Inventory Competition with Subscriptions Hyun-soo Ahn Stephen M. Ross School of Business, University of Michigan, Ann Arbor, MI, 48109, USA E-mail:
[email protected] Tel: 734-764-6862
Tava Lennon Olsen Olin School of Business, Washington University in St Louis, St. Louis, MO 63130-4899, USA
[email protected] Tel: 314-935-4732
1
Inventory Competition with Subscriptions
Abstract
This paper presents a model of competition in industries where some customers will purchase a good or service repeatedly (i.e., subscribe). We present a duopoly model that explicitly considers the customers’ repeat purchase behavior as well as the effect of product availability in an overlapping market. Two firms compete at the time of initial purchase through product availability and then later through subscription price. In each period, some customers buy the off-the-shelf product from traditional channels, and those who are happy with the experience and the offered subscription price will subscribe to purchase on a regular basis. Using a Markov game framework, we model this as a multi-period game and analyze the structure of any Markov perfect pure strategy equilibria. We also discuss how changes in the firm’s internal (or external) conditions as well as the number of subscribers affect the firms’ decisions and profit. A numerical study shows that taking into account both the effect of subscriptions and the degree of competition when making operational decisions can substantially improve a firm’s profit.
Keywords: Inventory Competition, Subscription, Multi-Period Game
2
1
Introduction
This paper presents a model of competition in industries where customers have the option to subscribe to a good or service after an initial consumption through traditional channels. In this type of marketplace, the ability to retain existing customers, and therefore secure a customer base and revenue, is imperative for a firm’s long-term survival and growth. We consider the lingering impact of attracting (or losing) customers for multiple periods and the efficacy of using operational strategies, such as product availability and subscription price, to affect customers’ behavior. Unlike research in which consumers form implicit and non-binding preference for a product or a brand in consumer product markets (see, e.g., Agrawal, 1996; Erdem and Kean, 1996; Noble and Gruca, 1999), we will focus on industries where the repeat purchase is made through some form of explicit subscription. Examples of products or services in this category include magazines, newspapers, Internet service providers, primary care health services, and media rental services (e.g., Netflix). Therefore, the price and service dynamics occur both at the time at which the initial trial purchase is made (where the product must be found to be available) as well as at the time at which the initial purchase is materialized into a subscription. The canonical example of a product fitting the above framework is a magazine or newspaper. In each month (or an equivalent time period), new customers will buy off-the-shelf product and such purchases often lead to subscriptions. At the operational level, the publisher must therefore decide how much inventory to stock for walk-in customers at newsstands (either by making the decision directly or inducing a distributor to stock to a desired level through a contract) and how to price subscription purchases to entice customers with a positive experience to commit. Results from empirical studies (c.f. Cecchetti, 1986; Willis, 2000) show that these two decisions are ones that the firm has control over and can alter more frequently than the newsstand price, which requires coordination at many levels to change. For some magazines, such stocking decisions are made directly by the publisher or its subsidiary. For example, Time Warner Inc. oversees distribution of its magazines through its subsidiary, Time Distribution Services. For other magazine companies, a single (or small number of) distributor(s) is (are) usually responsible for stocking and collecting all the company’s magazines available at retail channels in a fairly large local market. The distributor works as a middleman providing logistic service for publishers. Although the distributor is an agent on his/her own, many publishers now offer more sophisticated contracts, such as buy-back or revenue sharing contracts, which allow a publisher to induce the distributor to act in a desired way. For most magazines, when the publisher
3
decides the newsstand quantity it will then also commit to collect unsold past issues either directly or through a local distributor (Koschat et al., 2003). For most popular magazines, it is usual to see sales through subscription comprising a considerable portion of the publisher’s revenue. Some publishers offer a large discount (e.g., over 50% for Sports Illustrated and Cosmopolitan) for subscription purchases. According to data collected by the Audit Bureau of Circulations (ABC), approximately 85% of magazines were sold via subscriptions in 2002 (i.e., 305 million copies through subscription versus 54 million single-copy sales). The same data also shows that the subscription revenue (excluding advertising revenue, which also depends heavily on the number of subscribers) accounts for 70% of total circulation revenue for all ABC magazines. In addition to subscription revenue, for many publishers the first and foremost reason that the number of subscribers is important is its influence on advertising revenue. For example, the top five magazines in terms of circulation revenue in 2002, namely, TV Guide, People, Reader’s Digest, TIME, and Sports Illustrated, received on average 56% of gross revenue from advertising in 2001 (source: Magazine Publishers of America). Typically, the price for advertisement space is determined by the rate base, which is a price agreed upon when the magazine publisher maintains a certain circulation level over a given audit period. Failure to meet the circulation level often incurs costly penalties, such as makegood (e.g., giving out free advertising space) or explicit financial penalties (see, e.g., Lukovits, 2002). The fact that subscribers significantly outnumber single-copy sales makes subscribers the dominant influence on advertising price for many magazines. Since the rate is determined by the circulation level, a publisher with a larger number of subscribers is able to charge more than a publisher with a smaller number of subscribers for the same advertisement within the same category of magazine. In addition to subscription demand being much larger than single-copy demand, the variability of subscription-copy sales is often much smaller than that of single-copy sales. For example, the average and standard deviation of monthly subscribers for Good Housekeeping magazine in 2002 were 3.7 million and 28,500, respectively (i.e., a standard deviation equal to 0.7% of the average number of subscribers); while the standard devision of single-copy sales for the same period is 108,920 with an average of 939 thousand copies (i.e., a standard deviation equal to 11.6% of the average single-copy sales). Therefore, one way to secure a stable stream of advertisement revenue (and avoid penalties) is to convert occasional readers into subscribers, thereby reducing the variability of circulation demand. Our model considers how advertising revenue affects a firm’s operational decisions. 4
