Intertemporal Price Discrimination with Forward-Looking

Consumers: Application to the US Market for Console

Video-Games

Harikesh Nair1

September 2004

This version: February 2007

Forthcoming: Quantitative Marketing & Economics

Abstract

Firms in durable good product markets face incentives to intertemporally price discriminate,

by setting high initial prices to sell to consumers with the highest willingness to pay, and

cutting prices thereafter to appeal to those with lower willingness to pay. A critical

determinant of the profitability of such pricing policies is the extent to which consumers

anticipate future price declines, and delay purchases. I develop a framework to investigate

empirically the optimal pricing over time of a firm selling a durable-good product to such

strategic consumers. Prices in the model are equilibrium outcomes of a game played between

forward-looking consumers who strategically delay purchases to avail of lower prices in the

future, and a forward-looking firm that takes this consumer behavior into account in

formulating its optimal pricing policy. The model outlines first, a dynamic model of demand

incorporating forward-looking consumer behavior, and second, an algorithm to compute the

optimal dynamic sequence of prices given these demand estimates. The model is solved using

numerical dynamic programming techniques. I present an empirical application to the market

for video-games in the US. The results indicate that consumer forward-looking behavior has a

significant effect on optimal pricing of games in the industry. Simulations reveal that the profit

losses of ignoring forward-looking behavior by consumers are large and economically

significant, and suggest that market research that provides information regarding the extent

of discounting by consumers is valuable to video-game firms.

Keywords: durable-good pricing, forward-looking consumers, Markov-perfect equilibrium,

numerical dynamic programming, video-game industry.

JEL classification: C25, C61, D91, L11, L12, L16, L68, M31.

1 This paper is based on my dissertation. I thank my dissertation committee, Pradeep Chintagunta, Jean-Pierre Dubé, Günter

Hitsch, and Peter Rossi for their guidance. I am grateful to Ester Han, Karen Sperduti and Sima Vasa of the NPD group, and

R. Sukumar of IPSOS-Insight for their help in making available the data used in this research. I thank Dan Alderman of the

Microsoft Xbox group, and Norman Basch of Reservoir Labs for sharing with me their insights on the video-game industry. I

also received useful feedback from Tim Conley, Ulrich Doraszelski, Liran Einav, Wes Hartmann, Puneet Manchanda, Peter

Reiss, Alan Sorensen; two anonymous referees; and seminar participants at Berkeley, CMU, Columbia, Cornell, Dartmouth,

HKUST, ISB, MIT, Northwestern, Purdue, Stanford, UCLA, UConn, UMaryland, UPenn, UToronto, UWisconsin, Washington

St. Louis and Yale. All errors in the paper are my own. My contact information is: 518 Memorial Way, Graduate School of

Business, Stanford University, Stanford, CA 94305-501; Ph: 650-736-4256; Email: harikesh.nair@stanford.edu.

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1. Introduction

Firms selling durable good products typically face varying demand conditions over the life-cycle of

their products. Initial sales of the product are likely to arise from high valuation consumers who

subsequently exit the market after purchase. Firms thus face a relatively inelastic demand curve at

the time of introduction, and progressively face more elastic demand conditions as the product ages.

In such situations, firms have an inherent incentive to lower prices to “skim” the market, thereby

price discriminating over time. It can set high prices initially and extract the maximum surplus from

high valuation consumers, and then cut prices over time to sell to the low valuation consumers

remaining in the market. However, a significant problem to profiting from such intertemporal price

discrimination arises when consumers are forward-looking. Anticipating future price cuts, forward-

looking consumers may strategically delay their adoption, and purchase at low prices later. This

reduces the profitability of price skimming. Further, the more the firm skims the market, the more

the extent to which it can expect such strategic delay, since consumer expectations of future prices

may get revised based on observed price cuts. Given this complex consumer behavior, a challenging

issue for firms in real world markets is how they would choose the optimal profit-maximizing

sequence of prices for their durable-good products. A methodological challenge arises because both

consumer purchase decisions and firms’ pricing decisions are inherently dynamic. Existing empirical

frameworks are not well suited for analyzing such contexts since they have not dealt with dynamics

on both sides of the market in a satisfactory way. In this paper, I develop a framework that

accommodates these dynamics, and illustrate the value of this approach empirically in the context of

analyzing price discrimination in the market for video-games in the US. I use the model to calibrate

the extent to which forward-looking consumer behavior affects the ability of firm to implement

intertemporal pricing policies profitably. Additionally, I estimate how much firms in this market

should value market research that provides information regarding the nature of consumer forward-

looking behavior.

Starting with Coase (1972), a large theory literature (c.f. Stokey 1979, 1981; Bulow 1982;

Conlisk, Gerstner and Sobel 1984; Landsberger and Meilijson 1985; Gul, Sonneschein and Wilson

1986; Kahn 1986; Moorthy 1988; Narasimhan 1989; Besanko and Winston 1990; Balachander and

Srinivasan 1998; Desai and Purohit 1999) has analyzed the pricing of durable goods in markets with

forward-looking consumers. Broadly, the consensus from the theory is that forward-looking

consumer behavior hurts firm profits, and curtails monopoly power. The richness of the theoretical

literature contrasts sharply with the lack of empirical work in this area. To the best of my

knowledge, no previous work has analyzed the optimal dynamic pricing policy of a durable-good firm

facing forward-looking consumers in the context of an empirically specified demand and profit

system. The literature thus currently offers little practical guidance to managers for setting prices

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optimally in actual market settings. The theory literature has also worked with simple models that

yield tractable analytical solutions. The question is whether the main results regarding pricing are

robust to more flexible models that correspond better with the way we typically model demand

empirically. In particular, is the extent to which forward-looking behavior affects firms’ prices and

profits, economically significant in the context of real world markets? Is this an issue that managers

in these markets need to be concerned about? This paper attempts to shed light on these issues.

To this end, I start by developing a framework with which to analyze empirically the optimal

pricing policy of a firm selling a durable good to a set of heterogeneous consumers. Consumers are

strategic and make purchase decisions by trading off the utility from current purchase with the

value of buying the product at a lower price in the future. Consumers take into account that their

purchase decisions change the future state of the market, and thus, also change the future prices

charged by the firm. The firm realizes this and formulates its pricing policy taking the consumer

behavior into account. An equilibrium in the market obtains when the firm’s pricing policy is optimal

given consumer behavior, and consumer purchase behavior is optimal given prices. The equilibrium

is a fixed point of a multi-agent game between the monopolist and the consumers. The fixed point

requires each consumer type to maximize their expected utility subject to consistent perceptions of

the likelihood of future states for the firm, and firms to maximize expected payoffs based on

consistent perceptions on the likelihood of future consumer states. I present an algorithm to solve for

the sub-game perfect Nash equilibrium in prices in this model. Nested in the computation of the

equilibrium is the solution to both the consumer’s intertemporal adoption problem, as well as the

firm’s intertemporal pricing problem. In the context of this game, I restrict attention to equilibrium

pricing strategies that depend only on the current values of “payoff-relevant” state variables, which

are ensured to evolve according to a Markov process. In this sense, the equilibrium that I solve for is

Markov-perfect. In my empirical application, I find that such relatively simple equilibrium policies

are effective in explaining the key qualitative features of the data.

Since current pricing decisions impact future demand and profits, the firm’s optimal pricing

problem is inherently dynamic. Analytical solutions to the dynamic pricing policy exist for stylized

demand models (e.g. Besanko and Winston 1990, henceforth BW). However, to be of practical value,

the pricing problem has to be based on a realistic demand system. The use of a more elaborate

demand specification precludes solving the pricing problem analytically. Instead, I use numerical

dynamic programming techniques to solve for the equilibrium. The approach is able to handle most

demand systems, including the popular random-coefficients aggregate logit and probit models.

Relevant to this approach is a theoretical literature in marketing that has analyzed the pricing of

durable-goods subject to diffusion effects (e.g. Robinson and Lakhani 1975; Dolan and Jeuland 1981;

Kalish 1983, 1985; Mahajan, Muller and Kerin 1985; Krishnan, Bass, and Jain 1999). These models

generally assume that demand evolves exogenously over time (e.g. via a diffusion process), and

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unlike my framework, typically do not consider the intertemporal adoption problem faced by

consumers (an exception is Horsky 1990). This literature also typically solves for open-loop pricing

solutions, due to the difficulty in characterizing analytically the closed-loop solution. In contrast, the

proposed numerical dynamic programming policy delivers a closed-loop state-contingent pricing

strategy by construction. This makes it more suitable for managerial decision-making in real-world

contexts.

The analysis comprises two steps. In the first step, I present a method to obtain estimates of

demand parameters under forward-looking consumer behavior. The model of demand is derived from

an underlying dynamic discrete choice model that incorporates the intertemporal adoption problem

faced by forward-looking consumers. Consumers make rational forecasts about future price declines,

and adopt the product when the utility from current purchase exceeds the option value for waiting

for the next period. The demand model shares features with dynamic models of technology adoption

recently proposed in the literature, e.g., Melnikov 2000; Song and Chintagunta 2003; Erdem, Keane

and Strebel 2005; Gowrisankaran and Rysman 2006 (I discuss the relationship of my demand model

with these papers in the “Model” section.) The demand parameters are estimated by maximum

likelihood using a limited information approach. In the second step, I take these demand parameters

as given and solve numerically for the corresponding optimal pricing policies of the firms in the

sample. This two-stage approach has the advantage that the demand estimates obtained are

independent of supply-side restrictions on the nature of pricing conduct. Hence, I am able to

recommend optimal prices to managers, which would not be possible if restrictions from the optimal

pricing policy had been imposed in estimation. Further, the two-stage approach reduces the

computational burden of the estimator, since the equilibrium need not be solved to obtain the

demand parameters. The framework is intended to be used normatively by managers to obtain the

optimal sequence of prices over time for their products. The manager would first need to estimate the

proposed demand system from available data. Given demand parameters, the model can then be

used to simulate the optimal intertemporal pricing policy for his product.

To illustrate the economic and managerial implications of the framework, I present an

empirical application to the market for 32-bit video-games in the US. I choose this industry on

purpose since there is evidence that the element of dynamics that matters the most in this industry

is intertemporal price discrimination, rather than costs or competition. All video-games exhibit

consistent patterns of price cutting from their times of introduction. My interviews with managers in

the industry revealed that much of the motivation for this price cutting arises from the desire to sell

to the segment of high valuation “hardcore gamers” initially, and to cut prices over time to sell to the

“mass market”. This closely parallels the price discrimination incentive. On the cost-side, production

of video-games is characterized by a constant marginal cost structure. Marginal costs correspond to

royalty fees paid by the game manufacturer to the hardware console manufacturer (i.e. Sony), and

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also the costs of producing and packaging each CD-ROM title, both of which were constant over the

time-period of the data. Hence, falling marginal costs are unlikely to be an issue in pricing over time.

Further, competition from other games is also unlikely to be driving force behind falling prices.

Given the large number of games in the market (over 600 for the Sony Playstation alone), and the

fairly unique characteristics of each game, I find video-games to be weak substitutes for each other.

The observed price data also reveal that the rates at which prices fall are not explained by

competitive conditions in the market, a feature corroborated by managers in the industry.2 Hence,

this industry forms an almost ideal setting to study the value of intertemporal price discrimination

policies in practice. Given the features of the empirical application, I work with a monopolistic model

of pricing that ignores competitive considerations. Towards the end of the paper, I discuss how the

current framework could be extended to account for competition in other categories in which

substitution effects may be more important.

The data from this industry comprise aggregate retail sales and prices of all new video-

games compatible with the Sony Playstation released in the US market between October 1998 and

March 2000. A unique feature of the data is the inclusion of a large number of products (over 100),

along with the complete history of monthly sales and prices since introduction for each product. The

large number of products gives us the necessary cross-sectional variation to study pricing across

games. And the complete time-series enables us to model price dynamics over the entire lifecycle of

each product. The estimates for the video-game data imply that demand for games becomes

increasingly elastic with game-age, thereby generating incentives for firms to cut prices. Conditional

on these demand characteristics, I find that optimal pricing strategies for firms do indeed exhibit

“skimming”: an aggressive high initial price followed by steep discounting over time. To the extent

that the optimal policy explains observed price patterns, I am thus also able to provide some

empirical evidence for intertemporal price discrimination as an explanation for price cutting, which

has previously been lacking in the empirical literature.

I use the equilibrium solutions to explore the implications of consumer forward-looking

behavior on profits for the video-game firms in my sample. Specifically, I empirically measure the

extent to which forward-looking behavior by consumers reduces equilibrium firm profits. In the

context of the data, I find that such profit effects are large and economically important. I find that

that on average, the present discounted value of profits under myopic consumers is 172.2% higher

than under forward-looking consumers. This indicates that the effect of consumer forward-looking

behavior is significant in this industry.

2 Competition with used-games is also unlikely to be an issue. The trade-press suggests that used-games sales in the US

during the period 1998-2000 constituted only a small fraction of overall industry sales. For instance, FuncoLand, the only

used-game retailer in the US with a national presence, reports annual dollar sales of $0.16 million in 1998 (Serlin 1998). In

comparison, overall new video-game dollar sales in 1998 in the US was $5.5 billion (IDSA 2001).

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The results suggest that consideration of consumer dynamics is important for firms seeking

to set prices optimally in this industry. Firms can better gauge the extent of consumer forward-

looking behavior by conducting surveys that elicit consumer expectations of future prices or by

investing in market research that provides information regarding the extent of discounting by

consumers. How much should these firms be willing to pay to acquire this information? How worse

off would firms be if they instead set prices assuming no forward-looking behavior on the part of

their consumers? I obtain measures of this value by simulating the loss in profits incurred when

firms in the sample commit to pricing policies that incorrectly assume that forward-looking

consumers are myopic. I find that in the limiting case, relative to the situation where the true

consumer’s discount factor is 0.975 and firms are pricing optimally, average profits are 29.83% less

when firms commit to a pricing policy that incorrectly assumes that forward-looking consumers are

myopic. These profit losses are large and economically significant. I conclude that investing in

market research that provides information on consumer forward-looking behavior could provide

significant payoffs to firms in the industry. While these results are obtained in the context of video-

games, they can potentially inform similar analysis for comparable categories like entertainment

products (e.g. CD-s and DVD-s) and fashion goods (e.g. clothing and apparel), where similar pricing

issues are observed (e.g. Lazear 1986; Pashigian 1988).

The rest of the paper is organized as follows. Section 2 presents details of the demand and

pricing models. In the first part of section 2, I discuss the consumer’s choice rule, derive the implied

aggregate demand function and characterize the policy-relevant state variables for the firm’s

problem. In part two, I discuss the corresponding pricing model and equilibrium notion. Section 3

then discusses the empirical strategy and derives the likelihood function for estimation of the

dynamic model of consumer demand. Section 4 introduces the video-game data and discusses the

demand estimates. Section 5 discusses the pricing implications corresponding to these estimates of

demand. The last section concludes. Details of the numerical algorithm for computing the pricing

equilibrium and the likelihood function, and analysis related to the role of costs and competition in

the video-game market are presented in the appendices.

2. A model of dynamic pricing

2.1. Overview

This section lays out the details of the dynamic pricing model and the related notion of equilibrium. I

present a model of how a monopolistic firm selling a durable good product to a population of

heterogeneous consumers should set its optimal sequence of prices over time. The model takes

demand parameters as given. In the subsequent section, I discuss how these demand parameters

would be estimated from observed price and sales data. Demand estimation does not require the

6

assumption that observed prices in the data are set optimally by firms according to the dynamic

model presented here. Given demand estimates, I will use the dynamic pricing model developed in

this section to simulate optimal prices, and to generate normative policy recommendations.

The durability of the product results in three sources of intertemporal linkages in demand

that the firm has to account for in formulating its pricing policy. First, durability implies that

consumers who buy the good today will drop out of the market for the product in subsequent periods.

