International Macro Lecture 8. March 15, 2006

Transcription

1 Inernaional Macro Lecure 8 March 15, 2006

2 Speculaive Aacks Krugman: I Krugman model shows ha counries wih a ied echange rae pursuing epansionary moneary policy will have an aack on reserves, orcing hem o loa, ar beore reserves run ou. Sar wih money demand: M P And uncovered ineres. pariy: E i = i + E = Ae η i

3 Speculaive Aacks Krugman: II The cenral bank s balance shee jus says ha money demand is equal o money supply. In oher words, he amoun o money in he economy is equal o domesic credi plus he value, in domesic currency o oreign reserves: M = C + Addiionally, we assume ha he governmen is epands domesic credi over ime a a consan rae: Ċ C Then we can show ha oreign reserves decline over ime. E = γ

4 Speculaive Aacks Krugman: III Deriving he ime pah o oreign reserves: We can solve or he rae o change in oreign reserves by solving (where omega() is he raio o oreign reserves in high-powered money): C C C M = = C C C C... = = = γ M C M C ω ω γ γ γ = = = 1.

5 Speculaive Aacks Krugman: IV So, evenually, he moneary auhoriy will run ou o oreign reserves. However, beore ha, i will eperience an aack on he currency. Firs, we assume PPP. Thereore, he domesic price level is ied wih respec o he echange rae. Normalizing he oreign price level o 1, we ge: P = E Now, we deine he shadow echange rae ha would eis i he cenral bank sold all reserves and le he currency loa. However, when reserves are gone money demand is equal o domesic credi: M = Taking logs o money demand and subsiuing in, we ge: logc C log E = log A ηi

6 Speculaive Aacks Krugman: V Also, we know ha aer he currency is allowed o loa, given perec price leibiliy, he rae o depreciaion o he echange rae will be equal o he rae o domesic credi growh:. C C = γ = Thereore, rom uncovered ineres pariy:. E E i = i + γ Thus, our shadow echange rae equaion is: log C ( i + ) ln A = log E + η γ

7 Speculaive Aacks Krugman: VI Now, whenever he shadow echange rae is depreciaed relaive o he ied echange rae, selling shor domesic currency unil he cenral bank loas brings an ininie rae o reurn. Thereore, his can no happen. Since he governmen will no loa he currency unil i gives up all is reserves, aer he dae when he shadow echange rae equals he ied echange rae, he governmen can no hold any reserves. Also, beore such ime, speculaors earn a negaive reurn rom selling he domesic currency. Thereore, here will be a speculaive aack on he currency where by (1.) here will be a discree drop in reserves wih all remaining reserves dissappearing, (2.) he money supply will drop discreely, (3.) he echange rae will loa adjusing coninuously saring rom he dae when he shadow echange rae equals he ied echange rae.

8 Speculaive Aacks: Obseld I In he Krugman model, (1.) governmen wasn a sraegic acor and (2.) individual invesors were no sraegic acors. However, when governmen is a sraegic acor and here are many speculaors who are credi consrained, hen, (1.) here is a possibiliy ha even aer he irs dae when an aack is viable, i does no happen because invesors do no coordinae or i o happen even hough i all invesors coordinaed, i would be proiable i.e. muliple equilibria and (2.) wheher or no muliple equilibria eis depends upon he sae o he undamenals (or reserves) o he economy.

9 Speculaive Aacks: Obseld II Suppose here are a coninuum o speculaors wih payo equal o: i here is an unsuccesul aack and: e θ i he aack is succesul. The governmen ges a payo o zero i i does no deend he aack oherwise ges a payo o: v αθ

10 Speculaive Aacks: Obseld III Suppose here are a large inie number o speculaors. I a speculaor aacks, i mus be a ransacions ee wheher or no he aack is succesul. Thus a specualor ges a payo equal o: i here is an unsuccesul aack. However, i he aack is succesul, he invesors ge a benei greaer han : e > 0 The governmen ges a payo o zero i i does no deend he aack oherwise ges a payo o: θ n where hea is sae o undamenals and N is he number o poenial specualors and n is he number o acual ones.

11 Speculaive Aacks: Obseld IV Noe ha i will only be worhwhile or a speculaor o aack i he aack is succesul. Thereore, we can ideniy hree regimes: (1.) θ > N In his case, even i everyone aacks, he governmen will no give up he peg. Thereore, noone will ever aack. (2.) θ <1 In his case, even i one person aacks, he governmen will give up he peg. Thereore, everyone will aack.

