## 2.1    Equilibrium Mergers and Acquisitions Dynamics with Heterogeneous Firms

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---

### Model setup and primitives
- Time is continuous, t ∈ [0,∞).
- Unit-mass continuum of target firms producing products of heterogeneous quality, ε ∈ [εl, εh], with initial cross-sectional pdf g(·).
- Death and matching:
  - Each target firm dies and is replaced at a Poisson rate η > 0; replacement product quality drawn from g(·).
  - Each target firm is matched for M&A at a Poisson rate λ > 0; death and matching shocks independent and i.i.d.
- M&A outcome: meeting of firms with qualities ε′ and ε′′ produces one firm with product quality A + ε′ + ε′′ (A ≥ 0), and the other firm’s quality is drawn from pdf g.
- Negotiation: Pareto-optimal bargaining with symmetric bargaining powers upon matching.

### Dynamics of the endogenous product-quality distribution
- Kolmogorov Forward Equation (KFE) for endogenous pdf f(·):
  - ̇f(ε) = −(η+λ) f(ε) |exit + [η + λ/2] g(ε) |entry + (λ/2) ∫_{−∞}^{∞} f(ε′) f(ε − A − ε′) dε′ |mergers.
- Interpretation of KFE terms:
  - Exit term: removal via death and M&As at rate η + λ.
  - Entry term: arrival via death and one firm adopting a g-drawn quality after M&A; λ/2 reflects one firm adopting g per M&A.
  - Merger convolution term: firms ending up with ε = A + ε′ + ε′′ conditional on meetings.
- Solve convolution via characteristic function f̂(z) = ∫_{−∞}^{∞} e^{izε} f(ε) dε.

### Steady state and characteristic-function solution (Proposition 1)
- Characteristic-function KFE:
  - ̇f̂(z) = −(η+λ) f̂(z) + [η + λ/2] ĝ(z) + (λ/2) e^{iAz} [f̂(z)]^2.
- At steady state, explicit solution:
  - f̂(z) = [η+λ − r(η+λ)^2 − 2λ e^{iAz} [η + λ/2] ĝ(z)] / [λ e^{iAz}].
- Unique steady-state pdf:
  - f(ε) = (1/2π) ∫_{−∞}^{∞} e^{-iεz} f̂(z) dz.
- Moments obtainable from f̂ allow comparative statics.

### Moments and comparative statics (Proposition 2)
- First two moments at steady state:
  - i) strictly increasing in the matching rate, λ;
  - ii) strictly increasing in the synergy parameter, A.
- Recursive n-th moment (n ≥ 0):
  - E_f[ε^n] = (1/η) [η + λ/2] E_g[ε^n] + (1/η)(λ/2) ∑_{l=0}^{n−1} ∑_{m=0}^{min{n−l,n−1}} (n choose n−l−m,l,m) A^{n−l−m} E_f[ε^l] E_f[ε^m].
- First two moments explicitly:
  - E_f[ε] = (1/η) { [η + λ/2] E_g[ε] + (λ/2) A }.
  - E_f[ε^2] = (1/η^3) { η^2 [η + λ/2] E_g[ε^2] − λ/2 A^2 } + λ [η + λ/2]^2 [E_g[ε] + A]^2.

### Frictional goods market with endogenous market power
- Environment:
  - Two goods: continuum of seller-specific perishable products and a homogeneous perishable numéraire.
  - Flow ω > 0 of homogeneous buyers. Buyers meet sellers at rate α > 0; after meeting a seller, buyer exits the economy regardless of trade outcome.
  - Bilateral Kalai bargaining with seller-relative bargaining power θ ∈ [0,∞), endogenously determined by sellers' rent-seeking.
- Solution with u(y) = √y and c(y) = c0 y (c0 > 0):
  - Interior bargaining solution: ε u′(y*) = c′(y*) → y* = ε^2 / (4 c0^2).
  - Per-unit price p = [θ/(θ+1) + 1/(θ+1)] c0.
  - Buyer surplus per match: (1/(θ+1)) (ε u(y*) − c(y*)).
  - Seller surplus per match: (θ/(θ+1)) (ε u(y*) − c(y*)).
- Markup and market shares:
  - Seller markup μ(ε) ≡ p / c′(y*) − 1 = θ(ε) / [θ(ε) + 1].
  - Market share s(ε) = p(ε) y*(ε) / ∫ p(ε′) y*(ε′) dF(ε′) = [μ(ε) + 1] ε^2 / ∫ [μ(ε′) + 1] (ε′)^2 dF(ε′).
  - Herfindahl-Hirschman Index (HHI) = ∫ [s(ε)]^2 dF(ε).

