Stochastic dynamic pricing and advertising in isoelastic oligopoly models

Accepted Manuscript

Stochastic Dynamic Pricing and Advertising in Isoelastic Oligopoly Models Rainer Schlosser PII: DOI: Reference:

S0377-2217(16)30946-8 10.1016/j.ejor.2016.11.021 EOR 14100

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European Journal of Operational Research

Received date: Revised date: Accepted date:

3 December 2015 7 November 2016 9 November 2016

Please cite this article as: Rainer Schlosser, Stochastic Dynamic Pricing and Advertising in Isoelastic Oligopoly Models, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.11.021

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Highlights • We consider dynamic pricing and advertising differential games with stochastic demand. • We study finite as well as infinite horizon settings. • We derive uniquely determined stochastic feedback Nash equilibria in closed form.

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• We compare solutions of the stochastic model with its deterministic counterpart.

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• We find that the optimal expected profits can exceed the optimal profits of the deterministic model.

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Stochastic Dynamic Pricing and Advertising in Isoelastic Oligopoly Models Rainer Schlosser

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Hasso Plattner Institute, University of Potsdam, Germany

Abstract

In this paper, we analyze stochastic dynamic pricing and advertising differential games in special oligopoly markets with constant price and advertising elasticity. We consider the sale of perishable as well as durable goods and include adoption effects in the demand. Based on a unique stochastic feedback Nash equilibrium, we derive closed-form solution formulas of the value functions and the optimal feedback policies of all competing firms. Efficient simulation techniques are used to evaluate optimally controlled sales processes

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over time. This way, the evolution of optimal controls as well as the firms’ profit distributions are analyzed. Moreover, we are able to compare feedback solutions of the stochastic model with its deterministic counterpart. We show that the market power of the competing firms is exactly the same as in the deterministic version of the model. Further, we discover two fundamental effects that determine the relation between both models. First, the volatility in demand results in a decline of expected profits compared to the deterministic model. Second, we find that saturation effects in demand have an opposite character. We show that the second effect can be strong enough to either exactly balance or even overcompensate the first one. As a

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result we are able to identify cases in which feedback solutions of the deterministic model provide useful approximations of solutions of the stochastic model.

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Keywords:

1. Introduction

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pricing, advertising, stochastic differential games, oligopoly competition, adoption effects

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Markets have become more transparent, competitive, and agile, especially in ecommerce (e.g., Amazon, eBay). Firms have to deal with uncertain demand and oligopoly competition. Revenue management optimization models become increasingly important. Since firms typically have to use advertising channels to promote their products to attract more customers, setting prices and determining the level of advertising

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expenditures are two key marketing mix variables. Due to the digital transformation it is getting easier to adjust prices and advertising rates more frequently. In many online markets both controls can be adjusted in an automated way. While prices are adjusted according to dynamic pricing strategies, firms can also smoothly balance display advertising, e.g., using online banners etc. Applications can be found in a variety of contexts that involve perishable as well as durable goods. Examples of the sale of durable goods are, e.g., the depletion of exhaustible resources (oil, minerals), cf. Hotelling (1931), Stiglitz (1976). Further examples are technical devices or licenses. The goal of these Email address: [email protected] (Rainer Schlosser) Preprint submitted to European Journal of Operational Research

November 12, 2016

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”infinite horizon” use-cases is to coordinate dynamic pricing and advertising in a way to optimally liquidate inventory in a reasonable amount of time (taking discounting into account). Business settings with perishable goods are, e.g., the sale of event tickets (sports, arts), cf. Shapiro, Drayer (2012), Diehl et al. (2015), Laamanen (2013), or fashion goods, cf. Caro, Gallien (2012). Other use-cases can be found in transportation (airline tickets) or accommodation services (hotels). In ”finite horizon” problems the goal is to coordinate dynamic advertising and pricing decisions such that items are sold in an optimal way before the end of the time horizon.

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In various applications, the demand is time-dependent. For instance, the demand for fashion goods is typically decreasing, and airline or hotel reservation prices usually increase with time. However, most dynamic sales models only consider time homogeneous demand. Furthermore, the facts that in real-life demand is uncertain and the distribution of sales and profits play a prominent role motivate the study of dynamic sales models with stochastic demand.

In this paper, we study pricing and advertising oligopoly adoption models in a stochastic dynamic framework. We take perishable as well as durable goods into account. Moreover, we will consider different

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adoption effects, which are allowed to depend on the untapped market size. Our aim is (i) to derive optimal closed form policies, (ii) to evaluate optimally controlled sales processes over time, and (iii) to reveal differences as well as similarities of stochastic and deterministic versions of our model. The best way to sell products is a classical application of revenue management theory. The problem is closely related to the field of dynamic pricing, which is summarized in the books by Talluri, van Ryzin (2004), Phillips (2005), and Sethi, Thompson (2000).

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The survey by Chen, Chen (2015) provides an excellent overview of recent oligopoly pure pricing models. In the article by Gallego, Wang (2014) the authors consider a continuous time multi-product oligopoly for perishable goods. They also use optimality conditions to reduce the multi-dimensional dynamic pure pricing

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problem to a one dimensional one. The article by Gallego, Hu (2014) analyzes structural properties of more general oligopoly models for the sale of perishable products. In the stochastic models mentioned above no advertising effects and no adoption effects are included in the demand.

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Marketing literature mostly considers the sale of durable goods. A recent overview is given by Huang et al. (2012). Several deterministic pure advertising oligopoly models are analyzed, for instance, by Prasad, Sethi (2003) or Erickson (2009, 2011).

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Joint dynamic pricing and advertising models are less well studied. Classical oligopolistic pricing and advertising models are discussed in Teng, Tompson (1984), Dockner, Feichtinger (1986), Chintagunta et al. (1993), and Joergensen, Zaccour (1999). These models typically assume an infinite time horizon and

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time homogenous demand. Other more recent pricing and advertising models focus on, e.g., cooperative manufacturer-retailer models (Xie, Wei 2009, SeyedEsfahania et al. 2011), web content provider (Kumar, Sethi 2009), brand competition (Karraya, Martin-Herran 2009), generic advertising campaigns (Roma, Perrone 2010), sponsored search marketing (Ye et al. 2014), personalized marketing (Anderson et al. 2015) and time-varying demand (Feng et al. 2015). For the case of isoelastic demand and a monopolistic setting optimal pricing and advertising strategies are derived by Sethi et al. (2008) and Helmes et al. (2013). A duopoly version of the Sethi et al. (2008) model is discussed in Krishnamoorthy et al. (2010). Helmes, Schlosser (2015) extends the latter model by including more general adoption effects as well as an oligopolistic setting. Note, all these papers are 3

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characterized by deterministic demand. Stochastic dynamic pricing and advertising models are analyzed by MacDonald, Rasmussen (2010), Helmes, Schlosser (2013) and Schlosser (2015). However, these models consider the situation of a monopoly. The literature on stochastic dynamic pricing and advertising oligopoly models is limited. Since oligopoly models are characterized by equilibrium strategies, they are much more complex than monopoly models. To our knowledge, there are no publications that provide explicit solutions of stochastic dynamic joint pricing and advertising adoption models in a competitive setting. Our aim is to close that gap.

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This paper builds on the deterministic dynamic pricing and advertising oligopoly model by Helmes, Schlosser (2015), which analyzes an infinite horizon model in case of isoelastic demand. The main contribution of this paper is threefold. We extend the model by Helmes, Schlosser (2015) in the following three main points: (i) a stochastic setting, (ii) a finite horizon setting, and (iii) time-dependent model parameters. The assumption of constant demand elasticities is a common one. Isoelastic demand is widely accepted since it is used by economists, cf. Seaman (2006), McAfee, te Velde (2008), Chen, Roma (2011), and allows for theoretical solutions (see, e.g., papers mentioned above). It provides a good balance/mixture of closeness

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to reality and mathematical tractability.

