📈 economics

Robust Welfare under Imperfect Competition

This paper extends robust welfare analysis to imperfect competition by incorporating supply-side constraints, demonstrating that extremal welfare bounds under partial knowledge of pass-through and conduct parameters are achieved by inverse pass-through functions with a single price cutoff, thereby enabling the derivation of simple bounds for consumer surplus, producer surplus, total surplus, and deadweight loss.

Original authors: Konstantin von Beringe, Mark Whitmeyer

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Konstantin von Beringe, Mark Whitmeyer

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 you are a city planner trying to figure out how much a new tax on coffee will hurt the people buying it, the shops selling it, and the city's overall happiness.

Usually, economists try to solve this by drawing a perfect, smooth line connecting two points: the price and quantity of coffee before the tax, and the price and quantity after the tax. They assume they know exactly how that line curves. But in the real world, we don't know the exact shape of that line. It might be a gentle slope, a steep drop, or a weird curve. If you guess the wrong shape, your calculation of who wins and who loses could be completely wrong.

This paper, "Robust Welfare under Imperfect Competition," by Konstantin von Beringe and Mark Whitmeyer, offers a new way to solve this puzzle without guessing the exact shape of the line. Instead of trying to find the one true answer, they calculate the best-case and worst-case scenarios (the "bounds") that are mathematically possible given what we actually know.

Here is how they do it, using simple analogies:

1. The Two Snapshots (The Only Facts We Have)

The researchers start with just two photos of the market:

Photo A: Before the tax (Price $p_0$ , Quantity $Q_0$ ).
Photo B: After the tax (Price $p_1$ , Quantity $Q_1$ ).

They don't know what happened in between. The price might have jumped up immediately, or slowly crept up. The shops might have reacted differently.

2. The "Supply Side" Mystery (The Water in the Tub)

The paper introduces a new way to handle the "supply side" (the shops and their costs). They assume we don't know the exact rules of how shops pass the tax cost to customers, but we do know a range (a minimum and maximum) for how much of the tax gets passed on.

To find the worst and best outcomes, they use a clever trick called the "Bathtub Principle."

Imagine the demand for coffee is a bathtub that is deeper on the left (high demand) and shallower on the right (low demand).
You have a fixed amount of "water" (the total tax change) to pour into this tub.
To get the worst outcome for producers, you pour all the water into the deepest part of the tub first (where demand is highest).
To get the best outcome, you pour it into the shallowest part first.
The Result: The math shows that the extreme scenarios always happen when the tax is passed on at either the maximum rate or the minimum rate, with a single "cutoff point" where it switches. It's never a messy mix of rates; it's a clean switch from one extreme to the other.

3. The "Demand Side" Mystery (The Shape Shifter)

On the demand side (the customers), the authors build on previous work by Kang and Vasserman. They say: "We don't know the curve, but we know it follows certain rules (like it always slopes down)."

They found that even if the math gets complicated, the "worst" and "best" demand curves aren't weird, jagged lines. They are simple shapes that look like a staircase or a piecewise line that switches directions at most twice. They call this shape "EATS" (Endpoints with At most Two Switches).

Think of it like a hiker walking from point A to point B. The hiker can only walk at a "slow" speed or a "fast" speed. To get the most extreme result, they just switch between slow and fast a couple of times, rather than constantly changing their speed.

4. The "Market Power" Factor (The Lerner Index)

The paper also accounts for how much power the shops have. Are they in a fierce competition (like a gas station on a highway) or a monopoly (the only coffee shop in town)?

They use a "conduct parameter" (a dial from 0 to 1) to represent this power.
The math shows that the extreme welfare outcomes always happen when you turn this dial all the way to the lowest setting (perfect competition) or all the way to the highest setting (monopoly). You never need to check the middle settings to find the bounds.

The Final Recipe (The "Cookbook")

The paper concludes with a simple recipe for policymakers:

Take your two photos (Before and After).
Pick your ranges: Decide what is the minimum and maximum possible "pass-through" (how much tax is passed to customers) and the minimum and maximum "market power."
Run two scenarios:
- Scenario A: Assume the tax is passed on at the max rate up to a certain price, then the min rate.
- Scenario B: Assume the tax is passed on at the min rate up to a certain price, then the max rate.
Find the Demand Curves: For each scenario, find the "EATS" demand curve (the one that switches speed at most twice) that makes the result the highest or lowest.
Calculate: Plug these into the formulas to get the tightest possible range for Consumer Surplus, Producer Surplus, and Deadweight Loss (the lost efficiency).

