pricing and the news vendor problem a review with extensions in the newsvendor problem a decision maker facing random demand for a perishable product decides how much of it to stock for a single selling period this simple problem with its intuitively appealing solution is a crucial building block of stochastic inventory theory which comprises a vast literature focusing on operational efficiency typically in this literature market parameters such as demand and selling price are exogenous however incorporating these factors into the mode l can provide an excellent vehicle for examining how operational problems interact with marketing issues to influence decision making at the firm level in this paper we examine an extension of the newsvendor problem in which stocking quantity and selling price are set simultaneously we provide a comprehensive review that synthesizes existing results for the single period problem and develop additional results to enrich the existing knowledge base we also review and develop insight into a dynamic inventory extension of this problem and motivate the applicability of such models

dynamic pricing of inventory capacity with infrequent price changes we consider a problem of dynamically pricing a single product sold by a monopolist over a short time period if demand characteristics change throughout the period it becomes attractive for the company to adjust price continuously to respond to such changes i e price discriminate intertemporally however in practice there is typically a limit on the number of times the price can be adjusted due to the high costs associated with frequent price changes if that is the case instead of a continuous pricing rule the company might want to establish a piece wise constant pricing policy in order to limit the number of price adjustments such a pricing policy which involves optimal choice of prices and timing of price changes is the focus of this paper we analyze the pricing problem with a limited number of price changes in a dynamic deterministic environment in which demand depends on the current price and time and there is a capacity inventory constraint that may be set optimally ahead of the selling season the arrival rate can evolve in time arbitrarily allowing us to model situations in which prices decrease increase or neither we consider several plausible scenarios where pricing and or timing of price changes are endogenized various notions of complementarity single crossing property supermodularity and total positivity are explored to derive structural results conditions sufficient for the uniqueness of the solution and the monotonicity of prices throughout the sales period furthermore we characterize the impact of the capacity constraint on the optimal prices and the timing of price changes and provide several other comparative statics results additional insights are obtained directly from the solutions of various special cases

unimodality of price setting newsvendor s objective function with multiplicative demand and its applications abstract we present a general solution framework for the price setting newsvendor problem with a multiplicative stochastic demand under mild assumptions such as increasing price elasticity on the mean demand function and increasing generalized failure rate on the distribution of the random factor we first prove that both the profit function with respect to price and its derived function with respect to order quantity are quasi concave three applications are then studied under our solution framework 1 we consider a wholesale price only contract by which a manufacturer sets a wholesale price and a newsvendor determines an order quantity and the retail price and show that the manufacturer s profit function is unimodal with respect to retailing price or stocking factor under certain conditions 2 we consider a newsvendor problem in which the demand depends on both the retail price and the level of sales effort and the cost exerting the sales effort is proportional to the order quantity we prove that there exists a unique pair of price and sales effort levels that maximize the total profit this result is established under a set of mild assumptions on the demand and cost functions 3 we identify a property in the single period profit function that satisfies condition 1 of huh and janakiraman 2008 which in turn guarantees the optimality of s s policy for an infinite stationary dynamic inventory price control system with lost sales and fixed order costs finally the unimodality of the newsvendor problem with a general stochastic and price sensitive demand is studied

on sales effort and pricing decisions under alternative risk criteria abstract in the literature different risk criteria are proposed for risk averse decision making but it is often unclear which risk criteria should be used and what their differences may be this study investigates this problem when a risk averse make to order firm decides its pricing and sales effort two factors affecting demand we consider alternative risk criteria commonly used in the literature namely expected utility theory eut mean variance mv mean semideviation ms value at risk var conditional value at risk cvar and loss aversion la we find that all of these criteria lead to similar qualitative behaviors regarding how risk loss averse decisions deviate from risk neutral ones a particularly interesting result is that the form of demand randomness i e additive or multiplicative can completely reverse the effect of risk loss aversion in certain cases we also find that the risk aversion models i e eut mv ms var and cvar are essentially equivalent in inducing the same set of optimal solutions under adverse market states however loss aversion is a mild version of risk aversion in that the la model can only lead to a subset of the solutions generated by other models these results may not be valid for other applications such as the newsvendor pricing problem so we can conclude that the relationship between alternative risk criteria is case sensitive and should be evaluated more carefully

capacity pricing and production under supply and demand uncertainties with an application in agriculture abstract we consider capacitated joint lot sizing and pricing problems when supply and a price sensitive demand are uncertain in cases where there is excess capacity the decision maker has the option to rent her capacity we model the case where price and production quantity are determined before the yield is realized joint decisions as well as the case where the price is determined after the yield is known sequential decisions in the joint decisions case we introduce the concept of expected demand fill rate elasticity and characterize the conditions in which a one sided production and pricing policy is optimal we extend the results to the case when there is a fixed production cost and show that a two sided production and pricing policy is optimal we also investigate the conditions under which it is optimal for the decision maker to rent her capacity in addition we look at the case in which the decision maker is risk averse and analyze the effect of the risk attitude on the optimal price and production quantity we extend our model to the multi period case and the case where there is a lead time to acquire new capacity we show that a one sided production pricing and capacity planning policy is optimal in this situation as well we apply our model to the agricultural sector and present numerical examples using data from the california almond industry

