Reservation Value Research Articles

Optimal stopping problem with recall cost

This paper presents an optimal stopping problem with recall where some cost must be paid to accept the best offer y which has so far appeared. In this model, the optimal decision rule may have a property, called the Double Reservation Value (DRV) property, that for at least one current offer w there exist two different critical values y ∗ and y ∗ such that, if y ∗⩽y⩽y ∗ , continuing is optimal, or else stopping by accepting the better offer between the current offer w and the best offer y up to the previous point in time is optimal. This property is known to appear in the model with uncertain recall but not in the model with recall. However, we find that it may also appear in the latter model by introducing recall cost. This is one of the most distinctive results in this paper. We examine conditions for the optimal decision rule to have the DRV property in this model and its limiting behavior as the time horizon tends to infinity.

Auctions of Heterogeneous Objects

Simultaneous sealed bid auctions of heterogeneous objects are analyzed. Each bidder's reservation value for an object depends upon the other objects he obtains. Bidders' reservation values are common knowledge. In simultaneous first-price auctions, the set of Walrasian equilibrium allocations contains the set of pure strategy Nash equilibrium allocations which in turn contains the set of strict Walrasian equilibrium allocations. Hence, pure strategy Nash equilibria (when they exist) are efficient. Mixed strategy Nash equilibria may be inefficient. In simultaneous second-price auctions, any efficient allocation can be implemented as a pure strategy Nash equilibrium outcome if a Walrasian equilibrium exists. Journal of Economic LiteratureClassification Numbers: D44, D51.

Um Simples Mercado de Compra e Venda

We model a buying and selling market as a market game in which (a) all goods except money are identic and inclivisible, (b) agents' preferences can be described by a reservation value, and (c) all buyers are price-takers and can buy up to their quota of goods. We prove that equilibrium prices exist and support efficient allocations of the goods to the buyers. Furthermore, every such allocation is supported by any equilibrium price. Results of comparative statics are also obtained. We reformulate a dinamic mechanism of auction existent in the literature and show that this mechanism yields the minimum competitive equilibrium in a finite number of steps. We obtain sufficient conditions under which this mechanism is individually non-manipulable by the buyers.

A Hierarchical Bayes Model of Primary and Secondary Demand

Product design, pricing policies, and promotional activities influence the primary and secondary demand for goods and services. Brand managers need to develop an understanding of the relationships between marketing mix decisions and consumer decisions of whether to purchase in the product category, which brand to buy, and how much to consume. Knowledge about factors most effective in influencing primary and secondary demand of a product allows firms to grow by enhancing their market share as well their market size. The purpose of this paper is to develop an individual level model that allows an investigation of both the primary and secondary aspects of consumer demand. Unlike models of only primary demand or only secondary demand, this more comprehensive model offers the opportunity to identify changes in product features that will result in the greatest increase in demand. It also offers the opportunity to differentially target consumer segments depending upon whether consumers are most likely to enter the market, increase their consumption level, or switch brands. In the proposed hierarchical Bayes model, an integrative framework that jointly models the discrete choice and continuous quantity components of consumer decision is employed instead of treating the two as independent. The model includes parameters that capture individual specific reservation value, attribute preference, and expenditure sensitivity. The model development is based upon the microeconomic theory of utility maximization. Heterogeneity in model parameters across the sample is captured by using a random effects specification guided by the underlying microeconomic model. This requires that some of the effects are strictly positive. This is accommodated through the use of a gamma distribution of heterogeneity for some of the parameters. A normal distribution of heterogeneity is used for the remaining parameters. Gibbs sampling is used to estimate the model. The key methodological contribution of this paper is that we show how to specify a hierarchical Bayes continuous random effects model that integrates consumer choice and quantity decisions such that individual-level parameters can be estimated. Individual level estimates are desirable because insights into primary demand involve nonlinear functions of model parameters. For example, consumers not in the market are those whose utilities for the choice alternatives fall below some reservation value. The proposed methodology yields individual specific estimates of reservation values and expenditure sensitivity, which allow assessment of the origins of demand other than the switching behavior of consumers. The methodology can also be used to help identify changes in product features most likely to bring new customers into a market. Our work differs from previous research in this area as we lay the framework needed to obtain individual-level parameter estimates in a continuous random effects model that integrates choice and quantity. The methodology is demonstrated with survey data collected about consumer preferences and consumption for a food item. For the data available, a large response heterogeneity was observed across all model parameters. In spite of limited data available at the individual level, a majority of the individual level estimates were found to be significant. Predictive tests demonstrated the superiority of the proposed model over existing latent class and aggregate models. Particularly, significant gains in predictive accuracy were observed for the “no-buy” behavior of the respondents. These gains demonstrate that by structurally linking the choice and quantity models results in a more accurate characterization of the market than existing finite mixture approaches that model choice and quantity independently. We show that our joint model makes more efficient use of the available data and results in better parameter estimates than those that assume independence. Finally, the individual level demand analysis is illustrated through a simple example involving a $1.00 price cut. We demonstrate practical usefulness of the model for targeting by developing the demographic, attitudinal, and behavioral profiles of consumer groups most likely to increase consumption, enter the market, or switch brands because of a price cut decision.