### Sellers’ continuation utility and endogenous bargaining power
- Bellman/HJB at steady state (general):
  - ρ V(ε) = max_{θ ≥ 0} { −υ(θ) + (α ω / α) [θ/(θ+1)] (ε u(y*) − c(y*)) − η V(ε) + λ̃ ∫∫ (1/2)[V(ε′ + ε + A) + V(ε′′) − V(ε) − V(ε′)] dF(ε′) dG(ε′′) }.
  - υ(θ): rent-seeking cost; λ̃: Poisson arrival rate of an M&A deal for a seller who committed to M&A effort λ̃.
- With u(y) = √y, c(y) = c0 y, υ(θ) = υ0 θ:
  - (ρ + η) V(ε) = max_{θ ≥ 0} { −υ0 θ + ω [θ/(θ+1)] ε^2 / (4 c0) + λ̃ ∫∫ (1/2)[V(ε′ + ε + A) + V(ε′′) − V(ε) − V(ε′)] dF(ε′) dG(ε′′) }.
- First-order condition for interior solution (sufficiently large ε):
  - θ(ε) = sqrt[ ω / (4 c0 υ0) ] ε^{−1}.
- Endogenous markup:
  - μ(ε) = θ(ε) / [θ(ε) + 1] = [ sqrt( ω / (4 c0 υ0) ) ε^{−1} ] / [ sqrt( ω / (4 c0 υ0) ) ε^{−1} + 1 ].

### Steady-state markups, market shares, and HHI (Proposition 3)
- Markup closed form:
  - μ(ε) = 1 − 1/ε sqrt(4 c0 υ0 / ω).
- Market share expression (as given in source):
  - s(ε) = [2 ε^2 − sqrt(4 c0 υ0 / ω) ε^2 E_f[ε^2] − sqrt(4 c0 υ0 / ω) E_f[ε]] / [2 E_f[ε^2] − sqrt(4 c0 υ0 / ω) E_f[ε]].
- HHI (as given in source):
  - HHI = [4 E_f[ε^4] − 4 sqrt(4 c0 υ0 / ω) E_f[ε^3] + 4 c0 υ0 / ω E_f[ε^2]] / [2 E_f[ε^2] − sqrt(4 c0 υ0 / ω) E_f[ε]]^2.
- Comparative statics intuition:
  - Higher ε → higher μ(ε).
  - Aggregate markups increase with higher ω, lower υ0, and lower c0.

### Equilibrium value function (quadratic solution) (Proposition 4)
- Define K = sqrt(ω / c0). Assume supports of F and G bounded below so ε ≥ 2 sqrt(υ0) / K.
- Unique quadratic solution:
  - V(ε) = V0 + [K / (4 (ρ + η))] [ ( −4 sqrt(υ0) + (λ̃ / (ρ + η)) K E_f[ε + A] ) ε + K ε^2 ].
- Constant V0 satisfies:
  - (ρ + η) V0 = υ0 + (λ̃ / 8 (ρ + η)) K [ −4 sqrt(υ0) E_g[ε + A] + (λ̃ / (ρ + η)) K E_f[ε + A] E_g[ε + A] + K E_g[ε^2] + K A (2 E_f[ε] + A) ].
- This characterizes the dynamic value of a seller sustaining M&A effort λ̃.

### Endogenous M&A efforts, best response, and multiplicity
- Entrant sellers commit to λ̃ before realizing ε; flow cost χ(λ̃) twice continuously differentiable, strictly increasing. Quadratic search cost: χ(λ̃) = (1/2) χ0 λ̃^2.
- Entrant problem: max_{λ̃ ∈ [0, λ̄]} (ρ + η) E_g[V(ε)] − χ(λ̃).
- First-order condition for interior optimal λ̃ (best-response):
  - λ̃ = [ (1 + λ/(2η)) E_g[ε + 2A] / E_g[ε] − 2 sqrt(υ0)/K E_g[ε + A] + (1/2) E_g[ε^2] + (1/2) (1 + λ/η) A^2 ] × [ 4 (ρ + η) K^2 χ0 − (1/(ρ + η)) (1 + λ/(2η)) [E_g[ε + A]]^2 ]^{−1}.
- Second-order condition requirement:
  - χ0 > K^2 / [4 (ρ + η)^2] × (1 + λ/(2η)) [E_g[ε + A]]^2.
- Strategic complementarity:
  - E_f[ε] increases in λ, so higher λ raises entrants’ marginal benefit from λ̃, implying λ̃(λ) increasing in λ and potential multiple equilibria.

