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重 庆 理 工 大 学文 献 翻 译二级学院 计算机科学与工程学院 班 级 110030801 学生姓名 侯银川 学 号 11003080109译 文 要 求1、译文内容必须与课题（或专业）内容相关，并需注明详细出处。2、外文翻译译文不少于2000字；外文参考资料阅读量至少3篇（相当于10万外文字符以上）。3、译文原文（或复印件）应附在译文后备查。 译 文 评 阅导师评语（应根据学校“译文要求”，对学生外文翻译的准确性、翻译数量以及译文的文字表述情况等作具体的评价） 指导教师： 年 月 日3工资分布正如我们所说的，虽然它不是逻辑错误的认为F（W）作为外生的，对于许多应用来说是利益有模型内部确定的工资。因为原来的搜寻理论的一个关键因素是，一个给定的工人面临工资分配的劳动服务，即出现一个自然的问题是在什么条件下市场上相同的工人可能会产生一个非退化工资分配的均衡明白，有有两种意义，其中这个问题是利益。首先，由于基本模型表明，搜索是只有相关的，如果工资分配是简并，经济学家谁想到搜索是很重要也不得不制定工资差距的理论。其次，对于那些谁了经济学家工资差距为工人相同（或价格色散均匀不错）作为生活中的事实，我们希望在搜索摩擦会产生均衡，其中一价定律不成立。钻石（ 1971）是第一个解决这个问题的一个（产品市场，但问题是在劳动力市场上类似） 。尽管他的模型不会产生工资差距，要考虑他的分析是很有用的，一方面是因为它提供了洞察问题，因为它形多随后的分析。考虑了大量同质化的工人和企业，其中每个工人解决像一个在上一节，也就是说，他们随机抽取一个提供各个时期，在一个简单的搜索问题的模型已经失业收入等于b ，死于率&等每个企业要与劳动力规模技术作为唯一的输入，以恒定的回报边际产品p b。假设工资是由企业在工资设定发帖比赛：在每个周期开始时的坚定承诺工资，给予的其他选择，以最大限度地提高预期利润的工资。设F是CDF工资。一个纳什均衡接口需要每一个工资发布以正的概率赚取同样的利润，并没有其他的工资赚取任何更大的利润。钻石的发现是相当惊人的：有一个独特的平衡，在这种平衡的雇主设置相同的工资，等于失业的收入值：W = B 。证明很简单。鉴于工人是同质的，对于任何累积分布函数F他们都选择相同的保留工资WR 。显然，没有坚定的将张贴W WR ，因为他们可以聘请每一个工人，他们在联系W = WR 。要知道为什么它原来是W平衡= B，假设所有的企业都张贴W B，并考虑个别企业的情况。如果他们偏离，并提供工资比瓦特他们仍然会聘请每一个工人，他们满足（即略少，工人的保留工资也将下降无穷，因为工资上的损失小于等待另一段为要约成本W） 。因为这种说法仍然有效，只要为w B，独特的均衡工资必须等于B 。事实上，这仍然是真实的，如果我们放宽同质企业的假定边际产量的假设有所不同。得出的结论似乎相当严重：即使生产力跨越不同公司，搜索摩擦不产生工资的非退化分布，如果工人是同质的。不仅可以搜索理论不能帮助理顺在现实世界中找到的工资或价格的分散性，它甚至显得摇摇欲坠地面逻辑，因为在矿井基本模式是工资差距的存在这促使搜索在矿井第一名。许多研究人员随后发达的模型中，均衡工资或价格分布是简并，包括巴特斯（1977），Reinganum（1979），MacMinn（1980），伯德特 - 贾德（1983），罗伯（1985），阿尔布莱希特 - 阿克塞尔（1984），和伯德特 - 莫特森（1998）。我们回顾在本节几种选择。在继续之前，我们突出了钻石模型的一些特点这变成是非常重要的。首先，搜索摩擦的确切性质事项，例如，它可以不管搜索者有时会收到两份报价在一次。其次，即使工人有相同的生产力，事实证明，其他类型的异质性问题，因为我们将在下面看到。第三个特点就是假设公司岗位工资，而不是，比方说，在确定通过谈判工资，因为我们将在长在后面讨论。最后，我们认为有趣的是，要注意如何文学的演变：在早期的努力致力于试图找出其中有一些工资分散条件处于平衡状态，最近出现了使用这些模型已经很成功考虑到在数据的实际工资分布的各个方面。3.1 工人的异质性为了激励下面的各种型号，问自己：为什么会1希望找到工资差距？答案是，搜索摩擦产生一个自然的权衡：虽然每张贴在低利润更高的工资结果工人，这将有可能在增加，工人能进能出的速度，因为更多的工人将愿意接受这份工作在更高的工资。它原来，在钻石模型这种权衡实际上是不存在的，因为在平衡，如果你增加你的工资相对于其他公司有没有增加的速率可以聘请当所有工人具有相同的保留工资。鉴于此，一个自然的方法来生成工资分散体是允许的异质性在一些尺寸，这将产生异质性保留工资。该阿尔布雷希特 - 阿克塞尔模型假定一些工人失业收入B1等都有B2 B1。这意味着，对于一个给定的工资分布函数F会有两个不同的保留工资，说W1和W2 W1，钻石的结果最明显的泛化意味着没有公司会发布一个工资比W1或者W2等。但是，重要的是一些公司可以张贴W1和W2别人，只要这些暗示等于利润似乎是可能的。这似乎是可能的，在原则上，因为它们可以设置W = w1和只聘请B1的工人，或者设置W = W2和全部聘用工人。前者意味着每个工人，但工人的低到达率更高的利润，而后者则意味着工人较高的到达率，但每个工人的低利润。假设有大量的企业和工人，和规范企业的措施是1，而让工人的措施是：L = L1 + L2，其中，Lj为失业收入BJ措施。正如我们所说的，给定任意分布函数F，所有的1型工人选择保留工资w1和所有类型的2工人选择保留工资W2 W1，并在平衡所有的公司无论是发布W1或者W2。让成为企业发布W2的比例，要低于内生决定的。鉴于至多2工资张贴，F完全由W1，W2和归纳。但是请注意，F是工资在企业间的分布，一般不同于跨越工人工资的分配，我们将拭目以待正如上一节中，让速率工人接触的企业是和分离率是这两者都是外生现在。自企业有规模化生产技术，一个边际收益不变产品p B2，他们将采取所有接受他们的工资的工人。继钻石模型的逻辑，可以表明，在平衡最高工资张贴满足W2= B2。以确定W1，注意使b1工人同时接受W = W1和w= W2，其价值函数满足虽然他们可能会略有不同，这些仅仅是我们在上一节中所看到的特殊情况下适用，当F是两点分布。请注意，虽然B1工人接受W = W1，他们得不到任何资本收益这样做，因为W1是他们的保留工资，同样，他们苦于没有当从W1下岗资本损失。用W1（W1）= U 1和w 2= B2，我们发现我们现在知道w1和w2为基础的参数和功能，说WJ= WJ（）。在稳定状态下，失业率为1型和2由和定，而总失业。就业率可以写成，而总就业率接下来的步骤是分析.均方根值，假设谁最大限度地稳定国家利润，为了计算I我们首先推导出稳态工人在给定的坚定发帖WJ数，记NJ。由于总J型工人和每个时期企业之间的交往数为aLjuj，这也是J型工人对于一个给定公司的到达率，因为我们有标准化的企业数量为1。因此，W1企业招聘工人的速度aL1u1和w2企业增聘速度a。这当然是关键的权衡：支付更高的工资增加了招募工人人均支出利润。由于类型j的坚定失去每期新泽西州的工人，我们有N1=aL1u1/和n2=aU /。故后插入wj以及uj,由利润的差异这是成正比的正如我们所说的，平衡是工资分配，使得每个工资发布以正的概率赚取同样的利润，并没有其他的工资赚取任何更大的利润。鉴于一个分数的企业发布W2（&），其余后W1（&），任何平衡是完全特点&的值，该下述之一成立和T（1）0;和T（0）0;或者和T（0）=0。在第一种情况下，所有的企业都发布W2= W2（1）= B2;.第二所有的公司发布W1= W1（0）= B1。以及在第三种情况下一些公司发布W2= B2，而其他发布W1（B1，B2），但两者赚取同样的利润，II1=II2一个可以显示总存在一个唯一的平衡，当且仅当其中当生产率非常低所有的企业支付W1= B1，当它是非常高的全行支付W2= b2和当它是中间我们有工资分散体。如果需要的话，可以解决T（&）=0,&和替代以得出明确工资分配张贴。工资的工人各地的分布简单：EJ工人挣WJ。为& wR表示这一个工人会避免临界工资从有关活动，而不是冒失去他的工作，定义为K +U W(wC) = 0。显然，在平衡没有公司会张贴任何工资除WR或wc，并让是部分张贴更高的工资。关于工人们的贝尔曼方程 注意到工人们接受Wr，他们从做没有资本收益从W(wR) = U;同样的，他们将苦于没有资金损失失去工作。用K +U W(wC) = 0和W(wR) = U,我们得到 不同公司的工资稳态分布的特点是wR, wC 和，虽然再次这不同于跨越工人工资的稳态分布，因为低工资的企业失去他们的工作人员更频繁，因此是较小的。也就是说，全部的公司以相同的速度招聘，但是如果公司付的钱是wR工人离职的速度在，然而当公司付的钱为wc是工人离职的速度只有因此，和由此可见，稳态的利润为二类企业由下式给出一个现在可以显示成正比的公式如Albrecht-Axel模型，这是一个明显的代数问题是一个独特的平衡,当时，当且仅当P是在一些区域T(0) = 0 和T(1) = 0时，作为通过求解确定。对于非常低（高）P没有坚定的（每家公司）发现它有利可图，从而避免使得工人出现一些不良行为。所以独特的工资是wR(wC)；我们的工作分布从P中间分开。而Albrecht-Axel假设本质上异构的工人，犯罪模型假定在任何时期都会有不同的个体事后差异（有的则要犯罪的机会，有的不需要；一些人赚的是wr而一些人赚的是wc）。