# Taking firms’ margin targets seriously in a model of competition in supply functions

Denis Claude, Mabel Tidball

► To cite this version:

Denis Claude, Mabel Tidball. Taking firms’ margin targets seriously in a model of competition in supply functions. Journal of Economics and Management Strategy, 2025, 34 (1), pp.174-192. 10.1111/jems.12577. hal-04743649

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Submitted on 26 May 2025

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# Taking firms' margin targets seriously in a model of competition in supply functions

Denis Claude<fnref n="1" /> | Mabel Tidball<fnref n="2" />

## Correspondence

Denis Claude, Laboratoire d'Économie de Dijon (LEDi), Université de Bourgogne, 2 boulevard Gabriel, Dijon, France.

Email: denis.claude@u-bourgogne.fr

## Funding information

Agence Nationale de la Recherche, Grant/Award Numbers: ANR-10-LABX-11-01, ANR-19-CE26-0008-01; Conseil régional de Bourgogne-Franche-Comté, Grant/Award Number: Project DyIn; French Agence Nationale de la Recherche (ANR); LabEx Entreprendre, Grant/Award Number: ANR-10-LABX-11-01; RILLM, Grant/Award Number: ANR-19-CE26-0008-01; Region Bourgogne-Franche Comté

## Abstract

This article deals with the integration of industry-level markup targets into oligopoly theory. It proposes a behavioral competition model in which firms use the average cost-plus price to determine their supplies. Specifically, firms are assumed to increase (resp., decrease) their supplies as the market price rises over (resp., falls below) this reference price. The equilibrium market outcome lies between those corresponding to Bertrand and Cournot competition. It depends on the industry's margin target, which determines the slope of firms' supply functions. The more significant the markup target is, the lower are the firms' equilibrium supplies at any price level and the higher is the equilibrium market price. An industry-wide commitment to targeting a markup thus reduces competition in equilibrium. The reduction in competition is more pronounced than when firms commit to linear supply functions.

# 1 | INTRODUCTION

Cost-plus pricing is a lot like the romance novel genre, in that it's widely ridiculed yet tremendously popular. (Dholakia, 2018)

The *average cost-plus margin* pricing rule is an essential benchmark for businesses.<fnref n="1" /> Altomonte et al. (2015) evidence<fnref n="2" /> this by surveying a representative sample of 14,000 European firms. Among the respondents, 60% stated they have market power. Of these price-making firms, 75% revealed that they rely on markup pricing. Hence, a large proportion of firms price their products primarily by applying the following rule of thumb:

$$p = (1 + m)AC, \qquad (1)$$

where *p* denotes the price, *m* is a fixed profit margin expressed in percentage, and *AC* is the average cost of production.³ With the notable exception of Grant and Quiggin (1994), oligopoly theory has paid inadequate attention to actual pricing practices and, in particular, to average cost pricing. Grant and Quiggin (1994) assume that firms adhere strictly to rule (1) and adjust their respective markups *m* strategically so as to maximize their profits.⁴

<footnote n="1">¹Laboratoire d'Économie de Dijon (LEDi), Université de Bourgogne, Dijon, France</footnote>

<footnote n="2">²Centre d'Économie de l'Environnement – Montpellier (CEE-M), Université Montpellier, CNRS, INRAe, Institut Agro, Montpellier, France</footnote>

<footnote>This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.</footnote>

<footnote>© 2024 The Authors. Journal of Economics & Management Strategy published by Wiley Periodicals LLC.</footnote>
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While in some cases, it can be assumed that business firms are markup setters, in many cases, they follow industry practices. Business owners may know from business tradition, learn from training, or infer from cost surveys published by trade associations what the appropriate margin target is in their industry. Then, this common and exogenously defined margin target *m* is a characteristic of the industry from which firms may apprehend to deviate for fear of infuriating consumers, competitors, or both. Reaching this target, however, is a complex task. Indeed, it requires each firm to make production adjustments that simultaneously affect both sides of Equation (1), that is, the firm's cost of production and the market price.

The question arises as to how the industry's margin target, conceived as a market characteristic, can be incorporated into oligopoly theory. This article suggests a way forward. We assume that firms consider the markup price as a reference price and commit to conditioning their respective supplies on how the market price diverges from it. Building on this assumption, we develop a behavioral model of price or quantity setting by firms in an oligopoly market structure.

Different behavioral assumptions lead to diverging predictions regarding oligopoly pricing and, thus, contrasting market outcomes. The two classical conjectures are those of Cournot (1838) and Bertrand (1883). The Cournot firm strategically sets quantities, whereas the Bertrand firm sets prices. In either case, they do so while satisfying the Nash equilibrium condition that their predictions about competitors' equilibrium decisions are correct and thus come true at equilibrium. The resulting Cournot price is generally above the marginal cost, implying that firms enjoy an oligopoly rent in the form of abnormal profits. By contrast, the Bertrand price typically coincides with the marginal cost of production so that the oligopoly rent vanishes.

This article, like that of Grant and Quiggin (1994), belongs to a line of work that proposes to go beyond the price-quantity opposition by refining or elaborating firms' strategy sets. Bylka and Komar (1976) proposed to mix Cournot and Bertrand assumptions to define Cournot-Bertrand oligopolies in which quantity-setting firms face price-setting competitors.⁵ The conjectural variation literature (Bowley, 1924; Bresnahan, 1981) helped clarify that the profit-maximizing firm is indifferent as to the choice of its strategic variable. Indeed, whatever the strategic variable (price, quantity, or markup) is, the firm adjusts it so that marginal revenue equals marginal cost. What matters to the firm is the strategic variable chosen by rival firms. For example, when competitors adjust their prices rather than their quantities, the firm faces a more elastic residual demand curve and has incentives to produce more (Cheng, 1985).

The conjectural variation is the firm's conjecture about rival firms' reaction functions. Bresnahan (1981) defined the concept of consistent conjectural equilibrium by imposing a consistency requirement on conjectures. In addition, to the Nash equilibrium condition, he added the requirement that each firm conjecture about competitors' behavior matches their actual reaction functions (locally). Although widely used in empirical industrial organization, the conjectural variation approach has been dismissed as lacking game-theoretic foundations. Tirole (1988), for example, writes that:

> [a] static game is, by definition, a game in which each firm's choice is independent of its rivals' choices. By the very timing and information structure of the game, firms cannot react to one another. Thus, any conjecture about one's opponents' reaction that differs from no reaction is irrational. (Tirole, 1988, p. 244)

Grossman (1981) and Hart (1985) proposed the concept of “supply functions” as a theoretically valid alternative to “conjectural variations.” In their perspective, the choice of a supply function $S_i(\cdot)$ represents a commitment by firm $i$ to supply a quantity $q_i$ at any admissible price $p$, satisfying $q_i = S_i(p)$. A Nash equilibrium in supply functions, or *supply function equilibrium*, is a profile of supply functions that are mutual best responses. The supply function equilibrium determines the market outcome. Menezes and Quiggin (2023) showed that the equilibrium supply function coincides with the conjectural equilibrium under certain conditions, thus bridging theoretical and empirical research on oligopoly pricing.

