1 Introduction
1.1 Summary of results
2 The Model
We consider a vertical market with a monopolistic upstream manufacturer, distributor or service provider, henceforth supplier selling a homogeneous product (or resource) to downstream symmetric retailers. The supplier acts first (Stackelberg leader) and chooses a unique wholesale price for all retailers (linear pricing scheme without price differentiation), see also Tyagi (1999). The supplier produces at a constant marginal cost which we normalize to zero. This corresponds to the situation in which the supplier has ample quantity to cover any possible demand by the retailers and his lone decision variable is his wholesale price, or equivalently his profit margin, .
To study the market in which the supplier is less informed than the retailers about the retail demand level , we assume that after the supplier’s pricing decision but prior to the retailers’ order decisions, a value for is realized from a continuous cummulative distribution (cdf) , with finite mean and nonnegative values, i.e., . Equivalently, can be thought of as the supplier’s belief about the demand level and, hence, about the retailers’ willingnesstopay his price. We will write for the tail distribution of ,
for its probability density function (pdf), whenever such a pdf exists, and
for its support, with , and . The case is not excluded^{1}^{1}1Formally, this case contradicts the assumption that is continuous or nonatomic. However, it is allowed so that unnecessary notation is avoided. It should cause no confusion. and corresponds to the situation where the supplier is completely informed about the retail demand level.Given the demand realization , the total quantity that the retailers will order from the supplier for any possible wholesale price will be denoted by . Using this notation, and under risk neutrality, the supplier aims to maximize his expected profit function , which is equal to
(1) 
In general, depending on the form of second stage competition, the quantity may vary. In this paper, we focus on markets with linear demand, see Mills (1959), Petruzzi and Dada (1999) and Huang et al. (2013) among others, and allow for a wide range of competition structures between the retailers, see Table 1. All structures presented in Table 1 give rise – in equilibrium – to essentially the same and hence to the same objective function for the supplier. More importantly, in all these structures, the secondstage equilibrium between the retailers is unique and hence, is uniquely determined under the assumption that the retailers follow their equilibrium strategies in the game induced by each wholesale price (subgame perfect equilibrium concept).To ease the exposition, we restrict to retailers and generalize our results to arbitrary number of symmetric retailers in Subsection 3.3. Specifically, we assume that each retailer faces the inverse demand function
(2) 
for and . Here, denotes the potential market size (primary demand), the storelevel factor and the degree of product differentiation or substitutability between the retailers, see Singh and Vives (1984), Wu et al. (2012). As usual, we assume that . Each retailer’s only cost is the wholesale price that she pays to the supplier. Hence, each retailer aims to maximize her profit function , which is equal to
(3) 
Given the wholesale price set by the supplier and the realized value of the demand parameter , our model allows the retailers to engage in various forms of competition, the most common of which are summarized in Table 1. Here, denotes the positive part, i.e., and is a function of which denotes the equilibrium quantity that retailer orders from the supplier for realized demand and posted wholesale price . Since, there is no uncertainty on the side of retailers, is also the quantity that each retailer will sell to the market (in equilibrium).
Retail market structure  Retailer ’s order 

