# Chapter 6 Entry and Exit

## 6.1 Motivations

- By studying the entry decisions of firms, we can identify the profit function including the entry cost of firms.
- A profit function is the reduced-form parameter of the underlying demand function, cost function, and conduct parameters.
- The profit function can be identified without assuming a particular conduct.
- The parameter is informative enough to answer questions regarding the market structure and producer surplus.
- In the last chapter, we discussed the identification of conduct, in which we learned that the exogenous change in the number of firms in a market gives some information about the conduct.
- In the entry/exit analysis, we study the relationship between the market size, which exogenously changes the equilibrium number of firms, and the change in the market structure to infer the conduct.
- Entry and exit is not all about firms.
- The decision of launching a product is a sort of entry decision and the decision of abolishing a product is a sort of exit decision.
- The framework in this chapter can be applied to a wider class of problems.
- This chapter is mostly based on Steve Berry & Tamer (2006).

### 6.1.1 Entry Cost, Mode of Competition, and Market Structure

- Fixed and sunk entry costs and mode of competition are key determinants for market structure (Sutton, 2007).

- The tougher the mode of competition, the less firms can earn enough profit to compensate the entry cost.
- Therefore, the tougher the mode of competition, the number of firms in the market in the equilibrium cannot grow when the market size increases.

### 6.1.2 Exogenous and Endogenous Entry Cost

**Exogenous fixed and sunk entry cost**:- The cost of entry is the same across modes of entry.
**Endogenous fixed and sunk entry cost**:- The cost differs across modes of entry.
- For example, firms decide the quality of the product upon entering the market and the the cost of entry is increasing in the quality choice.
- If endogenous fixed and sunk entry cost is relevant, the entry cost to compete with the incumbent and have a positive profit will increase as the the number of incumbent firms increases.
- Therefore, the equilibrium firm number will be small and firm size will be large when endogenous fixed and sunk entry cost is relevant.

## 6.2 Monopoly Entry

### 6.2.1 Variable Profits and Fixed Costs

Consider a cross-section of markets, with one potential entrant in each market.

Profits in market \(i\) are given by: \[ \pi(x_i, F_i) \equiv v(x_i) - F_i. \]

\(v(x)\) is deterministic and \(F\) is random.

Typically, \(v(x)\) is interpreted as the variable profits and \(F\) as the fixed costs.

The potential entrant will enter market \(i\) if and only if: \[ F_i \le v(x_i). \]

The parameters of interest are \(v\) and the distribution of \(F\), \(\Phi\).

In general, \(F\) in a market can be correlated with \(x\).

### 6.2.2 Non-Identification

- Assume that \(F\) is independent of \(x\).
- Notice that a monotonic transformation of both sides of: \[ F_i \le v(x_i). \] will not change the entry probability.
- Therefore, without further restrictions, \(v\) and \(\Psi\) are at best identified only up to a monotonic transformation.

### 6.2.3 Restrictions for Identification.

- Keep assuming that \(F\) is independent of \(x\).
- Let \(p(x)\) is the observed entry probability at \(x\).

- \(v\) is known: Because \(v\) is the variable profit function, we can identify it directly from the demand function and cost function.
- Let \(z = v(x)\). Then \(\Psi\) is identified at \(z\) by: \[ \Psi(z) = \mathbb{P}\{F \le z\} = \mathbb{P}\{F \le v[v^{-1}(z)]\} = p[v^{- 1}(z)]. \]

- \(\Psi\) is known: Normalize the distribution of \(F\).
- Then, \(v\) is identified at \(x\) by: \[ v(x) = \Psi^{-1}[p(x)]. \]

- Impose shape restrictions on \(v\) (Matzkin, 1992):
- \(v(x)\) is homogeneous of degree 1 and there exists \(x_0\) such that \(v(x_0) = 1\). Then the functions \(v\) and \(\Psi\) are identified.
- Homogeneous of degree 1: \(v(z \cdot x) = z v(x)\).
- Let \(p(z, x_0)\) be the probability of entry at \(z\) and \(x_0\).
- Then, \(\Psi\) is identified at \(z\) by: \[ \Psi(z) = \mathbb{P}\{F \le z\} = \mathbb{P}\{F \le z v(x_0)\} = p(z, x_0). \]
- Then, \(v\) is identified by the second argument.

