Investment and Scale in Online Search: A Dynamic Model

Transcription

1 Investment and Scale in Online Search: A Dynamic Model Ángel Luis Lopez a Bita Shabgard b June 2017 Abstract We explore and develop a dynamic duopoly investment game for two competing search engines where firms invest in R&D to improve quality of services through time. To examine optimal solutions, we employ differential game. The open-loop Nash equilibrium is derived which corresponds to the symmetric steady state. The purpose of present paper is to identify the key factors that affect the development of market structure. Our findings show that under specific conditions, asymmetries between firms tend to decrease over time and quality and R&D investment converge to steady state equilibrium. Otherwise distance between firms increases and subsequently online search market reduces to monopoly. Next we analyse consequences of monopoly market through characteristic of optimal path along feasible trajectories. Particularly, we focus on policy s response to monopoly solutions by studying welfare s implications. Key words: Dynamic competition, Search engine, R&D investment, Differential game. a Lopez, Departament d Economia Aplicada, Universitat Autonoma de Barcelona, Spain. address: angelluis.lopez@uab.cat b Shabgard, Ph.D candidate of Applied Economics, Departament d Economia Aplicada, Universitat Autonoma de Barcelona, Spain. address: bita.shabgard@e-campus.uab.cat 1

2 Introduction: The markets for online search and online advertising are highly concentrated. In February 2017, Google had above 85% of the global market share and Bing had above 5%. Moreover, rapid growth of online search and significant amount of revenues that they generate establish the importance of the market. Online search and online advertising depend crucially on search engines. The truth is that search engines are critical gateways for users and businesses. On the one hand, search engines allow users to find what they are looking for on Internet, and on the other hand, search engines give businesses the opportunity of offering their products or special deals at the precise instant the user shows a special interest on a particular products or services. Present paper has two major objectives. First, we attempt to identify the key factors that affect the evolution of market structure in a model of dynamic competition between two search engines. Second, we analyse whether (and under which circumstances) asymmetries between firms tend to increase or decrease over time and whether increasing dominance with monopolisation is possible. The market characteristics that affect future profitability of firms have an influence on firm s decision to behave strategically. We take a differential game approach to analyse dynamic investment game where firms invest in R&D to improve quality of services over continuous time t [0, ). The focus is to characterize optimal path of quality and R&D investment along feasible trajectories. Search engine market presents some specificity that distinguish it from other markets. Consumers are not charged a fee to conduct queries and thus profit of search engine is only obtained by online advertising. Its revenue is amount to almost half of all online advertising revenues. Estimates from emarketer 1 indicate that advertising cost worldwide reaches $170 billion in 2015 with search ads accounting for $81 billion. The importance of the market for online search and online advertising relies not only on the significant amount of revenues that they generate but also on their implications on the traditional markets. Online shopping has become an increasingly common staple of life in the 21 st century. It offers a sense of ease and comfort, the ability to scroll quickly through several stores and actual sales, also saves precious time and allows shopping online any time during the day. Meanwhile, one of the remarkable features of the search engine is highly concentrated levels of market. Figure1 depicts market share of online search providers in February Figure1: Market share of search engine in February Yahoo; 4.30% Baidu; 4.23% Other; 1.04% Ask; 0.13% Bing; 5.04% Google-Global; 85.27%

3 The first highest market share belongs to Google and the second one belongs to Bing. Total market share of other engines such as Yahoo, Baidu, Ask and etc is less than 10% of global market share. Higher market share generates higher market power and dominant position. As figure1 shows, Bing is most powerful rival of Google. This paper aims to analyse a dynamic investment game in online search market. We explore and develop two different dynamic models, according to whether market is characterized as duopoly or monopoly. The optimal control for firms is to find R&D level that maximises the present value of payoff function, subject to the dynamic associated with quality in an infinite time horizon. In particular, our models reflect the following real facts: i) the quality of search engine s services evolves over time as it handles more user queries, ii) greater query is a consequence of faster innovation, iii) search engines will attract advertisers more easily as they get more queries. We consider market share of search engine,, as a function of levels of quality of search algorithms. Quality of search algorithm, encompasses precision search results and response time to process of consumer s query. Each firm can improve quality of its search algorithm by investing an amount of in R&D. Moreover, we assume that dynamic quality of the search algorithm of each firm depends on the stock of quality and the level of investment in R&D. The investment cost,, is a strictly convex function of the rate of investment. Furthermore, the marginal cost of investing in R&D decreases in the level of quality achieved by the algorithm. Revenue from online advertising, is some functions of each search engine s market share with. Hence firms invest in R&D so as to maximise its total discounted pay-off. As the level of quality of search services increases with the level of investment in innovation, and as the marginal cost of investing in R&D diminishes with level of quality, firms will have incentives to behave strategically in order to improve their future position. In present paper, we study two dynamic models. First, we analyse duopoly competition between two search engines, Google and Bing, by applying continuous-time differential game. We address the question whether asymmetries between search engines tend to increase or decrease over time. The dynamical behaviour of search engines studies in open-loop equilibrium which is corresponded to symmetric steady state. We find that under specific cases, asymmetries between firms decrease over time and systems converge to steady state equilibrium. It is worth stressing that whether the leader has incentives to invest more than the laggard, then this fact increases the distance between two search engines. Subsequently, asymmetries between firms increase and online search market reduces to monopoly. Second, in monopoly model, we present continuous-time differential game to analyse optimal control problem for Google that has superior technology. As a matter of fact, Google s search algorithm has significantly higher quality than its rivals. Our results imply that optimal quality and R&D investment converge to steady state through time under particular circumstances. Monopolist may offer services with low quality. Consumers can harm if they get useless results. We evaluate policy s response to monopoly solutions by studying 3

