# Heterogeneity of food safety standards and exporters' strategies: what conditions for the integration of developing country producers into international trade?

Lamia Rouached, Maria Angela Perito, Abdelhakim Hammoudi

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Lamia Rouached, Maria Angela Perito, Abdelhakim Hammoudi. Heterogeneity of food safety standards and exporters' strategies: what conditions for the integration of developing country producers into international trade?. 2025. hal-05302060

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# Heterogeneity of food safety standards and exporters' strategies: what conditions for the integration of developing country producers into international trade?

Lamia Rouached<fnref n="1" />, Maria Angela Perito², Abdelhakim Hammoudi³

¹ Laboratoire d'Économie et de Gestion Industrielle, École Polytechnique, ISG de Tunis, University of Tunis, Tunisia

²France Faculty of Bioscience and Technology for Food, Agriculture and Environment, University of Teramo, R. Balzarini 1, 64100, Teramo, Italy

³ University of Paris-Saclay, AgroParisTech, INRAE, PSAE, Palaiseau,

## Abstract

In this article, we discuss the role played by exporters of various international food chains in emergence of private standards and access of developing countries in international markets. We show that exporters' strategies can directly or indirectly affect the level of developing producer' participation (or exclusion) in export sectors. We show how the exporters adopt a strategy depending on market opportunities and the type of vertical relationships (upstream and downstream) along the international supply chains.

**Keywords:** food safety, heterogeneity, exporters strategies, international trade, developing countries' producers

JEL classification: Q18, L1, D81

## 1 Introduction

During the last decade, consumer perception on food safety has been increasing worldwide, following major food crises (Li et al., 2017). To re-establish consumer trust, food standards have become an increasingly powerful tool for the policy maker to ensure the food safety. Private operators (producers, processing, retailers...) seek likewise to meet consumer requirements in terms of food safety through the emergence of a plethora of private safety standards<fnref n="1" />. These latter standards can be even more demanding than public standards (see for

<footnote n="1">¹ 60% of world trade in food-related products is affected by at least one of food safety standards (Cadot and Gourdon, 2016).</footnote>
example Fulponi et al., 2006; Starbird, 2005). Several studies showed how standards in the international markets are heterogeneous and exporters, especially those of developing countries, faced with this heterogeneity (Maskus et al., 2013; Breinlich and Criscuolo, 2011; Helpman et al., 2008; Bernard et al., 2007). As pointed out by OECD (2007), the fruit and vegetable exporters from developing countries are confronted with public standards (GAPs, maximum residue levels of pesticides and traceability) and an extremely larger number of private standards. In the European market, for example, there are various types of safety and environment quality standards depending on the type of market (wholesalers, supermarkets, types of retailers, etc.) (OECD, 2007), countries (OECD, 2007)<fnref n="2" /> and governance of agro-food value chains (see e.g., Russo et al., 2014; Hammoudi et al., 2009; Minten et al., 2006). This heterogeneity has a great impact on private operators of various international value chains and modify their production and management policies, as well as the organization and strategy of the entire supply chain (Rouviere et al., 2010; Malorgio et al., 2016).

At the international level, exporters face markets with various characteristics: (i) market which is relatively not very demanding about the production practices; (ii) markets with more (or less) private standards (iii) markets characterized by more (or less) reliable control systems (Malorgio et al., 2016; Nait-Mohand et al., 2017). Exporters are therefore confronted on the market with the heterogeneity of standards as well as the heterogeneity of supply chains and operators. Moreover, there are several types of international chains depending on the structure, organization and governance of food safety (Lee et al. 2010; Henson and Jaffee, 2008; Gereffi et al. 2005; Rouviere et al., 2010). Lee et al. (2010), for example, show how the export system for fruit and vegetables from Africa to the UK has been progressively geared towards governance types based on the highest level of vertical co-ordination rather than on spot market transactions. Upstream managerial activities influence downstream food quality (Zúñiga-Arias et al, 2009) and the amount of food quality standards. Thus, as Lee et al. (2010) and Gereffi et al. (2005) point out, customers (supermarkets) have progressively developed stronger relationships with exporters from African countries. These relationships occur through increasingly stringent contracts, in particular with respect of obligation of means (good hygienic practices and good agricultural practices) and implementation of inspection and control systems for suppliers (Okello et al. 2011; Gereffi et al. 2005; Singh, 2002). A number of field studies have analysed the role of vertical coordination on the actors' performance in developing

<footnote n="2">² In general, British supermarkets require a high level of private standards. In Germany, the Netherlands and Switzerland, supermarket chains also require private standards, but these standards are less stringent or minimal (OECD, 2007, p. 21).</footnote>
countries (Briz et al., 2000; Malorgio et al., 2016)<fnref n="3" />. Other chains are vertically independent,
with local actors who take on their own adaptation costs: neither a direct contribution from
exporters in a contractual framework nor an indirect contribution through the purchase price
(i.e., a fair distribution of the value). Several studies show how this type of organization and
structure are characteristic of the spot market, in which safety governance is based mainly on
public standards (Jaffee, 2003).

Exporters are often key players in international value chains. They establish supply strategies and choose their marketing channels according to quantitative purpose (exporting the largest possible volumes) and qualitative purpose (high quality level of production practices and compliance with health and safety standards and regulations). Their choices (in terms of quantity and quality) have a direct impact on the risk of rejection of products at the boundaries of developed countries and exclusion of producers from the markets. This implies that exporters strategies may be compatible or incompatible with the general interest of developing countries: impact on the credibility of a sector/country and risk of bad reputation that should have an impact on the country's macroeconomic outcomes, added value shared with upstream operators, minimizing the exclusion of producers, etc. According to Jouanjean et al. (2012), for example, reputation plays an important role in driving border rejections: exporters with a previous reputation of non-compliance with food safety regulation increase the chances of being refused entry into the US market.

Despite the very rich scientific literature on standards, few theoretical works have been devoted to the strategic role played by exporters of various international food chains, in the access of developing countries products to international markets.

We try to fill this gap proposing a theoretical model of industrial economics. One of the most important goals of our work is to understand the impact of exporters' strategies on the performance of the agro-food export sectors of the developing countries. In particular, the proposed analysis allows us to understand how exporters' strategy (standard/market) can directly or indirectly affects the levels of participation (or exclusion) of producers in developing countries.

Moreover, our hypothesis is that the exporter's strategy (in terms of importing market, quantity and the choice of upstream suppliers) depends on market opportunities and the type of vertical

<footnote n="3">³ The degree of organization and coordination could contribute to the success of the compliance process of these export chains, and Morocco is generally taken as an example (see for example, Garcia Martinez and Poole, 2004; Hellin et al., 2008).</footnote>
relationships (upstream and downstream) along the supply chain. We highlight two types of exporter' profiles based on i) the impact of their strategies on the dynamics of upstream farming practices and ii) the characteristics of international chain in which the exporter belongs.