### 6.2.4 Homogeneous of Degree 1 Variable Profits

- Sufficient condition for the variable profit function to satisfy the stated condition:
- Demand is proportional to the population.
- Marginal cost is constant.
- Then, the variable profit is the market size \(z\) times the per-capita profit.
- We can normalize \(v\) at a market of size \(z = 1\) by choosing arbitrary \(x_0\).

## 6.3 Complete-Information Homogeneous Oligopoly Entry

### 6.3.1 Variable and Fixed Costs

- Timothy F. Bresnahan & Reiss (1991) pioneered the analysis of oligopoly entry models.
- We observe a cross-section of markets in which we observe the number of homogeneous firms and other market-specific characteristics.
- Let \(y_m\) be the number of firms in market \(m\).
- Let \(v_{y_m}(x_m)\) is the variable profit per firm in a market with number of firms \(y_m\) and the market characteristics \(x_m\).
- Let \(F_m\) be the market-specific fixed costs that are i.i.d. across markets with unknown distribution \(\Psi\).
- If there are \(y_m\) firms in market \(m\), the profit per firm in the market is: \[ \pi(y_m, x_m, F_m) \equiv v_{y_m}(x_m) - F_m. \]

### 6.3.2 Equilibrium Condition

The unique Nash equilibrium in the number of firms in market \(m\) is determined by the following equilibrium condition: \[ v_{y_m}(x_m) \ge F_m. \] \[ v_{y_m + 1}(x_m) < F_m. \]

Under this equilibrium, the probability of observing \(y\) firms in a market of type \(x\) is: \[ \mathbb{P}\{y = 0|x\} = 1 - \Psi[v_1(x)]. \] \[ \mathbb{P}\{y = 1|x\} = \Psi[v_1(x)] - \Psi[v_2(x)]. \] \[ \mathbb{P}\{y = 2|x\} = \Psi[v_2(x)] - \Psi[v_3(x)]. \] \[ \cdots \]

In other words, the probability of observing at least \(y\) firms in a market of type \(x\) is: \[ \mathbb{P}\{y \ge 1|x\} = \Psi[v_1(x)]. \] \[ \mathbb{P}\{y \ge 2|x\} = \Psi[v_2(x)]. \] \[ \mathbb{P}\{y \ge 3|x\} = \Psi[v_3(x)]. \] \[ \cdots \]

### 6.3.3 Identification under a Shape Restriction

- The identification argument for each \(y\) is the same as the monopoly entry case.
- For example, assume \(v_y(x) = z v_y(\tilde{x})\) and \(v_1(\tilde{x}_0) = 1\).
- Let \(P_y(z, \tilde{x})\) be the observed probability that the number of firms is no less than \(y\) in market of type \(z, \tilde{x}\).
- Then \(\Psi\) is identified at \(z\) by: \[ \Psi(z) = \mathbb{P}\{F \le z\} = \mathbb{P}\{F \le z v_1(\tilde{x}_0)\} = P_1(z, \tilde{x}_0). \]
- Then, the identification of \(v_y\) follows from: \[ v_y(\tilde{x}) = \frac{\Psi^{-1}[P_y(z, \tilde{x})]}{z}. \]

### 6.3.4 Log Likelihood Function

- The log likelihood of observing \(\{y_m\}_{m = 1}^M\) given \(\{x_m\}_{m = 1}^M\) is: \[ l(v, \Psi|\{y_m\}_{m = 1}^M, \{x_m\}_{m = 1}^M) = \sum_{m = 1}^M \log\{\Psi[v_{y_m}(x_m)] - \Psi[v_{y_m + 1}(x_m)]\}. \]
- This is a
**ordered**model in \(y_m\). - If \(\Psi\) is a normal distribution, it is called an
**ordered probit model**.

