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:

Assume now that capital is no longer fixed. Use the Lagrange method to calculate Jess's long-run total, average and marginal cost functions when r = 2 and w can take any value. Find her output supply as a function of p and w. Will Jess's firm exit the market for some dumpling price p? If she doesnt exit, what will be her labour demand assuming p = 3? Finally, use again your economic reasoning but no additional calculations to discuss how the transition from the short to the long run will aect the welfare of the three friends.