Innovation adoption by forward-looking social learners

Frick, Mira; Ishii, Yuhta Article Innovation adoption by forward-looking social learners Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Frick, Mira; Ishii, Yuhta (2024) : Innovation adoption by forward-looking social learners, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 4, pp. 1505-1541, https://doi.org/10.3982/TE4455 This Version is available at: https://hdl.handle.net/10419/320273 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 1505–1541 1555-7561/20241505 Innovation adoption by forward-looking social learners Mira Frick Department of Economics, Princeton University Yuhta Ishii Department of Economics, Pennsylvania State University We build a model studying the effect of an economy’s potential for social learning on the adoption of innovations of uncertain quality. Assuming consumers are forward-looking (i.e., recognize the value of waiting for information), we analyze how qualitative and quantitative features of the learning environment affect equilibrium adoption dynamics, welfare, and the speed of learning. Based on this, we show how differences in the learning environment translate into observable differences in adoption dynamics, suggesting a purely informational channel for two commonly documented adoption patterns: S-shaped and concave curves. We also identify environments that are subject to a saturation effect: Increased opportunities for social learning can slow down adoption and learning, and do not increase consumer welfare, possibly even being harmful. Keywords. Innovation adoption, social learning, informational free-riding, strategic experimentation, exponential bandits. JEL classification. D80, D83, O33. 1. Introduction Suppose a new product of uncertain quality, such as a novel elective medical procedure (e.g., Lasik eye surgery or bariatric weight-loss surgery) or a new movie, is released into the market. In recent years, the rise of online review sites, search engines, videosharing platforms, and social networking sites has greatly increased the potential for social learning in the economy: If other patients suffer a serious complication or many viewers enjoy the movie, this is more likely than ever to find its way into the public domain; and there are more and more people who have access to this common pool of consumer-generated information. Mira Frick: [email protected] Yuhta Ishii: [email protected] This paper is a revised version of chapters of our PhD dissertations at Harvard University. We are grateful to Drew Fudenberg, Attila Ambrus, Eric Maskin, and Tomasz Strzalecki for generous advice and encouragement. For helpful comments that significantly improved the paper, we thank three anonymous referees, as well as Nageeb Ali, Dirk Bergemann, Aislinn Bohren, Yeon-Koo Che, Thomas Covert, Martin Cripps, Ben Golub, Marina Halac, Johannes Hörner, Ryota Iijima, Boyan Jovanovic, Daniel Keniston, Daria Khromenkova, Scott Kominers, David Laibson, Greg Lewis, Chiara Margaria, César Martinelli, Stephen Morris, Giuseppe Moscarini, Pauli Murto, Aniko Öry, Luciano Pomatto, Sven Rady, Larry Samuelson, Heather Schofield, Jesse Shapiro, Ron Siegel, Andy Skrzypacz, Caroline Thomas, Jean Tirole, Chris Udry, Juuso Välimäki, Leeat Yariv, and numerous conference and seminar audiences. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4455 1506 Frick and Ishii Theoretical Economics 19 (2024) This paper builds a model studying the effect of an economy’s potential for social learning on the adoption of innovations of uncertain quality. A central ingredient of our model is that consumers are forward-looking social learners: In choosing whether to adopt an innovation, they recognize the value of delaying their decision to learn from other adopters’ consumption experiences.1We analyze how consumers’ delay incentives depend on qualitative and quantitative features of the learning environment, and how this affects equilibrium adoption, welfare, and the speed of learning. Our analysis has two main implications. First, qualitatively, we show how differences in the learning environment translate into observable differences in adoption dynamics. This implies a new, purely informational channel for two of the most commonly documented adoption patterns: S-shaped and concave curves. Second, quantitatively, we suggest caution in evaluating the impact of increases in the potential for social learning. We identify environments that are subject to a saturation effect, whereby beyond a certain level, increased opportunities for social learning can slow down adoption and learning, and do not improve consumer welfare (possibly even being harmful). In our model (Section 2), an innovation of fixed, but uncertain quality (better or worse than the status quo) is introduced to a large population of forward-looking consumers. Consumers are (ex ante) identical, sharing the same prior about the quality of the innovation, the same discount rate, and the same tastes for good and bad quality. At each instant t∈R+, consumers receive stochastic opportunities to adopt the innovation. A consumer who receives an opportunity must choose whether to irreversibly adopt the innovation or to delay his decision until the next opportunity. In equilibrium, consumers optimally trade off the opportunity cost of delays against the benefit of learning more about the quality of the innovation. Learning is summarized by a public signal process, representing news that is obtained endogenously—based on the experiences of previous adopters—and possibly also from exogenous sources (e.g., watchdog agencies, professional critics). To study the importance of quantitative and qualitative features of the news environment, we build on the exponential-bandit framework widely used in the literature on strategic experimentation (see Section 1.1): Individual adopters’ experiences generate public signals at a fixed Poisson rate that we use to quantify the potential for social learning. Qualitatively, as we interpret in Section 2.2, there is a natural distinction between bad news markets, where signal arrivals (breakdowns) indicate bad quality and the absence of signals makes consumers more optimistic about the innovation, and good news markets, where signals (breakthroughs) suggest good quality and the absence of signals makes consumers more pessimistic. Section 3analyzes and contrasts equilibrium adoption dynamics in bad and good news markets. As in many applications of Poisson learning, we focus on the stark but tractable case of perfect bad (respectively, good) news, where a single signal arrival conclusively indicates bad (respectively, good) quality. Thus, incentives are nontrivial only absent signals. As a preliminary step, Lemma 1shows that equilibrium incentives over 1Forward-looking social learning is well documented empirically, e.g., in the development economics literature studying the adoption of agricultural innovations (see Section 4.2). 