Model Assumptions & Parameter Sources

Empirically Estimated vs Assumed¶

✅ FROM LITERATURE (Empirical Estimates)¶

Own-Price Elasticities:

• Ticket demand: -0.49 to -0.76 Noll (1974)Scully (1989)

• General alcohol: -0.79 to -1.14 (not stadium-specific)

Consumption Rates:

• 40% of fans drink alcohol Lenk et al. (2010)

• Mean BAC: 0.057% among drinkers Lenk et al. (2010)

Externalities:

• Crime: 10% alcohol ↑ → 1% assault ↑, 2.9% rape ↑ Carpenter & Dobkin (2015)

• External costs: $0.48-$1.19/drink (1986$) Manning et al. (1991)

⚠️ ASSUMED (Not From Empirical Estimates)¶

Cross-Price Elasticity (Complementarity): 0.1

Source: ASSUMPTION, not empirical estimate

Rationale:

• Coates & Humphreys (2007) and Krautmann & Berri (2007) document that tickets and concessions are complements

• Both papers show teams price tickets in inelastic region to drive concession sales

• BUT: Neither provides specific cross-price elasticity estimate

What we know:

• Tickets and beer are complements (qualitative)

• Teams jointly optimize (strong evidence)

• Complementarity is “significant” (exact magnitude unknown)

Key Modeling Choice: Two-Way Complementarity

Recent theoretical work by Leisten (2025) explicitly analyzes this assumption:

• Leisten assumes: Beer prices do NOT affect ticket demand (one-way complementarity: tickets → beer)

• We assume: Beer prices DO affect ticket demand (two-way complementarity: tickets ↔ beer)

Both models predict beer ceilings cause ticket prices to rise, but through different mechanisms:

• Leisten: Complementarity discount term in FOC shrinks → tickets rise to restore markup

• Our model: Beer margin collapses → stadium shifts to tickets → higher tickets reduce attendance (limiting beer sales at bad margin)

Why we model two-way complementarity:

1. More realistic: fans likely consider total game cost including beer

2. Allows for substitution to pre-game drinking if stadium beer too expensive

3. Consistent with observed fan behavior (attendance drops when concession prices rise significantly)

4. Makes model symmetric and general

Calibration approach:

• Assume 10% beer price change → 1% attendance change

• Consistent with “weak to moderate” complementarity

• Conservative estimate (could be 0.2-0.3 if beer very important)

Sensitivity range:

• Leisten: 0.00 (one-way only)

• Low: 0.05 (beer minor part of experience)

• Base: 0.10 (current model)

• High: 0.30 (beer central to fan experience)

Monte Carlo analysis: Our uncertainty quantification samples cross_price_elasticity uniformly from [0.0, 0.3], spanning the full range from Leisten’s pure one-way model to strong two-way complementarity. This ensures our qualitative conclusions (tickets rise, consumption increases) are robust to this critical but unmeasured parameter.

Why 0.1 is reasonable for point estimates:

• Coates & Humphreys (2007) show teams sacrifice ticket revenue for concession profits, implying cross-effects matter

• If cross-elasticity were very high (>0.3), we’d expect stadiums to heavily subsidize beer to drive attendance—they don’t

• If cross-elasticity were very low (<0.05), teams wouldn’t care about beer prices’ effect on attendance—but they do (7th inning cutoffs, etc.)

• 0.1 represents a middle ground: beer matters, but isn’t the primary attendance driver

⚙️ CALIBRATED (To Match Observed Prices)¶

Demand Sensitivities (λ):

• Beer: 0.133 (calibrated so $12.50 is optimal)

• Tickets: 0.017 (calibrated so $80 is optimal)

These are NOT elasticities - they’re parameters in semi-log demand that produce realistic price levels.

Internalized Cost (α = 250):

• Calibrated to make observed prices profit-maximizing

• Reflects convex costs from crowd management, brand, experience

• Order of magnitude plausible but not directly measured

💭 EDUCATED GUESSES¶

Marginal Costs:

• Beer: $5.00 (materials + labor + overhead)

• Tickets: $3.50 (variable labor + cleaning)

Basis: Industry knowledge, reasonable cost accounting Not from: Yankees financial data (proprietary)

Impact on Results¶

Robust findings (insensitive to assumptions):

• Beer ceiling → tickets rise ✓

• Beer ceiling → stadium profit falls ✓

• Beer ceiling → consumption increases ✓

Uncertain magnitudes (sensitive to assumptions):

• Exact ticket price increase (3-6x multiplier)

• Exact welfare distribution

• Exact consumption levels

Critical assumption: Cross-elasticity 0.1

• If actually 0.2-0.3: Ticket response smaller, welfare effects different

• If actually 0.05: Current model reasonably accurate

Literature Gap¶

What we need but don’t have:

1. Empirical cross-price elasticity estimates between stadium tickets and beer

   • Could be estimated with panel data across stadiums

   • Or natural experiments (price changes)

2. Stadium-specific demand estimates

   • Yankees fans may differ from MLB average

   • NYC market effects

3. Quantified internalized costs

   • Actual crowd management costs per intoxication level

   • Brand value impact from incidents

Until then: Treat model as illustrative framework showing mechanisms, not precise predictions.