本章导读 本章是实证综述式的，围绕公平 (fairness) 展开。§4.1 利他 (altruism)：Becker (1981) 利他效用、Andreoni (1989) 不纯利他 (4.1)、Andreoni (1990) 暖光 (4.2)、Charness & Rabin (2002) 显式公平模型。§4.2 价格缺乏变动：Kahneman et al. (1986) 电话调查——成本下降/需求上升时涨价（降薪）被视为不公平。§4.3 互惠 (reciprocity)：4.3.1 礼物交换 Gneezy & List (2006)、Kube et al. (2013)（正互惠效应小而短暂）；4.3.2 罢工工人的怠工 Krueger & Mas (2004)（负互惠效应大，Firestone 轮胎，图 4.3–4.4）。§4.4 过程的重要性：4.4.1 分配正义 vs 程序正义（程序正义四因素）；4.4.2 民主决定的村庄项目满意度更高 Olken (2010)。§4.5 公平与收入：4.5.1 绝对工资 Stevenson & Wolfers (2008)（GDP↑→更幸福，图 4.5）、Mas (2006)（警察仲裁，图 4.6）；4.5.2 相对工资 Card et al. (2012)、Perez-Truglia (2020)（图 4.7）、Bertrand et al. (2015)（妻收入>夫→婚姻更易破裂，图 4.8–4.9）、Breza et al. (2018)。§4.6 社会压力：4.6.1 Asch (1951) 从众实验（图 4.10）、4.6.2 足球裁判偏袒 Garicano et al. (2005)（图 4.11）、4.6.3 同伴压力提升效率 Falk & Ichino (2006)、Mas & Moretti (2009)、4.6.4 社会规范与节能 Allcott (2011) OPOWER（图 4.12–4.13）。图 4.1–4.13 均已转述。

Becker (1981) 与 Andreoni 的利他模型 / altruism models of Becker (1981) and Andreoni Becker (1981) 通过引入如下效用函数正式讨论利他。对一个他人的利他：设主体 \(h\) 效用 \(U_h=U(Z_{1h},\dots,Z_{mh},\psi(U_w))\)，且 \(\frac{\partial U_h}{\partial U_w}>0\)——\(h\) 的效用不仅取决于自身消费 \((Z_{1h},\dots,Z_{mh})\)，还取决于配偶的效用 \(U_w\)。对多个他人的利他：\(U_h=U(Z_h;Z_1,Z_2,\dots,Z_p)\)，且 \(\frac{\partial U_h}{\partial Z_j}>0\)（\(j=1,\dots,p\)），\(Z_h\) 为 \(h\) 自身消费、\((Z_1,\dots,Z_p)\) 为 \(h\) 所关心的 \(p\) 个人的消费。Andreoni 的模型思想与 Becker (1981) 相似。Andreoni (1989) 不纯利他 (impure altruism)：主体求解 \(\max_{x_i,Y,g_i}U_i(x_i,Y,g_i)\) (4.1)，s.t. \(x_i+g_i=w_i-\tau_i\)、\(G_{-i}+g_i+T=Y\)，其中 \(x_i\) 私人品消费、\(Y\) 公共品消费、\(g_i\) 主体 \(i\) 对公共品的贡献、\(w_i\) 工资、\(\tau_i\) 所缴税、\(G_{-i}=\sum_{j\ne i}g_j\)、\(T=\sum_i\tau_i\)。主体通过两个渠道关心公共品：(1) 公共品经 \(Y\) 直接产生效用；(2) 贡献 \(g_i\) 本身单独给主体满足感。政府资金 \(T\) 的挤出效应：因 \(U_i(x_i,Y,g_i)\) 关于 \(Y\) 凹，更高的 \(T\) 会经 \(Y\) 降低 \(g_i\) 的边际价值，从而减少主体最优选择的 \(g_i\)。Andreoni (1990) 暖光 (warm glow，纯利己)：与 (4.1) 相同，但 \(U_i\) 只是 \((x_i,g_i)\) 的函数 \(U_i(x_i,g_i)\) (4.2)。主体只从自身贡献获得满足、不在意公共品最终总量；无挤出效应（政府资金不影响其效用）；因 \(U_i(x_i,g_i)\) 关于 \(g_i\) 凹，主体会对许多公共品（慈善基金）各贡献一点点，因为对主体而言"分散小额贡献"总优于"一次大额贡献"。Becker (1981) formally discusses altruism by introducing the following utility functions. Altruism towards one other person: agent \(h\)'s utility \(U_h=U(Z_{1h},\dots,Z_{mh},\psi(U_w))\) with \(\frac{\partial U_h}{\partial U_w}>0\) — \(h\)'s utility depends not only on his own consumption \((Z_{1h},\dots,Z_{mh})\) but also on the utility \(U_w\) of his spouse. Altruism towards multiple persons: \(U_h=U(Z_h;Z_1,Z_2,\dots,Z_p)\) with \(\frac{\partial U_h}{\partial Z_j}>0\) for \(j=1,\dots,p\), where \(Z_h\) is \(h\)'s own consumption and \((Z_1,\dots,Z_p)\) is the consumption of the \(p\) persons \(h\) cares about. Andreoni's models have ideas similar to Becker (1981). Andreoni (1989) impure altruism: an agent solves \(\max_{x_i,Y,g_i}U_i(x_i,Y,g_i)\) (4.1), s.t. \(x_i+g_i=w_i-\tau_i\) and \(G_{-i}+g_i+T=Y\), where \(x_i\) is private-good consumption, \(Y\) public-good consumption, \(g_i\) agent \(i\)'s contribution to the public good, \(w_i\) wage, \(\tau_i\) the tax paid, \(G_{-i}=\sum_{j\ne i}g_j\), \(T=\sum_i\tau_i\). The agent cares about the public good through two channels: (1) the public good directly generates utility through \(Y\); (2) the contribution \(g_i\) itself gives agent \(i\) satisfaction separately. Crowd-out effect of government fund \(T\): since \(U_i(x_i,Y,g_i)\) is concave in \(Y\), a higher \(T\) lowers the marginal value of \(g_i\) through \(Y\), which reduces the contribution \(g_i\) optimally decided by agent \(i\). Andreoni (1990) warm glow (purely egoistic): same as (4.1) except \(U_i\) is only a function of \((x_i,g_i)\), \(U_i(x_i,g_i)\) (4.2). The agent only feels satisfaction from his own contribution and doesn't care about the final amount of the public good; no crowd-out effect (the government fund won't affect his utility); since \(U_i(x_i,g_i)\) is concave in \(g_i\), the agent contributes a little to many public goods (charity funds), because it's always better to contribute small amounts to many pieces than a single big chunk.

