Denis Claude
email: denis.claude@u-bourgogne.fr

Mabel Tidball
email: mabel.tidball@inrae.fr

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

Keywords: supply function, markup pricing, price fixing, oligopoly

We introduce price-markup objectives into a model of supply function competition. We characterize the corresponding supply-function equilibrium and study its qualitative properties. Adherence to price-markup targets is conducive to reduced market competition and increased firm profitability. While pursuing such goals reduces social welfare, welfare never drops below the level corresponding to a Cournot oligopoly. Finally, we establish conditions under which consumer preference for fair pricing inhibits the industry's use of markups as a collusive device.

Introduction

Cost-plus pricing is a lot like the romance novel genre, in that it's widely ridiculed yet tremendously popular. [START_REF] Dholakia | When Cost-Plus Pricing Is a Good Idea[END_REF] Like it or not, markup (or cost-plus) pricing1 remains the most popular product pricing method. There is indeed ample survey evidence2 showing that most (price-making) firms set their product prices primarily by applying the following rule of thumb:

𝑝 = (1 + 𝑚)AC, ( 1
)
where 𝑚 is the gross profit margin and 𝐴𝐶 denotes the average production cost. In doing so, these firms set their prices at the average production cost plus some suitable markup.

The managerial literature has long recognized the empirical prevalence of this pricing routine. 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' [START_REF] Godin | Business Rules of Thumbs[END_REF]Conley 1987, cited in Phillips 2011) and hotels where the 'Dollar per Thousands' rule determines room rates [START_REF] O'neill | ADR rule of Thumb: Validity and Suggestions for its Application[END_REF]. [START_REF] Dholakia | When Cost-Plus Pricing Is a Good Idea[END_REF] claims he has known of companies that apply markups ranging from 5% to 800%. These examples underline that the applied markup varies widely in practice.3 By contrast, the economic profession welcomed initial survey evidence about markup pricing with skepticism.

In its defense, one may note that [START_REF] Hall | Price Theory and Business Behaviour[END_REF], 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 1952.4 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; i.e., 𝑝 = (1 + 𝑚 (𝜀)) AC, 𝑚 ′ (𝜀) < 0.

(2)

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.5

Such arguments combined with changes in economic methodology helped calm the marginalist controversy. Specifically, Herbert [START_REF] Simon | A Behavioral Model of Rational Choice[END_REF] '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 [START_REF] Friedman | The Methodology of Positive Economics[END_REF] '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 [START_REF] Grant | Nash equilibrium with mark-up pricing oligopolists[END_REF], new advances in the understanding of firm pricing practices have hardly changed the modeling of firms' behavior in oligopolistic markets.6

Recent studies confirm the prevalence of markup pricing. [START_REF] Altomonte | Average-cost pricing: Some evidence and implications[END_REF] provide survey evidence on pricing behavior from a representative sample of 14000 European firms. Among the respondents, 60% stated they have market power. Of these price-making firms, 75% revealed that they rely on markup pricing. Likewise, the Bank of England (Greenslade andParker, 2010, 2012) surveyed a representative sample of 2331 firms on the considerations guiding their pricing decisions for the main product they sell in the UK. 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 [START_REF] Rao | Pricing Objectives and Strategies: A Cross-Country Survey[END_REF], who surveyed a sample of firms operating in the USA, 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 [START_REF] Phillips | Pricing and Revenue Optimization[END_REF][START_REF] Nagle | The strategy and tactics of pricing, Always learning[END_REF]. In particular, risk-averse managers might choose a price that covers the average cost as protection against the risk of loss [START_REF] Shy | Industrial Organization: Theory and Applications[END_REF]. Secondly, companies think this pricing approach is fair. As already argued by [START_REF] Hall | Price Theory and Business Behaviour[END_REF], 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 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 [START_REF] Simon | Confessions of the Pricing Man[END_REF] 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 translates into implicit price contracts between companies and consumers. [START_REF] Okun | Prices and quantities[END_REF] 3Some pharmaceutical companies that distribute generic drugs are probably making even higher margins than this upper bound suggests. See, for example, [START_REF] Hill | Estimated costs of production and potential prices for the WHO Essential Medicines List[END_REF].

4For a history and discussion of this controversy, see [START_REF] Lee | The Marginalist Controversy and the Demise of Full Cost Pricing[END_REF], [START_REF] Mongin | The "Full-Cost" Controversy of the 1940s and 1950s: A Methodological Assessment[END_REF], and Nubbemeyer (2010) 5See, for example, [START_REF] Koutsoyiannis | Modern microeconomics[END_REF] and [START_REF] Langlois | A markup over average cost pricing rule for meat and poultry in the United States[END_REF][START_REF] Langlois | A markup over average cost pricing rule for meat and poultry in the United States[END_REF]. 6In macroeconomics, [START_REF] Rotemberg | Customer anger at price increases, changes in the frequency of price adjustment and monetary policy[END_REF]2011) and [START_REF] Eyster | Pricing Under Fairness Concerns[END_REF] have, however, proposed more sophisticated models of firm pricing behavior. These models aim to explain price rigidity by taking into account consumers' sensitivity to fair pricing. See Section 4.2.

elegantly formulated this idea using the image of an 'invisible handshake' between consumers and firms. [START_REF] Rotemberg | Customer anger at price increases, changes in the frequency of price adjustment and monetary policy[END_REF]2011) and [START_REF] Eyster | Pricing Under Fairness Concerns[END_REF] further explored how fairness considerations constrain pricing decisions.

Many contemporary authors also agree with [START_REF] Hall | Price Theory and Business Behaviour[END_REF]'s contention that the practice of markup pricing is fair to competitors. As noted by [START_REF] Hall | Price Theory and Business Behaviour[END_REF], 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 [START_REF] Phillips | Pricing and Revenue Optimization[END_REF], 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". [START_REF] Phillips | Pricing and Revenue Optimization[END_REF] 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 become apparent when comparing Equations ( 1) with (2). 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 [START_REF] Nagle | The strategy and tactics of pricing, Always learning[END_REF]:

"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." [START_REF] Nagle | The strategy and tactics of pricing, Always learning[END_REF] 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 [START_REF] Phillips | Pricing and Revenue Optimization[END_REF]. 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 firms have expectations regarding the fixed margin (relative to production cost) they should obtain on a given market and commit to explicit (or implicit) price contracts that constrain their pricing decisions.

With this purpose in mind, we consider a simple model of supply function competition in which firms hope to achieve a certain margin on the average cost of production. Models of supply function competition were introduced by [START_REF] Grossman | Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs[END_REF] and popularized by [START_REF] Klemperer | Supply function equilibria in oligopoly under uncertainty[END_REF]. They represent a natural avenue for modeling competition among firms that commit to fixed price-quantity schedules.7 A supply function describes the number of units of a good that a firm agrees to provide to consumers for any eligible price. As argued by [START_REF] Menezes | More competitors or more competition?[END_REF], this concept allows the construction of more elaborate strategy spaces than those usually encountered in oligopoly theory. It makes it possible to analyze how changes in the values of market parameters alter firms' strategy spaces and vice-versa. [START_REF] Menezes | More competitors or more competition?[END_REF] illustrate this point by examining the trade-offs that competition authorities face when measures designed to increase competition in the marketplace might push some competitors out of business. They find that changes in the number of market participants may (or may not) alter competition in the market depending on whether firms' supply schedules are aggressive (or not). In other words, the slope of firms' supply functions -a strategy space parameter -interacts with the number of firms -a market parameter -to determine the market outcome. 7Models of competition in supply functions have proven to be particularly suitable for modeling competition in electricity markets. See, for example, [START_REF] Baldick | Theory and Application of Linear Supply Function Equilibrium in Electricity Markets[END_REF] and [START_REF] Willems | Cournot versus Supply Functions: What does the data tell us?[END_REF].

In this paper, we follow the path set by [START_REF] Menezes | More competitors or more competition?[END_REF]. Accordingly, we assume that firms compete in supply functions and condition their supply upon their ability to achieve a given profit margin on average production cost. The wider the gap between the actual market price and that which would result from the markup pricing rule, the greater the quantity supplied. Inversely, when this gap narrows, supply contracts. This modeling approach makes it possible to analyze how firms' (profit) margin objectives alter market competition. We find that the higher the profit margin expectations, the more collusive the market outcomes. By committing to our augmented supply function, firms are able to maintain market power even when the market is highly competitive. Furthermore, they earn higher profits than when relying on standard linear supply function. Finally, since each firm profit is monotonically increasing in the profit margin 𝑚, producer surplus is also monotonically increasing in 𝑚. Consequently, firms have a collective incentive to increase without bound the profit margin they aspire to achieve. However, this increase would cause the supply curve to rotate around its y-intercept, so that the supply curve would become horizontal and Cournot's solution would be obtained. Finally, we find that collusion over a positive but small profit margin only occurs when consumers are antagonized by pricing practices they perceive as unfair.

The paper that most closely resembles ours is that of [START_REF] Grant | Nash equilibrium with mark-up pricing oligopolists[END_REF]. The authors assume that firms adhere strictly to the average cost pricing rule so that condition (1) always binds, and firms set their profit margins strategically. By contrast, we assume that the desired profit margin is constant, and firms adjust strategically the amount they produce to try to achieve it. Consequently, in our model, Condition 1 does not bind; i.e., the selling price is allowed to diverge from the markup price, and firms adjust quantities accordingly.

The structure of the paper is as follows. Section 2 introduces the model, which is solved in Section 3. Equilibrium quantities and the main comparative statics results are presented in Subsection 3.1 and 3.2. These are compared with the results obtained by [START_REF] Menezes | More competitors or more competition?[END_REF] in Subsection 3.3. Section 4 analyzes firms' incentives to agree on an industry price markup standard and explains how consumers' preferences for fair pricing contracts may limit applicable markup rates. Section 5 concludes the paper.

