Dynamic advertising and pricing with constant ...

Journal of Economic Dynamics & Control 37 (2013) 2814–2832

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Dynamic advertising and pricing with constant demand elasticities Kurt L. Helmes n, Rainer Schlosser Humboldt-Universität zu Berlin, Berlin, Germany

a r t i c l e in f o

abstract

Article history: Received 13 January 2012 Received in revised form 21 May 2013 Accepted 25 July 2013 Available online 17 August 2013

In this paper we analyze a stochastic dynamic advertising and pricing model with isoelastic demand. The state space is discrete, time is continuous and the planing horizon is allowed to be finite or infinite. A dynamic version of the Dorfman–Steiner identity will be derived. Explicit expressions of the optimal advertising and pricing policies, of the value function and of the optimal advertising expenditures will be given. The general results will be used to analyze the case of impatient customers. Furthermore, particular time inhomogeneous models and homogeneous ones with and without discounting will be examined. We will study the social efficiency of a monopolist's optimal policies and the consequences of specific subsidies. From a buyer's perspective, our analysis reveals that waiting – when looking at (immediate) expected prices – is never profitable should two or more units be available. But we will also prove that the sequence of average sales prices is monotone decreasing. Moreover, the techniques applied to solve the discrete stochastic advertising and pricing problem will be used to solve a related deterministic control problem with continuous state space. & 2013 Elsevier B.V. All rights reserved.

JEL classification: C61 D42 M37 Keywords: Dynamic advertising and pricing Optimal deterministic and stochastic control Dorfman–Steiner identity Constant demand elasticities

1. Introduction The theory of dynamic pricing of stochastic models in the general context of revenue management is well established, see Talluri and van Ryzin (2004) and the review papers Bitran and Caldentey (2003), Elmaghraby and Keskinocak (2003) and Shen and Su (2007); for additional references see Bitran and Mondschein (1993), McAfee and te Velde (2006), and the literature cited therein. The theory of how to set optimal advertising rates together with optimal prices when selling perishable products in a stochastic environment is yet still in its infancy, see MacDonald and Rasmussen (2009) for an excellent literature revue. However, there is an extensive literature on deterministic advertising models and combined advertising and pricing models, for instance, Nerlove and Arrow (1962), Bass (1969), and many others; see the surveys Mahajan et al. (1990), Feichtinger et al. (1994), and Bagwell (2007) for details and annotated bibliographies. These models typically consider the sale of durable goods, the introduction of new goods and the controlled evolution of, for example, the market share of a product with the objective to maximize overall profits. The Sethi (1983) model, is a well researched example of a deterministic model and a stochastic model of such kind. The state variable of the Sethi model represents the market share of a product which is driven by Brownian motion and controlled by the rate of advertising effort. The variation of the

n

Corresponding author. Tel.: þ49 30 20935744; fax: þ 49 30 20935745. E-mail addresses: [email protected] (K.L. Helmes), [email protected] (R. Schlosser).

0165-1889/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jedc.2013.08.004

K.L. Helmes, R. Schlosser / Journal of Economic Dynamics & Control 37 (2013) 2814–2832

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original model considered in Sethi et al. (2008) is an example of a deterministic model where the state is controlled by the marketing instruments advertising and price. At first glance the traditional revenue management problems, e.g. selling airplane tickets of a particular flight, and classical advertising models, e.g. increasing the market share of a newly introduced product, seem to deal with rather different kind of situations. It will be pointed out in Section 7 that several aspects of the well known models of both camps are but two sides of the same coin. Differences exist in the way state spaces are described and interpreted, the horizon of the control problems, whether or not unit costs are taken into account and, very important, the modeling of the stochastic environment. In the context of classical advertising models stochastic driving forces are often modeled by Brownian motion or more general stochastic diffusion processes. In revenue management the Poisson process and more general jump Markov processes play a prominent role. For perishable products, MacDonald and Rasmussen (2009) is a recent paper where simultaneous advertising and pricing decisions are analyzed in a stochastic environment. The stochastic environment is due to the random arrival of customers and their willingness to pay. MacDonald and Rasmussen consider the case when N units of a perishable product are to be sold over a finite horizon T. Their model assumes the initial stock of the product to decrease according to a Poisson type process with time homogeneous intensity λ. For each price value p Z 0 and advertising rate w Z 0 the intensity has the form λðp; wÞ ¼ a wδ emp , 0 r δ o1, where a and m are positive constants. The factor awδ reflects the (average) number of shoppers which are attracted by advertising to take a look at a product. In the pure pricing version of the model the average number of shoppers is assumed to be equal to the fixed number a. The second factor reflects the force (depending on price) which turn shoppers into buyers. Their intensity function λ is a combination of the classical willingness to pay function considered in Gallego and van Ryzin (1994) and the increasing concave function of the advertising rate with constant elasticity δ. For their model, which is an extension of the pure pricing model studied by Gallego and van Ryzin, MacDonald and Rasmussen derive closed form expressions of the optimal advertising and pricing policies as well as the value function. The value function is defined as the maximum of expected revenue minus expected advertising cost of all policies under consideration. The authors also evaluate a fixed-price, fixed-advertising heuristic which can be easily implemented and can also be used when the demand is stochastic. Their numerical analyses show the performance of this advertising and pricing heuristic for the model considered in MacDonald and Rasmussen (2009). Gallego and van Ryzin (1994) analyze dynamic pricing problems with general reservation price distributions for deterministic models as well as for stochastic ones. In particular, see Theorem 4.1 of Gallego and van Ryzin (1994), they prove the value function of a deterministic pricing problem to dominate the value function of the corresponding stochastic problem. In the special case of exponential demand (i.e. λðpÞ ¼ aemp ), they prove that optimal prices of a stochastic model (given in feedback form) are larger than the optimal prices of the corresponding deterministic problem, cf. Proposition 3 of Gallego and van Ryzin (1994). This property also holds in the case with advertising if demand is isoelastic, see below. For isoelastic models we derive analytical expressions of the value function of a deterministic problem and its stochastic counterpart. These analytical formulas make it possible to precisely evaluate the difference between deterministic and expected profits. In this paper we shall look at the situation when the time dependent jump intensity λ of the process of unsold units – a pure “death process” – is of the form λðt; p; wÞ ¼ aðtÞ wδ pε where aðtÞ 40 is a given deterministic function, ε 4 1 and 0 r δo 1. This intensity has two important characteristics. Since p is allowed to be any positive number, even arbitrarily small, the factor pε of λ implies that all items will be sold over any finite time interval ½t 0 ; T, 0 r t 0 o T, if the objective is to maximize the expected revenue and no unit cost term is taken into account. Moreover, not only is the advertising elasticity of λ constant, but the price elasticity is constant as well. The assumption of constant elasticities has far reaching implications. In the case of a stochastic (pure) pricing problem it has been pointed out by McAfee and te Velde (2008) that constant price elasticity of demand and zero unit costs imply a monopolist to set socially efficient prices. In the deterministic case, Stiglitz (1976) has actually shown that under the same assumptions the optimal pricing strategy coincides with the (perfectly) competitive price. The combination of isoelastic demand and zero (unit) cost implies that optimal sales intensities may be unbounded, and in the deterministic case as well as the stochastic case any initial stock will be always cleared over any given period. If advertising is endogenized, i.e. δ 40, we will prove that this property still holds true but that a monopolist will not choose socially efficient prices. However, in Section 6, we will describe mechanisms which lead to efficient policies by a monopolist. In particular, an adjustment which is a combination of a revenue tax and an advertising subsidy will be proposed which guarantees efficient prices and which is (even) self-financing. To analyze the problem we extend and generalize results obtained in McAfee and te Velde (2008) for the pure pricing model to the case with advertising. The dynamic (pure) pricing model has many applications, see Talluri and van Ryzin (2004) for a variety of examples. Applications of models which combine advertising and pricing include, for instance, onetime events which are individually advertised. Examples include ticket sales of singular sporting events (Olympic Games, Champions League soccer games, etc.), special concerts, shows and performances. Fish and fruit markets are other well known examples where combined advertising and pricing activities can be observed almost every day. Not only do prices typically drop close to the end of the sales period but the volume and the frequency of the “shouting” of sellers increases as well. From a general perspective, these examples are special cases of the situation when a decision maker is selling a given number of identical (or very similar) assets over a fixed period of time, and advertising as well as discounting is common. Specific industries include the business of new car dealers at the end of a (car)model year, or sectors of the real estate

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market. Common practices of realtors and developers, for instance, open houses, newspaper ads, etc., in conjunction with price variations are well known marketing activities. Furthermore, many second-hand businesses, e.g. used-car dealers, etc. face the situation captured by (1), cf. Jerenz (2008). The tactical question of dynamically setting prices and modulating advertising expenditures can be combined with the strategic problem of choosing an optimal capacity (optimal “order quantity” N). This newsvendor-like problem, cf. Petruzzi and Dada (1999), and see Choi (2012) for a recent collection of survey papers on the newsboy problem, will be briefly discussed in Section 3. The pure pricing model and the combined advertising and pricing model offer the possibility of studying and testing the evolution of various characteristics of these models. Examples of such studies include analyzing the evolution of yet unsold items over time, the dynamics of prices and advertising rates, the distribution of the time of the n-th sale as well the corresponding price and advertising distributions, etc. Within the class of admissible policies, see Section 2, we are going to identify the best feedback advertising and pricing strategy ðut Þt ≔ðwt ; pt Þt for which the expected (discounted) revenue, when selling N copies of the same product, minus the expected advertising expenditures, Z τ 4 T Jð0; N; ðut Þt Þ≔E eRðtÞ ððvðtÞpt cÞ λðt; pt ; wt ÞkðtÞwt Þ dtjXð0Þ ¼ N ; ð1Þ 0

