Knowing What Others Know: Coordination Motives in Information Acquisition Additional Notes

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1 Knowng Wha Ohers Know: Coordnaon Moves n nformaon Acquson Addonal Noes Chrsan Hellwg Unversy of Calforna, Los Angeles Deparmen of Economcs Laura Veldkamp New York Unversy Sern School of Busness March 1, 8 1 roof of Equlbrum Unqueness n he Beauy Cones Game Ths secon shows ha he equlbrum of he acon game n secon 1 of he man ex s unque. does ha by adapng an argumen frs made Angeleos and avan 7, proposons 1 and 3) o our envronmen. The dea of he proof s ha here s a socal planner problem such ha every equlbrum of our model s also a soluon o hs plannng problem. The plannng problem s srcly convex, meanng ha has a unque mnmum. Snce he plannng problem has a unque soluon and every equlbrum s a soluon o he plannng problem, he equlbrum of he model mus be unque. We begn by seng up some noaon for he proof. We le ˆp ) denoe he canddae equlbrum funcon characered by equaon 4) n he man ex, and wll make use of he fac ha s b. We le F ) denoe he pror dsrbuon of, wh densy f ). We le µ denoe he dsrbuon of he agens nformaon choces, and φ X ) he dsrbuon of observed sgnals, condonal on he sae. Togeher, µ and φ deermne he dsrbuon F )of nformaon ses, X), condonal on he sae. The agens poseror belefs condonal on are defned by he pdf ˆφ ) ˆ φ X ) f ) φ X ˆ) df ˆ). roposon 1 Le denoe he se of funcons p for whch { p ) 1 r) b + r p ) } df ) ˆφ ) d 1

2 for all bu a ero measure of ypes. Then, p, f and only f p ˆp, almos everywhere. Tha s, up o a measure ero perurbaon, he equlbrum sraeges are unquely characered by equaon 4) n he man ex. roof: Defne he funconal L p) p ) b ) df ) df ) r p ) df ) b ) df ) Sep 1 shows ha L p) s srcly convex n p, whch mples ha f p 1, p arg mn p L p), p 1 p almos everywhere. Sep shows ha arg mn p L p). Bu hen, p 1, p mples p 1 p almos everywhere, and he proposon hen follows from nong ha ˆp. Sep 1: L p) s srcly convex for all p. For arbrary funcons p 1 and p, and α, 1), noce ha αp α) p p α) p α, where p p 1. Afer some algebra, one obans: L αp α) p ) αl p 1 ) 1 α) L p ) α [L p α) ) L p 1 ) + 1 α) [L p α ) L p ) { ) } α 1 α) )) df ) r ) df ) df ) { α 1 α) V ar ) ) + 1 r) [E ) ) } df ), where E ) ) ) df ) and V ar ) ) )) df ) [E ) ). Moreover, he las nequaly s src, whenever p 1 ) p ) for a posve measure of s, mplyng ha for any p 1, p arg mn p L p), p 1 ) p ) for almos every. Sep : arg mn p L p). For arbrary p ) and δ ) and a scalar, L p + δ) L p) A δ) + B p, δ) { where A δ) δ )) df ) r B p, δ) ) } δ ) df ) df ) { δ ) p ) b ) df ) r δ ) p ) df ) ) } b df ) df )

3 Clearly, A δ) >, for all δ ) for whch δ )) df ) >.e. ha are dfferen from ero for a posve measure of ypes). Then, for any par p, δ), L p + δ) s mnmed a B p, δ) /A δ), and L p + δ) L p) B p, δ) /A δ). Therefore, p arg mn p L p) f and only f B p, δ), for every δ ). We can rewre B p, δ) as B p, δ) [ δ ) p ) df ) df ) δ ) 1 r) b + r p ) df ) df ) df ) δ ) p ) φ X ) ddµ ) df ) [ δ ) 1 r) b + r p ) df ) φ X ) ddµ ) df ) δ ) p ) φ X ) df ) ddµ ) δ ) [ 1 r) b + r p ) df ) φ X ) f ) dddµ ) Snce ˆφ ) φ X ) f ) / ˆ φ X ˆ) df ˆ), hs las expresson can be rewren as B p, δ) δ ) { p ) { p ) ˆ ˆ [ 1 r) b ˆ + r [ 1 r) b ˆ + r p ) df ˆ ) p ) df ˆ ) } ˆφ ˆ ) dˆ } ˆφ ˆ ) dˆ df ) df ) Therefore, p mples B p, δ) for all δ, and p arg mn p L p). For p /, seng [ δ ) p ) 1 r) b + r p ) df ) ˆφ ) d yelds B p, δ) δ )) df ) df ) >, whch mples p / arg mn p L p). φ X ) df ) ddµ ) General Equlbrum Foundaons for lannng Model n hs appendx, we derve mcro-foundaons for our dynamc plannng and prce adjusmen model from a fully specfed dynamc general equlbrum model. On he household sde, our formulaon follows he connuous me model Golosov and Lucas 7). On he frm sde however, here are dsnc dfferences, as Golosov and Lucas generae nomnal rgdes hrough menu coss of prce adjusmen, whereas here, hey arse from he frms cos of plannng. Tme s connuous and nfne. There s a measure 1 connuum of dfferen nermedae goods, ndexed by [, 1, each produced by one monopolsc frm usng labor as he unque 3