In our model, firms compete with product availability and perhaps also subscription price over overlapping demand for a finite number of periods. Product availability measures the number of copies of the magazine available in traditional channels (such as supermarkets, bookstores, newsstands). In each period, a new stream of customers arrive to buy a single unit of product and their initial preference toward a particular magazine is determined by an initial allocation scheme. Consumers who are satisfied with the experienced product quality (and the offered subscription price) will initiate a subscription from the following period. Dissatisfied customers will not subscribe. We assume that the firm’s revenue is two-fold: direct revenue collected from single-copy sales and subscriptions and indirect revenue collected from advertising space. Each individual firm’s objective is to maximize its expected discounted profit over a planning horizon. In this paper, we analyze how customers’ repeat purchases affect the firm’s choice of optimal product availability and subscription price under dynamic competition. There has been a stream of research papers which study the firm’s quantity choice under static or single-stage competition. Parlar (1988) and Wang and Parlar (1994) consider inventory decisions for two or more competing retailers selling substitutable products. They assume that initial demand for one retailer is independent from demand for another retailer, but that competition occurs over spill-over demand (as a consequence of one firm’s stock-out). Lippman and McCardle (1997) extend this work to general demand distributions by modeling demand for each retailer via an industry demand and an initial allocation mechanism that splits the realized industry demand into demand for each retailer. Among the related works in stochastic multi-stage games, Kirman and Sobel (1974) consider an oligopoly model where firms compete with price and quantity in each period and any portion of demand not immediately satisfied will be backlogged or lost in the infinite horizon. Under the assumptions that demand only depends on prices and there is no competition or substitution upon stock-outs at the retailer level, they show that there exists a Nash equilibrium and that the optimal stocking quantity follows a base stock policy. Bernstein and Federgruen (2004) consider a similar model, but with more specific assumptions on the demand distribution. They show that stationary prices and a base stock policy comprise a Nash equilibrium in the infinite horizon game when average profits are considered. Avsar and Baykal-G¨ ursoy (2002) consider dynamic competition of inventory under substitutable demand and show that there exists a Nash equilibrium which consists of stationary base-stock policies when a fraction of customers facing stock-out will switch retailers. Netessine et al. (2006) extend this work to the case with backlogging. Assuming a linear framework (i.e., unit costs and revenue are linear), they show that a stationary base stock policy is a Nash 5
equilibrium and characterize the conditions which the equilibrium must satisfy. See also Olsen and Parker (2007) for related work and other references. Our model is most closely related to Hall and Porteus (2000) (and its extension Liu et al., 2007) with a key difference. While they consider the negative feedback from customers to inferior service, there is no attraction that results from setting high service quality; we model the case that the firm can increase the market share or expected profit by providing a better service (i.e., increasing the product availability or lowering the subscription price). As a result, the firm’s action affects the revenue and future state of the competitor and the equilibrium strategy is jointly determined by both firms. The paper is organized as follows. Section 2 introduces our model and assumptions. The equilibrium definition and results are presented in Section 3. We show that there exists a Markov perfect Nash equilibrium such that the firm’s optimal stock quantity and proves the uniqueness of the equilibrium under certain assumptions. It also presents the related model when the firm sets subscription price along with quantity decision. In Section 4, we present results of our numerical study and discuss how factors affecting competition and subscriptions will change the likely outcomes of the game. We conclude the paper in Section 5.
2
Model and Preliminaries
In this section we describe our model, outline our assumptions, and give some preliminary results. In particular, we present a stochastic (sequential) game over a finite time horizon, where in each period two magazine companies compete at a single location. In the basic model the companies must only decide how much to stock in each period. This decision is based both on sales in the current period and also on future income to be realized due to subscriptions. Magazines are only marketable in the current period and so no inventory is kept across periods. We focus soley on quantity competition and leave exploration of subscription price competition to Section 3.1. Throughout the paper, we adopt conventional notation where we say increasing instead of nondecreasing and reserve strictly increasing for the strict case. Similarly we use decreasing instead of non-increasing. Firms We assume there are two firms (publishers) in the overlapping market. Firms are indexed by i and j and these indices are kept exclusively for firms so there is no confusion. We therefore drop the need to keep repeating that i, j = 1, 2; j 6= i and this is assumed as a given if we use these indices. Each firm seeks to maximize the expected discounted return over a finite planning horizon t = 1, . . . , T . 6
We assume that any revenue accrued in the future is discounted by a common discount factor α per period. Although all results hold for the non-stationary case, we have dropped the period index from parameters and cost functions for ease of exposition.
Customer Arrivals At each period t = 1, 2, . . . , T , Λ(t) customers will arrive to buy a single copy of magazine sold over the counter (i.e., Λ(t) represents aggregate industry demand during period t). Then, in each period, aggregate industry demand is split into initial demands of two firms via an allocation mechanism. That is, Λ(t) = Λ1 (t) + Λ2 (t), where Λi (t) is the initial demand seen by magazine i, i = 1, 2). We assume that industry demand at each period t follows a continuous distribution function on a finite support, [Lt , Ut ] where Lt ≥ 0 and Ut is bounded above. Initial demands for the two firms do not have to be independent, thus a number of models (as in Lippman and McCardle, 1997) can be considered. However, initial demands are independent of the firm’s inventory policies. By assuming demand to have a continuous distribution we are, in effect, assuming that demand is large enough such that a continuous approximation suffices.
Demand Substitution If firm 1 stocks out, then a portion of the excess demand is reallocated and bought from firm 2’s counter (and vice versa). If firms i and j stock quantities qi and qj , respectively, then net demand for magazine i during period t is as follows: Dit (uj ) = Λi (t) + γi max(Λj (t) + uj , 0), where uj = −qj and γi ∈ [0, 1] is the proportion of dissatisfied customers who want to substitute. Thus, the over-the-counter sales in period t are Qti (qi , uj ) = min(qi , Dit (uj )).
