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Games and Economic Behavior 45 (2003) 442–464www.elsevier.com/locate/ge

Monopolists and viscous demand

Roy Radnera,∗ and Thomas J. Richardsonb

a Stern School of Business, New York University, 44 West 4th St., New York, NY 10012, USAb Flarion Technologies, Inc., Bedminster, NJ 07921, USA

Received 17 February 2003

Abstract

We characterize the optimal dynamic price policy of a monopolist who faces “viscous” defor its services. Demand isviscous if it adjusts relatively slowly to price changes. We show that wthe optimal policy the monopolist stops short of achieving 100% market penetration, even wof the consumers have the same long-run willingness to pay for the service. Furthermore, forparameter values in the model, the price policy requires rapid oscillations of the price path. 2003 Elsevier Inc. All rights reserved.

JEL classification: C61; D11; D42

Keywords: Viscous demand; Monopoly pricing; Dynamic games; Bounded rationality

1. Introduction and summary

“Viscous demand” refers to a phenomenon common to many markets, in which deadjusts relatively slowly to changes in prices and quality. There are many reasothe viscosity of demand, depending on the type of good or service being sold, athe organization of the market. In this paper we focus on the case of a service, sa subscription to a magazine, newspaper, or long-distance carrier, that does nota major part of the customer’s budget. We also focus on responses to price chrather than to changes in marketing or quality, or the introduction of innovations. Rospeaking, the viscosity of demand can be explained by a model in which the conhas an attention budget, and only occasionally reconsiders the the decision about wto subscribe to the service. Thus in this case the viscosity of demand is due to a

* Corresponding author.E-mail address: [email protected] (R. Radner).

0899-8256/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0899-8256(03)00143-X

R. Radner, T.J. Richardson / Games and Economic Behavior 45 (2003) 442–464 443

suchspecial,mple,

eting in

upplierrtance,with

gh andan inture on(Chenthird

s haveratherthose

3.)ght bycribedd, seed thericingg, weA, the

til themarket

then ar, the

ed totly low

illatestration

on ismodely, onemodel.

somea in a

licy in

type of bounded rationality on the part of the consumers. (A more formal model ofconsumer behavior is presented in Radner, 2003.) Although this case is somewhatwe hope that the approach we introduce here will be helpful in other models. For exaone of the parameters of our model can be interpreted as the effectiveness of markinducing consumers to change behavior.

The presence of demand viscosity creates a temporary monopoly power for a sof the service. In such a case, the time path of a firm’s prices acquires added impoand the problem of optimal pricing becomes significantly more complex, comparedthe case of instantaneous demand response, even for the case of a monopolist.A fortiori,as one might expect, the dynamic-game-theoretic problems inherent in a thorourigorous treatment of oligopolistic markets are an order of magnitude more difficult ththe instantaneous-demand case. In fact, there is very little previous theoretical literathis topic. An exception is the group of three papers, (Rosenthal, 1982, 1986) andand Rosenthal, 1996), in which Robert W. Rosenthal (with Yongmin Chen in thepaper) analyzed some models of markets with ‘customer loyalties.’ These modelsome analogies with the present model of viscous demand, but their structure isdifferent. For this latter reason, we shall not attempt here to compare our results withof Rosenthal and Chen. (For a review of this and related literature, see Radner, 200

We focus in this paper on the case of a monopolist selling a service that can be boueach consumer at a fixed rate per unit of time, or not at all; the precise model is desbelow. (For an analysis of a corresponding model of duopoly with viscous demanRadner, 2003.) In spite of the simplicity of the situation described by the model, anrelatively small number of parameters, we shall see that the monopolist’s optimal pstrategy can be radically different for different parameter values. Roughly speakincan summarize our results as follows. In one region of the parameter space, call itmonopolist has a “target” market penetration,which is less than 100percent. If the initialpenetration is below the target, then the firm charges a minimum price (say zero) unmarket penetration reaches the target, and then switches to a price that stabilizes thepenetration at the target level. If the initial market penetration is above the target,price is charged that stabilizes the market penetration at the initial level. In particulaparameters will be in the region A if the firm’s discount rate is not too high comparthe inverse of the demand viscosity, and the price is bounded above by a sufficienbound.

In the complementary region of the parameter space, call it B, the firm’s price oscrapidly between its lower and upper bounds, maintaining its average market peneat a low level. In fact,strictly speaking there is no optimal policy in the usual sense; wemay say that in the “optimal” policy the price oscillates infinitely fast between its lowerand upper bounds! (We provide a precise meaning to this statement.) Such a situatihardly realistic, and provokes a reconsideration of the behavioral assumptions of thewhen it has these parameter values. In particular, if prices are oscillating very quicklwould not expect consumers to react so myopically as they do in the first postulatedFor example, one might expect (boundedly rational) consumers to forecast prices in“adaptive” manner, e.g., with a moving average of past prices. We formulate this idesecond model. Although a complete characterization of the monopolist’s optimal pothe face of consumers with adaptive expectations is not known, we show that

444 R. Radner, T.J. Richardson / Games and Economic Behavior 45 (2003) 442–464

n thendopic

may

he twolts, and

ions”l pricet the

totaling the

rential

namicter of. Thus

ret

ngence

(1) an optimal policy exists,(2) the maximum profit in the “adaptive expectations” model approaches that i

“myopic behavior” model as the speed of adaptation increases without bound, a(3) the optimal adaptive-expectations price policy is nearly optimal for the “my

behavior” model if the speed of adaptation is large enough.