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1507 time satisfy a single-crossing property: Absent signals, there is at most one transition from preference for adoption to preference for waiting, or vice versa, with a possible period of indifference in between. Building on this, Theorems 1and 2establish equilibrium existence and uniqueness under bad and good news.2Equilibrium adoption dynamics admit simple closed-form descriptions that are Markovian in current beliefs and in the mass of consumers who have not yet adopted. Under bad news, the unique equilibrium is characterized by two cutoff times 0 ≤ t∗ 1≤t∗ 2. Until t∗ 1, no adoption takes place and consumers acquire information only from exogenous sources; from t∗ 2on, all consumers adopt immediately when given a chance (absent breakdowns). If t∗ 1<t ∗ 2, then throughout (t∗ 1,t∗ 2)there is partial adoption:Only some consumers adopt when given a chance, with others free-riding on the information generated by the adopters, where the flow of adopters on (t∗ 1,t∗ 2)ensures indifference between adopting and delaying throughout this interval. A period of partial adoption arises in economies with a large enough potential for social learning, and with sufficiently patient and not too optimistic consumers; otherwise, there is no partial adoption. By contrast, the unique good news equilibrium is always all-or-nothing, featuring immediate adoption up to some time t∗and no adoption from t∗on (absent breakthroughs). Thus, regardless of the potential for social learning, consumers’ discount rate, or prior beliefs, there is never any partial adoption. This highlights a new distinction between the way in which bad and goods news learning affects consumers’ incentives. Specifically, as Section 3.3 explains, sustaining periods of indifference between immediate adoption and delay requires the prospect of receiving news that makes consumers (instantaneously) go from being willing to adopt to being unwilling to adopt; breakdowns have this effect, but breakthroughs do not. We highlight two implications of our analysis. Section 4.1 shows that, depending on the informational environment, our model generates two commonly documented adoption curves (e.g., Hoyer, MacInnis, and Pieters (2012), Keillor (2007)). Bad news equilibria with t∗ 1<t ∗ 2lead to the leading empirical pattern of S-shaped adoption: Absent breakdowns, the share of adopters increases convexly throughout the partial adoption phase (t∗ 1,t∗ 2), as convex growth ensures that, despite becoming increasingly optimistic, consumers remain indifferent between adopting and delaying; during the immediate adoption phase from t∗ 2on, adoption is concave, reflecting the gradual depletion of the population. In contrast, the all-or-nothing structure of good news equilibria (or bad news equilibria with t∗ 1=t∗ 2) leads to purely concave adoption curves. Section 4.2 considers increases in the potential for social learning. Proposition 1establishes a saturation effect: If learning is via bad news and the equilibrium features partial adoption, then such increases are (ex ante) welfare-neutral. Indeed, they are balanced out by an expansion of the period (t∗ 1,t∗ 2)of informational free-riding, which slows down the adoption of (both good and bad) products and has a non-monotonic effect on the speed of learning. More strongly, with heterogeneous consumers, increased opportunities for social learning can be Pareto-harmful (Remark 1). By contrast, in environments where equilibrium is all-or-nothing, increasing the potential for social learning is (essentially) always strictly beneficial and speeds up learning at all times. 2Uniqueness is in terms of aggregate adoption behavior. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1508 Frick and Ishii Theoretical Economics 19 (2024) 1.1 Related literature We study a model of innovation adoption with endogenous timing and social learning from public information. Related informational externalities and strategic delay incentives are analyzed in the literature on observational learning with endogenous timing;3 see, e.g., Chamley and Gale (1994) and, more closely related, Murto and Välimäki (2011), where players privately obtain Poisson signals about the quality of a risky project at a fixed exogenous rate until they choose to irreversibly exit to a safe outside option. A key difference is that in this literature, players hold private information about the state and draw inferences from others’ actions, whereas news in our model is public and derived from previous adopters’ experiences. Information aggregates in random bursts in these models rather than smoothly as in our setting, and the aforementioned papers do not derive adoption curves or study how they are shaped by the informational environment. Our public learning model builds on the framework of strategic experimentation with exponential bandits, originating with Keller, Rady, and Cripps (2005)andKeller and Rady (2010,2015) (for a survey, see Hörner and Skrzypacz (2017)). We depart in two main ways. First, we study irreversible adoption (i.e., exit to the risky arm), rather than allowing for continuous back-and-forth switching. Second, we assume a continuum of agents, who each have a negligible influence on public information. These departures entail a qualitative difference between bad and good news learning—the presence vs. absence of partial adoption regions—that has observable implications for adoption curves and is absent in the aforementioned papers, where the symmetric Markov equilibrium features a region of partial adoption/mixing under both bad and good news.4 Another implication of these departures is that, unlike strategic experimentation, our setting does not feature an “encouragement effect,” i.e., an incentive to increase current experimentation to drive up beliefs and induce more future experimentation by others. This yields new comparative statics that isolate the impact of informational free-riding: For example, in the bad news environment of Keller and Rady (2015), an increase in the number of players or signal informativeness makes players more willing to experiment at pessimistic beliefs, whereas the saturation effect in Proposition 1relies on the opposite effect. More recently, Laiho, Murto, and Salmi (2024) study informational free-riding incentives in a related model of collective experimentation with irreversible adoption and a continuum of (heterogeneous) agents, focusing, however, on Brownian news and learning from the stock rather than the flow of adopters.5 A large literature in economics, marketing, and sociology seeks to explain why innovations diffuse gradually and why S-shaped (and to a lesser extent concave) adoption 3A large literature studies observational learning/innovation adoption with exogenous timing (e.g., Banerjee (1992), Bikhchandani, Hirshleifer, and Welch (1992), Smith and Sørensen (2000), Herrera and Hörner (2013), Board and Meyer-ter-Vehn (2021)). Strategic delay incentives without social learning are at the center of the literature on wars of attrition (e.g., Maynard Smith (1974), Fudenberg and Tirole (1986), Anderson, Smith, and Park (2017)). 4Bonatti and Hörner (2017) study a different departure—unobservable actions—and find that this also leads to the symmetric equilibrium under bad vs. good news being in mixed vs. pure strategies. 5Fajgelbaum, Schaal, and Taschereau-Dumouchel (2017) study strategic investment timing by a continuum of agents whose investment produces Gaussian public signals about an evolving state. They show that this generates self-reinforcing episodes of high uncertainty and low investment. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1509 patterns are prevalent. We relate to other learning-based models of these phenomena.6 Unlike existing work that focuses on social learning by myopic consumers (e.g., Young (2009)) or forward-looking learning from exogenous signals (e.g., Jensen (1982)), we consider a model of forward-looking social learning. This allows us to provide a purely informational explanation of observed adoption patterns, whereas models with myopic consumers or exogenous signals require specific forms of consumer heterogeneity to generate S-shaped adoption.7Our saturation effect also hinges on the combination of forward-looking incentives and social learning, as under myopic or exogenous learning, a greater ease of information transmission is always beneficial. Our focus on the informational determinants of innovation adoption contrasts with work that combines informational and payoff externalities. Rob (1991)modelsentry into a new market, where the current number of firms in the market influences not only entrants’ learning about a demand parameter, but also their profits via the market price. Related to our bad news equilibrium, equilibrium entry is pinned down by a zero profits condition and is lower than socially optimal. He does not study how the informational environment affects entry dynamics or provide conditions for S-shaped growth, both of which would also depend on the inverse demand function. Bergemann and Välimäki (1997) obtain S-shaped adoption as a result of duopolistic competition between an established and a new seller in a model with reversible adoption and learning on both the buyer and the seller side. Initial adoption of the new product exceeds the social optimum in their model. Laiho and Salmi (2018) build on our model by incorporating monopoly pricing and consumer heterogeneity. 