ECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION

Transcription

1 ECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION Lecture 12: Dynamic Games of Oligopoly Competition Victor Aguirregabiria (University of Toronto) Toronto. Winter 2018 Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

2 Empirical dynamic games provide a framework to estimate these parameters Victor Aguirregabiria and perform () policy analysis. Empirical IO Toronto. Winter / 144 Introduction Dynamic Games: Introduction In oligopoly industries, firms compete in investment decisions that have returns in the future, involve substantial uncertainty, and can have important effects on competitors profits. - Investment in R&D, innovation. - Investment in capacity, physical capital. - Product design / quality - Market entry / exit... Measuring and understanding the dynamic strategic interactions between firms decisions (e.g., dynamic complementarity or substitutability) is important to understand the forces behind the dynamics of an industry or to evaluate policies. Investment costs, uncertainty play, and competition effects play an important role in these decisions. Structural estimation of these parameters is necessary for some empirical questions.

3 Examples of Empirical Applications Examples of Empirical Applications Competition in R&D and product innovation between Intel and AMD: Goettler and Gordon (JPE, 2011). Product innovation of incumbents and new entrants in the hard drive industry: Igami (JPE, 2017). Land use regulation (entry cost) and entry-exit and competition in the hotel industry: Suzuki (IER; 2013). Environmental regulation and entry-exit and capacity choice in cement industry: Ryan (ECMA, 2012). Subsidies to entry in small markets for the dentist industry: Dunne et al. (RAND, 2013); Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

4 Examples of Empirical Applications Examples of Empirical Applications [2] Fees for musical performance rights and the choice of format (product design) of radio stations: Sweeting (ECMA, 2013). Hub-and-spoke networks, route entry-exit and competition in the airline industry: Aguirregabiria and Ho (JoE, 2012). Dynamic price competition: Kano (IHIO, 2013); Ellickson, Misra, and Nair (JMR, 2012). Cannibalization and preemption strategies in the Canadian fast-food industry: Igami and Yang (QE, 2016). Demand uncertainty and firm investment in the concrete industry: Collard-Wexler (ECMA, 2013); Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

5 Examples of Empirical Applications Examples of Empirical Applications [3] Time-to-build, uncertainty, and dynamic competition in the shipping industry: Kalouptsidi (AER, 2014). Endogenous mergers: Jeziorski (RAND, 2014). Exploitation of a common natural resource (fishing): Huang and Smith (AER, 2014). Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

6 Outline Dynamic Games in oligopoly industries 1. Structure of empirical dynamic games 2. Data and Identification 3. Estimation 4. Dealing with unobserved heterogeneity 5. Environmenal Regulation and Competition: Ryan (2012) 6. Dynamic Product Positioting: Sweeting (2013) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

7 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Dynamic Games: Basic Structure Time is discrete and indexed by t. The game is played by N firms [potential entrants] that we index by i. Following the standard structure in the Ericson-Pakes (1995) framework, firms compete in two different dimensions: a static dimension and a dynamic dimension. We denote the dynamic dimension as the "investment decision". Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

8 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Dynamic Games: Basic Structure (2) Let a it be the variable that represents the investment decision of firm i at period t. This investment decision can be an entry/exit decision, a choice of capacity, investment in equipment, R&D, product quality, other product characteristics, etc. Every period, firms observed the state variables (e.g., their capital stocks) and compete in prices or quantities in a static Cournot or Bertand model. Let p it be the static decision variables (e.g., price) of firm i at period t. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

9 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Dynamic Games: Basic Structure (3) I start presenting a simple dynamic game of market entry-exit and "quality" choice where every period incumbent firms compete a la Bertrand. The dynamic investment decision a it {0, 1,..., J} represents the quality choice if a it > 0, and a it = 0 if the firm is not active in the market at period t. The action is taken to maximize the expected and discounted flow of profits in the market, E t ( r =0 δ r Π it+r ) where δ (0, 1) is the discount factor, and Π it is firm i s profit at period t. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

10 Structure of dynamic games of oligopoly competition Profit function Basic Framework and Assumptions The profits of firm i at time t are given by Π it = VP it FC it EC it IC it + SV it where: VP it represents variable profit; FC it is the fixed cost of operating; EC it is a one time entry cost IC it is an investment cost SV it is the exit value of scrap value Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

11 Structure of dynamic games of oligopoly competition Variable profit function Basic Framework and Assumptions The variable profit VP it is an "indirect" variable profit function that comes from the equilibrium of a static Bertrand game with differentiated product. Suppose that firm i has a marginal cost c i (a it, z t ), where z t is the a vector of exogenous state variables, and produces a product with quality v i (a it, z t ). Consumer utility of buying product i is u it = ν i (a it, z t ) α(z t ) p it + ε it, where ν i (.) and α(.) are functions, and ε it is a consumer-specific i.i.d. extreme value type 1 random variable. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

12 Structure of dynamic games of oligopoly competition Variable profit function (2) Basic Framework and Assumptions Under these conditions, the equilibrium variable profit of an active firm depends only on the number of firms active in the market. VP it = (p it c i (a it, z t )) q it where p it and q it represent the price and the quantity sold by firm i at period t, respectively. According this model, the quantity is: q it = H t 1{a it > 0} exp{v i (a it, z t ) α(z t ) p it } 1 + N j=1 1{a jt > 0} exp{v j (a jt, z t ) α(z t ) p jt } = H t s it where H t is the number of consumers in the market (market size) and s it is the market share of firm i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

13 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Variable profit function (3) Under the Nash-Bertrand assumption the first order conditions for profit maximization are: or q it + (p it c i (a it, z t )) ( α(z t )) q it (1 s it ) = 0 p it = c i (a it, z t ) + 1 α(z t ) (1 s it ) These N equations define a Bertrand equilibrium with prices pt = (p1t, p 2t,..., p Nt ) and market shares: s it = 1{a it > 0} exp{v i (a it, z t ) α(z t ) p it } 1 + N j=1 1{a jt > 0} exp{v j (a jt, z t ) α(z t ) p jt } Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

14 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Model: Variable profit function (4) It is simple to show that an equilibrium always exists. Equilibrium prices depend on the vector of product qualities of the active firms in the market (a t ), and on the exogenous variables z t : p it = p i (a t, z t ). Similarly, the equilibrium market shares s it is a function of (a t, z t ): s it = s i (a t, z t ). Therefore, the indirect or equilibrium variable profit of an active firm is: VP it = a it H t (p i (a t, z t ) c i (z t )) s i (a t, z t ) = a it H t θ VP i (a t, z t ) where θ VP i (.) is a function that represents variable profits per capita. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

15 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Model: Variable profit function (5) For most of the analysis below, I will consider that the researcher does not have access to information on prices and quantities. Therefore, we will treat {θ VP i (a t, z t )}, for each possible value of (a t, z t ), as parameters to estimate from the structural dynamic game. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

16 Structure of dynamic games of oligopoly competition Fixed cost Basic Framework and Assumptions The fixed cost is paid every period that the firm is active in the market, and it has the following structure (mode of entry-exit): [ ] FC it = 1{a it > 0} θ FC i (a it, z t ) + ε FC it (a it ) θ FC i (a it, z t ) is a function that represents the fixed operating cost of firm i if it produces a product with quality a it. z t is a vector of exogenous state variables that are common knowledge to all the firms. ε FC it (a it ) are a zero-mean shocks that is private information of firm i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

