Increasing willingness to pay in the food supply chain: a blockchain-oriented trust approach

Abstract

Food products' quality information is advertised on labels but do customers trust them? This study investigates how the consumers' Willingness-To-Pay (WTP) for food products can be increased by deploying managerial effort and advanced technologies, such as the Blockchain Technology (BT). Our model explains how revealing verified information about product quality throughout the supply chain will generate optimal consumers' WTP and maximise profit. At each echelon of a multi-echelon supply chain, a buyer holds a Bayesian belief about the quality of the input to be procured. This belief is shaped by the accuracy and veracity of the information about this quality. Managerial effort is required both to enhance quality as well as ensure full and verified information. We show why this effort must be made across the chain and how opportunistic behaviour may be circumscribed. Using empirically grounded analytics and real prices of olive oil intermediate produce from various official bodies, we show how the application of BT may be financially justified. This research shows how trust and WTP can be further enhanced through the use of BT and additional smart technologies in a supply chain, which may be projected on other supply chains of organic and sustainable food products.

KEYWORDS:

• Supply chains
• trust
• willingness to pay
• blockchain technology
• Bayesian belief

1. Introduction

Consumers' purchase behaviour for food products has considerably changed over this century. Demand for local and organic food production is increasing (Research, and Markets, 2022). Along the years, numerous studies highlight that consumers perceive organic products as healthier (e.g. Bruschi et al., 2015, Ditlevsen, Sandøe, and Lassen, 2019, Hoefkens et al., 2009, Prada, Garrido, and Rodrigues, 2017) and that these perceived health benefits are the major driving force behind organic food purchases (e.g. Britwum, Bernard, and Albrecht, 2021, Chen, 2009, Magnusson et al., 2003, Rana and Paul, 2020, Schleenbecker and Hamm, 2013, Tsakiridou et al., 2008). They are willing to pay a premium price for the benefits associated with organic food products (Kushwah et al., 2019, Schleenbecker and Hamm, 2013). Of course, information plays a crucial role in influencing both the acceptance of and the WTP for the organic claim (McFadden and Huffman, 2017, Teuber, Dolgopolova, and Nordström, 2016). This means that ensuring the authentication of product quality and origin is critical for all chain partners and for all food supply chains (Laddomada et al., 2013) so that it increases consumers' buying inclination (Nuttavuthisit and Thøgersen, 2015). This is even more so in the case of food safety: consumers are increasingly sensitive about the quality of agricultural products and the implementation of safe practices (Yawar and Kauppi, 2018). The WTP increases if the information about food products comes from consumer associations, less if it comes from the European Food Authority and not at all if it comes from industrial partners (Nocella, Romano, and Stefani, 2014, Yormirzoev, Li, and Teuber, 2020).

How can information about quality be made apparent in the food product for consumers to trust it? As we will show in the literature review, there is an abundant literature on how information can be collected along the supply chain about the quality of a food product and how this information is then relayed to the consumer. Food producers, transformers, and processed food manufacturers have invested in information systems to track quality and ensure full traceability from ‘farm to fork’. The confidence in such traceability systems and corresponding WTP is still lacking (A. Zhang, Mankad, and Ariyawardana, 2020).

The fact is that there are many steps between the farm and the fork. For each step along the way, a specialised firm takes care of the required transformation or logistic task. This division of each food supply chain into many independent firms looking after their own particular interest does not breed the expected trust and information exchange. Relying on public authorities' supervisory and law enforcing powers is not enough (Yinghua, Ningzhou, and Dan, 2018).

More recently, BT has been assumed to enhance the information exchange which should breed the expected trust (Chang, Iakovou, and Shi, 2020, A. Kumar, Liu, and Shan, 2019, Mendling et al., 2018). However, despite the recent hype about BT and ancillary technologies to increase the veracity of information in supply chains, no research has been able to justify the deployment cost so putting its application in doubt. If the investment in BT can not be justified, the supposed advantages disappear. In what follows, we view BT as part of a new ‘governance mechanism to organise collaborations in supply chains’ (Koh, Dolgui, and Sarkis, 2020, W. Wang, Lumineau, and Schilke, 2022, Zhu, Kouhizadeh, and Sarkis, 2022).

We pose the following three questions: How can information about quality be made apparent about a food product for consumers to trust it? Which information system can be sufficiently robust that consumers trust the corresponding information and that their WTP increases? There is already a large literature addressing such questions, which is why we prefer to dedicate our attention to the question below. Given the multiple actors involved, a common information exchange mechanism to coordinate them is needed to ensure optimal quality levels and trust, we wish thus to address the following one: What is the optimal combination of managerial and technological investment, e.g. in BT infrastructure, to build trust and increase WTP in a multi-echelon food supply chain?

Blockchain, the Internet of Things (IoT), sensor technologies, and the development of Industry 4.0 have matured to such an extent that their combination might be expected to now lead to new breakthroughs in food safety, traceability, and sustainability thanks to increased collaboration (for example, see Biswas et al., 2023, Iftekhar et al., 2020, S. Liu et al., 2022, Rejeb et al., 2021). We build upon this literature to present a model which should help in answering the above question. We will show that the managerial effort described in the preceding paragraph calls for the combination of such technologies with the continuous improvement of processes and information systems, as well as the constant training of operators and managers.

In this paper, building upon already proven results in the literature, we show that BT in the supply chain must be coupled with the right managerial effort to improve the consumers' WTP and that of the multiple partners throughout the chain. In this way, the extra cost of deploying BT would be justified.

We model the increase of the WTP in food products by a stylised game theoretic model of a supply chain to show how the consumer's WTP is increased as all upstream suppliers update truthfully the necessary information about the quality of the product through its different transformation phases. To do so, we model each level of the chain as a supplier-seller-buyer triad (Mena, Humphries, and Choi, 2013). The seller engages in effort to ensure the highest WTP on the part of the downstream buyer by informing about the product quality resulting from his transformation process as well the quality of the transformation processes of the upstream supplier. We show how WTP drops when one partner cheats on the quality of the intermediate produce or if the information about such quality can be doubted. We show that a constant effort is required from all partners to ensure that no doubt arises in the minds of buyers, be they intermediate customers or consumers. We validate the mechanism by applying it in a stylised analytic example using intermediate prices for raw material for different qualities in an olive oil supply chain.

Our contribution is both positive and normative (Bertrand and Fransoo, 2002). From a positive viewpoint, our model explains how and why some partners may opportunistically engage in deceiving the next level buyer about the quality of the intermediate products being sold (as so often happens, see olive oil fraud in Al-Zoubi, 2019). We show how consumers can be made to trust a particular supply chain's products and hence be willing to pay a higher price given that they trust the quality. From a decision-making stance, two contributions are presented: (i) the Bayesian updating mechanism proposed for the different players of the game about the quality of the product being sold as well as how this mechanism can be influenced by the seller's information and the quality of such information; (ii) how governance mechanisms coupled with the right sensor and BT can help all members in the SC achieve the highest consumer WTP. From a behavioural stance, our contribution provides two theoretical arguments. The first one is justifying the partners' effort both in ensuring the highest quality intermediate product and in sharing truthfully the relevant information to the other partners. The second one, a contrario, explains why, in the case of absence of or misleading information, one partner can behave opportunistically by adulterating or processing sloppily the intermediate product and thus capture an undue extra profit because of lower processing and effort cost.

The paper is organised as follows: we discuss how our contribution is placed with regard to two streams of literature in Section 2 before developing a model in Section 3, and illustrating the result in Section 4. We draw some conclusions and inferences for future research in Section 5.

2. Literature review

Several streams of literature are relevant to this work and are dealt with in different subsections. We first bring up in Section 2.1 how the WTP of the consumer and of intermediate partners in the supply chain is increased, specify the research gap and our contribution. We then highlight how researchers have shown how to harness BT to improve security, traceability, and information sharing in operations and supply chain management (Section 2.2).

2.1. The customer's willingness to pay for quality and sustainability

In the following we show how WTP has been shown to exist in both constructivist and positivist, normative literature.

2.1.1. WTP in quantitative surveys literature

Like Bresnahan (1987) and Berry (1994), we consider here that customers care about product quality, which is modelled as depending on product characteristics, some of which may be unobservable (Berry, 1994). In some cases, products may look similar but differ in customers' perceptions regarding quality, durability, status, service at the point of sale, or after-sales service. Customers maximise the utility of a product as a function of budget constraints (Hanemann, 2001) that hence directly impacts their WTP.

In particular, origin and quality are directly linked to the value of the product (Bánáti, 2011, Giraud and Halawany, 2006, Padilla et al., 2007, Santosa et al., 2013). Notably, WTP estimates are positively linked to customer trust in certified animal-friendly products (Nocella, Hubbard, and Scarpa, 2010), or organic coffee (Dionysis, Chesney, and McAuley, 2022).

They want to know if a product has been produced sustainably or through a high-quality process (Choe et al., 2009, Giampietri et al., 2018). In effect, trust in food supply chains covers a number of different concepts (Tejpal, Garg, and Sachdeva, 2013). The scandals concerning various foodstuffs in China in the 2010s have led to effort in real-time food tracing to enhance the safety assurance (Tian, 2018) and increase the value in the eyes of the consumers (Pang et al., 2015) through complete traceability (Chang, Tseng, and Chu, 2013), especially as compliance with food quality regulations is not enough to generate trust (Robinson and Ruth, 2020).

