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 Open Access
Optimal twostage pricing strategies from the seller’s perspective under the uncertainty of buyer’s decisions
 Martín Egozcue^{1},
 Jiang Wu^{2, 3}Email author and
 Ričardas Zitikis^{3}
https://doi.org/10.1186/s4048801700672
© The Author(s) 2017
 Received: 13 February 2017
 Accepted: 11 July 2017
 Published: 1 September 2017
Abstract
In Punta del Este, a resort town in Uruguay, realestate property is in demand by both domestic and foreign buyers. There are several stages of selling residential units: before, during, and after the actual construction. Different pricing strategies are used at every stage. Our goal in this paper is to derive, under various scenarios of practical relevance, optimal strategies for setting prices within twostage selling framework, as well as to explore the optimal timing for accomplishing these tasks in order to maximize the overall seller’s expected revenue. Specifically, we put forward a general twoperiod pricing model and explore pricing strategies from the seller’s perspective, when the buyer’s decisions in the two periods are uncertain: commodity valuations may or may not be independent, may or may not follow the same distribution, be heavily or just lightly influenced by exogenous economic conditions, and so on. Our theoretical findings are illustrated with numerical and graphical examples using appropriately constructed parametric models.
Keywords
 Decision theory
 Behavioral economics
 Uncertainty
 Strategy
 Twoperiod economy
 Background risk model
 Gamma distribution
MSC codes
 62P20
 91A55
 91A80
 91B24
Introduction
Commodity pricing has been a prominent topic in the literature, with various models and strategies suggested and explored. In this paper, motivated by a problem described next, we put forward and investigate (both theoretically and numerically) a general model for pricing within the twoperiod framework that naturally arises in the context of the motivating problem.
1.1 Motivating problem
In Punta del Este, a resort town in Uruguay, realestate property is in demand by both domestic and foreign buyers. As a recent example, the frequency distribution of buyers for certain highrise buildings was approximately as follows: 75% Argentineans, 10% Uruguayans, 9% Brazilians, and the remaining ones from the rest of the world (Chile, U.S.A., and so on). A few immediate observations follow. First, the ratio of domestic and foreign buyers varies depending on a number of factors, including economic, financial, and political. Second, it has been observed that the average foreign buyer is wealthier than the average domestic one, and thus tends to exhibit higher bidding prices. Furthermore, given the diversity of buyers, the prices are usually in the US dollars (USD), but some of the building costs such as salaries of workers are in the Uruguayan pesos (UYU).
To properly understand our problem, we need to describe the property development and selling processes. Namely, contracted by an investor, a construction company starts building, say, a residential tower. There are several stages of selling residential units: before, during, and after the actual construction of the tower. Different pricing strategies are used at every stage. It is frequently the case that, at least initially, the investor wishes to sell the units en masse and thus hires a realestate agent for several months. If the sale is not successful during this initial stage, then the units are put on sale individually, with no particular time horizon set in advance, and at a possibly different price, which could be higher or lower than the original price.
The goal that we set out in this paper is to derive, under various scenarios of practical relevance, optimal strategies for setting first and secondstage prices, as well as to propose the optimal timing for accomplishing these tasks, in order to maximize the overall seller’s expected revenue. In the next subsection, we give a brief appraisal of what we have accomplished in the current paper, with a related though brief literature review given in the following subsection.
1.2 Results and findings – an appraisal
First, in this paper we put forward a highly encompassing, yet tractable, model and explore optimal pricing strategies from the seller’s perspective when buyer’s realestate valuations and decisions in the two stages are uncertain: they can be independent or dependent, identically distributed, or stochastically dominate each other, be influenced by exogenous factors at various degrees, and so on. In particular, we shall see from our considerations and examples in the next section that the simultaneous pricing strategies yield higher expected revenues than those under the sequential pricing strategy.
