A differential game model of the marketing-operations interface

In the development of their dynamic strategies, the marketing and operations functions within a firm have differing objectives, and conflict between the two functions is common. The strategic interdependence involving marketing and operations decisions is modeled as a noncooperative differential game. Demand is assumed to be a function of price and advertising goodwill, and marketing controls price and advertising to maximize its discounted stream of revenue net of advertising costs. Backlogging is allowed, and operations controls production to minimize its discounted stream of production and backlog costs. A feedback Nash equilibrium is derived for the game, which allows a solution of the system of differential equations for goodwill and backlog, and is analyzed to study the nature of the dynamic strategies for price, advertising, and production.     

The impact of dynamic pricing on the economic order decision

This paper analyzes the impact of dynamic pricing on the single product economic order decision of a monopolist retailer. Items are procured from an external supplier according to the economic order quantity (EOQ) model and are sold to customers on a single market without competition following the simple monopolist pricing problem. Coordinated decision making of optimal pricing and ordering is influenced by operating costs – including ordering and inventory holding costs – and the demand rate obtained from a price response function. The retailer is allowed to vary the selling price, either in a fixed number of discrete points in time or continuously. While constant and continuous pricing have received much attention in the literature, problems with a limited number of price changes are rather rare. This paper illustrates the benefit of dynamically changing prices to achieve operational efficiency in the EOQ model, that is to trigger high demand rates when inventories are high. We provide structural properties of the optimal time instants when the price should be changed. Taking into account costs for changes in price, it provides numerical guidance on number, timing, and size of price changes during an order cycle. Numerical examples show that the benefits of dynamic pricing in an EOQ framework can be achieved with only a few price changes and that products being unprofitable under static pricing may become profitable under dynamic pricing.     

Vendor Management Inventory with consignment contracts and the benefits of cooperative advertising

Most of the cooperative advertising literature has focused on studying the effects of such programs considering marketing variables. This paper integrates production and inventory management with pricing and advertising considerations to assess the effects of cooperative advertising programs in bilateral monopolies. We consider a supply chain where a Vendor Managed Inventory (VMI) along with a consignment contract is implemented to coordinate the chain. We develop and solve a differential model for two games. The first one is a benchmark scenario where no cooperative advertising is offered, while the manufacturer offers the cooperative program in the second game. The main results show that cooperative advertising programs, usually considered as successful marketing initiatives, can be very difficult to implement in a supply chain undertaking a VMI policy with a consignment contract, in which operations and marketing interface is taken into account. A cooperative program mainly hurts the manufacturer’s profits, and can be profit-Pareto-improving only in a few cases. Although the retailer is generally willing to receive a support from the manufacturer, she can opt for a non-cooperative program when the largest part of the supply chain profits goes to the manufacturer. We developed several special cases to strengthen our findings.     

Optimal transfer pricing schemes for work averse division managers with private information

This paper models a decentralized firm under information asymmetry and effort disutility on the part of managers. We assume that managers choose efforts before observing some private information. However, after the effort choice managers receive private information on their cost parameters which they report to the headquarters. There exist many situations in which managers need to take efforts before obtaining private information; for example, the regular maintenance effort on the machine, the effort on R&D for reducing costs and the effort taken to build relationships with the supplier. Two models are considered in this paper based on the timing of acquisition of private information by the managers. We derive optimal coordination mechanisms to facilitate internal transactions for the models. The equilibrium outcome of this paper suggests that: 1) regardless of the timing of managers' information acquisition, the optimal output level under asymmetric information can have overproduction or underproduction when compared with the full information optimal output; 2) under certain demand conditions managers cannot receive any information rent benefit for their private information even if they have the option to renege on the contract after obtaining their private information.     

An oligopoly model of dynamic advertising competition

An oligopoly model is presented that allows the determination of feedback Nash equilibrium advertising strategies for an oligopoly. Analyses of symmetric and asymmetric oligopolies with the model show that unit contribution and advertising effectiveness have positive effects on a competitor’s own advertising and steady-state sales, while discount rate and decay rate have negative effects. An asymmetric analysis further shows that unit contribution and advertising effectiveness affect positively, and discount rate and decay rate negatively, a competitor’s rivals’ advertising, but have effects in opposite directions regarding rivals’ steady-state sales. The symmetric and asymmetric analyses also show that steady-state sales per competitor decline with the number of competitors in the oligopoly, while total oligopoly steady-state sales increase. The model is applied empirically to the triopolistic competition involving Anheuser-Busch, SABMiller, and Molson Coors in the beer industry.     

