Empirical Industrial Organization II
Part II: Estimating Static Games
Naoki Wakamori University of Tokyo
2017 S2 Term
Introduction (1/2)
▶ So far we have seen:
▶ Demand estimation
▶ Production function estimation
Introduction (1/2)
▶ So far we have seen:
▶ Demand estimation
▶ Production function estimation
▶ What is missing in these methodology?
▶ What kind of data do we need?
Introduction (2/2): Features of Entry/Exit Models
▶ Models of market entry in I.O. can be characterized by the following three main features:
1. Fixed cost: associated with being active in the market 2. Dependent variables: firms’ decisions to operate or not 3. Game structure: the payoff of being active in the market
depends on the number and the characteristics of other active firms in the market.
Introduction (2/2): Features of Entry/Exit Models
▶ Models of market entry in I.O. can be characterized by the following three main features:
1. Fixed cost: associated with being active in the market 2. Dependent variables: firms’ decisions to operate or not 3. Game structure: the payoff of being active in the market
depends on the number and the characteristics of other active firms in the market.
▶ What can we learn from market entry?
1. Identification of some parameters: The equilibrium entry conditions contain useful information of structural parameters: demand and cost parameters, in particular, fixed costs. 2. Treating market structure as endogenous: firms decisions
are interdependent. c.f. firms make their decisions independently in Olley and Pakes (1996).
Brief roadmap for this literature
▶ Bresnahan and Reiss (1991a, 1991b)
▶ a: The seminal paper which opened a new literature!
▶ b: Multiple equilibria and non-existence of pure-strategy eq.
▶ Complete information games with heterogeneous firms
▶ Berry (1992): heterogeneous but exogeneous firm characteristics
▶ Mazzeo (2002): heterogeneous and endogenous firm characteristics
▶ Incomplete information games
▶ Seim (2006) - heterogeneous, endogenous firm characteristics, and incomplete information
Bresnahan and Reiss (1991, JPE)
Basic idea of the model
▶ Consider a market m, m = 1, 2, · · · , M , which is isolated.
▶ There are N potential entrants in each market.
▶ Each firm i decides whether or not being active in the market.
▶ Πm(n) denotes the profit of an active firm in market m when there are n active firms.
▶ Πm(n) is strictly decreasing in n.
▶ If nm is an equilibrium number of firms in the market m, then it should satisfy the following condition:
Πm(nm) ≥ 0, and Πm(nm+ 1) < 0.
▶ Utilize this information to estimate the model.
Model (1/6): Profit function
▶ Total profit is equal to variable profit, Vm(n), minus fixed costs, Fm(n):
Πm(n) = Vm(n) − Fm(n).
▶ We do not observe prices nor quantities. However, we observe the number of firms in each market and market conditions.
Model (1/6): Profit function
▶ Total profit is equal to variable profit, Vm(n), minus fixed costs, Fm(n):
Πm(n) = Vm(n) − Fm(n).
▶ We do not observe prices nor quantities. However, we observe the number of firms in each market and market conditions.
▶ Key difference between variable profit and fixed costs: variable profits increase with market size where fixed costs do not.
Model (2/6): Behind the Profit Function
▶ Demand function:
Q= d(Z, P )S
▶ d(Z, P ): the demand function of a “representative consumer”
▶ Z: demographic variables affecting market demand
▶ S: the number of consumers (market size)
▶ Total cost function:
T C= [V C(q, W ) − F (W )]
▶ q: firm output
▶ W: exogenous variables affecting costs
▶ Profit function for a monopolist can be expressed as: Πm(1) = P1d(Z, P1)S − [V C(q1, W) + F (W )]
= [P1− AV C(q1, W)]d(Z, P1)S − F (W ).
Model (3/6): Variable Profit
Vm(n), an incumbent firm in market m when there are n active firms, is specified:
Vm(n) = Smvm(n)
= Sm(XDmβ− α(n)), where
▶ Sm: the market size
▶ v
m: the variable profit per capita
▶ XD
m: the vector of market characteristics that may affect the demand of the product, e.g., per capita income, age
distribution in market m
▶ β: a vector of parameters
▶ α(1), · · · , α(N ): parameters capturing the degree of competition. We expect α(1) ≤ α(2) ≤ · · · α(N ).
Model (4/6): Fixed Costs
Fm(n), an incumbent firm in market m when there are n active firms, is specified:
Fm(n) = XCmγ− δ(n) + εm,
where
▶ XC
m: the vector of observed market characteristics that may affect the demand of the product, e.g., rental price
▶ γ: a vector of parameters
▶ εm: an unobserved characteristics to econometricians but observable for firms
▶ δ(1), · · · , δ(N ): parameters - the interpretation is not completely clear. Allowing the fixed cost to depend on the number of firms in the market for robustness reasons.