这表明另一种版本的阿尔布雷希特和阿克塞尔事前的工人同质,但其中失业收入，或者更一般地，非市场的机会，为每个单独的进化根据一个 两状态马尔可夫过程，同时对B1的价值观和B2。无需经过的细节，应该能够看到一个公司可以提供一个低工资，吸引搜索与B的低电流值，但这些工人将离开，如果乙增加其价值。当然,这假设工资是固定的-但这是一个基本的假设所有的模型被认为是在这里。3.3 在职工作搜索在上述公司提出的两种模式可以支付更高的工资来增加流入或减少工人的外流。该伯德特 - 莫特森（1998）模型有两个，而是通过不同的机制：对在职搜索。在这个基本的模型中，工人是同类的和有推卸或犯罪活动没有机会，但雇用的工人继续采样新的供应和离开时，他们得到高于其目前的工资的报价。在前面一节中，让出价的比率达到0和1，而失业和就业分开，并假定每一个提供的价格都是随机的低于F(w).因为所有的失业工人他们是相同的常见保留工资,Wr.显然没有公司职位w低于这个共同的保留工资福wr,所以失业人员可以接受所有的工作，和失业率是为便于说明，我们用一些特殊的例子0=1=，哪一个意味着通过（18）wr=b，转回一般的情况。分析这个模型更复杂比前面已经被考虑到的，因为工人流入率坚定张贴工资w现在取决于整个工人支付工资的分配（因为他们可以吸引他们接触的任何员工的工资低于W）。我们需要做的第一件事要做的就是计算整个工人工资的稳态分布，G(w),得到一个工资分布F(W)通过公司得到。如上，这些不同是因为公司发布不同的工资聘用不同数量的工人。这个问题简单分析呈现出我们从个体公司来描述这个问题，为简单起见聚焦在这个例子上r到0(看到高斯1997的这个例子当ro)。稳定的公司利益状态W是。n(w)是一个稳定个工人数量。推断n(w)，为了计算N（W），只需注意，在公司支付w雇用工人的数量必须等于多少工人收入w由公司支付w数除以。假设值可变，那么我们将核查底线，这就意味着n(w)=G(w)(1-u)/f(w).因此后面加入u和G，后者是我们给的可变的值。再一次，平衡要求支付所有工资产生相同的利润，这至少是一样大，可以发布的任何其他工资的利润。一个平衡F可能看起来像什么?我们争论上面的公司，他们的职位薪资没有wp,因此这意味着下面是一些其他功能的平衡：（1）F不包含乱点；（2）一些公司付款恰好在w=b；（3）这里不会有空隙在支持F时。总之，支持F在b,w之间为了大的范围wp,和这里没有空隙或者乱点在支持的情况下。关键的下一步是使用该公司赚取高薪的所有工资，包括最低工资b等于利润的事实：因此，对于wb,w时，由于F(b)=0，我们可以得到等同于这个表达式在（25）中，我们还有一个等同的解决产量的表达式F（w）这是F的形状必须采取非均衡：它是独一无二的工资分配暗示等于利润支付所有工资。为了完整的描述这个F它只需要找的w的残余，这些我们可以简单得到从F（w）=1。在话，结果如下。首先，所有失业工人接受他们接受，因为所有的报价都高于第一次报价 WR= B。他们每一次拉升工资分配更好报价走来，因外裁员定期回失业，有一个由企业提供的工资的非退化分布函数F，和由工人赢得青睐,G,这是不同的，因为不同的工资意味着不同的号码工人：企业支付更高的工资吸引其他公司更多的工人，失去更少的工人给其他公司。因此，高工资的企业处于平衡状态时，尽管所有企业获得同样的利润。这个模型有一些有趣的扩张，首先，没有太多难解决在0不等于1时，这个结果是在wr是内部（在例子0=1我们知道wr=b）。为了确定wr，我们需要解决（18），那个可以明确地考虑到功能从（17）中。这个结果是强调这种泛化的原因之一是有趣,考虑到限制1到0或者r到无穷。将会发生什么，明确地解决F（w）=1为了在这个限制，最高的出价w等于wr,也是最低的出价因为没有公司出价wwr。因此，有一个单一的工资，w=wr。此外，（28）暗示在一个限制wr=b，和全部雇佣工人的价值是失业人员的收入，钻石解决方案是一个特例。其他有趣的结果来自限制 当1到无穷。这个结果意味着，w=p和G（w)=0为了全部的wp,因此，在这个限制内全部的工人赚的钱恰好是W=p。而且，当0到无穷的时候，失业率从u到0。因此,类似的东西有竞争力的解决方案出现-例如。所有的工人总是赚他们的边际产品-可以认为是极限情况作为搜索摩擦消失,在某种意义上,这两个0和1大。因此，这个模板生成竞争的结果和特殊情况时的钻石解决方案。你也可以让企业成为异质相对于模型的生产力页。给定类型的商行的数量有限，存在的分布每种类型支付的工资，以及所有与企业生产率P2付出比所有企业更大的工资与生产率P1 P2。因此，高生产率的企业必然最终大。这是一个重要的扩展，因为与常数p由（26）给出的工资分配具有密度增加，而这并不是人们看到的数据。用异源行，然而，F可以具有降低的密度，即使p的基础分布没有。此外，如图范登堡（2000），异构的企业可以有多重均衡（见下文基于相同的经济学一个简单的例子）。3.4 其他问题我们已审阅三种模式产生内源性的工资差距，当然，你可以将它们合并。例如，我们可以对在职搜寻和谁，相对于B不同，整合伯德特 - 莫特森与工人阿尔布雷希特 - 阿克塞尔。这实际上是重要的，其理由如下。在对在职搜索模型可以做会计了很好的工作工资的经验分布，至少有一次我们允许异类公司，但它确实较差的占个人就业的历史，特别是有问题的观察负的持续时间依赖（也就是说，危险率与失业率法术的长度减小） 。相比之下，阿尔布莱希特 - 阿克塞尔模式确实占负的持续时间依赖，因为失业池有工人不同流出率一个更好的工作，但在占工资数据表现不够理想。模型相结合的在职搜寻和工人异质性有更大的余地来考虑工资和失业数据（见文旦，罗宾和范登伯格1999）。人们还可以把在职/逃避框架-搜索和犯罪作品。在生产模型，公司的内源性o溜支付wwc,w,和剩下的1-o付wwr,w,其中WR是保留工资和wc的劝阻犯罪活动的关键工资。对分布WR,w和wC,W有没有间隙或质点，一个能解决他们的封闭形式o的函数。然而，有间w和WC中的间隙(因为通过增加W从wc-e到w您可以生成在在工人流出率的离散下降，增加利润)。我们可以有o=0或1的平衡，在这种情况下，事情看起来很像基本伯德特 - 莫特森模型，但是我们也可以有0o b. Assume that wages are set by rms in a wage- posting game : at the beginning of each period a rm commits to a wage, given the wages chosen by the others, to maximize expected prots. Let F be the cdf for wages. A Nash equilibrium F requires that every wage posted with positive probability earns the same prot, and no other wage earns any greater prot.Diamonds nding is rather striking: there is a unique equilibrium, and in this equilibrium all employers set the same wage, equal to the value of unemployment income: w = b. The proof is simple. Given workers are homogeneous, for any cdf F they all choose the same reservation wage wR . Clearly, no rm will post w wR as they can hire every worker they contact at w = wR. To see why it turns out that w = b in equilibrium, assume all rms are posting w b, and consider the situation of an individual rm. If they deviate and oer a wage that is slightly less than w they will still hire every worker they meet (i.e., workers reservation wage will also fall innitesimally since the loss in wages is less than the cost of waiting another period for an1 8 Obviously the most basic competitive model can generate wage dispersion for worker s who vary in product ivity; what is relevant in this context is wage dispersion acr oss oer s for a given worker type.oer of w). Because this