This supply function approach suffers from a major drawback, however. If we do not restrict the functional form of $S_i(\cdot)$, multiple supply function equilibria typically exist. Several approaches have been proposed to solve this problem. Klemperer and Meyer (1989) introduced demand uncertainty. Vives (1986) showed that capacity precommitments uniquely determine firms' supply functions in competitive markets. In his model, firms strategically set their production capacities before entering a competitive market. Each firm sets its production capacity to reach its efficient scale of operation, determining its long-term cost structure. Once a firm has set its production capacity, any attempt to exceed it incurs additional short-term costs, the magnitude of which depends on the flexibility of the technology. The equilibrium market outcome transitions from Cournot to Bertrand as one considers technologies exhibiting increasing degrees of flexibility. This result indicates that the shift from rigid to increasingly flexible technologies diminishes the strategic value of a capacity precommitment.
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Menezes and Quiggin (2012) restricted firms' strategies to linear supply functions with a common, exogenously given, fixed slope. In this latter approach, the slope of firms' supply functions can be interpreted as a measure of their competitive (or market) toughness in the spirit of d'Aspremont et al. (2007) and d'Aspremont and Dos Santos Ferreira (2009). When the slope parameter is changed, the model describes a continuum of market outcomes ranging from the Bertrand solution to the Cournot Solution.⁶ This comparative statics result concerning competitive toughness thus aligns with that of Vives (1986) concerning technology flexibility. This alignment is unsurprising as a less flexible production technology leads to a steeper marginal cost and, thus, relatively flat competitive supply functions, signaling a reduced competitive strength.

Supply function competition models are of clear interest for modeling the strategic interaction between traders who compete in supply schedules on financial and energy markets (particularly electricity and natural gas).⁷ They are also well suited to analyzing competitive environments where firms sign (legally binding) supply contracts with their customers. As Grossman (1981) noted, Cournot's conjecture that firms take their competitors' supply as given may reflect that supply contracts bind firms and consumers. On the contrary, Bertrand's conjecture that firms can capture customers from their competitors by offering lower prices might suggest an inability for firms and customers to enter into supply contracts.

Finally, competition in supply functions is no different from competition in reaction functions (Vives, 1999, p. 188). Supply function competition models can, therefore, be seen as reduced forms of two-stage games in which the market competition stage comes after a strategic commitment stage. During this initial stage, economic agents make strategic commitments designed to alter their competitors' reaction functions to their own advantage. Examples of such games can be found in the strategic delegation (Fershtman & Judd, 1987; Sklivas, 1987; Vickers, 1985) and strategic trade (Laussel, 1992) literatures.

In the above-mentioned contributions, supply functions are grounded in firms' rational decisions. In actual practice, however, they may have behavioral determinants, notably a behavioral bias towards average cost-plus pricing, which translates into the goal of achieving a fixed markup over average production costs. This paper relies on linear supply functions to embed information about this markup target into firms' strategies. To fix ideas, and for the sake of simplicity, we shall consider an industry consisting of identical firms that share a common, exogenously set, profit margin target m. Firms are assumed to consider the average cost-plus price as the reference price. They commit to increase (resp., decrease) the quantity supplied when the market price rises above (resp., falls below) this reference price. Namely, to capture the supply behavior of firm i, we adopt the following behavioral assumption:

$$q_i = \phi_i + \psi(p - (1 + m)AC(q_i)) \quad (2)$$

where the vertical intercept $\phi_i$ is firm i strategic choice variable and $\psi \ge 0$ is a sensitivity parameter. In the spirit of Klemperer and Meyer (1989), the value of parameter $\psi$ is interpreted as the result of “decisions about the size and structure of the organization, the organization's values, and the decision rules to be followed by lower-level managers [that] must be made in advance.” Capacity precommitments, underutilized human resources or enduring recruitment difficulties, employee compensation programs linking the wage to the output price, supply contracts signed with consumers, and competition for resources among the different divisions of the firm are among the many factors that determine the firm i's ability to react to prices changes and so the slope of its supply function. In line with Vives (1986), these factors can also be considered as determining the degree of flexibility of the technology the firm uses for production.

The plan of the paper is as follows. Section 2 provides background information on average-cost pricing. Section 3 presents the main characteristics of the oligopolistic market and shows that our behavioral assumption (Equation 2) implicitly defines the firms' supply schedules. These consist of linear supply functions having the same slope and parameterized by their intercepts. The slope of the supply functions, common to all firms, reflects the degree of market toughness. Three exogenous parameters determine its value: the sensitivity of supply to price ($\psi$), the curvature of the cost function ($v$), and the industry's margin target ($m$).

Section 4 rephrases the oligopoly game as a model of competition in supply functions and subsequently solves it. Within this model, there is a unique Nash equilibrium in the intercepts of the firms' linear supply functions. Comparative static analysis reveals that the competitiveness of the market outcome depends on the slope of firms' supply functions. It shows that the lower is the slope of firms' supply functions, the steeper are their equilibrium supply curves and, thus, the lower is the equilibrium industry output. Accordingly, a higher cost curvature or industry margin target, which implies relatively flat firm-level supply functions, reduces competition. Conversely, a more challenging market environment, which translates into steeper firm-level supply functions, leads to a relatively competitive market outcome. The market outcome coincides with the Cournot outcome whenever firms' supply functions are horizontal.
However, unless firms operate at a constant marginal cost, the Bertrand outcome is ruled out even in the toughest market environments.

Section 5 compares the solution of our *behavioral model* of oligopoly competition in which firms regard the markup price as a reference price with the solution of the standard model of competition in linear supply functions. It shows that the industry-wide commitment to target a markup *m* reduces market supply and yields higher industry profits. Furthermore, industry profits are increasing in the markup *m*. This increase in profits comes at the expense of consumers and results in a loss of social welfare. Section 6 concludes the paper.

## 2 | BACKGROUND ON AVERAGE COST PRICING

The managerial literature has long recognized the empirical prevalence of average cost pricing. It provides many illustrative examples, among which the most notables are restaurants where “food is marked up three times direct costs, beer four times and liquor six times” (Godin & Conley, 1987, cited in Phillips, 2011) and hotels where the “dollar-per-thousand rule”⁸ determines room rates (O'Neill, 2003). Dholakia (2018) claims he has known of companies that apply markups ranging from 5% to 800%. Some pharmaceutical companies that distribute generic drugs are probably making even higher margins than this upper bound suggests (Hill et al., 2018). These examples underline that the markup that is applied varies widely in practice.

By contrast, the economic profession welcomed initial survey evidence about markup pricing with skepticism. In its defense, one may note that Hall and Hitch (1939), in their seminal contribution on the topic, went as far as to suggest that the use of heuristic pricing methods is incompatible with profit maximization and, thus, provides a refutation of *Marginalism*. This challenging claim led to a heated debate known as the “Marginalist Controversy” over the period ranging from 1939 to 1952.⁹ In this tense context, survey evidence supporting the markup pricing hypothesis was often erroneously interpreted as questioning mainstream economics. However, subsequent research quickly pointed out that average cost pricing is consistent with profit maximization provided certain conditions are satisfied. For example, markup pricing is rational when firms pursue long-run profits. Indeed, in this case, the profit-maximizing price *p* is such that the profit margin *m* decreases with the elasticity of demand *ε*; that is,

$$p = (1 + m(\varepsilon))AC, \quad m'(\varepsilon) < 0. \qquad (3)$$

Thus, markup pricing and profit maximization lead to the same market price whenever firms base their expected profit margin *m* on an accurate estimation of the long-run price elasticity of demand *ε*.¹⁰

Such arguments combined with changes in economic methodology helped calm the marginalist controversy. Specifically, Herbert Simon's Simon (1955) “Behavioral Model of Rational Choice” offered a rationale for the reliance on rules-of-thumb in complex decision environments, emphasizing the virtues of pursuing satisfactory rather than optimal outcomes. Also, Milton Friedman's Friedman (1953) “Methodology of Positive Economics” prompted a methodological shift from the realism of assumptions to their predictive power. Once the marginalist controversy subsided, empirical evidence of average cost pricing continued to pile up without raising economists' eyebrows. Yet, with the notable exception of Grant and Quiggin (1994), new advances in the understanding of firms' pricing practices have hardly changed the modeling of firms' behavior in oligopolistic markets.