(1)
Subscription Mechanism The single copy and subscription prices for magazine i are assumed to be pi and si , respectively, and are fixed for each magazine (although in Section 3.1 we consider a model where firms compete both on stocking quantities and subscription prices). From those who bought the magazine over the counter last period a certain fraction will buy a subscription, which is in effect from the next period onwards. If Qi customers buy a magazine of type i then we assume that Si (Qi ) = ∆i Qi , where ∆i ∈ [0, 1], customers will subscribe. In other words, 7
(A2) Si (Qi ) is increasing and linear in Qi . We note that ∆i is either an exogenous constant between 0 and 1 or an explicit function of subscription and single copy prices such as ∆i = ki (pi − si )/pi with ki ∈ [0, 1], which reflects the case that customers are more likely to initiate subscriptions when a considerable discount is offered for subscription. In this case, notice that a fraction ki of the single-copy buyers will subscribe if si = 0 whereas there is no incentive to subscribe if si = pi .
Subscription Attrition Of those who have a subscription, a certain number will leave in each period. If ni customers have a subscription to type i magazine at the beginning of a period, then Ki (ni ) will continue to subscribe in the following period. In particular, we assume that: (A3) The subscription attrition function Ki (ni ) = (1 − βi )ni , where βi is an exogenous constant in [0, 1]. Define the function Ni (qi , uj , ni ) = Ki (ni ) + Si (Qti (qi , uj )).
(2)
Then if (qi , −uj ) are the stocking quantities for firms i and j respectively, in period t and ni is the number of people with a subscription at the start of period t, then Ni (qi , uj , ni ) is the number of people with a subscription at the start of period t + 1.
System Costs We let Ci (qi , ni ) be the one period holding, production, shipping and distribution costs (including a discount offered to a distributor) if qi units are produced for stocking the shelves and ni units are produced for subscription customers. We assume: (A4) The cost function Ci (qi , ni ) = cin qi + cis ni , where cin and cis are exogenous constants in [0, ∞) and represents the marginal cost for a newstand copy and a subscription copy, respectively. Subscription Revenue Let Ri (ni ) be the per period revenue earned by having ni subscribers to magazine i. We assume that (A5) The subscription revenue function Ri (ni ) = (si + ai )ni , where ai is an exogenous constant in [0, ∞) that represents the per subscriber advertising revenue. 8
Expected Revenue in period t, t = 1, . . . , T The expected payoff function in period t for firm i, git (qi , uj , ni ), therefore equals: git (qi , uj , ni ) = E pi Qti (qi , uj ) − Ci (qi , ni ) + Ri (ni ) ,
where qi and −uj are the quantities stocked by firms i and j, respectively, and ni is the current number of subscribers to magazine i.
End of Horizon Values We assume that the game continues for T periods and that in period T + 1 each firm evaluates its terminal value. Since most firms stay in business far longer than the planning horizon, it is reasonable to assume that the firm and its customer base are valuable. The terminal value for firm i is represented by ViT +1 (ni , mj ), where ni are the current number of subscribers to firm i and mj = −nj is the negative of the number of subscribers to firm j. We assume that (A6) The terminal value function ViT +1 (ni , mj ) = Vi ni + V˜i , where Vi is the value of a subscriber to firm i at time T + 1 and V˜i expresses the value of firm i independent of the number of subscribers. Stochastic Game Framework At each period t the firms simultaneously choose the number of magazines to stock. They have full information on magazines stocked and demand for each type for periods 1, . . . , t − 1. They also know the number of subscribers for both types of magazines for periods 1, . . . , t. They seek to maximize their discounted payoff over periods 1, 2, . . . T + 1, where the payoff in period T + 1 is the firm’s terminal value (defined above). Therefore if the expected payoff in period t, 1 ≤ t ≤ T , for firm i is represented by git (qit , −qjt , nti ), where qit and qjt are the quantities stocked by firms i and j in period t, respectively, and nti is the number of subscribers to magazine i in period t. Each firm i seeks to maximize E
"
T X
αt−1 git (qit , −qjt , nti )
+α
T
#
ViT +1 (niT +1 , −nTj +1 )
t=1
.
Notice that this objective depends on the behavior of the other firm and on stochastic demand.
3
Equilibrium Concept and Results
We restrict our attention to Markov Perfect (Nash) Equilibrium (MPNE) (see, e.g., Fudenberg and Tirole, 1991) for this game. First, because we believe firms often operate in Markovian fashion, 9
taking into account the current state for making decisions but little else. Second, because by doing so we may use inductive arguments, not possible in more general equilibria. We could also consider subgame perfect (Nash) equilibria where inductive arguments can be used but then one must condition on the history vector, rather than the state variables; however, we give conditions under which there exists a unique subgame perfect equilibrium, thus this restriction is without loss of generality. Under a Markov strategy, the firm chooses the action independently of the game except for the ones that are directly relevant to the current and future payoffs at the beginning of each subgame. In our case, since inventory carried-over is no use, the current state (consisting of the number of subscribers for each firm, nt = (nt1 , nt2 )) is sufficient for our purpose of defining each subgame. A (pure) strategy for firm i is represented by σi and is a sequence of maps from histories (or, in the case of Markov strategies, states nt ) to actions qit . Without loss of generality we may assume that the strategy space for q t = (q1t , q2t ) is [0, maxt Ut ] × [0, maxt Ut ], which is a compact sublattice of R2 (see, e.g., Topkis, 1998). A (pure) joint strategy is represented by σ = (σ1 , σ2 ). Because we only consider MPNE our strategies will be maps from the current state nt = (nt1 , nt2 ) σ (ns , −ns ) be the expected payoff for firm i from period s onwards, to stocking quantities q t . Let vi,s i j
1 ≤ s ≤ T + 1, when the current state is ns , under some MPNE σ. Then " T # X T +1 σ s s t−1 t t t t T T +1 T +1 vs (ni , −nj ) = E α gi (qi , −qj , ni ) + α Vi (ni , −nj ) , t=s