## 6.4 Complete-Information Heterogeneous Oligopoly Entry

### 6.4.1 Bivartiate Game with Heterogeneous Profits

- We observe a cross-section of markets in which there are two potential entrants.
- Let the profit of firm \(i\) in market \(m\) be: \[ \begin{split} \pi_{im}(x_{im}, y_{jm}, f_{im}) & \equiv v(y_{jm}, x_{im}) - f_{im}\\ &=v_{0i}(x_{im}) + y_{jm} v_{1i}(x_{im}) - f_{im}. \end{split} \]
- \(y_{im}\) and \(y_{jm}\) are the indicators of entry of firm \(i\) and \(j\) in market \(m\).
- \(x_{im}\) and \(x_{jm}\) are firm \(i\) and \(j\)’s characteristics in market \(m\).
- \(f_{im}\) and \(f_{jm}\) are the fixed costs of firm \(i\) and \(j\) in market \(m\).
- The second equation is without loss of generality because \(y_{jm}\) is a binary variable.
- The competitive effect of firm \(j\) on \(i\) and the effect o firm \(i\) on \(j\) are asymmetric.
- The parameters of interest are \(v_{0i}, v_{0j}, v_{1i}, v_{1j}\) and the joint distribution of \(f_{im}, f_{jm}\) conditional on \(x_{im}\) and \(x_{jm}\).
- Firms
**observe**both \(f_{im}, f_{jm}\) when they make decisions, but econometrician does not.

### 6.4.2 Sampling Assumption and Observations

- We have a random sample of observations on markets where every observation is an observable realization of an equilibrium game played between firm \(i\) and \(j\).
- Thus, we can observe:
- \(\mathbb{P}\{0, 0|x\}\): the probability that a market of type \(x\) has no firm.
- \(\mathbb{P}\{1, 0|x\}\): the probability that a market of type \(x\) has firm \(i\) but not firm \(j\).
- \(\mathbb{P}\{0, 1|x\}\): the probability that a market of type \(x\) has firm \(j\) but not firm \(i\).
- \(\mathbb{P}\{1, 1|x\}\): the probability that a market of type \(x\) has both firms.

### 6.4.3 Identification Assuming Pure-Strategy Equilibrium

Tamer (2003) considers the identification when there are only two potential entrants and the pure-strategy Nash equilibrium is assumed.

Assume that the data is from a pure-strategy equilibrium.

Then, the probabilities \(\mathbb{P}\{0, 0|x\}\) and \(\mathbb{P}\{1, 1|x\}\) are written as: \[ \mathbb{P}\{0, 0|x\} = \mathbb{P}\{f_{im} \ge v_{0i}(x_{im}), f_{jm} \ge v_{0j}(x_{jm})|x_{im}, x_{jm}\}. \] \[ \mathbb{P}\{1, 1|x\} = \mathbb{P}\{f_{im} \le v_{0i}(x_{im}) + v_{1i}(x_{im}), f_{jm} \le v_{0j}(x_{jm}) + v_{1j}(x_{jm})|x_{im}, x_{jm}\}. \]

Assume that \((f_{im}, f_{jm})\) are distributed independently of \((x_{im}, x_{jm})\) with a joint distribution \(F\).

Assume that \(v_{0i}(x_{im}) = z_{im} v_0(\tilde{x}_{im})\) and \(v_{0j}(x_{jm}) = z_{jm} v_0(\tilde{x}_{jm})\).

Assume that \(v_{0i}(\tilde{x}_0) = v_{0j}(\tilde{x}_0) = 1\).

Assume that \(v_{1i}\) and \(v_{1j}\) are non-positive.

Assume that \(z_{im}| z_{jm}, \tilde{x}_{im}, \tilde{x}_{jm}\) has a distribution with support on \(\mathbb{R}\) and similar for \(z_{jm}\).

Then, \(F\) is identified by: \[ \begin{split} \mathbb{P}\{f_{im} \ge z_{im}, f_{jm} \ge z_{jm}\} &= \mathbb{P}\{ f_{im} \ge z_{im} v_0(\tilde{x}_{0}), f_{jm} \ge z_{jm} v_0(\tilde{x}_{0})\}\\ &= \mathbb{P}\{0, 0|z_{im}, \tilde{x}_0, z_{jm}, \tilde{x}_0\}. \end{split} \]