4 npu no producon. There s a fnal consumpon good, whch s produced by a perfecly compeve fnal goods secor usng he connuum of nermedaes accordng o a Dx-Sgl CES echnology wh consan reurns o scale. On he consumpon sde, here s an nfnely-lved represenave household, who purchases he fnal consumpon good and supples labor o he nermedae frms. Fnally, here s a complee se of markes for conngen nomnal bonds. Markes are open connuously, and frms connuously adjus prces, bu hey only updae her nformaon nfrequenly. Money Supply rocess: The Logarhm of nomnal money supply follows an exogenous Brownan Moon wh no drf, and a dffuson parameer σ: d log M µd + σdz nomnal money njecons ake he form of lump sum axes or ransfers o he represenave household. Represenave Household: The represenave household s preferences are defned over he fnal consumpon good, labor supply, and real balances { C, n, M D / }, U E { [ C e ρ 1 ε M D 1 ε δn + log ) } d where ρ denoes he dscoun rae, he prce of he fnal consumpon good, and M d for nomnal balances. 1) he demand Le Q denoe he process for he shadow prce of nomnal cash flows, so ha an earnngs sream {D } s valued as E [ Q D d. The household s budge consran s hen [ M E Q C + R M D ) W n Π d ) where Π ndcaes he ncome from nomnal profs and lump sum money ransfers, and R denoes he nomnal neres rae, whch s mplcly defned by Q e Rd E Q +d ). The erm R M d hus represens he opporuny cos of holdng nomnal balances. The household chooses processes { C, n, M D } / o maxme 1) subjec o ), akng as gven he process {Q,, W, R, Π }. Fnal Good roducers: A large number of fnal goods producers uses he nermedae goods o produce he fnal oupu accordng o a consan reurns o scale echnology, whch s gven by he CES aggregaor [ 1 C c ) θ 1 θ 4 θ θ 1 d. 3)

5 Fnal goods producers maxme profs, akng as gven he marke prces of nermedae and fnal goods. For a oal demand Y of he fnal good by he household, a fnal goods prce, and npu prces, he demand for nermedae good by he fnal good secor s The fnal goods prce s gven by he Dx-Sgl aggregaor c c ) ) θ C. 4) [ 1 ) 1 θ d 1 1 θ. 5) nermedae Good roducers: Each nermedae good s produced by a sngle monopols frm, usng labor l as an npu, accordng o y Al α, 6) for some A >, and α 1. Frms nomnal profs n perod, no ncludng plannng coss, are her prce mes quany sold, mnus wages W ) mes labor: π c ) W c ) /A ) 1/α. 7) Each frm faces a fxed labor cos F, f hey decde o updae her nformaon or plan. The frm hen chooses s process of prces { }, and a process of plannng daes D ), where dd ) 1 f he frm decdes o plan a dae, and dd ) oherwse, o maxme s expeced dscouned profs E [ Q π d F Q W dd ) akng as gven he processes {Q,, W, Y }, and s dae- expecaons E [. Marke equlbrum: An equlbrum s characered by processes { Q,, W, R ; n, C, M D for he aggregae varables and { D ), }, for each, ha solve he frms and household s opmaon problem, and clear goods and labor markes: A every dae, M D M, C Y n he frm s problem), and labor supply n equals he oal labor demand for producon and plannng purposes. The followng proposon summares he soluon o he represenave household problem: roposon There exss a marke equlbrum, n whch he followng condons hold: ) Nomnal neres raes are consan: R R ρ + µ 1/σ. 5 8) }