will be maximized. In expression (1) k(t), v(t) are positive functions and R(t) is a nonnegative parameter function; τ is the first (random) time when all items are sold. The time dependent functions offer the possibility to analyze various extensions of the classical models. While the parametrization of (1) is not unique the chosen one is useful for many applications, see Rt Section 3. The function RðtÞ≔ 0 rðsÞ ds, r Z0, describes an accumulated discount rate, while k and v allow the modeling of tax or subsidy factors. The nonnegative constant c represents a cost whenever an item is sold. Besides the model with objective function (1) the cases with an additional terminal pay-off or penalty term representing a salvage value of unsold units, or a charge when unsold items have to be discarded are of interest. For the special case when c ¼0 and there is no terminal pay-off the decision problem can be solved analytically; the more general cases, when (i) c a 0 and/or (ii) the case with an additional pay-off at time T, can generally only be solved numerically. The fact that analytical solutions can be derived if c ¼0 is most useful for a better theoretical understanding of such market situations. Analytical solutions also enable analysts to evaluate numerical solution procedures designed to solve problems with general cost expressions. For instance, based on extensive studies it has become apparent that already straightforward discretization schemes of t, p and w, combined with simple-minded reduction ideas, guarantee good numerical results for problems where c is different from zero, and/or additional inventory costs are taken into account. The basic idea which underlies our analysis of the dynamic joint advertising and pricing problem can be explained by looking at the static version of the problem. The static case was considered in Dorfman and Steiner (1954) and, nowadays, is a classical textbook example. To this end, let qðp; wÞ be the demand function of the form a wδ pε . Imagine a firm which is trying to maximize its profit πðp; wÞ ¼ ðpcÞqðp; wÞw by setting the price p 4 0 and advertising expenditure w Z 0 appropriately when the marginal cost is c 4 0. If ε 4 1 and 0 rδ o1, the solution of a firm's advertising and pricing problem is given by p ¼ ε=ðε1Þc and w ¼ ða δ=ε p ðε1Þ Þ1=ð1δÞ ; the maximal profit π ¼ πðp; wÞ equals δ=ð1δÞ δ 1δ ε1 ðε1Þ=ð1δÞ ðε1Þ=ð1δÞ 1 π ¼ a1δ c : ε ε ε The way π depends on c will be exploited when analyzing a particular difference-differential equation in the dynamic setup. There, c will be replaced by an expression which represents an opportunity cost. Note, for the optimal advertising value w and the optimal price p the Dorfman–Steiner identity holds, i.e. the quotient of (optimal) advertising expenditure and (optimal) revenue equals the ratio of the two elasticities δ and ε: w δ ¼ : p qðp; wÞ ε Next, consider the dynamic version of this problem when c¼ 0 and N units of a perishable product are to be sold during the period ½0; T. Advertising rates w(t) and prices p(t), 0 r t rT, are to be chosen such that the sales process controlled by the policy ðwt ; pt Þ via the intensity λt is most profitable, i.e. we maximize the objective (1) with c¼0. For any n rN let Vðt; nÞ denote the value function of the advertising and pricing problem when n units are to be sold during the period ½t; T, i.e. Vðt; nÞ≔ sup fJðt; n; ðws Þvs ; ðps Þs Þg: ðws Þs ;ðps Þs

Obviously, the value function V satisfies the boundary conditions VðT; nÞ ¼ 0, 0 r n rN, and Vðt; 0Þ ¼ 0, 0 rt r T. It follows from the general theory of controlled jump Markov processes, for instance, Gihman and Skorohod (1979), Brémaud (1980) or Fleming and Soner (2006), that V satisfies the Bellman equation rðtÞVðt; nÞ ¼ V_ ðt; nÞ þ

sup

fλðt; p; wÞðvðtÞpΔVðt; nÞÞkðtÞwg;

ð2Þ

p 4 0;w Z 0

here, ΔVðt; nÞ≔Vðt; nÞVðt; n1Þ is the opportunity cost of having, over the same time span ðTtÞ, one unit less to sell. Since for each t o T and n 4 0 the maximization problem which is part of the Bellman equation is a special case of the static Dorfman–Steiner problem whenever ΔVðt; nÞ 4 0, we obtain the following expressions of the optimal pricing and advertising


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decisions pn ðt; nÞ and wn ðt; nÞ: pn ðt; nÞ ¼

ε 1 ΔV ðt; nÞ; ε1 vðtÞ

wn ðt; nÞ ¼

ε1 !1=ð1δÞ aðtÞ vðtÞ δ 1 : kðtÞ ε pn ðt; nÞ

ð3Þ

ð4Þ

Formulas (3) and (4) reveal the remarkable fact that optimal dynamic prices and optimal advertising rates are synchronized and inversely proportional to a power of each other. Using the explicit expressions of pn and wn one obtains, see Section 3, a dynamic Dorfman–Steiner identity in feedback form: For every t, 0 rt o T , and n, 1 r n r N, vðtÞ

kðtÞ wn ðt; nÞ δ ; λðt; pn ðt; nÞ; wn ðt; nÞÞ ε

pn ðt; nÞ

ð5Þ

i.e. the advertising rate is pointwise proportional to the revenue rate. This identity will be employed to give a simple proof of a formula of the optimal expected revenue, see Section 4, but is of course of its own interest. For a dynamic version of the Dorfman–Steiner identity in the context of deterministic advertising and pricing models see Schmalensee (1972). We would like to point out that (5) is comparable to the Merton ratio in portfolio analysis. While the Merton ratio indicates how the investor's wealth needs to be continuously (in time) rebalanced between a risky asset and a risk-free one, (5) suggests how price adjustments and advertising spending need to be coordinated at any instant. In the sequel, we shall only analyze deterministic intensities of the product form introduced above. Natural extensions of the advertising and pricing model could include intensities λ with additional random components or models of unknown parameters. Models with such general intensity functions will lead to control problems with partial information, see, for instance, Beneš et al. (1995), and will be considered in the future. This paper is organized as follows. A detailed formulation of the general time inhomogeneous model and the precise definition of admissible controls will be given in Section 2. In Section 3 we shall derive explicit formulas of the value function and the optimal advertising as well as pricing policies together with the generalized Dorfman–Steiner identity. Based on the explicit formulas it is easy to prove structural properties of the value function and the optimal policies of certain subclasses of the model. We shall also analyze two special cases of the general problem in greater detail, (i) the model with impatient customers and (ii) the discounted case with infinite horizon. In Section 4 we derive additional results concerning the optimal sales process, for instance, formulae of the total average revenue and the total average advertising expenditures. Moreover, a recursive formula of state probabilities of time homogeneous models and, following McAfee and te Velde (2008), a binomial approximation of the optimal state probabilities will be proposed. The explicit expressions derived in Section 3 can be used to efficiently simulate sales trajectories. Thus one is able to evaluate the statistical properties of many quantities of interest, e.g. the j-th sales time, the average number of unsold items at time t and the distribution of the total profit. In Section 5 we analyze the sequence of optimal expected prices of all sales and we show, for example, that this sequence is monotone decreasing. In light of this result it might come as a surprise that we shall also prove that at any time t it does not pay to wait if there are two or more units left to be sold; see Theorem 5.1. This result extends and qualifies a result by McAfee and te Velde (2008). In Section 6 we study, in the case with advertising, the question whether or not optimal prices chosen by a monopolist are socially efficient, cf. McAfee and te Velde (2008) for the pure pricing model. The main result of this section will be a characterization of subsidies and incentives such that a monopolist's actions become socially efficient. In Section 7 we analyze a particular deterministic advertising and pricing model which is closely related to the stochastic one. We study the case when a finite amount of an infinitely divisible product is sold with a view towards maximizing the corresponding profit. 2. Model formulation We begin with a precise formulation of the dynamic advertising and pricing problem. Assume the coefficients v(t) and k(t) to be positive bounded measurable functions. For any given δ, 0 r δo 1, assume aðtÞ1=ð1δÞ to be a positive function and r(t) to be a nonnegative one. Both functions are supposed to be integrable on any finite sub-interval of ð0; 1Þ. We separately define the set of admissible controls of finite horizon problems and infinite horizon ones. Let N A N and T 4 0 be given. From now on, we assume all functions pðt; nÞ, 1 rn rN, 0 rt r T, to be positive and measurable on ½0; T. The advertising controls wðt; nÞ are nonnegative measurable functions. Let λðt; nÞ≔λðt; n; w; pÞ≔aðtÞwðt; nÞδ pðt; nÞε , 1 rn rN, where ε 4 1 and 0 r δo 1 are given; we define λðt; 0; w; pÞ≔0. Throughout, the parameter γ≔ðεδÞ=ð1δÞ will play a prominent role. Pairs of functions ðwðt; nÞ; pðt; nÞÞ will be called admissible if – summarized in a nutshell – they satisfy certain integrability conditions. To be precise, if T o 1 the set of admissible controls of (1) is given by, ut ¼ ðwt ; pt Þ, U ðTÞ N ≔fðut Þt j Condition 2:1 is satisfiedg:

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Condition 2.1. Assume gðtÞ≔ðaðtÞvðtÞε =kðtÞδ Þ1=ð1δÞ to be integrable on ½0; T. Let r be a nonnegative integrable function on ½0; T. The control functions w ¼ ðwt Þt and p ¼ ðpt Þt are assumed to satisfy the following conditions: (a) w is a nonnegative and p is a positive measurable function on ½0; T f1; 2; …; Ng; (b) for each n, 1 r n r N, wð; nÞ, λð; nÞ and pð; nÞλð; nÞ are integrable functions on ½0; T. To define the set of admissible controls for the case when T ¼ 1 we formulate a set of slightly different conditions. Rt Condition 2.2. Let r be a positive integrable function on any ½0; t, 0 r t o1; RðtÞ ¼ 0 rðsÞ ds, and gðtÞeγRðtÞ is integrable on ½0; 1Þ. The control functions w ¼ ðwt Þt and p ¼ ðpt Þt are assumed to satisfy the following conditions: (a) w is a nonnegative and p is a positive measurable function on ½0; 1Þ f1; …; Ng. (b) λð; nÞ is integrable on any interval ½0; tÞ, t 40, 1 rn rN. (c) For any n, 1 r n r N, pð; nÞλð; nÞeRðÞ and wð; nÞeRðÞ are integrable functions on ½0; 1Þ. If T ¼ 1 the set of admissible controls of (1) is defined as ut ¼ ðwt ; pt Þ, U ð1Þ N ≔fðut Þt j Condition 2:2 is satisfiedg: Given any admissible control ut ¼ ðwt ; pt Þ, either for a finite horizon problem or an infinite horizon problem, let τ0 ≔0 and assume τj , 1 r j rN, to be recursively defined as τj ¼ τj1 þ Δj τ, where Δj τ is the j-th (conditionally on τj1 independent) interarrival time of the nonhomogeneous Poisson process with (control) intensity λ. Let ðX t Þt Z 0 denote the jump process with initial value Xð0Þ ¼ N which jumps from Nj þ 1 to Nj at the random times τj , 1 rj r N; once the process hits zero it stays at zero. Note, Conditions 2.1 and 2.2 imply that if u ¼ ðw; pÞ is admissible the objective value (1) will always be a finite ð1Þ n number. We call any unT A U ðTÞ resp., such that N , u1 A U N JðunT Þ ¼ sup fJðuÞg; ðTÞ u A UN