Then, push \(z_{jm} \to - \infty\) to get: \[ \begin{split} \mathbb{P}\{f_{im} \ge z_{im} v_0(\tilde{x}_{im})\} &= \lim_{z_{jm} \to - \infty} \mathbb{P}\{f_{im} \ge z_{im} v_0(x_{im}), f_{jm} \ge z_{jm} v_0(x_{jm})\}\\ &= \lim_{z_{jm} \to - \infty} \mathbb{P}\{0, 0|z_{im}, \tilde{x}_{im}, z_{jm}, \tilde{x}_{jm}\} \end{split} \]

Hence, \(v_{0}\) is identified by: \[ v_0(\tilde{x}_{im}) = \frac{F^{-1}[\lim_{z_{jm} \to - \infty} \mathbb{P}\{0, 0|z_{im}, \tilde{x}_{im}, z_{jm}, \tilde{x}_{jm}\}]}{z_{im}}. \]

The identification of \(v_{1i}\) and \(v_{1j}\) are similar.

### 6.4.4 Heterogeneous Independent Fixed Costs

- S. T. Berry (1992) considers several extensions to the homogeneous oligopoly models.
- Assume that the fixed costs are heterogeneous and independent across firms.
- c.f. In homogeneous oligopoly model, the fixed costs were perfectly correlated across firms in a market.

- Assume that the characteristics can be firm-specific (that can include market-specific characteristics).
- Assume that the variable profit function is homogeneous: \[ \pi_{y}(x_m, F_{im}) = v_y(x_{im}) - F_{im}. \]
- Assume that \(F\) is independent of \(x\).
- Suppose that we observe the
**number of potential entrants**in each market. - For example, in the airline industry, market is a city pair.
- The potential entrants into an airline city pair were those with some service out of at least one of the endpoints of the city pair.
- The variation in the number of potential entrants can be used to identify the model.
- Define: \[ \mu(x) = \mathbb{P}\{F_{im} < v_1(x)\}. \] \[ \delta(x) = \mathbb{P}\{F_{im} < v_2(x)\}. \]
- Suppose that we
**know**that there only two potential entrants into a market. - Among such markets, we have: \[ \mathbb{P}\{0, 0|x_1, x_2\} = [1 - \mu(x_1)] [1 - \mu(x_2)]. \] \[ \mathbb{P}\{1, 1|x_1, x_2\} = \delta(x_1) \delta(x_2). \]
- If we set \(x_1 = x_2 = x\), then we have: \[ \mathbb{P}\{0, 0|x, x\} = [1 - \mu(x)]^2. \] \[ \mathbb{P}\{1, 1|x, x\} = \delta(x)^2. \]
- They identify \(\mu\) and \(\delta\) at \(x\).
- Under shape restrictions, we can identify \(v\) and \(\Psi\) as well.

### 6.4.6 Inference Based On a Unique Prediction

- S. T. Berry (1992) develops a more general model built on the ideas above.
- The key for his analysis is that he ensures that
**there is a unique number of equilibrium entrants**. - This enables him to calculate the likelihood of observing a sequence of number of entrants, but at the cost of the generality of the underlying model.

### 6.4.7 Entry in the Airline Industry: One-shot Game

- Based on S. T. Berry (1992).
- A market = a city pair market at a single point in time.
- Consider a one-shot entry game that yields a network structure.
- At the beginning of the period, each firm takes its overall network structure as given and decides whether to operate in a given city pair
**independently**across markets.

### 6.4.8 Entry in the Airline Industry: Profit Function

- There are \(K_m\) potential entrants in market \(m\).
- Let \(y_m\) be a strategy profile.
- \(y_m = (y_{1m}, \cdots, y_{K_m m})'\), \(y_{im} \in \{0, 1\}\).
- The profit function for firm \(i\) in market \(m\): \[\begin{equation} \pi_{im}(y_m, f_{im}) = v_m\left(N_{m}\right) - f_{im}. \end{equation}\]
- \(N_{m} = \sum_{i = 1}^{K_m} y_{im}\).
- \(v_m\) is strictly decreasing in \(N_m\).