argument remains valid as long as w b, the unique equilibrium wage must be equal to b. Indeed, this is still true if we relax the assumption of homogeneous rms by assuming marginal products dier.The conclusion seemed rather severe: even if productivities dier acrossrms, search frictions do not yield a nondegenerate distribution of wages if workers are homogeneous. Not only could search theory not help rationalize the wage or price dispersion found in the real world, it even seemed on shaky ground logically, since in the basic model it is the presence of wage dispersion that motivates search in the rst place. Many researchers subsequently devel- oped models in which equilibrium wage or price distributions are nondegener- ate, including Butters (1977), Reinganum (1979), MacMinn (1980), Burdett- Judd (1983), Robb (1985), Albrecht-Axel (1984), and Burdett-Mortensen (1998). We review several alternatives in this section.Before proceeding, we highlight some features of the Diamond modelthat turn out to be important. First, the exact nature of the search friction matters; for example, it can matter if searchers sometimes receive two oers at once. Second, even if workers have identical productivities, it turns out that other types of heterogeneity matter, as we will see below. A third feature is the assumption that rms post wages, as opp osed to, say, determining wages through bargaining, as we will discuss at length later. Lastly, we think it is interesting to note how this literature has evolved: while early eorts were devoted to trying to nd conditions in which there was some wage dispersion in equilibrium, more recently there has been much success using these models to account for various aspects of the actual wage distributions in the data.3.1 Worker HeterogeneityTo motivate the various models that follow, ask yourself this: why might one expect to nd wage dispersion? The answer is that search frictions produce a natural trade-o: while posting a higher wage results in lower prot per worker, it will potentially increase the rate at which workers can be hired, since more workers will be willing to accept the job at a higher wage. It turns out that in the Diamond model this trade-o is actually non-existent, since in equilibrium if you increase your wage relative to other rms there is no increase in the rate at which you can hire when all workers have the same reservation wage. In view of this, a natural approach to generating wage dispersion is to allow for heterogeneity in some dimension that will generate heterogeneity in reservation wages.The Albrecht-Axel model assumes some workers have unemployment in- come b1 and others have b2 b1. This implies that for a given wage distrib- ution F there will be two dierent reservation wages, say w1 and w2 w1 . The obvious generalization of Diamonds result implies that no rm will post a wage other than w1 or w2. But, it seems possible that some rms could post w1 and others w2 , as long as these imply equal prot. This seems p ossible, in principle, since they can set w = w1 and hire only b1 workers, or set w = w2 and hire all workers. The former implies a higher prot per worker but a lower arrival rate of workers, whereas the latter implies a higher arrival rate of workers but lower prot per worker.19Suppose there are large numb ers of rms and workers, and normalize the measure of rms to be 1, and let the measure of workers be L = L1 + L2 where Lj is the measure with unemployment income bj . As we said, given any distribution F , all type 1 workers choose reservation wage w1 and all type 2 workers choose reservation wage w2 w1 , and in equilibrium all rms post either w1 or w2 . Let be the fraction of rms posting w2 , to be determined endogenously below. Given that at most two wages are posted, F is completely summarized by w1, w2 and . Note, however, that F is the distribution of wages across rms, which generally diers from the distribution of wages across workers, as we will