Before the Altomonte et al. (2015) study mentioned in Section 1, numerous other surveys had shown that markup pricing remains widely used. The Bank of England (Greenslade & Parker, 2010, 2012) surveyed a representative sample of 2331 firms on the considerations guiding their pricing decisions for the main product they sell in the United Kingdom. Business respondents reported that their pricing decisions are primarily driven by cost considerations: 25% stated they rely on constant markup prices and 33% on variable markup prices. The study of pricing behaviors in other parts of the world leads to similar lessons. In this regard, one notable study is that of Rao and Kartono (2009), who surveyed a sample of firms operating in the United States, Singapore, and India. They asked firms to select up to five pricing approaches from a list of 19 options. Again, cost-plus pricing was the most frequently selected option.

There are several possible explanations for the enduring popularity of this pricing approach. First, companies often view it as financially prudent (Nagle et al., 2014; Phillips, 2011). In particular, risk-averse managers might choose a price that covers the average cost as protection against the risk of loss (Shy, 2008, section 3.5.2). Second, companies think this pricing approach is fair. As already argued by Hall and Hitch (1939, p. 19), firms perceive this price as “the ‘right’ price, the one which ought to be charged.” They believe they owe this price to their consumers as well as their
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competitors. Further contributions in economics and management provide support for this idea as applied to consumers. The selling price must be deemed acceptable by consumers. As Simon (2015) notes, consumers have a cost-plus mentality: they assume that the selling price reflects production costs and, ultimately, value. This mindset imposes constraints on companies' pricing strategies. If the price deviates excessively from the costs of production, consumers will feel resentful and businesses will suffer a loss of reputation. Equity considerations then translate into *implicit price contracts* between companies and consumers. Okun (1981) elegantly formulated this idea using the image of an “invisible handshake” between consumers and firms. Rotemberg (2005, 2011), and more recently Eyster et al. (2020), have expanded the understanding of how such implicit contracts shape firms' pricing strategies, showing that they force firms to cut back on margins.

Many contemporary authors also agree with Hall and Hitch's (1939) contention that the practice of markup pricing is fair to competitors. Khaneman et al. (1986) summarized these ideas in a “dual entitlement principle” according to which both firms and consumers are entitled to fair terms (i.e., price or profit) of trade. As noted by Hall and Hitch (1939), the process of price formation is not purely decentralized. Quite to the contrary, firms use three different channels to coordinate their pricing decisions. First, they may resort to conventions and traditions in setting the margin rate (as is the case for the restaurant and hotel industries). Second, they may use the price set by a market leader as a guide in their pricing decisions. Finally, firms may coordinate their pricing decisions by relying on producer cost surveys published by trade associations. Following Phillips (2011), we may note that:

> [t]he cost-plus pricing approach appears to be objective and defensible. If all competitors in a market have similar cost structures, it would appear to be a reasonable way to ensure consistency with the competition.
> (Phillips, 2011)

Here, the consistency requirement is to be interpreted as relating to price moderation, coordination, and stability.

Despite its popularity, the markup pricing rule is inherently flawed. The error of cost-based pricing decisions becomes apparent when comparing Equation (1) with Equation (3). The price markup *m* should be linked to market demand, but cost-based pricing decisions ignore consumers' willingness to pay. If it is perfectly correct to imply that cost information has a role to play in pricing, cost information alone cannot be the basis of sound pricing. Indeed, the average cost of a good depends on the quantity that is produced. This point is made clear by Nagle et al. (2014):

> The mistake that cost-plus pricers make is not that they consider costs in their pricing, but that they select the quantities they will sell and the buyers they will serve before identifying the prices they can charge. They then try to impose cost based prices that may be either more or less than what buyers will pay. In contrast, effective pricers make their decisions in exactly the opposite order. They first evaluate what buyers can be convinced to pay and only then choose quantities to produce and markets to serve. (Nagle et al., 2014)

Markup pricing is a “pure” pricing approach like *value-based pricing*—where businesses price in line with perceived consumer value—or *market-based pricing*—where businesses price in keeping with competitive offerings (Phillips, 2011). All of these approaches are inadequate because they fail to recognize the multidimensional aspect of pricing decisions. Corporate pricing strategies account for both costs and margin, demand, and competition imperatives. In the remainder of this paper, we develop a model where all these imperatives contribute jointly to firms’ pricing decisions. Specifically, our purpose is to analyze strategic interaction among firms when their supply decisions are conditional on the market price reaching their reference price, which results from cost-plus pricing principles.

3 | THE MODEL

Our purpose is to build a simple model of supply function competition that explicitly captures firms' profit margin objectives. We will work under the following customary assumptions. On the supply side, there are $n \ge 2$ identical firms (indexed by $i = 1, 2, ..., n$) producing a single commodity. The quantity supplied by firm $i$ is $q_i$ and the overall quantity supplied is $Q = \sum_{i=1}^{n} q_i$. All firms share the same technology summarized by the cost function $C(q_i) = cq_i + \frac{v}{2}q_i^2$, with $c, v \ge 0$. On the demand side, there is a representative consumer with a utility function given by $U(Q, \xi) = aQ - \frac{b}{2}Q^2 + \xi$, where $a$ and $b$ are positive parameters, and $\xi$ is a numeraire good. Accordingly, the
inverse demand function is linear and given by $p(Q) = a - bQ$. Firms' profits are then given by $\pi_i := p(Q)q_i - cq_i - \frac{\nu}{2}q_i^2$, $i = 1, 2, ..., n$.

Now that we have specified the market structure, we will define the strategies of the firms, which is where our contribution lies. We assume that firms' supply is driven by a common and exogenously given margin target $m$ over the average cost. Specifically, firms are willing to supply more (resp., less) units of their goods as the market price rises above (resp., falls below) the marked-up price. For the sake of simplicity, we retain the following linear specification¹¹

$$q_i = \phi_i + \psi \left( p - (1 + m) \frac{C(q_i)}{q_i} \right), \quad i = 1, 2, ..., n, \qquad (4)$$

where the firm-specific intercepts $\phi_i$ are not restricted in sign¹² and the common slope $\psi$ is nonnegative.

In the above specification, the value of $\psi$ is assumed to be exogenously given. It may reflect internal organizational factors determining the supply's ability to react to price changes, such as incentive pay or organizational slack. By contrast, the vertical intercept $\phi_i$ is a strategic choice variable.

A convenient way to solve the oligopoly game defined by the behavioral assumption (4) is to rephrase it as a model of competition in linear supply functions. A linear supply function $S_i(p)$ represents a commitment by firm $i$ to choose price-quantity pairs $(q_i, p)$ satisfying

$$q_i := S_i(p) = \alpha_i + \psi p, \qquad (5)$$

where $\psi \ge 0$ is a parameter measuring market toughness. For firm $i$, choosing a supply function then amounts to choosing an intercept value $\alpha_i \in \mathbb{R}$.