17 Structure of dynamic games of oligopoly competition Fixed cost (2) Basic Framework and Assumptions There are two main reasons why we incorporate these private information shocks in the model. First, as shown in Doraszelski and Satterthwaite (2012), it is a way to guarantee that the dynamic game has at least one equilibrium in pure strategies. Second, they are convenient econometric errors. If private information shocks are independent over time and over players, and unobserved to the researcher, they can explain players heterogeneous behavior without generating endogeneity problems. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

18 Structure of dynamic games of oligopoly competition Entry cost Basic Framework and Assumptions The entry cost is paid only if the firm was not active in the market at previous period (entry-exit model): [ ] EC it = 1{a it > 0 & a it 1 = 0} θ EC i (a it, z t ) + ε EC it (a it ) θ EC i (a it, z t ) is a function that represents the entry cost of firm i if the initial product quality is a it. ε EC it (a it ) are private information shocks in the entry cost Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

19 Structure of dynamic games of oligopoly competition Investment cost Basic Framework and Assumptions There are also costs of adjusting the level of quality, or repositioning product characteristics. For instance, ( AC (+) AC ( ) IC it = 1{a it 1 > 0} θ (z t ) 1{a it > a it 1 } + θ (z t ) {a it < a it AC (+) i AC ( ) θi (z t ) and θi (z t ) represents the costs of increasing and reducing quality, respectively, once the firm is active. In this specification the adjustment costs are lump-sum. We could consider more flexible specifications with (asymmetric) linear, quadratic, and lump-sum ACs. i Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

20 Structure of dynamic games of oligopoly competition State variables Basic Framework and Assumptions The payoff relevant state variables of this model are: (1) the exogenous state variables affecting demand and costs, z t, and market size H t. For notational simplicity, we represent them in the vector z t (2) the previous qualities of all the firms a t 1 {a it 1 : i = 1, 2,..., N}; (3) the private information shocks {ε it : i = 1, 2,..., N}. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

21 Structure of dynamic games of oligopoly competition State variables (2) Basic Framework and Assumptions The specification of the model is completed with the transition rules of these state variables. (1) Exogenous state variables follow an exogenous Markov process with transition probability function F z (z t+1 z t ). (2) The transition of the qualitiy choices is trivial in this model. We could extend it to stochastic evolution. However, note that future returns of investment in quality is uncertain. (3) Private information shock ε it is i.i.d. over time and independent across firms with CDF G i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

22 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Timing of decisions and state variables In this example, firms dynamic decisions are made at the beginning of period t and they are effective during the same period. An alternative timing that has been considered in many applications is that there is a one-period time-to-build, i.e., the decision is made at period t, and entry costs are paid at period t, but the firm is not active in the market until period t + 1. This is in fact the timing of decisions in Ericson and Pakes (1995). All the results below can be easily generalized to this model with time-to-build. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

23 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium Markov Perfect Equilibrium Most of the recent literature in IO studying industry dynamics focuses on studying a Markov Perfect Equilibrium (MPE), as defined by Maskin and Tirole (Econometrica, 1988). The key assumption in this solution concept is that players strategies are functions of only payoff-relevant state variables. In this model, the payoff-relevant state variables for firm i are (y t, z t, ε it ). We use x t to represent the vector of common knowledge state variables, i.e., x t (y t, z t ). Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

24 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (2) Markov Perfect Equilibrium Let α = {α i (x t, ε it ) : i {1, 2,..., N}} be a set of strategy functions, one for each firm. A MPE is a set of strategy functions α such that every firm is maximizing its value given the strategies of the other players. For given strategies of the other firms, the decision problem of a firm is a single-agent dynamic programming (DP) problem. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

25 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (3) Markov Perfect Equilibrium Let V α i (x t, ε it ) be the value function of the DP problem that describes the best response of firm i to the strategies α i of the other firms. This value function is the unique solution to the Bellman equation: Πi α(a it, x t ) ε it (a it ) Vi α (x t, ε it ) = max a it +δ Vi α (x t+1, ε it+1 ) dg i (ε it+1 ) Fi α (x t+1 a it, x t ) where Π α i (a it, x t ) and F α i (x t+1 a it, x t ) are the expected one-period profit and the expected transition of the state variables, respectively, for firm i given the strategies of the other firms. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

26 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (4) Markov Perfect Equilibrium For the quality choice game, the expected one-period profit Πi α(a it, x t ) is: [ ] Πi α(a it, x t ) = H t Pr (α i (x t, ε it ) = a it x t ) θ VP i (a it, a it, z t ) a it [ θ FC i ] (a it, z t ) + (1 a it 1 ) θ EC i (a it, z t ) And the expected transition of the state variables is: F α i (x t+1 a it, x t ) = 1{y it+1 = a it } F z (z t+1 z t ) [ Pr (α j (x t, ε jt ) = y j,t+1 x t ) j =i ] Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

27 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (5) Markov Perfect Equilibrium A firm s best response function gives his optimal strategy if the other firms behave, now and in the future, according to their respective strategies. In this model, the best response function of player i is: αi (x t, ε it ) = arg max {v a i α (a it, x t ) ε it (a it )} it v α i (a it, x t ) is the conditional choice value function that represents the value of firm i if: (1) the other firms behave according to their strategies in α; and (2) the firm chooses alternative a it today and then behaves optimally forever in the future. vi α (a it, x t ) Πi α (a it, x t ) + δ V α i (x t+1, ε it+1 ) dg i (ε it+1 ) F α i (x t+1 a it, x t ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

28 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (6) Markov Perfect Equilibrium A Markov perfect equilibrium (MPE) in this game is a set of strategy functions α such that for any player i and for any (x t, ε it ) we have that: { } αi (x t, ε it ) = arg max v α a i (a it, x t ) ε it (a it ) it Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

29 Structure of dynamic games of oligopoly competition Conditional Choice Probabilities Conditional Choice Probabilities Given a strategy function α i (x t, ε it ), we can define the corresponding Conditional Choice Probability (CCP) function as : P i (a x) Pr (α i (x t, ε it ) = a x t = x) = 1{α i (x t, ε it ) = a} dg i (ε it ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

30 Structure of dynamic games of oligopoly competition Conditional Choice Probabilities Conditional Choice Probabilities (2) Since choice probabilities are integrated over the continuous variables in ε it, they are lower dimensional objects than the strategies α. For instance, when both a it and x t are discrete, CCPs can be described as vectors in a finite dimensional Euclidean space. In our entry-exit model, P i (1 x t ) is the probability that firm i is active in the market given the state x t. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

31 Structure of dynamic games of oligopoly competition Conditional Choice Probabilities Conditional Choice Probabilities (3) Under the additivity and iid assumptions on the ε it, it is possible to show that there is a one-to-one relationship between strategy functions α i (x t, ε it ) and CCP functions P i (a x t ). We also use Π P i and Fi P instead of Πi α and Fi α to represent the expected profit function and the transition probability function, respectively. From now on, we use CCPs to represent players strategies, and use the terms strategy and CCP as interchangeable. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

32 Structure of dynamic games of oligopoly competition MPE in terms of CCPs Conditional Choice Probabilities Based on the concept of CCP, we can represent the equilibrium mapping and a MPE in way that is particularly useful for the econometric analysis. This representation has two main features: (1) a MPE is a vector of CCPs; (2) a player s best response is an optimal response not only to the other players strategies but also to his own strategy in the future. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

33 Structure of dynamic games of oligopoly competition MPE in terms of CCPs (2) Conditional Choice Probabilities A MPE is a vector of CCPs, P {P i (a x) : for any (i, a, x)}, such that: ( { } ) P i (a x) = Pr a = arg max v P a i (a i, x) ε i (a i ) x i vi P (a i, x) is a conditional choice probability function, but it has a slightly different definition that before. Now, vi P (a i, x) represents the value of firm i if the firm chooses alternative a i today and all the firms, including firm i, behave according to their respective CCPs in P. The Representation Lemma in Aguirregabiria and Mira (2007) shows that every MPE in this dynamic game can be represented using this mapping. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