optimal ordering policies in response to a discount offer abstract sometimes supplier offers a temporary price discount to increase cash flow or decrease the inventory level of certain items thus the manufacturer may be able to improve the effectiveness of his inventory system by ordering a special quantity at this sale period in this paper economic order quantity eoq models with a discounted price are developed to obtain the optimal ordering policy during the sale period for five different cases a coincidence of sale period with replenishment time b non coincidence of sale period with replenishment time c sale period is longer than a cycle d discounted price as a function of the special ordering quantity and e incremental discount each case has its own characteristics of the sale period and the discounted price the objective is to take the maximum possible advantage from the discounted price by ordering a special quantity during the sale period the optimal ordering policy is obtained by maximizing the difference between the two costs regular eoq cost and special quantity cost during the sale period moreover a comparison of different discount scenarios is developed to sense the effect of different parameters on the ordering policies the annual gain obtained is linearly related to the discount and the on hand remnant inventory numerical analyses are provided to illustrate and testify the values of the optimal special quantity the analysis showed an impressive improvement in the effectiveness of the inventory system when a special order is placed during the sale period the optimal special quantity is driven for each case to visualize real life problems sensitivity analysis is also initiated to study the change in the total savings with respect to the variation of the special optimal quantity

relationships among three assumptions in revenue management this note discusses the relationships among three assumptions that appear frequently in the pricing revenue management literature these assumptions are mostly needed for analytical tractability and they have the common property of ensuring a well behaved revenue function the three assumptions are decreasing marginal revenue with respect to demand decreasing marginal revenue with respect to price and increasing price elasticity of demand we provide proofs and examples to show that none of these conditions implies any other however they can be ordered from strongest to weakest over restricted regions and the ordering depends upon the region

a fractiles perspective to the joint price quantity newsvendor model pricing and quantity decisions are critical to many firms across different industries we study the joint price quantity newsvendor model where only a single quantity and price decision is made such as a fashion or holiday product that cannot be replenished and where the price is advertised nationally and cannot be changed demand is uncertain and sensitive to price we develop a method for easily finding the optimal price and quantity that applies to more general cases than the usual one in which uncertainty is either additive multiplicative or a combination of the two we represent a quantity by its fractile of the probability distribution of demand for a given price we use a standard approach to approximating a given distribution with a finite number of representative fractiles and assume that these fractile functions are piecewise linear functions of the price we identify effects that are not usually seen in a joint price quantity newsvendor model for example although the optimal quantity is a decreasing function of the unit cost the optimal price can be nonmonotone in the unit cost and we shed insight into why we illustrate that using a simplified structure of demand uncertainty can result in substantially lower profits

an inventory system with demand dependent on both time and price assuming backlogged shortages abstract in this work we analyze an inventory model for items whose demand is a bivariate function of price and time it is supposed that the demand rate multiplicatively combines the effects of a time power function and a price logit function the aim is to maximize the profit per time unit assuming that the inventory cost per time unit is the sum of the holding shortage ordering and purchasing costs an algorithm is developed to find the optimal price the optimal lot size and the optimal replenishment cycle several numerical examples are introduced to illustrate the solution procedure

optimal pricing and inventory control policy with quantity based price differentiation a firm facing price dependent stochastic demand aims to maximize its total expected profit over a planning horizon in addition to the regular unit selling price the firm can utilize quantity discounts to increase sales we refer to this dual pricing strategy as quantity based price differentiation at the beginning of each period the firm needs to make three decisions replenish the inventory set the unit selling price if the unit sales mode is deployed and set the quantity discount price if the quantity sales mode is deployed or the combination of the two modes of sales we identify conditions under which the optimal inventory control policy and selling pricing strategy is well structured remarkably under a utility based demand framework these conditions can be unified by a simple regularity assumption that has long been used in the auction and mechanism design literature moreover sharper structural results are yielded for the optimal selling strategy we also examine the comparative advantag

