Modeling the effects of dated medical supplies donation on recipient countries

Most developed countries hold significant quantities of medical supplies in reserve for emergency response. Due to high handling costs and remote storage locations, such stocks are typically not used for day‐to‐day operations. In consequence, the expiry of reserve supplies (without use) is a significant problem. One possible remedy for such wastage is to donate large batches of dated supplies to developing nations, which often do without adequate medical supplies in their health systems. Here, we focus on personal protective equipment (PPE) and similar products, which have low risks associated with aging and safety. However, the international community is cautious about donating dated medical supplies, with the World Health Organization explicitly recommending against it. Issues of safety, while important, are not the primary concern as recent studies have shown expiration dates to be conservative. Instead, most concerns relate to macrolevel effects on the recipient country. Taking safety as given, we carefully model the incentives in the medical supplies donation supply chain, providing insights into the likely effects of corruption and impacts on the local industry. Overall, we find that the impact of donation is not monotone in the quality of donated products. In particular, dated donations whose quality is slightly lower than the quality of products in the local market are likely to be more beneficial than fresh donations; thus, we suggest that the international community reconsiders its stand on banning donations from dated reserves. We provide concrete guidelines for such donations and suggest a possible path for implementation of a donation program.

The U.S. healthcare system wastes a large quantity of medical supplies that are in usable condition; for example, Wan et al. (2015) estimated that just 4% of all hospitals nationwide waste at least $15.4 million of supplies annually.The large gap between deprived areas and developed countries suggests the possibility of donations.In this paper, motivated by the huge waste of medical products in developed countries, we focus on donations of PPE-type (personal protective equipment) products.
A key source for such donations could be the medical reserve stocks in developed countries.Medical reserve refers to the back-up medical supplies held by a government to prepare for emergencies.Encouraged by the World Health

Production and Operations Management
Organization (WHO), most developed countries hold such reserves, which typically consist of antiviral drugs, vaccines, syringes, disposable gloves, masks, and gowns.Because these products usually have a fixed use-by date and the likelihood of using them for emergencies during that shelf-life is relatively small, these reserves face severe issues with expiration (see below).Further, because some of these reserves have proved inadequate in the face of COVID-19, we would expect to see many countries rebuild their stocks after the pandemic has passed, and likely to even greater levels than seen previously (Dyer, 2020).This will further exacerbate issues of expiry in a few years' time.
Our focus in this paper is PPE (such as protective clothing, gloves, masks, and respirators) and similar products (like hand sanitizers and syringes).These products share similar features: They are protective, disposable, typically not for repeat use, and are usually labeled with a long fixed shelf life.They have fewer risks associated with aging as the WHO suggests PPE beyond the labeled expiration date "can still be effective at protecting health care providers" subject to the product being inspected before use (WHO, 2020), whereas drugs and vaccines may become unstable and dangerous with age.Further, there are no hazardous waste disposal issues for unused PPE, so there is no concern that the products are being donated to avoid disposal costs, as there can be with other types of medical supplies (e.g., Pinheiro, 2008).
PPE and similar products need to be replaced after a number of years of sitting in a reserve because eventually their seals are considered nonsterile.An inspection of the U.S. Department of Homeland Security (DHS') reserve stock reported 84% of examined hand sanitizer was expired with some by up to 4 years, and 200,000 respirators were beyond the 5-year manufacturer's guaranteed usability (US DHS, 2014).In preparation for the COVID-19 pandemic, Ontario, Canada, found roughly 55 million N95 masks and other medical equipment had passed their expiration date (Martell & Warburton, 2020).However, old sanitizer is likely more effective than diluted sanitizer and an old respirator will be more effective than no respirator.These reserve products, if used for a population that lacks adequate medical resources, could greatly improve healthcare outcomes.
Nevertheless, despite the huge potential social benefits, the option of donating aged reserve is not adopted in practice and dated reserve products are typically sent to landfill.The international community has been cautious, yet with inconsistent guidelines and regulations.In a prudential guideline, the WHO discourages donating expiring products, and requires that all donations "should have a remaining shelf-life of at least one year" when arriving in the recipient country (WHO, 2011).Meanwhile, although acknowledging the WHO's guideline, the American Food and Drug Administration (FDA) states that it is willing to donate expired medicine supplies on "a case-by-case basis" (FDA, 2011).
These seemingly contradictory stances do not deny the potential benefits of donation, especially when the recipient areas are extremely short of medical products.Caution regarding donating dated medical supplies can be attributed to four aspects: namely, efficacy-whether it is safe to use products close or past the labeled shelf life (e.g., Saha & Galper, 2013); ethics/fairness-whether such donations sufficiently consider the "rights and worth of different populations" (Pinheiro, 2008); corruption-corrupt bureaus may take advantage of donation to line their own pockets (e.g., Shelley, 2014); and adverse effects on the local industrydonations can fail to help the recipient areas by pushing local suppliers out (e.g., Moyo, 2009).
Efficacy arguments against using dated products are based on the assumption that expiring or expired products, as defined by the "expiry date" label, lose their potency and could be harmful.However, a product does not become unsafe immediately after the labeled expiry date.The labeled date is usually much shorter than the true shelf life, which is typically unknown and is estimated quite conservatively (Pomerantz, 2004).Convincing evidence has been produced by the Shelf Life Extension Program (SLEP), which found 88% of the tested product lots could be extended at least 1 year beyond their original expiration date for an average extension of 66 months (Lyon et al., 2006).To address the efficacy concern, we assume that the donated products will be used within a short enough time span (relative to the long lifetime of the products) that further in-country degradation is a second-order effect (and not modeled).
We do not consider the ethical issue of whether it is "fair" for a population to use dated medical supplies, that is, whether it treats people with sufficient dignity (e.g., Tomasini & Van Wassenhove, 2009).Comprehensively studying the fairness aspects would require discussions from a sociological perspective, which is beyond the scope of this study.Instead, we take a more pragmatic approach that currently we have large quantities of usable supplies being dumped in developed countries and large populations doing without suitable supplies in developing nations.We therefore ask whether the recipient nation would be materially better off with these supplies.
From the recipient's perspective, this paper investigates whether dated donation is beneficial when there is the potential for corruption and impacts on the local industry.In particular, we formulate a Stackelberg game between an official who is in charge of the donated supply and a local monopoly supplier who responds to the official's actions (i.e., whether the donation is provided free) and sets the market price.If the market price is forced too low, the supplier cannot cover the overhead cost and will exit the market.The official can be noncorrupt or corrupt, which could lead to different objectives: A noncorrupt official looks to maximize the region's healthcare welfare, which is modeled as a Cobb-Douglas function, whereas a corrupt official aims to maximize his own interests by pocketing the money from selling the donation.To focus on the impact of corruption resulting from the donation, we look at donation-related Production and Operations Management corruption, which emerges only when donation stocks are made available.We analyze and compare the region's healthcare welfare and the supplier's decision in three cases: the base case when there is no donation, a noncorrupt official with donation, and a corrupt official with donation.
We find that the value of the donations is not monotonic in their quality, and so dated donations could be more beneficial for the recipient country than fresh donations.Specifically, while very dated donations are certainly not beneficial, fresh donations could make the whole recipient country reliant on donation by driving out the local industry; and further, when coupled with corrupt officials, fresh donations could make the recipient country worse off compared to the case without donation.This is because donation products serve as an alternative to the local products and compete with the supplier.Whether the official is corrupt or not, donation forces the local supplier to lower the market price, and a high quality of donation could drive the supplier's price too low to maintain its operations and cause the supplier to exit the market, which could discourage any further attempts to develop local supply capacity.Our analysis suggests that dated donations of an appropriate quality could alleviate these problems and so provide a viable solution, reducing the adverse impact on the local industry and constraining the negative impact from corruption.
Overall, we recommend reconsidering the bans on donating dated medical products: Such bans may be relaxed if there are no additional safety and ethical issues, for example, when the products (such as PPE) pose low risks of harm and when such donations are well stored (e.g., in the reserve) and are not being donated to avoid disposal costs.There need to be strict requirements on the donation product: The product efficacy needs to be proven; and it is also important that the product quality is lower than that of the fresh product from local suppliers, otherwise the effect on the local market will be too detrimental.This could be achieved through rigorous testing and clear labeling of the product.Although our work shows that donation of slightly dated products can be beneficial, this must be accompanied by safeguards to ensure that other factors not explicitly modeled in the work, such as medical safety, ethical issues, and environmental concerns, are not compromised.For example, donation of dated products with adequate efficacy should be agreed upon by the recipient before these products are made available; otherwise, it could lead to a higher distrust between the recipient and the donor.
The rest of this paper is organized as follows.Section 2 reviews the literature.Section 3 presents our base model of the healthcare supply chain for the product being considered, and Section 4 models two cases where dated donation is provided with a noncorrupt manager and a corrupt manager, respectively.Then, Section 5 compares the three modeled cases and discusses the results.Finally, Section 6 concludes the paper and makes concrete recommendations for implementation of donation programs of dated reserve products.

