ECO102 Tutorial 01

ECO102 - Week 01.pdf
#tk

• Spam tutorials and lecture questions
• • Total population is $10000$
  • Population less than $15$ is $2000$ or $\frac{2000}{10000}=0.2\cdot 100=20.0\mathrm{\%}$
  • Population greater than $15$ is $8000$ so $80\mathrm{\%}$
  • Employed is $3000$ as $30\mathrm{\%}$
  • Unemployed are $2000$ as $20\mathrm{\%}$
  • a)
    • Employment Rate
  • b)
    • Unemployment Rate
  • c)
    • Participation Rate
• • (ECO102 - Week 01, p.3)

  • A Human Resources analyst collects data showing the hourly wages for the 10 auditors in the Accounting Department. Hourly Wage ($/hour) 10 15 20 30 60 Number of Auditors 4 2 2 1 1 a. The average income is this population is equal to $x_1$ dollars. b. The median income in this population is equal to $x_2$ dollars. c. The range of incomes in this population is equal to $x_3$ dollars. d. The RP10 Ratio for this population is equal to $x_3$ times. e. The RP20 Ratio for this population is equal to $x_4$ times. f. Provide a Lorenz Curve Diagram based on Income Quintiles and compute the Gini Coefficient for this Lorenz Curve.

  • $$\begin{array}{cccccc}\text{HW}& 10& 15& 20& 30& 60\\ \text{<a class="tag" onclick="toggleTagSearch(this)" data-content="\#A">\#A</a>}& 4& 2& 2& 1& 1\end{array}$$
  • Calculate the mean
    • $\mathrm{E}[X]=\frac{(4)(10)+(2)(15)+(2)(20)+30+60}{4+2+2+1+1}=20.0$
    • Mean
  • Median
    • Order data
    • $\begin{array}{cccccccccc}10& 10& 10& 10& \underbrace{15}& \underbrace{15}& 20& 20& 30& 60\\ & & & & \text{Median}& \text{Median}\end{array}$
    • Median is $\frac{15+15}{2}=15$
    • Median
  • Mode
    • $10$
    • Mode
  • Range of data
    • $60-10=50$
    • Range in Statistics
  • Find what the RP10 and RP20 are:
    • $$\begin{array}{cccccccccc}10& 10& 10& 10& 15& 15& 20& 20& 30& 60\end{array}$$
    • ${90}^{\text{th}}$ percentile:
      • $\frac{30+60}{2}=45.0$
    • ${10}^{\text{th}}$ percentile:
      • $\frac{10+10}{2}=10.0$
    • $R{P}_{10}=\frac{{90}^{\text{th}}}{{10}^{\text{th}}}$
    • $R{P}_{10}=\frac{45}{10}=4.5$
    • The ${90}^{\text{th}}$ percentile earns $4.5$ times more than the bottom $10$ percent.
    • $R{P}_{20}=\frac{{80}^{\text{th}}}{{20}^{\text{th}}}$
    • ${80}^{\text{th}}$ percentile:
      • $\frac{20+30}{2}=25.0$
    • ${20}^{\text{th}}$ percentile:
      • $10$
    • $R{P}_{20}=\frac{25}{10}=2.5$
    • The ${80}^{\text{th}}$ percentile earns $2.5$ times more than the bottom $20$ percent.
    • Percentile
  • In general to find Percentile:
    • $\text{Percentile}=\mathrm{\%}\text{ trying to find}\times (\underset{\text{Number of data points}}{\underbrace{n}}+1)$
  • Find Gini Coefficient
    • $$\begin{array}{cccccccccc}10& 10& 10& 10& 15& 15& 20& 20& 30& 60\end{array}$$
    • Break it into bins
    • $$\begin{array}{cccccc}\text{Class Interval:}& \text{People:}& \text{\% of People:}& \text{Cumulative Income:}& \text{Relative Income (\% of Income):}& \text{Cumulative Relative Income:}\\ [1,2]& 2& 20& 20& 0.1& 0.1\\ [3,4]& 2& 40& 20& 0.1& 0.2\\ [5,6]& 2& 60& 30& 0.15& 0.35\\ [7,8]& 2& 80& 40& 0.2& 0.55\\ [9,10]& \underset{―}{2}& 100& \underset{―}{90}& 0.45& 1\\ & 10& & 200\end{array}$$
    • Create a Lorenz Curve