see.As in the previous section, let the rate at which workers contact rms be and the separation rate be , both of which are exogenous for now. Sincerms have a constant returns to scale production technology with marginal product p b2, they will employ every worker that accepts their wage. Following the logic of the Diamond model, one can show that in equilibrium the highest wage p osted satises w2 = b2. To determine w1, note that b1 workers accept both w = w1 and w = w2 , and their value functions satisfyrU1=b1 + W1 (w2) U1 rW1 (w1)rW1 (w2)=w1w2 + U1 W1(w2 ) :Although they may look slightly dierent, these are merely special cases1 9 Albrect-Axel (1984) do not actually assume all rms earn t he same prot, but r ather that productivity p is distributed in som e way across rms, and look for a cuto p such that r ms with p p pay w = w2. On t his dimension, the model her e is actually more similar to the consumer search models in Diamond (1987) or Curtis and Wright (2001).of what we saw in the previous section that apply when F is a two-point distribution.Note that although b1 workers accept w = w1 they get no capital gain from doing so, since w1 is their reservation wage, and similarly they suer no capital loss when laid o from w1. Using W1(w1) = U1 and w2 = b2 , we nd(r + )b1 + b231w1 =:(20)r + + We now know w1 and w2 as functions of the underlying parameters and , say wj = wj (). In steady state, the unemployment rates for type 1 and 2 aregiven by u1 = and u2 = , while total unemployment is L1u1 + L2u2 .+The employment rates can be written e1 = 1 u1 , e2 = 1 u2 , while theaggregate employment rate is e = (L1e1 + L2 e2)=L.The next step is to analyze rms, who are assumed to maximize steadystate prot, = n(p w). To compute we rst derive the steady state number of workers at a given rm p osting wj , denoted nj . Since the total number of contacts between type j workers and rms per period is Lj uj , this is also the arrival rate of type j workers for a given rm, since we have normalized the number of rms to 1. Hence, w1 rms recruit workers at rateL1 u1 and w2 rms recruit at rate u. This is of course the key trade o:paying higher wages increases recruitment at the expense of prot per worker.Since a rm of type j loses nj workers per period, we have n1 = L1 u1=and n2 = u=. Hence,111 + =n (p w ) = L1p (r + )b1 + b2 #r + + L1L2 2 =n2(p w2) =after inserting wj as well as uj .+ + + (p b2);The dierence in prots is given byuL1 u12 1 =which is proportional to(p w2) (p w1); (21)T () = (r + + ) f(p b2) L1 + ( + )L2 (p b1)L1grL1 (b2 b1):(22)As we said, an equilibrium is a wage distribution such that every wage posted with positive probability earns the same prot, and no other wage earns any greater prot. Given that a fraction of the rms post w2() and the rest post w1(), any equilibrium is completely characterized by a value of such that one of the following holds: = 1 and T (1) 0; = 0 and T (0) 0; or0 1 and T () = 0. In the rst case, all rms post w2 = w2(1) = b2; in the second all rms post w1 = w1 (0) = b1 ; and in the third case some rms post w2 = b2 while others post w1 2 (b1 ; b2), but b oth earn the same prot,1 = 2.One can show there always exists a unique equilibrium , and 0 1if and only if p p wR denote the critical wage at which a worker would refrain from the activity in question rather than risk losing his job, dened by K + U W (wC) = 0. Clearly, in equilibrium no rm would post any wage other than wR or wC, and let be the fraction posting the higher wage. Bellmans equations for a worker arerU=b + W (wC) U rW (wR)rW (wC)=wR + KwC + W (wC ) U :Notice that although workers accept wR, they get no capital gain from doingso since W (wR) = U ; likewise, they suer no capital loss from losing the job.Using K + U W (wC ) = 0 and W (wR) = U , we havewC = b + (r + + )K=wR = b K + K= :The steady state distribution of wages across rms is characterized by wR, wC and , although once again this diers from the steady state distri- bution of wages