In this sense, the durable good monopolist creates his own competition: by selling more today, the

monopolist reduces his demand tomorrow. An optimal pricing policy for the firm should take this

“shrinking market” effect into account. Second, high-valuation consumers purchase the product, and

exit the market early. Their exit changes the distribution of consumer valuations in the market in

subsequent periods. The rate of exit of consumers is a function of the firm’s prices, and hence, the

evolution of heterogeneity in the market is endogenous to the firm’s pricing policy. Therefore, the

firm also needs to consider how current prices affect the distribution of heterogeneity in the future.

Finally, the firm’s current prices could also affect consumer expectations of future prices, which

shape product demand in the future.

The firm’s problem is to set a sequence of prices that incorporates these intertemporal

effects, and at the same time, enables it to extract the most surplus from its consumer base. Since

the valuation of any one consumer is unknown, perfect price discrimination is not a feasible strategy.

Instead, the firm can use time to discriminate among consumers, setting high prices initially to sell

to high valuation consumers, and cutting prices over time to appeal to the low valuation consumers

remaining in the market. A significant problem to “skimming” the market in this way however, is

that rational consumers may anticipate lower future prices, and strategically delay their purchases.

This incentive to delay is a function of consumer expectations of future prices, which are formed

endogenously with the prices chosen by the firm. To obtain the optimal set of prices, we must solve

for a sequence of equilibrium prices and consumer expectations, such that the firm’s prices are

optimal given consumer expectations, and such that the expectations are optimal given the firm’s

pricing policy.

The theory literature on durable good pricing has also emphasized the difference between

equilibria under commitment versus non-commitment (e.g. Stokey 1979 vs. Stokey 1981). The

literature has noted that when the firm can credibly commit to a pricing strategy, it would choose

not to cut prices over time, choosing the monopoly price in the first period and holding it fixed

thereafter. However, such commitment policies suffer from the fact that they are time-inconsistent:

once the first period passes, the monopolist faces an incentive to deviate from the chosen policy. In

my framework, I assume that the firm cannot credibly commit to a sequence of prices. Further, I

assume that consumers have rational expectations about the firm’s pricing policy. Thus, in essence, I

solve for the equilibrium pricing policy of a monopolist that lacks commitment power facing

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consumers with rational expectations. The non-commitment equilibrium I solve for is sub-game

perfect, and delivers a sequence of state-contingent prices that are time-consistent and optimal in

every period, and hence, are more managerially relevant. Besanko and Winston (1990) present a

stylized theory model of durable-good price equilibrium similar to this formulation.

To demonstrate how to solve for the equilibrium, I first discuss a very simple two period

model that captures the essence of the demand-side dynamics discussed above, and illustrates the

main features of the intertemporal price discrimination problem. I then present the full model that

can be used for analyzing the pricing problem in real-world empirical settings.

A simple two-period model

Following Besanko and Winston (1990), I consider a monopolistic firm selling a zero marginal cost

durable good product to a unit mass of consumers, with valuations uniformly distributed on [0, v ].

Both the firm and consumers share a discount factor δ, and live for two periods. Both consumer and

firm behavior is common knowledge: consumers understand the firm’s pricing problem and firms

understand the consumer’s purchase behavior. Consumers have rational expectations about the

firm’s pricing policy and, therefore, correctly predict the firm’s future prices in equilibrium. The

firm’s problem is to set prices p1 and p2 in periods 1 and 2. To solve for the optimal prices, we first

note that the marginal consumer in period 1, v1*, is indifferent between buying in period 1 and

waiting for period 2,

( )δ− = − ≥* *1 1 1 2 0ev p v p

For this forward-looking consumer, the value of waiting is a function of the expected price in period

2, 2ep . Assuming rational expectations, I set =2 2

ep p and obtain the marginal consumer in period 1

asδ

δ

−

−= 1 2*1 1

p pv . Correspondingly, as shown the figure below, valuations in period 2 lie the range [0, v1*),

and demand is, −*

1 2v p

v .

This simple set-up captures the three intertemporal demand linkages discussed above. In particular,

a decrease in the first period price reduces 2nd period demand; tightens the distribution of valuations

in the second period; and to the extent that p2 is a function of v1*, changes consumer expectations of

period 2 prices. In this simple example, v1* summarizes the mass of consumers remaining in the

market at the beginning of the next period. It is easy to see that it is the relevant state variable for

p2 0 v1* v

period 1 period 2

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the firm’s pricing problem, which is an intuition I utilize in setting up the full model below. The firm

sets prices in period 2 to maximize the flow of profits, and in period 1 to maximize the discounted

sum of profits:

( )

( ) ( )δ

δδ

−

−−−

−

= =

= + =

*1 2 1

2

**1 21

1

2 1 2 2

2 1

1 1 2 1 4 3

argmax

argmax

v p p

vp

vv pv v

v vp

p p p

p p p p

Equilibrium discounted profit is ( )δ

δπ−

−=1

4 3

v. We see that the equilibrium involves price cutting (i.e.

p2<p1), and has the feature that consumer expectations are optimal given prices, and prices are

optimal given expectations. It is also interesting to contrast these prices and profits to the situation

where consumers are myopic and do not care about future outcomes (i.e. have a discount factor of

zero). In this case, the marginal consumer in period 1 is simply =*1 1v p , and the corresponding prices

and profits are, = 1

2 2

pp , δ−= 2

1 4 3vp and δπ −= 4

v . It is easy to verify that the firm sets higher initial

prices and earns higher profits when consumers are myopic.

I explore the ideas illustrated in this simple model in detail in the full model below. While

capturing the underlying intuition, this simple model is inadequate for analyzing pricing in real-

world market situations. The full model considers the issues outlined above, but also incorporates

many features important for actual empirical applications, including longer time horizons, and more

realistic specifications of heterogeneity and demand.

2.2. Model framework

States and decisions

The firm’s objective to set a sequence of prices τ τ

∞

=tp that maximize the expected present discounted

value of profits from the product, ( )τ

τδ π

∞ −

=∑ ,t

tt t tE p S . The flow of profits in each period is a function of

the price, pt, and a vector of state variables, St. The firm is assumed to observe the state vector which

contains all relevant information required to decide the optimal price, p*(St). The vector St could

potentially include time since introduction, and also include functions of the entire history of the

game between the firm and consumers. For simplicity, I assume that a firm bases its current pricing

decision only on “payoff relevant” historic information, i.e. functions of history that only affects

current profits. Thus, I focus on Markov pricing strategies and solve the model for a stationary

Markov Perfect equilibrium. The resulting strategies are subgame perfect conditional on the current

state.

The state vector St consists of the following payoff-relevant variables: The mass of consumers

of each “type” in the market at the beginning of period t, Mrt, r = 1,..,R; and, the realization of a shock

to demand for the product in period t, ξt. As is explained in more detail below, in my model I assume

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that there exists a discrete finite number of consumer types in the population, each having its type-

specific valuation for the product. The specification of finite types is the aggregate analogue to the

latent-class models used for individual-level data (Kamakura and Russell 1989). Mrt indexes the

mass of type r in the market in time t. The demand shocks ξt, capture factors that affect demand in

period t, and are observed by the firm and consumers, but unobserved to the researcher.

The timing of the game between consumers and firms in the market is as follows. At the

beginning of period t, the firm observes the sizes of each segment in the potential market, Mrt. The

demand shock for the product in that period, ξt is then realized, and is observed by the firm. The firm

sets prices for that period conditioning on these state variables. Consumers observe the prices and

the realization of the demand shocks, and depending on their valuations, decide to buy the product

or delay purchase. Based on these decisions, aggregate demand for that period is realized at the end

of the period.

The profit function π(pt,St) is based on a chosen demand system. The proposed framework

can accommodate any demand system that generates profits of the form described above, and also

enables computation of the sizes of each type remaining in the market, Mrt. This includes most

discrete choice-based demand systems, including the popular random-coefficients logit and probit

model. Discrete choice based demand systems are appropriate for the durable good products I

consider, since consumers typically have unit demand for these products.

Demand function

I model demand using the random coefficients logit demand system. Past literature has used the

logit model to measure empirically the demand for high technology durables products (e.g., Melnikov

2000; Song and Chintagunta 2003; and Erdem, Keane and Strebel 2005; see also Chevaier and

Goolsbee 2005 who provide reduced-form evidence that durable-good consumers are forward-

looking). My model is different from these papers in that I allow for heterogeneity (while Melnikov

does not), and econometric errors in the demand function (while Song and Chintagunta do not), and

also base inference on aggregate data (while Erdem et. al. have access to individual-level adoption

data). In the model of demand, I treat each video-game as a separate market, explicitly ruling out

competitive pricing across games. In the empirical section below, I find supporting evidence for this

assumption as video-games are found to be very weak substitutes empirically. At the end of the

paper, I discuss the limitations of this assumption and some suggestions for future work. A recent

working paper by Gowrisankaran and Rysman (2006) considers a similar demand-model. This paper

uses an idea due to Melnikov to extend the monopoly demand-model presented here to include

substitution effects.

As noted above, I assume that there exists a finite number of discrete consumer types in the

population, each indexed by r, r = 1,..,R. I denote the population of consumers of type r in the

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potential market for the product in period t, by a continuum of mass Mrt. Each consumer faces an

intertemporal adoption problem that involves trading off the benefits from adopting the product

today, versus waiting for a potentially lower price tomorrow. In essence, the purchase decision is an

optimal-stopping problem. Let δc denote the consumer discount factor and ar denote the utility that

consumer r derives from the use of the product per period of consumption. The present discounted

utility of the service-flow from the product conditional on purchase is then αr = ar/(1-δc).3 A consumer

r’s conditional indirect utility from buying the product in period t is assumed to be:

α β ξ= − +rt r r t tu p (1)

where, βr is the price-sensitivity. The product and time-specific term, ξt, controls for any additional

product characteristics that are observed by consumers and influence their choices, but are

unobserved to the researcher. In the video-game context, ξt may capture such unobserved (to the

researcher) game-specific demand-shifters as box-office performance of co-branded movies linked to

the theme of the game and events related to celebrities (e.g. sports personalities) on whom game

characters are based. As explained further in the estimation section, ξt also serves as the

econometric error term in the estimation of demand.

The utility associated with no-purchase is more complex in the current durable goods

environment than in usual static discrete choice models. No-purchase captures the option value of

deferring purchase to a future period. I model the utility of not buying the product in period t, ur0t,

as the discounted expected value of waiting until period (t+1):

( )δ ε ε+ + = + − 0 , 1 0, 1 0max ,r t c t r t r t r t rtu E u u (2)

where εrt and εr0t are Type-1 Extreme Value disturbances that shift the consumers’ utility of no-

purchase in each period, assumed to be iid over time and over consumers.4 The expectation in (2) is

taken with respect to the distribution of future variables unknown to the consumer, conditional on

the current information, i.e. with respect to F(pt+1,ξt+1,εrt+1,εr0,t+1|pt,ξt,εrt,εr0t). Following Rust (1987),

the integration with respect to the extreme value error terms can be done analytically, and the

deterministic component of the consumer’s utility of waiting be expressed in terms of an “alternative-

specific” value function for waiting, Wr(pt,ξt). This value function satisfies the functional equation:

( ) ( ) ( )( ) ( )ξ δ α β ξ ξ ξ ξ+ + + + + + = − + + ∫ 1 1 1 1 1 1, log exp exp , , ,r t t c r r t t r t t t t t tW p p W p dF p p (3)

3 Motivated by the empirical application to video-games, depreciation in product quality, and the potential for resale in the

future are ignored. 4 For technical convenience I assume that εrt and εr0t are mean zero extreme value variates with location parameter -Γ, and

scale parameter 1 (where, Γ is Euler’s constant.) Assuming that the location parameter is -Γ rather than 0 (as is standard in

the literature) does not change the choice probabilities, but eliminates the euler’s constant from the value function equation in

(3) (see Rust 1992.)

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I now impose the assumption that consumers have rational expectations regarding future prices.

Hence, in equilibrium, consumers will correctly anticipate that the firm, when facing the future state

St+1, will set the price pt+1 = p(St+1). This implies that consumer expectations of future prices are

formed endogenously with the pricing policy chosen by the firm. Further, in equilibrium, the price

expectations of each consumer will be consistent with the pricing policy chosen by the firm. To solve

for the equilibrium, it is convenient to re-write the value function for waiting for consumers over the

states St,

( ) ( )( ) ( )( ) ( )δ α β ξ+ + + + = − + + = ∫ 1 1 1 1ln exp exp , 1,..,r t c r r t t r t t tW p W dF r RS S S S S (4)

The state variables St here purely reflect the consumer's beliefs about pricing. Intuitively, we

represent the value function Wr(St) as depending on factors that the consumer believes will drive

future pricing. Implicitly, we also assume that both consumers and firms understand and share the

same expectation of how the state of the market tomorrow, St+1, evolves given the state of the market

today, St. Consumer r will buy in period t if his utility from purchase exceeds that of waiting:

( )α β ξ ε ε− + > + −0r r t t r t r t rtp W S (5.1)

With the extreme value assumption on the errors εrt and εr0t, this decision rule implies logit

probabilities of purchase each period:

( )( )

( )( ) ( )α β ξ

α β ξ

− += =

+ − +

exp, , 1,..,

exp exp

r r t t

r t t

r t r r t t

ps p r R

W pS

S (5.2)

Integrating over the continuum of consumers, I obtain the following expression for the aggregate

demand for the product at state St,

( ) ( )=

=∑ 1, ,

R

t t rt r t trQ p M s pS S (6)

Evolution of states

I now discuss how the above demand system generates interdependence in demand over time.

Dependence in demand arises from dependence in the mass of each consumer type remaining in the

market each period. Unlike static models of discrete choice demand, the mass of consumers of each

type in the potential market, Mrt, is endogenous to the firm’s historic pricing behavior. Therefore, the

dynamics of pricing introduce a dynamic in the evolution of Mrt. To derive the evolution of Mrt, note

that each period, Mrt consumers of type r will purchase the product with probability sr(St,pt), and

drop out of the market. This leaves Mrt[1-sr(St,pt)] consumers in the market for the next period.

Hence, the size of type r in period (t+1) is Mrt[1-sr(St,pt)] plus any such consumers that enter the

market for the product in period (t+1). In the video-game context, I observe entry of such new

consumers. These correspond to new buyers of video-game consoles who subsequently enter the

12

market for all compatible video-games. To accommodate this aspect of the data, I specify that a mass

Nr of “new” consumers of type r enter the market each period. Then, the evolution of Mrt is given as,

( )+ = − + , 1 1 ,r t rt r t t rM M s p NS (7)

Equation (7) summarizes the effect of the history of prices, demand shocks and new consumer entry

on the demand side of the market. From equation (4) and (7), it is easy to see that demand, as well as

the evolution of the state variables, Mrt-s, are endogenously determined in equilibrium. The state

variables, M1,..,MR, depend on the corresponding probability of purchase, sr(St,pt), which in turn

depends on the evolution of the state variables through (4).

Denote the transition probability of a new state St+1 given state St and price pt, as F(St+1|St).

I denote the conditional density of ξ as Fξ(ξt+1|ξt). From equation (7), the market sizes Mrt evolve

deterministically. Given the current state M1t,..,MRt,ξt, the current value of the control, pt =

p*(M1t,..,MRt,ξt), and the future values of the state variables, Mr,t+1, are known. Hence, I need to

integrate the RHS of equation 4 only against Fξ(ξt+1|ξt). In my empirical work, I assume ξ is

distributed iid normal across games and time-periods. Hence, in computing the optimal policy

Fξ(ξt+1|ξt) = Ф(ξt+1;0,σξ), where Ф(.) denotes the cdf of a normal distribution.