This last point implies that in the adaptive-expectations model the optimal priceoscillate, but at a finite frequency.

In the remainder of this section, we summarize the assumptions and results for tmodels. In Section 2 we present a detailed statement of the models and the main resushow how the “myopic behavior” model is a limiting case of the “adaptive-expectatmodel. In Section 3 we derive the characterization of the value function and optimapolicy for the “myopic behavior” model. Section 4 contains further information abou“adaptive expectations” model, and the results of some numerical computations.

1.1. Summary of results for the main model

A monopolist sells a service. The population of consumers is a continuum, withmass one. Time is continuous. At each instant of time each consumer is purchasservice at a rate of either 0 or 1; in the latter case the consumer is called acustomer. (Notethat here and henceforth ‘customer’ refers to a customer of the service.)

For each timet 0, letQ(t) denote the mass of customers, andP(t) denote the priceof the service per unit time. The mass of customers evolves according to the diffeequation

Q(t) = f(P(t),Q(t)

), (1)

where

f (p,q) =λ(w − p)(1− q), p w,

−λ(p − w)q, p w,(2)

andλ andw are strictly positive constants. (We shall use upper case to denote dyvariables and lower case for static and ‘dummy’ variables.) The essential charac(1)–(2) is that customers do not respond instantaneously to a change in the pricesuppose that in some time interval the price is held constant at the levelp. If p < w thenthe firm will gain customers at a rate proportional to the difference,(w − p), and to themass of remainingnoncustomers, (1 − q). On the other hand, ifp > q , then the firm willlose customers at a rate proportional to(p − w) and to the mass of remainingcustomers.The constant of proportionality, lambda, is the inverse of the viscosity. We may interpw

as the consumers’ “long-run” or “static” willingness to pay for the service. (Note thatw ishere the same for all consumers.)

The monopolist’s instantaneous cost per unit of time att is

k + cQ(t),

wherek, c 0. The magnitude ofk (the fixed cost) does not affect the optimal pricipolicy, although it does influence whether the monopolist’s profit is positive or not. Hwe shall takek = 0.


ill

e thecies,al

sllatory,

profit,ose

meters,

s.

ry

The monopolist’s total discounted profit is therefore

V (Q0) =∞∫

0

e−ρt[P(t) − c

]Q(t)dt, (3)

whereQ0 = Q(0), andρ > 0 is the discount rate. Given the initial mass of customers,Q0,the monopolist wants to choose the price pathP(t) to maximize (3). For reasons that wbecome apparent below, we make the following assumptions:

0 P(t) p, (4a)

w,c < p. (4b)

(In place of (4a), one could modify (2) so thatf (p,q) = −∞ for p > p.)We shall present a complete solution to this problem in the next section. Sinc

problem is time invariant, by Blackwell’s Theorem we need only consider Markov polii.e., P(t) = φ[Q(t)] for some functionφ. There are essentially two candidate optimstrategies. The first is ‘oscillatory,’ and might be thought of as the ‘abnormal case’:

φ1(q) =

0, 0 q < a,

p, a < q 1,(5)

where

a =

w

2h

√ρ2 + 4λ(p − w)h + ρ − 2h

p − 2w, p = 2w,

λ(w − c2)

(ρ + 2λw), p = 2w,

(6)

h = ρ + λw + c

pλ(p − 2w). (7)

At q = a the policy is to oscillate betweenp and 0 so thatq = a is a stationary point. (Thiwill be made more precise in the next section.) The second candidate is is not osciand might be thought of as the ‘normal case’:

φ2(q) =

0, 0 q < σ,

w, σ q 1,(8)

σ = λ(w − c)

ρ + λ(w − c). (9)

Each of these strategies has associated to it a return, i.e., a discountedV 1(Q0),V

2(Q0), respectively. This is the main result of this paper: it is optimal to chowhichever of these two strategies gives the largest return. Depending on the paraeither one strategy dominates the other for allQ0, or there is a single valueQ0 = x suchthatφ1 dominates forQ0 > x andφ2 dominates forQ0 < x. Furthermore, ifQ0 ≷ x thenQ(t) ≷ x for all t under the optimal policy, so there is no switching between strategie

Remark 1. Call Q(t) the (market) penetration, and callσ the target penetration undepolicy φ2. If the initial penetration is strictly less than the target, then, under the policφ2,


. On. These

rarginal

ionf the

,odel.eep they.

ld or

. This

fast.ero. Int hasfinite’

the).

ch

the penetration will increase monotonically to the target, reaching it in finite timethe other hand, any penetration greater than or equal to the target is a steady stateconclusions hold even if the target does not satisfy (9), i.e., even if it is not optimal.

Remark 2. The optimal target, in (9), is decreasing in the marginal cost,c, so that, undepolicy φ2, the monopolist increases his steady-state penetration if he reduces his mcost.

Remark 3. Under policyφ1 the target penetration isa. If Q0 is larger thana then it isoptimal to charge a high price(p), losing customers until, in finite time, the penetratreachesa. Here we clearly see the optimal policy taking advantage of the viscosity ocustomer base.

Remark 4. As p → ∞ the target penetrationa in (5) goes to 0 likep(−1/2). Furthermorepolicyφ1 dominates policyφ2 for p large enough. Here we see a degeneracy in the mIn reality, too large a price would cause a mass exodus of the customer base. To kmodel tractable we letp be the price above which all customers leave instantaneousl

Remark 5. It is interesting to note that in the eventV 1(x) = V 2(x) for x ∈ (0,1) then ifQ0 > x, i.e., if the customer base is relatively small, then the optimal strategy is to hoincrease the customer base. If the customer base is large, i.e.,Q0 < x, then it is optimal totake advantage of its viscosity by overcharging.