LITERATURE REVIEW
Our work relates to three broad streams of literature: donation-related operations, the impact of donation, and corruption.In this section, we review the relevant work and discuss our contributions to the literature.First, our work contributes to the growing literature on donation-related operations, particularly for medical supplies donation from the recipient's perspective.One growing topic in donation of medical supplies is medical surplus recovery and allocation; see Zhang et al. (2020) for an overview.With known recipient valuations, Atasu et al. (2017) suggest that a provider-driven model can improve value provision to recipients, where the donor selects recipients and decides the type and quantity of donations.Zhang et al. (2020) consider private recipient valuations and design a mechanism to enable the donor to select the recipients to serve at each shipping opportunity.Essentially, research on medical surplus recovery studies a resource allocation problem without monetary transfer, where the main focus is matching uncertain supply of medical donations with recipients from a waiting list.In these studies, the matching problem is studied from the donor's perspective, yet the outcomes for donation recipients are not discussed.We take a different perspective and investigate the impact of providing medical supplies donations on the recipient, in line with the call for donations to take the recipient's actual needs into account (e.g., Berenguer & Shen, 2020).
A few studies of donation operations explicitly consider the decision or the outcome on the donation recipient's side.Natarajan and Swaminathan (2014) study the recipient's optimal procurement policy with funding constraints, where the timing of the receipt of previously promised funding from donor agencies is uncertain and constrained.Taylor and Xiao (2014) consider the subsidizing strategy in the distribution channel and investigate the impact of donor subsidy on recipients' access to malaria drugs, and find that the donor should subsidize purchases but not sales.Further, Taylor and Xiao (2019) look at consumer awareness changes in a donation program implemented through a commercial channel, and examine how the donor's optimal subsidy and utility should change with the loss of price control and the level of consumer awareness.While these studies provide important insights informing donor decisions to better help less developed areas, donation quality is not considered and bureaucratic corruption is not discussed.Different from these studies, our work investigates the impact of donation quality from the recipient's perspective, and looks at how donation can affect recipient welfare and the local supplier when confronted with noncorrupt versus corrupt officials.
Our work also contributes to the broader literature on the impact of donation on the recipients.While studies on this topic are relatively limited in the operations management literature, there are extensive discussions in development economics and public health.This body of literature has found that, despite the potential benefits, in-kind donations can pose Production and Operations Management a negative price effect, that is, large volumes of donations may shift the supply curve and put pressure on local producers to reduce prices (cf.Awokuse, 2011).Originating from food aid (Schultz, 1960), the price effect has been studied in several empirical investigations of donations of food (e.g., Tadesse & Shively, 2009), used clothing (Frazer, 2008), and shoes (Wydick et al., 2014), but not yet medical donations.While a negative price effect has been found in apparel and shoe donations, mixed findings have been reported for food donations, with the price effect supported in some but not other studies.The varying conclusions from these studies suggest the impact on the local market and the recipient's economy depends on the context and the type of aid.Tadesse and Shively (2009) find that the quantity of food donation influences the extent of the price effect.Specifically, the effect on local prices and local food production is detrimental when the donation shipments exceed the 10% threshold of local production; below this level the impacts appeared to be benign.
Compared to the work on the price effect, discussions about the impact of in-kind medical donations (new or dated) are more qualitative and narrative.Igoumenidis et al. (2013) discuss the impact of in-kind drug donations (as opposed to cash) on the recipient and suggest a localized and decentralized approach to manage the aid, so donations can bypass unnecessary procedures and encounter less corruption.Despite the concerns raised regarding dated donations, other voices support the donation of subpar medical supplies.From a medical ethics perspective, Saha and Galper (2013) suggest donations of products that are known to be stable past their expiry date can be made per recipient's request.Overall, though these studies touch the impact of donation quality, they use a qualitative rather than a quantitative approach, and the negative price effect has not been well studied for medical donations.As such, our work contributes by quantifying the price effect for one type of medical donation while looking into the impact of donation quality.
Finally, this work sits within the literature on corruption as we explicitly consider that corrupt members of the supply chain may divert the donated products for their own gains.Underdeveloped countries are particularly vulnerable to corruption and fraud in their healthcare sectors.Partly because the administrative systems are neither well developed nor transparent, leakages of public resources and misuse of medical supplies for private gain are common in these countries (Musau & Vian, 2008).There is evidence that donation provides a chance for corrupt bureaucrats in recipient countries to divert the products for repackaging and resale.For example, the Uganda Minister of Health estimated that half of the medicines accessed by the government for public healthcare were siphoned off (MeTA, 2009).Media reports also claim that medical supplies donated to North Korea are diverted by corrupt officials and sold at local markets (Kim et al., 2013) and that officials in Africa divert and sell donated medicines for their personal profit (Shelley, 2014).
Modeling the role of an official in the donation supply chain, we contribute to corruption studies related to government officials and public products.Due to the difficulties of incorporating all the cost, demand, and supply functions, it is challenging to build a model of bureaucratic corruption (Jain, 2001).Looking at government officials selling government property for personal gain, Sheiffer and Vishny (1993) find that weak governments usually experience high levels of corruption, which is very distortionary and costly.Though bureaucratic corruption is commonly perceived as bad practice, some suggest that corruption may improve efficiency and help growth, given the preexisting distortions in underdeveloped countries (cf.Bardhan, 1997;Mèon & Weill, 2010).Corruption is typically modeled as a principal-agent model in which the government or the citizens act as the principals and aim to reduce the corrupt behaviors of officials (e.g., Acemoglu & Verdier, 2000).Different from the principal-agent setting, Popa (2014) takes the view that corruption can benefit both the corrupt officials and individual citizens, and models corrupt interactions as markets in which public goods are sold.Our results suggest that, depending on the donation quality, corruption can hurt or benefit a recipient's healthcare welfare as well as the local supplier's profitability, and we characterize these conditions.As such, we extend the modeling of corruption to the donation supply chain setting, analyze the impact on the recipient, and obtain managerial insights from the structural results.