12 Speculaive Aacks: Obseld V (3.) N > θ > 1 In his case, i everyone aacks, hen i is raional or any speculaor o aack. However, i noone aacks, hen i is raional or any speculaor o no aack. Thereore, here are muliple equilibria. There are, in ac, wo pure sraegy equilibra, one where everyone aacks and one where noone aacks. In addiion, here is a symmeric equilibrium in mied sraegies where everyone aacks wih he same probabiliy and in equlibrium, he aack is succesul wih a cerain probabiliy (generically no equal o he probabiliy wih which speculaors mi). Noice ha his equilibrium goes away wih a coninuum o speculaors since hen he oucome o he aack can no be sochasic and hus speculaors can no be indieren.

13 Speculaive Aacks Morris & Shin: I We now generalize he Obseld model boh by allowing he coss o he governmen deending o vary wih he sae o undamenals as well as he percenage o people aacking ( α). Speculaors have uiliy: e ( θ ) > 0 The governmen has uiliy: v c( α,θ )

14 Speculaive Aacks Morris & Shin: II We make he ollowing uncional orm and parameer assumpions: (1.) Thea is bounded and ransacions coss o speculaion are posiive (2.)The coss o deense are increasing in he percenage o aackers and decreasing in undamenals: (3.) In he wors sae o undamenals, he cos o deending he currency eceeds he value even i no speculaors aack: (4.) I all speculaors aack, he coss ouweigh he values even in he bes sae: (5.) In he bes sae o undamenals, he loaing echange rae is suicienly close o he pegged level ha i is no worh he ransacions cos o speculae: [ ], 0 θ 0,1 > C C > 0, < 0, α θ C (0,0) > v C (1,1) > v > e (1)

15 Speculaive Aacks Morris & Shin: III Then, as wih he Obseld model, here is a riparie division o undamenals: (1.) I undamenals are suicienly bad, hen here will be no aack: θ s.. C 0, ( θ ) = v θ θ, here is always an aack (2.)The coss o deense are increasing in he percenage o aackers and decreasing in undamenals: θ s.. ( ) θ = e, θ θ, here is no aack (3.) For inbeween saes o he economy, here are always muliple equilibria:

16 Speculaive Aacks Morris & Shin: IV Now we add ha he sae is observed wih noise where every speculaor observes he ruh plus uniormly disribued error: [ θ ε θ + ε ] U, The governmen, which decides wheher or no o deend, moves aer aack decisions are made and hus observes boh he sae as well as he percenage who aack he currency.

17 Speculaive Aacks Morris & Shin: V We will now show ha equilibria are unique wih incomplee inormaion. We solve by backward inducion. Below θ, he governmen will no deend even i no speculaor aacks. Denoe by a( θ ) he maimum percenage o aacking speculaors or which he governmen will sill be willing o deend as a uncion o he sae. Noe ha a ( θ ) = α s.. c( α, θ ) = v so ha a is sricly increasing or hea greaer han θ. Also, deine he se o combinaions o percenage o aacking speculaor as uncion o he sae and an equilibrium sraegy: s θ,π ( )

18 Speculaive Aacks Morris & Shin: VI The relaion, s, can poenially be a correspondence. We can wrie s as: π ( ) s ( θ, π ) = π ( ) Where is he percenage o speculaors who aack given signal. Now we can deine he se o combinaions o percenage o aackers and undamenals such ha he governmen will no deend: A 1 2ε θ + ε θ ε ( π ) = θ s( θ, π ) a( θ ) { } d

19 Speculaive Aacks Morris & Shin: VII We wan o show ha s inersec a only once. In oher words, here eiss a value o hea, θ, such ha he governmen abandons he peg i and only i: θ θ We irs show hree lemmas and hen prove he main heorem: Uiliy is increasing in he aggressiveness o bidding I individuals ollow cuo simple cuo sraegies, uiliy is decreasing in he cuo There is a unique cuo such ha, in any equilibrium, all invesors aack i hey ge a or above he cuo and do no aack oherwise Then we conclude wih a proo o he main resul, ha he equilibrium is unique. Also, we prove ha even in he limi as he variance o noise goes o zero, equilibria remain unique.

20 Speculaive Aacks Morris & Shin: VIII Lemma 1: Proo: I he share o invesors aacking a any given signal level is higher, hen he share o invesors aacking given any signal and any underlying sae o he world is higher which means ha he se under which he governmen abandons he peg is larger: Then he uiliy o a speculaor is given by: ( ) ( ) ( ) ( ) ',, ' π π π π U U ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ', ,, ') (, ) ( π θ θ ε θ θ ε π ε ε π ε ε π u d e d e u A A = = + + ( ) ( ) ( ) ( ) ( ) ( ) ' ',, ' π π π θ π θ π π A A s s

21 Speculaive Aacks Morris & Shin: VIII ( ) Lemma 2: U k, I k is coninuous and sricly decreasing in k where: U ( k I ), k 1 = 0 < k k Proo: Given ha all agens ollow he above equilibrium sraegy, hen we can solve or s: s ε ( θ, I ) = ( θ k) k θ k -ε k -ε θ k + ε θ k + ε