2.1.2. WTP in mechanism design literature

The game theoretic literature on supply chains considers that a Bayesian mechanism is required so that a player in an incomplete information game can update a prior belief using available information on the actions of the other players in the game (Harsanyi, 2004). The actual purchase decision is based on the perceived quality and risk associated with the product, rather than on consumers' initial intention (Khor and Hazen, 2016). Choi et al. (2020) shows how an on-demand service platform will derive the risk aversion profiles of customers and hence WTP by using BT. In some instances, consumers rely on suppliers to help them decide on whether to buy a product. The process through which a supplier provides assistance in Özer, Subramanian, and Wang (2018) includes information sharing, advice provision or delegation: the better the information, the higher the WTP. Confirmation of this result is provided by Zhao, Dong, and Cheng (2018) when two firms compete and one has a higher quality product: the higher quality firm will prefer to disclose quality information. Both Guo (2009) and Guan and Chen (2015) discuss which of two competing manufacturers or retailers will disclose quality information to the consumer, whereas Guo and Zhao (2009) characterise which of two competitors will do so and in what order.

In a slightly different take on the same issue, an experiment has shown that when there is a large difference in service quality between firms, the social network information from feedback by other users increases the higher quality firm's market share, provides the lowest demand uncertainty and the fastest convergence to a steady-state market share between both firms (Davis, Gaur, and Kim, 2021). None of the above look into the process of building the case about the true quality of the product. The intrinsic quality of the product is supposed to preexist.

The literature is also extensive on how consumers learn from their own decisions and experience of product quality (Erev and Haruvy, 2016) and from information coming from social networks (Acemoglu et al., 2011, Besbes and Scarsini, 2018, Ifrach et al., 2019). Here we must distinguish between the B2C and B2B scenarios. In the B2C scenario, the final consumer may repeatedly choose among a set of suppliers when she is not well informed about the supplier quality level and will only converge slowly to the highest quality supplier through a Bayesian updating of her beliefs (Gans, 2002), whereas in the case of the B2B scenario the firm must also choose between suppliers (Sener et al., 2021). Here, in difference to our approach, the way the supplier will build the necessary product quality information is not in question.

The consumer is not the only echelon for considering WTP, purchasing managers are also willing to pay to assure compliance dimensions to ensure that suppliers are following sustainability standards (Goebel et al., 2018). This WTP is influenced negatively when standards are not met as evidenced in a number of annual reports in the food and textile industries (Inditex, 2020, Nestlé, 2018).

Our research extends this mechanism to all partners in the supply chain, including the consumer. Because of asymmetric information, a supply chain partner has to form a belief as to the supplier's product quality. Truthful information will help the Bayesian updating of the belief held by the buyer about the supplier's product quality, also named the Bayesian Mechanism (Cabral, 2005).

2.2. BT in operations and supply chain management

Transparency, visibility, and increased efficiency are the promises of BT when viewed from an operations or supply chain management viewpoint (A. Kumar, Liu, and Shan, 2019, K. Li, Lee, and Gharehgozli, 2023, M. Liu, Zhang, and Wu, 2023, Lohmer, Bugert, and Lasch, 2020). The database or ledger aspect of BT has advantages over traditional ways of storing data as it can be scaled up efficiently, reduces human and other transaction costs (Babich and Hilary, 2020, J. Wu and Yu, 2023). This is especially so now that major platforms have moved their consensus protocol from proof of work to proof of stake (eg, Ethereum in September 2022) which is less costly in energy (R. Zhang and Chan, 2020).

The consensus protocol named ‘Proof-of-Trust’ (PoT) consensus selects transaction validators based on the participants' trust values and Shamir's secret sharing algorithms (Q. Li and Christensen, 2019). The additional advantage is that the validators are part of the network in the supply chain. As such, they guarantee the veracity of the information. Recent blockchain-enabled supply chain pilot projects have shown how such information systems can be harnessed to promote visibility of the information about quality (Bai and Sarkis, 2020, M. Liu, Zhang, and Wu, 2023, Y. Wang, Huirong Chen, and Zghari-Sales, 2020). At present, many open-source communities or platforms are available for supply chains to build and customise BT-based solutions. Hyperledger Fabric is one example of such a private blockchain. Hyperledger Fabric is a global open-source collaboration that enables supply chains to develop decentralised applications.

The RAFT consensus algorithm, as discussed in the context of Hyperledger Fabric, offers significant advantages across various industries (Alexandridis et al., 2021). Its simplicity and ease of understanding streamline the development of blockchain networks. RAFT ensures efficient and quick operations, comparable to other consensus mechanisms like PAXOS, but with easier implementation. It provides robust fault-tolerance and safety, making it suitable for environments requiring reliable data management. Moreover, RAFT's design avoids the formation of multiple majorities during reconfiguration, enhancing the stability of the system. These features make RAFT a valuable tool for managing complex data systems in diverse fields, particularly where decentralised and fault-tolerant solutions are needed.

The decentralised approach of blockchain, which involves encrypting data 'key values' in the language of Hyperledger, and the consensus mechanism used for recording and sharing traceable and transparent data, are its key advantages when compared to other databases or information systems structures. While these advantages are significant, ensuring the accuracy of information (matching reality) is a concern that BT alone cannot guarantee. This is why we have proposed and taken into account complementary technologies, such as sensors and managerial effort, to address this challenge and harness the potential of blockchain to enhance WTP. Although some literature 'conceptually' solves the accuracy problem of information using blockchain and builds on this hypothesis, we believe that discussing the issue of trustworthy information to enhance customers' WTP, considering agency, opportunistic behaviours, and information asymmetry, is a more realistic approach.

The combination of BT with the corresponding and relevant technologies, be it sensors, IoT devices (Iftekhar et al., 2020, Z. Li et al., 2020, Z. Zhang et al., 2020), or secure communication networks, have been shown to provide a verifiable and traceable IoT network (Ben-Daya, Hassini, and Bahroun, 2017, Sidorov et al., 2019) enhancing the automated and trusted identification of physical objects, critical to their traceability (Balagurusamy et al., 2019, Casino et al., 2021). A streamlined trust model which simplifies data sharing and reduces computational, storage, and latency requirements while increasing the security of the IoT-based supply chain management has been presented in Al-Rakhami and Al-Mashari (2021).

Data security and integrity are guaranteed against replay attacks as well as physical attacks on sensors because they can report such attacks directly to the blockchain and because of cross-checking by sensors between themselves (for additonal reference about security, automatic transaction management, and offline-to-online data verification, see Capocasale et al., 2021, Giovanni, 2020, Krishnan, 2021, Lao et al., 2020, Mendling et al., 2018, Sanchez-Gomez et al., 2020, Y. Wang, Huirong Chen, and Zghari-Sales, 2020). The information thus validated will then be used by smart contracts (Tapscott and Tapscott, 2017) connected through oracles (Bakos and Halaburda, 2019, Dolgui et al., 2019, Kamilaris, Fonts, and Prenafeta-Boldù, 2019, Mao et al., 2018). These are drivers that allow firms to move from the physical to the digital world (for further information please see Agrawal, Sharma, and Kumar, 2018, Hawlitschek, Notheisen, and Teubner, 2018, Lao et al., 2020, Mendling et al., 2018, Meyer, Kuhn, and Hartmann, 2019, Savelyev, 2018).

To summarise, the above literature, mostly at the border of supply chain and information systems research, places its emphasis on the decentralised topology, security, information sharing, and traceability necessary in a supply chain through a proper implementation of BT and the corresponding IoT architecture combined with the necessary smart contracts (A. Kumar, Liu, and Shan, 2019). However, in all of the above, the return on investment of investing in BT is rarely addressed (Alkhudary, Brusset, and Fenies, 2020). Typically, authors only present how costs can be reduced (Chaudhuri et al., 2021, X.-Y. Wu, Fan, and Cao, 2023), or if managers intend to use BT (Wong et al., 2020). Most articles barely acknowledge that managers are often not aware of the benefits of implementing BT, nor the organisational difficulties of doing so (Oguntegbe, Paola, and Vona, 2022).

When looking at the supply chain game theoretic literature, a number of papers describe the advantages of BT. For example, Biswas et al. (2023) describe a game theory model where a proper implementation of the BT will increase traceability and so overcome distrust of a product by end-consumers, hence increasing sales. The model presents the conditions for an equilibrium where the cost of BT is quadratic in the level of traceability. How the end-customer is informed about the level of traceability is not explained.

In the same stream, other papers refer to the quality disclosure effect. Chod et al. (2020) shows how transparency through BT enables inventory verifiability, while Shen, Xu, and Yuan (2020) shows how secondhand products can be priced higher on e-marketplaces, thus helping, for example, brand-name companies obtain a quality disclosure effect (Shen, Dong, and Minner, 2021). In M. Liu, Zhang, and Wu (2023), customer WTP hinges on the belief about quality which is uniformly distributed between two bounds. These four papers consider that blockchain deployment immediately makes the information about quality common and truthful. In Shen et al. (2021), conditions where quality checks prevail over a blockchain system to detect counterfeit masks are investigated but do not show how the information about quality is verified before being locked into the blockchain system (ie, opportunistic behaviour can go undetected).

How a blockchain-enabled supply chain might increase the consumer's WTP, which could be a way to justify the investment in BT, at least in the food supply chain, was rarely investigated.

Finally, fighting counterfeit products by adopting BT to prove product origin and so increase trust in the end-customers' mind has been modelled in Niu et al. (2021) and Pun, Swaminathan, and Hou (2021). However, in Pun, Swaminathan, and Hou (2021), the government must subsidise the cost of BT; while in Niu et al. (2021) multinational firms will not want to participate in chain-wide BT deployment for cost and tax reasons.