Jðun1 Þ ¼ sup fJðuÞg

resp:;

u A U ð1Þ N

an optimal solution of the dynamic advertising and pricing problem. The Hamilton–Jacobi sufficient conditions, see Fleming and Soner (2006), Section 3.8, or Brémaud (1980), Section 7, will be used to verify that an optimal control exists for each control problem. For the problem under consideration we shall be looking for functions Vðt; nÞ, 0 rn rN, which are differentiable in t, 0 r t oT, and continuous up to the boundary such that the Bellman equation, see Section 2, is satisfied, and such that for every t and n the values wn ðt; nÞ, pn ðt; nÞ, cf. (3) and (4), are maximizers of the associated optimization problems. 3. An analytical solution of the model From now on we shall use the following abbreviations: δ=1δ δ εδ γ η γ1 γ1 and ξðt Þ≔g ðt Þ : η≔ ε ε γ γ

ð6Þ

Substituting for the optimal pn and wn of the Bellman equation (2) the right hand sides of (3) and (4) the Bellman equation turns into the nonlinear difference-differential equation 0 ¼ V_ ðt; nÞ þξðtÞ ΔVðt; nÞ1γ rðtÞVðt; nÞ

ð7Þ

with the “natural” boundary conditions described in Section 1. Since (7) has a similar structure as the Bellman equation of the pure pricing model, see McAfee and te Velde (2008), we look for a separable solution Vðt; nÞ which is the product of two factors. One factor should only depend on t, and the other one should be a sequence with index n: Vðt; nÞ ¼ αðtÞ βn :

ð8Þ

Hence, αðtÞ and βn need to satisfy η γ1 γ1 1 βn ðβn βn1 Þ1γ r ðt Þαðt Þ; 0 ¼ α_ ðt Þ þ g ðt Þ αðtÞ1γ γ γ 1γ 1 to hold, the resulting where αðTÞ ¼ 0, β0 ¼ 0, and βn 4 0, if n Z1. Assuming the identity ððγ1Þ=γÞγ1 β1 n ðβn βn1 Þ equation which determines α is the Bernoulli differential equation η α_ ðt Þ ¼ r ðt Þαðt Þg ðt Þ αðtÞðγ1Þ ð9Þ γ

_ Thus, the two with terminal condition αðTÞ ¼ 0. It can be solved using the substitution ZðtÞ ¼ αðtÞγ , and Z_ ðtÞ ¼ γ αðtÞγ1 αðtÞ. factors of the value function are αðtÞ ¼ ðη AðtÞÞ1=γ , where Z T eγRðsÞ gðsÞ ds; ð10Þ AðtÞ≔eγRðtÞ t


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and βn ≔βn ðγÞ≔βn ðε; δÞ, n Z 1, where β0 ≔0, and βn are the implicitly defined positive and monotone increasing solution values of the recursion βn ðβn βn1 Þγ1 ¼

γ1 γ

γ1

:

ð11Þ

~ ¼ MαðtÞ, M 40 fixed, where ðβ~ n Þn satisfies Note, α and β enjoy a remarkable scaling property of the form β~ n ¼ βn =M and αðtÞ ~ the recursion (11) with the right hand side multiplied by M γ while αðtÞ satisfies (9) with η~ ¼ η M γ . It is worthwhile to realize that actually both factors of the value function (8) are determined by or are related to a Bernoulli differential equation. While the “time-factor” α satisfies (9) the “space-factor” β is related to the Bernoulli differential equation FðxÞðF′ðxÞÞγ1 ¼ ððγ1Þ=γÞγ1 , x 4 0, Fð0Þ ¼ 0, which has the solution FðxÞ ¼ xðγ1Þ=γ . This relationship will be exploited in Section 7. The next two propositions summarize characteristic and asymptotic properties of βn and A(t). The proofs of these properties are given in the Appendix A or follow directly from the definition. The first two properties listed in Proposition 3.1 have already been proved in McAfee and te Velde (2008); property (iv), inequality ðnÞ, is essential for the proof of the main result of Section 6 and (v) is a nontrivial identity which can be proved by induction, cf. Helmes and Templin (2013). This proof by induction is quite technical and uninspiring. At the end of this section we shall see that this identity is an almost immediate consequence of the explicit formula of the value function of the discounted problem, cf. Theorem 3.1 below. Proposition 3.1. Let γ ¼ ðεδÞ=ð1δÞ. Identity (11) implies:

(i) There is a unique positive, strictly monotone increasing sequence ðβn Þn Z 0 which satisfies (11); γ=γ1 (ii) limn-1 βn =nðγ1Þ=γ ¼ 1, i.e. βn nγ1=γ 3 βn n for large n; (iii) The sequence ðβn Þn Z 0 is strictly concave in n; (iv) γ=ðγ1Þ 1=γ γ=ðγ1Þ ðnÞ ð1 þ βn Þ o 1 þ 1=γβn o ðβn =βn1 Þ1=ðγ1Þ ðnnÞ 1 γ=ðγ1Þ γ=ðγ1Þ 1=γ o ð1 þ βn1 Þ o 1 þ βn1 ; for all n Z2; γ 1=ðγ1Þ

(v) γ1=γ∑ni¼ 1 βni þ 1

1=ðγ1Þ

γ=ðγ1Þ

γ=ðγ1Þ

¼ βn ¼ ∑ni¼ 1 βni þ 1 ð∏ij ¼ 1 βnj þ 1 =ðβnj þ 1 þ 1=γÞÞ, for any n Z1.

In light of the close relationship between the nonlinear differential equation (11) and the associated Bernoulli equation for F(x) property (ii) of Proposition 3.1 can be thought of as a stability result of a nonlinear difference equation. The proof by McAfee and te Velde is somewhat technical. In the Appendix A we shall give a more elementary proof of (ii). The left most inequality of the chain of inequalities (iv) is an immediate application of the Bernoulli inequality with an exponent between zero and one, i.e. 1=γ. The other inequalities will be shown as part of the proof of (iv), see Appendix A. Note, ðnnÞ of (iv) γ=ðγ1Þ

γ=ðγ1Þ

to be bounded by 1 from which the boundedness (by one) of the sequence ðβn =nÞn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 follows, cf. (ii). For the special case γ ¼ 2 there is the simple recursion βn ¼ 2 βn1 þ βn1 þ2 ; the analysis of properties of

implies the increments of βn

this special sequence is straightforward. The product representation (8) of the maximum expected profit Vðt; nÞ when combined with the property that ðβn Þn is a concave sequence, see (6) (iii), expresses in analytical form one's intuition that marginal profit decreases in the number of units to be sold. Proposition 3.2. The time factor A has the following asymptotic properties: r-0 R T (i) AðtÞ⟶ t gðsÞ ds, T o1. (ii) For the time-homogeneous case,

8 r-0 > >⟶ vε 1=ð1δÞ 1eγrðTtÞ < Aðt Þ ¼ a δ T-1 > γr k > :⟶

ε 1=ð1δÞ a kvδ ðTt Þ; 1=ð1δÞ 1 vε : γr a kδ

The expression A(t) should be thought of as being a potential which quantifies the effective demand over the sales period ½t; TÞ. Our first theorem offers explicit formulae of the optimal policies and the value function of the general dynamic advertising and pricing problem. They follow from the derivation given above, cf. (8)–(11).

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Theorem 3.1. Under the conditions specified in Section 2, the Dorfman–Steiner identity (5) holds; the optimal advertising policy wn and the optimal pricing policy pn are given by, 0 r t o T, n Z1, γ !ðγ1Þ=γ ! ε=ðεδÞ γ=ðγ1Þ ðγ1Þ=γ hom: δ gðtÞ βn δ 1 v ε=ðεδÞ βγ1 n r ¼ 0 ε ¼ a ; w ðt; nÞ ¼ ε kðtÞ ε k AðtÞ Tt

n

pn ðt; nÞ ¼

!1=γ !1=γ δ=ðεδÞ hom: δ 1 AðtÞ δ 1 v δ=ðεδÞ Tt r ¼ 0 δ a ¼ : γ γ ε vðtÞ βγ1 ε k βnγ1 n

ð12Þ

ð13Þ

The value function V which represents the net present value of the firm at time t, 0 r t r T, n Z0, equals δ=ðεδÞ δ ε 1=ðεδÞ hom: δ δ δ δ v ¼ 0 V ðt; nÞ ¼ 1 βn AðtÞ1=γ r ¼ 1 a βn ðTtÞ1=γ : ε ε ε εδ k δ