2. Model 2.1 The game Time t∈R+is continuous. At time t=0, an innovation of unknown quality θ∈{G= 1, B=−1}and of unlimited supply is released to a continuum population of potential consumers of mass N0∈R>0. Consumers are ex ante identical. They have a common prior p0∈(0, 1)that θ=G, they are forward-looking with common discount rate r>0, and they have the same actions and payoffs, as specified below. At each time t, consumers receive stochastic opportunities to adopt the innovation. Adoption opportunities are generated independently across consumers and histories according to a Poisson process with exogenous arrival rate ρ>0.8Given an adoption 6Non-informational models (for surveys, see Baptista (1999), Geroski (2000)) include “epidemic” models (e.g., Mansfield (1961), Bass (1969)), “probit” models of heterogeneously evolving benefits to adoption (e.g., Davies (1979)), and models of pure payoff externalities (e.g., Jovanovic and Lach (1989), Farrell and Saloner (1986)). Wolitzky (2018) contrasts adoption levels of cost-saving vs. outcome-improving innovations in a model of learning from others’ outcomes. Che and Hörner (2018) take a mechanism design approach to incentivizing social learning about an innovation. 7In those models, agents adopt if and only if their beliefs exceed a cutoff. This precludes regions of convex adoption with identical agents, instead requiring specific distributions of heterogeneous priors/tastes. 8In the context of our motivating examples, stochastic adoption opportunities may represent, e.g., convenient times to take off work to undergo an elective surgery or a free evening to watch a movie. Section 5 discusses the case when ρ→∞. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1510 Frick and Ishii Theoretical Economics 19 (2024) opportunity, a consumer must choose whether to adopt the innovation (at=1) or to wait (at=0). If a consumer adopts, he receives an expected lump sum payoff of Et[θ], conditioned on information available up to time t, and drops out of the game.9If the consumer chooses to wait or does not receive an adoption opportunity at t, he receives a flow payoff of 0 until his next adoption opportunity, where he faces the same decision again. 2.2 Learning Over time, consumers observe public signals that convey information about the quality of the innovation. We employ a variation of the Poisson learning models used in the literature on strategic experimentation. Let ntdenote the flow of of consumers newly adopting the innovation at time t, which we define more precisely in Section 2.3.Conditional on the quality of the innovation being θ, public signals arrive according to an inhomogeneous Poisson process with arrival rate εθ+λθnt,whereλθ>0andεθ≥0are exogenous parameters that depend on θ. The signal process summarizes news events that are generated from two sources. First, the social learning term λntrepresents news generated endogenously, based on the experiences of other consumers. It captures a flow ntof new adopters each generating signals at rate λ.10 Thus, the greater the flow of consumers adopting the innovation at t, the more likely it is for a signal to arrive at t; hence, the absence of a signal at tis more informative the larger is nt. Second, we also allow for (but do not require) signals to arrive at a fixed exogenous rate ε, representing information generated independently of consumers’ behavior (e.g., by watchdog agencies or professional critics). As in many applications of Poisson learning, we focus for tractability on perfect news processes, where a single signal provides conclusive evidence of the quality of the innovation. Qualitatively, there is a natural distinction between two types of news environments. Learning is via perfect bad news (for short, bad news)ifεG=λG=0and εB=ε≥0, λB=λ>0; that is, the arrival of a signal (a breakdown)isconclusiveevidence that the innovation is bad. Learning is via perfect good news (for short, good news) if εB=λB=0andεG=ε≥0, λG=λ>0; that is, a signal arrival (a breakthrough)isconclusive evidence that the innovation is good. The nature of the news environment may be influenced by whether a bad or good quality innovation is more likely to generate newsworthy (e.g., extreme) payoff realizations. For example, an unsafe medical procedure may cause serious complications that are widely reported, but a safe procedure that performs as intended may not lead to newsworthy outcomes.11 Alternatively, the 9Irreversible adoption is natural for innovations such as medical procedures or movies, for which “consumption” is typically a one-time event, or for technologies with large switching costs. 10We obtain qualitatively similar results when the social learning component at time tis taken to depend on the stock, t 0nsds, rather than the flow of adopters at t.SeeSection5. 11More generally, suppose payoffs of the quality θinnovation are drawn (independently across consumers) from cumulative distribution function Fθ,where∞ −∞ ξdF θ(ξ)=θ. Suppose payoff realizations ξare newsworthy if and only if ξ≤ξor ξ≥ξfor some “extreme” low and high payoffs ξ < ξ, and that newsworthy payoffs generate public signals at some rate. Bad news learning assumes FB(ξ)>0=FG(ξ) 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1511 news environment may reflect reporting practices of the available social learning systems. For example, several movie review aggregator and streaming sites provide “best of” lists of new releases with the highest user ratings, but do not display “worst of” lists. Quantitatively, we use 0:=λN0as a simple measure of the potential for social learning in the economy, summarizing both the likelihood λwith which individual adopters’ experiences find their way into the public domain and the size N0of the population that can contribute to and access the common pool of information. Under bad news, consumers’ posterior on θ=Gpermanently jumps to 0 at the first breakdown, while under good news, consumers’ posterior on θ=Gpermanently jumps to 1 at the first breakthrough. Let ptdenote consumers’ no-news posterior, i.e., the belief at tthat θ=Gconditional on no signals having arrived on [0, t). Given a flow of adopters (nt), Bayesian updating implies12 pt=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ p0 p0+(1−p0)e−t 0(ε+λns)ds under bad news p0e−t 0(ε+λns)ds p0e−t 0(ε+λns)ds +(1−p0) under good news. (1) In particular, if (nt)is continuous in ton some open interval, then on this interval (pt) evolves according to the ordinary differential equation (ODE) ˙ pt=(ε+λnt)pt(1−pt)under bad news −(ε+λnt)pt(1−pt)under good news. Note that the no-news posterior is continuous. Moreover, it is increasing under bad news and decreasing under good news. 2.3 Equilibrium Our interest is in the aggregate adoption dynamics of the population. Thus, our equilibrium concept takes as its primitive the aggregate flow (nt)of new adopters and does not explicitly model individual consumers’ behavior. Given our focus on perfect news processes, incentives are nontrivial only in the absence of signals: Under bad news, no consumers adopt after a breakdown, while under good news, all remaining consumers adopt at their first opportunity after a breakthrough. We henceforth denote by ntthe flow of new adopters at tconditional on no signals up to time tand we define equilibrium in terms of this quantity. Capturing that aggregate adoption is predictable with respect to the public news process, we require (nt)to be a deterministic function of time. We consider all such functions that are feasible; that is, (nt)is right-continuous in tand nt∈[0, ρNt]for all t∈R+,whereNt:=N0−t 0nsds denotes the mass of consumers remaining in the game and FB(ξ)=FG(ξ)=1, i.e., bad innovations sometimes generate extreme low payoffs, but neither good nor bad innovations generate extreme high payoffs. Good news learning assumes FB(ξ)=FG(ξ)=0 and FB(ξ)=1>F G(ξ). 