MODEL DESCRIPTION
We consider an overall impoverished area/country where there is a shortage of medical products.Donations could help the area improve healthcare, but could also influence the local market and induce corrupt behaviors by officials processing the donated products.In this work, we model the impacts associated with donations of one type of medical product (e.g., PPE), which has a long shelf life and whose properties are deemed to be stable.The product can continue to be useful after the initial labeled dates, especially if it is well stored.For example, after being in storage for a long time, the strap and nose foam of a respirator may become less elastic, which can reduce the fit and seal and make the respirator less effective in keeping outside air from leaking around the edges of the respirator; but this just means the respirator is less likely to perform at its full potential (3M, 2016), not that it is totally ineffective.While such dated products are inappropriate to use in a setting where respirators are mandated, such respirators will improve health welfare in settings where they are not typically used.Indeed, PPE products past expiration dates are distributed and used for prolonged hours in developed countries under situations of desperate shortage (e.g., BBC News, 2020; Martha, 2020), and recently the CDC (Centers for Disease Control and Prevention) issued guidance explicitly endorsing the use of certain respirators from the reserve past their labeled shelf life when responding to the COVID-19 pandemic (CDC, 2020).

Production and Operations Management
Motivated by the above, we assume that the quality/effectiveness degrades as the product ages but at a very slow rate (Saha & Galper, 2013); thus, the quality of donated products is no higher than that of fresh products but does not change during the decision horizon.Another example of donated product fitting our assumptions is where new models are more effective than older models.If the donation quality is higher than the equivalent available the local market, then donation will surely outperform the local supplies and the potential supplier effects discussed in this paper will be even more than if quality is equal, though this is not explicitly modeled.The quality of the product indicates its healthcare value, and a higher quality will lead to higher healthcare welfare (if everything else is held equal), as detailed in Equation ( 1) below.
Several regions are contained in the impoverished area, and they can get the product from two different sources.One source is to buy from the supplier in the local market.Yet, the amount to purchase is subject to a budget constraint, so it is possible these regions cannot afford enough product to fulfill their healthcare needs.An alternative source is from donation, which is expected to ease the need for the product and improve healthcare; this humanitarian purpose motivates donations to these regions.
However, donations could be taken advantage of by corrupt officials, possibly undermining the expected positive effects.Meanwhile, the local supplier could react to donations and change the affordability of the product.As corrupt officials and the strategic supplier change the dynamics associated with donation, it is necessary to investigate if donation can fulfill the humanitarian purpose as expected.Though belonging to the same area/country, each regional administration oversees its regional healthcare issues and is able to secure donations from independent sources; as a result, the type and the amount of donations received by each region are relatively independent (Shaw et al., 2015).Thus, we can look at a typical region as a donation recipient and analyze the impacts of donation.While the region could receive donations intermittently, we consider the time span for a large donation to be used in the region (if freely available); the length of the time span will depend on the donation size but is generally sufficient for the local supplier to react and adjust prices accordingly.

The model and notation
Over the considered time span, the region has a healthcare budget T, which can be used to procure the product considered in this work as well as other healthcare products.With budget T, and subject to the market price p set by the supplier, the region decides how much product to buy from the market, in order to maximize the healthcare welfare of the population served by the region.Due to the budget constraint and/or high market price, the region may not be able to afford all necessary healthcare products.
Denote the region's purchasing amount as x s , and let v denote the quality of the fresh product, that is, the healthcare value provided per unit product.Purchasing x s units of product may or may not use all of the budget T. Thus, the healthcare welfare comes from two sources: the health-related utility from disposable money after (or not) purchasing the product and the healthcare improvement associated with accessing the product.Here, we use the term "healthcare welfare" to highlight the healthcare-related considerations.If we ignore donations, then the healthcare welfare, W(T; x s ), is represented in the Cobb-Douglas form as in Equation ( 1), with 0 <  < 1, (1) The parameter  reflects the preference of saving the budget for other healthcare products and services, relative to spending it on the product under consideration.A large  indicates the product has a relatively low priority in the healthcare system; and this is likely to be the case for the types of product considered in this paper, for example, PPE, as underdeveloped countries experience inadequate use of these products due to limited medical resources, as discussed in the introduction.
While the region decides the purchase quantity x s , the supplier will react by adjusting the selling price p. Modeling a monopoly supplier means we do not need to consider competition in the market, and thus provides a special case to clearly quantify the effect that donations can pose on the local industry; other market types would also experience similar effects to those studied in this paper.Due to the investment in infrastructure and the financial requirements to cover the overhead cost, the supplier is subject to a minimum viable price p > 0 in order to remain operational, that is, the supplier cannot sustain in the market if the selling price p is lower than p.Since the region's budget comes from various sources such as the local government's fiscal revenue and cash donations, the local supplier does not know the exact budget level T, but has a reasonable understanding of the market and thus knows budget T can be drawn from the cumulative distribution function Φ(⋅), for which T m represents the upper bound of the budget so Φ(T m ) = 1.Based on her estimation of the market, the supplier accordingly sets price p in order to maximize her expected revenue R(p).Increased prices will mean that the region can afford fewer units of the product, which is somewhat discretionary in the recipient area (Okwen et al., 2011).
On top of budget T, the region may receive an external donation of products, though the donated products could be dated, that is, if they come from the medical reserve.Let q denote the quality of dated donation, then 0 ≤ q ≤ v, as discussed in the third paragraph of Section 3. How the donation is processed, that is, how much of the donation indeed reaches the region, denoted by x d , impacts the healthcare welfare.We will refer to the decision maker processing the donation as "the manager."Such a person could be sitting anywhere (at a high level or a fairly low level) along the supply chain (within or outside the region) before the donation reaches the Production and Operations Management region.When processing the donation, a "corrupt" manager seeks to maximize his own interests by selling donation products in the black market, the objective function of which will be made concrete later; while a "non-corrupt" manager seeks to maximize the healthcare welfare so the objective function coincides with the region's welfare function W(⋅).Focusing on the effect of donation-related corruption, we do not consider corruption in operations other than those associated with donations, because such corrupt conduct would exist even when donation is not available.
As the donation provides extra resources for the region and thus changes the dynamics of the local market, the supplier will react by adjusting market price p, which will be discussed later in corresponding scenarios.In this aspect we assume complete information, such that the supplier can anticipate the response functions of the region and the manager; this standard modeling practice allows us to derive the supplier's decision and obtain meaningful insights.
Within the considered time frame, the region's total potential demand for the product is D, so x s + x d ≤ D holds.A fixed demand is assumed for simplicity and to single out the effects of corruption and donation.In a slight abuse of notation, we use x s and x d to denote the actual quantities, and x s (T, p) and x d (T, p) to denote the region's response function where the quantity is a function of budget T and price p.A summary of the notation is as follows.