We are interested here in answering the question of how end-customers will pay more for a food product because they are confident of the provably true qualities of that product and so pay for the cost of implementing the corresponding solution. We have not found any paper which addresses this question.

Having gone over a picture of the necessary technology to bring about the consumer's WTP, we describe in the next section how our model would apply it.

3. Model and analysis

After providing some background considerations about the motivation of the model in Section 3.1, in Section 3.2, we explain the way each buyer at each production step from raw material to consumer has the ability to buy the trusted quality product or an alternate one for which quality is not verified. In Section 3.3, we evaluate the optimal effort and optimal price at each level. We explain how adopting BT must be further supported by additional effort. We then extend the model to the overall chain in Section 3.4. We discuss the case when one or more members of the chain cheat in Section 3.5.

3.1. Motivation of the model

We consider the case of a food supply chain where partners, each one in charge of a transformation step, have come to the conclusion that the consumer will be willing to pay for a quality and sustainably produced food product. In the market, there are various alternatives which fail to prove to be of quality and/or to come from a sustainable supply chain. As is common in such settings, this quality level is enshrined in a charter describing in detail the entire production process from raw material to finished consumer product including all quality requisites. The more detailed and stringent the technical specifications, the higher the expected quality. The consumers' WTP increases with such expected quality. By contrast, the non-descript product lacks such quality specification. Without loss of generality, the price for the higher quality product is higher than for the non-descript one, while the production costs are similar and standardised to 0.

In an initial phase, the partners agree to belong to one single supply chain (see Figure 1). They also agree to abide by a governance mechanism including a code of conduct and commitment to produce the quality product (Gulati and Nickerson, 2002, Robson, Katsikeas, and Bello, 2008, Singh and Teng, 2016). They invest in the BT required so that all partners can register immutably on a shared registry all information pertaining to the quality of the intermediate and final product.

Figure 1. Sequence of events in the model: top left, all partners contribute, bottom, actions by each actor in turn. Top right: actions by the consumer.

Horizontal timeline giving the sequence of events in the model. Above the line, the actions by the partners: supply chain set up, governance mechanism agreed upon, investment in BT, below the line: production and quality information effort is deployed, the information about quality relayed to all partners. Above the line: consumer checks information, defines her WTP and pays.

In the production phase, they ensure that the sensors duly report the correct data to prove that the quality of the products corresponds to the one expected in the charter (Dolgui et al., 2019, Fadda et al., 2018, Lao et al., 2020, Tian, 2018, Y. Wu, Dai, and Wang, 2021).

All such information about quality is then shared with all other partners so that each can check that the quality of products bought from upstream partners corresponds to the expected one.

It is easily understood that a partner who has too few sensors and too little control over quality will not be able to provide as trustworthy information as one who can provide verified data for all the steps of the transformation process under her responsibility. When the data is patchy, with limited verification, the trustworthiness of the quality is lower. This impairs the buyer's WTP, as well as all the downstream partners' WTP.

Hence the consumer's WTP can be treated as a Bayesian belief in the true quality of the product on offer (Harsanyi, 2004). As a Bayesian belief, it can be modelled as a random variable $Z$ from a probability distribution $F\left(z\right)=P(Z\le z)$ with mean μ and standard deviation σ. This Bayesian belief is built from the consumer's information available to her. The quality standard enshrined in the quality charter that the supply chain partners abide by corresponds to a specific mean μ: the topmost quality will correspond to the highest μ, whereas a non-descript product will have the lowest one since the quality is not verifiable. Now, if the quality information available about the production process also demonstrates that all such standards have been respected, then the standard deviation of the belief is low: the customer has no doubt about that the quality of the product matches that promised by the charter. When she has doubts about the veracity of the information or that the quality of the product effectively reflects the expected one, she will build a Bayesian belief with a high standard deviation. She must estimate a value $\hat{z}$ for the price to pay.

The extra profit in the supply chain is generated by the consumer who is willing to pay a higher price for a food product for which the true quality is revealed than for a product of unknown quality (Chaudhuri et al., 2021). We thus target the final price of the food product as the objective to be maximised as a proxy for the utility derived by the consumer from buying a product of known and truthful quality.

Now, consider a partner in the chain who decides to opportunistically cut corners in terms of the quality she has to supply by, for example, adding some lesser quality (and cheaper) material in her product. To do so, she must also falsify or fail to report the true quality of the product to the database in the Blockchain. Conceivably, she may also obtain the connivance or complicity of other chain members to subvert the quality of the product. The purpose might be to obtain (and share) extra profit to the detriment of the unwary customer. This situation is modelled in Section 3.5.

For illustration purposes, let us consider the case of edible olive oil as highly representative of food supply chains. As can easily be verified, olive oil is sold with widely different prices, ranging in scale from 1 to 5 for the most commonly available brands (Devarenne, 2021). Such a supply chain can be represented in a simplified manner as composed of a farm where olives are harvested in olivars, a mill where the olives are pressed, a bottler who will bottle the bulk oil, and a retailer selling the oil bottles to the consumer (Figure 2).

Figure 2. Supply chain triads composed of a supplier, an actor transforming the product, and a buyer buying this transformed product.

3.2. Buying process in the olive oil supply chain echelon by echelon

We separate the chain into sub-parts each composed of a triad of three partners : a supplier who sells an unfinished product to an actor who transforms it and sells it on to the buyer (see Figure 2).

In this way, the olive oil supply chain can be decomposed in several triads (Mena, Humphries, and Choi, 2013) with the same characteristics: in each case an actor (a) buys a raw material or semi-finished product from a supplier (s) and sells it on to a buyer (b). We then have the following triads: farm-mill-bottler; mill-bottler-retailer; and finally, bottler-retailer-consumer. In the following, to describe the workings of the triads, we shall subscript the supplier with an s, the actor who buys from this supplier with an a, and the buyer who buys from the actor with a b.

A buyer has a need to fulfil and will pay different prices according to the type, the origin, or some other characteristic of the olive oil (see Figure 3). Let us consider here that the buyer can buy a specific quality, as described in a particular charter (Padilla et al., 2007, Parra-López et al., 2015). Hence, she is willing to pay $p_a$, the going price for this quality.

Figure 3. The buyer pays $p_a$ to the actor who pays the supplier $p_s$ for the intermediate product which she transforms. Alternately, the buyer can buy at price $p_o^b$ from a non-strategic supplier.

Decomposition of the choice available to the buyer between suppliers: The buyer pays pa to the actor who pays the supplier ps for the intermediate product which she transforms. Alternately, the buyer can buy at price pab from a non-strategic supplier.

Alternately, she also has the possibility of paying a price $p_0^b$ for an available non-descript product with $p_0^b<p_a$ from another seller. Note that this $p_0$ is available for every triad of the chain as will be explained in Remark 3.4.

Remark 3.1

The differentiating factor is the available information about quality. The buyer has a need to fulfil but has the choice of buying a product for which some information about quality is known or from an alternative source for which no information about quality is available. We do not tackle the strategic multi-supplier problem here (Elmaghraby, 2000, Laffont and Tirole, 1993, Stole, 1994).

We will denote ${\text{β}}_a$ as the effort deployed by the actor to build visibility and verifiability with $0\le {\text{β}}_a\le 1$. The cost of this effort is $T_a\left({\text{β}}_a\right)$. Without loss of generality, all other costs, such as warehousing, administration and marketing, are normalised to zero. For all triads described in the chain in Figure 2, the actor's objective function is (1) $\begin{array}{r}{\mathrm{\Pi }}_a\left({\text{β}}_a\right)=p_a\left({\text{β}}_a\right)-p_s\left({\text{β}}_s\right)-T_a\left({\text{β}}_a\right),\end{array}$(1) with $p_s\left({\text{β}}_s\right)$ the price of the product sold by the supplier as a function of her own effort to build trust ${\text{β}}_s$ at her level of the chain. Note that the effort to build trust is a decision variable for each chain member and taken myopically at each level.

The buyer's profit then becomes (2) $\begin{array}{r}{\mathrm{\Pi }}_b\left({\text{β}}_b\right)=p_b\left({\text{β}}_b\right)-p_a\left({\text{β}}_a\right)-T_b\left({\text{β}}_b\right)\end{array}$(2) if buying from the actor, and (3) $\begin{array}{r}{\mathrm{\Pi }}_b\left({\text{β}}_b\right)=p_b\left({\text{β}}_b\right)-p_0^b-T_b\left({\text{β}}_b\right)\end{array}$(3) if buying from the alternative supplier.

Remark 3.2

The cost of developing trust as well as the effort ${\text{β}}_a$ is only known by the actor. As we shall see below, if the buyer does not trust the quality of the product, she will not pay the price corresponding to that quality but only the price corresponding to the alternative no-name product available on the market. That is, the buyer's WTP will be at its lowest.

Remark 3.3

The way the trust between three levels in a supply chain is built and how it is related to the WTP of a buyer at each level has never been attempted before. The results that we outline here have never been presented either.

For the consumer, the utility derived from consuming olive oil of known quality is $u$ and, in monetary terms, (4) $\begin{array}{r}\mathrm{\Pi }\left(u\right)=u\left({\text{β}}_r\right)-p_b\left({\text{β}}_r\right),\end{array}$(4) where ${\text{β}}_r$ is the effort deployed by the retailer and, since she does not have to build trust, her utility increases with the trust in the quality of the retailer's product but must be higher than the economic utility of the price paid.