Flow of profits and value function

I now derive the profit function for the firm implied by the above demand system. Let c denote the

marginal cost of production of the product, which is assumed to be constant over time. The constant

cost assumption is consistent with the features the video-game industry, and is discussed further in

Appendix E. The flow of profit for each firm can be written as:

( ) ( )( )π = −, ,t t t t tp Q p p cS S (8)

Firms are assumed to be risk-neutral and set prices by maximizing the expected present discounted

value of future profits, where future profits are discounted using the constant discount factor δf. The

solution to the pricing problem is represented by a value function, V(St) which denotes the present

discounted value of current and future profits when the firm is setting current and future prices

optimally. The value function satisfies the Bellman equation:

( ) ( ) ( ) ( )ξπ δ ξ ξ+ +>

= + ∫ 1 10

max ,t

t t t f t t tp

V p V dFS S S (9)

The optimal pricing policy, p*(St), is stationary, and maximizes the value function:

( )( )

( )>

= 0

* argmaxt

t tp S

p VS S (10)

I now present the definition of equilibrium in this game.

Equilibrium

13

Definition: A Stationary Markov-perfect equilibrium in prices in this model is defined by a set of

waiting functions, Wr*(St), r = 1,..,R and price function, p*(St), such that equations (4-10) are

simultaneously satisfied at every St.5

The equilibrium is the fixed point of the game defined by (4-10). The equilibrium defined

above has three properties. First, equations (9) and (10) implies that in equilibrium, when faced with

state St, the firm’s pricing policy is a best response to consumer behavior at that state. At the same

time, equations (4) and (5) imply that when faced with a state St, and price p(St), consumers make

purchase decisions by maximizing intertemporal utility. Both consumers and firms take into account

the effect of their actions on the evolution of states in the market through equation (7). Finally,

equation (6) implies that the realized aggregate demand in state St is consistent with optimal

consumer purchase decisions at the corresponding optimal price p*(St). The fixed point that defines

equilibrium thus requires each consumer type to maximize their expected utility subject to

consistent perceptions of the likelihood of future states for the firm, and firms to maximize expected

payoffs based on consistent perceptions on the likelihood of future consumer states. With R = 2, the

optimal pricing policy is the solution to a three-agent decision problem between a monopolist and two

types of consumers. Consumers of each type take into account that their purchase decisions change

the future state of the market, M1t+1, M2t+1 (equation 7), and thus, also change the future prices

charged by the firm, p*(M1t+1,M2t+1,ξt+1).

Numerical solution

I present an algorithm to solve numerically for the fixed-point in Appendix A. The algorithm takes as

input consumer parameters (αr,βr, r = 1..,R) that are estimated separately in a first-stage demand

analysis (the details of this estimation are described in section 3 below.) The algorithm is a modified

version of policy iteration and proceeds in five steps: In step 1, I make a guess of the firm’s policy

function, p(n)(S). Given p(n)(S), in step 2, I obtain each consumer type’s value of waiting Wr(0)(S), r

=1,2 by solving equation 4. Given Wr(0)(S), I use (5) to obtain the new probabilities of purchase,

sr(S,p(n)(S)), r =1,2, which when inputted into (7), gives us the updated new value of the endogenous

states M’r’(0), r = 1,2. This ends step 3. Given M’r’(0), I solve equation 4 again to obtain an updated

guess of the value of waiting Wr(1)(S) (step 4). I iterate on steps 2-4 till |Wr(1)(S)- Wr(0)(S)| is

arbitrarily small. The final value, denoted as Wr(n)(S), r = 1,2, represents the guess of the equilibrium

values of waiting for the 2 consumer types at the current guess of the optimal policy, p(n)(S). This

completes one iteration of the solution to the consumer’s problem. In step 5, holding Wr(n)(S), r =1,2

5 Let ht = sτ,pτ, τ = 1,..,t-1 denote the history of the game till period t. A policy is Markov at time t if there is a function p*:

R+(R+1)→R+ such that pt(ht) = pt*(st) for all st, and f(st|ht) ≡ f(st |st-1,pt-1). A sequence of policies pt(ht), is said to be a subgame

perfect equilibrium if at any time t >0, V(pt(ht)) ≥ V(pt’(ht)), where pt’(ht) is any alternate policy. If a sequence of Markov

policies pt* is a subgame perfect equilibrium, the equilibrium is said to be Markov-perfect. A sequence of Markov policies pt*

is said to be stationary if pt* ≡ p* for all t ≥ 0.

14

fixed, I implement a “policy improvement” step for the firm by finding p(n+1)(S) that maximizes the

right-hand side of (9). I steps repeat steps 1-5 till |p(n+1)(S)- p(n)(S)| is arbitrarily small.

I solve for the optimal pricing policy using parametric policy iteration (Rust 1996, Benitez

Silva et al. 2000). The R+1 dimensions of the state space are discretized, and the firms’ value

function V(St), and the consumer waiting functions, Wr(St) are approximated as the tensor product of

Chebychev polynomial bases in each of the state dimensions. The polynomial approximation works

well since both the state space and the control are continuous. The algorithm was programmed in

Matlab©, and converged smoothly for the parameters I considered. Convergence of the numerical

procedure to a solution indicates that an equilibrium exists at those parameter values. Since

analytical solutions are unavailable, I am unable to formally state whether the converged

equilibrium is unique.6 I found that the converged solution was robust to initial guesses of the

pricing policy and the waiting function for the range of demand parameters and discount factors that

I expect to reasonably see with real world data. Thus, I did not find evidence for multiple equilibria.

It is however possible that multiple equilibria exist for other (boundary) values of the parameters.

Discussion

To justify some of the modeling assumption made in the previous sections, I spoke with managers

from the video game industry. I learned that the typical rules-of-thumb used for pricing share many

similarities to the proposed model. First, estimates are used to assess the evolution in the size of the

potential market. My interviews also revealed that managers revise game-prices periodically, cutting

prices if sales are low, and keeping prices high if realized sales are high. I interpret this heuristic as

indicating that the total sales of the game is an important state variable for the firm’s pricing

decision. This is roughly consistent with the model since the theoretical state variables, the segment

sizes, are a function of cumulative sales of the game until that time period. The model assumes that

managers know the distribution of consumer types and can, therefore, translate the observed sales of

the game into segment sizes, which form the “payoff relevant” state variables for the pricing decision.

Managers are also aware that high willingness-to-pay “hardcore gamers” sustain initial high prices,

which have to be lowered once the game becomes “main-stream”. This adherers to the notion of price

discrimination over time. As a reviewer pointed out, this heuristic thumb-rule for price cutting is

also consistent with managers cutting prices to reflect lower valuations of consumers for older

games. To the extent that such “novelty” effects are common across segments, these are captured by

the proposed model via the game-specific shocks to utility, ξjt-s. Since ξ is included as a state variable

6 BW establish analytically the existence and uniqueness of a subgame perfect Nash equilibrium in prices in the context of a

related, stylized demand model with uniformly distributed consumers and no uncertainty (i.e. no shocks to demand).

15

in the policy function, the pricing policy I solve for potentially reflects such exogenously declining

valuations of games.7

A relevant question here is whether it is the retailer, rather than the manufacturer (as in

this model), that is initiating the observed price-cuts in the data. For instance, it could be that

wholesale prices from the manufacturer are constant, and the retailer is cutting prices over time due

to reasons unrelated to intertemporal price discrimination. For example, falling retail prices could

arise from retailers rapidly clearing inventory of low-selling games to free up shelf-space for new

releases. Interviews with managers in the industry however, indicated that game manufacturers do

periodically initiate cuts to wholesales prices, which are mostly passed through to consumers by

retailers. Further, I found that the industry typically implements excess inventory return policies

within the manufacturer-retailer channel, whereby retailers can return unsold stocks of games back

to the manufacturer, making retail price-cutting to clear inventory a less compelling explanation.

Nevertheless, in the absence of retail-level inventory data, I am unable to rule out this explanation

completely. I thank an anonymous reviewer for pointing this out.

3. Empirical strategy and estimation

I now discuss my econometric specification and empirical strategy. The empirical strategy involves

two steps, the first dealing with estimation of demand-side parameters, and the second comprising

the solution of optimal prices given these parameters. I first describe the rationale behind this two-

step approach and the methodology for estimation of demand parameters. The results for optimal

pricing are presented in the subsequent sections.

The rationale for the empirical strategy and estimation method is derived from the intended

use of the model, which is primarily normative (see Hitsch 2006 for a similar argument). From the

perspective of the end-user of the model, viz. the manager setting prices, the estimation method

needs to address two issues. First, the model should deliver estimates of parameters indexing the

demand for his product. Second, the method should provide a way of accounting for consumer

expectations of future prices. Demand parameters are obtained using the maximum likelihood

estimator I describe below. The firm also knows its own historical pricing policy and therefore knows

the exact process generating prices. It can assume that consumers understand the true price

generating process, and thus obtain an estimate of consumer expectations. Or it can employ market

research and elicit survey data on the nature of consumer perceptions of future prices.

7 One way to explicitly account for such “novelty” effects may be to include a flexible function of the time since game-

introduction into consumer utility, so as to capture the decline in utility with the age of the game in a reduced form way (e.g.

Einav 2006, and the regressions presented in appendix D). One computational difficulty associated with introducing functions

of time in this manner is that “calendar time” then explicitly becomes a relevant state variable for the firm’s pricing problem.

This implies that we can no longer solve for a stationary pricing policy function. Further, one can get different pricing

patterns depending on the chosen functional form for the decay.

16

The estimation problem is more challenging for a researcher trying to infer demand

parameters from observed price and sales data. The researcher faces two challenges. First, it has to

address the fact that unobserved factors that shift demand over time may be correlated with

observed marketing mix variables like prices. While the firm observes the realization of these shocks

to demand each period, and can simply use it as data in estimation, the researcher does not. Hence,

unlike the firm, the researcher trying to estimate demand parameters faces an econometric

endogeneity problem. In principle, the researcher could estimate the demand model in (6) using

maximum likelihood, where ξ is the econometric error term. The concern is that manufacturers could

condition on the demand shocks ξ in setting prices causing observed prices to be correlated with the

demand shocks, and leading to an endogeneity bias. Second, the researcher does not know the exact

process generating observed prices, and hence cannot use it to proxy for consumer expectations.

Rather, he has to make an assumption about the process generating prices, and estimate the

parameters underlying the process from observed prices. Like the firm, an alternative strategy for

the researcher would be to obtain direct survey data on consumer expectations (e.g. Erdem et. al.

2005). But in many cases, including ours, such information may not be available.

To address both issues, I adopt a “limited information” maximum likelihood approach to

estimation. The method involves specifying a flexible joint density for prices and demand shocks, and

estimating its parameters jointly with the demand function. The specification of correlation between

the price process and the demand shocks helps control for potential endogeneity biases. The density

of future prices also serves to characterize consumer expectations. The approach is termed limited

information because the density of prices helps handle the correlation induced by ξ, but does not

provide any additional information about the demand parameters. The technique is analogous to the

approach of Villas-Boas and Winer (1999) and Yang, Chen and Allenby (2003; models 5/10 in Table

2) for household-data, and to the parametric control function approach of Petrin and Train, 2004 (see

also the discussion in Chintagunta, Dube and Goh, 2005). The empirical strategy I adopt is as

follows. I first use the observed sales and price data to estimate the demand function for each game.

Given the demand estimates, I solve numerically for each firm’s optimal pricing policy. Similar two-

stage approaches have been adopted in Benkard (2004), and Dubé, Hitsch and Manchanda (2005).

Alternative approaches

An alternative to this empirical strategy is to add restrictions from the optimal equilibrium pricing

strategy into the demand estimation procedure. This approach, while more efficient, is

computationally burdensome since the equilibrium has to be repeatedly solved for every guess of the

demand parameters. It also entails imposing the strong assumption of optimal pricing by firms in

demand estimation, which has the potential to bias the estimated parameters if observed pricing is

indeed, not optimal (I find some evidence of this in the results section.) Further, in this approach,

17

perfect firm rationality is imposed when estimating parameters; this leaves little room for testing

whether observed outcomes are optimal or for making normative policy recommendations, which are

the primary goals of this paper. A more agnostic strategy that does not impose optimality but still

retains the spirit of markov perfect pricing is to specify observed prices as a flexible function of the

theoretical state variables (viz. the sizes of the segments and the shock to demand), and to model

demand jointly with this pseudo-policy function. Ching (2005) has proposed this strategy in the

context of a static demand system for experience goods. Extending this approach to forward-looking

consumers is significantly more challenging.8 Further, the policy-function approximation tends to be

poor unless one uses rich enough polynomial basis functions for the state variables. As the number of

state variables increases, this entails estimating a large number of parameters, and also entails a

curse of dimensionality in approximation.

In contrast, the limited information approach that I adopt here is parsimonious, and

computationally simpler. The main disadvantage is it that requires taking a stand on the joint

density of prices and demand shocks. It is difficult to prove analytically that prices generated by the

optimal dynamic pricing model would imply observed prices that conform to the postulated form for

this assumed density (see the next section for some computational support for the validity of the

assumed density.) I view the limited information approach as a practical tractable solution to

demand estimation that exhibits some of the richness of the theory model (e.g. auto-correlated

prices.) With a sufficiently rich density for prices, I am able to explain over 90% of the variation in

observed prices. This density likely better captures the belief process of a real consumer than the

assumption that consumers can calculate the future market sizes of all segments. This latter

assumption is required when computing optimal prices since we need a model of how consumer

expectations are formed in equilibrium; however, when estimating demand, it seems more

reasonable to use the density of observed prices to proxy for consumer expectations.

In the remainder of this section, I first present the econometric assumptions. I then discuss

the limited information technique for estimating demand parameters. The subsequent sections

describe the data and results.

3.1. Econometric assumptions

The remaining econometric assumptions concern the specification of heterogeneity, and the choice of

the joint density for prices and demand shocks. These assumptions are motivated by the rationale

outlined above, and also by some of the specifics of the empirical application to the video-game

industry. To clarify the differences across games, I add the subscript j indexing each game to the

notation from this point onward.

8 With forward-looking consumers, the Jacobian terms in the associated likelihood function will involve numerical derivatives

of the consumer value function Wr(.) with respect to the demand shocks ξ. These tend to be unstable unless Wr(.) is

approximated very finely over the grid of ξ, which is computationally expensive.

18

Heterogeneity

Allowing for heterogeneity is important in this model, since it is the existence of differences in

valuations across consumers that generate the incentive for the firm to price discriminate over time.

As noted before, I adopt a random coefficients approach to model consumer heterogeneity, using a

discrete approximation. I assume that the distribution of price sensitivity, βr, and valuations, αrj, for

product j across the R consumer segments is,

( )( )( )

α βα β

α γ β β

= + +

, if = 1,

, if = 2,...,R

j

jr jr

j j r r

r

D r (11)

Here, segment 1 is treated as a “base” segment and the parameters of the other (R-1) segments are

expressed as deviations from the base segment (see Besanko, Dubé and Gupta 2003). In (11) above, I

have also assumed that the deviation of the rth segment’s valuation from the base segment can be

expressed in terms of the characteristics of the video-game, Dj. The vector of characteristics Dj is

time-invariant and includes a constant, and dummy variables for the game’s genre. Thus, a separate

game-specific fixed effect is estimated for the first (“base”) segment, and a genre-specific deviation

from the base segment is estimated for all other segments. I make this assumption for purposes of

model parsimony. For simplicity, I also assume a constant inflow of N new consumers into the

market, where N is computed as the mean new console sales per month in the data. I assume that a

proportion φr of the new consumers are of type r, where φr, r = 1,..,R are parameters to be estimated

from the data.9 Thus, in equation (7), I set Nr = φr N. In Appendix B, I discuss how this specification

would generate demand-side incentives for firms to cut prices over time. Finally, in equation (7), I

also need to specify Mr0, the initial sizes of each consumer segment in the market the product. I set

Mr0 = M0φr, where M0 is an initial market size for the product, computed from the data (see Appendix

C for details).