Competitive Equilibrium in an Exchange Economy with Indivisibilities

We analyze an exchange economy in which (i) all commodities except money are indivisible, (ii) agents' preferences can be described by a reservation value for each bundle of indivisible objects, and (iii) all agents are price-takers. We obtain a necessary and sufficient condition under which market clearing prices exist. Implications for market mechanisms are discussed.Journal of Economic LiteratureClassification Numbers: D51, C78.

A Reservation Value Approach to Harvest Scheduling with Stochastic Stumpage Prices

Abstract This article presents a formulation of the multistand harvest scheduling problem with stochastic stumpage prices, where demand constraints on the periodic yield of some products exist over a long planning horizon. The model incorporates explicit information about future yields and the possibility of a future high stumpage price simultaneously for each analysis area and prescription. The underlying price uncertainty is modeled as a stationary process with normally distributed deviations. A test case, which involves two products, sawtimber and pulp, is introduced to illustrate the formulation. Sawtimber is assumed to be sold on the open market, while a certain periodic pulp harvest is required by contract to supply a mill at a set price. By maximizing the sum of current revenues and future reservation values subject to yield constraints, the model optimally decides when to harvest immature stands in response to low sawtimber prices, while retaining mature areas for future high prices. Results show significant present value improvements and a much greater probability of feasibility compared to traditional formulations. In addition, because of the possibility of a future high price, some stands are retained to and beyond the culmination of mean annual increment, thereby creating a residual forest with a greater diversity of age classes and more mature timber than is the case with traditional formulations. For. Sci. 42(4):465-473.

Optimal Repeated Purchases When Sellers Are Learning about Costs

A buyer repeatedly purchases some good. Suppliers privately learn their cost only upon producing at least once. Efficiency would imply sampling sellers until one is found with cost lower than a (increasing with time) reservation value. Then the good would be permanently purchased from the seller with lowest cost. A sequence of second-price auctions with participation premia and entry fees is shown to be both efficient and optimal for the buyer. An alternative calls for price offers by informed sellers, with compensations that increase in the offer. In both cases, the buyer subsidizes competition through participation premia or compensations.Journal of Economic LiteratureClassification Numbers: D44, D83, L14.

The Long Side of the Market and the Short End of the Stick: Bargaining Power and Price Formation in Buyers', Sellers', and Balanced Markets

The determinants of bargaining power and price formation in a dynamic exchange market where new traders enter randomly over time are studied. When agents on the long side of the market possess the option to wait for the arrival of future partners, the terms of trade in the spot market must honor the value of this option. The equilibrium terms of trade are expressed in intuitive closed-form equations that highlight the distinct influences of short-run spot-market conditions and long-run market demographics. I. INTRODUCTION Few concepts in economics are as intuitive as the idea that traders on the long side of a market wind up holding the short end of the stick. That is, they receive little or none of the gains from trade. Consider, for instance, the classic setting in which b identical buyers each with unit demands for an indivisible homogeneous good face s identical sellers each with one unit of the good for sale. It is well-known that in a broad class of one-shot trading games, all of the surplus in this setting goes to the short side of the market; in equilibrium. That is, if b s, buyers bid the price up to their reservation value. Often, it is only when b = s that market institutions (the rules of the game) have any bearing on the equilibrium division of surplus. This classic result is somewhat brittle because of the discontinuity in the terms of trade when moving from positive to negative levels of excess demand. Its logic is familiar, however, from Bertrand competition, and it is very compelling. The option value to an agent frozen out of a trade in a one-shot market game is his reservation payoff of zero. Any outcome that confers nonzero surplus to a long-side agent cannot be sustained in equilibrium, because all of the long-side agents who would otherwise be frozen out possess a strict incentive to undercut.