34 Structure of dynamic games of oligopoly competition MPE in terms of CCPs (3) Conditional Choice Probabilities The form of this equilibrium mapping depends on the distribution of ε i. For instance, in the entry/exit model, if ε i is N(0, σ 2 ε ): ( v P P i (1 x) = Φ i (1, x) vi P ) (0, x) σ ε In the model with endogenous quality choice, if ε i (a) s are extreme value type 1 distributed: { v P } exp i (a, x) σ P i (a x) = ε { v a P a =0 exp i (a }, x) σ ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

35 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs Since vi P is not based on the optimal behavior of firm i in the future, but just in an arbitrary behavior described by P i (..), calculating vi P does not require solving a DP problem, and it only implies a valuation exercise. By definition: v P i (a i, x) = Π P i (a i, x) + δ x Vi P (x ) Fi P (x a i, x) where Π P i (a i, x) is the expected current profit. In the entry/exit example: [ ] Π P i (a i, x) = a i H Pr (a i x, P i ) θ VP i (a i ) θ FC i (1 x i )θ EC i a i = z P i (a i, x) θ i Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

36 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs where θ i is the vector of parameters: ( θ i = θ VP (a i ) for any a i, θ FC i ), θ EC i and z P i (a i, x) is the vector that depends only on the state x and on the CCPs at state x, but not on structural parameters. z P i (1, x) = (H Pr (a i x,p i ) for any a i, 1, (1 x i )) z P i (0, x) = (0, 0,..., 0) In the dynamic game with endogenous quality choice, we also have Π P i (a i, x) = z P i (a i, x) θ i Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

37 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs The value function Vi P (.) represents the value of firm i if all the firms, including firm i, behave according to their CCPs in P. We can obtain V P i V P i (x) = as the unique solution of the recursive expression: [ ] a P i (a i x) a i =0 z P i (a i, x) θ i + δ Vi P (x ) F P (x a i, x) x When the space X is discrete we can obtain Vi P as the solution of a system of linear equations: ] ] V P i = [ a P i (a i ) z P i (a i ) a i =0 = z P i θ i + δ F P V P i θ i + δ [ a a i =0 P i (a i ) F P i (a i ) V P i Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

38 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs Then, solving for V P i, we have: V P i = ( I δ F P) 1 z P i θ i = W P i θ i ( where Wi P = I δ F P) 1 z P i is a matrix that only depends on CCPs and transition probabilities but not on θ. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

39 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs Solving these expressions into the formula for the conditional choice value function, we have that: v P i (a i, x) = z P i (a i, x) θ i where: z P i (a i, x) = z P i (a i, x) + δ x F P i (x a i, x) W P i Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

40 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs The best response mapping in the space of CCPs becomes: ( { } ) P i (a x) = Pr a = arg max z P a i (a i, x) θ i ε i (a i ) x i For the entry/exit model with ε i N(0, σ 2 ε ): ( P i (1 x) = Φ [ z i P (1, x) z i P (0, x) ] θi σ ε ) In the model with endogenous quality choice with ε i (a) s extreme value type 1 distributed: { exp z i P (a, x) θ } i σ P i (a x) = { ε a a =0 exp z i P (a, x) θ } i σ ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

41 Data Data, Identification, and Estimation Data The researcher observes a random sample of M markets, indexed by m, over T periods of time, where the observed variables consists of players actions and state variables. For the moment, we consider that the industry and the data are such that: (a) each firm is observed making decisions in every of the M markets; (b) the researcher knows all the payoff relevant market characteristics that are common knowledge to the firms, x. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

42 Data Data, Identification, and Estimation Data With this type of data we can allow for rich firm heterogeneity that is fixed across markets and time by estimating firm-specific structural parameters, θ i. This fixed-effect approach to deal with firm heterogeneity is not feasible in data sets where most of the competitors can be characterized as local players, i.e., firms specialized in operating in a few markets. Condition (b) rules out the existence of unobserved market heterogeneity. Though it is a convenient assumption, it is also unrealistic for most applications in empirical IO. Later I present estimation methods that relax conditions (a) and (b) and deal with unobserved market and firm heterogeneity. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

43 Data Data, Identification, and Estimation Data Suppose that we have a random sample of M local markets, indexed by m, over T periods of time, where we observe: Data = {a mt, x mt : m = 1, 2,..., M; t = 1, 2,..., T } We want to use these data to estimate the model parameters in the population that has generated this data: θ 0 = {θ 0 i : i I }. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

44 Identification Data, Identification, and Estimation Identification A significant part of this literature has considered the following identification assumptions. Assumption (ID 1): Single equilibrium in the data. Every observation in the sample comes from the same Markov Perfect Equilibrium, i.e., for any observation (m, t), P 0 mt = P 0. Assumption (ID 2): No unobserved common-knowledge variables. The only unobservables for the econometrician are the private information shocks ε imt and the structural parameters θ. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

45 Data, Identification, and Estimation Identification (2) Identification First, let s summarize the structure of the dynamic game of oligopoly competition. Let θ be the vector of structural parameters of the model, where θ = {θ i : i = 1, 2,..., N}. Let P(θ) = {P i (a x, θ) : for any (i, a, x)} be a MPE of the model associated with θ. P(θ) is a solution to the following equilibrium mapping: for any (i, a i, x): { exp z i P (a i, x) θ } i σ P i (a i x,θ) = { ε a a =0 exp z i P (a, x) θ } i σ ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

46 Data, Identification, and Estimation Identification (3) Identification where the vector of values z P i (a, x) are z P i (a i, x) = z P i (a i, x) + δ x F P i (x a i, x) W P i and W P i = W P i = ( I δ F P) 1 z P i, and z P i (a i, x) is a vector with the different components of the current expected profit. For instance, in the entry-exit example: z P i (0, x) = (0, 0, 0,...0) z P i (1, x) = (H Pr (a i x,p i ) for any a i, 1, (1 x i )) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

47 Data, Identification, and Estimation Identification (4) Identification I will use the function Ψ i (a i, x; P, θ) to represent{ the best response or exp z i P (a i, x) θ } i σ ε equilibrium function that in our example is { a a =0 exp z i P (a, x) θ }. i σ ε Then, we can represent in a compact form a MPE as: P = Ψ(P, θ) where Ψ(P, θ) = {Ψ i (a i, x; P, θ) : for any (i, a, x)}. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

48 Data, Identification, and Estimation Identification (5) Identification Under assumptions ID-1 & ID-2, the equilibrium that has generated the data, P 0, can be estimated consistently and nonparametrically from the data. For any (i, a i, x): P 0 i (a i x) = Pr(a imt = a i x mt = x) For instance, we can estimate consistently Pi 0 (a i x) using the following simple kernel estimator: ( ) xmt x m,t 1{a imt = a i } K Pi 0 b (a i x) = ( ) n xmt x m,t K b n Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

49 Data, Identification, and Estimation Identification (6) Identification Second, given that P 0 is identified, we can identify also the expected present values z i P0 (a i, x) at the "true" equilibrium in the population. Third, we know that P 0 is an equilibrium associated to θ 0. Therefore, the following equilibrium conditions should hold: for any (i, a i, x), { } exp z Pi 0 i P0 (a i, x) θ0 i σ (a i x) = 0 ε { } a a =0 exp z i P0 (a, x) θ0 i σ 0 ε If there the components of z i P0 (a i, x) are not collinear, then these equilibrium conditions identify θ0 i σ 0. ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

