t Average (mean-field) market cumulative demand at time t P t Average (mean-field) market price at time t r Discount rate c (.) Unit production cost (.) Profit of the firm

Introduction
Dynamic pricing of new products has been widely examined in monopolistic [17,54,68,69] and oligopolistic settings [31,32]. Such settings, characterized by a finite number of firms with market power, can be modeled with the classical tools of optimal control and differential games. However, these tools do not allow to model competitive markets, where numerous firms without market power interact. Competitive markets are frequent, covering a wide range of products such as printers [48], city-bikes [60], books [20], home furniture [74] and electric vehicles [75]. They are particularly well suited to model interactions among (a large number of) sellers and buyers within digital environments, marketplaces, and peer to peer platforms [9,15]. Yet, "computational concerns have typically limited the analysis to industries with just a few firms, much less than the real-world industries at which the analysis is directed" [82]. Indeed, differential games already fail to provide tractable solutions when the number of competing firms is "moderate" [11,73]. Due to the technical complexity and the lack of appropriate tools, explicit analyses of competitive markets are still scarce (see the surveys by Chen and Chen [14] and Den Boer [29]). Mean field games (MFG) appear as a promising tool to investigate the behavior of markets with numerous firms. MFG have been developed by Lasry and (Fields medalist) Lions [61][62][63] in applied mathematics and independently by Huang et al. [44][45][46] in computer science. The MFG approach based on a "continuum limit" [63, p. 230] allows modeling the interactions of non-atomic firms, while the MFG equilibrium also represents a tractable solution approximating the oligopoly case [47,86].
This paper applies MFG to analyze the dynamic pricing policies of firms selling new products in competitive markets. The theoretical framework rests on a sales modeling à la [6], with diffusion and saturation effects on the demand side and learning effects on the supply side; each firm influences its sales with its pricing policy over time [13,17,31,32,54,59,68,69]. The literature on dynamic pricing of new products informs this research.
Studies on dynamic pricing originate from Bass [6], who models a monopoly's sales of a new product accounting for diffusion and learning effects. The influence of pricing on sales is modeled by Kalish [54] for the monopoly case and by Dockner and Jørgensen [31] for the oligopoly case. An extensive literature followed, which is regularly surveyed [13,29,32,53].
Extensions of the original model cover several areas. For instance, Danaher et al. [27] and Jiang and Jain [49] characterize the marketing-mix of successive generations of products. The integration of dynamic advertising highlights marketing-mix concerns [33,34,42,43,55], with stochastic generalization in Schlosser [71]. Inventory is addressed in Jørgensen et al. [52] and Chenavaz and Paraschiv [21]. Gutierrez and He [41] examine coordination between a retailer and a manufacturer for new product launchs, while Chutani and Sethi [23] address promotional issues. Further research has been conducted by Vörös [78,79], Chenavaz [18,19] and Chenavaz et al. [22] on product quality, Jing [50] on social learning, Rubel [70] on competitive entry, and Den Boer [30] on changing environments. While extensive previous research has examined dynamic pricing in monopoly and oligopoly markets, the research investigating competitive markets is less developed.
A recent stream of MFG applications appears as particularly insightful to studying pricing decisions for numerous firms. Yang and Xia [86] investigate a dual price setting, Wang and Huang [80,81] look at sticky prices, while [11] consider Bertrand and Cournot competition. Leduc et al. [64] study diffusion networks and Moon and Başar [67] look at risk sensitivity. Several researchers investigate electricity, with an emphasis on electrical vehicles [25,75,77], on electricity networks [4], on smart grids [3,57], and on oil prices [12]. Gomes et al. [39] consider more broadly commodities with electricity as a special case.
This paper contributes to the literature on dynamic pricing of new products by investigating competition with a MFG approach. The main results are as follows. We prove the existence and the uniqueness of a Nash-MFG equilibrium, deriving the properties of equilibrium pricing policies over time and across firms. Our MFG model allows to characterize analytically market heterogeneity in terms of prices and profits. Numerical simulations show that, when consumers are weakly sensitive to price differences between firms, market dynamics can lead to the emergence of two groups of firms in terms of pricing behavior, with contrasting sales and profit results. We further illustrate that consumers end up paying for the heterogeneity in firms' experience: higher differences in experience between firms result in higher prices and higher profits. It thus appears that a perfectly competitive market is compatible with large differences in pricing strategies and firms' profitability. This implication calls for a greater carefulness from administrative authorities investigating anti-competitive practices.

Dynamic Pricing of New Products
A major part of dynamic pricing research originates from the Bass [6] new product diffusion model. The cumulative demand x faced by a monopoly evolves at time t as dx/dt = (a + bx)(N − x), with a, b, N > 0. The current demand dx/dt increases with innovation a and imitation b and decreases with market saturation N − x, where N reflects the total expected market for the product. In the Bass model, demand remains "passive," in the sense that price exerts no role.
Robinson and Lakhani [69] enrich the analysis of new product diffusion by incorporating price sensitivity, allowing for a dynamic pricing examination in a monopoly setting. Demand depends on price p as: dx/dt = (a + bx)(N − x)e −dp , where d > 0 captures the price sensitivity of the demand. Firm's experience is modeled through the reduction of the unit production cost c with firm's cumulative (past) demand: c = c 0 (x 0 /x) α , where c 0 denotes firm's initial cost, x 0 its initial demand, and α ≥ 0 is a learning parameter. Kalish [54] generalizes this parametric formulation to a structural modeling, where demand writes dx/dt = f (x, p) and the unit cost c = c(x).
The extension of the monopolistic dynamic pricing to an oligopoly market with a finite number M of competing firms is provided by Dockner and Jørgensen [31]. The current demand of oligopoly i (i = 1, . . . , M) depends on its price p i , its cumulative demand x i , as well as the prices and cumulative demands of other firms: dx i /dt = f i (x 1 , . . . , x M , p 1 , . . . , p M ), while the firm's unit production cost depends only on its own cumulative past demand c i = c i (x i ). This general formulation allows modeling particular cases, such as multiplicatively separable demand dx i /dt = h i (x i )g i ( p 1 , . . . , p M ) and pricedifferential effect with g i ( p 1 , . . . , where p is the average market price and the differential-price sensitivity k > 0. The dynamic pricing model developed in this paper generalizes Dockner and Jørgensen [31], by transposing the oligopoly market with a finite number of atomic firms into a competitive market with a large (tending to infinity) number of non-atomic firms. Building on Bass [6], Robinson and Lakhani [69], and Kalish [54], we integrate diffusion, innovation, imitation, saturation, and experience effects. Both structural and parametric analyses of dynamic pricing in competitive markets are derived. The modeling of competitive markets is based on an MFG framework.