Charness 和 Rabin (2002)：显式的公平模型 / an explicit fairness model Charness & Rabin (2002) 提出更显式纳入公平的综合模型。设主体 \(B\) 偏好 \(U_B(\pi_A,\pi_B)=(\rho\cdot r+\sigma\cdot s+\theta\cdot q)\cdot\pi_A+(1-\rho\cdot r-\sigma\cdot s-\theta\cdot q)\cdot\pi_B\)，其中 \(\pi_A,\pi_B\) 为 \(A,B\) 的货币收益；\(r=1\) 若 \(\pi_B\ge\pi_A\)、\(r=0\) 若 \(\pi_B\le\pi_A\)；\(s=1\) 若 \(\pi_B<\pi_A\)、\(s=0\) 若 \(\pi_B\ge\pi_A\)；\(q=-1\) 若 \(A\) 行为不公平、\(q=0\) 否则。\(B\) 的效用本质上是 \(A\)、\(B\) 货币收益的加权平均，权重取决于谁获益更多、以及对手是否公平行事。实验博弈表明：主体更在意提高所有人的收益而非缩小收益差距；主体受互惠驱动（若他人不愿牺牲，自己愿为公平牺牲的意愿也更低，有时会惩罚不公平行为）。然而此类实验室实验的外部效度不强，因实验室环境与田野实践差异很大；后续各节聚焦来自精心设计的田野实验或观测数据的研究。Charness & Rabin (2002) propose a comprehensive model incorporating fairness more explicitly. Suppose agent \(B\) has preferences \(U_B(\pi_A,\pi_B)=(\rho\cdot r+\sigma\cdot s+\theta\cdot q)\cdot\pi_A+(1-\rho\cdot r-\sigma\cdot s-\theta\cdot q)\cdot\pi_B\), where \(\pi_A,\pi_B\) are the money payoffs of \(A\) and \(B\); \(r=1\) if \(\pi_B\ge\pi_A\) and \(r=0\) if \(\pi_B\le\pi_A\); \(s=1\) if \(\pi_B<\pi_A\) and \(s=0\) if \(\pi_B\ge\pi_A\); \(q=-1\) if \(A\) has behaved unfairly and \(q=0\) otherwise. \(B\)'s utility is basically a weighted average of \(A\)'s and \(B\)'s money payoffs, with weights depending on who gains more and whether the counterpart acts fairly. Experimental games show: agents care more about increasing the payoffs of all people than about reducing differences in payoffs; agents are motivated by reciprocity (they have lower willingness to sacrifice for fairness if others are unwilling to sacrifice, and sometimes tend to punish unfair behaviors). However, the external validity of this type of lab experiment is not strong, because the lab environment is very different from field practice; the next sections focus on studies that provide evidence from carefully designed field experiments or observational data.