1.2. Adaptive expectations

There is perhaps something unnatural about the solution described aboveunnaturalness lies in the implied behavior of the customer whenQ(t) = a and φ1

is optimal. From the customer’s perspective, prices apparently fluctuate infinitelyConsequently, customers flow to and from the service, the net flow being equal to zreality customers will not respond this way. Nevertheless, the ‘oscillating’ fixed poinsignificance. If some smoothing mechanism is introduced into the model then the ‘inoscillation may disappear and yet the basic character of the oscillation will remain.

Accordingly, we shall consider a modification of our model, which we shall calladaptive expectations model (following the terminology of Arrow and Nerlove, 1958The corresponding optimization problem will be called theγ -AR problem. Hereγ is anew parameter. We shall use∞-AR to refer to the problem described earlier in whiadaptive expectations are absent.

In the adaptive expectations model, we assume that

Q(t) = f(P (t),Q(t)

),

whereP (t) solves the differential equation

P ′(t) = γ(P(t) − P (t)

), P (0) = P0.

Here, as before,P(t) is the price set by the monopolist, and we assume thatP0 ∈ [0, p]andγ > 0.


sumeratlicity,ntial

prices,ean betails.

datlso

g

raised;be seen

lass of

orderfound

Here is a heuristic interpretation of the AR model. Suppose that when a conarrives the monopolist has a long pricing record. LetP (t) denote the average price ththe consumer ‘expects’ to pay until he next considers his decision problem. For simpsuppose that this record extends infinitely far into the past. A solution of the differeequation forP (·) is

P (t) = γ

∞∫s=0

e−γ sP (t − s).

In this representation, the consumer’s ‘expected price’ is a weighted average of pastin which the weights decline geometrically into the past. In such a representation, thP (t)

is said to respond to prices with a ‘distributed lag.’ A corresponding representation cderived if the consumer only takes account of a finite history of prices; we omit the de

We are not able to solve theγ -AR model explicitly forγ < ∞. However, we can anwill prove that in this case theγ -AR model admits a real-valued optimal solution. This, unlike the∞-AR model, infinitely fast oscillations are never required. We shall aprove that theγ -AR model approximates the∞-AR model in the sense thatPγ (t) isnearly optimal for∞-AR if γ is large enough. This will imply that the optimal pricinin theγ -AR model can require oscillation (at a finite rate) between high(p) and low (0)prices. Periods of low prices, i.e., sales, draw in customers. Subsequently prices arecustomers leave, but not instantaneously and profits are reaped. This behavior canin many markets.

2. Formulation and statement of main results

We shall now present a mathematically precise statement of our∞-AR problem. (Mostproofs are deferred to Section 3.) Letψ denote the set of measurable functions from[0,∞)

to [0, p]. GivenP ∈ ψ andQ0 ∈ [0,1] let Q(t) be the solution to

Q(t) = f(P(t),Q(t)

), Q(0) = Q0, (10)

where

f (p,q) :=λ(w − p)(1− q), p w,

−λ(p − w)q, p > w.

Define

V (Q0,P ) :=∞∫

0

e−ρt(P(t) − c

)Q(t)dt and V opt(Q0) := sup

P∈ψ

V (Q0,P ).

Our goal is to findV opt and, if possible, find a Markov policyP(t) = φ(Q(t)) thatachievesV opt. By this we mean a measurable functionφ : [0,1] → [0, p] such that ifQ(t)

solves (10) withP(t) formally set toφ(Q(t)) thenV (Q0, φ(Q(t))) = V opt(Q0). It willturn out that this is not possible in general and it is necessary to broaden the cadmissible controls to admit measure-valued controls (Gamkrelidze, 1978).

Typically in problems of this type one broadens the class of admissible controls into obtain existence results and then proves that a ‘regular’ optimal control can be


es ofo that

er

an

ols

iginal

edk

ealized

stricts

in the original class. This is usually achieved by virtue of certain convexity propertithe functional to be minimized. Our functional, it turns out, lacks these properties sfor some parameter valuesP(t) must be replaced by a measure-valued controlµt . We canthen find an optimal Markov policy of the formµt = φ[Q(t)].

2.1. The weak formulation: measure-valued pricing

Let µt be a family of probability measures on[0, p] depending on the paramett ∈ [0,∞). Letg(t,p) be a continuous function on[0,∞)×[0, p]. We define the function

h(t) := Eµt

(g(t,Pt )

),

wherePt denotes a random variable with lawµt . If h(t) is Lebesgue measurable forarbitrary continuousg then we sayµt, t ∈ [0,∞) is weakly measurable with respect tot .Such an object is said to be ageneralized control. We denote the set of generalized contrby Θ.

The weak formulation of the optimization problem is the following. Maximize

V (Q0,µt ) :=∞∫

0

e−ρt(Eµt (Pt ) − c

)Q(t)dt,

whereQ(t) solves

Q(t) = Eµt

(f

(Pt ,Q(t)

)), Q(0) = Q0.

Define

V opt(Q0) = supµt∈Θ

V (Q0,µt ).

There are a few fundamental results relating the weak formulation to the orformulation. The first is

V opt(Q0) = V opt(Q0).

