小结 / Summary 两种情形都能支持 \(w^\star\) 为竞争均衡。一言以蔽之：若 \(\mathbb{E}[\theta]Both cases can support \(w^\star\) as a competitive equilibrium. In a nutshell: if \(\mathbb{E}[\theta]

命题 12.1（存在性）/ Proposition 12.1 (existence) 在上述两条假设下，至少存在一个均衡工资 \(w^\star\)。Under the above two assumptions, there exists at least one equilibrium wage \(w^\star\).

证明 / Proof 定义 \(g(w)\equiv\mathbb{E}[\theta\mid r(\theta)\le w]\)，即在工资 \(w\) 下被雇佣工人的平均生产率。如上所述，竞争均衡工资满足 \(\mathbb{E}[\theta\mid r(\theta)\le w^\star]=w^\star\)，即 \(g(w)\) 与 45 度线的交点。Define \(g(w)\equiv\mathbb{E}[\theta\mid r(\theta)\le w]\), the average productivity of employed workers at wage \(w\). As above, a competitive-equilibrium wage satisfies \(\mathbb{E}[\theta\mid r(\theta)\le w^\star]=w^\star\), i.e. the intersection of \(g(w)\) with the 45-degree line.

由保留工资定义，\(g(r(\underline\theta))=\underline\theta\)。因假设 \(r(\theta)\le\theta\)，故 \(g(r(\underline\theta))=\underline\theta\ge r(\underline\theta)\)。若 \(g(r(\underline\theta))=r(\underline\theta)\)，则 \(w^\star=r(\underline\theta)\) 即为均衡工资，证毕；否则 \(g(r(\underline\theta))>r(\underline\theta)\)，即点 \((r(\underline\theta),\,g(r(\underline\theta)))\) 位于 45 度线上方。By the definition of the reservation wage, \(g(r(\underline\theta))=\underline\theta\). Since \(r(\theta)\le\theta\), we have \(g(r(\underline\theta))=\underline\theta\ge r(\underline\theta)\). If \(g(r(\underline\theta))=r(\underline\theta)\) then \(w^\star=r(\underline\theta)\) is an equilibrium and we are done; otherwise \(g(r(\underline\theta))>r(\underline\theta)\), i.e. the point \((r(\underline\theta),\,g(r(\underline\theta)))\) lies above the 45-degree line.

另一端，At the other end,

$$g\!\left(r(\bar\theta)\right)=\mathbb{E}\!\left[\theta\mid r(\theta)\le r(\bar\theta)\right]=\mathbb{E}[\theta]$$

而 \(g(\bar\theta)=\mathbb{E}[\theta]<\bar\theta\)，故点 \((\bar\theta,\,g(\bar\theta))\) 位于 45 度线下方。由于类型空间连续、\(g\) 连续，曲线 \(g(w)\) 必至少与 45 度线相交一次。\(\blacksquare\)while \(g(\bar\theta)=\mathbb{E}[\theta]<\bar\theta\), so the point \((\bar\theta,\,g(\bar\theta))\) lies below the 45-degree line. Since the type space is continuous and \(g\) is continuous, the curve \(g(w)\) must cross the 45-degree line at least once. \(\blacksquare\)

图 2 / Figure 2（情形 1，已转述 / Case 1, paraphrased） 横轴为工资 \(w\)、纵轴为类型 \(\theta\)。\(g(w)\) 是一条从 \((r(\underline\theta),\underline\theta)\) 出发、先位于 45 度线上方、随 \(w\) 增大逐渐趋于水平高度 \(\mathbb{E}[\theta]\) 的递增曲线；它与 45 度线在唯一一点 \(w^\star\in(r(\underline\theta),\,r(\bar\theta))\) 相交（从上方穿到下方）。对应的就业集为 \(\Theta(w^\star)\)，即一部分工人被雇佣。The horizontal axis is wage \(w\), the vertical axis type \(\theta\). \(g(w)\) is an increasing curve starting at \((r(\underline\theta),\underline\theta)\), lying above the 45-degree line at first and flattening toward height \(\mathbb{E}[\theta]\) as \(w\) grows; it crosses the 45-degree line at a unique point \(w^\star\in(r(\underline\theta),\,r(\bar\theta))\) (from above to below). The employed set is \(\Theta(w^\star)\), i.e. a fraction of workers are hired.