Kahneman et al. (1986)：公平约束价格 / fairness constrains prices 经济学研究稀缺资源的配置，而货币是清算市场的"润滑剂"。但出于公平考虑，某些市场不能用货币清算：如器官交换、投票、大学录取——无法用钱完成交易；此时改用随机抽签、排队等待时间、社会关系等非货币手段清算。例如 UNOS（美国器官共享网络）是管理美国器官共享的非营利组织，其评分系统按"需求、疗效、弱势"等类别给予加分。即便在能用货币清算的市场，也有些事因会造成不公平而不能做。Kahneman et al. (1986) 用电话调查研究此问题，发现：(1) 若利润受威胁，企业在消费品市场涨价（或在劳动市场降薪）被视为公平；(2) 若成本下降，企业维持价格（或维持工资）被视为公平；(3) 利用需求上升而涨价（或利用条件降薪）被视为不公平。批评：数据来自 1984 年 5 月至 1985 年 7 月对多伦多与温哥华居民的电话调查，代表性存疑。Economics is about the allocation of scarce resources, of which the lubricant is money. However, due to fairness concerns, certain markets cannot use money to clear: e.g. organ exchange, voting, college admission — money cannot complete the transaction; instead random lotteries, waiting times, social connections and other non-monetary measures clear the market. For instance, UNOS (United Network for Organ Sharing) is a non-profit organization managing organ sharing in the U.S., built on a scoring system that gives bonus points to factors classified into need, efficacy, and disadvantage categories. Even where money can clear markets, there are certain things firms cannot do because they cause unfairness. Kahneman et al. (1986) study this with telephone surveys, finding: (1) it is fair for a firm to raise price in the consumer market (or reduce wage in the labor market) if profits are threatened; (2) it is fair to maintain price (or maintain wage) if costs are reduced; (3) it is unfair to take advantage of higher demand by raising price (or reducing wage). Critique: the data are May 1984 – July 1985 telephone surveys of residents in Toronto and Vancouver, whose representativeness is questionable.

Gneezy & List (2006) 与 Kube et al. (2013)：正互惠效应小而短暂 / positive reciprocity is small and temporary Gneezy & List (2006)：田野实验，受试者从事两项不同任务——大学图书馆数据录入、上门募捐。发现：处理组（给予更高工资 USD 20 的"礼物"）的工人努力在最初几小时显著高于控制组（无礼物、承诺工资 USD 12 或 USD 10），但几小时后两组差异开始消失（图 4.1a 数据录入、图 4.1b 上门募捐）。总体而言，礼物处理组比无礼物控制组做得更差——意味着若不给任何人礼物，同样的预算能产生更好的结果。Kube et al. (2013)：田野实验，在德国通过海报招募受试者把大学图书馆编目作为一次性工作。设计：海报上的预期工资为 15 欧元；2 个月招募期约 200 人申请，随机选 30 人并再次提醒预期 15 欧元工资；到馆当天随机分三组——加薪组被告知涨到 20 欧元、基线组被告知维持 15 欧元、减薪组被告知降到 10 欧元。结果：减薪组（负礼物）效率显著低于另两组，而加薪组（正礼物）的效应不强（图 4.2）。这两篇论文表明人们对礼物馈赠确有一定程度的互惠。Gneezy & List (2006): a field experiment in which subjects work on two distinct tasks — data entry in a university library, and door-to-door fundraising. They find: workers' effort in the treatment group (with a "gift" of higher wage USD 20) is significantly higher than the control group (without gift, promised wage USD 12 or USD 10) in the first few hours, but the difference starts to disappear after a few hours (Figure 4.1a for data entry, Figure 4.1b for door-to-door fundraising). Overall, the gift treatment group does a worse job than the no-gift control group — meaning the same budget would generate better outcomes if none of the individuals were given gifts. Kube et al. (2013): a field experiment in which subjects are recruited by posters to catalog a university library as a one-time job in Germany. Design: the projected wage on the poster is 15 Euros; over a two-month recruiting period about 200 applicants applied, of which 30 were randomly selected and reminded again of the projected 15 Euros wage; upon arrival on the experiment day, subjects are randomly divided into three groups — the PayRaise group told of raising the wage to 20 Euros, the Baseline group told of a non-changing wage of 15 Euros, and the PayCut group told of reducing the wage to 10 Euros. Results: the PayCut group (negative gift) works significantly less efficiently than the other two groups, while the effect of PayRaise (positive gift) is not very strong (Figure 4.2). These two papers show that people do have a certain degree of reciprocity to gift givings.

图 4.1、4.2（已转述 / Figures 4.1, 4.2, paraphrased） 图 4.1（努力与礼物）：(a) 图书馆数据录入——横轴为时间（90、180、270、360 分钟），纵轴"每 90 分钟平均录入书数"；"礼物"组起步约 53、"无礼物"组约 41，但"礼物"组随时间下降并在数小时后与"无礼物"组趋同（约 40）。(b) 上门募捐——按"午餐前 / 三小时休息 / 午餐后"三时段画柱，"礼物"组仅在前期高于"无礼物"组、之后差异消失。图 4.2（工资与生产率）：(a) 各季度平均录入书数——加薪、基线、减薪三条线随时间上升，减薪组明显最低，加薪与基线接近；(b) 累积分布函数——减薪组的 CDF 整体左移（产出更低）。Figure 4.1 (effort and gift): (a) data entry in the library — the horizontal axis is time (90, 180, 270, 360 minutes), the vertical axis "average number of books entered per 90 minutes"; the "Gift" group starts around 53 and the "No Gift" group around 41, but the "Gift" group declines over time and converges with the "No Gift" group (around 40) after a few hours. (b) door-to-door fundraising — bars for three time blocks ("pre-lunch / 3-hour break / post-lunch"); the "Gift" group is higher than "No Gift" only early and the difference disappears later. Figure 4.2 (productivity as a function of wages): (a) average books entered per quarter — the PayRaise, Baseline and PayCut lines all rise over time, with the PayCut group clearly the lowest while PayRaise and Baseline are close; (b) cumulative distribution functions — the PayCut group's CDF is shifted left overall (lower output).