where the q t are chosen in accordance to σ. Then, for 1 ≤ t ≤ T , defining σ σ vi,t (ni , mj |qi , uj ) = git (qi , uj , ni ) + E[vi,t+1 (Ni (qi , uj , ni ), −Nj (−uj , −qi , −mj ))]
(3)
we have that, if −uj is chosen according to the MPNE σ, then σ vi,t (nti , −ntj ) = max vi,t (nti , −ntj |qi , uj ). qi
(4)
The following lemma shows that when each player follows an MPNE, the expected discounted payoff functions can be expressed as an affine function of the firm’s current state. Lemma 1 Suppose that, for some time t ∈ [1, 2, . . . , T ] and MPNE σ, σ vi,t+1 (ni , −nj ) = φt+1 ni + wit+1 i
(5)
where φt+1 and wi,t+1 are constants that do not depend on ni or nj . Let (nti , ntj ) be the state at i time t. Assume that firms 1 and 2 choose inventory levels of q t = (q1t , q2t ) at time t, consistent with 10
σ. Then the expected total discounted payoff from period t can be written as σ vi,t [nti , −ntj |qit , −qjt ] = φti nti + wit (q t )
where φti = (ai + si − cis ) + αφt+1 i (1 − βi ), t+1 t t wit (q t ) = −cin qit + pi E[Qi (qit , −qjt )] + αφt+1 i ∆i E[Qi (qi , −qj )] + αwi ,
wiT +1 = V˜i , and φTi +1 = Vi . Proof: The proof is primarily algebraic and is relegated to the Appendix. Lemma 1 provides the following result. Theorem 1 For a finite horizon problem, there exists a pure strategy MPNE independent of current state as follows. For all 1 ≤ t ≤ T , 1. If −cin + pi + α∆i φt+1 ≤ 0 and −cjn + pj + α∆j φt+1 ≤ 0, then both firms do not produce for i j over-the-counter sales, (qit∗ , qjt∗ ) = (0, 0). 2. If −cin + pi + α∆i φt+1 > 0 and −cjn + pj + α∆j φt+1 ≤ 0, then only firm i produces a non-zero i j quantity qit∗ such that P
Λi (t) + γi Λj (t) > qit∗
=
cin . pi + αφt+1 i ∆i
3. If −cin + pi + α∆i φt+1 > 0 and −cjn + pj + α∆j φt+1 > 0, then firm i (firm j) produces the i j quantity qit∗ (qjt∗ ) satisfying P Dit (qit∗ , −qjt∗ ) > qit∗ =
t t∗ cjn cin t∗ t∗ = . t+1 and P Dj (qj , −qi ) > qj pi + α∆i φi pj + α∆j φt+1 j
Proof: We prove the claim by induction starting from a subgame starting at period T . The proofs in other periods are identical to that of one period except that Vi is replaced by φTi −k+1 from Lemma 1. Details are relegated to the Appendix. In addition to these results, the form of the profit-to-go function implies the following result. Corollary 1 Theorem 1 implies the following
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1. The amount produced in each period by firm i is non-decreasing in the over-the-counter price, pi , the advertising revenue, ai , the subscription fraction, ∆i , (for constant subscription price si ), the subscription price, si , (for constant subscription fraction ∆i ), and the discount factor, α. 2. The amount produced in each period by firm i is non-increasing in the production and one period holding cost, ci , and the fraction who do not renew each period, βi . Although Theorem 1 guarantees the existence of a pure strategy subgame perfect equilibrium, it does not guarantee the uniqueness of the equilibrium. In fact, more than one equilibrium can exist, depending on the distribution of initial demand and allocation mechanism. However, under some mild regularity conditions, we can show that the equilibrium described above is in fact unique. Theorem 2 If P (Λ(t) ≤ x) is strictly increasing and continuous on its support and if Λi (t), i = 1, 2 are continuous and strictly increasing deterministic function of Λ(t) for all t, that is, there exists a deterministic function st (Λ(t)) such that Λi (t; d) = st (d) and Λj (t; d) = d − st (d) for d ∈ Λt , then there exists a unique MPNE. Proof: The result is proven inductively by noting that the industry demand level at which the initial allocation to firm i’s product is equal to the quantity ordered is uniquely defined by the industry demand distribution. Further, by assumption, the Λi (t) are continuous and strictly increasing functions, as are s(d) and d − s(d), therefore qit and qjt are uniquely determined. Details are relegated to the Appendix.
3.1
Extension to Quantity and Subscription Price Competition
Thus far, we have assumed that the firms are competing with quantities only. However, in the magazine industry, for example, the publisher often offers a different price for subscribers. To capture this, consider a finite horizon problem where firm i dynamically not only sets the production quantity to gain market share, but also sets the subscription price, sti , which is offered to new subscribers at the end of period t. We continue to assume that assumption (A1) - (A6) hold throughout the section. Further, we assume that ∆i is now a function of sti , written ∆i (sti ) = k
pi − sti . pi
Let Sit+1 (qit , −qjt , sti , Sit ) equal the subscription revenue that is received by firm i in period t + 1, when the stocking quantities in period t are (qit , qjt ), the subscription price for firm i in period t is 12
sti and subscription revenue in period t is Sit , then Sit+1 (qit , −qjt , sti , Sit ) = (1 − β)Sit + (sti − cis + ai )∆i (sti )Qti (qit , −qjt ). Here we assume that instead of a fixed proportion of customers departing each period, a fixed proportion of revenue is lost. While a more detailed model would include all customers and the subscription price they pay in order to calculate the next period’s revenue, such a model would clearly be intractable. Notice that the subscription revenue is not received until the following period. In other words, the firm receives revenue for selling the magazine in period t and possible future revenues from subscriptions in periods t + 1, t + 2, . . .. Thus, the expected payoff function in period t for firm i, git (qi , uj , si , Si ), equals: git (qi , uj , si , Si ) = E pi Qti (qi , uj ) − cin qi + Si ,
where qi and −uj are the quantities stocked by firms i and j, respectively, firm i’s subscription price is set at si , and subscription revenue expected in this period is Si . Analogously to assumption (A5) we assume that at period T + 1, the terminal value of firm i can be expressed as: ViT +1 (SiT +1 ) = Vi SiT +1 + V˜i ,
(6)
where Vi is the value of a unit of subscription revenue to firm i at time T + 1 and V˜i expresses the value of firm i independent of the subscription revenue. Under some MPNE σ, the payoff from period t onwards for firm i is σ vi,t (Sit , −Sjt )
=
max
qi ≥0,0≤si ≤pi
h
i σ Sit − cin qi + E[pi Qti (qi , uj ) + αvi,t+1 (Sit+1 (qi , uj , si , Sit ), −Sjt+1 (−uj , −qi , sj , Sjt ))] ,
where −uj , firm j’s stocking quantity, is chosen according to σ. Thus for period T , vi,T (SiT , −SjT ) =
max
qi ≥0,0≤si ≤pi
h
SiT − E[cin qi + pi QTi (qi , uj ) + α(Vi SiT +1 (qi , uj , si , SiT ) + V˜i ).