Second, we study the case when real estate costs are possibly denominated in different currencies, as is the case in our motivating problem and, in general, is an important and very common factor in developing countries where large fractions of building costs are denominated in foreign currencies. Hence, currency exchangerate movements influence optimal pricing decisions.
Third, our model provides conditions under which secondstage prices could be higher or lower than the firststage prices. This might, initially, be surprising because it is a common intuitive assumption that if a property is not sold during the first stage, then the property price should be reduced before commencing the second stage. As we shall see from our following considerations, however, the relationship between the two stage prices is much more complex: higher holding costs, currency exchange movements, or some type of dominance between the first and secondstage buyers’ bidding price distributions, could very much influence the secondstage price, thus possibly making it larger than that of the first stage, assuming of course that the property was not sold during the first stage.
Finally, our general model accommodates sellers with different shapes of their utility functions, such as those arising in Behavioral Economics (see, e.g., Dhami 2016). In general, while working on this project, we were considerably influenced by, and benefited from, research contributions by many authors, and the following brief literature snapshot highlights some of those that we have found particularly related to the present paper.
1.3 Related literature
House pricing from the seller’s and buyer’s perspectives has been studied by many authors. For instance, Quan and Quigley (1991), and Biswas and McHardy (2007) adopt the seller’s viewpoint in their research. Furthermore, Stigler (1962); Rothschild (1974); Gastwirth (1976); Quan and Quigley (1991); Bruss (2003) and Egozcue et al. (2013) explore the problem from the buyer’s perspective. Pricing under different seller’s risk attitudes has been studied in the real estate literature as well. For instance, seller’s risk neutral behavior has been researched by Arnold (1992; 1999) and Deng et al. (2012). Biswas and McHardy (2007) analyze optimal pricing for risk averse sellers. In addition, Genesove and Mayer (2001); Anenberg (2011) and Bokhari and Geltner (2011) study house price determination for sellers whose risk behavior follows the teachings of Prospect Theory (Kahneman and Tversky 1979).
Bruss (1998, 2003); Egozcue et al. (2013); Egozcue and Fuentes García (2015) and Wu and Zitikis (2017) apply a twoperiod model to determine optimal commodity (e.g., real estate, computer, etc.) prices that maximize the expected revenue, or minimize the expected loss. Some of the aforementioned works have been influenced by the twoenvelope problem, and in particular by the viewpoint put forward by McDonnell and Abbott (2009) and McDonnell et al. (2011). Furthermore, Titman (1985) considers a twoperiod model to analyze the optimal land prices when the condominium unit prices are uncertain. We also refer to Lazear (1986); Nocke and Peitz (2007); Heidhues and Koszegi (2014), and reference therein, for additional twoperiod pricing models for real estate.
The rest of this paper is organized as follows. In Section 2 we present several illustrative examples that clarify certain key aspects of our general model proposed in Section 3, such as sequential and simultaneous price settings, differing valuations and thus bid prices, costs associated with holding property unsold. In Section 4 we analyze the firststage selling probability, and in Section 5 we explore the more complex dynamic secondstage selling probability. In Section 6 we discuss modeling first and secondstage value functions and then use them to numerically illustrate our general model. Section 7 concludes the paper.
Sequential vs simultaneous price setting
In this section we discuss scenarios that clarify various aspects of the problem at hand. In particular, we shall see the difference between setting the two prices sequentially and simultaneously, and we shall also see how the two prices are influenced by considerations such as seeking certain gross or net profits, taking into account possibly different treatments of domestic and foreign buyers, and so on.
