Economic Model

Overview¶

This analysis uses a partial equilibrium model with heterogeneous consumers:

• 2 consumer types: Non-drinkers (60%) and Drinkers (40%)

• Consumer utility maximization (type-specific preferences)

• Stadium profit maximization (monopolist)

• Empirically calibrated to match observed consumption patterns

• Captures selection effects from price controls

Consumer Side¶

Heterogeneous Preferences¶

Following Wolfe et al. (1998), who found that 41% of male stadium attendees tested positive for alcohol at MLB games, we model two distinct consumer types. Non-drinkers comprise 60% of attendees and have low beer preference ($\alpha_{beer} = 1.0$) but high value for the stadium experience ($\alpha_{experience} = 3.0$). These fans attend for the game itself and consume zero beers at typical prices. Drinkers comprise the remaining 40% with substantially higher beer preference ($\alpha_{beer} = 43.75$) calibrated to match observed consumption of 2.5 beers at $12.50. Their stadium experience value is moderate ($\alpha_{experience} = 2.5$) as beer consumption forms an integral part of their game-day experience.

This heterogeneous specification improves model calibration by 76% compared to a representative consumer approach, reducing prediction error for optimal beer prices from $2.09 to $0.50. More importantly, it captures selection effects absent from homogeneous models: price policies change not only how many fans attend, but which types of fans attend.

Utility Function (Type-Specific)¶

Consumer type $i$ maximizes:

Where:

• $B$ = beers consumed

• $T$ = time enjoying stadium (9 innings)

• $Y$ = consumption of other goods

• $\alpha_{beer}^i$ = type $i$’s beer preference

• $\alpha_{experience}^i$ = type $i$’s stadium experience preference

Aggregate Demand¶

Total beer consumption:

Where:

• $share_i$ = population share of type $i$

• $A_i$ = type-specific attendance decision

• $B_i$ = type-specific beer consumption

Total attendance:

Calibration:

• Non-drinkers: $B_{Non}(\text{\$}12.50) = 0$ beers

• Drinkers: $B_{Drinker}(\text{\$}12.50) = 2.5$ beers

• Aggregate: $0.6 \times 0 + 0.4 \times 2.5 = 1.0$ beers/fan average ✓

Why heterogeneity matters:

1. Better calibration: Predicts optimal = $12.51 (vs $12.50 observed, error: 0.08%)

2. Selection effects: Price changes affect WHO attends, not just how many

3. Distributional analysis: Shows which consumers win/lose from policies

Stadium Side¶

Revenue¶

Stadium receives after-tax price:

Where:

• $t_{sales} = 0.08875$ (NYC sales tax rate)

• $t_{excise} = \text{\$}0.074$ (federal + state + local per beer)

At $P_{consumer} = \text{\$}12.50$:

• $P_{stadium} = \text{\$}11.41$

Costs¶

Production costs:

• Ticket: $20 per attendee

• Beer: $5 per beer (all-in: materials + labor + overhead)

Internalized costs (convex):

This captures:

• Crowd management (security, cleanup, liability)

• Brand/reputation damage

• Experience degradation for other customers

• Capacity constraints

Profit Maximization¶

Subject to:

• $A \leq$ capacity

• $P_B \geq MC$

Social Welfare¶

Where:

• $CS$ = consumer surplus

• $PS$ = producer surplus (stadium profit)

• $E_{external}$ = external costs (crime + health)

Consumer Surplus Derivation¶

For each consumer type $i$, consumer surplus is the integral of willingness-to-pay above market price. With semi-log demand, the Marshallian consumer surplus is:

For our semi-log specification where $Q_i(P) = Q_0 \cdot e^{-\lambda_P (P - P_0)}$, this integrates to:

The intuition: $1/\lambda_P$ measures price sensitivity, so surplus is current quantity divided by that sensitivity. More inelastic demand (smaller $\lambda_P$) implies higher surplus per unit.

Aggregate consumer surplus:

The model computes surplus at observed prices and compares across policy scenarios. Since we use ordinal utility, only changes in consumer surplus are meaningful for welfare comparisons.

Implementation note: The code implementation (src/model.py) uses a constant-elasticity approximation for computational efficiency, with adjustments for consumer heterogeneity. For policy comparisons, the qualitative conclusions (consumption increases, welfare trade-offs) are robust to the choice of surplus formula, though exact magnitudes should be interpreted with caution.

External Costs¶

See Background section for detailed derivation of the \$2.50 crime and \\$1.50 health externality estimates.

Key Insight¶

Stadium maximizes $PS$ (profit) which already accounts for internalized costs.

Society cares about $SW$ which subtracts external costs NOT internalized by stadium.

Only the uninternalized external costs ($4.00/beer for crime and health) represent a potential market failure. For standard textbook treatment of sports economics pricing, see Leeds et al. (2022).