E-Companion: Mathematical Proofs
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- Aldous Rogers
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1 E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth ttegy et nd ot uncton e dentcl co etle In ode to etlh the etence o ue-ttegy ymmetc equlum y udeneg nd Tole 99 t uce to how tht Π unmodl unde the condton Gven o ny Π tctly concve n nd ttn unque otmum t ES y uttuton Π cn e wtten uncton o only Π Π Π The t-ode devtve o Π wth eect to ] [ w Π ettng T Π we hve TS n T d d n TS d dn d d T T d d n n d dn d d T d d n d n d n / whch negtve n / decee n nd / d d
2 The t-ode devtve o T wth eect to chnge gn t mot once om otve to negtve Gven T > ence ethe T h unque oot t whch Π ttn otmum o the ot uncton monotone In ethe ce the uncton unmodl on the comct ttegy et The me nly le to etle y ymmety o tht unde the me condton etle ot uncton unmodl gven etle decon Pt DS Sytem et the ttegy et o ech etle n DS e { : [ w M ] [ ]} whee M lge otve nume y ml gument tht n the oo o t t uce to how tht Π unmodl unde the condton Gven Π concve n y uttuton we cn wte nd ttn the otmum t EST Π uncton o only Π Π Π It t-ode devtve wth eect to w [ ] Π ettng T we hve T n TS T n d T d I d / d then t-ode devtve o T chnge gn t mot once om otve to negtve h uot ] [ T / > nce > nd w T [ ] w [ ] o T coe eo then t h unque oot t whch etle ot uncton ttn the otmum; othewe the ot uncton monotone In ethe ce etle ot uncton unmodl etle ot uncton cn e nlyed y ymmety nd t unmodl unde the me condton ence the clm eoe contnung wth the oo we eent ome eult tht we wll lte ee to We dened nd n the mn tet whch cn e wtten :
3 d d Λ Λ d We dene nd o In emm E nd E we eent nd ove the oete o nd eectvely emm E h the ollowng oete: Gven [] ncee n on Gven > t decee n wth Poo o emm E Pt Gven [] o d o ] > nd > then tctly ncee n on It cn e veed tht nd So o Pt Gven d] d Snce > d nd d d > Theeoe decee n o > nd d nd d It ey to vey tht emm E: h the ollowng oete: 3
4 4 > concve nd ncee n on ; nd Λ > concve nd ncee n 3 lm Λ nd ] [lm lm 4 k ncee n on k Poo o emm E Pt > G whee d G et > d G g ence G tctly ncee nd tctly concve n on G nd d G G nd then We hve thu hown tht concve nd nceng n on IS Λ Λ > nce > IS Pt y denton IS Λ It t-ode devtve wth eect to IS Λ Λ d d Λ Λ d d d Λ Λ d d d d ξ > d ξ The outh equlty ollow y Intemedte Vlue Theoem wth ξ
5 The lt nequlty ollow nce ξ nd > ξ o ] So IS nd hence tctly ncee n 3 d thu concve nd nceng n Pt 3 lm lm Λ d lm { Λ d d} { Λ Λ } lm lm lm Λ { d} lm { } d lm [lm ] ence lm Pt 4 ~ k dene k k ~ k It t-ode devtve wth eect to : k y Pt Snce k then ~ k > nd hence k ncee n o To chctee the ymmetc equl n DS nd DS [] [] we dene: M M nd [ Λ ] E E In emm E3 we chctee ome oete o M emm E3: o ] nd [] M nd M nd M ncee n then: e concve wth unque oot n ; 5