across workers, since low wage rms lose their workers more2 1 One reason that jail t ime is interest ing in the model is the following. In the basic eciency wage model, the rm is supposed to punish a worker by laying him o when he is caught shir king, but this is not really in the r ms interest what is t he point of getting rid of a worker, only to search for another who will behave exactly the same? If an out side authority like the criminal just ice system exogenously t akes the worker out of the job t his issue does not come up.frequently and hence are smaller. That is, all rms recruit at the same rateLu, but rms paying wR lose workers at rate + while rms paying wC lose workers only at rate . Hence, nR = Lu=( + ) and nC = Lu=. It follows that steady-state prots for the two types of rms are given by:R R R + = n (p w ) =Lu p b + K K=LuC= nC(p wC) =p b (r + + )K= :One can now show that C R is proportional toT () = (p b) (r + )K= (r + 2)K K:(23) As in the Albrecht-Axel model, it is a matter of algebra to show that thereis a unique equilibrium, and 0 0). Steady state prot for a rm posting w is (w) = n(w) (p w), where n (w) is its steady state numb er of workers. To compute n (w), simply note that the number of workers employed at a rm paying w must equal the number of workers earning w divided by the number of rms paying w. Assuming dierentiability, which we will verify b elow, this means n (w) = G0 (w) (1 2 2 It can make a dierence in this model if, instead of matching with rms at random, one assumes balanced matching in the sense of Burdett-Vishnawath (1988b) i.e., if you ar e more likely to get an oer from a lar ger rm t han a smaller rm; see Robin-Roux (1998).2 3 Consider the following argument. Given any w, the number of wor kers employed at awage no greater than w is G (w) ( 1 u). This increases over time at rate uF (w), the rate at which unemployed worker s contact a rm paying less t han w, and decreases over time at rate G ( w) (1 u) + G (w) (1 u) 1 F (w), the r ate at which workers employed at less than w are terminated for exogenous reasons plus the rate at which t hey move torms paying more than w. Equating these ows and inserting u = =( + ) implies (24).u)=F 0 (w). ThereforeG0 (w) (w) =(1 u)(p w) =(p w)2 ;(25)F 0 (w)f + 1 F (w)gafter inserting u and G0, the latter of which we get from dierentiating (24).Again, equilibrium requires that all wages paid yield the same prot, which is at least as large as the prot from posting any other wage. What could an equilibrium F possibly look like? We argued ab ove that no rm posts w p, since this implies 0. Some other features of any equilibrium are the following: (1) F contains no mass points; (2) some rm pays exactly w = b; ; and (3) there can be no gaps on the support of F .24 Summarizing, the support of F is b; w for some upper b ound w 0, whereas t he decrease in pr ot per wor ker goes to 0 as goes to 0. To show (2), suppose t he lowest wage paid is w0 b. Then any rm paying w0 can increase pr ot by paying w = b, since it still at tracts and loses the same number of worker s (given there are no mass point s), which means n (b) = n (w0). Hence, the lowest wage paid is exactly b. To show (3), supposethere is an non-empty interval w0 ; w00 , with w0 b and some rm paying w00 but no rmpaying w 2 w0 ; w00 . Then the rm paying w00 can make strictly greater prot by payingw00 for some 0.workers, G, which are dierent since dierent wages imply dierent numbers of workers: rms paying higher wages attract more workers from other rms and lose fewer workers to other rms. Hence, high wage rms are larger in equilibrium, although all rms earn the same prot.The model has many interesting extensions. First, it is not much harder to solve with 0 = 1. The result isF (w) = + 11 s pw !1 p wR;(27)where now wR is endogenous (in the case 0 = 1 we knew wR = b). To determine wR, one needs to solve (18), which can be done explicitly given the functional form in (27). The answer is( + 1 )2 b + (0 1) 1pwR =( + 1) + (0 1) 1:(28)To highlight one reason why this generalization is interesting, consider the limit as either 1 ! 