Static Entry Jonathan Williams October 10, 2009 1 Introduction: Why study entry? Remember that when we were looking at questions like prices and competition we need to that market structure itself is a decision of rms, and hence is itself endogenous. I will show an example later on in which this will matter. 1. Market Structure and Prices. It was routine for economists to regress prices on measures of market structure (hhi) for sometime, see Borenstein (1989). However, market structure is clearly an outcome of individual rm s entry decisions which in turn determine prices and other market outcomes. Therefore, market structure is really something that should be modeled and accounted for or at a minimum treated as endogenous in any econometric analysis. However, we have already seen a number of papers that take the market structure as exogenous, Shepard (1991). 2. Is there too much or too little entry? Sometimes factors such as barriers to entry limit the number of rms in a market. For instance Microsoft has an application barrier to entry for Windows: to enter the OS market you need to convince developers to build new applications for your platform. One the other hand, the number of entrants might be above what is socially optimal, as discussed by Mankiw and Whinston RAND 1986. When I enter, I don t take into account the fact that by entering I lower the pro ts of my rival, hence there is a business-stealing externality to entry! 1
3. Innovation: Which types of products do rms choose to develop. For instance, will product di erentiation induce Microsoft and Nintendo to produce very di erent gaming platforms (XBox 360 and Wii). Product innovations can also be modeled as an entry game. Firms are interacting deciding what products to introduce and the actions or product introductions of their competitors impact their decision. 4. Endogeneity of Product Characteristics: When BLP look at demand for automobiles, they take product characteristics as exogenous. What happens if we try to gure out which type of cars lead to the highest pro ts for manufacturers? (Take the example of the minivan: it lled a gap between getting a van and a large sedan and yielded huge pro ts for Chrystler). Sutton Example: Often you will see people assume that something like the Her ndahl appropriately captures the concept of a more competitive market. John Sutton has an interesting example of why more concentrated markets are not necessarily more competitive. is: Suppose that demand is characterized by a Cournot model of competition, where demand P = a bq (1) and assume for that marginal costs are c(q) = c q. The xed cost of entering a market is F. A rm s pro ts as a function of the number of competitors is: Taking logs of this expression (for variable pro ts) we get: = 1 2 a c F (2) b (N + 1) log( var ) = ln( 1 ) + 2 ln(a c) + 2 ln(n + 1) (3) b We will use this expression to justify some of the functional forms used later on as being additively separable in the number of rms and other parameters, and the log form often used in these models. 2
So the number of rms in the market will be determined by the free-entry condition, i.e.: max N s:t: 2 1 a c F > 0: b (N + 1) Now think of the Bertrand model of competition, since the price is set to marginal cost for any number of rms (p=c), then rms will never be able to cover their xed costs of entry if they have a competitor. Thus the Bertrand model predicts either 0 or 1 rms in a market. This means that a Bertrand competitive market always has fewer rms than a Cournot market, while the Her ndahl would say that the Bertrand market has little competition while the Cournot market is more competitive. However, this result is only because of the entry process, not the toughness of product market competition. 2 Bresnahan-Reiss (1991, JOE) Bresnahan and Reiss introduced a simple discrete (binary) game that Timing: 1. Firms simultaneously decide to enter the market. 2. Firms play Cournot (or some type of di erentiated bertrand game) in quantities in the subgame. Note that there are multiple equilibria in these games. For instance, suppose rms payo s are the following: Out Enter Out 0,0 4,0 Enter 0,5-11,-10 Now lets consider a more general game that can be used for empirical inference. This is very similar to the game considered by Bresnahan and Reiss (1991). 3
Lets assume that rm s payo s can be represented by the simultaneous discrete system y 1m = 1[ 1 X 1m + 2 y 2m + 1m 0] y 2m = 1[ 2 X 2m + 1 y 1m + 2m 0] where X 1m and X 2m are rm-market speci c exogenous regressors that shift the payo s of each rm in a market. 2 and 1 measure the impact that one rm s entry has on another rm (in a cooperative game, this parameters may be greater than zero). The error terms allows our model to explain entry behavior that the exogenous characteristics can not. We will assume we observe X and y. The error terms are observed by rms (game of complete information), but not observed by the econometrician. We will use the idea of pure strategy Nash Equilibrium as an equilibrium concept (we can also consider mixed strategies, but it complicates things signi cantly). However, if we assume the error terms are private information to rms, then the appearance of rm s mixing (randomness in strategies) are actually pure strategies being played with knowledge of the private information by each rm (Seim 2006 uses this idea). The di culty with these types of games can be seen in the following gure: 4