Thus to ensure that incentive and participation constraints are met, (5) $\begin{array}{r}\{\begin{array}{l}{p}_a\left({\text{β}}_a\right)\ge p_s\left({\text{β}}_s\right)+T_a\left({\text{β}}_a\right),\\ p_b\left({\text{β}}_b\right)\ge p_a\left({\text{β}}_a\right)+T_b\left({\text{β}}_b\right),\\ u\left({\text{β}}_r\right)\ge p_r\left({\text{β}}_r\right)\end{array}\end{array}$(5) must hold.

Remark 3.4

The price for unknown quality products available at each echelon corresponds to the state at which the olives have been transformed as presented in Figure 2: the miller can buy ordinary olives at price $p_0^m$, the bottler can buy no-name olive oil at $p_0^{\mathrm{bo}}$, the retailer can buy the bottled olive oil at $p_0^r$, and, finally, the consumer would pay $p_0^c$ for a no-name bottle on the retailer's shelf. In such a case, obviously, the end product is not a high quality olive oil.

3.3. Finding the optimal price to pay and optimal effort for each actor in the chain

End-consumer utility increases with trust (Hanemann, 2001, Lancaster, 1979), and end-consumer WTP increases continuously with trust in the ability of the retailer to provide the expected quality (Santosa et al., 2013). In the same way, as retailers trust manufacturers, their WTP increases (N. Kumar, 1996).

The estimated distribution of the Bayesian belief Z follows a probability density function $f(.)$ and a cumulative density function $F(.)$ over a domain $[\underset{\_}{z},\overline{z}]$. We consider, without loss of generality, a domain such that $0<\underset{\_}{z}\le p_0^b\le p_a\le \overline{z}$ with $p_0^b$ as the alternative unknown quality product available to the buyer. We assume that this distribution has an Increasing Failure Rate (IFR : Barlow and Proschan, 1965), or is log-concave (Bagnoli and Bergstrom, 2005). This failure rate is given by $r\left(z\right)=f\left(z\right)/\overline{F}\left(z\right)$ and $r^{\prime }\left(z\right)\ge 0$, where $\overline{F}\left(z\right)=1-F\left(z\right)$. Positive-valued log-concave distribution functions feature IFR characteristics and include a large variety of statistical distribution functions such as the continuous uniform, the gamma, the Weibull, the modified extreme value, the truncated normal, and the log normal as characterised in Barlow and Proschan (1965). These functions, given that we do not consider negative values, are all log-concave (An, 1998, Bagnoli and Bergstrom, 2005).

We explore how this behaviour applies to the triads as already characterised in Figure 2. For all triads in the chain, the buyer maximises the expected profit (utility) ${\mathrm{\Pi }}_b$ from her purchase in terms of her WTP. If she trusts the actor to provide the true quality of olive oil, she will pay a price z with a probability $\overline{F}\left(z\right)$. If, on the contrary, she does not trust the retailer and hence believes that the true quality of the olive oil is lower than advertised, she will instead buy from the alternative retailer at price $p_0^b$ with probability $F\left(z\right)$.

Generalising for all the buyers' profit functions, from mill to retailer, the buyer's profit function changes from (2(2) $\begin{array}{r}{\mathrm{\Pi }}_b\left({\text{β}}_b\right)=p_b\left({\text{β}}_b\right)-p_a\left({\text{β}}_a\right)-T_b\left({\text{β}}_b\right)\end{array}$(2) ) and (3(3) $\begin{array}{r}{\mathrm{\Pi }}_b\left({\text{β}}_b\right)=p_b\left({\text{β}}_b\right)-p_0^b-T_b\left({\text{β}}_b\right)\end{array}$(3) ) to: (6) $\begin{array}{r}\underset{z}{max}{\mathrm{\Pi }}_b\left(z\right)=z\overline{F}\left(z\right)+p_0^{b}F\left(z\right).\end{array}$(6) We now enunciate the following Theorem (the proof is in Appendix).

Theorem 3.1

If the Bayesian belief distribution of f is IFR, there exists a unique maximum value $z^{\ast }$ representing the buyer's optimal WTP given his Bayesian belief of the quality of the product solution to the above objective function such that: (7) $\begin{array}{r}{z}^{\ast }-p_0^b=\frac{\overline{F}\left(z^{\ast }\right)}{f\left(z^{\ast }\right)}.\end{array}$(7)

Corollary 3.1

The optimal value corresponding to the WTP exists and is always higher than the outside price $p_0^b$ reflecting the belief held by the buyer that there is a non-zero probability that the actor may be selling a quality product such that it would be slightly better than what can be found on the market from an untrusted seller.

Corollary 3.2

When the buyer does not trust the supplier, the Bayesian belief distribution will have a mean of $p_0^b$ and a standard deviation of $\sigma =0$, so that $z^{\ast }=p_0^b$.

From the above, it appears clearly that this WTP evolves both with the mean and with the standard deviation of the probability distribution of the Bayesian belief. The buyer builds this distribution in terms of the trust inspired by the actor and the information available: trust but verify (Russian proverb used by Ronald Reagan). The more information about the quality is provided by the actor, the higher the mean of the random variable. Moreover, she will be considering that the potential distribution of such true quality cannot vary wildly: the higher the trust, the lower the standard deviation. So, the following Proposition can be enunciated.

Proposition 3.1

WTP is built upon an a priori Bayesian belief which has a probability distribution $F\left(z\right)$ with μ and σ as first and second moments. Hence we can relate the effort ${\text{β}}_a$ to build such WTP which stems from the shape of the Bayesian belief distribution function. This relationship is designated here by $G_a(.)$ and $H_a(.)$, as follows: (8) $\begin{array}{rl}\mu & =G_a\left({\text{β}}_a\right),\\ \sigma & =H_a\left({\text{β}}_a\right),\end{array}$(8) where $G_a(.)$ is a strictly increasing function, whereas $H_a(.)$ is a strictly decreasing one over the domain of the possible values for ${\text{β}}_a$, presumed to be in a closed set.

Corollary 3.3

When the buyer has an imprecise notion of the quality of the product, she will increase the variance of the Bayesian belief distribution. The higher the variance, the lower the estimate of the price the buyer is willing to pay $\hat{z}$. If the buyer expects the quality of the product to be low, she will lower the mean of the belief distribution. The actor's effort ${\text{β}}_a$ in trust building is to induce the buyer into increasing the mean and lowering the variance of the Bayesian belief distribution.

To link back to the system we are suggesting, if, for example, the buyer does not believe in the information shared on the blockchain by the actor as to the true quality, the former's WTP will be lower. The only case when the buyer is willing to pay $p_a$ is when information available indicates the advertised quality is the right one, $\mu =p_a$, and can be trusted, $\sigma =0$, hence $\hat{z}=p_a$.

This means that $H_a\left({\text{β}}_a\right)\ge 0$ and $p_0^b\le G_a\left({\text{β}}_a\right)\le p_a$. To understand in a more visual way how this works, we refer the reader to the numerical illustration in Figure 4. Therein, note that the optimal price that the buyer is willing to pay increases with the mean of the Bayesian belief distribution and decreases with the variance of such distribution.

Figure 4. Representation of the price the retailer is willing to pay in terms of the standard deviation of the Bayesian belief: the optimal price $z^{\ast }$ increases as the precision and confidence in the validity of the expected decreases (except when the variance is so large that the buyer is misled so leading to a higher optimal price as can be observed when $\sigma >7$).

Graph shows evolution of WTP in terms of standard deviation of the WTP for two means of the Bayesian belief distribution. Top curve is with μ=50, bottom with μ=35.

To the actor, the cost of building WTP is hypothesised as being increasingly costly, so that $T_a\left({\text{β}}_a\right)$ is a strictly increasing convex function, which is plausible due to decreasing marginal considerations.

Remark 3.5

Note that the case where an actor cheats or misrepresents the quality of the product is included in the model. Either μ will be lower or σ will be larger, thus reducing WTP. In the extreme, governance mechanisms to punish cheating can be triggered so that the actor is kicked out of the chain, or a penalty can be levied. We discuss the impact for the whole supply chain in Section 3.5.

3.4. Building WTP in the whole supply chain

We now extend the dynamics of the triplet of players developed above to the whole supply chain. We refer the reader to Figure 1: in the initial phase, the supply chain partners have decided to set up the chain, have invested in the corresponding specific assets, have trained the operators required by the BT, and set up governance mechanisms. The production phase is when all partners produce the bottled olive oil and engage in the necessary effort to ensure that the proper information about the oil is registered and shared between all chain members.

Obviously, if a consumer trusts the retailer on the quality of the product, she must trust by extension all the supply chain upstream to this retailer. Because we are in a case where the retailer represents a set of agents who can engage in a number of actions, this problem can be assimilated to the agency problem with a principal and multiple agents who can engage in multiple actions of which only a limited subset will be approximately incentive compatible for a transaction to take place because the outcomes of these actions constitute a compact space (Dütting, Roughgarden, and Cohen, 2020). If one upstream supplier to the retailer cheats or otherwise does not sell the expected quality raw or semi-processed material, then the final product cannot be said to comply with the quality expected by the consumer, thus leading to a breach of trust. As explained earlier, this breach of trust will lead the consumer, if she still wishes to buy from the retailer, to revise her belief by increasing the standard deviation of the distribution of that belief. In the worst case, as mentioned in Corollary 3.2, the alternative for the consumer is to buy a no-name product but at the price for a product of unknown quality. In this case, her WTP is null, as is her utility.