Differential game approach to pricing and advertising decisions

This study proposes a model to make concurrent decisions on dynamic pricing and advertising to maximise firms' profitability over an infinite time horizon in a duopoly market. To this end, the Nerlove-Arrow pricing and advertising model is designed in the presence of shifting costs in a dynamic duopolistic competition as a differential game. The Nash equilibrium solution is defined based upon a set of Hamilton–Jacobi–Bellman. Four scenarios are applied for economic interpretations and the efficacy of the model.     

Intrafirm trade, arm’s-length transfer pricing rule, and coordination failure

This paper demonstrates that uniform imposition of the arm’s-length principle on transfer pricing leads to coordination failure among countries in terms of economic welfare if the countries trade products in the form of intrafirm transactions by multinational firms (MNFs). To highlight this implication, we first show that imposition of the arm’s-length principle on an MNF induces it to transfer a product among subordinate divisions at marginal cost, i.e., the competitive price, which is consistent with the purpose of the principle. Nonetheless, if regulators in each country impose the principle on MNFs, all of the following economic welfare measures decrease compared with the situation where the principle is not imposed: (1) consumer welfare in each of the trading countries, (2) profit of each MNF, and thus (3) total world economic welfare. This result indicates that it is possible that enforcement of the principle has no positive effect at all in the world because economic welfare of all economic agents deteriorates when the principle is imposed. A numerical analysis demonstrates that this possibility arises in a broad range of circumstances, even including the situation where a giant economic world power and a small underdeveloped country mutually trade products. In these circumstances, an agreement among trading countries that no country imposes the arm’s-length principle may encourage Pareto improvement of the world economy.     

Transfer pricing and decentralized dynamic lot-sizing in multistage,multiproduct production processes

This paper presents the use of a transfer pricing system to coordinate business units in a Wagner–Whitin type model for a decentralized lot-sizing problem in a dynamic multistage, multiproduct environment. The paper includes two major proofs: (1) a transfer pricing system enabling optimal decentralized lot-sizing is characterized. The transfer pricing system can be interpreted as a generalization of the reciprocal method or step-down allocation method in cost location accounting; and (2) based on a Wagner–Whitin type theorem and the Kakutani fixed point theorem, it is shown that such a transfer pricing system exists.     

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.     

On dynamic programming equations for utility indifference pricing under delta constraints

In this paper we study the problem of utility indifference pricing in a constrained financial market, using a utility function defined over the positive real line. We present a convex risk measure −v(•:y) satisfying q(x,F)=x+v(F:u₀(x)), where u₀(x) is the maximal expected utility of a small investor with the initial wealth x, and q(x,F) is a utility indifference buy price for a European contingent claim with a discounted payoff F. We provide a dynamic programming equation associated with the risk measure (−v), and characterize v as a viscosity solution of this equation.     

Optimal pricing and advertising in a durable-good duopoly

This paper analyzes dynamic advertising and pricing policies in a durable-good duopoly. The proposed infinite-horizon model, while general enough to capture dynamic price and advertising interactions in a competitive setting, also permits closed-form solutions. We use differential game theory to analyze two different demand specifications – linear demand and isoelastic demand – for symmetric and asymmetric competitors. We find that the optimal price is constant and does not vary with cumulative sales, while the optimal advertising is decreasing with cumulative sales. Comparative statics for the results are presented.     

Optimal promotion planning—depth and frequency—for a two-stage supply chain under Markov switching demand

In considering the retailer–supplier supply chain, this paper analyzes how a retailer reasonably decides both the depth and frequency of the price discount promotion including or excluding a supplier’s inventory decision. Assuming that the promotion frequency used by the retailer is probabilistic, we model a promotion-inventory decision under an AR(1) demand with a Markov switching promotion regime. After obtaining the optimal promotion plan, our analysis also considers the behavior of the optimal promotion decision; the retailer’s price format selection, either an Every-Day-Low-Price policy (EDLP) or a Promotion policy (HiLo); and the impact of information sharing of promotion status on the system’s performance. Our results suggest that a retailer tends to overpromote if inventory cost is excluded in its promotion decision, that increasing the market share is a preferable action for both the retailer and the supplier, that total margin and price-elasticity play an important role in selecting the price format, and that the profitability for a supplier of sharing promotion information depends on the transition probabilities of the Markov switching regime.     

