From Maura:
The objective today was to talk about graduated income tax (the math being piecewise linear functions). We started with an easier example, looking at the tax on clothing in Massachusetts. In fact clothing is tax-free, as long as the item (not the total sale) costs less than $175. The “luxury tax” is 6.25% of the amount over $175. We did a few examples and constructed the graph.
Then on to the more interesting example: graduated income tax. I showed them the first two brackets and we did a couple of examples. Not too bad but definitely more difficult than the clothing examples. Then I put in the next bracket and gave them an example to work where income in $62,550. They really struggled to break the amount into the three pieces. I talked them through it, since most of them didn’t get it. They wanted to first subtract $8,025 (the maximum for the first income bracket) and then subtract $32,550 (the maximum for the second income bracket). We talked about why that didn’t work, then I decided to try to make the point with a clearer example. We looked at an income of $32,550. I think most of them saw that they should subtract $8,025 and tax that amount at 10%, then tax the leftover amount ($24,525) at 15%. That seemed to make sense. Then I asked: what if the income is $32,650? Most of them saw that the additional $100 would be taxed at the higher rate of 25% and that the rest of the work was already done. Then I asked: what if the income is $33,550? They saw the pattern: tax the additional $10,000 at the 25% rate and use what they already knew for the rest of it. I showed them the next income bracket and we talked a bit about how to handle it.
Then I showed them a clip from YouTube of “Joe the Plumber.” The clip showed the encounter that then-presidential candidate Barack Obama had with Joe Wurzelbacher in Ohio. During the conversation, Obama describes marginal tax increases for small business in terms of thousands of dollars, noting that “…from 250 down, your taxes are going to stay the same… from 250 to 300 or so, so for that additional amount you’d go from 36 to 39 percent.” We then calculated what the increase in taxes would be. If Joe the Plumber’s business earned $300,000, then he would be taxed on $50,000. Instead of being taxed at a rate of 36%, under Obama’s plan he would be taxed at 39%. The extra amount he would pay would be $1500. In the context of a $300,000 business, this is not a lot of money. (I got the idea of presenting this from a colleague in the QL world.)
Postscript: we assigned an interesting homework problem about the Obamas’ taxes but it turned out to be more complicated than we though and the numbers didn’t quite work. We’ll see if this is too frustrating for the students or if they are ok with the idea that sometimes the numbers don’t quite work out.
From Ethan:
Today was tax day.
Maura warned me that the class might have trouble grasping the tax bracket idea. Knowing that helped me minimize the trouble.
We listed all the taxes we could think of – sales, income, real estate, inheritance (“death tax” led to a brief digression on the psychological impact of the names you give things). I added social security. Never talked about VAT.
Started with sales tax. Sketched the graph: linear with slope 0.0625 (dollars of tax) / (dollar of purchase), intercept 0 (dollars). Then (at Maura’s suggestion) the luxury tax – none on clothing until the price reaches $250. Sketched the piecewise linear graph. Talked about why no sales tax on food or on clothing (up to a point). The answer to “why?” isn’t to be found in quantitative reasoning, it’s in social policy. Exempting necessities makes the tax more progressive. Some people think that’s good policy.
Social security tax is piecewise linear too: fixed rate up to about $112K, then 0 thereafter. We could make the computations. It’s regressive. The effective rate is half when your income is twice the cutoff.
With those examples done we worked through the brackets in the 2008 table in the book. One student pointed out that it was for single taxpayers, so not useful in the exercise on Obama’s 2008 return. That provided an opportunity to look at the historical brackets and rates (on the web, and in a spreadsheet linked from the book). We found the Bush tax cuts, and the sharp drop under Reagan, and the high rate during WWII (in those days when we thought a war was necessary we knew we had to pay for it …)
We finished with a discussion of adjusted gross income – and the kinds of adjustments available (mortgage interest, some medical expenses, union dues sometimes, charities). Noted that if you’re in the top bracket and you write a $100 check to a charity you’re really paying $65, the government $35.
One student noted that March is the home of two kinds of bracket madness – IRS and NCAA.
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