Econometrica

In the context of one‐to‐one matching markets, we study myopic‐farsighted stable sets , which are internally and externally stable when myopic agents consider immediate payoffs from their deviations, while farsighted agents anticipate counter‐deviations and consider final payoffs. We constructively prove the existence of a (rational expectations) myopic‐farsighted stable set, in which farsighted agents receive a single payoff while myopic agents may receive multiple payoffs. Our existence result extends to settings with enforcing coalitions of arbitrary size, yielding coalitional myopic‐farsighted stable sets , and to settings where not all members of an enforcing coalition must strictly gain, yielding myopic‐farsighted weakly stable sets . When all farsighted agents have unit demand, our results also extend to many‐to‐one matching markets. As a key corollary, we provide a foundation for the efficiency‐adjusted deferred acceptance algorithm by showing that its outcome constitutes a singleton myopic‐farsighted stable set when one side is farsighted and the other is myopic.

在一对一匹配市场框架下，我们研究近视-远见稳定集：当近视代理人仅考虑其偏离的即时收益，而远见代理人预期后续反偏离并考虑最终收益时，该稳定集满足内部和外部稳定性。我们构造性地证明了一个（理性预期）近视-远见稳定集的存在性，其中远见代理人获得单一收益而近视代理人可能获得多重收益。该存在性结果可扩展至具有任意规模执行联盟的情境，以及并非所有联盟成员必须严格获益的情境。当所有远见代理人具有单位需求时，结果还可扩展至多对一匹配市场。作为一个关键推论，我们为效率调整的延迟接受算法提供了理论基础。

Apprenticeships play a key role in enabling successful school‐to‐work transitions in many countries, but in the presence of imperfect information, the specificity of this type of training may entail important costs for those working outside their training fields. I study this issue in one of the most prominent training settings, the German apprenticeship system. Using administrative data and a broad occupational classification, I find that 40% of individuals work in occupations different from their training. I estimate the cost of mismatch using vacancy instruments and extend methodological approaches in high‐dimensional selection settings. Lacking training in one's occupation entails an average wage penalty of 14%, the equivalent of two years of work experience. The penalty increases with the task distance between training and occupation. My findings suggest that retraining is crucial to mitigate the adverse consequences from imperfect information in specialized training settings.

学徒制在许多国家促成成功的学校到工作过渡中发挥着关键作用，但在不完全信息条件下，此类培训的专用性可能对在培训领域之外工作的人带来重要成本。我在最具代表性的培训体系之一——德国学徒制中研究此问题。使用行政数据与广泛职业分类，我发现 40% 的个人从事与其培训领域不同的职业。我利用空缺职位工具变量估计错配成本，并扩展了高维选择情境下的方法论。缺乏自身培训职业带来平均 14% 的工资惩罚，相当于两年工作经验。该惩罚随培训与职业之间的任务距离增大而增加。

Tirole (1985) studied an overlapping generations model with capital accumulation and showed that the emergence of asset bubbles solves the capital over‐accumulation problem. His Proposition 1(c) claims that if the dividend growth rate is above the bubbleless interest rate (the steady‐state interest rate in the economy without the asset) but below the population growth rate, then bubbles are necessary in the sense that there exists no bubbleless equilibrium but there exists a unique bubbly equilibrium. We show that this result (as stated) is incorrect by presenting an example economy that satisfies all assumptions of Proposition 1(c) but its unique equilibrium is bubbleless. We also restore Proposition 1(c) under the additional assumptions that initial capital is sufficiently large and dividends are sufficiently small. We show through examples that these conditions are essential.

Tirole（1985）研究了一个包含资本积累的世代交叠模型，表明资产泡沫的出现解决了资本过度积累问题。其命题 1(c) 声称，如果股利增长率高于无泡沫利率但低于人口增长率，则泡沫是必需的，即不存在无泡沫均衡但存在唯一的泡沫均衡。我们通过一个满足所有假设但唯一均衡为无泡沫均衡的例子经济，证明该结果是不正确的。我们也在初始资本足够大且股利足够小的额外假设下恢复了命题 1(c)，并通过例子证明这些条件是必不可少的。

This paper studies how the risk of hold‐up affects procurement. I use data on the universe of solar power auctions in India. The Indian context allows clean estimates of counterparty risk, because solar plants set up in the same states, by the same firms, are procured in auctions intermediated by either risky states themselves or the trusted central government. I find that the counterparty risk of an average state increases solar prices by 10%. This risk premium sharply reduces investment, because demand for green energy is elastic. Contract intermediation by the central government eliminates the counterparty risk premium.

