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科罗拉多州Fictional小镇的市议会正在考虑采用新的税法，他们需要我们的帮助。目前，当地的所得税是一个统一的税种–每个人都要支付他们收入的相同百分比，无论收入是多少。市议会正在考虑将税收结构改为累进税，即那些收入较高的人支付其收入的较大比例的税收，而那些收入较低的人支付较小的比例。在这个项目中，你的小组将分析两个州的收入不平等情况，一个是统一税，另一个是累进税，并利用你的发现为科罗拉多州虚构市议会准备一份报告，其中包括你的小组关于他们是否应该改用累进税，或保持他们目前使用的统一税的建议。
调查

  1. 首先，你的小组必须选择两个你想用基尼指数进行比较的州。选择一个实行统一税的州和一个实行累进税的州。阿拉斯加、佛罗里达、内华达、南达科他、德克萨斯、华盛顿和怀俄明等州不征收所得税，所以你将不能选择这些州中的任何一个。科罗拉多州、伊利诺伊州、印第安纳州、马萨诸塞州、密歇根州、北卡罗来纳州、宾夕法尼亚州和犹他州实行统一税制，而其余各州则实行累进税制。
  2. 你的小组需要获得你所在州的收入五分位数。
  • 进入statisticsatlas.com，从地图上选择你的一个州。
  • 从右边的菜单中选择家庭收入。
  • 寻找标有家庭收入区间均值的图表。
  • 在你自己的表格中，记录每个收入五分位数的平均收入。在这个项目中，我们将忽略 “前5%”的数据点。
  • 对你的第二个州重复这个过程。

3. 接下来，我们需要从这些收入五分位数中建立洛伦兹曲线。这个过程类似于
这个过程与你在课堂上做的工作表相似。

  • 找到五分位数的总收入，并将平均收入转化为百分比。
  • 找出每个五分位数的累积收入百分比。在每个五分位数，将该五分位数占总收入的百分比与之前所有五分位数的百分比相加。
  • 对你的第二个州重复这个过程。

4. 我们将使用指数曲线来研究你们州的基尼指数。我们正在寻找一个形式为y=Aert的函数，它尽可能地接近洛伦兹曲线上的五分位点。

  • 请访问wolframalpha.com。(如果你喜欢使用Desmos，你可以，但不会提供事先写好的带有说明的Desmos页面）。)
  • 在搜索栏中输入指数拟合和你的洛伦兹曲线数据点。你需要在每个数据点周围加上{-}以及整个数据点，像这样。
    {{0.00, 0.00}, {0.20, 0.04}, {0.40, 0.13}, {0.60, 0.28}, {0.80, 0.52}, {1.00, 1.00}}
  • Wolfram Alpha会找到与你的数据最匹配的指数函数。记录这个函数。
  • 将你的指数曲线与数据点一起生成一个图表。这可以由Wolfram Alpha完成，但也欢迎你使用其他方法。
  • 对你的第二个状态重复这个过程。
  1. 现在
    基尼指数是你刚刚发现的指数曲线和 “平等线 “之间的面积与平等线下的总面积之比（即12，所以你可以认为基尼指数是洛伦兹曲线和平等线之间面积的两倍）。
    报告
    你们小组将负责写一份报告（应在11月16日星期五实验开始时提交），提交给科罗拉多州虚构市的市议会，总结你们对吉尼指数的调查结果。每份报告应包含以下内容。
  2. 为Fictional市议会总结你的结果的介绍，只包括对他们最重要的结果–关键是，你应该包括你的小组对他们税收结构的建议。
  3. 2. 描述用于建立洛伦兹曲线模型和计算基尼指数的过程。
  4. 3. 每个模型产生的洛伦兹曲线图的复印件，以及导致每个州的基尼指数的计算结果。(如有必要，计算结果可以手写）。)
  5. 4. 两个州的基尼指数的比较。税收结构是否是一个吉尼指数较高而另一个较低的唯一原因，或者其他原因可能导致两个州之间不平等的差异？你可能要做一些研究，因为其他因素，如贫困率、人口统计学和该州的主要产业也会影响收入不平等。
  6. 讨论一下你们小组对哪种税收结构更好的个人想法。没有 “正确 “的答案，但你的建议必须是合理的。如果你们组内有不同意见，请确保所有的声音都有机会在你们的报告中得到体现。同时更广泛地讨论建模的过程。你对吉尼指数和洛伦兹曲线技术有什么看法？你认为它在数学上是否有效？在经济上？在哲学上？