### Equilibrium search intensities and multiplicity (Proposition 5)
- Define ∆ and thresholds:
  - ∆ ≡ [ (ρ + 3η)/2 − 4η(ρ + η)^2 K^2 [E_g[ε + A]]^2 / χ0 ]^2 − 2η(ρ + η) [1 + (1/2)(E_g[ε^2] − A^2) − 2 sqrt(υ0) K E_g[ε + A] ] [E_g[ε + A]]^2 / χ0.
  - λ1 = max(0, 4η(ρ + η)^2 K^2 [E_g[ε + A]]^2 / χ0 − (ρ + 3η)/2 − sqrt(∆)).
  - λ2 = 4η(ρ + η)^2 K^2 [E_g[ε + A]]^2 / χ0 − (ρ + 3η)/2 + sqrt(∆).
- Existence and multiplicity:
  - If 0 ≤ λ̄ < λ2, unique symmetric equilibrium λ* = min{λ1, λ̄}.
  - If λ̄ = λ2, two symmetric equilibria λ* ∈ {λ1, λ2}.
  - If λ̄ > λ2, three symmetric equilibria λ* ∈ {λ1, λ2, λ̄}.
- Interpretation: coexistence of low-M&A/low-markup/low-quality and high-M&A/high-markup/high-quality equilibria.

### Stability of equilibria and regulatory implications (Proposition 6)
- Stability criterion: interior equilibrium λ stable iff derivative of best-response at equilibrium satisfies λ̃′(λ) < 1.
- If λ̃(·) strictly convex:
  - Low-intensity interior equilibrium (λ1) is stable.
  - Middle interior equilibrium (λ2) is unstable.
  - Corner equilibrium at λ̄ is stable if λ2 < λ̄.
- Formal condition: define
  - a = E_g[ε + 2A] / E_g[ε] + 2 sqrt(υ0)/K E_g[ε + A] − (1/2) E_g[ε^2] + (1/2) A^2,
  - b = (1/(2η)) E_g[ε + 2A] / E_g[ε] + (1/(2η)) A^2,
  - c = 4 (ρ + η) K^2 χ0 − (1/(ρ + η)) [E_g[ε + A]]^2,
  - d = −(1/(2η))(1/(ρ + η)) [E_g[ε + A]]^2.
  - If (b − a) c d − 2 d (b c − a d) − (b − a) d^2 λ (c + d λ)^3 > 0 for all λ ∈ (0, λ̄), then:
    - λ1 stable, λ2 unstable, λ̄ stable provided λ2 < λ̄.
- Regulatory implication:
  - Tightening antitrust (increasing χ0) can raise λ2 above λ̄, moving economy to unique stable low-activity equilibrium λ1 (low M&A activity, low markups, low product qualities).
  - Such transitions may be hard to reverse due to stability of λ1 — hysteresis in M&A activity, markups, and product quality.

---

### 2.4 Welfare

### Definition and decomposition of social welfare
- Flow social welfare definition:
  - ρW = ∫ [ −υ[θ(ε)] + ω max_y∈u(y) − c(y) + ω c0 y*(ε) ] dF(ε) − χ(λ),
  - where y*(ε) = arg max_y∈ u(y) − c(y).
- Term interpretations:
  - −υ[θ(ε)]: flow cost of rent-seeking by seller with quality ε.
  - ω max_y∈ u(y) − c(y): flow gains from trade (buyer + seller shares sum to 1).
  - ω c0 y*(ε): earnings of hand-to-mouth workers enabling production.
  - χ(λ): flow cost of M&A activities.

### Parametric simplification and equilibrium welfare (Proposition 7)
- With u(y) = √y, c(y) = c0 y, υ(θ) = υ0 θ, χ(λ) = 1/2 χ0 λ^2, flow welfare simplifies:
  - ρW = ∫ [ −υ0 θ(ε) + ω 4 c0 ε^2 + ω 4 c0 ε^2 ] dF(ε) − 1/2 χ0 λ^2.
- Proposition 7 — equilibrium welfare given λ:
  - Let K = √(ω / c0). Assume supports of F and G bounded below such that all ε ≥ 2 √(υ0 / K).
  - W(λ) = 1/ρ [ υ0 − K √(υ0) 2 E_f[ε] + K^2 / 2 E_f[ε^2] − 1/2 χ0 λ^2 ].
- Component interpretations:
  - 1/ρ [ υ0 − K √(υ0) 2 E_f[ε] ] < 0: welfare cost of market power (more negative when average product quality is higher).
  - 1/ρ [ K^2 / 2 E_f[ε^2] ]: welfare gain from trading surpluses and labor earnings.
  - − 1/ρ [ 1/2 χ0 λ^2 ]: direct flow cost of M&A search activity.

