x. The saver is John Riley 7 December 2016 Econ 401a Final Examination Sketch of answers 1. Choice over time Then Adding,

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1 John Riley 7 December 06 Econ 40a Final Eamination Sketch of answers Choice over time (a) y s, Adding, y ( r) s y s r r y y r r (b) The slope of the life-time budget line is r When r The initial optimum is rises the compensated consumption vector is C The saver is better off with a higher interest rate so the consumer has to be taed to consume at C Give the ta back and the consumer moves from C to Note that the substitution effect and (normal) income effects are reinforcing for commodity but offsetting for commodity C (c) For a borrower the consumer has to be subsidized at Taking the subsidy back reduces demand for both commodities in the normal case Now the two effects are reinforcing for commodity and offsetting for commodity / / y Ma { U( ) y } r r FOC

2 John Riley 7 December 06 p p : ; ( r) ( r) / / ( r) r r r W r r W ( r) W (, ) r r With three periods FOC ( r) ( r) / / / ( r) ( r) 4 r ( r) ( r) ( r) W r ( r) solve for Walrasian Euilibrium with production U( ) ln ln ln (a) U( ) ln ln ln 6ln ln ln ln 6ln U( ) Thus if U( y) U( ) then U( y) U( ) z 80 z, z 6 z (b) z z /4 / U( ) ln(80 z ) ln(6 z ) ln z z / / ln(80 z ) ln(6 z ) ln z ln z FOC U z 80z z 0 Therefore 80 z z and so z 8

3 John Riley 7 December 06 U z 4z z 0 Therefore 4z 4 8 z and so z 7, Also 4() 48 so 48 Profit p z z p z p z with / / p / / 4 4z z p p 0 z 4 / / 4z z p 48 p 0 z For the consumer p p p, p p p (7,,48) so we reuire that ()7 () ()48 This is satisfied So So p p State claims euilibrium prices and asset prices (a) U(, ) / / / U(, ) ( ) ( ) ( ) ( ) U(, ) (b) FOC p p p / / / / / /, p p p / / / For the representative consumer (64,5,9) Therefore

4 John Riley 7 December 06 / / / (64) (5) (9) ie 8p 5p p p p p p (,, ) is a WE price vector (c) The value of asset a pz (,, ) (40,0,) 64 a 8 8 is 5 The value of asset b pz (,, ) (4,5,6) 64 b 8 8 is 5 (d) If you own all of both assets you own the entire endowment (your optimal consumption) If you own assets worth % of the market portfolio you want to consumer % of and you can achieve this by buying % of each asset (e) From the answer to (d), you simply buy a share of the market portfolio ( a mutual fund) 4 Production and Cost After spending the fied cost the net budget is B 8 rz r z 4z 8z B 8 Ma { z z 4z 8z B 8} /4 / Ma { ln z ln z 4z 8z B 8} z 8z 4z 8z B8 From the first euation, z Therefore Therefore B8 z /4 / /4 B8 /4 ( B) z z z ( ) Suppose the firm wants to produce ˆ Choose z ˆB so that ˆ ( Bˆ ) Since ( ) with any smaller budget ˆ is not feasible Thus ˆB is the smallest budget C64 ( ) /4 B is strictly increasing, 4

5 John Riley 7 December 06 Therefore C64 4/ AC 64 4/ MC / 6 At the minimum AC MC and so 4/ 4 64 pmc 8 and so 5 Quantity competition p c 60 (a) profit is U (, ) ( p c) 60 U b 4 and so the best response is By the same argument 5 b 4 Mutual best responses b b 5 and 5 b b 4 Guess 4 b b and solve (,) p c 60 (c) U (, ) ( p c) 60 U ( ) b 4 and so the best response is Appealing to symmetry 5 ( ) 5 ( ) 5 (0,0,0) b b b b b b 4 4 (d) U (, ) ( p c) (5 ) b b 4 U * 6 7 5

6 John Riley 7 December 06 (e) When firm raises its output it understands that form will respond by lowering its output and that this will raise p So the marginal revenue is higher 6 All pay auction F( ) U( ) W( ) B( ) F( ) B( ) Method presented in course Consider a deviation from the best response in which a buyer always bids UD( ) F( ˆ ) B( ˆ ) B() ˆ Note that this is linear with slope His payoff is F() ˆ The graphs of the two functions are depicted It cannot be better to deviate so the two curves have the same slope at This argument holds for all ˆ ˆ So ˆ ˆ U( ) F( ) Alternative method Appeal to the Envelope Theorem as follows: can be written as b B() for some If the buyer bids b B() W( ) Pr{ } F( ) Any bid b Her epected payoff is therefore u(, ) W( )( B( )) For B() to be a best response, this takes on it maimum at U( ) Ma { W( )( B( ))} By the Envelope Theorem it follows that U( ) u(, ) W( ) her win probability is Remark: Note that the class method is actually a proof of the Envelope Theorem for the auction case 6

7 John Riley 7 December 06 (b) Therefore U( ) F( ) U( ) ( ) K The lowest type has a zero probability of winning so U() 0 K 0 Therefore U( ) ( ) K ( ) B( ) ( ) B( ) ( ) ( ) ( )( ( )) ( )( ) U( ) ( ) B( ) ( ) B( ) B( ) 0 (c) 4 Therefore B() 4 (d) U( ) ( ) K U( ) ( ) K K so 8 U( ) ( ) 8 U( ) W( ) B( ) F( ) B( ) ( ) B( ) ( ) B( ) U( ) ( ) 8 7 Auction with multiple units for sale (a) A bit sketchy but OK answer If I win I pay L, the lowest of the other bids This is independent of my bid I want to win if my value is higher than L and lose if my value is lower than L If I bid my value A perfect answer I win if and only if L 0 7

8 John Riley 7 December 06 (i) If I bid b the only time it matters is if the maimum of the other bids, m is between, ie bm If I had bid I would have won and made a profit of m (ii) If I bid b the only time it matters is if the maimum of the other bids, m is between, ie mb If I had bid I would have not won By bidding b my loss is m (b) The argument above is not based on the strategies of the other players so yes (c) If Pr{ } F( ) B() i is strictly increasing I lose to another buyer if his bid is higher than mine ie if his value is higher than mine This probability is Pr{ } F( ) i I win an item unless I lose to all three other buyers so my loss probability is probability is ( ) my win W( ) ( ) (d) My epected payoff is U( ) W( )( B( )) (*) By the argument in (6) U( ) W( ) A buyer with value 0 has a zero probability of winning so U(0) 0 U( ) W( ) d 0 (e) Solve for U() and substitute into (*) (f) In both cases U( ) W( ) d the seller must be as well and the win probability is the same So the buyers are indifferent 0 8