We work with a discretetime twoperiod economy: t=0 and t=1. Let X _{0} and X _{1} denote the amounts (i.e., bidding prices) that the buyer is willing to pay for the property during the initial (i.e., t=0) and subsequent (i.e., t=1) selling stages, respectively. Both X _{0} and X _{1} are random variables from the seller’s perspective, and thus we also view them in this way. For the seller, the task is to set an appropriate price p _{0} for the initial selling stage, and also an appropriate price p _{1} (which is usually different from p _{0}) for the following selling stage.
It is natural to think that the seller would tend to first set p _{0} that would result in a desired outcome such as the maximal expected profit during the initial selling stage, and then, if the sale fails, the seller would set p _{1} that would maximize the expected profit during the following selling stage. As we shall illustrate below, the two prices set in this sequential manner may not maximize the expected overall profit, and thus a sensible strategy for the seller who is not in a rush would be to set both p _{0} and p _{1} before commencing the initial selling stage. The above caveat ‘who is not in a rush’ is important because rushed decisions usually give rise to very different forces at play, such as willingness to set the price p _{0} low enough to ensure a very high probability of selling the property during the initial selling stage. There are of course many other scenarios of practical interest, but in this paper we concentrate on maximizing the expected (gross or net) profit.
The rest of the section consists of two subsections: the first one contains preliminary facts such as sequential and simultaneous pricing, and the second subsection discusses four scenarios that clarify (and justify) the complexity of our general model that we start developing in Section 3.
2.1 Preliminaries
2.1.1 Sequential price setting
2.1.2 Simultaneous price setting
Since Π _{0}(p _{0,max})+Π _{1}(p _{1,max}) is equal to Π(p _{0,max},p _{1,max}), which cannot exceed \(\Pi (p_{0}^{\text {max}},p_{1}^{\text {max}})\) by the very definition of \((p_{0}^{\text {max}},p_{1}^{\text {max}})\), the seller cannot be worse off by simultaneously setting the prices before commencing the initial selling stage.
Note 2.1
The simultaneous setting of prices can be viewed as a strategic decision, whereas setting the prices sequentially just before commencing the respective selling stages are tactical choices, which in view of the above arguments cannot outperform the strategic (i.e., simultaneous) one. Deciding on which of these alternatives, and when to make them, has been a prominent topic in the literature, particularly in enterprise risk management (e.g., Fraser and Simkins 2010; Segal 2011; Louisot and Ketcham 2014).
2.1.3 Gamma distributed bidding prices
Although our general model is not limited to any specific price distribution, in our numerical illustrative considerations, we assume that the prices follow the gamma distribution, which is a very reasonable assumption, extensively used in the literature (see, e.g., Pratt et al.1979; Quan and Quigley1991; Hong and Shum2006). In particular, Quan and Quigley (1991) characterize the density function of the reservation price of a group of selfselected buyers using this distribution. Hong and Shum (2006) apply the gamma distribution to model search costs, including time, energy and money spent on researching products, or services, for purchasing. There are numerous cases of using the gamma distribution when modeling insurance losses (e.g., Hürlimann2001; Furman and Landsman2005; Alai et al.2013).
2.2 Scenarios
2.2.1 Scenario A: identical X _{0} and X _{1}
with the earlier defined a _{0} and the gamma random variable G∼Ga(α,β). Naturally, if the property is not sold during the initial stage, then under Scenario A, in order to at least hope to be successful during the subsequent stage, the seller has no alternative but to reduce the price, and we shall see this clearly from our following mathematical considerations. We note at the outset, however, that other scenarios to be discussed below will show the possibilities of increasing secondstage prices and still be able to successfully sell the property.
Prices and profits when X _{0}=X _{1}=X with X=a _{0}+G and G∼Ga(α,β)
α  β  σ _{ G }  Π(p _{0,max},p _{1,max})  p _{0,max}  p _{1,max}  \(\Pi (p^{\max }_{0},p^{\max }_{1})\)  \(p^{\max }_{0}\)  \(p^{\max }_{1}\) 