Mean-Field Games
The classic approach to analyze dynamic games considers a finite number of players [35] whose decisions depend on the state of all players. Iyer et al. [47] note the weakness of this approach when the number of players increases. First, increased complexity of the equilibrium may make it untractable. 1 Indeed, the equilibrium requires solving a coupled system of nonlinear partial differential equations, with a value function assigned to each player [11]. Second, postulating that each player takes into account the behavior of all other players is a strong requirement raising plausibility concerns [2]; each player is thus assumed to track and integrate the state of all other players at any moment in time.
The MFG approach has been developed to deal with situations where the number of players is infinite (or tends to infinity) [44][45][46][61][62][63]. A mean-field denotes the distribution of a random variable, captured in the MFG model through different indexes, such as mean or variance. An MFG equilibrium is both simpler and more credible than an equilibrium with a finite number of players [1,2,47].
From a mathematical perspective, when the number of players is large or tends to infinity, the analytic properties of the asymptotic regime simplify the analysis. Such simplification based on the law of large numbers drives a MFG equilibrium. MFG are characterized by a coupled system of (only) two partial differential equations [7,10,40,83,84]. A backward Hamilton-Jacobi-Bellman equation captures the optimizing behavior of players; a forward Kolmogorov equation (also known as Fokker-Planck equation) accounts for the evolution of the mean field. Further, MFG equilibrium possesses the property of approximating 2 a Nash equilibrium when numerous players exert influence [10,11,28].
From a managerial perspective, the MFG approach reduces the amount of information required in the decision-making process. Indeed, the mean-field synthesizes all information about a variable's distribution relevant for players making a decision. Players take a meanfield as given and behave dynamically while fixing this mean-field [1]. As players react 1 Weintraub et al. [82, p. 1377] affirm that "most industries contain more than 20 firms, but it would require more than 20 million gigabytes of computer memory to store the policy function for an industry with just 20 firms and 40 states." 2 Technically, an MFG equilibrium approximates a Nash equilibrium with a finite but large number of players. The approximation process is the following. First is the optimization of the game with a finite number of players; second is the passage to the limit. Note that the steps are not commutative.
"only" to the mean-field, they disregard the situation and the actions of every other player (opposing a classical Nash approach, which entails much more sophisticated and less realistic players). MFG also enable players to form rational expectations. That is, players correctly anticipate the evolution of the mean-field (thanks to the backward equation). One of the strengths of MFG is providing a computational simplification at the macroscopic level, while still recognizing the microstructure of the problem. The power of the MFG tool has led to increasing applications [see the survey of Gomes and Saúde [38]].

Model Formulation
This paper applies MFG to analyze the dynamic pricing behavior of firms in competitive markets. We briefly discuss the main characteristics of the model, introduce the notations, and then describe the optimization problem of a firm and the MFG equilibrium.

Model Characteristics
• Market Structure We investigate a competitive market with a large number of firms pricing an homogeneous product to a large number of consumers over time. This setting is modeled as a dynamic non-atomic game with a continuity of firms and consumers. While our formal MFG modeling is continuous, the results approximate well competitive markets with a large number of firms [10,28]. The advantage of the continuous setting is to reduce problem complexity, making possible the computation of the equilibrium. Each firm sells a finite number of goods, making a finite profit. The market is competitive in the sense that no single firm influences market conditions and extracts rent from consumers. However, firms may price differently identical products due to different consumer locations and transportation costs (see also Yang and Xia [86,p. 91] for similar assumptions). • Firm Behavior Firms are not identical, being characterized by different initial levels of production experience. However, they all share the same market information, captured by the mean-field. More precisely, the firms are not assumed to be overly sophisticated in the sense of tracking the states and decisions of every other firm. Based on the law of large numbers, firms only react to the average variables of interest, which resume information about the competitors at market level. In this sense, the interactions between firms are mean-field. Yet, each firm is still smart, as it correctly anticipates 3 the collective behavior of all firms (captured by the mean-field), and sets the price accordingly. • Information Structure We assume complete information concerning the demand addressed to the firm. That is demand is deterministic. 4 Each firm is also cognizant of its production cost, the mean-field price (which aggregates market prices), and the mean-field cumulative demand (which aggregates cumulative demands at market level). Mean-field price and mean-field cumulative demand summarize all relevant information that the firm needs to define its pricing strategy.
• Dynamic Demand and Cost The model accounts for diffusion effects on the demandside and learning effects on the supply side, disregarding inventory. Each firm's demand depends on its own price and on its cumulative (past) demand (diffusion effect). It also depends on the collective behavior of the other firms through the mean-field price and mean-field cumulative demand. The unit production cost of each firm decreases with its experience, approximated by its own cumulative demand/production (learning effect). While diffusion is related to aggregate firms, experience is tied to a single firm. • Objective Function Each firm sells a homogeneous product over an infinite planning horizon. The dynamic pricing policy of each firm maximizes the intertemporal profit, taking into account the evolution of the demand. For mathematical tractability, all firms share the same demand and cost structure, although they differ in their initial cumulative demand (experience). This heterogeneity in firms' initial experience characterizes different pricing schemes. • Equilibrium Concept We model a competitive market, in which each firm makes profitmaximizing decisions over time. In this market, each firm sets its dynamic pricing policy, while believing that all other firms behave in the same way. Recall that the number of firms is infinite, and they optimize according to the mean-field price and mean-field cumulative demand. Thus, the strategic behavior of firms can be modeled with a MFG framework. The resulting outcome is a MFG equilibrium, that is, a Nash equilibrium for an infinite number of players.