本文研究敲竹杠风险如何影响采购。我使用印度太阳能拍卖市场的全量数据。印度背景允许对交易对手风险进行清晰估计，因为由相同公司在同一邦设立的太阳能电站，分别通过有风险的邦政府自身或可信赖的中央政府作为中介进行拍卖采购。平均邦政府的交易对手风险使太阳能价格上升 10%。由于绿色能源需求具有弹性，这一风险溢价显著减少了投资。中央政府提供的合约中介服务消除了交易对手风险溢价。

This paper studies the economic effects of rural‐urban migration on Brazilian cities. Using a shift‐share IV design, we show that, over a decade, drought‐induced immigration reduces informality, has no effect on unemployment, and increases the number of formal firms and jobs. Downward formal wage adjustments play a key role, as these long‐run effects are weaker in regions with stronger wage rigidity. In the short run, when wage rigidity is strongest, we replicate the informality‐increasing effects documented in the literature. We develop and estimate a model of firm dynamics and informality that rationalizes these results. The counterfactuals reveal that, in the short run, the informal sector absorbs the expanding labor force and acts as a “stepping‐stone” to formality for firms and workers. In the long run, however, it reduces the aggregate benefits from immigration by allowing the least productive firms to survive.

本文研究农村向城市移民对巴西城市的经济效应。利用偏移份额 IV 设计，我们发现，十年间干旱引发的移民减少了非正规性，对失业没有影响，并增加了正规企业和正规就业的数量。正规工资的向下调整发挥了关键作用，因为这些长期效应在工资刚性更强的地区更弱。短期内，当工资刚性最强时，我们复现了文献中记录的非正规性增加效应。我们构建并估计了一个企业动态与非正规性模型，为这些结果提供了理论解释。反事实分析表明，短期内非正规部门吸收了扩张的劳动力并充当「踏脚石」，但长期来看它因允许最低生产率的企业存活而减少了移民的总收益。

Certain extremum estimators have asymptotic distributions that are non‐Gaussian, yet characterizable as the distribution of the arg max of a Gaussian process. This paper presents high‐level sufficient conditions under which such asymptotic distributions admit a continuous distribution function. The plausibility of the sufficient conditions is demonstrated by verifying them in three examples, namely, maximum score estimation, empirical risk minimization, and threshold regression estimation. In turn, the continuity result buttresses several recently proposed inference procedures whose validity seems to require a result of the kind established herein. A notable feature of the high‐level assumptions is that one of them is designed to enable us to employ the Cameron–Martin theorem. In a leading special case, the assumption in question is demonstrably weak and appears to be close to minimal.

某些极值估计量具有非高斯的渐近分布，但可被刻画为高斯过程 arg max 的分布。本文提出了高水平充分条件，在此条件下这些渐近分布具有连续的分布函数。我们通过三个例子——最大得分估计、经验风险最小化和阈值回归估计——验证了这些充分条件的合理性。该连续性结果反过来支撑了几种近期提出的推断程序的效度。一个显著特征是，其中一个假设被设计为使我们能够使用 Cameron-Martin 定理；在一个重要的特殊情形下，该假设可证明是弱的且接近最小条件。

We cast the problem of communicating scientific uncertainty as one of reporting a posterior distribution on an unknown parameter to an audience of Bayesian decision‐makers. We establish novel bounds on the audience's regret when the analyst reports an approximation to a posterior that the audience treats as exact. Under a palatable restriction on the audience's decision problems, the bounds take an especially convenient form. Under a further restriction on the audience's priors, a bootstrap distribution can be used as a stand‐in posterior. We propose a practical recipe for checking whether a conventional statistical report (say, a normal parameterized by a point estimate and standard error) is a good approximation, and for improving the report if it is not. We illustrate our proposals using the articles in the 2021 American Economic Review that use a bootstrap for inference.

我们将沟通科学不确定性的问题建模为向贝叶斯决策者受众报告未知参数的后验分布。我们建立了分析师报告一个被受众视为准确的后验近似分布时，受众遗憾的新界限。在受众决策问题的一个可接受限制下，这些界限呈现出特别便利的形式。在受众先验的进一步限制下，自助法分布可被用作后验替代。我们提出了一种实用方法，用于检查传统统计报告是否为良好近似，并在不是良好近似时改进报告。

A buyer and seller bargain in continuous time over a good. Bargainers can be rational or committed to some fixed price. A rational buyer has a private value and outside option. If the set of buyer values and commitment types is rich and the probability of commitment vanishes, outcomes are partially consistent with the Coase conjecture: the seller chooses a price below the maximum of the lowest outside option and half the lowest value; the buyer immediately accepts or takes his outside option.

一个买方和一个卖方在连续时间中就一件商品讨价还价。议价者可以是理性的，也可以承诺某种固定价格。理性买方具有私人价值和外部选择。如果买方价值和承诺类型的集合足够丰富且承诺概率趋近于零，结果部分符合科斯猜想：卖方选择低于最低外部选择与最低价值一半中最大者的价格；买方立即接受或选择其外部选择。

We investigate how formularies used by pharmacy benefit managers (PBMs) affect equilibrium manufacturer rebates for branded drugs through tiering and exclusion. We develop a theoretical model of multidimensional contracting in which a PBM negotiates with drug manufacturers over menus of formulary‐contingent rebates and chooses a formulary. We then estimate consumer demand responses to tier placement for statins using claims data from Princeton University, a large employer contracting with a single PBM to offer prescription drug coverage to its employees. Combining the theoretical model with demand estimates and observed list prices, we quantify how allowing for differential tier placement and exclusion affect equilibrium rebates. Our predictions are consistent with available aggregate rebate data, and we find that allowing a PBM to place branded drugs on preferred‐ and non‐preferred tiers can substantially increase negotiated rebate payments.

