Please help with the following questions

Auction Basics Consider an auction environment that matches the symmetric independent private value-3 spec- ication discussed in class. This means that all of the bidders in the auction know their own value-3 for the item with certainty but they do not know the value held by other bidders. They do know that the values held by the other bidders are independently distributed according to a uniform distribution between 0 and 100. For the questions below, refer to the following specic value examples: # of Bidders Bidder 1 Bidder 2 Bidder 3 Bidder 4 Bidder 5 5 84 45 13 65 72 2 32 74 3 26 78 42 a. Explain the equilibrium bidding strategy for a second price auction. For each bidder in each of the example auctions compute the equilibrium bid in a second-price auction. Identify which bidder would win and what they would pay. b. Explain the equilibrium bidding strategy for a rst price auction (you don't have to give a full derivation). For each bidder in each of the example auctions compute the equilibrium bid in a rstprice auction. Identify which bidder would win and what they would pay. c. Assume that the actual bids placed by the bidders are as given in the table below. Find the winner and the price they would pay in both auction formats. Calculate the efciency achieved. # of Bidders Bidder 1 Bidder 2 Bidder 3 Bidder 4 Bidder 5 5 60 42 18 64 48 2 15 64 3 10 47 48 - - (:1. Looking at the results in parts a and b (ignore c) and from what you have learned in class, if you were conducting an auction in this environment, explain which auction format you would choose and why. Consider the following production function with three inputs: x = f (1, k,n) = 11/3 k1/3 n1/3 where x is the output, I is the labour input, k is the capital input, and n is the land input. a) Set up the cost minimization problem and solve for the first order conditions using the Lagrange Method. Let w be the wage rate, r the rental rate of capital, and v the rental rate of land. b) Using your answer in (a), find how much labour, capital and land would the firm use to produce x outputs, given w, r and v? What are these functions called? c) Using your answer in (b), find the minimum cost it takes to produce x outputs. What is this function called? d) Use your answer in (c) to derive the marginal cost function.3 Stackelberg (Leader-Follower) model . Sequential Move Game unlike simultaneous move game (like Cournot) . The Leader firm moves first followed by the Follower firm. . The Leader firm will have to take into account the consequences of actions taken by the Follower while choosing its own action. . Both firms know each other's costs. . Both firms know the market demand (values of the parameters of the inverse demand function. . There are two firms: call the Firm 1 (leader who moves first) and Firm 2 (follower who moves second) . Output produced by the two firms are given by Q1 and Q2 . Total output in the market Q = Q1 + Q2 . Market inverse demand: P = a - bQ = a - b(Q1 + Q2) . Marginal costs of production: C1, C2 II2 = (P - C2)Q2 = [a - b(Q1 + Q2)]Q2 - C2Q2 = aQ2 -bQ,Q2 -bQ3 - C2Q2 F.O.C. Q, ally = a - 2bQ2 -bQ1 - C2 = 0 Q2 =20 -2 (This is Firm 2's reaction function to Firm 1's quantity choice) Ill = (P - Ci)Q1 = [a - b(Q1 + Q2)]Q1 - CiQ1 = [a - b(Q1 + 40 -9)]Q1 -01Q1 =[8+9 - -C]Q1 F.O.C. all = 4+0 -bQ, - C1 =0 = Q* = 1-20+4 = 0; = 1-6 _21 - 4+201-30 26 2 4b Q* = Q; + Q; = 0-201+02 0+201-302 _ 20-401+202+a+201-302 2b 1b * = 30-20-C Problem 9 Find the P*, II;, II, and II* = II; + II? Problem 10 Compare and contrast the P*, II;, II; and II* = II; + II, when the firms are moving simultaneously (Cournot Oligopoly/Duopoly Model) Problem 11 Find the Qi, Q; and P* when firms share the mar- ginal cost (C1 = C2 = C) Problem 12 Let's say the inverse demand is given by P = 340-7Q and the costs are given by Ci = 10 and C2 = 12. Find the P*, Qi, Q;, Q*, II;, II; and II* = II, +