Technical Summary: Profiling and Endogenous Valuation

1. Problem and Motivation

This paper investigates the welfare implications of a monopolist possessing information about a buyer's private characteristics when the buyer's valuation is endogenous, determined by a costly pre-trade investment. While existing literature (e.g., Bergemann et al., 2015) establishes that information allows sellers to better price-discriminate and divide a fixed surplus, this study highlights a critical departure: when valuations depend on buyer actions, information affects both surplus creation and surplus division.

The central tension arises because information that helps a seller extract surplus can simultaneously undermine the buyer's incentive to create that surplus. The authors analyze a setting where a buyer can pay a privately known cost $c$ to increase their valuation from a baseline $L$ to a higher value $H$. The seller observes a signal about $c$ (profiling) before setting a price but does not observe the investment decision itself. The primary research question is: Which buyer-seller payoff pairs are attainable across all possible profiling structures?

2. Model and Methodology

The model features a risk-neutral monopolist and a unit-demand buyer with an initial valuation $L$. The buyer can invest at a private cost $c \sim F$ to raise their valuation to $H > L$. The investment gain is $\gamma = H - L$. It is assumed that $\gamma > \bar{c}$, meaning investment is socially efficient for all types.

Timing:

1. A signal structure (profiling mechanism) is announced.
2. The buyer observes their cost $c$ and decides whether to invest.
3. The seller observes a signal $s$ correlated with $c$ and sets a take-it-or-leave-it price $p \in \{L, H\}$.
4. The buyer accepts if their payoff is non-negative.

Methodological Approach:
The authors adopt the "surplus division" framework of Bergemann et al. (2015) but adapt it to endogenous valuations. Instead of assuming a fixed surplus, they characterize the set of attainable payoffs $(u_B, u_S)$ by identifying necessary and sufficient constraints. They utilize a Bayes-Nash equilibrium concept and construct specific signal structures (dual-cutoff equilibria) to demonstrate sufficiency.

3. Key Constraints and Results

3.1 Necessary Bounds on Attainable Payoffs

The paper identifies two fundamental constraints that any attainable payoff pair must satisfy:

1. Surplus Constraint (SC): The sum of payoffs cannot exceed the total surplus generated by the equilibrium investment level. If $\tau$ represents the cutoff type who is indifferent to investing, the total surplus is:
   $$SW(\tau) = \int_{\underline{c}}^{\tau} (H - c)f(c)dc + \int_{\tau}^{\bar{c}} \frac{c}{\gamma} L f(c)dc$$
   Thus, $u_B + u_S \leq SW(\tau)$.

2. Hold-up Risk Constraint (HC): The seller's payoff must be at least what she could earn by ignoring her signal and always charging the high price $H$. If the mass of investing types is $F(\tau)$, the seller can guarantee $F(\tau)H$. Thus:
   $$u_S \geq F(\tau)H$$
   This constraint imposes an upper bound on investment. If investment is too high, the seller's temptation to "hold up" the buyer by charging $H$ becomes irresistible, causing the buyer to unravel their investment incentive.

Proposition 1 establishes that these two constraints are sufficient for attainability. A pair $(u_B, u_S)$ is attainable if and only if there exists a cutoff $\tau$ such that:
$$SW(\tau) - u_B \geq u_S \geq F(\tau)H$$

3.2 The Pareto Frontier and Welfare Implications

The paper characterizes the Pareto frontier of the attainable set, revealing several counter-intuitive results compared to fixed-valuation models:

• First-Best Unattainability: The first-best surplus (where every type invests) is never attainable. Achieving full investment would require the seller to price at $H$, which would deter investment entirely.
• Strict Concavity and Welfare Trade-offs: The Pareto frontier is strictly concave with a slope steeper than $-1$($\bar{U}'_S(u_B) < -1$).
  • Key Finding: Increasing the buyer's payoff on the frontier strictly reduces total welfare.
  • Mechanism: As the investment cutoff $\tau$ increases (more investment), the seller's required payoff ($F(\tau)H$) grows faster than the total surplus created ($SW(\tau)$). To sustain higher investment, the seller must capture a disproportionately larger share of the surplus. Therefore, protecting the buyer requires discouraging investment to reduce the seller's hold-up temptation, even though investment is socially efficient.
• Information Value: Unlike fixed-valuation settings where information never hurts the seller, here more accurate profiling can strictly lower the seller's profit. Perfect profiling leads to a unique equilibrium with payoffs $(0, L)$, which is the seller's lowest individually rational payoff.

3.3 Attaining Payoffs: Dual-Cutoff Equilibria

Proposition 3 constructs a "dual-cutoff" equilibrium to attain any feasible payoff pair. This equilibrium uses two thresholds, $\phi$ and $\tau$ (where $\phi < \tau$):

• $\tau$ (Surplus Creation Margin): Types $c \leq \tau$ invest; types $c > \tau$ do not.
• $\phi$ (Surplus Division Margin):
  • Types $c \leq \phi$ are "mis-profiled" to receive price $L$ with probability 1, allowing them to retain the full investment gain ($\gamma - c$).
  • Types $\phi < c \leq \tau$ are profiled such that they are indifferent between investing and not, earning zero surplus, while the seller captures the investment gain.
  • Types $c > \tau$ do not invest.

This construction reveals that information about high-cost types (who are unlikely to invest) is beneficial as it prevents inefficient trade breakdowns. Conversely, information about low-cost types (who are likely to invest) is purely distributive and can weaken investment incentives.

3.4 Comparative Statics

Proposition 4 analyzes the effect of a mean-preserving spread in the distribution of investment costs. A more dispersed distribution (higher variance) expands the set of attainable payoffs. This occurs because a dispersed distribution allows a given mass of investment to be supplied by lower-cost types, relaxing the trade-off between surplus creation and division.

Proposition 5 extends the results to multiple investment options. If the highest-return option is socially efficient for all types, the attainable set remains structurally identical to the binary case, with the efficient option serving as the "high" valuation.

4. Significance and Claims

The paper claims to provide a complete characterization of the welfare implications of buyer profiling in markets with endogenous valuations. Its primary contributions are:

1. Entanglement of Creation and Division: It demonstrates that with endogenous valuations, surplus division and creation are inseparable. Information cannot freely divide a fixed pie; it alters the size of the pie itself.
2. The Hold-Up Risk Constraint: It identifies a novel constraint where the seller's ability to extract surplus is limited by the threat of expropriating the buyer's investment, which in turn limits the total surplus available.
3. Policy Implications: The paper argues that the policy question is not simply whether profiling should be banned or allowed, but whose information should be profiled.
   • Profiling low-cost (likely investing) types harms the buyer by increasing hold-up risk.
   • Profiling high-cost (unlikely investing) types can benefit both parties by sustaining trade.
4. Counter-Intuitive Welfare: It establishes that protecting buyer welfare may require discouraging investment (via less accurate profiling of low-cost types), even when that investment is socially efficient.

The authors conclude that in endogenous valuation environments, the "first-best" is unattainable, and the optimal information structure depends critically on balancing the seller's hold-up risk against the buyer's incentive to invest.