## 2 The model

### 2.1 A representation of the heterogeneity of producers and the characteristics of the production system of developing countries

We develop a vertical relationship model between a group of local producers in the developing country and an exporter. Each producer has an identical production capacity $q$. We assume that producers are heterogeneous and differ in their initial characteristics such as infrastructures, technical know-how, efforts and hygiene practice on the production site. These characteristics are aggregated on a synthetic index “equipment” in production practices reflecting the degree of good agricultural practice (GAP) initially applied by the producer. As in Giraud-Héraud et al. (2010), we suppose that producers differ according to their level of equipment, which is represented by the one-dimensional parameter $e$. We assume that the available production practices are uniformly distributed on the interval $[\underline{e}, \bar{e}]$, according to a uniform probability density function $f(e) \equiv 1$. For simplicity, we pose $\underline{e} = 0$ and $\bar{e} = 1$.

Giraud-Héraud et al. (2010) analyse the issue of interactions between food safety regulations and market strategies of private actors. In this perspective, the private strategy is only used to select suppliers of products compliant to the public standard. One limit of the paper is that it didn't consider future consequences of non-compliance in relationships in trade market. We propose to extend the field of analysis to the international context. Moreover, we introduce an additional step to the game. In the event of a finding of non-conformity during the control of the first period, a possible penalty is applied to the production of the following period. Our modeling work presents new perspectives in the face of the heterogeneity of national quality standards.

Exporting to the market of an importing country requires a minimum level of equipment, which corresponds to a minimum quality standard $e_1$ set by the importing market. We assume that countries are heterogeneous in terms of level requirement reflecting various private or public
standards imposed internationally. According to the chosen importing market, the standard
level $e_1$ is imposed by the exporter to his suppliers.

If a producer with equipment *e* comply with the minimum quality standard $e_1$, he has to pay a
fixed cost given by $Max\{0, F(e_1, e)\}$. More specifically, the compliance cost of a producer is
0 if his initial equipment *e* is greater than the standard $e_1$ and $F(e_1, e)$ if not (Fig. 1). The
compliance cost for an initial poor equipment increases in range between the minimum standard
$e_1$, and his initial equipment level *e*. The compliance cost function increases in $e_1$, decreases in
*e*, and we assume that is linear in $e_1$ and in *e*. Given that producers are heterogeneous in their
initial equipment, the minimum investment required to comply with the standard $e_1$ is different
from one producer to another<fnref n="4" />.

<figure><img src="image_2.png" /><figcaption>Figure 1 – Relationship model between local producer and exporter</figcaption></figure>

<footnote n="4">⁴ We assume that $F(e_1, e)$ is given by $F(e_1, e) = (e_1 - e)$.</footnote>
We assume that the compliance cost can be shared among stakeholders (producer group and exporter), and the exporter pays a level $\alpha$, with $0 \le \alpha \le 1$. A higher level of this parameter indicates a greater involvement of the exporter in the process of improving production practices.

### Representation of the risk of rejection associated with the production practices

We start from the idea that the compliance with a certain level of production practice does not guarantee the absence of rejection at the borders of the importing country. However, we deal with the assumption that result requirements are positively correlated to means requirements. While standards targeting production' processes are measures of means requirements (various national or international Good agricultural practices (GPA) guides, private standards such as GlobalGap or BRC, etc.), standards relating to the final product are characterized by result requirements (for example, microbiological thresholds, pesticides, heavy metals, etc.), generally imposed through regulatory measures at importing markets. In this regard, we assume that compliance with means requirements affects the quality of the final product and thus the likelihood of compliance with standards. Accordingly, the improvement of the production process through the adoption of an increasingly restrictive “means standard” reduces the probability of rejection of the product without, however, eliminating this risk. In this sense, the compliance with the minimum quality standard as a particular case of means requirements is not a sufficient condition for access to the importing market. Fulfilling this threshold is a condition of ex-ante participation in exports without prejudging the ex-post access.

To model the risk of rejection at the border of the importing country, we assign a level of risk to the equipment of each producer. We assume that the risk of rejection, for the producer (and his product), follows the probability of rejection and its distribution admits a probability density function $\sigma(e)$, supposed known to the producer and to the exporter. The function $\sigma(e)$ is decreasing in $e$ and given by$^5$ $\sigma(e) = (1 - e)$. The risk of rejection is linked to a given producer's equipment: better is the production practices of the producer, fewer is the risk of rejection of his product. That is, only the producer characterized by the best equipment ($e = 1$) has an individual risk of rejection equal to zero. Since the individual supply capacity of each producer is limited to $q$ and the supply is heterogeneous in equipment, the exporter has to buy from producers less equipped than the “ideal” equipment ($e = 1$) in order to increase the quantity sold on the market. Hence, there is a strictly positive total risk of rejection obtained from all selected producers which can be expressed as follows: $\bar{\sigma}$. On this basis and as regard

<footnote>$^5$ See Giraud-Héraud et al., (2010)</footnote>
to $f(e) \equiv 1$, the probability distribution of equipment changes if at least one of the producers improves its initial equipment.

## 2.2 Representation of downstream and vertical relationships

Consider an importing market where the demand depends on the exogenous market price P. The exporter is a price taker, and he is able to supply entire quantity bought from the producer $J$ as long as the product complies with the requirements of the importing market. The exporter buys the product at the intermediate price $\omega$, where $\omega = 0$.

We consider that each rejected product generates a marginal cost which may represent various costs associated with rejections: proportion of the rejection penalty paid by the exporter (possible transfer, partial or total of the rejection penalty from the importer to the exporter), estimated costs generated by a future delisting by the customer, proportion of the cost of destruction of the product or re-shipment paid by the exporter etc. We assume that this estimated cost, denoted by $\Gamma$ depends on the balance of power between the exporter and his customer and on the alternative supply choices available to the customer on the market (other potential suppliers of the product traded). This parameter is therefore an indicator of the power relationship in the supply chain, between the exporter and the importer. Let us denote by $X$, the total quantity ordered from producers by the exporter to cover the two periods, $X = 2Q$. Thus, in each period, the average risk associated with the supply of the producers selected by the exporter was designated by $\bar{\sigma}$.

For instance, if we consider $\Gamma$ as penalty associated to delisting in case of rejection, a high level of $\Gamma$ deals with a type of exporters more concerned about their reputation towards their customers. For such level of $\Gamma$, this type of exporters anticipates having to undergo future delisting associated with the potential rejection. This is the case, for example, for exporters who are involved in relatively durable relationships with big industrial food companies or with GPO. Alternatively, a low level of $\Gamma$ deals with a type of exporter whose potential loss associated to delisting is relatively low because their customers are mainly wholesale markets (see, Malorgio et al., 2016).