12Section 2.3 imposes measurability on (nt),sotheexpressionsin(1) are well defined. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1512 Frick and Ishii Theoretical Economics 19 (2024) at time t. Imposing nt≤ρNtensures that at each t,ntis consistent with all remaining Ntconsumers independently receiving adoption opportunities at rate ρ. Any feasible adoption flow (nt)induces a no-news posterior (pt)via (1). In equilibrium, we require that at each t,ntis consistent with optimal behavior by the remaining Ntforward-looking consumers: Consumers who receive an adoption opportunity at tconsider the expected payoff to adopting immediately, which is ut:=2pt−1 absent news, and optimally trade this off against the value to waiting, taking into account that future adoption evolves according to process (nt). Formally, define the value to waiting (Wt)associated with process (nt)to be the solution to the following Bellman equation at each t.13 Under bad news, Wt=∞ t ρe−(r+ρ)(s−t)pt+(1−pt)e−s t(ε+λnk)dk   prob. of no breakdown in [t,s) max{us,Ws}ds; that is, Wtis the expected discounted payoff to waiting until the next stochastic adoption opportunity s, and then adopting at this opportunity if and only if (i) there has been no breakdown and (ii) at the updated belief ps, the expected payoff to adopting usexceeds thenewvaluetowaitingWs. Under good news, Wt=∞ t ρe−(r+ρ)(s−t)1−pt+pte−s t(ε+λnk)dk   prob. of no breakthrough in [t,s) max{us,Ws} +pt1−e−s t(ε+λnk)dk   prob. of breakthrough in [t,s)ds; that is, Wtis the expected discounted payoff to waiting until the next adoption opportunity s, and adopting at this opportunity if either (i) there has been no breakthrough and, at the updated belief ps, the expected payoff to adopting usexceeds the new value to waiting Ws, or (ii) there has been a breakthrough. Definition 1. An equilibrium is a feasible adoption flow (nt)such that (i) Wt≥utfor all tsuch that nt<ρN t (ii) Wt≤utfor all tsuch that 0 <n t. Condition (i) says that if some consumers who receive an adoption opportunity at tdecide not to adopt, then the value to waiting Wtmust weakly exceed the expected payoff to immediate adoption ut. Similarly, (ii) requires that if some consumers adopt at time t, then the value to waiting must be weakly less than the payoff to immediate 13A unique solution exists by standard arguments (e.g., Theorem 3.3 in Stokey, Lucas, and Prescott (1989)). 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1519 Figure 2. Left: The S-shaped adoption curve under bad news conditional on no breakdowns (t∗ 1=0). Right: Concave adoption curves under good news (blue =breakthrough before t∗;yellow =breakthrough after t∗;pink=bad quality). Corollary 1. Bad News. In the unique equilibrium of Theorem 1,At=0for 0≤t<t ∗ 1, Atis strictly increasing and convex in tfor t∗ 1≤t<t ∗ 2,andAtis strictly increasing and concave in tfor t≥t∗ 2. If the first breakdown occurs at time t, adoption ceases from then on. Good News. In the unique equilibrium of Theorem 2,At=1−e−ρt is strictly increasing and concave for all t<t ∗. If there is a breakthrough prior to t∗,thenAt=1−e−ρt for all t. If the first breakthrough occurs at s>t ∗(which requires ε>0), then adoption comes to a temporary standstill between t∗and s, and for all t≥s,Atis strictly increasing and concave, and is given by 1−e−ρ(t∗+t−s). Thus, in bad news markets (Figure 2, left), the adoption curve exhibits an S-shaped (i.e., convex–concave) growth pattern whenever t∗ 1<t ∗ 2, where convex growth coincides with the partial adoption region (t∗ 1,t∗ 2). By contrast, in good news markets (Figure 2, right), adoption proceeds in (up to two) concave bursts. Concave adoption curves also arise in bad news markets with very optimistic and impatient consumers or little potential for social learning (so that t∗ 1=t∗ 2by Lemma 2). ThefactthattheconvexgrowthperiodofAtunder bad news coincides with the partial adoption region (t∗ 1,t∗ 2)is tied to consumer indifference in this region. Absent breakdowns, consumers grow increasingly optimistic about the quality of the innovation, which increases their opportunity cost of delaying adoption. To maintain indifference, the benefit to delaying adoption must then also increase over time. This is achieved by increasing the arrival rate of future breakdowns, which improves the odds that waiting will allow consumers to avoid the bad product. Since the arrival rate of information is increasing in the flow ntof new adopters, this means that ntmust strictly increase throughout (t∗ 1,t∗ 2), i.e., that Atis convex.25 By contrast, the concave growth regions under both bad and good news simply reflect the gradual depletion of the population when all consumers adopt immediately upon an opportunity.26 25This argument for convex growth does not rely on the linearity of λnt; it remains valid as long as the rate at which the bad product generates breakdowns at tis increasing in nt. 26If there is an inflow of new consumers of it=γNtat all t(i.e., the population size grows exponentially at rate γabsent adoption), then it can be shown that adoption is eventually concave if and only if the growth rate γis less than the rate ρof stochastic adoption opportunities. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1520 Frick and Ishii Theoretical Economics 19 (2024) As discussed in the Introduction, S-shaped adoption is documented for many innovations. Our model complements existing explanations (see Section 1.1)byidentifying a purely informational channel for this regularity: If there is a high enough chance that previous adopters’ experiences may reveal negative information about the innovation and consumers are forward-looking, then S-shaped adoption can arise due to some consumers strategically delaying adoption. This channel may be especially natural for innovations whose introduction is accompanied by substantial safety concerns, as may plausibly be the case for our motivating example of new medical procedures, where S-shaped adoption patterns are indeed commonly documented.27 Though less prevalent than S-shaped curves, concave adoption is another leading pattern documented in the marketing literature (e.g., Keillor (2007), pp. 51–61), with leisure-enhancing innovations such as movies, books, and games as examples. While our model abstracts away from many important product-specific forces, Corollary 1suggests some factors that could contribute to concave adoption. In particular, high levels of consumer impatience or optimism, or if social learning in these markets is predominantly via good news signals or their absence (as Section 2.2 suggested could be driven by features of the relevant review platforms). 