100  2  5  238.3249  238.3686  232.1596  243.6800  246.8699  236.3763 
25  0.5  10  229.9880  230.1152  220.5280  238.4024  244.7218  226.8302 
6.25  0.125  20  217.1578  217.3732  206.2535  229.8237  241.8907  213.1633 
2.78  0.056  30  207.4594  207.5638  200.5773  223.6389  240.9744  204.4544 
2.2.2 Scenario B: independent X _{0} and X _{1}
Obviously, p _{0,max} and p _{1,max} must be identical because X _{0} and X _{1} follow the same distribution, but there is of course no reason why \(p_{0}^{\max }\) and \(p_{1}^{\max }\) should be identical: the clear difference between the two will be seen from the following numerical example.
Prices and gross profits when X _{0} and X _{1} are independent and follow the distribution of a _{0}+G with G∼Ga(α,β)
α  β  σ _{ G }  Π(p _{0,max},p _{1,max})  p _{0,max}  p _{1,max}  \(\Pi (p^{\max }_{0},p^{\max }_{1})\)  \(p^{\max }_{0}\)  \(p^{\max }_{1}\) 

100  2  5  238.3595  238.3686  238.3686  244.1578  247.0367  238.3651 
25  0.5  10  230.0839  230.1152  230.1152  239.2700  244.9522  230.1226 
6.25  0.125  20  217.3058  217.3732  217.3732  230.9968  242.0798  217.3781 
2.78  0.056  30  207.5201  207.5638  207.5638  224.4556  241.0434  207.5595 
is achieved. Note from Table 1 that the values of all the four selling prices p _{0,max}, \(p^{\max }_{0}\), p _{1,max} and \(p^{\max }_{1}\) decrease when the standard deviation σ _{ G } (=σ _{ X }) increases. Note also that the bounds \(p_{0,\max }<p^{\max }_{0}\) and \(p^{\max }_{0}>p^{\max }_{1}\) hold. Furthermore, we always see the ordering \(p_{0,\max }<p^{\max }_{0}\) in Table 1, but the ordering of p _{1,max} and \(p^{\max }_{1}\) seems to depend on the value of σ _{ G }.
2.2.3 Scenario C: X _{1} stochastically dominates X _{0}
We see from previous two Tables 1 and 2 that neither sequential nor simultaneous secondstage selling prices are higher than the corresponding firststage prices: we always have p _{0,max}≥p _{1,max} and \(p_{0}^{\max }\ge p_{1}^{\max }\) in Tables 1 and 2. In practice, however, we often observe that after the failed initial sales, the sellers increase the prices and achieve successful results. There are several explanations of this phenomenon, and we shall next discuss one of them, with the other one making the contents of Scenario D below.
Prices and gross profits when X _{0}=a _{0}+G _{0} and X _{1}=a _{0}+b _{1} G _{1} with independent G _{0},G _{1}∼Ga(α,β) and varying parameter b _{1} values
α  β  σ _{ G }  Π(p _{0,max},p _{1,max})  p _{0,max}  p _{1,max}  \(\Pi (p^{\max }_{0},p^{\max }_{1})\)  \(p^{\max }_{0}\)  \(p^{\max }_{1}\)  

