A Comparison of Alternative Optimal Models of Advertising Expenditures: Stock Adjustment Vs. Control Theoretic Approaches

Abstract

A DVERTISING expenditures may well be regarded as a form of investment. Using this concept, Nerlove and Arrow (1962) examined an optimal advertising policy for the firm which maximizes present valued cash flow. More recently Gould (1970) extended the Nerlove-Arrow model by introducing an adjustment cost function. These studies are theoreltical analyses leaving out possible empirical applications.1 In this paper the Nerlove-Arrow model is applied by use of the usual stock adjustment formula to empirical data. In addition, since the stock adjustment model is made in an ad hoc fashion, a suboptimization model is presented as an attempt to derive an estimable equation directly from optimization behavior. This model is derived from control theoretic suboptimization procedures incorporating an adjustment cost function. Empirical results from both models are compared. Semiannual data of eight Japanese pharmaceutical companies from 1963 to 1970 are used for this study. It seems to have become the predominant practice in econometric studies of dynamic relationships to apply some form of distributed lags to the stock adjustment model Yt x(L) [x*t-xt1(1) where Yt and xt are respectively dependent and explanatory variables, and x*t is the desired level of xt. X(L) is the distributed lag operator function. In equation (1) sometimes x*t_1 is used in place of xt1. As an explicit form of the distributed lag operator function, X (L), various functional forms and estimation procedures have been suggested (Almon, 1965; DeLeeuw. 1962: Torgenson, 1966, Solow, 1960; Tsurumi, 1971). Of late formulations such as equation (1) above have been criticized on the grounds that the distributed lag function is chosen in an ad hoc fashion without a 'theoretical justification for the particular lag structures.2 To make equation (1) estimable, one derives a relationship between the desired variable, X*t, and observable variables by solving some optimization problems. The derived relationship, then, is taken to be the long-run equilibrium position which the economic system tries to approach by the adjustment equation (1). The distributed lag pattern of adjustment, X(L), is superimposed on the system independent of the optimization problem from which x*t is derived, and thus it is often hard to explain an estimated lag structure. One way to overcome the deficiencies of this stock adjustment approach to the distributed lag model is to construct a model incorporating adjusitment costs into a dynamic optimization problem solved by techniques of optimal control theory. Since Eisner and Strotz (1963) introduced the costs of adjustment to derive an investment function, there have been a number of contributions to the theory of invesitment incorporating the costs of adjustment (Chetty and Sankar, 1970; Gould, 1968; Lucas, 1967; Treadway, 1969). The earlier mentioned paper by Gould (1970) is an adaptation of cost of adjustment approach to advertising outlay. All these optimal models treat an infinite time horizon and some of them, e.g., Lucas (1967), Treadway (1969), and Gould (1970) are concerned with long-run equilibrium values hence leaving little room for short-run consideration. For systems whose dynamics are deterministic and completely known, it will be reasonable to optimize the objective function over a long period of time. In many practical situations, however, the dynamics of the system may be Received for publication March 2, 1972. Revision accepted for publication November 30, 1972. 'There are many empirical studies which analyse the cumulative effects of advertising expenditures on sales using some form of distributed lags (Bass, 1971; Nerlove and Waugh, 1961; Palda, 1965; Telser, 1962). This paper deals with the determination of advertising expenditures. 2 One of the most recent criticisms on this point was made by Nerlove in his Henry Schultz Memorial lecture to the Second World Congress of the Econometric Society, Cambridge, England, September, 1970.