Model (5/6): Interpretation of δ
There are several possible interpretations for why fixed costs may depend on the number of firms in the market:
▶ Entry Deterrence
Incumbents create barriers to entry.
▶ Firm Heterogeneity in Fixed Costs Late entrants are less efficient in fixed costs.
▶ Endogenous Fixed Costs
Rental prices or other components of the fixed costs, which is not included in XCm, may increase with the number of
incumbents (e.g. demand effect on rental prices). Therefore, we expect δ(1) ≤ δ(2) ≤ · · · δ(N ).
Model (6/6): Parameters Estimated
▶ Since both α(n) and δ(n) increase in n, Πm(n) declines with n.
▶ For n ∈ {0, 1, 2, · · · , N }
{nm= n} ⇔ {Πm(n) ≤ 0 and Πm(n + 1) < 0}
▶ The model has unique solution for any value of the exogenous variables and parameters.
▶ Parameters estimated:
θ= {β, γ, α(1), · · · , α(N ), δ(1), · · · , δ(N )}
Estimation (1/3): Distributional Assumption
▶ Assumption: εm is independent of (Sm, XDm, XCm) and it is i.i.d. over markets wish distribution N (0, σ).
▶ Alternatively, we could say: fixed costs are independently distributed across markets according to distribution of Φ(F |Sm, XDm, XCm).
▶ Suppose N firms are active in the market m: Πm(N ) = Vm(N ) − Fm(N ) − εm≥ 0
Πm(N + 1) = Vm(N + 1) − Fm(N + 1) − εm <0, or
Vm(N + 1) − Fm(N + 1) < εm ≤ Vm(N ) − Fm(N ).
Estimation (2/3): Ordered Probit
▶ Given the market structure, (Sm, XDm, XCm), probability of observing N firms should be:
Prob(nm = N |Sm, XDm, XCm)
Estimation (2/3): Ordered Probit
▶ Given the market structure, (Sm, XDm, XCm), probability of observing N firms should be:
Prob(nm = N |Sm, XDm, XCm)
= Prob(Vm(N + 1) − Fm(N + 1) < εm≤ Vm(N ) − Fm(N )
|Sm, X)
Estimation (2/3): Ordered Probit
▶ Given the market structure, (Sm, XDm, XCm), probability of observing N firms should be:
Prob(nm = N |Sm, XDm, XCm)
= Prob(Vm(N + 1) − Fm(N + 1) < εm≤ Vm(N ) − Fm(N )
|Sm, X)
= Φ(Vm(N ) − Fm(N )|Sm, X)
−Φ(Vm(N + 1) − Fm(N + 1)|Sm, X)
Estimation (2/3): Ordered Probit
▶ Given the market structure, (Sm, XDm, XCm), probability of observing N firms should be:
Prob(nm = N |Sm, XDm, XCm)
= Prob(Vm(N + 1) − Fm(N + 1) < εm≤ Vm(N ) − Fm(N )
|Sm, X)
= Φ(Vm(N ) − Fm(N )|Sm, X)
−Φ(Vm(N + 1) − Fm(N + 1)|Sm, X)
= Φ(Sm(XDmβ− α(N )) − XCmγ+ δ(N )|Sm, X)
−Φ(Sm(XDmβ− α(N + 1)) − XCmγ+ δ(N + 1)|Sm, X)
▶ This is identical to the ordered probit model!
Estimation (3/3): Objective function
▶ Objective function should be given by θˆ= arg max
θ M
∑
m=1
log[Prob(nm= Nm∗|Sm, XDm, XCm)].
▶ Most of econometrics software packages includes a command for the estimation of the ordered probit model and you can easily estimate the model.
Data (1/2): Isolated Market
▶ 202 “isolated markets” based on two criteria:
▶ Small: at least 20 miles (36 km) away from the nearest town of 1,000 people or more.
▶ Far away: at least 100 miles (160 km) away from cities with 100,000 people or more.
▶ See Figure 2 in page 986.
▶ Industry: Retail industries and professional services
▶ Tire dealers, Plumbers, movie theaters...
▶ Doctors, dentists, barbers...
▶ See Table 2 in page 987.
▶ The model estimated for each industry.
Data (2/2): Market Observable Variables
▶ Population variables (Sm):
▶ Town population
▶ Town population growth
▶ Commuters out of the county etc..
▶ Demographic variables (XDm, XCm):
▶ Elderly ration
▶ Per capita income
▶ Heating degree days
▶ Housing units etc..
▶ See Table 3.