Singh and Vives (1984)  
Cournot competition – product differentiation  
Bertrand competition – product differentiation  
Padmanabhan and Png (1997)  
Single retailer no/full returns  
Competing retailers (orders/price) – no returns  
Competing retailers (orders/price) – full returns  
Yang and Zhou (2006)  
Collusion between retailers – product differentiation 
The standard Cournot and Betrand outcomes arise as special cases of the above. In particular, for , the goods are independent and we have the monopoly solution . For , the goods are perfect substitutes with in Bertrand competition (at zero price) and in Cournot competition. All of the above are assumed to be common knowledge among the participants in the market (the supplier and the retailers).
3 Equilibrium analysis: supplier’s optimal wholesale price
We restrict attention to subgame perfect equlibria^{2}^{2}2Technically, these are perfect BayesNash equilibria, since the supplier has a belief about the retailers’ types, i.e. their willingnesstopay his price, that depends on the value of the stochastic demand parameter . of the extensive form, twostage game, in which for any wholesale price, , set by the supplier, the retailers order their equilibrium quantities, , with in the game induced by .
Under subgame perfection, uniqueness of the equlibrium strategies in all the retail market competitionstructures of Table 1 implies that, conditional on , the supplier correctly predicts the retailers’ orders (optimal responses) for choosing wholesale price . Hence, taking the expectation over all possible values of , (1) becomes . Based on the second column of Table 1, the aggregate quantity that the retailers will order from the supplier in equilibrium has the general form , for all competitionstructures, where is a suitable modelspecific constant. Thus, the supplier’s expected profit maximization problem becomes
(4) 
From the supplier’s perspective, we are interested in finding conditions such that the maximization problem in (4) admits a unique and finite optimal wholesale price, .
3.1 Deterministic Market
First, we treat the case in which the supplier knows the primary demand (deterministic market). According to the notation introduced in Section 2, this corresponds to the case . In this case and it is straightforward that . Hence, the complete information twostage game has a unique subgame perfect Nash equilibrium, under which the supplier sells with optimal price and each retailer orders quantity as determined by Table 1.
3.2 Stochastic Market
The equilibrium behavior of the market in which the supplier does not know the demand level (stochastic market) is less straightforward. Now, and the supplier is interested in finding an that maximizes his expected profit in (4). For an arbitrary demand distribution , may not be concave (nor quasiconcave) and, hence, not unimodal, in which case the solution to the supplier’s optimization problem is not immediate. To obtain a general unimodality condition, we proceed by differentiating the supplier’s revenue function , see also Leonardos and Melolidakis (2018). First, since is nonnegative, we write , for . Since, and is nonatomic by assumption, we have that
for any . With this formulation, both the supplier’s revenue function and its first derivative can be expressed in terms of the mean residual life (MRL) function of . In general, the MRL function of a nonnegative random variable
with cumulative distribution function (cdf)
and finite expectation, , is defined asand , otherwise, see, e.g., Shaked and Shanthikumar (2007); Lai and Xie (2006) or Belzunce et al. (2016). Using this notation, we obtain that and
(5) 
for . Based on (5), the first order condition (FOC) for the supplier’s revenue function is that or equivalently that . We call the expression
(6) 
the generalized mean residual life (GMRL) function, see Leonardos and Melolidakis (2018), due to its connection to the generalized failure rate (GFR) function , defined and studied by Lariviere (1999) and Lariviere and Porteus (2001). Its meaning is straightforward: while the MRL function at point measures the expected additional demand, given the current demand , the GMRL function measures the expected additional demand as a percentage of the given current demand. Similarly to the GFR function, the GMRL function has an appealing interpreation from an economic perspective, as it is related to the price elasticity of expected or mean demand, , see also Xu et al. (2010). Specifically,
(7) 
which implies that corresponds to the inverse of the price elasticity of expected demand. Hence, the FOC asserts that the supplier’s payoff must be maximized at the point(s) of unitary elasticity. Also, realistic problems must have a price elasticity that eventually becomes greater than , see Lariviere (2006). Consequently, for an economically meaningful analysis, we focus on distributions for which eventually becomes less than , i.e., distributions for which is finite. Observe that for a nonnegative random demand with continuous distribution and finite expectation , and hence .
Based on these considerations, it remains to derive conditions that guarantee the existence and uniqueness of an that satisfies the FOC and to show that this indeed corresponds to a maximum of the supplier’s revenue function. This is established in the following Theorem which is the main result of the present Section.
Theorem 3.1 (Equilibrium wholesale prices in the stochastic market).
Assume that the nonnegative demand parameter, , follows a continuous (nonatomic) distribution , with support .