The inequalityV opt(Q0) V opt(Q0) follows from the fact thatδP(t) ∈ Θ for P ∈ ψ . Hereδp is the Dirac delta function atp ∈ [0, p]. The inequalityV opt(Q0) V opt(Q0) can beproved by finding a sequencePn(t) converging “weakly” (in the sense of generalizcontrols) toµt and showing thatV (Q0,P

n(t)) → V (Q0,µt ). The advantage the weaformulation has over the original formulation arises from the fact thatΘ is a convex spacwhereasψ is not. General lower semicontinuity and compactness properties of genercontrols (Gamkrelidze, 1978) allow us to assert

Theorem 2.1. For any Q0 ∈ [0,1] there exists an optimal generalized control µoptt , i.e.,

V opt(Q0) = V(Q0,µ

optt

).

It turns out that our problem has a piecewise affine structure which allows us to reto a small subclass ofΘ. Given a probability measureµ on [0, p] there exist constant


d

is

m0 0,mp 0 satisfyingm0 +mp 1 such that, for any continuous functiong on [0, p]which is affine on[0,w] and on[w, p], we have

Eµ(g) = m0g(0) + (1− m0 − mp)g(w) +mpg(p).

Now, f (p,q) andp − c are piecewise affine in this sense (inp). Thus, any generalizecontrolµt can be replaced by a control

µt = m0(t)δ0 + (1− m0(t) − mp(t)

)δw + mp(t)δp

without alteringQ(t) or V (Q0, ·). (Strictly speaking, we should verifyµt ∈ Θ. It caneasily be verified thatm0(t) andmp(t) are measurable.) Define

Ω = (m0,mp): m0 0, mp 0, m0 + mp 1

.

We now have

Theorem 2.2. For any Q0 ∈ [0,1] there exists a measurable function (m0,mp) : [0,∞)→Ω satisfying

V opt(Q0) = V(Q0, µ

optt

),

where µoptt = m0(t)δ0 + (1−m0(t) −mp(t))δw + mp(t)δp .

We shall look for a Markov optimal policy, that isφ : [0,1] → Ω such that setting(m0(t),mp(t)) = φ(Q(t)) yields an optimal control. By Blackwell’s theorem thissufficient.

2.2. Statement of main result: optimal policy

We are now ready to present the optimal value functionV opt and an optimal policyφ.Define

D(x) = f (0, x)− f (p, x)

and the quadratic function

Q(x) := (λw)2(1− x)− (λw + ρ)xD(x) − c

pD2(x).

Lemma 2.3. Q has exactly one root a in [0,1] and sgn(Q(x)−a) = −sgn(x−a) on [0,1].

Proof. Since Q(0) = (λw)2(1 − c/p) > 0 and Q(1) = −(λw + ρ)(λ(p − w)) −c/pD2(1) < 0, we see thatQ has exactly one root in[0,1]. Remark. We have given a formula fora in (6).

We now define two possible value functionsV 1 andV 2. The functionV 1 is the valueassociated with the following policy:

φ1(q) :=

m0 = 1, mp = 0, q < a,

m0 = 1− m(a), mp = m(a), q = a,

m0 = 0, mp = 1, q > a,


ges

l

and the functionV 2 is associated with the policy

φ2(q) :=m0 = 1, mp = 0, q < σ,

m0 = 0, mp = 0, q σ,whereσ = λ(w − c)

λ(w − c) + ρ.

Closed-form expressions forV 1 andV 2 can be easily obtained. Letφopt be a policy whereφopt = φ1 whenV 1 >V 2 andφopt = φ2 whenV 2 > V 1 andφopt is eitherφ1 or φ2 whenV 1 = V 2. The following is the main result to be proved.

Theorem 2.4 (Optimality).The policy φopt is optimal and

V opt = max(V 1,V 2).

Furthermore, either V opt = V 1 or V opt = V 2 or there exists x ∈ (a,1) such that V opt = V 1

on [0, x] and V opt = V 2 on [x,1].

Remark. The functionV opt is C2 except atx where it fails to be differentiable.

2.3. Existence of optimal control for the γ -AR model

Establishing existence of optimal control functions for theγ -AR model is straightfor-ward. Letµt be the “weak” limit of a minimizing sequence fromψ . It follows easily thatµt is a weak minimizer. However,P(t) := E(µt) is obviously equivalent toµt with respectto theγ -AR model. This establishes existence.

An alternative proof that avoids generalized controls proceeds as follows. SincePγ (t)

is Lipshitz (with constantγ p) we can extract a minimizing sequence which converuniformly on [0, T ] for any T < ∞ to some Lipshitz functionP ∗

γ (t). We then defineP(t) = P ∗

γ (t) + (1/γ )dP ∗γ (t)/dt and verifyP(t) ∈ [0, p] for almost allt . It follows that

P(t) is optimal.

2.4. Approximation results

It is fairly easy to prove thatV optγ approximatesV opt∞ for largeγ .

Theorem 2.5. For any P (0) ∈ [0, p] we have

limγ→∞V

optγ

(Q0, P (0)

) = Vopt∞ (Q0).

Proof. If P(t) is aC∞ function then it is easy to prove

limγ→∞Vγ

(C0, P (0),P (t)

) = V∞(Q0,P (t)

).

