Optimal Advertising

Annals of Operations Research 88(1999)15–29

15

Optimal advertising with a continuum of goods★ Emilio Barucci a and Fausto Gozzib a

DIMADEFAS, Università di Firenze, Via C. Lombroso 6y17, I-50134 Firenze, Italy E-mail: [email protected] b

Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 2, I-56127 Pisa, Italy E-mail: [email protected]

In this paper, we present a model of optimal advertising with a continuum of goods differentiated by their vintage. The model is an infinite horizon infinite dimensional optimal control model and the firm advertises a continuum of goods. We prove that the goodwill of a good accumulated through advertising does not necessarily reach its maximum when it is launched onto the market: it can be a single peaked function of the good vintage. Keywords: optimal advertising, infinite dimension, infinite horizon, optimal control AMS subject classification: 90A11, 49J20; JEL C61, C62, E22

1.

Introduction

Following the classical paper by Nerlove and Arrow [12], extensive literature on the dynamics of optimal advertising has appeared, see Gould [7] and Jorgensen [9]. The optimal advertising problem for a firm maximizing the profits in the long run has been analyzed assuming that the consumer demand is a function of the price of the good and of the goodwill which summarizes the effect of past advertising on consumer behavior. Assuming that the firm sells a unique homogeneous good, the advertising problem has been handled by means of standard optimal control techniques (the firm’s profit over an infinite horizon is maximized subject to a controlled differential equation describing the goodwill accumulation). In what follows, we assume that the firm sells a continuum of goods; specifically, we assume that new goods are continuously launched by the firm onto the market. Therefore, at time t the firm sells a continuum ★

Thanks are due to the Vienna workshop participants for comments; the usual disclaimers apply.

© J.C. Baltzer AG, Science Publishers

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E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

of goods differentiated by their vintage s. By A(t, s), we denote the stock of goodwill for goods of vintage s accumulated at time t; a(t, s) represents the advertising rate at time t for the goods of vintage s and we assume that the advertising rate is nonnegative, a(t, s) ≥ 0, ∀t, s ≥ 0. We assume that the demand function for goods of vintage s is only affected by the goodwill specific to that vintage good and not by the goodwill of other vintage goods. The goodwill accumulation process is described by the following controlled dynamical system: ∂ A(t, s)  ∂A(t, s) + + µ A(t, s) = λ(s)a(t, s),  ∂t ∂s   A(t, 0) = λ(0) a(t, 0),    A(0, s) = A0 (s),

t ∈ (0, + ` ), s ∈ [0, s], t ∈ (0, + ` ),

(1)

s ∈ [0, s ],

where µ > 0 and s ∈(0, + ` ]. t stands for time (t ≥ 0) and s as for vintage (s ≥ 0); s′ > s means that goods indexed by s ′ are older than goods indexed by s. The maximum vintage considered in the analysis is s , it is assumed that goods older than s no longer have a market; s can also be infinite. λ(s) is the parameter describing the linear technology through which the goodwill is accumulated from advertising, λ(s) ≥ 0, ∀s ∈[0, s ]. We assume that advertising young goods is more productive than advertising old goods, i.e. λ′(s) ≤ 0, ∀s ∈[0, s ], and a single-peaked shape for λ(s) can also be assumed (i.e. λ first increasing in s and then decreasing). a(t, 0) represents the advertising rate at time t for a new good and also defines the boundary condition for the evolution of the goodwill; the goodwill for a new good is given by the instantaneous advertising rate multiplied by λ(0). A0 (s) represents the initial goodwill condition for goods with different vintage. The Partial Differential Equation (PDE) (1) generalizes the classical dynamical system describing the goodwill accumulation in the optimal advertising literature, i.e. A˙ (t) = a(t) – µA(t). With respect to this equation, the PDE (1) relates the flow of time to the vintage of the stock of goodwill. Leaving aside the decay rate component and the control in (1), the stock of goodwill at time t of a good of vintage s becomes, after the time period δt, the stock of goodwill of a good of vintage s + δt, i.e. A(t, s) = A(t + δt, s + δ t). The goodwill represents a proxy of the extent of the market for the good, e.g. the number of people who are aware of the good. The firm operates in a monopolistic regime, given the stock of goodwill at time t for a good of vintage s, and therefore the extent of the market to which the firm has access, A(t, s), the quota part of the people in the market buying the good and their demand is determined by the prices the firm establishes, p(t, s), s ∈[0, s ]. The demand for a good of vintage s per unit of goodwill is f (s, p(t, [0, s ])) (here p(t, [0, s ]) stands for the image of the map p(t, ·)), where we allow for a substitution among goods of different vintage (the demand for a good of a given vintage also depends on the prices of the goods with different vintage). This