Finally, given the difficulty past literature has noted in estimating discount factors (e.g. Song

and Chintagunta 2003), I do not attempt to estimate the discount factors for consumers and firms (δc

and δf). Instead, I set the discount factors δc and δf to 0.975. This value is lower than typically

9 This implies that we assume that the distribution of valuations for video-games in the population adopting the console is

stationary (e.g. Conlisk, Gerstner and Sobel 1984; Narasimhan 1989). Stationarity implies that the distribution of (αjr,βr) in

entering cohorts of consumers is constant over time. This assumption could be violated if consumers adopting the hardware

console later in the life-cycle also have correspondingly lower valuations for compatible games. Incorporating this effect would

require modeling the consumers’ joint decision to adopt the console and the set of compatible games, which is beyond the scope

of the current analysis. Given the short time series in the video-game data (1.5 years), I expect the stationarity assumption to

be a reasonable approximation. Further, I do not find any evidence in the data that games released later in the life-time of the

consoles have lower levels or higher rates of decline in prices. Reflecting my empirical application to this industry, I also

model the number of new consumers entering the market as exogenous to prices of game j. Given the large number of games

for each console (over 600 for the Sony PlayStation), prices of any one game do not tend to shift aggregate sales of the

hardware console. In the data, PlayStation console sales do not correlate significantly with prices of any one game. Further,

after controlling for hardware console prices and game availability, game prices do not significantly explain console sale

variation over time. This suggests that conditional on availability, to a first approximation, console sales can be treated

exogenous to prices of any one game.

19

assumed for monthly data, and is partly motivated by the increased computational cost of solving the

consumer’s intertemporal problem at larger discount values. The marginal cost, denoted by c, is

constant, and set to $12 per unit (see the discussion in Appendix E). In the remainder of this section,

I discuss the maximum likelihood estimator.

3.2. Maximum likelihood estimation

Discussion

I first provide a short general discussion of the limited information approach (Villas-Boas and Winer

1999). I then discuss specific details of its application to the current empirical context. Let denote

qt,ptTt=1 the sales/price data available for estimation. The analyst’s chosen model of demand predicts

aggregate demand for each period, qt = Dt(pt,ξt;Ω), as a function of prices, shocks to demand, ξ and

parameters, Ω. Denote the pdf of ξ as fξ (.), and the inverse of the demand function as ξt = Dt-

1(qt,pt;Ω). The distribution of errors, ξt induces a distribution on demand, which generates a

likelihood for the observed demand data:

( ) ( )( )ξ−

=Ω = Ω∏

1

1, ;

T

t t ttf D q p J (12)

where, J is the Jacobian. The concern for endogeneity biases arises since prices are set as a

function of demand, and hence are expected to be a function of ξt. Thus, p is correlated with ξ. Since

prices depend on predicted demand, the density of prices will contain information about the demand

parameters. Hence, the likelihood above must be augmented to include this information:

( ) ( ) ( )( )ξ ξξ −

=Ω = Ω Ω∏

1| t1

| ; , ;T

p t t t ttf p f D q p J (13)

To obtain fp|ξ(.), a full information approach would to assume a specific model of pricing conduct by

firms (e.g. profit maximization), and derive the implied density of prices by change of variables

calculus. This approach is not suitable in my context for three reasons: a) my primary goal is

normative, to recommend optimal prices to firms, which would not be possible if restrictions from the

optimal pricing policy are imposed in estimation; b) the density of prices implied by optimal profit

maximization behavior requires computation of the full dynamic pricing equilibrium for every guess

of the parameter vector, which hugely increases the computational burden of the estimator; and 3) as

has been pointed out in the literature (e.g. Chintagunta, Dube and Goh 2005), imposing restrictions

from the wrong pricing policy could potentially to bias estimated demand parameters.

Hence I adopt the alternative approach termed “limited information” by Villas-Boas and

Winer (1999). This approach is more agnostic about the nature of pricing conduct, and specifies a

process for prices as:

( ) ( ) ( )ηη η η ξ= + ≠, ~ . ; cov , 0t t t t t tp g fz (14)

20

such that the assumed density on η implies a density for prices. Here g(.) is a flexible polynomial,

and zt is a vector of variables that shift prices over time. In principle, z can include cost-side factors

and variables excluded from the demand-side that serve as instruments for prices. I then obtain the

joint probability of the data as,

( ) ( )( ) ( )( )η ξ ξξ −

=Ω = Ω Ω∏

1| t1

| , ; , ;T

t t t t ttf g f D q p Jz z (15)

which serves as the limited information likelihood. The main advantage of this approach is that it is

computationally simpler. The main disadvantage is that it entails making an assumption about the

process generating prices and ξ. In the remainder of this section, I discuss the specifics of the

application of the estimator to the video-game context.

Price process

As noted above, the limited information approach requires a specification for the joint density of

prices and demand shocks. A price process generated by the dynamic pricing model I described above

should satisfy two properties at minimum:

• The process has to be serially correlated over time.

• The process should allow for contemporaneous correlation of prices with the shock to demand

since equilibrium prices are a function of the realization of the demand shock.

I specify the following process for prices and demand shocks:

( ) ( )ξη η ξ σ σ ρ−== +∑ , 10

, , ' ~ 0,0, , ,K k

jt k j t jt jt jt pkp a p BVN (16)

where, BVN denotes a Bivariate Normal distribution. The process specifies demand shocks as mean

zero iid normal variates, and prices as conditionally normally distributed variates with mean equal

to a K-order polynomial in one-period lagged values. This process allows prices and demand shocks

to be contemporaneously correlated, and allows prices to be serially correlated over time in a flexible

way. The price process serves two roles on the demand-side. First, in computing the likelihood

function, it helps us incorporate the density of ξ correctly as prices covary with the level of ξ.

However, the cost is that we have to make a strong assumption on the form of the joint distribution

between prices and ξ. Second, under the assumption that consumers form expectations rationally, it

serves to characterize the distribution of future prices and demand shocks, required to compute the

value function for delaying purchase for each consumer.

Three aspects of the price process are noteworthy. First, the specification in (16) imposes

conditional independence in the sense of Rust (1987) on the transition of prices and ξ over time.10 I

10 That is, we have assumed in (16) that the transition density of (pj,t+1,ξjt+1)|pt,ξjt can be factored as fpξ(pt+1,ξt+1|pt,ξt) =

fξ|p(ξt+1|pt+1)fp(pt+1|pt). Implicitly, while p and ξ can be contemporaneously correlated, I make the assumption that the

realization of ξt does not contain any information about pt+1 or ξt+1 except through pt. The assumption implies that (a) ξ is iid

21

impose this for computational simplicity. As discussed in the “Likelihood” section below, conditional

independence allows us to integrate the unobserved shock ξ out the consumers value function for

waiting, so that Wr(.) is not a function of ξ. This reduces the cost of the nested fixed point

computation inherent in the estimation procedure considerably. Second, in this specification, lagged

prices essentially serve the role of instruments for current prices. This approach is valid if a) ξjt is iid,

and b) the pricing errors,η, are serially uncorrelated over time. Regressions of observed game prices

on polynomials in lagged prices strongly reject serial correlation in residuals (see Table 2.1 and 2.2.).

Lagged prices also explain over 88% of observed price variation (see Table 2.1 and the discussion in

section 4.2). Lack of other game-specific instruments for prices precludes my ability to explore

alternative approaches. Commonly used instruments for prices are not suitable for this category:

Constant marginal costs rule out cost-side instruments; lack of regional data rule out “Hausman”-

style instruments; and the fact that game attributes explain little or no price/sales variation rule out

the use of attributes as instruments, as in BLP (1995). Third, a final concern about this specification

is whether the supply-side pricing model I consider would generate prices that are Markovian as

assumed, i.e. a function of only 1-period lags. I am unable to prove analytically that this would

indeed be the case. However, to address this issue, I simulated prices from the supply side model for

a large number of hypothetical parameter values. In the simulated price data, I find almost always

that polynomials in 1-period lagged prices are able to explain over 90% of the variation in prices.

Further, I do not find evidence of serial correlation in the unobserved component of simulated prices

after controlling for lagged price effects. Finally, the assumption of time-invariant parameters in the

price process follows from the stationarity of the optimal pricing policy.

I now collect the set of parameters to be estimated in a vector α β λ β ϕ= = =Θ ≡ 1 2 0 , , , , , ,J R Kj j r r r r k ka

ξσ σ ρ, ,p .

Likelihood

I denote the observed data for game j, j = 1,.., J as (qj,pj) = (qj2,pj2,.., qjTj,pjTj) . The conditional

likelihood of the data is the joint probability of observing the vector (qj,pj) given Θ and the first

observations, qj1,pj1:

( ) ( )

( )

( ) ( ) ( )

ξη ξ η

ηξ η ξ η

ξ η

ξ η

− −

=

− − →=

− − →=

Θ = Θ

= Θ

= Θ Θ

∏

∏

∏

, 1 , 12

, 1 , 1 , ,2

, 1 , 1 , ,2

, ; , , ;

, , ;

, , ;

j

j

j

T

j j jt jt j t j tt

T

jt jt j t j t q pt

T

jt jtjt j t j t q pt

f f q p q p

f q p

f p q p f

q p

J

J

across time, and (b) pt+1 is a function of ξt only through pt. In principle, conditional independence can be relaxed (e.g. Keane

and Wolpin 1994), albeit at much larger computational cost.

22

( ) ( ) ( ) ηξ η ξ ηξ − − − →=

=

= Θ Θ − ∑∏ , 1 , 1 , 1 , ,02

, , ;jT

K kjt jt j t j t jt k j t q pk

t

f p q p f p a p J (17)

where, fξ|η(.) is the conditional distribution of ξ given η and Jξ,η→q,p is the Jacobian of the

transformation between the data and the econometric errors, ξ and η.

Computation of the likelihood function above requires two additional steps. First, note that

the state ξ is unobserved implying that it needs to be inferred from the observed demand data for

each step of the parameter search. Hence, we need a method to infer ( )ξ Θjt that equate predicted

and observed aggregate demand for every guess of Θ. Second, we need to compute the Jacobian,

Jξ,η→q,p. I discuss these steps in sequence below.

Computing ( )ξ Θjt

I compute ( )ξ Θjt by inverting the demand system numerically. Inversion of the demand system

requires computation of the value function for waiting for each consumer type. I first discuss how I

compute these value functions, and then discuss the details of the inversion procedure.

Value of waiting

The value function for waiting to period t+1 for a consumer of type r is defined recursively via

equation 3 in section 2. Recall that for computing the equilibrium in section 2, we reparametrized

the value of waiting for each consumer in terms of the states St, i.e. Wr(St). For estimation of demand

it is convenient to work directly with the value function defined in terms of prices and ξ: i.e.

Wrj(pjt,ξjt). Note that the Wrj(pjt,ξjt) functions cannot be solved for analytically, and hence, I

approximate these numerically over a grid of the state variables. A complication arises because the

inversion procedure I discuss below involves repeated evaluation of Wrj(.) for many trial values of ξ.

Since Wrj(pjt,ξjt) is a function of ξ, it has to be numerically interpolated over ξ for each of these trial

values. In practice, I found that the inversion for ξ tends to be numerically unstable unless I

approximated Wrj(pjt,ξjt) very finely over the space of ξ. This approximation is computationally

expensive since Wrj(pjt,ξjt) is specific to game j and consumer type r and has to be recomputed J*R

times for each guess of Θ. A significant computational saving is obtained however, by the assumption

that the process for prices and demand shocks satisfy Conditional Independence in the sense of Rust

(1987). The significance of this assumption is that it implies that ξ can be integrated out of the

option value of waiting for the consumer. 11 Hence, while the consumer’s utility of purchase is a

11 To see this, note that with logit errors, the option value of waiting satisfies the functional equation: Wrj(pjt,ξjt;Θ)

( ) ( )( ) ( )ξ

ξ

δ ξ ξ ξα β ξ ξ+ + + ++ + + +

= +− + Θ ∫ ∫ 1 1 1 11 1 1 1ln exp exp , ,, ;

c p jt jt jt jt jt jt

p

rj r jt jt rj jt jtf p p dp dp W p

( ) ( )( ) ( ) ( )ξ ξ

ξ

δ ξ ξ ξα β ξ ξ+ + + + ++ + + +

= +− + Θ ∫ ∫ 1 1 1 1 11 1 1 1ln exp exp ,, ;

c p jt jt jt jt jt jt

p

rj r jt jt rj jt jtf p p f dp dp W p .

Its clear that the RHS is not a function of ξjt, and hence, Wrj(pjt,ξjt;Θ) ≡ Wrj(pjt;Θ).

23

function of the realization of the demand shock, the value of delaying purchase for the next period is

not. Hence, we do not need to re-approximate the value function of waiting for every realization of ξ.

However, this computational saving comes at the cost of ruling out serial correlation in ξ, which is a

significant restriction.

With these assumptions, the functional equation defining the value of waiting for each

consumer type can be written as:

( ) ( ) ( )( )

( ) ( )

ξ

ξ ξ

δ α β ξ

φ ρσ σ ξ φ ξ σ ξσ ρ

+ + +

+ + + + +=

Θ = − + + Θ

× − − −

∫ ∫

∑

1 1 1

1 1 1 1 102

; log exp exp ;

1

1

rj jt c rj r jt jt jr jt

p

K kjt k jt p jt jt jt jtk

p

W p p W p

p a p dp d

(20)

where, φ(.) is the pdf of a standard normal distribution. The mapping defined in (20) is a contraction

(Rust 1987); hence iterating on (20) is guaranteed to converge to a unique solution for Wrj(pjt;Θ). As

in the supply-side model, I approximate Wrj(pjt;Θ) by Chebychev polynomials in prices. With this

approximation, computing each Wrj(pjt;Θ) takes about 2 seconds on a standard 3.2 GHz Pentium PC.

I now discuss the inversion procedure for recovering ( )ξ Θjt .

Inversion of the demand system

To make the exposition of the inversion procedure easier, I denote by pjt and ξξξξjt the vector of observed

prices and demand shocks for game j till (and including) period t. I stack the observed demand for

game j in a vector qj = qjtt=1,..,Tj, and the predicted demand for the game given Θ in a vector

Qj(pjTj,ξξξξjTj,Dj;Θ) = =Σ 1Rr Mrjt(pj,t-1,ξξξξj,t-1,Dj;Θ)srjt(pjt,ξjt,Dj;Θ)t=1,..,Tj. For each guess of the parameters Θ, I

first compute Wrj(pjt;Θ) as above; I then make an initial estimate of the demand shocks ( )

j

njTξ and

iterate on the expression,

( ) ( )( )+ = + − Θ( 1) ( ) ( )ln ln , , ;j j j j

n n njT jT j j jT jT jDξ ξ q Q p ξ (18)

across all games till convergence. This procedure is similar to the inversion proposed by Berry,

Levinsohn and Pakes (BLP) (1995) in the context of the aggregate logit demand model. The main

difference, is that unlike BLP, consumers in my model maximize intertemporal utility, implying that

the corresponding aggregate demands, Qj(pjt,ξξξξjt,Dj;Θ) are a function of the consumer’s value for

waiting each period. The computational burden of the estimation procedure is driven by the fact that

Wrj(pjt;Θ) is specific to game and consumer type and needs to be computed J*R times for every guess

of Θ.

Jacobian and Likelihood function

The Jacobian for the problem is derived in appendix D. Following Appendix D, I can write the joint

likelihood of the data for game j as,

24

( ) ( ) ( ) ( )

( ) ( )ξ ηξ

ξ ξ

− − −=

==

Θ Θ − Θ = Θ − Θ

∑∏

∑

| , 1 , 1 , 10

21

, , ;, ;

, ; 1 , ;

j

K kTjtp jt j t j t jt k j tk

j j Rt rjt rjt jt jt rjt jt jtr

f p q p f p a pf

M s p s pq p (21)

The joint (log) likelihood of the entire data is thus,

( ) ( )( )=Θ = Θ∑

1log , ;

J

j jjf q p

(22)

Identification

I now provide an informal discussion of identification in this model. The game-fixed effects are

identified from differences in the mean level of sales across games. Within game variation in prices

identifies the price parameters. Heterogeneity is identified from the structure of the model and the

rate of change in market shares over time in response to price changes. A simple example illustrates

this. Consider a game with potential market M, and sales q1 and q2 in periods 1 and 2. Let N new

consumers enter into its market every period, and prices be the same in both periods. Then observed

market shares in periods 1 and 2 are: s1 = q1/(M+N); s2 = q2/(M-q1+N). A model without heterogeneity

would predict that s1 = s2 = Pr(purchase), and would not be able to explain an observation s1 ≠ s2.