50 Data, Identification, and Estimation Identification (7) Identification For instance, in this logit example, we have that for (i, a i, x), ( P 0 ) ln i (a i x) [ ] θ 0 Pi 0 = z i P0 (a i, x) z i P0 (0, x) i (0 x) Define Y imt ln Then, ( ) P 0 i (a imt x mt ) Pi 0 (0 x mt ) and Z imt z i P0 (a imt, x mt ) z i P0 (0, x mt ). Y imt = Z imt θ 0 i σ 0 ε σ 0 ε And we can also write this system as, E (Z imt Y imt) = E (Z imt Z imt) θ0 i Under assumption ID-3: σ 0 ε. θ 0 i σ 0 ε = E (Z imtz imt ) 1 E (Z imty imt ) θ 0 Victor i Aguirregabiria () Empirical IO Toronto. Winter / 144

51 Data, Identification, and Estimation Identification (7) Identification Note that under the single-equilibrium-in-the-data assumption, the multiplicity of equilibria in the model does not play any role in the identification of the structural parameters. The single-equilibrium-in-the-data assumption is a suffi cient for identification but it is not necessary. Sweeting (2013) and Aguirregabiria and Mira (2012) present conditions for the point-identification of games of incomplete information when there are multiple equilibria in the data. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

52 Estimation Estimation I will describe the following estimators: 1. Two-step estimators with moment equalities 2. Two-step estimators with moment inequalities 3. Recursive K-step and NPL Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

53 Estimation (2) Estimation For the sake of concreteness, we consider the binary choice entry-exit game, where ε it (1) ε it (0) is iid N(0, σ 2 i ). The equilibrium mapping is: P i (a i x, θ i ) = Φ ( ] ) [ z i P (1, x) z i P θi (0, x) σ i For notational simplicity, we use θ i to represent θ i σ i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

54 Estimation Pseudo Likelihood Function For the description of the different estimators, it is convenient to define the following Pseudo Likelihood function: Q(θ, P) = M N T a imt m=1 i=1 t=1 + (1 a imt ) ln ([ ] ln Φ z i P (1, x mt ) z i P (0, x mt ) [ ([ ] 1 Φ z i P (1, x mt ) z i P (0, x mt ) θ i ) θ i )] This pseudo likelihood function treats firms beliefs P as parameters to estimate together with θ. Note that for given P, the function Q(θ, P) is the likelihood of a Probit model. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

55 Two-step methods Estimation Two-step methods Suppose that we knew the equilibrium in the population, P 0. Given P 0 we can construct the variables z i P0 (1, x mt ) z i P0 (0, x mt ) and then obtain a very simple estimator of θ 0. ^θ = arg max θ Q(θ, P 0 ) This estimator is root-m consistent and asymptotically normal under the standard regularity conditions. It is not effi cient because it does not impose the equilibrium constraints (only asymptotically). While equilibrium probabilities are not unique functions of structural parameters, the best response probabilities that appear in Q(θ, P) are unique functions of structural parameters and players beliefs. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

56 Estimation Two-step methods (2) Two-step methods The previous method is infeasible because P 0 is unknown. However, under the Assumptions "No-unobserved-market-heterogeneity" and "One-MPE-in-the-data" we can estimate P 0 consistently and at with a convergence rate such that the two-step estimator ^θ is root-m consistent and asymptotically normal. For instance, a kernel estimator of P 0 is: ( ) M P i 0 m=1 T xmt x t=1 a imt K b (x) = ( ) M m=1 T xmt x t=1 K b Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

57 Estimation Two-step methods Two-step methods: Finite sample properties (1) The most attractive feature of two-step methods is their relative simplicity. However, they suffer of a potentially important problem of finite sample bias. The finite sample bias of the two-step estimator of θ 0 depends very importantly on the properties of the first-step estimator of P 0. In particular, it depends on the rate of convergence and on the variance and bias of P 0. It is well-known that there is a curse of dimensionality in the NP estimation of a regression function such as P 0. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

58 Estimation Two-step methods Two-step methods: Finite sample properties (2) In particular, the rate of convergence of P 0 declines, and the variance and bias increase, very quickly as the number of conditioning regressors increases. In our simple example, the vector x mt contains only three variables: the binary indicators a im,t 1 and the (continuous) market size S mt. In this case, the NP estimator of P 0 has a relatively high rate of convergence an its variance and bias can be small even with relatively small sample. However, there are applications with more than two (heterogeneous) players and where firm size, capital stock or other predetermined continuos firm-specific characteristics are state variables. Even with binary state variables (a im,t 1 ), when the number of players is relatively large (e.g., more than 10)... Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

59 Estimation Recursive K-step estimator Recursive K-step estimator K-step extension of the 2-step estimator. Given an initial consistent (NP) estimator P 0, the sequence of estimators { θ K, P K : K 1} is defined as: θ K +1 = arg max Q (θ, P ) K θ where: ([ P i K (x) = Φ ] ) z P K 1 i (1, x) z P K 1 i (0, x) θk i Aguirregabiria and Mira (2002, 2007) present Monte Carlo experiments which illustrate how this recursive estimators can have significantly smaller bias than the two-step estimator. Kasahara and Shimotsu (2008) derive a second order approximation to the bias of these K-stage estimators. They show that, if the equilibrium in the population is stable, then this recursive procedure reduces the bias. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

60 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (BBL) Remember that: z P i (a i, x mt ) θ i is the value of choosing alternative a i today, given that firms behave in the future according to the probabilities in P. Then, the value V P imt is: V P imt = (1 P i (x mt )) [ z P i (0, x mt ) θ i ] + Pi (x mt ) [ z P i (1, x mt ) θ i ] = W P imt θ i where W P imt = (1 P i (x mt )) z P i (0, x mt ) + P i (x mt ) z P i (1, x mt ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

61 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (2) Let s split the vector of choice probabilities P into the sub-vectors P i and P i, P (P i, P i ) where P i are the probabilities associated to player i and P i contains the probabilities of players other than i. P 0 is an equilibrium associated to θ 0. Therefore, P 0 i is firm i s best response to P 0 i. This implies that for any P i = P 0 i the following inequality should hold: W (P0 i,p0 i) imt θ 0 i W (P i,p 0 i) imt θ 0 i We can define an estimator of θ 0 based on these (moment) inequalities. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

62 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (3) There are infinite alternative policies P i, and therefore there are infinite moment inequalities. For estimation, we should select a finite set of alternative policies. This is a very important decision for this class of estimators (more below). Let H be a (finite) set of alternative policies for each player. Define the following criterion function: R ( θ, P 0, H ) ( { min 0 ; i,m,t P H [ W (P0 i,p0 i) imt ] }) 2 W (P i,p 0 i) imt θ i This criterion function penalizes departures from the inequalities. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

63 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (4) Then, given an initial NP estimator of P 0, say ^P 0, we can define the following estimator of θ 0 based on moment inequalities (MI): ^θ = arg min θ i,m,t ( { P H min 0 ; [ W (^P 0 i,^p 0 i) imt ] }) 2 W (P i,^p 0 i) imt θ i There are several relevant comments to make on this MI estimator: 1 Point identification / Set identification 2 Properties (relative to two-step estimators using moment equalities) 3 Continuous dependent variables Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