In the general case, for the different actors in the chain, the profit functions can be established in terms of the effort deployed by each as (9) $\begin{array}{rl}{\mathrm{\Pi }}_f\left({\text{β}}_f\right)& =p_f^{\ast }\left({\text{β}}_f\right)-T_f\left({\text{β}}_f\right),\\ {\mathrm{\Pi }}_m\left({\text{β}}_m\right)& =p_m^{\ast }\left({\text{β}}_m\right)-p_f^{\ast }\left({\text{β}}_f\right)-T_m\left({\text{β}}_m\right),\\ {\mathrm{\Pi }}_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)& =p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-p_m^{\ast }\left({\text{β}}_m\right)-T_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right),\\ {\mathrm{\Pi }}_r\left({\text{β}}_r\right)& =p_r^{\ast }\left({\text{β}}_r\right)-p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-T_r\left({\text{β}}_r\right),\\ {\mathrm{\Pi }}_c& =u\left({\text{β}}_r\right)-p_r^{\ast }\left({\text{β}}_r\right),\end{array}$(9) where ${\mathrm{\Pi }}_c$ is the utility obtained by the consumer from consuming the olive oil and the star in superscript denotes the optimal price achieved because the WTP of each buyer in turn is at its highest. Each of those prices is evaluated using (7(7) $\begin{array}{r}{z}^{\ast }-p_0^b=\frac{\overline{F}\left(z^{\ast }\right)}{f\left(z^{\ast }\right)}.\end{array}$(7) ) in Theorem 3.1.

BT can provide the backbone along the supply chain for information to be shared but can not guarantee that the only true information is registered due to the ‘trust-frontier’ (Altmann et al., 2019, Glaser, 2017, Hawlitschek, Notheisen, and Teubner, 2018, 2020). The necessary condition for trust to emerge can only come from constant effort, captured in our model by β, by all members of the chain (${\text{β}}_f,{\text{β}}_m,{\text{β}}_{\mathrm{bo}},{\text{β}}_r$) in ensuring that the information is truthful and verifiable (Chaudhuri et al., 2021, Dolgui et al., 2019, Giovanni, 2020, Shermin, 2017).

The retailer has all the arguments to inform the consumer about that true and exact quality. The retailer can enhance this visibility effort by providing through a QR code label on the final product's container the full quality report (Bumblauskas et al., 2020).

In the case where the BT has been deployed throughout the supply chain and the required investment costs have been incurred (meaning that they are now sunk), the overall chain's total profit function in terms of the effort decisions $({\text{β}}_f,{\text{β}}_m,{\text{β}}_{\mathrm{bo}},{\text{β}}_r)$ of ongoing continuous effort developed by each chain member (farm, mill, bottler, retailer), using Equation (9(9) $\begin{array}{rl}{\mathrm{\Pi }}_f\left({\text{β}}_f\right)& =p_f^{\ast }\left({\text{β}}_f\right)-T_f\left({\text{β}}_f\right),\\ {\mathrm{\Pi }}_m\left({\text{β}}_m\right)& =p_m^{\ast }\left({\text{β}}_m\right)-p_f^{\ast }\left({\text{β}}_f\right)-T_m\left({\text{β}}_m\right),\\ {\mathrm{\Pi }}_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)& =p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-p_m^{\ast }\left({\text{β}}_m\right)-T_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right),\\ {\mathrm{\Pi }}_r\left({\text{β}}_r\right)& =p_r^{\ast }\left({\text{β}}_r\right)-p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-T_r\left({\text{β}}_r\right),\\ {\mathrm{\Pi }}_c& =u\left({\text{β}}_r\right)-p_r^{\ast }\left({\text{β}}_r\right),\end{array}$(9) ), can be evaluated as (10) $\begin{array}{rl}& \mathrm{\Pi }({\text{β}}_f,{\text{β}}_m,{\text{β}}_{\mathrm{bo}},{\text{β}}_r)=p_r^{\ast }\left({\text{β}}_r\right)-\sum _i{T}_i\left({\text{β}}_i\right),\\ & i\in \{f,m,\mathrm{bo},r\}.\end{array}$(10) Each member of the chain has a distinct cost function of the effort to build trust so that, at the retail level, the retailer can charge the optimal price in relation to WTP $p_r^{\ast }\left({\text{β}}_r\right)$ to the consumer, a higher price than in the case of a chain where BT has been deployed but information about the true quality of the product is not verified because the chain partners have not deployed effort to do so.

Given both Equations (9(9) $\begin{array}{rl}{\mathrm{\Pi }}_f\left({\text{β}}_f\right)& =p_f^{\ast }\left({\text{β}}_f\right)-T_f\left({\text{β}}_f\right),\\ {\mathrm{\Pi }}_m\left({\text{β}}_m\right)& =p_m^{\ast }\left({\text{β}}_m\right)-p_f^{\ast }\left({\text{β}}_f\right)-T_m\left({\text{β}}_m\right),\\ {\mathrm{\Pi }}_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)& =p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-p_m^{\ast }\left({\text{β}}_m\right)-T_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right),\\ {\mathrm{\Pi }}_r\left({\text{β}}_r\right)& =p_r^{\ast }\left({\text{β}}_r\right)-p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-T_r\left({\text{β}}_r\right),\\ {\mathrm{\Pi }}_c& =u\left({\text{β}}_r\right)-p_r^{\ast }\left({\text{β}}_r\right),\end{array}$(9) ) and (10(10) $\begin{array}{rl}& \mathrm{\Pi }({\text{β}}_f,{\text{β}}_m,{\text{β}}_{\mathrm{bo}},{\text{β}}_r)=p_r^{\ast }\left({\text{β}}_r\right)-\sum _i{T}_i\left({\text{β}}_i\right),\\ & i\in \{f,m,\mathrm{bo},r\}.\end{array}$(10) ), it is clear that the price of the olive oil must be maximised under the constraint of positive utility of the consumer and from the overall cost of the effort in building trust from below for extra profit to be captured by the chain partners. In this way, each partner in the chain can sell at a higher price if his downstream partner trusts the quality of the product being sold. For instance, if the farmer chooses to deploy an effort to increase the trust that the mill has in his product, she will be able to sell at a higher price than if she would not. Hence, each actor can choose to enhance this trust through the proper effort in monitoring and reporting quality so that the consumer may trust that the quality of the product is due to the overall care along the chain in ensuring the highest quality.

It is obvious that all the constraints of the chain members in Equation (5(5) $\begin{array}{r}\{\begin{array}{l}{p}_a\left({\text{β}}_a\right)\ge p_s\left({\text{β}}_s\right)+T_a\left({\text{β}}_a\right),\\ p_b\left({\text{β}}_b\right)\ge p_a\left({\text{β}}_a\right)+T_b\left({\text{β}}_b\right),\\ u\left({\text{β}}_r\right)\ge p_r\left({\text{β}}_r\right)\end{array}\end{array}$(5) ) must also be met. The chain members have an overall incentive to lower the total cost of this effort. There is no misalignment of incentives here as each member must also maximise her own profit function in terms of her effort to build WTP, but she must also reduce or minimise her own cost of effort.

This guaranteed quality is a factor for increased market share, increased profit for all, and maximised utility for the consumer.

The result achieved above is due to the visibility and corresponding WTP that the deployment of BT and of the ongoing trust enhancing effort provides to the different actors.

3.5. Opportunistic behaviour and effect in the chain

Building from Remark 3.5, in one period after having invested in trust-building effort, one actor may cheat or multiple actors in the chain may enter into a coalition to defraud the remaining members on the quality of the product sold. For this to be possible, data on the blockchain database must be inconsistent with the reality of the product's quality. Now, as part of their constant effort to control and ensure that digital information in the Blockchain is truthful and verifiable, the cheated partners should be able to observe the inconsistency between digital records and the true quality of the product (for how this is to be done, we refer the reader to Section 2.2).

In the following period, the buyer is now saddled with a tainted product which can no longer be sold as a quality product. If the buyer does nothing and sells on the tainted product, that loss ripples down to the other actors downstream from the cheater as the batch of tainted oil progresses in the chain since the WTP is now lower. When this tainted product comes to market, given the corresponding batch information available to the consumer, her WTP decreases. Further, as reported in Section 2.1 of the literature review, consumers will share information about the true quality further eroding WTP and market participation.

It is easily understood that no downstream partner from the opportunistic one should invest in trust-building effort $T_a\left({\text{β}}_a\right)$ for this batch (and possibly ulterior ones) since this effort cannot be recouped through the selling price and violates the participation constraints in Equation (5(5) $\begin{array}{r}\{\begin{array}{l}{p}_a\left({\text{β}}_a\right)\ge p_s\left({\text{β}}_s\right)+T_a\left({\text{β}}_a\right),\\ p_b\left({\text{β}}_b\right)\ge p_a\left({\text{β}}_a\right)+T_b\left({\text{β}}_b\right),\\ u\left({\text{β}}_r\right)\ge p_r\left({\text{β}}_r\right)\end{array}\end{array}$(5) ). The best strategy is not to sell the intermediate or final product through the regular channel but dump it on a buyer willing to pay the unknown quality price $p_0^b$ and forfeit the corresponding profit but at least keep the reputation and hence, the WTP intact.