22 Speculaive Aacks Morris & Shin: IX Denoe by ( k) s( k + ψ ( k) ) = a( k ψ ( k) ) ψ + Then he governmen abandons he peg on he inerval: [ 0, k + ψ ( k) ] In which case, he payo uncion or an aacking invesor is given by: k + ψ ( k ) 1 ( ) ( u k, I = ( )) k e θ dθ 2ε k ε e ( k) Using Lebniz rule, since we know ha is sricly decreasing in hea, we need only o show ha phi(k) is weakly decreasing in k. Bu ψ ( k) = ε i k θ ε, -ε < ψ ( k) < ε i k > θ ε

23 Thus, in equilibrium: Speculaive Aacks Toally diereniaing, we ge: Coninuing o solve, we ge: a' s a' ( k + ψ ( k) ) = a( k + ψ ( k) ) Morris & Shin: X ( k + ψ ( k) )( 1+ ψ '( k) ) ( θ ) ( k) = = 1 2 [ 1+ 2εa' ( θ )] ψ '( k) 2ε 2εa' ( θ ) [ 1+ 2εa' ( θ )] ψ ' = < 0 ( k) ψ 2ε ψ ' 2ε ( k)

24 Speculaive Aacks Morris & Shin: XI Thus we have shown he he uiliy o he cuo sraegy is decreasing in he cuo. Moreover, since he uiliy is given by an inegral, he uiliy uncion is diereniable and hus coninuous. This concludes our proo o lemma 2. Lemma 3: There is a unique cuo value o he signal such ha all invesors aack i hey receive a value lower han he cuo and do no aack oherwise.

25 Proo o lemma 3: Speculaive Aacks Morris & Shin: XII Firs noe ha or small enough k, he uiliy o he cuo sraegy is posiive and or large enough k, i is negaive: u ( k I ) > 0, k [ 0, θ ], u( k, I ) < 0, k > θ, k k By coninuiy o he uiliy uncion and he ac ha i is sricly decreasing in k, we know ha (1.) here eiss a level o k such ha he uiliy rom ollowing he cuo sraegy is equal o zero i.e. equal o he uiliy o no aacking and (2.) ha such a level o k is unique. Now we make he ollowing deiniions: = u(, I ) = 0, = in{ π ( ) < 1 }, = in{ π ( ) > 0},

26 We will show ha: Speculaive Aacks Morris & Shin: XIII = = Firs noe ha: This is due o: { 0 < π ( ) < 1} in{ 0 < π ( ) < } sup 1 Now i remains o show he reverse:

27 Speculaive Aacks Morris & Shin: XIV When pi()<1, hen here are a leas some invesors who weakly preer no o aack. Taking he limi, we ge: u (, π ( ) ) 0 limu(, π ( ) ) 0 u(, π ( ) ) 0 Bu: I ( ) > π ( ) So, rom lemma 1, we ge: u ( ) ( ),I u,i 0

28 Speculaive Aacks Morris & Shin: XV A symmeric argumen ges us: Thus, we have lemma 3. The unique sraegy ollowed by invesors is he cuo sraegy a : I ( ) We have igured ou he unique sraegy by invesors and by he governmen. I remains o show ha he equilibrium is unique.

29 Speculaive Aacks Morris & Shin: XVI The equilibrium sraegy or invesors is given by: We know ha: And s is sricly increasing over his range. Moreover, below his range, s is below a and above i, a is below s. This means ha a and s cross precisely once. ε θ ε θ > > + ( ) ( ) + + = , ε θ ε θ ε θ ε ε θ θ I s

30 Speculaive Aacks Two addiional noes: Morris & Shin: XVII There is a proo ha uniqueness o equilibria remains even as epsilon (and hus he variance o he noise) goes o zero. There is a misake in he proo by Morris and Shin in heir original AER paper. The correc proo is a noe in a subsequen AER ediion. Hellwig, Mukherji, and Tsyvinski have a paper which says ha he Morris and Shin resuls on uniqueness are dependen upon wheher or no he cenral bank has a discree or coninuous policy (like seing ineres raes versus jus deending) and wheher or no he signals are public or privae. This paper has a revise and resubmi righ now a AER.

31 Speculaive Aacks: Generaion III Eample: Aghion, Banerjee and Bachea (JET) Can eplain why currency crises covary wih recessions Deb denominaed in oreign currency I speculaors believe here will be an aack, hen he value o deb will go up, making irms closer o insolvency. Thus ineres raes will rise and capial will low ou. In order o resore low ineres raes, cenral bank will le he currency loa bu his raises he value o eernal deb, causing inabiliy o irms o borrow abroad and incompleed projecs (recession). However, i speculaors did no believe here would be an aack, hen none o his will happen muliple equilibria (Eas Asian Currency Crisis).