6 et the unque oot o M nd M M M nd Poo o emm E3 Pt e nd eectvely then y denton o M n E t t-ode devtve wth eect to dm d concve n y emm E decee n ncee n nd ncee n y umton So dm decee n nd hence M concve d M > nce > y umton M y contnuty thee et oot uch tht M The unquene o the oot gunteed y the concvty o M y denton o M dm d n E t t-ode devtve wth eect to decee n ncee n nd ncee n y umton So dm M d decee n nd hence M concve n > nce > y umton M y contnuty thee et t whch M The unquene o the oot gunteed y the concvty o M Pt M M > nd y Pt the unque oot o M then ollow y the concvty o M M M > nd y Pt the unque oot o M then ollow y the concvty o M We net contnue wth the oo o the mn eult Poo o Pooton 6
7 Pt et e the ymmetc equlum n DS Gven t the et ttegy o m to et y t-ode condton wth nd we cn get equton 8 nd 9 Suttutng 9 nto 8 nd mlyng we cn how tht te [ Λ ] whch y denton E gve M I monotone ncee then y emm E3 the unque oot to M Suoe tht etle chooe I the condton tted n emm t hold then etle ot uncton unmodl Snce t te the t-ode condton the otml decon o m y ymmety mut olve the et o the et-eone uncton y oth etle nd hence the equlum Pt et e the ymmetc equlum n DS Gven t otml o m to et y t-ode condton wth nd we get nd Suttutng nto nd mlyng we cn how whch y denton E gve M I ncee n then y emm E3 the unque oot to M te n Suoe etle chooe dented ove I the condton tted n emm t hold then etle ot uncton unmodl gven etle decon Snce the decon vecto te t-ode condton t the otml oluton o m y ymmety t the oluton to the et o et-eone uncton y oth etle nd the equlum o the emnng oo we ume tht thee et ue-ttegy h equlum n ce nd tock n oth DS nd DS ncee n nd > We wll lo ue the ct tht nd [] o Poo o Pooton In DS the unque oot to M y Imlct uncton Theoem we hve dened n E 7
8 8 M d d y emm E3 e M y emm E ncee n The t-ode devtve o wth eect to d So the ove devtve otve nd ncee n y The eected le cn e wtten [ ] S The t-ode devtve o nd S wth eect to e d d d d nd d d d ds ncee n nd ncee n nd o nd S ncee n Poo o Pooton 3 In DS the unque oot to M dened n E y Imlct uncton Theoem we hve ] [ M d d M y emm E3 nd ncee n y emm E It h een hown tht decee n et we how tht decee n when ucently lge d d d d decee n
9 When we let ; othewe thee mut et uch tht nd o nd > othewe et u{ : } then o o ll nd hence ncee n Ung 8 the equlum ce cn e wtten [ Λ ] It t-ode devtve wth eect to d { G d d } d ncee n o ncee n So ncee n Ung 8 the eected equlum le cn e wtten S [ Λ ] [ Λ ] It t-ode devtve wth eect to ds G d > [] nd d d I M M G M then 3 [ ] G G { G } Then [ ] { G } On the othe hnd > [] nd ; nd > 9
10 I then nd hence S ncee n Poo o Pooton 4 We t conde DS [] te M whee M The t-ode devtve o M M dened n E wth eect to M nd hence M decee n whch mle tht the unque oot to M decee n w It t-ode devtve wth eect to w Snce decee n decee n Totl tock q cn e wtten q q > nce > y umton nd decee n q then ncee n Totl le S cn e wtten S S G G
11 > > G emm E decee n o S ncee n whee the econd nequlty ollow om In DS [] te M whee M dened n E The t-ode devtve o M wth eect to M nd hence M / Snce M decee n o nd the unque oot to M decee n The oo to how decee nd q nd S ncee n e the me thoe o the coeondng eult n DS ; we omt the detl Poo o Pooton 5 y emm E decee n > Then y 9 ncee n Wth eogenou nd dentcl etl ce the tock nd le t one etle e eectvely q nd S ncee n o q ncee n S ncee n y emm E y the denng equton nd hence the tock nd le n DS when e dentcl to thoe n DS Poo o Pooton 6 Pt y the denton n [] g U ncee n o tht te whee g U conve concve n o g U concve g U > nd g U g U then h unque oot nd thee unque oluton to [] The oo to how tht [] h unque oluton ml y denng the oot o g Λ Pt In DS the unque oot to M dened n E
