This problem set tells the story of three friends Mary, Jack, who are consumers, and Jess, who owns a firm. Mary, and Jack have preferences over two goods, Dumplings and Free time. Each of them have T_i available hours each day with i = A; B. Jsss's firm produces dumplings using two inputs, Labour and Kapital with respective prices w and r: Denote the market price of dumplings as p:

1. [12 points] Mary's preferences over dumplings and free time are represented by the utility function u_A(D_A; F_A) = D_A + 2 lnF_A. She has no other income than the one she can obtain by selling part of her available time T_i as labour to Jess's company. Find the expression for Mary's budget constraint, with use of income on the one hand and source of income on the other. Use the Lagrange method to calculate her demand for dumplings and her labour supply as a function of T_A; p and w. Find Mary's optimal consumption of dumplings and labour supply for the case (p; w) = (1; 1) and T_A = 16.