Estimation Results
▶ See Table 4 in page 994. Notice
Vm(n) = Sm(XDβ− α(n)) where α(k) = α1− · · · − αk
Fm(n) = XCγ− δ(n) where δ(k) = γ1+ · · · + γk
▶ α(n) is increasing in n
▶ δ(n) (γ in the table) is increasing in n
Entry Threshold (1/2)
▶ Let S(n) be the minimum market size to sustain n firms in the market (it is called entry threshold):
SN = X
C
mγˆ− ˆδ(N )
XDmβˆ − ˆα(N ).
▶ See Table 5 Panel (A) in page 995.
▶ Druggist and Tire dealers require 500 people in town.
▶ Dentist and Doctors require 700-900 people.
▶ Plumber requires even more.
Entry Threshold (2/2): More Insight
▶ Let s
N = SNN be the per firm entry threshold.
▶ The ratio shows how would one additional entrant change the competitiveness.
▶ For example: Doctors industry
▶ For monopolistic case, it requires 880 people.
▶ For duopolistic case, it requires 3490 people in total.
Entry Threshold (2/2): More Insight
▶ Let s
N = SNN be the per firm entry threshold.
▶ The ratio shows how would one additional entrant change the competitiveness.
▶ For example: Doctors industry
▶ For monopolistic case, it requires 880 people.
▶ For duopolistic case, it requires 3490 people in total.
▶ Each firm requires 1745 people for break even profit.
▶ Ratio 1.98 implies the additional entrant change the competitiveness a lot!!
Entry Threshold (2/2): More Insight
▶ Let s
N = SNN be the per firm entry threshold.
▶ The ratio shows how would one additional entrant change the competitiveness.
▶ For example: Doctors industry
▶ For monopolistic case, it requires 880 people.
▶ For duopolistic case, it requires 3490 people in total.
▶ Each firm requires 1745 people for break even profit.
▶ Ratio 1.98 implies the additional entrant change the competitiveness a lot!!
▶ For trigopolistic case, it requires 5780 people in total.
▶ Each firm requires 1920 people for break even profit.
▶ Ratio 1.1 implies the additional entrant does not change the competitiveness.
▶ LR test support at least the null hypothesis of s4 = s5 for all industries.
Some Issues
▶ Are firms in each industry/market homogeneous?
▶ What is the intuition behind the identification of the effect of competition in this model? Are not there any endogeneity problems?
▶ How is the number of potential entrants chosen in the model? Are the estimates of the other parameters very sensitive to such the numbers of potential entrants?
▶ What if competition is not really at the level of local markets but global markets?
Bresnahan and Reiss (1991, JoE)
Multiple Outcomes (1/6): A Simple Example
▶ A problem of multiple equilibria is a typical in entry models (discrete games with interdependent choices).
▶ A simple example: two firms entry model
▶ One-shot game (static)
▶ Two potential entrants: firm 1 and 2
▶ Two actions: ‘0’ (Do not enter) or ‘1’ (Enter)
▶ Perfect information about each other’s profit depending on their actions: π1(D1, D2) and π2(D1, D2)
▶ Consider simultaneous Nash equilibrium
Multiple Outcomes (2/6): Payoff Matrix
▶ Profit is given by:
D2 = 0 D2 = 1 D1 = 0 (0, 0) (0, π2M) D1 = 1 (πM1 ,0) (π1D, π2D) assuming π1D < πM1 and πD2 < πM2 .
▶ Observing (D1, D2), we want to recover (π1, π2).
Multiple Outcomes (2/6): Payoff Matrix
▶ Profit is given by:
D2 = 0 D2 = 1 D1 = 0 (0, 0) (0, π2M) D1 = 1 (πM1 ,0) (π1D, π2D) assuming π1D < πM1 and πD2 < πM2 .
▶ Observing (D1, D2), we want to recover (π1, π2).
▶ We decompose firm’s profit into an observable and additively separable unobservable components:
πj =
0 if Dj = 0
πMj (x, zj) + ϵj if Dj = 1 and Dk = 0 πDj (x, zj) + ϵj if Dj = 1 and Dk = 1 where π is the observable profits depending on market observable, x, and firm-specific characteristics, zj.