      \begin{document}
        \begin{tikzpicture}[scale=0.08]
      	% Define the areas for the Gini Coefficient calculation
      	% Area B: The region under the Lorenz Curve
      	\fill[blue!10] (0,0) -- (20,10) -- (40,20) -- (60,35) -- (80,55) -- (100,100) -- (100,0) -- (0,0) -- cycle;
      	
      	% Area A: The region between the Line of Equality and the Lorenz Curve
      	\fill[orange!20] (0,0) -- (100,100) -- (80,55) -- (60,35) -- (40,20) -- (20,10) -- cycle;
      
      	% Draw the axes with labels
      	\draw[thick, ->] (0,0) -- (115,0) node[right] {Cumulative \% of People};
      	\draw[thick, ->] (0,0) -- (0,115) node[above] {Cumulative \% of Income};
      
      	% Line of Perfect Equality (Diagonal)
      	\draw[thick, dashed, gray] (0,0) -- (100,100);
      	\node[rotate=45, font=\small] at (50,56) {Line of Equality};
      
      	% Dashed projection lines from axes to the plotted points
      	\draw[thin, dashed, gray!60] (20,0) |- (0,10);
      	\draw[thin, dashed, gray!60] (40,0) |- (0,20);
      	\draw[thin, dashed, gray!60] (60,0) |- (0,35);
      	\draw[thin, dashed, gray!60] (80,0) |- (0,55);
      	\draw[thin, dashed, gray!60] (100,0) |- (0,100);
      
      	% Plot the Lorenz Curve
      	\draw[blue, thick, mark=*] plot coordinates {
      	  (0,0) (20,10) (40,20) (60,35) (80,55) (100,100)
      	};
      
      	% Labelling the axis ticks
      	\foreach \x in {20,40,60,80,100} {
      		\draw (\x, 1.5) -- (\x, -1.5) node[below, font=\tiny] {\x\%};
      	}
      	\foreach \y in {10,20,35,55,100} {
      		\draw (1.5, \y) -- (-1.5, \y) node[left, font=\tiny] {\y\%};
      	}
      
      	% Adding labels for Area A and Area B
      	\node[font=\Large\bfseries] at (45, 35) {A};
      	\node[font=\Large\bfseries] at (75, 20) {B};
      
      	% Title of the diagram
      	\node[font=\bfseries] at (50, 125) {Lorenz Curve and Gini Coefficient Analysis};
      
      	% Legend at the bottom
      	\begin{scope}[shift={(10,-25)}]
      	  \draw[blue, thick] (0,0) -- (10,0) node[right, black, font=\small] {Lorenz Curve};
      	  \draw[thick, dashed, gray] (50,0) -- (60,0) node[right, black, font=\small] {Equality Line};
      	\end{scope}
      
        \end{tikzpicture}
      \end{document}
      

    • Graph $\mathrm{\%}\text{ of people}\text{ vs.}\text{ Cumulative Relative Income}$.
    • Gini Coefficient
    • Area under the line of equality:
      • ${\int }_0^{100}x\text{differential}x=5000.0$
    • Area under the Lorenz Curve:
      • $100+200+100+150+400+700+200+1100+450=3400.0$
    • Shaded area: $5000-3400=1600.0$
    • $\text{Gini}=\frac{1600}{5000}=0.32$
• Frictional Unemployment is normal turnover
• Structural Unemployment is workers becoming obsolete