This paper is organized as follows. In Section 2, we describe the stochastic dynamic oligopoly model for the case of isoelastic demand. The state space contains the market size still to be captured. We allow several parameters to be time-dependent and include state-dependent adoption effects. Optimality conditions for optimal pricing and advertising strategies are derived from the system of Bellman equations for all firms. In Section 3, we consider durable goods. We assume time homogeneous demand and an infinite time

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horizon. Firm-specific production costs and price elasticities as well as specific adoption effects are included. In case of isoelastic demand, we solve the system of coupled Bellman equations. We derive the value functions of the different firms as well as the associated optimal feedback strategies. We identify a specific condition

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that guarantees the existence and uniqueness of a feedback Nash equilibrium. Moreover, we describe how different sales processes can be evaluated over time and studied in detail. In Section 4, we study the sale of perishable products in a time-dependent finite time horizon framework.

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We consider generalized adoption effects and assume that the price elasticities of the competing firms are the same. For this model, we derive explicit solution formulas of the optimal competitive feedback controls. The solution of the value function is separable and characterized by the market saturation, a function of time,

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and specific market power coefficients for each firm. To obtain economic insight into the complex interplay of time, market level, and model parameters we show how to evaluate optimal strategies of the oligopolistic model: efficient simulation techniques in continuous time as well as the computation of state probabilities

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are proposed. The closed form solution of the model also allows for various comparative statics. In Section 5, we analyze the relation of the solution of the stochastic model and its deterministic counterpart. While the market power in both models is the same it turns out that the impact of the size of the untapped market is differs.We find two fundamental effects: (i) the uncertainty in demand results in lower expected profits compared to the deterministic model; (ii) we find that saturation effects in demand have an opposite character. We show that the second effect can be strong enough to either exactly balance or even overcompensate the first one. As a result we are able to identify cases in which the deterministic feedback solution of the model provides a useful approximation of the solution of the stochastic model. Conclusions and managerial recommendations are offered in the final Section 6. 4

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2. Model Description In this section, we will describe the stochastic dynamic pricing and advertising oligopoly adoption model. The time horizon is denoted by T and can be finite or infinite. We assume that J firms are selling one type of product. For this product, there is a given market of size N , N < ∞, which is to capture. For each firm

j, 1 ≤ j ≤ J, we assume nonnegative unit costs cj which occur if firm j sells an article. The discount rate Rt rj (t) may be time-dependent as well as firm-specific; and we let Rj (t) := 0 rj (s)ds, 1 ≤ j ≤ J.

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Furthermore, we consider the stochastic control problem for the sale of units of different size. Using unit

shares of size h we assume that N is a multiple of h. If firm j sells a h share of the product at time t unit costs (e.g., production costs) h · cj , 0 ≤ t ≤ T , have to be paid. All quantities as price pj or production costs cj will still be given in units of size 1. Thus, assuming the positive revenue/tax parameter vj (t) the price of a share of size h is pj · h and a sale of a share h leads to a net revenue of vj (t) · pj · h − cj · h.

For each firm j we assume a positive time-dependent arrival intensity uj (t), 1 ≤ j ≤ J, 0 ≤ t ≤ T .

Differences in the arrival intensity of firms can, e.g., be ascribed to their different locations, cf. Ashtiani

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(2016). Firm j can use advertising to attract more customers. The impact factor of firm js advertising rate wj (t) is concave increasing and described by the power function wj δ , δ ≥ 0. The advertising elasticity δ is the same for all firms. The advertising expenditures are given by the cost rate kj (t) · wj a , where a > δ

and kj (t) is a given positive parameter function. For firm j the price elasticity of demand is assumed to be constant in time and equal to εj > 1, 1 ≤ j ≤ J. A list of variables and parameters can be found at the end of the Appendix, see Table A.2.

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Motivated by market saturation and diffusion effects, we will take adoption effects into account. Letting P the number xj represent the amount sold by firm j; the corresponding number of items left is y = N − j xj .

We assume that the adoption effect is the same for all firms and depends on the market size y still to be

captured. I.e. the distribution of the individual sales figures is not specifically included in the model. We

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define the rate of sales for firm j by the jump intensity, 0 ≤ t ≤ T , 1 ≤ j ≤ J, pj > 0, wj > 0, −εj

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δ λ(j) y (t, pj , wj ) := uj (t) · wj · χy · pj

,

(j)

which on average leads to the sale of λy /h unit shares of size h per unit of time. When h is smaller than the reference size 1 we have a higher number of jumps that are of smaller size. For every time t and (j)

(j)

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market situation y each firm j has to decide on a price pj and an advertising rate wj . We call strategies (j)

(pt , wt )t admissible if they belong to the class of Markovian feedback policies; i.e. pricing pt (j)

advertising wt

≥ 0 decisions may only depend on time t and state y.

> 0 and

(j)

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The random number of articles, which are sold by firm j up to time t are denoted by Xt . I.e. at P (j) time t the market size still to capture is given by Yt = N − j Xt . The state space is given by S (h) :=

{0, h, 2h, ..., N − h, N }. The dynamic of the controlled sales process is characterized as follows. For given

pricing and advertising policies of the different firms the sale of a single item corresponds to the earliest (j)

jump of J different non-homogeneous Poisson processes with rate λYt (t, pj , wj ), h ≤ Yt ≤ N , 0 ≤ t < T .

The end of sale τ is the first point in time when all items are sold, i.e., τ := min {t : Yt = 0} ∧ T ; for all 0≤t≤T

t ≥ τ we let λ := 0. Each firm wants to determine an admissible strategy that maximizes the expected profits during the sales period starting at time 0 and ending at time T (resp. τ ). Thus, each firm j maximizes,

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j = 1, ..., J,     τ ∧T  (j)  (j) (j) Z   λ t, p (t), w (t) Y Y Y (j) t t t (j) (j) a · vj (t) · pYt (t) · h − cj h − kj (t) · wYt (t)  dt |Y0 = N  . (1) E e−R (t) ·  h 0

In contrast to monopolistic models there are several firms that are competing for the same market. Hence, if an article is sold by firm j the expected future profits of all firms are affected. The individual value (j)

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functions Vy (t) of the J firms conditioned on state y (respectively the market size Yt = y) at time t are mutual dependent; they describe the best present value of expected future profits for equilibrium strategies. A good theoretical basis of differential games is given by the book by Joergensen, Zaccour (2004). To solve the problem, cf. (1), i.e., to identify a feedback Nash equilibrium, we want to determine the value functions (j)

Vy (t) which are characterized by the following system of associated Bellman equations, cf. Br´emaud (1980), 0 ≤ t < T , h ≤ y ≤ N , j = 1, ..., J,

sup pj >0,wj ≥0

n   o (j) a λ(j) y (t, pj , wj ) · vj (t) · pj − cj − ∆Vy (t)/h − kj (t) · wj

−∆Vy(j) (t)/h · (j)

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rj (t) · Vy(j) (t) = V˙ y(j) (t) +

X

∗(m)

λ(m) y (t, pt

∗(m)

, wt

(2)

)

m6=j

with the natural boundary conditions V0 (t) = 0 for all 0 ≤ t ≤ T . In case of a finite time horizon, i.e., (j)

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T < ∞, for all 0 ≤ y ≤ N < ∞, the additional boundary conditions Vy (T ) = 0, 1 ≤ j ≤ J, have to be (j)

(j)

(j)

satisfied. Firm js opportunity costs are given by ∆Vy (t) := Vy (t) − 1{y>0} · Vy−h (t), 1 ≤ j ≤ J. The

necessary optimality conditions, cf. (2.2), for the optimal feedback prices (per unit of size 1) and advertising

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rates of firm j are, 1 ≤ j ≤ J, 0 ≤ t < T , h ≤ y ≤ N ,

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py∗(j) (t) =

and

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wy∗(j) (t)

=



  εj · cj + ∆Vy(j) (t)/h vj (t) · (εj − 1)

vj (t) · uj (t) · δ −ε +1 · χy · p∗(j) (t) j y kj (t) · εj · a

1  a−δ

(3)

.