Notations
We consider a setting with continuous time t and infinite horizon. Assuming away inventory, demand equals sales, which equal production. Firm's Cumulative Demand, State Variable The cumulative demand of a firm at time t is denoted x t . When prior conditions concerning firm's cumulative demand y at prior time τ (τ ≤ t) need to be specified, we use the notation x τ,y t (to be understood as cumulative demand at time t of a firm that faced cumulative demand y at prior time τ ).
Firm's Price, Control Variable The pricing strategy of the firm consists in setting, at each time t, its price p t for the product. The price is the unique control variable in the model.
Cost of Production Accounting for experience, the unit production cost c(x) depends on firm's cumulative demand. The unit production cost is assumed to be constant or decreasing: A higher production experience allows to produce at lower cost. Formally, this corresponds to the condition: c (x) ≤ 0 for all x > 0. The learning effect considered in this model is based on an autonomous learning, due to the repetition of the task, as opposed to induced learning, due to investment in process improvement [51,58,65].
Market Structure: Mean-Field Cumulative Demand At the beginning of the planning period, t = 0, firms have different initial levels of production experience x 0 . To account for this aspect, the firms are further indexed by ω ∈ . The initial cumulative demand x 0 is thus a function of ω described as a (positive) variable X 0 : ( , F , P) → R + . Similarly, one can define X t as the (positive) variable reflecting cumulative demand x t at time t. One can associate to X t its probability law μ t (x) on R + and its average X t = E [X t ]. X t corresponds to the average (mean-field) cumulative demand at market level at time t.

Remark 3.1
The MFG model assumes the existence of an infinity of firms. In order to have something meaningful for the demand addressed to each firm, one must also assume the existence of an infinity of consumers. Otherwise, as any firm is negligible with respect to the sum of all others, the sales of any firm would be zero. The total market N (total units to sell) and the total profit at market level are therefore considered infinite. However, each firm makes, individually, a finite profit and sells a finite number of units. Formally, this implies X t ≤ N 0 , where N 0 represents the (mean) market potential per firm defined (in the limit) as the total units to sell on the market divided by the total number of firms. 5 Market Structure: Mean-Field Price At a given time t, different firms may also charge different prices. The mean-field price P t is the (positive) variable ω → P t (ω), with the average (mean-field) price P t = E [P t ].

Remark 3.2
For a given firm ω at time t, the notations distinguish between the mean-field price P t (ω) and the price set by the firm p t (ω). Indeed, a firm ω charging the mean-field price P t (ω) may decide to optimize its profit by changing its price to p t (ω). Note, however, that although p t (ω) may differ from P t (ω) on the optimization path, the two prices are necessarily equal at equilibrium. Recall that P t and p t refer to price distributions, which vary across firms. More precisely, P t refers to the (mean-field) distribution of the prices observed on the market and p t to the distribution of prices announced by the firms. When the market is at equilibrium, the two distributions must coincide. Indeed, by definition, an equilibrium is obtained when no player has an incentive to deviate from his equilibrium strategy. In particular, if a firm ω charging the mean-field price P t (ω) has an individual incentive to deviate from the mean-field price P t (ω) and to change its price to p t (ω) = P t (ω), the market cannot be at equilibrium. One can also note that if the price distributions p t and P t coincide in law, all the moments of p t and P t must coincide, 6 in particular the mean, 7 which formally translates into the condition Current Demand The (current) demand addressed to a firm at time t, dx t /dt, depends on its past demand, its price, and the market conditions, summarized by the mean-fields. In our model, the firms only know the mean tendencies of the mean-fields at time t, namely the average (mean-field) cumulative demand X t and the average (mean-field) price P t . Formally, the demand at t writes 8 dx Intertemporal Profit, Discounted Payoff Let r ≥ 0 be the discount factor. The intertemporal profit (or present value of the profit stream over the planning period) of a firm choosing the price p t over time and starting at time τ from cumulative demand y writes: where we abuse the notation by writing x t for x τ,y t . To simplify the writing, in the situations where no confusion is possible, we omit X t and P t and only write (τ, y, p t ). 5 Note that inequality X t ≤ N 0 is only true in average and does not imply x t ≤ N 0 . This means that a particular firm may sell more than N 0 units. However, if some firms sell more than N 0 , some other firms sell necessarily less to compensate at market level. 6 The reciprocal is also true: given the price distribution P t , if the price distribution p t is optimal and any moment of p t and P t coincide, then we have an equilibrium. 7 Note that the condition of mean equality (i.e., E[ p t ] = E[P t ]) is necessary for the equilibrium, but not sufficient. If only the means coincide one cannot conclude to a MFG equilibrium because this condition alone does not ensure that the distribution P t of prices observed on the market coincides with the distribution p t of prices announced by the firms. 8 A more general formulation of the type dx t /dt = h(x t , p t ; μ t , P t ) is also possible. It corresponds to the case where the firms have more information about the mean-field distributions of cumulative demands and prices.

Firm Optimization Problem
The optimization problem of each single firm consists in maximizing the intertemporal profit (3.2) over the planning period [τ, ∞) by setting at each time t ≥ τ the price p t , while taking into account the demand dynamics given in (3.1).
Optimal Profit The optimal profit of the firm is the largest profit available among all admissible pricing rules. Formally, this corresponds to: Optimal Price If it exists, an optimal price p * t is a price that satisfies * (τ, y) = (τ, y, p * t ). Note that p * t is not necessarily unique.

MFG Equilibrium
To distinguish between optimal quantities (i.e., the best strategy of a firm taking as given the behavior of the other firms, summarized by the mean-field) and equilibrium quantities (i.e., where each firm plays its best strategy generating an equilibrium), we maintain the superscript notation * for the first and employ the notation † for the latter. With these notations, an MFG equilibrium is reached when:

Remark 3.3
A MFG game implies several players, yet, our game specification disregards the number of players. Indeed, we analyze here a limiting regime with an infinite number of firms. The resulting mean field equilibrium is defined regardless of a finite number of firms (in mathematical terms P({ω 1 , ω 2 , . . . , ω l }) = 0, ∀ω 1 , ω 2 , . . . , ω l ∈ and ∀l ∈ N). In addition, with numerous firms, the fluctuations of cumulative demand and product price are expected to "average out" (see Adlakha et al. [2, Subsection 2.2] for a discussion). Consequently, each firm may ignore the specific cumulative demand and price of every other particular firm. Instead, each firm simply sets its price based on the distribution μ t of the cumulative demand X t of all other firms and on the mean-field price P t .

Model Resolution
We now solve the problem by specifying the function h expressed in (3.1). We consider the particular case where firm's demand at t depends on multiplicatively separable functions of market conditions (X t and P t ) and price ( p t ). Demand (3.1) becomes with g ( p t ) < 0 for all p t > 0, and other suitable hypotheses on f and g, which will be made precise later. The class of multiplicative separable demand functions has been widely used in dynamic pricing literature [14,29,31,32,54,69]. The condition g ( p t ) < 0 states that the demand addressed to the firm decreases with price.
Parametric Example An example of demand function can be obtained with the functional forms f (X t , P t ) = (a + bX t )(N 0 − X t )e k P t and g( p) = e −(d+k) p . In this case, the demand is given by where innovation a > 0, imitation b > 0, average market per firm N 0 > 0, price sensitivity d > 0, and differential-price sensitivity k > 0. Example (4.2) generalizes parametric demand functions from prior research [6,31,54,69].