Multiple Outcomes (3/6): Profit function specification
D2 = 0 D2 = 1 D1 = 0 (0, 0) (0, πM2 ) D1 = 1 (π1M,0) (π1D, πD2 )
▶ The threshold conditions supporting the possible entry outcomes:
Market Outcome N Conditions on Profits
Multiple Outcomes (3/6): Profit function specification
D2 = 0 D2 = 1 D1 = 0 (0, 0) (0, πM2 ) D1 = 1 (π1M,0) (π1D, πD2 )
▶ The threshold conditions supporting the possible entry outcomes:
Market Outcome N Conditions on Profits No firms 0 π1M + ε1<0 and πM2 + ε2 <0
Multiple Outcomes (3/6): Profit function specification
D2 = 0 D2 = 1 D1 = 0 (0, 0) (0, πM2 ) D1 = 1 (π1M,0) (π1D, πD2 )
▶ The threshold conditions supporting the possible entry outcomes:
Market Outcome N Conditions on Profits No firms 0 π1M + ε1<0 and πM2 + ε2 <0 Firm 1 Monopoly 1 0 < πM1 + ε1 and π2D+ ε2 <0
Multiple Outcomes (3/6): Profit function specification
D2 = 0 D2 = 1 D1 = 0 (0, 0) (0, πM2 ) D1 = 1 (π1M,0) (π1D, πD2 )
▶ The threshold conditions supporting the possible entry outcomes:
Market Outcome N Conditions on Profits No firms 0 π1M + ε1<0 and πM2 + ε2 <0 Firm 1 Monopoly 1 0 < πM1 + ε1 and π2D+ ε2 <0 Firm 2 Monopoly 1 πD1 + ε1 <0 and 0 < π2M + ε2
Multiple Outcomes (3/6): Profit function specification
D2 = 0 D2 = 1 D1 = 0 (0, 0) (0, πM2 ) D1 = 1 (π1M,0) (π1D, πD2 )
▶ The threshold conditions supporting the possible entry outcomes:
Market Outcome N Conditions on Profits No firms 0 π1M + ε1<0 and πM2 + ε2 <0 Firm 1 Monopoly 1 0 < πM1 + ε1 and π2D+ ε2 <0 Firm 2 Monopoly 1 πD1 + ε1 <0 and 0 < π2M + ε2
Duopoly 2 0 < π1D+ ε1 and 0 < πD2 + ε2
▶ (Need to copy these conditions to blackboard)
Multiple Outcomes (4/6): Graphical Expression
−πM1 −π1D
−πM2
−π2D
ε1
ε2
▶ Multiple equilibria: either firm 1 or 2 could be a monopolist.
Multiple Outcomes (4/6): Graphical Expression
−πM1 −π1D
−πM2
−π2D
ε1
ε2
No firms
▶ Multiple equilibria: either firm 1 or 2 could be a monopolist.
Multiple Outcomes (4/6): Graphical Expression
−πM1 −π1D
−πM2
−π2D
ε1
ε2
No firms
Both firms
▶ Multiple equilibria: either firm 1 or 2 could be a monopolist.
Multiple Outcomes (4/6): Graphical Expression
−πM1 −π1D
−πM2
−π2D
ε1
ε2
No firms
Both firms
Firm 1 Monopoly
▶ Multiple equilibria: either firm 1 or 2 could be a monopolist.
Multiple Outcomes (4/6): Graphical Expression
−πM1 −π1D
−πM2
−π2D
ε1
ε2
No firms
Both firms
Firm 1 Monopoly Firm 2 Monopoly
▶ Multiple equilibria: either firm 1 or 2 could be a monopolist.
Multiple Outcomes (4/6): Graphical Expression
−πM1 −π1D
−πM2
−π2D
ε1
ε2
No firms
Both firms
Firm 1 Monopoly Firm 2 Monopoly
Firm 1 or 2 Monopoly
▶ Multiple equilibria: either firm 1 or 2 could be a monopolist.
Multiple Outcomes (5/6)
▶ We cannot construct the log likelihood function. Why?
Multiple Outcomes (5/6)
▶ We cannot construct the log likelihood function. Why?
▶ Define the mutually exclusive outcome indicators:
▶ Y1= 1(D1= 0, D2= 0)
▶ Y2= 1(D1= 1, D2= 0)
▶ Y3= 1(D1= 0, D2= 1)
▶ Y4= 1(D1= 1, D2= 1)
what we can observe in the data.
Multiple Outcomes (5/6)
▶ We cannot construct the log likelihood function. Why?
▶ Define the mutually exclusive outcome indicators:
▶ Y1= 1(D1= 0, D2= 0)
▶ Y2= 1(D1= 1, D2= 0)
▶ Y3= 1(D1= 0, D2= 1)
▶ Y4= 1(D1= 1, D2= 1)
what we can observe in the data.
▶ Assuming ϵ
j ∼ N (0, 1), likelihood for each outcome should be:
▶ Pr(Y1= 1) = Φ(−πM1 )Φ(−π2M)
▶ Pr(Y2= 1) ≤ [1 − Φ(−πD1)]Φ(−πM2 )
▶ Pr(Y3= 1) ≤ Φ(−πM1 )[1 − Φ(−πD2)]
▶ Pr(Y4= 1) = [1 − Φ(−πD1)][1 − Φ(−πD2)]
▶ Without additional information, we cannot estimate the model.
