Papers

We study social welfare rules with strict individual and social preferences. A preference profile is dictatorial for a rule if the social preference coincides with some agent's preference and no agent with a different preference can change the social preference unilaterally. We establish that (i) a rule is dictatorial if and only if all of its profiles are dictatorial; and (ii) if a rule satisfies unanimity and independence (of irrelevant alternatives), then all its profiles must be dictatorial. These results constitute a proof of Arrow's Impossibility Theorem.

We study the strategic design of learning algorithms in a canonical continuous-time pricing game. We introduce a meta-game in which firms select learning rates for a gradient dynamic, evaluating payoffs as the discounted sum of profits accrued along the entire learning path. We uncover a fundamental dichotomy driven by stage-game incentives: if the stage game exhibits strategic substitutability, firms unambiguously prefer the fastest possible algorithms to rapidly exploit a gradually adjusting opponent. Under strategic complementarity, however, an excessively fast algorithm accelerates the rival's competitive response, destroying transitional profit margins. Even though the competitive price is a strictly dominant action in the underlying stage game, we prove that firms optimally design sluggish algorithms, extracting surplus during a prolonged convergence to the competitive equilibrium.

An agent learns from data that many parameters of her model explain equally well. Berk's theorem states that her belief concentrates on the parameters that best fit the data, but does not select between them. We show that, with a continuum of parameters, the posterior concentrates on the best-fitting parameters with the smallest local learning coefficient. We propose a definition of narratives, subsets of the best-fitting parameters on which the learning coefficient is constant. Narratives are characterized by the number of coincidences they require (complexity) and how exactly each must hold (robustness). When the agent's actions affect her data, every long-run action is a singular equilibrium, a best reply to a belief supported on the least complex and most robust narratives. Singular equilibrium refines Berk-Nash and self-confirming equilibrium by restricting overly complex or knife-edge off-path beliefs, and its strict uniform version characterizes uniformly stable actions.

Evidence that pricing algorithms collude comes from markets in which every seller runs the same algorithm. We ask what happens when they do not. Four language models from four providers, each at its cheapest tier, price in a four-firm logit Bertrand market without communication, in every homogeneous, two-by-two and fully mixed composi- tion (11 cells, 20 runs, 200 periods). Collusion is a property of the model: Claude and Gemini markets reach 72 to 79 percent of the monopoly rent, DeepSeek markets 24 percent, and GPT markets none, though GPT prices drift above the monopoly level rather than toward competition. Mixing does not reduce collusion by itself. Markets containing Gemini, which opens at the highest price and settles highest, are more col- lusive than the homogeneous markets they are built from; markets containing Claude, which opens lower and follows its rivals down, are less so; and only the fully mixed market is significantly less collusive than the average homogeneous one. Stability de- pends on the least stable participant: two GPT firms suffice to keep any market from converging. Inside mixed markets the rent is shared in a transitive order, DeepSeek over Claude over Gemini over GPT, that inverts the anchor ranking. The model that raises the price collects the least of the rent, as the price-leadership model predicts.

Standard theories trace information asymmetry to privately held information, heterogeneous signals, or costly information acquisition. We identify a distinct mechanism: otherwise identical rational agents observing a common Markov state at asynchronous times can hold different predictive distributions because their information differs in age. When the transition kernel separates information ages, independent Bernoulli observation arrivals make such divergence recur infinitely often almost surely, even with identical arrival probabilities. When agents instead choose costly updating intensities, independent update realizations can regenerate heterogeneous information ages among ex ante identical agents. In a stationary Gaussian AR(1) environment, the consequences depend on common staleness as well as the age gap: holding the gap fixed, older information raises individual uncertainty while reducing expected squared forecast disagreement and the value of certifying relative freshness. This generates stale consensus: low disagreement despite high uncertainty. With multiple ordered ages, costly certification supports partial-unraveling equilibria in which fresher types certify and sufficiently stale types pool. Under geometric information ages, more frequent updating weakly lowers the smallest and largest equilibrium disclosure cutoffs wherever positive-certification equilibria exist. The framework has implications for credit, insurance, financial markets, and multi-agent AI.

