Technical Summary: Topping Up and Optimal Redistribution

1. Problem Statement

This paper investigates the impact of "topping up"—the ability of recipients of in-kind transfers to supplement subsidized consumption with private-market purchases—on the optimal design of redistribution programs. While governments frequently redistribute resources in kind (e.g., food stamps, housing vouchers, childcare subsidies), programs vary significantly in whether they permit recipients to combine public benefits with private spending.

The central tension addressed is between screening and flexibility. Allowing topping up expands consumer choice sets but weakens the social planner's ability to screen consumers based on their demand types. Specifically, if high-demand consumers can access subsidized allocations intended for low-demand consumers and then top up their consumption privately, the planner loses the ability to use low quantities as a screening device. Conversely, prohibiting topping up requires monitoring or enforcement, which may be costly, but preserves screening power.

The paper asks three specific questions:

1. How does topping up affect the decision of when a social planner should intervene (the extensive margin)?
2. When does topping up change the structure of the optimal mechanism (the intensive margin)?
3. How does topping up alter the comparative statics of optimal redistribution with respect to changes in redistributive priorities?

2. Methodology and Framework

The authors develop a mechanism-design model featuring a unit mass of consumers with heterogeneous demand for a homogeneous good.

• Consumer Environment: Consumers have quasilinear utility $U = \theta v(q) - t$, where $\theta$ is a private type representing demand intensity, drawn from a distribution $F$. They have access to a perfectly competitive private market with constant marginal cost $c$.
• Social Planner: The planner offers an alternative, potentially nonlinear, price schedule. The planner cannot make lump-sum cash transfers within the mechanism but can subsidize the good. The planner's objective is to maximize weighted total surplus, where consumers have heterogeneous welfare weights $\omega(\theta)$ (representing redistributive priority) and public funds have an opportunity cost $\alpha$.
• Participation Structures:
  • No Topping Up: Consumers must choose between the planner's allocation and the private market. This generates standard participation constraints (Individual Rationality, IR) relative to the laissez-faire utility.
  • Topping Up: Consumers can purchase from the planner and then supplement with the private market. This imposes a pointwise lower-bound constraint on total consumption: $q(\theta) \geq q^{LF}(\theta)$, where $q^{LF}$ is the laissez-faire quantity.

Solution Approach:
The authors solve two distinct mechanism-design problems with type-dependent outside options.

1. With Topping Up: The problem involves a pointwise lower-bound constraint. The authors use a Lagrangian approach and generalized ironing methods (adapting Amador and Bagwell, 2013) to characterize the optimal allocation.
2. Without Topping Up: The problem involves participation constraints that can be reformulated as majorization constraints. Unlike linear programs where solutions are extreme points, the authors' objective is strictly concave. They explicitly characterize the relevant Lagrange multipliers in terms of primitives to solve the problem, avoiding reliance on extreme-point methods.

3. Key Contributions and Results

A. The Extensive Margin: When to Intervene

The paper characterizes the sufficient statistics that determine whether the social planner should deviate from laissez-faire.

• Without Topping Up: Intervention is optimal if and only if the maximal pointwise welfare weight exceeds the opportunity cost of public funds: $\max_{\hat{\theta}} \omega(\hat{\theta}) > \alpha$.
• With Topping Up: Intervention is optimal if and only if the maximal upper-tail average welfare weight exceeds the opportunity cost: $\max_{\hat{\theta}} E[\omega(\theta) \mid \theta \geq \hat{\theta}] > \alpha$.

Implication: Since the average of a set cannot exceed its maximum, topping up weakens the incentive to intervene. The planner can only justify intervention if the average priority of those who would mimic a targeted type (and top up) is high enough.

• Positive Correlation: If welfare weights increase with demand ($\omega' > 0$), the maximum equals the upper-tail average. Topping up has no effect on the intervention decision.
• Negative Correlation: If welfare weights decrease with demand ($\omega' < 0$), the upper-tail average is strictly lower than the maximum. Topping up reduces the scope for intervention, potentially rendering it suboptimal in environments where it would be optimal without topping up.

B. The Intensive Margin: Structure of the Optimal Mechanism

The paper characterizes how the optimal mechanism changes when intervention occurs.

• Positive Correlation: Topping up does not alter the optimal mechanism. The mechanism features a "lower-cutoff" structure: low types are left to the private market, while high types receive a nonlinear subsidy (potentially including a free public option if average weights are sufficiently high). Self-targeting ensures no leakage.
• Negative Correlation: Topping up significantly alters the mechanism.
  • Without Topping Up: The planner can use low quantities to screen. High types must forgo private consumption to access subsidized low quantities. This allows for a mechanism that serves low types (high priority) while excluding high types.
  • With Topping Up: High types can access subsidized low quantities and top up privately. This weakens screening, forcing the planner to reduce the scope of the free public option and serve fewer types. The optimal allocation rotates: consumption is reduced for low types (the intended beneficiaries) and increased for high types relative to the no-topping-up case.

C. Comparative Statics

The paper analyzes how optimal redistribution responds to changes in welfare weights.

• Pointwise Increases: Under positive correlation, increasing welfare weights expands redistribution monotonically. Under negative correlation with topping up, the effect is also monotonic. However, without topping up under negative correlation, a pointwise increase in welfare weights can lead to a non-monotonic effect: allocations for high-demand consumers may decrease as the planner distorts quantities further to redirect surplus to the bottom.
• Mean-Preserving Spreads (Increased Informativeness):
  • Positive Correlation: Increased dispersion in welfare weights (making demand more informative) expands redistribution.
  • Negative Correlation with Topping Up: Increased dispersion reduces redistribution. The leakage effect (high types mimicking low types and topping up) dominates the gain in statistical informativeness.
  • Negative Correlation without Topping Up: Increased dispersion can expand the scope of a free public option, as the planner can better utilize low quantities for screening.

4. Significance and Claims

The paper claims to provide the first general characterization of optimal in-kind redistribution in the presence of private market access and topping up.

• Policy Relevance: The results offer a precise rationale for when topping up is benign versus costly. If redistributive priorities align with demand (e.g., disability care where higher need implies higher consumption), topping up imposes no screening costs and should be allowed. If priorities run counter to demand (e.g., housing assistance where lower income implies lower ability to pay), topping up creates significant screening costs, reducing both the likelihood of intervention and the effectiveness of the program.
• Methodological Contribution: The paper demonstrates that mechanism-design problems with type-dependent outside options (specifically majorization constraints and pointwise lower bounds) remain tractable. By explicitly characterizing Lagrange multipliers and utilizing generalized ironing, the authors derive global comparative statics that are often difficult to obtain in nonlinear mechanism design.
• Theoretical Insight: The paper clarifies that the "screening value" of low quantities is the critical friction. Topping up destroys this value, forcing the planner to rely on blunt instruments (like free public options) or reducing intervention entirely.

In summary, the paper argues that the design of in-kind redistribution cannot be separated from the rules governing private market supplementation. The optimal policy depends fundamentally on the correlation between the social value of redistribution and the market demand for the good.