📈 economics

Simple vs. Optimal Congestion Pricing

This paper analyzes the performance gap between static and dynamic congestion pricing, demonstrating through theoretical models and real-world data that simple static tolls achieve roughly 80–90% of the optimal revenue while incurring only a modest increase in system costs, thereby validating their practical viability despite the theoretical superiority of dynamic pricing.

Original authors: Devansh Jalota, Xuan Di, Adam N. Elmachtoub

Published 2026-03-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Devansh Jalota, Xuan Di, Adam N. Elmachtoub

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a busy morning commute as a giant, chaotic game of "Musical Chairs," but instead of chairs, there are spots on a bridge or in a city center, and instead of music stopping, the traffic just keeps piling up.

This paper asks a very practical question: Do we need a super-complex, real-time computer system to manage traffic tolls, or is a simple, flat fee (like a fixed price tag) good enough?

Here is the breakdown of their findings using everyday analogies.

The Two Approaches: The "Smart Chef" vs. The "Simple Menu"

The researchers compare two ways to charge drivers to reduce traffic:

The "Smart Chef" (Dynamic Pricing): This is the theoretically perfect system. Imagine a chef who watches the kitchen in real-time. If the stove is crowded, they instantly raise the price of ordering a steak. If the kitchen is empty, they lower it. They adjust the price every second based on exactly how many cars are on the road.
- The Problem: This requires super-computers, sensors everywhere, and drivers who can handle their toll bill changing every minute. It's expensive and confusing to implement.
The "Simple Menu" (Static Pricing): This is what cities like New York and London actually use. It's like a restaurant with a fixed menu: "Breakfast costs $10, no matter what time you come in." It's easy to understand, easy to collect, and doesn't need a super-computer.
- The Fear: Critics worry that because the price never changes, it's "dumb" and wastes money or fails to stop traffic effectively.

The Experiment: Testing the "Simple Menu"

The authors ran simulations using two famous traffic models:

The Bottleneck Model: Think of a single-lane bridge (like the San Francisco-Oakland Bay Bridge) where everyone has to squeeze through one point.
The City Model (MFD): Think of a whole city grid (like Manhattan) where traffic jams happen because the whole network gets clogged, not just one bridge.

They asked: If we use the "Simple Menu" (Static Toll) instead of the "Smart Chef" (Dynamic Toll), how much money do we lose? How much worse does the traffic get?

The Big Reveal: The "Simple Menu" is Surprisingly Good

The results were a pleasant surprise for city planners who want to keep things simple:

1. The Revenue Gap is Tiny
Even with a fixed price, cities can collect 80% to 90% of the money they could have made with a perfect, real-time system.

Analogy: Imagine a lemonade stand. The "Smart Chef" would charge $1.00 when it's hot and $0.50 when it's cloudy, maximizing profit. The "Simple Menu" just charges $0.80 all day. The study shows the Simple Menu makes almost as much money as the Smart Chef, without the headache of constantly changing signs.

2. The Traffic Cost is Manageable
The "Simple Menu" does cause slightly more traffic jams than the perfect system, but usually only by 8% to 20%.

Analogy: If the perfect system gets you to work in 30 minutes, the simple system might get you there in 35 minutes. It's annoying, but it's not a disaster.

3. The "Public Transit" Safety Net
The study found that static tolls work best when there is a good alternative, like a subway or bus.

Analogy: If the subway is fast and cheap, drivers are willing to pay the toll to avoid the traffic. If the subway is terrible, drivers will just pay the toll and sit in traffic anyway, making the toll less effective. In cities with good subways (like NYC), the simple toll works very well.

The Trade-Off: Money vs. Time

The paper highlights a fundamental trade-off:

If you want to maximize Revenue: You might need a slightly more complex system, but a simple toll gets you 90% of the way there.
If you want to minimize Traffic: A perfect dynamic system is best, but a simple toll is still "good enough" for most real-world situations.

The Real-World Verdict

The authors looked at real data from the San Francisco Bay Bridge and New York City.

New York City: Because the subway is so popular, the current $9 flat toll is actually very close to the "perfect" price. It captures nearly all the benefits of a complex system.
San Francisco Bay Bridge: The current $8.50 toll is also very effective, though the gap between "simple" and "perfect" is slightly wider here because fewer people take the train compared to NYC.

Conclusion: Don't Overthink It

The paper's main message is: Stop trying to build a sci-fi, real-time pricing machine.

While a perfect, dynamic system sounds great in theory, the "Simple Menu" (static tolls) is robust, easy to explain to the public, and performs almost as well as the complex version. The extra cost and confusion of building a "Smart Chef" system aren't worth the tiny gains in efficiency.

In short: A simple, fixed toll is like a reliable, comfortable pair of sneakers. A complex, dynamic toll is like a pair of high-tech, self-lacing shoes. The sneakers get you to the finish line almost as fast, cost less, and you won't trip over the laces.

1. Problem Statement

Congestion pricing is a proven mechanism for mitigating traffic inefficiencies, yet the theoretically optimal implementation—dynamic, state-dependent tolls (Pigouvian tolls)—is often impractical due to high informational, computational, and cognitive burdens on users and planners. Consequently, real-world deployments (e.g., New York City, London, San Francisco-Oakland Bay Bridge) rely on simple, static tolls (fixed fees).

The central research question is: What is the performance gap between simple static tolls and optimal dynamic tolls in terms of revenue generation and system welfare (total system cost)? Specifically, how much revenue and efficiency are sacrificed when using static pricing?