### 6.4.9 Entry in the Airline Industry: Profit Function

- The common term is assumed to be: \[ v_m(N) = x_m' \beta + h(\delta, N_m) + \rho u_{m}, \]
- \(x_m\) is the observed market characteristics, \(h(\cdot)\) is a function that is decreasing in \(N_m\), say, \(- \delta \ln (N_m)\).
- \(u_m\) is the market characteristics that is observed by firms but not by econometrician.
- The firm-specific term: \[ f_{im} = z_{im}' \alpha + \sigma u_{im}, \]
- \(z_{im}\) is the observed firm characteristics.
- A scale normalization: \(\sigma = \sqrt{1 - \rho^2}\) \(\Rightarrow var(\rho u_m + \sigma u_{im}) = 1\).

### 6.4.10 Entry in the Airline Industry: Likelihood Function

- The observed part: \[ r_{im}(N) = x_m' \beta - \delta \ln (N_m) + z_{im}' \alpha. \]
- The unobserved part: \[ \epsilon_{im} = \sqrt{1 - \rho^2} u_{im} + \rho u_{m}. \]

### 6.4.11 Is the Equilibrium Number of Firms Unique?

- Either of the following conditions are sufficient:
- No firm-level unobserved heterogeneity: \(\rho = 1\).
- No market-level unobserved heterogeneity: \(\rho = 0\).
- The order of entry is predetermined, for example, the most profitable firms enter first.
- The incumbent firms enter first.
- Under either of the above assumptions, simulate the equilibrium number of firms in each market and match with the data.

## 6.5 Multiple Prediction

- If we further generalize the model, we will suffer from the problem of
**multiple prediction**. - First, even in S. T. Berry (1992)’s framework, the identity of entrants were not uniquely predicted.
- Second, if we allows for the asymmetric competitive effects, we will not have the unique number of equilibrium entrants.
- Third, the uniqueness of the equilibria may not hold once we allow for the mixed-strategy Nash equilibria.
- If we do not have the unique prediction on the endogenous variables, we cannot write down the likelihood function.

### 6.5.1 Multiple Equilibria in Bivariate Game with Heterogenous Profits

- Return to Tamer (2003)’s bivariate game with heterogeneous profits.
- Consider the pure-strategy Nash equilibrium when \(f_{im}, f_{jm}\) are realized.
- \((0, 0)\) is a pure-strategy Nash equilibrium if: \[ f_{im} \ge v_{0i}(x_{im}); \] \[ f_{jm} \ge v_{0j}(x_{jm}). \]
- \((1, 1)\) is a pure-strategy Nash equilibrium if: \[ f_{im} \le v_{0i}(x_{im}) + v_{1i}(x_{im}); \] \[ f_{jm} \le v_{0j}(x_{jm}) + v_{1j}(x_{jm}). \]
- \((0, 1)\) is a pure-strategy Nash equilibrium if: \[ f_{im} \ge v_{0i}(x_{im}) + v_{1i}(x_{im}); \] \[ f_{jm} \le v_{0j}(x_{jm}). \]
- \((1, 0)\) is a pure-strategy Nash equilibrium if: \[ f_{im} \le v_{0i}(x_{im}); \] \[ f_{jm} \ge v_{0j}(x_{jm}) + v_{1j}(x_{jm}). \]
- In a certain region of \(f_{im}, f_{jm}\), there are multiple equilibria.

### 6.5.2 Multiple Prediction in Bivariate Game with Heterogenous Profits

- In the orange region, we have multiple pure-strategy equilibria.
- If we allow for mixed-strategy equilibria, the region of \(f_{im}, f_{jm}\) on which there are multiple equilibria will increase.
- In this example, we can write the likelihood of the number of equilibrium entrants:
- \(2\): green area.
- \(1\): yellow, orange, and red areas.
- \(0\): blue area.

- However, we cannot write the likelihood of the strategy profile.
- \((1, 1)\): green area.
- \((0, 1)\): \(\ge\) red area, \(\le\) orange and red area.
- \((1, 0)\): \(\ge\) yellow area, \(\le\) orange and yellow area.
- \((0, 0)\): blue area.