6 ) The nomnal wage rae s proporonal o M : W δrm ) 1/ε. ) Real demand s gven by C RM v) The sae-prce process s Q 1/ R) e ρ M ) 1, where denoes he Lagrange mulpler on he Household s budge consran. roof. The household s frs-order condons w.r.. C, M and n sasfy e ρ C ε Q, e ρ M 1 R Q and e ρ δ W Q. From hese hree condons, ), ) and v) follow mmedaely. We herefore jus need o show ha he equlbrum nomnal neres rae s ndeed consan. R sasfes Q e R d E Q +d ). Usng he FOC for M, we have E Q +d /Q ) e ρd E R M / R +d M +d )). We conjecure and verfy) ha R R ndeed solves hs condon: n ha case, E R M / R +d M +d )) E M /M +d ) e µd+1/σ d. Therefore he condon for R becomes 1 e Rd e ρd e µd+1/σ d, or, afer akng logs, R ρ + µ 1/σ These properes follow from our assumpons ha ) he dsuly of labor s lnear, ) preferences for real balances are logarhmc, and ) nomnal spendng shocks follow a Brownan moon whou mean reverson). Snce hese properes do no rely n any way on he exac form of he labor demand or ndvdual prcng processes on he nermedae frm s sde, hey drecly apply also o our model n whch here are plannng, nsead of prce adjusmen coss. From here on, we wll assume ha equlbrum nomnal wages, sae prces and real demand are governed by he above processes. We focus on he nermedae frm s prcng and plannng problem. rcng and lannng Decsons: The updang decsons ake place as descrbed n he man ex. n wha follows, we wll use he same noaon for nformaon ses and expecaons. Subsung he sae prce process, he real demand, and he nomnal wage rae no he nermedae frm s prof funcon, he frms perod- profs, no ncludng nformaon coss, and valued a he prce of nomnal cash flows Q, are Q π [ RM ) ) 1/ε 1 ) ; M, e ρ 1 θ δ ) 1/αε) ) RM θ/α A 1/α. 9) Therefore, he full model s counerpar o he frm s reduced form objecve equaon 9 of he man ex) s gven by E { [ RM ) 1/ε 1 ) e ρ 1 θ δ A 1/α RM 6 ) 1/αε) ) θ/α d δf e ρ dd ) 1) }.

7 wh he average prce gven by 5). Wh hs srucure, our equlbrum defnon from he man ex apples dencally. Full-nformaon rce Under full nformaon, he opmal prce n perod s ) 1 δ θ/α 1 θ+θ/α RM A 1/α ) 1 r r, 11) θ 1 where r ε 1 1 α) /α 1 + θ 1 α) /α. 1) We normale A so ha he nal consan erm s equal o 1. Takng logarhms, we fnd an expresson for log ), whch mrrors equaon 1) from he man ex: log 1 r) log RM ) + r log. Moreover, 5) s approxmaed by log 1 log d. Second-order approxmaon: We conclude hs appendx by showng ha he reducedform formulaon consdered n he man ex s a second-order approxmaon o he full general equlbrum formulaon consdered here. We ake a consan frs erm) and subrac from he frm s objecve 8). Maxmng 8) s equvalen o mnmng E [ Q π ) ; M, ) π )) ; M, d + F Q W dd ). 13) The las negral erm represens nformaon coss. Subsung n he formulas for W and Q from proposon 1, becomes δf e ρ dd ). Usng a second-order Taylor expanson of 9) n he frs erm, we have: e ρ ) Q π ; M, π ; M, ) ) ) 1/ε 1 ) [ RM 1 θ ) 1 θ 1 { g M, ) 1 where ) 1 θ θ 1 θ/α δ A 1/α RM [ ) θ/α } 1 RM g M, ) ) 1/ε θ α+θ αθ ) 1/αε) ) [ θ/α ) θ/α 1 The lneared frs-order condon ells us ha under full nformaon, R M. Therefore, f he shocks are small, he economy s close o he full-nformaon economy and g M, ) 1. 7

8 Defnng x log log and usng a second-order Taylor expanson around x, we have ) 1 θ [ 1 θ 1 ) θ/α 1 θ 1 [ θ/α 1 e 1 θ)x 1 e θ/αx θ/α θ 1 [1 θ + θ/α x ) Thus, he frm s objecve s approxmaed by { E e ρ θ 1 [1 θ + θ/α log log ) d δf } e ρ dd ). 14) Ths mrrors he objecve n he plannng model equaon 4 of he paper), once we defne he plannng cos as δf/ 1 θ + θ/α)θ 1)). Comparave Sacs Fnally, we examne he relaonshp beween he srucural parameers and wo key parameers n he reduced-form model of he man ex ha deermne prce rgdy and updang frequency. One s r, he complemenary n prce-seng, defned n equaon 1). There are hree underlyng srucural parameers ha deermne complemenary. Frs, relave rsk averson ɛ ncreases complemenary r/ ɛ > ). Second, he elascy of subsuon θ ncreases complemenary r/ θ > ). Fnally, he rae of dmnshng margnal reurns o labor α ncreases complemenary ff θ > 1/ɛ. The second key deermnan of updang frequency s he cos of nformaon. n hs mcrofounded model, ha cos s a labor cos and herefore vares over me wh he wage. The plannng cos s ncreasng n α and decreasng n θ, as long as here are dmnshng reurns α < 1 and elascy θ > 1. Therefore, f θ > 1/ɛ, hen ncreases n α ncrease complemenary and ncrease he plannng cos, boh of whch make updang less frequen and prces more scky. Bu he effec of changes n elascy θ are ambguous because hey ncrease complemenary bu decrease plannng coss. References [1 Angeleos, George-Maros and Alessandro avan 7), olcy wh Dspersed nformaon, workng paper. [ Golosov, Mke, and Rober Lucas 7), Menus Menu Coss and hllps Curves, Journal of olcal Economy, 115,