Substituting in QTi (qi , uj ) and SiT +1 (qi , uj , si , SiT ) we have
i
vi,T (SiT , −SjT ) SiT − cin qi + αV˜i = max E[pi QTi (qi , uj ) + αVi [(1 − βi )SiT + (si − cis + ai )∆i (si )QTi (qi , uj )] qi ≥0,0≤si ≤pi = (1 + αVi (1 − βi ))SiT + αV˜i + max[−cin qi + (pi + αVi (si − cis + ai )∆i (si ))E[min{qi , DiT (qi , uj )}]].
qi ,si
13
Now arg maxs (s − cis + ai )∆i (s) = (cis − ai + pi )/2 and αVi E[min{qi , DiT (qi , uj )}] ≥ 0 so vi,T (SiT , −SjT ) = (1 + αVi (1 −
βi ))SiT
k(cis − ai + pi )2 T ˜ + αVi + max −cin qi + pi + αVi E[min[qi , Di (qi , uj )]] . qi 4pi
Notice that the above implies that the best response si can be chosen independently of sj and qiT , qjT . Therefore, we have not lost any realism by assuming (like we did) that si is set for each stocking location. Under the linear framework, the expected discounted profit function is an affine function of the state variable Sit as stated in the following lemma. Lemma 2 Suppose that, for some time t ∈ [1, 2, . . . , T ] and i ∈ 1, 2, σ vi,t+1 [Sit+1 , −Sjt+1 ] = φt+1 Sit+1 + wit+1 i
(7)
where φt+1 and wi,t+1 are constants that don’t depend on (Sit , Sjt ). Assume that firms 1 and 2 i choose inventory levels of q t = (q1t , q2t ) and subscription prices of st = (st1 , st2 ) at time t, consistent with an MPNE σ. Then the expected payoff from period t under a policy (q t , st ) can be written as σ vi,t [Sit , −Sjt ] = φti Sit + wit (q t )
where φti = 1 + αφt+1 i (1 − βi ), t+1 t t t wit (q t ) = −cin qit + pi E[Qti (qi , −qj )] + αφt+1 i (si − cis + ai )∆i (si )E[Qi (qi , −qj )] + αwi ,
wiT +1 = V˜i , and φTi +1 = Vi . Therefore, the optimal strategy is independent of the current state. Proof: The proof is analogous to Lemma 1.
Following a similar inductive proof used in Theorem 1, we now show that there exists a subgame perfect pure strategy equilibrium in quantity and subscription price competition for the finite horizon problem. Theorem 3 For a finite horizon problem, there exists a subgame perfect pure strategy equilibrium cis −ai +pi t∗ independent of the firm’s current market share such that (st∗ ∨ 0, i , sj ) = ( 2
and 14
cjs −aj +pj 2
∨ 0)
1. if −cin + pi + α∆i φt+1 ≤ 0 and −cjn + pj + α∆j φt+1 ≤ 0, then both firms do not produce for i j over-the-counter sales (i.e., (qit , qjt ) = (0, 0)), or 2. if −cin + pi + α∆i φt+1 > 0 and −cjn + pj + α∆j φt+1 ≤ 0, then only one firm produces the i j non-zero quantity qit > 0 such that P Λi (t) + γi Λj (t) > qit =
cin , or pi + αφt+1 i ∆i
3. if −cin + pi + α∆i φt+1 > 0 and −cjn + pj + α∆j φt+1 > 0, then firm i (firm j) produces the i j quantity qit (qjt ) satisfying: P Dit (qi , −qj ) > qit =
t cjn cin t . t+1 and P Dj (qj , −qi ) > qj = pi + α∆i φi pj + α∆j φt+1 j
Proof: The proof mirrors Theorem 1 and is provided in the Appendix. Notice that the optimal subscription price increases in cost per unit and price of a single copy, but decreases in advertisement revenue per subscriber. The result is intuitive as a higher advertisement revenue implies a greater value per a subscriber, therefore the firm has an incentive to sweeten the offer. On the other hand, when the price of a single copy increases, the firm can attract customers without decreasing subscription price (because the subscription mechanism is determined by the relative price difference between the two offers), therefore the subscription price indeed increases. For a sufficiently high advertising revenue, the firm is even willing to offer subscriptions at no charge. Another interesting fact is that the equilibrium subscription price is independent of the competitor’s price. This is because in our model customers decide whether to subscribe or not only after actually experiencing the product. We expect that the result would be quite different if we allow that customers can subscribe to another magazine after negative experience from the one purchased. This is left as the subject of future research.