If an optimal price for the supplier exists, then satisfies the fixed point equation
(8) 
If the generalized mean residual life (GMRL) function, , of is strictly decreasing and is finite, then in equilibrium, the supplier’s optimal price exists and is the unique solution of (8). In this case, , if , and , otherwise.
Proof.
Since for , the sign of the derivative is determined by the term and any critical point satisfies . Hence, the necessary part of the theorem is obvious from (5) and the continuity of . For the sufficiency part, it remains to check that such a critical point exists and corresponds to a maximum under the assumptions that is strictly decreasing and . Clearly, is continuous and . Hence, starts increasing on . However, the limiting behavior of and hence of as approaches from the left, may vary depending on whether is finite or not. If is finite, i.e. if the support of is bounded, then . Hence, eventually becomes less than 1 and a critical point that corresponds to a maximum exists without any further assumptions. Strict monotonicity of implies that this is unique. If , then an optimal solution may not exist because the limiting behavior of as may vary, see Example 3.3 or Bradley and Gupta (2003). In this case, the condition of finite second moment ensures that . In particular, as we show in Leonardos and Melolidakis (2018), if the GMRL function of a random variable with unbounded support is decreasing, then if and only if is finite. This establishes existence. Uniqueness follows again from strict monotonicity of which precludes intervals of the form that give rise to multiple optimal solutions.
To prove the second claim of the sufficiency part, note that is equivalent to . Then, the DGMRL property implies that for all , hence . In this case, and hence is given explicitly by , which may be compared with the optimal of the complete information case. On the other hand, if , then for all , which implies that must be in . ∎
The economic interpretation of the sufficiency conditions in part (b) of Theorem 3.1 is immediate. By (7), demand distributions with the DGMRL property are precisely distributions that exhibit increasing price elasticity of expected demand. By Leonardos and Melolidakis (2018), Theorem 3.2, finiteness of the second moment is required to ensure that the expended demand will eventually become elastic, even in the case of unbounded support. In sum, part (b) characterizes in terms of their mathematical properties, demand distributions that model linear markets with monotone and eventually elastic expected demand. These conditions are derived in a broad probabilistic context and apply to distributions that may neither be absolutely continuous (do not possess a density) nor have a connected support.
Remark 3.2.
Strict monotonocity can be relaxed to weak monotonicity in the statement of Theorem 3.1 without significant loss of generality. This relies on the explicit characterization of distributions with MRL functions that contain linear segments which is given in Proposition 10 of Hall and Wellner (1981). Namely, on some interval if and only if for all . If is unbounded, this implies that has the Pareto distribution on with scale parameter . In this case, , see Example 3.3, which is precluded by the requirement that . Hence, to replace strict by weak monotonicity – but still retain equilibrium uniqueness – it suffices to exclude distributions that contain intervals with in their support, for which for all .
Example 3.3 (Pareto distribution).
The Pareto distribution is the unique distribution with constant GMRL and GFR functions over its support. Let be Pareto distributed with pdf , and parameters and (for we get , which contradicts the basic assumptions of our model). To simplify, let , so that , , and . The mean residual life of is given by and, hence, is decreasing on and increasing on . However, the GMRL function is decreasing for and is constant thereafter, hence, is DGMRL. Similarly, for the failure (hazard) rate is decreasing, but the generalized failure rate is constant and, hence, is IGFR. The payoff function of the supplier is
which diverges as , for and remains constant for . In particular, for , the second moment of is infinite, i.e. , which shows that for DGMRL distributions, we may not drop the assumption that the second moment of is finite, for part (b) of Theorem 3.1 to hold. Contrary, for , we get as the unique optimal wholesale price, which is indeed the unique fixed point of .
3.3 General case with identical retailers
To ease the exposition, we restricted our presentation to identical retailers. However, the present analysis applies to arbitrary number of symmetric retailers for most competitionstructures that are given in Table 1. This relies on the fact, that these structures give rise to a unique secondstage equilibrium, and hence, to a well defined expected profit function for the supplier, as in (4), under the subgame perfect equilibrium concept, for any . Moreover, the total quantity that is ordered from the supplier depends on only up to a scaling constant. Thus, the approach to the supplier’s expected profit maximization in the firststage remains the same independently of the number of secondstage retailers.