我们研究药品福利管理机构（PBM）使用的处方集如何通过分级和排除影响品牌药的均衡回扣。我们构建了一个多维合约理论模型，其中 PBM 与制药商就处方集相关的回扣菜单进行谈判并选择处方集。我们利用普林斯顿大学的索赔数据估计消费者对他汀类药物处方集分层的需求响应。将理论模型与需求估计和观察到标价结合，我们量化了允许差异化分级和排除如何影响均衡回扣。我们的预测与可获得的加总回扣数据一致，允许 PBM 将品牌药置于优选和非优选分级可显著增加协商回扣支付。

We analyze marital matching on income using an extremely rich Dutch data set containing all income tax files over seven years. We develop a novel methodology that directly extends previous contributions to allow for highly flexible matching patterns. Investigating all marriages that took place between 2013 and 2019, we find that marital patterns are particularly intriguing. While a majority of couples match assortatively, a small but significant minority display negative assortative matching. We also show that standard approaches, which consider all married couples using current incomes (as opposed to pre‐marriage incomes used in our approach), may generate misleading conclusions.

我们利用一个包含七年所得税文件的极其丰富的荷兰数据集分析婚姻匹配中的收入因素。我们开发了一种新颖的方法，直接扩展了先前的研究，允许高度灵活的匹配模式。考察 2013 年至 2019 年间的所有婚姻，我们发现婚姻模式尤为有趣：虽然大多数夫妇是正向匹配，但少数但显著比例的夫妻表现出负向匹配。我们还表明，使用当前收入（相对于本文使用的婚前收入）考虑所有已婚夫妇的标准方法可能产生误导性结论。

Biopharmaceuticals advance health and economic growth. Unlike central bargaining abroad, the United States uses private firms, pharmacy benefit managers (PBMs), to manage medicine access and spending. Yet, PBMs' roles in advancing efficiency are understudied. Ho and Lee model PBMs' use of tiered formularies, lists of covered medicines, without the use of proprietary data on the price concessions drug makers offer for preferred placement. They find PBMs' use of tiered formularies generate significant payor savings through competition. Complementing Ho and Lee, Feng and Maini (2024) model how patient demand inertia limits PBMs' ability to extract price concessions from drug makers which consequently erodes the efficacy gains of PBM formularies. Conti, Frandsen, Powell, and Rebitzer (2021) model formulary auctions where PBM size drives payor savings, but also spurs endogenously set high list prices, reducing patient access. Future economic research should focus on PBM market entry and vertical integration, pharmacy steering, and effects on innovation.

生物制药推动健康和经济增长。与海外集中谈判不同，美国使用私营企业——药品福利管理机构（PBM）管理药品获取和支出。然而，PBM 在提升效率方面的作用尚未被充分研究。Ho 和 Lee 发现 PBM 使用分级处方集通过竞争产生了显著的支付方节省。Feng 和 Maini（2024）建模了患者需求惯性如何限制 PBM 提取价格让步的能力。Conti、Frandsen、Powell 和 Rebitzer（2021）建模了处方集拍卖。未来经济研究应关注 PBM 市场进入和垂直整合、药房导引以及对创新的影响。

We apply a revealed preference approach to administrative data from Washington to show that constraints on work hours are widespread: Workers have limited discretion over hours at a given employer, and there is substantial mismatch of workers who prefer long hours to employers that provide short hours. Voluntary job transitions imply a ratio of the marginal rate of substitution of earnings for hours to the wage rate of 0.5–0.6 for prime‐age workers. The average absolute deviation between observed and optimal hours is about 15%, and low‐wage workers face particularly acute constraints on hours. On average, observed hours tend to be less than preferred levels, and workers would require a 12% higher wage with their current employer to be as well off as with an employer offering ideal hours. These findings suggest that hour constraints are an equilibrium feature of the labor market because long‐hour jobs are costly to employers.

我们运用显示性偏好方法对华盛顿州行政数据进行分析，表明工时约束普遍存在：工人在给定雇主处对工时的自由裁量权有限，且偏好长工时的工人与提供短工时的雇主之间存在严重的错配。自愿工作转换意味着壮年工人收入对工时的边际替代率与工资率之比为 0.5-0.6。实际工时与最优工时之间的平均绝对偏差约为 15%，低工资工人面临的工时约束尤为严重。平均而言，实际工时往往低于偏好水平，工人需要当前雇主提高 12% 的工资才能与提供理想工时的雇主所带来的福利持平。这些发现表明，工时约束是劳动力市场的均衡特征，因为长工时工作对雇主而言成本较高。