The model

Our purpose is to build a simple model of supply function competition that explicitly captures firms' profit margin aspirations. We will work under the following customary assumptions. On the supply side, there are 𝑛 ≥ 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 = 𝑛 𝑖=1 𝑞 𝑖 . All firms share the same technology summarized by the cost function 𝐶 (𝑞 𝑖 ) = 𝑐𝑞 𝑖 + 𝑣 2 𝑞 2 𝑖 , with 𝑐, 𝑣 ≥ 0. On the demand side, there is a representative consumer with a utility function given by 𝑈 (𝑄, 𝜉) = 𝑎𝑄 -𝑏 2 𝑄 2 + 𝜉, where 𝑎 and 𝑏 are positive parameters, and 𝜉 is a numeraire good. Accordingly, the inverse demand function is linear and given by 𝑝(𝑄) = 𝑎 -𝑏𝑄. Firms' profits are therefore given by 𝜋 𝑖 := 𝑝(𝑄)𝑞 𝑖 -𝑐𝑞 𝑖 -𝑣 2 𝑞 2 𝑖 , 𝑖 = 1, 2, . . . , 𝑛. Now that we have specified the market structure, we move on to defining firms' strategies. We assume that firms compete in supply functions. [START_REF] Grossman | Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs[END_REF][START_REF] Hart | Imperfect Competition in General Equilibrium: an overview of recent work[END_REF], proposed models of supply function competition as alternatives to models of either price or quantity competition. In their approach, a supply function 𝑆 𝑖 (.) represents a commitment by firm 𝑖 to choose price-quantity pairs ( 𝑝 𝑖 , 𝑞 𝑖 ) satisfying 𝑞 𝑖 = 𝑆 𝑖 ( 𝑝 𝑖 ). Each firm selects its supply function 𝑆 𝑖 (.) to maximize its profit, taking as given competitors (equilibrium) supply functions. The set of Nash equilibria in supply functions then determines possible market outcomes.8

The difficulty with this modeling approach is well known. If firms are allowed to select arbitrary supply functions, a profusion of Nash equilibria exists so that models of supply functions fail to predict the market outcome. Different avenues have been explored to address this multiplicity issue. Adding demand uncertainty [START_REF] Klemperer | Supply function equilibria in oligopoly under uncertainty[END_REF] or incomplete information [START_REF] Vives | Strategic Supply Function Competition With Private Information[END_REF] has been shown to limit the number of Nash equilibria. In this paper, to ensure the existence of a unique solution, we assume that firms choose among linear supply functions as [START_REF] Menezes | More competitors or more competition?[END_REF] and [START_REF] Delbono | On the properties of linear supply functions in oligopoly[END_REF] do. Namely, we restrict our attention to linear supply functions of the form:9 8For an introduction to models of competition in supply functions, see Vives (1999, chapter 7). 9Delbono and Lambertini (2015) actually consider the slightly more general form

𝑆 𝑖 ( 𝑝) = 𝛼 𝑖 + 𝛽 𝑖 𝑝, (3) 𝑆 𝑖 ( 𝑝) = 𝛼 𝑖 + 𝛽𝑝, ( 5
)
where 𝛽 is a parameter describing how intense competition is and 𝛼 𝑖 is the strategic variable which allows firm 𝑖 to shift supply upward or downward. This assumption ensures the existence of a unique Nash equilibrium in supply schedules.

We will now specify further the form that firms' supply functions take in our model. We assume that firms use the markup price as a reference when making their supply decisions. More precisely, each firm conditions its supply to the difference between the market price and the price that would result from applying the markup pricing rule. The wider the divergence is, the greater the quantity supplied. Inversely, when this gap narrows, supply contracts. This supply behavior is captured by assuming that 10

𝑞 𝑖 = 𝜙 𝑖 + 𝜓 ( 𝑝 -(1 + 𝑚) 𝐴𝐶 (𝑞 𝑖 )) , ( 6
)
where the intercept 𝜙 𝑖 is not restricted in sign,11 the slope 𝜓 and the gross margin 𝑚 are non-negative, and 𝐴𝐶 (𝑞 𝑖 ) = 𝐶 (𝑞 𝑖 )/𝑞 𝑖 denotes the average cost of production.

Note that the above behavioral assumption addresses the shortcomings of the standard average cost pricing routine. Firms do not behave on the false assumption that either i.) they can base their prices exclusively on costs considerations and ii.) they know their production costs before knowing the actual demand. In our model, firms set quantities, and the interaction of supply and demand determines the market price. Therefore, the market price depends on all the market parameters; the most relevant for our current purpose being the expected profit margin (𝑚) and the degree of market competition (𝛽).

Let us now reformulate Equation ( 6) in the general form 𝑞 𝑖 = 𝑆 𝑖 ( 𝑝), with 𝑆 𝑖 ( 𝑝) given by Equation (5). By substituting the average cost expression, we obtain the following equation:

𝑞 𝑖 = 𝜙 𝑖 + 𝜓 𝑝 -(1 + 𝑚) 𝑐 + 𝑣𝑞 𝑖 2 , 𝑖 = 1, 2, . . . , 𝑛. (7)
Solving for 𝑞 𝑖 yields

𝑞 𝑖 := 𝑆 𝑖 ( 𝑝) = 2𝜙 𝑖 -2𝜓𝑐(𝑚 + 1) 𝜓(𝑚 + 1)𝑣 + 2 + 2𝜓 𝜓(𝑚 + 1)𝑣 + 2 𝑝, 𝑖 = 1, 2, . . . , 𝑛, (8)
which we rewrite as 𝑆 𝑖 ( 𝑝) = 𝛼 𝑖 (𝜙 𝑖 , 𝜓; 𝑚, 𝑣) + 𝛽 (𝜓; 𝑚, 𝑣) 𝑝, (9) by using the following change of variables:

𝛽 (𝜓; 𝑚, 𝑣) = 2𝜓 𝜓(𝑚 + 1)𝑣 + 2 , 𝛼 𝑖 (𝜙 𝑖 , 𝜓; 𝑚, 𝑣) = 2 (𝜙 𝑖 -𝜓(𝑚 + 1)𝑐) 𝜓(𝑚 + 1)𝑣 + 2 , 𝑖 = 1, 2, . . . , 𝑛. ( 10
)
Note that firms' supply behaviors becomes more aggressive when the value of 𝛽 (𝜓; 𝑚, 𝑣) increases because, for the same market equilibrium price, their supply is higher as their supply function is steeper. Conversely, a decrease in 𝛽 (𝜓; 𝑚, 𝑣) leads to less aggressive market behaviors. It will prove helpful to understand how a change in the values of 𝜓 and 𝑚 affects the supply behavior of firms to interpret the model and its results. With this in mind note that which allows firms to strategically set 𝛽 𝑖 instead of 𝛼 𝑖 . They show that the Nash equilibrium of the game is the same irrespective of whether firms select the slope 𝛽 𝑖 or the vertical intercept 𝛼 𝑖 of the supply function. [START_REF] Delbono | Ranking Bertrand, Cournot and supply function equilibria in oligopoly[END_REF] use the same specification but allow for product differentiation. Finally, [START_REF] Ciarreta | Supply function vs. quantity competition in supergames[END_REF] and [START_REF] Delbono | Ranking Bertrand, Cournot and supply function equilibria in oligopoly[END_REF] simply assume that

𝑆 𝑖 ( 𝑝) = 𝛽 𝑖 𝑝. ( 4
)
10Replacing the average cost with the marginal cost in Equation ( 6) would not change the conclusion of the analysis. The parameter that plays the most significant role in our analysis is 𝑚.

11As we will see below, firm 𝑖's supply is always non-negative at the equilibrium, whatever the sign of the intercept 𝜙 𝑖 . Hence, it is not necessary to introduce explicit positivity constraints on 𝑞 𝑖 and/or 𝜙 𝑖 . lim

𝜓→0 𝛽 (𝜓; 𝑚, 𝑣) = 0, 𝜕 𝛽 (𝜓; 𝑚, 𝑣) 𝜕𝜓 = 4 (2 + 𝜓 (1 + 𝑚) 𝑣) 2 > 0, lim 𝜓→+∞ 𝛽 (𝜓; 𝑚, 𝑣) = 2 𝑣 (1 + 𝑚) , (11) lim 𝑚→0 𝛽 (𝜓; 𝑚, 𝑣) = 2𝜓 2 + 𝑣𝜓 , 𝜕 𝛽 (𝜓; 𝑚, 𝑣) 𝜕𝑚 = - 2𝑣𝜓 2 (2 + 𝜓 (1 + 𝑚) 𝑣) < 0, lim 𝑚→+∞ 𝛽 (𝜓; 𝑚, 𝑣) = 0. (12)
Firms' supply functions are horizontal whenever 𝜓 is zero, or the profit margin target 𝑚 is arbitrarily large. In this case, the market will be uncompetitive, and Cournot's solution will prevail. When 𝜓 increases, or alternatively when 𝑚 decreases, the supply behavior of firms becomes more aggressive. However, unless the marginal cost of production is constant (i.e., 𝑣 = 0), the slope of supply functions is finite so that they may be steep but never vertical. Finally, we have the following remark, which will be helpful for comparison purposes.

Remark 1. By assuming a constant marginal cost of production and no profit margin, we arrive at the same supply behavior as that posited by [START_REF] Menezes | More competitors or more competition?[END_REF]. Indeed, if 𝑚 = 𝑣 = 0 then Equation ( 10) reduces to12

𝑞 𝑖 := 𝑆 𝑖 ( 𝑝) = 𝛼 𝑖 (𝜙 𝑖 , 𝜓; 0, 0) + 𝛽 (𝜓; 0, 0) 𝑝, (13) = 𝜙 𝑖 + 𝜓 ( 𝑝 -𝑐) , ( 14
)
From now on we omit the arguments in the functions 𝛼 𝑖 and 𝛽 whenever this does not lead to any confusion. These arguments will only be specified when it is necessary to distinguish the solution of our general model from the particular case treated by [START_REF] Menezes | More competitors or more competition?[END_REF].

We assume that each Firm 𝑖 sets strategically the intercept 𝛼 𝑖 of its supply function. Consequently, we now express price and quantities as a function of the strategy chosen by Firm 𝑖, 𝛼 𝑖 , and the profile of strategies chosen by all other firms 𝛼 -𝑖 = (𝛼 1 , 𝛼 2 , . . . , 𝛼 𝑖-1 , 𝛼 𝑖+1 , . . . , 𝛼 𝑛 ). Suppose that firm 𝑖's supply function is given by Equation ( 5), 𝑖 = 1, 2, . . . , 𝑛. Then, the overall industry' supply function is

𝑛 𝑖=1 𝑆 𝑖 ( 𝑝) = 𝑛 𝑖=1 𝛼 𝑖 + 𝑛𝛽𝑝. ( 15
)
By replacing total output with overall industry supply in the inverse demand function, we obtain the equation 𝑝 = 𝑎 -𝑏 𝑛 𝑖=1 𝛼 𝑖 + 𝑛𝛽𝑝 , which can be solved for the market price

𝑝 (𝛼 𝑖 , 𝛼 -𝑖 ) = 𝑎 -𝑏 𝑛 𝑖=1 𝛼 𝑖 1 + 𝑛𝑏𝛽 , ( 16
)
with partial derivative given by 𝜕 𝑝 (𝛼 𝑖 , 𝛼 -𝑖 )

𝜕𝛼 𝑖 = - 𝑏 1 + 𝑛𝑏𝛽 . ( 17
)
Expressed in terms of supply functions, the profit of firm i can then be written as follows

𝜋 𝑖 (𝛼 𝑖 , 𝛼 -𝑖 ) = 𝑝 (𝛼 𝑖 , 𝛼 -𝑖 ) -𝑐 - 𝑣 2 𝑆 𝑖 ( 𝑝) 𝑆 𝑖 ( 𝑝) , ( 18
) = 1 2 (𝛼 𝑖 + 𝛽𝑝 (𝛼 𝑖 , 𝛼 -𝑖 )) {2 ( 𝑝 (𝛼 𝑖 , 𝛼 -𝑖 ) -𝑐) -𝑣 (𝛼 𝑖 + 𝛽𝑝 (𝛼 𝑖 , 𝛼 -𝑖 ))} , 𝑖 = 1, 2, . . . , 𝑛. (19)
Having completed the model, we can turn to the analysis of firms' equilibrium supply behavior.