图 3、图 4 / Figures 3, 4（已转述 / paraphrased） 图 3（情形 2）：\(g(w)\) 整体位于 45 度线下方，仅在起点 \(r(\underline\theta)=\underline\theta\) 处与之相切/相交，\(w\) 增大时趋于水平高度 \(\mathbb{E}[\theta]\)；唯一均衡 \(w^\star=r(\underline\theta)\)。图 4（情形 3）：\(g(w)\) 呈先上后平的 S 形，与 45 度线相交三次，交点横坐标依次为 \(w_1^\starFigure 3 (Case 2): \(g(w)\) lies entirely below the 45-degree line, touching/crossing it only at the start \(r(\underline\theta)=\underline\theta\) and flattening toward height \(\mathbb{E}[\theta]\) as \(w\) grows; the unique equilibrium is \(w^\star=r(\underline\theta)\). Figure 4 (Case 3): \(g(w)\) is S-shaped (rising then flat), crossing the 45-degree line three times at \(w_1^\star

情形 3 的均衡选择 / Equilibrium selection in Case 3 把竞争均衡工资集合记为下式。在一定条件下，只有最高的 \(w_3^\star=\max_{w^\star} CE\) 能被达成：逻辑是任何企业总有动机把工资抬高到 \(w^\star\in(w_2^\star,\,w_3^\star)\) 以吸引到全部工人——此处每个工人都能带来严格为正的剩余价值 \(g(w^\star)-w^\star>0\)，于是企业不断略微加价 \(\varepsilon\)，直到 \(w^\star=w_3^\star\)。Denote the set of competitive-equilibrium wages by the expression below. Under certain conditions only the highest, \(w_3^\star=\max_{w^\star} CE\), can be reached: any firm always has an incentive to raise the wage to some \(w^\star\in(w_2^\star,\,w_3^\star)\) to attract all workers — there each worker brings a strictly positive residual value \(g(w^\star)-w^\star>0\), so firms keep adding a small \(\varepsilon\) until \(w^\star=w_3^\star\).

命题 12.2（MWG 13.B.1）/ Proposition 12.2 (MWG 13.B.1) 若 \(\bar w^\star>r(\underline\theta)\)，且存在 \(\epsilon>0\) 使得对所有 \(w'\in(\bar w^\star-\epsilon,\,\bar w^\star)\) 都有 \(g(w')>w'\)，则在每个 SPNE 中，都有两家或更多企业开出 \(\bar w^\star\)，且所有满足 \(r(\theta)\le\bar w^\star\) 的工人都接受这些报价。If \(\bar w^\star>r(\underline\theta)\) and there exists \(\epsilon>0\) such that \(g(w')>w'\) for all \(w'\in(\bar w^\star-\epsilon,\,\bar w^\star)\), then in every SPNE two or more firms offer \(\bar w^\star\) and all workers with \(r(\theta)\le\bar w^\star\) accept these offers.

证明 / Proof 分三步。In three steps.

第一步：每家企业在均衡中利润为零。 假设某企业 \(i\) 盈利为正，即它开出工资 \(\bar w\) 满足 \(\mathbb{E}[\theta\mid r(\theta)\le\bar w]>\bar w\) 且有工人接受。那么对任何企业 \(j\) 只有两种可能：要么 \(j\) 开出更低工资而无人接受，要么 \(j\) 开出与 \(i\) 相同的工资。无论哪种情况，\(j\) 都可偏离、开出 \(\bar w+\epsilon\) 抢走 \(i\) 的全部工人，从而获得严格为正的利润。故企业不可能有正利润。当然，SPNE 中企业也不可能负利润，因为它们总可选择高于 \(r(\bar\theta)\) 的工资、一个工人都不雇。Step 1: every firm makes zero profit in equilibrium. Suppose firm \(i\) makes positive profit, i.e. it offers a wage \(\bar w\) with \(\mathbb{E}[\theta\mid r(\theta)\le\bar w]>\bar w\) and some workers accept. Then for any firm \(j\) there are only two possibilities: either \(j\) offers a lower wage that no worker takes, or \(j\) offers the same wage as \(i\). In either case \(j\) could deviate to \(\bar w+\epsilon\) and steal all of \(i\)'s workers, making strictly positive profit. Hence firms cannot make positive profit. Of course, in an SPNE firms cannot make negative profit either, as they can always pick a wage above \(r(\bar\theta)\) and hire no one.

