Internet Appendix for Political Cycles and Stock Returns

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1 Inerne Appendx for Polcal Cycles and Sock Reurns ĽUBOŠ PÁSTOR and PIETRO VERONESI November 6, 07 Ths Inerne Appendx provdes he proofs of all proposons n Pásor and Verones 07 Proof of Proposon a Governmen workers Consder an agen who decdes a me o be a governmen worker From he governmen s budge equaon, for a gven ax rae τ, oal ax receps avalable a me + are gven by ax + τ Y j,+ dj τ e µ j+ε j,++ε + dj G τg e ε+ m E e µ j j I, j I j I where we used he law of large numbers j I e µ j+ε j,+ dj m E e µ j+ε j,+ j I m E e µ j j I E e ε j,+ j I m E e µ j j I Explong he balanced budge resrcon, he consumpon of a governmen worker s C no + τg e ε+ m E e µ j j I m A When γ, he expeced one-perod uly a he me of he vong decson s E U C no τ γ + τ G γ E e γ ε + E e µ j j I γ γ m m γ A We mmedaely see ha f and only f E U C no + τ H > E U C no + τ L τ H > τ L Smlarly, when γ, hen he uly funcon s log, and we oban E U C no + τ log τ + E log G e ε+ m E e µ j j I log m,

2 so ha he concluson holds b Enrepreneurs Consder now he consumpon of an enrepreneur, under he assumpon ha m agens decde o be enrepreneurs n equlbrum Each enrepreneur sells θ shares and reans θ shares of hs own company All shares are one-perod clams o he nex-perod dvdend, ne of axes: M, E π,+ Y,+ τ, where π,+ s he equlbrum sochasc dscoun facor and τ s he ax rae decded a he elecon a me Each enrepreneur uses he shares sold a me o purchase clams from oher enrepreneurs Le N j denoe he fracon of frm j purchased by enrepreneur a me and le N 0 be he enrepreneur s long or shor poson n he bond The enrepreneur s budge consran s θm j N j M j dj + N 0, where we normalze he prce of bonds o one Snce here s no neremporal consumpon/savng choce, he value of a bond a me s ndeermnae We assume s equal o one and acs as he numerare If agen chooses o be an enrepreneur, hs consumpon a me + for gven τ s C + θy,+ τ + N j Y j,+ τ dj + N 0 From Proposon A below, N 0 j I e 0 and Nj θ µ, so ha Rk I eµ k d C + τg e µ e ε + θe ε,+ + θ A3 Then, for γ, he expeced uly of an enrepreneur s E U C yes τ γ G γ,+ τ e γ µ E e γ ε + θe ε,+ + θ γ γ We clearly have f and only f Smlarly, f rsk averson γ, hen τ γ G γ e γ µ E e γ ε + E θe ε,+ + θ γ γ E U C yes,+ τ L > E U C yes,+ τ H τ L < τ H E U C yes,+ τ log τ + log G e µ + E log e ε + θe ε,+ + θ and he same concluson holds QED

3 Proof of Proposon The argumen s analogous o ha n Pásor and Verones 06 Consder γ, ax rae τ k, and le I k be he equlbrum se of enrepreneurs and m k be he equlbrum mass of enrepreneurs For any agen, V,yes > V,no f and only f E U C yes + τ k, m k > E U C no + τ k, m k Usng expressons A and A3, we oban τ k γ G γ e γ µ E e γ ε + E θe ε,+ + θ γ γ > τ k γ G γ E e µ j j I L γ m k γ E γ m k e γ ε + Deleng common erms, akng logs, and re-arrangng, we oban µ + log E θe ε,+ + θ γ γ τ k > log + log E e µ τ k j j I L m k + log, m k or τ µ > K k k log + log E e µ τ k j j I L m k + log m k log E θe ε,+ + θ γ γ We now derve m k and E e µ j j I L From he defnon of m k and he dsrbuon of skll µ N µ, σ µ, we oban In addon, m k K k φ µ ; µ, σ µ E e µ j j I k m k dµ Φ K k ; µ, σ µ eµ+ σ µ e µ j φ µj ; µ, σ µ dµj K k Φ K k ; µ + σ µ, σµ Therefore, subsung n he expresson for K k, we oban τ K k k log + µ + Φ K k ; µ + σ τ k σ µ µ + log, µ σ Φ µ k ; µ, σ µ log E θe ε,+ + θ γ γ 3 m k A4