To avoid unnecessary notation, we restrict attention to classic Cournot competition. Formally, let , with denote the set of symmetric retailers. A strategy profile (retailers’ orders from the supplier) is denoted by with and . Assuming linear inverse demand function , the payoff function of retailer , for , is given by . Under these assumptions, the second stage corresponds to a linear Cournot oligopoly with constant marginal cost, . Hence, each retailer’s equilibrium strategy, , is given by , for . Accordingly, in the first stage, the supplier’s expected revenue function on the equilibrium path is given by . Hence, it is maximized again at if the supplier knows or at if the supplier only knows the distribution of . Based on the above, the number of secondstage retailers affects the supplier’s revenue function only up to a scaling constant and Theorem 3.1 is stated unaltered for any . The other forms of competition admit similar generalizations provided that the retailers remain symmetric.
4 Market Efficiency
We now turn to study the effect of upstream demand uncertainty on the efficiency of the vertical market. Unless stated otherwise, and to avoid unnecessary notation, we consider classic Cournot competition with linear demand and arbitrary number of competing retailers in the second stage, see also Subsection 3.3. After scaling to , this implies that the equilibrium order quantities are for each and any wholesale price . The supplier’s optimal wholesale price, , is given by Theorem 3.1.
4.1 Probability of notrade
Markets with incomplete information are usually inefficient in the sense that trades that are profitable for all market participants may actually not take place. In the current model, such inefficiencies appear as values of for which a transaction does not occur in equilibrium under incomplete information, although such a transaction would have been beneficial for all parties involved, i.e., supplier, retailers and consumers.
If , then the retailers buy units and there is an immediate stockout. Hence, for a continuous distribution of , the probabilitiy of notrade in equilibrium under incomplete information is equal to . To study this probability as a measure of market inefficiency, we restrict attention to the family of DMRL distributions, i.e., distributions for which is nonincreasing. In this case, we have
Theorem 4.1.
For any demand distribution with the DMRL property, the probability of notrade in the equilibrium of the stochastic market cannot exceed the bound . This bound is tight over all DMRL distributions.
Proof.
Expressing the distribution function in terms of the MRL function, e.g. see Guess and Proschan (1988), we get . Hence, by the DMRL property and the monotonicity of the exponential function, it follows that . Since , we conclude that
. If the MRL function is constant, as is the case for the exponential distribution, see
Example 4.2, then all inequalities above hold as equalities, which establishes the second claim of the Theorem. ∎Examples 4.3 and 4.2 highlight the tightness of the notrade probability bound that is derived in Theorem 4.1. Example 4.4 shows that this bound cannot be extended to the class of DGMRL distributions. The conclusions are summarized in Figure 1.
Example 4.2 (Exponential distribution).
Let , with , and pdf . Since , for , the MRL function is constant over its support and, hence, is both DMRL and IMRL but strictly DGMRL, as , for . By Theorem 3.1, the optimal strategy of the supplier is . The probability of no transaction is equal to , confirming that the bound derived in Theorem 4.1 is tight. Thus, the exponential distribution is the least favorable, over the class of DMRL distributions, in terms of efficiency at equilibrium.
Example 4.3 (Beta distribution).
This example refers to a special case of the Beta distribution, also known as the Kumaraswamy distribution, see
Jones (2009). Let with , and pdf . Then, and for . Since the MRL function is decreasing, Theorem 3.1 applies and the optimal price of the supplier is . Hence, as . This shows that the upper bound of in Theorem 4.1 is still tight over distributions with strictly decreasing MRL, i.e., it is not the flatness of the exponential MRL that generated the large inefficiency.Example 4.4 (Generalized Pareto or Pareto II distribution).
This example shows that the bound of Theorem 4.1 does not extend to the class of DGMRL distributions. Let , with pdf and cdf , with . For the parametrization and , with , the cdf becomes . Moreover, , since for any . Hence, by a standard calculation, , which shows that is DGMRL but not DMRL. In this case, and , which shows that the probability of a stockout may become arbitrarily large for values of close to . The “pathology” of this example relies on the fact that as .
4.2 Division of realized market profits
If the realized value of is larger than , then a transaction between the supplier and the retailers takes place. In this case, we measure market efficiency in terms of the realized market profits. Specifically, we fix a demand distribution (which satisfies the sufficiency conditions of Theorem 3.1) with support and a realized demand level and compare the individual realized profits of the supplier and each retailer between the deterministic and the stochastic markets. For clarity, we summarize the necessary quantities in Table 2.
Upstream Demand for the Supplier  
Uncertain  Deterministic  