第二步：最高报价工资是竞争均衡工资。 由第一步，要么最高报价工资 \(\bar w=\max\{w_i,\ i=1,2,\dots\}\in CE\)，要么 \(\bar wStep 2: the highest offered wage is a competitive-equilibrium wage. By Step 1, either the highest offered wage \(\bar w=\max\{w_i,\ i=1,2,\dots\}\in CE\), or \(\bar w

第三步：两家企业都开出 \(\bar w^\star\) 是 SPNE。 若只有一家企业开出 \(\bar w^\star\)，它将获得零利润；因为对所有 \(w'>\bar w^\star\) 有 \(\mathbb{E}[\theta\mid r(\theta)\le w']Step 3: both firms offering \(\bar w^\star\) is an SPNE. If only one firm offers \(\bar w^\star\), it earns zero profit; and since \(\mathbb{E}[\theta\mid r(\theta)\le w']\bar w^\star\), any upward deviation is unprofitable either. \(\blacksquare\)

注 12.1 / Remark 12.1 命题 12.2 中的条件是说：在 \(\bar w^\star\) 的左侧邻域内 \(g\) 正从上方穿过 45 度线（如图 4 所满足）。这排除了 \(g\) 只与 45 度线相切而不穿过的"均衡"（见图 5 转述）。The condition in Proposition 12.2 says that just to the left of \(\bar w^\star\), \(g\) is cutting down through the 45-degree line (as Figure 4 satisfies). This rules out a "equilibrium" where \(g\) only touches the 45-degree line without crossing (see the paraphrase of Figure 5).

图 5 / Figure 5（已转述 / paraphrased） "被排除的相切"：\(g(w)\) 在某点恰好与 45 度线相切（在 \(\mathbb{E}[\theta]\) 高度附近自下方触及又回落），不构成命题 12.2 所要求的"从上穿下"型均衡。"Ruled-out tangency": \(g(w)\) is tangent to the 45-degree line at some point (touching it from below near height \(\mathbb{E}[\theta]\) and falling back), which does not constitute the "cross from above" equilibrium required by Proposition 12.2.

注 12.2 / Remark 12.2 这为"市场为何能在某种社会意义上达到最优结果"提供了一个博弈论的正当性论证。同时也说明：我们无法靠政府干预在有效结果处实现帕累托改进。具体而言，设想政府强制每个工人都为某企业工作，再让企业按"工资减保留工资之差"补偿那些新进入市场的工人；然而要为此融资，就必须向已在岗的工人征税，反而使他们境况变差。This provides a game-theoretic justification for why we expect the market to reach the socially best outcome. It also shows we cannot achieve a Pareto improvement at the efficient outcome through government intervention. Concretely, suppose the government mandates that everyone works for a firm, and then to make those newly entered workers better off you pay them the difference between the wage and the reservation wage; however, to fund this you must tax the already-employed workers, who are thus made worse off.

设定 / Set-up（基于 JR 2011, Advanced Microeconomic Theory, 3rd Ed.） 消费者风险（类型）：\(\pi\in[\underline\pi,\bar\pi]=\Pi\subseteq[0,1]\)，其累积分布函数为 \(F(\pi)\)，表示消费者发生损失（出险）的概率。若发生损失，损失额为 \(L\)；若不发生，损失为零。工人（消费者）知道自身的出险概率 \(\pi_i\)，企业不知道个体概率，只知道类型分布 \(F\)。消费者风险厌恶，具有严格凹的 VNM 效用 \(U(\cdot)\)，初始财富为 \(y\)。企业风险中性；暂设只有一份合约——全额保险。若企业能以高于精算公平价 \(\pi L\) 的价格出售，则获利。Consumer's risk (type): \(\pi\in[\underline\pi,\bar\pi]=\Pi\subseteq[0,1]\) with c.d.f. \(F(\pi)\), the probability that a consumer suffers a loss (an accident). If the loss happens it equals \(L\); if not, the loss is zero. Workers (consumers) know their own accident probability \(\pi_i\); firms do not know individual probabilities, only the type distribution \(F\). Consumers are risk-averse with strictly concave VNM utility \(U(\cdot)\) and initial wealth \(y\). Firms are risk-neutral; for the moment assume a single contract — full insurance. If a firm can sell at a price above the actuarially fair price \(\pi L\), it makes money.