Multiple Outcomes (6/6): Potential Solutions
1. Additional Conditions
Assumptions for equilibrium selection and refinements 2. Aggregating Outcomes - Bresnahan and Reiss (1991)
Focus on the aggregated outcome, N, not individual decision 3. Timing - Sequential Entry - Berry (1992)
It guarantees a unique solution, but requires additional data 4. Incomplete Information [Modern Standard] - Seim (2006)
Introducing incompleteness of information structure 5. Partial Identification - Pakes, Porter, Ho and Ishii (2015)
Giving up point identification...
Berry (1992, Econometrica)
Overview (1/4): A Structure of the Game
▶ A simple two-stage game in each market m:
1. Kmpotential firms choose to be “in” or “out” of the market 2. When choosing “in” in the first stage, they play some game
that determines post-entry profits
Overview (1/4): A Structure of the Game
▶ A simple two-stage game in each market m:
1. Kmpotential firms choose to be “in” or “out” of the market 2. When choosing “in” in the first stage, they play some game
that determines post-entry profits
▶ Berry made two assumptions for timing in the first stage:
▶ The most profitable firms move first
▶ Incumbents move first with more profitable incumbents moving before other incumbents, and then more profitable entrants move before other entrants
Overview (1/4): A Structure of the Game
▶ A simple two-stage game in each market m:
1. Kmpotential firms choose to be “in” or “out” of the market 2. When choosing “in” in the first stage, they play some game
that determines post-entry profits
▶ Berry made two assumptions for timing in the first stage:
▶ The most profitable firms move first
▶ Incumbents move first with more profitable incumbents moving before other incumbents, and then more profitable entrants move before other entrants
▶ A profit function for firm j in market m as πjm= vm(N (s)) − ϕjm+ εjm.
Overview (2/4): Result from the Specification
▶ Result: Given the profit function, a function v(N ) that is strictly decreasing in N , and a vector ϕjm, all pure strategy Nash equilibria in market m involve a unique number, N∗, of entering firms.
Overview (3/4): How to Estimate the Model
▶ Basic idea: minimize the prediction error νm = Nm− E[nm|Xm, Zm, θ], where
E[nm|Xm, Zm, θ] =
∫
· · ·
∫
n∗mdF(ε1m, ε2m,· · · , εKm,m)
▶ Estimate the model using the method of moments: θˆ= arg min
θ
1 M
M
∑
m=1
(Nm− E[nm|Xm, Zm, θ])2.
Overview (4/4): Useful Simulation Technique
▶ We use simulation technique: E[nm|Xm, Zm, θ] ≈ 1
S
∑
s
n∗,sm (εs1m, εs2m,· · · , εsKm,m; θ)
where the equilibrium number of firms, n∗,sm, can be calculated for a random draw of (εs1m, εs2m,· · · , εsKm,m).
▶ Moreover, we calculate n∗,sm via n∗,sm(εs, θ) = max
0≤n≤Ki[ n | #(k|πjm(n, εs) ≥ 0) ≥ n], which is the largest integer n such that at least n firms are profitable in an equilibrium.
General Framework
This part largely comes from the slides by Pat Bajari
A Simple Example (1/6): Setup
▶ Data contains on a cross section of markets, m = 1, 2, · · · , M
▶ In each market, two firms, i = 1, 2, make their entry decisions
▶ Let ai,t = 1 and ai,t = 0 denote entry and no entry, respectively.
A Simple Example (1/6): Setup
▶ Data contains on a cross section of markets, m = 1, 2, · · · , M
▶ In each market, two firms, i = 1, 2, make their entry decisions
▶ Let ai,t = 1 and ai,t = 0 denote entry and no entry, respectively.
▶ Economic theory suggests that profits depend on demand, costs, and the number of competitors:
▶ popm is population of market m (demand)
▶ distim is a distance from the nearest plant (cost)
▶ a−i,t denotes entry by the competitor
A Simple Example (2/6): Setup
▶ The profit of firm i is defined as uim(aim, a−im) =
{αpopm+ βdistim+ δa−it+ εim, if aim= 1
0, otherwise.
▶ ε
im is private information - incomplete information games!
A Simple Example (2/6): Setup
▶ The profit of firm i is defined as uim(aim, a−im) =
{αpopm+ βdistim+ δa−it+ εim, if aim= 1
0, otherwise.
▶ ε
im is private information - incomplete information games!