Remark 4.1
One can show that the term N 0 − X t in Eq. (4.2) has the same interpretation as market saturation condition in the Bass model. Bass states that the total market sales are always inferior to the total market potential N , which translates formally for an oligopoly market with M firms (i = 1, . . . , M) through the condition: M i=1 x i t ≤ N . This is equivalent (when dividing both terms by M and passing to the limit) to inequality X t ≤ N 0 . In fact, Eq. (4.2) is the limit of the Bass model (see equation 3.3.2, page 326 in Dockner and Jørgensen [31], with g(x) as in equation 3.3.5, page 327) when the number of firms M tends to ∞.
For a firm starting at time τ from sales y with the demand function (4.1), the intertemporal profit (3.2) writes being the precise meaning of x t .

Modeling Hypothesis
Assuming f , g ≥ 0 are differentiable, we formulate the following modeling hypotheses: Hypothesis (H1) originates from the strict concavity of the Hamiltonian with respect to price (i.e., the control variable). This second-order condition guarantees a unique solution (provided its existence). It means that marginal revenue is a non-decreasing function of price. (H2) lim p→∞ pg( p) = 0 (thus, in particular, lim p→∞ g( p) = 0 and lim p→∞ p + g( p) g ( p) = ∞). Hypothesis (H2) is a technical condition that eliminates the possibility for firms to make infinite profit by selling zero products at an infinite price (as a limit process for p → ∞). (H3) X t , P t and Z X t ,P t = ∞ 0 e −rt f (X t , P t )dt are all bounded. Hypothesis (H3) rules out the possibility for firms to earn infinite profits. We will show later that this assumption is always satisfied at equilibrium. Where there is no ambiguity, we write Z instead of Z X t ,P t .
(H4) For any v ≥ 0, the equation Hypothesis (H4) guarantees finite sales for all firms (consequence of Remark 3.1). This technical assumption about the function f in Eq. (4.1) imposes that, irrespective of the mean price, the market is saturated when time tends to the (infinite) planning horizon. The quantity N 0 can be interpreted, in economic terms, as the average market potential per firm (see Remark 3.1).  (7)]. Examples of demand functions satisfying (H1) include the exponential function g( p) = e −dp , the linear function g( p) = N − dp, and the isoelastic function g( p) = G p γ , with γ > 1 and G > 0.

Remark 5.2
The hypotheses formulated in this section, except (H1), are specific to the MFG modeling and are necessary for deriving optimal and equilibrium conditions.

Remark 5.3
Hypothesis (H4) is satisfied, for instance, with the parametric demand function (4.2). This hypothesis, used to prove the existence of an equilibrium, may be weakened. For instance when in the long limit t → ∞ the market ends up in some saturation state N − 0 < N 0 for all price levels v, the proof can be adapted and the equilibrium will also end up at market saturation N − 0 .

Dynamic Pricing: Firm Optimum
We first consider a given firm and investigate its dynamic pricing strategy over time. Recall that while all firms share the same pricing rules, they may differ in their initial experience, reflected by the initial cumulative demand x 0 .

Lemma 5.4
Suppose that the average (mean-field) cumulative demand X t and average (mean-field) price P t are given and known to the firm. If f , g ≥ 0 differentiable and (H1)-(H3) hold, then: For r > 0 the optimal price p * t decreases monotonically over time. 3. For r = 0 the optimal price p * t is constant over time. 4. For any t, the optimal profit * (t, x) increases with respect to x and lim x→∞ Proof The proof of Lemma 5.4 is given in "Appendix A".  [65] focus on product quality improvement, whereas we analyze unit production cost reduction, and (2) these authors model both autonomous and induced learning, whereas we restrict our analysis to autonomous learning. Note, however, that point 3 of Lemma 5.4 is consistent with the result of Kalish [54,Equation (8), page 141], which also reports a constant price for the undiscounted case.
Besides generalizing the context of application of prior results, the MFG framework allows to address the role of the initial advantage in firm behavior. Point 4 of Lemma 5.4 reveals that the optimal profit increases with the initial cumulative demand, stressing the importance of the initial advantage. Indeed, a firm with larger initial cumulative demand starts the planning period with a lower production cost. This firm can then always make pricing decisions at least as beneficial as a firm with a lower initial advantage. That is to say, the initial advantage of a firm persists over time. Note that because experience cannot decrease the unit production cost below zero, the optimal profit remains bounded even for arbitrary large initial cumulative demands.
The findings in Lemma 5.4 allow to reconcile the theoretical reflection on the role of past sales (or cumulative production) in prior research and the empirical evidence of declining price patterns, in the context of competitive markets. Krishnan et al. [59, p. 1650] best describe the opposition between these ideas: "the research predominantly suggests that the optimal price path should be largely based on the sales growth pattern. However, in the real world we rarely find new products that have this pricing pattern. We observe either a monotonically declining pattern or an increase-decrease pricing pattern that does not seem close to the sales path." Thus, our work provides, for the particular case of competitive markets, theoretical support for empirical observations of declining pricing paths.

Remark 5.5
The proof of Lemma 5.4 makes use of the property that the results hold if an exogenous temporal effect-modeled by a time function, for instance l(t)-affects demand in a multiplicative way. Consequently, the (qualitative) results of Lemma 5.4 hold if demand (4.1) writes dx t dt = f (X t , P t )g( p t )l(t), where the temporal effect l(t) accounts for a product life cycle effect, such as advertising, fashion, quality improvement, and the like.