ð14Þ

Note, Theorem 3.1 provides the analytical underpinning of the claim, see Introduction, that any initial capacity N will always be fully liquidated over any finite sales period, no matter how large N or how small T, T o1, will be. The formulas (12) and (13) reveal that close to the end of the sales period, i.e. t-T, optimal pricing policies tend to zero and optimal advertising strategies simultaneously diverge. Remark 3.1. If T o1 formulae (12) and (13) imply that ðwn ; pn Þ is an admissible policy. For the infinite horizon case with discounting we can take the limit of the expression A(t) for T-1 and deduce the formulas of an optimal pair of controls in U ð1Þ N . Remark 3.2. If δ ¼ 0, then expressions (12) and (14) provide the solution formulas of the pure pricing model. Whenever a deterministic advertising rate function w(t) is exogenously specified, its impact on the evolution of the (price) optimized sales process can be quantified. To do so, the arrival rate of the pure pricing model will be multiplied by w(t) and, for the mathematical analysis, advertising expenses will not be relevant. This simple observation ensures that results of this paper on impatient customers, etc. can be applied in the context of revenue studies of the hospitality industry, airline industries and so an. Remark 3.3. For the time-homogeneous case without discounting, i.e. aðtÞ a, vðtÞ v, kðtÞ k and rðtÞ 0, the formulas given in Theorem 3.1 simplify. For any t and for “large” n (see Proposition 3.1 (ii)): Vðt; nÞ const V nðγ1Þ=γ ðTtÞ1=γ ; pn ðt; nÞ const p ððTtÞ=nÞ1=γ ; and

wn ðt; nÞ const w ððTtÞ=nÞðγ1Þ=γ ;

i.e., the ratios of left hand sides and right hand sides converge to 1 if n-1. These formulas show that the value function V behaves asymptotically like a homogeneous Cobb-Douglas function of degree one in the variable “units to sell” n and the variable “time-to-go” ðTtÞ. The elasticity with respect to n, ðTtÞ resp., is approximately ðγ1Þ=γ, 1=γ resp. The latter result quantifies the saying “Time is Money” for the problem under consideration. The optimal advertising and pricing policies are approximately homogeneous functions of degree zero in the two variables. Since we have explicit formulas for V, wn and pn structural properties of these functions can be easily deduced. It is well known that even in more general cases the following properties – as functions of the inventory level n – are satisfied. Proposition 3.3. If the assumptions of Theorem 3.1 are satisfied, then for any t: (i) Vðt; nÞ is a concave and increasing function in n, (ii) pn ðt; nÞ is a decreasing function, while wn ðt; nÞ is an increasing function in n.

As far as the variable t is concerned the functions V, pn and wn enjoy similar properties. For any t, 0 rt r T, let V ðt; nÞ≔eRðtÞ Vðt; nÞ denote the net present value of Vðt; nÞ at time zero. Note, V ð0Þ ðt; nÞ is obviously a decreasing function in t, while – in general – Vðt; nÞ is not. The fact that Vðt; nÞ is concave (in n) guarantees the existence of an optimal solution of the following newsvendor-like problem: Let Γ; 0 oΓ, denote the cost of one unit of capacity. Consider the following “economic order quantity” problem: ð0Þ

max fVð0; nÞΓng≕zn :

n A N0


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Formulae (14) and (11) yield the following ordering policy, ( ) γ1 γ1 ðγ1Þ=γ n ðηAð0ÞÞ ; i:e: n ¼ max n Z0jβn r γΓ the largest value for which the change in expected profit is still nonnegative if we increase the decision variable from n to n þ1. The corresponding optimal zn is given by zn ¼ βnn ðηAð0ÞÞ1=γ Γnn ; η is defined by (6). Using the asymptotic formula of βn , see Proposition 3.1 (ii), the expression of nn simplifies if n is large. In the homogeneous case we obtain γ1 γ 1 γ1 γ1 ηaT; and zn r ηaT: nn γΓ γ γΓ The difference between the upper bound and zn can be large if n is small; whenever n is large this difference is small. These expressions offer the following insight into the modified newsvendor problem. The optimal capacity and the optimal value are approximately proportional to the length of time that a dynamically operating newsvendor intends to sell his stock. The optimal (approximate) inventory value nn decreases superlinearly in the unit capacity cost Γ. Proposition 3.4. If the assumptions of Theorem 3.1 are satisfied, and n Z1: For the time-inhomogeneous model without discounting, i.e. rðtÞ 0, (i) Vðt; nÞ ¼ V ð0Þ ðt; nÞ is a decreasing function in the variable t. If the time derivative of A1=γ is a decreasing function in t then V is concave in t. (ii) If vðtÞ1 AðtÞ1=γ is a decreasing function in t then pn is decreasing in t as well. (iii) If gðtÞ=kðtÞ AðtÞ1 þ 1=γ is an increasing function in t then wn is increasing in t. For the time-homogeneous model with discounting, i.e. a, v, k, r are constants, (i) Vðt; nÞ is concave in t while, in general, V ð0Þ ðt; nÞ is not; both functions are increasing in a, v and decreasing in k, r. (ii) pn ðt; nÞ is decreasing in t; it is increasing in a, v and decreasing in k and in r. (iii) wn ðt; nÞ is increasing in t; it is increasing in a, v, r and decreasing in k.

Figs. 1 and 2 illustrate properties of V and the optimal feedback controls pn and wn. Note, the concavity of V in n reflects the property that the opportunity cost ΔVðt; nÞ decreases with higher stock values. The properties of pn and wn can be observed daily at, for instance, fruit markets. Shortly before closing prices decrease and the “shouting” increases. The properties of optimal pricing and advertising strategies as described by Propositions 3.3 and 3.4 are well known among practitioners. The theory highlights and makes precise the fact that the time evolution and the state dependency of both policies are inversely related to each other. The general model (1) is characterized by the time dependent parameters a(t), r(t), v(t) and k(t) which influence the demand intensity, the discount rate and the effective revenue and advertising rates. Compared to the time homogeneous models considered in Gallego and van Ryzin (1994) and MacDonald and Rasmussen (2009), or the time inhomogeneous pricing model in McAfee and te Velde (2008), model (1) allows more complex applications and interpretations. For instance, the intensity parameter a(t) can be used to describe product life cycles, seasonal effects or impatient customers. Moreover, due to the discounting parameter r(t) it is possible to analyze problems over an infinite horizon. Finally, the revenue factor v(t) and the advertising factor k(t) can be interpreted as surcharges and/or subsidies. Their impact will be analyzed in greater detail in Section 7 in the context of social efficiency.

Fig. 1. Value function V ðt; nÞ; T ¼1, N¼ 10, ε ¼ 1:8, δ ¼ 0:2, a ¼ k ¼ v ¼ 1 and r ¼ 0. The top curve shows V ðt; 10Þ and the bottom one shows V ðt; 1Þ.

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Fig. 2. Optimal pricing policy pn and optimal advertising policy wn; T ¼ 1, N ¼10, ε ¼ 1:8, δ ¼ 0:2, a ¼ k ¼ v ¼ 1 and r ¼0. The top graph of the window on the left represents pn ðt; 1Þ while the graph at the bottom shows pn ðt; 10Þ; the right window shows, from top to bottom, wn ð; 10Þ; ⋯; wn ð; 1Þ.

There are parameter transformations of the standard model (1) which result, for instance, in model formulations without the explicit discounting term eRðtÞ ; there are other transformations which produce a “normal” form of the model, i.e. vðtÞ kðtÞ 1. Certain characteristics, however, often lack a natural interpretation in these transformed models. Moreover, when reversing the transformation, and the correct form of the original value function and optimal policies need to be derived with care. The standard parametrization (1) together with Theorem 3.1, however, make it easy to analyze the impact of different parameter choices and to study the effect of individual parameter variations. In the sequel, let R symbolize a given set of parameter functions a(t), v(t), k(t) and r(t). The functions are assumed to satisfy the conditions specified in ~ ~ refer to a second set of such functions denoted by aðtÞ, ~ ~ Theorem 3.1. Let R vðtÞ, kðtÞ and r~ ðtÞ. Let V R , pR and wR , V R~ , pR~ and ~ wR~ resp., denote the value function and optimal policies if R, R resp., is given. By inspection, see Theorem 3.1, the following identities hold, 0 r t rT, 0 rn rN: !ðγ1Þ=γ ~ ~ wR~ ðt; nÞ gðtÞ kðtÞ AðtÞ ¼ ~ AðtÞ wR ðt; nÞ gðtÞ kðtÞ

and

!1=γ ~ pR~ ðt; nÞ vðtÞ AðtÞ ¼ : ~ AðtÞ pR ðt; nÞ vðtÞ

ð15Þ

~ in greater detail. First, we examine the case of impatient customers. Let We consider two particular choices of R and R ~k k, v~ v and r~ r but assume aðtÞ ~ ¼ φðtÞaðtÞ, where φ is a positive, monotone decreasing function defined on ½0; T such that φð0Þ ¼ 1. The function φ reflects growing impatience of customers over time which results in a reduced intensity of customers willing to buy at a given price p. Time dependent accumulated discount expressions are special examples of such functions φ. The next proposition is an immediate implication of the identities (15). Proposition 3.5. Impatient customers imply lower optimal revenues and reduced optimal prices. For all n Z 1, the optimal advertising policy wR~ dominates wR at the beginning of the sales period ½0; T but drops below wR at the end if, for example, φðTÞ ¼ 0: wR~ ð0; nÞ w ~ ðT; nÞ 414 R ; wR ð0; nÞ wR ðT; nÞ the notation T-refers to taking the limit from the left. As a second application of Theorem 3.1 it is most instructive to consider the time-homogeneous infinite horizon problem with discounting. We start analyzing how the value function depends on r rðtÞ 4 0, if T ¼ 1. Since the optimal polices are stationary the optimal advertising rates and optimal prices are constant values between consecutive sales. Their values only depend on the number of unsold units prior to the corresponding sale. If T is finite the identities (15) and the corresponding ratio of the value function can be used to study the effect of variations of r for the cases with discounting and without discounting. Proposition 3.6. For the time inhomogeneous infinite horizon problem with constant discount rate r the optimal advertising rates wð1Þ and the optimal prices pð1Þ n n , 1 rn rN, are independent of t and are given by wð1Þ ¼ n

ε=ðεδÞ 1=γ δ g ðγrÞðγ1Þ=γ βn ε k

and

pð1Þ ¼ n

δ=ðεδÞ 1=γ δ 1 g 1=ðγ1Þ βn : ε v γr

and pð1Þ are the limit values of the initial prices and rates of a finite horizon problem when the Note, the values wð1Þ n n length of the horizon T tends to infinity. Thus, these expressions provide, for instance, (upper) bounds on initial prices, or suggest values of asking prices. In this context, the present value of a firm with N units to sell, while operating until all items are sold, equals ððη gÞ=ðγ rÞÞ1=γ . Hence, the elasticity of the value with respect to r is 1=γ, and optimal (feedback) prices are decreasing functions of r while advertising rates are increasing ones. The infinite horizon problem with constant discount rate r will again be taken up in the next section.


