7. Consider a 2 x 2 pure exchange economy with two consumers, Andy and Beth, or {A, B}, and two goods {1,2}. Andy has endowment e4 = (1,4) and Beth has endowment e = (4,1). Andy's preferences can be represented by the utility function vy = \\/Z 41245 and Beth's preferences can be represented by the utility function up = min {zg;, zp2}. (a) Find all Walrasian Equilibria. (5 marks) (b) Now we introduce a firm owned equally by the two agents with production set Y ={y eR?|y 2. (5 marks) (c) Find the Walrasian Equilibrium. (Hint: start by finding equilibrium prices and then both agents' demand). (6 marks) (d) Return to the economy without production in part (a). Find the Pareto Set and use this to put this scenario into the form of a bargaining problem: define a pair (S, d) where S is the set of feasible utility pairs and d is the disagreement point (no trade). lllustrate (S,d) on a diagram. (6 marks) (e) Find: (i) the Nash Bargaining Solution and (ii) the Subgame Perfect Equilibrium of the Ultimatum game with Andy as proposer. For each, find both the utilities and allocation. (6 marks) 7. Consider a 2 x 2 pure exchange economy with two consumers, Andy and Beth, or { A, B}, and two goods {1,2}. Andy has endowment e = (1,4) and Beth has endowment eB (4,1). Andy's preferences can be represented by the utility function uy = /T 1742 and Beth's preferences can be represented by the utility function ug = min {z g, 2} (a) Find all Walrasian Equilibria. (5 marks) Answer: The unique Walrasian Equilibrium is p = (1,1), x4 = x5 = (g, g) Students get 2 marks for finding this Walrasian Equilibrium; a third mark for working justifying why it it an equilibrium; while the remaining two marks are for showing that there are no other Walrasian Equilibria. This can either be done by writing out consumer demands as a function of prices and solve for market clearing and/or splitting up into 3 cases: i) p1 > p2 leads to excess demand of good 2; ii) p; 2. (5 marks) S Answer: Students are encouraged to draw a diagram to help them out with the working. If answers are wrong, they can still get partial credit for working. Students should find 0 BL 2 When p; 2, the firm's profit maximising output is to produce nothing leaving us with the same economy as in a) but then ij; >2=p1 > j:d hence no Walrasian Equilibria by our solution to a). Give 3 marks for y (p) and a further 2 marks for showing no Walrasian Equilibrium in those 2 cases. Find the Walrasian Equilibrium. (Hint: start by finding equilibrium prices and then both agents' demand). (6 marks) Answer: By process of elimination using part b) the Walrasian Equilibrium prices must be p = (2,1). Andy's demand is x4 (p) = p1+4p2 p1+4pe :2,up>1 This has solution (u4,up) = (3,2) and the allocation giving this is x4 = (3, 3), -~