b _{1}=0.5  100  2  5  238.2429  238.3686  218.6924  241.1509  243.4476  218.6442 
25  0.5  10  229.9149  230.1152  214.1065  235.7860  240.6705  214.1065  
6.25  0.125  20  217.1670  217.3732  207.1570  228.2490  238.4532  207.1570  
2.78  0.056  30  207.4754  207.5638  202.3953  223.0999  238.8913  202.3953  
b _{1}=1.1  100  2  5  238.3830  238.3686  242.3547  241.4216  243.7144  220.6309 
25  0.5  10  230.1187  230.1152  233.4259  240.3086  246.1226  233.3253  
6.25  0.125  20  217.3363  217.3732  219.6358  231.6847  242.9417  219.6334  
2.78  0.056  30  207.5313  207.5638  208.8733  224.8072  241.2413  208.8733  
b _{1}=2.1  100  2  5  238.6217  238.3686  282.7481  241.4216  243.7144  220.6309 
25  0.5  10  230.4780  230.1152  267.7370  250.7337  252.1044  248.5686  
6.25  0.125  20  217.6737  217.3732  245.0225  241.6766  253.5593  244.0345  
2.78  0.056  30  207.6767  207.5638  226.2926  229.9190  247.4248  225.7422 
We see from Table 3 that for every noted value of b _{1}, the prices p _{0,max} and p _{1,max} decrease when the standard deviation σ _{ G } (=σ _{ X }) increases, but the pattern of \(p^{\max }_{0}\) and \(p^{\max }_{1}\) is unclear. Note also from the table that the ordering \(p_{0,\max }<p^{\max }_{0}\) always holds, but various orderings hold between the secondstage prices p _{1,max} and \(p^{\max }_{1}\). Furthermore, we see that when b _{1}=0.5, we have p _{0,max}>p _{1,max} and \(p^{\max }_{0}>p^{\max }_{1}\), but when b _{1}=1.1 and b _{1}=2.1, we have p _{0,max}<p _{1,max} and \(p^{\max }_{0}>p^{\max }_{1}\).
2.2.4 Scenario D: cost of holding the property
Based on Scenario C, when the seller guesses that the buyer might be willing to pay a large price during the secondstage selling stage, the price in the second stage can be set larger and still the maximal expected gross profit achieved.
Prices and profits for various holding cost c _{1} values when the bidding prices X _{0} and X _{1} are independent and follow the distribution of a _{0}+G with G∼Ga(α,β)
α  β  σ _{ G }  Π(p _{0,max},p _{1,c,max})  p _{0,max}  p _{1,c,max}  \(\Pi (p^{\max }_{0,c},p^{\max }_{1,c})\)  \(p^{\max }_{0,c}\)  \(p^{\max }_{1,c}\)  