(4)

For each firm the optimal advertising rates are a function of the optimal prices, which itself are a function of the corresponding opportunity costs. Note, a feasible solution of (2) with positive feedback controls implies (j)

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cj + ∆Vy (t) > 0, 1 ≤ j ≤ J, cf. (3). The derivation of (3) and (4) as well as the second order conditions

are given in the Appendix. Furthermore, the optimality conditions imply the Dorfman-Steiner identity on the grid y ∈ S (h) \{0} ∗(j)

kj (t) · wy

vj (t) · ∗(j)

where λy

(j)

(t) := λy



∗(j)

t, py

∗(j)

(t), wy

∗(j) py (t)

·

a

(t)

∗(j) λy (t)

≡

δ , a · εj

(5)

 (t) , 1 ≤ j ≤ J. Plugging the optimality conditions (3) and (4)

into the system of Bellman equations (2) we obtain the following nonlinear system of difference-differential equations, 0 ≤ t < T , h ≤ y ≤ N , 1 ≤ j ≤ J, 6

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=

V˙ y(j) (t)

(j)

ηj (t) + · γj

∆Vy (t) cj + h

!−γj +1

(j) (m) ∆Vy (t) X γm − 1 ∆Vy − · · ηm (t)· cm + h γm h m6=j

where γj := (aεj − δ)/(a − δ), 1 ≤ j ≤ J, and a−δ ηj (t) := · δ

ε

δ · uj (t) vj (t) j · a/δ a · εj kj (t)

!−γm

a

· χya−δ

(6)

a

· χya−δ ,

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rj ·

Vy(j) (t)

a ! a−δ  γ −1 εj − 1 j · . εj

The derivation of (6) is given in the Appendix. To determine equilibrium strategies for different versions of our initial oligopoly problem, cf. (1), we want to construct a solution of the system of value functions, which are characterized by (6) and its natural boundary conditions. In the next two sections we will consider

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two special cases of the model described in this section. While in Section 3, we will consider durable goods, in Section 4 we will analyze the sale of perishable products.

3. Oligopoly Models with Infinite Horizon

In this section we will solve the stochastic dynamic oligopoly problem described in Section 2 with infinite

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time horizon, i.e., T = ∞. The consideration of durable goods is typical in the marketing literature. We assume that N units of a product can be sold in the long run. In new product diffusion models the adoption

effect plays a prominent role, cf. for instance the classical deterministic models by Bass (1969) or Sethi

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(1973).

In this section we assume the special adoption function χy = y (a−δ)/a , which expresses a saturation effect in the demand, cf. the Sethi model. Moreover, all model parameter are assumed to be time homogeneous,

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i.e., we will use the positive constants cj , uj , vj , kj and rj , 1 ≤ j ≤ J. The firm specific unit costs cj may be used to mirror production or extraction costs as well as tax or shipping costs. As a special case of (1)

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each firm now considers the maximization of  τ  Z       (j) (j) (j) (j) (j) (j) a E  e−r ·t · λYt pYt (t), wYt (t) /h · vj · pYt (t) · h − cj · h − kj · wYt (t) dt |Y0 = N  .

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0

In time homogeneous models with durable goods the value functions Vy (j) (t) ≡ Vy (j) are independent

of time and the associated system of Bellman equations, cf. (2), as well as the optimality conditions (3) -

(4) for the optimal controls simplify. For each firm j there is only one boundary condition V0 (j) = 0. The system (6) turns into the system of difference equations, cj ≥ 0, εj > 1, 1 ≤ j ≤ J,

rj · Vy(j)



ηj = · γj

(j)

∆Vy cj + h

!−γj +1

(j)

∆Vy − h

(m) X γm − 1 ∆Vy · · η m · cm + γm h m6=j

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!−γm 

a

 · χya−δ ,

(8)

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where the parameter ηj (t) = ηj and γj = (aεj − δ)/(a − δ), 1 ≤ j ≤ J, are defined as in Section 2.

The model described above is a stochastic version of the deterministic oligopoly problem solved in

Helmes, Schlosser (2015). The Nash equilibrium of their deterministic model is based on specific market power coefficients, which as we will see, also appear in the solution of our stochastic oligopoly model. Motivated by Helmes, Schlosser (2015), Section 4, we try a separable ansatz for the value functions;

Vy(j) := αj · y. (j)

Thus, the associated opportunity costs are ∆Vy a/(a−δ)

adoption function we have χy

(j)

:= Vy

(j)

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we consider individual market power coefficients αj ≥ 0 and a linear dependence of the market saturation, 1 ≤ j ≤ J, 0 ≤ y ≤ N , 0 ≤ t ≤ T , (9)

− Vy−h = αj · h. Further, due to the specific

= y and the system (8) turns into, 1 ≤ j ≤ J,

 −γm  X γ − 1 η m j −γ +1  · y. · ηm · cm + α(j) rj · αj · y =  · (cj + αj ) j −αj · γj γm

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

m6=j

Hence, the state dependence vanishes and the system above is equivalent to, j = 1, ..., J, rj =

X γm − 1 ηj −γ +1 −γ · (cj + αj ) j /αj − · ηm ·(cm + αj ) m . γj γm

(10)

m6=j

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Following the results of Helmes, Schlosser (2015), Lemma 2.1, the system of equations (10) uniquely determines the αj coefficients and there is a positive solution if and only if the condition

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1>

X

j=1,..,J:cj =0

(1 − 1/γj )

(11)

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is satisfied. The next theorem summarizes the solution of the infinite horizon oligopoly problem. Theorem 3.1. Let T = ∞ and χy = y (a−δ)/a . Let (αj )1≤j≤J be the unique solution of (10). If condition

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(11) is satisfied then problem (7) has the following unique solution, 1 ≤ j ≤ J, 0 ≤ y ≤ N :

Let T < ∞, N < ∞, and h > 0. If the condition 1 > J · (1 − 1/γ) is satisfied then problem (7) has the

following solution, 0 ≤ y ≤ N , 1 ≤ j ≤ J, 0 ≤ t < T : (j)

Vy

optimal prices

(j) py

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expected profits

(j)

= αj · y,

=

εj vj ·(εj −1)



· (αj + cj ), 1/(a−δ) (j) −εj +1 · py · y 1/a ,

vj ·uj ·δ kj ·εj ·a (j) (j) (j) λy (py , wy )

optimal advertising rate

wy =

optimal sales rates

λy := = zj · y, P (j) λy := λy = Z · y, where

total sales rate

(j)

j

Proof. See Appendix and derivation above.

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where zj := P Z := zj . j

γj −1 γj

· ηj · (cj + αj )−γj ,

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Remark 3.1. The market power of the different firms is independent of the market saturation und the adoption effect. Comparing the results of the stochastic and the deterministic version of the model, cf. (7), we observe that the corresponding value functions are surprisingly the same. Note, the market power coefficients as well as the state-dependent market factor are identical. I.e. in expectation the firms profits are equal to the optimal profits of the deterministic model. In the following, we want to analyze the stochastic component of the model in greater detail. First,

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we will compute the probability distribution of the realized market shares of the individual firms, i.e., the probability that at time t firm j has already sold xj items. The explicit formulas for the probabilities P [Xj (t) = xj ], 1 ≤ j ≤ J, are given in the following lemma. The result is based on the state probabilities P [Yt = y] of the untapped market is of size y.

(j)

Lemma 3.1. Let the rate of sales of firm j of the form λy

= zj · y, 1 ≤ j ≤ J, and let Z :=

P

j

zj .

Then the state probabilities that y units, y = h, 2h, ..., N , are not yet sold by time t, 0 ≤ t < ∞, i.e.,

q(t, y) =

N/h y/h

!

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q(t, y) = P [Yt = y], are given by a Binomial distribution B(N, D(t)), where D(t) := e−t·Z : · D(t)y/h · (1 − D(t))

(N −y)/h

.

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The probabilities of realized sales P [Xj (t) = xj ] of each firm j are given similarly by the Binomial
P distribution B(N, Dxj (t)), where Dxj (t) := (1 − D(t)) · zj / i zi = 1 − e−t·Z · zj /Z.

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Furthermore, we have E [Y (t)] = N · e−t·Z and, for each firm j, 1 ≤ j ≤ J,

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Proof. See Appendix.


E [Xj (t)] = (N − E [Y (t)]) · zj /Z = N · 1 − e−t·Z · zj /Z.

Note, the expected evolution of market shares exactly coincides with the corresponding optimal evolution in the deterministic model, cf. Helmes, Schlosser (2015), Sec. 4. However, as we will see below the differences

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of single realizations of sales trajectories can be large. To evaluate random realizations of sales processes we will use simulations in continuous time.

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Proposition 3.1. Let h > 0 such that the initial value N is a multiple of h. Let (optimal or suboptimal) P (j) (j) sales rates of the firms j be given by λy , j = 1, ..., J, and let λy := j λy . The beginning of the period, the system is in state y, 0 ≤ y ≤ N , is denoted by τy .