64 Estimation Estimation using Moment Inequalities MI Estimator: Point / Set identification This estimator is based on exactly the same assumptions as the 2-step moment equalities (ME) estimator. We have seen that θ 0 is point identified by the moment equalities of the ME estimators (e.g., by the pseudo likelihood equations). Therefore, if the set H of alternative policies is large enough, then θ 0 is point identified as the unique maximizer of R ( θ, P 0, H ). However, it is very costly to consider a set H with many alternative policies. For the type of sets H which are considered in practice, R ( θ, P 0, H ) does not have a unique maximizes and therefore θ 0 is set identified. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

65 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML Why to use an estimator that only set-identifies θ 0 when we have alternative estimators which point identified θ 0? The Moment Inequalities (MI) estimator should have other advantages. Let s start examining which ARE NOT the advantages. The MI estimator is not more robust than the 2S-PML estimator. Both estimators are based on exactly the same model and assumptions. Asymptotically, the MI estimator is less effi cient than the 2S-PML estimator. The effi cient 2S-PML has lower asymptotic variance than the MI estimator, even as the set H becomes very large. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

66 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML Computationally, the MI estimator is more costly than the 2S-PML estimator. The main computational cost of implementing the MI and ME estimators comes from obtaining the vectors of values {Wimt P }. The 2S-PML estimators compute {W P imt } only once: at the estimated W ^P 0 imt. Instead, the MI estimator has to calculate also W (P i,^p 0 i) imt at the different alternative policies in H. Furthermore, the MI estimator needs an algorithm for set optimization. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

67 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML There are two potential advantages of MI against 2S-PML estimator: 1 finite sample bias; 2 estimation of models with continuous decision variables. In terms of finite sample bias, the MI estimator may have lower bias than ME estimators. Exploiting the information in the alternative policies might be useful to reduce the bias. However, there is no evidence on this point. This may dependent very much on the choice of the set H. Typically, H contains only a few alternative policies (e.g., 5, 10, 20). The selection of these alternative policies should be very careful, and there should be some intuition of how the inequalities associated with an alternative policy can help to identify a particular parameter or group of parameters. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

68 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML BBL show that, when combined with simulation techniques to approximate the values {Wimt P }, this method can be easily extended to the estimation of dynamic games with continuous decision variables. In fact, the BBL estimator of a model with continuous decision variable is basically the same as with a discrete decision variable. The ME estimator of models with continuous decision variable may be more complicated. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

69 Dealing with the curse of dimensionality Dealing with the curse of dimensionality Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

70 Dealing with the curse of dimensionality Simulation-Based Estimation (1) Simulation-based estimation Though two-step methods (with either PML or MI) are computationally much cheaper than full solution-estimation methods, they are still impractical for applications where the dimension of the state space X is very large, e.g., a discrete state space with millions of points or a model in which some of the observable state variables are continuous. To deal with this problem, Hotz, Miller, Sanders and Smith (REStud, 1994) proposed an estimator that uses simulation techniques to approximate the values z P imt (a i ) or similarly the vector of values W P imt. In the context of dynamic games, Bajari, Benkard and Levin (BBL) have proposed to used this simulation and have extended it to models with continuous decision variables. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

71 Dealing with the curse of dimensionality Simulation-Based Estimation (2) Simulation-based estimation Remember that: z P imt (a i ) a i z P imt + E ( s=1 β s a im,t+s z Pim,t+s (1) x mt, a imt = a i ) The expectations E (.) are taken over all the possible future paths of actions and state variables conditions on (x mt, a imt = a i ) and conditional on future behavior P. The simulator of z P imt (a i ) are obtained by replacing the true expectations E (.) by Monte Carlo approximations to these expectations. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

72 Dealing with the curse of dimensionality Simulation-Based Estimation (3) Simulation-based estimation For every value of x mt in the sample and every choice alternative a i (in the sample or not), we consider (a i, x mt ) as the initial state for player i and then we use the probabilities in P, and the transition probabilities of x, to generate R simulated paths of future actions and state variables from period t + 1 to t + T (i.e., T periods ahead). We index simulated paths by r {1, 2,..., R}. The r th simulated path associated with the initial state (a i, x mt ) is {a (r,a i ) im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T } Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

73 Dealing with the curse of dimensionality Simulation-Based Estimation (4) Simulation-based estimation A simulated path {a (r,a i ) follows. im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T } is obtained as Given (a i, x mt ), we use the transition probability function F x (. a i, x mt ) to obtain a random draw x (r,a i ) m,t+1. Given x (r,a i ) m,t+1, we use the choice probability P i (x (r,a i ) m,t+1 ) to obtain a random draw a (r,a i ) i,m,t+1. Given (a (r,a i ) i,m,t+1, x(r,a i ) m,t+1 ), we use the transition probability function F x (. a (r,a i ) i,m,t+1, x(r,a i ) m,t+1 ) to obtain a random draw x(r,a i ) m,t+2. And so on. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

74 Dealing with the curse of dimensionality Simulation-based estimation Simulation-Based Estimation (5) Then, given the simulated paths {a (r,a i ) im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T }, we construct the simulator of z P imt (a i ) as: z P,sim imt (a i ) = a i z P imt (1)+ 1 R The simulator of W P imt is W P,sim imt [ R T β j a (r,a i ) im,t+j z P i r =1 j=1 ( ) ] 1, x (r,a i ) m,t+j = (1 P i (x mt )) z P,sim imt (0) + P i (x mt ) z P,sim imt (1) If the DP problem has finite horizon, or if T is large enough such that the approximation error associated with the truncation of paths is negligible, then these simulators are unbiased. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

75 Dealing with the curse of dimensionality Simulation-Based Estimation (6) Simulation-based estimation This simulators can be used either for ME or for MI estimation. The simulation-based ME estimator is the value of θ that solves the system of equations: ( { ( ) }) E H(x mt ) a imt Φ z^p 0,sim imt θ i for any i, t = 0 The simulation-based MI estimator is the value (or set of values) of θ that minimizes the criterion function: ( { [ ] }) 2 min 0 ; W (^P 0 i,^p 0 i),sim imt W (P i,^p 0 i),sim imt θ i i,m,t P H Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

76 Dealing with the curse of dimensionality Simulation-Based Estimation (7) Simulation-based estimation These estimators are consistent only as the number of simulated paths, R, goes to infinity. Note that there are three sources of error in (W ^P 0,sim imt 1 Estimation error: because ^P 0 = P 0 W P0 imt ): 2 Simulation error: because the expectation is not taken over all the possible future histories but only over the simulated paths. 3 Approximation error: because T = (3) can be negligible if T is not too small or β too close to one. (1) can be very important: curse of dimensionality in NP estimation. (2) can be very important. Even with millions of simulated histories we may have a very small proportion of all possible histories. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

77 Dealing with the curse of dimensionality Counterfactual Experiments with an Estimated Model Counterfactual Experiments with Estimated Model (1) One of the most attractive features of structural models is that they can be used to predict the effects of new policies or changes in parameters (counterfactuals). However, this a challenging exercise in a model with multiple equilibria. The data can identify the "factual" equilibrium. However, under the counterfactual scenario, which of the multiple equilibria we should choose? Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

78 Dealing with the curse of dimensionality Counterfactual Experiments with an Estimated Model Counterfactual Experiments with Estimated Model (2) Different approaches have been implemented in practice. Select the equilibrium to which we converge by iterating in the (counterfactual) equilibrium mapping starting with the factual equilibrium P 0 Select the equilibrium with maximum total profits (or alternatively, with maximum welfare). Homotopy method: Aguirregabiria and Ho (2007) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

