\qrsection[taxes]{Federal payroll taxes}

\subsection*{Income tax}

Taxes are a part of life (the only other certainty is death),
so it's only common sense to learn how they work. In 
\sref*{directproportion} we studied sales taxes. 
Cities and states collect them; they are computed
as a percentage of the purchase price. In this section we'll explain
two important federal taxes that depend on your income, not on how you
spend it. 

Federal income tax is not simply a proportion of your income.
It's a \emph{progressive graduated tax}\index{progressive
  tax}\index{graduated tax}. When you make more money you not only pay
more tax, some of your income may be taxed at a higher rate. 
Table~\ref{table:2019taxtable} shows the 2019 \emph{tax brackets}
for single taxpayers. 

\begin{table}
\centering
\begin{tabular}{r@{\ --\ }lS[table-format=2.1]}
\toprule
\multicolumn{2}{c}{Bracket (\$)} & {Marginal Tax Rate (\%)} \\ 
\midrule
0  &  9,700 & 10 \\
9,701  &  39,475 & 12 \\
39,476  &  84,200  & 22 \\
84,201  &  160,725 & 24 \\
160,726  &  204,100  & 32 \\
204,101 & 510,300 & 35  \\
510,301 & & 37 \\
\bottomrule
\end{tabular}
\caption{2019 single taxpayer brackets and rates}
\tablesource{Public information.}
\label{table:2019taxtable}
\end{table}

That tells you that the first \$9,700 of
your income is taxed at 
10\%. If you make exactly that much, you pay \$970 in
tax. If you make more, the extra income is taxed at a higher rate ---
you have moved to a higher \emph{tax bracket}\index{tax bracket}.
For example, if you make between \$9,700 and \$39,475 you will pay
\$970 for the first \$9,700 and
12\% of
the amount you earn over \$9,700. If you earn more than
\$39,475 you start paying at a 22\% rate on the extra.

Let's try an example.  If your taxable income is \$50,000, then your
total tax is 
\begin{align}\label{eq:fiftyK}
\text{total tax} & =  0.10 \times \$9,700
+ 0.12 \times (\$39,475 - \$9,700) \\
& \quad
+ 0.22 \times (\$84,281 - \$39,475) \notag \\
& =  \$970.00 + \$3573.00 + \$2315.50 \notag \\
& =  \$6858.50 \ . \notag
\end{align}

Note carefully that when you are in a higher tax bracket the higher
rate applies only to the extra income. The taxpayer in this
example is in the 22\% bracket, but that rate applies only to her
earnings in that bracket.

Figure~\ref{fig:2019taxbrackets} explains this rule in another way.

%Table~\ref{table:2019taxtable2} explains this rule in another way.

%\begin{table}
%\centering
%\begin{tabular}{
%rrrcl
%}
%\toprule
%\multicolumn{2}{c}{Taxable income} & & \\
%{more than} & {less than} & \multicolumn{3}{c}{Tax} \\
%\midrule
%\$0 & \$9,700 & && 10\% of taxable income \\
%\$9,700 & \$39,476 & \$970 & $+$ & 15\% of the excess over \$9,075 \\
%\$36,900 & \$89,350 & \$5,081 & $+$ & 25\% of the excess over \$39,600 \\
%\$89,350 & \$186,350 & \$18,194 & $+$ & 28\% of the excess over \$89,350 \\
%\$186,350 & \$405,100 & \$45,354 & $+$ & 33\% of the excess over \$186,350 \\
%\$405,100 & \$406,750 & \$117,541 & $+$ & 35\% of the excess over \$405,100 \\
%\$406,750 &  & \$118,189  & $+$ & 39.6\% of the excess over \$406,750 \\
%\bottomrule
%\end{tabular}
%\begin{csmr}[2014 single taxpayer tax calculation\label{table:2019taxtable2}]
%K. P. Erb,
%IRS Announces 2018 Tax Brackets, 
%Forbes.com (October 31, 2013),
%\url{www.forbes.com/sites/kellyphillipserb/2013/10/31/irs-announces-2014-tax-brackets-standard-deduction-amounts-and-more/}
%\access{August 12, 2015}.
%\csmrcomment{public tax bracket data}
%\end{csmr}
%\end{table}
%