Since there existP(t) ∈ C∞ such thatV opt∞ (Q0) V∞(Q0,P (t)) + ε, whereε > 0 isarbitrary, we have limγ→∞ V

optγ (Q0, P (0)) V

opt∞ (Q0). The opposite inequality is triviasinceVγ (Q0, P (0),P (t)) = V∞(Q0, P (t)).


ndi-

ields

icals” hasfer the, 1985;

SupposePγ (t)γ=1,2... is a sequence of optimal pricing strategies for the initial cotion Q0, P (0). By weak compactness of generalized controls there existsµt , a generali-zed control which is the “weak” limit of some subsequence ofPγ (t)γ=1,2,.... It followsthat µt is optimal for the non-adaptive problem. To see this, integrating by parts y|∫ ∞

0 e−ρtQγ (t)Pγ (t)dt| C for some constantC. Hence,∣∣∣∣∣∞∫

0

e−ρtQγ (t)(Pγ (t) − Pγ (t)

)dt

∣∣∣∣∣ C

γ. (11)

Applying this inequality we obtain

limγ→∞V

optγ

(Q0, P (0)

) = limγ→∞V∞

(Q0, P (t)

) = V∞(Q0,µt ).

We conclude thatµt is optimal from Theorem 2.5. Thus, we have proved

Theorem 2.6. Any “weak” cluster point (in the sense of generalized controls) of optimalPγ (t) as γ → ∞ is optimal for the ∞-AR problem.

3. Main result

In this section we prove our solution to the∞-AR problem.

3.1. The value function: viscosity solutions

For ξ, q,p ∈ R define

G(ξ, q,p) := ξf (p,q) + (p − c)q and F(ξ, q) := supp∈[0,p]

G(ξ, q,p).

For future reference we also define

G(ξ, q,m0,mp) := ξ(m0f (0, q)+ mpf (p, q)

)+ (

(1− m0)w + mp(p − w) − c)q.

Further, foru ∈ R, we define the ‘Hamiltonian’

H(q,u, ξ) := ρu − F(ξ, q).

Formally, the value functionV opt satisfies the Hamilton–Jacobi–Bellman equation,

H(q,U,Uq) = 0, (HJB)

whereU is a function on[0,1]. This equation does not have a solution in the classsense, in general. Fortunately, in the last 15 years the theory of “viscosity solutionbeen developed. We shall not define the concept of a viscosity solution here, but reinterested reader to Fleming and Soner, 1993; Lions, 1985; Lions and SougandisCrandall et al., 1992. The main point is the following

Theorem 3.1. V opt is the unique viscosity solution to (HJB).


ionstheons)points

85;

Any classical, i.e.C1, solution to (HJB) is a viscosity solution, hence must beV opt.For our problemV opt will not beC1 in general and the classical theory of such equatis insufficient. It turns out thatV opt is locally a classical solution except possibly atpoint x. Only for this point are we required to verify non-classical ‘viscosity’ conditiof the solution. It turns out that a continuous piecewiseC1 function which satisfies (HJBin the classical sense on those (closed) intervals determined by the finite number ofwhere the function is not locallyC1 is automatically the viscosity solution; see Lions, 19Lions and Sougandis, 1985.

3.2. Proof of main result

Define the function

H 1(q) := m(q)p − c

ρq, wherem(q) := f (0, q)

f (0, q)− f (p, q).

If we choosem0 = 1−m(q) andmp = m(q). Thenq is a stationary point under(m0,mp),and the corresponding value atq is H 1(q), i.e.,

∞∫0

e−ρt((1− m0)w + mp(p − w) − c

)q dt = H 1(q).

Let us also define

H 2(q) := (w − c)q

ρ.

This is the value under the stationary policym0 = 0,mp = 0. Note thatV 1(a) = H 1(a)

andV 2(q) = H 2(q) for q ∈ [σ,1].We shall consider the policies

φ′b :=

m0 = 1, mp = 0, q < b,

m0 = 1− m(b), mp = m(b), q = b,

m0 = 0, mp = 1, q > b,

and the value functionV (q,φ′b). Note thatV (b,φ′

b) = H 1(b) andV 1 = V (φ′a).

Lemma 3.2. For b = a we have V (φb) < V 1 and, furthermore, V 1 H 1 with equalityholding only at a.

Proof. We claim that

sgn

(∂

∂bV

(q,φ′

b

)) = sgn(a − b). (12)

The claim proves the lemma since

V 1(q)− H 1(q) =a∫ (

∂

∂bV

(q,φ′

b

))db

q


0)

and

V 1(q)− V(q,φ′

b′) =

a∫b′

(∂

∂bV

(q,φ′

b

))db.

We now prove the claim. In general, we have

V(q,φ′

b

) = e−ρT (q)H 1(b)+T (q)∫0

e−ρt (1q>bp − c)Q(t)dt,

whereT (q) = mint : Q(t) = b. If q b then e−λ(p−w)T (q)q = b and if q b thene−λwT (q)(1− q) = 1− b. Assume for convenience thatq = b. We obtain

∂

∂bV

(q,φ′

b

) = e−ρT (q) ∂

∂bH 1(b) + e−ρT (q)

(−ρH 1(b) + (1q>bp − c)b) ∂

∂bT (q).

Now, (∂/∂b)T (q) = 1/f (0, b) or (∂/∂b)T (q) = 1/f (p, b) according toq < b or q > b.By calculating

∂

∂bH 1(b) = λwpf (0, b)− λwbD − cD2

ρD2

and noting that

1

f (p, b)

(−ρH 1(b)+ (p − c)b) = 1

f (0, b)

(−ρH 1(b)− cb) = −pb

D(b),

we obtain

∂

∂bV

(q,φ′

b

) = pe−ρT (q)

ρD2(b)Q(b).