2. Methodology

The authors analyze this problem using two canonical modeling frameworks that represent dominant real-world congestion scenarios:

A. Vickrey's Bottleneck Model with an Outside Option

Structure: A single capacity-constrained facility (e.g., a bridge) with a fixed service rate $\mu$ . Users choose between driving (incurring queuing delay and tolls) and a public transit alternative with a fixed cost.
Assumptions: Deterministic demand, homogeneous users (initially), and a public transit outside option that prevents infinite queuing.
Analysis: The authors derive closed-form solutions for revenue-optimal static and dynamic tolls. They characterize the equilibrium user behavior (mode choice and departure times) under these policies.

B. Macroscopic Fundamental Diagram (MFD) Urban System Model

Structure: A city-scale extension where congestion is state-dependent. The system's throughput (service rate) is a function of the total vehicle accumulation (density) within the urban area, typically modeled via a triangular MFD.
Assumptions: Triangular MFD (linear free-flow, capacity drop after a critical threshold), fixed trip distances, and a public transit outside option.
Analysis: The authors characterize the dynamic revenue-optimal policy and derive a closed-form expression for revenue under static tolls, acknowledging that static tolls can induce operation on the congested branch of the MFD.

C. Performance Metrics

The study evaluates policies based on two primary metrics:

Revenue ( $R$ ): Total toll payments collected.
Total System Cost ($SC$): The aggregate travel burden (waiting time + schedule delay + generalized travel costs) excluding toll payments (as tolls are transfers).

3. Key Contributions

A. Characterization of Revenue-Optimal Policies

While prior literature focused heavily on welfare-optimal (system-cost minimizing) tolls, this paper provides the first rigorous characterization of revenue-optimal policies in these settings:

Bottleneck Model: The authors show that computing revenue-optimal static and dynamic tolls reduces to solving single-variable quadratic programs with closed-form solutions.
- Dynamic Policy: Takes a trapezoidal shape, maintaining the system at the throughput-maximizing capacity (zero queuing) but shifting the toll profile vertically compared to welfare-optimal tolls.
- Static Policy: Derived as a function of the cost difference between transit and driving ( $z_T - z_C$ ).
MFD Model: The dynamic revenue-optimal policy maintains the system at the critical accumulation level ( $n_c$ ) to maximize throughput. For static tolls, they derive a complex closed-form revenue expression involving logarithmic terms.

B. Theoretical Performance Guarantees

The paper establishes worst-case bounds on the performance of static revenue-optimal tolls relative to dynamic benchmarks:

Revenue:
- In the Bottleneck Model, static revenue-optimal tolls achieve at least 50% of the dynamic optimal revenue across all parameter regimes, and at least 66% in many practically relevant regimes.
- In the MFD Model, the guarantees are slightly weaker but still robust: static tolls achieve at least 66% of dynamic optimal revenue in practically relevant regimes.
System Cost:
- When public transit is a viable alternative (i.e., the cost difference $z_T - z_C$ is not excessively high), static revenue-optimal tolls incur at most twice the minimum achievable system cost in both models.
- The paper notes that if transit is extremely unattractive, the system cost ratio can become unbounded, but this is an edge case not typical of major cities with viable transit.

C. Numerical Validation with Real-World Data

The authors validate their theory using two case studies:

SF-Oakland Bay Bridge (Bottleneck Model): Calibrated with traffic counts and BART ridership data.
New York City Congestion Relief Zone (MFD Model): Calibrated with subway ridership, taxi data, and road network geometry.

4. Key Results

Theoretical Findings

Revenue vs. Welfare Trade-off: There is a fundamental trade-off. In scenarios where the ratio of maximum system throughput to desired arrival rate is close to 1 (e.g., Bay Bridge), static tolls capture high revenue but may incur higher system costs. In scenarios with high transit mode share (e.g., NYC), static tolls capture slightly less revenue but achieve system costs very close to the dynamic optimum.
Non-Monotonicity: In the bottleneck model, the performance gap between static and dynamic tolls is non-monotonic with respect to the "discomfort multiplier" (transit attractiveness). Performance dips when transit is moderately worse than driving but improves when transit is significantly worse (as users self-select out of the system or the toll structure aligns better).

Empirical Findings (Numerical Experiments)

Revenue Performance: Static revenue-optimal tolls achieve 80–90% of the dynamic optimal revenue in both case studies for practically relevant parameters.
System Cost Performance: Static tolls incur only 8–20% higher total system cost than the minimum achievable.
Current Implementation: The authors found that the currently implemented static tolls in NYC ($9) and the Bay Bridge ($8.50) correspond to parameter regimes where static pricing captures nearly 98% of the dynamic optimal revenue and increases system cost by only 3% compared to the theoretical minimum.
Revenue vs. System-Cost Optimal Static Tolls: The study reveals that the static toll that maximizes revenue is often nearly identical to the static toll that minimizes system cost, suggesting that policymakers do not need to choose between revenue and welfare objectives when setting static tolls.

5. Significance and Implications

Justification for Static Pricing: The paper provides strong theoretical and empirical evidence that the simplicity of static tolls does not come at a massive cost. Static pricing captures the vast majority of the benefits of complex dynamic pricing, making it a highly effective policy tool.
Policy Guidance: For cities implementing congestion pricing, the results suggest that investing in complex, real-time dynamic tolling infrastructure may yield diminishing returns compared to well-calibrated static tolls.
Revenue Focus: By shifting the focus to revenue maximization (often a primary goal for funding transit), the authors show that revenue-optimal static tolls are robust and perform well on welfare metrics as well.
Worst-Case Guarantees: The theoretical bounds are interpreted as worst-case scenarios. Since the analysis abstracts from the real-world friction of dynamic pricing (e.g., user confusion, communication costs), the practical case for static tolling is even stronger.

In conclusion, the paper demonstrates that simple static congestion pricing is "near-optimal" for both revenue generation and system efficiency in real-world settings, validating the operational strategies of major global cities.