Suppose we allow for two types of consumers such that type 1 has higher valuations and thus a

higher probability of purchase. Let the probability that a consumer, drawn at random from the

population of potential buyers, is of type 1 be φ1 and φ2 in periods 1 and 2. Then, predicted market

shares in period i = 1,2, si = φiPr(purchase|type1)+(1-φi)Pr(purchase|type2). If we see in the data

that s1 > s2, we infer that φ2 < φ1, i.e. that the market in the second period is composed of more

consumers of type 2. If in addition, prices are lower in the second period, the extent to which s1 > s2

tells us more about the extent to which there is heterogeneity.12 Finally, price expectations are

pinned down by the assumption that consumers have rational expectations and understand the

process generating prices.

4. Data and estimation results

4.1. Overview of market

I first present a brief overview of the console video-game market along with the main stylized

features of the data. I then describe the demand estimates and pricing results.

The video-game game industry is a two-sided market comprising a hardware-side (consoles),

and a software-side (games). On the hardware-side, platform providers (like Sony, Nintendo and

Sega, the main players in the 32/64 bit generation) develop and sell consoles, and charge royalty fees

12 I thank Wes Hartmann for suggesting this example. Chintagunta (1999) (section 3), and Song and Chintagunta (2003)

(section 4.1) provide Monte Carlo evidence for parametric identification of heterogeneity in static and dynamic logit-based

aggregate demand systems respectively, with 1 brand and an outside good.

25

to firms producing software. Software firms, mainly independent publishers, develop games for one

or more consoles, and pay royalty fees to the hardware manufacturers for every game unit sold.

Traditionally, the hardware has been sold at or below cost, subsidizing the sales of the software,

which in turn accounts for most of the profits. In 1999, software revenues from video games in the

US totaled $7.4 billion, more than any other entertainment industry (Williams 2002).

My focus here is on the software-side. I use data on sales and prices of all new video-games

compatible with the Sony Playstation released in the US market between October 1998 and March

2000. The sample includes the complete history of aggregate retail sales and prices of a total of 102

Sony-compatible games since their date of introduction. The data were collected by NPD Techworld

using scanners linked to over 80% of the consumer-electronics retail ACV in the US.13

4.2. Data

The two main stylized features of the data are as follows:

• Prices of all games start high at about the same level and then almost always fall over time. On

average, prices fall by 4.2% ($1.75) every month, though the rate of decline in prices is higher

during the later phase of each game’s life-cycle.

• Unit sales of games start high and then fall over time. While there is wide variance in the level

of sales, on average, sales fall by about 10.1% (1982 units) every month for a given game.

These two features of the data, viz. declining prices, and declining sales are consistent with my

model of dynamic price skimming in a durable good market. The motivation to inter-temporally price

discriminate explains why prices are cut over time. Market saturation, arising from the exit of one-

time purchasers from the potential market for each game, explains the declining sales paths.

Further descriptive statistics of the sales and price data grouped by game-age (i.e., the time since

introduction) are presented in Table 1.1. Both prices and sales of games are falling over time. In

appendix D, I present an extensive analysis of the data that suggests that competition among games

is not the main driver of these patterns in this industry. In particular, I find that a) cross-price

effects across games are very low, indicating that games are not very substitutable for one another;

these hold after accounting for potentially strategic behavior by game-manufacturers who may

release games so as to minimize cannibalization from similar games existing in the market, b), the

pricing predictions from a demand model that ignores substitution effects are comparable to ones

that explicitly account for these effects, c) intertemporal price effects within narrowly defined game

genres are statistically insignificant, d) entry of hit games do not have significant effects on sales and

prices of games within the genre, and e) the rates at which prices fall are independent of competitive

conditions in the market.

13 I do not consider games released after March 2000, to avoid pricing issues related to expectations that game publishers

could have had about the release of the next generation Playstation 2 console in October 2000.

26

There is wide variance in the unit sales of the games. The least successful game in the data

(“Rat Attack”) had 3,515 units sold in 9 months in the market, while the most successful game

(“Driver”) had a total of 1,071,853 units sold in 10 months in the market. I estimate a full set of fixed

effects for each game that capture these large differences in the level of sales. As noted in section

3.1., I also use genre-fixed effects to model demand. The distribution of genres among the 102 games

in the sample is presented in Table 1.2. The genre classifications correspond to standard industry

definitions and were provided by the market research firm, NPD Techworld. I now discuss the

demand estimates.

4.3. Estimates of demand and price process parameters

I first discuss how I choose the order of the polynomial in lagged prices (K) in the price process (12). I

start by sequentially adding higher order terms to regressions of observed prices across games on

lagged values. Table 2.1 presents the results. Table 2.1 indicates that most of the variation in prices

is explained by a 1st order lagged price term, and that the design matrix is not invertible after K = 3.

With a 3rd order polynomial, I am able to explain about 90% of the variation in observed prices,

suggesting that this is a reasonable specification for these data. Based on this analysis, I set K = 3.

These regressions also reject serial correlation in the unobserved component of the process

generating observed prices. Recall that this was important in using lagged prices as instruments. To

test whether these are driven by pooling across genres Table 2.2. reports the same regressions split

by genres. These regressions also do not find evidence for persistence in the pricing errors (η in

equation 12).

The parameters of the price process as well as the demand parameters are jointly estimated

by maximum likelihood. I determine the number of customer segments (R) to include in the demand

specification by adding segments until one of the segment sizes is not statistically different from zero

(Besanko, Dubé and Gupta 2003). The data identify two segments. Though not reported, the

estimates for the three-segment model yielded several insignificant parameters, including the

probability of membership in the third segment. Demand estimates for the two-segment case are

presented in Table 3.

Referring to Table 3, segment 1 corresponds to roughly 42% of the potential market at the

time of introduction of games. While not reported, the full set of game-intercepts estimated for

segment 1 lie in the range (-0.5,-3.3). I do not find much evidence for endogeneity concerns, since the

estimated correlation between prices and demand shocks (ρ) is small, and statistically insignificant.

The parameters for segment 2 are estimated as deviations from that for segment 1. Segment 2

consumers are seen to be significantly more price sensitive than segment 1 consumers on average

(price sensitivity -0.017 for segment 1 compared to -0.113 for segment 2). Compared to segment 1

consumers, they also have a significantly lower preference for shooting, sports and racing games.

27

Overall, game firms face a more elastic demand curve from segment 2 consumers (mean price

elasticity of -3.28 across all games and time-periods), than from segment 1 (mean price elasticity -

0.27). I also explore how the proportion of the two segments in the market varies over time. Recall

from the discussion in section 2.1 that those that value a game highly relative to its prices will

purchase the game early and drop out of the market earlier. Hence, we would expect the proportion

of the more elastic segment 2 consumers to increase with the age of the game. Figure 1 presents a

plot of the proportion of segment 2 consumers in the market, computed at the final parameter

values, averaged across games in the data. Consistent with intuition, I find that the proportion of

segment 2 consumers in the market, averaged across all games in the data, rises from about 58% at

game-age 0 (i.e., the time of introduction), to about 92% at game-age 17. This indicates a shift in the

market towards lower valuations as games get older, generating incentives to cut prices. Figure 1

also indicates that rate at which segment 1 consumers drop out of the market varies by game. For

example, high valuation segment 1 consumers buy early and drop out of the market much faster for

“Baseball 2000” compared to “Big Air”.

I now explore how the price elasticity of demand varies across games and across time

periods. I find significant variation in elasticities across games and across time. The most price

elastic game, “Fisherman’s Bait”, has an average price elasticity of -3.29 over its life-cycle, while the

least price elastic game in the data, (“Armored Core”) has an average price elasticity of -1.27 over its

life-cycle. To explore the variation in elasticities across time, I present estimates of intertemporal

price elasticities by game age in Table 4. For each game-age t, these elasticities represent the effect

of a 1% increase in the period-t price, on demand in periods t+τ (τ = 0, 1,..), holding all other prices

fixed, averaged across all the games in the data. Note that a 1% increase in price in period t has two

effects. First, it reduces the probability that consumers of either segment buy and drop out of the

market for the next period, thus increasing the demand for the game in subsequent periods. Second,

it changes the mix of consumers available in the potential market in subsequent periods. In

particular, segment 1 consumers are less price elastic, and therefore, more likely to delay purchase

at the higher price in period t. These consumers are more likely to remain in the market for the game

in the subsequent periods, thus increasing the level of demand at all period t+τ prices. The net effect

of an increase in price is therefore to decrease current demand, and to increase demand in future

periods. Table 4 indicates that the intertemporal price elasticities are significant, and that rational

firms would take the persistence in price effects into account in formulating their pricing policy.

Figure 2 presents the distribution of own price elasticities of demand across games by the

age of the game, computed at the final parameter values. The demand elasticity for game j of age t,

ηjt, is computed as the % change in the demand for game j in the tth period after its introduction, due

to a 1% change in its price in that period. Figure 2 indicates that price elasticities are increasing (i.e.

becoming more negative) on average with the age of the game. This reflects the endogenous increase

28

in the proportion of the more price elastic segment 2 consumers in the market, and implies that

firms should optimally cut prices over time. Table 4 and figure 2 also indicate that some of the mean

elasticities are <1 (in absolute magnitude). In appendix F, I illustrate in the context of a simple 2-

period model that a forward-looking durable-good monopolist would not optimally set prices in the

inelastic portion of the demand curve. This suggests that conditional on the assumed demand

structure and rational expectations on the part of consumers, the sequence of observed prices is not

consistent with optimal price setting by game manufacturers (in particular, observed prices are too

high; I compare observed prices to the optimal prices implied by the model below.) This also provides

added motivation for not imposing restrictions from optimality in estimating demand parameters.

5. Pricing implications

I now consider the optimal pricing policy corresponding to the above estimates of demand. The

optimal price maximizes the present discounted value of current and future profits, where future

profits depend on current prices through the endogenous evolution of valuations and market sizes.

Using the demand estimates as an input, I solve numerically the firm’s dynamic pricing problem for

each game. I check whether the solution implies that price-cutting is the optimal strategy. I then use

the equilibrium solution to explore the implications of the level of consumer patience (as summarized

by the consumer discount factor) on the firm’s pricing policy. In this comparative “dynamics”

exercise, I numerically investigate the effect of changes in the consumer discount factor on the level

and slope of the equilibrium price paths. I then explore the implications of consumer forward-looking

behavior on profits for the video-game firms in the sample and quantify the extent to which it affects

equilibrium firm profits. Finally, I use the model structure to estimate the value of information

about consumer forward-looking behavior to the firms in my sample. I measure this value as the loss

in profits incurred when firms commit to pricing policies that incorrectly assume that forward-

looking consumers are myopic. I find that such profit losses are large and economically important.

5.1. Equilibrium pricing policy

I first describe the important qualitative features of the equilibrium pricing policy. Figures 3 and 4

present the pricing policy p*(S) and value function V(S) for the game “Akuji: The Heartless”. These

plots are representative of those obtained for the games in the sample. Looking at figure 3, we see

that the optimal pricing policy implies declining prices. Optimal prices are decreasing in the sizes of

both segments; and since the sizes of both segment 1 and segment 2 decreases over time, the firm

would cut prices. We also observe that prices fall at a faster rate when there are fewer consumers of

either segment in the market. Hence, the optimal rate of price cutting for the firm is increasing with

the age of the product. Figure 3 also shows the implications of the relative proportions of the two

29

segments for the rate at which prices decline. In general, we see that the rate of price cutting is

driven more by segment 1 consumers. In particular, exit of segment 1 consumers from the market

induces the firm to apply larger price cuts than the exit of a proportionate mass of segment 2

consumers. This is intuitive since segment 1 consumers have higher valuations for the game relative

to segment 2 consumers. Hence, relative to this segment, their exit from the market results in larger

declines in the average valuations in the market.

The value function for the game is presented in figure 4. As expected, the value function is

increasing in the size of both segments. Further, it increases at a faster rate as the number of

segment 1 consumers increases, as compared to that of segment 2. This is intuitive, since segment 1

consumers are more price inelastic than segment 2 and are hence more “valuable” to the firm.

Figure 5 presents the equilibrium value functions for waiting Wr(S), r = 1,2, for consumers

belonging to segments 1 and 2 for this game. To make these comparable to the demand-side, I plot

these over the range of observed prices in the data, rather than over the space of the two segment

size state variables. Figure 5 indicates that the value of waiting is increasing over time both

segments. This is a reflection of the fact that given a declining price path, the value of waiting is

higher at lower prices. Consistent with our intuition, we also see that the more elastic segment 2

consumers have a higher incentive to wait at all prices. Figure 5 also plots the purchase hazard, i.e.

the consumer’s probability of purchase in each period, as a function of prices. Recall that the

probability of purchase is a function of the realization of the shocks to demand. To emphasize the

dependence on prices, I present mean purchase hazards averaged over 1000 draws of ξ ~ N(0,σ2ξ) at

each price. Figure 5 indicates that the equilibrium purchase hazards for both segments increase as

prices fall. Hence, while the incentive to wait is higher at lower prices, the value of purchase is also

higher, and the net effect is an increase in purchase probabilities as prices decline over time.

5.2. Observed vs. predicted prices

I now compare the predicted prices from the equilibrium pricing model to those observed in the data.

My objective is to verify whether the demand estimates, which were obtained without imposing

assumptions about the pricing policy, combined with the equilibrium solution, can explain the

qualitative pattern of declining prices in the data. Table 5 shows the observed and predicted prices

averaged across all the games in the sample by game age (i.e. time since introduction). Table 5

indicates that the dynamic policy is able to explain the declining slope of the price schedules in the

data. Further, Table 5 indicates that predicted equilibrium prices are lower on average than

observed values. The empirical regularity in the data is that video-games release at about $42 (mean

$42.21, std. $3.08, cf. Table 1.) The price simulations in Table 5 indicate an optimal initial price of

about $36.5, indicating that initial prices seen in the data are too high.

5.3. Impact of consumer forward-looking behavior

30

I now use the equilibrium solution to explore main empirical question posed in the beginning of the

paper: specifically, the implications of consumer forward-looking behavior on the pricing policy and

associated profits for the firms in the sample. To develop some intuition on the effect of consumer

forward-looking behavior on equilibrium prices, I first discuss how the firm’s pricing policy would

change for various values of the consumer’s discount factor, δc, where δc = 0 corresponds to the case of

myopic consumers. The firm’s discount factor, δf, is fixed at 0.975 for all computations.

Figure 6 presents the optimal pricing policy plotted as a function of the two state variables

M1 and M2 for various values of δc. We note the following from figure 6:

• Equilibrium price levels are decreasing in the consumer’s rate of time preference. In general,

the higher the consumer’s discount factor, the lower are the equilibrium prices. The pricing

policy with myopic consumer prescribes much higher prices than those with forward-looking

consumers (δc > 0).

• The rate at which prices fall over time (i.e. the slope of the pricing policy with respect to M1

and M2) is decreasing in the consumer’s rate of time preference. The higher the consumer’s

discount factor, the lower the rate of fall of equilibrium prices. The pricing policy with myopic

consumer prescribes a steeper fall in prices than with forward-looking consumers (δc > 0).

The numerical predictions based on the demand estimates are consistent with the theory literature

that has documented a reduction in the level of equilibrium prices and profits with an increase in the

level of consumer patience.14 To measure empirically whether these profit effects are economically

significant for the games in the sample, I compare simulated equilibrium profits when firms faces

forward-looking versus myopic consumers.