The base case
We first consider the base case when donation is not available, and take this as a benchmark.When there is no donation, the manager cannot conduct donation-related corruption, and the region only needs to make one decision about x s , that is, how much to buy from the supplier.The game has two stages: First, anticipating the region's response function x s (T, p), the supplier sets price p; and second, the region decides x s given the market price p.To solve this, x s (T, p) can be derived by maximizing the healthcare function W(T; x s ) in Equation (1); then with x s (T, p) and the budget distribution Φ(T), the supplier sets p to maximize her expected revenue where E[⋅] stands for the expectation.
The solution is straightforward: Whether and how much to buy depends on the region's budget level T, and the price p depends on the budget distribution, as per Proposition 1.A superscript b is used to indicate the results for the base case, and a superscript * indicates the optimal solution.

The market price p
It is not surprising that the amount of product purchased, x b * s , depends on the budget level.With the Cobb-Douglas welfare function, the region spends a fixed proportion, 1 − , of its budget until it reaches all D products.The amount of product the region purchases is proportional to D and the proportion is T∕T m .That means if the realization of the budget resolves around the very low end of the distribution, then a region would only access very few products; and a region can only enjoy all D products if its budget is at the upper bound T m .This comes with the implicit assumption that the quantity ordered could be any fraction of the demand; allowing only integers for the order quantity is not likely to generate additional results but will make the analysis more complex.
Setting price p b * , the supplier can exploit the market: She sets the price high enough so that only regions with the maximum budget T m can afford to fulfill demand D, and thus extracts all the surplus from the market.This is intuitive because the supplier acts like a monopolist.While a monopolist hurts market efficiency in general, the harm is more evident in the studied context, an underdeveloped area in need of the scarce resource.Recall that the supplier has a minimum viable price p.To study the impact of donation on the supplier, we assume p < p b * ; otherwise the supplier will not operate in the market even when there is no donation.

MODEL ANALYSIS
In this section, we consider situations when donation products are provided.If the manager is corrupt (as is discussed in Section 4.2), donated products may be diverted into the black market and generate a charge for the region to receive them: In that case, the unit price of the donated product is denoted by w.Compared to the base case, the total amount obtained by the region is x s + x d , and the resultant healthcare welfare changes to Equation (2) to account for both the fresh and the donation products.Before investigating the impact of donation, it is helpful to understand how the region should make purchasing decisions from a healthcare welfare perspective.Putting aside whether the manager is corrupt or not, to maximize the region's

Production and Operations Management
healthcare welfare is to solve the following problem: s (T, w, p) and x h d (T, w, p) be the region's response functions of the quantity to buy from the supplier and the donation, respectively; we will drop some or all of the arguments on x h s (⋅) and x h d (⋅) when doing so causes no confusion.Define w t = pq∕v, and this is the threshold for w beyond which x h d (T, w) = 0 for all T, that is, buying the donation product is uneconomic relative to buying the fresh product.Then Proposition 2 gives the results.
Proposition 2. The region's response functions depend on the market price p, donation price w, and budget T as follows: 1.For w ≥ w t , x h d (T) = 0 and x h s (T) is as per Proposition 1. 2. For 0 < w < w t , (3) 3.For w = 0, x h s + x h d = D, and (4) With the region's response functions in Proposition 2, the supplier and the manager will respond by setting p and w, respectively, based on their objectives.Next we discuss two scenarios: when the manager is noncorrupt and when the manager is corrupt.These results will assume no constraint for the local supplier's capacity.Intuitively, one would expect having a supply constraint to both decrease the healthcare welfare and increase the pressure on the local industry.
Therefore, the results give an upper bound for a region's welfare and a lower bound for the adverse effect on the local industry.

Noncorrupt manager
In this section, we consider how donation could impact the region by putting the possibility of corruption aside.While the region's purchasing decision follows Proposition 2, a noncorrupt manager will decide w so as to maximize the region's healthcare welfare W(T; x s , x d ), as in Equation ( 2).This objective function implies that any revenue raised from the donation, that is, wx d , will be taken from the healthcare system and used for the region's other expenditures, given the noncorrupt manager does not keep it.Alternatively, the noncorrupt manager could add the revenue back into the region's healthcare budget; in this case, the objective function becomes (T − px s )  (vx s + qx d ) 1− .We can show that the noncorrupt manager's optimal decision is the same under these two objective functions.A superscript n is used to indicate the results for the noncorrupt manager case.
Let w n * be the optimal price a noncorrupt manager would set for the donation, then w n * = 0 regardless of the market price p, as is shown in Proposition 3.This is because charging for the donation will influence the amount of products accessed by the region.Recall that the healthcare welfare comes from two sources: the product and the health-related utility from the disposable money.Charging would reduce the quantity of products affordable for the region and so reduce the product-related welfare, while the amount of disposable money remains the same if the revenue is returned to the healthcare budget, or decreases if not.Therefore, a noncorrupt manager will not charge for the donation but instead distribute it for free.This serves the humanitarian purpose of donations and conforms to the common expectation.
When there are free and sufficient donations, the region will top up the total quantity of products to its demand D, since free donated products can always improve the healthcare welfare.Accordingly, the supplier's revenue function is different from when no donation is available, which motivates her to adjust the market price p.Specifically, in the case where the equilibrium price p n * is no lower than the minimum viable price p, the supplier would like to maximize her expected revenue function below, where x n s (T, p) equals x h s as per Proposition 2 Part 3: D pD dΦ(T). (5)

Production and Operations Management
To derive the pricing decision for the supplier, we make a further assumption so that the model is more tractable.Assume that the budget level T follows a uniform distribution between 0 and T m , that is, Φ(T) = T∕T m for 0 ≤ T ≤ T m ; then we can show the supplier's expected revenue function R n (p) is concave in price p and so the optimum is achieved when the first-order condition (FOC) is satisfied.Proposition 3 gives the pricing decision results and shows the donation quality q influences the supplier's price p n * .This in turn influences the composition of the products accessed by the region and the healthcare welfare achieved, as outlined in Proposition 4.