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

17

formalization of the agent demand is general and it incorporates the agent’s budget constraint. The firm’s profit at each instant of time is the integral over the domain of the goods [0, s ] of the returns the firm realizes from each good minus two different kinds of costs: advertising costs, described by the unit cost q(s), q(s) ≥ 0, ∀s ∈[0, s ], and adjustment costs. Adjustment costs are a quadratic function of the advertising rate with a coefficient β(s), β(s) ≥ 0, ∀s ∈[0, s ]. For new goods, the firm bears an extra quadratic cost with coefficient β 0 , β 0 > 0; with β(0), we represent adjustment costs for advertising in new goods and with β 0 we represent an innovation cost. Innovation costs can be explained considering the fact that one has to pay to launch a new product; we assume that this extra cost has only to be paid for new goods. The instantaneous profit at time t becomes s

⌠ 2  [ A(t, s) f(s, p(t, [0, s ]))p(t, s) − q(s)a(t, s) − β(s)a (t, s)]ds ⌡

(2)

0

and the entrepreneur’s objective function becomes +`

− ρt J( A0 ; a, p) = ⌠ e ⌡ 0

 2 − β0 a (t, 0) 

s

⌠ [ A(t, s) f( s, p(t, [0, s ]))p(t, s) − q(s)a(t, s) − β(s)a 2 (t, s)]d s dt +  ⌡ 

(3)

0

over all state–control triples {A, a, p} which are solutions in a suitable sense of equation (1), with a(t, s) ≥ 0, ∀t ≥ 0, ∀s ≥ 0. We study the problem without positivity constraints on A. The reason for this is that we focus our attention on a neighborhood of the long-run equilibrium, which is assumed to be strictly positive. The optimization problem (3) in p and a can be decoupled solving the static maximization in (2) with respect to p given A, and then studying the original dynamical problem with respect to a. From the first optimization problem, we have in an abstract way that the profit per unit of goodwill can be described by a time invariant function of s, γ (s) = max p(t , [ 0 s, ]) f (s, p(t, [0, s ]))p(t, s), and the maximization problem (3) becomes +`

⌠ e− ρt J( A0 ; a) =  ⌡ 0

s

 ⌠ [A(t, s)γ (s) − q(s)a(t, s) − β(s)a 2 (t, s)]d sd t (4)  − β 0a 2 (t, 0) +  ⌡   0

over all state–control couples {A, a} which are solutions in a suitable sense of equation (1), with a(t, s) ≥ 0, ∀t ≥ 0, ∀s ≥ 0. The shape of γ (s) and of q(s) and of β(s) is assumed to be similar to the one of λ(s): γ ′(s), q′(s), β ′(s) ≤ 0, ∀s ∈[0, s ]; it is less expensive to advertise vintage goods rather than new goods, and a unit of goodwill

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E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

for new goods is more profitable than a unit of goodwill for vintage goods. If s = + ` , then we assume that γ (s), β(s), q(s) are integrable in [0, + ` ) and so they go to zero as s → + ` . The problem can be studied also assuming time-dependent parameters. We prove the existence of the optimal advertising policy and of a long-run stationary equilibrium. The long-run stationary goodwill equilibrium is either a strictly decreasing function of the vintage or a single-peaked function first increasing and then decreasing with a maximum; this depends on the parameters of the model. In the latter case, we have that the goodwill for a good (or if one wants the extension of the market for a good) does not reach its maximum when it is launched onto the market, but only after some length of time. The paper is organized as follows. In section 2, we present the state equation through the semigroups language. In section 3, we analyze the optimal control problem. In section 4, we study the optimal advertising trajectory and the long-run equilibrium. In the appendix, we present the technicalities needed in our analysis. 2.