79 Dealing with the curse of dimensionality Counterfactual Experiments with an Estimated Model Counterfactual Experiments: Homotopy method Let θ be the vector of structural parameters in the model. An let Ψ(θ, P) be the equilibrium mapping such that an equilibrium associated with θ can be represented as a fixed point: P = Ψ(θ, P) The model could be completed with an equilibrium selection mechanism: i.e., a criterion that selects one and only one equilibrium for each possible θ. Suppose that there is a "true" equilibrium selection mechanism in the population under study, but we do not know that mechanism. Our approach here (both for the estimation and for counterfactual experiments) is completely agnostic with respect to the equilibrium selection mechanism. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

80 Dealing with the curse of dimensionality Counterfactual Experiments with an Estimated Model Counterfactual Experiments: Homotopy method We only assume that there is such a mechanism, and that it is a smooth function of θ. Let π(θ) be the (unique) selected equilibrium, for given θ, if we apply the "true" selection mechanism. Since we do not know the mechanism, we do not know π(θ) for every possible θ. However, we DO know π(θ) at the true θ 0 because we know that: and both P 0 and θ 0 are identified. P 0 = π(θ 0 ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

81 Dealing with the curse of dimensionality Counterfactual Experiments with an Estimated Model Counterfactual Experiments: Homotopy method Let θ 0 and P 0 be the the population values. Let (^θ 0, ^P 0 ) be our consistent estimator. We do not know the function π(θ). All what we know is that the point (^θ 0, ^P 0 ) belongs to the graph of this function π. Let θ be the vector of parameters under a counterfactual scenario. We want to know the counterfactual equilibrium π(θ ). Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

82 Dealing with the curse of dimensionality Counterfactual Experiments with an Estimated Model Counterfactual Experiments: Homotopy method A Taylor approximation to π(θ ) around our estimator ^θ 0 implies that: ) ) (^θ π(θ ) = π (^θ 0 + π 0 ( ) ( θ ) θ θ ^θ 0 + O 2 ^θ 0 ) (^θ = ^P 0 + π 0 ( ) ( θ ) θ θ ^θ 0 + O 2 ^θ 0 To get a) first-order approximation to π(θ ) we need to know π (^θ 0 θ. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

83 Dealing with the curse of dimensionality Counterfactual Experiments with an Estimated Model Counterfactual Experiments: Homotopy method ) We know that π (^θ 0 = Ψ(^θ 0, ^P 0 ), and this implies that: ) ( ) π (^θ 1 0 θ = I Ψ(^θ 0, ^P 0 ) Ψ(^θ 0, ^P 0 ) P θ Then, π(θ ) = ( ^P 0 + I Ψ(^θ 0, ^P 0 ) P ) 1 Ψ(^θ 0, ^P 0 ) ( ) ( θ ) θ θ ^θ 0 + O 2 ^θ 0 ( ) 1 ( ) Therefore, ^P 0 + I Ψ(^θ 0,^P 0 ) Ψ(^θ 0,^P 0 ) P θ θ ^θ 0 approximation to the counterfactual equilibrium P. is a first-order Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

84 Environmental Regulation in the Cement Industry: Ryan (2012) Empirical Application Outline Stephen Ryan (2012): "The Costs of Environmental Regulation in a Concentrated Industry," Econometrica. 1. Motivation and Empirical Questions 2. The US Cement Industry 3. The Regulation (Policy Change) 4. Empirical Strategy 5. Data 6. Model 7. Estimation and Results Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

85 Environmental Regulation in the Cement Industry: Ryan (2012) Empirical Questions Motivation and Empirical Questions Most previous studies that measure the welfare effects of environmental regulation (ER) have ignored dynamic effects of these policies. ER has potentially important effects on firms entry and investment decisions, and, in turn, these can have important welfare effects. This paper estimates a dynamic game of entry/exit and investment in the US Portland cement industry. The estimated model is used to evaluate the welfare effects of the 1990 Amendments to the Clean Air Act (CAA). Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

86 Environmental Regulation in the Cement Industry: Ryan (2012) US Cement Industry (1) The US Cement Industry For the purpose of this paper, the most important features of the US cement industry are: (1) Indivisibilities in capacity investment, and economies of scale (2) Highly polluting and energy intensive industry (3) Local competition, and highly concentrated local markets Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

87 Environmental Regulation in the Cement Industry: Ryan (2012) US Portland Cement Industry (2) The US Cement Industry Indivisibilities in capacity investment, and economies of scale Portland cement is the binding material in concrete, which is a primary construction material. It is produced by first pulverizing limestone and then heating it at very high temperatures in a rotating kiln furnace. These kilns are the main piece of equipment. Plants can have one or more kilns (indivisibilities). Marginal cost increases rapidly when a kiln is close to full capacity. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

88 Environmental Regulation in the Cement Industry: Ryan (2012) US Cement Industry (3) The US Cement Industry Highly polluting and energy intensive industry The industry generates a large amount of pollutants by-products. Second largest industrial emitter of Sulfure Dioxide (SO2) and Carbon Dioxide (CO2), and a major source of NOx (Nitric oxide and Nitrogen Dioxide) and particulates. High energy requirements and pollution make the cement industry an important target of environmental policies. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

89 Environmental Regulation in the Cement Industry: Ryan (2012) US Cement Industry (4) The US Cement Industry Local competition, and highly concentrated local markets Cement is a commodity diffi cult to store and transport, as it gradually absorbs water out of the air rendering it useless. Transportation costs per unit value are large. This is the main reason why the industry is spatially segregated into regional markets. These regional markets are very concentrated. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

90 Environmental Regulation in the Cement Industry: Ryan (2012) The Regulation / Policy Change The Regulation (Policy Change) The Clean Air Act (CAA) is a (the) main environmental Act in US. The 1990 ammedment was a major revision. It has been the most important new environmental regulation affecting this industry in the last three decades. It added new categories of regulated emissions. Cement plants were required to undergo an environmental certification process. Environmental permits of operation. This regulation may have increased sunk costs, fixed operating costs or even investment costs in this industry. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

91 Environmental Regulation in the Cement Industry: Ryan (2012) Evaluation of Policy Effects The Regulation (Policy Change) Previous evaluations of these policies have ignored effects on entry/exit and on firms capacity investment. They have found that the regulation contributed to reduce marginal costs and therefore prices. Positive effects on consumer welfare and total welfare. Ignoring effects on entry/exit and on firms investment could imply an overestimate of these positive effects. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

92 Environmental Regulation in the Cement Industry: Ryan (2012) Empirical Strategy (1) Empirical Strategy Specify a model of the cement industry, where oligopolists make optimal decisions over entry, exit, production, and investment given the strategies of their competitors. Estimate the model for the cement industry using a 20 year panel and allowing the structural parameters to differ before and after the 1990 regulation. Changes in cost parameters are attributed to the new regulation. The MPEs before and after the regulation are computed and they are used for welfare comparisons. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

93 Environmental Regulation in the Cement Industry: Ryan (2012) Preview of Empirical Results Empirical Strategy Amendments roughly doubled sunk costs of entry, to $35M. The larger entry cost reduced net entry and the number of plants over time, increasing market power. Amendments led to higher investment by incumbents, but lower aggregate market capacity. Consumer welfare decreased 25% due to lower entry and increased market power (approx. $1.2B). Static analysis would ignore the benefits of increased market power on incumbent firms, and welfare effect could have wrong sign. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

94 Environmental Regulation in the Cement Industry: Ryan (2012) Data (1) Data Period: 1980 to 1999 (20 years); 27 regional markets. Index local markets by m, plants by i and years by t. Data = {S mt, W mt, P mt, n mt, q imt, i imt, s imt } S mt = Market size W mt = Input prices (electricity prices, coal prices, natural gas prices, and manufacturing wages) P mt = Output price n mt = Number of cement plants q imt = Quantity produced by plant i s imt = Capacity of plant i (number and capacity of kilns) i imt = Investment in capacity by plant i Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