Proposition 3. If the manager is noncorrupt, then:
1. the donation price w n * = 0, 2. if donation quality q ≤ qn , the supplier's price p n * = ; otherwise, the supplier exits the market, where the threshold qn is the highest donation quality q such that One important observation from Proposition 3 is that the supplier could be driven out of market by high-quality donation products.This is due to the supplier's reaction to the donation: As shown in Proposition 4, the supplier lowers her price for fresh products as donation quality q increases, because donation breaks her monopoly power.A donation quality q > qn would drive the supplier's price below her minimum viable price p and force the supplier to leave the market.Even when the supplier can remain in the market (i.e., if the donation quality q ≤ qn ), the supplier does not benefit from her adjusted pricing strategy as her price and expected revenue decreases with donation quality q. , W n (T) is noncontinuous at q = qn .Specifically, the right limit W n (T, qn +) is smaller than the left limit W n (T, qn −); and W n (T) increases with donation quality q for q > qn .
If the supplier is out of the market, then regions intending to purchase from the supplier will have to turn to donation.As indicated by the expression of the product composition (x n * s , x n * d ) (which is in the Supporting Information), regions with a high budget will still be willing to buy from the supplier even when the donation quality q is high and approaches the fresh product quality v, yet there is no way for them to do so if the supplier exits the market.Having the supplier out of market makes the whole recipient country reliant on donation.To make matters worse, the healthcare welfare decreases when regions intending to buy from the supplier switch to donation, because taking the donation is a suboptimal action for them.As shown in Proposition 4, Item 3, if the supplier exits the market, for regions with a budget T > qT m v−q+2q , the healthcare welfare is lower than it would be if the supplier stays in the market.
As such, the analysis indicates donation could be a doubleedged sword.While donation seemingly improves access to the product, it could drive out the local supplier and fail to improve healthcare welfare as expected.The results confirm the fear that free donations can hurt the local industry and make the recipient country too reliant on foreign aid (Moyo, 2009;Oxfam, 2005).Having donations drive out the local supplier could discourage the development of local production capacity and distribution network, and thus hinder the long-term development of the local community.This is in line with the impression that many underdeveloped countries do not appear to make much headway, even when they receive massive amounts of donations and assistance.In line with the views Moyo outlines in her book (Moyo, 2009), we provide a model-based analysis showing that free donations of a good quality may well be the reason for that.
Due to the adverse effect on the local supplier, the recipient region's healthcare welfare is not continuously increasing with donation quality, and donations of a very high quality do not necessarily improve welfare more than donations of a lower quality.This leads to our key observation for this section.
Observation 1. Donations of an average quality, which is somewhat lower than the fresh quality but still reasonably good, could ease the adverse effect on the supplier but also significantly improve healthcare welfare.This observation follows from two results in Proposition 4.
(1) The threshold quality for the supplier to remain in the market, qn , is lower than the fresh quality v.As discussed, donation of a quality lower than the threshold quality qn (and thus lower than the fresh quality v), that is, q < qn < v, could keep the supplier in the market.We note that the threshold qn decreases with the supplier's minimum viable price p.A high minimum price p leads to a low threshold qn and indicates donation of a low quality could push the Production and Operations Management supplier out of market; this is unlikely because a high minimum viable price, especially when it gets close to the monopoly price, would make it hard for the supplier to make sufficient profit even when donation is not available.As such, a likely and also preferable situation is that the supplier's minimum viable price p is of a medium to low level (compared to the monopoly price 1− D T m ).In this case, the supplier can stay in the market with donation products of a reasonably high quality because the threshold quality qn is high (while still lower than the fresh quality).( 2) The healthcare welfare W n (T) is concavely increasing with donation quality q in several cases; particularly when the budget level is low, that is, T < qT m v−q+2q , and when the product has a relatively low priority in the healthcare system, that is,  ≥ 0.5, as with the typical products considered in this paper.The concave welfare function W n (T) indicates donations could be of a slightly lower quality (compared to the fresh products), without reducing the healthcare welfare too much.Note that low-budget poor regions are the ones that need donations the most; the humanitarian purpose of donation can be fulfilled, as long as the donation quality is high enough to ensure the welfare of these poor regions increases to a satisfactory level.
While the above is derived when the noncorrupt manager's objective coincides with the healthcare welfare function, some may argue that a noncorrupt manager should care about overall social welfare, including for the supplier who contributes to the local economy.The solution of donation quality being slightly lower than the fresh quality provides a viable and practical approach for a noncorrupt manager who cares about both the supplier's revenue and the healthcare welfare.The ideal level for donation quality could be determined by jointly considering the supplier's price and the impact on the healthcare welfare.

Corrupt manager
In this section, we consider the case when the manager is corrupt.The corrupt manager would divert the donation for sale on the black market (e.g., Kim et al., 2013;Shelley, 2014), and thus would seek to maximize his personal earnings by setting price w for the donation product.Since Proposition 2 shows the donation cannot be sold if w > w t , the viable price range for donation is w ≤ w t .Anticipating the corrupt manager's decision of donation price w, the supplier would set market price p accordingly.As previously, we assume full information, as is relatively standard, and considering partial information is beyond the scope of this paper.Proposition 5 presents how the supplier's and the corrupt manager's decisions depend on donation quality q, and outlines the properties of these decisions and the welfare function.A superscript c is used for the results with a corrupt manager.
Proposition 5.If the manager is corrupt, let qc be the highest donation quality q such that p c (q) ≥ p where p c (q) is the sup-plier's response function, then the supplier exits the market if q > qc .
1.The threshold qc decreases with the supplier's minimum viable price p.