c _{1}=20  100  2  5  238.2367  238.3686  238.5186  241.0633  243.3410  238.5186 
25  0.5  10  229.8538  230.1152  230.4159  234.9445  239.6038  230.4159  
6.25  0.125  20  216.9603  217.3732  217.9019  225.1370  234.1515  217.9019  
2.78  0.056  30  207.2347  207.5638  208.1217  217.2595  230.1206  208.1216  
c _{1}=100  100  2  5  237.7470  238.3686  239.3381  238.2509  239.9507  239.3381 
25  0.5  10  228.9377  230.1152  232.1152  229.8364  233.0915  232.1152  
6.25  0.125  20  215.5916  217.3732  221.1499  216.9273  222.2088  221.1500  
2.78  0.056  30  206.1066  207.5638  212.1154  207.4254  212.7300  212.1154  
c _{1}=150  100  2  5  237.4428  238.3686  240.2109  237.6034  239.1895  240.2109 
25  0.5  10  228.3730  230.1152  234.0368  228.6428  231.6018  234.0368  
6.25  0.125  20  214.7622  217.3732  225.3874  215.1202  219.6026  225.3874  
2.78  0.056  30  205.4343  207.5638  218.7960  205.7393  209.6705  218.7952 
We see from the table that for all specified values of c _{1}, the sequentially set selling prices follow the order p _{0,max}<p _{1,c,max}, which is the opposite of what we have seen in the previous scenarios. In the case of simultaneously set prices, we have \(p^{\max }_{0,c}>p^{\max }_{1,c}\) for the two smaller costs c _{1}=20 and c _{1}=100, with the opposite ordering \(p^{\max }_{0,c}<p^{\max }_{1,c}\) in the case of the cost c _{1}=150. The sequentially set selling prices in the initial stage are always smaller than the corresponding simultaneously set prices, that is, the ordering \(p_{0,\max }<p^{\max }_{0,c}\) holds throughout the entire table. The reported in Table 4 numerical values of the selling prices p _{1,c,max} and \(p^{\max }_{1,c}\) are very similar.
The general model
where c _{0} is, e.g., the property development cost evaluated during the initial selling stage. (By definition, x _{+}=x when x≥0, and x _{+}=0 when x<0.) If, however, X _{0}<p _{0}, then the buyer rejects the offer and makes the second (and final) attempt to buy the property at a later time, which is generally unknown and thus treated as a random variable, which we denote by T.
Note 3.1
There are of course situations when T is prespecified and thus deterministic, say T=1. For example, Wu and Zitikis (2017) consider a twoperiod economy with t=0 standing for the Black Friday promotion period and t=1 for the Boxing Day promotion period. In this paper, we let T be random, with specific choices of a distribution and parameter values provided in Note 6.1 at the end of this paper.
Let X _{ T } be the amount of money that the buyer is willing to pay at time T>0 during the second selling stage. Conditionally on T, the price X _{ T } is a random variable from the seller’s perspective. Let p _{1} be the price set by the seller some time prior to commencing the second selling stage (the price can be set as early as the time of setting the initial price p _{0}). Analogously to the initial decision making, if X _{ T }≥p _{1}, then the property is sold and the seller’s profit is v _{ T }(p), where v _{ T } is a value (or utility) function, perhaps different from v _{0}, but usually such that v _{ T }(p)≤p for all p≥0. For example, v _{ T }(p)=(p−c _{ T })_{+}, where c _{ T } is, e.g., the costs of property development and holding it unsold at time T. We shall provide specific details on the structure of c _{ T } later in this paper.
would be maximal, where F _{ T } is the cdf of T. The seller may have various goals to achieve, and our following considerations can be adjusted accordingly. When deriving Eq. (23), which involves conditioning on T, we have assumed that the events X _{ t }≥p _{1} and T=t are independent and in this way obtained the probability P[X _{ t }≥p _{1}∣X _{0}<p _{0}].
Henceforth, we shall make a number of other simplifying yet practically sound assumptions, so that the technicalities would not be too complex.
The initialstage selling probability
To assess the probabilities P[X _{0}≥p _{0}] and P[X _{ t }≥p _{1}∣X _{0}<p _{0}] on the righthand side of Eq. (23), we need to specify appropriate models for the random variables X _{ t }, t≥0. Their distributions may involve population heterogeneity, as our motivating example shows, which we take into consideration. Specifically, we assume that the population of potential buyers consists of two groups: domestic buyers (D) permanently residing in Uruguay and foreigners (A) wishing to make investments.
Note 4.1
We have reserved F for denoting cdf’s, as is usually the case in the literature, and so use A to denote foreign buyers. This notation also reflects the fact that most of the foreign property buyers in Punta del Este are Argentineans.
Since economic and financial considerations of the two types of buyers are usually different, the structures of the corresponding random variables are also different. In this section we concentrate on the probability P[X _{0}≥p _{0}] and thus specify the structure of X _{0}. For this, we first note the forces that give rise to the amount of money X _{0} that the buyer (domestic or foreign) is willing to pay for the property during the initial selling stage.