### Numerical calibration and quantitative findings
- Calibration parameters:
  - η = 0.065
  - ρ = 0.04
  - A = 0.06
  - χ0 = 1800
  - K = 17.2
  - υ0 = 0.04
- Matching empirical moments (stable interior equilibrium λ1):
  - λ1 = 0.1625 (effective merger arrival rate)
  - E_f[μ(ε)] = 0.3498 (average net markup)
  - V_f[μ(ε)] = 0.12 (variance of markups)
  - average profitability = 0.144
- Comparative numerical results (summary of Figure 3):
  - High search-intensity equilibrium associated with higher average quality, larger quality variance, larger markups, and higher consumer welfare (consumer welfare tied to goods’ quality).
  - Relationship between search intensity and HHI is non-monotonic because markups rise with search intensity but not as much as average quality; concentration can decrease.
  - λ = 0 leads to the lowest social welfare and consumer welfare in the numerical panels.
  - Firm profits from markups incentivize M&A search; M&A improves product quality distribution and can make consumers better off despite larger markups when product quality distribution improves (first-order stochastic dominance).

### Comparative statics and equilibrium multiplicity (numerical illustrations from Figure 4)
- Two interior candidate equilibria λ1 and λ2:
  - λ1: low M&A equilibrium — only stable interior equilibrium in example.
  - λ2: high M&A equilibrium — unstable, self-fulfilling equilibrium.
- Comparative statics examples:
  - Search cost χ0:
    - λ1 decreases with χ0.
    - λ2 increases with χ0.
  - Synergy benefit A:
    - λ1 increases with A.
    - λ2 decreases with A.
- General pattern: λ1 and λ2 often respond with opposite signs to parameter changes due to complementarity sustaining λ2.

### Policy-relevant implications and counterfactuals
- Interpreting χ0 as antitrust stringency:
  - More stringent antitrust (higher χ0) raises effective cost of concluding acquisitions.
  - Example cited: blocked acquisition of T-Mobile USA by AT&T led to a breakup fee of around $4.2 billion (illustrative in source text).
- Policy risk:
  - Raising search costs can reduce equilibrium search intensity and push the economy into low-search equilibrium λ1 with lower welfare.
  - High-search equilibria can generate higher welfare via improved product quality distribution; ignoring dynamic search incentives when designing antitrust policy can reduce welfare.
- Additional insight:
  - Markups are not necessarily inefficient—they can provide incentives for costly M&A that enhance product quality and welfare; eliminating markups could remove search incentives and reduce welfare.

### Welfare conclusion and policy recommendation
- The model yields closed-form expressions and quantification to weigh market-power costs versus efficiency-enhancing effects of M&A.
- Key finding: when the higher search-intensity equilibrium exists, it leads to higher welfare because efficiency gains from M&A can dominate market-power distortions.
- Policy recommendation: antitrust policies should account for dynamic effects on search intensities; otherwise they risk inducing a transition to a low-search, low-welfare equilibrium.
- Suggested future research: study how different markup specifications could reshape dynamic incentives and welfare trade-offs.

*Source: wpiea2025239-source-pdf - 2.1    Equilibrium Mergers and Acquisitions Dynamics with Heterogeneous Firms; 2.4 Welfare*

### 2.1    Equilibrium Mergers and Acquisitions Dynamics with Hetero-

### 2.1    Equilibrium Mergers and Acquisitions Dynamics with Heterogeneous Firms

### Model setup and primitives
- Time is continuous, t ∈ [0,∞).
- There is a unit-mass continuum of target firms (sellers) producing products of heterogeneous quality, ε ∈ [εl, εh], with initial cross-sectional pdf g(·).
- Death and matching:
  - Each target firm dies and is replaced at a Poisson rate η > 0; replacement product quality is drawn from g(·).
  - Each target firm is matched for M&A at a Poisson rate λ > 0; death and matching shocks are independent and i.i.d. across firms.
- M&A outcome: an M&A between firms with initial qualities ε′ and ε′′ results in one firm with product quality A + ε′ + ε′′ (A ≥ 0 captures synergy), and the other firm ends up with a new product quality drawn from pdf g.
- Negotiation protocol: following a matching shock, a Pareto-optimal bargaining procedure with symmetric bargaining powers immediately results in a deal.