In this setting, a chain $\Phi(\Gamma, \alpha)$ is characterized by the set of parameters $(\Gamma, \alpha)$. These parameters are known by the exporter when he takes his strategic decisions: choice of an importing country (i.e., a certain level of standard) and a quantity to sell in this market.
The capacity of the upstream producers is given by $Jq$. Let $\tilde{e}$ denote the equipment threshold at which producers are selected by the exporter. Insofar as the probability distribution of the upstream producers is uniform, we can deduce the supply function as $(1 - \tilde{e})$. Under these conditions, the difference between supply and demand leads to the following equation $Q = Jq(1 - \tilde{e})$, with $Q \le Jq$. The expression of the threshold $\tilde{e}(Q)$ is thus given by:

$$ \tilde{e}(Q) = 1 - \frac{Q}{Jq} \qquad (1) $$

Let us define the benchmark, as the situation where no standard is required to enter the importing market ($e_1 = 0$). In this referenced situation, the supply capacity initially complains with the standard $e_1$ and the production capacity corresponds to $Jq$. The exporter selects the producers located between 0 and 1. As noted previously, producers are uniformly distributed over the interval $[0,1]$, so when $e_1 = 0$ and according to the density function, $f(e)$, for $\tilde{e} < e < 1$, $f(e) = \frac{1}{1-\tilde{e}}$

Knowing that $\tilde{e} = \tilde{e}(Q)$, in the benchmark situation, the average risk of rejection $\tilde{\sigma}(Q)$ is given by:

$$ \bar{\sigma}(Q) = \int_{\bar{e}}^{1} \sigma(e)f(e)de = \frac{Q}{2Jq} \qquad (2) $$

Thus, the average risk of rejection is an increasing function of the quantity $Q$ ordered by the exporter.

## 2.3 The game

We consider that the exporter and the producer (or a group of producers) play a game which is defined by:

* Stage I. The exporter chooses the importing market (i.e., a targeting standard level $e_1$).

* Stage II. The exporter chooses a quantity $X$ of product to be ordered from producers (or members of producers group) and propose to buy this quantity with price $\omega$ and to finance with a proportion $\alpha$, the compliance cost of the producers initially non-compliant with the defined standard. The total quantity $X$ will serve the two periods.
* Stage III. At the border of the importing country, checks are carried out in each period.

Products are rejected or not and if the products are rejected, the exporter recovers the *P* value. Moreover, delisting can be applied if the products are rejected in the first period.

## Typologies of exporter supply strategies and effects on rejection risk

Consider $Q = Jq(1 - \tilde{e})$ the quantity ordered by the exporter and totally sold on the importing market in case of non-rejection. Consider $\tilde{Q} = Jq(1 - e_1)$, the available supply, initially compliant with the standard $e_1$. Let us call this quantity, "eligible capacity". It may be noted that when the standard $e_1$ is very restrictive, the eligible capacity shrinks. Thus, supply is equal to the quantity ordered by the exporter when he selects all producers initially compliant with the standard $e_1$ ($\tilde{e} = e_1$). Using (1), one can check that $\tilde{e} \ge e_1$ when $Q \le \tilde{Q}$. In other words, if $Q \le \tilde{Q}$, the eligible quantity is sufficient to satisfy the exporter's demand. In this case, there is no need to involve producers in a compliance process. Otherwise ($Q \ge \tilde{Q} \Leftrightarrow \tilde{e} \le e_1$), the exporter's demand exceeds the eligible capacity. In order to meet the demand $Q$ of the destination country (i.e., to complete the eligible capacity), the exporter has to complete the available eligible quantity by selecting some producers that are not equipped (located between $\tilde{e}$ and $e_1$). These producers are required to engage in the compliance process where the exporter covers a proportion $\alpha$ of the cost.

The types of the exporter's supply strategy can thus be defined according to their impact on the upstream supply in terms of equipment improvement. These types are illustrated in the following Definition 1.

**Definition 1.** An exporter's strategy is called "Neutral effect strategy" (NES) if no producer has to improve his equipment in order to enter the destination market ($Q \le \tilde{Q}$), this condition is due to the good equipment of the producers. Similarly, it is called "Positive effect strategy" (PES) if at least one producer has to improve his equipment in order to enter the importing market ($Q > \tilde{Q}$).

Definition 1 illustrates the link between the exporter's choice in terms of the quantity ordered and the effect of this choice on the equipment change in the upstream supply chain, taking into account the standard level. Only sufficiently large enough quantity may contribute to the emergence of PES. The exporter's choice of quantity may or may not affect the statistical distribution of producers over the interval $[\tilde{e}, 1]$ and then, the average risk of rejection. In the
case of NES, the distribution of equipment over the interval [ẽ, 1] remains unchanged and the average risk of rejection is given by (2). In the alternative case of PES, the distribution of the equipment over the interval [ẽ, 1] is given by:

$$g(e) = \begin{cases} \frac{e_1 - ẽ}{1 - ẽ} & \text{if } e = e_1 \\ \frac{1}{1 - ẽ} & \text{if } e_1 < e \le 1 \end{cases} \quad (3)$$

with $\int_{ẽ}^{1} g(e) d_e = 1$. It is clear that over the interval [0, ẽ], $g(e) = 0$.

Considering the equation (1) and system (3), the total risk of rejection $\bar{\sigma}(e_1, Q)$ can be written as:

$$\bar{\sigma}(e_1, Q) = \int_{e}^{1} \sigma(e) h(e) d_e = \begin{cases} \frac{Q}{2jq} & \text{if } Q \le \hat{Q} \\ (1 - e_1) \left[1 - \frac{jq}{2Q}(1 - e_1)\right] & \text{if } Q > \hat{Q} \end{cases} \quad (4)$$

Where $h(e) \equiv f(e)$ if $Q \le \hat{Q}$ and $h(e) \equiv g(e)$ if $Q > \hat{Q}$

Using (4), one can verify that the total risk of rejection is an increasing function of the quantity Q expressed by the exporter for a given $e_1$. An increase in the quantity ordered implies selecting producers who are less equipped compared to the ideal equipment $e = 1$ which leads to the increase in the risk of rejection. At a given quantity, the risk is independent of the standard in the context of a neutral strategy and decreasing on $e_1$ when the exporter adopts a positive strategy.

## Total compliance costs

It might be necessary to distinguish between the two types of the exporter's strategy (positive and neutral). The positive strategy, contrary to the neutral strategy, induces additional compliance costs. As we have already mentioned, such a strategy requires that each producer is located between ẽ and $e_1$ perform an adaptive effort the cost of which is measured by the function $F(e_1, e)$. The total cost F induced by the compliance process of all producers located between ẽ and $e_1$ is given by:

$$F(e_1, ẽ) = \begin{cases} 0 & \text{if } Q \le \hat{Q} \\ \frac{J}{1 - ẽ} \int_{ẽ}^{e_1} F(e_1, e) d_e & \text{if } Q > \hat{Q} \end{cases} \quad (5)$$
Now, using relation (1), the total compliance cost F can be expressed as a function of the standard $e_1$ and the ordered quantity Q such as:

$$F(e_1, Q) = \begin{cases} 0 & \text{if } Q \le \hat{Q} \\ \frac{[Q - Jq(1 - e_1)]^2}{2Qq} & \text{if } Q > \hat{Q} \end{cases} \qquad (6)$$

In the interval of quantity where the positive strategy is ordered, expression (6) shows that the total compliance cost $F(e_1, Q)$ is increasing in $Q$ and in $e_1$. Remember that the compliance cost is partially borne by the exporter in proportion $\alpha$.

