7.1 The Nerlove-Arrow Advertising Model
7.1.2 Solution by the Maximum Principle
While Nerlove and Arrow (1962) used calculus of variations, we use Pontryagin’s maximum principle to derive their results. We form the current-value Hamiltonian
H=π(G)−u+λ[u−δG] (7.8)
with the current-value adjoint variableλsatisfying the differential equa- tion
λ˙ =ρλ− ∂H
∂G = (ρ+δ)λ− dπ
dG (7.9)
and the condition that
t→+∞lim e−ρtλ(t) = 0. (7.10) Recall from Sect.3.6 that this limit condition is only a sufficient condi- tion.
The adjoint variable λ(t) is the shadow price associated with the goodwill at timet. Thus, the Hamiltonian in (7.8) can be interpreted as the dynamic profit rate which consists of two terms: (1) the current net profit rate (π(G)−u) and (2) the value λG˙ =λ[u−δG] of the goodwill rate ˙G created by advertising at rate u. Also, Eq. (7.9) corresponds to the usual equilibrium relation for investment in capital goods; see Arrow and Kurz (1970) and Jacquemin (1973). It states that the marginal opportunity cost λ(ρ+δ)dt of investment in goodwill, by spending on advertising, should equal the sum of the marginal profitπ(G)dtfrom the increased goodwill due to that investment and the capital gaindλ:= ˙λdt on the unit price of goodwill.
We use (3.108) to obtain the optimal long-run stationary equilibrium or turnpike {G,¯ u,¯ λ¯}.That is, we obtain λ= ¯λ= 1 from (7.8) by using
∂H/∂u= 0.We then setλ= ¯λ= 1 and ˙λ= 0 in (7.9) to obtain
7.1. The Nerlove-Arrow Advertising Model 229
π( ¯G) =ρ+δ. (7.11)
In order to obtain a strictly positive equilibrium goodwill level ¯G, we may assumeπ(0)> ρ+δ and π(∞)< ρ+δ.
Before proceeding further to obtain the optimal advertising policy, let us relate (7.11) to the equilibrium condition for ¯Gobtained by Jacquemin (1973). For this we defineβ = (G/S)(∂S/∂G) as the elasticity of demand with respect to goodwill. We can now use (7.3), (7.5), (7.6), and (7.9) with ˙λ= 0 and ¯λ= 1 to derive, as you will in Exercise 7.3,
G¯
pS = β
η(ρ+δ). (7.12)
The interpretation of (7.12) is that in the equilibrium, the ratio of good- will to sales revenuepS is directly proportional to the goodwill elasticity, inversely proportional to the price elasticity, and inversely proportional to the cost of maintaining goodwill given by the marginal opportunity cost λ(ρ+δ) of investment in goodwill.
The property of ¯G is that the optimal policy is to go to ¯G as fast as possible. If G0 < G,¯ it is optimal to jump instantaneously to ¯G by applying an appropriate impulse at t= 0 and then set u∗(t) = ¯u= δG¯ for t > 0. If G0 > G,¯ the optimal control u∗(t) = 0 until the stock of goodwill depreciates to the level ¯G, at which time the control switches tou∗(t) =δG¯ and stays at this level to maintain the level ¯Gof goodwill.
This optimal policy is graphed in Fig.7.1 for these two different initial conditions.
Of course, if we had imposed an upperbound M >0 on the control so that 0 ≤ u ≤ M, then for G0 < G,¯ we would use u∗(t) = M until the goodwill stock reaches ¯G and switch tou∗(t) = ¯u thereafter. This is shown as the dotted curve in Fig.7.1.
Problem (7.7) is formulated with the assumption that a dollar spent on current advertising increases goodwill by one unit. Suppose, instead, that we need to spendmdollars on current advertising to increase good- will by one unit. We can then define u as advertising effort costing the firm mu dollars, and reformulate problem (7.7) by replacing [π(G)−u]
230 7. Applications to Marketing
Case :
Figure 7.1: Optimal policies in the Nerlove-Arrow model
in its integrand by [π(G)−mu].In Exercise 7.4, you are asked to solve problem (7.7) with its objective function and the control constraint replaced by
0≤maxu≤M
J =
∞
0 e−ρt[π(G)−mu]dt
, (7.13)
and show that the equilibrium goodwill level formula (7.11) changes to π( ¯G) = (ρ+δ)m. (7.14) With ¯G thus defined, the optimal solution is as shown in Fig.7.1 with the dotted curve representing the solution in Case 2: G0<G.¯
For a time-dependent Z, however, ¯G(t) = G(Z(t)) will be a func- tion of time. To maintain this level of ¯G(t), the required control is
¯
u(t) =δG(t) + ˙¯¯ G(t).If ¯G(t) is decreasing sufficiently fast, then ¯u(t) may become negative and thus infeasible. If ¯u(t)≥0 for all t, then the opti- mal policy is as before. However, suppose ¯u(t) is infeasible in the interval [t1, t2] shown in Fig.7.2. In such a case, it is feasible to set u(t) = ¯u(t) for t≤t1; att=t1 (which is point A in Fig.7.2) we can no longer stay on the turnpike and must setu(t) = 0 until we hit the turnpike again (at point B in Fig.7.2). However, such a policy is not necessarily optimal.
For instance, suppose we leave the turnpike at point C anticipating the infeasibility at point A. The new path CDEB may be better than the old path CAB. Roughly the reason this may happen is that path CDEB is “nearer” to the turnpike than CAB. The picture in Fig.7.2illustrates
7.1. The Nerlove-Arrow Advertising Model 231 such a case. The optimal policy is the one that is “nearest” to the turn- pike. This discussion will become clearer in Sect.7.2.2, when a similar situation arises in connection with the Vidale-Wolfe model. For further details; see Sethi (1977b) and Breakwell (1968).
The Nerlove-Arrow model is an example involving bang-bang and impulse controls followed by a singular control, which arises in a class of optimal control problems of Model Type (b) in Table3.3that are linear in control.
Nonlinear extensions of the Nerlove-Arrow model have been offered in the literature. These amount to making the objective function non- linear in advertising. Gould (1970) extended the model by assuming a
Figure 7.2: A case of a time-dependent turnpike and the nature of opti- mal control
convex cost of advertising effort, which implies a marginally diminishing effect of advertising expenditures. Jacquemin (1973) assumed that the current demand functionS in (7.2) also depends explicitly on the current advertising effort u. In Exercise 11.6, you are asked to analyze Gould’s extension via the phase diagram analysis introduced in Chap.11. The analysis of Jacquemin’s extension is similar.