Characterization of equilibrium supply functions

We solve the market game for a Nash equilibrium in linear supply functions, 𝑆 𝑖 ( 𝑝) = 𝛼 𝑖 + 𝛽𝑝. In doing so, recall that we assume that firms strategically choose the vertical intercept of their supply schedules. Selecting a supply function 𝑆 𝑖 (.) for a firm 𝑖 then reduces to setting the value of 𝛼 𝑖 . Formally, a profile of strategies 𝛼 ★ 1 , 𝛼 ★ 2 , . . . , 𝛼 ★ 𝑛 is an equilibrium in supply functions if the following condition holds

𝛼 ★ 𝑖 ∈ argmax 𝛼 𝑖 𝜋 𝑖 𝛼 𝑖 , 𝛼 ★ -𝑖 , ( 20
)
for all 𝑖 = 1, 2, . . . , 𝑛. Assuming an interior solution, the first-order conditions for profit maximization write as

𝜕𝜋 𝑖 𝜕𝛼 𝑖 = 0, 𝛽 𝜕 𝑝 𝛼 ★ 𝑖 , 𝛼 ★ -𝑖 𝜕𝛼 𝑖 + 1 𝑝 𝛼 ★ 𝑖 , 𝛼 ★ -𝑖 (1 -𝑣 𝛽) -𝑣𝛼 ★ 𝑖 -𝑐 + 𝛼 ★ 𝑖 + 𝛽𝑝 𝛼 ★ 𝑖 , 𝛼 ★ -𝑖 𝜕 𝑝 𝛼 ★ 𝑖 , 𝛼 ★ -𝑖 𝜕𝛼 𝑖 = 0, (21)
𝑖 = 1, 2, . . . , 𝑛. Turning to the second order optimality conditions, we have

𝜕 2 𝜋 𝑖 𝜕𝛼 2 𝑖 = -((𝑏𝛽(𝑛 -1) + 1) (𝑏𝛽(𝑛 -1)𝑣 + 𝑣 + 2𝑏)) 𝜕 𝑝 𝛼 𝑖 , 𝛼 ★ -𝑖 𝜕𝛼 𝑖 2 < 0, 𝑖 = 1, 2, . . . , 𝑛, ( 22
)
where we used the fact that 𝜕 2 𝑝/𝜕𝛼 2 𝑖 = 0. Hence, each firm 𝑖's profit function is strictly concave in its strategic variable 𝛼 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 (21). We obtain the following proposition.

Proposition 1. There exists a unique Nash equilibrium in supply schedules. Equilibrium supply functions are identical and given by 𝑆 ★ 𝑖 ( 𝑝) = 𝛼 ★ + 𝛽𝑝, with

𝛼 ★ = (𝑎 -𝑐) -𝑏𝛽 2 (𝑛 -1) (𝑎𝑣 + 𝑏𝑐𝑛) + 𝛽(𝑏(𝑎(𝑛 -2) -2𝑐𝑛 + 𝑐) -𝑎𝑣) (𝑏𝛽(𝑛 -1) (𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) , ( 23
)
and

𝛽 := 𝛽 (𝜓; 𝑚, 𝑣) = 2𝜓 𝜓 (𝑚+1) 𝑣+2
, for all 𝑖 = 1, 2, . . . , 𝑛.

Proof. See Appendix A. Now, by successively replacing 𝛼 ★ 𝑖 = 𝛼 ★ in Equations ( 16), ( 9), ( 15) and ( 18), we find the following equilibrium price-quantity pairs and profits:

𝑞 ★ = (𝑎 -𝑐) (𝑏(𝑛 -1) 𝛽 + 1) 𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣 > 0, 𝑝 ★ = 𝑏(𝑛 -1) 𝛽(𝑎𝑣 + 𝑏𝑐𝑛) + 𝑎(𝑏 + 𝑣) + 𝑏𝑐𝑛 𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣 > 0, ( 24
)
𝑄 ★ = 𝑛(𝑎 -𝑐) (𝑏(𝑛 -1) 𝛽 + 1) 𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣 > 0, 𝜋 ★ = (𝑎 -𝑐) 2 (𝑏(𝑛 -1) 𝛽 + 1) (𝑏(𝑛 -1)𝑣𝛽 + 2𝑏 + 𝑣) 2 (𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) 2 > 0. (25)
In subsection (3.1), we use Equation ( 10) to rewrite the above solution in terms of the initial parameters of the model (𝜙, 𝜓). Then, in subsection (3.2), we consider the two limit cases where supply functions are either horizontal (𝜓 = 0) or relatively steep (𝜓 → +∞).13 Finally, in subsection (3.3) we compare it to the solution obtained by [START_REF] Menezes | More competitors or more competition?[END_REF] which corresponds to the special case of our model for which 𝑚 = 𝑣 = 0. In the remainder of this paper, the superscript ★ , ℓ and 𝑐 are used to denote the general solution, the solution of [START_REF] Menezes | More competitors or more competition?[END_REF] and the Cournot solution.

13From Equation (10, note that 𝛽 (𝜓; 𝑚, 𝑣) is monotonically increasing in 𝜓 and has limit 𝛽 (+∞; 𝑚, 𝑣) = 2 𝑣 (1+𝑚) . Consequently, the slope of firms' supply functions steepens as 𝜓 increases. However, it remains finite unless the marginal cost of production is assumed to be constant (in which case firms' supply functions would be vertical).

Supply function equilibrium when firms have a price-markup target

Assume that firms' supply behavior is given by Equation ( 6). Then, their respective equilibrium supply functions are given by 𝑆 ★ ( 𝑝) = 𝛼 ★ + 𝛽𝑝 and market equilibrium quantities are given by:

𝑞 ★ = (𝑎 -𝑐)(2𝑏𝜓(𝑛 -1) + 𝜓(𝑚 + 1)𝑣 + 2) 2𝜓𝑏 2 (𝑛 -1)𝑛 + 𝜓𝑏𝑣((𝑚 + 3)𝑛 + 𝑚 -1) + 2𝑏(𝑛 + 1) + 𝑣(𝜓(𝑚 + 1)𝑣 + 2) , ( 26
)
𝑝 ★ = 𝑎𝜓𝑣(𝑏(𝑚 + 2𝑛 -1) + 𝑚𝑣 + 𝑣) + 2𝑎(𝑏 + 𝑣) + 𝑏𝑐𝑛(2𝑏𝜓(𝑛 -1) + 𝜓(𝑚 + 1)𝑣 + 2) 2𝜓𝑏 2 (𝑛 -1)𝑛 + 𝜓𝑏𝑣((𝑚 + 3)𝑛 + 𝑚 -1) + 2𝑏(𝑛 + 1) + 𝑣(𝜓(𝑚 + 1)𝑣 + 2) , ( 27
)
𝜋 ★ = (𝑎 -𝑐) 2 (2𝑏𝜓(𝑛 -1) + 𝜓(𝑚 + 1)𝑣 + 2)(𝜓𝑣(2𝑏(𝑚 + 𝑛) + 𝑚𝑣 + 𝑣) + 4𝑏 + 2𝑣) 2 2𝜓𝑏 2 (𝑛 -1)𝑛 + 𝜓𝑏𝑣((𝑚 + 3)𝑛 + 𝑚 -1) + 2𝑏(𝑛 + 1) + 𝑣(𝜓(𝑚 + 1)𝑣 + 2) 2 . ( 28
)
We obtain the following comparative statics results:

𝜕𝑞 ★ 𝜕𝑚 ≤ 0, 𝜕𝑄 ,★ 𝜕𝑚 ≤ 0, 𝜕 𝑝 ★ 𝜕𝑚 ≥ 0, 𝜕𝜋 ★ 𝜕𝑚 ≥ 0, ( 29
)
𝜕𝑞 ★ 𝜕𝜓 ≥ 0, 𝜕𝑄 ★ 𝜕𝜓 ≥ 0, 𝜕 𝑝 ,★ 𝜕𝜓 ≤ 0, 𝜕𝜋 ★ 𝜕𝜓 ≤ 0. ( 30
)
Note that the above inequalities hold with equality whenever 𝜓 = 0, 𝑣 = 0 or 𝑚 becomes arbitrarily large.

An increase in the margin rate results in a reduction in sales volume. However, this quantity effect is more than offset by the increase in the selling price, so that the equilibrium profit increases. It is interesting to compare these comparative statics results with those which result from a change in the value of 𝜓. Recall that the parameter 𝜓 measures the degree of competitiveness of firms on the market. An increase in the degree of competitiveness of firms naturally leads to an increase in sales volumes and, in turn, a decrease in price. This time, however, the price effect outweighs the quantity effect, so that firms profits falls. Finally, it should be noted that comparative statics results in (29) are diametrically opposed to those in (30). In the face of aggressive competition (𝜓 > 0), firms may increase their profits by increasing their benchmark margin rate. Hence, committing to pursue average-cost-plus pricing is a remedy for aggressive competition. Finally, one finds that individual sales volumes, market price and firms profits decrease monotonically with the number of competitors:

𝜕𝑞 ★ 𝜕𝑛 < 0, 𝜕𝑄 ★ 𝜕𝑛 < 0, 𝜕 𝑝 ★ 𝜕𝑛 < 0, 𝜕𝜋 ★ 𝜕𝑛 < 0. ( 31
)
Let us now consider the two limit cases of the model, which correspond respectively to a relatively uncompetitive and comparatively competitive market.

Limit cases: horizontal and steep supply functions

To begin with, let us consider the least competitive market situation. Recall that firms' supply behavior coincides with Cournot's behavior when 𝛽 (𝜓; 𝑚, 𝑣) is zero; i.e., when 𝜓 is negligible or 𝑚 becomes arbitrarily large. Hence, we have:

lim 𝑚→+∞ 𝜋 ★ = 𝜋 𝑐 = lim 𝜓→0 𝜋 ★ , lim 𝑚→+∞ 𝑝 ★ = 𝑝 𝑐 = lim 𝜓→0 𝑝 ★ , lim 𝑚→+∞ 𝑞 ★ = 𝑞 𝑐 = lim 𝜓→0 𝑞 ★ . ( 32
)
We note that the Cournot equilibrium can be obtained by replacing 𝛽 (𝜓; 𝑚, 𝑣) = 0 in Equations (24-25). We obtain:

𝑞 𝑐 = 𝑎 -𝑐 𝑣 + 𝑏 (𝑛 + 1) , 𝑄 𝑐 = 𝑛 (𝑎 -𝑐) 𝑣 + 𝑏 (𝑛 + 1) , 𝑝 𝑐 = 𝑎 (𝑏 + 𝑣) + 𝑏𝑐𝑛 𝑣 + 𝑏 (𝑛 + 1) , 𝜋 𝑐 = (𝑎 -𝑐) 2 (2𝑏 + 𝑣) 2(𝑣 + 𝑏 (𝑛 + 1)) 2 . ( 33
)
Now we consider the opposite case where the market is relatively competitive. Firms behavior is most aggressive when 𝜓 becomes arbitrarily large. For this limiting case, the equilibrium price/quantity pair is given by: lim