This paper develops a general framework for dynamic mechanism and information design with AI agents, encompassing optimal contracting and integrating evolving hidden information and hidden action with partial verification, technological authorization, and multi-agent interaction. It nests conventional dynamic screening, hidden-action contracting, and conditional dynamic information design, while allowing reporting, action, and information-use decisions to interact through a common continuation mechanism. We establish a dynamic revelation principle and a recursive characterization of implementability in which truthful reporting and post-report obedience are jointly determined through continuation incentives. In quasilinear optimal-contracting environments, dynamic envelope and integral-monotonicity methods yield a verification-adjusted dynamic virtual-surplus representation and identify three distinct margins: dynamic information rents, verification rents, and the implementation cost of hidden action. These margins generate a branched hierarchy separating conventional dynamic screening, hidden-action contracting, and their joint problem. We also study the joint design of monitoring, information use, and incentives: more informative monitoring capacity expands the designer's opportunity set, but full disclosure need not be optimal, and information use can itself alter hidden-action incentives. With multiple agents, correlated information, peer discipline, coupled incentives, joint feasibility, and continuation opportunities create additional strategic interactions. Although motivated by AI-agent environments, the framework applies more broadly to dynamic agency settings with similar economic features.

Can a regulator promise both a risk ceiling and a development deadline when safety productivity is unknown? A ceiling below the final model's unprotected hazard requires a minimum stock of safety knowledge, so both promises hold only if weak research can be ruled out. Learning first and then replaying development, with spare compute in safety, comes close to that minimum. In a calibration with only state risk, constant safety yield, and full knowledge transfer, it needs only 3.3 percent more productivity than the necessary bound. Customer services pay for the guarantee. Rules that fix compute allocation fix dates, not risk.

Can noisier threshold selection produce a group with higher mean true value? We answer this question by varying the scale of a fixed assessment error. Holding the selection rate fixed, less noise yields a first-order stochastically dominant distribution of selected values. At a fixed threshold, however, more noise can raise the selected mean even as selection loss, which weights selection mistakes by the distance of true value from the threshold, increases. Under joint Gaussianity, we characterize when the selected mean rises or falls with noise. In treatment selection, noise alone can generate subgroup differences in participants' average gains despite identical population gain distributions.

Climate policy changes production costs and can reshape the geography of production. We study this spatial channel in a focused model where mobile industrial activity responds to local attractiveness, agglomeration strengthens attractiveness, and carbon policy lowers the profitability of emissions-intensive locations. The model yields an explicit carbon-policy threshold separating spatial instability from stability. When emissions intensity rises with local industrial density, spatial concentration raises aggregate emissions even with unchanged total industrial activity. Embedding these dynamics in a two-jurisdiction policy game creates a wedge between regional and collective incentives: each government values the industrial activity located within its own territory, while aggregate welfare internalizes the effect of policy on economy-wide concentration. With transferable utility, cooperation can select a policy on the stable side of the threshold while the non-cooperative policy lies on the unstable side. A numerical exercise illustrates this regime change. European regional evidence, based on industrial emissions and gross value added at NUTS2 level, documents a positive emissions-activity association and identifies the empirical quantities needed to measure relocation and concentration effects. The analysis links climate cooperation to the spatial organization of production and establishes a benchmark for policy-induced industrial relocation.

One-to-one matching without replacement is a classical approach to constructing comparable treated and control samples in the design of observational studies. It pairs each treated unit with a distinct control while minimizing a covariate distance objective. With continuous covariates, the matched pairs generally remain inexact, which contributes to bias in downstream analysis. Its key theoretical properties, such as the resulting imbalance between matched pairs and when it is negligible to support valid inference, remain unclear. In this paper, we analyze one-to-one matching based on $d$-dimensional, continuous covariates with a quadratic covariate-distance objective. First, we find that when $d\leq 3$, under standard conditions on the propensity score ensuring abundant control samples near each treated sample, the imbalance (difference between within-group averages) is root-$n$ negligible uniformly over the family of smooth functions with a common first- and second-order derivative bound. However, such balance is subject to a dimension restriction, as we construct examples in which the imbalance is root-$n$ non-negligible when $d=4$ and dominates root-$n$ rate when $d>4$. Second, we show that when $d\leq 3$, the matched design allows valid Wald-type and bootstrap inference for the average treatment effect on the treated, distributional treatment effects, and quantile treatment effects. Thus, the same outcome-blind matched design supports various downstream inferences without having to tailor the design to the targets. Finally, paired randomization inference based on the matched design is asymptotically valid in the super-population sense for $d\leq 3$ but can fail when $d=4$. We corroborate the theoretical results with numerical experiments.