### Dynamics of the endogenous product-quality distribution
- The endogenous pdf f(·) of product quality evolves according to the Kolmogorov Forward Equation (KFE):
 ̇f(ε) = −(η+λ) f(ε) |exit
   + [η + λ/2] g(ε) |entry
   + (λ/2) ∫_{−∞}^{∞} f(ε′) f(ε − A − ε′) dε′ |mergers.
- Key interpretation of KFE terms:
  - Exit term: removal via death and M&As at rate η + λ.
  - Entry term: arrival via death and one firm adopting a g-drawn quality after M&A; λ/2 reflects one firm adopting g per M&A.
  - Merger convolution term: accounts for firms that end up with quality ε = A + ε′ + ε′′ conditional on meetings.
- To solve the convolution, use the characteristic function f̂(z) = ∫_{−∞}^{∞} e^{izε} f(ε) dε.
- Proposition 1 (Steady state solution):
  - Characteristic-function KFE:
    ̇f̂(z) = −(η+λ) f̂(z) + [η + λ/2] ĝ(z) + (λ/2) e^{iAz} [f̂(z)]^2.
  - At steady state, explicit solution:
    f̂(z) = [η+λ − r(η+λ)^2 − 2λ e^{iAz} [η + λ/2] ĝ(z)] / [λ e^{iAz}].
  - One-to-one correspondence between f̂ and cumulative distribution; unique steady-state pdf f(ε) = (1/2π) ∫_{−∞}^{∞} e^{-iεz} f̂(z) dz.
- Moments via characteristic function allow comparative statics.

### Moments and comparative statics
- Proposition 2 (Moments of the steady state distribution):
  - At steady state, the first two moments of product quality are:
    - i) strictly increasing in the matching rate, λ;
    - ii) strictly increasing in the synergy parameter, A.
  - Recursive formula for n-th moment (n ≥ 0):
    E_f[ε^n] = (1/η) [η + λ/2] E_g[ε^n] + (1/η)(λ/2) ∑_{l=0}^{n−1} ∑_{m=0}^{min{n−l,n−1}} (n choose n−l−m,l,m) A^{n−l−m} E_f[ε^l] E_f[ε^m].
  - First two moments explicitly:
    - E_f[ε] = (1/η) { [η + λ/2] E_g[ε] + (λ/2) A }.
    - E_f[ε^2] = (1/η^3) { η^2 [η + λ/2] E_g[ε^2] − λ/2 A^2 } + λ [η + λ/2]^2 [E_g[ε] + A]^2.

### Frictional goods market with endogenous market power
- Environment:
  - Two goods: continuum of seller-specific products (perishable, produced in bilateral meetings) and a homogeneous perishable numéraire.
  - Flow ω > 0 of homogeneous buyers. Buyers meet sellers at rate α > 0; after meeting a seller, buyer exits the economy regardless of trade outcome.
  - Bilateral Kalai bargaining with seller-relative bargaining power θ ∈ [0,∞), which will be endogenously determined by sellers' rent-seeking.
- Trade primitives and solution with u(y) = √y and c(y) = c0 y (c0 > 0):
  - Interior bargaining solution: ε u′(y*) = c′(y*); with u(y)=√y and c(y)=c0 y this yields y* = ε^2 / (4 c0^2).
  - Per-unit price p = [θ/(θ+1) + 1/(θ+1)] c0 = [θ(ε)/(θ(ε)+1) + 1/(θ(ε)+1)] c0 (emphasizing θ(ε)).
  - Buyer surplus per match: (1/(θ+1)) (ε u(y*) − c(y*)).
  - Seller surplus per match: (θ/(θ+1)) (ε u(y*) − c(y*)).
- Markup and market shares:
  - Seller markup μ(ε) ≡ p / c′(y*) − 1 = θ(ε) / [θ(ε) + 1].
  - Market share s(ε) = p(ε) y*(ε) / ∫ p(ε′) y*(ε′) dF(ε′) = [μ(ε) + 1] ε^2 / ∫ [μ(ε′) + 1] (ε′)^2 dF(ε′).
  - Herfindahl-Hirschman Index (HHI) = ∫ [s(ε)]^2 dF(ε).

### Sellers’ continuation utility and endogenous bargaining power
- Sellers discount at rate ρ > 0. Bellman/HJB expression for continuation utility V(ε) at steady state:
  ρ V(ε) = max_{θ ≥ 0} { −υ(θ) + (α ω / α) [θ/(θ+1)] (ε u(y*) − c(y*)) − η V(ε) + λ̃ ∫∫ (1/2)[V(ε′ + ε + A) + V(ε′′) − V(ε) − V(ε′)] dF(ε′) dG(ε′′) }.
  - Here υ(θ) is rent-seeking cost; λ̃ denotes Poisson arrival rate of an M&A deal for a seller who committed to M&A effort λ̃.
- With u(y) = √y, c(y) = c0 y, υ(θ) = υ0 θ (υ0 > 0), the HJB simplifies to:
  (ρ + η) V(ε) = max_{θ ≥ 0} { −υ0 θ + ω [θ/(θ+1)] ε^2 / (4 c0) + λ̃ ∫∫ (1/2)[V(ε′ + ε + A) + V(ε′′) − V(ε) − V(ε′)] dF(ε′) dG(ε′′) }.
- First-order condition for interior solution yields endogenous bargaining power (for sufficiently large ε):
  θ(ε) = sqrt[ ω / (4 c0 υ0) ] ε^{−1}.
- Endogenous markup then:
  μ(ε) = θ(ε) / [θ(ε) + 1] = [ sqrt( ω / (4 c0 υ0) ) ε^{−1} ] / [ sqrt( ω / (4 c0 υ0) ) ε^{−1} + 1 ].