A Simple Example (3/6): An Equilibrium
▶ Suppose σi(aim= 1) is probability that i enters market m
▶ In a Bayesian-Nash equilibrium, agent i must make best response to σ−i(a−it= 1)
▶ Therefore i’s decision rule must be:
aim= 1 ⇔ α · popm+ β · distim+ δ · σ−i(a−it= 1) + εim>0
▶ Assuming Type I extreme value distribution for error terms, we have
σi(aim= 1) = exp(α · popm+ β · distim+ δ · σ−i(a−im = 1)) 1 + exp(α · popm+ β · distim+ δ · σ−i(a−im = 1))
A Simple Example (4/6): An Equilibrium
Equilibrium - two equations with two unknowns
σ1(a1m = 1) = exp(α · popm+ β · distim+ δ · σ2(a1m= 1)) 1 + exp(α · popm+ β · distim+ δ · σ2(a1m= 1)) σ2(a2m = 1) = exp(α · popm+ β · distim+ δ · σ1(a2m= 1))
1 + exp(α · popm+ β · distim+ δ · σ1(a2m= 1))
A Simple Example (5/6): Two-Step Estimator
First, estimate ˆσi(ai|popm, dist1m, dist2m) using a flexible method.
▶ This is the frequency that entry is observed empirically.
▶ We recover agents’ equilibrium beliefs by using the sample analogue.
▶ Given the first stage, agent i’s decision rule is estimated as aim= 1 ⇔ α · popm+ β · distim+ δ · ˆσ−i(a−it= 1) + εim>0
▶ Therefore, the probability that i choose to enter the market is: σi(aim= 1) = exp(α · popm+ β · distim+ δ · ˆσ−i(a−im = 1))
1 + exp(α · popm+ β · distim+ δ · ˆσ−i(a−im = 1)) which is identical to a standard conditional logit model!
A Simple Example (5/6): Two-Step Estimator
Second, estimate (α, β, γ) using maximum likelihood
L(α, β, γ) =
M
∏
m=1 2
∏
i=1
Prob(ait = 1|α, β, γ, ˆσ−i)1{qi,m=1}
· (1 − Prob(ait= 1|α, β, γ, ˆσ−i))1{qi,m=0} where
Prob(ait= 1|α, β, γ, ˆσ−i) =
exp(α · popm+ β · distim+ δ · ˆσ−i(a−im = 1)) 1 + exp(α · popm+ β · distim+ δ · ˆσ−i(a−im= 1))
A Simple Example (6/6): Some Comments
▶ Computationally tractable and accurate.
▶ Easy to include unobserved heterogeneity.
▶ Can be generalized easily, including dynamic models.
General Model (1/9): Setup
▶ Players, i = 1, . . . , n and actions ai∈ {0, 1}.
▶ This two-strategy assumption can be relaxed
▶ Let A = {0, 1}n and a = (a1, . . . , an).
▶ Let s ∈ S denote a vector of state variables.
▶ sis common knowledge and observed by econometricans too
▶ Per-period utility is given by
ui(a, s, εi) = Πi(ai, a−i, s) + εi
▶ Preference shock εi is private information
▶ Πi(ai, a−i, s) is mean utility
▶ Πi(0, a−i, s) = 0 is assumed for normalization
General Model (2/9): Setup
▶ Define the choice specific value function: Πi(ai = 1, s) =∑
a−i
σ−i(a−i|s)Πi(ai = 1, a−i, s)
which can be viewed as the expected utility from choosing ai = 1, excluding the preference shock (εi)
General Model (2/9): Setup
▶ Define the choice specific value function: Πi(ai = 1, s) =∑
a−i
σ−i(a−i|s)Πi(ai = 1, a−i, s)
which can be viewed as the expected utility from choosing ai = 1, excluding the preference shock (εi)
▶ The optimal decision rule must satisfy:
ai= 1 ⇔ Πi(ai = 1, s) + εi >0.
General Model (2/9): Setup
▶ Define the choice specific value function: Πi(ai = 1, s) =∑
a−i
σ−i(a−i|s)Πi(ai = 1, a−i, s)
which can be viewed as the expected utility from choosing ai = 1, excluding the preference shock (εi)
▶ The optimal decision rule must satisfy:
ai= 1 ⇔ Πi(ai = 1, s) + εi >0.
▶ Type I extreme value distribution enables us to have σi(ai = 1|s) = exp(Πi(ai = 1, s))
1 + exp(Πi(ai= 1, s))
General Model (3/9): Identification
▶ The model is identified if we can find Πi(ai, a−i, s) that uniquely rationalize σi(a|s).
▶ Π
i(ai, a−i, s) is a non-parametric function of (ai, a−i, s).
▶ Assume that the error terms εi(ai) are distributed i.i.d. with Type I extreme value across actions and agents.