We study differences in economic outcomes by perceived skin tone among African Americans using full-count U.S. decennial census data from the late-19th and early-20th centuries. Comparing children coded as "Black" or "Mulatto" by census enumerators and linking these children across population censuses, we first document large gaps in educational attainment and income among African Americans with darker and lighter perceived skin tones. To disentangle the drivers of these gaps, we identify all 35,992 families in which enumerators assigned same-gender siblings different Black/Mulatto classifications. Relative to sisters coded as Mulatto, sisters coded as Black had lower educational attainment, were less likely to marry, and had less-educated, likely lower-earning husbands. These patterns are consistent with more severe contemporaneous discrimination against African American women with darker perceived skin tones. In contrast, we find similar educational attainment, marital outcomes, and incomes among differently-classified brothers. Men perceived as African Americans of any skin tone faced similar contemporaneous discrimination, consistent with the "one-drop" racial classification rule that grouped together individuals with any known Black ancestry. Lower incomes for African American men perceived as having darker skin tone in the general population were driven by differences in opportunities and resources that varied across families, likely reflecting the impacts of historical or family-level discrimination.

Many of the questions now put to large language models have no correct answer to score against: what a policy is worth, which option a user should choose, how to weigh competing values. Stated-preference economics has faced this problem for decades. It judges survey responses without knowing the true value, through a framework of validity and related concepts: content, construct, and criterion validity, reliability, incentive compatibility, and consequentiality. We argue that this framework is a general method for evaluating language models, and we set out what each concept means for LLM evaluation. We demonstrate the approach using a published water-quality stated preference economic valuation survey (Vossler et al. 2023) administered to six models. In this economic application, the validity tests take the form of predictions from economic theory: demand should slope down, and willingness to pay should respond to the scope of the good and to income. The tests separate the models sharply. Two older models fail the most basic test at a household income level of \$75,000, and the two newest pass every test of theoretical validity we can score, but diverge on convergent validity. Passing validity tests shows that a model's answers are coherent, not that they are correct.

We introduce a mean-reverting stochastic differential equation with a three-component stochastic term and show that it generates a hierarchy of steady-state (stationary) distributions. At the top level, the hierarchy is described by a modified-Beta distribution, while one- and two-parameter reductions produce compact-support, power-law-tailed, and exponential-type limiting families within a single stochastic framework. We then construct two generalized extensions of this hierarchy. In the first, the power transformation is applied directly at the level of the stochastic differential equation; in the second, the same transformation is applied only after the stationary modified-Beta hierarchy has been obtained. While these two procedures agree on important lower branches and limiting cases they generally differ at the top level. The generalized hierarchy is therefore not unique: nonlinear transformation and stationary-state reduction do not commute. For both routes, we derive the probability density and cumulative distribution functions, express their parameters in terms of the underlying stochastic dynamics, clarify the relations among their limiting cases, and compare the resulting families with the traditional Generalized Beta framework.

The Guaranteed Utility Mechanism (GUM) was introduced to implement efficiency in Guaranteed Utility Equilibrium (GUE) in dynamic social choice problems. We show that this mechanism also has a previously unknown connection to risk minimization. Fixing an efficient decision policy, we study variance-minimizing budget-balanced transfer rules in the class of GUE-implementations of truthful reporting. A symmetrized version of GUM (sym-GUM) minimizes the sum of the agents' utility variances under truthful reporting when there is no public state. When a public state is present, a slight modification of this rule yields the minimizer for the same objective. Sym-GUM, however, does not minimize transfer risk. The minimizer for that problem is instead the symmetrized version of a closely related GUM-like transfer rule. In both problems, the minimizing truthful total transfers are unique up to agent-specific constants summing to zero and events of probability zero.