Equation (4) implicitly defines a supply function of the form (5). The following lemma aims to characterize this function. Assume that firm $i$ adjusts its supply according to the behavioral rule (4).¹³ Then, firm $i$'s supply function is

$$S_i(p) = \alpha_i(\phi_i) + \beta p, \qquad (8)$$

where

$$\alpha_i(\phi_i) = \frac{2(\phi_i - \psi(m+1)c)}{\psi(m+1)v + 2}, \quad \beta = \frac{2\psi}{\psi(m+1)v + 2}. \qquad (9)$$

*Proof.* By substituting the average cost expression and solving the implicit equation for $q_i := S_i(p)$. ☐

It is interesting to compare the supply function we have just obtained with Equation (5). The main difference is that, in our model, the technology parameters ($c$ and $v$) and the target margin ($m$) determine both the supply function's intercept and slope.

Note that the intercept $\alpha_i(\phi_i)$ is monotonically increasing in $\phi_i$, so each strategy choice $\phi_i$ determines a unique vertical intercept, and vice versa. Then, from the firm's perspective, choosing $\alpha_i$ or $\phi_i$ is equivalent. Parameter $\beta$ plays an analogous role to parameter $\psi$ in supply function (5), indicating the degree of market toughness. A significant difference, however, is that in our model, cost curvature and the common markup target $m$ determine the degree of market toughness in conjunction with $\psi$. Note that

$$\lim_{\psi \to 0} \beta = 0, \quad \frac{\partial \beta}{\partial \psi} = \frac{4}{(2 + \psi(1 + m)v)^2} > 0, \quad \lim_{\psi \to +\infty} \beta = \frac{2}{v(1 + m)}, \qquad (10)$$

$$\lim_{m \to 0} \beta = \frac{2\psi}{2 + v\psi}, \quad \frac{\partial \beta}{\partial m} = -\frac{2v\psi^2}{(2 + \psi(1 + m)v)} < 0, \quad \lim_{m \to +\infty} \beta = 0, \qquad (11)$$

$$\lim_{v \to 0} \beta = \psi, \quad \frac{\partial \beta}{\partial v} = -\frac{2(m+1)\psi^2}{((m+1)v\psi + 2)^2} < 0, \quad \lim_{v \to +\infty} \beta = 0. \qquad (12)$$
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Hence, an increase in the target margin (m) or a steeper cost function (v) mitigates market toughness (as measured by $\psi$).

Let $(S_i(p), S_{-i}(p))$ be a profile of supply functions, where $S_{-i}(p)$ is the vector listing the supply functions chosen by the $n-1$ firms competing with firm $i$. A supply function equilibrium¹⁴ is a profile of supply functions

$$ (S_i^*(p), S_{-i}^*(p)) := (\alpha_1^* + \beta p, \alpha_2^* + \beta p, \dots, \alpha_n^* + \beta p), \quad (13) $$

such that each firm's supply function is the best reply to the supply functions of other firms:

$$ \alpha_i^* = \arg\max_{\alpha_i} \pi_i(S_i(p), S_{-i}^*(p)), \quad i = 1, 2, \dots, n. \quad (14) $$

Equilibrium strategy profiles determine possible market outcomes.

The set of supply function equilibria encompasses Cournot and Bertrand equilibria as limit cases. If $\beta$ is equal to zero, then firm $i$'s supply function becomes flat and $q_i = \alpha_i$. Accordingly, choosing strategically a supply function $S_i$ amounts to choosing the quantity $q_i$ so that the supply function equilibrium coincides with the Cournot equilibrium. In our model, this is the case if $\psi = 0$ or if $m$ becomes arbitrarily large.

On the contrary, as market competition becomes more intense (i.e., $\psi$ increases), the slope of firm $i$'s supply function increases (i.e., $\beta$ increases). Eventually, it may become vertical as $\psi$ becomes arbitrarily large if firm $i$ has a constant marginal cost of production ($v = 0$), in which case the supply function equilibrium coincides with the Bertrand equilibrium. For comparison purposes, in the remainder of this article, we will use the superscripts c and b to denote Cournot's and Bertrand's behaviors and market outcomes.

Finally, note that by replacing $\beta$ with $\psi$ in the definition of the equilibrium profile (13), we obtain the equilibrium strategy profile corresponding to the case where firms' supply schedules are given by (5); that is,

$$ (S_1^\ell(p), S_2^\ell(p), \dots, S_n^\ell(p)) := (\alpha_1^*|_{\beta=\psi} + \psi p, \alpha_2^*|_{\beta=\psi} + \psi p, \dots, \alpha_n^*|_{\beta=\psi} + \psi p). \quad (15) $$

Having completed the model, we can turn to the analysis of firms' equilibrium supply behavior.

## 4 | CHARACTERIZATION OF EQUILIBRIUM SUPPLY FUNCTIONS

Let us express firm $i$'s profit as a function of $\alpha_i$ and $\alpha_{-i} = (\alpha_1, \alpha_2, \dots, \alpha_{i-1}, \alpha_{i+1}, \dots, \alpha_n)$. If firm $i$'s supply is given by (8) for all $i = 1, 2, \dots, n$, then the total supply is

$$ \sum_{i=1}^{n} S_i(p) = \sum_{i=1}^{n} \alpha_i + n\beta p. \quad (16) $$

Substituting this quantity into the market clearing condition then gives the implicit equation for the market price; $p = a - b(\sum_{i=1}^{n} \alpha_i + n\beta p)$. Upon solving, we obtain

$$ p(\alpha_i, \alpha_{-i}) = \frac{a - b \sum_{i=1}^{n} \alpha_i}{1 + nb\beta}. \quad (17) $$

Firm $i$'s profit then writes as

$$ \pi_i(\alpha_i, \alpha_{-i}) = p(\alpha_i, \alpha_{-i})S_i(\alpha_i, p(\alpha_i, \alpha_{-i})) - C\{S_i(\alpha_i, p(\alpha_i, \alpha_{-i}))\}, \quad (18) $$

$i = 1, 2, \dots, n$, and the decision problems (14) become $\alpha_i^* \in \operatorname{argmax}_{\alpha_i} \pi_i(\alpha_i, \alpha_{-i}^*)$, $i = 1, 2, \dots, n$. Assuming an interior solution, the first-order condition for firm $i$'s profit maximization is
$$ \frac{\partial p(\alpha_i^*, \alpha_{-i}^*)}{\partial \alpha_i} S_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*)) + p(\alpha_i^*, \alpha_{-i}^*) \frac{dS_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))}{d\alpha_i} = C' \{S_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))\} \frac{dS_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))}{d\alpha_i}, \quad (19) $$

i = 1, 2, ..., n. This condition basically says that firm i should set $\alpha_i$ so that marginal cost equals marginal revenue. Turning to the second-order optimality conditions, we have

$$ \frac{\partial^2 \pi_i}{\partial \alpha_i^2} = -((b\beta(n-1) + 1)(b\beta(n-1)v + v + 2b)) \left( \frac{\partial p(\alpha_i, \alpha_{-i}^*)}{\partial \alpha_i} \right)^2 < 0, \quad i = 1, 2, \dots, n. \quad (20) $$

Hence, the profit function of each firm is strictly concave in its strategic variable $\alpha_i$ and the existence of a unique symmetric Nash equilibrium is ensured. This equilibrium is obtained by solving the system of first-order optimality conditions (19).