▶ We CANNOT nonparametrically identify both error terms Πi(ai, a−i, s)
General Model (4/9): Identification
▶ We first use the “Hotz-Miller” inversion: σi(ai = 1|s) = exp(Πi(ai = 1, s))
1 + exp(Πi(ai = 1, s)) ⇒
Πi(ai= 1, s) = log(σi(ai= 1|s)) − log(σi(ai= 0|s))
▶ Intuitively, we can infer the choice specific value function Πi(ai= 1, s) from the observed choice probabilities, σi(ai= 1|s).
General Model (5/9): Identification
▶ Identification requires inversion of the following system: Πi(ai= 1, s) =∑
a−i
σ−i(a−i|s)Πi(ai = 1, a−i, s), ∀i = 1, . . . , n
▶ For a fixed s, there are n × 2n−1 unknowns, whereas there are only n equations.
▶ In general, this system cannot be inverted and the model is underidentified!
General Model (6/9): Identification
One way to identify the system is to impose exclusion restrictions
▶ Consider a partition s = (si, s−i)
▶ Suppose Π(ai, a−i, s) = Π(ai, a−i, si) depends only on si i.e. profit of firm i excludes distance of other agents, etc..
▶ Then we can rewrite the system as Πi(ai, si, s−i) =∑
a−i
σ−i(a−i|si, s−i)Πi(ai, a−i, si)
▶ By varing s−i we increase the number of equations but no the number of unknowns
General Model (7/9): 3-Step Nonparametric Estimation
Step 1: Estimation of Choice Probabilities
▶ There are m = 1, . . . , M repetitions of the game with actions ant states (ai,m, si,m)
▶ In the first step, estimate ˆσi(ai|s) of σi(ai|s) using flexible method
▶ sieve logit
▶ splines, orthogonal polynomials etc..
General Model (8/9): 3-Step Nonparametric Estimation
Step 2: Inversion
▶ Perform the empirical analogue of the Hotz-Miller inversion Πˆi(ai= 1, sm) = log(ˆσi(ai= 1|sm)) − log(ˆσi(ai= 0|sm)) which gives estimates of the choice specific value function.
General Model (9/9): 3-Step Nonparametric Estimation
. Step 3: Recovering the Structural Parameters
▶ We “invert” our system of equations to estimate Πi(ai, a−i, s)
▶ Choose Πi(ai, a−i, s) which minimize the following weighted least square problem:
1 M
M
∑
m=1
ˆΠi(ai, sm) −∑
a−i
ˆ
σ−i(a−i|si, s−i)Πi(ai, a−i, si)
2
w(m, si)
▶ Note that ˆΠi(ai, sm) and∑a−iˆσ−i(a−i|si, s−i)Πi(ai, a−i, si) are from the first and second steps
▶ The weights w(m, si) can be kernel weights: K( sim− si
h )
which measures the distance between sim and si
Seim (2006, RAND)
Seim (2006) (1/15): Introduction
Katja Seim (2006): “An Empirical Model of Firm Entry with Endogenous Product-type Choices,” RAND Journal of Economics, vol. 37(3), pp. 619-640.
▶ As well as dealing with an important question, the paper has methodological contributions:
▶ Relaxing the assumption of isolated markets
▶ Endogenizing product characteristics (i.e., location choice)
▶ Introducing incomplete information in firms’ profits
Seim (2006) (2/15): Model (1/8) Location Choice
▶ Consider the market m ∈ {1, 2, · · · , M }.
▶ In each market m, there exists a set of Fm potential entrants.
▶ The number of potential entrants is known to all firms.
▶ All firms simultaneously choose whether or not to enter market m and where to locate.
▶ The set of possible locations in market m is indexed by l= 0, 1, 2 · · · , Lm.
▶ A firm enter at most 1 location in market m.
▶ A decision of firm f is denoted by df = (0, 0, · · · , 1, 0, · · · , 0), where the decision not enter the market is denoted by l = 0.
Seim (2006) (3/15): Model (2/8) Profit
▶ Upon entry, firm f ’s payoff in location l in market m is: Πmf l = Xml β+ ξm+ h(Γml , nm) + ϵmf l,
where
▶ Xm: a vector of demand and cost characteristics specific to location l in market m.
▶ ξm: an unobservable exogenous market-level characteristics
▶ εm
f l: the idiosyncratic component of firm f ’s profits from operating in location l.
▶ h(Γm
l , nm): the effect on profits due to competition
▶ Γm
.l: a matrix of competitive effects (TBE)
▶ nm: a vector containing the number of firms in each of the Lmlocations in market m
Seim (2006) (4/15): Model (3/8) Assumptions
1. Assumption I (Independent symmetric private values): Players’ profitability types (ϵm1 ,· · · , ϵmFm) are private information to the players and are iid draws from the distribution G(·).