Knowing the price P, the total compliance cost F given by (6) and the total risk of rejection given by (4), the exporter's profit is:

$$\pi_1(e_1, Q) = [1 - \bar{\sigma}(e_1, Q)][2 - \bar{\sigma}(e_1, Q)\Gamma]PQ - \alpha F(e_1, Q) \qquad (7)$$

This function consists of three expressions: $2[1 - \bar{\sigma}(e_1, Q)]$ which is the expected unit margin associated to a non-rejection case over the first or a second period. The second expression, $[1 - \bar{\sigma}(e_1, Q)][\bar{\sigma}(e_1, Q)\Gamma]$ which represents the expected unit loss of delisting and $\alpha F(e_1, Q)$ characterizes the proportion of the total compliance cost financed by the exporter in the positive strategy case. In other words, the exporter faces a trade-off between the effort to obtain a quantity $Q$, in accordance with the requirements of the destination market, and the potential loss associated to the risk of rejection.

## Exporter profile, rejection and producer exclusion

In this section, we consider the exporter's supply strategy and we focus on its effects on the risk of rejection and on the exclusion rate of upstream producers.

First, we determine the import's optimal supply strategy ($Q^*(e_1)$) for a given value of the standard $e_1$. We then analyze the influence of the characteristics of the supply chain $\Phi(\Gamma, \alpha)$ on $e_1$. We show how the potential loss associated with delisting, affects the type of strategy chosen by the exporter (see definition 1) and in particular, affects its motivation to contribute to the compliance process of the producers. More precisely, we define a critical loss threshold $\hat{\Gamma}$ below which the exporter has an interest in engaging in the equipment improvement process. The expression of this critical threshold is given by:
$$ \hat{\Gamma} = \frac{8e_1}{1+e_1(2-3e_1)} \qquad (8) $$

We can thus define the emergence condition of each type of procurement strategy (positive or neutral) according to the exporter profile and its position in relation to the threshold expressed by (8). Proposition 1 illustrates the condition determining either positive strategy or neutral one.

*Proposition 1. The exporter chooses a passive strategy if and only of $\Gamma \ge \hat{\Gamma}$ (otherwise, he chooses a positive strategy).*

Proposition 1 shows that only for relatively low levels of potential loss in delisting, the exporter has an interest in contracting beyond the eligible capacity and then sharing the costs associated with the improvement process of production practices<fnref n="6" />.

The threshold $\hat{\Gamma}$ denotes that exporter setting up the positive strategy is sensitive to the variation of the parameter $e_1$. Hence, the impact of these parameters on the exporter trade-off between the risk of rejection and the compliance costs associated with a quantity $Q$ should also be taken into account. In particular, we show that the threshold is positive, increases (and convex) in $e_1$. The range on positive strategy (all the selected producers that help exports comply) could be widened if these variables changed.

Let us now consider the effect of the standard on the exporter's procurement strategy. As long as the targeted markets have relatively low-standard ($\Gamma \ge \hat{\Gamma}$), the eligible capacity exceeds the quantity $Q$ which the exporter has to sell. Then, the exporter has no interest in contracting beyond this limit. As such, whatever the level of the standard, the exporter orders the same quantity $(2Jq \frac{(2+\Gamma-\sqrt{4-2\Gamma+\Gamma^2})}{3\Gamma})$, constant in $e_1$ and favors the best producers (namely, the least risky) since the loss in delisting remains relatively high. In the opposite case where the delisting loss is relatively low ($\Gamma \le \hat{\Gamma}$), the eligible capacity is insufficient to meet the quantity requested $Q$ of the final market. Thus, the exporter is involved in the process of improving practices. As a result, the quantity ordered depends on the level of the standard $e_1$.

We show (see annex 1) that the optimal quantity is not monotonic function of $e_1$. Moreover, the production capacity is not exceeded when $e_1 \le \bar{e}_1$. The variation of the quantity associated with the positive strategy can be understood through the outcome of two opposite effects. Indeed, at a given quantity $Q$, a higher standard implies:

* a lower risk of rejection and, consequently, a higher expected margin and a lower expected loss of delisting. These incite the exporter to increase the quantity ordered

<footnote n="6">⁶ All mathematical details and proofs of the different propositions are given in the Annex 1.</footnote>
when targeting increasingly demanding markets (*effect i*). This first effect is sensitive
to variations in parameters *P* and Γ. More precisely, the higher the price *P* is, higher the
loss associated with delisting Γ is.

* an increase in the total financing *F* (proportionately borne by the exporter in a fraction *α*) required by the compliance of the initially under-equipped producers and thereby, meeting the quantity *Q* for each period. This situation encourages the exporter to reduce the quantity ordered when targeting increasingly demanding markets (*effect ii*). This second effect is sensitive to variations in the proportion *α* (by unit produced) of the amount *F* supported by the exporter.

In sum, the model parameters, such as market price and sector characteristics play an important
role in the trade-off outcome of these two effects. For instance, *effect i* dominates *effect ii*,
leading to an increasing quantity with respect to *e*₁, when the price is high, the loss of delisting
is high, or the proportion of the compliance cost assumed by the exporter is low.

Let us qualify the increasing reaction of quantity with the standard as "risky", due to the effect
it may have on the risk of rejection. The relationship between rejection risk and standard is
given by the following Proposition 2.

*Proposition 2. The risk of rejection does not necessarily decrease in e₁.*

Proposition 2 shows that compliance with increasingly demanding standards does not
necessarily yield a reduction in the risk of rejection at the border of the destination country.

We show that when the exporter adopts a neutral strategy ($\Gamma \ge \hat{\Gamma}$), the quantity ordered is constant with the standard so that no producer modifies his equipment to reach the exporters market of destination. It follows that the risk of rejection does not depend on the level of the standard.

Now, when the exporter adopts the positive strategy ($\Gamma \le \hat{\Gamma}$), it may be in his best interest to
increase the quantity ordered *Q* regarding the standard when the loss of referencing is high.
Indeed, as the exporter targets relatively high-standard markets, he will be more motivated to
implement a compliance process including a growing number of producers. Market
opportunities (represented by high price *P*) or a high loss associated to delisting are
characteristics (represented by high threshold $\tilde{r}$), which matter in this compliance process.
However, these motivations push the exporter to increase the quantity in relation to the level of
the standard, can be slowed down by the effect on the rejection risk. In particular, this "risky"
behavior of the exporter may generate an increase in the rejection risk with regard to the level of the standard. Two effects explain this result:

* On the one hand, for a given quantity, encouraging the compliance of producers' practices with relatively high-standard markets reduces the risk of rejection (effect 1),

* On the other hand, for a given standard, the exporter's strategy of increasing the quantity increases the risk of rejection (effect 2).

Here, the two effects work against one another so that when effect 2 outweighs effect 1, the risk of rejection increases.

Conversely, when the exporter decreases the quantity ordered under the active strategy (with regards to the standard), the joint action of increasing the standard and reducing the quantity ordered leads to a reduction in the risk of rejection.

We now examine the consequences of exporter's type on producer exclusion. We show that the exporter' strategic behavior also affects the rate of exclusion of upstream producers, as displayed in the following Proposition 3.

**Proposition 3.** The rate producer exclusion does not necessarily increase in $e_1$.

According to Proposition 3, targeting markets with increasingly restrictive standards does not systematically imply that this market is unaffordable.