To prevent opportunistic behaviour, the supply chain partners should adopt the governance mechanism which matches the characteristics of the relationships at different levels (Ghosh and Fedorowicz, 2008, Kittilaksanawong, 2016). We argue that in the context of a supply chain which intends to set up an information system linking all partners to share information about quality, such governance mechanisms must be set up at the same time to deal with all the possible issues related to quality at the different levels. For example, if a batch lacks the proper quality, the actions necessary for its withdrawal from the chain must be scripted and the corresponding cost attributed.

In case of opportunistic behaviour occurring in the production phase, the governance mechanism that the partners set up in the initial phase for just such a case is triggered. Depending on the severity of the case and the balance between value creation and appropriation (Kittilaksanawong, 2016), punishment by exclusion from the chain of the cheating actor is possible (the game is a repeated one, Axelrod, 1981). In a multi-period setting, even if the same partner is still in the chain, trust will have been reduced and so will the WTP of the downstream partner buying from the opportunistic one for a number of periods. This entails that, once trust building has started, the best strategy for all actors is to maintain the effort and receive $p_a^{\ast }\left({\text{β}}_a\right)$ in every period (it is reasonable to take into account the discounted future profits as described in Fudenberg and Maskin, 1986).

To show this, we model the corresponding behaviour as a Nash equilibrium multi-period game with five players (the four of the chain plus the consumer), where a cheating strategy is observable. In any triad, suppose that actor a can cheat, then in the period where he may cheat (11) $\begin{array}{rl}{\mathrm{\Pi }}_a\left(\text{β}\right)& =\delta [p_a-p_s({\text{β}}_s\left)\right]\\ & +(1-\delta )\left[p_a\right({\text{β}}_a)-p_s({\text{β}}_s)-T_a({\text{β}}_a\left)\right]\end{array}$(11) where $\delta \in \{0,1\}$, the decision to cheat or not. Clearly, in this period, the actor's profit is higher if $\delta =1$ and he obtains a ‘short-run gain’ in the sense of Shapiro (1983). However, in the next period, this cheating action is now observed by the other actors, including the consumer, because the lower quality has been recorded on the immutable ledger. This means that all the downstream actors of the chain will refuse to buy from the cheater from that next period on. Hence, over n periods, if he decides to cheat in period 1<k<n, we can write the cheater's profit function as (12) $\begin{array}{rl}{\mathrm{\Pi }}_a({\text{β}}_a,{\delta }_k)& =(k-1)\left[p_a\right({\text{β}}_a)-p_s({\text{β}}_s)-T_a({\text{β}}_a\left)\right]\\ & +\left[p_a\right({\text{β}}_a)-p_s({\text{β}}_s\left)\right]+(n-k)p_0^b.\end{array}$(12) Since $p_0^b<p_a\left({\text{β}}_a\right)-T_a\left({\text{β}}_a\right)$, it is clear that the decision to cheat is detrimental to the actor's multi-period profit and that his Pareto improving strategy is to stick with a trust-building strategy (Lahno, 2004). It is easy to demonstrate also that each of the supply chain partners' profit is reduced because of the consumer's decision to stop buying from this supply chain.

4. Numerical illustration

This illustration is divided in three parts: in the first we show how a buyer (in this first part, the retailer) will build her optimal price she is willing to pay. In the second, we illustrate this Bayesian belief held by each buyer can be modified by managerial effort. In the third part, using actual price data for the intermediate products in the extra virgin olive oil supply chain, we illustrate the model for each player in the chain and the overall supply chain extra profit.

4.1. Evaluation of the optimal price

Let us identify in this illustration the actor of our model as a bottler and the buyer as a retailer. Suppose that the bottler wants to sell to the retailer. As a bottler, he has access to several sources of olive oil. Suppose he behaves opportunistically, he can choose to mix the premium product with a given quantity of a product of lesser value, which will then enable her to increase his margin to the detriment of the quality of the product sold on to the retailer. Let us consider the following parameters for a litre of olive oil. (13) $\begin{array}{r}{p}_0^r=10,z=50,\end{array}$(13) where $z$ is the quality product value and $p_0^r$ is the standard quality alternative available widely on the market. In a supply chain without deployment of BT, under the assumption of asymmetric information, two belief situations may be distinguished. In the first, the retailer may take at face value the statement of the bottler about the premium quality of the product and yet not trust him entirely. In the second, she may consider that the information about the quality is at best dubious. We deal with both situations as follows.

In the first case, the retailer will build her belief of the true value of $\hat{z}=50$ a distribution function with the corresponding mean and standard deviation. Let us consider here an IFR distribution such as a truncated normal distribution function and $Z\sim \mathcal{N}(\mu =50,\sigma =10)$. In this case, according to (7(7) $\begin{array}{r}{z}^{\ast }-p_0^b=\frac{\overline{F}\left(z^{\ast }\right)}{f\left(z^{\ast }\right)}.\end{array}$(7) ), we have $z^{\ast }=40.93$. In the case of dubious quality, let us suppose that she estimates $\hat{z}=\mu =30$, and has a lower belief of the true quality, so that she will assume that the estimate has a broader distribution $Z\sim \mathcal{N}(\mu =30,\sigma =15)$. This leads to $z^{\ast }=29.41$. These values are the prices that the retailer is willing to pay the bottler (her WTP) based upon her belief of the true quality of the olive oil in the two situations of belief. In Figure 4 we plot, in two different hypotheses of standard deviations, the retailer's WTP when her estimate of the true quality of the olive oil varies from the one with minimum value $p_0^r$ to the maximum quality. We see that the willingness increases with her estimate of the quality but never reaches the price she would pay if she were certain of the quality (dashed line). Note also that, by construction, the lower bound of this WTP is the price for the alternative product with no guarantee of quality (here set at $p_0^r=10$). The upper bound for μ is when an optimal value can no longer be found solving (7(7) $\begin{array}{r}{z}^{\ast }-p_0^b=\frac{\overline{F}\left(z^{\ast }\right)}{f\left(z^{\ast }\right)}.\end{array}$(7) ), in our case $\mu =57$ when $\sigma =15$ and $\mu =64$ when $\sigma =5$ does not admit an optimal $z^{\ast }$.

4.2. Evaluation of the optimal effort by each chain player

Let us now suppose that the partners in the chain have proceeded with the initial phase as illustrated in Figure 1. The production phase starts and trust-building effort can take place. We now present an illustration of Proposition 3.1: how the trust-building effort of the bottler improves the retailer's WTP. To do so, we must characterise the functions $H_a$ and $G_a$ which represent the connection between the effort and the first and second moment of the Bayesian belief distribution function f of the retailer.

We present two cases, in the first, the functions are linear, and in the second polynomial. $\begin{array}{rlrl}& \text{linear}& & \text{polynomial}\\ G\left({\text{β}}_{\mathrm{bo}}\right)=& 53{\text{β}}_{\mathrm{bo}}-3& G\left({\text{β}}_{\mathrm{bo}}\right)=& 50{\text{β}}_{\mathrm{bo}}^3-3\text{β}+12,\\ H\left({\text{β}}_{\mathrm{bo}}\right)=& -15{\text{β}}_{\mathrm{bo}}+16& H\left({\text{β}}_{\mathrm{bo}}\right)=& -10{\text{β}}_{\mathrm{bo}}^2+11.\end{array}$Using a truncated Normal distribution for the Bayesian belief with $\mu \left(\text{β}\right)=G\left(\text{β}\right)$ and $\sigma \left(\text{β}\right)=H\left(\text{β}\right)$, we see in Figure 5(a) how optimal $z^{\ast }$ is obtained from the trust-building effort. The evolution of the bottler's effort thus illustrates Corollary 3.3.

Figure 5. Representation of the WTP price $z^{\ast }$ in terms of the effort β. (a) when the relationships between the effort and the Bayesian belief distribution parameters are linear and (b) when the relationships between the effort and the Bayesian belief distribution parameters are polynomial.

When the relationships between the effort and the Bayesian belief distribution parameters are linear, WTP is convex upwards. In the case of a polynomial relationship (i) the structure of the behaviour is decreasing and then increasing; (ii) it reaches the upper bound when βbo→1, unlike the linear case.

In the second case, the relationships are polynomial and represented in Figure 5(b):

We note that (i) the structure of the behaviour is decreasing and then increasing; (ii) it reaches the upper bound when ${\text{β}}_{\mathrm{bo}}\to 1$, unlike the linear case.

Next, we explore the impact of the cost of effort $T_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)$ on the bottler's profit function. We use a suitably strictly increasing concave function for the cost of building trust such as (14) $\begin{array}{r}{T}_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)=25{\text{β}}_{\mathrm{bo}}^2+5{\text{β}}_{\mathrm{bo}}.\end{array}$(14) Assume that $p_m\left({\text{β}}_m\right)=12$ so that the profit function in (1(1) $\begin{array}{r}{\mathrm{\Pi }}_a\left({\text{β}}_a\right)=p_a\left({\text{β}}_a\right)-p_s\left({\text{β}}_s\right)-T_a\left({\text{β}}_a\right),\end{array}$(1) ) becomes (15) $\begin{array}{r}{\mathrm{\Pi }}_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)=p_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)-12-12{\text{β}}_{\mathrm{bo}}^2-5{\text{β}}_{\mathrm{bo}},\end{array}$(15) with $p_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)$ solving the retailer's optimal WTP from Theorem 3.1. The corresponding profit function is presented in Figure 6. In this illustration, the bottler should either provide no effort at all or maximise it. Witness how profit increases even with a strongly convex cost $T_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)$. Note that even in this specific instance where it would appear that the bottler is better off by shirking, such behaviour would backfire as the bottler would be punished and obtain a loss in the following period.