The state equation

In what follows, we study the state equation of our model in an infinite-dimensional setting through the semigroups language. For notations, symbols and some preliminaries, see the appendix. Fix s ∈(0, + ` ] and consider a controlled dynamical system whose behavior is described by the partial differential equation (1) in the strip [0, + ` ) × [0, s ] (when s = + ` , we will substitute everywhere the intervals [0, s ] and (0, s ] with [0, + ` ) and (0, + ` ), respectively), where a : [0, + ` ) × [0, s ] → R+ is the control function and A : [0, + ` ) × [0, s ] → R is the state function. We want to express this Partial Differential Equation (PDE) as an Ordinary Differential Equation (ODE) in the Hilbert space L2 = L2 (0, s ). In the following, we will often omit the variable s: A(t), a(t) will denote the elements A(t, ·), a(t, ·) ∈L2; we will employ the variable s only when it is needed to avoid misunderstandings. Consider the following linear closed operator A on L2 associated with the PDE (1): D( A) = { f ∈H 1 : f (0) = 0}, A f (s) = – f ′(s) – µ f (s). (5) A semigroup T (t) is associated to the PDE (1); T (t) and its properties are described in the following proposition. Proposition 1. The operator A is a linear closed dissipative operator on L2 and generates a strongly continuous semigroup T (t) given by [T (t) f ] (s) = e− µ t f (s − t)I{s ≥ t} for t ∈[0, s ], and [T (t) f ]( s) = 0, ∀s ∈[0, s ], if t > s . The resolvent set of A contains the half plane {Re η > – µ}. For Re η > – µ , f ∈H, s ∈(0, + ` ], we have

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

+`

19

s

⌠ e− ηt [T (t) f ] (s)d t = ⌠ e− (η + µ) (s − σ ) f (σ )dσ , [ R(η; A) f ] (s) =   ⌡ ⌡ 0

s ∈ [0, s ].

0

*

Let A be the adjoint operator of A. The following proposition can be stated about A*. Proposition 2. The operator A* is given by D(A * ) = { f ∈ H 1 : f (s ) = 0},

[A* f ] (s) = f ′(s) − µ f (s),

when s < + ` . When s = + ` , we have that lims → + ` f (s) = 0 for f ∈H 1 (0, + ` ), so that D( A* ) = H 1, [ A* f ] (s) = f ′(s) − µ f( s). The operator A* is the generator of the strongly continuous semigroup T * (t) (which is the adjoint of T (t)) on L2. When s < + ` , T * (t) is given by [T * (t) f ] (s) = e − µt f (s + t) I{s ≤ s − t} for t ∈[0, s ] and [T * (t)] f (s) = 0, ∀s ∈[0, s ], if t > s . The resolvent set of A* always contains the half plane {Re η > – µ}. For Re η > – µ, f ∈H, s ∈(0, + ` ], R(η; A * ) is given by +`

s

⌠ e− ηt [T * (t) f ] (s)d t = ⌠ e − (η + µ )(σ − s) f (σ )dσ , [R(η; A ) f ] (s) =   ⌡ ⌡ *

0

s ∈ [0, s ].

s

For a proof of the above propositions, see e.g. Pazy [13, pp. 7, 11, 41, 44]. We now want to write the solution of the state equation (1). To handle the control problem with the boundary condition A(t, 0) = λ(0)a(t, 0), the control strategy has to be well defined at s = 0 for every t ≥ 0. We will assume that a ∈L ` (0, + ` ; R) × L ` (0, + ` ; L2 ), which allows us to take care of the initial condition at s = 0. This choice allows us to use strategies measurable in t and piecewise continuous (right continuous with left limit) in s. First, let a(t, 0) ≡ 0 for every t ≥ 0. Then, as usual (see e.g. [13, section 4.2], we define the mild solution of (1) as the continuous function A : [0, + ` ) → L 2, t

⌠ T (t − τ )B a(τ )dτ . A(t) = T (t)A0 +  λ ⌡

(6)

0

The solution of (1) can be written in integral form when a(t, 0) ≡/ 0 by a standard procedure, see Bensoussan et al. [2]. Denoting by w(t, ·) the unique element of H 1 such that

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E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

∂w (t, s) + µw(t, s) = 0, ∂s

w(t, 0) = λ(0) a(t, 0)

(it is easy to check that w(t, s) = w0 (s)λ(0)a(t, 0), where w0 (s) = e – µs ), then the mild solution of (1) becomes t  t  ⌠ ⌠   A(t) = T (t)A0 − A  T (t − τ )w(τ )dτ +  T (t − τ )Bλ a(τ )dτ , (7)  ⌡  ⌡  0  0 which, by standard calculations, can be written more explicitly as A(t, s) = e− µt A0 (s − t)I{s ≥ t} − e− µs λ(0) a(t − s, 0)I{s < t} t ∧s

⌠ e − µr λ(s − r)a(t − r, s − r)dr . + ⌡

(8)