Krueger & Mas (2004)：负互惠效应大 / negative reciprocity has large effects 2000 年 8 月，Firestone 召回 1,440 万条轮胎，这些轮胎造成 271 人死亡、800 多人受伤。事件背景是 Firestone 位于 Decatur 工厂的一场工人罢工：1994 年 7 月，Decatur 工厂把工作班次从 8 小时增至 12 小时、并把新雇员工资削减 30%；工人随即罢工，工厂引入替代工人。1995 年 5 月，工人因弹尽粮绝被迫妥协、接受工厂要求、返岗工作，但此时显然心怀不满。1996 年 12 月，工厂与工人达成协议，工人此后重新变得满意。Krueger & Mas (2004) 利用"问题轮胎与罢工同期在 Decatur 厂生产"这一事实研究本案。发现：罢工期间在 Decatur 厂生产的轮胎导致索赔的概率是其他厂轮胎的 15 倍（图 4.3）。他们还论证质量问题并非由替代工人造成，因为在替代工人工作期间（图中红框）生产的轮胎引发的索赔数远小于之前与之后（图 4.4）。这意味着工人在自认遭受不公时，会通过生产中的怠慢表现出负互惠。In August 2000, Firestone recalled 14.4 million tires that had caused 271 fatalities and more than 800 injuries. The background was a workers' strike at Firestone's plant in Decatur: in July 1994 the Decatur plant increased the work shift from 8 hours to 12 hours and cut wages for new hires by 30%; the workers went on strike and replacement workers were brought in. In May 1995 the workers had to compromise and accept the plant's demands because they ran out of money, so they returned to the plant and started to work, but they were definitely not happy at that point. In December 1996 an agreement was reached between the plant and the workers, and the workers became happy again after that. Krueger & Mas (2004) look into this case and exploit the fact that the problematic tires were produced in the same period of the strike in the Decatur plant. They find: the tires made in the Decatur plant during the labor dispute were 15 times more likely to cause claims than tires produced in other plants (Figure 4.3). They also argue that the quality issue was not caused by replacement workers, because during the replacement workers' working period (in the red square in the figure) the tires produced incurred a much smaller number of claims than before and after (Figure 4.4). This implies that workers do have negative reciprocity by being careless in production when they think they are treated unfairly.

图 4.3、4.4（已转述 / Figures 4.3, 4.4, paraphrased） 图 4.3（失效时轮胎使用年限）：左半（Decatur 厂）与右半（Joliette & Wilson 厂）对比，横轴为轮胎使用年数、纵轴"累积索赔率 (ppm)"，按生产年份（1991–1999）画多条线；Decatur 厂在 1994、1995 年线的累积索赔率远高于其他、也远高于其他厂——劳动争议期间生产的轮胎质量明显更差。图 4.4（Decatur 厂按月生产轮胎的索赔数）：横轴为月份（1994–1997），纵轴索赔数；上方标注"提出让步 / 罢工开始 / Decatur 本地投票返岗 / 工会与厂方达成新合约"等时间点；红框圈出替代工人工作期，该期索赔数明显低于罢工前后——质量恶化对应的是心怀不满的正式工，而非替代工。Figure 4.3 (age of tire at failure): a left half (Decatur plant) vs. a right half (Joliette & Wilson plants); the horizontal axis is the age of the tire in years and the vertical axis "cumulative claims rate (ppm)", with multiple lines by production year (1991–1999); the Decatur plant's 1994 and 1995 lines have far higher cumulative claims rates than the others and than the other plants — tires produced during the labor dispute are clearly of worse quality. Figure 4.4 (number of claims on tires produced in Decatur by month): the horizontal axis is month (1994–1997), the vertical axis the number of claims; annotated at the top with timeline events ("concessions demanded / strike begins / Decatur local votes to return to work / union and BFS settle on new contract"); a red square marks the replacement workers' period, during which the number of claims is markedly lower than before and after — the quality deterioration corresponds to the unhappy regular workers, not the replacement workers.

互惠的总体结论 / Overall lesson on reciprocity 综合 Gneezy & List (2006)、Kube et al. (2013)、Krueger & Mas (2004) 的结论：正互惠的效应短暂且小；负互惠的效应相对较大。该议题其他近期工作的综述见 DellaVigna et al. (2016)。The overall lesson from Gneezy & List (2006), Kube et al. (2013), and Krueger & Mas (2004): positive reciprocity has temporary and small effects; negative reciprocity has relatively large effects. See DellaVigna et al. (2016) for a good summary of other recent works on this topic.

4.4.1 分配正义与程序正义 / Distributive and Procedural Justice 分配正义 (distributive justice) 是最终结果或资源分配的公平。程序正义 (procedural justice) 是导向最终结果或资源分配的过程的公平。程序正义由以下因素决定：1. 过程控制 (process control)——表达意见或呈交证据的机会；2. 中立性 (neutrality)——决策中的诚实、无偏与据实；3. 可信度 (trustworthiness)——决策者动机的纯粹；4. 体面 (standing)——礼貌与对权利的尊重。Distributive justice is the fairness of the final outcome or distribution of resources. Procedural justice is the fairness of the process that leads to a final outcome or distribution of resources. Procedural justice is determined by: 1. Process control — the opportunity to express an opinion or present evidence; 2. Neutrality — honesty, unbiasedness and factuality in decision making; 3. Trustworthiness — the pureness of the decision maker's motives; 4. Standing — politeness and respect for rights.