4 welfare s implications. Results of welfare analysis show whether regulators aim to maximise welfare in order to avoid abuse of dominant power, optimal path does not exist. The paper is structured as follow. Section_1 presents and discusses related literatures. Section_2 characterizes duopoly game with two firms, Google and Bing. General model for each firm is qualified. We demonstrate open-loop Nash equilibrium of the game. Optimal path of the system is studied. The qualitative effects of optimal solution are reviewed by variation of some interested parameters. Section_3 lays out monopoly model. We examine qualitative behaviour of the dynamic system and determine the optimal path. Then we analyse how optimal quality and R&D investment behave in response to a change in some parameters. In the following, welfare analysis and regulators problem are discussed. Section_4 provides some concluding remarks. Appendices gather the proofs. 1) Related Literatures: In this section, we provide a revision of relevant literatures on competition between search engines. These literatures deal with four agents: consumers, content providers, advertisers and search engine. Consumers use search engine to conduct the search queries. Content providers are the web sites which publish information on the internet and search engine indexes them. Advertisers get access consumers through search engine to make profit. Figure 2 shows a diagram of the market. Search engine connects indirectly consumers, content providers and advertisers. Figure2: Four-sides of search engine market Consumers Search Engine Content Providers Advertisers Most of the existing relevant papers study properties of the market in terms of two-sided market by applying static competition model or dynamic competition model. Our analysis considers a dynamic differential game where search engines invest in R&D to improve their quality. Search engine market has high levels of innovation and R&D investment. In recent years, Google and Microsoft appertain to 10 most innovative companies in the world 3. Figure 3 displays R&D investments of Google and Microsoft in United States during

5 R&D Investments ($Bn) Figure3: R&D investments of Google and Microsoft Year Google Microsoft The main aim of present paper is to determine circumstances that optimal path of quality and R&D investment converge to the steady state equilibrium through time. The novelty is to investigate optimal solutions by using continuous time differential game. This paper is related to some works on market structure and increasing dominance in the search engine market. However, the online search is rather new but search engine market has been restructured by new entrances with offering higher quality of services. In early stages of the market, there were five major search engines: Yahoo, Lycos, Excite, Infoseek, Altavista, and Yahoo was dominant in the market. Each search engine was able to cover small part of the total web space. In the middle of the 1998, the structure of the market decomposed whereas some of them do not exist anymore. In 1996 Google was launch and within a few years, the company with superior search quality improved the search relevance and became leader in online search and advertising. Google used the new search algorithm which delivered reasonably accurate results. In 2000 Yahoo partnered with Google and let Google improved their organic (unpaid) results. In 2009 Microsoft launched Bing. While Yahoo lost most of its relevances, Bing became formidable competitor of Google. In recent years Google is leader and Bing is standing as its most powerful rival. Gandal (2001) analyses competition in the oligopoly search engine market between five firms, Yahoo- Lycos- Excite- Infoseek and Altavista, in an empirical study. Some specifications of the industry such as number of unique visitors, longevity of search engine and number of dead links are applied to his growth model. Results show that early entrants have an advantage in the market but negative coefficient of longevity declares that this advantage decreases over time. Next, he develops the discrete choice model for product differentiation in order to examine robustness of the growth model s results. Decreasing first mover advantage over time recommends that there are low barriers to entry in the market which cause to make strong competition. Also his results are consistent with the fact that Yahoo can hold his position if it extends innovation and offers superior services. In another study, Pereira (2002) evaluates the effect of consumer prices on competition for price-comparison search engines in a partial equilibrium search model. He deals with three parts of the market; vendors, consumers and search engines. Model has four main characteristics: i) search engines make profit from vendors, ii) products are branded and consumers have distinctive preferences between various brands, iii) vendors can have quite 5