Bidding Off the Wall: Why Reserve Prices May Be Kept Secret

This note shows by means of an example that in a common-value auction a seller with a random reservation value can increase her ex ante expected profits by following a policy of conducting an auction in which her reserve price is kept secret compared to an auction in which the reserve price is announced. By keeping the reserve price secret, the seller is able to encourage greater participation from the bidders and can, therefore, increase the linkage of the price paid to the value of the purchased object. Journal of Economic Literature Classification Numbers: D82, D44.

Consignment contracting

Consignment is a unique contract form in which a middleman takes possession of a good owned by a seller, promotes its sale to buyers, and is compensated with a share of sales revenue. If no sale occurs possession reverts back to the seller. An alternative to consignment is a middleman contract in which the good is sold to a dealer. Dealers have incentive to choose the expected joint surplus maximizing level of promotional activity, while consignment agents will generally underinvest. This creates a strong presumption favoring the use of dealers. The analysis finds that if valuations are subjective, then highly optimistic sellers will tend to use consignment. Consignment also has a role in resale markets when sellers have relatively large reservation values. Dealers will not intermediate when expected net sales revenue is smaller than the seller's opportunity cost. On the other hand consignment gives sellers a chance to either make profitable sales or retain the good at its reservation value.

THE OPTIMAL STOPPING PROBLEM IN WHICH THE SUM OF THE ACCEPTED OFFER'S VALUE AND THE REMAINING SEARCH BUDGET IS AN OBJECTIVE FUNCTION

The paper deals with an optimal stopping problem with a finite planning horizon where an available search budget, the total amount of money that can be invested in search activities throughout the planning horizon, is limited and where both the probability of an offer being obtained at each point in time and the probability distribution function of an obtained offer's value may depend on the search cost invested at that time. The objective is to maximize the expected present discounted value of the sum of the accepted offer's value and the remaining search budget at that point. The optimal decision strategy for the problem consists of the two decision rules: an optimal investment rule, prescribing how much of the search budget to invest in search activities of each point in time and an optimal stopping rule, prescribing how to stop the search with accepting an offer. The present paper examines the properties of the optimal decision strategy analytically and numerically. The most interesting and important properties revealed here are the two: (1) In a recall model, the optimal stopping rule has a reservation value property, and if the given search budget tends to infinity, the reservation value becomes time-independent, implying that the optimal stopping rule has a myopic property, and (2) in both of a recall model and a no recall model, the optimal investment does not always become monotone in the amount of search budget remaining then, possibly increasing or decreasing drastically with its very slight change.

Feedback Thinning Policies for Uneven-Aged Stand Management with Stochastic Prices

Abstract This paper presents an optimization model and a numerical method for determining feedback thinning rules for uneven-aged stand management when the stumpage price forecast is a stochastic process. Periodic thinning takes place over a finite time horizon, and the objective is to maximize expected present value. The thinning rules are based on a continuous function that maps the relationship between harvest intensity and observed stand value. The parameters of the function include its slope, location, and shape. The location parameter is defined as the stand reservation value, which is the minimum stand value required for harvest. Optimal parameter values are determined using stochastic simulation. Numerical results are presented using a stage-structured model for uneven-aged stands of California white fir (Abies concolor [Gord. & Glend.] Lindl. (Iowiana [Gord.])). Stumpage price is the sum of a deterministic constant and a normally distributed random error. With these assumptions, optimal feedback thinning policies have significantly higher expected present values than do thinning policies that ignore price fluctuations. However, feedback policies have greater volatility in harvest revenue over time. Sensitivity analysis shows that the expected present value of optimal feedback management increases as the level of price risk increases. For. Sci. 36(4):1015-1031.