Figure 6. Representation of the actor's profit function in terms of the trust-building effort ${\text{β}}_a$.

the curve is concave and exhibits a minimum around an effort βa=0.64.

4.3. Stylised numerical study of an extra virgin olive oil supply chain

To support the above calculations, we have collected information about actual costs for producing the different intermediate products or the cost of retailing a bottle of extra virgin olive oil (see Table 1). The price of a kilogramme of olives cost 2.75 €/kg (average weighted price in 2012 in Spain). The average cost of crushing olives is 0.03 €/kg (Spain). The average volume of a kg of olive oil is 1.12/litre. The bulk olive oil can be bought for 3.88 €/litre (average cost of imported bulk olive oil in France in 2019 Autentika Global, 2020). As for the crushing, the bottling cost including bottle, label and carton cases is approximately 0.35 €/litre. The wholesale price of a bottle to the retailer is approximately 4.5€/l which then retails for 7.5€/l. Those are presented in Table 1. We evaluate the overall profit of the chain using results from Section 3.4. We consider that all chain members do act in a way to maximise WTP (that is, β is maximal).

Table 1. Approximate selling price, transformation costs of Extra Virgin Olive Oil, optimal price and profit (€/litre) in the European market in 2020 (Autentika Global, 2020, Barjol et al., 2015, Devarenne, 2021).

	Selling price range €/litre
Chain partner	$p_0^i$	$z^{\ast }$	Effort cost	Transformation cost	Profit per level
Farm	2.45	4.66	0.61	2.4	1.65
Mill	3.88	7.39	0.97	0.03	1.72
Bottler	4.6	8.76	1.15	0.05	0.17
Retailer	7.5	14.29	1.87	1.5	2.34

To represent the cost of managerial effort in a plausible manner, we propose to evaluate it as a cost to be divided into litres of final product (that is, litres of olive oil) for each echelon. Traditionally, such cost of effort function should be convex to represent the fact that improving quality has an increasing marginal cost. We adapt the cost of building trust from Equation (14(14) $\begin{array}{r}{T}_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)=25{\text{β}}_{\mathrm{bo}}^2+5{\text{β}}_{\mathrm{bo}}.\end{array}$(14) ) so that $T_i\left({\text{β}}_i\right)=\left(p_0^{i2}/2\right){\text{β}}_i^2$, $i\in \{f,m,\mathrm{bo},r\}$. The Bayesian belief distribution is evaluated as a truncated normal distribution with the mean and variance being functions of effort as in (8(8) $\begin{array}{rl}\mu & =G_a\left({\text{β}}_a\right),\\ \sigma & =H_a\left({\text{β}}_a\right),\end{array}$(8) ) and using the corresponding prices for the alternative price for non-descript product in each echelon using the $p_0^i$, $i\in \{f,m,\mathrm{bo},r\}$, from Table 1 so that $G\left({\text{β}}_i\right)=p_0^i{\text{β}}_i^3-\left(p_0^i/5\right){\text{β}}_i+p_0^i$ and $T\left({\text{β}}_i\right)=-\left(p_0^i/5\right){\text{β}}_i^2+p_0^i$. We obtain the optimal selling prices in the second column of Table 1. In the same way, when we consider that the effort at each level is ${\text{β}}_i=1$, $i\in \{f,m,\mathrm{bo},r\}$ and that the effort cost function at each level is $T_i\left(\text{β}\right)=(p_0^i/2{)}^2{\text{β}}^2$, the profit by each partner before transformation cost can be evaluated from Equation (9(9) $\begin{array}{rl}{\mathrm{\Pi }}_f\left({\text{β}}_f\right)& =p_f^{\ast }\left({\text{β}}_f\right)-T_f\left({\text{β}}_f\right),\\ {\mathrm{\Pi }}_m\left({\text{β}}_m\right)& =p_m^{\ast }\left({\text{β}}_m\right)-p_f^{\ast }\left({\text{β}}_f\right)-T_m\left({\text{β}}_m\right),\\ {\mathrm{\Pi }}_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right)& =p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-p_m^{\ast }\left({\text{β}}_m\right)-T_{\mathrm{bo}}\left({\text{β}}_{\mathrm{bo}}\right),\\ {\mathrm{\Pi }}_r\left({\text{β}}_r\right)& =p_r^{\ast }\left({\text{β}}_r\right)-p_{\mathrm{bo}}^{\ast }\left({\text{β}}_{\mathrm{bo}}\right)-T_r\left({\text{β}}_r\right),\\ {\mathrm{\Pi }}_c& =u\left({\text{β}}_r\right)-p_r^{\ast }\left({\text{β}}_r\right),\end{array}$(9) ). We present in the last column of Table 1 the profit made by each level net of transformation cost.

This numerical example comforts the results obtained in Dionysis, Chesney, and McAuley (2022) where participants in the study were willing to pay between 5 and 30% more for an organic coffee with blockchain guaranteed traceability as opposed to a traditionally certified organic coffee.

5. Discussion and implications

This paper complements prior literature on the applications of BT infrastructure can realise interorganisational processes, increase trust among partners, and achieve traceability in food supply chains (A. Kumar, Liu, and Shan, 2019). Our contribution spans two levels. The first is a managerial level: it shows how managers can improve the management of their supply chains. We cover this in the next section. On the second level, this paper presents new insights in how WTP and BT can be combined to improve overall coordination in a supply chain.

5.1. Managerial insights

On the first, our contribution is in showing how such technology has to be complemented by managerial effort and can be paid for because consumers are willing to pay more for products from such supply chains (Dionysis, Chesney, and McAuley, 2022, showed that properly informed consumers are willing to pay more).

It follows from the discussion presented in Section 2.2 that the deployment of a BT solution requires substantial outlays. But, as said in W. Wang, Lumineau, and Schilke (2022), ‘managers should consider blockchains as an important strategic tool to organise collaborations, (…) [they] should consider the joint use of different approaches to mitigate collaborative hazards and enhance efficiency.’ This technology is still in the process of developing a common definition among both managers and researchers. Quickly changing vocabulary and evolving protocols represent challenges when describing policies. The deployment of a substantial number of sensors within an IoT network generates data which then have to be graded in terms of veracity (verification of the sensors), completeness (no lost or unrecorded data points), and certified by oracles. Such oracles are then used in a cascade of smart contracts so that the data, essential to establish trust among the supply chain players, are considered trustworthy. Given the amount of data, the number of smart contracts and oracles, such trustworthiness can not be a binary result but will be based on a score-based evaluation of the products and players involved (Al-Rakhami and Al-Mashari, 2021).

Governance mechanisms have to be deployed and carefully designed ex ante to achieve coordination between the actors and ensure proper deployment and use of the corresponding BT outlays. In particular, effort has to be deployed throughout the chain to ensure the evolution of the BT to follow the evolution of the market, the suppliers, and the retail channels. Some of the necessary effort that will have to be deployed includes the training of technical operators to monitor the calibration and operation of field sensors, as well as updating the programs (W. Wang, Lumineau, and Schilke, 2022). The outlays in information systems and sensors require more than ongoing maintenance. Managers will have to be trained in ensuring that the reports and monitoring software are indeed operational and correctly understood and manipulated by the relevant operational staff (as has been implemented in Y. Wang, Huirong Chen, and Zghari-Sales, 2020). Smart contracts will also warrant careful consideration (A. Kumar, Liu, and Shan, 2019). When stepping back, we venture to say that such routines, processes and skill sets are akin to those necessary in total quality management and would provide a specific competitive advantage as in the dynamic capabilities approach of the resource-based view of the firm (Zollo and Winter, 2002).

Progress towards trustworthy food products has stalled because of the heterogeneity in managerial sophistication of partners in any food supply chain, whether in the case of BT (Mathivathanan et al., 2021), or in the case of regulatory frameworks (Yinghua, Ningzhou, and Dan, 2018) based on information technology (Harland et al., 2007, A. Zhang, Mankad, and Ariyawardana, 2020), this is why we believe that our two-pronged approach provides a possible way forward. Given the incentives to cut corners and the variety of players in a food supply chain, food scandals can always happen, whatever the sophistication of the technology used. Hence we strongly suggest that managers complement the BT by setting up ex ante governance mechanisms, and ex post continuously oversee and control the whole supply chain so that consumers are informed about and are willing to pay for the quality they are expecting. In this way, the possibility of cheating is weened out by the combination of strong oversight and risk of penalties (which can reach exclusion of the cheater from the chain). The present research provides clear indications to help managers in this endeavour and also opens up new avenues in supply chain management modelling as we new describe.

5.2. Scientific results

On the second level, our model establishes in a normative and prescriptive manner how managerial effort at each level (a) ensures that the information about quality of a product is trustworthy, (b) presents the evidence to the intermediate buyers of the semi-finished product and to the consumer, (c) leads to a higher WTP for all the members of the chain, and (d) generates a higher profit for all of them. We show how each actor is responsible for the effort required to provide evidence of the quality of the product at her level. Her profit also increases insofar as her effort includes verifying that her partners upstream and downstream do their part so that the final product benefits from the consumer's highest WTP as shown in the numerical illustration using true prices of the intermediate products.

We model how effort and WTP are positively related. This relationship is based on the modification of the a priori Bayesian belief of each buyer in the chain (including the consumer) about the quality of the intermediate (final) product sold. Two outcomes are presented, one when information is true and WTP is high, the other when opportunistic behaviour by one or several actors entails loss of WTP and profit.