Dynamic Pricing: Market Equilibrium
We now analyze the dynamic pricing policies across firms, deriving equilibrium characteristics. For tractability, we focus on the non-discounted case with far-sighted firms (r = 0). This case is of particular interest in light of the current economic conditions in Europe, 9 where the European Central Bank opted since March 2016 for a policy of zero interest rates, which renders economic actors far-sighted (see Remark 5.10 for the discounted case r > 0). Theorem 5.6 hereafter describes the equilibrium. It also gives information regarding the evolution of market distribution of cumulative revenues and cumulative sales over time. The Gini index and the Lorenz curve (see Gini [37] and Lorenz [66] for a definition) are retained as particularly useful tools to characterize market inequalities.
. Proof The proof of Theorem 5.6 is given in "Appendix B." Theorem 5.6 completes extant literature on dynamic pricing, informing on pricing policies across firms on competitive markets. We reveal both the existence and the uniqueness of an equilibrium for the pricing policy of far-sighted firms (point 1). At the equilibrium, the average market price P t remains constant over the planning horizon (point 2a). However, firms do not charge this average price. Instead, each firm sets its own (constant over time) optimal price, which depends on initial cumulative demand x 0 . The MFG setup enables addressing the impact of differences in initial conditions on firm behavior. Theorem 5.6 captures the following market characteristic: firms (1) operating in a competitive market (no single firm enjoys market power), and (2) sharing the same optimizing rules, but (3) starting from a different initial advantage, set different prices for the same product. Due to differences in initial cumulative demands and thus in optimal prices, cumulative demand, which varies over time, depends on each firm, leading to market heterogeneity that gets stronger over time.
Besides mean tendencies, the MFG approach allows to analyze market evolution in terms of distributions, capturing heterogeneity. The MFG equilibrium can thus be characterized through the dynamics of differences between firms over time. The use of the Gini coefficient and Lorentz curve allows to provide an aggregate picture of the differences between the firms and their evolution over time. Theorem 5.6 claims that, on a competitive market, the variance of cumulative demand increases with time (point 2b), translating into greater variance of past cumulative revenues (point 2c). More precisely, this means that differences in experience between firms increase over time. This evolution is also true for cumulative revenue. Moreover, the inequality in the distribution of X † t (measured through the Gini coefficient) must have the same qualitative behavior over the whole planning horizon. Indeed, the Gini coefficient of X † t is either monotonically increasing or decreasing (point 2d), depending on the dispersion of the initial cumulative demand and on the dispersion of initial prices. The analysis of competitive markets in terms of heterogeneity and inequalities, allowed by the MFG approach, complements classical results from oligopoly markets, giving new insights about firms' dynamic pricing strategies for new products.

Corollary 5.7 At equilibrium, the variance of past cumulative demand
Proof The proof of Corollary 5.7 is in "Appendix C". Proof The proof of Corollary 5.8 is in "Appendix C". Corollaries 5.7 and 5.8 give additional results concerning the MFG equilibrium characterization in terms of variance, in complement to Theorem 5.6. The evolution of variance is monotonic for past and future cumulative demand (Corollary 5.7), as well as for future cumulative revenue (Corollary 5.8). More precisely, the increases in experience differences between firms may be explained by the initial distribution of cumulative demands X 0 rather than sales opportunities near market saturation. The same is true for future cumulative revenues. This suggests that when the market is close to saturation, the opportunities for sales and thus for revenues are low. These findings are direct consequences of points 2b and 2c of Theorem 5.6.

Corollary 5.9
Let † (x) be the optimal equilibrium profit given by the point 1 of Theorem 5.6 (see also footnote 13). Then † (x) has at most a countable number of non-differentiability points. When the number of non-differentiability points of † (x) is finite the firms can be separated into a finite number of groups, each corresponding to an interval I k =]y k , y k+1 [, k = 1, . . . K . Firms with initial cumulative demand x 0 ∈]y k , y k+1 [ have a unique optimal equilibrium price p † (x 0 ) which is a continuous function of x 0 . For any k ≤ K , firms with x 0 = y k have two optimal equilibrium prices, corresponding to the limit of optimal prices inside ]y k , y k +1 [ and ]y k −1 , y k [.

Proof
The proof of Corollary 5.9 is in "Appendix C". Corollary 5.9 highlights an interesting additional property of the equilibrium, which is characterized by the emergence of groups of firms with different strategies regarding the pricing of the product. The firms within a group have similar pricing strategies, while different groups are characterized by different pricing strategies. Numerical simulations in Sect. 6 will illustrate this result for two groups.

Remark 5.10 Equations (A.5)-(A.6) and (A.7) from "Appendix A"
can be used to define an equilibrium for r > 0. The technical considerations are more complex, and a more powerful fixed point theorem is required (for instance the Kakutani-Glicksberg-Fan or Tychonoff's fixed point theorem, as in Yang and Xia [86]). 10 Note that in this case only the existence of the equilibrium is guaranteed by the theoretical result, and not the uniqueness, which is notoriously difficult to obtain and requires additional hypothesis (see for instance, Carmona and Delarue [10], Vol. 1, Chapter 3.4). On the other hand, we expect an average equilibrium price P † t decreasing over time and a variance of cumulative demand V(X † t ) increasing over time. The other points of Theorem 5.6 need a careful analysis.

Numerical Results
This section presents the numerical simulations to illustrate the theoretical results in Sect. 5 and further explores the properties of the equilibrium.

Parameters
The parameters used for the numerical analysis are as follows: 10 Indeed, when the price of each firm is a time-dependent curve, the difficulty comes from the fact that the MFG will be a Nash equilibrium between an infinity of curves (whose mean values X t and P t represent the mean-fields). The following fixed point procedure is used: Starting with the mean-fields X t and P t , we compute the (optimal) price of any firm through the critical point equations. By averaging, we obtain the novel mean-fields which should coincide with the initial datum X t and P t . The result by Kakutani-Glicksberg-Fan guarantees that a fixed point exists if the space of curves is regular enough (the required mathematical concept is the compactness).
• The discount factor is null r = 0, implying far-sighted firms.
• The initial distribution X 0 of firms' cumulative demand is a log-normal distribution 11 of parameters mean −3 and standard deviation σ = 1/10. This implies X 0 = 0.05 and V(X 0 ) = 2.5e −5 . Figure 1 (left) plots this distribution. In their seminal article on numerical simulations of dynamic pricing, Robinson and Lakhani [69,Equations (13) and (14)] consider a demand price sensitivity d = 0.35. Parameter d = 0.4 is slightly higher in our case, but the differential price parameter k = 0.2 lowers its effect. There is large agreement that a < b [49,56,76], which our parameters verify. Numerical simulations in Kiesling et al. [56, Figure 1, p. 187] use a range from 0.3 < a + b < 0.5 and empirical estimates for cellular phones in Jiang and Jain [49, (15), p. 1118], the cost learning parameter is α = 0.4. That is, the experience curve is such as the unit production cost declines by 25% each time the cumulated demand doubles.