### Steady-state markups, market shares, and HHI
- Proposition 3 (Steady-state Markups and Market Concentration):
  - μ(ε) = 1 − 1/ε sqrt(4 c0 υ0 / ω).
  - s(ε) = [2 ε^2 − sqrt(4 c0 υ0 / ω) ε^2 E_f[ε^2] − sqrt(4 c0 υ0 / ω) E_f[ε]] / [2 E_f[ε^2] − sqrt(4 c0 υ0 / ω) E_f[ε]]  (expressed in the source text as s(ε) = 2ε^2 − q 4c0υ0 ω ε^2 E_f[ε^2] − q 4c0υ0 ω E_f[ε], with denominator 2E_f[ε^2] − q 4c0υ0 ω E_f[ε]).
  - HHI = [4 E_f[ε^4] − 4 sqrt(4 c0 υ0 / ω) E_f[ε^3] + 4 c0 υ0 / ω E_f[ε^2]] / [2 E_f[ε^2] − sqrt(4 c0 υ0 / ω) E_f[ε]]^2 (as given in the source).
- Intuition and comparative statics:
  - Higher product quality ε → higher markup μ(ε) due to endogenous rent-seeking capturing larger trade surplus.
  - Aggregate markups increase with higher ω (arrival of buyers), lower υ0 (lower rent-seeking cost), and lower c0 (lower marginal production cost).

### Equilibrium value function (quadratic solution)
- Define K = sqrt(ω / c0). Assume supports of F and G are bounded below so ε ≥ 2 sqrt(υ0) / K.
- Proposition 4 (Equilibrium value function):
  - Unique quadratic solution: V(ε) = V0 + [K / (4 (ρ + η))] [ ( −4 sqrt(υ0) + (λ̃ / (ρ + η)) K E_f[ε + A] ) ε + K ε^2 ].
  - Constant V0 satisfies:
    (ρ + η) V0 = υ0 + (λ̃ / 8 (ρ + η)) K [ −4 sqrt(υ0) E_g[ε + A] + (λ̃ / (ρ + η)) K E_f[ε + A] E_g[ε + A] + K E_g[ε^2] + K A (2 E_f[ε] + A) ].
- This expression characterizes the dynamic value of a seller sustaining an M&A effort level λ̃.

### Endogenous M&A efforts, best response, and multiplicity
- Entrant sellers commit to M&A effort λ̃ before realizing ε (drawn from G); flow cost χ(λ̃) with χ twice continuously differentiable, strictly increasing. Quadratic search cost assumption: χ(λ̃) = (1/2) χ0 λ̃^2.
- Entrant optimization: max_{λ̃ ∈ [0, λ̄]} (ρ + η) E_g[V(ε)] − χ(λ̃), subject to V(ε) in Proposition 4.
- First-order condition for interior solution (best-response / optimal λ̃):
  λ̃ = [ (1 + λ/(2η)) E_g[ε + 2A] / E_g[ε] − 2 sqrt(υ0)/K E_g[ε + A] + (1/2) E_g[ε^2] + (1/2) (1 + λ/η) A^2 ] × [ 4 (ρ + η) K^2 χ0 − (1/(ρ + η)) (1 + λ/(2η)) [E_g[ε + A]]^2 ]^{−1}.
  - Second-order condition requires:
    χ0 > K^2 / [4 (ρ + η)^2] × (1 + λ/(2η)) [E_g[ε + A]]^2.
- Strategic complementarity:
  - The cross-sectional average quality E_f[ε] increases in aggregate M&A intensity λ (Equation (4)), so higher λ raises entrants’ marginal benefit from λ̃, making λ̃(λ) increasing in λ.
  - This generates strategic complementarity among entrant sellers and potential multiple equilibria.
- Proposition 5 (Equilibrium search intensities):
  - Define ∆ ≡ [ (ρ + 3η)/2 − 4η(ρ + η)^2 K^2 [E_g[ε + A]]^2 / χ0 ]^2 − 2η(ρ + η) [1 + (1/2)(E_g[ε^2] − A^2) − 2 sqrt(υ0) K E_g[ε + A] ] [E_g[ε + A]]^2 / χ0.
  - Let λ1 = max(0, 4η(ρ + η)^2 K^2 [E_g[ε + A]]^2 / χ0 − (ρ + 3η)/2 − sqrt(∆)).
  - Let λ2 = 4η(ρ + η)^2 K^2 [E_g[ε + A]]^2 / χ0 − (ρ + 3η)/2 + sqrt(∆).
  - Then:
    - If 0 ≤ λ̄ < λ2, there exists a unique symmetric equilibrium with M&A intensity λ* = min{λ1, λ̄}.
    - If λ̄ = λ2, there exist two symmetric equilibria with λ* ∈ {λ1, λ2}.
    - If λ̄ > λ2, there exist three symmetric equilibria with λ* ∈ {λ1, λ2, λ̄}.
  - Interpretation: equilibria with low M&A activity, low markups, and low product qualities can co-exist with equilibria with high M&A activity, high markups, and high product qualities.