Seim (2006) (4/15): Model (3/8) Assumptions
1. Assumption I (Independent symmetric private values): Players’ profitability types (ϵm1 ,· · · , ϵmFm) are private information to the players and are iid draws from the distribution G(·). 2. Assumption II : h(Γml , nm) =∑k=1Lm γklnmk.
▶ Competitors’ effects are additively separable across locations
▶ The incremental impact on payoffs of an additional firm in a given location is constant
3. Assumption III : γkl= γk′l= γb, if Db≤ dmkl and dmk′l <Db+1, where ( Db, Db+1) denote cutoffs that define a distance band.
Seim (2006) (4/15): Model (3/8) Assumptions
1. Assumption I (Independent symmetric private values): Players’ profitability types (ϵm1 ,· · · , ϵmFm) are private information to the players and are iid draws from the distribution G(·). 2. Assumption II : h(Γml , nm) =∑k=1Lm γklnmk.
▶ Competitors’ effects are additively separable across locations
▶ The incremental impact on payoffs of an additional firm in a given location is constant
3. Assumption III : γkl= γk′l= γb, if Db≤ dmkl and dmk′l <Db+1, where ( Db, Db+1) denote cutoffs that define a distance band.
Assumption II and III implies:
h(Γml , nm) =
B
∑
b=1
γbNbl, where Nbl =∑
k
Iklnk.
Seim (2006) (5/15): Model (4/8) An Example
▶ Consider the payoffs to locate in cell 7.
▶ Then h(·, ·) is given by
h(·, ·) = [γ0N7+ γ1(N4+ N5+ N8)
γ2(N1+ N2+ N3+ N6+ N9)].
Seim (2006) (6/15): Model (5/8) Equilibrium Conjecture
From now, let omit the superscript m.
▶ Due to imperfect information about other firms’ profitability, a firm form an expectation of its profit when locating at l:
E[Πf l] = Xlβ+ ξ +
B
∑
b=1
E[Nbl] + ϵf l
= E[ ¯Πf l] + ϵf l,
▶ The probability that competitor g chooses location l is: pgl(dgl= 1|ξ, X, E, θ)
= Pr(E[ ¯Πgl] + ϵgl ≥ E[ ¯Πgk] + ϵgk, ∀k ̸= l, ∀g ̸= f ),
▶ The expected number of firms entering each distance band b collapses to
E[Nbl] =∑
k
IbklE[nk] =∑
k
Ibkl(E − 1)pgk+ Ib=0.
Seim (2006) (7/15): Model (6/8) Equilibrium Conjecture
▶ Assuming Type I EV disturbance, the probability that firm g chooses location l is:
pgl = ∑Lexp(E[ ¯Πgl])
k=1exp(E[ ¯Πgk])
▶ In a symmetric Bayesian Nash equilibrium, every firm has the same equilibrium conjecture of its competitors’ location choices, namely pg= pf = p∗:
p∗l = exp( ¯Πl(p
∗, X,E, θ))
∑L
k=1exp( ¯Πk(p∗, X,E, θ))
∀l = 1, · · · , L.
Seim (2006) (8/15): Model (7/8) An Example
▶ Consider the payoffs to locate in cell 7.
▶ The profit, E[ ¯Π7], is given by E[ ¯Π7] = ξ + X7β+ γ0
+(E − 1)[γ0p∗7+ γ1(p∗4+ p∗5+ p∗8) γ2(p∗1+ p∗2+ p∗3+ p∗6+ p∗9)].
Seim (2006) (9/15): Model (8/8) Equilibrium
▶ The probability of entering the market is given by:
Pr(entry) =
exp(ξ)[∑Ll=1exp(E[ ¯Πl(p∗, X,E, θ)])] 1 + exp(ξ)[∑Lk=1exp(E[ ¯Πk(p∗, X,E, θ)])] .
▶ As a consequence, the expected number of entrants is simply E = F · Pr(entry).
which enables us to solve for ξ.
Seim (2006) (10/15): Nested ML (1/2)
▶ Two set of parameters of interest:
▶ θ1= {β, γ}: The parameters in the profit function.
▶ θ2= (µ, σ): The parametric for ξ, ξ ∼ N (µ, σ2).
▶ Three step estimation procedure:
1. Given (θ1, θ2), solve for p∗mvia fixed point algorithm:
p∗l = exp( ¯Πl(p
∗, X,E, θ))
∑L
k=1exp( ¯Πk(p
∗, X,E, θ)) ∀l = 1, · · · , L.
2. Given E, F, and p∗, solve for ξ using
ξ= ln(E) − ln(F − E) − ln ( L
∑
l=1
exp( ¯Πgl(X, p∗,E, θ1)) )
.
3. Search for (θ1, θ2) which maximizes the log likelihood, L(θ1, θ2), given in the next slide.