0 or ! 1. To see what happens, solve F (w) = 1 explicitly forw = p !2 + 1(p wR):In the limit, the highest wage oered wis equal to wR, which is also thelowest wage oered since no rm ever oers w wR. Hence, there is a singlewage, w = wR. Moreover, (28) implies that in the limit wR = b, and so all employers oer workers their value of unemployment income. The Diamondsolution emerges as a special case.Another interesting result comes from taking the limit of 1 q p w G(w) =as 1 ! 1.The result impliespwR1 q p wpwRw = p and G(w) = 0 for all w p.Hence, in the limit all workers earn exactly w = p. Moreover, as 0 ! 1,the unemployment rate u becomes 0. Hence, something that resembles thecompetitive solution emerges i.e., all workers always earning their marginal product can be thought of as the limiting case as the search frictions vanish, in the sense that both 0 and 1 get large. Thus, the model generates both the comp etitive outcome and the Diamond solution as special cases.One can also let rms b e heterogenous with respect to productivity p in the model. Given a nite numb er of rm types, there is a distribution of wages paid by each type, and all rms with productivity p2 pay a greater wage than all rms with productivity p1 p2. Thus, higher productivityrms necessarily end up larger. This is an important extension because with constant p the wage distribution given by (26) has an increasing density, which is not what one sees in the data. With heterogenous rms, however, F can have a decreasing density, even if the underlying distribution of p does not. Additionally, as shown in van den Berg (2000), with heterogeneous rms there can be multiple equilibria (see below for a simple example based on the same economics).3.4 Other IssuesWe have reviewed three models that generate endogenous wage disp ersion, and of course, one can combine them. For example, we can have on-the- job search and workers who dier with respect to b, integrating Burdett- Mortensen with Albrecht-Axel. This is actually important, for the following reasons. The on-the-job search model can do a good job of accounting for the empirical distribution of wages, at least once we allow heterogeneousrms, but it does less well in accounting for individual employment histories. Especially problematic is observed negative duration dependence (i.e., hazard rates that decrease with the length of unemployment spells). In contrast, the Albrecht-Axel model does a better job of accounting for negative duration dep endence since the unemployment pool has workers with dierent outow rates, but does less well in accounting for the wage data. Models that combine on-the-job search and worker heterogeneity have greater scope to account for both wages and unemployment data (see Bontemps, Robin and van den Berg1999).One can also integrate the on-the-job search and crime/shirking frame- works. In the resulting model, an endogenous fraction of rms pay w 2 wC ; w, and the remaining 1 pay w 2 wR; w , where wR is the reser- vation wage and wC the critical wage that dissuades criminal activity. The distributions on wR ; w and wC; w have no gaps or mass p oints, and one can solve for their closed forms as a function of . However, there is a gap between w and wC (since by increasing w from wC to w you can generate a discrete drop in the rate at which workers ow out, increasing prot). We can have = 0 or 1 in equilibrium, in which case things look a lot like the basic Burdett-Mortensen model, but we can also have 0 1. Indeed, the model can have multiple equilibrium values of , although only when criminals are actually sent to jail, and not just to unemployment.25We also want to mention that a slightly dierent version of any of the above models can be formulated, which gives very similar results but is based on a dierent vision of employers. Rather than having a