4.4.2 民主决定的村庄项目满意度更高：Olken (2010) / Higher satisfaction in democratically decided village projects Olken (2010) 用 2005 年 9 月至 2006 年 1 月间 49 个印尼村庄的数据，研究民主在项目选择中的作用。作者发现：基于代表的会议（不太民主）与基于直接选举的公投（更民主）产生相似的项目选择；但更民主的方式带来显著更高的满意度、对项目更深的了解、更高的感知收益、以及村民更高的报告贡献意愿。因此结论是：决策中的直接参与极大提升了幸福感与正当性 (legitimacy)。Olken (2010) uses data from 49 Indonesian villages between September 2005 and January 2006 to study the effect of democracy in project choices. The author finds: representative-based meetings (less democratic) and direct election-based plebiscites (more democratic) yield a similar project choice; but the more democratic way generates much higher satisfaction, increased knowledge about the project, greater perceived benefits, and higher reported willingness to contribute among villagers. Therefore it is concluded that direct participation in decision making greatly improves happiness and legitimacy.

Stevenson & Wolfers (2008) 与 Mas (2006)：绝对财富 / absolute wealth Stevenson & Wolfers (2008) 用 2006 年 Gallup 世界民调，证明绝对财富在决定幸福中的重要性。图 4.5a（国家间）：GDP 人均更高的国家往往人民更幸福。图 4.5b（国家内及国家间）：在一国之内，财富与幸福之间的正相关依然成立（每个点的箭头斜率即该国内部"实际 GDP 人均—幸福"的相关）。主要结论：绝对财富水平对幸福很重要。Mas (2006) 用 1978–1996 年新泽西州警察工会与市政当局之间涉及薪酬争议的最终报价仲裁 (FOA) 数据，检验警察绩效是否受仲裁结果影响（脚注：FOA 指诉诸法院前的私下争议解决机制，仲裁员接收争议双方提交的最终报价、并通过从两个报价中择一作出有约束力的裁决，通常涉及公共部门工会）。发现：在警察工会输掉仲裁后的数月里，警察努力的代理指标（逮捕率、量刑长度）下降、犯罪报案数上升；最终工资偏离警察工会诉求越多，效应越强（图 4.6）。Stevenson & Wolfers (2008) use the Gallup World Poll of 2006 to demonstrate the importance of absolute wealth in determining happiness. Figure 4.5a (between countries): countries with higher GDP per capita tend to have happier people. Figure 4.5b (within-country and between countries): on a within-country level the positive correlation between wealth and happiness still holds (the slope of the arrow at each point is the within-country correlation between real GDP per capita and life satisfaction). The main takeaway: the absolute wealth level matters for happiness. Mas (2006) uses data on final offer arbitration (FOA) involving compensation disputes between the New Jersey police union and municipalities between 1978 and 1996 to test whether police performance is affected by the outcome of arbitration (footnote: FOA refers to the private dispute-resolving mechanism before going through the court; the arbitrator receives the final offers submitted by the disputing parties and makes binding decisions by choosing from the two offers, typically involving public-sector unions). The author finds: in the months after the police union loses the arbitration, the proxies of police effort (arrest rate, sentence length) decrease and the number of crime reports increases; the further the deviation of the final wage from the police union's demand, the stronger the effect (Figure 4.6).

图 4.5、4.6（已转述 / Figures 4.5, 4.6, paraphrased） 图 4.5（生活满意度与实际 GDP 人均）：(a) 国家间——横轴为实际 GDP 人均（对数刻度），纵轴生活满意度，散点呈明显正斜率（富国更幸福），拟合相关约 0.82。(b) 国家内及国家间——在每个国家内画一束箭头，箭头斜率为该国内部的财富—满意度相关，多数为正——国家内部也是越富越满意。图 4.6（仲裁前后的警察绩效）：横轴为"距仲裁的月数"（−23 到 +23），纵轴"每 10 万人的清案数 (clearances)"；以仲裁月为界，"仲裁裁给工会"线在仲裁后维持/上升，"仲裁裁给雇主"线在仲裁后明显下降——工会输掉仲裁后警察绩效下滑。Figure 4.5 (life satisfaction and real GDP per capita): (a) between countries — the horizontal axis is real GDP per capita (log scale), the vertical axis life satisfaction, scatter with a clear positive slope (richer countries happier), fitted correlation about 0.82. (b) within-country and between countries — a bundle of arrows is drawn within each country, the arrow slope being the within-country wealth-satisfaction correlation, mostly positive — within a country, richer is also more satisfied. Figure 4.6 (police performance around arbitration): the horizontal axis is "months since arbitration" (−23 to +23), the vertical axis "clearances per 100,000 capita"; with the arbitration month as the divider, the "arbitrator ruled for union" line maintains/rises after arbitration while the "arbitrator ruled for employer" line clearly drops after arbitration — police performance declines after the union loses the arbitration.

