Exercise 1.1.75

Question

In a certain country, income tax is assessed as follows. There is no tax on income up to \$10,000. Any income over \$10,000
is taxed at a rate of 10\%, up to an income of \$20,000. Any income over \$20,000 is taxed at 15\%.
\begin{enumerate}[label=(\alph*)]
    \item Sketch the graph of the tax rate $R$ as a function of the income $I.$
    \item How much tax is assessed on an income of \$14,000? On \$26,000?
    \item Sketch the graph of the total assessed tax $T$ as a function of the income $I.$
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item From the exercise text, we can conclude that if the income $I$ is smaller than $\$10,000,$ the tax rate is equal to $0\%.$ If the income $I$ is between $\$10,000$ and $\$20,000$, the tax rate is equal to $10\%.$ If the income is over \$20,000, the tax rate is 15\%. We plot the \emph{marginal} tax rate $R$ (i.e., not the overall tax rate) as follows:
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=30000, ymin=0, ymax=20,ytick={0,10,15}, xtick={10000,20000}, scaled ticks=false, tick label style={/pgf/number format/fixed}, xlabel={$I, \$$}, ylabel={$R, \%$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:10000, samples=2, mark=none, color=blue, ultra thick] {0}; 
         \addplot[domain=10000:20000, samples=2, mark=none, color=blue, ultra thick] {10};
         \addplot[domain=20000:30000, samples=2, mark=none, color=blue, ultra thick] {15};
         \addplot[mark=*, only marks, fill=white] coordinates {(10000,10) (20000,15)};
         \addplot[mark=*, only marks, fill=blue] coordinates {(10000,0) (20000,10)};
    \end{axis}
    \end{tikzpicture}
    \end{figure}

\item Since \$14,000 is in the interval $[10000,20000]$, the total assessed tax is equal to $10\%\cdot 14,000=1,400\$.$ Since \$26,000 is in the interval $[20000,+\infty]$, the total assessed tax is equal to $10\%\cdot 10,000+15\%\cdot 6,000=1,900\$.$

\item Total assessed tax $T$ is equal to $0$ if the income $I$ is smaller or equal to $10,000.$ In the interval $I\in [10000, 20000],$ the value of $T$ increases. When $I$ is over \$20000, the value of $T$ continues increasing, but with a smaller angle coefficient.\newline
    Using the information from the exercise text, we have:
    \begin{align*}
    \forall I\in [0, +\infty):\quad 
    T(I) &=\begin{cases}
        0, &\text{if } I\leq 10000\
        0.1\cdot (I-10000), &\text{if } 10000<I\leq 20000\
        0.1\cdot 10000+0.15\cdot (I-20000), &\text{if } 20000<I
        \end{cases}\[10pt]
        &= \begin{cases}
        0, &\text{if } I\leq 10000\
        0.1\cdot (I-10000), &\text{if } 10000<I\leq 20000\
        1000+0.15\cdot (I-20000), &\text{if } 20000<I
        \end{cases}
    \end{align*}
    We can sketch the graph after representing the graph numerically:
    \begin{table}[H] \centering 
 \begin{tabular}{|*{12}{c|}} 
 \hline 
 $I$ & 0 & 10000 & 15000 & 20000 & 21000 & 27000\ \hline 
 $T(I)$ & 0 & 0 & 500 & 1000 & 1150 & 2050\ \hline
 \end{tabular} 
 \end{table}
Now we are able to sketch the graph of the function $T:$
\begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=30000, ymin=0, ymax=3000,xtick={10000,20000,30000}, scaled ticks=false, tick label style={/pgf/number format/fixed}, xlabel={$I, \$$}, ylabel={$T, \$$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:10000, samples=2, mark=none, color=red, ultra thick] {0}; 
         \addplot[domain=10000:20000, samples=2, mark=none, color=red, ultra thick] {0.1*x-1000)};
         \addplot[domain=20000:30000, samples=2, mark=none, color=red!60!blue, ultra thick] {1000+0.15*(x-20000)};
    \end{axis}
    \end{tikzpicture}
    \end{figure}
    
\end{enumerate}

Exercise 1.1.75

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Exercise 1.1.75

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