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4. Additional results In this section we shall exploit the fact that there is an explicit expression of the optimal sales intensity λn of the optimally controlled advertising & pricing model, cf. (6), (10) and Proposition 3.2, viz. λn ðt; nÞ≔aðt Þpn ðt; nÞε wn ðt; nÞδ ¼

where

gðtÞ γ=ðγ1Þ β ; AðtÞ n

ð0Þ _ gðtÞ AðtÞ A_ ðtÞ ¼ þ γ r ðt Þ ¼ ð0Þ ; AðtÞ AðtÞ A ðtÞ

Að0Þ ðt Þ≔eγRðtÞ Aðt Þ:

ð16Þ

ð17Þ

Formula (16) together with the dynamic Dorfman–Steiner identity (5) makes if possible to characterize and compute several quantities of interest, e.g. the total expected revenue and associated advertising expenditures, etc. Furthermore, for the time-homogeneous case there is a recursive way to compute the state probabilities qðt; nÞ≔P½X n ðtÞ ¼ n, where ðX n ðtÞÞt denotes the (optimal) process of the number of units still to be sold. Theorem 4.1. Let X n ð0Þ ¼ N. In the time-homogeneous case, i.e. v, k, a are positive constants and r Z0, the probabilities qðt; nÞ satisfy the recursion, 0 r t rT, 1 r n rN1, qðt; nÞ ¼

βγ=ðγ1Þ Z t γ=ðγ1Þ γrðTsÞ βn 1eγrðTtÞ n γ=ðγ1Þ 1e βn þ 1 qðs; n þ 1Þ ds; γr γr 0 γ=ðγ1Þ

where qðt; NÞ ¼ ð1e Þβ N 1eγrT γrðTtÞ

.

Proof. See Appendix A. Knowing the state probabilities qðt; nÞ is important for inventory management as well as for model evaluation. With qðt; nÞ at hand one can quantify the fluctuations of inventory levels about the average number of unsold items at any time t. Moreover, one can compute quantiles and design tests which will help in the process of model calibration. For general time dependent functions v(t), k(t), a(t) and r(t) which satisfy the conditions spelled out in Theorem 3.1 there is a most useful approximation of qðt; nÞ, see Proposition 4.1, below. Exploiting ideas of McAfee and te Velde (2008) we describe a binomial approximation of qðt; nÞ for the time inhomogeneous case, and an efficient way to simulate trajectories of the process ðX n ðtÞÞt . The approximation is very accurate if both n and T–t are “large”. It yields a simple way to approximate, for instance, the average inventory level. Proposition 4.1. For the general advertising and pricing problem let A(t) be defined by (10); we put, see (17), R t A_ ð0Þ ðsÞ R t gðsÞ Að0Þ ðtÞ ds ds Dðt Þ≔e 0 AðsÞ ¼ e 0 Að0Þ ðsÞ ¼ ð0Þ : A ð0Þ N

n Then, qðt; nÞ q~ ðt; nÞ≔ n DðtÞ ð1DðtÞÞNn , and E½X n ðtÞ N DðtÞ. Proof. See Appendix A. In light of Proposition 3.1 (ii), the approximation of qðt; nÞ by binomial distributions is kind of natural in the time homogeneous case without discounting. If the intensity depends linearly on the state variable n then the inventory process becomes a Yule process and the qðt; nÞ are binomially distributed. The Yule process is well known from the mathematical theory of evolution, see Feller (1968, Chapter 17),. The next theorem deals with the total optimal expected advertising expenditures of a firm as well as with the expected revenue of the business. The formulas are expressed in terms of the value function, see Theorem 3.1, and are derived by exploiting the dynamic Dorfman–Steiner identity (5). The theorem is supportive of the time-honored practice of setting the advertising budget equal to a fixed percentage of the expected revenue. R T 4 τ RðsÞ Theorem 4.2. Consider the model described in Theorem 3.1. We define W ðTÞ e kðsÞwn ðs; XðsÞÞ ds, i.e. the optimal N ≔ 0 R T 4 τ RðsÞ ðTÞ (random) advertising expenditure when N units are to be sold over the time interval ½0; T. Let U N ≔ 0 e vðsÞpn ðs; XðsÞÞ dXðsÞ denote the (random) revenue of the business. Then h i δ V ð0; N Þ E W ðTÞ ¼ N εδ

and

h i ε E U ðTÞ V ð0; NÞ: ¼ N εδ

Proof. Since for every t and n Z1 the Dorfman–Steiner identity (5) holds the following equation is satisfied along any trajectory ðXðtÞÞt : ε kðt Þwn ðt; X ðt ÞÞ ¼ vðt Þpn ðt; X ðt ÞÞλn ðt; X ðt ÞÞ δ

2824


3

εδ kðt Þwn ðt; X ðt ÞÞ ¼ vðt Þpn ðt; X ðt ÞÞλn ðt; X ðt ÞÞkðt Þwn ðt; X ðt ÞÞ: δ

Integrating over ½0; T 4 τ and taking expectation yields (i); similarly, we obtain (ii).

□

Remark 4.1. While it is possible to derive these formulas without using the DS-identity such a proof is fairly involved and rather complicated. The almost trivial proof of the result illustrates the power of the dynamic version of the Dorfman– Steiner identity. This power is also illustrated by the fact that Theorem 4.2 can be strengthened: the results hold for any initial time t0, 0 rt 0 o T, and stock level n, 1 r n rN. Remark 4.2. The formulas of Theorem 4.2 yield formulas for the infinite horizon case by simply taking the limit when T-1, cf. Remark 3.1. To study the distribution of the optimal (random) profit of the general time inhomogeneous model one can rely on simulation studies. There is an efficient way – in the pure pricing context already sketched in McAfee and te Velde (2008) – to efficiently simulate trajectories of ðX n ðtÞÞt . Based on the simulated trajectories it is straightforward to provide good estimates of all quantities (related to X n ðtÞ)of interest. The key idea of the efficient simulation algorithm is the observation that realizations of the random times of all sales can be generated as transformed values of independent uniformly distributed random values on ½0; 1. To this end, let τj , 1 r jr N, again denote the (random)time of the j-th sale; let τ0 ≔0 and γ=ðγ1Þ define θj ≔βNj þ 1 . By definition, the conditional probability that the j-th sale takes place before t, t rT, while the ðj1Þ-th sale happened at s, 0 r s ot, is given by Z t Ru λn ðy;Nj þ 1Þ dy n P½τj rtjτj1 ¼ s ¼ e s λ ðu; Njþ 1Þ du s Rt n λ ðu;Nj þ 1Þ du ¼ 1e s : ð18Þ ð0Þ Since the optimal intensity λn ðt; nÞ can be written as λn ðt; nÞ ¼ θNn þ 1 ðA_ ðtÞÞ=ðAð0Þ ðtÞÞ integrating λn , cf. (18), yields !θj Að0Þ ðtÞ : P½τj rt τj1 ¼ s ¼ 1 ð0Þ A ðsÞ

ð19Þ

θ1 j

Hence, ððAð0Þ ðτj ÞÞ=ðAð0Þ ðτj1 ÞÞÞθj is (conditionally) uniformly distributed on ½0; 1. Thus, Að0Þ ðτj Þ Z j Að0Þ ðτj1 Þ, where ð0Þ Z j Uð0; 1Þ, holds for 1 r j rN. Since eγRðtÞ gðtÞ ¼ A_ ðtÞ is positive the function t↦Að0Þ ðtÞ ¼ eγRðtÞ AðtÞ has an inverse ð0Þ1 function A ðtÞ which is positive and strictly monotone decreasing. Using the existence of the inverse function of Að0Þ and the strong Markov property of the sales process Xn the distributional properties of ðτj Þj are a consequence of (19). Proposition 4.2. Let ðZ j Þ1 r j r N be independent uniformly distributed random variables on ½0; 1. Let τ0 ¼ 0 and define τj , 1 rj rN, recursively, θ1

τj ¼ Að0Þ1 ðZ j j Að0Þ ðτj1 ÞÞ: Then the joint distribution of this sequence of isotone random variables is identical to the joint distribution of the jump times of the optimally controlled processes Xn. In particular, τN oT a.e. δ

In the time homogeneous case Að0Þ1 ðyÞ can be explicitly expressed by an elementary formula. Let g≔ða ðvε =k ÞÞ1=ð1δÞ . Note, for 0 r y rAð0Þ ð0Þ ¼ g=rγð1erγT Þ the inverse function is given by 1 rγ Að0Þ1 ðyÞ ¼ ln y þ erγT : γr g

Remark 4.3. In the time homogeneous case Proposition 4.2 yields the recursion,

θ1 1 1eγrðTτj1 Þ ; 1 r j rN: τj ¼ τj1 ln 1 1Z j j γr If T ¼ 1 and r 40 this expression simplifies to τj ¼ τj1 þ

θ1 1 1 j ln Z j j lnðZ i Þ; ¼ ∑ θ1 γr γr i ¼ 1 i

if r ¼0 and T o 1 we obtain ! j θ1 θ1 j i τj ¼ τj1 þ ðTτj1 Þ 1Z j ¼ T 1 ∏ Z i : i¼1


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Fig. 3. A few simulated optimal price trajectories and associated advertising trajectories of the finite horizon problem; T ¼1, N¼ 10, ε ¼ 1:8, δ ¼ 0:2, a ¼ k ¼ v ¼ 1 and r ¼0.