This paper extends Goodwin's (1967) predator-prey growth-cycle model to incorporate the negative impact of Opioid Use Disorder (OUD) on labor productivity. Using U.S. state-level data for 1998–2019, we first document that deteriorations in the labor share are followed by rising drug-induced mortality. We then build this link into the model through a novel discrete-choice mechanism in which the probability of OUD decreases with the wage share, so when real wages fail to keep pace with productivity growth, opioid use rises. This behavioral health channel feeds back into the economy through a productivity damage function. Applying the existence part of the Andronov-Hopf bifurcation theorem, we show that the resulting three-dimensional nonlinear system admits a limit cycle, as confirmed by our numerical simulations. Along the cycle, rising OUD paradoxically raises the employment rate, as falling productivity requires more labor per unit of output; across steady states, however, higher OUD prevalence is associated with a lower employment rate, consistent with evidence that opioid use disorder reduces labor force participation. Moreover, our numerical experiments show that OUD sensitivity to income widens fluctuations, while productivity sensitivity to OUD counterintuitively compresses them through a self-correcting feedback that speeds the recovery of the wage share. Policies that weaken the link between income and OUD, such as stronger social safety nets, stabilize the resulting dynamics.

When voters evaluate alternatives sequentially, an anchoring bias makes their ballots depend on the order in which alternatives are presented. For approval ballots, no non-constant voting rule is robust to this dependence. We show that ranked ballots change the picture sharply: a rank-based rule is robust to the presentation order if and only if it is tops-only. Robustness to anchoring is thus available, but only at the price of ignoring everything a voter reports beyond her favourite alternative.

A planner wants to elicit partial information about a subject's preference relation. Preferences are grouped into “types" relevant to the planner's purpose. Can the planner identify the subject's type through menu-choice data or experiment, without distinguishing preferences within the same type? We provide a complete answer by characterizing identifiable type spaces. The characterization uses a novel geometric condition, called face consistency, which governs distinctions within and across square and hexagonal faces of the permutohedron. For every identifiable type space, we explicitly construct the unique irreducible identifying experiment from its boundary. This experiment is optimal among all identifying experiments under any cost criterion for which adding menus cannot reduce cost. Using the equivalence between identifiability and elicitation with acts, as shown in Azrieli et al. (2021), we study symmetric policies requesting unranked shortlists that contain only acceptable products, include every product preferred to a selected one, and are nonempty whenever an acceptable product exists. Only two such policies are elicitable with acts: requesting the favorite acceptable product or the entire acceptable set.

We take a revealed-preference approach to study the welfare effects of price discrimination. An analyst observes price and quantity data before and after a monopolist segments the market. The analyst knows neither the aggregate demand curve nor the information available to the monopolist. We characterize the set of observable price-quantity pairs that are rationalizable by some market segmentation. We then obtain sharp bounds on the change in consumer surplus arising from some segmentation. Finally, we provide conditions under which the data identifies an unambiguous increase (decrease) in consumer surplus.

Centralized admissions platforms grant priority based on self-reported facts such as residence, sibling status, or family income, and a false claim can cost another applicant her seat. Verifying every claim before matching is expensive, yet checking only admitted applicants afterward is unsafe: a false claim by an applicant who ends up unmatched may already have triggered a chain of rejections. We show how a platform can check far fewer claims without changing any outcome. A claim matters only if it can move its applicant across a school's admission cutoff, and we give an exact, easily computed test for when this happens. Checking only such claims, and only when they could change a school's tentative choice, reproduces every application, rejection, and assignment of deferred acceptance under full verification. When claims are truthful, checking the highest claimed priority first minimizes checks among such policies at each school decision, and the number of checks never exceeds the school's capacity. Truthful reporting of preferences and eligibility is a weakly dominant strategy, so incentives deliver the conditions under which this audit order is optimal. In simulated markets, the policy cuts verification by 76 percent relative to checking every claim upfront and by 36 percent relative to checking each claim when it first reaches a school. The design principle is to verify decisions, not files.

We introduce a class of demand models for market-level data. The models can be estimated by linear instrumental variables regression while accommodating substitution patterns far richer than the logit and nested logit models they embed. They are built from closed-form inverse market share functions through a generator analogous to McFadden's generalized extreme value generating function, but acting on market shares. Constructive results allow arbitrary nesting structures, including overlapping nests and partial membership, yielding inverse-share analogs of generalized extreme value models. The class is consistent with utility maximization and strictly larger than the class of regular additive random utility models.

What should a macroeconomically relevant climate index contain? The composition is inherently ambiguous, and aggregation choices affect structural inference. We introduce the Assemblage VAR, which jointly estimates nonnegative aggregation weights and VAR parameters by maximizing the system likelihood-gain criterion, effectively outsourcing aggregation to observed macroeconomic dynamics. Two variants operate in component-space and rank-space, reweighting named subcomponents or emphasizing regions of the cross-sectional distribution. Applied to disaggregated U.S. climate data from the Actuaries Climate Index and NOAA, VARs using the assembled climate measures yield contractionary impulse responses that are substantially larger than those estimated using fixed-weight benchmarks. Weights emphasize high-wind variables and distributional tails over slow-moving components such as sea level.