4
Numerical Study
Although the structural results in the previous section shed some light into the likely outcome of a multi-stage game, the equilibrium alone does not provide clear insights into how the operational policy and the firm’s long-run profit are affected by customers’ repeat purchases and the duopoly competition. To this end, we conducted a numerical study to obtain further insights on how competition and customers’ repeat purchases through subscriptions affect the firm’s decisions and profit. In particular, we investigate the following questions: (1) how does the optimal policy differ under 15
different parameters and cost structures? (2) How much does the firm benefit from considering both competition and the effects of subscriptions? (3) What would be the likely outcome under asymmetric competition and how does one firm’s inefficiency affect the competitor’s operational policy? To simplify the effort of searching for the subgame perfect equilibrium without losing significant qualitative insights, we restrict our study to the case of quantity competition with stationary parameters, a deterministic splitting function, and a linear framework. Further, we use a discretized demand function instead of a continuous demand function in our numerical study. We also assume that all end-of-horizon values are zero in order to isolate end-of-horizon effects. We first conducted a numerical study to compute the equilibrium and the expected discounted profit for two symmetric firms under various scenarios and horizon lengths. The results of four scenarios are reported. For all four scenarios we assume that α = 0.95, β = 0.1, cin = cis = 1.5, si = 1, γ = 0.8, and Λ ∼ U nif orm(0, 20). The price of a single copy (pi ), the advertising revenue (ai ), and the rate of initiating a subscription after an initial purchase (∆i ) are as follows: (3, 1, 2/3) for Scenario 1, (3, 3, 2/3) for Scenario 2, (0.5, 3, 2/3) for Scenario 3, and (0.5, 3, 0.1) for Scenario 4, respectively. Compared to Scenario 1, Scenario 2 represents a case with a higher advertising revenue, while Scenarios 3 and 4 represent cases where the publisher can only generate profit by attracting subscribers. While selling at loss may look like a rather unconventional situation, it is not uncommon in the magazine or newspaper industry where advertising revenue and non-pecuniary product quality may provide an incentive to sell at a very low price at newsstands to attract potential subscribers (sometimes lower than the subscription price itself). For example, the subscription price of The New York Times Sunday edition is indeed higher than the newsstand price in many midwest and western states. Scenarios 3 and 4 are examples of this category and are identical except for the rate that customers initiate a subscription after an initial purchases (∆ =
2 3
in scenario 3 versus
∆ = 0.1 in scenario 4). Figures 1 and 2 show the optimal stocking quantity under the equilibrium, the value of a single subscriber, and the fixed value of the firm for each of the four scenarios. The optimal stocking quantity and expected profits under Scenario 2 are the largest since the marginal revenue from both newsstand sales and advertising are the highest among all scenarios. For all scenarios, the optimal quantity, the value of a single subscriber, and the fixed value of a firm, all increase as the length of the planning horizon increases. The optimal stocking quantity and profit increase in the planning horizon because the value of generating an additional subscriber (compared with 16
Figure 1: The optimal order quantity over time. the potential cost of overstocking) increases in the number of remaining periods. This trade-off is manifested in scenario four where the immediate profit of selling a single copy is negative. When the number of remaining periods is small, there is not enough revenue collected from subscriptions to offset the initial loss occurred at a newsstand, therefore it becomes optimal not to sell any single copies. On the other hand, for a sufficiently long planning horizon, T ≥ 8, it is optimal to sell single copies at loss and remunerate excess cost through subscriptions and advertising revenue. Observe that, although the value of a single subscriber is identical under Scenarios 2, 3, and 4, the expected profits are quite different. In these models, once a customer begins a subscription, the aggregate behavior is identical for all three scenarios, thereby generating the same expected profit. However, the profit of a single copy and the probability that a customer initiates a subscription are different in each scenario, and this yields the different expected profits. In order to quantify the additional benefit that a firm can capitalize on by considering both the effect of competition and the effect of customer subscription, we compare the optimal policies with two heuristic policies, namely a monopoly policy and a myopic policy. In the monopoly policy, the firm acknowledges the effects of subscriptions, but ignores the effects of competition. As a result, in each period the firm believes that all realized demand would prefer its firm, not the competitor. In other words, in each period, the firm tries to find the optimal quantity under the assumption that the opponent will not participate and therefore it will be awarded with the full market share, which is all of its own demand plus the spill-over demand). In each period, the order-up-to level is
17
Figure 2: The value of a single subscriber and the fixed value of a firm. computed by solving Mo qit = arg max qit ≥0
"
(ai +h si − cis )Ni (t) − cin qit
i t t +E pi Qi (t) + αvi+1,t (Ni (t + 1), Nj (t + 1)) qi , −qj = 0
#
.
In the myopic policy, the firm acknowledges the effect of competition for single copy sales, but ignores the potential profit generated from subscribers when determining product availability. My The quantity under the myopic policy (denoted by qit ) is equivalent to the equilibrium quantity
obtained in Lippman and McCardle (1997). We report how the optimal quantity fares under these two heuristics in Figure 4 and Table 1. For all our examples, the optimal quantity is less than or equal to the quantity under the monopoly policy and greater than or equal to the quantity under the myopic quantity. In fact, it can easily be shown that q M y ≤ q ∗ ≤ q M o as long as the value of a single subscriber from the next period and onward is positive (i.e., φt+1 ≥ 0). As demonstrated in Table i 1, considering competition and the effect of subscription can significantly enhance the publisher’s profit. The fact that the myopic policy can considerably underperform in many cases explains, in part, why most publishers prefer to manage newsstand stocks instead of selling directly to retailers. If retailers were in charge of quantity decisions, the results would be very similar to the myopic policies as retailers could not benefit from revenues from subscription and advertisement. On the other hand, the results also show that the publisher must be aware of the degree of competition and try to avoid excess inventory by taking account of effective demand for its product. Failure to do so results in excess inventory with little benefit. Table 1 also shows that the myopic policy (which stocks less) outperforms the monopoly policy (which stocks more than it should) when the profit from a single copy sale is high and the profit 18
Figure 3: Optimal, myopic, and monopoly policies under scenario 2.
