Alice (A) and Bob (B) are the only people living in a village. They consume only two goods, cider (C) and dumplings (D). Anne has an initial endowment of 10 bottles of cider and 30 dumplings. Bob has an initial endowment of 50 bottles of cider and 50 dumplings. Alice's utility function is uA(CA,DA) = 9ln(CA) + 10(DA), where CAand DArepresent her consumption of cider and dumplings, respectively. Bob's utility function is uB(CB,DB) = CBxDB, where CBand DBdenote his consumption of cider and dumplings, respectively.

1. a)Find the competitive equilibrium, i.e. the price ratio, of this exchange economy and the resulting equilibrium allocation.
2. b)Find the expression of the contract curve for this economy and use your answer to check that the equilibrium allocation you found in (b) is indeed Pareto optimal.
3. c)Find Alice's and Bob's offer curves (DAas a function of px/pyfor Alice and DBas a function of px/py) and use them to show your answer to (b) is correct.
4. d)Draw this economy in an Edgeworth box with cider in the horizontal axis and Alice in the lower left corner. Include the equilibrium allocation you found in (a), the contract curve you found in (b) and the offer curves you found in (c).