Assume you are managing the company producing two different goods, x and y. Undertaken market research segmented the customer population into 5 different groups. Each group is identified by specific preference over the bundles of goods consisting of x and y. Preference of representative consumer from customer segment i (i = 1,2,3,4,5) is described by respective utility function Ui(x,y), x 0; y 0. The utility functions Ui(x,y), (i=1,2,3,4,5) look as follows:

1. U1(x,y) = 2(3x+5y)^0.5;
2. U2(x,y) = - 10 / (3x+5y)^2;
3. U3(x,y) = 0.5e^(3x+5y);
4. U4(x,y) = 4(5x+2y)^2;
5. U5(x,y) = (5x+2y / (5x+2y)^0.5) + 1

Your company is a price-taker for both products and provides goods in any amounts demanded by the consumers for given prices. The price for x is given by Px = 15 and price for y equals 10, i.e., Py = 10. Assume that for each segment the firm is aware of the number of consumers Ni and average disposable income Mi representative consumer of type i can spend for purchasing goods x and y. Then, please, get done the following tasks and explain the way you do it:

1) Determine the structure of optimal consumption bundle under given prices for x and y for each type of representative consumer.

2) If disposable incomes are given by M1 = 300: M2 = 180: M3 = 360; M4 = 420; M5 = 540, then what amount of good x and good y will be purchased by each representative consumer i when Px = 8 and Py = 10.

3) Under given prices for goods and disposable incomes of representative consumers Mi (i = 1,2,3,4,5), the size of each consumer segment is given by number of consumers of each type N1 = 3000; N2 = 4800; N3 = 4000; N4 = 2000; N5 = 1200. Then, calculate total quantities of good x and good y that will be supplied by your company to the market.