Since this formula is continuous atq = b, it holds there also. By Lemma 2.3, Eq. (1holds.

We now define the policies

φ′′r :=

m0 = 1, mp = 0, q < r,

m0 = 0, mp = 0, q r.

Lemma 3.3. For all r ∈ [0,1] we have V (φ′′r ) V 2. Furthermore, V 2 H 2 with equality

holding only on [σ,1].

Proof. For q maxσ, r we haveV (q,φ′′r ) = V 2(q). For q < maxσ, r we have

V 2(q) − V (q,φ′′r ) = ∫ σ

r(∂/∂sV (q,φ′′

s ))ds. If q < r then V (q,φr) = e−ρT H 2(r) +∫ T

0 e−ρt (−c)Q(t)dt whereQ(t) solvesQ(t) = λw(1 − Q(t)), Q(0) = q , andT is givenby Q(T ) = r. Hence, in this case we have


e

o

n

above,is

gen-

∂

∂rV (q,φr) = −e−ρT ∂

∂rH 2(r) + e−ρT

(−ρH 2(r) − cr) ∂

∂rT

= e−ρT ρ + λ(w − c)

λρ(1− r)(σ − r).

Forq r, we have∂V (q,φ′′r )/∂r = 0, henceV (φ

′′r ) V 2. To prove the last statement w

noteH 2(q) = V (q,φ′′q).

Lemma 3.4. V 1 satisfies (HJB) at q if and only if V 1(q) H 2(q).

Proof. Let

F 1(q,m0,mp) := G(V 1q (q), q,m0,mp

).

For V 1 to satisfy (HJB) on[0, a) it is sufficient to show that∂F 1(q,m0,mp)/∂m0 0 and ∂F 1(q,m0,mp)/∂m0 ∂F 1(q,m0,mp)/∂mp. The first condition reduces tV 1q (q)f (0, q) wq . Since on[0, a] we haveρV 1(q) = V 1

q (q)f (0, q)− cq , the condition

further reduces toρV 1(q) (w− c)q , orV 1(q) H 2(q). Similarly, the second conditioreduces toV 1

q (q)(f (0, q) − f (p, q)) pq which reduces toV 1(q) H 1(q). Since the

second condition is satisfied, according to Lemma 3.2, we see thatV 1(q) H 2(q) isnecessary and sufficient to guarantee (HJB).

For q = a, the same argument holds but we require∂F 1(q,m0,mp)/∂m0 =∂F 1(q,m0,mp)/∂mp, orV 1(a) = H 1(a) which holds by Lemma 3.2.

For q ∈ (a,1), we require∂F 1(q,m0,mp)/∂mp 0 and ∂F 1(q,m0,mp)/∂mp ∂F 1(q,m0,mp)/∂m0. Note that hereV 1 satisfiesρV 1(q) = V 1

q (q)f (p, q) + (p − c)q .A calculation shows that the two conditions reduce to the same conditions as thosenamelyV 1(q) H 2(q) andV 1(q) H 1(q), respectively. By Lemma 3.2, the proofcomplete. Lemma 3.5. V 2 satisfies (HJB) at q ∈ [0, σ ) if and only if V 2(q) H 1(q).

Proof. Define

F 2(q,m0,mp) := G(V 2q (q), q,m0,mp

).

ForV 2 to satisfy (HJB) atq ∈ [0, σ ) it is necessary and sufficient that∂F 2(q,m0,mp)/∂m0

0 and ∂F 2(q,m0,mp)/∂m0 ∂F 2(q,m0,mp)/∂mp . SinceV 2 satisfiesρV 2(q) =V 2q (q)f (0, q) − cq , here these conditions reduce toV 2(q) H 2(q) andV 2(q) H 1(q)

respectively, exactly as in Lemma 3.4. By Lemma 3.3, the first condition holds ineral. Lemma 3.6. V 2 satisfies (HJB) on [σ,1] if and only if σ 1/2.


t

Proof. Here we require∂F 2/∂m0 0 and∂F 2/∂mp 0 whereF 2 is as in the proof ofLemma 3.5. SinceV 2(q) = H 2(r) these conditions easily reduce toq σ andσ 1/2,respectively. The first condition is given, thus, the proof is complete.Lemma 3.7. If σ 1/2 then V 1 V 2 with strict inequality unless a = σ = 1/2.

Proof. Assumeσ 1/2 and consider the policies (assumer σ )

φ′′′r (q) :=

m0 = 1, mp = 0, q < σ,

m0 = 0, mp = 0, q ∈ [σ, r],m0 = 0, mp = 1, q > r.

Forq ∈ [0, σ ] we haveV (q,φ′′′r ) = V 2(q). Forq > σ we have

V(q,φ′′′

σ

) − V 2(q) = −q∫

σ

(∂

∂rV

(q,φ′′′

r

))dr.

Since, here,

V(q,φ′′′

r

) = e−ρT H 2(r) +T∫

0

e−ρt (p − c)Q(t)dt,

whereQ(t) solvesQ(t) = −λ(p − w)Q(t), Q(0) = q andT is given byQ(T ) = r, weeasily obtain

∂

∂rV

(q,φ′′′

r

) = e−ρT

(w − c

ρ− 1

λ

) 0.

Thus,V (φ′′′σ ) V 2 with strict inequality unlessσ = 1/2.