6 different prices at their homepages and at a search engine, iv) vendors inform rival s prices at a search engine. He considers a market which opens only for one period with some specifications like; a finite number of search engines and vendors, large number of consumers and branded search goods. Vendors can only join one search engine. Consumers face no monetary cost, but face utility loss and access cost in order to access to a search engine s site. Author classifies consumers in 3 groups: strong loyals which have preferred brand and strong valuation of good, weak loyals which have preferred brand and weak valuation of good, switchers which have no preferred brand. The timing is as follows: First: search engines set submission fees. In oligopoly online search market, search engine sets lower submission fee than in the monopoly market in order to index larger number of vendors. Second: vendors decide by which search engine is indexed that offers the largest non-negative profit. Third: vendors set price at their homepages. Fourth: vendors set price at the search engine in which they are indexed. Fifth: consumers decide to buy. The results rely upon the price distribution at search engine. Strong royals buy at the vendor s page of preferred brand. Weak loyals compare the price of good at search engine plus access cost to search engine with price at vendor s page and choose the cheaper one. Switchers buy at search engine where they expect to pay lowest price. Prominent works have focused on the quality of services and market share of search engine. Mukhopadhyay, Rajan and Telang (2003) analyse quality competition between incumbent and entrant in the search engine market. Quality is interpreted as capability of search engine to provide results that satisfy a consumer s desire. Authors design a dynamic game in two periods. Their game has some characteristics amongst i) demand of using a search engine is assumed the same in each period and it is a linear increasing function of quality D(q)=b.q, where q is the quality and b > 0. ii) Cost is a strictly increasing function of quality, C(q). iii) Consumers face zero prices for using online search services. In the first period, only incumbent exists in the market and chooses the level of quality. Consumers have only one choice in this period. In the second period, entrant enters to the market with newest technology. The cost function of entrant is where, - If then entrant makes benefit from cost advantage. Incumbent has advantage of brand loyalty in the second period. Brand loyalty is captured by parameter, -. The proportion of old consumers remains loyal to incumbent regardless of quality. The fraction (1 ) of old consumers plus new ones choose search engine in terms of higher quality of services. Search engines run different algorithms for the same query. Consequently, they provide different results which encourage consumers of high quality search engine visit lower quality one. The results of the residual demand in the vertical differentiation model demonstrate that low quality search engine can remain in the market. White (2013) argues that providing high quality results of consumer s queries have benefits and costs for search engine. He studies three versions of a static model in order to clarify the forces deriving the choice of search quality. These forces relate to online advertisement. In the first version of the model, quality of services does not consider and search engine behaves as a platform to connect consumers and advertisements. In the second version of the model, search engine must make costly investment in order to improve quality. The interaction 6

7 between high quality organic (unpaid) results and advertisements are overlooked. Third version of the model has comprehensive insight into the online search market. It perceives the fact that paying advertisers should compete with organic results in the same market. High quality of organic results induces consumers pay less attention to paid advertisements on the right side of the result s page. This sort of competition may seem search engine has little incentive to display high quality organic results. Although it does not eliminate the fact that high quality organic results absorb attention of consumers and bring higher market share and profits. He concludes that search engine s profit relies extremely on the type of unpaid results. He suggests that for online query, search engine can show relevant informational links instead of showing links of additional merchants. Kim and Tse (2012) develop a differential game model in order to study competition between two heterogeneous search engines, inferior firm with knowledge-sharing services and superior firm without knowledge-sharing services. Knowledge sharing services let knowledge to share amongst consumers of search engine by freely asking and answering questions. Inferior search engine faces with four sides in the market; searchers, answerer members, advertisers and database of an answer. Answerer members answer the questions of searcher in order to gain reputation of search engine. In open-loop equilibrium, each firm solves its optimal control problem. The problem of inferior search engine is maximising { } Under some constraints such as: i) growth of consumers who visit inferior search engine and superior search engine, ii) growth of answerer members, iii) growth of advertisers, iv) growth of the database, v) growth of online content providers, vi) the amount of online content in particular language changes. The problem of superior search engine is maximising { } under same constraints. The variables of models are described in table1. Table1: description of variables The number of searchers who visit search engine i. The number of answerer members of search engine1 s knowledge sharing services. The number of advertisers who use search engine i. The periodic fee that search engine i charges to advertisers. Search engine i s periodic marginal costs of maintaining searchers. Search engine i s periodic marginal costs of maintaining advertisers. Search engine 1 s periodic marginal costs of maintaining its answerer members. Authors consider two different scenarios to study market share for inferior search engine with knowledge-sharing services. The results show that inferior search engine can survive in the market with advantage of knowledge sharing services even if it has small amount of online content. The point is that the inferior search engine s database draws attention of searchers 7

