Find the Gini index for the given Lorenz curve. $L(x)=x^{2}$

Find the Gini index for the given Lorenz curve. $L(x) = x^3$

Symmetry is an important concept in mathematics. In prior units of Core-Plus Mathematics, you examined geometric shapes and graphs of functions in terms of their symmetries. Symmetry also applies to matrices, but only to square matrices. A square matrix is said to be symmetric if it has reflection (or mirror) symmetry about its main diagonal. (Recall that the main diagonal of a square matrix is the diagonal line of entries running from the top-left to the bottom-right corner.) So, a square matrix is symmetric if the numbers in the mirror-image positions, reflected in the main diagonal, are the same. For example, consider the three matrices below. Matrices A and B are symmetric, but matrix C is not symmetric.

$$A = \left[ \begin{array} { l l l l } { 0 } & { 1 } & { 0 } & { 1 } \\ { 1 } & { 0 } & { 1 } & { 1 } \\ { 0 } & { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 0 } & { 0 } \end{array} \right]$$

,

$$B = \left[ \begin{array} { c c c c } { 25 } & { 3 } & { 4 } & { 5 } \\ { 3 } & { 36 } & { 6 } & { 7 } \\ { 4 } & { 6 } & { 9 } & { 8 } \\ { 5 } & { 7 } & { 8 } & { 10 } \end{array} \right]$$

,

$$C = \left[ \begin{array} { l l l l } { 0 } & { 0 } & { 1 } & { 1 } \\ { 1 } & { 0 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 1 } & { 0 } \end{array} \right]$$

. a. Identify two square matrices from this lesson. b. Which of the following matrices are symmetric? For those that are not symmetric, explain why not. i. The pottery matrix, ii. The degree-of-difference matrix, iii. The movie matrix, iv. The loan matrix, v. The nonshooting-performance-statistics matrix, vi. The mutual-friends matrix. c. For those matrices in Part b that are symmetric, what is it about the situations represented by the matrix that causes the matrix to be symmetric? d. Create your own symmetric matrix with four rows and four columns. i. Compare the first row to the first column. Compare the second row to the second column. Do the same for the remaining two rows and columns. ii. Make a conjecture about the corresponding rows and columns of a symmetric matrix. e. Test your conjecture from Part d on the symmetric matrices you identified in Part b.

If the Lorenz curve for the income distribution for a given year is $L(x)=x^p$, use integration to find if the Lorenz curve for this country is a simple formula for the corresponding Gini coefficient.

Suppose a study indicates that the distribution of income for professional baseball players is given by the Lorenz curve $y = L_1(x),$ where $L_{1}(x)=\frac{2}{3} x^{3}+\frac{1}{3} x$ while those of professional football players and basketball players are $y = L_2(x)$ and $y = L_3(x),$ respectively, where $L_{2}(x)=\frac{5}{6} x^{2}+\frac{1}{6} x$ and $L_{3}(x)=\frac{3}{5} x^{4}+\frac{2}{5} x.$

For the consumers' demand function D(q): (a) Find the total amount of money consumers are willing to spend to obtain $q_0$ units of the commodity. (b) Sketch the demand curve and interpret the consumer willingness to spend in part (a) as an area. $D(q)=40 e^{-0.05 q}$ dollars per unit; $q_0 = 10$ units

In a certain state, it is found that the distribution of income for lawyers is given by the Lorenz curve $y=L_{1}(x),$ where $L_{1}(x)=\frac{4}{5} x^{2}+\frac{1}{5} x$ while that of surgeons is given by $y=L_{2}(x),$ where $L_{2}(x)=\frac{5}{8} x^{4}+\frac{3}{8} x.$ Compute the Gini index for each Lorenz curve. Which profession has the more equitable income distribution?

Find the interest earned at $4.15 \%$, compounded continuously, for 4 years for a continuous income stream with a rate of flow of f(t)=1,000-250 t.

Product mix for maximum profit. A firm produces two types of earphones per year: $x$ thousand of type $A$ and $y$ thousand of type $B$. If the revenue and cost equations for the year are (in millions of dollars)

$$\begin{align*} R(x,y)&=2x+3y\\ C(x,y)&=x^2-2xy+2y^2+6x-9y+5 \end{align*}$$

determine how many of each type of earphone should be produced per year to maximize profit. What is the maximum profit?

**Productivity.** A consulting firm for a manufacturing company arrived at the following Cobb–Douglas production function for a particular product: $$ N(x,y)$=50x^{0.08}y^{0.2}$ In this equation, $x$ is the number of units of labor and $y$ is the number of units of capital required to produce $N(x,y)$ units of the product. Each unit of labor costs $\$40$ and each unit of capital costs $\$80$. (A) If $\$400,000$ is budgeted for production of the product, determine how that amount should be allocated to maximize production, and find the maximum production. (B) Find the marginal productivity of money in this case, and estimate the increase in production if an additional $\$50,000$ is budgeted for the production of the product. $$

Money is transferred continuously into an account at the constant rate of $\$ 5,000$ per year. The account earns interest at the annual rate of $5 \%$ compounded continuously. How much will be in the account at the end of 3 years?

The research department of a manufacturing company arrived at the following Cobb-Douglas production function for a particular product: $N(x, y)=10 x^{0.6} y^{0.4}$ In this equation, x is the number of units of labor and y is the number of units of capital required to produce N(x, y) units of the product. Each unit of labor costs $\$ 30$ and each unit of capital costs $\$ 60$. (A) If $\$ 300,000$ is budgeted for production of the product, determine how that amount should be allocated to maximize production, and find the maximum production. (B) Find the marginal productivity of money in this case, and estimate the increase in production if an additional $\$ 80,000$ is budgeted for the production of the product.

In a study conducted by a certain country's Economic Development Board regarding the income distribution of certain segments of the country's workforce, it was found that the Lorenz curve for the distribution of income of college professors is described by the function

$$g(x) = \dfrac{1}{3}x\sqrt{1+8x}$$

Compute the coefficient of inequality of the Lorenz curve.

A Lorenz curve, y = f(x), represents the actual income distribution in the country. In this model, x represents percents of families in the country and y represents percents of total income. The model y = x represents a country in which each family has the same income. The area between these two models, where 0 ≤ x ≤ 100, indicates a country’s “income inequality.” The following lists percents of income y for selected percents of families x in a country. [x: 10 y: 3.35] [x: 20 y: 6.07] [x: 30 y: 9.17] [x: 40 y: 13.39] [x: 50 y: 19.45] [x: 60 y: 28.03] [x: 70 y: 39.77] [x: 80 y: 55.28] [x: 90 y: 75.12] (a) Use a graphing utility to find a quadratic model for the Lorenz curve. (b) Plot the data and graph the model. (c) Graph the model y = x. How does this model compare with the model in part (a)? (d) Use the integration capabilities of a graphing utility to approximate the “income inequality.”