The simple expressions of Að0Þ1 make it possible – in particular for the infinite horizon case – to efficiently simulate sales trajectories and to study various distributional aspects of the advertising and pricing problem. It is again instructive to look at the discounted problem over an infinite horizon in greater detail. We assume that r and a are positive constants. For a given stock level N let s1 o s2 o …o sN , sj A ½0; 1Þ, denote the random times of the first sale, the second sale, etc. of the optimal sales process. It follows from Theorem 3.1 and Proposition 3.2 – allow T-1 – that the optimal price of the j-th item to be sold is given by, cf. Proposition 11, ε 1 ηg 1 1=γ ≔ ; pð1Þ j εδv rγ θj

1 rj r N:

This (finite)sequence of prices is strictly monotone increasing. By definition, the expected present value E½U ð1Þ N of the total revenue ð1Þ rsj v ∑N e can be expressed as the sum of the product of “averaged discount factor” times price multiplied by v, j ¼ 1 pj ε 1 ηg 1 1=γ rsj E e : j ¼ 1 εδ v rγ θ j

h i N ¼v ∑ E U ð1Þ N

Using the formula of the value function and Theorem 4.2 we can also express the left hand side as h i ηg 1=γ ¼ βN : E U ð1Þ N rγ These two equations imply the identity N

1=γ

βN ¼ ∑ θj

E½ersj ;

ð20Þ

j¼1

from which the identity (v) of Proposition 3.1 can be derived, see Appendix A. If the horizon is infinite optimal price trajectories will always be increasing step functions. The jumps, of course, occur at random times. If, however, the horizon is finite randomness together with the definite deadline T cause price fluctuations. Fig. 3 displays a few simulated price trajectories and associated advertising trajectories which illustrate some of their characteristic properties. Given the stylized market environment described by (1) and (16) prices are typically more volatile at the end of the selling period. Price trajectories are piecewise continuous paths consisting of segments of the graphs of the family of functions pn ð; nÞ, see Fig. 2. Between sales, prices are decreasing while advertising rates are increasing. Right after each sale there is a price jump and a rate drop. Depending on how quickly the initial stock level N decreases individual prices can be high. Due to the random nature of the purchasing times price fluctuations can be “tame” or rather “wild”, see Fig. 3, left panel. The last segment of each price trajectory will be a part of the graph of pn ðt; 1Þ. Thus, although decreasing in time, in principle, the last price can be any number in the interval ð0; pð0; 1ÞÞ. The two windows of Fig. 3 nicely display the coordinated evolution of optimal advertising spending and price adjustments. The advertising trajectories are kind of a mirror image of the associated price trajectories. Thus, upward, downward resp., price movements go together with downward, upward resp., advertising spending; the two marketing activities affect the sales intensity in a synchronized manner.

5. Waiting and expected prices In this section we want to examine if at any time t (assuming the optimal policy is applied) an individual customer should buy or, instead, wait for better prices to come. The principle which we will use when evaluating waiting is based on the slope of the (conditional) expected prices pn, i.e. the derivative of the function s↦E½pn ðs; X s ÞjX t ¼ n, s 4 t, at t. We shall say that waiting pays off if this slope is non-positive. Obviously, if the expected price asked at time t þh, h 40, is higher than the current price waiting is not profitable. Since a sale takes place (approximately)with probability λðtÞh within time t and

2826


t þ h this comparison is equivalent to the strict inequality pn ðtÞ o E½pðt þ h; X ðt þ hÞ ÞjXðtÞ ¼ nÞ 3

pn ðtÞ oλn ðt; nÞhpn1 ðt þ hÞ þ ð1λn ðt; nÞhÞpn ðt þhÞ þoðhÞ:

ð21Þ

Letting h-0 we see that this strict inequality follows from p_ n ðtÞ o λn ðt; nÞðpn1 ðtÞpn ðtÞÞ;

ð22Þ

but (22) is equivalent to 1 1 _ gðtÞ γ=ðγ1Þ 1=ð1γÞ 1=ðγ1Þ 1=ðγ1Þ β o AðtÞγ βn1 βn A ðt ÞAðtÞð1γÞ=γ βn γ AðtÞ n γrðtÞAðtÞ 1 ðγ þ 1Þ=ðγ1Þ 1=ðγ1Þ 1=ðγ1Þ o βn : ⟺ 1 βn1 βn gðtÞ γ

ð23Þ

For all positive parameter functions a(t), k(t), v(t) and discount rates r(t) the inequality γrðtÞAðtÞ 1 1 1 r gðtÞ γ γ trivially holds. Since 1 γ=ðγ1Þ 1=ðγ1Þ 1=ðγ1Þ β o βn βn1 1 γ n is equivalent to inequality ðnÞ of Proposition 3.1 (iv) (23) is always satisfied, and the following statement is a qualification of Theorem 5.1 in McAfee and te Velde (2008), see also Nerlove and Arrow (1962). Theorem 5.1. Waiting, in the sense defined above, is never profitable if the inventory level is larger than or equal to 2. The situation when n¼ 1 is also unfavorable for a customer since he might not be able to buy at all should he wait for any period of length h, h 4 0. For n 41 it follows from Theorem 5.1 that for any time interval ½t 0 ; T, 0 r t 0 o T, the (conditional, i.e. there is at least one unit left) expected prices on ½t 0 ; T are increasing for a while. But since all prices are bounded from above by pðt; 1Þ we also know that the (conditional) expected prices are always decreasing to zero if t-T (see Fig. 4). Thus, for risk prone buyers waiting a “long enough” period of time can be profitable; but these customers have to deal with the increasing risk of not being able to purchase an paper at all. Fig. 4 shows the plot of pn ðt; 1Þ, the plot of the average price charged (conditioned that at time t there is still at least one unit left to be sold) and the plot of the run-out probability qðt; 0Þ for two values of r, viz r ¼0 and r¼ 1 (second window). Numerical computations show the graph of the function of (conditional)expected prices, depending, of course, on the chosen parameters, always has a unimodal shape. For different parameter values and, in particular, specific choices of time dependent functions a(t) the hump of E½pt jX t 4 0 can be more pronounced than the one displayed in Fig. 4. Furthermore, if N is large and r is small the time a customer has to wait until lower prices can be expected is usually close to T (see Fig. 4). Therefore, it depends on an individual's evaluation of expected prices and the probability of getting no paper at all whether or not the customer is going to wait or not. Since one is able to compute (see Theorem 4.1), or approximate (see Proposition 4.1) or simulate (cf. Proposition 4.2) the run-out probabilities qðt; 0Þ as well as the (conditional) expected prices Eðpn ðt; X ðt ÞÞX ðt Þ 4 0Þ ¼

N 1 ∑ qðt; nÞpn ðt; nÞ; 1qðt; 0Þ n ¼ 1

individual buyers can rationalize their “waiting” strategies. Phrased in a simplified way, waiting is only profitable for customers who are kind of indifferent about being able to buy an paper or not. Consequently, for such last minute-customers it would be optimal to wait almost until time T.

Fig. 4. The price envelope pn ðt; 1Þ, (conditional) expected prices and run-out probabilities as functions of time for two values of r, viz. r ¼0 and r ¼1 (window on the right); the other parameters are T ¼ 1, N ¼ 10, ε ¼ 1:8, δ ¼ 0:2, a ¼ k ¼ v ¼ 1.


2827

The statement of Theorem 5.1 should be compared with the following result. It states that in the finite horizon case without discounting, with discounting resp., the finite sequence of optimal expected sales prices, the expected discounted sales prices resp., is monotone decreasing. At first glance this result seems to contradict the statement about “waiting”. Taking a closer look at both statements reveals the differences between the two results. If the inventory is larger or equal to two then there can be, within any (short)interval, a sale accompanied by an increase in price. Theorem 5.2, however, is about the optimal spacing between consecutive sales and the optimal expected prices of the individual sales of the optimally controlled inventory process. Theorem 5.2. Assume that the conditions of Theorem3.1 hold, and let pj ≔pn ðτj ; Nj þ 1Þ denote the optimal (random) price of the j-th sale. The sequence of expected discounted sales prices ðE½eRðτj Þ pj Þ1 r j r N is monotone decreasing. Proof. In light of (19) and by Theorem 3.1, the quotient of any two consecutive discounted sales prices can be written as, γ=ðγ1Þ θj ≔βNj þ 1 , 1 r ir N, !ðθj þ 1 Þ=ðγθj þ 1 Þ eRðτj þ 1 Þ pj þ 1 θj 1=γ Að0Þ ðτj þ 1 Þ ¼ : θj þ 1 eRðτj Þ pj Að0Þ ðτj Þ Since ðAð0Þ ðτj þ 1 Þ=Að0Þ ðτj ÞÞθj þ 1 is uniformly distributed, see (19), we obtain " " # h i eRðτj þ 1 Þ pj þ 1 τj E eRðτj þ 1 Þ pj þ 1 ¼ E eRðτj Þ pj E eRðτj Þ pj ! 1=γ h i θj 1 ¼ E eRðτj Þ pj : 1 θj þ 1 1 þ γθj þ 1 According to Proposition 3.1 (iv) the first factor of the product is less than one.

