Sofia Gomes
5 min
This paper presents a collection of three distinct microeconomic problem sets, covering consumer theory, game theory, and mechanism design. The author systematically works through the mathematical requirements for rational choice, the strategic interaction between an incumbent and an entrant, and the optimal screening of consumer types by a monopolist.
The first section examines the consistency of consumer choices. By applying the Weak Axiom of Revealed Preference (WARP), the author evaluates whether observed choice sets can be represented by a utility function. The analysis highlights that for a utility function to exist, preferences must be complete and transitive, and the choice environment must be sufficiently rich to allow for consistent preference ordering.
The second section models a game between an incumbent firm and a potential entrant. Using both pure and mixed strategy Nash equilibria, the author explores how an incumbent might deter entry. The analysis extends to Perfect Bayesian Equilibrium (PBE) in a signaling game, where the incumbent's strategy depends on its private type. The author demonstrates how off-path beliefs are critical in sustaining specific pooling or separating equilibria.
The final section addresses a classic mechanism design problem: a monopolist attempting to extract surplus from consumers with different valuations (types). The author derives the First-Best solution (where the monopolist has full information) and the Second-Best solution (where the monopolist must provide incentive compatibility to prevent high-type consumers from mimicking low-type consumers). The results show that the monopolist must leave informational rents to the high-type consumer, leading to a distortion in the quality offered to the low-type consumer.
Sam: And if the firm could identify individuals directly, it would just move to first-degree discrimination and the distortions disappear entirely.
Alex: Yes—full observability collapses the incentive compatibility constraints. The distortions are a symptom of the information gap, not of the technology or the cost structure. [[RP_SECTION:market-exclusion-and-welfare|Market Exclusion and Welfare]]
Sam: Are there conditions where the firm just drops the low tier altogether?
Alex: Yes. If the informational rent the firm must pay the high type—plus the cost of the distortion—exceeds the profit extractable from the low type, the firm rationally excludes that segment. So a market heavily skewed toward high-value users might simply have no entry-level product. Not because it's unprofitable to serve those consumers in isolation, but because serving them makes the high-type extraction problem more expensive.
Sam: That's a meaningful welfare implication. The low-type consumers aren't just getting a degraded product—they might get no product at all.
Alex: And that's where the real-world limitations of the model start to bite. The framework assumes preferences are one-dimensional and ordered. Real consumer preferences are multi-dimensional. When that's true, the standard menu construction breaks down—there's no clean way to rank types, and the incentive compatibility constraints multiply in ways that make the optimization problem analytically intractable. [[RP_SECTION:data-and-observational-gaps|Data and Observational Gaps]]
Sam: Which is presumably why firms don't actually solve this from first principles—they use behavioral data instead.
Alex: Exactly. Real-time behavioral data lets firms approximate individual willingness to pay directly, which relaxes the incentive compatibility constraints. You're moving toward first-degree discrimination without calling it that. The menus get less rigid, the distortions shrink, and the firm captures more surplus.
Sam: But total welfare doesn't necessarily improve—it just redistributes. The consumer surplus that the current model leaves on the table gets absorbed by the firm.
Alex: That's the core tension. Better observation increases efficiency in the sense that fewer units are undersupplied, but it eliminates the informational rent that currently protects the consumer. Whether that's a welfare improvement depends entirely on your distributional weights.
Sam: So the trajectory of pricing strategy is less about designing better menus and more about closing the observational gap that makes menus necessary in the first place.
Alex: That's a precise way to put it. And it reframes what looks like a technical pricing problem as fundamentally a question about information asymmetry—who knows what about whom, and what that knowledge is worth. The distortions we see in tiered products aren't accidents or oversights. They're the equilibrium outcome of a well-specified information problem. Understanding that mechanism is what lets you predict where those distortions will appear, how large they'll be, and when they'll disappear entirely.
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