Why This Matters

Instead of saying, "If we assume the demand curve is a straight line, the tax costs $10 million," this paper says, "Given that we only know the start and end points, and that the tax pass-through is between X and Y, the cost must be somewhere between $8 million and $12 million."

It gives a safety net. It tells policymakers, "No matter how the market actually behaved in the middle, the truth cannot be outside these boundaries." This makes policy decisions more robust because they don't rely on a single, potentially wrong guess about the shape of the economy.

1. Problem Statement

The paper addresses the challenge of conducting welfare analysis for discrete (non-marginal) policy changes (specifically tax changes) when the underlying supply and demand structures are only partially known.

The Gap: Traditional empirical welfare analysis often relies on strong functional-form assumptions (e.g., linear or CES demand) to interpolate between two observed equilibrium snapshots: $(p_0, Q_0, \tau_0)$ and $(p_1, Q_1, \tau_1)$ . These assumptions can lead to misleading conclusions if the true functional form differs.
The Objective: To derive sharp, robust bounds for Consumer Surplus (CS), Producer Surplus (PS), Total Surplus (TS), and Deadweight Loss (DWL) without assuming a specific parametric form for supply or demand. Instead, the authors rely on:
1. Demand-side shape restrictions (e.g., bounds on elasticity or slope).
2. Supply-side intervals for feasible pass-through rates ( $\rho$ ) and conduct parameters ( $\kappa$ , representing market power).

2. Methodology and Framework

The authors extend the robust demand-side framework of Kang and Vasserman (2025) (KV) by integrating a supply-side analysis. The methodology proceeds through three main analytical steps:

A. Model Setup and Decomposition

Data: Two snapshots around a specific tax change $\Delta \tau > 0$ , resulting in price change $\Delta p > 0$ and quantity change $\Delta Q < 0$ .
Assumptions:
- Demand: Inverse demand $P(Q)$ is strictly decreasing.
- Supply (Pass-through): The equilibrium price schedule $p(\tau)$ is absolutely continuous with a pass-through rate $\rho(\tau) = dp/d\tau$ bounded by $[\rho_L, \rho_U]$ . This defines an interval for the inverse pass-through $u(p) = d\tau/dp \in [\alpha, \beta]$ , where $\alpha = 1/\rho_U$ and $\beta = 1/\rho_L$ .
- Supply (Conduct): The market structure is captured by the Bresnahan (1982) conduct identity:
  $p - \tau - MC(Q) = \kappa(Q)[-P'(Q)Q]$
  where $\kappa(Q) \in [\kappa_L, \kappa_U]$ . This nests perfect competition ( $\kappa=0$ ), Cournot ( $\kappa \approx 1/n$ ), and monopoly ( $\kappa=1$ ).
Key Decomposition: The change in Producer Surplus ( $\Delta PS$ $Δ P S$ ) is decomposed into three terms:
1. Transfer from Consumers: $-\int Q(p) dp$ (which is $\Delta CS$ ).
2. Tax Burden: $-\int Q(p)u(p) dp$ (the "inverse pass-through term").
3. Markup/Conduct Effect: $-\int \kappa(Q)Q(p) dp$ .

B. Supply-Side Optimization (The "Bathtub" Principle)

The authors isolate the term $\int Q(p)u(p) dp$ as the sole dependence on the unknown pass-through schedule $u(p)$ .

Constraint: $u(p)$ must satisfy $\alpha \le u(p) \le \beta$ and $\int u(p) dp = \Delta \tau$ .
Optimization: Since demand $Q(p)$ $Q (p)$ is decreasing, the authors apply a Hardy-Littlewood rearrangement argument.
- To maximize the integral (worst-case for producers), the high value of $u$ ( $\beta$ ) should be placed where $Q(p)$ is largest (low prices).
- To minimize the integral, the high value of $u$ should be placed where $Q(p)$ is smallest (high prices).
Result: The extremal inverse pass-through functions are "BATS" (Boundary-value At a Threshold Switch). They take only the endpoint values $\alpha$ $α$ and $\beta$ $β$ , separated by a single price cutoff $p^*$ $p^{*}$ .
- $\beta$ -BATS: $u(p) = \beta$ for $p < p^*$ , then $\alpha$ .
- $\alpha$ -BATS: $u(p) = \alpha$ for $p < p^*$ , then $\beta$ .