4.2 The effect of increased opportunities for social learning Next, we consider an increase in the potential for social learning 0:=λN0, capturing either a greater ease of information transmission (e.g., due to the introduction of new social networking platforms) or a larger community of consumers. We ask how this affects welfare, learning, and adoption dynamics. Again, informational free-riding in the form of partial adoption has important implications. Indeed, under bad news, an economy’s ability to harness its potential for social learning is subject to a saturation effect:If the equilibrium features partial adoption, then further increases in the potential for social learning are welfare-neutral, cause learning to slow down over certain periods, and decrease adoption levels at all times. Formally, we fix all other parameters and study the effect of increasing 0on ex ante equilibrium welfare W0(0), no-news posteriors p0 t, and ex ante expected adoption levels At(0,G)and At(0,B)conditional on good and bad quality, respectively. We assume that the original potential for social learning 0is such that there is partial adoption, i.e., t∗ 1(0)<t ∗ 2(0); under the conditions in Lemma 2, this is the case whenever 0 is large enough. Proposition 1. Consider learning via bad news. Fix r,ρ,ε,andp0.If0is such that t∗ 1(0)<t ∗ 2(0), then an increase in the potential for social learning to ˆ 0> 0has the following effect: 27See, e.g., the adoption data for bariatric surgery in Buchwald and Oien (2009,2013). 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1521 (i) Welfare Neutrality. We have W0(ˆ 0)=W0(0). (ii) Non-Monotonicity of Learning. There exists t>t ∗ 2(0)such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p0 t=pˆ 0 tif t≤t∗ 2(0)(learning is equally fast under 0and ˆ 0) p0 t>pˆ 0 tif t∗ 2(0)<t<t(learning is slower under ˆ 0) p0 t<pˆ 0 tif t>t(learning is faster under ˆ 0). (iii) Slowdown of Adoption. For all tand θ=B,G, we have At(0,θ)≥At(ˆ 0,θ), with strict inequality for all t>t ∗ 1(0). We prove Proposition 1in Appendix A.6. The idea behind (i) is as follows. Since the equilibrium features partial adoption at 0, the same is true when the potential for social learning increases to ˆ 0. Moreover, both the time t∗ 1at which adoption begins and the posterior pt∗ 1at t∗ 1are the same under 0and ˆ 0.28 Since consumers strictly prefer to delay at all t<t ∗ 1, and are indifferent between delaying and adopting at t∗ 1,exante welfare under both 0and ˆ 0then corresponds to the expected payoff to waiting until t∗ 1and adopting at t∗ 1absent breakdowns. Thus, W0(ˆ 0)=W0(0).29 This welfare neutrality result contrasts with the cooperative benchmark where consumers coordinate on socially optimal adoption levels. In the latter case, increased opportunities for social learning are strictly beneficial and for any p0>1 2,thefirst-best (complete information) payoff of ρ r+ρp0can be approximated in the limit as 0→∞.30 The result also contrasts with myopic social learning or forward-looking exogenous learning, where welfare necessarily increases in response to more informative signals (even if consumers are heterogeneous).31 Points (ii) and (iii) further illuminate the forces behind welfare neutrality. By (ii), an increase in 0affects learning dynamics in a non-monotonic manner. Thus, the impact on a consumer’s expected payoff varies with the time tat which he obtains his first adoption opportunity. If t≤t∗ 2(0), his expected payoff is the same under 0and ˆ 0.If t∈(t∗ 2(0),t), he is worse off under ˆ 0, because in case the innovation is bad, he is less likely to have found out by then than under 0.32 Finally, if t>t, he is better off under ˆ 0. Depending on ˆ 0,tadjusts endogenously to balance out the benefits, which arrive at times after t, with the costs incurred at times (t∗ 2(0),t). 28Indeed, as we saw in Section 3.2,t∗ 1is the first time at which the posterior exceeds the threshold p= ε+r ε+2rand learning up to t∗ 1is purely via the exogenous news source. 29Related welfare neutrality results can arise in mixed equilibria in other games; e.g., in certain static public goods provision games, the equilibrium welfare/provision probability of the public good can be independent of the number of players. 30Frick and Ishii (2023) (Supplement C) show the cooperative benchmark is all-or-nothing, with no (resp. immediate) adoption below (resp. above) a cutoff belief pSO. Equilibrium adoption displays two inefficiencies: (i) it starts too late (pSO <p t∗ 1); (ii) once it starts it is initially too low (if t∗ 1<t ∗ 2). 31To define ex ante welfare with myopic consumers, assume that consumers’ payoffs are discounted at some arbitrary rate r>0, but consumers behave myopically. 32The fact that learning on (t∗ 2(0),t)is slower under ˆ 0than 0reflects that the flow of adopters under 0jumps up at t∗ 2(0)(due to the transition from the partial adoption to immediate adoption regions), whereas under ˆ 0, partial adoption continues until t∗ 2(ˆ 0)>t ∗ 2(0). 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1522 Frick and Ishii Theoretical Economics 19 (2024) Figure 3. The effect of increased opportunities for social learning on the adoption of a good product under bad news ( ˆ 0> 0). Similarly, by (iii), an increase in 0strictly decreases the adoption At(0,G)of good products (which is harmful), but also decreases the ex ante expected adoption At(0,B) of bad products (which is beneficial), and welfare neutrality obtains because these forces balance out in equilibrium. Figure 3illustrates that the strict slowdown in the adoption of good products is due to two effects: On the extensive margin, the increase in 0pushes out t∗ 2, i.e., prolongs free-riding; on the intensive margin, the increase drives down the growth rate of Atat all t<t ∗ 2(0). Point (iii) yields new testable implications relative to existing models of innovation adoption, suggesting, for example, that the fraction of adopters may grow more slowly in larger communities. Broadly consistent with this, Bandiera and Rasul (2006)study the adoption of a new crop by farmers in Mozambique and find that farmers whose network includes many adopters may be less likely to adopt initially themselves; thus, in equilibrium, larger networks of farmers should feature lower percentages of adoption.33 Finally, the logic behind the saturation effect relies crucially on partial adoption/informational free-riding. If under bad news, 0is so low that there is no partial adoption in equilibrium, then increasing 0is strictly beneficial (see Frick and Ishii (2023), Supplement B.1). Likewise, there is no saturation effect under good news (see Frick and Ishii (2023), Supplement B.2): Since equilibrium adoption is all-or-nothing, increasing the potential for social learning speeds up learning at all times, which strictly improves welfare (provided ε>0).34 Remark 1. Proposition 1shows that increasing 0is welfare-neutral under bad news. More strongly, if consumers have heterogeneous discount rates, then increasing the potential for social learning can lead to Pareto decreases in ex ante welfare. To illustrate, 33In related work, Munshi (2004) finds that in rice-growing regions in India, where (due to more heterogeneous plot conditions) social learning is less feasible than in wheat-growing areas, farmers are more likely to experiment with a new crop than their counterparts in wheat-growing areas. 34Even under good news, increasing 0increases welfare only if this affects agents’ preference for adoption vs. delay at some histories. If ε=0, agents weakly prefer to adopt at all histories (note ut=Wtfor all t≥t∗as pt=ps=1 2for all t≥t∗); hence, W0(0)=ρ r+ρ(2p0−1)is independent of 0.Ifε>0, increasing 0improves welfare by leading more agents to adopt only after a breakthrough (ut<W tfor all t>t ∗and t∗ is decreasing in 0). 