The supplier's price p c * decreases with donation quality q
for q ≤ qc .3. The donation price w c * increases with donation quality q if q ≤ min{q c , v 2+1 }, and decreases with donation quality 4. The donation price w c * is noncontinuous at q = qc and increases to for q > qc . 5. For a given budget T, the healthcare welfare function W c (T) increases with donation quality q for q ≤ qc .6.The welfare function W c (T) is noncontinuous at q = qc : the right limit W c (T, qc +) is smaller than the left limit W c (T, qc −); and W c (T) increases with quality q for q > qc .
Proposition 5 shows that the supplier will leave the market if the donation quality q is high (i.e., q > qc ), which is similar to the noncorrupt manager case.Responding to the supplier's exit, the donation price and the welfare function are not continuous anymore, which will be further discussed later.When the supplier is in the market, that is, q ≤ qc , both the corrupt manager's and the supplier's pricing decisions are continuous, but with different monotonicity.Based on Proposition 5, Items 2 and 3, as donation quality q increases, the supplier's price continues to decrease, while the corrupt manager's price for donated products first increases when q ≤ min{q c , v 2+1 } and then decreases until quality q reaches qc .
The case when quality q ≤ min{q c , v 2+1 } is intuitive, as a better quality leads to a higher price for donation product and drives down the price for fresh product.As a result, a low quality will keep the donation price low yet allow the market price to be high, so that the case q = 0 resembles the base case where there is no donation available.This suggests that if donation quality is low, the corrupt manager cannot make welfare worse than the base case, when considering donation-related corruption only.Obviously we do not want donations with a very low quality, because that does not serve the purpose of providing aid.
It appears surprising that when v 2+1 < q ≤ qc , the price for donated products decreases with donation quality, since one would naturally expect a higher price for better quality.This happens because of the competition between the supplier and the corrupt manager, which resembles a "price war."Donation being available breaks the supplier's monopoly power and directly competes with the supplier's products.Once donation quality is above a certain threshold, that is, q > v 2+1 , as donation quality gets higher, the difference between donation and fresh products gets smaller and they become more homogenous, so the competition gets more intense and forces both sellers to lower their prices.

Production and Operations Management
The decrease in donation price stops once donation quality q becomes higher than the threshold qc and so the supplier is driven out.In this case, the corrupt manager replaces the supplier as the monopoly provider and charges the monopoly price , which is the price the supplier charges in the base case.This means the healthcare welfare is the same as the base case if donation quality is the same as the fresh quality, and will be lower if the donation quality is lower.As such, if the donation quality is high (i.e., q > qc ), then the corrupt manager makes the situation worse than the base case in at least two aspects: by driving out the supplier and hurting the local industry, and by reducing the healthcare welfare (compared to the base case) unless q = v.
The above discussion indicates that the healthcare welfare achieved in the corrupt case becomes equal to that achieved in the base case when donation quality q = 0 and q = v, yet for different reasons and with different implications.A corrupt manager makes donations of a very high quality (i.e., q > qc ) more harmful as they lead to lower healthcare welfare than the base case, on top of driving out the local supplier.One way to mitigate such negative impact is by only allowing donations of a quality lower than the threshold qc .Similar to the noncorrupt case, we note that the quality threshold qc depends on the supplier's minimum viable price p, so a preferable situation is when p is at a medium to low level (relative to the monopoly price ), enabling the supplier to remain in the market with a reasonably high donation quality.
While these results are derived with the model setup so that the supplier and the corrupt manager compete for the product market, another possibility is that they collude, set a nonzero price that they both agree on, and then split the earnings in some fashion, particularly when q > qc : The corrupt manager may want to collude and keep the supplier in the market to cover his selling behavior, thereby avoiding being questioned on selling donations if the supplier is out of market.Though collusion is possible, it is less attractive for the supplier to collude as donation quality gets lower.This is because as the gap between the two types of products gets larger, the supplier will have more motivation to compete with the donation product that is of a lower quality.Thus, donations of a quality slightly lower than the fresh product, as discussed in Section 4.1, could not only alleviate the adverse effect on the local industry, but also restrict the harm from the corrupt manager and mitigate the risk of collusion.This leads to our second key observation.
Observation 2. Donations of a reasonable quality (below fresh but still of significant value) continue to be a viable solution when the manager is corrupt; mitigation measures are of use to ensure an appropriate quality level for donated products.
The corrupt manager will have a preference for donation quality since it affects his earnings.Not surprisingly, the corrupt manager will want donations of a high quality, that is, q > qc , as this will enable him to pocket high earnings by driving out the supplier.However, if the donation quality q is below the threshold qc as suggested above, the corrupt manager could prefer a low quality, particularly if qc > v

2𝛼+1
; this is because the manager's earnings decrease with donation quality q once q ≥ v 2+1 .As a result, if the corrupt manager could choose the quality of the donation, he would want , where he can charge the highest price and earn the highest personal gain.Note that the threshold v 2+1 depends on the parameter , which reflects the preference for the product in the healthcare welfare function.For the typical products considered in this paper that tend to have a large , the threshold v 2+1 can be low; and the value is bounded by v 3 when  approaches 1, where the healthcare administration puts little attention on the product.While we suggest an allowable donation quality slightly lower than the threshold qc , it is important to prevent the corrupt manager from purposely delaying the diversion of donations and waiting for the quality to deteriorate to his desired level.
Since donation quality plays a key role in the studied effects, preventative mitigation measures are necessary to ensure the quality level is appropriate.To make the quality lower than the threshold quality, the donation products can be clearly labeled with the approaching or passed expiry dates and the projected effectiveness.Yet, due to the possibility that a corrupt manager waits for donation products to degrade, it is not clear whether any other regulatory actions taken by the donors, such as clearly labeling when the donation should be distributed by, can mitigate this form of corruption.In-country studies would need to be carried out to examine whether this corruption effect is likely to occur in practice, which is clearly beyond the scope of this paper.

COMPARISONS AND DISCUSSIONS
In this section, we compare the results of the three cases: base case, noncorrupt manager, and corrupt manager, and summarize the results in Proposition EC.3 in the Supporting Information.We then discuss how the three cases compare with each other in terms of the supplier's price and the healthcare welfare, and summarize the results for low-priority products, for example, PPE, as considered in this paper.

The market price
In this subsection, we discuss how the supplier's price compares across the three cases, as in Proposition EC.3, Item 1, in the Supporting Information, and in Figure 1.When donation is available, the supplier sets the market price lower than the base case, no matter whether the manager is corrupt or not, that is, p n * ≤ p b * and p c * ≤ p b * .This is not surprising since donation provides an extra source of products, thus competing with the supplier; and that is why high-quality donation could drive out the supplier.The comparison between the noncorrupt and the corrupt case is not intuitive as it depends on donation quality q.If q > (1 − 2)v, the supplier's price is higher in the corrupt case than in the noncorrupt case.This is because the corrupt manager selling the donated products will ease the pressure for the supplier to lower market price, as donation products on sale are not as competitive as free donation products provided by the noncorrupt manager.

Production and Operations Management
Yet, if donation quality q ≤ (1 − 2)v, then the corrupt manager pushes the supplier to lower her price more than when the manager is noncorrupt.This surprising result is due to the condition q ≤ (1 − 2)v: For it to hold under a nonnegative donation quality q, the parameter  needs to be lower than 0.5 (see Figure 1a), which indicates that a region places considerable weight on quality in its welfare function.As such, donations of a low quality do not directly compete with fresh products in the noncorrupt case: Low-quality donation will primarily attract low-budget regions while high-budget regions will still buy fresh products, so the supplier can charge a relatively high price.Since the corrupt manager will set a low donation price to attract low-budget regions, this leaves regions that plan to buy a combination of fresh and donation products with a smaller amount to spend on fresh products, and thus forces the supplier to lower her price compared to the noncorrupt case.This suggests the corrupt manager hurts the local supplier more than the noncorrupt manager if the donation quality is lower than (1 − 2)v.