### Stability of equilibria and regulatory implications
- Stability criterion: an interior equilibrium λ is stable iff derivative of best-response at equilibrium satisfies λ̃′(λ) < 1.
- If the best-response function λ̃(·) is strictly convex:
  - The low-intensity interior equilibrium (λ1) is stable.
  - The middle interior equilibrium (λ2) is unstable.
  - The corner equilibrium at λ̄ is stable if λ2 < λ̄.
- Proposition 6 (Stability): define
  - a = E_g[ε + 2A] / E_g[ε] + 2 sqrt(υ0)/K E_g[ε + A] − (1/2) E_g[ε^2] + (1/2) A^2,
  - b = (1/(2η)) E_g[ε + 2A] / E_g[ε] + (1/(2η)) A^2,
  - c = 4 (ρ + η) K^2 χ0 − (1/(ρ + η)) [E_g[ε + A]]^2,
  - d = −(1/(2η))(1/(ρ + η)) [E_g[ε + A]]^2.
  - If (b − a) c d − 2 d (b c − a d) − (b − a) d^2 λ (c + d λ)^3 > 0 for all λ ∈ (0, λ̄), then:
    - equilibrium at λ1 is stable;
    - equilibrium at λ2 is unstable;
    - equilibrium at λ̄ is stable provided λ2 < λ̄.
- Regulatory and welfare implications highlighted in the source:
  - If the best-response is strictly convex and χ0 is small so that λ2 < λ̄, the economy may reside in the maximal equilibrium λ̄ (high M&A activity, high markups, high product qualities).
  - Tightening antitrust (increasing χ0) can raise λ2 above λ̄, causing the economy to move to the unique stable low-activity equilibrium λ1 (low M&A activity, low markups, low product qualities).
  - This transition may be hard to reverse: if the policy is later relaxed, the economy can remain trapped in the low-activity equilibrium because λ1 is stable. Thus, M&A search complementarities imply potential hysteresis and unintended persistent effects of antitrust policy changes on M&A activity, markups, and product quality.

*Source: wpiea2025239-source-pdf - 2.1    Equilibrium Mergers and Acquisitions Dynamics with Heterogeneous Firms*

### 2.4    Welfare

### 2.4 Welfare

### Definition and decomposition of social welfare
- Flow social welfare is defined as
  ρW =
  ∫ [ −υ[θ(ε)] + ω max_y∈u(y) − c(y) + ω c0 y*(ε) ] dF(ε) − χ(λ),
  where y*(ε) = arg max_y∈ u(y) − c(y).
- Interpretation of terms inside the integral:
  - −υ[θ(ε)]: flow cost of rent-seeking activities by a seller of product quality ε.
  - ω max_y∈ u(y) − c(y): flow gains from trade (buyer + seller shares sum to 1; no θ(ε) coefficient inside this term).
  - ω c0 y*(ε): earnings of hand-to-mouth workers enabling production.
- Term outside the integral:
  - χ(λ): flow cost of M&A activities (search cost).
- Under the parametric assumptions u(y) = √y, c(y) = c0 y, υ(θ) = υ0 θ, and χ(λ) = 1/2 χ0 λ^2, flow social welfare simplifies to
  ρW = ∫ [ −υ0 θ(ε) + ω 4 c0 ε^2 + ω 4 c0 ε^2 ] dF(ε) − 1/2 χ0 λ^2,
  and, using sellers’ optimal θ(ε), leads to an explicit equilibrium welfare expression (Proposition 7).

