Economic literature treats price as the traditional competition variable. However, Schmalensee (1976, p. 493) argued that prices change infrequently, and that firms compete mainly through non-price variables. The most important non-price competition variables are advertising and product variation. Although there are many works dealing with non-price competition, they have not devoted much attention to the multi-period effect of these variables and the models have been mainly static, or a sequence of static models. (Further discussion of this argument can be found in Kydland (1977), in which dynamic dominant player models are discussed.) In an earlier paper, Arrow and Nerlove (1962) pointed out that advertising expenditure should be. treated in the same way as investment in a durable good. They assumed that there is a stock of goodwill that determines the current demand. This stock of goodwill summarizes the advertising in the past and, like capital stock, depreciates over time. Arrow and Nerlove analysed the optimal advertising strategy for the monopolistic firm. In this paper we extend the Arrow-Nerlove model to oligopolistic competition, in which the firms compete among themselves via goodwill. The game we describe takes place over time and discounted profit is used as a criterion. The current advertising of any player affects future demand, and therefore the model has an intertemporal dependence structure. Extending an assumption by Schmalensee (1978), in Section I we allow market shares to depend on the firm's goodwill and not merely on current advertising. Using this definition of market share, we discuss in Section II the optimal advertising policy for a single firm. In Section III we prove the existence of a unique stationary equilibrium point for oligopolistic competition. Since we allow firms to have different cost functions, the equilibrium point is not a symmetric one; i.e. different firms have different market shares and different goodwill at the equilibrium point. In Section IV we discuss this asymmetric solution by explaining the relation between production cost and market shares. The impact of the number of firms on the advertising expenditure is discussed in Section V. We prove that the firms' advertising expenses tend to decline as the number of firms increases. However, if one of the firms is a leading firm in the industry and has a market share above one-half, we show that this firm might increase its advertising as other firms enter into the industry. As an example, we discuss in Section VI the duopoly case and offer an explicit statement of the market shares at the equilibrium point. In Section VII we analyse an industry consisting of identical firms, and explore the influence of the interest rate and the depreciation rate on the firms' goodwill and advertising policy.