Simplifying this, we obtain the following set of PE allocations: (x E ; x W ) 2

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2 Answer for Q (a) ( pts:.5 pts. for the de nition and.5 pts. for its characterization) The de nition of PE is standard. There may be many ways to characterize the set of PE allocations. But whichever way is taken, eventually the following system of equations is obtained. x s;e; = x s ;W (MRS for the state-contingent goods are equalized) x s;e x s;w x s;e + x s;w = 4 (resource constraint for banana at s ) x s;e + x s;w = (resource constraint for banana at s ) Simplifying this, we obtain the following set of PE allocations: (x E ; x W ) f(4; ) ; (4 ( ) ; ( )) j [; ]g : (b) ( pts.) By the rst welfare theorem, the Arrow-ebreu equilibrium allocation must be Pareto-e cient. Hence the equilibrium price ratio p p ; which is equal to MRS x s ;E x = x s ;W s ;E x ; must be s ;W : Thus a banana at s must be twice more expensive as a banana at s : Given this equilibrium price ratio, it is easy to derive the following optimal consumption vectors: x s ; ;E x s ;E = x s ; ;W x s ;W = (; ) ; which of course satisfy the market clearing conditions. If p p (c) ( pts.) For general u i ; we have u i(x s ;i) u i(x s ;i) = p p in equilibrium for i = E; W. = ; then the consumers would obtain a full insurance, i.e. x s ;i x s ;i = for i = E; W: If p p > ; then x s ;i x would be even strictly larger than for i = E; W: s ;i Hence the total consumption of banana at state s is at least as large as the total consumption of banana at state s if p p =. But then the markets would not clear as the total supply of banana is larger at s ; so this is a contradiction. Hence the equilibrium price ratio p p must be strictly less than : (d) (.5 pts.) This is because... The Arrow-ebreu equilibrium allocation is unique in this economy with log utility function. The market is complete with two independent assets for two states. The set of Radner equilibrium allocations coincides with the set of A- equilibrium allocations in a complete market.

3 (e) (.5 pts.) Since this is an economy with one good, there will be no trade in period given any strictly positive price of banana in each state. Let s normalize the price of banana in each state to. In A- equilibrium, consumer E exchanges bananas at s for 5 bananas at s : Given the equivalence between the Radner equilibrium and the A- equilibrium, consumer E must buy 5 units of Asset B and sells units of Asset A (this is the only way as this is a complete market case). Of course consumer W must sell 5 units of Asset B and buy units of Asset A. This asset trading in period is feasible if q A q = B ; where q k is the equilibrium price of asset k = A; B: So one possible Radner equilibrium would be x E ; x W ; p s ; p s ; qa ; B q = ((; ) ; (; ) ; ; ; ; ) 3

4 Answer for Q (a) ( pts: pt for writing down the problem correctly and pt for the conditions) The maximization problem is The optimality condition is max xk u (x ) + u (x ) + 3 u (x 3 ) s:t: p x + p x + p 3 x 3 w k u (x k ) p k = for k = ; ; 3 p x + p x + p 3 x 3 = w (b) (3.5 pts.) e ne x k = x k a and w = w (p + p + p 3 ) a > : Note that the objective function can be transformed into a CES function: px + p x + p x 3 and the budget constraint becomes p x +p x +p 3 x 3 w : So the solution is Hence we have x k (p; w) = a + x k (p; w ) = p k p k p + p + w : p 3 p + p + (w (p + p + p 3 ) a) p 3 It is clearly a normal k (p;w) k(p;w) k (p; u) = = > k > for any k: Note k(p;u) x j (p; w) > (Slutsky equation) for any j 6= k: Hence p B j p + p + p 3 p k C A w p k B p + p + p j p + p + A w B p 3 p + p + p 3 p k So any pair of goods are substitutes, not complements. C A + = p k p + p + p 3 (c) (.5 pts.) Let p k k u (x k ) = be the rst order condition for x k for the cost minimization problem. i erentiating this with respect to p j ; j 6= k, C A w + p j p + p + w p 3 C A 4

