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Properties of the Cobb-Douglas Production Func- tion Suppose iPhones are produced using three inth factors combined in a Cobb- Douglas production function Q : KGLEILCS where K is capital and LU and L5 are unskilled and skilled labour. 1. What assumptions on a, l), and c must be true for this production function to exhibit constant returns to scale? . Given your answer in part a), explain why this production function ex- hibits diminishing returns to an}r single factor (Hint: Use deriVatiVes) . Now assume a + c = l, and b = 0 (Le. no unskilled labour is used in production). Derive the relative factor demand function (%(%)) under the standard HeckscherOhlin assumptions where K/L is a function of 10/1" and :1. Dene the cost share of each factor (share of the total cost of each factor) as CK = 1\";er and c1, = w. Using the assumption and your result from 3., show that these cost shares are a function of a. Are they also a function of relative factor prices? . Now suppose iPhones take some a number of input factors to manufacture. The production function now takes the following form: 62:11:13? i=1 (a) What assumption on the ms must be true for this production to exhibit increasing returns to scale (i.e. increasing each input by some amount results in an increase in the output by a greater factor). Prove it using the production function. $8. Felix and Oscar are playing a simplified version of poker. Each makes an initial bet of 8 dollars. Then each separately draws a card, which may be High or Low with equal probabilities. Each sees his own card but not that of the other. Then Felix decides whether to Pass or to Raise (bet an additional 4 dollars). If he chooses to pass, the two cards are revealed and compared. If the outcomes are different, the one who has the High card collects the whole pot. The pot has 16 dollars, of which the winner himself contrib uted 8, so his winnings are 8 dollars. The loser's payoff is -8 dollars. If the outcomes are the same, the potissplitequallyand each getshis8dollars back (payoff 0). If Felix chooses Raise, then Oscar has to decide whether to Fold (con- cede) or See (match with his own additional 4 dollars). If Oscar chooses Fold, then Felix collects the pot irrespective of the cards. If Oscar chooses See, then the cards are revealed and compared. The procedure is the same as that in the preceding paragraph, but the pot is now bigger. (a) Show the game in extensive form. (Be careful about information sets.) If the game is instead written in the normal form, Felix has four strat- egies: (1) Pass always (PP for short), (2) Raise always (RR), (3) Raise if his own card is High and Pass if it is Low (RP), and (4) the other way round (PR). Similarly, Oscar has four strategies: (1) Fold always (FF), (2) See al- ways (SS), (3) See if his own card is High and Fold if it is Low (SF), and (4) the other way round (FS).U2. Suppose that three risk-neutral bidders are interested in purchasing a Princess Beanie Baby. The bidders (numbered 1 through 3) have valua- tions of $12, $14, and $16, respectively. The bidders will compete in auc- tions as described in parts (a) through (d); in each case, bids can be made in $1 increments at any value from $5 to $25. (a) Which bidder wins an open-outcry English auction? What are the final price paid and the profit to the winning bidder? (b) Which bidder wins a second-price, sealed-bid auction? What are the final price paid and the profit to the winning bidder? Contrast your answer here with that for part (a). What is the cause of the difference in profits in these two cases? (c) In a sealed-bid, first-price auction, all the bidders will bid a positive amount (at least $1) less than their true valuations. What is the likely outcome in this auction? Contrast your answer with those for parts (a) and (b). Does the seller of the Princess Beanie Baby have any clear reason to choose one of these auction mechanisms over the other? (d) Risk-averse bidders would reduce the shading of their bids in part (c); assume, for the purposes of this question, that they do not shade at all. If that were true, what would be the winning price (and profit for the bidder) in part (c)? Does the seller care about which type of auction she chooses? Why