Weighted Shapley values represent asymmetry through positive weights and ordered priority classes. We give these parameters a preference interpretation in Roth's framework of lotteries over positions in transferable-utility games. Under common calibration, nullity, ordinary risk neutrality, unanimity efficiency, and unanimity positivity, position values in unanimity games are probability equivalents. Luce conditioning of these probabilities characterizes the Kalai-Samet family and is equivalent to partnership consistency. We separate conditioning into transitivity of revealed priority, pairwise determination of support, and invariance of positive odds. Two-player observations admit a weighted Shapley extension exactly when revealed priority is transitive and within-class odds satisfy a product rule. The extension is unique: pairwise comparisons identify the priority classes and weight ratios within each class. Larger unanimity games provide additional restrictions on supports and shares. We give constructive proofs, establish independence of the five axioms, and explain the scope and limits of the elicitation interpretation.

Data collection is often costly or logistically demanding, limiting both the questions researchers can pursue and how precisely they can answer them. Prediction-powered inference (PPI) can reduce the amount of data needed for precise parameter estimation by combining labeled data with machine learning predictions. However, ignoring dependence within clusters can produce confidence intervals that cover the true parameter less often than their nominal rate. We introduce Cluster-Robust PPI++, which provides standard errors in closed form and asymptotically valid confidence intervals under arbitrary dependence within independent clusters, requiring no bootstrap or resampling. Our central contribution is to accommodate partially labeled clusters, a common empirical setting in which clusters contain both labeled and unlabeled units. As units are dependent within clusters, partially labeled clusters violate the independence assumption of PPI++. We also show how precision increases depend on the labeling design, and derive a cluster-aware power tuning rule that minimizes asymptotic variance. In an application to television news, standard PPI++ confidence intervals have coverage below 60%, whereas Cluster-Robust PPI++ can achieve nominal 95% coverage.

This paper develops novel smooth tests for structural changes in the innovation copula of multivariate dynamic models. We characterize deviations from copula constancy through a collection of generalized Fourier coefficients and test their joint significance. Under the null hypothesis, estimation of the dynamic parameters and the unknown marginals has no first-order estimation effect on the proposed statistics. Consequently, the feasible tests based on estimated residuals are asymptotically equivalent to their infeasible counterparts based on the unobserved innovations, yielding a desirable oracle property. Our proposed tests are asymptotically $\chi^2$-distributed and possess nontrivial power against local alternatives that approach the null at the parametric rate. To enhance the practicability of our methods, we further develop a data-driven procedure that automatically selects the truncation orders of the basis expansions. Unlike existing methods based on empirical copula processes or kernel smoothing, our tests require neither computationally intensive bootstrap procedures nor bandwidth selection. Extensive simulations demonstrate the satisfactory empirical size and power of our proposed tests. In particular, the data-driven test delivers substantial power gains under sparse alternatives, while remaining competitive under dense alternatives. Applications to exchange rates and stock returns further illustrate the practical usefulness of the proposed methods.

Program evaluations routinely find larger returns to investment in low-skill children, while structural models of skill formation typically find that skill and investment are complements. I show that these findings are not in tension: they measure different objects. The Beckerian schedule of returns (BSR) is the elasticity of output with respect to an input, holding all inputs at fixed values; average marginal returns (AMR) average that elasticity over the joint distribution of inputs. Revisiting Cunha, Heckman, and Schennach (2010)'s model of child skill formation on the Children of the NLSY79, I find that skill and investment are complements, but that low-skill children receive so much less investment that their AMR to investment is higher in early childhood — a pattern that reverses at later ages. The framework also provides a natural way to sign bias from curvature in log-linear models with interaction terms, which conflate the BSR and AMR and accurately estimate neither; adding quadratic terms both reduces this bias and often strengthens estimated interaction effects. Methodologically, I model discrete proxies with an ordered probit measurement system — treating them as continuous biases AMR estimates and fails to detect that low-skill children have higher AMR than high-skill children in early childhood — and introduce effective factors, a scaling system that allows latent factors to take different effective scales when producing different skills in the same period. Finally, conditional on maternal permanent income, current household income has no predictive power for parental investment, undermining the exogenous variation on which the standard control-function approach relies and cautioning a descriptive interpretation of my estimates.