Realized Profits in Equilibrium  
Supplier  
Retailer  
Aggregate 
We are interested in addressing the following questions: First, how do the supplier’s (retailers’) realized profits compare between the stochastic and the deterministic market? Second, how does retail competition and demand uncertainty affect the supplier’s (retailers’) share of realized market profits? Third, how does the level or retail competition – number of retailers – affect supplier’s profits in both markets? The answers are summarized in the following Theorem which follows rather immediately from Table 2. To avoid technicalities, we assume throughout that is large enough, so that (e.g. ).
Theorem 4.5.
Let denote a demand distribution with support , the respective optimal wholesale price in the stochastic market, and , with , a realized demand level, for which trading between supplier and retailers takes place in both the stochastic and the deterministic market. Let, also, and denote the supplier’s share of realized profits in the stochastic and deterministic markets respectively. Then,

, with equality only for . In particular, for any .

decreases in the realized demand level .

is independent of the demand level .

is higher than for values of , equal for , and lower otherwise.

and both increase in the level of retail competition.
Finally, each retailer’s profit in the stochastic market, , is strictly higher than her profit in the determinstic market for all demand levels and less otherwise, with equality only for .
Proof.
By Table 2, we have that: (i) if and only if which holds with strict inequality for all values of , except for for which the quantities are equal. The second part of statement (i) is immediate. For (ii) , and for (iii) . Now, (iv) and (v) directly follow from the previous calculations. Finally, if and only if which holds with strict inequality for all values of and with equality for . ∎
The statements of Theorem 4.5 are rather intuitive. The supplier is always better off if he is informed about the retail demand level, (i). Moreover, he captures a larger share of the realized market profits for lower values of realized demand (but not lower than the notrade threshold of ) in the stochastic market, (ii), whereas his share of profits is constant (with respect to the demand level) in the deterministic market, (iii). Yet, there exists an interval of demand realizations, namely , for which the supplier’s profits, albeit less than in the deterministic market, represent a larger share of the aggregate market profits in the stochastic market, (iv). In any case, retail competition benefits the supplier, (v). Finally, and abiding to intuition, each retailer makes a larger profit for higher realized demand values in the case that the supplier prices under uncertainty. These observations confirm the existence of conflicting incentives regarding demandinformation disclosure between the retailers and the supplier.
4.2.1 Market performance: aggregate profits
We next turn to the comparison of the aggregate market profits between the deterministic and the stochastic market. As before, we fix a demand distribution (which is again assumed to satisfy the sufficiency conditions of Theorem 3.1) with support , and evaluate the ratio of the aggregate realized market profits in the stochastic market to the aggregate market profits in the deterministic market. To study market performance under the two scenarios, we need to evaluate the combined effect of two competing forces: demand uncertainty and retail competition. For a realized demand , there is a stockout and the realized aggregated profits are equal to . In this case, the stochastic market performs arbitrarily worse than the deterministic market and the ratio is equal to for any number of competing retailers. Hence, for a nontrivial analysis, we study the question of how good can the stochastic market perform when compared to the deterministic market. We have the following^{3}^{3}3Again, to avoid technicalities, and to ensure that for any , we assume that is large enough, e.g. ..
Theorem 4.6.
Let denote a demand distribution with support , the respective optimal wholesale price in the stochastic market, and , with , a realized demand level, for which trading between supplier and retailers takes place in both the stochastic and the deterministic market. Let, also, denote the ratio of the aggregate realized profits in the stochastic market to the aggregate profits in the deterministic market. Then,

for , if , if and if .