pricing and revenue management the value of coordination the integration of systems for pricing and revenue management must trade off potential revenue gains against significant practical and technical challenges this dilemma motivates us to investigate the value of coordinating decisions on prices and capacity allocation in a stylized setting we propose two pairs of sequential policies for making static decisions on pricing and revenue management that differ in their degree of integration hierarchical versus coordinated and their pricing inputs deterministic versus stochastic for a large class of stochastic price dependent demand models we prove that these four heuristics admit tractable solutions satisfying intuitive sensitivity properties we further evaluate numerically the performance of these policies relative to a fully coordinated model which is generally intractable we find it interesting that near optimal performance is usually achieved by a simple hierarchical policy that sets prices first based on a nonnested stochastic model and then uses these prices to optimize nested capacity allocation this tractable policy largely outperforms its counterpart based on a deterministic pricing model jointly optimizing price and allocation decisions for the high end segment improves performance but the largest revenue benefits stem from adjusting prices to account for demand risk this paper was accepted by yossi aviv operations management

inventory management under price based and stockout based substitution abstract we examine a stochastic inventory and pricing problem for a firm that sells two vertically differentiated products the demands for the two products are determined by total random market size and the customers net utility from buying the two products which is determined by the products quality attributes the individual quality valuation unknown to the firm and the selling prices in case the preferred product is out of stock customers may be willing to buy a substitute instead if their net utility is non negative therefore we analyze an inventory and pricing model considering price based and stockout based substitution we show that the demand function is not continuous in price by decomposing the profit function into different price regimes we are able to derive closed form expressions for the stockout based substitution rates upward and downward substitution and the optimal inventory levels under exogenous pricing under endogenous pricing we find that the profit function is not necessarily unimodal however we show that a unique solution exists for the integrated price and inventory problem under price based substitution only numerical results reveal that not considering stockout based substitution i leads to lower profit margins for high quality products and ii may cause severe supply demand mismatches throughout the entire assortment finally we show the performance of two approximated pricing policies

optimal ordering and pricing policy for demand functions that are separable into price and inventory age abstract we formulate and analyze two models for determining the optimal pricing order quantity and replenishment period for items whose demand function is separable into components of price and inventory age the first model assumes a multiplicative demand function we provide conditions which are satisfied by most common price dependent demand functions to reduce the three variable profit maximization problem into a single variable problem which can be solved using an efficient line search method next we show that a genuine additive model cannot exist and instead suggest and analyze a pseudo additive model however this model is more limited than the multiplicative model in its ability to incorporate various combinations of price and inventory age effects and reduction of the maximization problem into a single variable problem is more complicated except in the case of a linear price effect which is further analyzed for both models we show that the optimal solution satisfies the first order condition for equilibrium under a monopoly with a modification that includes inventory holding costs we solve numerical examples to illustrate the solution procedures

an elasticity approach to the newsvendor with price sensitive demand we introduce a measure of elasticity of stochastic demand called the elasticity of the lost sales rate which offers a unifying perspective on the well known newsvendor with pricing problem this new concept provides a framework to characterize structural results for coordinated and uncoordinated pricing and inventory strategies concavity and submodularity of the profit function as well as sensitivity properties of the optimal inventory and price policies are characterized by monotonicity conditions or bounds on the elasticity of the lost sales rate these elasticity conditions are satisfied by most relevant demand models in the marketing and operations literature our results unify and complement previous work on price setting newsvendor models and provide a new tool for researchers modeling stochastic price sensitive demand in other contexts

do fundamentals explain the behaviour of the swedish real effective exchange rate this study examines the long run relationship between the real effective exchange rate and its fundamental determinants and derives a real effective equilibrium exchange rate for the swedish krona our results indicate that the krona was severely overvalued in late 1992 when the fixed exchange rate regime was abandoned by the end of 2000 the krona was undervalued by approximately 5 percent given the prevailing economic conditions arithmetic examples of suitable sek eur conversion rates are calculated under various assumptions to provide a guideline if sweden were to adopt the euro in the future

combined pricing and inventory control under uncertainty this paper addresses the simultaneous determination of pricing and inventory replenishment strategies in the face of demand uncertainty more specifically we analyze the following single item periodic review model demands in consecutive periods are independent but their distributions depend on the item s price in accordance with general stochastic demand functions the price charged in any given period can be specified dynamically as a function of the state of the system a replenishment order may be placed at the beginning of some or all of the periods stockouts are fully backlogged we address both finite and infinite horizon models with the objective of maximizing total expected discounted profit or its time average value assuming that prices can either be adjusted arbitrarily upward or downward or that they can only be decreased we characterize the structure of an optimal combined pricing and inventory strategy for all of the above types of models we also develop an efficient value iteration method to compute these optimal strategies finally we report on an extensive numerical study that characterizes various qualitative properties of the optimal strategies and corresponding optimal profit values