It is known that completeness of the coalitional ($C$-)desirability relation in a simple game does not imply the existence of a weighted-majority representation. We show that weightedness is implied in an approximate sense. Call an $n$-player simple game $\varepsilon$-weighted if, for some weight vector in the $(n-1)$-simplex, any losing coalition weight exceeds any winning coalition weight by at most $\varepsilon$; let $\varepsilon^{\ast}$ be the smallest such $\varepsilon$. We establish an upper bound on $\varepsilon^{\ast}$ in a general $C$-complete game, and deduce that, if all classes of equally desirable players become large asymptotically but the number of classes is bounded, then $\varepsilon^{\ast}=O(1/n_{\min})$ for the minimal player class size $n_{\min}$, and the bound is tight. Importantly, the minimal attainable fraction $\delta$ of coalitions mislabeled by a weighting rule also tends to $0$ when $n_{\min}\rightarrow\infty$ and the number of player classes is bounded, with $\delta=O(1/\sqrt{n_{\min}})$.

I estimate firm-level capital-, labor-, and material-augmenting productivity for Chinese manufacturing, 1998–2008, and the effect of state-owned enterprise (SOE) privatization on each. Results show technological change is strongly biased toward capital and labor. Private firms operate above SOEs on capital and labor, but below on materials. Privatization raises labor-augmenting productivity right after the transition, a front-loaded gain that fades on the full panel but persists among 1998 incumbents. It leaves capital-augmenting productivity no higher and modestly lowers materials. The SOE sector's catch-up runs through restructuring of the plants the state kept, with exit of overstaffed plants adding to the labor gain.

We show that distinct pure-strategy Nash equilibria typically have different utilitarian welfare values in sufficiently rich classes of $C^2$ games with open strategy sets. Here, typical means both generic and finitely prevalent, while rich means that the class admits a finite-dimensional family of perturbations that can independently move payoff values and first derivatives in any direction at any two distinct strategy profiles (for example, polynomial perturbations of degree at most three). Consequently, a welfare-maximizing equilibrium, if one exists, is unique. The result extends to each fixed $C^1$ payoff aggregator whose derivative with respect to the payoff vector is everywhere nonzero. In the full $C^2$ utility space, we also treat scalar criteria depending on strategies, payoffs, and payoff derivatives, subject to a nonvanishing condition in payoff-level or cross-player derivative directions. Finally, adjoining arbitrarily small perturbations from a suitable finite-dimensional space makes welfare separation typical in the augmented family, with separation holding for almost every perturbation of each fixed base game.

Businesses regularly offer gift card programs to drive customer spending and increase engagement. A central question is how much incremental revenue these programs generate, and which channels drive it most efficiently. Measuring the incremental revenue associated with a gift card program is a challenging problem in causal inference, requiring a firm to infer how much each customer would have spent if they never received a gift card. Observational data on past customer purchasing behavior reveal possession of a gift card only when a customer makes a purchase, thus leaving a customer's treatment status systematically censored. In this paper, we develop a novel data fusion approach to overcome this missing data challenge. We identify and estimate incrementality by combining a large observational dataset with a smaller experimental dataset from a different population. Our approach relies on a mild transferability condition, which posits that the conditional relative treatment effect of gift card receipt on the decision to purchase is invariant across the two populations. We develop a flexible, machine learning-based estimator for the incremental revenue and establish its asymptotic normality. We apply our estimator across both first- and third-party channels through which Airbnb distributes gift cards, finding heterogeneity in incrementality across segments of the population. In particular, we find not only that third-party channels are more incremental than first-party ones, but also that "self-gifters" (i.e., customers likely to have purchased their own gift cards) are more incremental than the broader population.

The answers to many economic questions depend on the slope and curvature of demand. Estimating demand for differentiated products entails a trade-off between flexibility and scalability. Existing nonparametric approaches face a curse of dimensionality in the number of products. I develop a scalable nonparametric approach using market-level data that overcomes this curse through economically motivated restrictions embedded in many standard models. Simulations and an application to the U.S. beer market demonstrate flexibility in demand slope and curvature, with consequential differences in counterfactual price responses. Estimation in the application takes seconds, versus several hours for a random coefficient nested logit model.