### 6.5.3 Inference Based on Moment Inequalities

- According to the theory, we have: \[ \mathbb{P}\{0, 0|x\} = \mathbb{P}\{f_{i} \ge v_{0i}(x_{i}), f_{j} \ge v_{0j}(x_{j})\} \equiv H(0, 0|x). \]

\[ \mathbb{P}\{1, 1|x\} = \mathbb{P}\{f_{i} \le v_{0i}(x_{i}) + v_{1i}(x_{i}), f_{j} \le v_{0j}(x_{j}) + v_{1j}(x_{j}) \} \equiv H(1, 1|x). \]

\[ \mathbb{P}\{0, 1|x\} \ge \mathbb{P}\{f_{i} \ge v_{0i}, f_{j} \le v_{0j}\} + \mathbb{P}\{f_{i} \ge v_{0i} + v_{1i}, f_{j} \le v_{0j} + v_{1j}\} \equiv \underline{H}(0, 1|x). \]

\[ \mathbb{P}\{0, 1|x\} \le \underline{H}(0, 1|x) + \mathbb{P}\{v_{0i} + v_{1i} \le f_{i} \le v_{0i}, v_{0j} + v_{1j} \le f_{j} \le f_{0j}\} \equiv \overline{H}(0, 1|x). \]

\[ \mathbb{P}\{1, 0|x\} \ge \mathbb{P}\{f_{j} \ge v_{0j}, f_{i} \le v_{0i}\} + \mathbb{P}\{f_{j} \ge v_{0j} + v_{1j}, f_{i} \le v_{0i} + v_{1i}\} \equiv \underline{H}(1, 0|x). \]

\[ \mathbb{P}\{1, 0|x\} \le \underline{H}(0, 1|x) + \mathbb{P}\{v_{0} + v_{1i} \le f_{i} \le v_{0i}, v_{0j} + v_{1j} \le f_{j} \le f_{0j}\} \equiv \overline{H}(1, 0|x). \]

- The parameters should satisfy the moment conditions: \[ \mathbb{P}\{0, 0|x\} - H(0, 0|x) = 0; \] \[ \mathbb{P}\{1, 1|x\} - H(1, 1|x) = 0; \] \[ \min\left\{\mathbb{P}\{0, 1|x\} - \underline{H}(0, 1|x), 0\right\} = 0; \] \[ \max\left\{\mathbb{P}\{0, 1|x\} - \overline{H}(0, 1|x), 0\right\} = 0; \] \[ \min\left\{\mathbb{P}\{1, 0|x\} - \underline{H}(1, 0|x), 0\right\} = 0; \] \[ \max\left\{\mathbb{P}\{1, 0|x\} - \overline{H}(1, 0|x), 0\right\} = 0; \]
- We can estimate the parameters with the GMM method using the above modified moment conditions.
- Inference based on the moment inequalities are found in Andrews & Soares (2010).
- Ciliberto & Tamer (2009) study the entry exit of airlines when there are heterogeneous competitive effects using the above approach.

## 6.6 Incompelete-Information Heterogenous Oligopoly Entry

### 6.6.1 Bivariate Case

- There are two potential entrants.
- Let the profit of firm \(i\) in market \(m\) be: \[ \begin{split} \pi_{im}(x_{im}, y_{jm}, f_{im}) & \equiv v(y_{jm}, x_{im}) - f_{im}\\ &=v_{0i}(x_{im}) + y_{jm} v_{1i}(x_{im}) - f_{im}. \end{split} \]
- \(y_{im}\) and \(y_{jm}\) are the indicators of entry of firm \(i\) and \(j\) in market \(m\).
- \(x_{im}\) and \(x_{jm}\) are firm \(i\) and \(j\)’s characteristics in market \(m\).
- \(f_{im}\) and \(f_{jm}\) are the fixed costs of firm \(i\) and \(j\) in market \(m\).
- The second equation is without loss of generality because \(y_{jm}\) is a binary variable.
- The competitive effect of firm \(j\) on \(i\) and the effect of firm \(i\) on \(j\) are asymmetric.
- The parameters of interest are \(v_{0i}, v_{0j}, v_{1i}, v_{1j}\) and the joint distribution of \(f_{im}, f_{jm}\) conditional on \(x_{im}\) and \(x_{jm}\).
- Firm \(i\)
**observe**both \(f_{im}\) but**not**\(f_{jm}\) when it makes decision. - Firm \(j\)
**observe**both \(f_{jm}\) but**not**\(f_{im}\) when it makes decision. - Econometrican does not observe either of them.
- The joint distribution of \(f_{im}, f_{jm}\) is \(F\) (independent of \(x\)).