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(i) Let (Uy )h≤y≤N be independent uniformly distributed random variables on the unit interval [0, 1]. Let

τN := 0 and recursively define (τy )h≤y≤N as follows, y = h, 2h, ..., N , τy−h := τy − ln (Uy ) /λy · h.

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(ii) Let πj denote the probability that a given sale will be realized by firm j. Let, for any y = h, 2h, ..., N , (j) λy

denote the jump rate of firm j in state y. Then, 1 ≤ j ≤ J, πy(j) := λ(j) y /

X

i=1,..,J

9

λy(i) .

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Note, in case of optimal policies, cf. Theorem 3.1, we particularly obtain πj ≡ zj /Z. (iii) Based on simulated times of sale (τy )y and firm assignments we have the accumulated net revenues (t) and the advertising expenditures W (j) (t) of firm j up to time t, 0 ≤ t ≤ T , U (j) (t) =

X

y=0,h,2h,...,N −h 0<τy ≤t, τy of typej

  (j) e−rj ·τy · h · vj · py+h − cj

and

W

(j)

(t) =

X

y=h,...,N τy ≤t

min(τ Z y−h ,t) τy

a

e−rj ·s · kj · (wy(j) ) ds =

X

y=h,...,N τy ≤t

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U

(j)

(j) a

 kj · (wy )  −rj ·τy − e−rj ·min(τy−h ,t) . · e rj

The cumulated profits are given by G(j) (t) := U (j) (t) − W (j) (t), 1 ≤ j ≤ J.

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Proof. See Appendix.

The simulation approach described in Proposition 3.1 is based on exponentially distributed waiting times between the sales. The (total) rate of sales consists of J time homogeneous state-dependent Poisson rates for the different firms. The formulas of the realized net revenues U (j) , and expenditures W (j) follow their natural definitions. The simple recursion (13) and the trivial simulation of the transition probabilities (14) make it possible to simulate a large number of sales trajectories of all firms and to study various aspects

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of the oligopolistic model in great detail. In this way the expected evolution of prices and sales as well as profits of all firms can be approximated. Particularly it can be analyzed how the distribution of the profits G(j) is affected by different model parameters. The following example illustrates this possibility. The market

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situation is an example with three competing firms with heterogeneous price elasticities / brand images. Example 3.1. Let χy = y (a−δ)/a , a=2, δ=1, N =100 and J=3. Let ~ε=(1.7, 1.8, 1.9), c=u=10, v=k=1,

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r=0.1 and consider two different sizes of h, i.e., h=1 and h=5. In this example the leading firm 1 with the best brand image, i.e., the smallest price elasticity, can expect

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the largest total profits and the highest market share. Firm 3, the one with the highest price elasticity faces the smallest expected profits and the smallest market share. Example 3.1 is a stochastic version of the deterministic Example 4.2 given in Helmes, Schlosser (2015). The expected profit of the three firms is the ~N =(79.46, 42.55, 23.03). The time-invariant prices are same as in the deterministic model and given by V

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p~=(26.22, 23.46, 21.60); the market expected market shares are ~z/Z=(48.24%, 31.65%, 20.11%). All these

quantities are also independent of the share size h. However, the advertising rates of the firms, the realized market shares and of course the distribution of

profits is affected by h. The step functions in Figure 1 represent simulated trajectories of advertising rates and accumulated sales of firm 2 for two different values of h, i.e., h=1 and h=5. The smooth line in Figure 1a and 1b indicates the expected evolution. On average optimal advertising rates are convex decreasing in time; the expected number of sales is concave increasing. Furthermore, we can observe the impact of the share size h. If h is smaller the trajectories in Figure 1 are less volatile, which has a major effect on the distribution of profits. 10

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X2 HtL

w2 HtL

35

2.0

30 25

1.5

20

1.0

15 5 20

40

60

80

100

120

140

t

20

40

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10

0.5

60

80

100

120

140

t

Figure 1: Simulated trajectories of advertising rates of firm 2 (left window Fig. 1a) and accumulated sales for firm 2 (right window Fig. 1b) in case of h=1 and h=5; Example 3.1.

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For different values of h, Table 1 illustrates the dependence of the standard deviation of (i) the optimal profits and (ii) the number of items sold. The numbers are based on 100,000 simulations, which have been evaluated until the end of sale τ . As expected the standard deviation of profits and the number of sales, respectively, decreases when h gets smaller. Note, due to the advertising expenditures profits can also be negative. In the cases when h is large the standard deviation of profits can even exceed its corresponding expected value. Hence, the randomness in the model can have significant effects on profits. (2)

σ(Gτ ) 3.43 7.62 16.86 23.99 53.51 75.91

(3)

σ(Gτ ) 2.34 5.29 11.78 16.69 37.17 52.79

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(1)

σ(Gτ ) 5.06 10.98 24.77 35.24 78.83 110.48

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h 0.02 0.1 0.5 1 5 10

(1)

σ(Xτ ) 0.68 1.57 3.50 5.03 11.12 15.76

(2)

σ(Xτ ) 0.66 1.47 3.32 4.67 10.37 14.73

(3)

σ(Xτ ) 0.57 1.28 2.83 4.03 8.95 12.67

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Table 1: Comparison of the standard deviation of profits of three firms for different share sizes h, Example 3.1 (heterogeneous price elasticities).

4. Oligopoly Models with Finite Horizon

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In this section, we consider a stochastic dynamic oligopoly problem for a perishable product with finite horizon T and general state-dependent adoption effects. As a central assumption we let cj = 0, 1 ≤ j ≤ J.

Note, in many applications there are no shipping costs and production costs have to be paid in advance

(fruits, flowers, fish, etc). Furthermore, we assume the price elasticities of all firms to be identical, i.e., εj = ε, 1 ≤ j ≤ J. If the product is standardized (groceries) then different firms (discounter) usually have

similar price elasticities such that this assumption can be justified as well. Moreover, we assume the discount rates rj (t) = r(t) to be the same for all firms, 1 ≤ j ≤ J, 0 ≤ t ≤ T < ∞. The arrival rates ~u, the revenue parameters ~v , and the advertising cost parameter ~k are assumed to contain a common time-dependent parameter and a firm-specific factor, i.e., we let uj (t) := u(t) · u ˜j , 11

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vj (t) := v(t) · v˜j and kj (t) := k(t) · k˜j . Under these assumptions, as a special case of (1), for each firm j we

consider the maximization of, 1 ≤ j ≤ J,

 τ ∧T  Z   (j) (j) (j) δ −ε+1 a E e−R(t) · u(t) · u ˜j · wYt (t) · χYt · v(t) · v˜j · pYt (t) − k(t) · k˜j · wYt (t) dt |Y0 = N  ,

(15)

0

(6) simplifies to

Rt 0

r(s)ds. Following Section 2 the associated system of difference-differential equations 

(j)

∆Vy (t) r(t) · Vy(j) (t) = V˙ y(j) (t) + ηj (t) · h

−γ+1

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where R(t) :=

(m) (j) ∆Vy (t) X ∆Vy − · (γ − 1) · ηm (t)· h h m6=j

−γ



a

 · χya−δ , (16)

where γj = γ and the parameter ηj (t), 1 ≤ j ≤ J, are defined as in Section 2. To solve the system (16)

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again we try a separable ansatz for the value functions. The approach is based on the separation of (i) an individual market power coefficient αj ≥ 0, 1 ≤ j ≤ J, (ii) a general time dependence f (t), 0 ≤ t ≤ T , and (iii) a state-dependent market factor βy , 0 ≤ y ≤ N , i.e.,

Vy(j) (t) := αj · f (t) · βy .

(17)

(j)

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Thus, the associated opportunity costs are ∆Vy (t) := αj · f (t) · (βy − βy−h ) and under the assumptions

above the system (16) turns into the system, 1 ≤ j ≤ J, h ≤ y ≤ N , 0 ≤ t ≤ T < ∞,

−γ+1 a αj · f (t) · (βy − βy−h ) · χya−δ h  −γ a αj · f (t) · (βy − βy−h ) X αm · f (t) · (βy − βy−h ) − · (γ − 1) · ηm (t)· · χya−δ . h h

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r(t) · αj · f (t) · βy = αj · f˙(t) · βy + ηj (t) ·



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m6=j

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Using simple algebra we obtain the equivalent system 

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r(t) · f (t) = f˙(t) + f (t)−γ+1 · ηj (t) · αj −γ −

X

m6=j

(γ − 1) · ηm (t) · αm

−γ 

Using the following identity for the state dependence, 0 ≤ y ≤ N , γ−1

((βy − βy−h ) /h)

a

· βy = χya−δ

12





βy − βy−h · h | 1

⇔

−γ+1 {z !