□

6. Social efficiency In this section we are going to examine whether the optimal pricing and advertising strategies of a monopolist are socially efficient. More precisely, we are going to identify conditions and model characteristics such that the optimal policies yield efficient prices. To this end, we consider the welfare maximizing variant of model (1). The objective function involves the expected consumer surplus (ECS) as well as the producer surplus (PS). For the special model under consideration the sum of the price p and the expected consumer surplus ECS(p) is given by Z 1 xεxε1 ε p p ¼ pþ dx ¼ ε1 ε1 pε p for any t, from which the expression ECSðpÞ ¼ p=ðε1Þ follows. The producer surplus equals PSðt; pÞ ¼ vðtÞp. Let Sðt; nÞ denote the value function of the welfare maximizing advertising and pricing problem. The function S satisfies the following Bellman equation together with the boundary conditions, SðT; nÞ ¼ 0, 0 r n rN, and Sðt; 0Þ ¼ 0, 0 rt r T: _ nÞ þmaxfaðtÞwδ pε ððvðtÞ þ ðε1Þ1 ÞpΔSðt; nÞÞkðtÞwg: rðtÞSðt; nÞ ¼ Sðt; p;w

ð24Þ

~ ~ ¼ fvðtÞ; ~ Eq. (24) reveals that the welfare problem is a particular case of model (1) with parameters R kðtÞ; aðtÞ; rðtÞg, where ~ ~vðtÞ≔1=ðε1Þ þ vðtÞ and kðtÞ≔kðtÞ. Hence, the following results concerning social efficiency are a consequence of Theorem 3.1. Proposition 6.1. If either the revenue parameter v(t) or the advertising parameter k(t) is time dependent then n n n (i) in general, a firm's strategies are not efficient, i.e. wnR ðtÞ a wR ~ ðtÞ and pR ðtÞ a pR ~ ðtÞ; this is even the case if δ ¼ 0, i.e. in the pure pricing model; (ii) socially efficient actions by a monopolist will be guaranteed if his revenue parameter is adjusted to vadj ðtÞ≔vðtÞ þ ðε1Þ1 , for instance, by an appropriate subsidy. If ε gets bigger then the subsidy gets smaller.

Adjusting the parameter v, see above, can be interpreted in different ways. One possibility is to think of lowering a valueadded tax, i.e. increasing v. The adjustment described in Proposition 6.1 (ii) is a special one. Additional adjustments of time independent parameters vadj and kadj which yield socially efficient strategies wnadj and pnadj are suggested by (12)–(14). The reason is that these identities allow models with different parameters to be compared with each other. For time ~ s. above, as well as Radj ¼ fvadj ; kadj ; aðtÞ; rðtÞg, independent coefficients v and k, and for parameter sets R, ! !ε=ðεδÞ δ=ðεδÞ pnadj ðt; nÞ wnadj ðt; nÞ vadj k~ vadj k~ δ¼0 ¼ ¼ 1 and : pn~ ðt; nÞ wn~ ðt; nÞ kadj v~ kadj v~ R

R

2828


Hence, a monopolist's actions are socially efficient whenever vadj v~ 1=ðε1Þ þ v : ¼ k kadj k~

ð25Þ

Proposition 6.2. For the pure pricing model a firm's prices are socially efficient for all time independent revenue parameters v; see McAfee and te Velde (2008) for the case without discounting and v ¼1. If δ 4 0, (25) specifies a whole family of pairs ðvadj ; kadj Þ of positive parameters which imply socially efficient actions. Possible feasible pairs are appropriate combinations of, for instance, tax decreases and subsidies of advertising activities, or increased charges on advertising spending combined with tax decreases. The different expected transfer payments PTR and WTR (transfer payments to the firm have positive sign, payments by the firm have negative sign) due to the adjustment of v and k depend on vadj and kadj. They amount to, cf. Theorem 4.2, V adj ≔V adj ð0; NÞ,

ε V ; P TR ¼ 1v=vadj E U adj ¼ 1v=vadj εδ adj

δ V : W TR ¼ k=kadj 1 E W adj ¼ k=kadj 1 εδ adj It follows from (13), see Theorem 3.1, and (25) that for any such pair ðvadj ; kadj Þ the value Vadj is proportional to vadj, i.e. V adj ¼ const vadj , const 40. Simple algebra shows the difference of a firms profit and the sum of expected transfer payments ð0Þ is constant. Pairs ðvð0Þ ; k Þ for which expected transfer payments are compensating, i.e. TRadj ≔P TR þ W TR ¼ 0, and prices are adj adj socially efficient are of special interest. Such pairs satisfy (25) as well as the equation v δ k δk ; ð26Þ 1 ⟺ kadj ¼ 1 vadj ε kadj ε v=vadj ðεδÞ ð0Þ

ð0Þ ; kadj Þ which where vadj and kadj are positive. Elementary calculus implies that there is a unique pair of positive numbers ðvadj satisfies both conditions as long as v exceeds δ=ððεδÞ ðε1ÞÞ, viz., δ ε=ðεδÞ ð0Þ ð0Þ and kadj ¼ k 1 : vadj ¼ v ðεδÞðε1Þ 1 þ vðε1Þ

Our next theorem summarizes the various results obtained in this section. Theorem 6.1. Let v and k be positive constants.

(i) A firm's strategy becomes socially efficient by choosing vadj 1=ðε1Þ þ v ; vadj ; kadj 4 0: k kadj (ii) For all efficient regulations, the sum of a firms adjusted expected profit and the total expected transfers TRadj as well as the total expected consumer surplus ðTECSÞ is constant. The latter amounts to

1 ε E U adj ¼ V adj : TECS ¼ ðε1Þvadj ðε1ÞðεδÞvadj The optimal total expected welfare S equals TECS þ V adj TRadj . The ratio V=V adj satisfies !1=ðεδÞ 1=ðεδÞ δ V vε kadj vε ð25Þ 1 ¼ ε ¼ : δ δ V adj vadj ð1=ðε1Þ þvÞ k vadj (iii) Let v 4δ=ððεδÞðε1ÞÞ. The expected transfer costs are zero if δ ε=ðεδÞ ð0Þ and k : ≔v ≔k 1 vð0Þ adj adj ðεδÞðε1Þ 1 þ vðε1Þ ð0Þ

This adjustment requires a revenue tax ðvð0Þ ovÞ and an advertising subsidy ðkadj okÞ. adj Note, there are socially efficient adjustments when the firm actually receives money, and other such adjustments when the firm has to pay money. An extreme case is the following one. If ðvadj ; kadj Þ goes to zero while (25) holds then (almost all of) a monopolists profit is taxed away! 7. The deterministic model with continuous state space In this final section we consider the advertising and pricing problem when a firm is selling a finite amount of an infinitely divisible product in a deterministic environment. The problem is closely related to the stochastic model considered in the


2829

previous sections. To be specific, we consider a deterministic control problem on ½0; T with continuous state space R þ ¼ fx A Rjx Z0g. The evolution of the state x(t)is characterized by, xð0Þ ¼ N A R, w(t)and p(t)admissible controls, ( λðt; pðtÞ; wðtÞÞ; t oτ _ ¼ xðtÞ 0; t Zτ; where τ≔infftjt A ½0; T; xðtÞ ¼ 0g. Again, cf. Section 2, admissible controls are supposed to satisfy conditions such that the control problem, see below, is well defined and the sufficient optimality conditions can be applied; see Fleming and Soner (2006, Chapter 1), for the technical details. Controls are to be chosen such that Z T 4τ eRðtÞ ðvðtÞ pt λðt; pt ; wt ÞkðtÞ wt Þ dt ð27Þ 0

is maximized. The Bellman equation for the value function Vðt; xÞ – we denote by V_ , V' resp., the time derivative, space derivative resp., of V – is given by V_ ¼

max faðtÞpε wδ ðvðtÞpV′ÞkðtÞwrðtÞVg;

p 4 0;w Z 0

together with the terminal condition VðT; xÞ ¼ 0, N Zx Z 0, and the boundary condition Vðt; 0Þ, 0 r t rT. Assuming V′ðt; xÞ to be positive whenever x 40, t rτ, the necessary optimality conditions of the 2-dimensional nonlinear optimization problem in p and w yield, cf. (3) and (4), 1=ð1δÞ ε aðtÞvðtÞδ ^ ðt; xÞ ¼ ^ xÞðε1Þ V′ðt; xÞ and w pðt; : ð28Þ p^ ðt; xÞ ¼ vðtÞðε1Þ kðtÞε ^ xÞ and wðt; ^ xÞ actually are unique maximizers for each t and x. Using these expressions Elementary analysis shows that pðt; ^ and p^ the Bellman equation turns into the 1st order nonlinear partial differential equation, cf. (7), for w η γ1 γ1 0 ¼ V_ þ g ðt Þ ðV′Þ1γ r ðt Þ V ð29Þ γ γ with the boundary conditions described above. We can solve this PDE by mimicing the derivation of the value function in the case of a discrete state space and a stochastic evolution of the initial inventory. From the mathematical point of view, the difference between the two cases amounts to replacing the nonlinear difference equation for the β-factor by a Bernoulli equation, cf. Section 3. The following proposition summarizes these observations. Proposition 7.1. Eq. (29) has a unique C1-solution Vðt; xÞ which is separable in t and x, i.e. Vðt; xÞ ¼ αðtÞ βðxÞ, where αðtÞ ¼ ðη AðtÞÞ1=γ , cf. Theorem 3.1, and the function βðxÞ, x 40, is a solution of the Bernoulli equation, βð0Þ ¼ 0, βðxÞ1=ðγ1Þ β′ðxÞ ¼ 11=γ:

The solution of this differential equation is given by βðxÞ ¼ xðγ1Þ=γ . The solution formula of the value function Vðt; xÞ yields ^ and p^ in feedback form. These formulas are identical to (12)–(14) except that the term βn needs to be the optimal controls w ^ xÞ; wðt; ^ xÞÞ ¼ gðtÞ=AðtÞ xðtÞ, and the optimal trajectory of the replaced by xðγ1Þ=γ . These formulas imply that λðt; pðt; deterministic control problem satisfies the ODE, see (17), ð0Þ _ xðtÞ A_ ðtÞ ¼ ð0Þ ; xðtÞ A ðtÞ

xð0Þ ¼ N:

ð30Þ

Hence, by Proposition 4.1 the optimal inventory level, as a function of time, is given by xðt Þ ¼ xð0ÞDðt Þ ¼ N