\figfile{2019taxbrackets.png}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{\thefigurefilename}
\begin{csmr}[2019 tax brackets\label{fig:2019taxbrackets}]
  K. Phillips Erb,
New: IRS Announces 2018 Tax Rates, Standard Deductions, Exemption
Amounts And More,
Forbes, March 7, 2018,
\url{www.forbes.com/sites/kellyphillipserb/2018/03/07/new-irs-announces-2018-tax-rates-standard-deductions-exemption-amounts-and-more/\#6bb2ab533133}
\access{February 7, 2019}.
\end{csmr}
\end{figure}
\figfile{}

%Figure 2019taxbrackets.png
%https://www.forbes.com/sites/kellyphillipserb/2018/03/07/new-irs-announces-2018-tax-rates-standard-deductions-exemption-amounts-and-more/#6bb2ab533133

\begin{teacher}
In the tax rate brouhaha in the 2008 election we recall reading a
story about a dentist who complained that he would need to be careful
not to let his income exceed \$250,000 --- where candidate Obama drew a
no-new-taxes line --- lest his overall tax rate increase. If you find
the story let us know and we'll turn it into an exercise.
\end{teacher}

The first graph in Figure~\ref{fig:GraduatedTax1019cropped}
from the spreadsheet \link{GraduatedTax2019.xlsx}
shows that the dependence of tax on income is \emindex{piecewise linear}
--- built from pieces of straight lines that become steeper as income
increases.  

\figfile{GraduatedTax2019cropped.pdf}
\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{\thefigurefilename}
\caption{2019 single taxpayer tax liability}
\figsource{Charts from an Excel spreadsheet we built.}
\label{fig:GraduatedTax1019cropped}
\end{figure}
\figfile{}

The second graph shows the \myindex{effective tax rate}
--- the percentage of your income you pay in federal income tax. In
Equation~\ref{eq:fiftyK} we found that the total tax on a taxable income of
\%50,000 was  \$6858.50.
The effective tax rate is 
$\$6858.50 / \$50,000 = 13.72\%$. 
This is a weighted average of the three bracket rates
10\%, 12\% and 22\%, with weights the amount of income taxed at each rate.
The effective tax rate is less than the rate in your top bracket
because you pay at a
lower rate on the first part of your income.
The effective rate does not reach 35\% until about \$2 million in
income --- well into the top 37\$ bracket.

In fact, the actual effective tax rate is lower than this for
wealthier households because income tax is collected only on 
income from wages and earnings. Income from capital gains ---
returns on investment --- is taxed at a lower
rate.
Figure~\ref{fig:FederalTaxRateByIncome2007} shows the effective
federal tax rate by total household income for the year 2007. 
The effective tax rate for the wealthiest
households was just 16.6\% --- less than half the 35\% rate for the
top bracket that year.

\figfile{FederalTaxRateByIncome2007cropped.pdf}
\begin{figure}
\centering
\includegraphics[width=4in]{\thefigurefilename}
\begin{csmr}[Effective federal tax rate by income, 2007\label{fig:FederalTaxRateByIncome2007}]
Data from G. Anrig,
10 Reasons to Eliminate the Tax Break for Capital Gains,
The Century Foundation (October 20, 2011),
\url{tcf.org/blog/detail/10-reasons-to-eliminate-the-tax-break-for-capital-gains}
\access{August 4, 2015}.
\end{csmr}
\end{figure}
\figfile{}

\subsection*{Social security}

Social security tax payroll deductions show up labelled
\index{FICA| see {Social Security}}
``FICA'' on your pay stub. That acronym is from
the ``Federal Insurance Contributions
Act''. Those taxes pay for \myindex{Social Security} and
\myindex{Medicare}.

In 2019 the starting tax rate was 6.2\% for Social Security and 1.45\% for
Medicare. The Social Security tax is collected only on
the first \$132,900 of your earnings. Up to that income level the
combined rate is 7.65\%.

When your earnings exceed $132,900$ you pay no more Social Security
tax, but you continue to pay Medicare tax at the 1.45\% rate. When
your income reaches \$200,000 the Medicare rate increase to 2.35\% on
the amount over \$200,000.