In this section and throughout the rest of this paper, background risk models will play an important role. There are two major classes of such models: additive and multiplicative. For applications and discussions of additive models in Economic Theory, we refer to Gollier and Pratt (1996) and references therein, and to problems in Actuarial Science, we refer to Furman and Landsman (2005,2010); Tsanakas (2008), and references therein. Our current research is essentially based on the multiplicative model, which has been extensively explored and utilized in the literature (see, e.g., Tsetlin and Winkler2005; Franke et al.2006,2011; Asimit et al.2016; references therein). It is worth noting that a number of important parametric multiplicative models incorporate elements of both Pareto and gamma distributions, and we refer to Asimit et al. (2016); Su (2016) and Su and Furman (2017) for details and further references.
4.1 General considerations
Consider first the population of domestic buyers. Suppose that, initially, their buying decisions are based on individual considerations detached from all the exogenous factors, such as the overall economic situation. Let Y _{0D } be the amount of money (i.e., valuation) that the buyer thinks is affordable and worthy to pay, based on the aforementioned personal considerations. We call Y _{0D } the endogenous domestic valuation.
This is the amount of money (i.e., valuation) that the domestic buyer can afford, and is willing, to pay for the property during the initial selling stage.
which is the amount that the foreign buyer is willing to pay during the initial selling stage, where Y _{0A } is the corresponding endogenous valuation.
Note 4.2
Throughout this paper we assume that the random variables Y _{0D }, Y _{0A }, and Z _{0} are independent, which is a reasonable assumption as we argue next. Indeed, suppose that Y _{0D } and Y _{0A } are dependent. This would suggest that we have not properly separated the exogenous information from the individual valuations of the domestic and foreign buyers, thus contradicting the above description of the endogenous valuations Y _{0D } and Y _{0A }.
In the next subsection, we specialize formula (31) into a practically sound scenario based on the gamma distribution, under which we subsequently explore the expected profit Π(p _{0},p _{1}) numerically and graphically (Section 6 below).
4.2 Specific modelling
Note 4.3
Bound (33) is satisfied for the parameter choices that we shall specify in Note 6.2 below.
in our numerical research later in the paper, with appropriately chosen shape γ _{0}>0 and rate δ _{0}>0 parameters. For specific parameter choices, we refer to Note 6.2 at the end of this paper.
The secondstage selling probability
In this section, we express the probability P[X _{ t }≥p _{1}∣X _{0}<p _{0}] in terms of underlying quantities at every time instance t>0. We accomplish this task in a similar way to that for P[X _{0}≥p _{0}] in the previous section.
5.1 General considerations
5.2 Specific modelling
where, for the sake of simplicity, we have assumed that the distribution of the exogenous valuation adjustment Z _{ t } does not change with time t, that is, Z _{ t }∼Ga(α _{0},β _{0}) for all t≥0.
 1)
When p _{0}=a _{0}, it is reasonable to assume that there is not anyone wishing to wait until the second selling stage, and thus q _{ t }(a _{0},z)=0 for every exogenous valuation adjustment z.
 2)
When p _{0}=+∞, no one wishes to buy during the initial selling stage, and thus q _{ t }(+∞,z) should look like q _{0}(z). Hence, we let q _{ t }(+∞,z) be the survival function 1−H _{ t }(z) for a cdf H _{ t }(z) on the interval [0,∞). Just like in the case of t=0, a good model for the cdf H _{ t } is the gamma cdf \(F_{\gamma _{t},\delta _{t}}\) with shape γ _{ t }>0 and rate δ _{ t }>0 parameters, which may depend on t.
 3)
It is reasonable to assume that q _{ t }(p _{0},z) is an increasing function of p _{0}, because larger prices during the initial selling stage would suggest that more domestic buyers are deferring their purchases until the second selling stage.
For specific parameter choices, we refer to Note 6.3 at the end of this paper.
Value functions and a numerical exploration
To make formula (23) actionable, in addition to the already discussed probabilities P[X _{0}≥p _{0}] and P[X _{ t }≥p _{1}∣X _{0}<p _{0}], we need to specify appropriate models for the value functions v _{0}(p _{0}) and v _{ t }(p _{1}).
6.1 Value function v _{0}(p _{0})
We already have a model for v _{0}(p _{0}) given by Eq. (22), but in view of our motivating example, an adjustment to this function needs to be made. Namely, property prices in Punta del Este, Uruguay, are predominantly in the US dollars, while property development costs are partially in the Uruguayan pesos and partially in the US dollars. In general, the costs are mainly due to land, design and development, materials, labor costs and subcontracts. Those that are in the Uruguayan pesos are labor costs (i.e., salaries of Uruguayan workers) and they can, for example, be around 30% of the structure’s costs, that is, of the total cost minus the land cost. Therefore, we can say that, for some ν∈(0,1), the percentage ν100% of the total cost is in the Uruguayan pesos and the rest (1−ν)100% is in the US dollars.
Strictly speaking, the exchange rates are unknown in advance, and thus predicted values need to be used. It is very likely, however, that the prices p _{0} and p _{1} are set just before commencing the initial selling stage, and thus the value of ε _{0} can be reasonably assumed known, and thus v _{0} defined in Eq. (43) becomes deterministic and fully specified.
6.2 Value function v _{ t }(p _{1})
where N _{0,1} denotes the standard normal random variable. For specific parameter choices, we refer to Note 6.4 at the end of this paper.
6.3 A numerical illustration and parameter choices
Note 6.1