In other words, the overall chain-wide profit is linked to the consumer's WTP. Such a result and such a model have never yet been described. In the present model, the information about the quality of a product is linked to the consumer's WTP and to the corresponding managerial effort by the actors. In general, in literature, two unproven statements are assumed: (i) WTP increases only with user experience, and (ii) information about the quality of a product is always trustworthy. As ample evidence both in scientific literature and the general press can attest, consumers are sensitive to the information about the quality of the food products they buy, and do not take for granted that such information is trustworthy.

Our paper contributes to the stream of literature on the use of game theory and Bayesian behaviour to multi-level supply chains, an avenue which was suggested in Agi, Faramarzi-Oghani, and Hazır (2020).

The present study suffers from the usual limitations due to the use of a crude representation of the possible strategies that the players can apply in a stylised supply chain. The results would of course have been of higher value had we been able to collect the true values of the cost of the full deployment of a chain-wide blockchain and IoT system. As far as we know there are no full deployment of BT in an olive oil supply chain, only partial implementations. The costs for the fours intermediate products are also only averages of various types of different qualities. They were chosen as representative values for such products to give an idea of the scale.

This research could be usefully extended by looking into the actual deployment of a BT information systems which would include all echelons of a food supply chain, from the producer to the retailer. As in the case of the idealised case here, such a chain might include a transformation step performed by a manufacturer and a retailer before reaching the final consumer. Another interesting research avenue might be to further investigate the Bayesian belief updating mechanism in play when a buyer has to evaluate the true quality of a product which has gone through several echelons in a chain.

Acknowledgment

We wish to thank here Anna Nagurney who most kindly encouraged us when we presented this paper at the CLAIO (Conferencia Latino Americana en Investigación de Operaciones) XXI Latin-Iberoamerican Conference on Operations Research in Buenos Aires, Argentina, on December 12, 2022.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors confirm that the URL of the sources of the data supporting the findings of this study are available within the article and in the references. The corresponding freely available data have been obtained from Autentika Global (2020), Barjol et al. (2015), and Devarenne (2021).

Additional information

Notes on contributors

Xavier Brusset

Xavier Brusset holds a Ph.D. in Management Science from the Louvain Catholic University and an Habilitation à Diriger des Recherches from Paris Ouest Nanterre La Défense university. He has been teaching Logistics and Supply Chain Management at the Toulouse Business School, ESSCA and now SKEMA since 2009. His research focuses on the relationship between the supply chain partners and the impact of information on their behaviour. His findings have been published in academic journals such as the Journal of Operations Management, European Journal of Operational Research, the International Journal of Production Economics, Computers and Industrial Engineering, RAIRO Operations Research, la Revue Française de Gestion Industrielle (French Industrial Management Review). He edited and co-authored a textbook about business cases in distribution. Previously, Xavier also worked in financial markets and created in Argentina a web-based platform of information sharing and logistic services between shippers and carriers (WebLogistix). Every year, he organises or co-organises a Colloquium on European Research in Retailing (CERR). He is a recognised expert assessing the European Union in its research project funding. He is on the editorial board of the journal Logistics Research and a regular guest editor of the International Journal of Retail & Distribution Management.

Aseem Kinra

Aseem Kinra heads the professorship in Global Supply Chain Management at the University of Bremen. Prof. Kinra received his bachelor at the Delhi University followed by a master's degree in business administration. He went on to accomplish an M.Sc. in Economics and Business Administration, followed by a Ph.D. in Supply Chain Management at the Copenhagen Business School, where he graduated in 2009. He worked as an as an Assistant and later as an Associate Professor at the Copenhagen Business School, where he also led the Graduate Diploma programme in Supply Chain Management. His research has previously appeared in journals such as International Journal of Production Economics, International Journal of Production Research, International Journal of Physical Distribution and Logistics Management and the International Journal of Operations and Production Management.

Hussein Naseraldin

Dr. Hussein Naseraldin is a Senior Lecturer in the Department of Industrial Engineering and Management at Braude College of Engineering in Karmiel, Israel. He is the head of the Master in Science Program in Industrial Engineering and Management. He earned his Ph.D. from the Faculty of Data and Decision Sciences (previously, Industrial Engineering and Management) at the Technion – Israel Institute of Technology in 2006. In his graduate studies, he focused on the integration of strategic and operational decisions in a supply chain context. Prior to his current position, he spent three years as a post-doctoral fellow at the Haskayne School of Business at the University of Calgary, the Rotman School of Management at University of Toronto, and in the Department of Statistics at the University of Haifa. His research interests include policies and approaches to improve the performance of the supply chain utilising a myriad of innovative technologies, e.g. 3D Printing, AI, and Digital Twin. Before completing his graduate studies, he worked in the industry for eight years in various operational and managerial positions, the last of which was Quality Assurance Manager.

Rami Alkhudary

Rami Alkhudary, Ph.D., is an Assistant Professor at the University of Paris-Panthéon-Assas, where he also earned his Ph.D. in Management Science, focusing on the utilisation of blockchain technology in supply chains. In recognition of his research, he received the university's Thesis Award in Management Science in 2021. His current research centres on emerging technologies in supply chain and project management. Rami's work has gained acclaim at various international conferences, notably winning the Best Research Paper Award at PROLOG 2019. His publications have been featured in prestigious journals such as the International Journal of Project Management, Marketing Letters, European Business Review, Supply Chain Forum, International Journal of Retail & Distribution Management, Journal of Consumer Behaviour, Information Processing and Management, and Harvard Business Review France. Additionally, Rami actively contributes as a reviewer for numerous international journals and conferences.

Appendix. Proof of existence of a unique $z^{\ast }$ in Theorem 3.1

This proof has first been established in Brusset and Cattan-Jallet (2009) and also used in Brusset (2014) and Brusset and Agrell (2017). The first differential of the expected buyer's profit function in terms of the threshold level z is written (A1) $\begin{array}{r}\frac{\mathrm{\partial }{\mathrm{\Pi }}_b\left(z\right)}{\mathrm{\partial }z}=f\left(z\right)(p_0^b-z)-F\left(z\right)+1.\end{array}$(A1) For threshold z to be a maximising one in terms of profit to the buyer, we must have as $\frac{\mathrm{\partial }{\mathrm{\Pi }}_b\left(z\right)}{\mathrm{\partial }z}=0$, $z\in [\underset{\_}{Z},\overline{Z}]$, $\frac{{\mathrm{\partial }}^2{\mathrm{\Pi }}_b\left(z\right)}{\mathrm{\partial }{z}^2}<0$, $z\in [\underset{\_}{Z},\overline{Z}]$. The first-order condition (FOC) is (A2) $\begin{array}{r}{p}_0^b-z=-\frac{\overline{F}\left(z\right)}{f\left(z\right)},\end{array}$(A2) and as second differential, under the restriction that $f\left(z\right)\ne 0$, (A3) $\begin{array}{r}\frac{{\mathrm{\partial }}^2{\mathrm{\Pi }}_b\left(z\right)}{\mathrm{\partial }{Z}^2}=(p_0^b-Z)f^{\mathrm{\prime }}\left(Z\right)-2f\left(Z\right)<0.\end{array}$(A3) If both conditions have to be realised, then, replacing ${\pi }_0-Z$ by its value in (A2(A2) $\begin{array}{r}{p}_0^b-z=-\frac{\overline{F}\left(z\right)}{f\left(z\right)},\end{array}$(A2) ) in (A3(A3) $\begin{array}{r}\frac{{\mathrm{\partial }}^2{\mathrm{\Pi }}_b\left(z\right)}{\mathrm{\partial }{Z}^2}=(p_0^b-Z)f^{\mathrm{\prime }}\left(Z\right)-2f\left(Z\right)<0.\end{array}$(A3) ), we must verify that (A4) $\begin{array}{r}{f}^{\mathrm{\prime }}\left(Z\right)\frac{F\left(Z\right)-1}{f\left(Z\right)}-2f\left(Z\right)<0.\end{array}$(A4) Since $f\left(Z\right)$ is positive for all Z in the range $[\underset{\_}{Z},\overline{Z}]$, we can restate this inequality as (A5) $\begin{array}{r}{f}^{\mathrm{\prime }}\left(Z\right)\left(F\right(Z)-1)-2f^2\left(Z\right)<0.\end{array}$(A5) However, we have assumed that the distribution of Z is IFR which means that the failure rate $r\left(Z\right)=f\left(Z\right)/\overline{F}\left(Z\right)$ is weakly increasing for those values of Z for which $F\left(Z\right)<1$. Then the first differential of the function r, which is written (A6) $\begin{array}{r}\frac{\mathrm{\partial }r\left(Z\right)}{\mathrm{\partial }Z}=\frac{f^{\mathrm{\prime }}\left(Z\right)(1-F(Z\left)\right)+f(Z{)}^2}{(1-F(Z){)}^2}\end{array}$(A6) must be positive or null, so (A7) $\begin{array}{r}\frac{\mathrm{\partial }r\left(Z\right)}{\mathrm{\partial }Z}\ge 0\Rightarrow f^{\mathrm{\prime }}\left(Z\right)\left(F\right(Z)-1)-f(Z{)}^2\le 0.\end{array}$(A7) This last condition is stronger than the one spelt in (A5(A5) $\begin{array}{r}{f}^{\mathrm{\prime }}\left(Z\right)\left(F\right(Z)-1)-2f^2\left(Z\right)<0.\end{array}$(A5) ) because $(Z{)}^2>0$. $◼$