𝜓→+∞ 𝑞 ★ = (𝑎 -𝑐) (2𝑏(𝑛 -1) + (𝑚 + 1)𝑣) 2𝑏 2 (𝑛 -1)𝑛 + 𝑏𝑣((𝑚 + 3)𝑛 + 𝑚 -1) + (𝑚 + 1)𝑣 2 > 0, (34) lim 𝜓→+∞ 𝑝 ★ = 𝑏𝑣(𝑎(𝑚 + 2𝑛 -1) + 𝑐(𝑚 + 1)𝑛) + 𝑎(𝑚 + 1)𝑣 2 + 2𝑏 2 𝑐(𝑛 -1)𝑛 2𝑏 2 (𝑛 -1)𝑛 + 𝑏𝑣((𝑚 + 3)𝑛 + 𝑚 -1) + (𝑚 + 1)𝑣 2 > 0. ( 35
)
The price-markup over marginal cost is defined by

𝑝 ★ -𝐶 ′ 𝑞 ★ = 𝑝 ★ -𝑐 + 𝑣𝑞 ★ = 𝑏(𝑎 -𝑐) 𝑏(𝑛 -1) 𝛽 (𝜓; 𝑚, 𝑣) (𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣 , ( 36
)
and lim

𝜓→∞ 𝑝 ★ -𝐶 ′ 𝑞 ★ = 𝑏(𝑚 + 1)𝑣(𝑎 -𝑐) 2𝑏 2 (𝑛 -1)𝑛 + 𝑏𝑣((𝑚 + 3)𝑛 + 𝑚 -1) + (𝑚 + 1)𝑣 2 ≥ 0. ( 37
)
It is equal to zero if, and only if, 𝑣 = 0; in which case,

lim 𝜓→∞ 𝑝 ★ -𝐶 ′ 𝑞 ★ | 𝑣=0 := lim 𝜓→∞ 𝑝 ★ -𝑐 = 0. ( 38
)
and the Bertrand solution 𝑝 ★ = 𝑐 obtains. Finally, we find lim

𝜓→+∞ 𝜋 ★ = 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 ≥ 0. (39)
Unless the marginal cost of production is constant (as in [START_REF] Menezes | More competitors or more competition?[END_REF], the equilibrium price margin, and therefore profits, are strictly positive, whatever the degree of aggressiveness of firms on the market.

We have the following proposition:

Proposition 2. Unless firms operate a constant marginal cost technology, the equilibrium price-markup ( 𝑝 ★ -𝐶 ′ (𝑞 ★ )) remains strictly positive as 𝜓 → +∞.14

Now that we have characterized the solution of our model, two questions arise. First, is it desirable for firms to adopt a profit margin target? Second, would this adoption harm social welfare? We consider these two questions successively below.

The desirability of a margin target

To know if setting a profit margin target is beneficial for firms, it is first necessary to calculate the equilibrium prices, quantities and profits in the absence of such a goal. This is equivalent to determining the equilibrium values of the standard model of competition in linear supply functions with quadratic production costs. In line with this, we replace Equations 𝛽 (𝜓; 𝑚, 𝑣) in Equations ( 24) and ( 25) by

𝛽| ( 𝜓;𝑚,𝑣)=( 𝜓;0,0) := 𝛽 (𝜓; 0, 0) = 𝜓. ( 40
)
So doing, we obtain the following equilibrium values:

𝑞 ℓ = (𝑎 -𝑐) (𝑏(𝑛 -1)𝜓 + 1) 𝑏(𝑛 -1)𝜓(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣 > 0, 𝑝 ℓ = 𝑏(𝑛 -1)𝜓(𝑎𝑣 + 𝑏𝑐𝑛) + 𝑎(𝑏 + 𝑣) + 𝑏𝑐𝑛 𝑏(𝑛 -1)𝜓(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣 > 0, ( 41
)
𝑄 ℓ = 𝑛(𝑎 -𝑐) (𝑏(𝑛 -1)𝜓 + 1) 𝑏(𝑛 -1)𝜓(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣 > 0, 𝜋 ℓ = (𝑎 -𝑐) 2 (𝑏(𝑛 -1)𝜓 + 1) (𝑏(𝑛 -1)𝑣𝜓 + 2𝑏 + 𝑣) 2 (𝑏(𝑛 -1)𝜓(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) 2 > 0, ( 42
)
for all 𝑖 = 1, 2, . . . , 𝑛.

14Obviously then, we also have lim 𝜓→+∞ 𝑝 ★ -𝐴𝐶 𝑞 ★ > 0.

In order to obtain more compact expressions, we define a function Φ(𝑥) = 𝑏 ((𝑛 -1)(𝑏𝑛 + 𝑣)𝑥 + (𝑛 + 1)) + 𝑣. Also, the comparison of equilibrium quantities is facilitated if it is observed that: 𝛽 (𝜓; 0, 0) -𝛽 (𝜓; 𝑚, 𝑣) = 𝜓 2 (𝑚 + 1)𝑣/(𝜓(𝑚 + 1)𝑣 + 2) > 0.

(43)

Building on this observation, we obtain :

𝑝 ℓ -𝑝 ★ = - 𝑏 3 (𝑛-1)𝑛{ 𝛽 (𝜓; 0, 0) -𝛽 (𝜓; 𝑚, 𝑣)}(𝑎-𝑐)
Φ(𝛽 ( 𝜓;0,0) )Φ(𝛽 ( 𝜓;𝑚,𝑣) )

< 0, 𝑞 ℓ -𝑞 ★ = 𝑏 2 (𝑛-1) (𝛽 ( 𝜓;0,0) -𝛽 ( 𝜓;𝑚,𝑣) ) (𝑎-𝑐)

Φ(𝛽 ( 𝜓;0,0) )Φ(𝛽 ( 𝜓;𝑚,𝑣) ) > 0, ( 44
)
and

2( 𝜋 ℓ -𝜋 ★ ) (𝑎-𝑐) 2 = - 1 Φ(𝛽 ( 𝜓;0,0) ) 2 Φ(𝛽 ( 𝜓;𝑚,𝑣) ) 2 𝑏 3 (𝑛 -1) 2 (𝛽 (𝜓; 0, 0) -𝛽 (𝜓; 𝑚, 𝑣)) (𝑏𝛽 (𝜓; 𝑚, 𝑣) (2𝑛(𝑏𝑛 + 𝑣) -𝑣) + 𝑏𝛽 (𝜓; 0, 0) (2𝑏(𝑛 -1)𝑛𝛽 (𝜓; 𝑚, 𝑣) (𝑏𝑛 + 𝑣) + 2𝑛(𝑏𝑛 + 𝑣) -𝑣) + 2(𝑏𝑛 + 𝑏 + 𝑣))} ≤ 0. ( 45
)
From Equation ( 44), note that 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 marketclearing price. Equation ( 45) shows that this price increase more than compensates for the decrease in sales so that firms' profits increase. We have the following proposition:

Proposition 3. If 𝜓 > 0 then 𝜋 ★ > 𝜋 ℓ . Firms achieve higher profits by conditioning their offer on a given margin target and market price rather than allowing their offer to vary only with the market price.

We conclude this section with a comparative study of levels of social well-being.

Price-markup targets and social welfare

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

SW := CS + PS = 𝑏 2 𝑄 2 + 𝑛 𝑖=1 𝜋 𝑖 , ( 46
) = 𝑛𝑞(𝑎 -𝑐) - 1 2 𝑛𝑞 2 (𝑏𝑛 + 𝑣). ( 47
)
Let us now evaluate social-welfare at the supply function equilibrium. By plugging 𝑞 ★ into Equation ( 47), we obtain

SW ★ = 𝑛(𝑎 -𝑐) 2 (𝑏(𝑛 -1) 𝛽 + 1) (𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏(𝑛 + 2) + 𝑣) 2 (𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) 2 . ( 48
)
To compare social welfare levels, it is useful to note that

𝛽 (𝜓; 0, 𝑣) ≥ 𝛽 (𝜓; 𝑚, 𝑣) ≥ 𝛽 (0; 𝑚, 𝑣) := 0. ( 49
)
The slope of standard linear supply functions is higher than the slope of markup-pricing supply functions, which is itself higher than the slope of Cournot supply functions (i.e. zero). In other words, competition is less intense when firms engage in supply function competition and have a margin target rather than no margin target. However, the intensity of competition will always be higher than in the case of a Cournot oligopoly. By differentiating social welfare with respect to 𝛽, we obtain :

𝜕SW ★ 𝜕 𝛽 = 𝑏 3 (𝑛 -1)𝑛(𝑎 -𝑐) 2 (𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) 3 > 0, ( 50
)
where 𝛽 := 𝛽 (𝜓; 𝑚, 𝑣) = 2𝜓 𝜓 (𝑚+1) 𝑣+2 . Hence, social welfare increases as firms' supply behavior becomes more aggressive. Given the ranking in Equation ( 49), we can state the following proposition.

Proposition 4. Social welfare levels rank as follows: SW

ℓ ≥ SW ★ ≥ SW 𝑐 .
Social welfare is always higher when firms engage in supply function competition than when they compete à la Cournot. However, social welfare will be maximum only if no margin target appears in firms' supply functions.

Collusive price markup

So far, we have assumed a fixed and given price markup. This assumption may be justifiable for fragmented industries where considerations of profitability and equity have long crystallized into a conventional price markup around which market participants agree. However, this is less plausible for a concentrated industry where collusion opportunities arise since a limited number of firms supply the market. Consequently, we now investigate the possibility for firms to collude by setting the markup 𝑚 strategically. In doing so, we consider two scenarios that differ in how consumers respond to an increase in the price markup. In the first, this increase leaves consumers indifferent, while in the second, it infuriates them so much that they reduce their demand.

Markup-insensitive Consumers

Consumers have a cost-plus mentality [START_REF] Simon | Confessions of the Pricing Man[END_REF]: they expect product prices to reflect production costs while agreeing that it is legitimate for business firms to make money. Most of the time, firms' pricing practices agree with this mindset. For one thing, they imply a relatively small price markup that consumers welcome with leniency or a larger one that is sufficiently discreet to go unnoticed. For another, consumers may be uncertain as to what markup level is legitimate and thus acceptable. Then, as long as the price does not seem extravagant, they may forgive and forget quite noticeable price markups. The example of restaurants where hefty prices are charged for wine is well documented.15 Consumers may perceive those prices as necessary for restaurants to break even. High-tech companies enjoy a similar tolerance from consumers who recognize that the production of cutting-edge products deserves an extra reward.16 We start with the analysis of these situations where consumers are relatively insensitive to price markups.

Suppose that consumer demand is independent of the price markup 𝑚. Then, the comparative statics results presented in the previous section hold. Unless firms compete à la Cournot (𝜓 = 0) or operate a constant return to scale technology (𝑣 = 0) , from Equation ( 29), notice that 𝜕𝜋 ★ 𝑖 /𝜕𝑚 > 0 for 𝑚 finite and lim 𝑚→+∞ 𝜕𝜋 ★ 𝑖 /𝜕𝑚 = 0 for all 𝑖 = 1, 2, . . . , 𝑛. Observe that the industry has incentives to increase 𝑚 without bound. However, from Equation (10) note that the slope of the supply function is decreasing in 𝑚 and eventually reaches zero as 𝑚 becomes arbitrarily large -in which case the supply-function equilibrium replicates the standard Cournot equilibrium. We are now in a position to establish the following proposition.