The actual rules are a little more
complicated. First, the tax applies only to wages. Other income (like
stock dividends or interest) are not subject to this tax. Second, the
real rates are twice the quoted amounts, but  your employer is
required to pay half. If you're self-employed you pay it all.


If you earn \$500,000 your FICA tax is 
%
\begin{align*}
  0.062 \times \$132,9000 & + 0.0145 \times \$200,000
   + 0.0235 \times (\$500,000 - \$200,000) \\
 & = \$18,190.
\end{align*}



Since you pay no Social Security on wages over \$132,900 
the percentage of your earnings collected for FICA taxes
decreases as your earnings increase even though you still pay for
Medicare. So 
FICA taxes are \emph{regressive}\index{regressive tax}.
Up to \$139,700 the effective rate is 7.45\%.
For \$500,000 the effective rate is just
$\$18,190/\$500,000 = 3.64\%$. For higher incomes, the effective rate
is even smaller. At huge incomes 
It levels off at the top Medicare rate of 2.35\%.

Figure~\ref{fig:socialsecurity} from spreadsheet 
\link{SocialSecurityTax2019.xlsx} shows the amount of FICA tax paid and
the effective tax rate as a function of FICA earnings.
Social Security taxes are 
\emph{regressive}\index{regressive tax} ---
the effective rate decreases as earnings increase,

\figfile{SocialSecurityTax2019cropped.pdf}
\begin{figure}
\centering
  \includegraphics[width=4in]{\thefigurefilename}
\caption{FICA (Social Security and Medicare) tax}
\label{fig:socialsecurity}
\end{figure}

\subsection*{Income tax history}

The first federal income taxes were collected in 1913.
For complex historical, legal and political reasons that required a
constitutional amendment, the 16th, ratified in 1909 after the Supreme
Court rejected earlier attempts to collect such a tax.

Figure~\ref{fig:toptaxrate} shows the significant fluctuations in the
rate for the top bracket through 1919. They have been near their
historic lows since the late 1980's. 

To assess the social and economic
impact of the changes in income tax rates would require much more time
and knowledge than we can offer here. 
The spreadsheet \link{Federalindividualratehistory.xlsx} contains a
complete history of income tax brackets and rates from the inception
of the income tax in 1913 through its hundredth anniversary in 2013,
in both dollars current in each year and adjusted for inflation (2012
dollars). 


\figfile{toptaxrate.jpg}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{\thefigurefilename}
\begin{csmr}[Historical top tax bracket rate\label{fig:toptaxrate}]
  Bradford Tax Institute,
  History of Tax Rates: 1913 – 2019,
  \url{bradfordtaxinstitute.com/Free_Resources/Federal-Income-Tax-Rates.aspx}
  \access{February 2, 2019}.
\end{csmr}
\end{figure}
\figfile{}

Even in years when Congress does not revise the tax code, the
IRS routinely adjusts the brackets (not the rates) to take inflation
into account. If that were not done then salaries increased by
inflation would move people into higher brackets even when their
increased wages did not correspond to increased purchasing power.

Figure~\ref{table:2018taxtable} shows the brackets for the tax year
2018. You can check that the 2019 brackets are all just about 2\%
larger than these.

\figfile{2018taxbrackets.jpg}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{\thefigurefilename}
\begin{csmr}[2018 tax brackets\label{fig:2018taxbrackets}]
  K. Phillips Erb,
New: IRS Announces 2018 Tax Rates, Standard Deductions, Exemption
Amounts And More,
Forbes, March 7, 2018,
\url{www.forbes.com/sites/kellyphillipserb/2018/03/07/new-irs-announces-2018-tax-rates-standard-deductions-exemption-amounts-and-more/\#16f3fe9f3133}
\access{February 7, 2019}.
\end{csmr}
\end{figure}
\figfile{}

%Social security began in 1935 as a program to guarantee some financial
%security to older retired workers. It has grown over the
%years. Medicare was added in 2965.
%

%Figure 2019taxbrackets.png
%https://www.forbes.com/sites/kellyphillipserb/2018/03/07/new-irs-announces-2018-tax-rates-standard-deductions-exemption-amounts-and-more/#6bb2ab533133