Note that $p^* := p(\alpha_i^*, \alpha_{-i}^*)$, $q_i^* := S_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))$ and¹⁵

$$ \frac{dS_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))}{d\alpha_i} := 1 + \beta \frac{\partial p(\alpha_i^*, \alpha_{-i}^*)}{\partial \alpha_i}, \quad (21) $$

so that Conditions (19) can be rewritten as

$$ (p^* - C'(q_i^*)) = -\phi q_i^*, \quad \text{where} \quad \phi = \frac{\frac{\partial p(\alpha_i^*, \alpha_{-i}^*)}{\partial \alpha_i}}{1 + \beta \frac{\partial p(\alpha_i^*, \alpha_{-i}^*)}{\partial \alpha_i}}. \quad (22) $$

Condition (22) covers different forms of market conduct ranging from Cournot to Bertrand competition. If $\beta = 0$, then $q_i := S_i(p)|_{\beta=0} = \alpha_i$. As a consequence, firm i's strategy set is reduced to quantity choices and it behaves like a Cournot competitor. We can then check that $\phi|_{\beta=0} = -b := P'(Q)$ so that the condition (22) yields the familiar Cournot condition: $P(Q) + P'(Q)q_i = C'(q_i)$. Conversely, firm i acts as a Bertrand-type firm whenever the slope of its supply function $S_i'(p) = \beta$ becomes arbitrarily large. Indeed, we then have $\lim_{\beta \to \infty} \phi = 0$ so that condition (22) yields

$$ p^b = C'(q_i^b), \quad (23) $$

that is, the familiar prescription to price the product at marginal cost. Finally, since the slope of firm i's supply function $S_i'(p) = \beta$ ranges from zero to infinity, the equilibrium price-cost margin $p^* - C'(q_i^*)$ varies from the level corresponding to Cournot competition to that associated with Bertrand competition.

Now, by replacing $C'(q_i^*) = c + \nu q_i^*$ and $\partial p(\alpha_i, \alpha_{-i})/\partial \alpha_i = -b/(1 + nb\beta)$ into Conditions (22), we obtain

$$ p^* = c + h q_i^*, \quad (24) $$

with

$$ h := \frac{b + \nu(1 + b(n-1)\beta)}{1 + b(n-1)\beta}, \quad (25) $$
<figure><img src="image_7.png" /></figure>

i = 1, 2, ..., n. By summing over i = 1, 2, ..., n, and performing some manipulations, we obtain the following condition:

$$Q^* := \sum_{i=1}^{n} q_i^* = \frac{n(p^* - c)}{h(\beta)}. \qquad (26)$$

The equilibrium price $p^*$ must satisfy this condition, along with the market clearing condition, $p = a - bQ$. Solving the equation $p^* = a - bQ$ for $p^*$, we obtain

$$p^* = \frac{ah + bnc}{h - bn}. \qquad (27)$$

On the basis of the foregoing, we arrive at the following proposition.

**Proposition 1.** There is a unique Nash equilibrium in linear supply functions. This equilibrium is symmetric, implying that all firms choose the same supply function $S(p) = \alpha^* + \beta p$, where

$$\alpha^* = \frac{(a-c) - b\beta^2(n-1)(av + bcn) + \beta(b(a(n-2) - 2cn + c) - av)}{(b\beta(n-1)(bn + v) + bn + b + v)}. \qquad (28)$$

The corresponding equilibrium price, quantities and profits are

$$q^* = \frac{(a-c)(b(n-1)\beta + 1)}{b(n-1)\beta(bn + v) + bn + b + v} > 0, \quad p^* = \frac{b(n-1)\beta(av + bcn) + a(b + v) + bcn}{b(n-1)\beta(bn + v) + bn + b + v} > 0, \qquad (29)$$

$$Q^* = \frac{n(a-c)(b(n-1)\beta + 1)}{b(n-1)\beta(bn + v) + bn + b + v} > 0, \quad \pi^* = \frac{(a-c)^2(b(n-1)\beta + 1)(b(n-1)v\beta + 2b + v)}{2(b(n-1)\beta(bn + v) + bn + b + v)^2} > 0. \qquad (30)$$

*Proof.* Replacing $h$ by its expression into Equation (27) yields the value of the equilibrium price $p^*$. Then, from Equation (24), we find that $q_i^* = (p^* - c)/h$. Replacing $p^*$ and $h$ by their respective expressions into this equation yields identical equilibrium quantities $q_i^*(p^*) := q^*$ for all $i = 1, 2, ..., n$. Equilibrium industry output then is $Q^* := \sum_{i=1}^n q_i^*(p) = nq^*$. Finally, it must be the case that $S_i^*(p^*) := \alpha_i^* + \beta p^* = q_i^*$. Solving this last equation for $\alpha_i^*$ yields Equation (28). □

The remainder of this section is organized as follows. Section 4.1 uses Equation (9) to rewrite the above solution in terms of the initial parameters of the model ($\phi, \psi$). Section 4.2 considers the two limit cases where supply functions are either horizontal ($\psi = 0$) or relatively steep ($\psi \to +\infty$). Section 4.3 shows how the market outcome ranks compared to Cournot and Bertrand outcomes with respect to industry profits and social welfare.

## 4.1 | The equilibrium market outcome

Assume that the supply behavior of the $n$ firms is given by Equation (4). Then, their respective equilibrium supply functions are given by $S^*(p) = \alpha^* + \beta p$ and market equilibrium quantities are given by

$$q^* = \frac{(a-c)(2b\psi(n-1) + \psi(m+1)v + 2)}{2\psi b^2(n-1)n + \psi b v((m+3)n + m - 1) + 2b(n+1) + v(\psi(m+1)v + 2)}, \qquad (31)$$

$$p^* = \frac{a\psi v(b(m + 2n - 1) + mv + v) + 2a(b + v) + bcn(2b\psi(n - 1) + \psi(m + 1)v + 2)}{2\psi b^2(n - 1)n + \psi b v((m + 3)n + m - 1) + 2b(n + 1) + v(\psi(m + 1)v + 2)}, \qquad (32)$$
<figure><img src="image_8.png" /></figure>

$$ \pi^* = \frac{(a-c)^2(2b\psi(n-1) + \psi(m+1)v + 2)(\psi v(2b(m+n)+mv+v) + 4b+2v)}{2(2\psi b^2(n-1)n + \psi b v((m+3)n + m - 1) + 2b(n+1) + v(\psi(m+1)v+2))^2} \quad (33) $$

We obtain the following comparative statics results (see Appendix A):

$$ \frac{\partial q^*}{\partial m} \le 0, \quad \frac{\partial Q^*}{\partial m} \le 0, \quad \frac{\partial p^*}{\partial m} \ge 0, \quad \frac{\partial \pi^*}{\partial m} \ge 0, \quad (34) $$

$$ \frac{\partial q^*}{\partial \psi} \ge 0, \quad \frac{\partial Q^*}{\partial \psi} \ge 0, \quad \frac{\partial p^*}{\partial \psi} \le 0, \quad \frac{\partial \pi^*}{\partial \psi} \le 0, \quad (35) $$

$$ \frac{\partial q^*}{\partial v} \le 0, \quad \frac{\partial Q^*}{\partial v} \le 0, \quad \frac{\partial p^*}{\partial v} \ge 0, \quad \frac{\partial \pi^*}{\partial v} \ge 0. \quad (36) $$

The intuition for these results is straightforward. Figure 1 illustrates how the equilibrium changes when m increases. In this figure, market demand and firm i's equilibrium supply curve $p^*(q)$ (see Equation 24) are drawn in the price-quantity plane. The equilibrium point ($p^*, q^*$) is the point at which these two lines intersect. When the margin target m increases, only the slope of firm i's equilibrium supply curve changes. Note that

$$ \frac{\partial h}{\partial \beta} = - \frac{b^2(n-1)}{(1 + b(n-1)\beta)^2} < 0, \quad (37) $$

so that, from (11), we have $\partial h/\partial m = \frac{\partial h}{\partial \beta} \frac{\partial \beta}{\partial m} > 0$. The increase in the margin target has the effect of increasing the slope of firm i's equilibrium supply curve. As a result, the line $p^*(q)$ rotates counterclockwise and the equilibrium point moves in the northwest direction. Along the way, firm i's equilibrium output level decreases. Similar reasoning helps explain equilibrium comparative statics results with respect to v and $\psi$.

Finally, it should be noted that comparative statics results in (34) and (36) are diametrically opposed to those in (35). In particular, in the face of aggressive competition (i.e., a relatively large $\psi$ value), firms may increase their profits by increasing their margin target. Hence, committing to pursue average cost-plus pricing helps mitigating market toughness. This effect is illustrated in Figure 2 for a numerical example, using arbitrarily chosen parameter values.

<figure><img src="image_9.png" /><figcaption>FIGURE 1 The slope of equilibrium supply curves determines the ranking of equilibrium prices and quantities. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>
<figure><img src="image_10.png" /></figure>

Figure 2a shows isosupply curves in the $m - \psi$ plane. Let us consider point $x_1$ on the level curve corresponding to an (individual) equilibrium supply of nine units of the good. At this point, the margin target is $m = 10\%$, and the degree of market toughness is $\psi \approx 0.95$. The red arrow from $x_1$ shows the increase in the industry's margin target $m$ required to incite each firm to reduce equilibrium output by 0.1 units. Everything else being equal, moving from $x_1$ to $x_2$ necessitates increasing $m$ from 10% to 81%. Figure 2b,c shows isoprice and isoprofit curves in the $m - \psi$ plane. They allow one to visualize the effect of the increase in the target margin on the equilibrium price and the equilibrium profit. Moving from point $x_1$ to point $x_2$ causes the equilibrium price to increase from 13 to 13.3 and the equilibrium profit from 76.5 to 78.76. Finally, the vertical arrow from point $x_2$ to point $x_3$ indicates the increase in market toughness that restores initial equilibrium production, price, and profit levels. Everything else being equal, the value of parameter $\psi$ would need to increase from 0.95 to 1.4.

Finally, one finds that individual sales volumes, market prices, and firms profits decrease monotonically with the number of competitors:

$$ \frac{\partial q^*}{\partial n} < 0, \quad \frac{\partial Q^*}{\partial n} < 0, \quad \frac{\partial p^*}{\partial n} < 0, \quad \frac{\partial \pi^*}{\partial n} < 0. \qquad (38) $$

Let us now consider the two limit cases of the model, which correspond respectively to Cournot and, possibly, to Bertrand competition.

## 4.2 | Two limit cases

### 4.2.1 | Cournot competition

Recall that for $\psi = 0$, or $m \to +\infty$, we have $\beta = 0$. Each firm supply function then collapses to $q_i = \alpha_i$, which implies that firms compete à la Cournot. From Equation (24), firm i's equilibrium supply function boils down to

$$ p^*|_{\beta=0} := p^c = c + (b + v)q, \qquad (39) $$

and we have

$$ \lim_{m \to +\infty} \pi^* = \pi^c = \lim_{\psi \to 0} \pi^*, \quad \lim_{m \to +\infty} p^* = p^c = \lim_{\psi \to 0} p^*, \quad \lim_{m \to +\infty} q^* = q^c = \lim_{\psi \to 0} q^*, \qquad (40) $$

with

$$ q^c = \frac{a-c}{v+b(n+1)}, \quad Q^c = \frac{n(a-c)}{v+b(n+1)}, \quad p^c = \frac{a(b+v)+bcn}{v+b(n+1)}, \quad \pi^c = \frac{(a-c)^2(2b+v)}{2(v+b(n+1))^2}. \qquad (41) $$

### 4.2.2 | The possibility of Bertrand competition

Recall that $\partial\beta/\partial\psi > 0$ and note that

$$ \beta = \frac{2\psi}{\psi(m+1)v + 2} \le \frac{2}{v(m+1)} := \beta^{\infty}. \qquad (42) $$

Unless $v = 0$, the value of $\beta$ is limited by a finite upper bound $\beta^\infty$. This upper bound rules out the possibility of Bertrand competition unless firms have a constant marginal cost technology. We can state the following proposition.

**Proposition 2.** Unless firms operate a constant marginal cost technology, the equilibrium price markup over marginal cost ($p^* - C'(q^*)$) remains strictly positive when $\psi$ becomes arbitrarily large.¹⁶
<figure><img src="image_11.png" /></figure>

<caption>FIGURE 2 Equilibrium level curves. Parameter values are $a = 40, b = 1, \psi = 1, c = 0, n = 3$, and $v = 1$. (a) Isosupply curves:</caption>

$$ \{(m, \psi) \in \mathbb{R}_+^2 | q^*(m, \psi) = b\}, \text{ (b) isoprice curves:} $$

$$ \{(m, \psi) \in \mathbb{R}_+^2 | q^*(m, \psi) = b\}, \text{ and (c) isoprofit curves:} $$

$$ \{(m, \psi) \in \mathbb{R}_+^2 | \pi_i^*(m, \psi) = b\}. \text{ [Color figure can be viewed at} $$

wileyonlinelibrary.com]

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>
<figure><img src="image_15.png" /></figure>

This is illustrated in Figure 1. The supply curve $p^{\infty} := c + h^{\infty}q$ corresponds to the upper bound $\beta^{\infty} = 2/(v(m + 1))$. Unless $v = 0$ and $\psi \to \infty$, we have $p^{\star}(q) \neq p^{b}(q)$ and the (red) dashed area is nonempty. Then, the equilibrium price markup $p^{\star} - C'(q^{\star})$ is strictly positive, implying the existence of an oligopoly rent.

The market outcome is characterized by

$$q^{\infty} := \lim_{\psi \to +\infty} q^{\star} = \frac{(a-c)(2b(n-1) + (m+1)v)}{2b^2(n-1)n + bv((m+3)n + m - 1) + (m+1)v^2} > 0, \quad (43)$$

$$p^{\infty} := \lim_{\psi \to +\infty} p^{\star} = \frac{bv(a(m + 2n - 1) + c(m + 1)n) + a(m + 1)v^2 + 2b^2c(n - 1)n}{2b^2(n - 1)n + bv((m + 3)n + m - 1) + (m + 1)v^2} > 0, \quad (44)$$

$$\pi^{\infty} := \lim_{\psi \to +\infty} \pi^{\star} = \frac{v(a-c)^2(2b(m+n) + mv + v)(2b(n-1) + (m+1)v)}{2(2b^2(n-1)n + bv((m+3)n + m-1) + (m+1)v^2)^2} \geq 0. \quad (45)$$

## 4.3 | Social welfare

Social welfare is defined as the sum of consumer and producer surplus. Formally, we have

$$SW := CS + PS = \frac{b}{2}Q^2 + \sum_{i=1}^{n} \pi_i, \quad (46)$$

$$= nq \left( (a - c) - \frac{1}{2}q(bn + v) \right). \quad (47)$$

Let us now evaluate social welfare at the supply function equilibrium. By plugging $q^*$ into Equation (47), we obtain

$$SW^* = \frac{n(a-c)^2(b(n-1)\beta + 1)(b(n-1)\beta(bn + v) + b(n+2) + v)}{2(b(n-1)\beta(bn + v) + bn + b + v)^2}. \quad (48)$$

By differentiating social welfare with respect to $\beta$, we obtain

$$\frac{\partial SW^*}{\partial \beta} = \frac{b^3(n-1)n(a-c)^2}{(b(n-1)\beta(bn + v) + bn + b + v)^3} > 0, \quad (49)$$

where $\beta = \frac{2\psi}{\psi(m+1)v+2}$. Hence, social welfare increases as firms' supply behavior becomes more aggressive. Furthermore, since

$$\infty \geq \beta^{\infty} \geq \beta \geq \beta^c := 0, \quad (50)$$

social welfare levels rank as follows:

$$SW^b \geq SW^\star \geq SW^c. \quad (51)$$

Namely, the market performance of the equilibrium, measured in terms of social welfare, lies between those associated with Bertrand and Cournot outcomes. We can now compare the solution of our behavioral model with that resulting from competition in linear supply functions.
5 | COMPARISON WITH THE STANDARD MODEL OF SUPPLY FUNCTION COMPETITION

We compare the outcomes of supply function competition according to whether firms commit to the behavioral supply functions (4) or the standard linear supply functions (5). Let $h^l = h|_{\beta=\psi}$, then firm i's equilibrium supply curve is

$$p^l = c + h^l q_i, \quad i = 1, 2, \dots, n. \tag{52}$$

The corresponding equilibrium market outcome is

$$q^l = \frac{(a-c)(b(n-1)\psi + 1)}{b(n-1)\psi(bn + v) + bn + b + v} > 0, \quad p^l = \frac{b(n-1)\psi(av + bcn) + a(b+v) + bcn}{b(n-1)\psi(bn + v) + bn + b + v} > 0, \tag{53}$$

$$Q^l = \frac{n(a-c)(b(n-1)\psi + 1)}{b(n-1)\psi(bn + v) + bn + b + v} > 0, \quad \pi^l = \frac{(a-c)^2(b(n-1)\psi + 1)(b(n-1)v\psi + 2b + v)}{2(b(n-1)\psi(bn + v) + bn + b + v)^2} > 0. \tag{54}$$

Observe that

$$h - h^l = \frac{b^2(m+1)(n-1)v\psi^2}{(b(n-1)\psi + 1)(2b(n-1)\psi + (m+1)v\psi + 2)} > 0 \tag{55}$$

and

$$\frac{\partial}{\partial m}(h - h^l) = \frac{2b^2(n-1)v\psi^2}{(2b(n-1)\psi + (m+1)v\psi + 2)^2} > 0. \tag{56}$$

This indicates that the equilibrium supply curve is steeper than in the standard model of competition in supply functions, and the difference in slope is greater the higher m is.

On the basis of the foregoing analysis, we can now rank all equilibrium outcomes. Let $h^c = (b + v)$, $h^b = v$, and $h^l = h|_{\beta=\psi}$. Since $h^c \ge h \ge h^l \ge h^b$, and because the equilibrium is always where the equilibrium supply function intersects the residual (inverse) demand curve, we have $q^b \ge q^l \ge q^* \ge q^c$ and $p^b \le p^l \le p^* \le p^c$. This is illustrated in Figure 1.

Comparing profits we obtain

$$\operatorname{sgn}(\pi^l - \pi^*) = \operatorname{sgn}\{(\beta - \psi)(\beta(2b^2(n-1)n\psi(bn + v) + b(2n(bn + v) - v)) + \psi(2bn(bn + v) - bv) + 2(bn + b + v))\} = -1. \tag{57}$$

Each firm restricts its equilibrium output level when it targets a predetermined profit margin goal. The resulting reduction in total industry supply, in turn, leads to an increase in the equilibrium price. Equation (57) shows that this price increase more than compensates for the decrease in sales so that firms' profits increase. We conclude that it is preferable for firms to compete with behavioral rather than linear supply functions. We have the following proposition:

**Proposition 3.** If $\psi > 0$ then $\pi^* > \pi^l$. Firms achieve higher profits by conditioning their offer on deviations from the average cost-plus price than by relying on standard linear supply functions.

We can use Figure 3 to illustrate how increasing the industry's target margin improves profitability. Let the initial target margin be $m_1$ and $p_1^*(Q)$ be the corresponding equilibrium industry supply curve. The market demand and supply curves are drawn in Figure 3a. Equilibrium industry output is the level of production $Q_1^*$ at which the supply curve $p_1^*(Q)$ intersects the inverse demand function. For that level of production, the market price is $p_1^*$. Total industry profit is given by the sum of areas B and C. As shown in Figure 3b, an increase in the industry's target margin causes the supply curve to rotate counterclockwise. As a result, the equilibrium point moves along the inverse market demand curve towards the northwest and the industry's equilibrium supply contracts. This shift leads to an increase in the
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>FIGURE 3 Target margin and social welfare. (a) Social welfare and the oligopoly deadweight loss and (b) welfare implications of an increased target margin. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

market price, an increase in the size of area B, and a reduction in the size of area C. The net effect on producer surplus, measured by the difference $(B_2 + C_2) - (B_1 + C_1)$, is positive; in other words, increasing the industry's margin target has resulted in higher industry profits.

We conclude this section with a comparative study of social welfare levels. Recall that $\partial SW^*/\partial \beta > 0$ and

$$ \beta^e := \psi \ge \beta \ge \beta^c := 0. \qquad (58) $$

Building on this, we can now complete the preliminary ranking given by Equation (51). We have the following proposition:

**Proposition 4.** Social welfare levels rank as follows: $SW^b \ge SW^e \ge SW^* \ge SW^c$.

The industry-wide commitment to target a profit margin m reduces social welfare. However, social welfare never falls below the Cournot level.

# 6 | CONCLUSION

Many firms remain loyal to the markup pricing rule despite its various flaws. Even those who do not rely on this pricing heuristic generally agree that it constitutes a convenient benchmark for assessing whether the selling price is satisfactory (i.e., sufficient, fair, and acceptable for consumers and competitors) and determining whether supply adjustments are desirable. Frequently, firms know the suitable markup to apply over the average cost of production in their industry and can agree (or coordinate) on this markup at the industry level. Then, the industry-wide markup is a characteristic of the industry that impacts the market outcome. However, existing oligopoly pricing models fail to take it into consideration.

In this paper, we proposed a behavioral model of oligopoly competition that accounts for firms' markup targets. We assumed that firms consider the average cost-plus price as a reference price. Specifically, we posited that firms increase (resp., decrease) their supplies as the market price rises above (resp., falls below) this price. Under this behavioral assumption, firms' strategies consist of linear supply functions with the same slope but adjustable intercepts. Since cost
<figure><img src="image_18.png" /></figure>

curvature, markup target, and market toughness parameters jointly determine the slope of firms' supply functions, choosing a strategy for the firm boils down to strategically setting the intercept of its supply function.

We characterized the Nash equilibrium for an industry composed of *n* firms. We compared the corresponding market outcome with models of Cournot, Bertrand and supply function competition. The equilibrium market price always lies between the Cournot and the Bertrand price. It may approach the Bertrand price asymptotically as firms' supplies become increasingly aggressive, but only if firms' marginal cost of production is constant. Further comparative statics analysis made it possible to elucidate the strategic role played by the industry-wide markup target. Raising this markup target increases the slope of firms' (inverse) supply functions, implying a lower supply at any price level. Due to the reduction in supply, the equilibrium price rises and approaches the Cournot level asymptotically as the markup target becomes arbitrarily large. Social welfare then drops because the increase in profits fails to compensate for the loss in consumer surplus.

An industry-wide commitment to targeting a markup reduces competition in equilibrium. The reduction in competition is more pronounced than when firms commit to standard linear supply functions (as in Menezes & Quiggin, 2012). Indeed, the behavioral mode of play we have proposed results in a lower supply, a higher price and higher industry profits at the equilibrium. These extra profits come at the cost of a reduced consumer surplus and social welfare.