=1

a

· βy −1 · χya−δ . }

1

∆βy = h · χyε−1 · βyγ−1

(18)

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the system simplifies to r(t) · f (t) = f˙(t) + f (t)−γ+1 ·

u(t) · v(t)

ε

!a/(a−δ) 

· η˜j · αj −γ −

δ/a

k(t)

|

X

m6=j



(γ − 1) · η˜m · αm −γ , {z

(19)

}

!

= r˜

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 a/(a−δ) a/δ γ−1 where η˜j := (a − δ)/δ · δ · u ˜j · v˜jε /(a · ε · k˜j ) · (1 − 1/ε) , j = 1, ..., J, is independent of time. From Helmes, Schlosser (2015), Section 3, we already know, that as long as 1 > J · (1 − 1/γ) the special

market power system of equations, cf. (19),

η˜j · αj −γ −

X

m6=j

(γ − 1) · η˜m · αm −γ = r˜,

where r˜ > 0 is an arbitrary chosen positive constant, has a unique solution, which can be expressed explicitly αj (˜ ηj , r˜) = r˜−1/γ ·



 −1/γ J 1 · 1+ . η˜j · γ γ/(γ − 1) − J

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by

(20)

Using the solution (20), the system (19) reduces to the single differential equation ε

r(t) · f (t) = f˙(t) + f (t)

·

u(t) · v(t) k(t)

δ/a

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−γ+1

!a/(a−δ)

· r˜,

f (T ) = 0,

(21)

which finally determines the time dependence of the value functions of all J firms, cf. (17). The unique

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solution of the Bernoulli differential equation (21) is given by 

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f (t; r˜) = r˜1/γ · eγ·R(t) ·

ZT t

ε

u(s) · v(s)

e−γ·R(s) · γ ·

δ/a

k(s)

!a/(a−δ)

(j)

1/γ

ds

.

(22)

Note, from (20) and (22) it follows that Vy (t) = αj ·f (t)·βy does not depend on the choice of r˜. I.e. (18),

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(20) and (22) together uniquely determine the Nash equilibrium. Furthermore, in the time homogeneous case we obtain hom

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f (t; r˜) =



u · vε k δ/a

a/(a−δ)

1 − e−γ·r·(T −t) · r˜ · r

!1/γ

.

Letting r → 0 (the undiscounted case) or T → ∞ (infinite horizon) asymptotically we have 1/γ  1 − e−γ·r·(T −t) r˜ · r

r→0

→

T →∞

→

γ · (T − t)

1/γ

(˜ r/r)

(j)

1/γ

· r˜1/γ

r˜=1

→

r˜=r

→

γ · (T − t) 1.

1/γ

(23)

Formula (23) shows that the solution for Vy (t) can also be extended to the infinite horizon problem.

13

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Remark 4.1. The time dependence of the value functions is independent of the competition, the market saturation und the adoption effect. Letting T → ∞ and r˜ = r the time-dependent component of the value function converges to 1 and we obtain the infinite horizon solution derived in Helmes, Schlosser (2015), Sec. (j)

3. Moreover, in the undiscounted case (r = 0, T < ∞) using r˜ = 1 the time component of Vy (t) amounts to γ · (T − t)1/γ , which is a concave decreasing function in time t.

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In the next theorem we summarize the solution of the finite horizon oligopoly problem. Theorem 4.1. Let T < ∞, N < ∞, and h > 0. If the condition 1 > J · (1 − 1/γ) is satisfied then problem (15) has the following solution, 0 ≤ y ≤ N , 1 ≤ j ≤ J, 0 ≤ t < T : (j)

expected profits

Vy (t) := αj · f (t) · βy ,

optimal prices

py (t) =

(j)

total sales rate

(j)



f (t)·ε vj (t)·(ε−1)

−ε+1

(j)

vj (t)·uj (t)·δ kj (t)·ε·a

· ∆βy /h, 1/(a−δ)

· χy · py (t) ,   a/(a−δ) ε P (j) ·Z · λy (t) := λy (t) =f (t)−γ · u(t)·v(t) k(t)δ/a wy (t) =

j

optimal sales rates

= αj ·

(j)

g(t) ˙ g(t)

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optimal advertising rate

ε·∆Vy(j) (t)/h vj (t)·(ε−1)

h·βy ∆βy ,

g(t) := e−R(t) · f (t), by := h/˜ r · βy /∆βy , a   a−δ
γ−1 u ˜j ·˜ vjε δ · a·ε · ε−1 zj := η˜j · (γ − 1) · αj −γ , and η˜j := a−δ . δ · ε ˜ a/δ

λy (t) =

· zj · by ,

where

kj

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Proof. For the derivation of g(t), zj and by see Appendix.

(j)

In the following the representation of the (optimal) individual rates of sales λy , cf. Theorem 4.1, will be

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used to evaluate the optimal feedback policies over time. The following proposition shows how to efficiently

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simulate realizations of the finite horizon oligopoly model considered in this section. The continuous time simulation approach is based on realizations of the time points of sales, which are sold by a certain company. (j)

˙ · zj · by . Let Proposition 4.1. Let the (optimal) individual rates of sales be of the form λy (t) = −g(t)/g(t) τy denote the beginning of the period the system is in state y.

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(i) In any state y for the next total sale, i.e., considering all firms for all 0 ≤ s < t ≤ T we have P (τy−h ≤ t |τy = s) = 1 − (g(t)/g(s))

Z·by /h

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, Z·b /h and (g(τy−h )/g(τy )) y τy conditioned on τy is U (0,1) uniformly distributed.

(ii) Since g is decreasing in t for any given state y starting at time τy the simulation of time τy−h is

determined by a random variable U ∼ U (0,1) via the recursive relation g(τy−h ) = U h/(Z·by ) · g(τy ), hom

where

In the time-homogeneous case we have g(t) = u · v ε /k δ/a


1/(ε−δ/a)

1 the recursion simplifies to τy−h = − γ·r ln U γ/(Z·by /h) · e−γ·r·τy

14

τN := 0.



1/γ e−γ·r·t − e−γ·r·T · r˜/r , and

− e−γ·r·T + e−γ·r·T . ·

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(iii) If a product was sold, the probability that is was sold by firm j is given by, j = 1, ..., J, , τy−h X Z zj λ(m) . y (s)ds = Z m

τZy−h

λ(j) y (s)ds

τy

τy

(iv) Steps (ii)-(iii) can be repeated until all items are sold. All simulated/realized sales times τy are realized up to time t: U (j) (t) =

X

(j)

all 0<τy ≤t, τy saleoftype j

W

(j)

(t) =

X

allτy ≤t

e−R(τy ) · h · vj (τy ) · py+h (τy )

min(τ Z y−h ,t)

Proof. See Appendix.

e−R(s) · kj (s) · wy(j) (s)ds.

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and

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assigned to one of the J firms. On this basis, we can evaluate the revenues and expenditures of each firm

τy

The simulation technique described in Proposition 4.1 is based on the individual optimal sales rates for the J firms. Due to the separable structure for each fixed state the time dependence of all J rates is identical. Thus the point in time when the state changes, i.e., some product is sold can be easily simulated, cf. (ii). The selling firm can be simulated using ratios of the sales intensities, cf. (iii). The resulting revenues and

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expenditures for all firms can be evaluated by (iv).

Since f (T ) = g(T ) = 0 from Proposition 4.1 (i)-(ii) it follows that τ0 ≤ T , i.e., all products will be sold

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within the time horizon. Thus, a simulated sales process always consists of N/h points in time, where each

of these single recursively simulated sales times is assigned to a special firm. Hence, such a sequence can be used to evaluate the realized processes of the evolution of the market to capture Y and the accumulated ~ In this context, particularly the individual price trajectories of all firms can be studied. In a similar sales X.

