婚姻市场随机模型的唯一均衡与替代效应
Unique equilibria and substitution effects in a stochastic model of the marriage market
摘要：
Siow（2006）提出了一个婚姻市场模型，该模型允许每个参与者的偏好中存在随机同分布的噪声。随机性是麦克法登型的，它允许一个明确的决议的均衡偏好概率。本文的目的是证明由此得到的均衡婚姻分布的唯一性，并找到它的封闭形式的表示。这使我们能够推导出这种分布对外生偏好和总体参数的平滑依赖关系，并建立各种替代效应的符号、对称性和大小，从而促进比较静态。例如，我们表明，在这个模型中，任何给定类型的男性人口的增加都会导致每种类型的单身男性的增加，以及每种类型的单身女性的减少。我们发现，增加的数量的男性的一个给定的类型增加了均衡转移支付这些男人给他们的配偶，也增加了比例的男性选择保持未婚。这些性质的验证有助于证实模型的有效性。此外，我们做出了可以检验的意想不到的预测：$i$型未婚者相对于$j$型男性或女性总数波动的百分比变化在这个模型中形成了一个对称正定矩阵$r_{ij}=r_{ji}$，从而满足$|r_{ij}| \le (r_{ii}r_{jj})^{1/2}$这样的界限。一路上，我们给出了一个新的证明的存在的平衡，严格凸的变分原理和一个简单的估计，而不是一个不动点定理的基础上。不动点方法的存在部分，我们的结果已经探索了其他\cite{CSS} \cite{Dag} \cite{Fox}，但要复杂得多，既不产生唯一性，也不比较静态，也没有一个明确的代表性的解决方案。
英文摘要：
Choo-Siow (2006) proposed a model for the marriage market which allows for random identically distributed noise in the preferences of each of the participants. The randomness is McFadden-type, which permits an explicit resolution of the equilibrium preference probabilities. The purpose of this note is to prove uniqueness of the resulting equilibrium marriage distribution, and find a representation of it in closed form. This allows us to derive smooth dependence of this distribution on exogenous preference and population parameters, and establish sign, symmetry, and size of the various substitution effects, facilitating comparative statics. For example, we show that an increase in the population of men of any given type in this model leads to an increase in single men of each type, and a decrease in single women of each type. We show that an increase in the number of men of a given type increases the equilibrium transfer paid by such men to their spouses, and also increases the percentage of men of that type who choose to remain unmarried. The verification of such properties helps to substantiate the validity of the model. Moreover, we make unexpected predictions which could be tested: the percentage change of type $i$ unmarrieds with respect to fluctuations in the total number of type $j$ men or women turns out to form a symmetric positive-definite matrix $r_{ij}=r_{ji}$ in this model, and thus to satisfy bounds like $|r_{ij}| \le (r_{ii}r_{jj})^{1/2}$. Along the way, we give a new proof for the existence of an equilibrium, based on a strictly convex variational principle and a simple estimate, rather than a fixed point theorem. Fixed point approaches to the existence part of our result have been explored by others \cite{CSS} \cite{Dag} \cite{Fox}, but are much more complicated and yield neither uniqueness, nor comparative statics, nor an explicit representation of the solution.
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