is maximized for for , for which it is equal to . Moreover, converges to as for any .

increases in the level of competition for demand levels and decreases thereafter.
Proof.
By Table 2, a direct substitution yields that for . Taking the derivative of the ratio with respect to yields which shows that the ratio is increasing , and decreasing thereafter. In particular, the values of for which the ratio is larger than or equal to are , for , for and for . This establishes (i) and after some trivial algebra, also (ii). To obtain (iii) we take the partial derivative of with respect to . ∎
Theorem 4.6 asserts that there exists an interval of realized demand values, whose upper bound depends on the number of competing retailers, for which the stochastic market outperforms (in terms of aggregate profits) the deterministic marke, (i). The effect of increasing retail competition on the aggregate profits of the stochastic market is twofold. First, the range (interval) of demand values for which the ratio of aggregate profits exceeds is reduced to a single point as competition increases (). Second, for larger values of realized demand, the ratio converges to as . This shows that uncertainty on the side of the supplier is less detrimental for the aggregate market profits when levels of retail cometition are low. In particular, for , the aggregate profits of the stochastic market remain strictly higher than the profits of the deterministic market for all large enough realized demand levels. As competition increases this remains true only for lower (but still above the notrade threshold) demand levels. However, as competition increases, the ratio becomes arbitrarily low in the limit, i.e., for higher demand realizations.
The statements of Theorem 4.6 are illustrated in Figure 2. Here but the picture is essentially the same for any choice of demand distribution that satisfies the sufficiency conditions of Theorem 3.1 and for which is large enough, i.e., .
5 Comparative Statics
The closed form expression of (8) facilitates a comparative statics and sensitivity analysis on the demand distribution’s paremeters via the rich theory of stochastic orders, see Shaked and Shanthikumar (2007), Lai and Xie (2006) and Belzunce et al. (2016). Because in equilibrium, both the total quantity that will be sold to the market and the retail price are monotone in the wholesale price , we restrict henceforth attention to the response of in varying distribution characteristics.
To obtain a meaningful comparison between different markets, we assume throughout equilibrium uniqueness and hence, unless stated otherwise, we consider only distributions for which Theorem 3.1 applies^{4}^{4}4Since the DGMRL property is satisfied by a very broad class of distributions, see Banciu and Mirchandani (2013), Kocabıyıkoğlu and Popescu (2011) and Leonardos and Melolidakis (2018), we do not consider this as a significant restriction. Still, since it is sufficient (together with finitenes of the second moment) but not necessary for the existence of a unique optimal price, the analysis naturally applies to any other distribution that guarantees equilibrium existence and uniqueness.. First, we introduce some notation. Let be two nonnegative random variables with MRL functions and , respectively, such that for every . Then, we say that is less than in the mean residual life order, denoted by . This ordering plays a key role in the present model. Specifically, by (7), we have that for any if and only if for any , i.e., if and only if the price elasticity of expected demand in market is less than the price elasticity of expected demand in market for any wholesale price . This motivates the following definition: we will say that market is less elastic than market , denoted by , if for every . Based on the above, if and only if . Using this notation, the following Lemma captures the importance of the characterization in (8).
Lemma 5.1.
Let be two nonnegative, continuous and strictly DGMRL demand distributions with finite second moments. If is less elastic than , then the supplier’s optimal wholesale price is lower in market than in market . In symbols, if , then .
Proof.
In short, Lemma 5.1 states that the supplier charges a lower price in a more elastic market. Although trivial to prove once Theorem 3.1 has been established, it is the key to the comparative statics analysis in the present model. Indeed, combining the above, the task of comparing the optimal wholesale price for varying demand distribution parameters – such as market size or demand variability – essentially reduces to comparing demand distributions (markets) according to their elasticities or equivalently according to their MRL functions. Such conditions can be found in Shaked and Shanthikumar (2007) and provide the framework for the subsequent analysis.
5.1 Market Size
We start with the response of the equilibrium wholesale price to transformations that intuitively correspond to a larger market. Unless otherwise stated, we assume that the random demand is such that it satisfies the sufficiency conditions of Theorem 3.1 and hence that the supplier’s optimal wholesale price exists and is unique.
5.1.1 Reestimating Demand
Let denote the random demand in an instance of the market under consideration. Let denote a positive constant and an additional random source of demand that is independent of . Moreover, let denote the equilibrium wholesale price in the initial market and the equilibrium wholesale prices in the markets with random demand and respectively. How does compare to and ?
While the answer for is rather straightforward, see Theorem 5.2 below, the case of is more complicated. Specifically, since DGMRL random variables are not closed under convolution, see Leonardos and Melolidakis (2018), the random variable may not be DGMRL. This may lead to multiple equilibrium wholesale prices in the market, irrespectively of whether is DGMRL or not. To deal with the possible multiplicity of equilibria, we will write to denote the set of all possible equilibrium wholesale prices. Here, denotes the MRL function of a demand distribution, e.g., . To ease the notation, we will also write , when all elements of the set are less or equal than all elements of the set .
Theorem 5.2 largely confirms the intuition that wholesale prices are higher in the larger and markets. However, it also reveals that this is not always the case, and particularly for the market, that it hinges on additional, more restrictive assumptions for and .
Theorem 5.2.
Let be a nonnegative and continuous demand distribution with finite second moment.