xed number of rms that meet workers at some constant rate and all hire as many as they can get, suppose that to attract workers rms have to post vacancies, which is costly. Each vacancy can be lled by at most 1 worker. We assume that each employer can post only one vacancy, but allow entry by rms.26 Although we will go into models with entry in much more detail below, it is worth introducing the idea here to show how the endogenous wage distribution models can be recast in an alternative form. Mortensen (2000) does so for the Burdett-Mortensen model, and here we will do it for the Albrecht-Axel model.The basic setup is the same as Albrecht-Axel. Thus, any equilibrium wage distribution has a fraction of vacancies posting w2 = b2 and the remaining+1 posting w1 as given by (20), and the unemployment rates are u1 = +and u2 = . However, here we need to be more careful with the arrivalrates. As will be discussed below, to determine arrival rates in general onecan assume a matching function that maps the number of searching workers and rms, u and v , into the total number of meetings, m = m(u; v), but for now consider the special case m(u; v) = A minfu; vg. Assume that rms enter, or post vacancies, as long as expected prot exceeds the xed cost of entry, k. Clearly, we will have v L as long as the cost of entry is not too high, since rms make positive prot when v = L. Given v L we know v u; so the arrival rate for workers is the xed constant = A, and the arrival rate for rms is u=v . The rate at which a rm meets type j workers2 5 See Burdet t-Lagos-Wright (2000) . Intuitively, multiple equilibria can ar ise as follows: Suppose mor e rms are paying above wC . This makes t he value of search higher , so worker s ar e more reluctant to commit a cr ime because there is more to lose from spending time in jail. This makes it cheaper t o pay above wC , and hence we can have multiple equilibria. If , however, workers do not have t o spend any time in jail, but simply become unem ployed when caught , this cannot happen.2 6 Actually, it is equivalent to assum e t hat a xed number of employer s can each post asmany vacancies as they like, since only the total number of vacancies will be determ ined here. What is important is that posting vacancies is r equired to generate meetings and that this activity is costly.vis therefore Lj uj .Let Vj and Jj be the value functions for a given rm searching for a workerand matched with a worker (V for the value of a vacancy and J for the valueof a lled job), given it p osts wj. Since a rm posting w1 hires only type 1 workers,rV1 =vL1u1(J1 V1)rJ1 = p w1 + (J1 V1):The entry condition says that if any rms paying w1 enter at all then we must have V1 = k. Inserting this and solving we getrV1 = rk =L1u1vp w1 rk :r + Repeating the exercise for any rm posting w2, if rms enter at w2 , we must haverV2 = rk = uvp w2 rk :r + Entry by both types implies rV1 = rk = rV2, orT () = u(p w2) L1u1 (p w1) L2u2rk = 0:(29) Comparing this with (22) from Albrecht-Axel, we see that the equilibriumfunction in the two models reduces to exactly the same thing for either smallr or k. Hence, for small r or k the equilibrium is the same, except for the interpretation: now posting a high wage does not mean the rm becomes larger, but that it recruits faster.To close this section, we reiterate that every model discussed involves wage posting by rms, and the rm agrees to employ every worker it contacts at that wage independent of their characteristics. If the rm could condition wage oers on worker type it is simple to get wage dispersion. For example, in the Albrecht-Axel environment it is an equilibrium for rms to pay wj = bj to any type j worker who shows up (a generalized version of the Diamond result). One way to think of rms posting wages conditioned on worker type is as an extreme bargaining assumption where rms get to make take-it-or- leave-it oers to anyone who shows up. If we go to the other extreme and give workers all