AðtÞ γRðtÞ hom eγrt eγrT Tr o¼ 10 e ¼N ¼ Að0Þ 1eγrT

N N t: T

ð31Þ

_ Eq. (30) captures the property that at any time t the relative rate xðtÞ=xðtÞ of the optimal inventory trajectory equals the ð0Þ relative rate A_ ðtÞ=Að0Þ ðtÞ of the sales potential. For the case without discounting the optimal inventory level decreases linearly with rate N=T from N to zero over the interval ½0; T. For the discounted case the selling rate is higher at the beginning of the sales period and decreases over time, see Fig. 5. When the horizon is infinite the inventory level N decreases ^ exponentially to zero on ½0; 1Þ at rate γr. The explicit solution of the inventory level x together with the feedback form of w and p^ provide solution formulas of the (optimal)open-loop controls w and p as functions of t, viz., ε=ðεδÞ δ gðtÞ N ðγ1Þ=γ ðγ1ÞRðtÞ ^ ðt; xðt ÞÞ ¼ w ðt Þ≔w e ; ε kðtÞ Að0Þ δ=ðεδÞ ε ηAðtÞ 1=γ δ 1 Að0Þ 1=γ RðtÞ p ðt Þ≔p^ ðt; xðt ÞÞ ¼ ¼ e ; vðtÞðε1Þ xðtÞ ε vðtÞ N where xðtÞ ¼ NðAðtÞ=Að0ÞÞeγRðtÞ , cf. (28) and (31). Hence, for time homogeneous models with discounting, i.e. r 40 and T o1 or T ¼ 1, the optimal advertising rate w is an exponentially decreasing function with rate ðγ1Þr. The optimal prices

2830


Fig. 5. Optimal inventory paths (first window) and corresponding optimal open-loop price trajectories pðtÞ; r¼ 0 vs. r ¼1, and T ¼ 1 vs. T ¼ 1. The other parameters are N ¼10, ε ¼ 1:8, δ ¼ 0:2, a ¼ k ¼ v ¼ 1.

increase exponentially with rate r, exactly counterbalancing inflation. Note, if r ¼0 and T o1 the optimal advertising rate and the optimal price are constant over time! The second window of Fig. 5 shows the optimal open-loop controls p for three parameter pairs (r,T), viz. ð0; 1Þ, ð1; 1Þ and ð1; 1Þ. The trajectories pðtÞ, see Fig. 5, are to be compared with the expected prices of the stochastic model, see Fig. 4. The graphs illustrate the fact that the time evolution of expected optimal prices in the stochastic case is similar to the evolution of the trajectories pðtÞ, expect when t gets close to T. The (natural) deviation of the price trajectories at the end of the sales period is due to the discrete state space of the stochastic model and the continuous state space in the deterministic one. The fixed advertising and pricing policy of the deterministic model specifies a sub-optimal pair of strategies ðwt ; pt Þ ðw; pÞ of the stochastic model considered in Section 3. Like in the case of Gallego and van Ryzin pure pricing model and the advertising and pricing model analyzed in MacDonald and Rasmussen (2009), we can compute the gap between the optimal value, see Theorem 3.1, and the value corresponding to the sub-optimal controls. The value of the expected revenue minus the expected advertising cost for this sub-optimal policy pair can be computed using well known results for a standard Poisson process, see MacDonald and Rasmussen (2009) for detailed expressions. To conclude, we like to point out a close relationship between some classical marketing problems concerning the optimal control of the market share of a durable product and the problem considered in this section. This relationship reveals additional applications of our deterministic model. Let N ¼1, and think of the controlled inventory process x(t) to represent the untapped market share of a (durable) product. The model considered in this section is a basic one; it illustrates this relationship. General deterministic adoption problems with state dependent dynamic are analyzed in Helmes et al. (2013). Differential game extensions of these models will be analyzed in Helmes and Schlosser (2013).

Acknowledgments The authors would like to express their appreciation to the referees for their careful reading of the original paper and valuable comments. Appendix A Proof of Proposition 3.1. 1=ðγ1Þ

ðβn βn1 Þ ¼ ðγ1Þ=γ. Since βn , n Z1, is positive and ðγ1Þ=γ is positive too, the strict (i) By definition, βn inequality βn 4 βn1 , n Z 1, follows from the defining identity. (ii) We recall the property, see the paragraph following Proposition 3.1, γ=ðγ1Þ

βj

j

r1 ⟺

j γ=ðγ1Þ

βj

Z 1;

j Z 1:

Moreover, for any given n Z1, the function x↦1x I ½1;n þ 1 ðxÞ, I ½1;n þ 1 ðxÞ the indicator function of the interval, is dominated from above by the step function which has the value 1=i on the subinterval ½i; i þ1, 1 r ir n, and zero 1=ðγ1Þ else. Taking the telescoping sum of all differences βj βj1 ¼ ððγ1Þ=γÞβj we obtain for any n Z1, γ1 n γ1 n γ=ðγ1Þ 1=γ γ=ðγ1Þ 1=γ 1=γ ∑ β ∑ jβj ¼ j βn ¼ γ j¼1 j γ j¼1 1=γ Z γ1 n 1 γ1 n þ 1 1 1=γ ∑ Z Z dx ¼ ðn þ 1Þðγ1Þ=γ 1; γ j¼1 j γ 1 x from which (ii) follows. 1=ðγ1Þ (iii) According to (i) the sequence ðβn Þn Z 0 is strictly increasing for all γ 41. Since βn ðβn βn1 Þ ¼ γ=ðγ1Þ, n Z1,


2831

and the right hand side of this identity is a positive constant while the first factor of the left hand side of the identity is positive and strictly increasing in n, the difference ðβn βn1 Þ is positive and strictly decreasing in n. (iv, (n)) The identity (11) can be written as γ1 γ=ðγ1Þ βn ; ðA:1Þ βn1 ¼ βn 1 γ and (A.1) will be repeatedly applied in the sequel. By (A.1), inequality (iv, ðnÞ) is equivalent to 1 γ=ðγ1Þ ðγ1Þ γ1 γ=ðγ1Þ βn 1 þ βn 4 1 : γ γ

ðA:2Þ γ=ðγ1Þ

Next, we introduce the real valued function φðxÞ≔ðγ1Þx þ ð1 þxÞðγ1Þ , x Z0, and we think of x≔ð1=γÞβn 40 as a particular argument. It follows by elementary calculus that φ has a unique global minimum at x¼0 with value 1, from which (A.2) follows. (iv, (nn)) Using (A.1) once more, (iv, ðnnÞ) is equivalent to γ1 γ=ðγ1Þ γ=ðγ1Þ γ=ðγ1Þ βn o 1 þ βn1 : 1 γ Thinking of ðβn1 Þ=βn ≕y A ð0; 1Þ as a particular argument and using the relation ðγ1Þx ¼ 1y the last inequality is equivalent to y γ=ðγ1Þ o 1 þ γxy γ=ðγ þ 1Þ . This inequality is a particular case of the following functional inequality, 0 r y r1: γ ð1yÞ≕ΦðyÞ; 1 ryγ=ðγ1Þ þ γ1 which is strict if y is an element of the open interval (0,1). Note, the function Φ is monotone decreasing on ½0; 1 and takes on a value bigger than one if y ¼0, and equals 1 if y ¼1; this proves inequality ðnnÞ. (v) By definition, see end of Section 4, each τj , 1 r j rN, is the sum of independent exponential random variables Δi τ ¼ τi τi1 , 1 ri rN, τ0 ≔0, with intensity μi ≔ðrγ=gÞθi . Simple algebra shows that the Laplace transform of τk can be expressed as γ=ðγ1Þ

j βNi þ 1 θi ¼ ∏ γ=ðγ1Þ : θ þ1=γ i β i¼1 i ¼ 1 Ni þ 1 þ 1=γ j

∏

Together with (20) this equation implies Proposition 3.1 (v).

□

Proof of Theorem 3.1. We would like to solve 0 ¼ α_ ðt Þ þ g ðt ÞηγαðtÞ1γ r ðt Þαðt Þ with the terminal condition αðTÞ ¼ 0. Rewriting the differential equation in the form αðtÞrðtÞαðtÞ _ ¼ gðtÞðη=γÞαðtÞ1γ and using the substitution ZðtÞ ¼ αðtÞγ we get the RT _ equivalent ODE Z ðtÞγrðtÞZðtÞ with theterminal condition ZðTÞ ¼ 0. Since its solution is ZðtÞ ¼ eγRðtÞ t eγRðsÞ ηgðsÞ ds ¼ ηgðtÞ 1=γ R T we obtain αðtÞ ¼ ZðtÞ1=γ ¼ ηeγRðtÞ t eγRðsÞ gðsÞ ds . □ Proof of Theorem 4.1. By definition, qð0; NÞ ¼ 1 and hom q_ ðt; NÞ ¼ λn ðt Þqðt; NÞ ¼

γr γ=ðγ1Þ β qðt; N Þ: 1eγrðTtÞ n

Hence, for n ¼N,

qðt; NÞ ¼ e

0

λ ðs;NÞ ds hom

γ=ðγ1Þ

βN

¼e

Rt

¼e

Rt

γ=ðγ1Þ γr β 0 1eγrðTsÞ N

ðlnð1eγrðTtÞ Þlnð1eγrT ÞÞ

ds

1eγrðTtÞ ¼ 1eγrT

βγ=ðγ1Þ N

:

For 0 r n rN1, qðt; nÞ satisfies the linear nonhomogeneous ODE γr γ=ðγ1Þ γ=ðγ1Þ βn þ 1 qðt; n þ 1Þβn qðt; nÞ q_ ðt; nÞ ¼ 1eγrðTtÞ whose solution can be written as qðt; nÞ ¼

βγ=ðγ1Þ Z t γ=ðγ1Þ 1 γrðTsÞ βn 1eγrðTtÞ n γ=ðγ1Þ 1e βn þ 1 qðs; n þ 1Þ ds: γr γr 0

□

~ N þ 1Þ : 0 and let qðt; ~ nÞ, 1 r n rN, denote the solution of the system of ODEs Proof of Proposition 4.1. Let qðt; ð0Þ A_ ðtÞ q~_ ðt; nÞ ¼ ð0Þ ððn þ 1Þq~ ðt; n þ1Þnq~ ðt; nÞÞ A ðtÞ

2832


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