Alternative Complementarity Specifications - Beer Price Controls at Yankee Stadium

Theoretical Foundation: Leisten (2025)¶

Leisten (2025) provides rigorous theoretical analysis of beer price controls at stadiums:

Key result: Under log-concavity of demand, beer price ceilings cause ticket prices to rise.

His model:

Ticket demand: $q_x(p_x)$
Concession demand: $q_y = q_x(p_x) \cdot q_y(p_y)$ (multiplicative)
Assumption: Beer prices do NOT directly affect ticket demand (one-way complementarity)

First-order conditions:

p_y = -\frac{q_y(p_y)}{q_y'(p_y)}

(1)

p_x = -\frac{q_x(p_x)}{q_x'(p_x)} - p_y q_y(p_y)

(2)

When beer price ceiling $Z$ binds:

\frac{dp_x}{dZ} = \frac{Zq_y'(Z) - q_y(Z)}{\frac{q_x(p_x)q_x''(p_x)}{q_x'(p_x)} - 2}

(3)

Sign depends on: $2q_x'(p_x)^2$ vs $q_x(p_x)q_x''(p_x)$

Under log-concavity: $q_x q_x'' < q_x'^2$ , so Leisten proves $\frac{dp_x}{dZ} < 0$ (tickets rise when ceiling tightens).

Our extension: We allow two-way complementarity ( $A(P_T, P_B)$ ), which is more general but requires assuming the cross-elasticity magnitude.

Current Specification (Two-Way Multiplicative)¶

Functional form:

A(P_T, P_B) = A_0 \cdot e^{-\lambda_T(P_T - P_0^T)} \cdot \left(\frac{P_B}{P_0^B}\right)^{-\epsilon_{cross}}

(4)

Where $\epsilon_{cross} = 0.1$

Properties:

Cross-price elasticity: $\frac{\partial \ln A}{\partial \ln P_B} = -0.1$
10% beer price increase → 1% attendance decrease
Symmetric: effect scales with price level
Log-concave: Semi-log form satisfies Leisten’s condition

Citation: Standard in single-equation demand Varian (1992); two-way extension beyond Leisten (2025)

Alternative Specifications¶

1. Almost Ideal Demand System (AIDS)¶

Reference: Deaton & Muellbauer (1980)

Form: Derived from utility maximization

w_i = \alpha_i + \sum_j \gamma_{ij} \ln p_j + \beta_i \ln(x/P)

(5)

Where:

$w_i$ = budget share of good $i$
$\gamma_{ij}$ = cross-price effects (estimated)
Symmetry: $\gamma_{ij} = \gamma_{ji}$

Cross-price elasticity:

\epsilon_{ij} = \frac{\gamma_{ij}}{w_i} - \delta_{ij}

(6)

Advantages:

✅ Theory-consistent (utility-derived)
✅ Flexible (Engel curves, substitution patterns)
✅ Testable restrictions (symmetry, homogeneity)

Disadvantages:

❌ Requires panel data (multiple markets/times)
❌ Need variation in both ticket AND beer prices
❌ Computationally intensive

Typical estimates: Cross-elasticities range -0.5 to +0.5 for food items Deaton & Muellbauer (1980)

2. CES Utility Function¶

Reference: Arrow et al. (1961)

Form:

U = \left[\alpha B^\rho + (1-\alpha) T^\rho\right]^{1/\rho}

(7)

Where:

$\rho$ relates to elasticity of substitution: $\sigma = 1/(1-\rho)$
$\sigma < 1$ : Complements
$\sigma > 1$ : Substitutes
$\sigma = 1$ : Cobb-Douglas (independent)

Implied cross-elasticity:

\epsilon_{TB} = (\sigma - 1) \cdot \frac{P_B B}{E}

(8)

Advantages:

✅ Micro-founded (utility maximization)
✅ Single parameter ( $\\sigma$ ) controls substitution
✅ Nests Cobb-Douglas, Leontief (perfect complements)

Disadvantages:

❌ Restrictive (constant $\\sigma$ across price levels)
❌ Requires calibration or estimation

Typical range: $\sigma = 0.2$ to $0.8$ for complements

3. Translog Demand¶

Reference: Christensen et al. (1975)

Form: Second-order flexible functional form

\\ln q_i = \\alpha_i + \\sum_j \\beta_{ij} \\ln p_j + \\gamma_i \\ln y

(9)

Where $\\beta_{ij}$ are cross-price terms.

Advantages:

✅ Very flexible (no restrictive functional form)
✅ Can nest AIDS, Cobb-Douglas
✅ Captures non-linearities

Disadvantages:

❌ Many parameters to estimate
❌ May violate regularity conditions

4. Linear Interaction¶

Form:

A = A_0 - \\alpha P_T - \\beta P_B - \\gamma P_T \\cdot P_B

(10)

Where $\gamma > 0$ for complements.