The problem comes from the fact that the model does not predict a unique outcome for some values of the parameters and unobservables. This is called an incoherent econometric model, since the observables, parameters, and unobservables do not map to a unique outcome or dependent variable. If we were thinking about estimating the model by maximum likelihood, we need a unique prediction for the probability of each outcome. The probabilities of some outcomes can be predicted uniquely while others can only be bounded. Pr(1; 1jX) = Pr( 1m 1 X 1m 2 ; 2m 2 X 2m 1 ) Pr(0; 0jX) = Pr( 1m 1 X 1m ; 2m 2 X 2m ) Pr(( 1m ; 2m ) 2 R 1 (X; )) Pr(1; 0jX) Pr(( 1m ; 2m ) 2 R 1 (X; )) + Pr(( 1m ; 2m ) 2 R 2 (X; )) where 2 R 1 (X; ) = 4 ( 1m ; 2m ) : ( 1m 1 X 1m ; 2m 2 X 2m )[ 3 5 ( 1m 1 X 1m 2 ; 2 X 2m 2m 2 X 2m 1 ) R 2 (X; ) = [( 1m ; 2m ) : ( 1 X 1m 1m 1 X 1m 2 ; 2 X 2m 2m 2 X 2m 1 )] Therefore, if we make no assumption on equilibrium selection (which equilibrium occurs in the region of multiplicity) our model will predict unique probabilities for the 5
outcomes (1,1) and (0,0) while only predicting bounds for the probabilities of the (1,0) and (0,1) outcomes. There are a couple ways to solve the multiplicity issue and we will discuss di erent applications that use each method: 1. Re ning the set of equilibria (equilibrium selection mechanism), i.e. picking out an equilibrium which seems more plausible. In particular, suppose it is the case that the most pro table rm is always the rst rm to enter. This leads to the following relationship between parameters and entry. This leads to rm 1 entering alone in the region of multiplicity if 1 X 1m + 1m 2 X 2m + 2m This essentially divides the region of multiplicity in half and assumes whichever rm is more pro table will enter the market. 2. Partial Identi cation: we can also use the bounds that the model predicts for the equilibrium to infer something about our parameters. This allows us to avoid making potentially very unreasonable assumptions regarding selection of equilibria. However, the tradeo is that we might not be able to point identify the parameters of interest ( 1 and 2 as well as any parameters of the distribution). Tamer (2003) shows that with enough variation in the exogenous variables, the parameters of the model are point identi ed without any equilibrium selection assumptions. We observe the probability of each equilibrium in our data. Our model predicts upper and lower bounds on the probability of each equilibrium outcome conditional on the exogenous characteristics and a value of the parameters of the payo function. We then want to make sure that the observed probability of each outcome falls in between the upper and lower bounds of that equilibrium outcome predicted by our model (penalize those outcomes that are not in between the bounds). Pr(( 1m ; 2m ) 2 R 1 (X; )) d Pr(1; 0jX) Pr(( 1m ; 2m ) 2 R 1 (X; )) + Pr(( 1m ; 2m ) 2 R 2 (X; )) 6
Ciliberto and Tamer (2003) use this methodology to model entry in the airline industry. We will talk about how they implement 3. Focus on a Prediction of the Model that is Unique: Looking a the number of rms that enter, a feature which is pinned down across di erent equilibria. In these simple models, even in the regions of the unobservable space that does not give a unique prediction as to the identify of the rm that enters, the number of rms that enters is always unique. This is the approach taken by Bresnahan and Reiss (1991) and Berry (1992). 4. Model of equilibrium selection: Following the ideas of Bajari, Hong and Ryan (2009) we can think about modeling how the equilibrium is selected in regions of multiplicity. For example, we could write the probability of an equilibrium, y, as Z Pr(yjX; ) = S(yjX; )df ( 1m ; 2m ) Z Z = S(yjX; )df ( 1m ; 2m ) + S(yjX; )df ( 1m ; 2m ) R 1 (X;) R 2 (X;) Z Z = df ( 1m ; 2m ) + S(yjX; )df ( 1m ; 2m ) R 1 (X;) R 2 (X;) Region of Unique Outcome Region of Multiplicity This equilibrium selection function, S(yjX; ), can then be modeled and estimated with the rest of the model. The only restriction on this function is that it is between zero and one, since it is a probability (could use multinomial logit function). Notice this approach is di erent from 1 in that we let the data tell us what the equilibrium selection function is rather than apriori choosing it as has been done often in the literature. 3 Bresnahan and Reiss (1991, JPE) 3.1 Introduction They look at 5 di erent retail and professional industries. Use data on geographically isolated monopolies, dupolies and oligopolies (so no town 7