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The city council of the small town of Fictional, Colorado is considering adopting a new tax code, and they need our help. Currently, the local income tax is a flat tax–each individual pays the same percent of their income regardless of what that income is. The city council is considering changing the tax structure to a progressive tax, where those who earn more pay a larger percent of their income in taxes, and those who earn less pay a smaller percent. In this project, your group will analyze income inequality in two states, one with a flat tax and another with a progressive tax, and use your findings to prepare a report for the city council of Fictional, Colorado with your group’s recommendation about whether they should switch to a progressive tax, or keep the flat tax they currently use.
Investigation

  1. To begin, your group must choose two states that you would like to compare using the Gini index. Choose one state with a flat tax and one with a progressive tax. The states of Alaska, Florida, Nevada, South Dakota, Texas, Washington, and Wyoming don’t collect income tax, so you won’t be able to choose any of those states. Colorado, Illinois, Indiana, Massachusetts, Michigan, North Carolina, Pennsylvania, and Utah have a flat tax, while the remaining states have a progressive tax.
  2. Your group needs to obtain the income quintiles for your states.
    • Go to statisticalatlas.com and select one of your states from the map.
    • From the menu on the right choose Household Income.
    • Look for the chart labeled Household Income Interval Means.
    • In your own table, record the mean income for each income quintile. We will ignore the “top 5%” data point for this project.
    • Repeat this process for your second state.
  3. Next we need to build the Lorenz curves out of these income quintiles. This process is similar
    to the worksheet you did in class.
    • Findthetotalincomeofallthequintiles,andconverteachmeanincomeintoapercentage.
    • Find the cumulative income percentage of each quintile. At each quintile, sum that quintiles percentage of total income with the percentages of all previous quintiles.
    • Repeat this process for your second state.
  4. We will investigate the Gini Index of your states using an exponential curve. We are seeking a function of the form y = Aert which is as close as possible to the quintile points on the Lorenz curve.
    • Go to wolframalpha.com. (If you prefer to use Desmos, you may, but a specific pre- written Desmos page with instructions will not be provided).
    • Enter exponential fit followed by your Lorenz curve data points, into the search bar. You will need to surround each data point with {·} as well as the entire set of data points, like this:
    {{0.00, 0.00}, {0.20, 0.04}, {0.40, 0.13}, {0.60, 0.28}, {0.80, 0.52}, {1.00, 1.00}}
    • Wolfram Alpha will find the best fitting exponential function to your data. Record this function.
    • Generate a graph of your exponential curves along with the data points. This can be done by Wolfram Alpha, but you are welcome to use other methods.
    • Repeat this process for your second state.
  5. Now
    Gini Index is the ratio of the area between the exponential curve you just found and the “line of equality”, to the total area under the line of equality (which is 12 , so you may think of the Gini Index as twice the area between the Lorenz curve and the line of equality).
    Report
    Your group will be responsible for a report (due Friday, Nov. 16 at the beginning of lab) to be delivered to the city council of Fictional, Colorado that summarizes the result of your investigation into the Gini index. Each report should contain the following:
  6. An introduction summarizing your results for the city council of Fictional, including only the results most important to them–critically, you should include your group’s recommendation about their tax structure.
  7. A description of the process used to model the Lorenz curves and calculate the Gini index.
  8. Copies of the graphs of the Lorenz curves generated by each model, and a calculation leading to the Gini index of each state. (Calculations may be handwritten if necessary).
  9. Comparison of the Gini Index for the two states. Is the tax structure the only reason why one Gini index is higher and the other is lower, or could other reasons contribute to the difference in inequality between the two states? You may have to do some research, as other factors such as poverty rate, demographics, and the major industries in the state can also affect income inequality.
  10. A discussion of your group’s personal thoughts on what tax structure is better. There’s no “correct” answer, but your recommendation must be justified. If there is dissent within your group, please make sure all voices have a chance to be heard in your report. Also discuss the modeling process more broadly: What is your opinion on the Gini Index and Lorenz Curve technique? Do you think it is valid mathematically? Economically? Philosophically?

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