HELP me solve the following questions.

1. This question is about some basic properties of projection operators. Remember from class that if W is a subspace of R" which has the orthonormal basis {1'51\" . . ,ii'P}, then the (orthogonal) projection onto W is given by multiplication by the matrix P = U UT, where U = [t'il . . . 113,] is the n x p matrix whose columns are the basis vectors of W. Also remember that UTU = I, and that any vector :i." can be written in exactly one way in the form =t+5 oemsewi. By the way, be careful of the terminology here. The matrix P of an orthogonal projection is not generally an orthogonal matrix. Remember that an orthogonal matrix S is one such that S\"1 = ST. In particular, it must be invertible. The only projection which is invertible is I. (Don't blame me; I didn't invent the terminology!) (a) Show that P2 = P and PT = P. (b) Dene Q = I P. Show that Q2 = Q, QT = Q, and P0 = QP = 0, the zero matrix. (c) Explain why Col(P) = Coi(U). (For this, remember that the column space of a matrix A is the same as the range of the linear transformation whose matrix is A. That is, the vectors in Col(A) are precisely the vectors of the form A5.) (d) Show that if a e Col(P) and 5' 6 (301(0), then a. i; = o. (Hint: Remember that a. 5' = 5T3.) (e) Explain why :i.' = Pf + inf. This means that 001(0) = Col(P)-". (Remark: It is true that if P is any n x 11. matrix such that P2 = P and PT = P, then P is the matrix of an orthogonal projection. If all you have is the P2 = P, then P is still a projection, but it doesn't have to be an orthogonal projection. That is, instead mapping vectors onto a subsPace in a direction perpendicular to the subspace, it will instead map in some oblique direction.) (1') Let A be any n x p matrix with linearly independent columns. The p x p matrix ATA must then be invertible. (You don't need to explain why this is true, but you should at least think about it and try to gure out a reason why it's true.) Dene R = A(ATA)'1AT. Show that 32 = R and RT = R. (Remark: In light of the previous remark, then, R is an orthogonal projection. In fact, it's the projection onto the column space of A. The difference here is that while the columns of A are a basis for the column space, they aren't necessarily an orthonormal basis.) (3) Show that S = I - 2P is an orthogonal matrix. (Not an orthogonal projection!). 2) Suppose there are two firms A and B that sell an identical product (consumers view them as perfect substitutes). Total demand for the product is Q" - 100 - 1 P. where Q) is measured in 1000s. The firms have the same costs: CA= 500+20QA + Qx (Cahas the same form with Q:). Both firms know all of this information. First, let's look at the Coumot/Nash equilibrium: a) Calculate the marginal revenue of firm A, and set MRA=MCA to find firm A's optimal production. Remember, Q' = On+0s. This will be a partial derivative: you're taking the derivative of total revenue with respect to Qu. so treat Qs as a constant. The profit maximizing level of On will be a function of On - this is A's "best response function." b) Repeat these calculations for firm B. Use these two best response functions to find the Cournot equilibrium in this market: the quantity each firm will produce, the price of the good, and each firm's profit. Again, decimal values for ( are OK since Q is in 1000s. Now suppose that the two firms collude: they form a cartel and cooperatively set their price and quantity to maximize joint profits, splitting production and profit equally. c) First, calculate the marginal revenue and marginal cost curves for the cartel together. MR is based on total market demand for the good, and is with respect to the total quantity produced by the two firms. Similarly, the cartel's costs are the sum of each firm's cost, assuming each firm produces half of the cartel's output: C - CA + Ca. bur QA - On - 10*. MC will be different than in 2a. d) Use these to calculate the total level of production, the amount produced by each firm, the price of the good, and the profits of the firm. Compare these results to the Cournot values from 2b. e) Take the quantity firm B produces when the firms collude, and plug it into fin A's best response function from 2b. What value do you get? Explain the economics of this value: what does it imply about the stability of the cartel? 3) This problem is a game theory version of the same ideas at work in 2c. Suppose that two firms sell identical output. Each firm has two possible actions: it can produce a low level of output, or a high level of output. If both firms produce a low level of output, a high price results, and each firm gets $5 million in profits. If both firms produce a high level of output, a lower price results, and each firm only earns $3 million. If one firm produces low output but the other firm produces high output, the firm producing more output will get a large benefit from the other firm restricting quantity and driving up the price: the high-quantity firm will earn $6 million, while the low-quantity firm will earn only $2 million. These payoffs are a simplified version of the Cournot story in question 2.Markets with asymmetric information can be modeled as games of incomplete Information, resulting in Bayesian Nash equilibrium outcomes with inefficient trade outcomes. Harsanyi's purification theorem suggests that mixed-strategy equilibria in games of complete information can be thought of as representing pure-strategy Bayesian Nash equilibria of games with heterogeneous players. Exercises 12.1 Chicken Revisited: Consider the game of chicken in Section 12.2.1 with the parameters R = 8. H = 16, and L = 0 as described there. A preacher, who knows some game theory, decides to use this model to claim that moving to a society in which all parents are lenient will have detrimental effects on the behavior of teenagers. Does equilibrium analysis support this claim? What if R =8, H = 0, and L = 16? 12.2 Cournot Revisited: Consider the Cournot duopoly model in which two firms, I and 2, simultaneously choose the quantities they supply, q, and 92. The price each will face is determined by the market demand function p(91. 92) = a - b(91 + 92). Each firm has a probability & of having a marginal unit cost of cr and a probability 1 - u of having a marginal unit cost of CH. These probabilities are common knowledge, but the true type is revealed only to each firm individually. Solve for the Bayesian Nash equilibrium. 12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong ($) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that army's general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s