4 Defne µ k K k µ and explo he properes of he normal dsrbuon o oban or τ K k k µ log τ k + Φ K k σ µ + log µ; σ µ, σ µ Φ K k µ; 0, σ µ γ log E θe ε,+ + θ γ, τ µ k k log τ k whch defnes he equaon o solve Fnally, + Φ µ k σ µ + log ; σ µ, σ µ Φ µ k ; 0, σ µ log E θe ε,+ + θ γ, γ m k φ µ ; µ, σ µ dµ Φ K k µ; 0, σ µ Φ µ k ; 0, σ µ K k When γ, we nsead have ha holds f and only f E U C yes + τ k, m k > E U C no + τ k, m k log τ k + E log G e µ e ε + θe ε,+ + θ > log τ k G e + E log ε+ m k E e µ j j I m k Deleng common erms and re-arrangng, we fnd τ µ > K k k m k log + log + log E τ k m k e µ j j I E log θe ε,+ + θ The same argumen as above esablshes m k Φ µ k ; 0, σµ, where τ µ k k log + Φ µ k ; σ µ, σµ τ k σ µ + log Φ µ k ; 0, σ µ E log θe ε,+ + θ 4

5 QED Corollary The equlbrum mass of enrepreneurs m k s decreasng n a he ax rae τ k, b rsk averson γ for γ > c dosyncrac volaly σ, and d he degree of marke ncompleeness θ Proof of Corollary : a The mplc funcon defned by µ k F τ k, µ k s clearly ncreasng n τ k as we can see by akng he oal dervave dµ k F τ k, µ k dτ k + F τ k τ k, µ k µ k dµ k From he oal dervave, one obans he mplc funcon heorem dµ k dτ k Fτ k,k k τ k Fτk,K k K k > 0, as F τ k, µ k / τ k > 0 and F τ k, µ k / µ k < 0 Tha s, hgher axes ncrease he hreshold and decrease he mass of enrepreneurs m k Φ µ k, 0, σ µ b Frs, consder he funcon The frs dervave s U γ U γ γ log E θe ε,+ + θ γ γ log E θ e ε + γ E θ e ε + γ log θ e ε + γ E θ e ε + γ Defne X θ e ε + γ for convenence, and facor ou / γ > 0 o oban U E X log X γ log E X γ E X As X > 0 and he funcon f X XlogX s convex f X /X > 0, from Jensen s nequaly we have E X log X > E X log E X Therefore, U E X log X γ log E X γ E X E X log E X < log E X 0 γ E X Noe ha hs proof holds for any γ 5

6 We now defne now he mplc funcon µ k F γ, µ k where we emphasze γ raher han τ We hen have dµ k F γ, µ k dγ + F γ, µ k dµ k γ µ k From he oal dervave, one obans he mplc funcon heorem dµ k dγ Fγ,µ k γ Fγ,µk µ k > 0, as we have shown F γ, µ k / γ U γ > 0 and F γ, µ k / µ k < 0 Tha s, hgher rsk averson ncreases he hreshold µ k Hence dµ τ k, γ dγ > 0 Thus, hgher γ decreases he mass of enrepreneurs m k Φ µ k, 0, σ µ, ha s, Then c Le U θ γ dm k dγ < 0 U θ γ log E θe ε,+ + θ γ E θe ε,+ + θ γ E γθe ε,+ + θ γ e ε,+ E θ e ε,+ + γ e ε,+ E θ e ε,+ + γ < 0, whch holds f and only f whch holds f and only f E θ e ε,+ + γ e ε,+ < 0, Cov θ e ε,+ + γ, e ε,+ + E θ e ε,+ + γ E e ε,+ < 0, whch holds f and only f Cov θ e ε,+ + γ, e ε,+ < 0 Because θ e ε,+ + γ s decreasng n ε,+ and e ε,+ s ncreasng n ε,+, he resul follows 6