Card et al. (2012) 与 Perez-Truglia (2020)：信息透明扩大幸福差距 / information transparency widens the happiness gap Card et al. (2012)：加州大学系统已把员工工资设为公开信息，但许多人并不知情。作者设计田野实验，处理组收到一封提醒"该信息可在 Sacramento Bee 上查到"的邮件（脚注：这是一个发布相关工资信息的网站），控制组不收提醒；并控制了溢出效应（核实控制组确实不知信息可得）。发现：低于中位工资的工人比控制组有更低的薪酬与工作满意度、更可能去找新工作；高于中位工资的工人则不受影响。这表明工作满意度取决于相对工资。Perez-Truglia (2020)：2001 年挪威税务机关把税务记录公开，人人可见他人收入。作者利用此自然实验，用 1985–2013 年幸福/生活满意度调查数据，研究信息透明对贫富之间幸福/满意度差距的影响。发现：信息透明使贫富之间的幸福差距扩大 29%、生活满意度差距扩大 21%；作者称部分效应与自我认知和相对收入的变化有关（图 4.7）。其中"幸福 (\(U^{self}\))—收入 (\(r\)) 梯度"为 \(\frac{\partial U_i^{self}}{\partial r}\)，梯度越高意味贫富差距越大；蓝线为互联网接入较少（即获取收入数据较少）的群体、红线为接入较多的群体——人们对相对工资了解越多，贫富之间的幸福差距越大。Card et al. (2012): the University of California system has made its employees' wages public information, but many people were not aware of it. The authors design a field experiment in which the treatment group receives an email reminding them that the information is available on the Sacramento Bee (footnote: a website that publishes the wage information of interest), while the control group receives no such reminder; they also control for spillover effects (verifying the control-group subjects were indeed unaware of the information availability). They find: workers below median salary have lower pay and job satisfaction than the control group and are more likely to look for new jobs; workers above median salary are unaffected. This suggests job satisfaction depends on the relative wage. Perez-Truglia (2020): in 2001 the Norwegian tax authority made tax records publicly available, so everyone can see everyone else's income. The author exploits this natural experiment and uses survey data of happiness and life satisfaction between 1985 and 2013 to study the effect of information transparency on the gap in happiness/life satisfaction between rich and poor. The author finds: information transparency increases the happiness gap between rich and poor by 29% and the life-satisfaction gap by 21%; the author claims some of the effects relate to changes in self-perception and relative income (Figure 4.7). Here the "happiness (\(U^{self}\))-income (\(r\)) gradient" is \(\frac{\partial U_i^{self}}{\partial r}\), and a higher gradient implies a larger gap between poor and rich; the blue line represents the group with less internet access (i.e. the income data), and the red line the group with more access — the more information people have on relative wages, the larger the happiness gap between poor and rich.

图 4.7（挪威两群体幸福差距变化，已转述 / Figure 4.7, paraphrased） 横轴为年份（约 1985–2013，标注 85/87、89/91、…、09/11/13），纵轴"幸福—收入梯度的变化"。红线（互联网接入较多）在 2001 年税务记录公开（图中竖线）之后明显上升并高于蓝线（接入较少），表明信息更透明后贫富之间的幸福差距进一步扩大。The horizontal axis is year (about 1985–2013, labeled 85/87, 89/91, …, 09/11/13), the vertical axis "change in the happiness-income gradient". The red line (more internet access) clearly rises above the blue line (less access) after the 2001 public release of tax records (the vertical line in the figure), indicating that the happiness gap between poor and rich widens further once information becomes more transparent.

Bertrand et al. (2015) 与 Breza et al. (2018)：相对工资与性别 / relative wages and gender Bertrand et al. (2015) 研究两性间相对工资对婚姻结果的影响。发现：妻子工资高于丈夫的夫妻更不容易形成或维持，体现为图 4.8 中在 \(50\%\) 处的不连续骤降。图 4.8 用 1984–2004 年美国收入与项目参与调查 (SIPP) 中 73,564 对年轻已婚夫妇的收入数据：相对收入（妻子收入 ÷ 家庭总收入）大于 \(0.5\) 的夫妻显著更不易维持，离婚率在 \(0.5\) 处骤升（图 4.9，图 4.9 用加拿大 1983–2006 年纵向行政数据）。Breza et al. (2018) 在印度做田野实验，研究工资不平等对工作努力的影响。发现：在给定绝对工资水平下，薪酬不平等使产出下降 0.24 个标准差、出勤率下降 12%；这些效应随时间逐渐增大；但若生产率差异对工人清晰可观测，则薪酬不平等没有上述效应。Bertrand et al. (2015) study the effect of relative wage between genders on marriage outcomes. They find: couples with the wife's wage higher than the husband's are less likely to form or exist, shown in the discontinuous drop at \(50\%\) in Figure 4.8. Figure 4.8 uses data on income among 73,564 young married couples between 1984 and 2004 from the U.S. Survey of Income and Program Participation (SIPP): couples with relative income \(\dfrac{\text{wife's income}}{\text{total household income}}>0.5\) are sharply less likely to exist, and divorce rates go up sharply at \(0.5\) (Figure 4.9, which uses Canadian Longitudinal Administrative Data 1983–2006). Breza et al. (2018) run a field experiment in India to study the effect of wage inequality on working efforts. They find: for a given absolute wage level, pay inequality reduces output by 0.24 standard deviations and reduces attendance by 12%; such effects gradually increase over time; however, if the productivity differences are clearly observable to workers, the pay inequality has no above effects.

