Optimal Goodwill Model with Consumer Recommendations and Market Segmentation

We propose a new dynamic model of product goodwill where a product is sold in many market segments, and where the segments are indicated by the usage experience of consumers. The dynamics of product goodwill is described by a partial di erential equation of the Lotka{Sharpe{ McKendrick type. The main novelty of this model is that the product goodwill in a segment of new consumers depends not only on advertising e ort, but also on consumer recommendations, for which we introduce a mathematical representation. We consider an optimal goodwill model where in each market segment the control variable is the company's advertising e orts in order to maximize its pro ts. Using the maximum principle, we numerically nd the optimal advertising strategies and corresponding optimal goodwill paths. The sensitivity of these solutions is analysed. We identify two types of optimal advertising campaign: `strengthening' and `supportive'. They may assume di erent shapes and levels depending on the market segment. These experiments highlight the need for both researchers and managers to consider a segmented advertising polic

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs). These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states. The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts. Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators

How do loyalty programs affect goodwill? An optimal control approach

This paper examines the long-term impact of loyalty programs on a company’s profit and reputation among customers, and with different durations of product use. We analyze how the launch of loyalty programs may change the profitability of optimal advertising activities. The basis of this study is a modified goodwill model where the market is segmented according to usage experience. The main novelty is the role of loyalty programs and consumer recommendations in the creation of product goodwill, and also their influence on optimal advertising. The dynamics of goodwill are described by a partial differential equation. The firm maximizes the sum of discounted profits by choosing a different advertising campaign for each market segment. For a high-quality product, we observe that there is a trade off between the loyalty program and optimal advertising strategies. For a low-quality product, the loyalty program causes more profitable companies to invest heavily in additional advertising efforts.The authors gratefully acknowledge financial support from the National Science Centre in Poland. Decision number: DEC-2011/03/D/HS4/04269

On the Mitra-Wan Forest Management Problem in Continuous Time

The paper provides a continuous time version of the well known discrete time Mitra-Wan model of optimal forest management, where a forest is harvested to maximize the utility of timber flow over an infinite time horizon. Besides varying with time, the state variable (describing available trees) and the other parameters of the problem vary continuously also with respect to the age of the trees. The evolution of the system is given in terms of a partial differential equation and later rephrased as an ordinary differential equation in an infinite dimensional space. The paper provides a classification of the behavior of optimal and maximal programs when the utility function is linear, convex, or strictly convex and the discount rate is positive or null. Formulas are provided for modified golden-rule configurations (uniform density functions with cutting at the ages that solve a Faustmann problem) and for Faustmann policies, and the optimality or maximality of such programs is discussed. In all different sets of data, it is shown that the optimal (or maximal) control is necessarily something more general than a function, i.e. a positive measure. In particular, in the case of strictly concave utility and null discount, when the Faustmann policy is not optimal, it is shown that optimal paths converges over time to the golden rule configuration, while in the case of strictly concave utility and positive discount the Faustmann policy is shown to be not optimal, contradicting the corresponding result in discrete time

An age-structured model for a distributive channel

Negli ultimi anni, modelli con struttura per età sono stati studiati in diversi ambiti, da problemi di approviggionamento a ottimizzazione di coperture vaccinali e controllo di epidemie. Il motivo è che la struttura per età è tra la più semplici da studiare in una popolazione, dato che l'età evolve linearmente col tempo. Questo lavoro di tesi introduce una struttura per età nel lavoro di Buratto e Grosset, 'Advertising & Promotion in a marketing channel'. Nello specifico, il modello è un gioco differenziale a tempo infinito e lineare nello stato, con due giocatori, il produttore e il rivenditore di una certa merce, ognuno dei quali cerca di massimizzare il proprio guadagno; ciascuno di loro agisce tramite il proprio controllo, cioè la pubblicità e la promozione, rispettivamente. Il produttore può decidere di farsi carico di parte delle spese del rivenditore. Si è trovata una soluzione ottimale in senso 'catching up' per due espressioni dei profitti marginali arbitrariamente scelte, e tale soluzione è stata studiata tramite analisi di sensibilità. Nell'ultimo capitolo, il modello è stato modificato, prevedendo la possibilità che le persone nel pubblico interagiscano tra loro, parlando dell'azienda e del prodotto. Per le stesse espressioni dei profitti marginali, è stata trovata una nuova soluzione ottimale in senso 'catching up', ed è stata paragonata alla precedente in termini di guadagno per il produttore. Una ricerca più approfondita è richiesta per introdurre e studiare un meccanismo nella promozione del rivenditore, finalizzato a prevenire la possibilità di un impatto negativo del termine di interazione prima introdotto

Optimal advertising strategies with age-structured goodwill

The problem of a firm willing to optimally promote and sell a single product on the market is here undertaken. The awareness of such product is modeled by means of a Nerlove–Arrow goodwill as a state variable, differentiated jointly by means of time and of age of the segments in which the consumers are clustered. The problem falls into the class of infinite horizon optimal control problems of PDEs with age structure that have been studied in various papers either in cases when explicit solutions can be found or using Maximum Principle techniques. Here, assuming an infinite time horizon, we use some dynamic programming techniques in infinite dimension to characterize both the optimal advertising effort and the optimal goodwill path in the long run. An interesting feature of the optimal advertising effort is an anticipation effect with respect to the segments considered in the target market, due to time evolution of the segmentation. We analyze this effect in two different scenarios: in the first, the decision-maker can choose the advertising flow directed to different age segments at different times, while in the second she/he can only decide the activation level of an advertising medium with a given age-spectrum