0

In the following, we will denote by A(t; A0 , a) the mild solution of (1) for given data A0 , a. We will omit the data, simply writing A(t), when no confusion is possible. The expression of the state A(t) in (7) is made up of three components. The first component represents the effect on the state variable at time t of the initial state point. The second component represents the effect of A(τ, 0) = λ(0)a(τ, 0) ≠ 0, 0 < τ ≤ t, on the state at time t. The third component describes the effect of advertising at time τ, 0 < τ < t, on A(t). Given the framework described above, equation (1) can be written in the following differential form: A′(t) = A[ A(t) − w(t)] + Bλ a(t),

A(0) = A0 ,

(9)

which does not make sense in the space L 2. In fact, the term – A w(t) = – A w0 λ(0)a(t, 0), which is the effect of A(t, 0) = λ(0)a(t, 0) ≠ 0 on A′(t), is not a function in the space L2. More precisely, – A w0 is a distribution and is equal to the Dirac delta function, so we can write – A w0 λ(0)a(t, 0) = δ0 λ(0)a(t, 0). By using extrapolation spaces as described in Nagel and Sinestrari [11], it is possible to give a meaning to equation (9) in a suitable space of distributions when a ∈L ` (0, + ` ; R) × L ` (0, + ` ; L2 ); we omit the proof for brevity. We only mention that to give a meaning to equation (9) in this general framework, we have to define suitable extensions of the operator A and of the corresponding semigroup T . In the rest of the paper, these extensions will be denoted by the same symbols, for simplicity. 3.

The optimal control problem

Let us assume that β 0 ∈R, γ, β and q are bounded elements of H 2. Moreover, we make the following assumption.

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

21

Assumption 1. (i)

For every s ∈[0, s ], we have γ (s ) = 0, γ (s), q(s) ≥ 0 and β 0 , β(s) ≥ ε > 0 for a given ε > 0.

(ii) γ ′(s) ≤ 0, q′(s) ≤ 0, β ′(s) ≤ 0. We consider the functions g : L 2 → R, l : H 1 → R, s

⌠ γ (s) A(s)ds = 〈γ , A〉 , g( A) =  L2 ⌡ 0 s

⌠ [− q(s)a(s) − β(s)a 2 )s)]d s − β a 2 (0) = − 〈q, a〉 2 − 〈 B a, a〉 2 − β a 2 (0), l(a) =  0 β 0 L L ⌡ 0

where Bβ : L 2 → L 2 is the continuous linear operator defined as [Bβ f ](s) = β(s) f (s), s ∈[0, s ]. As pointed out in the previous section, we assume that the control strategy def a belongs to the set U = {a ∈L ` (0, + ` ; R) × L ` (0, + ` ; L2 ); a ≥ 0}. The optimal control problem becomes +`

⌠ e − ρτ [g( A(τ )) + l(a(τ ))]dτ J( A0 ; a) =  ⌡

(P) maximize the functional

(10)

0

over all control trajectories a ∈U , where A is the corresponding mild solution of the state equation (1). A control strategy a * ∈U is an optimal strategy if J(A0 ; a * ) ≥ J(A0 ; a), ∀a ∈U . A state–control pair (A*, a* ) is an optimal pair if a* is an optimal control strategy and A* is the corresponding state trajectory. The value function of the problem is defined as υ (A0 = supu ∈ U J(A0 ; a). For p ∈D(A * ), the current value Hamiltonian F0 is given by def

F0 ( a, p) = − a(0) 〈w0 , A* p〉 L2 + 〈Bλ a, p〉L 2 + l(a)

(11)

and the maximum value Hamiltonian H0 is given by H0 ( p) =

sup

a ∈R × L2 , a ≥ 0

F0 (a, p).

(12)

Following Fleming and Soner [6], the current value Hamiltonian is defined as F1 (a, p, A) = F0 (a, p) + g(A) – hA, A * pi L2 . To describe the solution of the model, we introduce the function s ⌠ e− (ρ + µ) (σ − s )γ (σ )dσ, γ (s) = [ R(ρ; A * )γ ] (s) =  ⌡

(13)

s

where γ (s) is the discounted return associated with a unit of goodwill for goods of vintage a. As time passes, the return for a unit of goodwill of goods of vintage σ ≥ s,

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E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

γ (σ ), is associated with a unit of goodwill for goods of vintage s. We remember that the connection between the vintage of a good and the flow of time is δs = δt. This means that a good of vintage s will be of vintage σ > s after the time period σ – s, and the discounted return associated with a unit of goodwill for goods of vintage s for being of vintage σ after the time period σ – s is e – ρ(σ –s) γ (σ ); meanwhile, a unit of goodwill is exponentially decreased and amounts to e – µ(σ –s). Therefore, the discounted return of a unit of goodwill for goods of vintage s for being of vintage σ after the time period σ – s becomes e –( ρ +µ)( σ – s) γ (σ ), which is the integrand of (13). After some tedious calculations, the functional J can be rewritten as follows: +`

J(A0 , a) = 〈γ , A0 〉 L2 where

⌠ e− ρt F(a(t))d t, +  ⌡

(14)

0

F(a) = F0 (a, γ ) = − 〈 A* γ , w0 〉λ(0) a(0) − β0 a 2 (0) + 〈− q + Bλ γ , a〉 L2 − 〈Bβ a, a〉L2 .

