This paper provides a thorough second‐best welfare analysis of the standard two‐stage model of R&D/product market competition with R&D spillovers. The planner's solution is compared to the standard non‐cooperative scenario, the R&D cartel, and the cartelized research joint venture (or joint lab). We introduce the notion of a social joint lab, as a way for the planner to avoid wasteful R&D duplication. With no spillovers, the non‐cooperative scenario, the joint lab, and the second‐best planner's solutions coincide. However, with spillovers, all three scenarios yield R&D investments that fall short of the socially optimal level. To shed light on the role of the spillover level on these comparisons, we observe that the gaps between the market outcomes and the planners solutions widen as the spillover parameter increases. Finally, we establish that a social planner and a social joint lab solutions may be achieved starting from any of the three scenarios by offering firms respective suitably weighted quadratic R&D subsidization schedules.

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The paper investigates the optimal allocation between defensive and offensive advertising efforts in a dynamic, growing market in which two companies are competing for market share. The study described in this paper extends the existing literature on dynamic advertising competition by considering a market that is in the growth phase and by including the heterogeneous decay rate of market share. A modified Lanchester is employed to describe the dynamics of market share by model. The goal of companies operating in this domain is to maximize their profits over a finite decision horizon. Based on the differential game approach the Markovian Nash strategies for offensive and defensive advertising activities are determined. Additionally, an analysis of the extent to which this solution is sensitive to changes in potential market and the rate of customer churn is made.

We propose a new optimal control model of product goodwill in a segmented market where the state variable behaviour is described by a partial differential equation of the Lotka-Sharp-McKendrick type. In order to maximize the sum of discounted profits over a finite time horizon, we control the marketing communication activities which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of heterogeneous electronic word of mouth. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we describe a numerical algorithm to find the optimal solution. Finally, we examine several examples on the optimal goodwill model and discover two types of marketing strategies.