8 who keen to ask questions online. In other scenario, market share of inferior firm can be increased by policy of keeping database closed to queries from other search engines. In the later work of Kim and Tse (2014), they look at static competition between inferior and superior search engines under knowledge sharing services. They introduce profit of search engines under different strategies. First: if there is not knowledge sharing service in the market, then the profit of inferior firm and superior firm are and, respectively. is the quality of firm i where i=1,2. Term is the probability that searchers first search at engine 2 is not satisfactory. Second: If inferior search engine introduces a knowledge-sharing service with Q 1 active questioners and K 1 answerers, then the profit of the inferior search engine is: { where is the searcher s demand of inferior search engine and c is the cost of introducing the knowledge-sharing service. Third: If both search engines introduce the knowledgesharing service, then the profit of inferior and superior are: { { where is the searcher s demand of superior search engine with Q 2 active questioners and K 2 answerers. The aim of the search engine is to maximise its profit under knowledge-sharing services. According to the different scenarios, authors show that if the amount of information is limited, both firms should adopt closed knowledge-sharing to queries from other search engines. If the amount of online information is increased, the best strategy to maximise profit of inferior firm is to choose closed knowledge-sharing and the best strategy for superior is to choose open knowledge-sharing services. Another stream of the literature looks at the analysis of concentration, dominance and social welfare. Zhao and Tsc (2011) develop a two-sided market model in which one side is the information seekers and the other side is the information providers and investigate industry structure and competition between search engines. They describe utility of consumers regarding to quality of the search engine (Q) such as usability of the site and response time of processing a user s query, capability of search engine to find the relevant web pages that users looking for (R) and the amount of web pages indexed by the search engine (M). The transportation cost for consumer is the time spent searching and browsing the results. Therefore the net benefit of consumers for using the search engine is where and are positive and constant. Parameter captures the percentage of a consumer s value that belongs to the indexed web page. The utility of advertisers relies on 8

9 capability of search engine to find the relevant webpages that consumers looking for (R), the number of consumers (X) and the number of advertisers (K). P is the price advertisers pay to search engines. Therefore the net benefit of the advertisers is where is positive and constant. The model also explains behaviour of consumers to use multiple search engines for the same queries. They argue that search engines compete in the market by improving search technology such as usability improvement or reduce response time and increase the size of indexed webpages in order to generate higher market power. As a result, the search engine market may reduce to monopoly. Argenton and Prufer (2012) develop discrete choice model for product differentiation to study structure of the search engine market. They consider some additional specifications of the market: incomplete information of consumers about quality of search engine, some degree of horizontal product differentiation and network externalities. Network externality in the search market is sorted results of consumer s queries by search engine and it translates as higher search quality perceived by consumers. Authors build a model for three competitive search engines; Google, Bing and Yahoo. They define market share of firm i as where is a quality of j s search engine and, i,j {1,2,3} and i j. The quality is interpreted as a quality perceived by consumers. Cost of quality production is C= where is the amount of past queries run on j s search engine. The objective of the search engine is to maximise its profit which is where is exogenous advertising revenue and is the fixed cost. The results show that firm 1 produces higher quality and therefore has higher market share whereas firm 3 is the weakest search engine in the market and sooner or later exit the market. Therefore the structure of the market reduces to duopoly. In duopoly each firm solves: * + If is sufficiently high, the profit of firm 2 is negative and in the long run it exits from the market. Therefor market characterizes as monopoly. Their results show that Google is the market leader. Pollock (2009) and Lianos and Motchenkova (2013) investigate dominant power and its effect on social welfare. To explain the high levels of concentration in the market, Pollock employs a basic static model with the key specifications of the search engine market. The structure of the model is based on four agents; consumer, advertiser, content provider, and search engine. He considers a market with N search engines each offering different quality, * +. If quality of services is zero, then firm is not active in the 9

10 market. Utility of consumer by using the search engine i is an increasing function of quality. He assumes consumers have identical taste for quality. Furthermore each consumer uses only one search engine which gives him highest utility. Search engine s demand is a function of own quality (q i ) and quality of rivals (q j where ). Advertising revenue is a function of quality of search engine and consumer s demand for search engine. Total costs of search engine encompass consumer cost (c) and advertising cost (c i ) which are function of quality, number of consumers and amount of advertising. Profit of search engine is defined as; =. The results show that firm with high fixed cost, low marginal cost and with pure quality competition for consumers (zero prices and consumers homogenous) is likely to be dominant in the market. He brings up that threat of entry is reliable and contestability can occur even though fixed cost is high. He also conducts social welfare in the monopoly model. Monopoly s quality is larger than welfare maximising s quality if advertising revenues exhibit increasing returns to the number of search engine consumers. Consequently, under this situation search engine does not require regulation. In contrast, if quality of monopoly digresses from optimal provision in order to increase advertising revenue then it delivers inefficiently low search quality. Whether search engine is able to display most organic (unpaid) relevant results of query, then advertisers and search engine are substitute which has negative direct effect on engine s revenue. Or if search engine is able to provide information that prevent consumers from using some products then reduces advertising revenue from that query. Consequently, quality of monopoly is under-provision of social optimum quality. In this case some forms of regulation are required. Other work by Lianos and Motchenkova (2013) develop a model of two-sided market, consumers and advertisers. Their target is to analyse concentration and dominance in the search engine market. In the monopoly platform, authors study possibility of abuse of dominant power in the market as for example by Google. Search engine can increase utility of two-sided with investing amount of k to improve quality of services. They define utility of each sided market as a function of the amount of other group members, quality and charged price by search engine for each sided. Therefore utility of consumers is given by: And utility of advertisers is given by: where and are the amount of members of consumers and advertisers, respectively. Consumers are served for free in the search engine and advertisers face positive charged price by search engine. Parameter captures benefit of consumers in terms of interacts with advertisers which can be positive or negative. Parameter captures benefit of advertisers in terms of interacts with consumers which assume to be positive. Parameter represents the quality improving innovation efforts. They consider members of each group as an increasing function of utility. Monopoly incurs a per-agent cost for advertisers and for consumers which is assumed to be zero. Cost of investing in innovation is F(k). Therefore, profit of search engine in the monopoly market is defined as 10