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way realized revenues U and advertising expenditures W of each firm, cf. (iv), can be evaluated. The profit process for a firm j amounts to G(j) (t) := U (j) (t) − W (j) (t). Using many simulations expected evolutions

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of sales, prices as well as profit distributions can be approximated.

Example 4.1. Let χy = y/N + 4 · y/N · (1 − y/N ), N =100, J=3, ε = 1.2, a=2, δ=1, c=0, v=k=1,

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r=0.01, h=1, ~u = (10, 20, 30) and different horizons T =1, 5, 10.

For the case T =10 the left window of Figure 2 shows four simulated price trajectories of firm 2. The

smooth line illustrates the expected evolution of the prices E (p2 (t)). The window on the right, cf. Figure 2b, depicts the accumulated sales trajectories of firm 2 and the corresponding expectation E (X2 (t)). We observe that the deviations of individual (stochastic) brand evolutions from their expected evolution can be large. That implies that profit variations can be substantial. Figure 3a illustrates how the expected evolution of prices of all three firms, cf. Example 4.1, are affected by different time horizons, cf. T =1, 5, 10. Figure 3b shows the corresponding expected advertising rates. Firm 3 − the one with the highest arrival rate − sets the highest prices and advertising rates. While the 15

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p 2 HtL

X2 HtL

40

7

30

5

6 4

20

3 2

10 4

6

8

10

t

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2

1

t

2

4

6

8

10

Figure 2: Simulated trajectories of prices of firm 2 (left window Fig. 2a) and accumulated sales for firm 2 (right window Fig. 2b) in case of T =10; Example 4.1.

expected advertising rates are decreasing in time, the prices are of increasing-decreasing shape, caused by

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the adoption effect, which corresponds to a Bass model with innovation factor 1 and imitation factor 4. If the time horizon becomes shorter the initial advertising rates are; the rest of the time they are lower. The prices are smaller when T gets smaller and in turn, the market saturation is faster. The expected profits ~ (T =5) = (22.87, 61.57, 109.89), ~ (T =1) = (7.39, 19.90, 35.51), V of the competing firms decrease in T , cf. V N N (T =10) ~ = (36.62, 98.57, 175.91). Further studies show, that the size h hardly affects the evolution of V N

optimal controls.

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EH p j HtLL 12 10

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8 6 2

2

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4 4

6

8

EHw j HtLL 10 8 6 4 2

10

t

2

4

6

8

10

t

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Figure 3: Expected prices (left window Fig. 3a) and advertising rates (right window Fig. 3b) for all three firms for different time horizons T , T =1, 5, 10; Example 4.1.

Remark 4.2. The model allows the following managerial implications: (i) Optimal price paths are mainly determined by market power and adoption effects. Advertising paths,

however, are hardly affected by adoption effects. (ii) The model includes market power. There are often changes in the market due to trends, shocks,

or disruptive innovations (location (u), brand image (ε), technology (c), financing costs/discount rate (r), exit/entry of competitors (J)). The model allows to quantify associated changes in market power and hence provides insight how to react to these changes. (iii) The closed form solution of our model allows for various comparative statics. 16

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5. Asymptotic Properties and Relation to the Deterministic Model In this section, we want to study the relation between the solution of the stochastic differential game and the deterministic one. We will show that the value function of the deterministic model can be an upper and a lower bound for its stochastic counterpart; in this context, the adoption effect will play a prominent role. In this section, for simplicity, we assume the model of the previous section with infinite time horizon and time homogeneous demand. −εj

continuous state space be given by λ(j) (pj , wj , y) := uj · wj δ · pj

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Let the sales intensity of the deterministic version of the model, cf. Helmes, Schlosser (2015), with · ψ(y), j = 1, ..., J, where ψ(y) is a

continuous positive (adoption) function. The Nash-equilibrium of the deterministic model is characterized by the system of value functions, j = 1, ..., J, y ∈ [0, N ], where

γ β(y) :=  · γ−1

Zy 0

ψ(s)

is the unique solution of the Bernoulli equation, 1

1 ε−1

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Vjdet (y) := αj · β(y),



1

β 0 (y) = β(y) γ−1 · ψ(y) ε−1 ,

(γ−1)/γ

ds

β(0) = 0,

(24)

(25)

cf. Helmes, Schlosser (2015), Lemma 2.2. To compare the function β(y), cf. (24), and the corresponding sequence βy , cf. (18), we will determine qualitative and asymptotic properties of βy . The results are

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summarized in the following lemma.

Lemma 5.1. Let h > 0 be fixed and y ∈ S (h) := {0, h, 2h, ..., N − h, N }. Let cj = 0 and let ε, a, δ be the

same constants for all firms. Let the firm-specific parameters rj , uj , vj and kj be constants. Let χy be a

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positive sequence, which is defined on S (h) .

(i) There exists a unique nonnegative strictly monotone increasing sequence (βy )y with β0 = 0, which

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satisfies the difference equation (18).

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(ii) If χy is not increasing on the h-grid {0, h, 2h, ..., N − h, N } then βy is strictly concave. q 2 /4 + h · χy 1/(ε−1) . (iii) For the special case γ = 2 we have the recursion βy = βy−h /2 + βy−h βy (γ−1)/γ y→∞ y

(iv) If χy = 1 then for large y we have lim

= 1; i.e. βy ≈ y

γ−1 γ

γ

⇔ βy γ−1 ≈ y.

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Proof. See Appendix.

The solution formulas of the value functions of the deterministic model and the stochastic model are (h)

characterized by the function β(y) and the sequence βy , respectively. They can be easily compared. Figure 4 illustrates the convergence of the β (h) -sequences (h & 0) to the continuous β-function. For

three adoption functions ψ(y), viz. ψ(y) = y, ψ(y) = 1 and ψ(y) = y (a−δ)/a , where a=2, δ=1 and ε=1.2, the three solid lines are the graphs of the solutions of the Bernoulli differential equation (25). In addition, (h) each line shows sequences βy , y ∈ S (h) , where h ∈ {0.5, 1, 2}. If h=2, the values are market by the large (green) dots; the small (red) dots mark the β (h) -values if h=0.5, and the dots between correspond to h=1. 17

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HhL

Β y , Β H yL Ψ H yL= y

14 12

Ψ H yL= y0.5

10 8 6 2 0

2

4

6

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Ψ H yL=1

4

8

10

(h)

y

Figure 4: Sequences βy and functions β(y) for different h = 0.1, 0.5, 1, 2, and three adoption functions ψ(y) = χy , y ≥ 0 (a=2, δ=1, ε=1.2; γ = 1.4).

(h)

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In the case ψ(y) = χy = y (a−δ)/a for all h the sequences βy , y ∈ S (h) , coincide with the identity (the

diagonal β(y) = y in Figure 4) on S (h) , cf. (18). Hence, the (expected) profits of the stochastic and the deterministic model are identical. In the case without any adoption effect (ψ(y) = χy = 1) the β sequence converges from below to the continuous solution. The fact that the expected profits of the stochastic model can even exceed the profits of the deterministic model is illustrated by the case when the adoption effect is ψ(y) = χy = y.

(h)

Using the convergence of implicit Euler schemes it can be shown that for h → 0 the sequence βy

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converges to the solution of the Bernoulli differential equation β 0 (y) = β(y)1/(γ−1) · ψ(y)1/(ε−1) , β(0) = 0, cf. (25).

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Remark 5.1. The value functions of the stochastic and the deterministic model have the following properties: (h)

(i) If the size of shares h decreases then the expected profits Vj

(y) of firm j tend to Vjdet (y), the

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corresponding value function of the deterministic model, j = 1, ..., J. (h)

(ii) If no adoption effects are involved then Vjdet (y) exceeds Vj

(y) for all y and h. (h)

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(iii) In case of the adoption effect ψ(y) = χy = y (a−δ)/a , we have Vjdet (y) = Vj (iv) If strong saturation effects in the demand are involved then

(h) Vj (y)

can exceed Vjdet (y). (h)

Property (i) is an immediate implication of the asymptotic relation between βy

AC

(y).

and β(y), cf. Figure 4,

and the fact that the market power coefficients of the stochastic and the deterministic model are characterized by the same system of equations, cf. Remark 3.1. The statements (ii)-(iv) reveal that adoption effects are critical for the relation between the solution of the stochastic and the deterministic model. The results obtained have also interesting implications for practical applications since it is common to use deterministic model solutions as a heuristic solution to be applied in related stochastic models, cf., e.g., Schlosser (2015). Note, typically deterministic models are much easier to solve. The optimal feedback controls of the deterministic model – which are characterized by the associated value functions – can be used as a heuristic policy in stochastic models. We find that this approach works fine if the number of 18

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items/shares left to sell is large. However, if the number of items/shares left to sell is small our results show that the accuracy of this approach strongly depends on the presence of adoption effects. We find that if demand is increasing with the number of items sold in the market the heuristic can be bad (profits are overestimated by the deterministic solution). One the other hand, if there is strong saturation in demand the heuristic can also be bad (profits are underestimated). Hence, our result shows that this widely accepted approach has to be used with care when the number of items/shares left to sell is small.