Proposition 5. Suppose that firms are able to collude by setting strategically a common price-markup 𝑚. If consumer demand is insensitive to 𝑚, then price-markup expectations become so large that the Cournot solution obtains.

Hence, supply function competition in concentrated industries will likely lead to Cournot competition. This finding supports the thesis that markup pricing is popular because it favors collusion.17

Markup-sensitive consumers

While consumers understand that businesses need to be profitable, they also demand to be treated fairly. Perceptions of unfair pricing practices can upset consumers, harm business goodwill, and reduce profits. In the past, one way for consumers to express their dissatisfaction was to participate in sporadic local protests and boycotts [START_REF] Friedman | Consumer Boycotts in the United States, 1970-1980: Contemporary Events in Historical Perspective[END_REF]. One early and well-documented example is the 1902 New York City kosher meat boycott which saw immigrant Jewish women protesting against the 'Meat Trust' decision to increase kosher meat price from 14 to 18 cents a pound.18 Instead, today when consumers feel wronged, they pour their anger 15See, for example, Amspacher and Amspacher, 2011. 16In this regard, the example of Apple's iPhone pricing is exemplary. According to [START_REF] Tilley | Apple's IPhone Pices Soar, but Not Profit Margins[END_REF], gross profit margins for Apple smartphones are estimated at over 60%, well above industry standards. So although Apple's sales accounted for less than 20% of all smartphone sales in 2017, they helped it capture 87% of the overall market profit.

17See the quotation of [START_REF] Phillips | Pricing and Revenue Optimization[END_REF] given in the introduction. [START_REF] Simon | Confessions of the Pricing Man[END_REF] also notes that this pricing method "[...] minimizes price competition for the product and encourages the sellers to compete on aspects other than price. Cost-plus pricing can create a de facto price cartel , with the resulting stability and predictability. All these factors explain why this method is so popular."

18For a historical account, see [START_REF] Hyman | Immigrant Women and Consumer Protest: The New York City Kosher Meat Boycott of 1902[END_REF] out on social media, giving it a global audience. Even giant tech companies such as Apple Inc had to apologize to appease upset consumers. For example, the 2007 decision to cut the price of the iPhone by $200 just two months after it hit the market created such a backlash that the company had to offer early adopters a $100 credit to use on the Apple Store. Consumers' assertiveness is not always sufficient to mitigate price unfairness, especially when the price elasticity of demand is low. The pricing of pharmaceuticals demonstrates this. The pharmaceutical industry defends patented drugs prices increases by arguing they are justified by growing R&D costs or higher value for the patient. However, these arguments fail to explain why generic drugs have seen their price increase steadily over the last two decades [START_REF] Furlow | Public outcry over US generic drug price hikes[END_REF]. For example, [START_REF] Hill | Estimated costs of production and potential prices for the WHO Essential Medicines List[END_REF] estimate that some generic drugs featured in the WHO essential drugs list sell in the UK at more than 100 times their production cost. As previously noted, [START_REF] Hall | Price Theory and Business Behaviour[END_REF] believed that the popularity of average cost pricing stems from its perception as a fair pricing practice to both consumers and firms. Also, [START_REF] Okun | Prices and quantities[END_REF] proposed to explain the relative stickiness of prices when demand suddenly increases by implicit price contracts among consumers and firms. [START_REF] Khaneman | Fairness as a Constraint on Profit Seeking: Entitlements in the Market[END_REF] 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. The introduction of consumer preference for fair prices in theoretical modeling is more recent. [START_REF] Rotemberg | Customer anger at price increases, changes in the frequency of price adjustment and monetary policy[END_REF]2011) assumes that consumers expect some benevolence from firms and are upset whenever this expectation is not met. Since firms internalize such reactions into their decision-making process, they moderate their pricing practices. [START_REF] Rotemberg | Customer anger at price increases, changes in the frequency of price adjustment and monetary policy[END_REF] shows that firms will pass cost increases only partially onto their consumers. Also, [START_REF] Rotemberg | Customer anger at price increases, changes in the frequency of price adjustment and monetary policy[END_REF] highlights they will follow third-degree price discrimination based on income rather than demand elasticity. Finally, [START_REF] Eyster | Pricing Under Fairness Concerns[END_REF] embed consumer preference for a fair price in a New Keynesian model. The authors assume that consumers are upset by prices they regard as unfair. However, they cannot observe firms' marginal costs of production. So the only option left for them is to infer these costs from prices. When doing so, they are assumed to overestimate the role price markups play in observed price increases. [START_REF] Eyster | Pricing Under Fairness Concerns[END_REF] shows that firms will cut back on their margins to moderate price increases. For the limited purpose of this paper, we will assume here that consumers' willingness to pay for the product is decreasing in the price-markup 𝑚. This assumption will suffice to show that the industry may choose a positive but small markup value. Suppose that consumers are antagonized by an increase in the gross margin rate. Then, firm i's profit may be written as

𝜋 ★ 𝑖 = (𝑎 (𝑚) -𝑐) 2 Γ(𝛽)
where Γ(𝛽) = (𝑏(𝑛 -1) 𝛽 + 1) (𝑏(𝑛 -1)𝑣 𝛽 + 2𝑏 + 𝑣)

2 (𝑏(𝑛 -1) 𝛽(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) 2 > 0, (51)
and with 𝑎 ′ (𝑚) < 0.

(52)

We have

𝑑𝜋 ★ 𝑖 𝑑𝑚 = 2𝑎 ′ (𝑚) (𝑎 (𝑚) -𝑐) Γ (𝛽) (-) + (𝑎 (𝑚) -𝑐) 2 (+} Γ ′ (𝛽) (-) 𝛽 ′ (𝑚) (-) (+) , ( 53
)
where

Γ ′ (𝛽) = - 𝑏 3 (𝑛 -1) 2 (𝑏𝛽𝑛 + 1) (𝛽𝑏(𝑛 -1) (𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) 3 < 0, ( 54
)
𝛽 ′ (𝑚) = - 2𝑣𝜓 2 ((𝑚 + 1)𝑣𝜓 + 2) 2 < 0. ( 55
)
There are two cases in which it is straightforward to determine whether a change in the gross profit margin 𝑚 leads to higher profits or not. The first one is already familiar. As shown above, if 𝑎 ′ (𝑚) = 0, the first term on the right-hand side of Equation ( 53) vanishes so that firms' equililbrium profit are monotonically increasing in 𝑚.

Figure 1 illustrates the case corresponding to the previous proposition where companies have the incentive to coordinate on a positive but small profit margin. For the chosen numerical values, the profit-maximizing markup is 𝑚 ★ ≈ 4.63/100.

Conclusion

Overall, businesses remain loyal to the markup pricing rule despite its many defects. Even when they do not rely on this pricing heuristic, they regard it as a convenient benchmark for judging whether the selling price is satisfactory and whether supply adjustments are desirable. Based on this observation, we have introduced a new supply function whereby the firm commits to increasing its supply as the selling price rises and deviates from the markup price. We have shown that this function is equivalent to a linear supply function whose y-intercept and slope are functions of the firms' profit margin objectives and cost parameters. Therefore, it coincides with the supply function considered by [START_REF] Menezes | More competitors or more competition?[END_REF] whenever the firm has a zero profit margin target and a constant marginal cost technology. We characterized the equilibrium in supply functions for an industry composed of n firms. We performed a comparative statics analysis to investigate how the profit margin standard in the industry alters market supply, profitability, and social welfare. We found that the higher the industry margin target, the less aggressive firms' equilibrium supply behavior becomes. The resulting reduction in supply leads to an increase in the equilibrium price, which results in higher equilibrium profits. However, the increase in profits fails to compensate for the loss in consumer surplus so that overall social welfare drops. Comparative statics analysis made it possible to elucidate the strategic role played by the profit margin targets of business firms. We showed that firms achieve higher profits when they condition their offers on the deviation of the selling price from the marked-up price rather than on the selling price itself (as is the case in [START_REF] Menezes | More competitors or more competition?[END_REF]. We also analyzed how the equilibrium profit of firms varies with the standard margin level in the industry depending on whether or not unfair pricing practices antagonize consumers. When consumers exhibit a low sensitivity to fair pricing issues, we showed that firms' profits increase monotonously with the industry's profit margin standard and reach their Cournot levels when it becomes arbitrarily large. Therefore, firms have incentives to price-fixing by switching from supply function competition to quantity competition. The same conclusion may obtain in the more plausible context where consumers get upset when firms aim at excessive profit margins. In particular, this will be the case if firms use a constant return to scale technology or have a low capacity to adjust supply. However, there are many situations where consumers' preference for fair treatment is enough to discipline the behavior of firms and cause them to compete in supply functions with relatively small markup over costs. 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 [START_REF] Anderson | Supply Function Equilibrium in Electricity Spot Markets with Contracts and Price Caps[END_REF]. However, the duopoly case with asymmetric costs lends itself to analysis. Second, we assumed a given and fixed industry profit margin standard. It would be interesting to study how this standard 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 standard is for a given industry.

A Appendix

A.1 Proof of Proposition 1

Replacing 𝑝(𝛼 𝑖 , 𝛼 -𝑖 ) and 𝜕 𝑝 (𝛼 𝑖 , 𝛼 -𝑖 ) /𝜕𝛼 𝑖 by their expressions ( 16) and ( 17) into Condition (21), and solving the resulting equation with respect to 𝛼 𝑖 , we get firm 𝑖's inclusive19 reaction function:

𝛼 ★ 𝑖 = Φ + Ψ 𝑛 𝑖=1 𝛼 ★ 𝑖 , 𝑖 = 1, 2, . . . , 𝑛. ( 62
)
where

Φ = - 𝑎𝑏𝛽 + 𝑎 (𝑣 𝛽 -1) (𝑏(𝑛 -1) 𝛽 + 1) + 𝑐 (𝑏(𝑛 -1) 𝛽 + 1) (𝑏𝑛𝛽 + 1) (𝑏𝑛𝛽 + 1) (𝑏(𝑛 -1)𝑣𝛽 + 𝑏 + 𝑣) , ( 63
) Ψ = 𝑏 (𝑏𝛽 + (𝑣𝛽 -1) (𝑏(𝑛 -1) 𝛽 + 1)) (𝑏𝑛𝛽 + 1) (𝑏(𝑛 -1)𝑣𝛽 + 𝑏 + 𝑣) . ( 64
)
Summing over all 𝑖 yields the fixed-point equation Finally, substituting (63) and ( 64) into (67) and rearranging terms we obtain the y-intercept given by Equation (23).

A.2 Comparative statics

We have (69)

Now, since (from Eq. 10)

𝜕 𝛽 𝜕𝑚 = -2𝜓 2 𝑣 /(𝜓(𝑚 + 1)𝑣 + 2) 2 ≤ 0, and 𝜕 𝛽 𝜕𝜓 = 4/(𝜓(𝑚 + 1)𝑣 + 2) 2 > 0, (70)

we obtain the comparative statics results ( 29)-( 30).