11 . The aim of the firm is to obtain the profit maximizing price and qualityimproving investment. In the next step, they examine welfare s implication and find socially optimal price and socially optimal level of quality-improving investment by maximizing welfare. Finally, they compare two results and conclude that monopolist search engine market has lower investment and charged higher price to advertiser than social optimum. 2) Dynamic Optimization for Duopoly: Though Google is broadly viewed as the most popular search engine, there are some alternative search engines that can be used. The last and strongest competitor of Google is Bing, owned by Microsoft and rebranded of MSN search in Bing has the second highest market share after Google in the world. Response time of Bing to search query is strictly comparable with Google (Zhao and Tse, 2011), while both search engines have different search algorithms (Moffitt, 2014). Consumers are capable to compare results from different search engines. If Bing can offer more relevant results than Google can, then consumers easily switch and join Bing. Search engines are trying to prevent consumers from switching. Therefore they invest in R&D in order to improve quality of services over time. In this section, we mainly concentrate on market for Google and Bing. We characterize a duopoly investment game and determine open-loop Nash equilibrium. Then we go through to study whether asymmetries between firms tend to increase or decrease over time. Next, we conduct a comparative static analysis and illustrate our results. 2-1) Model We build a dynamic duopoly model where firms invest in R&D in order to improve quality of services. Each firm indicates a path of R&D investment and thus an induced path of quality concerning to maximise his discounted pay-off. There are 2 firms indexed by i=1, 2, each providing a search online services with different qualities. Assume firm 1 has superior technology. Consumers are uniformly distributed on the segment [0,1], whereas firm 1 is located at 0 and firm 2 is located at 1. Each consumer makes one query in one of the two search engines. Cost incurred from using the search engine is zero for consumers. Utility of consumer derives from using search engine is a function of quality of search engine,, and is given by: = where i {1,2} Parameters are positive. We characterize the market share of firm i with regard to consumer s utility. Market share of firms, and, are defined as: 11

12 where variable captures the degree of substitutability between two search engines. We define that measures the distance between qualities of firms, Market share of each firm can describe as a function of. Therefore, we can rewrite where is positive. Firms invest continuously the amount of in R&D. Investment cost of each firm,, is a strictly convex function of its R&D investment. Furthermore, the marginal cost of investing in R&D decreases by the level of quality achieved by the algorithm. Investment cost of firms are given by and : = ( ) = ( ) where,, are common to both firms and positive constant over time. Profit of each firm in the online advertising, gets by: with 0. Profit of firm i is strictly concave function of its market share; = + > 0. Substituting market share of each firm in profit function, we obtain: = = where, and. We omit how firms compete for online advertising and assume that there exists a unique equilibrium to such competition for advertising game. The pay-off to firm i is given by: 12

13 Each firm s pay-off depends on its own quality as well as quality of its rival. The dynamic associated with distance between qualities is a function of quality and level of investment in innovation. It is qualified by following equation: (1) with. We notice that the stock of quality may have negative or positive effect on its growth, according to whether or. In order to explain the game, we follow Fershtman and Eitan (1983) and define firm i s strategy,, which belongs to the set: Definition1: The strategy space for firm i is =*, ), -, )+ where R&D investment,, takes its value in the interval, -. If then, investment cost function goes to infinity,. The stages of the game are as follow: First, given, each firm invests in R&D, u t i, in order to improve search algorithms, thereby incurring cost. Second, due to the distance between qualities amongst firms which evolves according to its dynamic, consumers choose the firm they want to join, resulting in market shares, from which firms make some revenues from online advertising. Each firm aims to maximise the discounted pay-off flow under the constraint given by dynamic quality and initial conditions. The objective of each firm is: = s.t where i {1,2} and i y(0)=, u i (0)= Firms discount future pay-off at rate which is assumed to be positive constant and equal across firms. Suppose that both firms must formulate their strategies at the beginning, chosen paths for the game, and this strategy cannot be changed once the system evolves. The solution concept is the open-loop differential game in which firm i maximises its pay-off,, subject to dynamic quality constraint and given for each. 2-2) Open-Loop Equilibria: Open-loop strategy is defined as Definition2: The open-loop strategy space for firm i is: ={, ) } 13