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Remark 5.2. From our model, we learn that there are two fundamental effects: (i) If demand is stochastic there is a trend that expected results are lower compared to the deterministic model.

(ii) Saturation effects in demand have an opposite character. The second effect can be strong enough to overcompensate the first one.

To this end, the fundamental effects highlighted in Remark 5.2 can balance each other, i.e., when there is a certain level of saturation in demand deterministic models can be used to approximate stochastic models

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– even if the number of shares left to sell is small. 6. Conclusion

We extended the deterministic infinite horizon oligopoly model solved in Helmes, Schlosser (2015) to a stochastic dynamic pricing and advertising model, where finite as well as infinite horizons can be considered. Due to these extensions, the sale of perishable and durable goods can be analyzed in greater detail. Moreover,

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we are able to study differences and similarities between stochastic and deterministic oligopoly models with adoption effects.

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The solution of the model, i.e., the feedback Nash equilibrium is based on a multi-dimensional fixed point, which is unique. Necessary and sufficient conditions for the existence of an equilibrium are obtained. The associated solution is of separable structure; the value functions (i.e., the expected profits) of the different firms are composed of (i) a common state-dependent market factor, (ii) a common time factor, and (iii) the

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individual market power of the competing firms. The market power coefficients depend on the number of competing firms and their firm-specific model parameters but they are independent of the time and the state

CE

dependence. The time dependence of the value functions is characterized by the discount factor, the arrival rates, and the length of the time horizon. The state-dependent market factor depends on the size of a share, the size of the untapped market, and the adoption effect. The market power and the time dependence of the value functions are independent of these effects.

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Moreover, we compared the (expected) profits of stochastic models and their deterministic counterpart.

It turned out that in the stochastic and the deterministic model the individual market power of the firms as well as the time dependence equals. The only component that changes if the deterministic version of the model is considered is the market factor. If no adoption effects are taken into account, the value function of the deterministic model dominates the value function of the stochastic model. However, our results show that as soon as adoption effects are included, the deterministic model does not necessarily yield an upper bound for the expected profits; surprisingly they can be smaller or larger than the optimal profits of the deterministic model. We find that the usual gap between the value functions of deterministic and stochastic models can be compensated if the 19

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demand is decreasing with realized sales (demand saturation). This result is less known in the literature since dynamic pricing models that include adoption effects are usually deterministic. As a result we are able to identify cases in which deterministic feedback solutions of the model provide a useful approximation of solutions of the stochastic model. We also verified that if the number of shares to be sold is large, the solution of deterministic models can be used to approximate the expected evolution of the sales processes of related stochastic models. However, we show that the deviation of realized and expected quantities, such as prices, advertising rates, market

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shares, and particularly profits can be significant.

The simulation techniques as well as the computation of state probabilities make it possible to directly evaluate different scenarios over time; in particular, the evolution of prices and market shares of different firms can be compared. Our examples show that the evolution of prices and advertising rates can be of various shape. The expected evolution of optimal prices is strongly affected by the adoption effect and the individual market power of the competing firms.

The closed form solution of our model allows for various comparative statics. Our results allow studying

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the complex interplay of specific adoption effects, competitors’ brand images, financing costs, technology, and location factors as well as their impact on optimal pricing and advertising strategies and the associated profit distributions. Hence, the results can also be used to identify competitive advantages. Further, in practical applications several model parameters are often reestimated or can change due to market developments or disruptive innovations. Thus, it is very helpful to know how to react to these changes. The model allows to quantify the change in the market power or demand potential, and hence provides insights how to react to

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these changes using pricing and advertising controls.

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Appendix A. Section 2

Derivation of the structure of the system of difference differential equations (6)

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Consider the system of Bellman equations, cf. (2), and let, 1 ≤ j ≤ J, −εj

Ky(j) (t, pj , wj ) := uj (t) · wj δ · pj

  · χy · vj (t) · pj − cj − ∆Vy(j) (t)/h − kj (t) · wj a .

(i)

∂Ky(j) (t,pj ,wj ) ! =0 ∂pj

∂Ky(j) (t,pj ,wj ) ! =0 ∂wj

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(ii)

CE

The necessary optimality conditions for the optimal controls of firm j, 1 ≤ j ≤ J, are given by (j)

⇔

vj (t) · pj − cj − ∆Vy (t)/h = vj (t) · pj /εj ,

⇔

δ · uj (t) · χy · pj

−εj +1

· vj (t)/εj = a · kj (t) · wj a−δ .

(j)

For optimal controls the expression Ky (t, pj , wj ) amounts to   (j) a Ky(j) (t, pj , wj ) = λ(j) y (t, pj , wj ) · vj · pj − cj − ∆Vy (t)/h − kj (t) · wj

kj (t) · a = · δ



δ a  a−δ   a−δ +1 vj (t) · uj (t) · δ δ · uj (t) · vj (t) −εj+1 −εj +1 · χy · pj − kj (t) · · χy · p j kj (t) · εj · a a · εj · kj (t) a   −εj +1  a−δ (j) = ηj (t) · χy · cj + ∆Vy (t)/h ,

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a−δ δ

λ(j) y (t)

= uj (t) ·

= uj (t) ·



wy(j)δ

· χy ·

j p(j)−ε y

vj (t) · uj (t) · δ kj (t) · εj · a

·



δ·uj (t) a·εj

= uj (t) ·

δ  a−δ

a a−δ

· χy

·



·

vj (t)εj kj (t)a/δ

a  a−δ

·



εj −1 εj

γj −1

(j)

. The sales rates λy (t) =

vj (t) · uj (t) · δ −εj +1 · χy · p(j) y (t) kj (t) · εj · a



δ  a−δ

  εj (j) · cj + ∆Vy (t)/h vj (t) · (εj − 1)

−εj · χy · p(j) y (t)

δ(−εj +1) −εj a−δ

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ε a−δ

j and ηj (t) := where γj = a−δ   (j) (j) (j) λy t, py (t), wy (t) amount to

 −γj a = (γj − 1) · ηj (t) · cj + ∆Vy(j) (t)/h · χya−δ ,

where γj − 1 = (aεj − a)/(a − δ).

(A.1)

Plugging in the optimal controls into the system of Bellman equations, cf. (2), we obtain X

  (m) (m) λ(m) t, pt , wt y

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rj (t) · Vy(j) (t) = V˙ y(j) (t) + Ky(j) (t, pj , wj ) − ∆Vy(j) (t)/h ·

m6=j

a   −εj +1  a−δ (j) (j) ˙ = Vy (t) + ηj (t) · χy · cj + ∆Vy (t)/h

(j)  −γm a ∆Vy (t) X · (γm − 1) · ηm (t) · cm + ∆Vy(m) (t)/h · χya−δ . h

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−

m6=j

Section 3

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Derivation of the optimal rates of sales, cf. Theorem 3.1 Using the formula (A.1) above for the optimal rates of sales we obtain:

PT

λ(j) y =

−1  −γ γ a ε −1 a−δ = (γj − 1) · ηj · αj −γ · (χ(h) · ∆βy(h) /h = zj · χy j · (βy(h) ) γ−1 . y ) | {z } | | {z } {z }

CE (h)

zj

(h)

(h)

h·βy /∆βy

(h)

(h)

h·βy /∆βy

(j)

(h)

(h)

= y we obtain λy = zj · y and the total rate of sales λy = Z · h · βy /∆βy

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With βy

ja   δ−ε a εj a − a a−δ · ηj · cj + ∆Vy(j) /h · χya−δ a−δ

Proof of Lemma 3.1 For T = ∞ and the total rate λy :=

P

j

= Z · y.