the bargaining power then the equilibrium has a single wage, w = p. In principle, given heterogeneity, any bargaining solution intermediatebetween take-it-or-leave-it oers by the rm and take-it-or-leave-it oers by the worker can generate wage dispersion, as we will see later.274 Two-Sided SearchThe preceding section focused on the issue of how to endogenously gener- ate an equilibrium with wage dispersion. This necessitated introducing rms into the analysis. Any such model is an example of a two-sided search model, since it considers behavior on both sides of a match. While the wage disper- sion literature tended to take the Diamond model as its starting point, this formulation is really just one of many possibilities. Some of the key choices that one makes in writing down any two-sided search problem concern what determines the rate at which meetings occur, how the output of a match is determined, and how the parties decide on compensation. In the Diamond model, meeting rates were taken to be exogenous, the output of a match was a homogeneous good, and compensation was determined through wage posting. Each choice is one of several alternatives available to a modeler, and generally the best option depends upon the issue. In this section we provide an overview of several of these options.4.1 Nontransferable UtilityUsually economists model employment relationships under the assumption of transferable utility: the worker and rm together produce some output which is to be somehow divided between the parties. However, it is also clear that there can be aspects of employment relationships that may not t this description all that well, such as how one gets along with ones boss, where2 7 If we assum e rm heterogeneity with respect to p then it is easy to generate wage dispersion if worker s m ake take-it-or-leave-it oers, subject to w pj (since a rm could always r ej ect ). Of course, t here may be no w pj at which the worker pr efer s employment to continued search. Call a rm ac tive if there is positive pr obability a worker will become employed there. Let the fraction of rms have p = p2 and 1 have p = p1 p2 . Assume worker s sample randomly from the set of active rms at r ate . One can show there is an equilibr ium where all rms are active (so some worker s get w = p1 while other s get w = p2 ) i p2 pA = r+ + p1 r+ . There is also an equilibr ium with only p2r +rms active i p2 pB = r+ p1 . These equilibr ia coexist in the nonem pty r egionwherepB p2 pA, along with an equilibrium in which a fraction of p1 r ms are active.This multiplicity is in the same spirit as van den Berg (2000).the job is located, and so on. Therefore some models of the labor market explore the implications of nontransferable utility. Moreover, very many of the search-based models of the marriage market also assume that utility is nontransferable. Hence, we begin here with a model in which the output from a match is entirely nontransferable. We later discuss the case in which there is a mix of transferable and non-transferable components.Consider an economy with a large group of workers of measure Lw and a large group of employers with measure Le. Each unmatched employer is searching for a single unemployed worker, and vice-versa. For simplicity, assume for now that Le = Lw = 1. Then, given a worker can only match with one employer and vice-versa, the number of unemployed workers u isalways equal to the number of rms with a vacancy. Also, assume for now that all individuals are ex ante identical. In particular, workers all produce output y if employed, but each worker and employer have idiosyncratic tastes concerning who they are matched with. So, although there is no such thing as an objectively better worker or employer, any individual may prefer one match over another.As in Burdett-Wright (1998), we formalize this by assuming that in arandom meeting, the payos to the worker and employer are given by zw and ze respectively, where zw and ze are (