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1523 suppose ε=0 and introduce a single (mass 0) impatient agent with discount rate ri>r into the population.35 Then, under the assumptions in Proposition 1, increasing λto ˆ λis welfare-neutral for the original population, but makes this impatient agent strictly worse off. Indeed, since the patient agents are initially indifferent between adopting and delaying, the impatient agent adopts upon his first opportunity absent breakdowns in both environments. By the non-monotonicity of learning in Proposition 1,thereexists some time t>t ∗:=t∗ 2(λ)such that learning is strictly slower under ˆ λbetween t∗and t, butfasterfromtimeton (and learning is equally fast under λ,ˆ λup to t∗). For patient agents, the costs of the early deceleration in learning and the benefits of the later acceleration exactly balance out. However, the impatient agent is hurt, because relative to a patient agent, he weights the early costs more heavily than the later benefits.  5. Concluding remarks This paper develops a model of innovation adoption when consumers are forwardlooking and learning is social. Our analysis isolates the effect of purely informational incentives on aggregate adoption dynamics, learning, and welfare. We highlight how qualitative and quantitative features of the learning environment shape these incentives, most importantly by determining whether or not there is informational free-riding in the form of partial adoption. The presence or absence of partial adoption has observable implications, suggesting a novel channel for two widespread adoption patterns: S-shaped and concave curves. Moreover, partial adoption has important welfare implications, entailing that increased opportunities for social learning need not benefit consumers and can be strictly harmful. Below, we briefly comment on some modifications and extensions of our model. Adoption opportunities. We assumed that consumers receive adoption opportunities at an arbitrarily large but finite Poisson rate ρ. This avoided technical issues related to defining strategies and continuation payoffs when agents can move continuously and adoption processes can feature mass points. The key qualitative implication of a finite ρin both the bad and good news equilibrium is to generate concave adoption regions. To illustrate what happens as ρ→∞, suppose ε=0andp0>1 2.Under bad news, the immediate (i.e., concave) adoption phase disappears as ρ→∞.InFigure 1,limρ→∞ N∗(p)=0forallp<1, so region I vanishes.36 Thus, by Theorem 1,there is an initial partial adoption phase with flow of adopters nt=r(2pt−1) λ(1−pt)and, in the limit as ρ→∞, this phase continues all the way until the finite time t∗ 2at which the population is fully depleted.37 Under good news, Theorem 2implies that for any finite ρ, equilibrium is all-or-nothing with cutoff posterior ps=1 2,butasρ→∞, the time t∗it takes to 35Section 4.3 of Frick and Ishii (2015) instead considered a small mass of impatient consumers. 36Intuitively, if there is any positive mass Mtof immediate adopters, then it is strictly beneficial to wait an instant, as the cost of delaying the decision by an instant is negligible (of order dt) relative to the probability (1−pt)(1−e−λMt)of observing a breakdown and avoiding the bad product. 37To see why t∗ 2is finite, note that the ODE for partial adoption implies nt=r λ 2p0−1 e−rtp0−(2p0−1),whichtends to ∞by the finite time t=1 rln p0 2p0−1. We also note that parts (i) and (iii) of Proposition 1remain valid as ρ→∞, but the non-monotonicity of learning in part (ii) no longer arises in the limit, because the acceleration/deceleration in learning occurs during the immediate adoption phase. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1524 Frick and Ishii Theoretical Economics 19 (2024) reach psabsent news tends to 0. Thus, the initial concave adoption region approximates a single mass point of M0=1 λln p0 1−p0adopters, where M0is such that absent news, the belief jumps down to ps.Hence,asρ→∞, the good news equilibrium approximates an initial burst of partial adoption (followed by a second burst if there is a breakthrough), but a drawn-out region of partial adoption can still only arise under bad news.38 Learning from the stock of adopters. In our model, the social learning component of the signal arrival rate at time t,λnt, depends only on the flow ntof new adopters. This effectively assumes that adopters can generate signals only once, at the time of adoption, approximating settings where the probability of receiving signals about the quality of the innovation (e.g., complications from a new medical procedure) depreciates rapidly from the time of adoption. In contrast, for some durable goods, it may be more natural to let signals at tarrive at rate λSt,whereSt:=t 0nsds represents the stock of adopters, capturing that adopters can generate signals repeatedly over time. This would produce similar results. Specifically, similar arguments yield the existence and uniqueness of equilibrium under both bad and good news. The good news equilibrium is again all-or-nothing, while, for appropriate parameters, the bad news equilibrium again features a partial adoption region with behavior pinned down by the indifference condition St=r(2pt−1) λ(1−pt)−ε λ. Finally, the partial adoption region again exhibits convex growth in adoption levels.39 More general signal processes. As in many applications of Poisson learning, we have focused for tractability on conclusive bad or good news signals. While a careful investigation of more general signal processes is beyond the scope of this paper, the analysis extends readily to hybrid environments with two types of conclusive Poisson signals: bad news and good news signals with respective arrival rates λBntand λGnt.Inparticular, if λB>λ G, the equilibrium is analogous to Theorem 1. Some of our insights also extend beyond environments with conclusive signals. For example, we note that partial adoption relies crucially on the possibility of news events that trigger discrete downward jumps in beliefs (although such events need not conclusively signal bad quality). Without such events (e.g., when learning is based on inconclusive good news Poisson 38If, instead, each consumer’s first adoption opportunity arrives at rate ρ<∞, but subsequent adoption opportunities arrive continuously, the good news equilibrium is still all-or-nothing as in Theorem 2,except that the cutoff belief limρ→∞ ps=ε+r ε+2ris greater than ps(if ε>0). The bad news equilibrium is qualitatively unchanged: Under suitable parameters, there is an initial partial adoption region with convex adoption growth (which continues until the stock of consumers who have received a first adoption opportunity is depleted); from then on, the remaining consumers adopt immediately at their first opportunity (leading to concave growth). However, the non-monotonicity of learning in Proposition 1(ii) no longer arises, as the flow of adopters now features a downward jump at the transition from partial to immediate adoption. 39Indeed, as in Section 3.2, indifference requires the benefit of avoiding a bad product when a breakdown occurs ((1−pt)(λSt+ε)) to equal the cost of delaying adoption absent news (r(2pt−1)). Since consumers grow more optimistic absent news, this has two implications throughout the indifference region: (i) beliefs ptincrease convexly, as the growth rate of ptequals the instantaneous probability of a breakdown ( ˙ pt pt= (1−pt)(λSt+ε)), which must increase over time to balance out the increasing cost of delay; (ii) the stock of adopters St=S(pt)increases convexly as a function of pt, to ensure that breakdowns arrive at a rate that counterbalances the convex growth (with respect to pt)oftheratio r(2pt−1) (1−pt)between the cost of delay and the probability of facing a bad product. Combining (i) and (ii), it follows that St—and,hence,adoption levels—increases convexly over time. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1525 or Brownian signals), a similar logic as in Section 3.3 implies that there cannot be continuous regions of partial adoption, because a consumer who is willing to adopt cannot instantaneously acquire decision-relevant information (see Frick and Ishii (2023), Supplement D).40 Appendix:Proofs A.1 Preliminary lemmas The following five lemmas will be used throughout the Appendix. For any feasible adoption flow (nt),wedenoteby(Wt)the corresponding no-news value to waiting and denote by (pt)the no-news posterior, without making explicit the dependency on (nt). Lemma A.1. For any feasible adoption flow (nt),thecorresponding(Wt)and (pt)are continuous in t. The proof is immediate from the definitions of ptand Wtin Sections 2.2 and 2.3. Lemma A.2. Suppose that (ns)is an equilibrium and that Wt<2pt−1for some t>0. Then there exists ν>0such that (Wτ)is continuously differentiable in τon the interval (t−ν,t+ν)and for all τ∈(t−ν,t+ν), ˙ Wτ=r+ρ+(εG+λGρNτ)pτ+(εB+λBρNτ)(1−pτ)Wτ −ρ(2pτ−1)−pτ(εG+λGρNτ)ρ ρ+r. Proof. Suppose Wt<2pt−1forsomet>0. Since (Wτ)and (pτ)are continuous in τ (Lemma A.1), there exists ν>0suchthatWτ<2pτ−1forallτ∈(t−ν,t+ν). Because (ns)is an equilibrium, this implies that nτ=ρNτfor all τ∈(t−ν,t+ν).Thus,nτis continuous at all τ∈(t−ν,t+ν).ThenWτis continuously differentiable in τfor all τ∈(t−ν,t+ν),as Wτ= t+ν τ ρe−(ρ+r)(s−τ)pτe−s τ(εG+λGnx)dx −(1−pτ)e−s τ(εB+λBnx)dxds +e−(r+ρ)(t+ν−τ)pτe−t+ν τ(εG+λGnx)dx +(1−pτ)e−t+ν τ(εB+λBnx)dxWt+ν + t+ν τ ρe−(ρ+r)(s−τ)pτ1−e−s τ(εG+λGnx)dxds +e−(r+ρ)(t+ν−τ)pτ1−e−t+ν τ(εG+λGnx)dxρ ρ+r. 40In contrast, Laiho, Murto, and Salmi (2024) obtain partial adoption/gradualism in a model with Brownian learning from the stock of adopters and continuous adoption opportunities. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1526 Frick and Ishii Theoretical Economics 19 (2024) The derivative of Wτcan be computed using Ito’s lemma for processes with jumps. Given perfect Poisson learning, the derivation is simple and we provide it for completeness. As above, for any ∈(0, t+ν−τ),wecanrewriteWτas Wτ= τ+ τ ρe−(ρ+r)(s−τ)pτe−s τ(εG+λGnx)dx −(1−pτ)e−s τ(εB+λBnx)dxds +e−(r+ρ)pτe−τ+ τ(εG+λGnx)dx +(1−pτ)e−τ+ τ(εB+λBnx)dxWτ+ + τ+ τ ρe−(r+ρ)(s−τ)pτ1−e−s τ(εG+λGnx)dxds +e−(r+ρ)pτ1−e−τ+ τ(εG+λGnx)dxρ ρ+r. Sincethisistrueforall∈(0, t+ν−τ), the right-hand side of this identity, which we denote R, is continuously differentiable with respect to and satisfies d dR≡0. Taking the limit as →0 and since ˙ Wτ=lim→0d dτ Wτ+by continuous differentiability, we then obtain ˙ Wτ=r+ρ+(εG+λGnτ)pτ+(εB+λBnτ)(1−pτ)Wτ −ρ(2pτ−1)−pτ(εG+λGnτ)ρ ρ+r. Plugging in nτ=ρNτyields the desired expression. Lemma A.3. Suppose that (nτ)is an equilibrium and that Wt>2pt−1for some t>0. Then there exists ν>0such that (Wτ)is continuously differentiable in τon the interval (t−ν,t+ν)and for all τ∈(t−ν,t+ν), ˙ Wτ=r+pτεG+(1−pτ)εBWτ−pτεG ρ ρ+r. Proof. The proof follows the same lines as that of Lemma A.2. Lemma A.1 again implies that if Wt>2pt−1, then there exists ν>0suchthatWτ>2pτ−1forall τ∈(t−ν,t+ν). By the definition of equilibrium, nτ=0forallτ∈(t−ν,t+ν). Hence, Wτsatisfies Wτ=e−r(t+ν−τ)pτe−εG(t+ν−τ)+(1−pτ)e−εB(t+ν−τ)Wt+ν +pτ t+ν τ εGe−(εG+r)sρ ρ+rds and, thus, is continuously differentiable in τ. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1527 To compute the derivative, note again that for any ∈(0, t+ν−τ), Wτ=e−rpτe−εG+(1−pτ)e−εBWt++pτ τ+ τ εGe−(εG+r)sρ ρ+rds. Differentiating both sides with respect to and taking the limit as →0, ˙ Wτ=r+pτεG+(1−pτ)εBWτ−pτεG ρ ρ+r, as claimed. Lemma A.4. Suppose (nt)is an equilibrium under bad news. Suppose ε>0or p0>1 2. Then limt→∞ pt=μ(ε,0,p0)and limt→∞ Wt=ρ ρ+r(2μ(ε,0,p0)−1),where μ(ε,0,p0):=⎧ ⎨ ⎩ 1if ε>0 p0 p0+(1−p0)e−0if ε=0. Proof. Suppose first that ε>0. Then trivially pt→1ast→∞. Since for any t, ρ ρ+r(2pt−1)≤Wt≤ρ ρ+r, this implies that limt→∞ Wt=ρ ρ+r,asclaimed. Now suppose ε=0andp0>1/2. Note that Wt≤2pt−1forallt. Indeed, suppose Wt>2pt−1forsomet.IfWs>2ps−1foralls≥t,thenWt=0, contradicting Wt>2pt− 1≥2p0−1>0. Thus, we can find s>tsuch that Ws=2ps−1andWs>2ps−1forall s∈(t,s). This implies ns=0foralls,and,hence,Wt=e−r(s−t)Ws=e−r(s−t)(2ps−1)= e−r(s−t)(2pt−1), again contradicting Wt>2pt−1>0. Let N∗:=limt→∞ t 0nsds =suptt 0nsds ≤N0.Letp∗:=limt→∞ pt=suptpt.Forany ν>0, we can find t∗such that whenever t>t ∗,thene−λt t∗nsds >1−ν. Because 2pt−1≥ Wtfor all t, we can then write the value to waiting at all t>t ∗as Wt= ∞ t ρe−(r+ρ)τpt−(1−pt)e−λτ tnsdsdτ ≤ρ r+ρpt−(1−pt)(1−ν). By optimality, Wt≥ρ ρ+r(2pt−1)for all t, so by combining, we have ρ ρ+r2p∗−1≤lim t→∞ infWt≤lim t→∞ supWt≤ρ r+ρp∗−1−p∗(1−ν). Sincethisistrueforallν>0, it follows that lim t→∞ Wt=ρ r+ρ2p∗−1, which is strictly less than 2p∗−1, so for all tsufficiently large we must have 2pt−1> Wt.Thenforalltsufficiently large, we have nt=ρNt.Thus,N∗=N0and, therefore, p∗=μ(ε,0,p0). 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1528 Frick and Ishii Theoretical Economics 19 (2024) Lemma A.5. Suppose that learning is via bad news. Suppose that ε=0and p0≤1 2.Then the unique equilibrium satisfies nt=0for all t. Proof. Suppose that (ns)is an equilibrium and suppose, for a contradiction, that t∗ 1:= inf{t:nt>0}<∞.Pickt≥t∗ 1such that nt>0. By right-continuity of (ns),wehavenτ>0 for all τ>tsufficiently close to t. This implies ∞ t∗ 1 ρe−(r+ρ)(s−t)pt∗ 1−(1−pt∗ 1)e−s t∗ 1λnkdkds > ρ r+ρ(2pt∗ 1−1)≥2pt∗ 1−1, (6) where the second inequality holds because pt∗ 1=p0≤1 2. The integral on the left-hand side is the expected payoff at time t∗ 1to adopting at the first opportunity in the future, conditional on no breakdown having occurred prior to this opportunity. By optimality of the value to waiting, this is weakly less than Wt∗ 1.Hence,(6)impliesWt∗ 1>2pt∗ 1−1. By continuity of (Ws)and (ps), it follows that for all s≥t∗ 1sufficiently close to t∗ 1,Ws> 2ps−1and,hence,ns=0, contradicting the definition of t∗ 1. This leaves nt=0foralltas the only candidate equilibrium. In this case, Wt=0≥ 2p0−1=2pt−1forallt, so this is indeed an equilibrium. A.2 Proof of Lemma 1 Good News. Suppose first that learning is via good news. Step 1: Wt=2pt−1=⇒ Wτ≥2pτ−1forallτ≥t. Suppose Wt=2pt−1atsome time tand suppose, for a contradiction, that at some time s>t,wehaveWs<2ps−1. Let s∗:=sup{s<s :Ws=2ps−1}. By continuity, s∗<s ,Ws∗=2ps∗−1, and Ws<2ps−1foralls∈(s∗,s).Thenby Lemma A.2, the right-hand derivative of Ws−(2ps−1)at s∗exists and satisfies lim s↓s∗ ˙ Ws−2˙ ps=r(2ps∗−1)+ps∗(ε+λρNs∗)r ρ+r>0. This implies that for some s∈(s∗,s)sufficiently close to s∗,wehaveWs>2ps−1, which is a contradiction. Step 2: Wt>2pt−1=⇒ Wτ>2pτ−1forallτ>t. Suppose, for a contradiction, that there exists s>tsuch that Ws=2ps−1. Let s∗:=inf{s>t:Ws=2ps−1}. By continuity, s∗>t,Ws∗=2ps∗−1, and Ws>2ps−1foralls∈(t,s∗).Notethatps∗≥1 2, because Ws∗is bounded below by 0. Moreover, by Lemma A.3, the left-hand derivative of Ws−(2ps−1) at s∗exists and is given by lim s↑s∗ ˙ Ws−2˙ ps=r(2ps∗−1)+ps∗r ρ+rε. If ε>0, this is strictly positive, implying that for some s∈(t,s∗)sufficiently close to s∗, we have Ws<2ps−1, which is a contradiction. If ε=0, then for all s∈(t,s∗),wehave ps∗=psand Ws=e−r(s∗−s)Ws∗=e−r(s∗−s)(2ps∗−1)≤2ps∗−1. Thus, Ws≤2ps−1, again contradicting Ws>2ps−1. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1535 Proof. Define Ht:=pt∞ 0(ε+λnt+τ)e−(ετ+t+τ tλnsds)ρ r+ρe−rτ dτ. Thus, Htrepresents a consumer’s expected payoff to waiting at time tgiven that from t on he adopts only if there has been a breakthrough and given that the population’s flow of adoption follows (ns). By optimality of Wt,wemusthaveHt≤Wtfor all t.Forany posterior p∈(0, 1),let H(p,0 ):=p∞ 0 εe−ετ ρ r+ρe−rτ dτ =pερ (ε+r)(r+ρ). That is, H(p,0 )represents a consumer’s expected payoff to waiting at posterior p,given that he adopts only once there has been a breakthrough and given that breakthroughs are only generated exogenously. Note that by definition of t∗,nt>0 if and only if t<t ∗. This implies that H(pt,0 )< Htif t<t ∗and H(pt,0 )=Ht=Wtif t≥t∗;moreover,2pt−1≥Wtif t<t ∗and 2pt−1≤ Wtif t≥t∗. Finally, note that ps:=(ε+r)(r+ρ) 2(ε+r)(r+ρ)−ερ has the property that 2p−1≤H(p,0 ) if and only if p≤ps. Combining these observations, if t<t ∗,then2pt−1≥Wt≥Ht>H (pt,0 ),sopt> ps.Ift≥t∗,then2pt−1≤Wt=H(pt,0 ),sopt≤ps,asclaimed. A.6 Proof of Proposition 1 Fix r,ρ,ε,p0. Suppose 0is such that t∗ 1(0)<t ∗ 2(0). By the proofs of Theorem 1 and Lemma 2,thismeansthatCondition1is satisfied, p0<p ,and0> 0,where 0:=max{∗(p0),∗(p)}as in the proof of Lemma 2.Consideranyˆ 0> 0. A.6.1 Proof of part (i) (welfare neutrality) Write 1 0:=0and 2 0:=ˆ 0,withcorresponding cutoff times ti 1and ti 2, value to waiting Wi t, and no-news posteriors pi tfor i=1, 2 (by the proof of Theorem 1, these quantities depend on λi,Ni 0only through i 0). Since t1 1<t 1 2and 2 0> 1 0> 0, Lemma 2implies t2 1<t 2 2. Moreover, by the proof of Lemma A.7,wehavemax{p0,p}=p1 t1 1 =p2 t2 1 . Because ni t=0forallt<t i 1for both i=1, 2, this implies that t1 1=t2 1=t1.ThenW2 t1=2p2 t1−1=2p1 t1−1=W1 t1. Since there is no adoption until t1,wehaveWi 0=e−rt1pt1 p0Wi t1for i=1, 2, whence W1 0=W2 0,asclaimed. A.6.2 Proof of part (ii) (non-monotonicity of learning) We first prove the following lemma. Lemma A.11. Suppose that ˆ 0=ˆ λˆ N0> 0=λN0> 0, with corresponding equilibrium flows of adoption (ˆ nt)and (nt).Then (i) t∗ 1(0)=t∗ 1(ˆ 0) (ii) 0<t ∗ 2(0)<t ∗ 2(ˆ 0) (iii) for all t<t ∗ 2(0),λnt=ˆ λˆ nt. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1536 Frick and Ishii Theoretical Economics 19 (2024) Proof. For (i), note that by the proof of Lemma A.7,timet∗ 1under both 0and ˆ 0is pinned down by the condition max{p0,p}=p0 t∗ 1(0)=pˆ 0 t∗ 1(ˆ 0). Because up to time t∗ 1, learning is purely exogenous under both 0and ˆ 0, this implies t∗ 1(0)=t∗ 1(ˆ 0). For (ii) and (iii), note first that by Lemma 2,wehavet∗ 2(ˆ 0),t∗ 2(0)>0. Let t∗ 2= min{t∗ 2(ˆ 0),t∗ 2(0)}. Then because t∗ 1(0)=t∗ 1(ˆ 0), the ODE in Corollary A.1 implies that at all times t<t ∗ 2,wehavep0 t=pˆ 0 t=pt. By Lemma A.6, this implies that for all t<t ∗ 2, λnt=ˆ λˆ nt. (11) Note that (11) implies that t∗ 2=0−t∗ 2 0 λntdt < ˆ 0−t∗ 2 0 ˆ λˆ ntdt =ˆ t∗ 2. Because p0 t∗ 2=pˆ 0 t∗ 2, Lemma A.7 implies that t∗ 2=t∗ 2(0)<t ∗ 2(ˆ 0).Fromthisand(11), it is then immediate that λnt=ˆ λˆ ntfor all t<t ∗ 2(0). Now we prove part (ii) of Proposition 1. By Lemma A.11,t∗:=t∗ 2(0)<t ∗ 2(ˆ 0),λnt= ˆ λˆ nt,andp0 t=pˆ 0 tfor all t≤t∗, which proves the first claim of part (ii). For the second claim of part (ii), we note that there exists some ν>0suchthatatall times t∈(t∗,t∗+ν),wehavep0 t>pˆ 0 t. To see this, we prove the following inequality for the equilibrium corresponding to 0: lim t↑t∗λnt<lim t↓t∗λnt. (12) That is, there is a discontinuity in the equilibrium flow of adoption at time t∗. Indeed, because nt=ρNtfor all t≥t∗and by continuity of Nt, feasibility implies that limt↑t∗λnt≤ limt↓t∗λnt. Suppose for a contradiction that limt↑t∗λnt=limt↓t∗λnt:=λnt∗.Thenλnt∗= ˆ λˆ nt∗.Moreover,forallt>t ∗,wehaveλnt=ρt∗e−ρ(t−t∗), which is strictly decreasing in t. On the other hand, ˆ λˆ ntsatisfies ˆ λˆ nt=⎧ ⎪ ⎨ ⎪ ⎩ r(2ˆ pt−1) (1−ˆ pt)−εif t∈[t∗,t∗ 2(ˆ 0)) ρt∗ 2(ˆ 0)e−ρ(t−t∗ 2(ˆ 0)) if t≥t∗ 2(ˆ 0). Thus, for t∈[t∗,t∗ 2(ˆ 0)),ˆ λˆ ntis strictly increasing in t. This implies that ˆ λˆ nt>λn tfor all t∈[t∗,t∗ 2(ˆ 0)).Hence,by(1), pˆ 0 t∗ 2(ˆ 0)>p 0 t∗ 2(ˆ 0), which by Lemma A.7 implies ˆ t∗ 2(ˆ 0)=∗pˆ 0 t∗ 2(ˆ 0)> ∗p0 t∗ 2(ˆ 0)> t∗ 2(ˆ 0). This yields that for all t≥t∗ 2(ˆ 0), ˆ λˆ nt=ρe−ρ(t−t∗ 2(ˆ 0)ˆ t∗ 2(ˆ 0)>ρe −ρ(t−t∗ 2(ˆ 0)t∗ 2(ˆ 0)=λnt. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1537 Thus, ˆ λˆ nt>λn tfor all t>t ∗and, hence, pˆ 0 t>p 0 tfor all t>t ∗. This implies Wˆ 0 t∗>W0 t∗, which is a contradiction, because we have Wˆ 0 t∗=2pˆ 0 t∗−1=2p0 t∗−1=W0 t∗. This proves that limt↑t∗λnt<limt↓t∗λnt.Hence, lim t↓t∗ ˆ λˆ nt=lim t↑t∗ ˆ λˆ nt=lim t↑t∗λnt<lim t↓t∗λnt. Thus, there exists some ν>0suchthatˆ λˆ nt<λn tfor all t∈[t∗,t∗+ν). Together with the fact that p0 t∗=pˆ 0 t∗, this implies that p0 t>pˆ 0 tfor all t∈(t∗,t∗+ν), proving the second claim. Finally, for the third claim of part (ii), observe first that there exists some t>t ∗such that p0 t=pˆ 0 t. If not, then by continuity of beliefs, p0 t>pˆ 0 tfor all t>t ∗and we have Wˆ 0 t∗<W0 t∗, again contradicting Wˆ 0 t∗=W0 t∗=2pt∗−1. Then t:=sup{s∈(t∗,t):p0 s> pˆ 0 s}exists, with t>t ∗by the second claim. Further, by continuity, p0 t=pˆ 0 t,which implies t 0λnsds =t 0ˆ λˆ nsds. This yields t<ˆ t, which implies that ˆ λˆ nt>λn tfor all t>t. Indeed, if t≥t∗ 2(ˆ 0),thisisobvious.Ift∈(t∗,t∗ 2(ˆ 0)),thenwemusthaveλns<ˆ λˆ ns for some s<t, which implies that λns<ˆ λˆ nsfor all s∈(s,t∗ 2(ˆ 0)), because Nis strictly decreasing and ˆ nis strictly increasing on this domain. To see that we also have λns< ˆ λˆ nsfor all s≥t∗ 2(ˆ 0), note that from the above, pˆ 0 t∗ 2(ˆ 0)>p 0 t∗ 2(ˆ 0), which as above implies that ˆ t∗ 2(ˆ 0)=∗pˆ 0 t∗ 2(ˆ 0)> ∗p0 t∗ 2(ˆ 0)> t∗ 2(ˆ 0). Hence, ˆ λˆ nt>λn tfor all t>t. Thus, in either case, pˆ 0 t>p 0 tfor all t>t. A.6.3 Proof of part (iii) (slowdown of adoption) Adoption of Good Products. By Lemma A.11,t∗ 1(0)=t∗ 1(ˆ 0)=:t∗ 1and λnt=ˆ λˆ ntfor all t∈(t∗ 1,t∗),wheret∗:=t∗ 2(0). Then for all t<t ∗, nt N0 =λnt 0 =ˆ λˆ nt 0 ≥ˆ λˆ nt ˆ 0 =ˆ nt ˆ N0 , with strict inequality for all t∈(t∗ 1,t∗). Therefore, At(0,G)≥At(ˆ 0,G)for all t<t ∗, with strict inequality for all t∈(t∗ 1,t∗). Finally note that for all t≥t∗,nt=ρNtand so At(0,G)=At∗(0,G)+1−e−ρ(t−t∗)1−At∗(0,G) At(ˆ 0,G)≤At∗(ˆ 0,G)+1−e−ρ(t−t∗)1−At∗(ˆ 0,G), where the second inequality follows from feasibility. Because At∗(0,G)>A t∗(ˆ 0,G), it follows that At(0,G)>A t(ˆ 0,G)for all t>t ∗ 1,asclaimed. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1538 Frick and Ishii Theoretical Economics 19 (2024) Adoption of Bad Products. Recall that At(λ,N0,B)denotes the expected proportion of adopters at time tconditional on θ=B. That is, letting (nt)denote the associated equilibrium, we have At(λ,N0,B):= t 0 (ε+λnτ)e−τ 0(ε+λns)dsτ 0 ns N0dsdτ +e−t 0(ε+λns)ds t 0 ns N0ds = t 0 nτ N0e−τ 0(ε+λns)ds dτ, where the final equality follows from integration by parts. Moreover, from the Markovian description of equilibrium in the proof of Theorem 1, it is easy to see that this expression depends on λand N0only through 0=λN0, so we can denote it by At(0,B). Then we can assume without loss of generality that 0and ˆ 0are of the form 0=λN0and ˆ 0=ˆ λN0, i.e., that the two environments have the same population size N0. Let (nt)and (ˆ nt)be the equilibrium under λand ˆ λ,respectively.Givenanarbitrary strictly positive adoption flow (ms)and t>0, note that the map λ→ t 0 mτe−τ 0(ε+λms)ds dτ is strictly decreasing in λ. Since ˆ 0> 0> 0,wehavet∗ 1(0)=t∗ 1(ˆ 0)=:t∗ 1,andsowe get that for all t>0, t 0 nτe−τ 0(ε+λns)ds dτ ≥ t 0 nτe−τ 0(ε+ˆ λns)ds dτ, (13) with strict inequality for all t>t ∗ 1. We now show that t 0 nτe−τ 0(ε+ˆ λns)ds dτ ≥ t 0 ˆ nτe−τ 0(ε+ˆ λˆ ns)ds dτ. Together with (13), this implies the desired conclusion that At(ˆ λN0,B)≤At(λN0,B)for all t>0, with strict inequality for all t>t ∗ 1. To prove this, suppose for a contradiction that there exists some t>0suchthat t 0 nτe−τ 0(ε+ˆ λns)ds dτ < t 0 ˆ nτe−τ 0(ε+ˆ λˆ ns)ds dτ. (14) 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Innovation adoption 1539 Note that by the above result for good products, N0Aτ(λ,G)=τ 0nsds ≥τ 0ˆ nsds = N0Aτ(ˆ λ,G)for all τ≥0andso,forallt≥0, t 0 εe−τ 0(ε+ˆ λns)ds dτ ≤ t 0 εe−τ 0(ε+ˆ λˆ ns)ds dτ. (15) Inequalities (14)and(15) together imply t 0 (ε+ˆ λnτ)e−τ 0(ε+ˆ λns)ds dτ < t 0 (ε+ˆ λˆ nτ)e−τ 0(ε+ˆ λˆ ns)ds dτ. This is equivalent to 1−e−t 0(ε+ˆ λns)ds<1−e−t 0(ε+ˆ λˆ ns)ds, which contradicts t 0nsds ≥t 0ˆ nsds. References Anderson, Axel, Lones Smith, and Andreas Park (2017), “Rushes in large timing games.” Econometrica, 85, 871–913. [1508] Bandiera, Oriana and Imran Rasul (2006) “Social networks and technology adoption in northern Mozambique”. Economic Journal, 116, 869–902. [1522] Banerjee, Abhijit V. 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