near enough to introduce substitutes), so market can be considered isolated from other markets. They study the relationship between the number of rms in a market, market size, and competition. They nd that competitive conduct changes quickly as the number of incumbents increases. In markets with ve or fewer incumbents, almost all variation in competitive conduct occurs with the entry of the second or third rm. Once the market has between three and ve rms, the next entrant has little e ect on competitive conduct. 3.2 Model BR are going to choose to use the idea # 3 for getting around the non-uniqueness problem in an entry game. Consider a market that has the demand function Q = d(z; P )S(Y ) where d(z,p) is the demand of a representative consumer, Z are demand shifters, P are the vector of prices, and S(Y) is the number of consumers. Z and Y are going to be demographic shifters that determine whether a consumer is going to be active in the market and how much they will puchase. On the cost side, they assume that rms incur xed costs of F(W) and marginal costs of MC(q,W) where W represents exogenous variables a ecting costs and q is the rm speci c output. With a xed cost, the rms will now have a U-shaped average cost curve. They will represent average variable costs by AVC(q,W) 8
The ideal situation would be to be able to measure the rate at which oligopoly margins decline toward zero with the number of rms, or we would ideally like to observe how quickly the breakeven price-cost margins M N = P N rms, N, increases. MC(q N ) fall as the number of Since (at this point), estimating margins was not a yet a reality for oligopoly industries, they use "entry thresholds" to reveal something about the e ect of entry First, lets compare a monopoly and competitive entry thresholds. A monopolist breaks even when or 1 (S 1 ) = [P 1 AV C(q 1 ; W )]d(z; P 1 )S 1 F = 0 S 1 = F [P 1 AV C(q 1 ; W )]d(z; P 1 ) which tells us that the break even market size for a monopolist, S 1 ; is just the ratio of xed costs to variable pro ts. Thus, the higher xed costs or the lower the variable pro ts per customer, the larger the market size must be to support a monopolist. Therefore, this is the "entry threshold" for a monopolist. S 4 = 4F [P 4 AV C(q 4 ; W )]d(z; P 4 ) For 4 rms, The numerator is now 4 times greater, but we would expect that the denominator has decreased which means that S 4 should be more than 4 times greater than S 1 due to the shrinking margins. If we denote the entry threshold for a competitive market, s 1, which is just xed costs divided by competitive variable pro ts S N s 1 = lim N!1 N which tells us how many customers are needed to support a competitive rm. little s N is the size of the market that is required to support each rm with the margins they would earn with N rms active (each rm s part of the market). The 9
The ratio of the competitive to monopolist threshhold, s1 s 1, then measures the fall in variable pro ts per customer between a monopoly and a competitive market. This measure is bounded below by 1, since a monopolist s variable pro ts are higher than those of a competitive rm (measure equals one if they are equal to each other). Between monopoly and competition is an oligopoly market structure. The analysis above suggests that we can use this type of idea to demonstrate how quickly variable pro ts of rms move towards the competitive variable pro ts if we have information on the entry thresholds for di erent numbers of rms. This can be done by looking at the ratio, s1 s N, which is the ratio of N- rm oligopoly variable pro ts to the competitive pro ts. Therefore, it tells us how quickly margins move towards the competitive margins and how the market size (entry thresholds) must expand to support a given number of rms. Just to see an example of what we are going to do. Suppose that we observe (estimate) that it takes 2,000 customers to support a monopolist. Suppose we also observe that the market becomes perfectly competitive when each rm has 4,000 customers (s 1 = 4; 000). These two entry thresholds bracket the range of oligopoly thresholds we should observe, since their margins will be somewhere between a monopolists and a competitive rms (need more consumers than a monopolist but fewer than a competitive rm to cover xed costs). Suppose further that the fourth entrant is enough to get perfect competition, then we should observe (estimate) S 4 = 4x4; 000 = 16; 000 so that the fourth rm will only enter if the market size exceeds 16,000. It is also the case that s1 s 4 = 1: Alternatively, suppose that the fourth rm is part of a cartel, so it will enter when it covers its xed costs at the monopoly price, that is, when the market has 4x4,000=8,000 customers. Notice, in this case, fewer customers are required when competition is less intense. Therefore, our ratio s1 s 4 = 2, is now higher and it captures the fact that with less intense competition, a smaller market size is required to support the same number of rms. Therefore, we can interpret increases in this ratio across industries as a sign 10
