Many Models of Inequality

（长尾-帕累托分布-幂律分布）We start by summarizing some empirical regularities. First, in all countries at all times, the distribution of income has an elongated tail, with many low-income people and a small percentage of people who earn large incomes. Historically,income distributions were calibrated to lognormal distribution or Pareto distributions. Recently, more granular data reveals the tail to be longer than lognormal, though not quite that described by a power law. Wealth distribution is similarly skewed.

Cobb-Douglas production model 柯布-道格拉斯生产模型

（生产函数模型）Our first model extends the Cobb-Douglas production model to include two types of labor: educated and uneducated. The wage paid to a type of labor depends on the relative supply of that type and on technology.

Output depends on physical capital (K), educated labor(S), and uneducated labor (U) as follows:
产出取决于物质资本（K）、受过教育的劳动力（S）和未受过教育的劳动力（U），具体关系如下：
$$
Output=AK^{\alpha}S^{\beta}U^{\gamma}
$$
The parameters A, α, β, and γ capture the technology and the relative value of the three types of labor. The relative market wage for high- and low-skilled workers is:
参数 A、α、β 和 γ 反映了技术水平以及三种生产要素的相对价值。高技能与低技能工人的相对市场工资为：
$$
Wage_{S}=[\frac{U}{S}]\times[\frac{\beta}{\gamma}]\times Wage_{U}
$$

（技术变革&供需关系）Cause of inequality: Technological changes that favor educated workers increase β and decrease γ. These changes, along with increases in the supply of low-skilled workers, increase inequality.

positive feedback model 正反馈模型

The model assumes that technologies - the internet and smartphones in particular - have made us more connected and more influenced by the choices of others. That behavior (communication) creates a positive feedback and more inequality. We model socially influenced economic choices by reframing the preferential attachment model as a model that links positive feedbacks to talent.

（社交&社会影响）Cause of inequality: Increased connectedness increases social influences, creating a positive feedback.
互联互通程度的提高增强了社会影响力，从而形成了正反馈循环。

spatial voting model 空间投票模型

CEO pay is determined by a vote of a compensation committee. In the United States, compensation committees include many current and former CEOs,who prefer higher pay, as well as compensation experts (X). Other countries include workers (W) on compensation committees, resulting in a median voter who prefers much lower pay.

（利益共同体）Cause of inequality: CEOs determine their own pay through capture. Increases in the pay of any one CEO shifts preferences toward higher pay for all CEOs.
CEO通过“俘获”机制决定自己的薪酬。任何一位CEO薪酬的增加，都会使所有CEO的薪酬偏好向更高水平偏移。

observation: rate of return on capital exceeds the growth rate of capital

It comes from Thomas Piketty’s best-selling book Capital in the Twenty-First Century. This is less a model than an observation.

（消费率、税率、资本回报率、资本存量）It follows that in an economy that consists of workers who earn wages and capitalists who earn income from rents, the share of income going to capitalists will increase. To be a bit more formal, the rate at which capital increases will depend on three rates: the consumption rate, the tax rate, and the return on capital. Consumption depends on the level of capital.
资本的增长率取决于三个比率：消费率、税率和资本回报率。消费水平取决于资本存量。

The economy consists of workers and capitalists. The wages of workers increase at a rate g, the growth rate in the economy. The capitalists have wealth W_t at time t and earn return r (net of taxes) and consume a constant amount A. The income of capitalists will increase faster than that of workers if and only if:
$$
r-\frac{A}{W_{t}}>g
$$
Cause of inequality: In a market economy, the rate of return on capital exceeds the overall rate of growth (r > g). Capitalists with large accumulations of wealth spend a small proportion of their income from capital on consumption, so their share of total income increases over time.
在市场经济中，资本回报率超过了整体经济增长率（r > g）。拥有大量财富积累的资本家将其资本收入中用于消费的比例很小，因此他们在总收入中所占的份额会随时间推移而不断增加。

assortative mating

A family’s income depends on the incomes of both partners. If a low-income person marries a high-income person, then that marriage will contribute toward equalizing income distributions. If high-income people marry other high earners, then income disparities will increase.

（门当户对）Cause of inequality: Increases in the number of educated women, increased pay for workers with higher levels of education, and assortative mating (the tendency for people to marry others of the same income level) result in an increase in household-level income inequality.
受过高等教育的女性数量增加、高学历劳动者薪酬上涨，以及同质婚配（即人们倾向于与收入水平相近的人结婚）的普遍化，共同导致了家庭层面收入不平等的加剧。

Intergeneration Income (Wealth) Dynamics - Markov model

（财富转移概率-社会流动性）The population can be divided into four income (or wealth) categories with equal numbers of people. We can estimate the transition probability that an individual (or family) in one category moves to another category within a generation, as shown in the figure below. More equal transition probabilities correspond to greater social mobility.

（社交、知识、风险、教育、遗产）Cause of inequality: Social skills, tacit knowledge, attitudes toward risk and education, and bequests reduce mobility between income classes.
社交技能、隐性知识、对风险和教育的态度，以及遗产继承等因素，降低了不同收入阶层之间的流动性。

Persistent Inequality (Durlauf)Schelling Segregation Model + Local Majority Model

（社区）Individuals belong to income classes and segregate residentially by income.Individuals allocate a portion of their income to education, resulting in positive spillovers that increase with community income level. The future income of a child living in community C depends on her innate ability, spending on education, and spillovers. The contributions of education and spillovers depend on the level of income within the community, I_C.
$$
Income_{C}=F(ability,education(I_{C}),spillover(I_{C}))
$$

（聚类效应）Cause of inequality: Children who grow up in low-income neighborhoods receive fewer educational opportunities and economic spillovers.
在低收入社区长大的孩子获得的教育机会和经济溢出效应较少。