Solve all questions,thanx.

Problem 3. Multivariate constrained maximization. (19 points) Consider the following maxi- mization problem: at. per + pyy = M. with 0 0,y > 0. Does your solution for a* and y" satisfies these constraints? What assumptions you need to make about pr, py and M so that a* > 0 and y* > 0? (1 point) 5. Write down the bordered Hessian. Compute the determinant of this 3x3 matrix and check that it is positive (this is the condition that you need to check for a constrained maximun) (3 points) 6. As a comparative statics exercise, compute the change in a* as pr wries. In order to do so, use directly the expressions that you obtained in point 3, and differentiate r* with respect to p. Does your result make sense? That is, what happens to the quantity of good r* consumed as the price of good a increases? (2 points) 7. Similarly, compute the change in a* as py varies. Does this result make sense? What happens to the quantity of good I* consumed as the price of good y increases? (2 points) 8. Finally, compute the change in c* as M varies. Does this result make sense? What happens to the quantity of good * consumed as the total income M increases? (2 points) 9. We have so far looked at the effect of changes in pr,py; and M on the quantities of goods consumed. We now want to look at the effects on the utility of the consumer at the optimum. Use the envelope theorem to calculate Ou(a* (pr.py. M).y* (pr.py; M))/Opr. What happens to utility at the optimum as the price of good a increases? Is this result surprising? (2 points) 10. Use the envelope theorem to calculate Ou(* (pr,py; M).y* (pr.py; M))/OM. What happens to utility at the optimum as total income M increases? Is this result surprising? (2 points)One day you lose your wallet. In it, you had 500 and some valuables that others cannot use, such as a few old photos. It will cost you another 500 to get new copies of the photos and replace the other valuables. Consequently, the wallet is worth 1,000 to you. Fortunately, someone finds your wallet. She opens it and sees that it contains 500. She thinks that if she keeps the money and throws the wallet away, she will get 500. However, if she returns it to you she might get a reward. After all, it is worth 1,000 to you. Suppose you give her either 600 in reward or nothing. We can represent this game as in Figure E.7.2. Figure E.7.2 Finder Keep Return wallet wallet Owner Rewar Do not 60 0 1,000 a) What is the name of the method used to find the subgame perfect equilibrium? b) Which is the subgame perfect equilibrium in Figure E. 7.2? c) Is the equilibrium efficient or not? Why or why not? d) Can you think of a way to change the structure of the game, such that a better equilibrium will arise?Problem 2. Multivariate unconstrained maximization. (13 points) Consider the following maxi- mization problem: max /(x, 7:a,b) = ax' - r + by -y . 1. Write down the first order conditions for this problem with respect to r and y (notice that a and b are parameters. you do not need to maximize with respect to them). (1 point) 2. Solve explicitely for r* and y* that satisfy the first order conditions. (1 point) 3. Compute the second order conditions. Under what conditions for a and b is the stationary point that you found in point 2 a maximum? (2 points) 4. Astime that the conditions for a and b that you found in point 3 are met. As a comparative static exercise, compute the change in y* as a varies. In other words, compute dy* / da. Compute it both directly using the solution that you obtained in point 2 and using the general method presented in class that makes use of the determinant of the Hessian. The two results should coincide! (3 points) 5. We are interested in how the value function f(x*(a. b);y*(a.b) ) varies as a varies. We do it two ways. First. plug in x*(a, b) and y*(a, b) from point 2 into f and then take the derivative of f(x*(a. b);y* (a. b)) with respect to a. Second, use the envelope theorem. You should get the sameExercise 7.1.1 For a game (in the game theoretic sense), we need to specify the players. What else needs to be specified? What is the difference between a normal-form game and an extensive-form game? Define in words what a dominant strategy is. What is a payoff-matrix? 7.2 Games on Normal Form Exercise 7.2.1 Two individuals, A and B, who like each other, have arranged a date. They will meet either at a pop concert or at a techno party. However, they have not decided on which of the two. A prefers techno whereas B prefers pop. However, they both prefer being at the same event as the other to going alone to the pop concert or to the techno party. Suppose they cannot communicate, and therefore must decide separately. Then the game can be represented as in Figure E. 7.1. The worst outcome is that they end up alone at their least preferred event. The best outcome for A is that they both go to the techno party, but that is only the second best outcome for B. The best outcome for B (and the second best for A) is that they both go to the pop concert. Figure E.7.1 B Techno Pop Techno 10, 9 2, 2 A Pop 0, 0 9, 10 a) What is a Nash equilibrium? Give a definition in words. b) Find all Nash equilibria in the game