a general equilibrium model for industries with price and service competition this paper develops a stochastic general equilibrium inventory model for an oligopoly in which all inventory constraint parameters are endogenously determined we propose several systems of demand processes whose distributions are functions of all retailers prices and all retailers service levels we proceed with the investigation of the equilibrium behavior of infinite horizon models for industries facing this type of generalized competition under demand uncertainty we systematically consider the following three competition scenarios 1 price competition only here we assume that the firms service levels are exogenously chosen but characterize how the price and inventory strategy equilibrium vary with the chosen service levels 2 simultaneous price and service level competition here each of the firms simultaneously chooses a service level and a combined price and inventory strategy 3 two stage competition the firms make their competitive choices sequentially in a first stage all firms simultaneously choose a service level in a second stage the firms simultaneously choose a combined pricing and inventory strategy with full knowledge of the service levels selected by all competitors we show that in all of the above settings a nash equilibrium of infinite horizon stationary strategies exists and that it is of a simple structure provided a nash equilibrium exists in a so called reduced game we pay particular attention to the question of whether a firm can choose its service level on the basis of its own input characteristics i e its cost parameters and demand function only we also investigate under which of the demand models a firm under simultaneous competition responds to a change in the exogenously specified characteristics of the various competitors by either i adjusting its service level and price in the same direction thereby compensating for price increases decreases by offering improved inferior service or ii adjusting them in opposite directions thereby simultaneously offering better or worse prices and service

a note on probability distributions with increasing generalized failure rates distributions with an increasing generalized failure rate igfr have useful applications in pricing and supply chain contracting problems we provide alternative characterizations of the igfr property that lead to simplify verifying whether the igfr condition holds we also relate the limit of the generalized failure rate and the moments of a distribution

coordinating inventory control and pricing strategies with random demand and fixed ordering cost the finite horizon case we analyze a finite horizon single product periodic review model in which pricing and production inventory decisions are made simultaneously demands in different periods are random variables that are independent of each other and their distributions depend on the product price pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered the objective is to find an inventory policy and a pricing strategy maximizing expected profit over the finite horizon we show that when the demand model is additive the profit to go functions arek concave and hence an s s p policy is optimal in such a policy the period inventory is managed based on the classical s s policy and price is determined based on the inventory position at the beginning of each period for more general demand functions i e multiplicative plus additive functions we demonstrate that the profit to go function is not necessarilyk concave and an s s p policy is not necessarily optimal we introduce a new concept the symmetrick concave functions and apply it to provide a characterization of the optimal policy

a risk averse newsvendor model under marketing dependency and price dependency abstract in this paper we analyze a single period inventory model where a risk averse newsvendor makes a joint decision on ordering pricing and marketing under the cvar risk measure criterion the newsvendor can exert marketing effort to enhance market demand we assume that the newsvendor s marketing effort affects the scale of the demand distribution and hence to model the marketing dependent demand as a multiplicative form we characterize sufficient conditions that guarantee existence and uniqueness of the optimal solutions under some mild assumptions on demand distribution expected demand function and marketing effort cost which are widely used in the related literature we show that the optimal order quantity increases in the newsvendor s marketing effort but the optimal price remains unchanged whether the demand is marketing dependent or not additionally we explore impact of exogenous parameters such as unit order cost salvage value and the newsvendor s risk aversion coefficient on the optimal decision variables and find that the optimal marketing effort always keeps the same changing direction with the optimal marketing independent order quantity as the exogenous parameters change this suggests that impact of the parameters on the optimal order quantity gets strengthened in the case of marketing dependent demand

coordinating inventory control and pricing strategies for perishable products we analyze a joint pricing and inventory control problem for a perishable product with a fixed lifetime over a finite horizon in each period demand depends on the price of the current period plus an additive random term inventories can be intentionally disposed of and those that reach their lifetime have to be disposed of the objective is to find a joint pricing ordering and disposal policy to maximize the total expected discounted profit over the planning horizon taking into account linear ordering cost inventory holding and backlogging or lost sales penalty cost and disposal cost employing the concept of l concavity we show some monotonicity properties of the optimal policies our results shed new light on perishable inventory management and our approach provides a significantly simpler proof of a classical structural result in the literature moreover we identify bounds on the optimal order up to levels and develop an effective heuristic policy numerical results show that our heuristic policy performs well in both stationary and nonstationary settings finally we show that our approach also applies to models with random lifetimes and inventory rationing models with multiple demand classes

optimal inventory policy for a perishable item with demand function sensitive to price and time abstract we formulate a model for determining the optimal pricing order quantity and replenishment period for perishable items with price dependent and time dependent demand the items have a fixed shelf life and the demand rate decreases linearly in the selling price and polynomially over the time after replenishment until it vanishes either at the reservation price or at expiration time we prove that the three variable profit maximization problem can be reduced into a single variable problem in which the variable is the duration of the replenishment period we show that the profit function is strictly pseudo concave and provide means of obtaining the optimal policy three numerical examples are presented to demonstrate the model accompanied by a sensitivity analysis