Market Equilibrium
To compute the equilibrium, 40,000 independent samples are drawn from the distribution specified by μ 0 . Then, an iterative procedure computing the fixed point of the mapping 12 E is performed. This yields the equilibrium price p † (·), plotted in Fig. 1(middle) (with the corresponding equilibrium value Z † = 58.26). The intertemporal profit 13 † (x 0 ) can be computed and is pictured in the right panel of Fig. 1.
At the beginning of the planning horizon, firm's experience (reflected by x 0 ) determines the unit production cost c(x 0 ), which affects firm's price and profit. Figure 1(middle) illustrates that price p † (x 0 ) decreases with initial cumulative demand; higher experience is associated with lower prices. A discontinuity in price is observed at x s 0 = 0.0529. For x 0 < x s 0 , the price declines up to 17.70. For x 0 > x s 0 , the price is almost independent of the initial cumulative demand (stable at around 4.66). No firm charges the average market price P † 0 14.79. Figure 1(right) shows that profit † (x 0 ) increases with initial cumulative demand; higher experience allows for higher profit. A jump in the derivative of ∂ x 0 † (x 0 ) occurs at the same value x s 0 as the discontinuity of p † (·). For x 0 > x s 0 , the profit presents a clear increase with 11 A log-normal (or Gibrat) distribution accounts for the multiplicative product of numerous independent and identically distributed variables, which are additive on a log scale. 12 The definition of the mapping E appears in the proof of point 1 from Theorem 5.6 in "Appendix B". 13  . Since x s 0 is not well defined in this case, it is not plotted the initial cumulative demand, suggesting that differences in experience translate into strong differences in profit. For x 0 < x s 0 , differences in experience do not impact significantly the profit, which is almost constant and low, albeit positive. Numerical results inform that no firm exits the market: The profit of all firms is positive, even if very low for the majority.
Until now, our simulations focussed on the case of a demand that is more sensitive to the absolute price p t than to the difference, p t − P t , between the price and the average market price (d > k). Staple products do not fall within this setting. Indeed, for such products, an opposite situation can be expected (d < k), with more sensitivity of the demand to price differences (relative price effect), than to absolute prices (absolute price effect). Figure 2 addresses the case of sample products by providing simulation outputs for the limit case d = 0, where the consumers are only concerned by the difference between the price of the product and the average market price, disregarding completely the absolute level of prices. We observe that, in this case, only one homogenous group of firms exists on the market. All firms act as experienced firms and propose prices very close to the average market price. However, while the prices are very similar, the differences in firms' production experience still translate in profit differences between the firms, as higher experience translates into a lower unit production cost. Note that results similar to those illustrated in Fig. 2 are obtained for all the situations where the relative price effect is stronger than the absolute price effect (d < k), suggesting that the number of groups on the market depends on the nature of the product. In the rest of the paper, we examine in further details the two-group case (d = 0.4), which is more interesting to understand, as it is conterintuitive relative to prior literature on dynamic pricing.

Experienced Versus Inexperienced Firms
The experience point x s 0 generates a threshold with respect to both price p † (x 0 ) (Fig. 1  middle) and profit † (x 0 ) (Fig. 1 right). This suggests the co-existence of two groups of firms: those below and those above the experience threshold x s 0 . We further qualify these two groups as inexperienced and experienced firms, respectively. The asymmetry in the behavior of firms around the threshold x s 0 implies that the two groups apply different pricing policies: inexperienced firms, representing 73.45% of the market, charge high prices that generate a low (but positive) profit; in contrast, experienced firms charge lower prices and derive higher profits. Pricing behavior allows to distinguish the two groups of firms more accurately than profit. Indeed, Fig. 1 reveals, at the threshold, a discontinuity (a jump) in price (Fig. 1  middle) opposing a continuity in profit (Fig. 1 right): inexperienced firms set prices far above the average market price (i.e., 17.70 or higher), while experienced firms set prices far below the average market price (i.e., 4.66 or lower). Figure 1 provided a static picture of firms' situation at t = 0. However, the threshold separating the two groups (which equals x s 0 = 0.0529 at initial time t = 0) increases over time (as all firms, including inexperienced ones, win experience over time). To illustrate the evolution of the two groups, we plot in Fig. 3(left) the (equilibrium) distribution of cumulative demands at different moments in time. We observe that the two groups of firms expand differently, which leads to a bimodal distribution. The distance between the two modes increases over time, reflecting the increase of the experience gap between the groups. Inexperienced firms do not evolve much: Due to a large production cost, they need to charge more, consequently selling less. Their cumulative demand remains low, and the unit production cost holds almost constant over time. In contrast, experienced firms face increasing demand and detach from the group of inexperienced firms. Due to a low production cost, they can charge less, maximizing profit by selling more. Their cumulative demand increases over time, which reinforces the decrease of their unit production cost. The distinction between inexperienced and experienced firms persists over time: no firm switches from one group to the other. Also, price, current demand, and profit appear relatively more homogeneous for inexperienced firms and more heterogeneous for experienced firms. It is interesting to note that high-profit firms on the competitive market benefit from a rent not tied to market power, but to initial experience. Figure 3(right) pictures the current demand dx t /dt at different moments in time. We observe that a difference exists between the current demand faced by the group of unexperienced firms (left) and the current demand faced by the group of experienced firms (right). The group of unexperienced firms does not evolve much. The firms in this group face a demand close to 0 at all times (but strictly positive). Consumers neglect the group of unexperienced firms and buy from the group of experienced firms. Figure 3(right) shows that the demand for the group of experienced firms is much higher and evolves over time. This evolution follows qualitatively market evolution: It first increases, then decreases. If one focusses on a given moment in time (t = 5, for instance), one observes that the individual demand addressed to each firm in the experienced group is rather similar. However, the profits of the firms are not similar, because the initial experience of each firm impacts the corresponding production cost.