C. Demand-Side Optimization

With the supply-side extrema fixed, the problem reduces to finding extremal demand curves $Q(p)$ subject to shape restrictions (e.g., bounds on elasticity).

Weighted Integral: The welfare objective becomes a weighted integral of the demand curve: $\int \phi(s) A^{-1}(G(s)) ds$ , where $\phi$ is an induced weight function.
Case 1: One-Signed Weight: If $\phi$ does not change sign, the extremal curves are the standard KV curves (pointwise maximal/minimal).
Case 2: Single-Crossing Weight: If $\phi$ $ϕ$ changes sign exactly once, the authors introduce a new class of extremal curves called EATS ("takes Endpoints with At most Two Switches").
- These curves have derivatives that switch between the lower and upper bounds of the shape restriction at most twice.
- The optimal curve is constructed by "gluing" the highest feasible curve on the interval where the weight is positive and the lowest feasible curve where the weight is negative, meeting at a specific threshold value.

3. Key Contributions

Integration of Supply and Demand Robustness: Unlike previous work that focused solely on demand uncertainty, this paper provides a unified framework that simultaneously bounds welfare under uncertainty in both supply (pass-through and conduct) and demand (shape).
Characterization of Extremal Supply Schedules: The paper proves that the worst-case supply schedules are simple step functions (BATS) determined by a single price cutoff, rather than complex functional forms.
Introduction of "EATS" Demand Curves: The authors identify that when welfare weights change sign (a common occurrence in imperfect competition), the extremal demand curves are not the standard KV curves but rather "EATS" curves with at most two switches in slope.
Transparent Role of Market Power: The framework explicitly links the Lerner index ( $L = \kappa / |\eta|$ ) to welfare bounds. It shows that lower welfare bounds correspond to higher conduct (more market power) combined with specific pass-through schedules, and vice versa.

4. Main Results

The paper establishes a "Cookbook" for calculating sharp welfare bounds:

Theorem 5.1: Global welfare bounds are attained by candidates with the following structure:
1. Inverse Pass-through ( $u$ ): Takes the BATS form (either $\alpha$ -BATS or $\beta$ -BATS).
2. Conduct ( $\kappa$ ): Takes an endpoint value ( $\kappa_L$ or $\kappa_U$ ).
3. Demand ( $Q$ ): Corresponds to a curve where the transformed derivative $f$ is an EATS function (switching at most twice between bounds).
Welfare Bounds Formulas:
- Producer Surplus (PS): Bounded by combinations of the BATS schedules and the conduct endpoints. Specifically, the lower bound uses the conduct upper bound ( $\kappa_U$ ) and the pass-through schedule that maximizes the tax burden on producers.
- Deadweight Loss (DWL): Bounded by the sum of the extremal tax burden terms and the conduct term, minus the fixed tax revenue.
- Computational Tractability: The problem reduces to a one-dimensional optimization over the "switching point" (the value of transformed demand at the sign-change of the weight) for the EATS curves.

5. Significance and Implications

Policy Relevance: The paper provides regulators and policymakers with a rigorous method to assess the robustness of welfare conclusions (e.g., "Is this tax beneficial?") without relying on arbitrary functional form assumptions. It quantifies how much the conclusion depends on the degree of market power and the elasticity of supply.
Empirical Application: The "Cookbook" approach allows researchers to plug in observed data and credible intervals for pass-through and conduct (derived from external moments or industry studies) to generate tight, defensible bounds for CS, PS, and DWL.
Theoretical Advancement: It bridges the gap between partial identification literature (Manski) and structural welfare analysis, demonstrating that even with significant uncertainty, welfare objects can be bounded by simple, canonical functions (BATS and EATS).
Market Power Insights: The results clarify that the impact of market power on welfare bounds is not monotonic in a simple way; it interacts with the pass-through schedule. For instance, high market power combined with low pass-through can yield different welfare outcomes than high market power with high pass-through.

In summary, von Beringe and Whitmeyer provide a mathematically rigorous and practically applicable toolkit for welfare analysis in imperfectly competitive markets, replacing untestable functional form assumptions with verifiable interval restrictions on economic behavior.