I start by sampling 50 sets of parameters from the empirical distribution of estimated game-

specific fixed effects (α1j) and segment-2 deviations (λ) in Table 3. Each parameter set thus

represents a hypothetical new game. For each parameter set, I solve for the corresponding pricing

policies under the assumption that consumers are forward-looking (δc = 0.975) or myopic (δc = 0).

Assuming that each game is in the market for 15 months, for each set, I then simulate 20 vectors of

demand shocks of dimension 15X1 from N(0,σ2ξ) and compute demand and profits. I report the

average (across parameter sets and demand shocks) difference in the present discounted value of

profits between the myopic and forward-looking consumer cases, in Figure 7. Figure 7 indicates that

the present discounted value of profits under myopic consumers is 172.2% higher than under

forward-looking consumers. The result indicate that consumer forward-looking behavior has a

significant effects on profits in this industry.

14 The results also indicate that both the slope and level of prices contain information about the consumer discount factor. In

particular, it suggests that, while computationally burdensome, imposing the equilibrium pricing policy in a joint model of

demand and pricing will be informative in pinning down the consumer’s discount factor. This is relevant given the difficulties

that past literature has documented in reliably estimating the consumer’s discount factor from observed data.

31

5.4. Comparing profits under myopic and forward-looking consumers

The results above suggest that information regarding the true consumer discount factor is valuable

to firms in formulating their pricing policy. Managers may want to invest in market research that

provides information about the distribution of discount rates in the population. I use the model

structure to estimate the potential value of this information to the firms in the sample. I estimate an

upper bound on this value as the loss in profits incurred when the firm commits to a pricing policy

that incorrectly assumes that forward-looking consumers are myopic.

Analogous to BW, I consider a situation where consumers are actually forward-looking and

patient, but the firm incorrectly commits to a pricing policy assuming a lower willingness to wait on

the part of consumers. In terms of the model, I consider the profits to the firm when the actual

consumer discount factor 1

ccδ δ= , but the firm commits to a pricing policy under the assumption that

δ δ δ= <2 1

c cc . A limiting case is the situation where consumers are actually forward-looking, but the

firm sets prices under the incorrect assumption that they are myopic.

As before, I start by sampling 50 sets of parameters from the empirical distribution of

estimated game-specific fixed effects (α1j) and segment-2 deviations (λ) in Table 3. For each set, I

first solve for the equilibrium pricing policy and profits at the consumer’s discount factor (δc = 0.975).

Assuming that each game is in the market for 15 months, for each set, I then simulate 20 vectors of

demand shocks of dimension 15X1 from N(0,σ2ξ) and compute demand and profits to the firm when

the actual consumer discount factor δ = 0.975c , but the firm commits to a pricing policy under the

assumption that δ δ= <2 0.975cc . I consider the following values of 2

cδ = 0.9,0.75,0.5,0 , where 2

cδ = 0

corresponds to the belief that consumers are myopic. For each case, I compute demand according to

equation (12), using the equilibrium solution of the waiting function at δc = 0.975; that is using

Wr*(M1,M2,ξ|δc = 0.975), r = 1,2. Thus, I compute demand under forward-looking consumer behavior

corresponding to a discount factor of 0.975. I compute the corresponding prices using the equilibrium

solution of the pricing function at δc = 2

cδ ; that is using p*(M1,M2,ξ|δc = 2

cδ ).

Figure 8 presents the average (across parameter sets and demand shocks) change in profits

relative to the situation where the true consumer’s discount factor is 0.975, and the firm is pricing

optimally (i.e. setting prices according to p*(M1,M2ξ|δc = 0.975)). I find that on average, the PDV of

profits is 1.41% less when the firm commits to a pricing policy under the belief that the consumer

discount factor is 0.9; 8.91% less when the firm commits to a pricing policy under the belief that the

consumer discount factor is 0.75; 23.64% less when the firm commits to a pricing policy under the

belief that the consumer discount factor is 0.5; and 28.83% less when the firm commits to a pricing

policy under the belief that consumers are myopic. I conclude that investing in market research that

32

provides information on consumer forward-looking behavior could provide significant payoffs to firms

in the industry.

6. Conclusions

I focus on the problem of setting prices over time for a product that is durable. Durability implies

that consumers who buy the product are not in its market in subsequent periods. Hence firms face a

shrinking market and lower average willingness-to-pay for the product over time. This generates an

incentive to “skim” the market, by starting at high prices and lowering these over time. Skimming

enables the firm to intertemporally price-discriminate by selling to high-valuation consumers at high

prices early, and to low-valuation consumers at low prices later. However, a problem is that the

incentive to cut prices in the future may be anticipated by forward-looking consumers, who could

then delay their purchases. How would firms in real-world markets set prices optimally in such

settings? I present a practical empirical approach to address this question. I model prices as

equilibrium outcomes of a game played between forward-looking consumers who strategically delay

purchases to avail of lower future prices, and a forward-looking firm that takes this consumer

behavior into account in formulating its pricing policy. The model incorporates first, a method to

infer estimates of demand under such consumer behavior, and second, an algorithm to compute the

optimal sequence of prices given these demand estimates The primary aim of the framework is

normative, as an input to managers to obtain the optimal sequence of prices over time for their

products. The manager would first need to estimate the proposed demand system from available

data. Given demand parameters, the model can then be used to simulate the optimal intertemporal

pricing policy for his product. The model also predicts the extent to which consumer forward-looking

behavior affects equilibrium firm profits, and quantifies the associated profit losses to firms of

ignoring this aspect of consumer behavior. The pricing policies I consider are relevant for a range of

industries including technology products, “creative” goods (e.g. CD-s, video-games, books), and

fashion-products that have durable-like features.

The model is applied to data on sales and prices of video-games compatible with the Sony

Playstation. The observed empirical regularity of declining prices for every new game, combined with

a constant marginal cost structure, makes this industry an almost ideal setting to study these

intertemporal pricing issues. I estimate demand for games using a limited information maximum

likelihood approach. My approach is to estimate demand parameters in a first-stage without

imposing restrictions from any particular model of pricing conduct. This enables me to make

normative predictions for optimal pricing using a supply-side model that take these estimates as

input.

33

The results reveal that, conditional on estimates of demand recovered from the data, the

optimal pricing strategy for the firm does indeed exhibit price-cutting. I use the equilibrium solution

to explore the implications of consumer rationality on profits for the video-game firms in the sample.

For the firms in the sample, I find that the extent to which consumer forward-looking behavior

reduces equilibrium firm profits is large and economically important.

The results suggest that information regarding the true consumer discount factor is valuable

to firms in formulating their pricing policy. I use the model structure to estimate the potential value

of this information to the firms in the sample. I estimate an upper-bound on this value as the loss in

profits incurred when the firm commits to a pricing policy that incorrectly assumes that forward-

looking consumers are myopic. I find that in the limiting case, relative to the situation where the

true consumer’s discount factor is 0.975 and firms is pricing optimally, average profits are 29.83%

less when it commits to a pricing policy that incorrectly assumes that forward-looking consumers are

myopic.

A limitation of the model is that I do not consider competitive effects in the solution of the

pricing policy. As described in section 1, this was motivated by the nature of the video-game industry

in which games are very imperfect substitutes for each other. Given the small substitution effects

that I see in the data, I expect the qualitative nature of the pricing results from a model with

competition to be fairly similar to the monopolistic case I consider. The model will have to be

extended to account for competition in other categories in which substitution effects may be more

important. However, the solution of a dynamic equilibrium model of pricing in industries like video-

games with over 600 products is still far beyond current computational capabilities.

Finally, given the computational difficulties in repeatedly solving for the equilibrium, I have

adopted a two-stage approach in which demand parameters are estimated without imposing

restrictions from the optimal pricing policy. Recall that this had the advantage that the demand

parameters are independent of specific assumptions on the nature of pricing conduct. However, the

disadvantage is a loss in efficiency in parameter estimation. If this is a concern, researchers can

consider joint estimation of both demand and supply models, albeit at a larger computational cost.

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Appendix A: Numerical computation of the equilibrium

I solve for the equilibrium numerically using policy iteration. The algorithm is summarized below:

i. Discretize the state space into Gs points, and choose a tolerance value ε.

ii. Let S denote an M1,M2 ,..,MR,ξ combination. Choose guesses for the optimal policy, p(n)(S),

and the consumer’s equilibrium waiting functions Wr(n,k)(S), r = 1,..,R.

iii. Given, Wr(n,k)(S), compute ( ) ( )( ) ( )

( ) ( ) ( ) ( )

α β ξ

α β ξ

− +

− += =

+,

, , 1,..,

nr r

n k nr r r

pn kr W p

es r R

e e

S

S SS .

iv. Given ( ) ( ),n krs S , set up evolution of endogenous state variables for each segment as:

( ) ( ) ( )' , ,1 , 1,..,n k n kr r r rM M s N r Rϕ = − + = S

v. Given ( )' ,n krM , r = 1..,R, solve consumer’s problem to compute the new guess of equilibrium

waiting functions for the R segments, ( ) ( )+ =, 1 , 1,..,n krW r RS :

( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )

α β ξ ξ ξδ ξ ξ

+− ++ = + = ∫

' , ' , ' ,, 1 ' ,' ' '1 1,.., , ,.., ,, 1 'log , 1,..,n k n k n kn k n k

r r r rRp M M W M Mn kr cW e e dF r RS

vi. Iterate on (iii)-(v) till ( ) ( ) ( ) ( ) ε+ − < =, 1 , , for 1,..,n k n kr rW W r RS S . This gives the equilibrium

waiting functions given the guess of the pricing policy p(n)(S).

vii. Solve for firm’s value function V(n) that satisfies (policy valuation step):

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )π δ ξ ξ ξ= + ∫', ' 'n n n

fV p V f dS S S S , where

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( )

α β ξ

α β ξ

π

+

+ +

=

− ++

− +

= − −

=

= =+

∑

, 1

, 1 , 1

1

, 1

,

, 1, ..,

nr r

n k nr r r

n n

Rn k n kr r

r

pn k

r W p

p Q p c F

Q M s

es r R

e e

S

S S

S S S S

S S

S

viii. Given V(n) compute the improved policy using:

( ) ( ) ( ) ( ) ( ) ( ) ( )π δ ξ ξ ξ+

>

= + ∫1

0

argmax , ' ' 'n nf

p

p p V f dS S S , with ( ) ( )np S as starting value.

ix. If ( ) ( ) ( ) ( ) ε+ − <1n np pS S , stop, and set p*(S) = p(n+1)(S), and Wr*(S) = Wr(n,k+1)(S), r = 1,..,R; else

go back to (ii) with p(n+1)(S) and Wr(n,k+1)(S) as initial guesses.

I approximate the firm’s value function V(S), the consumer’s waiting functions Wr(S), and the pricing

policy p(S), using the tensor product of a Chebychev polynomial basis of order 5 in each state

dimension (Judd 1988, chapter 6, provides a discussion of Chebychev approximation methods). The

state space is discretized using 15 points in each dimension and the state points chosen as the

corresponding Chebychev zeros. I allow for the complete set of interactions between polynomial

terms of the segment size M1,M2 ,..,MR state variables and ξ for approximating p(S) and Wr(S), and

set interactions between M1,M2 ,..,MR and ξ to zero in approximating V(S). Once the corresponding

37

Chebychev parameters are computed, the functions are trivially interpolated to other parts of the

state space. The integral in step (v) is computed using Monte-Carlo integration using 30 draws, and

the integral in step in (ix) is computed using Gauss-Hermite quadrature using 8 nodes.

Appendix B: Incentives to cut prices

Equation (7) shows how the sizes of each segment in the potential market evolve over time. If the

number of new consumers is low relative to the existing market size, this generates a shift in the

distribution of heterogeneity in the potential market toward lower valuations, implying that a firm

that takes this into account should optimally cut prices over time. To see this, consider the

probability that a consumer chosen at random in period t+1 belongs to segment r, µr,t+1, given the

corresponding probability µrt for period t. Let ηjr ≡ (αjr,βr) denote the parameters characterizing

segment r and let wrjt = 1 denote the event that consumer type r chooses to not purchase game j in

period t, and waits for period (t+1). We first note that the probability that a consumer chosen at

random from among those that waited for time t+1 belongs to segment r, is given as:

( )( )

( ) ( )

( ) ( )

( ) ( )( )( ) ( )( )

ξ ξ α β ξ

ξ ξ α β ξ

η η

η η η η

η η η η

µ

µ

µ

=

+

− +

− +

=

= =

=

= = =

= = =

=

=

+=

+

∑

∑

1

, 1

, ,

, ,

1

Pr & 1

Pr 1

Pr 1 Pr

Pr 1 Pr

jr rt

rt

rt r r

rt r r

r rt t t t r r t t

r rt t t t r r t t

R

r

r t

W p W p prt

R W p W p prtr

w

w

w

w

e e e

e e e

Noting that the share of new consumers in time (t+1) is λt+1 = ( )+=∑ , 11

R

r trN M , the distribution of

valuations in the entire potential market for the game in period (t+1) is given by the mixture:

( )µ λ µ λ ϕ++ + += − +, 1, 1 1 11 rr tr t t t

It is easy to see that the probability that a consumer belongs to a given segment r, falls at a faster

rate ( )µ µ+ +, 1 , 1r t r t for segments with higher valuations (αr) and lower price sensitivities (βr). Hence, if

λt is low, over time, the mix of consumers in the potential market will be composed of those with

lower valuations and higher price sensitivities, and the firm would cut prices.15

Appendix C: Computation of market sizes

To estimate demand, I also need to develop measures of the initial market size for each game (M0 in

equation 7). A firm using the model is likely to have information on the initial market potential for

its product, or can obtain a measure of it through market research. However, as a researcher, I need

to somehow infer this from the observed data. A simple option is to use the entire installed base of

the hardware at the time of introduction of the game as the initial market. A problem with this

approach is the large size of the hardware installed base relative to the total sales of each game. In

the data, the average installed base of Sony Playstation consoles at the time of game introduction

was 16.5 million − while the maximum sales across all games in the data stood at around 1 million.

The corresponding market shares are very small (of the order of 1E-5), and result in large negative

15 However, if the proportion of new consumers is high enough, we see that it is possible that the probability that a consumer

belongs to high-valuation segment does not fall over time. Then, the optimal pricing policy for the firm is cyclic: to keep prices

high for many periods to sell mainly to high-valuation new consumers, and then cut price to clear the market of the low-

valuation consumers (Conlisk, Gerstner and Sobel 1984; Narasimhan 1989).

38

fixed effects for each game when estimating demand. This effectively implies that even at the time of

exit from the market, the size of each segment in the potential market for the game would remain

virtually unchanged from the time of introduction. The state variables for the firm (i.e. the segment

sizes) therefore do not change over time, and all variation in prices will have to be explained by

changes in the realized demand shock ξ. To fix this issue, I adopt an alternative method to infer the

size of each game’s potential market prior to estimating demand.

Specifically, the diffusion literature starting with Bass (1969) shows how it is possible to

infer the size of the potential market of a product from data on the sales path. I use the Bass

diffusion model to estimate the market size for each game. An approximation to the discrete-time

version of the model implies an estimation equation in which current sales are related linearly to

cumulative sales, and (cumulative sales)2. Specifically, letting qjt and Qjt denote the sales and

cumulative of game j in month t respectively, I estimate the pooled regression:

2jt j jt jt jtq a bQ cQ υ= + + +

Given the regression coefficients, the Bass model implies that the market sizes for each game are

given as j j j

M a p= , where pj is the positive root of the equation: pj2 +pjb +ajc = 0. The mean market

size so computed is 1.52 million, which corresponds to an average market share of 0.0802 across

games and months.16 The market shares are then used for estimating demand.