19Here, we use the terminology of Wolfstetter (1999, page 91). For more details on the solution procedure, see [START_REF] Szidarovszky | A New Proof of the Existence and Uniqueness of the Cournot Equilibrium[END_REF].

A.3 Proof of Proposition 8

In order to simplify the proof of Proposition 8, it is convenient to define a new variable Now, we rewrite the industry's decision problem in terms of the new variable 𝜇. We begin with condition (58) whose satisfaction guarantees strictly positive profits. We solve Equation ( 71) for 𝑚 and plug its value into condition (58) to obtain

Ω := ā + 𝜌 1 + 2 (𝜇 -1) 𝑣𝜇𝜓 -𝑐 > 0, ( 73
)
This equation is satisfied provided that Using Equation (74), the above expression for 𝑛 3 may be written as a function of 𝜇. This yields the more compact Equation ( 61). Finally, it can be easily checked that 𝑛 3 > 𝑛 2 .

We now are in a position to establish Proposition (8). To this end, we must consider two cases separately. First, if 𝑛 > 𝑛 3 then 𝐴 ℎ > 0 for all ℎ = 1, 2, 3, 4. We have P (𝜇) > 0 and thus 𝜕𝜋 ★ 𝑖 /𝜕𝜇 > 0. Since marginal profit is monotonically increasing in 𝜇, maximum profit is reached for 𝜇 ★ = μ. Using Equation ( 71) we obtain 𝑚 ★ = 0.

Second, if 𝑛 < 𝑛 3 then we have P (𝜇) | 𝜇=0 > 0 and lim 𝜇→∞ P (𝜇) = -∞. Because P (𝜇) is continuous, there exists a unique value of 𝜇 that satisfies the equation P ( μ) = 0. Indeed, since 𝐴 3 < 0, the parabola P ′ (𝜇) = 3𝐴 3 𝜇 2 + 2𝐴 2 𝜇 + 𝐴 1 opens downwards. This implies that P (𝜇) first increases reaching a maximum at 𝜇 = μ, and then decreases. Finally, since μ > 𝜇, it follows that the solution is 𝜇 ★ = min { μ, μ} .

Figure 1 :

 1 Figure 1: The price markup as a collusion device. Parameter values are 𝑎 = 40, 𝑏 = 1, 𝜓 = 1, 𝑐 = 0, 𝑛 = 3, 𝑣 = 1, 𝜌 = 1.

  Equation (66) back into the (inclusive-) reaction functions (62), we obtain firm 𝑖's equilibrium intercept:

  𝑏 3 (𝑛-1) 2 (𝑎-𝑐) 2 (𝑏𝑛𝛽+1) (𝑏 (𝑛-1) 𝛽 (𝑏𝑛+𝑣)+𝑏𝑛+𝑏+𝑣) 3 𝜕𝛽 𝜕𝑘 .

  𝜇 > 2𝜌 2𝜌 + 𝑣 (𝑎 -𝑐 + 𝜌) 𝜓 := 𝜇. (74)Here, we take note that 𝜇 ∈ [0, 1]. We proceed by rewriting Firm 𝑖's profit as a function of 𝜇. By plugging 𝛽(𝜓; 𝑚, 𝑣) = 𝜇.𝜓 into Equation (25), we obtain𝜋 ★ 𝑖 (𝜇) = Ω 2 (𝑏𝜇(𝑛 -1)𝜓 + 1) (𝑏𝜇(𝑛 -1)𝑣𝜓 + 2𝑏 + 𝑣) 2(𝑏𝜇(𝑛 -1)𝜓(𝑏𝑛 + 𝑣) + 𝑏𝑛 + 𝑏 + 𝑣) 2 . (𝑏 + (𝑏𝑛 + 𝑣) (1 + 𝑏 (𝑛 -1) 𝜇𝜓)) P (𝜇) = 0, (77)where P (𝜇) = 𝐴 3 𝜇 3 + 𝐴 2 𝜇 2 + 𝐴 1 𝜇 + 𝐴 0 , and𝐴 0 = 4𝜌(2𝑏 + 𝑣) (𝑏𝑛 + 𝑏 + 𝑣) > 0, (78)𝐴 1 = 4𝑏(𝑛 -1) 𝜌𝜓 𝑏 2 (5𝑛 + 1) + 3𝑏(𝑛 + 2)𝑣 + 3𝑣 2 > 0, (79)𝐴 2 = 2𝑏 2 (𝑛 -1) 2 𝜓 𝑏𝑣𝜓(-𝑎 + 𝑐 + 6𝑛𝜌 + 5𝜌) + 6𝑏 2 𝑛𝜌𝜓 -2𝑏𝜌 + 6𝜌𝑣 2 𝜓 , (80)𝐴 3 = 2𝑏 3 (𝑛 -1) 2 𝜓 2 𝑏𝑛(𝑣𝜓(-𝑎 + 𝑐 + 2𝑛𝜌 -3𝜌) -2𝜌) + 2(𝑛 -1) 𝜌𝑣 2 𝜓 . (81)We now study the properties of the marginal profit curve (77). To begin with, note that the signs of coefficients 𝐴 2 and 𝐴 3 depends on 𝑛. Indeed, we have 𝐴 2 > 0 iff𝑛 > 𝑏𝑣𝜓(𝑎 -𝑐 -5𝜌) + 2𝑏𝜌 -6𝜌𝑣 2 𝜓 6𝑏𝜌𝜓(𝑏 + 𝑣) := 𝑛 2 . (82) Also, we have 𝐴 3 > 0 iff 𝑛 > 4𝑏𝜌𝑣 2 𝜓(𝑣𝜓(-𝑎 + 𝑐 + 𝜌) -2𝜌) + (𝑏𝑣𝜓(𝑎 -𝑐 + 3𝜌) + 2𝑏𝜌) 2 + 4𝜌 2 𝑣 4 𝜓 2 4𝑏𝜌𝑣𝜓 + 𝑏𝑣𝜓(𝑎 -𝑐 + 3𝜌) + 2𝑏𝜌 -2𝜌𝑣 2 𝜓 4𝑏𝜌𝑣𝜓 := 𝑛 3 .

12Actually,[START_REF] Menezes | More competitors or more competition?[END_REF] normalize the y-intercept. As noted by[START_REF] Delbono | On the properties of linear supply functions in oligopoly[END_REF] this normalization is useless.

CEE-M Working Papers / Contact : laurent.garnier@inrae.fr  RePEc https://ideas.repec.org/s/hal/wpceem.html  HAL https://halshs.archives-ouvertes.fr/CEE-M-WP/

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 Dyln.

The second one is more interesting. Observe that Γ is independent of 𝑚 whenever 𝛽 is independent of 𝑚. This is the case if firms operate a constant return of scale technology (𝑣 = 0), or if the market is little competitive (𝜓 = 0). In both cases, an increase in the gross margin rate 𝑚 unambiguously leads to reduced profits. Indeed, Equation (53) reduces to:

We have the following two propositions:

Proposition 6 (Collusive choice of profit margin with constant marginal costs). Suppose that the gross price margin antagonizes consumers and the technology of production is characterized by a constant marginal cost.

If firms have the possibility to collude by choosing the industry gross margin strategically then they prefer to abandon any margin aspiration; i.e., they set 𝑚 ★ = 0.

Proposition 7 (Collusive choice of profit margin with flat supply functions). Suppose that the gross price margin antagonizes consumers and supply functions are horizontal (𝜓 = 0) implying that firm supply is not aggressive. If firms have the possibility to collude by choosing the industry gross margin strategically then they prefer to abandon any margin aspiration; i.e., they set 𝑚 ★ = 0.

Except for these two special cases, it is impossible to sign Expression (53) without specifying further 𝑎(𝑚).

Here, to avoid multiplicity, we assume that consumer demand decreases linearly with the industry gross margin rate 𝑚; i.e.,

The industry would never choose a gross margin rate 𝑚 causing so much resentment from consumers that they give up consuming its product. Plugging 𝑎 = ā -𝜌𝑚 into Equation ( 26), we find that demand becomes zero if

or, equivalently, 𝑚 > ( ā -𝑐) /𝜌 := m. Accordingly, the industry never chooses a markup higher than m.

We now show that the optimal margin rate for the industry can be positive or zero depending on the number of firms operating in the industry. Plugging 𝑎 = ā -𝜌𝑚 into Equation ( 25), Firm 𝑖's equilibrium profit becomes

For convenience, we define

In Appendix A.3, we prove that marginal profit is monotonically decreasing in 𝑚 if

in which case, the industry sets 𝑚 ★ = 0. Otherwise, there may exist a positive markup 𝑚 ★ that maximizes profits. We are in a position to state the following proposition.

Proposition 8. If the number of firms is relatively large, 𝑛 > 𝑛 3 , then the industry prefers to abandon any expectation of margin; i.e., 𝑚 ★ = 0. Otherwise, the industry may collude over a positive markup; i.e., 𝑚 ★ ∈ (0, m).

Proof. See Appendix A.3.

CEE-M Working Papers


Wasif Razzaq

Christophe A Serra
email: ca.serra@unistra.fr

Leandro Jacomine

Delphine Chan-Seng

Microfluidic Elaboration of Polymer Microfibers from Miscible Phases: Effect of Operating and Material Parameters on Fiber Diameter

Keywords: microfluidics, microfiber diameter, photopolymerization, capillary number

Background: fiber diameter is one of the most important morphological parameters which drives the applications of microfibers. This creates a need for the development of processes capable of producing a large variety of microfibers with a given diameter. To this regards, microfluidic spinning has recently emerged as an outstanding and simple technique for the production of micro-and nanofibers with controllable size and morphology.

Methods: herein, microfibers were produced from (macro)monomers or prepolymers (core phase) by in situ photoirradiation using a capillary-based microfluidic device and a miscible sheath phase of various viscosity. The effects of the flow rate of both phases as well as the viscosity of the sheath fluid, the capillary dimensions and the monomer volume fraction in core phase were thoroughly studied.

Significant findings: by calculating the capillary number ratio from the ratios of sheath to core flow rate and viscosity, an empirical relationship which perfectly predicts the microfiber diameter as a function of monomer volume fraction, the capillary number ratio and capillary inner diameter but independent of its outer diameter is extracted. This result paves the way to the continuousflow production of microfibers with well-controlled morphological characteristics.