We now recall the policiesφ′b defined above. Sinceσ 1/2 we haveH 1(σ ) H 2(σ ),

henceV (φ′σ ) V (φ′′′

σ ). The analysis in the proof of Lemma 3.2 shows thatV (φ′a)

V (φ′σ ) with strict inequality unlessa = σ . SinceV 1 = V (φ′

a), the proof is complete. Lemma 3.8. If σ,a 1/2 then V 2 V 1 with strict inequality unless a = 1/2.

Proof. Consider the policies

φ′′′′r (q) :=

m0 = 1, mp = 0, q < a,

m0 = 0, mp = 0, a q r,

m0 = 0, mp = 1, r < q.

Sincea 1/2, we haveH 2(a) H 1(a), henceV (φ′′′′a ) V 1 and the inequality is stric

unlessa = 1/2. As in Lemma 3.7, forq > r we obtain

∂

∂rV

(q,φ′′′′

r

) = e−ρT

(w − c

ρ− 1

λ

) 0.

HenceV (φ′′a ) = V (φ′′′′

1 ) V (φ′′′′a ). By Lemma 3.3, we haveV 2 V (φ′′

a ). This completesthe proof.


if

n bel

ell’s

Lemma 3.9. Assume a < σ . Then there exists at most one point x ∈ [a,1] such thatV 1(x) = V 2(x). If q ∈ [a,1] satisfies q < x then V 1(q) > V 2(q) and if q > x thenV 2(q) > V 1(q). For q ∈ [0, a) we have sgn(V 1(q) − V 2(q)) = sgn(V 1(a) − V 2(a)).

Proof. We shall prove the last statement first. Forq ∈ [0, a] we have V 1(q) =e−ρT (q)V 1(a) andV 2(q) = e−ρT (q)V 2(a) whereT (q) is given by e−λwT (q)(1 − q) =1− a. ThusV 1(q)/V 2(q) = V 1(a)/V 2(a).

If σ 1/2 thenV 1 > V 2 by Lemma 3.7. We assumeσ > 1/2. Letq ∈ [σ,1], then wehaveV 2(q) = (w−c)q/ρ andV 1

q (q) = (−ρV 1(q)+ (p−c)q)/λ(p−w)q . It follows that

if V 2(q) = V 1(q), then(w − c)/ρ = V 2q (q) > V 1

q (q) = 1/λ. We can now conclude that

V 1(x) = V 2(x) for anyx ∈ [σ,1] thenV 1(q) > V 2(q) for q ∈ [σ,x) andV 1(q) < V 2(q)

for q ∈ (x,1].Now, letq ∈ [a,σ ] and assumeV 2(q) V 1(q). Then,

V 2q (q) = ρV 2(q)+ cq

f (0, q) ρV 1(q)+ cq

f (0, q).

SinceV 1q (q) = (ρV 1(q)− pq + cq)/f (p, q), we haveV 2

q (q) V 1q (q) if

ρV 1(q)+ cq

f (0, q) ρV 1(q)− pq + cq

f (p, q),

which reduces toV 1(q) H 1(q). Furthermore, equality can hold only ifV 2(q) =V 1(q) = H 1(q), which impliesq = a. The Lemma now follows from Lemma 3.2.

We are now ready to prove the main result.

Proof of Theorem 2.4. Let V = max(V 1,V 2). In the caseV = V 1 or V = V 2 we seethatV is sufficiently regular1 to proceed along classical lines. BothV 1 andV 2 are easilyverified to beC1. Lemmas 3.2 to 3.7 show that ifV = V 1 orV = V 2 thenV satisfies (HJB)everywhere on[0,1]. Classical “verification theorems” (Fleming and Soner, 1993) caapplied proving thatV = V opt. Since the policyφopt achievesV , it is optimal. The classicaverification theorems can be extended to prove optimality ofV even whenV = V 1 andV = V 2. However, we can also quote the viscosity theory: sinceV is piecewiseC1 (byLemma 3.9),V is the viscosity solution to (HJB) and it is therefore equal toV opt.

4. Analysis and simulation of the γ -AR model

In this section we collect some ideas concerning the analysis of theγ -AR model. Sincewe do not have a closed-form solution, we shall be somewhat informal. By BlackwTheorem we can restrict our attention to stationary policiesP = φ(Q, P ).

1 V 1 andV 2 are in fact bothC2 but we require onlyC1.


i–

e

e the

. In

ss

ctions8).

r

ed

set,

AssumingVoptγ = V

optγ (Q0, P0) is C1 at (Q0, P0), it satisfies the Hamilton–Jacob

Bellman equation there,

ρV (q, p) = supp∈[0,p]

V1f (p, q)+ V2γ (p − p) + q(p − c)

,

where Vi denotes partial differentiation with respect to theith argument. Since thexpression inside the brackets is linear inp, we see that we can assumep = φ(Q0, P0) ∈0, p without loss of generality. However, in so doing we may be forced to generaliz

notion of the solution to ˙P = γ (P − P ) to admit weak solutions: if(Q(t), P (t)) tracks

a boundary betweenφ = p andφ = 0 thenP(t) ‘oscillates’ between 0 andp. In thiscase it is possible to redefineφ along the boundary so that a classical solution admitsfact, if for some optimalP(t) we haveP(t) ∈ (0, p), Q(t) = 0 for t ∈ (a, b) then we canapply the calculus of variations to

V(Q(t), Q(t), Q(t)

) =∞∫

0

e−ρt(P(Q,Q, Q) − c

)Q(t)dt

to obtain a differential equation forQ on (a, b). (This is possible since we can expreboth P andP as functions ofQ,Q, andQ.) This will in turn determineP(t) on (a, b).Furthermore, such a ‘tracking boundary’ betweenφ = 0 and φ = p determines asolution to this differential equation. Simulations indicate that the optimal value funis piecewiseC1 in general. Corresponding to the pointx in the∞-AR solution there arisea 1-dimensional curve along which the gradient ofV

optγ appears discontinuous (see Fig.