α _{∗}=4 and β _{∗}=4
Note 6.2

a _{0}=200

α _{0D }=20 and β _{0D }=0.6

α _{0A }=30 and β _{0A }=0.4

α _{0}=β _{0}=4

φ _{0}=0.2

γ _{0}=10 and δ _{0}=0.1
Note 6.3

a _{ t }=200 (=a _{0})

φ _{ t }=0.2

ρ=0.1

α _{ tD }=20 (=α _{0D }) and β _{ tD }=0.6 (=β _{0D })

α _{ tA }=30 (=α _{0A }) and β _{ tA }=0.4 (=β _{0A })

α _{ t }=β _{ t }=4 (=α _{0}=β _{0})

η _{ t }=8 and θ _{ t }=1

γ _{ t }=10 and δ _{ t }=0.1
Note 6.4

ν=0.3

c _{0,UYU} ε _{0}=150 and c _{0,USD}=150

μ=0 and σ=1
The proposed model has been developed to facilitate wellinformed decisions, and the reallife example has guided us in every step of the model development. The model has, inevitably, turned out to be complex. Hence, at this initial stage of our exploration, we have prioritized certain aspects of the research according to their relevance in terms of policy implications, in order to keep considerations within reasonable space limits. The timing of price setting has perhaps been the most significant aspect that is affecting all the other ones. The dependence between the twostage pricing decisions and the influence of the systematic (or background) risk has been among the other important aspects. The exchange rate fluctuations, though very important, have nevertheless been given a lesser attention in the present paper, due to a justifiable reason. Namely, a detailed exploration of this aspect with due mathematical care of its various issues such as change points, heteroscedasticity, and other nonlinear structures manifesting naturally in financial stochastic models would require considerable space. Our use of the simple geometric Browning motion, instead of a more complex and realistic process, has also been influenced by space considerations. Nevertheless, to give an initial idea about the influence of the mean μ and the volatility σ, we have produced a set of graphs in Fig. 7.
Summary
Motivated by a real problem, we have proposed a general twoperiod pricing model and explored various pricing strategies from the seller’s perspective. Our model takes into account such practical considerations as the facts that the buyer’s valuations, which are random from the seller’s perspective, in the two periods may or may not be independent, may or may not follow the same distribution, and so on. We have seen in particular that the seller’s simultaneouspricing strategies yield higher expected revenues than the sequentialpricing strategies. Our general model allows for the possibility of commodity costs being denominated in different currencies, and thus being impacted by currency exchangerate movements. The model also takes into account various endogenous and exogenous factors, such as personal seller’s and buyer’s considerations, general economic conditions, different seller’s utility or value functions. We have illustrated our theoretical findings both numerically and graphically, using appropriately constructed multiplicative background models that easily take into account various specific elements of the motivating problem.
Endnote
^{1} The mean of this gamma distribution is α/β and the variance is α/β ^{2}.
Declarations
Acknowledgments
We are indebted to the two anonymous referees for incisive comments and suggestions that guided our work on the revision, and we also sincerely thank the editors for their patience. The research of the first author (ME) has been partially supported by the Agencia Nacional de Investigación e Innovación of Uruguay. The second and third authors (JW and RZ) have been supported by the Natural Sciences and Engineering Research Council of Canada. The second author (JW) also gratefully acknowledges a generous travel award by the organizers of the International Conference on Statistical Distributions and Applications, Niagara Falls, Canada, where preliminary results of the present paper were presented in the section on Dependence Modeling with Applications in Insurance and Finance organized by Edward Furman, whom we sincerely thank for the invitation.
Authors’ contributions
The authors, ME, JW, and RZ, with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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