of less competitive outcomes. So, if we were to estimate that s 4 = 3; 810; we should interpret this as though the market is nearly competitive with 4 rms, since s1 s 4 = 1:05. The econometric model of BR is very simple and based on two behavioral assumptions: 1. Firms that Enter make Positive Pro ts (N m ; X m ) + " m > 0 (4) 2. If an extra rm entered it would make negative pro ts: (N m + 1; X m ) + " m < 0 (5) where (N; X m ) is the observable component of pro t depending on demand factors X m and the number of identical competitors in a market N, while " m are unobserved components of pro tability common to all rms in a market. Assume market level shocks " m have a normal distribution with zero mean and unit variance. The probability of observing a market X m with N m plants is the following: Pr(N m jx m ) = [ (N m + 1; X t m)] [ (N m ; X m )]1(N m > 0) where (:) is the cumulative distribution function of the standard normal. If we parameterize the pro t function as (N m ; X m ; ), we can estimate the parameters of the model via Maximum Likelihood, where the likelihood is the following: MY L() = Pr(N m jx m ; ) (6) m=1 BR (1991) parameterize the pro t function as (N m ; X m ) = S(Y; )V N (Z; W; ; ) F N (W; ) 11
where variable pro ts are V N (Z; W; ; ) = [Z; W ] XN m i i=1 where market size is S(Y; ) = Y and xed costs are N m X F N (W; ) = W 0 i=1 i This formulation tells us that the entry thresholds can then be calculated as s Nm = W 0 [Z; W ] which are the estimates reported in Table 5. P Nm i=1 i P Nm i=1 i 3.3 Data and Results Need a great deal of variation in the size of the markets to get any variation in the number of rms serving the market, so this rst gure is important. 12
Need to show that there is variation in the number of professionals in these towns that needs to be explained. It is now up to the model to tease out what portion of this is captured by the model and what portion is picked up by the error terms. It is always important to adequately describe one s data. If you look, there is a lot of variation in demographic variables which will be great for providing helpful variation for identifying entry thresholds (a ects demand and market size). 13
This table summarizes their basic results. These estimates imply di erent entry thresholds, which is what we are really interested in. 14
Here they tell us what their model implies about competition or how margins (variable pro ts) change as rms enter (intensity of competition). 15
Results imply that a monopoly druggist or tire dealer requires about 500 people to set up shop in a town. The thresholds for other professions tend to be higher, with plumbers requiring the largest market size. Results also imply that the ratio of per rm entry thresholds decline with N (right side of table 5). However, the decline largerly stops after N=3. This tells us that after the third rm, margins have basically stopped declining (would imply a ratio of 1). 16
4 Ciliberto and Tamer (2009, Ecmtra) 4.1 Introduction provides a practical method to estimate the payo functions of players in complete informaiton, static, discrete game. allows for general forms of heterogeneity across players without making equilibrium selection assumptions or focusing on a unique prediction of rms (number of players) identi ed feature of the model are a set of parameters that are consistent with the empirical choice probabilities that are consistent with the data apply this methodology to investigate the empirical importance of rm heterogeneity in the airline industry nd competitive e ects of low-cost carriers and southwest are very di erent from legacy carriers then use parameter estimates to look at e ect of the Wright ammendment being repealed. nd that number of markets served out of Dallas Love airports increases signi cantly 4.2 Empirical Speci cation They assume that the pro t function of rm i in market m is given by: im = S m i + Z im i + W im i + X j6=i i jy jm + X j6=i Z jm i jy jm + im where im is unobserved to the econometrician but known to every player, a game of complete information. X = fs; Z; W g: Let X denote the entire collection of exogenous covariates, 17