5 we k(p;u) u = u same sign for all k 6= : This implies that, for each k(p;u) has the (d) (3 pts:.5 pts each) (Normal good): The rst order condition is k u (x k ) p k = for k = ; ; 3: If increases, then every consumption must decreases (for xed p). Then, as w increases, must decrease and every consumption must increase so that the budget constraint is binding. Hence every good is a normal good. (Substitutes): Suppose k(p;u) < for some j and k 6= j: k (p;u) < for any k 6= j by (c): We also know is not positive because S is negative semi-de nite (cost function is concave). Then this is a contradiction because h k (p; u) cannot decrease for every k as p j increases. Hence it must be the case k(p;u) for every j and k 6= j: That is, any pair of the goods must be substitutes for an additive utility function. 5

6 Answer for Q3. (a) ( points) Since the signal is ` for either quality level, and hence the price p = E[j`; ex()] does not depend on actual quality (but only on buyers beliefs over quality) the worker chooses low quality for any c >. Thus, p` = p h =. From now on assume >. (b) (3 points) Before observing s, buyers believe that quality is high with probability p = F (c ): After observing s = H, they know that quality is high, p h = E[jh; c ] = After observing s = L, Bayes rule implies p Pr(hjH) = : p Pr(hjH) + ( p ) Pr(hjL) {z } = p` = E[j`; c p Pr(`jH) ] = p Pr(`jH) + ( p ) Pr(`jL) = = F (c )( ) F (c )( ) + ( F (c )) = ( )F (c ) F (c ) = F (c ) ; which increases in c. Intuitively, the higher the prior belief of high quality, the higher the posterior belief after signal `, since the market is now attributing this bad signal more to bad luck and less to low quality. (c) (3 points) The bene t of producing high quality is that it increases the probability of a high signal from to, and a high signal raises the price by E[jh; c ] E[j`; c ]. Thus the bene t of a high signal is given by the RHS of (). In equilibrium, this must equal the cost of producing high quality for the threshold type c, the LHS. Since p` increases in c while p h is constant, equal to, the RHS decreases in c. The LHS clearly increases in c. Thus they cross at most once, and hence the equilibrium threshold c is unique. (d) ( points) The RHS of () (E[jh; c ] E[j`; c ]) = ( ( )F (c ) F (c ) ) = F (c ) F (c ) rises in ; intuitively, a rise in increases both the likelihood () that producing high quality results in the h signal, and the informativeness (and hence the value) of this signal p h p`. To restore equality in (), c has to rise, increasing the LHS and decreasing the RHS. Intuitively, when signal quality rises, more types (with higher costs) can be incentivized to produce high quality. 6

7 Answer for Q4. (a) ( point) By backward induction, the chain store will accomodate entry, and hence the entrant enters. (b) ( points) The entrants strategy is clearly a best response to the chain store. As for the chain store, the critical question is whether he is willing to ght entry today (cost u F = ), to secure a monopoly position for the future (bene t u O = ): ( )u F u O, i.e. u F =(u O + u F ) = =. Now assume that entrant-t observes only the outcome, i.e. O,F, or A, of period t (and assume for convenience that the outcome in period t = was O). Also assume that 6= =. (c) (3 points) The entrants strategy is clearly a best response to the chain store. As for the chain store: If last period s outcome was O or F, and the entrant (unexpectedly) enters the condition for the chain store to follow his strategy and ght is as in part (a): ( )u F u O, or u F =(u O + u F ) = =. The chain store must be su ciently patient to "defend his reputation by ghting entry". But if last period s outcome was A, and the entrant enters, the condition for the chain store to follow is strategy and accomodate is the reverse: ( )u F u O, or u F =(u O + u F ) = =. The chain store must be su ciently impatient to "accept the punishment phase rather than ghting to restore his reputation". These conditions cannot both hold since we ruled out the knifeedge case = u F =(u O + u F ) = =. (d) (4 points) For the entrant (after any history) the expected cost of being fought (cost v F = ) must equal the expected bene t of being accomodated (bene t v A = ), i.e. pv F + ( p)v A =, i.e. p = v A =( v F + v A ) = =. The chain store (after entry in period t) must be indi erent between (i) ghting in t and having entrant t + stay out, and (ii) accomodating in t, having entrant t + enter with probability q, and if so ght entry; both plans have entrants t t + stay out. Plan (i) yields utility u F + u O (in periods t and t + ). Plan (ii) yields utilty ( qu F + ( q)u O ). Equating yields u F = q(u F + u O ), i.e. q = u F =((u O + u F )) = =(). 7