14 It is a function of time and initial conditions. Two firms determine their strategies at the beginning of the game. Nash equilibrium for the game is a pair of open-loop strategies of both firms which characterizes by necessary and sufficient conditions correspond to the sense of open-loop solutions. At open-loop equilibrium, firms do not consider the impact of changes of quality on their strategy. The game starts at t=0 with initial condition y(0)= and u i (0)= (i=1,2). Let us characterize individually optimal path of R&D investment by applying Pontryagin Maximum Principle. For every given initial R&D investment of firm 2, firm 1 faces the problem of maximizing:, (( ) )- s.t. y(0)=, u 1 (0)= and given u 2 (0)= and similarly, For every given initial R&D investment of firm 1, firm 2 faces the problem of maximizing:, (( ) )- s.t. y(0)=, u 2 (0)= and given u 1 (0)= The Nash equilibrium in the market is a pair of open-loop strategies that simultaneously solves the two optimization problems. Definition3: Open-loop Nash equilibrium solution for a dynamic game is a pair of open-loop strategies X such that for every, where is the optimal open-loop strategy for firm i 4. To obtain optimal R&D investment, the corresponding Pontryagin function for firm 1,, and for firm 2,, are qualified as = (( ) ) ( ) 4 For more details see Reinganum 1982, Fershtman and Kamien 1987, Driskill and Mccafferty

15 = (( ) ) ( ) where with * + is the co-state variable associated to. shows marginal worth of the raise in distance between qualities at time t when moving along the optimal path of each firm. We expect that is positive. We apply necessary and transversality conditions in order to get open-loop strategy. The first order condition and the adjoint equation of Pontryagin function related to firm i is: i) = ii) Current value of Hamiltonian function is defined as: (, ) = where is obtained by the first order condition. The sufficient condition of the optimal path is transversality conditions, the role of terminal conditions on the adjoint variables, which are: iii) (, ) = 0, =0 From above, we immediately derive that optimal R&D investment for each firm is given by and, respectively: ( ( )) (2) ( ( )) (3) Lemma 1: Pay-off of each firm is concave in terms of quality and optimal R&D investment. Proof: See Appendix A. As our analyses show, Google and Bing face with different R&D path in the market,. If is an optimal control solution for firm i, then adjoint condition of the Pontryagin Maximum Principle declare the dynamic of co-state variable for firm i: (4) (5) with,, ( ). Equations (2) and (3) determine that depends on and. The dynamic optimal R&D is obtained by differentiating with respect to time t. Though takes the form: = (6) = (7) Dynamic optimisation of firm i s problem yields simultaneous solution of shown in proposition 1. and.this is 15

16 Proposition 1: Quality-R&D system for firm 1 is derived: {. /. / (8) and for firm 2 is derived: {. /. / (9) where. /, and Proof: See Appendix A.. In order to study qualitative behaviour of the game, we focus on stationary properties of the systems. The steady state curve for firm i is then the set of points and that satisfy. By imposing the stationary conditions to the dynamic systems (8) and (9), we have following: Lemma 2: There is steady state open-loop Nash equilibrium ( ) that is symmetric. = = (, ) = (0, ) (10) Proof: See Appendix A Firms optimal investment strategies are given by which are same for both firms at equilibrium,. It shows that at equilibrium, firm1 and firm2 symmetrically invest in R&D to maximise their pay-off. At steady state equilibrium, firms have identical qualities,. Since the steady state curve is linear, there is unique steady state which is symmetric. Steady state of R&D investment is positive according to the following conditions: Or D < 0 ( ) and. D > 0 ( ) and. We conduct analysis of stability to figure out under which circumstances the path of quality and R&D converg to stationary point over time. As regards of positive conditions, we analyse whether steady state is stable. Lemma 3: In duopoly model, steady state equilibrium qualifies as saddle point if. Proof: See Appendix A. 16

17 As our analyses establish, under condition of, symmetric steady state,, is a saddle. To contain comprehensive insights into the solution of the systems, we apply theoretical technique. Proposition 2: The solution of the system for firm 1 and firm 2 is: { { (12) where,, and are eigenvalues that are explained in proof of lemma 3.,, and are constant. Proof: See Appendix A Stable paths: We use simulation technique to understand the complexity of the game. Proof of lemma 3 in Appendix A determines that in the case of, systems have saddle steady state. The trajectories given by the eigenvectors of the negative eigenvalues are forward asymptotic to the saddle while the others are backward asymptotic. First, we assume parameter is negative which means the effect of stock of qualities on its growth is negative. Figure (4) and figure (5) depict phase diagram for firm1 and firm 2, respectively 5. The dotted vertical and horizontal lines separate the regions with negative R&D investment and/or negative quality from region with positive R&D investment and quality. We are interested in investigating the dynamic system in the region with positive quality and R&D investment 6. Therefore we eliminate to study the other regions. Point S specifies the saddle steady state. It is the intersection of loci and in figure (4) and intersection of and in figure (5). Worthwhile to express that in the line of or, for any value of y or on this line, y or cannot be change. The small lighter arrows indicate the direction of trajectories over time. Notice that according to initial conditions, firms converge or diverge from steady state. Stable paths are indicated by L 1 and L 2 for firm 1 and 2, respectively. L 1 is corresponding to initial conditions and and given 7. 5 Parameter values are =1, k=1.5, =-0.4, 1, = -0.9, =0.5, b=1, c=0.5, =1, r=0.1,f=1,. 6 We suppose firm1 has superior technology. Therefore should be positive. 7 The initial conditions are found by t=0 in system