Table 1: Optimal, myopic, monopoly polices (T=15.) Sc. No. 1 2 3 4
Description ai low (1), pi high (3) ai high (3), pi high (3) ai high (3), pi low (0.5) ai high (3), pi low (0.5) ∆ = 0.1
q∗ 7 9 9 3
qM o 12 15 15 5
qM y 5 5 0 0
Optimal 171.2 711.5 532.5 7.85
Monopoly 87.0 (51%) 561.9 (79%) 400.7 (75%) 2.12 (27%)
Myopic 160.2 (93 %) 589.9 (83 %) 0.0 (N/A) 0.0 (N/A)
φ 3.4 17.1 17.1 17.1
per subscriber is low; while the monopoly policy performs well when the profit generated from a subscriber is considerable. The latter is because the cost of overstocking is offset by revenue from additional subscribers when subscription and advertising revenues are high. This example shows that the myopic policy may prevent the firm from entering into a potentially profitable market if the return is not immediate. For example, in Scenario 3, the publisher produces nothing and receives zero profit under the myopic policy as the immediate profit for a single copy is negative (compared with q M o = 15 and w = 532.5 under the monopoly policy). This example clearly shows that accounting for the value of customer when making operational decisions can significantly affect a firm’s profit. We conducted an additional numerical study to investigate the change in the equilibrium as one or more parameters change. We report the case with ci (i.e., the unit production cost of a magazine) changing by comparing the stock level under the three policies (equilibrium, monopoly, and myopic) and the corresponding profits. Figure 4 shows that, as ci increases, the stocking policy, the value of a subscriber, and the expected profit decrease in each of three policies. The results for other parameters are similar and complementary to Corollary 1 and are therefore omitted.
19
Figure 4: The value of a single subscriber and the fixed value of a firm.
Figure 5: The change in initial preference to magazine 1
Thus far we have reported the result of our numerical study when two competing firms are symmetric. However, in reality, it is unlikely that the conditions for both firms are identical. In what follows, we investigate and report the outcome of games when the conditions under which one firm competes are different from those for a competitor. In particular, we are interested in whether a firm can capitalize on the opportunity given by the inefficiency of a competitor as the conditions (such as production cost or advertising revenue) for one firm deteriorates. To this end, we conducted a numerical study by changing one parameter while fixing all others. We report the initial preferences case in Figure 4 and the subscription attrition rate case in Figure 6. As the customer’s initial preference for magazine 1 increases (i.e., the preference for a competitor’s magazine decreases), both the optimal stocking quantity and expected profit of the firm increase at an almost linear rate; those of the competitor decrease. In this case, one firm’s loss in efficiency is immediately captured by increased demand for the other firm and the results follow
20
Figure 6: The change in subscription attrition rate immediately. Figure 6 demonstrates a case under which the one firm’s efficiency loss is not immediately capitalized by the competitors. The optimal stocking policy and expected profit change very little for a wide range of subscription attrition rates for magazine 2. An upward shift of the optimal stocking quantity and profit can be observed only when the attrition rate becomes extremely high. While this seems quite unintuitive, the fact that qi and −qj are strategic complements provides a plausible explanation. Note that the change in the attrition rate of magazine 2 does not change the profitability or initial demand for magazine 1. Therefore, the strategy adopted by firm 1 remains the same so long as its effective demand remains unchanged. This explains why there is no change in firm 1’s performance while both the profit and stocking quantity of firm 2 drops sharply. However, as the attrition rate increases, the value of a single subscriber for magazine 2 decreases. When it becomes too small, the profit from a magazine 2 subscriber also decreases, and firm 2 cuts its stocking quantity. Once firm 2 reduces its quantity and spill-over demand for firm 1 increases, firm 1 increases its stocking quantity (as a strategic complement) and the profit increases as well, although the profit per customer has not changed.
5
Conclusions
We have considered a dynamic model of competition in a market where customers have the option to subscribe to the good or service. In many examples, including magazines, a failure to retain a customer often results in a lingering loss of revenues because some of the lost demand would have become loyal customers who purchase the product through subscription. Using a stylized model, we show that a considerable gain can be realized by explicitly taking 21
account of the effects of subscriptions as well as the degree of competition (represented by spillover). Our results quantify the value of retaining customers through subscription and how potential revenue from subscribers affects the firm’s operational policies. Our results illustrate that a policy that explicitly considers revenues from subscription will cause the firm to stock more (compared with a policy that does not) even in cases when selling through traditional channels results in loss of revenue. However, the model we consider is still simple in many aspects, and our model gives rise to relevant open issues that need to be investigated further. One important extension is to analyze the impact of inventory level on single copy demand. As a significant portion of single copy sales is driven by impulse shopping behavior, the inventory level (relative to those of competitors) might influence single copy demand in each period (Koschat, 2006). This effect might increase the desired order-to-level even further when the interaction between stock level and demand is significant. Another important extension is to analyze the case where revenue, cost, or change in the number of subscribers are non-linear functions of subscription size and operational policy. In the magazine industry, building and maintaining a large number of subscribers is crucial to a publisher’s profit as the size of the subscription pool often leads to economies of scale (as the marginal production and distribution costs decrease) and improved revenue structure (as the greater market clout allows the publisher to charge higher advertising and subscription rates). One might think that one can capture some of these crucial non-linear dynamics in a stochastic game framework by restricting to Markov strategies. However, these cases appear to be extremely challenging even with MPNE because one must show convexity or related properties of the best response function (see, Vives, 2006), which is necessary step in the inductive argument. Other possible extensions to this work include adding the quality of product (service) provided in addition to product availability and incorporating retail price competition along with other operational decisions. Future research should also include modeling switching costs, incorporating different classes of customers, and modeling more realistic customer choice behavior.
Appendix Proof of Lemma 1 It is easy to see the result holds for T . Suppose that the result holds for periods t + 1 and both firms follow an MPNE σ from period t + 1 and onwards. Then, σ σ t t vi,t [nti , −ntj |qit , −qjt ] = git (qit , −qjt , ni ) + αE vi,t+1 [Ki (nti ) + Si (Qt+1 i (qi , −qj ))] 22
Applying (A2)-(A5), we get
σ σ [(1 − βi )nti + ∆i ∗ Si (Qi (qit , −qjt ))] vi,t [nti , −ntj |qit , −qjt ] = (ai + si − cis )nti − cin qit + pi E[Qi (qit , −qjt )] + αE vi,t+1
Using the induction hypothesis, the above equation is further simplified to σ vi,t [nti , −ntj |qit , −qjt ]
∆i E[Qi (qit , −qjt )]) + wit+1 = (ai + si − cis )nti + αφt+1 ((1 − βi )nti − cin qit + pi E[Qi (qit , −qjt )] + α φt+1 i i
(1 − βi ) nti − cin qit + pi E[Qi (qit , −qjt )] + αφt+1 = ai + si − cis + αφt+1 ∆i E[Qi (qit , −qjt )] + αwit+1 i i = φti nti + wit (q(t))
Since φti is independent of q t the optimal strategy is independent of the current state.