8 John Riley Micro comp answers 4 July 7 Answer to question 5 (a) point The firms know that any worker s value is at least the lowest value. Therefore r( α) m( α) = + α =. In a separating PBE every signal is chosen by type. Therefore ( q( α), r( a)) = ( q( α), m( α)) = ( q(),) Also for any zˆ < q( α), the wage offer rz ( ˆ) mα ( ) =. Thus q( α ) is a best response if and only if q( α ) = and so ( q( α), r( α ) = (,). Therefore U ( α ) =. (b) 3 points qx ( ) q( ) arg Max{ u(, q( x), r( x) = r( x) C(, q( x)) = + x } x A( ) q( ) U ( ) = + A( ) Envelope Theorem A ( ) q( ) A ( ) U ( ) = = ( + U( )) A( ) A( ) A( ) d [ A ( ) U ( )] = A ( ) U ( ) + U ( ) A ( ) = A ( )( + ) d Case (i) A( ) / =, A ( ) = / (One student observed that Cq (,) is not defined at =. However C(, ) = for all >. Thus the natural assumption is that the function is continuous so C (,) =. To be complete I should have noted that.) d [ U ( )] = ( + ) = + d / / / / U( ) = + + k. / / 3/ 3 IC mapping K (, u) = u / / 3/ 3 U ( ) = + > U O ( ) 3

9 John Riley Micro comp answers 4 July 7 Case (ii) A( ) = +, A ( ) = Method : d [( + ) U ( )] = ( + ) d Method d [( + ) U ( )] = ( + ) d Then ( ) ( ) ( ) + U = + + IC mapping K (, u) = ( + ) u ( + ) k K ( ) ( ) + U = + + = ( + ) u K (c) points k U ( ) = / U ( ) = + 3 Since U () = it follows that k =. Then Method : U ( ) = ( + ) + + k Since U () = it follows that k =. Then U ( ) = Method : Since U () =, K = 4. Then ( + ) U ( ) = U ( ) = + ( + )

10 John Riley Micro comp answers 4 July 7 (d) point Method U () = U ( ) U ( ) = + 3 Therefore is better for low types And is better for high types Method U ( ) U ( ) = 3 ( + ) = [ 3( + ) ] + This is positive for low types and negative for high types. (e) points The equilibrium level sets of the IC mappings are depicted below. The graph of U ( ) is the level set K (, u) =. The graph of U ( ) is the level set K (, u) =. Consider U ˆ ( ) on the level set depicted K ˆ (, u) = k > U( ) = Max{ U ( ), Uˆ ( )} is incentive compatible. u K (, u) = kˆ K (, u) = K (, u) = ˆ 3

11 John Riley Micro comp answers 4 July 7 Along the equilibrium level set for technology U ( ) = + 3 Along this line K (, U ( )) = ( + )( + ) ( + ) 3 d K (, U ( )) = ( + ) ( + ) = 3( ) d Thus the indifference maps touch at ˆ =. See the figure below. U( ) = Max{ U ( ), U ( )} is incentive compatible This is the Pareto dominant separating PBE u K (, u) = k K (, u) = K (, u) = = ˆ Then high types (( )use technology while low types us technology. (f) point U ( ) = U ( ) so equilibrium payoffs are unchanged. PBE for types below to stay out. Bonus point 4

12 John Riley Micro comp answers 4 July 7 But these types are indifferent between staying out and signaling so low types signaling on [,a] and staying out on [a,] is also a PBE. 5