图 4.8、4.9（相对收入分布，已转述 / Figures 4.8, 4.9, paraphrased） 图 4.8（美国相对收入分布）：横轴为"妻子赚取的家庭收入份额"（0 到 1），纵轴为占比 (fraction)；分布在 \(0.5\) 处出现明显的不连续下跌——恰好妻子收入略高于丈夫的夫妻数量异常偏少。图 4.9（加拿大相对收入与离婚）：横轴为相对收入，左纵轴为相对收入占比（柱）、右纵轴为各相对收入区间的离婚百分比（线）；离婚率线在相对收入超过 \(0.5\) 后整体上行、并在接近 \(1\) 处明显抬升——妻子收入占比越高、离婚越多。Figure 4.8 (distribution of relative income, US data): the horizontal axis is "share of household income earned by the wife" (0 to 1), the vertical axis the fraction; the distribution shows a clear discontinuous drop at \(0.5\) — the number of couples where the wife earns just slightly more than the husband is abnormally low. Figure 4.9 (relative income and divorce, Canada data): the horizontal axis is relative income, the left vertical axis the fraction of relative income (bars), the right vertical axis the percentage of divorce per relative-income bin (line); the divorce-rate line trends up after relative income exceeds \(0.5\) and rises noticeably near \(1\) — the higher the wife's income share, the more divorce.

4.6.1 Asch (1951) 从众实验 / Asch conformity experiments Asch (1951) 记录了用于理解从众行为的实验。基本设计：有两组人，每组被问右侧图中哪条线与左侧图中那条线长度相同（图 4.10）。在控制组，没有"托儿"故意给错误答案；在实验组，有数名"托儿"率先齐刷刷地给出明显错误的答案。结果：控制组错误率低于 1%；实验组中超过 75% 的真实参与者在 12 次关键试次中至少有一次跟随给出相同的错误答案（脚注：关键试次指托儿故意齐给错误答案的试次）。Asch (1951) records the experiments conducted to understand conformity behavior. The basic idea: there are two groups, and each group is asked which line on the right-hand picture matches in length with the line on the left-hand picture (Figure 4.10). In the control groups, no actor intentionally gives wrong answers, while in the experiment groups several actors intentionally give the obvious wrong answer first uniformly. Result: the control-group error rate is less than 1%; in the experiment group over 75% of real participants at least give the same wrong answer once out of the 12 critical trials (footnote: critical trials are the ones in which actors intentionally give uniform wrong answers).

4.6.2 社会压力下足球裁判的偏袒：Garicano et al. (2005) / biased soccer referees under social pressure 足球比赛中裁判对终场补时拥有自由裁量权。Garicano et al. (2005) 用欧洲主要职业足球联赛（20 支球队、1994–1995 赛季 380 场、1998–1999 赛季 380 场）数据研究职业裁判在主场比赛中的行为（这两个时期有不同的奖励方案）。发现：裁判系统性地偏袒主队——主队落后时延长补时、主队领先时缩短补时（图 4.11）；当赢球奖励上升时，裁判表现出更高的偏袒；人群中主队球迷规模越大，裁判偏袒越强。In soccer games, referees have discretion over the extra time at the end of the game. Garicano et al. (2005) use data from main professional soccer leagues in Europe (20 teams, 380 games in the 1994-1995 season and 380 games in the 1998-1999 season) to study the behavior of professional referees in home games (these two periods have different reward schemes). They find: referees systematically favor home teams by lengthening the game when the home team is behind and shortening the game when the home team is ahead (Figure 4.11); when the rewards of winning the game increase, the referees display a higher level of bias; the larger the size of home fans in the crowd, the stronger the referee favoritism.

图 4.10、4.11（已转述 / Figures 4.10, 4.11, paraphrased） 图 4.10（Asch 实验）：左图一条参照线，右图三条候选线 A、B、C，要求选出与参照线等长者；任务本身极易，但同伴齐给错答会诱使真被试从众。图 4.11（裁判对主队的偏袒）：横轴为"比分差（主队落后到领先）"（$<-2,-2,-1,0,1,2,3,>3$），纵轴"补时分钟数"；当主队落后 1 球时补时最长（约 4 分钟），当主队领先 1 球时补时最短（约 2 分钟）——裁判按对主队有利的方向调节补时。Figure 4.10 (Asch experiment): a reference line on the left and three candidate lines A, B, C on the right, with the task of choosing the one equal in length to the reference; the task itself is trivially easy, but peers uniformly giving the wrong answer induces real subjects to conform. Figure 4.11 (referee favoritism for the home team): the horizontal axis is the "score margin (home team behind to ahead)" ($<-2,-2,-1,0,1,2,3,>3$), the vertical axis "minutes of added extra time"; added time is longest (about 4 minutes) when the home team is behind by 1 and shortest (about 2 minutes) when the home team is ahead by 1 — referees adjust extra time in the direction favorable to the home team.