95 Environmental Regulation in the Cement Industry: Ryan (2012) Data (2) Data USGS Minerals Yearbook - Market-level data for prices and quantities - 27 markets covering United States market-year observations - Energy prices, labor inputs from Dept. Energy Portland Cement Association Plant Information Survey - Every plant in United States Kiln-level data on capacity and production plant-year observations Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

96 Environmental Regulation in the Cement Industry: Ryan (2012) Model (1) Model Regional homogenous-goods market. Every period, incumbent firms compete in quantities in a static equilibrium (Cournot) subject to their capacity constraints. They also decide entry-exit, and investment in capacity (time-to-build). Firms invest in future capacity and this decision is partly irreversible (and therefore dynamic). Incumbent firms also make optimal decisions over whether to exit. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

97 Environmental Regulation in the Cement Industry: Ryan (2012) Demand and Variable Costs Model Inverse demand curve (iso-elastic): log P mt = α mt + 1 ɛ log Q mt Production costs: C (q imt ) = (MC + ω imt ) q imt { qimt +CaPCOST 1 s imt s imt = installed capacity q imt /s imt = degree of capacity utilization ω imt = idiosyncratic shock in MC MC, CaPCOST and ν are parameters. } ( ) 2 qimt > ν ν s imt Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

98 Environmental Regulation in the Cement Industry: Ryan (2012) Costs of Capacity Investment Model Investment costs ( ) IC imt = I {i imt > 0} θ (+) 0 + θ (+) 1 i imt + θ (+) 2 iimt 2 ( ) +I {i imt < 0} θ ( ) 0 + θ ( ) 1 i imt + θ ( ) 2 iimt 2 Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

99 Environmental Regulation in the Cement Industry: Ryan (2012) Sunk Costs and Scrap Value Model Entry cost EC imt = 1 {s imt = 0 and i imt > 0} ( ) SUNK + ε EC imt Scrap value SV imt = 1 {s imt > 0 and i imt = s imt } ( ) SCRaP + ε SV imt Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

100 Environmental Regulation in the Cement Industry: Ryan (2012) Model State variables: Dynamic decisions Vector of state variables: s mt = {α mt, W mt, s imt : i = 1, 2,..., n mt } Incumbent firm: V (s mt, ε imt ) = max i imt {π(i imt, s mt ) + β E t (V (s mt+1, ε imt+1 ))} Potential entrant: V e (s mt ) = max {0 ; π e (s mt ) + β E t (V (s mt+1, ε imt+1 ))} Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

101 Environmental Regulation in the Cement Industry: Ryan (2012) Markov Perfect Equilibrium Model Strategy / investment functions: i imt = i(s mt ) = i(α mt, W mt, s imt : i = 1,...) Given other firms strategy functions, each firm chooses a strategy to maximize its intertemporal value. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

102 Environmental Regulation in the Cement Industry: Ryan (2012) Estimation Demand Curve Estimation and Results Includes local market region fixed effects (estimated with 19 observations per market). Instruments: local variation in input prices. The market specific demand shocks, α mt, are estimated as residuals in this equation. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

103 Environmental Regulation in the Cement Industry: Ryan (2012) Estimation Variable Costs Estimation and Results From the Cournot equilibrium conditions. Firm specific cost shocks, ω imt, are estimated as residuals in this equation. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

104 Environmental Regulation in the Cement Industry: Ryan (2012) Estimation and Results Reduced Form Estimation Investment Strategy Assumption: i imt = i(α mt, W mt, s imt, s imt ) = i ( ) α mt, W mt, s imt, s jmt j =i Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

105 Environmental Regulation in the Cement Industry: Ryan (2012) Estimation using Moment Inequalities Estimation using Moment Inequalities (BBL) Remember that: z P i (a i, x mt ) θ i is the value of choosing alternative a i today, given that firms behave in the future according to the probabilities in P. Then, the value V P imt is: V P imt = (1 P i (x mt )) [ z P i (0, x mt ) θ i ] + Pi (x mt ) [ z P i (1, x mt ) θ i ] = W P imt θ i where W P imt = (1 P i (x mt )) z P i (0, x mt ) + P i (x mt ) z P i (1, x mt ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

106 Environmental Regulation in the Cement Industry: Ryan (2012) Estimation using Moment Inequalities Estimation using Moment Inequalities (2) Let s split the vector of choice probabilities P into the sub-vectors P i and P i, P (P i, P i ) where P i are the probabilities associated to player i and P i contains the probabilities of players other than i. P 0 is an equilibrium associated to θ 0. Therefore, P 0 i is firm i s best response to P 0 i. This implies that for any P i = P 0 i the following inequality should hold: W (P0 i,p0 i) imt θ 0 i W (P i,p 0 i) imt θ 0 i We can define an estimator of θ 0 based on these (moment) inequalities. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

107 Environmental Regulation in the Cement Industry: Ryan (2012) Estimation using Moment Inequalities Estimation using Moment Inequalities (3) There are infinite alternative policies P i, and therefore there are infinite moment inequalities. For estimation, we should select a finite set of alternative policies. This is a very important decision for this class of estimators (more below). Let H be a (finite) set of alternative policies for each player. Define the following criterion function: R ( θ, P 0, H ) ( { min 0 ; i,m,t P H [ W (P0 i,p0 i) imt ] }) 2 W (P i,p 0 i) imt θ i This criterion function penalizes departures from the inequalities. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

108 Environmental Regulation in the Cement Industry: Ryan (2012) Estimation using Moment Inequalities Estimation using Moment Inequalities (4) Then, given an initial NP estimator of P 0, say ^P 0, we can define the following estimator of θ 0 based on moment inequalities (MI): ^θ = arg min θ i,m,t ( { P H min 0 ; [ W (^P 0 i,^p 0 i) imt ] }) 2 W (P i,^p 0 i) imt θ i There are several relevant comments to make on this MI estimator: 1 Point identification / Set identification 2 Properties (relative to two-step estimators using moment equalities) 3 Continuous dependent variables Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

109 Environmental Regulation in the Cement Industry: Ryan (2012) Simulation-Based Estimation (1) Simulation-based estimation Though two-step methods (with either PML or MI) are computationally much cheaper than full solution-estimation methods, they are still impractical for applications where the dimension of the state space X is very large, e.g., a discrete state space with millions of points or a model in which some of the observable state variables are continuous. To deal with this problem, Hotz, Miller, Sanders and Smith (REStud, 1994) proposed an estimator that uses simulation techniques to approximate the values z P imt (a i ) or similarly the vector of values W P imt. In the context of dynamic games, Bajari, Benkard and Levin (BBL) have proposed to used this simulation and have extended it to models with continuous decision variables. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

110 Environmental Regulation in the Cement Industry: Ryan (2012) Simulation-Based Estimation (2) Simulation-based estimation Remember that: z P imt (a i ) a i z P imt + E ( s=1 β s a im,t+s z Pim,t+s (1) x mt, a imt = a i ) The expectations E (.) are taken over all the possible future paths of actions and state variables conditions on (x mt, a imt = a i ) and conditional on future behavior P. The simulator of z P imt (a i ) are obtained by replacing the true expectations E (.) by Monte Carlo approximations to these expectations. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

111 Environmental Regulation in the Cement Industry: Ryan (2012) Simulation-Based Estimation (3) Simulation-based estimation For every value of x mt in the sample and every choice alternative a i (in the sample or not), we consider (a i, x mt ) as the initial state for player i and then we use the probabilities in P, and the transition probabilities of x, to generate R simulated paths of future actions and state variables from period t + 1 to t + T (i.e., T periods ahead). We index simulated paths by r {1, 2,..., R}. The r th simulated path associated with the initial state (a i, x mt ) is {a (r,a i ) im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T } Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