A Relative Experience Threshold
The threshold x s 0 separating inexperienced and experienced firms is relative, as opposed to absolute. Indeed, removing the inexperienced firms (below x s 0 = 0.0529) from the market allows to compute a new equilibrium where, again, two separate groups of firms arise. We also observe in the new equilibrium inexperienced firms with a high-price strategy and experienced firms with a low-price strategy. The experience threshold separating the two groups of firms almost doubles to 0.1072, and the proportion of inexperienced firms slightly decreases to 64.20%. Thus, after removing inexperienced firms from the market, a new set of inexperienced firms appears in a lower though still dominant proportion compared to the initial market. If, once again, the emergent group of inexperienced firms (below x s 0 = 0.1072) is removed from the market, the threshold increases to x s 0 = 0.1556 and the remaining firms still split between inexperienced (high price) firms (51.07%) and experienced (low-price) firms (48.93%). Alternatively, if in the initial distribution we remove the experienced firms from the market (firms above x s 0 = 0.0529) and recompute the equilibrium, then the market again splits into two groups of inexperienced and experienced firms. The separating threshold decreases to x s 0 = 0.0340, and the proportion of inexperienced firms changes to 76.69%. To conclude, distinct groups of inexperienced (high-price) and experienced (low-price) firms arise in a competitive market. This finding is stable, and dropping either the group of inexperienced or experienced firms does not change the matter: Both groups of firms endogenously reappear on the market. Figure 4 illustrates the findings of Theorem 5.6 regarding variance. Consistent with point 2b of Theorem 5.6, Fig. 4(left) shows that the variance of cumulative demand increases over time, reflecting an increase of the gap between experienced and inexperienced firms. Figure 4(middle) illustrates how this effect translates into increased variance of past cumulative revenue over time (point 2c Theorem 5.6). Experienced firms succeed better than inexperienced firms in transforming their production experience into higher revenue. Figure 4 also informs that greater dispersion in cumulative demand (left) and in past cumulative revenue (middle) does not necessarily generate greater dispersion in cumulative profit (right). The dispersion of cumulative profit increases at the beginning of the planning horizon and falls at the end. This is consistent with the fact that while inexperienced firms face a regular increase in their profit, some experienced firms may accept losses at the beginning of the planning period in order to reduce their unit production cost, fully benefiting from the experience effect in later stages. At the end of the planning horizon, the dispersion of cumulative profit vanishes, as opportunities to make profit on a saturated market are low for all firms.

Dispersion in Firms' Behavior Over Time
We further look at the evolution of the Gini coefficient G X † t . Recall that Theorem 5.6 (point 2d) demonstrated analytically that the evolution of the Gini coefficient associated to X † t is monotonic with respect to time. In our case, we observe an increasing pattern. The Gini coefficient associated to the cumulative demand G X † t varies from 0.0560 at the beginning of the planning period (t = 0) to 0.6995 at the end of the planning period. Note that the condition for an increasing Gini coefficient identified in Theorem 5.6 (point 2d) is satisfied as the Gini coefficient associated to prices G g(P † 0 ) equals 0.7342. The observed pattern of evolution of the Gini coefficient means that, in our case, the inequality between the firms in terms of demand reinforces over time. This result is consistent with intuition, as the group of unexperienced firms faces a very low demand that does not evolve much over time (see Fig. 3), while the group of experienced firms faces a high and increasing over time demand. Thus, as time passes, the relative part of demand corresponding to unexperienced versus experienced firms is decreasing, while the size of the two groups remains stable over time. This pattern is consistent with an increasing over time Gini coefficient. The same type of argument could be invoked to justify a similar evolution, with increasing demand inequalities, for other two-group markets. The reverse pattern, with a decreasing over time Gini coefficient for the cumulative demand, can be observed on a market where the price inequalities between firms are lower than the inequalities in experience (see Theorem 5.6 point 2d). This situation is more likely to be observed on one-group markets where firms practice similar prices (see the case d = 0 in Fig. 2).

Firm Heterogeneity
The parameter σ measures firms' heterogeneity with respect to initial cumulative demand, reflecting differences in production experience, and thus in the unit production cost. A smaller σ implies greater homogeneity among firms and a larger σ greater heterogeneity. To explore the role of firm heterogeneity on competitive markets, we represent in Fig. 5 the relation between σ and several market characteristics. Figure 5(top left) illustrates the increase of the average market price P † 0 with firm heterogeneity: Prices are low for homogeneous firms and high for heterogeneous firms. Interestingly, buyers end up by paying for market heterogeneity in production experience. That is, consumers bear the cost of firms' heterogeneity by being charged more on average. For firms, higher prices turn into greater average profit † (X 0 ) (Fig. 5 bottom left). In particular, when firm heterogeneity increases (σ ranging from 0.01 to 1), the average market price increases  Figure 5 informs that firm heterogeneity also causes dispersion, both in terms of prices (top right) and profits (bottom right). Thus, in a competitive setting, differences in unit production cost are enough to explain variations in price and profit. In line with the classical view in competitive markets, when all firms are identical (σ = 0), all firms make the same profit, which is null (bottom left and right). Yet, challenging the classical view, all prices are not identical (top right of Fig. 5 shows a positive variance of prices even with σ = 0). Consequently, the existence of price differences does not necessarily reflect consumer price discrimination; it may be explained by differences in firms' levels of experience. Figure 6 illustrates how the discontinuity of the price at x s 0 changes with firm heterogeneity σ . We observe that the increase in price and profit variance (observed in Fig. 5) is not induced by a larger price discontinuity. The relationship between σ and the price gap between the two groups is not linear. In fact, a U-shaped relationship appears: When firm heterogeneity increases, the price discontinuity first decreases, then increases.

Conclusion
This article explored the dynamic pricing policy of new products in a competitive market with an infinity of non-atomic firms. A MFG framework allowed to characterize the optimal pricing schemes. Using Brouwer's fixed point theorem, we provided analytic results based on the structural (opposing parametric) properties of a general (nonlinear) demand model. Firm heterogeneity σ and statistics of price discontinuity at x s 0 Results exposed that, in a competitive market, price policies exhibit a constant or decreasing trend, which conforms prior findings on monopolistic and oligopolistic markets. Focusing on the case of far-sighted firms, we proved the existence and the uniqueness of an MFG equilibrium, further characterized in terms of price and profit heterogeneity. Analytic results further showed the existence of one or more groups of firms on the market, characterized by different pricing strategies. Our numerical investigations illustrated that the number of groups may depend on consumer behavior. We analyzed in detail a market where consumers were more sensitive to relative than to absolute prices. In this case, two groups of pricing strategies emerged: more experienced firms adopted a low price strategy, while less experienced firms adopted a high price strategy. Furthermore, our analysis showed that greater market heterogeneity in production experience causes greater average market price and profit, as well as higher dispersion. Eventually, with homogeneous firms, the profit of each firm is null, but we still find price dispersion.
From a more general perspective, our research highlighted that different pricing policies or profit streams of firms may not be enough to assess anti-competitive behaviors. Regarding price, we showed that there is no need of group collusion, dumping, limit pricing, and the like to observe different pricing schemes, and that setting a higher or lower price does not necessarily reflect firms' market power. A greater dispersion in price, and thus in profit, may conform to the dynamics of a perfectly competitive market where firms differ in initial experience. Counter-intuitively, classic signals of anticompetitive behavior-based on price and profit-may prove inaccurate in competitive markets. This observation calls for a reexamination of the assessment of anticompetitive practices, enriching the debate on the regulation of managerial policies.
Future research may expand our work in several ways. First, in this paper, we focussed on pricing decisions of far-sighted firms. While this case fits well the current situation in European Union, the investigation of dynamic pricing behavior for non-far sighted firms could be an interesting enrichment of our analysis. Moreover, to better characterize the dynamic pricing behavior of competitive firms, stochasticity could also be modeled in complement to the deterministic setting we analyzed. Indeed, our setting only provided information about the mean tendencies. A stochastic modeling would provide complementary information regard-ing deviations from the mean. Also, this paper used a multiplicative-separable demand. Future research may also take into account a more general demand function, joint in its arguments.