The exclusion rate is measured here by the minimum level of equipment $\tilde{e}$ from which producers are selected by the exporter. From the equalization of supply and demand ($Q = Jq(1 - \tilde{e})$), the choice of quantity ordered $Q$ results in a proportion of selected producers from the $J$ producers (or the same way, in a proportion $\tilde{e}$ of the excluded producers).

Let us first consider the neutral strategy. The level of equipment required is denoted $\hat{e}$. In this case and, as already seen, the optimal procurement strategy involves an order that is less than or equal to the eligible capacity ($Q^*(e_1 = \hat{e}) = \hat{Q}(e_1 = \hat{e})$) so that all (and only) initially compliant producers are selected ($\tilde{e}(e_1 = \hat{e}) = e_1$). The value of this critical level of equipment can be written as:

$$ \hat{e} = \frac{-4+\Gamma+2\sqrt{4+\Gamma(-2+\Gamma)}}{3\Gamma} \qquad (10) $$

First of all, we show that in the context of a neutral strategy ($\Gamma \ge \hat{\Gamma}$), the level of demand on the exporter market has no effect on the proportion of producers selected by the exporter, which is thus constant in $e_1$. As mentioned above, it is straightforward to see that the exclusion rate is given by $\tilde{e} = \hat{e}$.
Turning to the positive strategy, two extreme cases are to be distinguished. When the quantity ordered increases in order to satisfy increasingly demanding markets (high prices coupled with a high loss associated with delisting), the exclusion rate of upstream producers decreases in $e_1$. This result can be explained by the exporter interest to increase the quantity ordered, which is reflected in his involvement in bringing a growing proportion of producers into compliance. One can verify that within this framework: $\tilde{e} \le \hat{e}$. Thus, as soon as the standard exceeds the critical threshold by triggering the active strategy, the exporter selects a proportion of compliant producers in $e_1$ which is higher than the proportion selected in a neutral strategy.

Alternatively and still under the active strategy, when the exporter decreases the quantity ordered, the producer exclusion rate increases in $e_1$. Thus, exceeding the threshold of the standard, the exporter selects a decreasing proportion of producers (with respect of $e_1$ which is lower than the one selected in the passive strategy).

## Strategic choice of the exporter market

In this section, we analyze the best exporter market that the exporter may choose at stage I of the game. We examine how this strategic choice changes according to the parameters of the model. We thus determine the conditions such that an active or passive strategy emerges at this stage of the game. We give the standard level chosen by the exporter in the following Proposition 4.

**Proposition 4.** If $P < \tilde{P}$, the exporter targets the least demanding market ($e_1^*=\hat{e}$) and sets up a passive strategy. If $P > \tilde{P}$, the exporter targets the most demanding market ($e_1^*=1$) and adopts an active strategy.

If the loss associated with delisting $\Gamma$ is relatively low, the profit of the exporter is maximized in $e_1 = \hat{e}$. In this case, the exporter targets the least demanding market selects the best producers (i.e. least risky) by adopting a neutral strategy. We now analyze the effect of this choice on the producer exclusion. As mentioned above, when $P < \tilde{P}$ the threshold $\tilde{e}(e_1)$ at which producers are selected is first constant in $e_1$ if $P \ge \tilde{P}$ and increases in $e_1$ if not.

Therefore, the choice of the least demanding market generates a minimum level of exclusion in relation to all other possible markets. In this case, the exporter prefers targeting a relatively low
standard market and selling large quantities, rather than getting involved in the compliance of farms. Although this strategy results in a minimum exclusion rate, it involves a relatively high risk of rejection.

On the other hand, if the price P (coupled with a high loss associated with delisting Γ) is relatively high $P > \tilde{P}$, the profit of the exporter is maximized in $e_1 = \text{Inf}\{\bar{e}, 1\}$. In this case, he targets the most demanding market and implements an active strategy by becoming involved in the compliance process of the under-equipped producers. Accordingly, the threshold $\tilde{e}(e_1)$ is constant in $e_1$ if $\Gamma \ge \hat{\Gamma}$ and decreases in $e_1$, $\Gamma < \hat{\Gamma}$. Thus, the choice of the most demanding market leads to the minimum level of exclusion with regard to all possible markets. It is easy to verify that the minimum level of equipment corresponding to the optimal target $e_1^* = 1$ is given by $\tilde{e}(e_1^* = 1) = 1 - \frac{q}{\alpha}P$. We can verify that $\tilde{e}(e_1^* = 1) \le \hat{e}, \forall P \ge \tilde{P}$.

In this context, the exporter prefers to target the most demanding market even if this requires his commitment to participate in the compliance of initially under-equipped producers. The choice of the market with the highest requirement allows him to minimize the rejection risk while supplying the maximum quantity. This results in a minimal exclusion rate.

Although both strategies imply a minimal exclusion rate in all possible markets, it should be noted that only targeting the most demanding one, generates minimal exclusion coupled with improved production practices at the upstream level (positive strategy). Moreover, one can verify that targeting the most demanding market induces a lower exclusion rate compared to the exclusion rate associated with the choice of the least demanding market:

$$\tilde{e}(e_1^* = 1) < \tilde{e}(e_1^* = \hat{e}) \quad \forall P \ge \tilde{P}.$$

It is important to note that the emergence of one or the other strategy depends on the parameters of the model: price, proportion of the price paid to producers, loss associated with dereferencing and proportion of compliance costs. It particularly depends on the influence of these parameters on the trade-off between the reduction of rejection risk and the exporter participation in the upstream producers' compliance effort. In particular, two situations may emerge:

*Situation 1.* The exporter targets the least demanding importing country and selects initially equipped producers while simultaneously inducing a minimum exclusion rate among all the other possible market. This case occurs when the price is relatively low, the loss associated with delisting is low and the proportion of compliance cost paid by the exporter is high. This case concerns, in particular, exporters who supply wholesale markets or exporters who are not linked
to customers through contractual and sustainable relationships. Under these conditions, it is in
the interest of the exporter to choose low demanding import markets and to benefit from a large
initially eligible supply capacity. This implied being able to sell large quantities without being
involved in bringing producers into compliance.

*Situation 2*. Exporter targets the most demanding import market and accepts to be involved in an upstream producers' compliance process. This situation results when the price is relatively high, the loss associated to delisting is high (exporters linked to customers by contractual relationships) and the proportion of the compliance cost borne by the exporter is low. Moreover, we show that in this context, the exporter's strategic choice induces a minimal exclusion rate in regards to all the possible markets. In addition, the exclusion rate is lower than that generated in the *situation 1*. This result allows us to highlight how targeting destination markets with high standards does not necessarily reduce the participation level of producers in developing countries. This point contrasts the results of the literature that attempts to show how integration into high value-added international chains may not be compatible with the interests of developing countries in terms of producer inclusion rates (Anders, Caswell, 2009; Hoekman and Nicita, 2011; Tran, et al., 2012; Wilson et al., 2003), especially for small-scale farmers, especially in developing countries (Asfaw et al. 2010).

In addition, when the exporter targets the most demanding market, the minimum level of exclusion is associated with a minimization of the rejection risk due to improved producers' practices.

