1. Write down the Plarmer's problem, as well as the Lagrangian. (Hint: There should be two constraints because aggregate consumption of each good cannot exceed the economy's endowment.) Label the multipliers $1 and 652. [5 points] 2. What are the rst order conditions of the Planner's problem? (Hint: There should be 6.). [5 points] 3. Find the solution to the Planner's problem and label the quantitim ($131193) and (3%, ypB) [5 points] 4. If we compare the rst order conditions of the Planner's problem to the rst order conditions of a corresponding competitive equilibrium (CE), we can obtain a relationship between (rm, (by, )1, p1, and pg so that the CE and the Planner's problem give us the same solution. This is called decentralizing the Plarmer's problem. Decentralize the Planner's problem. (That is, nd the equilibrium prices and the value of A so that the two problems have the same solution.) [5 points] 5. Can you relate this to the two welfare theorems? [5 points] Question 3: Finding NE [30 points] Consider the following 5 X 5 game between Player 1 (row) and Player 2 (column). The rst element in any square represents the payoff of Player 1 and the second element, the payoff to Player 2. F G H 1 J A 3,5 2,2 3,6 6,1 7,7 B 3,4 3,3 7,2 6,2 3,4 0 2,4 1,3 5,3 3,2 4,2 D 1,5 0,5 1,5 2,3 3,6 E 6,5 5,2 4,4 3,1 2,5 1. Specify the strategies for Player 1 and Player 2, respectively. See if you can nd any strategies for Player 1 that are strictly dominated by another; do the same for Player 2. Eliminate these strictly dominated strategies, and consider the reduced game. Repeat this same step of eliminating strictly dominated strategies in the reduced game. Continue this process until you can reduce the game no further, i.e., no player has a strictly dominated strategy. [10 points] 2. At this point, you should have arrived at a 2 X 2 game. Final all the Nash equilibria for this game, including pure and mixed strategy ones. [10 points] 3. Instead of the simultaneous game, consider transforming the above 2 X 2 game into a sequential one. Player 1 acts in the rst stage given the same two options as in the 2 X 2 game. After that, player 2 acts in the second stage. Player 2 is also endowed with the same two options as in the 2 x 2 game for each decision point following player 1's choice. All nal payoffs are the same as specied before. For example, as in the original 5 X 5 game, player 1 choosing A and player 2 choosing F leads to payoffs of (8, 5). Draw the game tree. Specify the available strategies for each player. Find all the subgame perfect equilibria. [10 points] Question 4: Oligopoly [graded for completion] Consider an industry comprised of three identical rms faced with a linear cost function given by: C(91) = \"1731 for 1' = 1, 2, 3. Let inverse market demand be given by: P0?) = a _ an where Q = 111 + (12 + 13