Mathematics for Management -- Supplementary Electronic Materials

Net Excess Profit & Lorenz Curves

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1. Suppose that \(t\) years from now, one investment will be generating profit at the rate of \(P'_1(t) = 50 + t^2\) hundred GEL per year, while a second investment will be generating profit at the rate of \(P'_2(t) = 200 + 5t\) hundred GEL per year. For \(15\) years the rate of profitability of the second investment exceed that of the first. Compute the net excess profit for these first \(15\) years.

\( 1522.33 \) hundred GEL \(1601.20\) hundred GEL \(1687.50\) hundred GEL \(1931.27\) hundred GEL

2. Suppose that \(t\) years from now, one investment will generate a profit at a rate of \(P'_1(t) = 100 + t^2\) hundred GEL per year, while a second investment will generate a profit at a rate of \(P'_2(t) = 220 + 2t\) hundred GEL per year. Compute the net excess profit in investing in the second plan until the second plan is no longer more profitable than the first plan. (Round to the nearest number.)

\( 975 \) hundred GEL \( 1008 \) hundred GEL \( 1132 \) hundred GEL \( 1251 \) hundred GEL

3. Suppose that when it is \(t\) years old, a particular industrial machine generates revenue at the rate \(R'(t) = 5000 - 20 t^2\) GEL per year and that operating and servicing costs related to the machine accumulate at the rate \(C'(t) = 2000 + 10 t^2\) GEL per year. After \(10\) years the profitability of the machine begins to decline. Compute the net earnings generated by the machine over these first \(10\) years.

\( 10000 \) GEL \( 15000 \) GEL \( 20000 \) GEL \( 25000 \) GEL

4. In a Lorenz diagram for income, the line of equality shows

the most equitable income distribution. how unequally incomes are distributed. how much redistribution occurs. the income distribution if everyone received the same income.

5. The farther away a Lorenz curve for income is from the line of equality, the

more equally wealth is distributed. more equally income is distributed. less equally income is distributed. None of the above.

6. In a country, the Lorenz curve is given by \(L(x) = 3x - 2 + 2(1-x)^{3/2}\). Find the Gini Index.

\( 0.2 \) \(0.4 \) \( 0.5 \) \( 0.7 \)

7. In a country, the Lorenz curve is given by \(L(x) = \frac{5}{6} x^2 + \frac{1}{6} x\). Find the Gini Index.

\( \frac{5}{18} \) \( \frac{5}{9} \) \( \frac{10}{18} \) \( \frac{10}{9} \)

8. If \(S\) is the area below the Lorenz curve, then, by re-interpreting the formula we gave in the lecture, the Gini Index \(GI\) is calculated using the following equation:

\( GI = 2S - 1 \) \( GI = 2S + 1 \) \( GI = 1 - 2S \) \( GI = 1 + 2S \)

9. A Gini Index closer to zero reveals

a more unequal distribution and a Lorenz curve closer to the diagonal line. a more equal distribution and a Lorenz curve closer to the diagonal line. a more unequal distribution and a Lorenz curve further from the diagonal line. a more equal distribution and a Lorenz curve further from the diagonal line.

10. Economist N calculates income inequality based on the Gini Index and the incomes of all individuals in the society. Economist M calculates income inequality based on the Gini Index and the incomes of all families in the society (simply adds up all family members' incomes before constructing the Lorenz curve). Which of the following is true?

N's Gini Index is likely to be smaller than M's. M's Gini Index is likely to be smaller than N's. The two Gini Index will be the same. None of the other statements is true.

Your grade is: __ of 10

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Solution: 1c; 2b; 3c; 4a; 5c; 6b; 7a; 8c; 9b; 10b

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.