(j)

λy we can assume that the state probabilities are characterized

by the differential equation q˙y (t) = λy+h · qy+h (t) − λy · qy (t) with the boundary conditions qN (0) = 1 and qy (0) = 0 ∀y < N . From Theorem 3.1 we have λ(h) (y) = Z · y, and we obtain q˙y (t) = Z · (y + h) · qy+h (t) − Z · y · qy (t).

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(A.2)

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The solution of the differential equation above is the Binomial solution N/h

qy (t) =

y/h

!

· D(t)y/h · (1 − D(t))

(N −y)/h

,

˙ where D(t) := e−Z·t ; that particularly implies −D(t)/D(t) = Z. Differentiating the candidate with N/h

∂ q˙y (t) = ∂t N/h

=

y/h

!

y/h

!

y/h

· D(t)

· (1 − D(t))

˙ · D(t) · D(t)(y−h)/h · (1 − D(t))

(N −y)/h

!

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respect to t yields:

(N −y−h)/h

(y − D(t) · N ) /h.

We check that the last expression equals the right-hand side of the differential equation (A.2):

= Z · D(t)

· (1 − D(t)) N/h y/h

!

(N −y−h)/h

·

N/h y/h

!

(h · (N/h − y/h) · D(t) − y · (1 − D(t))) (N −y−h)/h

˙ · D(t) · D(t)(y−h)/h · (1 − D(t))

· (y − N · D(t)) .

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=

y/h

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Z · (y + h) · qy+h (t) − Z · y · qy (t)

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Since the number of sales is Binomial distributed it follows E [Y (t)] = N · e−t·Z and for each firm j we
have E [Xj (t)] = 1 − e−t·Z · N · zj /Z, 1 ≤ j ≤ J. Proof of Proposition 3.1 (i)

(i) Let τN := 0. By definition, the conditional probability that in state y a sale (of size h) takes place

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before time t, t ≤ T , while the last sale happened at s = τy , 0 ≤ s < t, equals

CE

P (τy−h ≤ t |τy = s) =

Zt s

e−λy /h·(u−s) · λy /h

du = 1 − e−λy /h·(t−s) ;

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i.e. the waiting times τy−h − τy are  Exp(λy /h) exponentially distributed. Thus, for all states we obtain τy−h := τy − ln (U ) /λy · h = τy + ln U −h/λy , U ∼ U (0,1). Section 4

Proof of Theorem 4.1 (j)

(j)

With Vy (t) := αj · f (t) · βy follows py (t) =

αj ·ε·f (t) vj (t)·(ε−1)

· ∆βy /h. For the optimal single rates we obtain:

  a−δ a εa − a · χya−δ · ηj (t) · ∆Vy(j) (t)/h a−δ δ−εa

λ(j) y (t) =

−γ

= (γ − 1) · ηj (t) · (αj · f (t) · ∆βy /h) 22

a

· χya−δ

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ε

−γ

= f (t)

u(t) · v(t)

·

k(t)

δ/a

a ! a−δ

a
−γ . · η˜j · (γ − 1) · αj −γ · χya−δ · ∆βy /h

a

Hence, with by := χya−δ · (∆βy /h)−γj /˜ r = h/˜ r · βy /∆βy , η˜j :=

a−δ δ

zj := η˜j · (γ − 1) · αj−γ the optimal single sales rates can be written as

·



u ˜j ·˜ vjε a/δ ˜ k j

·

δ a·ε

a  a−δ

·


ε−1 γ−1 ε

and

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 a/(a−δ) ε δ/a −γ λ(j) · r˜ · u(t) · v(t) /k(t) · zj · by , y (t) = f (t)

i.e. they are separable in (i) time, (ii) firm and (iii) a state-dependent factor. Using the function (j)

g(t) := e−R(t) · f (t) the single rates can be written as λy (t) = g(t)/g(t) ˙ · zj · by since, from the Bernoulli  a/(a−δ) ε δ/a differential equation (21) with r(t) · f (t) = f˙(t) + u(t) · v(t) /k(t) · r˜ · f (t)−γ+1 it follows

=



ε

δ/a

u(t) · v(t) /k(t)

a/(a−δ)

f (t)

−γ+1

· r˜ · f (t)

ε

=

u(t) · v(t) k(t)

δ/a

a ! a−δ

· r˜ · f (t)−γ .

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Proof of Proposition 4.1

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r(t) · e−R(t) · f (t) − e−R(t) · f˙(t) r(t) · f (t) − f˙(t) −g(t) ˙ = = g(t) f (t) e−R(t) · f (t)

(i) Let τN := 0. By definition, the conditional probability that in state y a sale (of size h) takes place

≤ t τy = s) = P

Zt s

e

−

Ru P s

j

(j) λy (z)/hdz X

λ(j) y (u)/hdu

j

−

=1−e

Rt P s

j

(j) λy (u)/hdu

(A.3)

(j)

PT

P (τy−h

ED

before time t, t ≤ T , while the last sale happened in s = τy , 0 ≤ s < t, is given by

The total density λy (t) :=

CE

follows, 0 ≤ s < t ≤ T ,

AC

P (τy−h

j

˙ · Z · by . Hence, from (A.3) λy (t) can be written as λy (t) = −g(t)/g(t)

Rt

− ≤ t τy = s) = 1 − e s

λy (u)/hdu

=1−e

Rt Z·by /h· s

g(u) ˙ g(u)

du

Z·b /h

= 1 − eZ·by /h·ln(g(t)/g(s)) = 1 − (g(t)/g(s)) y , Z·b /h where (g(τy−h )/g(τy )) y τy is U (0,1) uniformly distributed, cf. Helmes, Schlosser (2013). Thus,

g(τy−h ) = U 1/(Z·by /h) · g(τy ), U ∼ U (0,1), holds for all states. Note, since g is strictly decreasing in t the function g has an inverse. The assertions (ii) - (iv) are directly implied. Section 5 Proof of Lemma 5.1 (i) Existence and uniqueness: The sequence is implicitly defined by, β0 := 0, 23

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βy /h − βy 1/(1−γ) · χy 1/(ε−1) = βy−h /h . | {z } | {z } | {z } | {z }

incr.inβy

decr.inβy

const.inβy

const.inβy

Since for given βy−h the left hand side is strictly monotone increasing in βy the value of βy is (recursively) p uniquely determined ∀y ≥ h. Furthermore, since χy > 0 we have βh = h · χh 1/(ε−1) > 0 and from

βy 1/(γ−1) · (βy − βy−h )/h = χy 1/(ε−1) > 0, ∀y ≥ h, it follows that βy > βy−h ≥ 0, ∀y ≥ h, i.e., the sequence

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is positive and strictly increasing.

(ii) In (i) we have shown that ∀γ > 1 (βy )y≥0 is strictly increasing in y. From βy 1/(γ−1) ·(βy − βy−h )/h = χy 1/(ε−1) | {z } | {z } | {z } >0

>0

>0

it follows that βy − βy−h is strictly decreasing in y ∀γ > 1, if χy is not increasing in y.

1/(ε−1)

(positive) solution is unique, since βy > βy−h .

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(iii) If γ = (aε − δ)/(a − δ) = 2 then ∆βy /h = βy 1/(γ−1) · χy 1/(ε−1) yields βy2 − βy · βy−h = h · χy q 1/(ε−1) 2 , starting from β0 := 0. Note, the /4 + h · χy and we obtain the recursion (βy )1,2 = βy−h /2 ± βy−h (iv) If χy = 1 then for large y we have βy → y (γ−1)/γ , see McAfee, te Velde (2008).

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time horizon initial market size time number of competing firms size of a unit share state, untapped market size items sold by firm j discount rate of firm j unit costs of firm j tax parameter of firm j advertising cost parameter arrival intensities/effectiveness

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T N t J h y Xj rj cj vj kj uj

wj pj χy λ εj δ Uj Wj Vj αj f (t) βy

price of firm j advertising rate of firm j adoption effect sales intensity of firm j price elasticity of firm j advertising elasticity realized net revenues realized advertising expenditures value function of firm j market power of firm j time effect state-dependent market effect

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Table A.2: List of variables and parameters.

AC

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