Cross-price elasticity:

\\epsilon_{TB} = -\\frac{\\beta + \\gamma P_T}{A} \\cdot P_B

(11)

Advantages:

✅ Simple, interpretable
✅ Easy to estimate (linear regression)

Disadvantages:

❌ Can produce negative quantities
❌ Elasticity changes with price level
❌ No utility foundation

5. Nested Logit (Discrete Choice)¶

Reference: McFadden (1978)

Form: For attendance decision

P(attend) = \\frac{e^{V_{attend}}}{e^{V_{attend}} + e^{V_{not\\_attend}}}

(12)

Where $V_{attend}$ depends on both ticket and beer prices.

Advantages:

✅ Microfounded (random utility)
✅ Handles discrete choices naturally
✅ Rich substitution patterns

Disadvantages:

❌ Complex estimation (maximum likelihood)
❌ Requires individual-level data

How Economists Evaluate Specifications¶

1. Theoretical Consistency¶

Question: Does specification come from utility maximization?

Evaluation criteria:

Slutsky symmetry: $\\frac{\\partial q_i}{\\partial p_j} = \\frac{\\partial q_j}{\\partial p_i}$ (compensated)
Homogeneity: Doubling all prices and income → no change
Adding up: Budget shares sum to 1

Rankings:

✅ AIDS, CES: Fully consistent
⚠️ Current (multiplicative): Partial (not derived from single utility)
❌ Linear: Not theory-consistent

2. Empirical Fit¶

Metrics:

R² or pseudo-R²
AIC/BIC (information criteria)
Out-of-sample prediction
Residual diagnostics

Data requirements:

Panel data (multiple markets, times)
Price variation
Exogenous price changes (instruments)

3. Flexibility vs Parsimony¶

Trade-off:

AIDS: Very flexible (many parameters)
CES: Parsimonious (single $\\sigma$ )
Current: Very simple (single $\\epsilon_{cross}$ )

Evaluation:

Use information criteria (AIC/BIC)
Test nested models (likelihood ratio)
Check if additional parameters improve fit

4. Plausibility of Estimates¶

Bounds checking:

For complements: $\epsilon_{cross} < 0$
Typical range: -0.1 to -2.0
Strong complements (cars/gas): -1.6
Weak complements: -0.1 to -0.3

Our choice (0.1):

At low end of plausible range
Implies weak complementarity
Conservative assumption

5. Policy Robustness¶

Question: Do policy conclusions change with specification?

Evaluation:

Simulate under different specifications
Check if directional effects robust
Quantify sensitivity of key outcomes

Comparison to Empirical Literature¶

Food Complements¶

Meat & vegetables: Cross-elasticity varies by study
Typically -0.1 to -0.5

Transportation¶

Cars & gasoline: -1.6 (strong complements)
Public transit & auto: +0.5 to +0.8 (substitutes)

Entertainment¶

Movie tickets & popcorn: No published estimates found
Theme park admission & food: No estimates found

Stadium tickets & beer: NO PUBLISHED ESTIMATES

Recommendation for This Analysis¶

Current Approach (Multiplicative with ε=0.1)¶

Pros:

✅ Simple, transparent
✅ Directionally correct (negative)
✅ Conservative (weak complementarity)
✅ Easy to adjust in sensitivity analysis

Cons:

❌ Not derived from utility
❌ No empirical validation
❌ Arbitrary functional form

Better Approaches (If Data Available)¶

Estimate AIDS model with Yankees panel data
- Vary prices across games/seasons
- Estimate full demand system
- Get cross-elasticity from data
Use car/gasoline analogy
- Both are “required + optional” like tickets/beer
- Cross-elasticity -1.6 as benchmark
- Test sensitivity to -0.5, -1.0, -1.6
Survey-based calibration
- Ask fans willingness to attend with/without beer
- Discrete choice experiment
- Estimate cross-effect structurally

For This Project¶

Keep current approach but:

✅ Document it’s ASSUMED (done)
✅ Run Monte Carlo over plausible range 0.05-0.30 (done)
✅ Cite analogous contexts (car/gas: -1.6)
✅ Show sensitivity of conclusions

Add to references:

Deaton & Muellbauer (1980) for AIDS framework
Varian (1992) for demand theory
McFadden (1978) for discrete choice
Empirical examples (cars/gas)

References¶

Leisten, M. (2025). Twitter Thread: Economic Analysis of Beer Price Controls at Yankee Stadium. Twitter/X. https://x.com/LeistenEcon/status/1990150035615494239
Varian, H. R. (1992). Microeconomic Analysis (3rd ed.). W.W. Norton & Company.
Deaton, A., & Muellbauer, J. (1980). An Almost Ideal Demand System. American Economic Review, 70(3), 312–326.
Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-Labor Substitution and Economic Efficiency. Review of Economics and Statistics, 43(3), 225–250.
Christensen, L. R., Jorgenson, D. W., & Lau, L. J. (1975). Transcendental Logarithmic Utility Functions. American Economic Review, 65(3), 367–383.
McFadden, D. (1978). Modeling the Choice of Residential Location. In A. Karlqvist, L. Lundqvist, F. Snickars, & J. Weibull (Eds.), Spatial Interaction Theory and Planning Models (pp. 75–96). North-Holland.