### 6.6.2 The Equilibrium Strategy and Belief

- The equilibrium strategy becomes a step function that decreases in a threshold: \[ y_{im} = 1\{f_{im} \le t_{im}\}. \] \[ y_{jm} = 1\{f_{jm} \le t_{jm}\}. \]
- Suppose that firm \(i\) believes that \(f_{jm}\) has a distribution of \(G_{jm}^{im}\) and firm \(j\) believes that \(f_{im}\) has a distribution of \(G_{im}^{jm}\).
- Usually, we assume a common and objective prior: \(G_{jm}^{im}(\epsilon_{jm}) = F(\epsilon_{jm}|\epsilon_{im})\) and \(G_{im}^{jm}(\epsilon_{im}) = F(\epsilon_{im}|\epsilon_{jm})\).
- When firm \(j\) follows strategy \(t_{jm}\), the expected payoff for firm \(i\) to enter is: \[ [v_{0i}(x_{im}) - f_{im}][1 - G_{jm}^{im}(t_{jm})] + [v_{0i}(x_{im}) + v_{1i}(x_{im}) - f_{im})] G_{jm}^{im}(t_{jm}). \]
- Thus, the threshold \(t_{im}\) is determined by: \[ [v_{0i}(x_{im}) - t_{im}][1 - G_{jm}^{im}(t_{jm})] + [v_{0i}(x_{im}) + v_{1i}(x_{im}) - t_{im})] G_{jm}^{im}(t_{jm}) = 0. \]
- In the same way, the threshold \(t_{jm}\) is determined by: \[ [v_{0j}(x_{jm}) - t_{jm}][1 - G_{im}^{jm}(t_{im})] + [v_{0j}(x_{jm}) + v_{1j}(x_{jm}) - t_{jm})] G_{im}^{jm}(t_{im}) = 0. \]
- This system of equations can have multiple solution.

### 6.6.3 Equilibrium Selection and Likelihood

- If we specify the equilibrium selection rule \(\mathbb{P}(t_{im}, t_{jm}|x_{im}, x_{jm}, f_{im}, f_{jm})\), then we can specify the likelihood of observing \((0, 0)\), \((0, 1)\), \((1, 0)\), and \((1, 1)\).
- If we do not, we will only have bounds on the likelihood of observing \((0, 0)\), \((0, 1)\), \((1, 0)\), and \((1, 1)\).
- Another way is to assume that
**the same equilibrium \(t_{im}^*(x_{im}, x_{jm}), t_{jm}^*(x_{im}, x_{jm})\) holds across markets**. - Then, the across market relative frequency of entries conditional on \(x_{im}, x_{jm}\) gives the estimates of the entry probabilities: \[ \widehat{G}_{im}(x_{im}, x_{jm}) \approx G_{im}^{jm}[t^*_{im}(x_{im}, x_{jm})], \] \[ \widehat{G}_{jm}(x_{im}, x_{jm}) \approx G_{jm}^{im}[t_{jm}^*(x_{im}, x_{jm})], \] for each \(x_{im}, x_{jm}\).
- The estimated distribution has to solve: \[ \widehat{G}_{im}(x_{im}, x_{jm}) = \mathbb{P}\{[v_{0i}(x_{im}) - f_{im}][1 - \widehat{G}_{jm}(x_{im}, x_{jm})] + [v_{0i}(x_{im}) + v_{1i}(x_{im}) - f_{im}] \widehat{G}_{jm}(x_{im}, x_{jm}) > 0\}. \] \[ \widehat{G}_{jm}(x_{im}, x_{jm}) = \mathbb{P}\{[v_{0j}(x_{jm}) - f_{jm}][1 - \widehat{G}_{im}(x_{im}, x_{jm})] + [v_{0j}(x_{jm}) + v_{1j}(x_{jm}) - f_{jm}] \widehat{G}_{im}(x_{im}, x_{jm}) > 0\}. \]
- We find parameters \(v_{0i}, v_{0j}, v_{1i}, v_{1j}, F\) that solve this system of equations.
- The “same equilibrium” assumption is hardly justified, but is often used in the empirical studies.