买方期望损失与均衡 / Buyers' expected loss and equilibrium 由临界条件可直接得出存在逆向选择：愿意购买的恰是出险概率高的人。广义地说，逆向选择指未获信息的一方得到的，正是已获信息的一方最不想要的对手；有利选择则相反——最谨慎的人最想买保险，最不愿出险。关键函数为下式 \(g(p)\)，即购买者的期望损失；它类比于劳动力市场里的 \(g(w)\)，并因逆向选择而递增。竞争均衡 \(p^\star\) 满足 \(p^\star=g(p^\star)\)（见其后公式）。The cutoff immediately implies adverse selection: those willing to buy are exactly the high-probability types. Broadly, adverse selection is when the uninformed side gets the counterparties the informed side least wants; advantageous selection is the opposite — the most cautious people most want insurance and least want accidents. The key function is \(g(p)\) below, the expected loss from buyers; it is analogous to \(g(w)\) in the labor market and is increasing due to adverse selection. The competitive equilibrium \(p^\star\) satisfies \(p^\star=g(p^\star)\) (formula after).

类比劳动力市场的结论 / Results analogous to the labor market (a) 若以价格 \(\underline\pi L\) 出售，则人人都会购买。(b) 低风险类型也会购买，因为有一标准结论：给定可买精算公平的保单，主体会选择完全投保。(c) 高风险类型会购买，因为 \(p=\underline\pi L<\bar\pi L\)。综合言之，竞争均衡可达 \(p^\star=\mathbb{E}[\pi]\,L\) 且所有类型都购买。(a) If insurance is offered at price \(\underline\pi L\) then everyone buys it. (b) Low-risk types also buy, by the standard result that given the option to buy an actuarially fair policy, agents fully insure. (c) High-risk types buy because \(p=\underline\pi L<\bar\pi L\). In sum, the competitive equilibrium can be \(p^\star=\mathbb{E}[\pi]\,L\) with all types purchasing.

注 12.3 逆向选择与有利选择 / Remark 12.3 Adverse vs Advantageous Selection 逆向选择：最愿意为保险付费的人，往往也是风险最高的人。有利选择：当（边际）购买者的边际成本高于全体购买人群的平均成本时出现，因为低风险消费者属于"边际内"（infra-marginal）。在有利选择市场，你有时会得到相对于有效结果过多的产出；这与逆向选择相反——后者相对于有效结果产出过少。这两类模型的另一个有趣特征是：在逆向选择模型中，需求曲线与边际成本曲线之间的距离就是风险溢价；而在有利选择模型中这不成立，需求曲线与边际成本曲线之间的距离是扣除任何固定成本后的风险溢价。Adverse selection: those willing to pay the most for insurance also have the highest risks. Advantageous selection occurs when the marginal cost of the (marginal) buyer is higher than the average cost of the buying population, because low-risk consumers are infra-marginal. In advantageous markets you can sometimes have too much output relative to the efficient outcome; this contrasts with adverse selection where you can have too little output relative to the efficient outcome. Another interesting feature of these models is that in adverse-selection models the distance between the demand curve and the marginal-cost curve is the risk premium; in advantageous-selection models this is not true — that distance is the risk premium net of any fixed costs. (References: Einav, Finkelstein, and Cullen (2010); Einav, Finkelstein (2011).)

图 6、图 7 / Figures 6, 7（已转述 / paraphrased） 两图横轴均为"投保比例 / Fraction Insured"，纵轴为价格 \(p\)。图 6（逆向选择）：需求曲线（Demand）向下倾斜；平均成本曲线 AC 与边际成本曲线 MC 均向下倾斜（更多人投保＝拉入更低风险者，拉低平均与边际成本），且 MC 在 AC 下方。均衡比例（需求与 AC 相交处）小于有效比例（需求与 MC 相交处），即投保不足。图 7（有利选择）：需求曲线向下倾斜；AC 与 MC 均向上倾斜，MC 在 AC 上方。均衡比例（需求与 AC 相交）大于有效比例（需求与 MC 相交），即投保过多。In both figures the horizontal axis is "Fraction Insured" and the vertical axis is price \(p\). Figure 6 (Adverse Selection): the Demand curve slopes down; both the average-cost curve AC and the marginal-cost curve MC slope downward (more insured = pulling in lower-risk people, lowering average and marginal cost), with MC below AC. The equilibrium fraction (where Demand meets AC) is less than the efficient fraction (where Demand meets MC) — under-insurance. Figure 7 (Advantageous Selection): the Demand curve slopes down; both AC and MC slope upward, with MC above AC. The equilibrium fraction (Demand meets AC) is greater than the efficient fraction (Demand meets MC) — over-insurance.