Uncertainty and Delay in Bargaining

This paper investigates the relationship between uncertainty and delay of agreement in the one-sided offer bargaining model with two-sided uncertainty where the seller makes offers. We construct a weak stationary equilibrium in which different types of the seller charge different prices in every period. We completely characterize the separating equilibrium by three regularity conditions, and show that the time interval between offers converges to zero, the seller's initial price offer in a separating equilibrium converges with probability 1 to the lowest valuation of the buyer if and only if the gain from trading is common knowledge. We have witnessed a proliferation of sequential bargaining models that, with a few remarkable exceptions (Fudenberg and Tirole (1983), Cramton (1984) and Chatterjee and Samuelson (1987)), are based on models with one-sided uncertainty. The importance of studying bargaining models with one-sided uncertainty is that despite their highly special informational structure, we can derive valuable insights into the role of uncertainty in the delay in reaching an agreement, while keeping the computational exercise tractable. However, the assumption of one-sided uncertainty essentially excludes the situation where the buyer's actual reservation value is lower than the seller's actual reservation value, though they are still engaged in the bargaining process. The primary objective of this paper is to investigate the role of uncertainty about the gain from trading in a sequential bargaining model with two-sided incomplete information. In the models with one-sided incomplete information, important studies (Fudenberg, Levine and Tirole (1985), and Gul, Sonnenschein and Wilson (1986)) are often based on weak stationary equilibria where the equilibrium strategy of each player only depends upon the posterior conjecture about the other player and the previous period's offer. Although the importance of two-sided uncertainty and the weak stationary equilibrium is well recognized, previous research on sequential bargaining models has not combined the two features within the same framework. Studies on the models under two-sided uncertainty have not focused on weak stationary equilibria. For example, Cramton (1984) computes a specific sequential equilibrium with interesting properties, in which the equilibrium strategies of both players depend upon histories in a rather complex fashion. On the other hand, it is only in models under one-sided uncertainty that the weak stationary equilibrium has been extensively studied. This paper computes and analyzes the weak stationary equilibria in a model under two-sided uncertainty in order to investigate the effect of uncertainty on the delay in agreement. We will extend the results from previous studies of one-sided uncertainty models. The computation of the equilibrium in models with two-sided uncertainty is generally very complicated. Furthermore, we must identify a sensible outcome if more than one equilibrium exists in the model. To simplify the analysis, we choose a standard bargaining

Variability and the Duration of Search

In sequential search models the variability of the offer distribution affects the duration of search in different ways. To agents searching for the highest offer, increased variability raises the reservation value (as shown in Kohn and Shavell 1974) which raises the expected duration of search; increased variability, however, also raises the likelihood that extreme offer values are observed which lowers the expected duration of search. Previous studies have examined the net effect of variability on the expected duration of search for specific cases. Venezia (1980) demonstrates that changes in variability have no net effect on the expected duration of search, but this result is shown for a normal distribution in the context of a finite horizon search model without recall and without search costs or discounting. Lippman and McCall (1986), on the other hand, proves that increases in variability raise the expected duration of search in an infinite horizon context with discounting. This paper uses a standard search model without learning about the offer distribution that encompasses the models of the above authors. It shows how their specific results follow from the one more general principle which establishes that a proportionate mean preserving spread of the offer distribution will raise, not change, or lower the expected duration of search depending on whether the sum of implicit and explicit search costs is positive, zero, or negative.

The Rate of Convergence to Efficiency in the Buyer's Bid Double Auction as the Market Becomes Large

A trader who privately knows his preferences may misrepresent them in order to influence the market price. This strategic behaviour may prevent realization of all gains from trade. In this paper, trade in a simple market with an explicit rule for price formation is modelled as a Bayesian game. We show that the difference between a trader's bid and his reservation value is maximally O(1/m) where m is the number of traders on each side of the market. Competitive pressure as m increases thus quickly overcomes the inefficiency private information causes and forces the market towards an efficient allocation.

Bilateral trade with the sealed bid k-double auction: Existence and efficiency

For k in the unit interval, the k-double auction determines the terms of trade when a buyer and a seller negotiate transfer of an item. The buyer submits a bid b and the seller submits an offer s. Trade occurs if b exceeds s, at price kb + (1 − k) s. We model trade as a Bayesian game in which each trader privately knows his reservation value, but only has beliefs about the other trader's value. Existence of a multiplicity of equilibria is proven for a class of trader's beliefs. For generic beliefs, however, these equilibria are shown to be ex ante inefficient.

Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions

We consider auctions for a single indivisible object, in the case where the bidders have information about each other which is not available to the seller. We show that the seller can use this information to his own benefit, and we completely characterize the environ- ments in which a well chosen auction gives him the same expected payoff as that obtainable were he able to sell the object with full information about each bidder's willingness to pay. We provide this characterization for auctions in which the bidders have dominant strate- gies, and for those where the relevant equilibrium concept is Bayesian Nash. In both set-ups, the existence of these auctions hinges on the possibility of constructing lotteries with the correct properties. WE CONSIDER the situation in which an agent, the seller, possesses one indivisible unit of a good to which he attaches no value. But the good has value to a number of potential buyers, and its transfer to one of them would increase social welfare. In particular, the transfer to the buyer with the highest valuation maximizes social welfare. In this paper, we completely characterize environments in which the seller can design an auction that will enable him to capture for himself the full increase in social welfare induced by the transfer of the good to the bidder with the highest willingness to pay. If the seller had full information about the reservation prices of potential buyers, his optimal selling strategy would be very simple. He would announce a price equal or very close to the highest reservation value. The optimal strategy for the bidder with the highest evaluation would be to accept the offer. (Note that we are treating a situation in which the seller can commit himself to a price.) As a result of the exchange, the utility of the seller increases by the full amount of the increase in social welfare, and he has been able to fully extract the surplus. In many circumstances, however, a seller has only imperfect knowledge of the buyers' willingnesses to pay. In this case, he must find some mechanism, or auction, which will enable him to maximize his benefit from the sale of the object. The auction literature starts with this observation and shows how the seller can, by an astute choice of auction, extract the largest possible fraction of the surplus. In general, the literature has shown that this proportion is strictly less than one. In some circumstances, the bidders will have information about each other which is not available to the seller. For instance, in auctions for petroleum drilling rights, bidders know the results of geological tests which they have

Development of computerized vehicle scheduling system for specialized transportation: delaware case

A computer‐aided vehicle scheduling system is developed for Delaware's state‐wide specialized transportation system for elderly and handicapped persons. This paper presents the model, computer program package, system implementation, evaluation and the lessons learned from the project. The project which spanned two and a half years consists of five phases: (1) examine the manual method of vehicle scheduling (prior to the computerization); (2) develop a scheduling model; (3) develop a computer program package for the operation of the model; (4) install and operate the system; and (5) evaluate the system performance. The scheduling model consolidates passengers by block of time, origin and destination zones and along the direction of vehicle travel, first; second, minimizes the empty vehicle travel; and, lastly, assigns drivers to route considering assignment priority among the types of drivers and passengers. The product of the model is a set of driver log‐sheets which specify the sequence of passenger pick‐ups and drop‐offs. The computer program package includes necessary auxiliary data management functions such as registration of trip reservation, operating statistics and parameter value changes, as well as the execution of the model. It also allows the dispatcher to override the computer generated log‐sheet based on his discretion. The system has been operational for more than one year and it has brought about a number of changes in the operation and the role of the dispatcher. A before‐and‐after comparison of the operation and the lessons learned are also shown.

The budgeting problem of a searching firm

This paper explores factors to be taken into consideration in determining the optimal exploration budget for a mining company readying itself for detail work within a mineral province. Optimal strategy is shown to be a combination of results from recent conventional “reservation value” stopping rules and optimal expenditure models appearing earlier in the literature.

Distributions of the Duration and Value of Job Search with Learning

Expected value maximizing sequential search rules can be expressed in terms of reservation values. In search with learning the reservation value at any stage of the search is unknown until that stage is reached. Thus calculating ex ante (and subsequent) probabilities of search duration and the offer accepted is difficult if these probabilities are expressed in terms of reservation values. This paper shows, for a wide class of learning procedures, how re-expressing these probabilities in terms of fixed points allows their direct calculation and, thereby, calculation of the expected value of adaptive search. Examples and comparative statics results are presented.