Optimal pricing in a dynamic duopoly game model

This paper deals with optimal pricing of new products over a finite planning period in a duopolistic market. Modelling saturation effects and no cost-side learning effects optimal pricing strategies for different kinds of demand functions are determined. In this direction the paper extends some results known from the monopolistic case. It turns out, that the optimal prices are decreasing functions of time, where the prices at each moment of time are higher than the marginal costs. Thus the optimal pricing strategies can be characterized as skimming policies.

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Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit der optimalen Preisgestaltung von neuen Produkten in einer endlichen Planungsperiode unter der Annahme einer duopolistischen Marktform. Das Modell beschränkt sich auf die Analyse von Sättigungseffekten, wobei es möglich ist, optimale Preispfade für verschiedene Klassen von Nachfragefunktionen herzuleiten. Die Ergebnisse der Arbeit stellen Verallgemeinerungen des Monopolfalles dar. Durch die Anwendung von quantitativen Lösungsmethoden wird gezeigt, daß die optimalen Preispfade fallende Funktionen der Zeit sind. Dadurch, und durch das Resultat, daß die Preise stets größer sind als die Grenzkosten, stellen die optimalen Preisstrategien Skimming Politiken dar.

     

基于合作博弈的组装供应链转移定价策略

研究了由n个供应互补的原料产品的上游供应商和一个组装生产最终产品的下游组装商组成的集团公司内部的转移价格问题。其中下游组装商面临的是一个价格敏感型的需求。每个上游供应商可以独立决策其销售价格,并决定是否以谈判的价格出售原料产品给下游组装商。供应链系统内的所有供应商和组装商都可以自由决定彼此之间是否进行合作。结果表明,在大联盟结构下,供应链的利润是最高的。为了分配大联盟结构下供应链系统总的利润,本文构造了特征函数具有超模性质的合作博弈,并利用Shapley值分配方法给出了整个供应链系统内部的转移价格。     

Risk-sensitive dynamic pricing for a single perishable product

We show that the monotone structures of dynamic pricing for a single perishable product under risk-neutrality are preserved under risk-sensitivity with the additive general utility and atemporal exponential utility functions. We also show that the optimal price is decreasing over the degree of risk-sensitivity under the exponential class of both additive and atemporal utility functions.     

Pure equilibria in a simple dynamic model of strategic market game

We present a discrete model of two-person constant-sum dynamic strategic market game. We show that for every value of discount factor the game with discounted rewards possesses a pure stationary strategy equilibrium. Optimal strategies have some useful properties, such as Lipschitz property and symmetry. We also show value of the game to be nondecreasing both in state and discount factor. Further, for some values of discount factor, exact form of optimal strategies is found. For β less than , there is an equilibrium such that players make large bids. For β close to 1, there is an equilibrium with small bids. Similar result is obtained for the long run average reward game.     

On a dynamic fractional game

Consider a two-person zero-sum game constructed by a dynamic fractional form. We establish the upper value as well as the lower value of a dynamic fractional game, and prove that the dual gap is equal to zero under certain conditions. It is also established that the saddle point function exists in the fractional game system under certain conditions so that the equilibrium point exists in this game system.     

Two-period dynamic versus fixed-ratio pricing in a capacity constrained duopoly

This paper analyzes the impact of dynamic and fixed-ratio pricing policies on firm profits and equilibrium prices under competition. Firms that have equal inventories of perfectly substitutable and perishable products compete for customer segments that demand the product at different times. In each period, customers first purchase from the low price firm and then from the high price firm up to their inventories, provided the prices are lower than the maximum they are willing to pay. The main conclusions of this paper are as follows: although dynamic pricing is a more sophisticated policy than fixed-ratio pricing, it may lead to decreased equilibrium profits; under both pricing policies, one firm assumes the role of a low-cost high-output firm while the other assumes the role of a high-cost low-output firm; and, the supply demand ratio has more impact on the outcome of the competition than the heterogeneity in consumer reservation prices.     

Some quantitative homogenization results in a simple case of interface

Following a framework initiated by Blanc, Le Bris and Lions, this article aims at obtaining quantitative homogenization results in a simple case of interface between two periodic media. By using Avellaneda and Lin’s techniques, we provide pointwise estimates for the gradient of the solution to the multiscale problem and for the associated Green function. Also we generalize the classical two-scale expansion in order to build a pointwise approximation of the gradient of the solution to the multiscale problem (up to the interface), and, adapting Kenig, Lin and Shen’s approach, we obtain convergence rates.