112 Environmental Regulation in the Cement Industry: Ryan (2012) Simulation-Based Estimation (4) Simulation-based estimation A simulated path {a (r,a i ) follows. im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T } is obtained as Given (a i, x mt ), we use the transition probability function F x (. a i, x mt ) to obtain a random draw x (r,a i ) m,t+1. Given x (r,a i ) m,t+1, we use the choice probability P i (x (r,a i ) m,t+1 ) to obtain a random draw a (r,a i ) i,m,t+1. Given (a (r,a i ) i,m,t+1, x(r,a i ) m,t+1 ), we use the transition probability function F x (. a (r,a i ) i,m,t+1, x(r,a i ) m,t+1 ) to obtain a random draw x(r,a i ) m,t+2. And so on. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

113 Environmental Regulation in the Cement Industry: Ryan (2012) Simulation-based estimation Simulation-Based Estimation (5) Then, given the simulated paths {a (r,a i ) im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T }, we construct the simulator of z P imt (a i ) as: z P,sim imt (a i ) = a i z P imt (1)+ 1 R The simulator of W P imt is W P,sim imt [ R T β j a (r,a i ) im,t+j z P i r =1 j=1 ( ) ] 1, x (r,a i ) m,t+j = (1 P i (x mt )) z P,sim imt (0) + P i (x mt ) z P,sim imt (1) If the DP problem has finite horizon, or if T is large enough such that the approximation error associated with the truncation of paths is negligible, then these simulators are unbiased. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

114 Environmental Regulation in the Cement Industry: Ryan (2012) Simulation-Based Estimation (6) Simulation-based estimation This simulators can be used either for ME or for MI estimation. The simulation-based ME estimator is the value of θ that solves the system of equations: ( { ( ) }) E H(x mt ) a imt Φ z^p 0,sim imt θ i for any i, t = 0 The simulation-based MI estimator is the value (or set of values) of θ that minimizes the criterion function: ( { [ ] }) 2 min 0 ; W (^P 0 i,^p 0 i),sim imt W (P i,^p 0 i),sim imt θ i i,m,t P H Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

115 Environmental Regulation in the Cement Industry: Ryan (2012) Simulation-Based Estimation (7) Simulation-based estimation These estimators are consistent only as the number of simulated paths, R, goes to infinity. Note that there are three sources of error in (W ^P 0,sim imt 1 Estimation error: because ^P 0 = P 0 W P0 imt ): 2 Simulation error: because the expectation is not taken over all the possible future histories but only over the simulated paths. 3 Approximation error: because T = (3) can be negligible if T is not too small or β too close to one. (1) can be very important: curse of dimensionality in NP estimation. (2) can be very important. Even with millions of simulated histories we may have a very small proportion of all possible histories. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

116 Dynamic Product Positioning in Differentiated Product Markets Empirical Application Andrew Sweeting (2013): "Dynamic Product Positioning in Differentiated Product Markets," Econometrica. 1. Motivation and Empirical Questions 2. The Commercial Radio Industry 3. The Regulation (Policy Change) 4. Empirical Strategy 5. Data 6. Model 7. Estimation and Results Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

117 Motivation Motivation and Empirical Questions Public/Fiscal policies can have an important impact on firms decisions on which type of products to produce, e.g., on vertical and horizontal product differentiation. Some policies may clearly favor some products. But also, "apparantely neutral" policies such as taxes (lump-sum, per-unit, or ad-valorem) that apply equally to all products, can also affect firms product selection. This paper studies the effect of the 2009 US Performance Rights Act on the formats of commercial radio stations. This Act requires radio stations to pay fees for musical performance rights. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

118 Motivation and Empirical Questions Empirical Questions and Contributions Long-run effects of this policy, taking into account firms incentives to reposition the characteristics of their products. Identification of the maginute of sunk costs of introducing new products and of product repositioning. Estimation of a dynamic game of oligopoly competition with endogenous product choices. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

119 The US Commercial Radio Industry Commercial Radio Industry (1) Differentiated product: Radio stations are classified in different categories or formats: **** Two-sided market: Most revenue for stations comes from advertisers. But advertisers demand is based on listeners demand. Local competition: The industry has many local markets, each with its own local stations. Regulated entry because spectrum constraints: FCC grants licenses. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

120 The Regulation (Policy Change) The Regulation / Policy Change In most broadcast industries in the world, stations should pay copyright fees for performing music. The same rules applies tin TV, etc. Surprisingly, US radio stations did not have to pay these fees until The 2009 US Performance Rights Act, established these fees. Interestingly, the fee is determined as a percentage of the station advertising revenue, and does not depend on the amount of music the station plays. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

121 Data Data Data from 102 radio markets during the period (2 observations per year). 2, 375 stations and 16, 566 station-period observations. For each station-period observation: - Format (20 formats aggregated in 7); - Station owner; - Band (AM or FM); - Coverage of station signal; - Audience share by demographic group (18 groups); - Advertising revenues. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

122 Data (2) Data Outside alternative based on a maximum of 6 hours per day of radio listening (average is 2.5 hours/person-day). Market demographics from the population census: 18 mutually exclussive demographic groups based on Age (3 grous) x Ethnic group (3 groups) x gender. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

123 Data Some Interesting Summary Statistics Substantial amount of format switching: on average, each period, 3.2% of stations switch between formats. [Important to identify sunk costs of format switching]. No substantial differences in frequency of format switching in small markets (2.9%) and big markets (3.3%), despite firm revenue is substantially different in these markets. Why? Switching stations tend to have low markets share, and their market share tends to increase after the switch. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

124 Model Model Data from 102 radio markets during the period (2 observations per year). 2, 375 stations and 16, 566 station-period observations. For each station-period observation: - Format (20 formats aggregated in 7); - Station owner; - Band (AM or FM); - Coverage of station signal; - Audience share by demographic group (18 groups); - Advertising revenues. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

125 Innovation and creative destruction (Igami, 2017) Introduction Innovation and creative destruction (Igami, 2017) Innovation, the creation of new products and technologies, necessarily implies the "destruction" of existing products, technologies, and firms. In other words, the survival of existing products / technologies / firms is at the cost of preemting the birth of new ones. The speed (and the effectiveness) of the innovation process in an industry depedns crucially on the dynamic strategic interactions between "old" and "new" products/technologies. Igami (JPE, 2017) studies these interactions in the context of the Hard-Disk-Drive (HDD) industry during Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

126 Innovation and creative destruction (Igami, 2017) Introduction HDD: Different generations of products Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

127 Innovation and creative destruction (Igami, 2017) Introduction Adoption new tech: Incumbents vs. New Entrants Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

128 Innovation and creative destruction (Igami, 2017) Introduction Adoption new tech: Incumbents vs. New Entrants Igami focuses on the transition from 5.25 to 3.5 inch products. He consider three main factors that contribute to the relative propensity to innovate of incumbents and potential entrants. Cannibalization. For incumbents, the introduction of a new product reduces the demand for their pre-existing products. Preemption. Early adoption by incumbents can deter entry and competition from potential new entrants. Differences in entry/innovation costs. It can play either way. Incumbents have knowledge capital and economies of scope, but they also have organizational inertia. Victor Aguirregabiria () Empirical IO Toronto. Winter / 144

129 Innovation and creative destruction (Igami, 2017) Data Market shares New/Old products Victor Aguirregabiria () Empirical IO Toronto. Winter / 144