Appendix A: Proof of Lemma 5.4
Proof General considerations When the overall dynamics X t and P t is given, the situation enters, formally, the setting of Kalish [ . 2 and hypothesis (H1) holds, we obtain L ( p) > 0. L is thus a strictly increasing function. Its image is [ g(0) g (0) , ∞[. Since g ≥ 0 and g ( p) ≤ 0, we get g (0) ≤ 0. Thus, the image of L contains R + . P

Proof of points 2 and 3
For r > 0, the optimum price p * satisfies as in Kalish [54] and Clarke et al. [24]: which together with hypothesis (H1) provide the point 2 of the conclusion. For r = 0, one obtains dp * t dt L ( p * t ) = 0 thus p * t is constant (point 3 of the conclusion).

Proof of point 1
We introduce the Hamiltonian The optimal price p * will maximize H with respect to p when λ is the adjoint state λ * ; recalling that at the optimal solution, λ * (t) = ∂ x * (t, x t ). With the market conditions given, the maximization of (A.3) is related to the maximization of p → ( p − β)g( p) (with the particular case of interest β = c(x) − λ). Its derivative is g( p) + ( p − β)g ( p) which is positive when β ≥ L ( p) or, equivalently, L −1 (β) ≥ p (for β in the domain of L ). Thus ( p − β)g( p) increases and then decreases; it has a unique maximum attained at L −1 (β). Going back to the maximization of Eq. (A.3) it results that the optimal price satisfies p * , which shows that the optimal price is unique when * (t, x) is differentiable with respect to x everywhere (and not necessarily so otherwise).

Proof of point 4
Intuitively the profit * is increasing with respect to x because a firm with higher initial sales can use the same strategy as a firm with lower initial sales, but its cost will be lower, which allows for higher profit. The rigorous transcription of this idea is as follows: consider two firms with initial sales x 1 0 and x 2 0 , with x 1 0 ≤ x 2 0 , p some pricing strategy as in ( Denote H (β) = max p≥0 ( p − β)g( p). The optimal profit * is the unique viscosity solution (see Crandall and Lions [26], Bressan and Piccoli [8] and Bardi and Capuzzo-Dolcetta [5] for more details) of the following Hamilton-Jacobi-Bellman equation: (A. 6) In addition, at any point x t where ∂ x * (t, x t ) exists the optimal price p * t of a firm with cumulative sales x t at time t only depends on t and x t and satisfies: Thus, hypothesis (H2) and (H3) allow bounding the profit for x 0 → ∞. For x 0 → 0 the optimal price can be unbound (since c(x 0 ) may tend to ∞ for x 0 → 0) but the profit will certainly be finite (being inferior to any profit for fixed x 0 > 0), which proves the second part of point 4 of the conclusion.

Appendix B: Proof of Theorem 5.6
Proof General considerations Point 3 of Lemma 5.4 gives that for r = 0, the optimal price p * of a representative firm is constant. Given X t , P t (not necessarily at equilibrium) the (constant) optimal price p * of a firm maximizes the profit [obtained from Eq. where I c (·) refers to a primitive of the cost function c; that is, I c (y) = c(y), for any y > 0. Using the definition of Z = Z X t ,P t , which is fixed and integrating (4.1) with respect to time, yields x ∞ = x 0 + Zg( p). To determine the optimal price p * = p Z , * (x 0 ) that maximizes Eq. (B.1), we define the profit function J : Note that J is increasing with respect to x 0 . By differentiating J with respect to p, we obtain Z ( p * g ( p * ) + g( p * )) = c(x 0 + Zg( p * ))Zg ( p * ). Divide now both terms by Zg ( p * ), to obtain p * + g( p * )/g ( p * ) = c(x 0 + Zg( p * )), which can also be written as L ( p * ) = c(x 0 + Zg( p * )). Hence, after inverting the function L and indicating the explicit dependence on x 0 , we obtain: p Z , * (x 0 ) = L −1 c(x 0 + Zg( p Z , * (x 0 ))) ≤ L −1 (c(x 0 )) .

(B.3)
Note that at this time p Z , * (x 0 ) is not necessarily unique. However, we can show that p Z , * (x 0 ) is decreasing with respect to x 0 ; indeed consider x 1 0 < x 2 0 . The optimum price † t −X 0 ) 2 E[g( p † (X 0 ))] 2 which is increasing because X † t − X 0 is positive increasing. The conclusion on the Lorenz curve follows from the remark that cumulative revenue P † 0 (X † t −X 0 ) is the product of a time-dependent real constant X † t −X 0 and a time-independent random variable .

Proof of point 2d
Recall that the Gini coefficient of a real variable B is Thus (B.13) But since y 1 − y 2 and g( p † (y 1 )) − g( p † (y 2 )) have the same sign and > 0 we obtain: and after some computations, hence the conclusion follows.
Proof The proof of Corollary 5.8 is similar to that of Point 3 of Theorem 5.6 in "Appendix B." Proof of Corollary 5.9 Proof In the General considerations of the proof of Theorem 5.6, we shown that the optimal equilibrium price p † (x 0 ) is strictly decreasing with respect to x 0 , has left and right limits (which are optimums). The monotonicity of p † (·) implies (by the Darboux-Froda theorem) that it can only have at most a countable number of discontinuities. Each such discontinuity corresponds to a non-differentiability point of † (·), which proves the claims of the Corollary.