Appendix D: Derivation of the Jacobian

I compute the Jacobian, Jξ,η→q,p as follows. Suppose we can invert the aggregate demand function

qjt = ( ) ( )ξ=

Θ =∑ 1,

R

jt rjt rjt jt jtrQ M s p in (6) to obtain ξjt. Denote this inversion mapping G1. Denote the

inversion of the price process to obtain ηt as G2. That is,

( )ξ

η

−−

−=

= Θ ≡

= − ≡∑

1, 1 1

, 1 20

, , ;jt jt j t jt

K kjt jt k j tk

Q q q p G

p a p G

By definition, the Jacobian is:

ξ η

ξ ξ

ξ

→

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= =

∂ ∂ ∂ ∂

= ∂ ∂

1 1

, ,2 2 0 1

jt jt t jt t jt

q pjt jt

jt jt

G q G p q p

G q G p

q

J

To compute the derivative, let G = ( )ξ=

Θ∑ 1, ;

R

rjt rjt jt jtrM s p - qjt = 0. By the implicit function theorem,

( ) ( )ξ ηξ

ξ ξ ξ→

=

∂ ∂ −= ∂ ∂ = − = −

∂ ∂ Θ − Θ ∑, ,

1

1

, ; 1 , ;

jt

jt jtq p Rjt rjt rjt jt jt rjt jt jtr

G qq

G M s p s pJ

Appendix E: Falling costs and competition

I discuss the role of declining marginal costs and increased competition in explaining price declines

of the video-games. I first consider the falling-cost explanation. While economies-of-scale in the

production of consoles imply that cost-related considerations play a role on the hardware-side, this is

not the case on the software-side. The cost structure of video-games involves a fixed cost of game

development and constant marginal costs thereafter. The latter correspond to royalty fees paid by

16 More sophisticated estimation approaches for the Bass model (e.g. non-linear least squares) gave comparable estimates for

the market potential.

39

the game manufacturer to the hardware platform provider, and also the costs of producing and

packaging each CD-ROM title. The royalty fee for the 32-bit Sony Playstation compatible games in

the data was pre-announced and held fixed at $10 by Sony throughout the life-cycle. Further,

Coughlan (2001) reports that production/packaging costs for 32-bit CD-ROM games remained

roughly constant at $1.5 per disc. Thus, I rule out falling costs and experience curve effects as a

motive for price-cutting.

I now consider the role of competition. Several features of the data indicate that competition

alone is inadequate in explaining the declining path of prices. In particular, I find that a) cross-price

effects across games are very low, indicating that games are not very substitutable for one another;

these hold after accounting for potentially strategic behavior by game-manufacturers who may

release games so as to minimize cannibalization from similar games existing in the market, b), the

pricing predictions from a demand model that ignores substitution effects are comparable to ones

that explicitly account for these effects, c) intertemporal price effects within narrowly defined game

genres are statistically insignificant, d) entry of hit games do not have significant effects on sales and

prices of games within the genre, and e) the rates at which prices fall are independent of competitive

conditions in the market. I discuss these in sequence below.

Small cross-price effects and comparable margins between nested-logit, multinomial-logit

and binary-logit specifications

We can expect substitution effects in the video-game market to be small for two reasons. First, there

are a large number of games in the market. Between October 1998 and March 2000, there was an

average of 665 titles available in the market per month for the Sony Playstation console. Second,

each game is fairly unique, having its own distinct features, characters and idiosyncrasies: other

than the genre-membership, there are few common tangible attributes by which to measure game

“quality”. A priori, the large number of games in the market and the wide differentiation of game

titles suggest that video-game titles are imperfect substitutes for each other.

We can test these formally by measuring cross-price effects among games. A concern is that

cross-price substitution effects may be understated if game-manufacturers release games

strategically so as to minimize cannibalization from similar games existing in the market. To

address this, I estimate specifications similar to Einav (2006) that tries to control for the endogeneity

of release times of games. This specification is a static nested-logit model of demand with nests

corresponding to the video-game genres. The benchmark specification is:

( ) ( ) ( )α λ β σ ξ= + − + + +0 |

ln lnjt t j j jt jt g jts s t r p s

where, t indexes month, rj is the release date of game j, pjt is the price, sjt is the market share of game

j in month t, s0t is the share of the outside good and sjt|g is the share of units sales of the game within

its genre, g. A large σ indicates strong correlation in utilities of games within genre g; a small σ

closer to zero indicates little within-genre correlation. The larger the σ, the larger the cross-price

effects among games within each genre. The parameter λ captures the rate of decay of game sales

from introduction.

The concern regarding introduction timing arises here because game manufacturers may be

unlikely to release a new game in periods with very high-quality games or with games that are very

similar. If “quality” or “similarity” of other games within the genre is included in the unobservable

component of demand in period t, this may make rj endogenous and λ biased. I can address this issue

fully by including game-fixed effects. By including a full set of fixed effects, all variation in demand

arising from aspects of game-quality is already controlled for. To estimate the above model, I also

need instruments for the within-genre share and prices. I use the number of games available in each

genre each month as an instrument for within-genre share. More games within each genre are likely

associated with more intense competition, and therefore should be negatively related to the within

genre share. Analogous to Einav (2006), the identifying assumption here is that the number of games

in a genre in a month is not correlated with the part of decay pattern that is specific to game j (ξjt). I

use lagged prices as instruments for current period prices. These instruments are admittedly

40

imperfect: I present specifications with and without including instruments for prices to demonstrate

that these are not fully driving the results.

I estimate the nested-logit specification on the sample of new games in my data. Results are

presented in Table E1. OLS and 2SLS specifications with and without including instruments for

prices, as well as adding quadratic and cubic polynomials in age (i.e. t-rj), are reported. In general,

the price coefficient goes up in absolute magnitude after instrumenting. The within-genre correlation

σ is close to zero after instrumenting. The value of σ does not change if I drop lagged prices from the

instrument matrix. These regressions suggest that within a genre, games are not perceived to be

very close substitutes by consumers.

I repeat the same regressions for multinomal-logit and nested logit specifications. These are

presented in Table E2. To compare whether accounting for substitution effects as well as within-

genre effects makes a qualitatively significant difference for pricing, I compare the percentage

markups corresponding to these specifications. Using the estimates for each model, I first compute

markup-s using the first-order conditions corresponding to static profit-maximization (e.g. for the

nested logit model, the mark-up is as in equation 33 in Berry, 1994.) While these are only static

estimates, strong substitution effects if present, are likely to result in large differences between the

markups corresponding to the multinomial versus the binomial logit specifications. If substitution

effects are strong within genre, these differences are likely to be even larger between the nested

versus the binomial logit specifications. The top panel of Figure E1 presents the histogram of the

difference in % markups between the nested-logit and the binary logit models, across all game-

months within each genre. The bottom panel presents the corresponding plots for the multinomial-

logit and the binary logit models. I find that the differences are small. These results suggest that the

primary aspect of substitution is with the outside good – i.e. whether to buy now or to delay

purchase; the binary logit demand model I have used captures this reasonably. Leaving out the

substitution effects, while limiting, does not seem a priori to result in huge biases in predicting

pricing policies for this industry.

Small Intertemportal price-effects

I now test whether there is evidence in the data for intertemporal substation across games. Such

intertemporal price effects would imply that lagged prices of games within the same genre would

have a significant effect on current demand. Presumably, a low price yesterday for say, action games

would attract consumers who were on the lookout for such games, take them out of the market for

this genre for awhile, and thus reduce demand for action games today. Likewise, a high price

yesterday, may likely leave more potential buyers of that genre in the market, thus raising its

demand today. To check for some evidence of these effects in the data, I estimate OLS specifications

of binary-logit models in which lagged prices of other games within the same genre are included as

repressors. The results are presented in Table E3. Full sets of game and age-fixed effects are

included. Columns [1-4] adds the lagged minimum, median and maximum prices within the genre

(excluding the focal game) as regressors. These variables are not significant. In columns [5-6] I find

the best-selling game (across all months in the data) in the genre, and add its lagged price as a

regressor. These are not significant. These suggest that intertemporal cross-price effects may not be

a first-order issue for these data.

Small Intertemportal substitution effects

Do sales/prices fall when a hit game is to be released? To test this empirically, I first find the best

selling game within each genre across all the months in the data. I then include the number of

months to release of this game as a regressor into binary-logit specifications of demand. Table E4

presents estimates of this model. I find that the included variable is not significant. The last columns

of table 1E allows for this effect to vary by genre. I do not find a significant effect on demand. These

results suggest that that current demand is not significantly shifted down as best-sellers are closer

to being released. To check whether prices decline at a faster rate when a hit game is closer to being

released, I also estimate regressions of prices on game-fixed effects, game-age and game-age*Tg,

where Tg is the number months to the release of the best-seller in that game’s genre’s. A significant

negative coefficient on game-age*Tg would indicate that the release affects the rate of decline of

41

prices. This regression gave a coefficient of -1.7 on age (t = -2.96) and 0.0444 on age*Tg (t = 0.193),

i.e. not significant; further, the interactions with genre fixed effects were not significant. To

summarize, I do not find much evidence of sales or prices falling in anticipation of hit-game releases

into the market.

Rate of fall of prices not affected by competitive conditions

I now explore whether the rate at which prices fall is affected by the degree of competition in the

market. The number of games in the market is increasing over the months in the data, and hence

the market is getting more competitive over time. Hence, if game manufacturers are responding to

the increased competition by cutting prices, I should find that the rate at which prices fall is higher

during the later months in the data. Testing this however, is confounded by the fact that games

released in the later months in the data are also likely to be of better “quality”, and hence, less likely

to cut prices. I resolve the issue as follows. Stacking the data across all games and months, I regress

prices on game-fixed effects, game-age (i.e. time since introduction), and interactions of game-age

with month fixed effects. Thus, I measure the rate at which prices fall (as the coefficient on age)

while fully controlling for quality using game-fixed effects. Statistically significant interactions

between game-age and month fixed effects will indicate that the rate at which prices fall is affected

by completive conditions in the market. Table E5 (columns 1-4) shows the results from the

regression. Controlling for game-age and game-quality, I find that the interactions are not

significant. In columns 5 and 6, I also report on the results from regressing the change in prices on

month fixed effects, where as before, I control for game-quality using game-specific intercepts. Again,

I find that changes in prices are not statistically significantly explained by month-specific effects.

Further, the month fixed effects explain a negligible percentage (less than 1%) of the variation in

price cuts. Finally, in other specifications (not reported) I allow the effect of age to be genre-specific

and interact the age of the game with month and genre-fixed effects. The interactions of age with

month-fixed effects are not significant for each genre in this specification.

Plots

A visual examination of pricing patterns of incumbent games in response to entry provides

additional evidence that is consistent with these results. Figure E2 plots the prices of the three

games that were introduced into the “Action-oriented Racing” genre during the time-frame of the

data. “Castrol Honda Spr. Bike” by Electronic Arts is the first entrant, followed by “Monaco Grand

Prix” from UBI Soft, and “Championship Motorcross” from THQ. We see that neither the entry by

the potentially close substitute, “Monaco Grand Prix”, not its price cuts till month 58 triggers a price-

cut in “Castrol Honda Spr. Bike”. While both games cut prices post month 58, these price cuts do not

induce the newer entrant, “Championship Motorcross” to lower prices. Figure E3 presents plots of

prices and sales for the “Role Playing Game” genre. We see that prices of “Legend of Legaia”, the

best-selling incumbent in the genre, and “'Shadow Madness”, one of least selling games, do not drop

in response to the entry by “Jade Cocoon” and “Star Ocean: Second Story”. This, in spite of the fact

that “Star Ocean: Second Story” is the best seller in the genre after entry. The pattern suggests that

that competition from newer and better games is not the driving force behind falling prices.

Appendix F: Pricing by a forward looking firm facing a “shrinking market”

Consider a two period model. My goal is to show that a forward looking (“dynamic”) monopolist

would be even more likely than a “static” monopolist to set prices in the elastic region of the demand

curve. This result depends on the fact that in the durable good context, a higher price by the firm

today is likely to result in a higher demand tomorrow (since fewer consumers buy today and hence,

more are in the market for the product tomorrow.) In other contexts, e.g. dynamic monopoly pricing

under goodwill effects or brand-loyalty, or under learning-by-doing (e.g. Tirole 1988, section 1.1.2.1 &

1.1.2.2), low prices today increases demand (or lowers costs) for the firm in the future. For these

cases, the dynamic monopoly has an incentive to set prices that are lower than the corresponding

myopic case, and could optimally price in the inelastic region of the demand curve.

42

I consider a two period model that incorporates the essence of the shirking market induced

by the durable nature of the good. Let D(p) denote the demand for the game given price p such that

D’(p)<0. Let c denote the marginal cost, δ the firm’s discount factor, and p1 and p2 denote the prices

in the two periods. To capture the shrinking market effect, I assume that given the first period price

p1, the firm faces a residual demand curve in the second period that is shifted downward by a factor

α(p1), such that 0 ≤ α(.) ≤ 1, and α’(.) > 0. Thus, the higher the firm’s first period price, the more will

be the residual demand in the next period. The firm’s problem is to choose prices p1 and p2 that

maximize discounted profits:

( ) ( ) ( ) ( ) ( ) ( )1 2 1 1 2 1 2,p p p c D p p c p D pπ δ α= − + − (a1)

The first-order conditions imply that the optimal prices satisfy:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

δ α

≥

− = − +

− = −

2* * * * * * *1 1 1 1 2 1 2

0

* * *2 2 2

' ' ' '

'

p c D p D p p D p D p D p

p c D p D p

(a2)

The optimal prices for the myopic case are: *1p c− ( ) ( )* *

1 1'D p D p= − , ( ) ( )* * *2 2 2'p c D p D p− = − . We see

that if the firm is forward looking, it has an added incentive to keep p1 higher than the myopic price

by a value ( ) ( ) ( ) ( )* 2 * * *

1 2 1 2' 'p D p D p D pδ α . Further, the more the firm cares about the future (i.e. the

higher the δ) the higher will be the first period price. Intuitively, the firm takes into account the fact

that a lower period 1 price would cut into period 2 profits, and this reduces the firm’s incentive to set

the period 1 price as low as the myopic case.

Denote the elasticity of demand in period 1, ( ) ( )* * *1 1 1'D p p D p as η1. Since costs are non-

negative, (a2) implies:

( ) ( ) ( ) ( ) ( ) ( )δ α = + + ≥ 2

* * * * * * *1 1 1 1 2 1 2' ' ' ' 0c p D p D p p D p D p D p

Some algebra yields:

( ) ( ) ( ) ( )η δ α

++ −

≤

≤ − + ≤ −

2* * * *

1 1 2 1 2

0

1 ' ' 1p D p D p D p

implying the price skimming monopolist would always price on the elastic region of the demand

curve.

43

Table E1: Nested Logit estimates

OLS 2SLS with Instruments for

within-nest share and prices

2SLS with Instruments for

within-nest share only Variables

Param tstat Param tstat Param tstat Param tstat Param tstat Param tstat Param tstat Param tstat Param tstat

Age -0.09 -8.03 -0.13 -5.77 -0.19 -5.11 -0.22 -3.68 -0.24 -3.70 -0.25 -3.53 -0.19 -7.74 -0.26 -6.77 -0.28 -5.36

Age^2 0.00 1.78 0.01 2.39 0.00 1.76 0.01 1.16 0.00 2.83 0.01 1.02

Age^3 0.00 -2.07 0.00 -0.74 0.00 -0.47

Price -0.04 -7.85 -0.04 -8.05 -0.04 -7.81 -0.07 -2.07 -0.05 -2.38 -0.05 -2.29 -0.06 -7.27 -0.06 -7.29 -0.06 -7.13

σ 0.52 22.75 0.52 22.56 0.52 22.69 0.03 0.37 0.01 0.09 0.01 0.12 0.05 0.51 0.01 0.13 0.02 0.17

Notes: Nobs = 1189. Dependant variable is log(sjt/s0t), where sjt is the market share of game j in period t, and s0t is the market share of the outside good. The

size of the potential market is fixed as the cumulative installed base of PS1 consoles each month. σ is estimated as the coefficient on log(sj|g,t), where sj|g,t is

the share of game j within genre g in month t. Game-fixed effects estimated in all specifications, but not reported. Instruments for within-nest share are #

of games within genre; for prices are lagged prices. First stage regressions of endogenous variables on instrument matrix: prices (R2 = 0.67; F = 20.96);

within-nest share (R2 = 0.13; F = 179.52).