(15)

Concerning the existence of the optimal control, we have the following result, proved in the appendix. Proposition 3. There exists only one optimal strategy a * for problem (P). The optimal strategy a * does not depend on the initial state A0 and time t, and is given by a * (0) =

λ(0)γ (0) [λ(s)γ (s) − q( s)]+ , a * (s) = , ∀s > 0; 2β0 2β(s)

(16)

a * satisfies the following Maximum Principle: F0 (a * , γ ) =

sup

a ∈R × L2 , a ≥ 0

F0 (a, γ ) = H0 (γ )

(17)

and is continuous in s if and only if the following condition is satisfied: λ(0)

γ (0) 1 = [λ(0)γ (0) − q(0)] > 0. 2β 0 2β(0)

(18)

The model being characterized by constant returns to scale with respect to the goodwill, the optimal control strategy is constant over time. About a * (0), we have that it is an increasing function of λ(0), γ (s), ∀s ∈[0, s ], decreasing in ρ, µ and β 0 . Thanks to assumption 1, a * is a function of L 2 and is differentiable almost everywhere for s > 0. From the optimal control expression, it follows that a * > 0 if and only if λ γ – q > 0, i.e. advertising for goods of a specific vintage is strictly positive if and only if the associated discounted return multiplied by the technology factor λ is strictly larger than the unit advertising cost. In the intervals where this happens, ceteris paribus, a

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

23

larger discount factor ρ or a larger decay rate µ lead to a lower level of advertising; the same thing happens if β(s) is increased. On the other hand, if λ(s), γ (s), s ∈(0, s ) are increased, then the level of advertising becomes larger. In the general case, nothing can be said a priori about (a * )′(s), which depends on the behavior of γ (s), q(s) and β(s); when (λγ )′(s) ≤ 0 and β(s) and q(s) are constant, then (a * )′(s) ≤ 0. Since γ ′(s) ≤ 0 implies γ ′(s) ≤ 0, then (λ γ )′(s) ≤ 0 when λ and γ are decreasing. In the above analysis, we have considered the case λ′(s) ≤ 0, γ ′(s) ≤ 0, ∀s ∈ [0, s ]. This assumption does not take into account the so-called learning or experience effect, i.e. the productivity of goodwill increases as time passes because of externalities, learning by doing, etc. However, λ (s), γ (s) first increasing in s and then decreasing can be easily introduced in our setting. 4.

Optimality conditions, the long-run equilibrium

In this section, we study the optimality conditions for the problem and then we show the existence of a stationary long-run equilibrium. The maximum value Hamiltonian H0 can be split into two parts due to the presence of the boundary control term and to the linearity of the problem. For p ∈D(A * ), the Hamiltonian function H0 ( p) can be written as H0 ( p) = H01 (〈w0 , A* p〉 L2 ) + H02 ( p), where we have set, for α ≤ 0 and p ∈L 2, H01(α) = sup [− rλ(0)α − β 0r 2 ] = r ∈R +

λ2 (0)α 2 , 4β 0

H02 ( p) = sup [〈a, Bλ p − q〉L 2 − 〈Bβ a, a〉 L2 ] = a ∈L2

1 4

〈B1 β (Bλ p − q)+ , (Bλ p − q)+ 〉L2 .

a≥ 0

Since a * maximizes the current value Hamiltonian F0 , it can easily be checked that it satisfies a * (0) = D H01 (〈w0 , A *γ 〉 L2 ) = DH01 (γ (0)),

a* (s) = D H02 (γ ) (s);

s ∈ (0, s ].