If is DGMRL and is a positive constant, then .

If is DMRL and
is a nonnegative, continuous random variable with finite second moment and independent of
, then , i.e., for any equilibrium wholesale price of the market.
Proof.
The proof of part (i) follows directly from the preservation property of the order that is stated in Theorem 2.A.11 of Shaked and Shanthikumar (2007). Specifically, since is the MRL function of , we have that , for all , with the inequality following from the assumption that is DGMRL. Hence, which by Lemma 5.1 implies that .
Part (ii) follows from Theorem 2.A.11 of Shaked and Shanthikumar (2007). The proof necessitates that is DMRL and hence requiring that is merely DGMRL is not enough. Since, is DMRL, we know that for all . Together with , this implies that , for all . Hence, , which implies that in this case, is a lower bound to the set of all possible wholesale equilibrium prices in the market. ∎
5.1.2 Preservation of Market Size & Wholesale Price
Next, we turn attention to operations on demand distributions that preserve the order and hence the order of wholesale prices. Specifically, let denote two different demand distributions, such that . In this case, we know by Lemma 5.1 that . We are interested in determining transformations of that preserve the order and hence the ordering . Again, to avoid technicalities, we assume that satisfy the sufficiency conditions of Theorem 3.1.
Theorem 5.3.
Let denote two nonnegative, continuous and strictly DGMRL demand distributions, with finite second moments, such that .

If is a nonnegative, IFR distribution, independent of and , then .

If is an increasing, convex function, then .

If for some , then .
Proof.
Part (i) follows from Lemma 2.A.8 of Shaked and Shanthikumar (2007). Since the resulting distributions may not be DGMRL nor DMRL, the setwise notation is necessary. Part (ii) follows from Theorem 2.A.19. Equilibrium uniqueness is retained in the transformed markets, , since the DGMRL class of distributions is closed under increasing, convex transformations, see Leonardos and Melolidakis (2018). Finally, part (iii) follows from Theorem 2.A.19. However, the DGMRL class is not closed under mixtures and hence, in this case, the market may have multiple equilibria, which necessitates, as in part (i), the setwise statement for the wholesale equilibrium prices of the market. ∎
If instead of , and are ordered in the stronger hazard rate, order, i.e., if for all , denoted by , then part (i) of Theorem 5.3 remains true by Lemma 2.A.10 of Shaked and Shanthikumar (2007), even if is merely DMRL (instead of IFR).
Although Theorems 5.3 and 5.2 are immediate once
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