This problem set tells the story of four friends Alice, Bob, Carol, who are consumers, and Fanny, who owns a firm. Alice, Bob and Carol have preferences over two goods, Dumplings and Eggs. Fanny's firm produces eggs. In what follows you should use mi for i = A, B, C to denote the income of each of the three consumers and p; for j = D, E to denote the prices of each of the two goods. Similarly, d; and e; denote the consumption of dumplings and eggs respectively by consumer i = A, B, C. Fanny uses two (non-fixed) inputs, called 1 and 2, to produce eggs; the price of these inputs is denoted by w; for j = 1, 2. (a) Assume N = 6:1 = (14 points] Bob's preferences are represented by the utility function ub(dB, eb) (N +1) ln(db) + (eb). Use the Lagrange method to find Bob's demand of dumplings and eggs as a function of his income and the prices of the two goods (Note: Try first to find the demand for eggs and take the positive root of the equation you will obtain). What would be Bob's demand for dumplings and eggs if the price of dumplings is 0.1, the price of eggs is PE = 2 and his budget is mb = 9? PD =