18 Under these initial conditions dynamic system of firm1 converges to open-loop steady state equilibrium over time. L 2 is corresponding to initial conditions and and given 8. Under these initial values, dynamic system of firm 2 converges to open-loop steady state equilibrium. The lighter trajectories show unstable path. The small black arrows show directions of trajectories in each region. The comprehensive overview of the game under assumptions of and is described in figure (6). Equilibrium lines are indicated by black lines for firm 1 and red lines for firm 2. As our simulation declares, under above conditions, firm1 has higher initial R&D investment than firm 2. L1 has negative slope regards to R&D investment and L2 has positive one. R&D investment of firm1 decreases while R&D investment of firm 2 increases over time. 8 The initial conditions are found by t=0 in system

19 Consequently, distance between firms reduces and quality and R&D investment converge to equilibrium through time. The intuition behind these results is whether laggard can create a much better technology, and then asymmetries decrease. As we assumed, quality of each firm increases with its level of investment in innovation. Regarding the case ( ), we can consider that effect of stock of qualities on is negative,. Under this condition we characterise the behaviour of firm1 in the neighbourhood of the saddle steady state. Proposition 3: Characteristic of the phase diagram of firm 1 under assumptions of and : Region I: Region II: Region III: Proof: See Appendix A. Under, we can assume that stock of has positive impact on,. Based on this assumption, Proposition 4 summarizes direction of trajectories in each region for firm 2. Proposition 4: Characteristic of the phase diagram of firm 2 under assumptions of and. Region I: Region II: Region III: 19

20 Proof: See Appendix A. It is immediate to check the alternative case which is ; the effect of stock of quality on its growth is positive. In this respect, we conduct simulation of dynamic systems. Figures (7) and (8) show the dynamical behaviours of firm1 and firm 2, respectively 9. The dotted vertical line separates the region with negative quality. Point S indicates saddle steady state. The equilibrium lines are shown by black lines. and are stable paths which converge to steady state along time. is corresponding to the initial conditions, and given. is corresponding to the initial conditions, and given. The lighter trajectories show unstable paths. Figure (9) depicts complete overview of the game with. The lighter grey trajectories indicate unstable path for firm 1. The light pink trajectories indicate unstable path for firm 2. S is the saddle steady state equilibrium that is symmetric for firm 1 and firm 2. Equilibrium lines are indicated by black lines for firm 1 and red lines for firm 2. 9 Parameter values are =1, k=2, =0.1, 1, = -0.9, =0.5, b=1, c=0.5, =1, r=0.9,f=1,. 20

21 In the case of, firm 2 has higher initial R&D investment than firm1. As figure (9) shows, R&D investment of firm 2 decreases while R&D investment of firm1 increases over time and they meet at equilibrium point. As our simulation show, increasing investment by firm1 and decreasing investment by firm2 lead to decrease the distance amongst firms that does not support the main assumptions of the model. To remind that we have assumed firm1 has superior technology. To this end, we consider is negative. We can conclude that under condition of and appropriate initial conditions, asymmetries between search engines tend to decrease over time and systems converge to stationary equilibrium. Notably, optimal path must satisfy the necessary and transversality conditions of Pontryagin Principles. Lemma 4 clarifies optimal paths of game. Lemma 4: The stable paths (L 1, L 2 ) which satisfy all main assumptions of model, are optimal paths. Proof: See Appendix A. Our analyses explain that under assumption of, steady state is an unstable node. Regardless of initial values of state and control variables, all trajectories get away from stationary point, S''. It shows in figures (10) and (11) 10. In this case, firms can not reach steady state equilibrium and there is not any stable path. As a result, asymmetries between firms increase over time and firm1 with superior technology may become monopolist. We characterize behaviour of the systems (8) and (9) when there is an unstable equilibrium. It establishes by following propositions. Proposition 5: characteristic of firm 1 s phase diagram with unstable equilibrium and under assumption of : Region I: 10 Parameter values are =1, k=1.1, =-0.4, 1, = -0.9, =0.5, b=1, c=0.5, =1, r=0.1,f=1,. 21

22 Region 2: Region 3: Proof: See Appendix A Proposition 6: characteristic of firm 2 s phase diagram with unstable equilibrium and under assumption of : Region I: Region 2: Region 3: Proof: See Appendix A. 22