Future research could extend this work in several directions. First, we recall that our model assumed identical firms. Solving an *n*-firm game of supply function competition with asymmetric firms is difficult (Anderson & Xu, 2005). However, the duopoly case with asymmetric costs lends itself to analysis. Second, we assumed that the industry margin target does not impact consumer demand. However, an excessive margin target can upset consumers and reduce demand. Therefore, it would be interesting to examine whether consumer's preference for fair treatment is enough to force the industry to seek moderate markups. Third, we assumed a given and fixed industry profit margin target. It would be worthwhile to study how this target emerges in the first place. One may think of small businesses, endowed with limited computational skills and market experience, trying to learn from their rivals' reactions what a suitable profit margin target is in their industry.

## ACKNOWLEDGMENTS

We thank anonymous reviewers for their insightful comments and suggestions, which helped to improve the paper. We also thank Flavio M. Menezes and John C. Quiggin for their encouragement and for bringing to our attention Menezes and Quiggin (2020, 2021). An earlier version of this paper was presented to the “Workshop in Memory of Professor Ngo Van Long” (Cirano, Montreal, Canada, September 2022), and the “70th Congress of the French Economic Association” (Dijon, France, June 2022). Finally, we also thank Ivan Ledezma for helpful discussions. Any errors are the sole responsibility of the authors. The authors acknowledge the financial support of the French Agence Nationale de la Recherche (ANR). Mabel Tidball received funding from the LabEx Entreprendre (ANR-10-LABX-11-01). Denis Claude received funding from the project RILLM (ANR-19-CE26-0008-01). Denis Claude also thanks the Region Bourgogne-Franche Comté for financial support under project DyIn.

## ORCID

Denis Claude http://orcid.org/0000-0001-7901-5122

Mabel Tidball https://orcid.org/0000-0002-5576-0977

## ENDNOTES

<footnote n="1">¹ In the remainder of this paper, we will use the expression “markup pricing” as a synonym of “average cost-plus (markup) pricing.”</footnote>

<footnote n="2">² For additional evidence, see Greenslade and Parker (2012), Greenslade and Parker (2010), Rao and Kartono (2009), and Hall and Hitch (1939).</footnote>

<footnote n="3">³ For illustration purposes and with no loss of generality, we focus exclusively on average cost-plus pricing in this paper. The conclusions of our research would be unchanged if, instead, we considered marginal cost-plus pricing (see, footnote 13). Moreover, most business firms find it more challenging to quantify their marginal cost than their average cost, so that average cost-plus pricing is more prevalent.</footnote>

<footnote n="4">⁴ See eq. (4), p. 247, in Grant and Quiggin (1994).</footnote>

<footnote n="5">⁵ For a survey on the Cournot–Bertrand oligopoly, the reader is referred to Tremblay and Tremblay (2019).</footnote>

<footnote n="6">⁶ The properties of the supply function equilibrium in the linear case are further studied in Delbono and Lambertini (2015, 2018).</footnote>
<figure><img src="image_19.png" /></figure>

⁷ For an introduction to how supply function competition is applied to the analysis of energy markets, the reader can refer to Holmberg and Newbery (2010).

⁸ The decades-old dollar-per-thousand rule states that the average nightly room rate should increase by one dollar for every $1000 investment in room values.

⁹ For a history and discussion of this controversy, see Lee (1984), Mongin (1992), and Nubbemeyer (2010).

¹⁰ See, for example, Koutsoyiannis (1979) and Langlois (1989, 1993).

¹¹ Replacing the average cost with the marginal cost in Equation (4) would not change the conclusion of the analysis. The parameter that plays the most significant role in our analysis is m.

¹² As we will see below, firm i's supply is always nonnegative at the equilibrium, whatever the sign of the intercept $\phi_i$. Hence, it is not necessary to introduce explicit positivity constraints on $q_i$ and/or $\phi_i$.

¹³ Suppose firms use marginal cost instead of average cost to guide their output decisions so that their supply is given by

$$q_i = \phi_i + \psi(p - (1 + m)C'(q_i)), \quad i = 1, 2, \dots, n. \quad (6)$$

Then, firm i's supply function is $\tilde{S}_i(p) = \tilde{\alpha}_i(\phi_i) + \tilde{\beta}p$, where

$$\tilde{\alpha}_i(\phi_i) = \frac{\phi_i - \psi(m+1)c}{\psi(m+1)v + 1}, \quad \tilde{\beta} = \frac{\psi}{\psi(m+1)v + 1}. \quad (7)$$

Observe that the comparative static properties of $\tilde{\beta}$ with respect to m, $\psi$, and v are identical to those of $\beta$. As our qualitative results depend exclusively on these properties, they extend to this alternative scenario.

¹⁴ For an introduction to models of competition in supply functions, the reader is referred to Vives (1999, chap. 7). See also Menezes and Quiggin (2020, 2021) for an alternative treatment based on the concept of the strategic industry supply curve.

¹⁵ Take note that

$$\frac{dS_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))}{d\alpha_i} := \frac{\partial S_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))}{\partial \alpha_i} + \frac{\partial S_i(\alpha_i^*, p(\alpha_i^*, \alpha_{-i}^*))}{\partial p(\alpha_i, \alpha_{-i})} \frac{\partial p(\alpha_i^*, \alpha_{-i}^*)}{\partial \alpha_i}.$$

¹⁶ Obviously then, we also have $\lim_{\psi \to +\infty} (p^* - AC(q^*)) > 0$.

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**How to cite this article:** Claude, D., & Tidball, M. (2025). Taking firms' margin targets seriously in a model of competition in supply functions. *Journal of Economics & Management Strategy*, **34**, 174–192.
https://doi.org/10.1111/jems.12577

## APPENDIX A

We have

$$ 
\begin{aligned}
\frac{\partial q^*}{\partial k} &:= \frac{\partial q^* \partial \beta}{\partial \beta \partial k} = \frac{b^2(n-1)(a-c)}{(b(n-1)\beta(bn+\nu) + bn + b + \nu)^2 \partial k}, \quad \frac{\partial Q^*}{\partial k} := n \frac{\partial q^* \partial \beta}{\partial \beta \partial k} \\
&= \frac{nb^2(n-1)(a-c)}{(b(n-1)\beta(bn+\nu) + bn + b + \nu)^2 \partial k},
\end{aligned}
\quad (A1) $$

$$ 
\begin{aligned}
\frac{\partial p^*}{\partial k} &:= \frac{\partial p^* \partial \beta}{\partial \beta \partial k} = - \frac{b^3(n-1)n(a-c)}{(b(n-1)\beta(bn+\nu) + bn + b + \nu)^2 \partial k}, \quad \frac{\partial \pi^*}{\partial k} := \frac{\partial \pi_i^* \partial \beta}{\partial \beta \partial k} \\
&= - \frac{b^3(n-1)^2(a-c)^2(bn\beta + 1)}{(b(n-1)\beta(bn+\nu) + bn + b + \nu)^3 \partial k}.
\end{aligned}
\quad (A2) $$

Now, since (from Equation 9)

$$ \frac{\partial \beta}{\partial m} = - \frac{2\psi^2\nu}{(\psi(m+1)\nu+2)^2} \le 0, \quad \frac{\partial \beta}{\partial \psi} = \frac{4}{(\psi(m+1)\nu+2)^2} > 0, \quad \frac{\partial \beta}{\partial \nu} = - \frac{2(1+m)\psi^2}{(2+(1+m)\nu\psi)^2} < 0, \quad (A3) $$

we obtain the comparative statics results (34)–(36).