The Maximum Principle can be stated as in Barucci and Gozzi [1]. We recall it here in its Hamiltonian form. For similar results, see Cannarsa and Tessitore [4], Gozzi and Tessitore [8], Fattorini [5], Lasiecka and Triggiani [10] and the appendix for a sketch of the proof. Theorem 4. Let (A*, a * ) be the optimal pair for problem (P). Then there exists a function p * ∈L ` (0, + ` ; L 2 ) such that (A*, p * ) is a mild solution of the following system (which only makes sense in integral form):

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E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

 A′(t) = A A(t) − D H01(〈w 0 , A* p(t)〉 L2 ) Aw 0λ(0) + Bλ D H02 ( p(t));  A(0) = A0 ,    p′ (t) = [ρ − A* ] p(t) − γ ,

(19)

satisfying the transversality condition lim e − ρt p(t) = 0.

t → +`

(20)

The solution (A*, p * ) of the system (19)–(20) is unique and is given by p *(t) = γ ; A*(t) = T (t) A0 − λ(0)a * ( 0 ) T [ (t) − I ]w 0 − R(0; A) [T (t) − I ]Bλ a* , where a * is given by (16). The system (19) has only one stationary equilibrium point (A ` , p` ), A` = R(0; A) Bλ a * + λ(0) a *(0) w0 , p ` = R(ρ; A* )γ = γ . The transversality condition (20) above implies the classical transversality condition lim t → + ` e –ρt hA(t), p(t)i = 0. γ can be interpreted as the costate variable since we have that p * (t) = R(ρ, A* )γ = γ , ∀t ≥ 0; the marginal value associated by the optimal control a * to the state along the optimal trajectory A* is the discounted return γ . The only solution of (19) that satisfies the transversality condition is characterized by a constant p(t) : p(t) = R(ρ; A* )γ = γ , t ≥ 0. This means that (A, p) = (A, γ ) in the phase space L 2 × L 2 is the stable manifold of the stationary equilibrium point (A ` , p` ) of system (19). If p0 = γ , then for every A0 > 0 the solution of system (19) converges to (A ` , p` ) as t → + ` . In Barucci and Gozzi [1], it is shown in a similar framework that the Turnpike Property holds, i.e. the optimal state–costate trajectory of the finite horizon problem belongs to a neighborhood of the optimal trajectory for the infinite horizon problem (A ` , p` ) for a given period of time. The long-run stationary equilibrium is given by s

⌠ e− (ρ + µ) (σ − s )γ (σ )dσ, p ` (s) = γ (s) = R(ρ; A )γ (s) =  ⌡ *

s

A` (s) = R(0; A)Bλ a *(s) + λ(0) a *(0) w0 (s) s

=⌠  e − µ(s − σ )λ(σ )a *(σ )dσ + λ(0)a * (0)e − µs , ⌡ 0

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

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where a * is given by (16). In what follows, we assume that λ(0) γ (0) – q(0) > 0, the discounted return associated with advertising in a new good, λ(0) γ (0), is larger than the unit advertising cost, q(0). Let us remark that we have studied the problem without explicitly imposing a positivity constraint on the state variable; in the following, we restrict our attention to the analysis of a stationary equilibrium (A ` , p` ) characterized by a positive solution, i.e. A ` (s) > 0. The optimal advertising path obtained from the optimal control problem supplies two interesting pieces of information: the optimal advertising path and the optimal long-run stock of goodwill. Because our model is linear-quadratic, the optimal advertising policy is constant over time. The analysis of the long-run stationary equilibrium confirms the analysis developed in the literature for the finite dimensional case. In that setting, a well-established result states that the optimal stock of goodwill is a decreasing function of the discount factor and of the interest rate. The result is confirmed in the infinite-dimensional setting. For every s ∈(0, s ), we have that A ` (s) is decreasing in β 0 , ρ, µ, β(σ ) and q(σ ) with σ ∈[0, s), and increasing in λ(σ), γ (σ ) with σ ∈[0, s ]. Note that if a * (s) = 0 for s ∈(s1 , s2 ), then in such an interval A ` (s) is strictly decreasing in s; more precisely, A ` (s) = A ` (s1 )e – µ(s –s1), ∀s ∈(s1 , s2 ). Let us remark that A ` (0) = λ2(0) γ (0) 2β 0 , therefore we have that the goodwill for new goods is decreasing in µ, ρ and β 0 , and increasing in λ(0) and γ (s), ∀s ∈[0, s ]. This reasoning can be replicated in a neighborhood of 0 + of s. From easy calculations, we obtain the following proposition. Proposition 5. Let assumption 1 be satisfied. Then the optimal control a * is continuously differentiable on (0, s ) (possibly discontinuous at s = 0). The function A ` belongs to H 1 and its derivative A′` is continuously differentiable out of s = 0. A ` is the unique solution of the equation A′(s) + µ A(s) = λ(s)a* (s), s > 0;

A(0) = λ(0) a* (0) =

λ2 (0)γ (0) . 2β 0

(21)

It follows that A`′ (s) = − µe −µs λ(0)a * (0) + λ(s)a* (s) − µR(0; A) (λa * ) (s) def

and, defining A′` (0 + ) = lims → 0 + A′` (s), A`′ (0 + ) = − =

µλ 2 (0)γ (0) λ(0)γ (0) − q(0) + λ(0) 2β0 2β(0)

λ2 (0)γ (0)  1 µ  q(0)λ(0) − .   − 2 β0  2β(0)  β(0)

(22)

The shape of A ` (s) can have different characterizations. The following proposition can be stated, see the appendix for a proof.