They allow the e ect of each rm on others to di er, rather than just the number of competitors. Also, allow characteristics of the rm to a ect their competitive impact on other rms. If there are k rms, then there are 2 k possible equilibrium outcomes. The equilibrium outcome y, can be represented by a 2 k vector of zeros and ones. As we showed last time, with no equilibrium selection mechanism, the model only predicts bounds on the probability of each equilibrium outcome. Pr( 2 R 1 (X; )) Pr(yjX) Pr( 2 R 1 (X; )) + Pr( 2 R 2 (X; )) Z Z Z df () Pr(yjX) df () + df () R 1 (X;) R 1 (X;) R 2 (X;) These bounds can then be written in vectorized format for every possible X and a value of the parameters,, as 2 3 2 H 1(; 1 X) 6 : 7 4 5 6 4 H1 2k (; X) Pr(y 1 jx) : Pr(y 2 kjx) 3 2 7 5 6 4 H 1 2(; X) : H 2k 2 (; X) 3 7 5 or in vector format H 1 (; X) Pr(yjX) H 2 (; X) H 1 (; X), the lower bound, represents the probability that the model predicts a particular market structure as the unique equilibrium. H 2 (; X) contains in addition, the probability mass of the region where there are multiple equilibria. 4.3 Estimation CT (2009) estimate their model by ensuring that the observed probabilities of each equilibrium outcome for every X, lies between the upper and lower bounds predicted by the model. The way they do this is by searching for the that minimizes Q() = Z kpr(yjx) H1 (; X)k + kpr(yjx) H 2 (; X)k + dfx (X) 18
which simply penalizes lower (upper) bounds predicted by the model that exceed (are less than) the observed probability of each equilibrium outcome. They then integrate across all the X s to calculate their objective function. In reality, they calculate a sample objective function where they choose N points out of the distribution of df x (X) and calculate the sample analog to the objective function as where g Q N () = NX Pr(yjX d im ) i=1 H 1 (; g X im ) + Pr(yjX d im ) H 2 (; g X im ) + d Pr(yjX im ) is a consistent estimate of the true probability of each equilibrium outcome (ideally done semi or non parametrically). of the estimation procedure. This is essentially the rst stage g H 1 (; X im ) is a simulated estimate of the lower bound predicted by the model for each of the equilibrium outcomes for a particular X, X i. The simulation is necessary, because calculating analytically or numerically, the mass for which the model predicts each equilibrium outcome is nearly impossible (particularly as the number of players grows and the dimensions of the distribution grows). The simulation procedure requires us to calculate an upper and lower bound for each equilibrium probability for every X, for a particular guess of the parameter values. The procedure is as follows: 1. Draw R simulations of the rm unobservables r, r = 1:::R. We will store these draws and hold them xed throughout the optimization procedure. It is easy to allow for correlation in these draws if each rm s draw is random normal by transforming the error terms using the cholesky decomposition of the variance covaraniance matrix. 2. For a particular value of X, a particular draw of the error term r, and your initial guess of the parameter vector,, calcuate the vector of rm s pro ts for a particular set of entry decisions, y j (for some j = 1::::2 k ). (y j ; X im ; r ; ) 19
3. If each rm is earning non-negative pro ts, then the outcome (y j ) is an equilibrium ( rms that don t enter earn zero pro ts, rms that enter earn positive pro ts) (y j ; X im ; r ; ) 0 4. If this equilibrium is unique (for no other y j is it true that (y j ; X im ; r ; ) 0), then add 1 R not unique, add 1 R to the lower and upper bound for this outcome. If the equilibrium is to only the upper bound for this outcome. 5. Steps 2 through 5 are then repeated for every X (i = 1:::N), r (r = 1:::R), and each equilibrium outcome, y j (j = 1:::2 k ). 6. Calculate the objective function, g QN (), using your estimates simulations estimates of H 1 (; gx im ) and H 2 (; gx im ). d Pr(yjX im ) and 7. Repeat Steps 1 through 6 as you search for a value of that minimizes g QN (). 4.4 Results The authors include a couple sets of results to show how allowing for the competitive e ects of each rm to di er from others alters the results 20
The rst column restricts the competitive e ects to be the same across all carriers. This essentially reduces the model to that of Berry (1992, Ecmtra). The other results are intuitive, the larger presence a carrier has at the airports, the more likely they are to serve the market, while the higher their costs, the less likely they are to serve that market. The second column allows for di erent competitive e ects of one rm on another. However, it does not interact these competitive e ects with any characteristics of the rms (airport presence). The third column is the same as the second except they allow the e ect of the controls 21
to di er by the type of rm. The fourth column groups rms together as LAR (large), LCC (low-cost), and WN (southwest) and allows the e ects on one another to be di erent. In the next set of results, they allow the competitive e ect to be di erent depending on characteristics of the rm that are speci c to that market If you look across the columns, what you see is that the rms have a negative e ect on one another and this negative e ect is larger if the rm has a larger presence in that market. 22