Introduction

Microfibers have attracted a lot of attention due to i) their large surface area to volume ratio, ii) their diverse properties arising from the great variety of materials they are made of, and iii) their ability to assemble into 3D complex structures and foldability. [START_REF] Du | Multifunctional micro/nanoscale fibers based on microfluidic spinning technology[END_REF] These benefits enable polymer microfibers to have an excellent potential in many applications such as biomedicine, [START_REF] Angelozzi | Composite ECMalginate microfibers produced by microfluidics as scaffolds with biomineralization potential[END_REF][START_REF] Takei | Novel technique to control inner and outer diameter of calcium-alginate hydrogel hollow microfibers, and immobilization of mammalian cells[END_REF][START_REF] Lin | Microfluidic synthesis of microfibers for magnetic-responsive controlled drug release and cell culture[END_REF][START_REF] Haynl | Microfluidics-produced collagen fibers show extraordinary mechanical properties[END_REF][START_REF] Sun | Assembly of alginate microfibers to form a helical structure using micromanipulation with a magnetic field[END_REF] fiber optics, [START_REF] Blue | Recent advances in optical fiber devices for microfluidics integration[END_REF] sensors, [START_REF] Liu | Polyaniline nanofiber wrapped fabric for high performance flexible pressure sensors[END_REF][START_REF] Yang | Conductive cotton fabrics for motion sensing and heating applications[END_REF] and water treatment. [START_REF] Korina | Polymer fibers with magnetic core decorated with titanium dioxide prospective for photocatalytic water treatment[END_REF][START_REF] Jurecska | Intensification of wastewater treatment with polymer fiber-based biofilm carriers[END_REF] Different approaches have been employed to produce micro-and nanofibers such as melt spinning, [START_REF] Cui | A Novel Nano/Micro-Fibrous Scaffold by Melt-Spinning Method for Bone Tissue Engineering[END_REF] wet spinning, [START_REF] Nechyporchuk | Wet Spinning of Chitosan Fibers: Effect of Sodium Dodecyl Sulfate Adsorption and Enhanced Dope Temperature[END_REF] draw spinning, [START_REF] Liao | Draw-Spinning of Kilometer-Long and Highly Stretchable Polymer Submicrometer Fibers[END_REF] macromolecular assembly, [START_REF] Loo | From short peptides to nanofibers to macromolecular assemblies in biomedicine[END_REF] and electrospinning, [START_REF] Lin | Direct Fabrication of Highly Nanoporous Polystyrene Fibers via Electrospinning[END_REF] techniques relying on the physical mechanism of solidification to produce fibers, i.e. the starting raw material is a polymer solution.

As such they suffer from limitations regarding i) the nature of the material employed and ii) the morphologies of fibers that can be achieved. [START_REF] Duboin | Cusps, spouts and microfiber synthesis with microfluidics[END_REF] Hence, it is difficult to produce fibers with diverse morphologies and with a broad range of materials. Recently, microfluidic spinning has shown a great potential for the production of microfibers with diverse compositions, morphologies, and surface functionalities. This technique consists in stretching a stream of monomer or polymer solution (core phase) by an immiscible or miscible solution (continuous or sheath phase) inside a microfluidic device. Due to the small size of the device microchannel (ca. 100 µm), laminar flow is commonly achieved affording reproducibility and stability to the flow; two highly desirable features required for the production of fibers with given diameters and morphologies. By manipulating the flow inertia (i.e. individual solution flow rates), solution viscosity, interfacial tension between the core and sheath phases and taking possibly advantage of gravity forces, fibers with diverse morphologies such as grooved, flat, core-shell, hollow, and Janus can be easily produced. The choice of material becomes broader due to the panel of solidification methods, i.e. photopolymerization, ionic and chemical crosslinking, solvent exchange, and solvent evaporation. Fiber surface functionalization is also possible by encapsulation method or in situ chemistry. [START_REF] Du | Multifunctional micro/nanoscale fibers based on microfluidic spinning technology[END_REF] Surface to volume ratio is one of the foremost fiber parameters that drive their applications. Since the latter is inversely proportional to the fiber diameter, [START_REF] Singha | Effects of fiber diameter distribution of nonwoven fabrics on its properties[END_REF] many properties such as mechanical, [START_REF] Young | Strong dependence of mechanical properties on fiber diameter for polymer-nanotube composite fibers: Differentiating defect from orientation effects[END_REF][START_REF] Kim | Effect of fiber diameter on surface morphology, mechanical property, and cell behavior of electrospun poly(ε-caprolactone) mat[END_REF] cell adhesion and proliferation, [START_REF] Kim | Effect of fiber diameter on surface morphology, mechanical property, and cell behavior of electrospun poly(ε-caprolactone) mat[END_REF] biomimicking extracellular matrix, [START_REF] Chen | Role of fiber diameter in adhesion and proliferation of NIH 3T3 fibroblast on electrospun polycaprolactone scaffolds[END_REF] optical extinction capacity, [START_REF] Yang | Engineering surface and optical properties of TiO 2-coated electrospun PVDF nanofibers via controllable self-assembly[END_REF] and filtration performance [START_REF] Chang | The effect of fibre diameter on filtration and flux distribution -relevance to submerged hollow fibre modules[END_REF] depend on the fiber diameter.

Although, fiber diameter is a crucial parameter, the literature scarcely and partially reports on all the possible parameters responsible for controlling the fiber diameter in microfluidic spinning.

Most reports address only the effect of sheath/continuous fluid and core/disperse fluids flow rates independently [START_REF] Jeong | Hydrodynamic microfabrication via"on the fly" photopolymerization of microscale fibers and tubes[END_REF][START_REF] Hu | Hydrodynamic spinning of hydrogel fibers[END_REF] or in the form of sheath to core flow rate ratio (Qs/Qc) [START_REF] Thangawng | UV polymerization of hydrodynamically shaped fibers[END_REF][START_REF] Nakajima | Stimuli-responsive hydrogel microfibers with controlled anisotropic shrinkage and cross-sectional geometries[END_REF][START_REF] Thangawng | A simple sheath-flow microfluidic device for micro/nanomanufacturing: fabrication of hydrodynamically shaped polymer fibers[END_REF][START_REF] Venkateshwarlu | Effects of capillary number and flow rates on the hydrodynamics of droplet generation in two-phase cross-flow microfluidic systems[END_REF]. Liu et al. observed the effect of the capillary diameter on the fiber diameter using the thermoinitiated polymerization induced phase separation technique [START_REF] Liu | Polymerization-induced phase separation fabrication: A versatile microfluidic technique to prepare microfibers with various cross sectional shapes and structures[END_REF]. But there is no literature in which researchers interpreted these results into some mathematical form to predict the fiber diameters.

Herein, we developed an empirical relationship which can predict the fiber diameter in relation with others operating and materials parameters for the case where the monomer and its polymer are miscible with the continuous phase. The investigation of operating parameters was not only limited to flow rates and capillary diameter, but the effect of viscosity of sheath fluid and monomer volume fraction in core phase was also investigated. Two monomers, tri(propylene glycol) diacrylate (TPGDA), poly(ethylene glycol) diacrylate (PEGDA), and one prepolymer, UV-curable adhesive NOA 89, were used as reference materials for the synthesis of fibers using in situ photoirradiation in a capillary-based microfluidic device. All results were used to extract an empirical correlation which predicted the final microfiber diameter upon variation of the capillary number ratio, monomer volume fraction, and internal capillary diameter.

Results and Discussion

The fibers were produced using a capillary-based microfluidic device (Fig. 1) involving two phases: 1) the core fluid (Φc) becoming the fiber upon photopolymerization and 2) the sheath fluid (Φs) consisting in poly(ethylene glycol) (PEG), whose flow rates are Qc and Qs respectively. PEG was used due to its miscibility with core fluid, commercial availability, high tunable viscosity with its molecular weight which prevents fast diffusion of Φc into Φs, and reasonable shearing force to produce thinner fibers. Φc comprised a diacrylate (TPGDA, PEGDA, or NOA 89) in ethanol in the presence of Irgacure 369 as photoinitiator.

Effect of the flow rate ratio Qs/Qc

The effect of sheath and core flow rates on the fiber diameter was investigated showing a decrease of the fiber diameter as the flow rate of Φs increased due to the increase in the shearing and stretching force exerted by Φs over Φc. On the other hand, the diameter increased with an increase in the flow rate of Φc, because of the widening of the jet due to the increase in the volume of Φc injected. [START_REF] Hwang | Microfluidic chip-based fabrication of PLGA microfiber scaffolds for tissue engineering[END_REF][START_REF] Wu | A novel synthetic microfiber with controllable size for cell encapsulation and culture[END_REF][START_REF] Pullagura | Microfluidics-based on-demand generation of nonwoven and single polymer microfibers[END_REF][START_REF] Zhou | Continuous fabrication of near-infrared light responsive bilayer hydrogel fibers based on microfluidic spinning[END_REF] To observe the overall impact of the flow rate on the fiber diameter, the effect of Qs/Qc on the fiber diameter instead of individual flow rates was studied using PEG 300 as Φs. In

Effect of viscosity ratio

The effect of viscosity ratio on the fiber diameter was investigated. While keeping the viscosity of Φc constant (3.14 cP) for all the experiments, the viscosity of Φs varied in the following order: PEG 400/PEG 600 (101.11 cP) > PEG 400 (82.11 cP) > PEG 300 (61.35 cP) > PEG 200 (42.03 cP). The fiber diameter was affected by both Qs/Qc and viscosity ratio (ηs/ηc) (Fig. 4). Higher values of Φs viscosity (ηs) produced fibers of lower diameter for the same Qs/Qc attributed to an increased shearing force induced by a more viscous Φs resulting into the production of thinner core jet. Fig. 4 seemed to indicate that for a given Φs, the fiber diameter followed a linear decrease with the increase in ηs/ηc. Hence, ηs/ηc was also another dimensionless parameter that controlled the fiber diameter.

Capillary number ratio (Cas/Cac)

Capillary number is defined as:

Ca = η v/𝛾𝛾 (1)
where η and v are the viscosity and velocity of the fluid respectively, and 𝛾𝛾 the interfacial tension between the fluids in contact. It compares the viscous force (proportional to the shear stress) to the interfacial force (proportional to the interfacial tension) and appears as an appropriate dimensionless number to account for the effects of the fluid velocity (or flow rate) and viscosity as stressed out in the above two previous studies. Two capillary numbers can be defined, Cas= ηsvs/𝛾𝛾 and Cac= ηcvc/𝛾𝛾 for Φs and Φc respectively. For a given set of sheath and core fluids, the interfacial tension was the same and constant for both capillary numbers. Thus, one can define the capillary number ratio as: [START_REF] Serra | A predictive approach of the influence of the operating parameters on the size of polymer particles synthesized in a simplified microfluidic system[END_REF] Ca

s Ca c = η s .v s η c .v c (2)
This ratio can simultaneously accommodate the effects of ηs/ηc and Qs/Qc. The latter can be expressed as a function of the velocity ratio (vs/vc) since the inner cross sections of the capillary and outlet tubing were kept constant for all experiments. The sheath fluid velocity (vs) was evaluated by subtracting from the outlet tubing inner cross section the capillary outer cross section.

To correlate all the data presented in the previous sections in the form of dimensionless numbers, the fiber diameter (Df) was arbitrarily normalized by the inner capillary diameter (DI.D). After calculating the values of vs/vc from the investigated Qs/Qc, the capillary ratios (Cas/Cac) was accessed from Equation 2. The variations of the normalized fiber diameter (Df/DI.D) as a function of this capillary ratio (Fig. 5) was plotted for all the data presented in Fig. 4. All data fell down to a unique master curve highlighting the decrease of Df when the capillary number ratio was increased. The slight deviation for PEG 200 could be due to its lower viscosity allowing higher dilution of Φc into the bulk Φs. This trend complied with the separate reduction in diameter observed with an increase of Qs/Qc and ηs/ηc (Fig. 3 and 4 respectively). The same trend was reported for the production of polymer microparticle and oily droplets particles. [START_REF] Serra | A predictive approach of the influence of the operating parameters on the size of polymer particles synthesized in a simplified microfluidic system[END_REF][START_REF] Chiarello | Generation of oil droplets in a non-newtonian liquid using a microfluidic T-junction[END_REF] For the determination of a useful empirical correlation between dimensionless numbers allowing the prediction of Df as a function of operating and materials parameters, the data from Fig. 5 were represented in a logarithm plot (Fig. 6) (blue curve). From the linear variation of Ln(Df/DI.D) with Ln(Cas/Cac), the following correlation was extracted by linear regression:

𝐷𝐷 𝑓𝑓 𝐷𝐷 𝐼𝐼.𝐷𝐷 = 𝐾𝐾 0 ( 𝐶𝐶𝑎𝑎 𝑠𝑠 𝐶𝐶𝑎𝑎 𝑐𝑐 ) -0.63 (3)
with K0 = 3.71.

Positive values of Ln(Df/DI.D) in Fig. 6 indicated fibers which diameters were greater than the inner capillary diameter while the negative values indicated smaller diameters. It was observed that a wide range of diameters, smaller or greater than the size of capillary diameter can be obtained. In the above-mentioned experiments, K0 is a constant but could possibly vary with the capillary dimensions. 3).

Effect of outer and inner capillary diameter

The To assess more precisely this dependence, the different y-intercepts (i.e. Ln(K0)) values extracted from Fig. 6 were plotted as a function of Ln(DI.D) (Fig. 7) showing a linear dependence between Ln(K0) and Ln(DI.D). This dependence can be expressed by the following equation:

𝐿𝐿𝐿𝐿(𝐾𝐾 0 ) = 1.06 𝐿𝐿𝐿𝐿(𝐷𝐷 𝐼𝐼𝐷𝐷 ) -2.82 (4)
which can be rewritten as:

𝐾𝐾 0 = 𝑒𝑒𝑒𝑒𝑒𝑒 -2.82 𝐷𝐷 𝐼𝐼𝐷𝐷 1.06 (5)
Introducing 𝐾𝐾 0 into Equation 3 and taking the average value of the slopes in Fig. 6 (-0.65) gave:

𝐷𝐷 𝑓𝑓 𝐷𝐷 𝐼𝐼.𝐷𝐷 = 𝐾𝐾 1 𝐷𝐷 𝐼𝐼.𝐷𝐷 1.06 ( 𝐶𝐶𝑎𝑎 𝑠𝑠 𝐶𝐶𝑎𝑎 𝑐𝑐 ) -0.65 (6)
with 𝐾𝐾 1 = 0.06 µm -1 .

Extracting Df from above equation resulted in:

𝐷𝐷 𝑓𝑓 = 𝐾𝐾 1 𝐷𝐷 𝐼𝐼.𝐷𝐷 2.06 ( 𝐶𝐶𝑎𝑎 𝑠𝑠 𝐶𝐶𝑎𝑎 𝑐𝑐 ) -0.65 (7)

Effect of monomer volume fraction

The effect of Φc composition was investigated by varying the monomer volume fraction (Xv,m, Equation 8) from 10 to 80%.

𝑋𝑋 𝑣𝑣,𝑚𝑚 = 𝑉𝑉 𝑚𝑚 𝑉𝑉 𝑚𝑚 +𝑉𝑉 𝑠𝑠 (8)
where Vm and Vs are the monomer and solvent (ethanol) volumes in core phase respectiely The logarithm of Df followed a linear increase with respect to the logarithm of the volume fraction (Fig. 8) which was expressed as:

𝐷𝐷 𝑓𝑓 = 𝐾𝐾 2 𝑋𝑋 𝑣𝑣,𝑚𝑚 0.47 (9)
with 𝐾𝐾 2 = 66 µm.

As the monomer volume fraction increased, a larger portion of the monomer core phase volume was polymerized resulting in a thicker fiber. Experiments with Xv,m above 80% led to clogging due to the buckling effect resulting from the fast polymerization induced by the high concentration of monomer.

By replacing 𝐾𝐾 1 (Equation 7) by the right-hand side expression of Equation 9, the overall correlation could be expressed as:

𝐷𝐷 𝑓𝑓 = 𝐾𝐾 3 𝑋𝑋 𝑣𝑣,𝑚𝑚 0.47 𝐷𝐷 𝐼𝐼.𝐷𝐷 2.06 ( 𝐶𝐶𝑎𝑎 𝑠𝑠 𝐶𝐶𝑎𝑎 𝑐𝑐 ) -0.65 (10)
𝐾𝐾 3 was determined to be equal to 0.12 µm -1 by minimizing the mean squared error between all experimental Df measured and the fiber diameters returned by Equation 10 under same experimental conditions (Fig. S3). Equation 10

Effect of material system

Equation 10 was obtained with one specific material system, i.e. TPGDA as Φc and PEG as Φs.

To check the relevancy of this empirical relationship, two additional Φc were investigated:

PEGDA with a molecular weight of 400 g/mol and NOA 89 while keeping the same Φs (PEG 300). It was observed that if the slope remained the same irrelevantly of the material system studied, 𝐾𝐾 0 value (and subsequent 𝐾𝐾 3 ) changed noticeably (Table 1). The change was moderate for TPGDA and NOA 89 while it was much more pronounced for PEGDA. This must be related to the core phase viscosity (ηc, Table 1) which is twice higher for PEGDA than for the other two core phases investigated. Thus, one can make the hypothesis that PEGDA monomer diffuses less in the sheath phase. As a result, Df will be thinner which will be expressed by a lower diameter and thus a lower y-intercept.

Monodispersity and reproducibility of fiber diameter

The robustness of the process was proved by evaluating the monodispersity and reproducibility of Df. The coefficient of variation (CV), defined as the ratio between the standard deviation σ (Equation 12) of all the diameters measured and the average fiber diameter Df,avg, was evaluated (Fig. S5):

𝐶𝐶𝐶𝐶 = 𝜎𝜎 𝐷𝐷 𝑓𝑓,𝑎𝑎𝑣𝑣𝑎𝑎 � (12)
CV is generally reported to check against the monodispersity of micron-sized droplets or particles.

It is commonly admitted that CVs below 5% are the signature of monodisperse objects. It was extended to fibers and the monodispersity was evaluated by measuring the diameter at different random locations along the fiber length for different sheath and core phase flow rates and for a capillary with I.D = 50 µm and O.D = 150 µm. It remained below 5% in all situations indicating that the developed process allowed producing monodisperse fibers (Fig. S5a).

As for the reproducibility, it was evaluated by determining the average diameter of fibers obtained for three batches with the same operating and material parameters carried out on different days.

The results (Fig. S5b) indicated that whatever the sheath phase employed, the returned CVs were always lower than 5% highlighting the strong reproducibility of the process.

Conclusion

Polymer microfibers of two different monomers and one photocurable adhesive with diameters as low as 23 µm were produced by in situ photopolymerization using a capillary-based coaxial microfluidic device and a miscible PEG sheath fluid. The impact of different operating and material parameters such as volume fraction of monomer in core phase, flow rate ratio of sheath to core fluid, of the sheath fluid and dimensions of the capillary on the resulting fiber's diameter were studied. The diameter of the fiber was inversely related to the flow rate ratio and viscosity of the sheath fluid. The size of the fiber could also be varied by changing the inner diameter of the capillary but was not affected by the outer diameter. From all the data collected, an empirical relationship which precisely predicted the fiber diameter as a function of the capillary's inner diameter, monomer volume fraction and capillary number ratio of sheath to core fluid was successfully extracted. Furthermore, the coefficient of variation of the fibers' diameter was consistently less than 5% thus highlighting a robust and reproducible process for the production of monodisperse fibers.

Fig. 1 .

 1 Fig. 1. (A) Capillary-based microfluidic device used to produce microfibers: Φs delivered from

aFig. 2 .

 2 Fig. 2. Effect of Qs/Qc on the fiber diameter considering the effect of either the sheath flow rate

Fig. 3 .

 3 Fig. 3. Fiber diameter vs. Qs/Qc for different Φs (I.D = 50 µm, O.D = 150 µm, Φc =

Fig. 4 .

 4 Fig. 4. Fiber diameter vs. sheath to core fluids viscosity ratio using PEG 200, PEG 300, PEG 400,

Fig. 5 .

 5 Fig. 5. Master curve presenting the variation of the diameter ratio (Df/DI.D) with respect to the

Fig. 6 .

 6 Fig. 6. Ln-Ln graph of fiber diameter variation with respect to Cas/Cac for the experimental data

  effect of the outer capillary diameter (O.D) on Df was assessed using O.D of150, 190, 280 and 365 µm, while keeping the inner diameter (I.D) equal to 50 µm (Qc = 2.5 µL/min, while varying Qs from 275 to 700 µL/min). The logarithm of Df followed a linear variation with respect to the logarithm of Qs/Qc irrelevantly of O.D (Fig.S2). Furthermore, the linear regression equation was the same as the one found for the blue curve (I.D = 50 µm, Fig.6). It was concluded that the capillary outer diameter did not affect Df in the size range investigated. Inversely to the production of droplets, whose diameter is fixed when they detach from the capillary tip,[START_REF] Serra | A predictive approach of the influence of the operating parameters on the size of polymer particles synthesized in a simplified microfluidic system[END_REF] fiber diameter is mainly set downstream to the capillary tip when the shear stresses of Φs and Φc equilibrate. The effect of the O.D was irrelevant. Similarly, the effect of I.D was investigated using capillaries with the same O.D (150 µm) but different I.D (30, 40, 50 and 75 µm) with the same conditions for Qc and Qs as the previous study conducted with a constant I.D. Varying I.D resulted in a set of parallel lines (same slopes around -0.63) but exhibited different y-intercepts as a function of I.D (Fig. 6) indicating that I.D played a role on the resulting Df through the K0 parameter of Equation 3.

Fig. 7 .

 7 Fig. 7. Variation of the y-intercept (Fig. 6) with respect to logarithm of I.D (O.D = 150 µm).

Fig. 8 .

 8 Fig. 8. Variation of the fiber diameter with respect to the monomer volume fraction (I.D = 50 µm,

Table 1 .

 1 Values of the slope, y-intercept for Ln(Df/DI.D.) vs. Ln(Cas /Cac) plots (Fig.S4) and viscosity using different core phases (I.D and O.D of 50 µm and 150 µm respectively).

	Core phase	Slope	y -intercept	K0 (µm -1 )	ηs (cP)
	TPGDA	-0.628	1.31	3.71	3.14
	NOA 89	-0.688	1.24	3.46	3.5
	PEGDA	-0.622	0.80	2.23	7.04

Acknowledgments

Authors thank W. Drenckhan for discussions and A. Collard for all technical aspects. Authors are grateful to P. Allgayer for mechanical engineering support. WR would like to acknowledge the Higher Education Commission of Pakistan (HEC) for his Ph.D. fellowship.