**Conclusion**

Many studies show how technical standards and regulations can constitute a barrier to trade, sometimes even larger than that generated by tariffs and quantitative restrictions (Laird and Yeats, 1990; Vogel, 1995; Jongwanich, 2009). Some authors, conversely, indicate that standards have no impact at all on exports from developing countries (Xiong and Beghin; 2013). Some studies illustrate that standards and their systems of compliance can play in important role in governing global value chains (Islam, 2008; Tallontire et al. 2011).

Previous studies, particularly the paper by Schuster and Maertens (2015), highlight the importance of firm characteristics in understanding export performance. However, our results indicate that differences in performance among developing countries, in terms of their access to international markets, may depend on the typology of international value chains by which these countries enter into international competition. The difficulties for these countries in
achieving adequate and lasting economic performance may be due to the operators to access higher international added value chains<fnref n="7" />. International value chains can be differentiated by the types of stakeholders involved and by the characteristics of target markets (volume of exports, nature and food safety requirement of safety regulations in force). There are also differences according to the nature of upstream vertical relationships, potential customers (wholesale markets, purchasing centers for different retailers, regional markets, etc.). Exporters generally take into account the heterogeneity of safety regulations and the heterogeneity of border controls among the various importing countries than can be targeted. They also have to consider the level of market prices in these countries and risk product rejection at the borders of these countries given the level of both borders controls and Maximum residues limits (MRL) regulation.

This framework allows us to understand the exporter' choice of the target market in regard to the level of food safety requirement and how this choice can or cannot lead to a standard compliance process in supply chains of developing countries. This process depends on the one hand, on the nature of exporters and exporters/producers relationships (the sharing of risk, costs and value) and, on the other hand, on the prices observed in the international markets.

The majority of economic literature explains producer exclusion in export chains in developing countries by prohibitive standards compliance costs. We show through our model that this cost-based approach (including traditional cost-benefit analysis) may not be sufficient to understand and explain this phenomenon. It may be necessary to have a strategic “dynamic” view in the sense that the stakeholders’ behavior (market choice, level of cost-sharing and value-sharing within the upstream and downstream supply chain) can favor or disfavor producer participation. In particular, when the market price is high, exporters can have the advantage of benefiting from this opportunity by selling a large quantity of their product. The lower the exporters’ contribution compliance cost is, the higher the quantity is. This leads the exporter to select a maximum number of producers, with the challenge of getting all the non-compliant producers to meet the food safety requirements. This case corresponds to the scenario where producer exclusion is reduced because the exporter incentive to increase the quantity delivered to some international markets.

<footnote n="7">⁷ A number of studies noted that some types of international chains largely contribute to bringing the productivity and the quality of production practices in developing countries. This improvement of productivity and quality is notably induced by standards' compliance process imposed within these international chains (Maertens and Swinnen, 2006). The revenue of the major exporters of the developing countries in these chains seems to be higher than the revenue of those who are involved in "traditional" chains (Maertens and Swinnen, 2006). However, the smaller producers are relatively excluded from these chains and the value sharing within this chain is not systematically fair.</footnote>
The emergence of this strategy allows us to illustrate how the integration of producers in the international markets can be achieved by implementing an upstream production practices process and a long term positioning of producers in international chains. On the contrary, when the international price market is relatively low, or the type of vertical relationships is such that it implies an unfavorable balance of the exporters' bargaining power, the former targets a less demanding exporter market by selecting the most efficient local producers.

The exclusion therefore depends on a multitude of factors that can affect exporters' choices and therefore the compliance of the potentially excluded producers. For example if the market price is high, the proportion of compliance costs and the risk of referral are equally high; the exporter's approach is to support a high offer to take advantage of the market opportunity (low rate of exclusion). This leads to the involvement of a large number of producers and the necessity for them to comply with the required standards.

In our results, the classic notion of exclusion doesn't explain in the developing countries the operators' strategies on the international markets. The exclusion risk does not only depend on the level of the standard required (and thus from the compliance costs), but also on the type of vertical relationships (sharing of compliance costs and rejection costs) and opportunities in the importer market and from the type of the vertical relationships that is present in the supply chain and in the typologies of importing markets.

**Acknowledgments:**

*This work is carried out under the project ValueTeam - ARIMNet2 with the financial support of the Ministry of Higher Education and Scientific Research (Tunisia).*

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## Annex 1. Risk of rejection

We determine the risk of rejection $\bar{\sigma}(e_1, \tilde{e})$ as a function of the equipment threshold $\tilde{e}$ from which producers are selected by the exporter.

If $Q \le \hat{Q}$, ($\tilde{e} \ge e_1$), the statistical distribution of producers over the interval $[\tilde{e}, 1]$ is given by $h(e) = f(e)$, where $f(e) = \frac{1}{1-\tilde{e}}$ where $\int_{\tilde{e}}^{1} f(e) de = 1$. The risk $\bar{\sigma}(\tilde{e})$ is then given by:

$$ \bar{\sigma}(\tilde{e}) = \int_{\tilde{e}}^{1} \sigma(e) f(e) de = \frac{1}{2}(1 - \tilde{e}) \quad (A1) $$

If $Q > \hat{Q}$, ($\tilde{e} < e_1$), the statistical distribution of producers over the interval $[\tilde{e}, 1]$ is given by $h(e) = g(e)$, where $g(e)$ is given by (3). The risk $\bar{\sigma}(e_1, \tilde{e})$ is then given by:

$$ \bar{\sigma}(\tilde{e}) = \int_{\tilde{e}}^{1} \sigma(e) f'(e) de = \frac{1}{2(1-\tilde{e})} (1 - e_1) (1 + e_1 - 2\tilde{e}) \quad (A2) $$

By using (A1), (A2) and (1), one can obtain the expression of the risk of rejection according to the quantity Q and the standard $e_1$ given by (4).

## Annex 2. Proof of Propositions

### Proposition 1:

The exporter's maximization program is written by:

$$ \max_{Q} \pi_1(e_1, P, Q) $$

From (4), (6) and (7), one can deduce:
$$ \text{If } Q \le \hat{Q} \text{ then } Q_1^* = 2Jq \frac{2+\Gamma-\sqrt{4-2\Gamma+\Gamma^2}}{3\Gamma} \qquad (A3) $$

By using (A3), one can verify ex-post that $Q_1^* \le \hat{Q} \Leftrightarrow \Gamma \ge \hat{\Gamma}$ where

$$ \hat{\Gamma} = \frac{8e_1}{1+e_1(2-3e_1)} \qquad (A4) $$

By using (A4), one can easily verify that the threshold $\hat{\Gamma}$ is increasing$^8$ and is convex in $e_1$.

$$ \text{If } Q > \hat{Q} \text{ then } Q_2^* = Jq\psi(e_1) \qquad (A5) $$

Where $\psi(e_1)$ is given by$^9$:

$$ \psi(e_1) = (1 - e_1) \sqrt{\frac{2r - \Gamma(1 - e_1)^2}{2r + 4\Gamma e_1(1 - e_1) - 8e_1}} \quad \forall \underline{\Gamma} < \Gamma \le \bar{\Gamma} $$

With

$$ \left| \begin{array}{l} \underline{\Gamma} = \frac{4e_1 - r}{2e_1(1 - e_1)} \\ \bar{\Gamma} = \frac{2r}{(1 - e_1)^2} \end{array} \right. \qquad (A6) $$

Another constraint is to be defined. It concerns the order between these thresholds: $\underline{\Gamma} < \bar{\Gamma}$.

By using (A5), one can verify ex-post that $Q_1^* \ge \hat{Q} \Leftrightarrow \Gamma \le \hat{\Gamma}$ where $\hat{\Gamma}$ is given by (A4).

This allows us to prove the result announced in proposition 1.

Note that there is a capacity constraint, the exporter cannot ask for more than the total group of producers can produce.

Therefore, $Q_2^* \le Jq$ if $\Gamma \ge \hat{\Gamma}$ with:

$$ \dot{\Gamma} = \frac{2e_1[4 - r(2 - e_1)]}{(1 - e_1)((1 - e_1)^3 + 4e_1)} \qquad (A7) $$

This new threshold of $\Gamma$ can be written as:

<footnote>$^8$ The threshold $\hat{\Gamma}$ is positive $\forall e_1 \in [0,1]$ and $\hat{\Gamma} < 1$ for $e_1 < (\sqrt{15} - 3)/6$.</footnote>

<footnote>$^9$ It is to be noted that $\bar{\Gamma} > \underline{\Gamma}$ holds when: $\begin{cases} \forall e_1 \in [0,1] & \text{if } r \in [0.71; 3.73] \\ \forall e_1 < e'_1 \text{ or } e_1 > e''_1 & \text{if } r < 0.71 \text{ or } r > 3.73 \end{cases}$</footnote>

<footnote>Where $e'_1 = \frac{4Pq-3\alpha-\sqrt{(4Pq-3\alpha)^2-16\alpha Pq}}{9}$ and $e''_1 = \frac{4Pq-3\alpha+\sqrt{(4Pq-3\alpha)^2-16\alpha Pq}}{9}$</footnote>
$$ \dot{\Gamma} = \frac{(1 - e_1)^3 \bar{\Gamma} + 4e_1 \underline{\Gamma}}{(1 - e_1)^3 + 4e_1} $$

Let us now study the direction of variation of the optimal quantity ordered by the exporter.

Using (A3), we verify that the quantity $Q_1^*$ is constant on $e_1$. Using (A5) and (A6), we can study the direction of variation of the quantity $Q_2^*$ as a function of $e_1$.

$$ \operatorname{sign} \frac{\partial Q_2^*(e_1, \Gamma, r)}{\partial e_1} = \operatorname{sign} \zeta(e_1, \Gamma, r) $$

Where $\zeta(e_1, \Gamma, r) = -2 \Gamma^2 e_1^4 + \Gamma(-6 + 5\Gamma)e_1^3 + \Gamma(10 + 2r - 3\Gamma)e_1^2 + (4r - 2\Gamma - 2r\Gamma - \Gamma^2)e_1 + 4r - 2\Gamma - 2r^2 + \Gamma^2$

(A8)

From (A8), if $\zeta(e_1, \Gamma, r) > 0$, $Q_2^*$ increases according to $e_1$ (and inversely).

Let us now study the sense of variation the second derivative of $\zeta(e_1, \Gamma, r)$ according to $e_1$.

$$ \frac{\partial \zeta(e_1, \Gamma, r)}{\partial e_1} = -8 \Gamma^2 e_1^3 + 3\Gamma(-6 + 5\Gamma)e_1^2 + 2\Gamma(10 + 2r - 3\Gamma)e_1 + 4r - 2\Gamma - 2r\Gamma - \Gamma^2 $$

(A9)

So that the second derivative of $\zeta(e_1, \Gamma, r)$ according to $e_1$ is given by

$$ \frac{\partial^2 \zeta(e_1, \Gamma, r)}{\partial e_1^2} = -24 \Gamma^2 e_1^2 + 6\Gamma(-6 + 5\Gamma)e_1 + 2\Gamma(10 + 2r - 3\Gamma) > 0 $$

(A10)

$$ \forall e_1 < \vec{e}_1 = \frac{15\Gamma - 18 + \sqrt{324 + 3\Gamma(-20 + 32r + 27\Gamma)}}{24} $$

(A11)

It is easy to check that if $r > 4$, $\vec{e}_1 > 1$. In this case, $\frac{\partial \zeta(e_1, \Gamma, r)}{\partial e_1}$ increases according to $e_1$. Moreover, $\frac{\partial \zeta(e_1, \Gamma, r)}{\partial e_1} \rvert_{e_1=0} > 0$ so $\zeta(e_1, \Gamma, r)$ increases with $e_1$. Let's calculate $\zeta(e_1, \Gamma, r) \rvert_{e_1=0} = 4r - 2\Gamma - 2r^2 + \Gamma^2 > 0$ under the case $r > 4$. To conclude, when $r > 4$, than $Q_2^*$ decreases according to $e_1$. Consider now the case $r \le 4$, the function $\frac{\partial \zeta(e_1, \Gamma, r)}{\partial e_1}$ increases if $e_1 < \vec{e}_1$. (otherwise, it decreases). Although, $\frac{\partial \zeta(e_1, \Gamma, r)}{\partial e_1} \rvert_{e_1=0} > 0$, $\zeta(e_1, \Gamma, r) \rvert_{e_1=0} > 0$ if $r > \tilde{r} = \frac{\Gamma(2+\Gamma)}{2(2-\Gamma)}$. Consequently, if $r > \tilde{r}$, $Q_2^*$ decreases if $e_1 < \vec{e}_1$. Otherwise, it increases according to $e_1$.

To summarize, $Q_2^*(e_1, \Gamma)$ is not a monotonic function in $e_1$. It increases or decreases as a function of $e_1$ depending on the value of $r$ (or $P$ relative to $\alpha/q$) so that,
* If $P < \bar{P} = \frac{\alpha}{4q}$, then $Q_2^*$ decreases according to $e_1$.

* If $\frac{\alpha}{4q} < P < \frac{\alpha}{\bar{r}q}$, then $Q_2^*$ is convex in $e_1$.

* If $P > \bar{P} = \frac{\alpha}{\bar{r}q}$, then $Q_2^*$ increases according to $e_1$.