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

26

Proposition 6. If (λa * )′(s) ≤ 0, ∀s ∈(0, s ), then we have • A`′ (0 + ) ≤ 0 ⇒ A`′ (s) ≤ 0, ∀s ∈ [0, s ]; • A`′ (0 + ) > 0 ⇒ A` (s) is single-peaked, there exists a point s0 ∈(0, s ) such that A′` (s0) = 0, A′` (s0) > 0 for s < s0 and A′` (s) ≤ 0 for s > s0 . If the parameters of the model generate a flow of advertising decreasing in s, ≤ 0 and λ′ ≤ 0, then the optimal goodwill A ` (s) can only be monotonically decreasing or single peaked with a maximum in s0 , increasing for s < s0 and decreasing for s > s0. From the analysis of (22), it turns out that if λ(0)γ (0) – q(0) À 0 and λ(0), β 0 are sufficiently high, then A′` (0 + ) > 0 and A ` (s) is single peaked with a maximum in s0 . So, if a unit of goodwill for a new good is highly profitable, advertising in new goods is highly productive and there is a high innovation cost, then we have a singlepeaked goodwill shape. The goodwill for a good reaches its maximum, not necessarily when it is launched onto the market, but after a length of time. Let us remark that A′` (0 + ) is decreasing in µ, q(0), and β(0), while it is increasing in β 0 . With regard to the other parameters, the sensitivity analysis depends on the sign of the term (1 β(0)) – (µ β 0 ). If this term is negative, then A′` (0 + ) is negative and decreasing in λ(0), γ (s), ∀s ∈[0, s ] and increasing in ρ. If (1 β (0)) – (µ β 0 ) is positive, then A′` (0 + ) is increasing in γ (s), ∀s ∈[0, s ] and λ(0) (after a threshold) and decreasing in ρ. If µ is high, then the optimal stock of goodwill is strictly decreasing in s, as the goodwill quickly depreciates, and it is not worthwhile to wait and to advertise a new good after a while without paying the innovation cost. The same thing happens for the discount factor. If the entrepreneur heavily discounts future returns, then there is no reason to waste time. If the parameters are such that a * (s) is first increasing and then decreasing, then the analysis changes slightly. The single-peaked feature of the longrun stock of goodwill is reinforced and we may also have that A ` (s) has a minimum and then a maximum. (a * )′(s)

Appendix In this appendix, we present some technical results useful in the analysis and we give proofs of some of the propositions. We begin by recalling some basic mathematical definitions and results that we used throughout the paper. We refer the reader to, for example, Brezis [3]. Given s ∈(0, + ` ], we will denote by L 2 (0, s ) or, when no confusion is possible, simply by L 2 , the space of Lebesgue measurable functions f : (0, s ) → R such that ∫0s | f (s)| 2ds < ` . The scalar product in this space will be denoted by h ·,· i L2 . Moreover, we will denote by H n (0, s ) (or simply H n ), n = 1, 2,…, the Sobolev space of functions f ∈L 2 such that the nth distributional derivative of f still belongs to L 2.

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

27

We denote by L ` (0, + ` ; L 2 ) (respectively, L ` (0, + ` ; R)) the space of all functions f : (0, + ` ) → L 2 (respectively, R) that are bounded and measurable. Finally, we will denote by C 0 ([0, s ]) (respectively, C p0 ([0, s ])) or simply by C 0 (respectively,, C p0 ), the space of all continuous (respectively, left continuous with right limit) functions f : (0, s ) → R, endowed with the usual norm. If s = + ` , then we include square integrability of the function in the above definitions. Given f ∈L 2, we will denote by f + the positive part of f. For g ∈L 2, Bg denotes the multiplication operator by the function g, i.e. Bg : L2 a L2 , [Bg f ] (s) = g(s) f (s). Finally, given a set O , Rn, the symbol IO will denote the indicator function of O, i.e. a function that is equal to 1 in O and 0 outside. Proof of proposition 3. First we prove that a * given by (16) is the unique optimal control for our problem. To this end, we observe that by rewriting the functional J, the problem (P) reduces to maximizing the functional +`