4.6.3 同伴压力提升工作效率：Falk & Ichino (2006) 与 Mas & Moretti (2009) / increased work efficiency under peer pressure Falk & Ichino (2006)：2002 年在苏黎世（瑞士）的两周期实验，海报招募学生做一次性 4 小时工作（把信件塞入信封），报酬 0.7 USD。控制组单独完成任务；处理组与另一人并排工作（配对处理）。发现：处理组学生比控制组塞更多信封（221 对 190）；配对内的产出标准差小于配对间，表明人们倾向与同伴在产出上互相匹配。Mas & Moretti (2009)：用 2003–2006 年某全国连锁超市的扫描仪级数据，对 370 名收银员的生产率作精确高频度量，研究社会压力对生产率与努力的影响。发现：在班次中引入一名高生产率收银员会带来正的生产率溢出——平均生产率提高 10%、工人努力提高 1.75%；某工人的努力与"面对他的工人（他被这些工人看到）"的在场与速度正相关，而与"他面对的工人（他看到这些工人但这些工人看不到他）"无关；工人对"班次重叠更频繁的同事"反应更强。结论：个体的努力可由社会压力与相互监督诱发。Falk & Ichino (2006): an experiment over a two-week period in 2002 in Zurich (Switzerland), in which students were recruited by poster to do a one-time 4-hour job (stuffing letters into envelopes) with payment of 0.7 USD. The control group does the task individually, while the treatment group does the task with another person working next to them (pair treatment). They find: students in the treatment group stuff more envelopes than the control group (221 to 190); the standard deviations of output are smaller within pairs than between pairs, which suggests people try to match with their pair in output. Mas & Moretti (2009): using scanner-level data from 2003 to 2006 from a national supermarket chain with an accurate high-frequency measure of the productivity of 370 cashiers, they study the effect of social pressure on productivity and effort. They find: there are positive productivity spillovers from introducing a high-productivity cashier into the shift — a 10% increase in average productivity and a 1.75% increase in workers' effort; a given worker's effort is positively related to the presence and speed of workers who face him (workers who can see him), but not related to the presence and speed of workers whom he faces (he sees them but they don't see him); workers respond more strongly to the presence of co-workers whose shifts overlap more frequently. The conclusion is that the efforts of individuals can be induced by social pressure and mutual monitoring.

4.6.4 社会规范与节能：Allcott (2011) / social norms and energy conservation OPOWER 是一个节能项目，向家庭寄送报告，把其用电量与邻居比较。例如报告会告诉某家庭：你的用电比所有邻居或高效邻居都多，以此施加社会规范压力。Allcott (2011) 用一项涉及明尼苏达州 80,000 户家庭、2008 年 1 月至 2009 年 8 月的田野实验数据。发现：该月度项目使能耗相对基线下降约 2%；报告的效应在期间内衰减、但在收到下一期报告后恢复，总体效应是持久的（图 4.12）；用电量更高的家庭更易被"助推"去节能（图 4.13）。结论：非价格的助推 (nudge) 能有效降低家庭用电，其效应相当于 11%–20% 的电价上涨。OPOWER is an energy-conservation program that sends out letter reports to households to compare their power usage with that of their neighbors. For example, the report might tell a household that they are using more energy than all neighbors or efficient neighbors, to impose some pressure from social norms. Allcott (2011) uses data from a field experiment involving 80,000 households in Minnesota between January 2008 and August 2009. They find: the monthly program reduces energy consumption by about 2% relative to baseline; the effects of the report decay during the period but restore upon receiving the next period's report, and the effect is overall permanent (Figure 4.12); households with higher usage of energy are more likely to be nudged to save energy (Figure 4.13). The conclusion is that a non-price nudge can effectively reduce energy usage by households, which is equivalent in effect to an 11%-20% electricity price increase.

图 4.12、4.13（已转述 / Figures 4.12, 4.13, paraphrased） 图 4.12（高频处理效应）：横轴为时间（2009、2010 各月），纵轴"平均处理效应（相对控制组，% 用电）"；季度报告与月度报告的处理效应均为负值（用电下降），效应在两期报告之间略有回升、收到新报告后再次下压，整体维持在 −1% 到 −3% 区间。图 4.13（不同用电分位的处理效应）：横轴为"基线用电的十分位"（1 到 10），纵轴处理效应；曲线整体为负且随分位上升而更负——用电越多的家庭被节能助推的效果越大（从约 0% 降到约 −6%）。Figure 4.12 (high-frequency treatment effects): the horizontal axis is time (months of 2009, 2010), the vertical axis "average treatment effect (relative to control, % electricity)"; the treatment effects of both quarterly and monthly reports are negative (electricity falls), the effect rebounds slightly between reports and is pushed down again upon receiving a new report, overall staying in the −1% to −3% range. Figure 4.13 (treatment effect by deciles of energy usage): the horizontal axis is "decile of baseline usage" (1 to 10), the vertical axis the treatment effect; the curve is negative overall and becomes more negative with higher deciles — households using more energy are nudged to save more (from about 0% down to about −6%).