### Proposition 7 — Equilibrium welfare given λ
- Let K = √(ω / c0).
- Assume the supports of F and G are bounded below such that all ε ≥ 2 √(υ0 / K).
- Then equilibrium welfare is
  W(λ) = 1/ρ [ υ0 − K √(υ0) 2 E_f[ε] + K^2 / 2 E_f[ε^2] − 1/2 χ0 λ^2 ],
  where E_f[ε] and E_f[ε^2] are given by (4) and (5), respectively.
- Interpretation of the components:
  - 1/ρ [ υ0 − K √(υ0) 2 E_f[ε] ] < 0: welfare cost of market power (more negative when average product quality is higher because high-quality firms can charge larger markups).
  - 1/ρ [ K^2 / 2 E_f[ε^2] ]: welfare gain from trading surpluses and labor earnings (rises with product quality dispersion and average quality).
  - − 1/ρ [ 1/2 χ0 λ^2 ]: direct flow cost of M&A search activity.

### Numerical calibration and quantitative findings
- Parameter values used in the numerical example:
  - Exit shock η = 0.065
  - Discount rate ρ = 0.04
  - Efficiency gains A = 0.06
  - Search cost χ0 = 1800
  - Reduced-form parameter K = 17.2
  - Rent-seeking cost ν0 = 0.04
- With these choices, the stable interior equilibrium (the equilibrium with λ1) matches four empirical moments from Cavenaile, Celik and Tian (2021):
  - λ1 = 0.1625 (effective merger arrival rate)
  - E_f[μ(ε)] = 0.3498 (average net markup)
  - V_f[μ(ε)] = 0.12 (variance of markups)
  - average profitability = 0.144
- Comparative numerical results (Figure 3 summary):
  - High search-intensity equilibrium is associated with:
    - higher average quality
    - larger quality variance
    - larger markups
    - higher consumer welfare (consumer welfare tied to goods’ quality)
  - Relationship between search intensity and market concentration (HHI) is non-monotonic due to concavity of markups:
    - markups rise with search intensity, but do not rise as much as average quality, so concentration can decrease.
  - λ = 0 leads to the lowest social welfare and consumer welfare in panels (e) and (f) of Figure 3.
  - Firm profits from markups incentivize M&A search; these M&A activities enhance product quality distribution and can leave consumers better off despite larger markups when product quality distribution improves (first-order stochastic dominance).

### Comparative statics and equilibrium multiplicity
- Two interior candidate equilibria λ1 and λ2 arise:
  - λ1: low M&A equilibrium — the only stable interior equilibrium in the example.
  - λ2: high M&A equilibrium — an unstable, self-fulfilling equilibrium sustained by search complementarities.
- Comparative statics examples (Figure 4 summary):
  - Search cost χ0:
    - λ1 decreases with χ0 (more costly search → lower λ1).
    - λ2 increases with χ0 (due to self-fulfilling nature: higher χ0 requires even higher coordinated λ to sustain λ2).
  - Synergy benefit A:
    - λ1 increases with A (higher A → larger M&A surplus → higher λ1).
    - λ2 decreases with A (because higher A allows achieving the same surpluses with lower λ in the coordinated high-λ equilibrium).
- General pattern: λ1 and λ2 often respond with opposite signs to parameter changes because λ2’s existence and level depend on mutual expectations (complementarities).

### Policy-relevant implications and counterfactuals
- Interpretation of search costs as an indicator of antitrust policy stringency:
  - More stringent antitrust policies (e.g., lengthier pre-merger filings, break-up fees) increase the effective cost of concluding an acquisition, captured by higher χ0.
  - Empirical example of merger costs: acquisition of T-Mobile USA by AT&T that was blocked led to a breakup fee of around $4.2 billion (cited in text).
- Potential policy risk:
  - Policies that raise search costs (i.e., make M&A more costly) can reduce equilibrium search intensity and may push the economy into the low-search equilibrium λ1 with lower welfare.
  - Because high-search equilibria can generate higher welfare via improved product quality distribution and gains from trade, ignoring dynamic incentives on search intensities when designing antitrust policy may produce negative welfare consequences.
- Additional insight:
  - Markups in this model are not necessarily inefficient; they can provide necessary incentives for firms to engage in costly M&A activities that enhance product quality and welfare. Removing markups could eliminate the incentive to search for M&A partners, potentially reducing welfare.

### Conclusion (welfare perspective)
- The model provides closed-form expressions and quantification to weigh opposing welfare effects:
  - Market-power costs (markups) vs. efficiency-enhancing effects of M&A (improved quality, greater trade gains and labor earnings).
- Key finding: when the higher search-intensity equilibrium exists, it leads to higher welfare because efficiency gains from M&A can dominate market-power distortions.
- Policy recommendation: antitrust policies should account for dynamic effects on search intensities; otherwise they risk inducing a transition to a low-search, low-welfare equilibrium.
- Suggested avenue for future research: study how different markup specifications could reshape these dynamic incentives and welfare trade-offs.

*Source: A Search-Based Theory of Mergers and Acquisitions, Working Paper No. WP/2025/239*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025239-source-pdf.pdf_