⌠ e− ρ t F (a(t), γ )d t J0 (a) =  0 ⌡

(23)

0

subject to (1). The functional J0 (a) does not depend on the initial datum A0 ; moreover, by the definition of the Hamiltonian H0 given in (12), it is easy to check that +`

⌠ e− ρt H (γ )d t = 1 H (γ ). J0 (a) ≤  0 0 ⌡ ρ 0

If we find an element ∈R × L such that a ≥ 0 and (17) is satisfied, then the control strategy a(t) = a * for every t ≥ 0 is optimal. Now we can consider the map G : R × L 2 → R, G(r, a) = λ(0)γ (0)r − β 0r 2 + 〈a, Bλ γ − q〉 L2 − 〈 Bβ a(t), a(t)〉L2 . a*

2

By construction, F(a) = G(a(0), a), G is strictly concave, weakly upper semicontinuous and coercive, while the set {a ∈R × L 2, a ≥ 0} is closed and convex in L 2. Then, on this set, G has a unique maximum point given by r = r* =

λ(0)γ (0) , 2β0

[λ(s)γ (s) − q(s)] + 2β(s)

a(s) = a* (s) =

and the value of the maximum is s

max

r ∈R a ∈L2, a ≥ 0

+ 2 λ2 (0)γ 2 (0) ⌠ ([λ(s)γ (s) − q(s)] ) d s. G(r, a) = H0 (γ ) = + ⌡ 4β 0 4β(s) 0

E. Barucci, F. Gozzi y Optimal advertising with a continuum of goods

28

When the compatibility condition is satisfied, the control strategy a * is obviously continuous. Vice versa, assume that the compatibility condition is not satisfied. Then by considering a sequence of controls (an )n ∈N such that a n (0) =

λ(0)γ (0) ; 2β0

an (s) =

1 [λ(s)γ (s) − q(s)]+ , 2β(s)

 1  s ∈  , s  n 

(it is enough to connect in a smooth way the points 0 and 1 n), we can see that we still have 1 lim J0 (an ) = sup J0 (a) = H0 ( R( ρ, A * )γ ). n→ +` ρ a ∈U So, if there exists an optimal control strategy a * (t, s), then it still has to satisfy the “maximum principle” (17). But this implies that a * is given by (16) and so, since (18) is not satisfied, it is not continuous with respect to s at s = 0. u Sketch of proof of theorem 4. For the first part, it is enough to verify that the costate satisfies (19)–(20). Uniqueness of the solution of (19)–(20) follows by observing that γ is the only solution p of the second equation of (19) that also satisfies (20). The other claims easily follow by recalling that the semigroups T and T * given in propositions 1 and 2 are characterized by an exponential decay rate. Proof of proposition 6. Setting z(s) = A′(s), s ∈(0, s ], the following equation can be obtained from (21): def

z ′(s) + µz(s) = (λa * )′ (s);

z(0) = A ′(0 + ).

(24)

First, let A′(0 + ) ≤ 0. Let s0 be a maximum point of z. If s0 ∈(0, s ), then we have µz(s0 ) = (λ a * )′(s0 ) ≤ 0. If s0 = 0, then z(s0 ) = A′` (0 + ) ≤ 0 and if s0 = s < + ` , then z ′(s0 ) ≥ 0, which implies µz(s0 ) ≤ (λ a * )′(s0 ) ≤ 0 (if s = + ` , we use that lims →+ ` A′` (s) = 0 and then the same argument). It follows that A′(s) = z(s) ≤ 0 for every s ∈(0, s ). This gives the first part of the claim. For the second part, we recall that z(0) > 0 and z(s ) ≤ 0 (use that lim s → + ` A′` (s) = 0 when s = + ` ). This implies that there exists a first point s0 ∈(0, s ] such that z(s0 ) = 0. This point cannot be s since in this case, we would have A′` ≥ 0 on (0, s ] and, from (21), we would also have A` ( s ) =

λ(s ) * a (s ) ≤ 0, µ

which is impossible. We finally prove that z(s) ≤ 0 for s > s0 . By contradiction, there exists a maximum point s1 ∈(0, s ) such that z(s1 ) > 0. But this is impossible by reasoning, as in the first part of the proof. u References [1]

E. Barucci and F. Gozzi, On investments in a vintage capital model, Research in Economics 52 (1998)159–188.

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[2] [3] [4]

[5] [6] [7] [8] [9] [10]

[11]

[12] [13]

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