Ho and Lee (2017) — Structural Model with Nash-in-Nash Bargaining · 14 California HSAs
Logit demand over hospital networks, with diagnosis-specific utility: \(\sigma^H_{i,j,k,m} = \frac{\exp(\delta_i + \sum \beta \cdot z \cdot v)}{\sum_{h'} \exp(\delta_{h'} + \sum \beta \cdot z \cdot v)}\)
Logit demand over three insurers, with premium and WTP: \(\hat{u}^M = \delta_{j,m} + \alpha^\varphi \cdot (\varphi_j / 60) + \sum \alpha^W \cdot \text{WTP} + \alpha^K \cdot \text{dist}\)
CalPERS-insurer Nash bargaining with logit markup: \(\tau^\varphi \cdot \log(1 - s) + (1 - \tau^\varphi) \cdot \log(\text{GFT})\)
Hospital system-insurer Nash bargaining: \(p = c + \frac{\tau_j}{1 - \tau_j} \cdot \frac{\text{GFT}^j}{D^H}\)
Simulate the insurer removal counterfactuals from Table VII of the paper:
| Metric | Baseline | Counterfactual | Δ % | |
|---|---|---|---|---|
| Premium ($/yr) | ||||
| Hosp Price ($/admission) | ||||
| Market Share (%) | ||||
Computing sweep... Converged in iterations · Methodology Details →
Two-Level Demand System: Hospital choice (Eq 8) is a logit model over networks, with interactions for teaching, nurse staffing, for-profit status, service lines (cardiac, imaging, cancer, labor), and distance (Table A.III). Insurer choice (Eq 9–10) is logit over three plans, with premium sensitivity (\(\alpha^\varphi\) from Table A.IV), age-sex-specific WTP for hospital quality (from diagnosis-specific logsums), and Kaiser distance effects.
Berry (1994) Inversion: Hospital fixed effects \(\delta_i\) and insurer fixed effects \(\delta_{j,m}\) are recovered from observed market shares using the logit inversion: \(\log s_j - \log s_0 = \bar{u}_j\). This ensures the synthetic model reproduces observed baseline enrollment exactly.
Nash-in-Nash Bargaining: Premiums follow CalPERS-insurer Nash bargaining (Eq 5, implemented via Eq 12): the bargaining weight \(\tau^\varphi\) enters via \(g(\tau) = \tau / (\tau + \kappa(1-\tau))\). Hospital prices follow hospital system-insurer Nash bargaining (Eq 6) with gains-from-trade that depend on enrollment leverage, network structure, and the ability to recapture volume through competing networks.
Premium Pass-Through (Eq 6, term i): When \(\tau^\varphi\) increases and premiums rise, hospital prices increase by only ~1.5% per 1% premium change. This small elasticity reflects CalPERS bargaining capturing most of the insurer surplus increase, leaving hospitals with limited leverage.
Network/Recapture Effect (Eq 6, term iv): When an insurer is removed, hospitals that served both networks lose their ability to threaten recapture. This reduction in leverage explains the large negative price effect when Blue Cross is removed (−9.4% for Blue Shield hospitals that overlap with Blue Cross).
Equilibrium Solver: The solver uses fixed-point iteration on premiums, with hospital prices updated periodically. Convergence is rapid (typically <100 iterations) because premium pass-through is small, allowing premiums and prices to decouple. Market shares are updated each iteration via logit with implied fixed effects from Berry inversion.
Data Sources: Bargaining parameters (\(\tau^\varphi\), \(\tau_j\), \(\eta_j\)) from Table V; hospital demand parameters from Table A.III; insurer demand from Table A.IV; market enrollment from Table II; hospital characteristics and costs calibrated to Table I summary statistics.