Table E2: Multinomial logit and binomial logit specifications

Multinomial Logit

OLS 2SLS with Instruments for prices

Param tstat Param tstat Param tstat Param tstat Param tstat Param tstat Age -0.203 -15.707 -0.267 -10.483 -0.283 -6.379 -0.249 -3.687 -0.240 -3.789 -0.249 -3.549

Age^2 0.004 2.932 0.007 1.033 0.004 1.761 0.008 1.151

Age^3 0.000 -0.444 0.000 -0.733

Price -0.060 -9.208 -0.064 -9.669 -0.064 -9.581 -0.087 -2.220 -0.053 -2.361 -0.045 -2.294

Binomial Logit

Age -0.204 -16.016 -0.270 -10.749 -0.280 -6.390 -0.249 -3.712 -0.240 -3.812 -0.247 -3.539

Age^2 0.004 3.056 0.006 0.890 0.004 1.800 0.007 1.001

Age^3 0.000 -0.273 0.000 -0.544

Price -0.059 -9.262 -0.064 -9.753 -0.064 -9.681 -0.086 -2.210 -0.051 -2.310 -0.046 -2.335

Notes: Nobs = 1189. Dependant variable is log(sjt/s0t), where sjt is the market share of game j in period t, and s0t is the market share of the outside good. The

size of the potential market is fixed as the cumulative installed base of PS1 consoles each month. Game-fixed effects estimated in all specifications, but not

reported. Instruments for prices are lagged prices. First stage regressions of prices on instrument matrix: prices (R2 = 0.67; F = 20.96).

44

Table E3: OLS regressions of binary-logit specifications to test for intertemportal demand effects

[1] [2] [3] [4] [5] [6]

Param t-stat Param t-stat Param t-stat Param t-stat Param t-stat Param t-stat

Pricet -0.043 -6.917 -0.044 -6.967 -0.043 -6.927 -0.044 -6.991 -0.038 -5.695 -0.038 -5.750

Minimum price in genre(t-1) 0.006 1.266 0.011 1.088 -0.005 -0.707

Median price in genre(t-1) -0.003 -0.583 -0.010 -1.605 -0.007 -1.086

Maximum price in genre(t-1) -0.0002 -0.031 0.002 0.360 0.015 1.641

Price of best selling game in genre(t-1) -0.002 -0.346 -0.002 -0.253

R2 0.752 0.752 0.752 0.753 0.798 0.799

N 1183 991

Notes: Dependant variable is log(sjt/s0t), where sjt is the market share of game j in period t, and s0t is the market share of the outside good. The size of the

potential market is fixed as the cumulative installed base of PS1 consoles each month. Game-fixed effects as well as fixed-effects for game-age are

estimated in all specifications, but not reported.

Table E4: OLS regressions to test whether game demand changes significantly with time-to-release of best-selling video-game in its genre

Param tstat Param tstat Param tstat Param tstat Param tstat Param tstat Age -0.570 -4.867 -0.610 -4.131 -0.721 -2.888 -0.510 -4.297 -0.600 -3.940 -0.700 -2.781

Age^2 0.007 0.447 0.045 0.637 0.015 0.942 0.050 0.694

Ag^3 -0.003 -0.554 -0.003 -0.501

Price -0.178 -23.477 -0.178 -23.368 -0.177 -23.014 -0.181 -25.008 -0.181 -24.788 -0.180 -24.419

1Tg -0.029 -0.219 -0.010 -0.075 0.008 0.054

Genre interactions

Tg * Action 0.089 0.647 0.140 0.949 0.151 1.009

Tg * Fighting 0.031 0.208 0.065 0.428 0.089 0.558

Tg * Racing -0.036 -0.202 -0.020 -0.111 0.006 0.033

Tg * Shooter 0.161 0.139 0.172 0.148 0.185 0.159

Tg * Sports 0.994 0.614 1.043 0.644 1.091 0.671

R2 0.2925 0.2933 0.2945 0.2978 0.3013 0.3023

Notes: 1Tg denotes the number of months to release of best selling game within a video-game’s genre. Nobs = 198. Dependant variable is log(sjt/s0t), where

sjt is the market share of game j in period t, and s0t is the market share of the outside good. The size of the potential market is fixed as the cumulative

installed base of PS1 consoles each month. Game-fixed effects estimated in all specifications, but not reported. Insufficient observations of

Family/entertainment game-months to include an interaction for that genre.

45

Table E5: Regressions exploring rate at which prices fall across months

Dependant variable: Price Dependant variable: Price(t) – Price(t-1) Variable Param t-stat Param t-stat Variable Param t-stat

Age -1.701 -54.331 -2.367 -1.271

Age*Dec-98 -0.151 -0.081 Dec-98 -1.316 -0.868

Age*Jan-99 -0.127 -0.07 Jan-99 -3.607 -1.454

Age*Feb-99 0.301 0.165 Feb-99 -2.249 -1.547

Age*Mar-99 0.493 0.27 Mar-99 -2.568 -1.795

Age*Apr-99 0.591 0.323 Apr-99 -2.673 -1.89

Age*May-99 0.513 0.28 May-99 -3.279 -1.33

Age*Jun-99 0.343 0.187 Jun-99 -3.725 -1.655

Age*Jul-99 0.277 0.15 Jul-99 -3.304 -1.37

Age*Aug-99 0.460 0.25 Aug-99 -1.853 -1.331

Age*Sep-99 0.457 0.248 Sep-99 -2.769 -1.695

Age*Oct-99 0.475 0.257 Oct-99 -2.776 -1.011

Age*Nov-99 0.399 0.216 Nov-99 -3.196 -1.32

Age*Dec-99 0.514 0.278 Dec-99 -2.012 -1.46

Age*Jan-00 0.614 0.332 Jan-00 -1.997 -1.449

Age*Feb-00 0.663 0.358 Feb-00 -2.303 -1.671

Age*Mar-00 0.729 0.394 Mar-00 -1.900 -1.378

R2 0.82 0.8278 0.0997

Number of obs. 1189 1189 1087

Notes: Full set of 102 game-specific fixed effects estimated in all regressions (no constant).

46

Figure E1: Percentage markups and substitution effects

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

10

20

30

40

Action-0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

10

20

30

40

Fighting

-0.08 -0.06 -0.04 -0.02 0 0.020

20

40

60

Racing-0.08 -0.06 -0.04 -0.02 0 0.020

10

20

30

40

50

Shooter

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

20

40

60

80

100

Sports-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

20

40

60

Family/Child-Ent

Differences in % markup-s between Nested logit and Binary Logit specifications

-0.04 -0.03 -0.02 -0.01 0 0.010

10

20

30

Action-0.04 -0.03 -0.02 -0.01 0 0.010

10

20

30

40

Fighting

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

20

40

60

Racing-0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

10

20

30

40

50

Shooter

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

20

40

60

80

Sports-0.05 -0.04 -0.03 -0.02 -0.01 0 0.010

20

40

60

80

100

Family/Child-Ent

Differences in % markup-s between Nested logit and Multinomial Logit specifications

47

Figure E2: Prices of Action Oriented Racing Games introduced for the PlayStation console between Oct 1998 and March 2000

Figure E3: Prices and sales of Role Playing (RPG) Games introduced for the PlayStation console between Oct 1998 and March 2000

48

Table 1.1: Prices and unit sales by age of game

Prices Unit sales Age of game

Mean Std. Dev. Mean Std. Dev.

1 $42.21 $3.08 26,270.8 65,111.5

2 $41.17 $3.36 18,053.9 32,867.2

3 $39.84 $4.46 17,799.8 41,004.9

4 $37.56 $5.67 15,287.2 41,888.7

5 $34.87 $6.47 10,675.3 21,209.2

6 $32.79 $7.38 7,269.2 12,768.6

7 $30.62 $8.12 9,586.9 27,126.1

8 $29.09 $7.78 5,622.9 9,656.5

9 $27.14 $7.29 4,593.9 5,966.8

10 $25.90 $7.30 5,026.4 5,955.2

11 $24.66 $6.93 4,029.4 5,847.1

12 $23.41 $6.98 3,367.5 5,546.0

13 $22.35 $6.69 6,921.4 16,370.2

14 $21.17 $6.21 12,653.4 36,131.6

15 $19.62 $4.54 5,844.7 9,846.2

16 $19.64 $4.72 4,832.7 9,116.7

17 $19.51 $5.01 3,191.9 7,209.1

18 $20.32 $6.01 448.5 332.6

Table 1.2: Distribution of games by genre

Genre Action Fighting Racing Shooter Sports Family/Children's entertainment Other1

Proportion

of games 10.34% 12.45% 13.71% 12.03% 19.18% 13.04% 19.26%

Notes: 1Composed of Adult, Arcade, Adventure, Simulation, Strategy, and Role Playing games.

49

Table 2.1: Price process: regressions of prices on lagged values

Param t-statistic Param t-statistic Param t-statistic

Constant 0.265 0.772 4.891 5.077 6.499 2.955

price(t-1) 0.941 92.913 0.604 9.104 0.416 1.722

price(t-1)2 0.006 5.131 0.012 1.467

price(t-1)3 -0.0001 -0.813

R2 0.8821 0.8908 0.8909

Box-Pierce-Ljung

Statistic2 1.3978 1.1107 1.0877

Nobs 1087

Notes: 1Design matrix not of full rank after adding 4th order lagged price term.2Tests against the null of no first-order auto-correlation in residuals in the

presence of lagged dependent variables (Ljung and Box, 1978). The corresponding critical value, χ2(1) at the 1% level is 6.635 (i.e. the null is not rejected).

Table 2.2: Testing for autocorrelation in pricing errors by genre

GENRE: Action Fighting Racing Shooter Sports Family/Children’s

Entertainment

param t-stat param t-stat param t-stat param t-stat param t-stat param t-stat

Constant 11.414 0.790 11.552 1.500 3.101 0.697 2.177 0.384 15.479 1.355 13.382 1.675

price(t-1) 0.334 0.206 -0.187 -0.241 0.661 1.139 0.896 1.332 -0.271 -0.229 -0.170 -0.183

price(t-1)2 -0.002 -0.025 0.034 1.384 0.006 0.274 -0.005 -0.179 0.028 0.722 0.027 0.793

price(t-1)3 0.000 0.358 0.000 -1.273 0.000 -0.087 0.000 0.351 0.000 -0.431 0.000 -0.459

Box-Pierce-Ljung

Statistic1 6.64E-04 4.65E-06 0.0333 0.3904 0.0422 2.1804

Breush-Godfrey Statistic2 0.8838 0.8718 0.5316 1.6815 0.632 2.5078

R2 0.8659 0.8991 0.9292 0.8786 0.8805 0.8777

Nobs 111 137 148 132 209 147

Notes: Tests against the null of no first-order auto-correlation in residuals in the presence of lagged dependent variables (1Ljung and Box, 1978; 2Godfrey,

1978). The corresponding critical value, χ2(1) at the 1% level is 6.635 (i.e. the null is not rejected).

50

Table 3: Demand estimates (2-segment solution)

Variable Parameter t-statistic

Segment 1*

Price (β1) -0.017 -3.326

Segment 2

Constant (λ) -0.088 -1.175

Price (β2) -0.116 -2.177

Genre fixed effects**

λAction 0.001 0.294

λFighting -0.045 -0.366

λRacing -0.021 -2.257

λShooter -0.012 -3.014

λSports -0.122 -2.163

λFamily/Children's entertainment -0.020 -0.380

ln(φ2/(1- φ2))*** 0.340 3.376

Price process

Constant (a0) 5.159 3.694

price(t-1) (a1) 0.616 3.499

price(t-1)2 (a2) 0.027 1.128

price(t-1)3 (a3) 0.000 0.987

Price std. deviation (σp) 3.213 4.355

Demand shock std. deviation (σξ) 1.349 6.038

Corr. coefficient (ρ) 0.054 0.383

Log-likelihood -16423.75

Number of observations 1087

*Full set of 102 game-specific intercepts estimated for segment 1, but not reported.**”Other” genre

is the base. *** implies φ2 = 0.58

Table 4: Price elasticities averaged by game age

Notes: 1 should be interpreted as, a 1% increase in prices in period 1 reduces period 1 demand by

0.525%, increases period 2 demand by 0.125%, period 3 demand by 0.111% etc.

On

demand in

period:1

Average (across games) effect of a % change in price in period:

1 2 3 4 5 6 7 8 9 10 11 12

1 -0.525

2 0.125 -0.713

3 0.111 0.139 -1.292

4 0.946 0.120 0.245 -1.962

5 0.844 0.180 0.241 0.337 -2.465

6 0.794 0.185 0.234 0.336 0.324 -2.680

7 0.799 0.942 0.139 0.383 0.281 0.353 -2.588

8 0.829 0.893 0.119 0.164 0.240 0.230 0.265 -2.588

9 0.772 0.792 0.113 0.154 0.174 0.268 0.250 0.262 -2.530

10 0.763 0.733 0.146 0.145 0.158 0.160 0.237 0.178 0.197 -2.452

11 0.794 0.637 0.866 0.129 0.147 0.135 0.127 0.154 0.159 0.170 -2.277

12 0.795 0.559 0.785 0.123 0.139 0.128 0.116 0.122 0.150 0.159 0.137 -2.179

51

Table 5: Observed and predicted prices across games

Observed prices Predicted prices Age of game

Mean Std. Dev. Mean Std. Dev.

1 $42.21 $3.08 $36.55 $12.18

2 $41.17 $3.36 $36.40 $10.63

3 $39.84 $4.46 $34.55 $8.87

4 $37.56 $5.67 $32.40 $6.96

5 $34.87 $6.47 $28.85 $7.38

6 $32.79 $7.38 $26.41 $8.14

7 $30.62 $8.12 $25.66 $8.10

8 $29.09 $7.78 $23.13 $5.69

9 $27.14 $7.29 $21.56 $5.11

10 $25.90 $7.30 $20.75 $5.67

11 $24.66 $6.93 $20.09 $4.91

12 $23.41 $6.98 $18.86 $5.02

13 $22.35 $6.69 $16.34 $6.00

14 $21.17 $6.21 $15.85 $5.09

15 $19.62 $4.54 $13.57 $4.32

16 $19.64 $4.72 $11.54 $5.64

17 $19.51 $5.01 $10.09 $5.23

18 $20.32 $6.01 $9.99 $4.78

Figure 1: Proportion of segment 2 consumers in market by game age

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Figure 2: Distribution of price elasticities of demand by game age

Notes: Figure 2 shows the distribution of price elasticities of demand across games by the age of the

game, computed at the final parameter values. The demand elasticity for game j of age t, ηjt, is

computed as the % change in the demand for game j in the tth period after its introduction, due to a 1%

change in its price in that period. The plot shows the distribution of ηjt across all j for each t. The

boxplot should be read as follows: e.g. 1-month after introduction, the mean price elasticity across games is around -0.5, with (-0.25,-0.75) as the 5th and 95th percentiles.

Figure 3: Equilibrium pricing policy – “Akuji: The Heartless”

Figure 4: Value function – “Akuji: The Heartless”

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Figure 5: Option value of waiting, and purchase hazards for segments 1 & 2 for game “Akuji:

The Heartless”

Figure 6: Optimal pricing policy with forward-looking consumers at different values of the

consumer’s discount factor

δ = 0.975

δ = 0.0

δ = 0.90

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Figure 7: Profits under myopic consumers relative to those under forward-looking consumers

PDV of profits under myopic consumers is 172.2% higher (8.41 vs. 22.9 M)

Figure 8: Effect of incorrect beliefs of firm about consumer forward-looking behavior on profits