5.2
The healthcare welfare The comparison of the healthcare welfare is illustrated in Figure 2. We first compare the noncorrupt case with the base case, which shows the same pattern in both Figure 2a,b.If the donation quality q is smaller than the threshold qn so the supplier remains in the market, then, with a noncorrupt manager, donation improves healthcare welfare for all regions compared to the base case: It always holds that W n (T) ≥ W b (T), where equality occurs when q = 0.This especially helps poor regions that desperately need the product, as they gain access through the donated products and thus buy less fresh products compared to the base case.Regions with a high budget also have increased welfare because they can afford more fresh products given that the free in-kind donation pushes the supplier's price down.
It is interesting to observe from Proposition EC.3, Item 3, in the Supporting Information, that donation could make a high-budget region's healthcare welfare lower than that in the base case when the supplier is driven out.This happens for regions with budget T > qT m v  − 1− when donation quality q > qn .Note that the threshold qT m v  − 1− can be higher than the budget upper bound T m when donation quality q is high, so it is possible that all regions have their healthcare welfare improved compared to the base case, as shown in both Figure 2a,b.Yet, the reasons for the welfare improvement could be different from when donation quality is low.While poor regions continue to improve welfare through accessing free donation, welfare improvement (if any, compared to the base case) for high-budget regions comes from the budget saved from not buying the product.This is because high-budget regions are forced to turn to donated products when the supplier is driven out, even though they may still want to buy from the supplier.
The comparison between the corrupt manager case and the base case also depends on the donation quality q.Similar to the noncorrupt manager case, donation under a corrupt manager improves healthcare welfare for all regions if donation quality q ≤ qc so the supplier can remain in the market.Despite the manager being corrupt, donation enables regions with a low budget to access the product through donation, and also pushes the supplier to reduce the price, so regions with a high budget can afford more fresh products.Yet, if the donation quality is so high that the supplier exits the market, then the healthcare welfare becomes lower than in the base case for all regions (in both Figure 2a,b), unless the donation quality is the same as the fresh quality.This is because the corrupt manager replaces the supplier to become the monopoly provider in the market and sells donated products whose quality is no higher than the fresh quality.This leads to our key observation below.
Observation 3. Compared to the base case, donation improves a region's healthcare welfare if the supplier remains in the market; but could reduce healthcare welfare when donation quality is high enough that the supplier is driven out of the market, especially if the manager is corrupt.
When comparing the corrupt and noncorrupt cases, as in Proposition EC.3, Item 5, in the Supporting Information, it is not surprising that corruption is harmful, with healthcare welfare in the corrupt case being lower than that in the noncorrupt case for most regions.However, it is unexpected that some high-budget regions can have a higher welfare in the corrupt case.This happens when the donation quality is in two ranges, but for different reasons.(1) If the donation quality q ≤ min{q n , qc , (1 − 2)v}, the supplier can remain in the market in both the noncorrupt and the corrupt cases, and the condition q ≤ (1 − 2)v means high-budget regions have higher welfare in the corrupt case (see Figure 2a).This is because, as discussed in Section 5.1, the corrupt manager pushes the supplier to lower her price further than the noncorrupt manager if q ≤ (1 − 2)v.So, high-budget regions that continue to buy from the supplier enjoy a lower price in the corrupt case and thus achieve a higher welfare.(2) The second range qn < q < qc indicates that the supplier is pushed out of the market when the manager is noncorrupt but stays in the market when the manager is corrupt (see both Figure 2a,b).This makes high-budget regions' welfare lower in the noncorrupt case because they would have benefited from buying from the supplier but are forced to turn to donation.
While a high-budget region's welfare can be higher in the corrupt case, the corrupt manager always makes a low-budget region's welfare lower than in the noncorrupt case.This further confirms the harm caused by corruption: Diverting and selling donation products hurt poor regions that need help the most; the inequality of resources in the recipient country is deepened, as it enables rich regions to take advantage of poor regions.This is summarized in the next key observation.
Observation 4. Compared to the noncorrupt manager, the corrupt manager reduces the healthcare welfare in most cases, but could cause an increase in the healthcare welfare for some high-budget regions if donation quality q is within certain ranges.
Note that if the parameter  is small, then a high donation quality could satisfy the condition q ≤ (1 − 2)v, under which the corrupt case leads to a lower market price and higher welfare for high-budget regions than the noncorrupt case.In other words, donation of a relatively high quality could enable the corrupt manager to hurt the local supplier and benefit rich regions at the cost of poor regions if the product has a small parameter  and thus a high priority in the healthcare system.This suggests the negative impacts of the corrupt manager are greater for high-priority products, such as vaccines and critical equipment, for which a very high quality is essential.For such critical and high-priority products, donation of a quality that is below fresh but still of significant value may not be a good solution, especially when it is hard to tell whether the manager is corrupt.As explained in the introduction, we mainly consider low-priority products like PPE and similar products.Thus, in the next subsection, we focus on low-priority products and summarize the results that particularly apply to them.

For low-priority products
For typical low-priority products where the parameter  is high, that is,  > 0.5, the comparison between the corrupt and the noncorrupt cases is more straightforward, as

Production and Operations Management
shown in Figures 1b and 2b.First, the corrupt case has a higher donation-quality threshold below which the supplier can remain in the market, that is, qc > qn always holds.This is because the supplier's minimum viable price p < p b * and p

2D
. Second, the supplier's price is higher in the corrupt case.This is because q > (1 − 2)v always holds since (1 − 2)v is negative.Third, the corrupt case leads to a lower healthcare welfare in most cases, except for some highbudget regions if donation quality q is between the quality thresholds in the corrupt and noncorrupt cases, that is, if the supplier is driven out in the noncorrupt case but not the corrupt case.Overall, these results suggest that for low-priority products, despite hurting the healthcare welfare compared to the noncorrupt case, the corrupt manager does not add to the adverse effect on the local supplier; specifically, with a donation quality that can keep the supplier in the market with a noncorrupt manager, the supplier can stay in market even if the manager is corrupt.
Meanwhile, while Proposition EC.3, Item 3, in the Supporting Information, indicates the noncorrupt case could lead to lower welfare than the base case, this is less likely to happen for low-priority products.This is because  − 1− is larger than 1 and increases with : So long as donation quality q and the parameter  are not too low, the threshold qT m v  − 1− is higher than T m , which means all regions can achieve higher welfare in the noncorrupt case than the base case.Further, as the comparison between the corrupt case and the base case does not depend on the parameter , all regions achieve a higher welfare in the corrupt case (compared to the base case) if the supplier can remain in the market, but get a lower welfare if the supplier exits.
As such, for low-priority products with the parameter  larger than 0.5, a pragmatic solution could be donations of a quality lower than the quality threshold in the noncorrupt case (and thus lower than the fresh quality).There are at least three benefits.The first is improved healthcare outcomes: The recipient's healthcare welfare is higher than when there is no donation, no matter whether the manager is corrupt or not.The second is alleviating the adverse effects on the local industry: The supplier can remain in the market no matter whether the manager is corrupt or not, and the corrupt manager does not make the adverse effect on the local supplier worse.The third benefit is constraining the negative consequences from corruption: Though the corrupt case leads to lower healthcare welfare than the noncorrupt case, all regions suffer, thus eliminating the chance that high-budget regions take advantage of low-budget regions when the manager is corrupt.