23 2-3) Comparative statics: We consider the case in which steady state is a saddle point. The purpose of this subsection is to examine how optimal R&D investment behaves in response to a change in some interested parameters in equilibrium. The theoretical method is based on taking appropriate derivatives and studies their signs. If we differentiate steady state equilibrium with respect to, we get: = (. / ) The sign of effect of on R&D investment at equilibrium distinguishes with the sign of.. / /. Since at equilibrium is positive 11, if. / dominates, then has positive impact on R&D equilibrium. The direct impact of distance in the levels of R&D investment across firms on is captured by parameter. Derivative of with respect to, we get = (. / ) By rising, equilibrium R&D investment increases when. / proposition states results.. The following Proposition 7: At symmetric equilibrium, if marginal profit with respect to quality,. /, is bigger than then parameters and have similar effect on equilibrium R&D investment. { } { } *(. / )+ Proof: See Appendix A. Parameter, discount rate, has negative effect on Pay-off over time. If we differentiate equilibrium R&D investment with respect to r, we get: = (. /) If. /, then. In this case, if discount rate increases, equilibrium R&D investment decreases. The other interested parameter is parameter of market share on profit. Derivative with respect to, we have: 11 From = f+ > 0 and, we get at steady state equilibrium. which captures the effect 23

24 iff. That means if the effect of R&D investment on is bigger than then rises when rises. 2-4) Main Results: Based on our analyses, under specific circumstances, asymmetries between firms decrease over time. Industry converges to the steady state and firms symmetrically invest in R&D. Otherwise increasing dominance with monopolization is possible. 3) Dynamic Optimization for Monopoly: One of the considerable features of the search engine market is high levels of concentration. Google boasts an impressive market share in United States and western countries for more than a decade. According to Statista 12, Google has 81.42% share of desktop searches in United States, 86.27% in United Kingdom, 90.24% in Germany, 92.92% in France, 94.03% in Spain and 93.11% in Italy as of October Google with new smart results, prominent features and high usability is evolving into dominance over the market. Due to the massive investment in the search infrastructures, Google has better quality than its rivals. Advanced technology of Google enabled it to process consumer s query rapidly. As for example, Google instant is a feature that allows Google to predict query and show results when users type which leads to have faster searchers and smarter prediction. All these factors afford Google to give better services to consumers. As our analysis in duopoly model shows, market may evolve into monopoly. In this section, first we describe the basic assumptions of the monopoly model. A dynamic differential game is developed in order to obtain equilibrium path. We study whether R&D investment and quality converge to steady state equilibrium. Then we conduct comparative statics of equilibrium when value of some interested parameters changes. Second, we study the implications of welfare in search engine market. 3-1) Model: Having established evidence for the dominant position of Google, we present continuoustime differential game in order to analyse optimal control problem for monopolist. Suppose consumers are uniformly distributed on the segment [0,1] and there is only one firm in the market which provides search online services. Quality of services at period t

25 is positive. Search engine does not charge consumers for using its services. The utility that a given consumer derives from using search engine is a linear function of quality of services which is: = with > 0 and 0. Market share 13 of search engine is a linear function of utility of consumers which is given by: = ( ) To simplify we assume, = Search quality evolves over time with its level of investment in innovation, movement is governed by following equation: = +k. Quality (13) where. By this assumption we are imposing that the level of R&D investment decreases by improving quality in the long-run. is the direct impact of stock of quality on quality growth over time. Firm obtains some revenues from online advertising. Suppose that online advertising revenue is strictly concave function of market share. A higher market share means a higher number of queries from which it can bring more online advertising profits. Monopolistic search engine s profit is given by: = with > 0 and. We can rewrite as: = m + where m=, =. Firm invests the amount,, in R&D so as to improve its search algorithm. We assume that the marginal cost of investing in R&D decreases in the level of quality achieved by the algorithm, which in turn depends on the total number of received queries and on the amount of investment. Investment cost function of monopolist is a strictly convex function of and is given by: with = ( ) The pay-off is defined as: 13 Market share can be bigger than one but does not go infinity because of R&D investment cost. 25

26 The optimal control for monopolist is to find control that maximise the present value of payoff function, subject to the dynamic quality and control constraint which must hold all the time, 0, and initial condition for quality, y(0)=y 0, t 0. Max u dt s.t. = +k and u(t) y(0)=y 0 where, ) is the firm s discount parameter. 3-2) Optimal Solutions: In order to solve the dynamic optimization problem, we form the Pontryagin function: = m + (( ) ) + ( +k ) is the co-state variable in current value. It measures the marginal worth of raise in quality at time t when moving along the optimal path. should be positive. Pontryagine s maximum principles state: i) = ii) = - iii) H(y, λ) = 0 and y=0 Current value of Hamiltonian function is defined as: H * (, ) =. Adjoint condition and transversality conditions capture the dynamical behaviour of market between beginning and infinite planning horizon. The following lemma establishes the optimal R&D investment. Lemma 5: Optimal R&D investment for monopolist is given by terms of quality and R&D investment. = ( ( )) Proof: See Appendix B. and payoff is concave in (14) If is an optimal control solution then the second condition of the Pontryagin Maximum Principle declares the dynamic marginal worth of the raise in quality: = (15) where,. /, 26. Equation (14) explains that depends on quality and co-state variable. Time derivative of optimal R&D,, takes the form: