Surplus-Elasticity Frontier in Screening

• The surplus-elasticity frontier is defined as a set of allocations that maximizes screening by using pointwise comparisons of sold-alone surplus and demand elasticity.
• The framework leverages comonotonic preferences and ordered demand curves to construct optimal menus that remain valid across diverse type distributions and welfare weights.
• It applies robustly to practical problems such as bundling, taxation, sequential screening, and regulation, ensuring distribution-agnostic and optimal mechanism design.

Searching arXiv for the cited papers and closely related work to ground the article. The surplus-elasticity frontier is a selection principle for optimal screening in arbitrary allocation spaces with comonotonic preferences. In "Screening Frontiers" (Yang, 23 Feb 2026), a principal screens an agent over an arbitrary set of allocations $X$, and a subset of allocations $X^\star\subseteq X$ is called a surplus-elasticity frontier when off-frontier allocations are jointly dominated in sold-alone demand level and elasticity, while frontier allocations can be ordered so that higher demand curves are more inelastic. Under quasilinear preferences, vertically ordered types, and redistributive welfare weights, any such frontier is an optimal menu; if incremental demand curves along the frontier are also ordered by their elasticities, the frontier is optimal even among stochastic mechanisms. The same paper emphasizes that the frontier is agnostic to type distributions and redistributive welfare weights, and applies the framework to bundling, taxation, sequential screening, selling information, and regulation (Yang, 23 Feb 2026).

1. Formal setup and primitive objects

The baseline environment has a one-dimensional compact type space and an arbitrary compact allocation space: $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$ with compact metric allocation space $\mathcal{X}\ni x$. Preferences are quasilinear,

$u(x,p;t)=v(x,t)-p,$

where $v(x,t)$ is strictly increasing and differentiable in $t$, and $v_t(x,t)$ is continuous on $\mathcal{X}\times\mathcal{T}$ (Yang, 23 Feb 2026).

The paper’s “comonotonic” assumption is the vertical-ordering condition that higher types have weakly higher values for every allocation. Formally,

$$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$

Equivalently, for any fixed $X^\star\subseteq X$0, the random variable $X^\star\subseteq X$1 is increasing in the first-order stochastic dominance order with respect to $X^\star\subseteq X$2 (Yang, 23 Feb 2026).

When an allocation $X^\star\subseteq X$3 is sold alone, the inverse demand curve is

$X^\star\subseteq X$4

where $X^\star\subseteq X$5 is the measure of types buying. The corresponding demand function is

$X^\star\subseteq X$6

where $X^\star\subseteq X$7 solves $X^\star\subseteq X$8. Elasticity is defined on inverse demand with respect to quantity: $X^\star\subseteq X$9 A higher $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$0 denotes a more elastic curve at $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$1, and in the price domain the usual demand elasticity $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$2 coincides with $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$3 when $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$4 (Yang, 23 Feb 2026).

The same framework links sold-alone surplus and demand. Defining

$$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$5

pricing at $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$6 yields demand $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$7, while $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$8 is the $$\mathcal{T}:=[\underline{t},\overline{t}],\quad t\sim F \text{ with density }f>0,$$9-quantile of $\mathcal{X}\ni x$0. The distribution of $\mathcal{X}\ni x$1 across types therefore induces the sold-alone demand curve, and the dispersion of $\mathcal{X}\ni x$2 governs elasticity through $\mathcal{X}\ni x$3 and $\mathcal{X}\ni x$4 (Yang, 23 Feb 2026).

2. Surplus order, elasticity order, and the frontier definition

The frontier is defined through two pointwise partial orders on allocations. The surplus order is

$\mathcal{X}\ni x$5

which is equivalent to

$\mathcal{X}\ni x$6

The elasticity order is

$\mathcal{X}\ni x$7

equivalently,

$\mathcal{X}\ni x$8

(Yang, 23 Feb 2026).

A compact subset $\mathcal{X}\ni x$9 is a surplus-elasticity frontier if two requirements hold. First, every off-frontier allocation is jointly dominated: for any $u(x,p;t)=v(x,t)-p,$0, there exists $u(x,p;t)=v(x,t)-p,$1 such that

$u(x,p;t)=v(x,t)-p,$2

Second, the frontier itself forms a chain: there exists an index set $u(x,p;t)=v(x,t)-p,$3 and labeling $u(x,p;t)=v(x,t)-p,$4 such that for all $u(x,p;t)=v(x,t)-p,$5,

$u(x,p;t)=v(x,t)-p,$6

so higher demand curves on the frontier are more inelastic (Yang, 23 Feb 2026).

The paper also defines incremental demand curves along the frontier. For $u(x,p;t)=v(x,t)-p,$7,

$u(x,p;t)=v(x,t)-p,$8

If along the frontier $u(x,p;t)=v(x,t)-p,$9, for all $v(x,t)$0,

$v(x,t)$1

then the frontier is a strong frontier (Yang, 23 Feb 2026).

This definition separates two screening margins. Surplus comparisons identify allocations that are pointwise better in sold-alone value, while elasticity comparisons identify allocations that are more or less effective at controlling information rents. A plausible implication is that the frontier summarizes the trade-off between surplus creation and rent extraction in a primitive, distribution-free way.

3. Optimality theorems and mechanism-theoretic logic

The main theorem states that under quasilinear preferences with comonotonic types and redistributive welfare weights $v(x,t)$2 that are continuous, non-increasing in $v(x,t)$3, and satisfy $v(x,t)$4, any surplus-elasticity frontier $v(x,t)$5 is an optimal menu: $v(x,t)$6 (Yang, 23 Feb 2026).

The proof sketch in the paper proceeds in three steps. The first is reconstruction by sold-alone dominance. Given any mechanism $v(x,t)$7, each assignment $v(x,t)$8 is replaced by a two-point lottery over adjacent frontier elements $v(x,t)$9 that gives type $t$0 the same truth-telling payoff but generates a demand curve that single-crosses that of $t$1 from below. This preserves IR and all downward IC constraints and keeps the principal’s objective unchanged; the resulting lotteries are stochastically ordered (Yang, 23 Feb 2026).

The second step is downward sufficiency. In one-dimensional screening with increasing differences, satisfying downward IC and IR is sufficient because transfers can be adjusted to satisfy upward IC without lowering the objective. The paper states this as a theorem for any type distribution and any redistributive welfare weights, provided the allocation set is compact and ordered and $t$2 has strict increasing differences (Yang, 23 Feb 2026).

The third step is purification. When the stochastic allocation set is stochastically ordered and the principal has generalized quasilinear preferences, randomization is unnecessary: for any fully IC mechanism with stochastically ordered lotteries, there exists a deterministic fully IC mechanism that weakly improves the principal’s objective (Yang, 23 Feb 2026).

These steps yield robust optimality of deterministic frontier menus. The paper’s economic interpretation is that surplus comparisons pin down what to offer absent private information, while elasticity comparisons capture dispersion of values and thus information rents. A frontier simultaneously steepens high-surplus options and dominates off-frontier options on both margins (Yang, 23 Feb 2026).

A second theorem concerns stochastic mechanisms. If $t$3 is a strong frontier, then

$t$4

so the frontier is optimal even among stochastic mechanisms (Yang, 23 Feb 2026). The paper’s intuition is that ordered incremental elasticities validate “demand profile” pricing: each upgrade can be priced at its monopoly quantity independently, and monotone rearrangement guarantees implementability in a one-dimensional environment.

A common misconception is that such menu characterizations should depend on the type distribution or on a specific welfare criterion. In this framework, the frontier is defined by pointwise comparisons of $t$5 and $t$6, which are independent of $t$7 and $t$8. The same frontier therefore remains optimal for all type distributions $t$9 and for any redistributive $v_t(x,t)$0 satisfying the stated restrictions, from profit maximization with $v_t(x,t)$1 to social welfare maximization with $v_t(x,t)$2 (Yang, 23 Feb 2026).

4. Generalized frontiers, construction, and a worked example

Because the surplus and elasticity orders are partial orders, a strict frontier need not exist. The paper therefore defines a generalized frontier. A subset $v_t(x,t)$3 is a generalized frontier if $v_t(x,t)$4 can be totally ordered by some $v_t(x,t)$5, there exists a compact $v_t(x,t)$6 totally ordered by the induced stochastic order, every $v_t(x,t)$7 is covered by $v_t(x,t)$8 in the sense that some $v_t(x,t)$9 has $\mathcal{X}\times\mathcal{T}$0 weakly single-crossing $\mathcal{X}\times\mathcal{T}$1 from below at any $\mathcal{X}\times\mathcal{T}$2, and increasing differences hold along $\mathcal{X}\times\mathcal{T}$3 in the sense that $\mathcal{X}\times\mathcal{T}$4 is strictly decreasing in $\mathcal{X}\times\mathcal{T}$5 for all $\mathcal{X}\times\mathcal{T}$6 (Yang, 23 Feb 2026). Any generalized frontier is an optimal menu, and with pointwise ordered demand curves and ordered incremental elasticities it is also optimal among stochastic mechanisms.

The paper gives a practical procedure for identifying a frontier from a candidate set $\mathcal{X}\times\mathcal{T}$7. First, compute inverse demand profiles and elasticities: $\mathcal{X}\times\mathcal{T}$8 When feasible, one uses the type-space equivalents

$\mathcal{X}\times\mathcal{T}$9

Second, remove any $$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$0 for which there exists $$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$1 with both $$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$2 and $$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$3. Third, check whether the remaining set can be totally ordered so that adjacent elements satisfy higher surplus and lower elasticity. Fourth, compute incremental inverse demand and incremental elasticity for adjacent pairs and verify that incremental elasticities are decreasing along the chain. Fifth, if no chain exists, seek a generalized frontier by building a stochastically ordered family of lotteries that covers each $$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$4 via single-crossing (Yang, 23 Feb 2026).

The minimal worked example has types $$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$5, features $$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$6, and valuation

$$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$7

Then

$$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$8

and

$$\text{for all }x\in\mathcal{X},\quad v_t(x,t)>0\text{ and }t\mapsto v(x,t)\text{ is strictly increasing}.$$9

Increasing $X^\star\subseteq X$00 raises $X^\star\subseteq X$01 pointwise and increases $X^\star\subseteq X$02, while increasing $X^\star\subseteq X$03 raises $X^\star\subseteq X$04 pointwise and decreases $X^\star\subseteq X$05 (Yang, 23 Feb 2026).

Among all $X^\star\subseteq X$06, the chain

$X^\star\subseteq X$07

satisfies the frontier conditions, and the paper states that incremental elasticities along $X^\star\subseteq X$08 are ordered, decreasing in $X^\star\subseteq X$09, so this is a strong frontier, optimal even allowing lotteries, with pricing via the demand-profile method (Yang, 23 Feb 2026). Under profit maximization, the paper also gives a corollary: if a strong frontier is differentiable in its index and marginal inverse demand has interior monopoly quantities, then the demand-profile method is valid and the frontier is a minimal optimal menu.

5. Applications across screening and regulation

The framework yields distribution-agnostic solutions by translating rich allocation spaces into sold-alone demand level and elasticity comparisons (Yang, 23 Feb 2026).

In optimal bundling, let goods be $X^\star\subseteq X$10, bundles $X^\star\subseteq X$11, and valuations $X^\star\subseteq X$12 monotone in inclusion and strictly increasing in $X^\star\subseteq X$13. A nested menu $X^\star\subseteq X$14 spans a frontier if

$X^\star\subseteq X$15

and every off-menu bundle is dominated by some larger menu bundle in the same log-slope sense. Then $X^\star\subseteq X$16 is a surplus-elasticity frontier and is optimal for all $X^\star\subseteq X$17 and redistributive $X^\star\subseteq X$18. If adjacent increments also satisfy the ordered log-slope condition for $X^\star\subseteq X$19, the frontier is strong and optimal among lotteries (Yang, 23 Feb 2026). The paper further states a special case: under stochastic ratio-monotonicity and a condition on welfare weights, pure bundling is optimal among stochastic mechanisms.

In optimal taxation with ordeals, labor choice is $X^\star\subseteq X$20, ordeal is $X^\star\subseteq X$21, disutility is $X^\star\subseteq X$22 strictly decreasing in ability $X^\star\subseteq X$23, and net utility is $X^\star\subseteq X$24. Two conditions imply that “no costly screening” is optimal, meaning only $X^\star\subseteq X$25 is offered: ordeal harm declines with ability,

$X^\star\subseteq X$26

and the net surplus without ordeal is log-supermodular while ordeal weakens take-up elasticity: $X^\star\subseteq X$27 The paper restates the latter condition in elasticity form through take-up elasticity to a marginal tax cut (Yang, 23 Feb 2026).

In sequential screening, a seller contracts before the buyer observes ex-post value $X^\star\subseteq X$28, with ex-ante type $X^\star\subseteq X$29 and conditional distribution $X^\star\subseteq X$30 that is FOSD-increasing in $X^\star\subseteq X$31. A simple nonrefundable posted price is optimal if

$X^\star\subseteq X$32

so dispersion of ex-post values increases with ex-ante types on the log scale (Yang, 23 Feb 2026).

In selling information, with a uniform prior on a finite state space and type-dependent indirect utility $X^\star\subseteq X$33 that is convex in posteriors, a menu of truth-or-noise signals $X^\star\subseteq X$34 is optimal in symmetric environments if $X^\star\subseteq X$35 is strictly increasingly convex in $X^\star\subseteq X$36. Conversely, if $X^\star\subseteq X$37 is increasingly concave in $X^\star\subseteq X$38, selling full information is optimal. In symmetry, truth-or-noise signals are Blackwell-monotone in $X^\star\subseteq X$39 and form a generalized frontier via single-crossing coverage (Yang, 23 Feb 2026).

In regulation of a data-rich monopolist, the regulator designs a menu of pricing rules for a firm that can price discriminate using consumer data $X^\star\subseteq X$40, with interdependent costs $X^\star\subseteq X$41 depending on firm type $X^\star\subseteq X$42. Let total surplus be $X^\star\subseteq X$43. If $X^\star\subseteq X$44 is log-submodular, uniform pricing regulations are optimal; if $X^\star\subseteq X$45 is log-supermodular, discriminatory pricing regulations are optimal (Yang, 23 Feb 2026). For the example $X^\star\subseteq X$46, the paper gives

$X^\star\subseteq X$47

and states that uniform pricing is optimal under advantageous selection or mild adverse selection, while severe adverse selection calls for discrimination plus taxation.

6. Relation to classical screening, dispersive order, and matching-based reinterpretations

The framework is positioned against classical screening results associated with Myerson and Rochet–Choné. The paper states that traditional results rely on virtual surplus and regularity in one dimension, and on additivity plus convex programming in multidimensional settings, whereas the surplus-elasticity frontier focuses on two primitives of sold-alone markets—surplus levels and demand elasticities—and operates with arbitrary allocation spaces under comonotonic types, without assumptions on $X^\star\subseteq X$48 or $X^\star\subseteq X$49 (Yang, 23 Feb 2026). This suggests a shift from distribution-specific regularity conditions toward pointwise comparisons of primitives.

The elasticity order is linked to dispersion on the log scale: $X^\star\subseteq X$50 The paper uses this relation to connect rent extraction to log-level dispersion and to generalize ratio-monotonicity conditions in bundling (Yang, 23 Feb 2026).

The same section also states the main limitation. Frontiers may not exist because pointwise surplus and elasticity orders are partial. The generalized frontier relaxes the requirement through covering by randomized demand curves and restores robust optimality. Comonotonic preferences are essential; environments with non-ordered types or non-monotone preferences require different tools (Yang, 23 Feb 2026).

A related but distinct use of the term appears in the synthesis of "Identification of Matching Complementarities: A Geometric Viewpoint" (Galichon, 2021). That paper does not define a “Surplus-Elasticity Frontier” explicitly. Instead, the synthesis constructs the concept from its geometric and entropy-based identification framework as the feasible set of elasticities of match probabilities to shocks in types or surplus that are consistent with equilibrium optimality, feasibility, and shape restrictions on complementarities (Galichon, 2021).

In the transferable-utility matching model of that synthesis, a matching matrix $X^\star\subseteq X$51 lies in a compact convex polytope $X^\star\subseteq X$52, stable assignment maximizes $X^\star\subseteq X$53, and the support function is

$X^\star\subseteq X$54

With Shannon entropy,

$X^\star\subseteq X$55

the regularized welfare problem yields the logit or gravity form

$X^\star\subseteq X$56

with $X^\star\subseteq X$57 chosen to match marginals. Elasticities of log match probabilities to surplus shocks then satisfy feasibility-constrained linear systems, and the synthesis interprets the convex envelope of these elasticity patterns, further restricted by complementarity conditions such as $X^\star\subseteq X$58, as a surplus-elasticity frontier in matching (Galichon, 2021). This is not the same object as the frontier in screening, but it uses the same pairing of surplus structure and elasticity constraints to delimit feasible and optimal responses.

Taken together, these two lines of work give the term two technically precise meanings. In screening, the surplus-elasticity frontier is a compact ordered subset of allocations that characterizes optimal menus. In the matching synthesis, it is a constructed geometric object: the set of elasticity matrices consistent with feasibility, KKT conditions, and shape restrictions. The common element is the use of surplus comparisons and elasticity comparisons as primitive organizing devices for optimal allocation under informational or equilibrium constraints.