Our simulations results were computed as follows. To each discrete pointz in the(Q, P )

space we assign a valuev(z) and an price,p(z). We letQ andP evolve according to theidifferential equations with initial conditionz andP = p(z) over a short intervalt ∈ [0, ε]so that(Q(ε), P (ε)) is still close toz. We then evaluate,

ε∫0

eρt(p(z) − c

)Q(t)dt + v

(Q(ε), P (ε)

),

where v(Q(ε), P (ε)) is computed by locally interpolatingv. For each pointz wecompute the above for all possible choices ofp(z) and then assign tov(z) the maximumattained among the choices forp(z). Initializing v = 0 and iterating the above describcomputation over allz we obtain an increasing sequence ofv which converge to anapproximation toV opt

γ .

In view of the results above we restrictp(z) ∈ 0, p except on the setP = w wherewe admitp(z) = w. The latter is admitted since we expect stationary points along thisin view of our convergence results. (We experimented with allowingp(z) ∈ 0,w, p ingeneral, and the solutions presented here were only slightly modified withp(z) = w in aneighborhood of the tracking boundary, as we would expect.)


ote

2b inllatory

Table 1Parameter values for simulation

Parameter set γ

λ ρ c w p a b c

1 0.5 0.1 0.0 0.6 2.0 1.0 2.0 5.02 0.5 0.1 0.0 0.5 2.0 2.0 5.0 3

(a)

(b)

Fig. 1. Value functions for parameter sets 1 (a) and 2 (b).

Table 1 lists various sets of parameter values which we have used for simulation.Figure 1 shows bothV 1 andV 2 for the∞-AR problem parameter sets 1 and 2. N

that for parameter set 1V 2 >V 1 while for parameter set 2, the two functions cross.We have plotted optimal vector fields for parameter sets 1a, 1b, 1c, 2a, and

Figs. 2–6. The optimal price is overlaid at each point in the discrete space. Osci


Fig. 2. Optimal vector field for parameter set 1a.

Fig. 3. Optimal vector field for parameter set 1b.


Fig. 4. Optimal vector field for parameter set 1c.

Fig. 5. Optimal vector field for parameter set 2a.


of thepears

c, the

1to

sions.ere atws of

Fig. 6. Optimal vector field for parameter set 2b.

limit cycles can be seen in Figs. 5 and 6, corresponding to the oscillating solution∞-AR problem arising in parameter set 2. Surprisingly, such a limit cycle also apin Fig. 2, corresponding to parameter set 1a. This indicates that small values ofγ favoroscillating limit behavior. In Figs. 3 and 4, corresponding to parameter sets 1b and 1oscillating limit is absent. This is to be expected from our approximation results.

The limit cycle in Fig. 5 (parameter set 2a) has larger amplitude inq than that of Fig. 6(parameter set 2b), as we would expect from our approximation results. In the limitγ → ∞this amplitude goes to 0.

All of the vector fields have some stable stationary set along the lineP = w. This isalso to be expected from our approximation results.

Figs. 7 and 8 plot the value function for two extreme values ofγ for parameter setsand 2. Here we see the smoothness of the value function and also the convergenceV

opt∞ .

Acknowledgments

The authors thank Hsueh-Ling Huynh and Peter Linhart for many viscous discusMuch of the research on which this paper is based was done while the authors wAT&T Bell Laboratories. The views expressed here do not necessarily reflect the vieAT&T.


1b.

(a)

(b)

Fig. 7. Optimal value functions for parameter set 1: dependence onγ . (a) Parameter set 1a. (b) Parameter set


2b.

. Ind.

ential

g.wski.

(a)

(b)

Fig. 8. Optimal value functions for parameter set 2: dependence onγ . (a) Parameter set 2a. (b) Parameter set

References

Arrow, K.J., Nerlove, M., 1958. A note on expectations and stability. Econometrica 26, 297–305.Chen, Y., Rosenthal, R.W., 1996. Dynamic duopoly with slowly changing customer loyalties. Int. J

Organ. 14, 269–296.Crandall, M.G., Ishii, H., Lions, P.L., 1992. User’s guide to viscosity solutions of second order partial differ

equations. Bull. Amer. Math. Soc. 27, 1–67.Fleming, W.H., Soner, H.M., 1993. Controlled Markov Processes and Viscosity Solutions. Springer-VerlaGamkrelidze, R.V., 1978. Principles of Optimal Control Theory. Plenum, New York. Translated by U. Mako


amic

cosity

6.

Lions, P.L., 1985. Optimal Control and Viscosity Solutions. Recent Mathematical Methods in DynProgramming. Springer-Verlag.

Lions, P.L., Sougandis, P.E., 1985. Differential games, optimal control and directional derivatives of vissolutions of Bellman’s and Isaacs’ equations. SIAM J. Control Optim. 23, 566–583.

Radner, R., 2003. Viscous demand. J. Econ. Theory. In press.Rosenthal, R.W., 1982. A dynamic model of duopoly with customer loyalties. J. Econ. Theory 27, 69–76.Rosenthal, R.W., 1986. Dynamic duopoly with incomplete customer loyalties. Int. Econ. Rev. 27, 399–40

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