Proof of Theorem 1 We prove the claim by induction. From Lemma 2, it is easy to show that the result holds for period T . Suppose that the subgame consisting of the final k ≥ 1 periods has a pure strategy equilibrium and the expected payoff function for period T − k + 1 satisfies (5). Now consider a subgame of the final k + 1 periods (i.e., the subgame starting from period T − k). Using the induction hypothesis and the fact that Lemma 1 implies that the optimal decision from period T −k +1 is independent of strategy used in period T − k and that the expected payoff from period T − k + 1 can be replaced by an affine function. Thus, the k + 1 period extended game has an equivalent one-stage reduced form game and the proof is complete if we show that the optimal action for firm i at period T − k follows the statement in the reduced game. For brevity, let t = T − k. From Lemma 1, the derivative of firm i’s payoff from period t and onwards with respect to qit , when firm j produces qjt and the state is (ni , nj ), can be written as σ (n , −n |q t , −q t ) ∂ vi,t i j i j t t t t = −cin + (pi + α∆i φt+1 i )P [Di (qi , −qj ) > qi ]. t ∂qi
(8)
Case 1. −cin + pi + α∆i φt+1 ≤ 0 and −cjn + pj + α∆j φt+1 ≤ 0] i j Since
σ (n ,−n |q t ,−q t ) ∂ vi,t i j i j ∂qit
≤ 0, qit∗ = 0 independent of qjt .
Case 2. −cin + pi + α∆i φt+1 > 0 and −cjn + pj + α∆j φt+1 ≤0 i j Since qjt∗ = 0 and is independent of qit , Dit (qit , −qjt∗ ) = Λi (t) + γi Λj (t). From the continuity of the demand distribution, there exists a qit such that P (Dit (qit , −qjt ) > qit ) = P (Λi (t) + γi Λj (t) > qit ) =
cin . pi + αφt+1 i ∆i
The concavity of the payoff function immediately follows from the fact that pi + α∆i φt+1 > 0. i
23
Case 3. −cin + pi + α∆i φt+1 > 0 and −cjn + pj + α∆j φt+1 >0 i j Suppose that (qit∗ , qjt∗ ) satisfy the first order condition. Then, the result immediately follows from the fact that P [Dit (qi , qjt∗ ) > qi ] is decreasing in qi for a given qjt∗ .
Proof of Theorem 2 We prove the claim by the induction. For the induction basis we first consider the final stage subgame at period T . As in Theorem 1, the proof will be divided into three cases. Case 1. −cin + pi + αVi ∆i ≤ 0 and −cjn + pj + αVj ∆j ≤ 0. Since Λ(T ), Λi (T ), and Λj (T ) are continuous, qi > 0 strictly decreases the profit. Thus, qiT ∗ = 0 is the unique best response, independent of firm j’s response. Therefore, (qiT ∗ , qjT ∗ ) = (0, 0) is the unique equilibrium.
Case 2. −cin + pi + αVi ∆i > 0 and −cjn + pj + αVj ∆j ≤ 0. Since −cjn + pj + αVj ∆j ≤ 0, qjT ∗ = 0. Since Λi (T ) and Λj (T ) are both continuous deterministic functions in Λ(T ), we can rewrite the net demand for firm i as a continuous function of Λ(T ): Λi (T ) + γi Λj (T ) = (1 − γi )sT (Λ(T )) + γi Λ(T ). From the continuity of Λ(T ), qiT ∗ = min{q : P (Λi (T ) + γi Λj (T ) > q) =
cin pi +α∆i Vi }
is the unique best
response to qjT ∗ = 0. Case 3. −cin + pi + αVi ∆i > 0 and −cjn + pj + αVj ∆j ≤ 0. Let (qiT ∗ , qjT ∗ ) be the equilibrium proposed in case 3 of Theorem 1. For uniqueness, let dTi and dTj be a solution of the equations, sT (dTi ) = qiT ∗ and dTj − sT (dTj ) = qjT ∗ , respectively. Note that dTi (dTj ) is the industry demand level at which the initial allocation to firm i’s product is equal to the quantity ordered, and such a quantity is uniquely defined by the industry demand distribution. Since Λi (T ) and Λj (T ) are continuous and strictly increasing, s(d) and d − s(d) are also strictly increasing in d. Thus, qiT ∗ and qjT ∗ are unique. To proceed induction, suppose that the subgame consisting of the final k ≥ 1 periods has a unique pure strategy subgame perfect equilibrium. Consider a subgame of the final k + 1 periods (i.e., the subgame starting from period T − k). The induction hypothesis and Lemma 1 imply that the revenue to go from period T − k + 1 can be replaced by an affine function, and that the k + 1 period subgame has an equivalent one-stage reduced form game. The uniqueness can be shown by following the argument used for the subgame starting in period T except that Vi is replaced by 24
φt+1 i .
Proof of Theorem 3 From Lemma 2, if the induction hypothesis holds, then the optimal strategy is independent of the t current state. Further, since φTi +1 ≥ 0 we can show inductively that αφt+1 i E[Qi (qi , −qj )] ≥ 0 and
s∗i = arg maxs [(s − cis + ai )∆i (s)] = (cis − ai + pi )/2. Since the choice of sti has no influence on φtj or wjt , the proof mirrors Theorem 1.
Acknowledgments We are grateful to Martin Cripps at the Olin School of Business, Washington University in St. Louis, Martin Lariviere at the Kellogg School of Management, Northwestern University, William Lovejoy at the University of Michigan Business School, and Matthew Sobel at Case Western University for discussions and helpful comments on this topic. This work was supported in part by NSF grant DMI-0245382.
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