13 John Riley Micro comp answers 4 July 7 Answer to question 6 (a) points dr Along a level set r = B(, q) = k. Hence the slope is = p(, q). This is increasing in. dq For type, the extra value of choosing ( qr, ) rather then B(, q) B(, qˆ ) p(, x) dx q =. qˆ ( qr ˆ, ˆ), where q> qˆ Is The extra cost is r rˆ The extra benefit is increasing in (this is the SCP). If type ˆ chooses ˆq the extra value of switching to any q B( ˆ, q) B( ˆ, qˆ) = p( ˆ, x) dx r rˆ for all q> qˆ qˆ q > qˆ is lower than the extra cost. For any lower type the demand price is smaller. Thus for all q < ˆ, q B(, q) B(, qˆ) = p(, x) dx< r rˆ for all q> qˆ Thus lower types will either choose qˆ ( qr ˆ, ˆ) or a plan with a lower q. (b) points U( ) = B(, q( )) r( ). Therefore E[()] r = E[ B(, q)] E [ U()] E [ U( )] = U( ) f( ) d = U( α) + U ( )( F( ) d F( ) U( α) U ( )( ) f( ) d f ( ) = + 6

14 John Riley Micro comp answers 4 July 7 Therefore U( α) U ( ) f( ) d h( ) = + U ( ) = U ( α) +E [ ] h( ) U ( ) E[()] r = E[ B(, q)] E [ ] U( α) (.) h( ) We obtain the marginal information rent by appealing to the Envelope Theorem for IC allocations For all x Θ buyer must prefer ( q( ), r( )) to ( qx ( ), rx ( )). Therefore FOC arg Max{ u(, q( x), r( x)) = B(, q( x)) r( x)} x U( ) = B(, q( )) r( ) B (, qx ( )) r( x) p(, qx ( )) q( x) r( x) x = = at x =, i.e. r ( ) = p(, q( )) q ( ) = ( + q( )) q ( ) (.) Appealing to the Envelope Theorem Therefore B U ( ) = (, q( )). B [ r( )] [ B( E = E, q)] E [ (, q)] U( ) h( ) α To maximize revenue choose a mechanism where the participation constraint for the lowest type is binding. Then U ( α ) =. Therefore where E[()] r = E [ R V (, q)] (.3) V B R (, q) B( =, q) (, q) h( ) (.4) (c) points 7

15 John Riley Micro comp answers 4 July 7 p(, q) = q V R (, q) = ( + ) q q q ( ) q q h( ) = + h( ). For the uniform case F( ) = =. Therefore h( ) f( ) R V (, q) ( ) q q = +. MR V (, q) = + q. The level set for marginal revenue is depicted below. Note that (,) is on the level set q V MR (, q) = c < MR(, q) = MR(, q) = c > MR V (, q) = Figure 6 : Level sets for virtual marginal revenue Point wise maximization. V MR (, q) = + q = c = so point wise maximization yields q ( ) = This is increasing. Therefore it is IC and so is maximizing for the designer. (d) points 8

16 John Riley Micro comp answers 4 July 7 Let R( q ) be the cost of a q pack. Then And so Therefore r( ) = R( q( )) r ( ) = R ( q) q ( ) r ( ) R ( q) = q ( ) Appealing to the FOC r ( ) R ( q) = = p(, q( )) = + q( ) q ( ) Since q =, = R ( q) = q q R( q) = q 4 q + k Since ( q(), r ()) = (,) it follows that k =. (e) points With b > h( ) =, b b < < h( ) =, b Thus h( ) is increasing with a discontinuity at = b. The level set for Virtual marginal revenue is depicted below. q V MR (, q) = c < MR V (, q) = 9

17 John Riley Micro comp answers 4 July 7 Implementation as price a q pack R( q ). The optimal q( ) has an upward discontinuity at qb ( ). Types are separated so r( ) = R( q( )) must also have an upward discontinuity. This is depicted below. The level set for type b touches the function R( q ). r Level set for type b r( ) = R( q( )) q ( b) qb ( ) q Figure 6 3 Implementation as a q pack