**Proposition 2:**

Using (4), the total risk of rejection is expressed such as

$$ \bar{\sigma}(e_1) = \begin{cases} If \Gamma \ge \hat{\Gamma} & \frac{2+\Gamma-\sqrt{4-(2-\Gamma)\Gamma}}{3\Gamma} \\ If \Gamma \le \hat{\Gamma} & (1-e_1)[1-\frac{Jq}{2Q_2^*(e_1)}(1-e_1)] \end{cases} \quad (A12) $$

We can easily verify that the function $\bar{\sigma}(e_1)$ is constant if $\Gamma > \hat{\Gamma}$. If $\Gamma < \hat{\Gamma}$,

$$ \frac{\partial \bar{\sigma}(e_1)}{\partial e_1} = -1 + \frac{\hat{Q}}{Q_2^*(e_1)} + Jq \frac{(1-e_1)^2}{2Q_2^*(e_1)} \frac{\partial Q_2^*(e_1)}{\partial e_1} \quad (A13) $$

Note that if $\Gamma < \hat{\Gamma}$, $\frac{\hat{Q}}{Q_2^*(e_1)} < 1$ so that if $\frac{\partial Q_2^*(e_1)}{\partial e_1} < 0$ than $\frac{\partial \bar{\sigma}(e_1)}{\partial e_1} < 0$. Now, let's consider the case $\frac{\partial Q_2^*(e_1)}{\partial e_1} > 0$, the equation (A13) can be written as:

$$ \frac{\partial \bar{\sigma}(e_1)}{\partial e_1} = -1 + \frac{Jq(1-e_1)}{2Q_2^*(e_1)} \left[ 2 + \frac{(1-e_1)}{Q_2^*(e_1)} \frac{\partial Q_2^*(e_1)}{\partial e_1} \right] = -1 + \tau(e_1) $$

$$ \tau(e_1) = -(1-e_1) \frac{\partial \left[ \frac{\sqrt{2\alpha + 4Pq\Gamma e_1(1-e_1) - 8Pq e_1}}{\sqrt{4\alpha - 2Pq\Gamma(1-e_1)^2}} \right]}{\partial e_1} $$

The sign of $\frac{\partial \bar{\sigma}(e_1)}{\partial e_1}$ is ambiguous. Two opposite effects are to be considered. The result depends on the value of $\tau(e_1)$ derivative relatively to the unit. It is straightforward that $\tau(e_1)$ is an increasing function according to $e_1$. One can check that $\begin{cases} \tau(e_1 = \hat{e}_1) > 1 \\ \tau(e_1 = \bar{e}_1) < 1 \end{cases}$

Where $\hat{e}_1 = \frac{-4+\Gamma+2\sqrt{P^2q^2(-2+\Gamma)^2+2\alpha Pq}}{3\Gamma}$ which is a solution of $Q_2^* = \hat{Q}$ and $\bar{e}_1$ which is a solution of $Q_2^* = \bar{Q}$.

**Proposition 3:**

The rate producer exclusion $\tilde{e}(e_1)$ is written by:

$$ \tilde{e}(e_1) = \begin{cases} If \Gamma \ge \hat{\Gamma} & \hat{e} \\ If \Gamma \le \hat{\Gamma} & 1 - \psi(e_1) \end{cases} \quad (A15) $$
The level of $\hat{e}$ is given by equation (10) and its expression is deduced directly from equation (A3).

One can easily verify that $\tilde{e}(e_1)$ is constant in $e_1$ over the interval if $\Gamma \ge \hat{\Gamma}$. When $\Gamma \le \hat{\Gamma}$, the exclusion rate varies in the opposite direction than the optimal quantity. It decreases in $e_1$ if $Q_2^*$ increases, and conversely.

**Proposition 4:**

The profit of the exporter $\pi_1(e_1)$ is given by:

$$ \pi(e_1) = \begin{cases} If \Gamma \ge \hat{\Gamma} & 2Jq \frac{(-2-2\Gamma+\sqrt{4-(2-\Gamma)\Gamma})(2+\Gamma-\sqrt{4-2\Gamma+\Gamma^2})(-2-5\Gamma+\sqrt{4-(2-\Gamma)\Gamma})}{27 \Gamma^2} \\ If \Gamma \le \hat{\Gamma} & [1-\bar{\sigma}(e_1, Q)][2-\bar{\sigma}(e_1, Q)\Gamma]PQ - \alpha F(e_1, Q) \end{cases} \quad (A16) $$

Using (A16), we verify that the profit of the exporter is constant in $e_1$ if $\Gamma > \hat{\Gamma}$. Let us now consider the case $\Gamma < \hat{\Gamma}$,

$$ \frac{\partial \pi(e_1, Q)}{\partial e_1} = \frac{\partial \pi(e_1)}{\partial e_1} + \frac{\partial \pi(Q)}{\partial Q} \frac{\partial Q}{\partial e_1} \quad (A17) $$

By using the envelope theorem (from the first order condition of the profit maximization in $Q$), $\frac{\partial \pi(Q)}{\partial Q} = 0$ so that $\frac{\partial \pi(e_1, Q)}{\partial e_1} = \frac{\partial \pi(e_1)}{\partial e_1}$

Note that

$$ \operatorname{Sign} \frac{\partial \pi(e_1)}{\partial e_1} = \operatorname{Sign}[-r + (1 - e_1)^2 \Gamma + (1 - e_1)(2 + (-1 + 2e_1)\Gamma) \sqrt{\frac{2r - \Gamma(1 - e_1)^2}{2r + 4\Gamma e_1(1 - e_1) - 8e_1}}] $$

Which is the sign of $\vartheta(e_1)$ given by : $\vartheta(e_1) = [-\Gamma(4 + \Gamma(-4 + 2r + \Gamma))e_1^4 + 4\Gamma((-2 + \Gamma)^2+2r\Gamma)e_1^3 + (8r(1 + 2\Gamma) - 6(\Gamma - 2)^2\Gamma - 2r\Gamma(4 + 5\Gamma))e_1^2 + (4(r^2(2 - 3\Gamma) + (\Gamma - 2)^2\Gamma + r(-4 + \Gamma(4 + \Gamma))e_1 - 2r^3 + 8r(1 - \Gamma) + 4\Gamma r^2 - (\Gamma - 2)^2\Gamma) \quad (A18) $

Using (A18), we verify that:

If $\Gamma > 2/3$, we distinguish the following cases:

If $\dot{r}$, $\vartheta(e_1)$ is increasing with $e_1$.

If $r > \dot{r}$, $\vartheta(e_1)$ has a local minimum in $e_1$.

$$ \text{Where } \dot{r} = \frac{-4+\Gamma(4+\Gamma)+\sqrt{16+\Gamma(-64+\Gamma(88+\Gamma(13\Gamma-48)))}}{6\Gamma-4} $$

If $\Gamma < 2/3$, $\vartheta(e_1)$ is increasing with $e_1$.

Let us calculate now the sign of $\vartheta(e_1)$ for two particular extreme values ($e_1 = \hat{e}$) and $e_1 = \bar{e}$
$$ \vartheta(e_1)|_{e_1=\hat{e}} > 0 \text{ if } r < \hat{r} $$

$$ \text{Where } \hat{r} = \frac{2}{9}(8 - \Gamma + \sqrt{4 + \Gamma(\Gamma - 2)} + \frac{-4+2\sqrt{4+\Gamma(\Gamma-2)}}{\Gamma}) $$

Furthermore,

$$ \vartheta(e_1)|_{e_1=\bar{e}} < 0 \text{ if } r > \bar{r} \\ \bar{r} = 2 + \bar{e}^2 \Gamma $$
Consequently, if $\Gamma > \hat{\Gamma}$, the profit of the exporter is constant in $e_1$. If $\Gamma < \hat{\Gamma}$, $\pi_1(e_1)$ is increasing in $e_1$ when $r < \hat{r}$ (i.e. $P > \frac{\alpha}{\hat{r}q}$) and is decreasing in $e_1$ when $r > \bar{r}$.