CONCLUSIONS AND IMPLEMENTATION
This paper models the impact of donating dated PPE-type medical supplies from a developed country's medical reserve.We find that donation quality plays a key role in the healthcare welfare, the effect on the local supplier, and the impact of corruption.In particular, we find that the value of donation is not monotone in quality.The healthcare outcome does not always increase with the donation quality: Recipient regions could experience a sharp decrease in healthcare welfare if donation quality goes above certain threshold where the supplier is driven out of the market; and this happens no matter whether the manager is corrupt or not.
As such, our results suggest it helps to provide donations of a quality that is slightly lower than the quality threshold determined when the manager is noncorrupt (and thus lower than the fresh quality).Dated stocks that are proved to be stable with rigorous testing can and should be donated when they are close to or just past the labeled shelf life, while donations of dated stocks that are unstable or whose quality deteriorates quickly with time should be avoided.Low-quality donation does not help with the healthcare welfare when the manager is noncorrupt, and can produce more harm than good when coupled with a corrupt manager by hurting both poor regions and the local supplier.However, donating products that are of a quality higher than the quality threshold or even higher than the quality of the local supplies, while possibly good for short-term health outcomes when the manager is noncorrupt, is likely to be detrimental in the long run by driving the local supplier out of business.This argues for making donation available that are clearly stamped as dated, in order to indicate their quality is lower than the fresh.
Given these results we recommend that the WHO reconsiders its stance on such donations.In situations with no additional concerns about safety and ethics, they could ease their strict policy of totally abandoning dated donations, especially for products confirmed to be stable long after the initial shelf life.It would, of course, be prudent to run a pilot program with a single product first to see if there are any unanticipated adverse effects.Given that 84% of the DHS' reserve stock of hand sanitizer has previously been reported to have passed its expiry date (US DHS, 2014) and impoverished countries are experiencing shortages of the product and using diluted hand-wash (Ider et al., 2012), this might be an obvious product to start with.Masks, needles, and syringes are other obvious candidates for donation.
As these donations come from the reserve, it would also help avoid expiration and dumping of the reserve stock.Two other ways to reduce expiration in the reserve are stock rotation (e.g., Zhou & Olsen, 2017) and shelf life extension (Courtney et al., 2009).However, such solutions are not always effective.Routinely rotating old reserve supply to regular daily operational use in the healthcare system (and then replacing with new stocks) may be uneconomic due to high handling costs and the possible remote location of the reserve.The SLEP in the United States, which extends the shelf life of the reserve products so they can be stored for a longer period, is a reactive way to delay expiration: Products still need to be replaced when they reach the extended shelf life.Instead, donating these stocks would use them effectively for the vast population in need of medical resources.When it comes to implementation, it is important to prevent dumping Production and Operations Management and making inappropriate donations, and the donor needs to discuss and seek consent from the recipient nation prior to making any dated donations.
While we have discussed the negative effects of donation on local suppliers, we have not specifically explored how to alleviate this side effect.As our results highlight the influence of donation quality, it is left for future research to explore other mechanisms to constrain the side effects and support local suppliers.Also, we have not discussed what lobbying medical supply companies might engage in if such a program was suggested.One possibility to avoid such lobbying would be to ask a large medical supplies company to actually run the program.They would undoubtedly be quite diligent in replacing dated supplies in the host country's reserve (since that is direct income), and therefore might be willing to donate the funds available to distribute the dated supplies.As mentioned earlier, they would be in the best position to ensure that the donations go to regions that are doing without the product (and therefore reduce issues of local supply replacement).Further, in the long run, this may improve demand from the impoverished country as using this product becomes the expected norm.Finally, eliminating local middlemen from the would also reduce the opportunities for corruption.
While our models are relatively simple, we believe they capture the key trade-offs faced in these scenarios.One limitation is that our model assumes full information.While this assumption is relative standard and allows us to derive meaningful results, future work considering partial information could enrich the story.We frame quality as age dependent, and it is left for future research to capture more dimensions of the quality variable.Our model considers a particular type of medical supplies, such as PPE and similar products, which are purchased and used by regions to fulfill their healthcare needs, and thus is not applicable to general donation items bought by end-user consumers.As our model focuses on a typical region, considering other forms of corruption before donation reaches the region could make an interesting separate study in the future, as would the consideration of donation of products for emergency (rather than routine) healthcare needs.

A C K N O W L E D G M E N T S
The authors are very grateful to the senior editor and the anonymous reviewers for their in-depth reading and valuable comments, which helped us improve the quality of this contribution.
Open access publishing facilitated by The University of Auckland, as part of the Wiley -The University of Auckland agreement via the Council of Australian University Librarians.

v
the quality of fresh supplies from the local market, q the quality of donated supplies, with 0 ≤ q ≤ v, T the amount of total budget available for the region, Φ(⋅) the cumulative distribution function of budget level T, T m the upper bound of possible budget level, with Φ(T m ) = 1, D the total potential demand for the product, p the market price charged by the supplier, p the supplier's minimum viable price, x s the quantity purchased from the supplier for the region, x d the quantity of donated products that reach the region,  the coefficient in the healthcare welfare function W(T, x s ).
The threshold qn decreases with the supplier's minimum viable price p.If 0 < p < 1− D T m , then 0 < qn < v. 2. If donation quality q ≤ qn , then the supplier can remain in the market: (a) The supplier's price p n * decreases with donation quality q.(b) For a given budget T, the healthcare welfare function W n (T) increases with donation quality q.(c) The welfare function W n (T) is always concave in donation quality q for T ≤ Regarding the continuity of the welfare function W n (T): (a) For all T ≤ qT m v−q+2q , W n (T) is continuous and increases with donation quality q.(b) For all T > qT m v−q+2q

F
I G U R E 1 Comparison of prices in different scenarios [Color figure can be viewed at wileyonlinelibrary.com]

F
Comparison of welfare in different scenarios [Color figure can be viewed at wileyonlinelibrary.com]