12 y emm E Λ y denton g M g [ ] nd [] Snce g U g nd M dmt unque oot g M g mle tht et ~ w Snce ncee n ~ ncee n ~ ~ nd ~ mle tht Tht nd cn e veed y comng the coeondng denng equton Pt 3 lyng Imlct uncton Theoem to d d g U we hve u [ ] g U y emm E y Pt g u concve nd dmt unque oot g u g u > nd g u then nd hence ncee n y denton [] It t-ode devtve wth eect to d d G d n The oo to how tht ncee n ml Pt 4 ollow y comng [] wth - Tht ncee n mle tht ncee Poo o Pooton 7 Pt y emm E [] y the denng equton n 9 nd o eogenou nd dentcl etl ce eult tock nd le e hghe n DS thn n DS Pt Wth endogenou etl ce n DS nd DS nd e the unque oot to M nd M dened eectvely n E nd E y emm E nd U
13 E decee n nd > Λ Gven > nd [] M [ Λ ] y emm E3 oth M M M M nd M e concve nd hve unque oot > M M o mle y 8 nd the equlum etl ce n the two ytem cn e eeed y [ Λ ] nd Snce Λ nd ncee n mle [ Λ Pt Gven > when ung 8 nd the equlum tock n DS nd DS cn e wtten q q q q > wth y Pt Snce o y emm E ncee n Then we hve > nd hence q > q ] 3
14 Ung 8 nd equlum le n DS nd DS cn e wtten S S S S [ ] > mle > nd hence S > S When y Pt Ung 9 nd equlum totl tock n DS nd DS e q nd q q q et then q q The t-ode devtve o wth eect to nd d d > mle q > q Smlly ung 9 nd totl le n DS nd DS e: S nd S S S ~ ~ ~ et then S S ~ The t-ode devtve o wth eect to ~ ~ d G d Snce G nd > ~ then mle tht S > S 4
15 Poo o Pooton 8 Pt We t ketch the oo to how / ymmetc equlum et n the DS ytem wth endogenou ech ntenty Sml to the oo o emm we how tht unde tted condton whee M lge otve nume Gven Π unmodl on { : [ w M ] [ ]} Π tctly concve n Whle n elct eeon o otml oluton not vlle t cn e veed to e nceng nd concve n y uttuton we cn wte Π devtve wth eect to to e w [ ] Π ettng T we hve T Π uncton o nd how t t-ode T n / T n d nd I / then the t-ode devtve o T chnge gn t mot once om otve to negtve h the uot on [ gven T / > nce > nd w T w [ ] o T coe eo then t h unque oot t whch etle ot uncton ttn the otmum; othewe the ot uncton monotone In ethe ce etle ot uncton unmodl nly o etle ot uncton ollow y ymmety nd we cn how t unmodl unde the me condton ence unde the tted condton thee et ymmetc E n DS wth endogenou ech ntenty whch cn e otned y t-ode condton The oo o unquene ml to tht o Pooton nd we omt the detl Pt Suoe when the ech ntenty eogenou nd te I oety In th oo we dd uct on the quntte wth eogenou ech ntenty y emm E3 the equlum ety tock n DS wth eogenou ech ntenty otned the unque oot to M y t the equlum ety tock n DS wth endogenou ech ntenty cn e otned the unque oot to M w Θ Θ non-negtve n nceng uncton y denton o tht 5
16 Wth M M gven whch mle tht y nd 6 w et w nd w ncee n wth mle q Then the tock n DS wth eogenou nd endogenou ech ntente e eectvely q q nd q q q Then mle tht q q et S Then the le n DS wth eogenou nd endogenou ech ntente e eectvely S S nd S S S G So mle tht S S 6
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