7 d Consder U σ γ log E θe ε,+ + θ γ now as a funcon of σ We wan o show ha as σ ncreases, U σ decreases Defne X θe ε,+ + θ θ e ε,+ +, so ha U σ γ log E X γ Because hs s a concave funcon of X, he resul s shown f he cdf F X x; σ s a meanpreservng spread of F X x, σ 0 wh σ > σ 0 Frs, noe ha E X Consder now he cdf F X x; σ PrX < x Prθ e ε,+ + < x Pr ε,+ < log + x θ Le η N0, so ha ε σ + ση Thus F X x; σ Pr σ + ση < log + x θ Pr η < σ + x θ σ log θ Φ σ + x θ σ log θ σ+ σ log x θ θ e η π dη We now show ha f σ > σ 0, hen F X x; σ, θ s a mean-preservng spread of F X x; σ 0, θ We already know he wo dsrbuons have he same mean Therefore, he clam s shown f for every x, x ha s, f for every x, he funcon H σ; x s ncreasng n σ We can wre Therefore, H σ; x σ H σ; x x F X x; σ dx > x x x Φ σ + σ log Φ x σ+ σ F X x; σ 0 dx, x θ σ + x θ σ log θ log x θ θ log x θ θ θ e η π dηdx dx dx e σ+ σ π x θ σ log θ 7 dx

8 Consder he change of varable so ha or Moreover, whch mples and Thus, H σ; x σ x ε σ + σ log x θ θ dε σ x θ dx, x θσdε dx θe εσ σ x θ, θe εσ σ σdε dx θe σε σ + θ x log x θ θ e σ+ σ π x θ σ log θ, dx θe σε σ + θ e ε π σ log e σε σ θe σε σ σdε θe σε σ + θ e ε π ε σ σ θe σε σ σdε θe σε σ + θ θ e σε σ e ε σ εdε π θe σε σ + θ θ e σ e ε +σε σ εdε π We have e ε +σε e ε σε e ε σε+σ σ e ε σε+σ σ e ε σ + σ Therefore, Defne H σ σ θe σε σ + θ e ε σ θ σ εdε π η ε σ dη dε η + σ ε 8

9 o ge H σ; η σ If η hen θe ση + σ + θ and hence θe ση + σ + θ e η θ ηdη π H σ; σ e η θ ηdη E η 0 π Moreover, he funcon Lη H σ; η σ s monooncally decreasng n η because Therefore, Hσ;η σ L θ e η «θe ση + σ + θ θe ση + σ + θ e η θ ηdη π θe ση + σ + θ < 0 π > 0 for every η I follows ha he dsrbuon under a hgher σ s a meanpreservng spread of a dsrbuon under a lower σ Thus, every concave funcon s decreasng n σ, and so s U σ QED Proof of Proposon 3: We know from Corollary ha boh µ τ L, γ and µ τ H, γ are ncreasng n γ, and we have µ τ L, γ < µ τ H, γ Le γ be such ha 05 Φ µ L γ, 0, σµ Then, for γ > γ, we have m L < 05 Clearly, also m H < m L So, regardless of he ax rae H or L, he maxmum mass of enrepreneus s below 05, and herefore τ L canno wn When γ > γ, he unque equlbrum mus be τ H As all agens expec hs o be he case, he equlbrum mass of enrepreneus s m m H Φ µ H γ, 0, σµ Smlarly, le γ such ha 05 Φ µ H γ, 0, σ µ Then for γ < γ, we have m H > 05 Clearly, also m L > m H, ha s, even under hgh axes, he majory of agens are enrepreneurs Therefore, τ H canno wn, and he unque equlbrum has τ L As all agens expec hs o be he case, he equlbrum mass of enrepreneurs s m m L Φ µ L γ,0, σµ For γ < γ < γ, he above argumens mply ha boh equlbra can be suppored QED 9

10 Proposon A In equlbrum, The sochasc dscoun facor s π,+ e γε + Enrepreneurs nves N j e µ θ k I eµ kd ; N0 0 n socks and bonds, respecvely 3 Asse prces are M τ e µ γ σ G 4 The aggregae marke value s M P τ e γ σ E e µ I G m Φ µ k; τ e γ σ σ, σ G m Φ µ k; 0, σ 5 The expeced rae of reurn on each sock s gven by E R e γ σ Proof of Proposon A The clams follow from Corollary C a - e n he echncal appendx of Pásor and Verones 06 The only dfference s ha T and oal producon s mulpled by G, whch s known a me and herefore does no change any calculaons The expresson for E e µ j j I s n equaon A4 QED Proof of Proposon 4 The proof follows from E R e γ σ beng unformly ncreasng γ, he fac ha γ > γ selecs a hgh-ax equlbrum, γ < γ selecs a low-ax equlbrum, and for nermedae γ he hgh-ax equlbrum s seleced wh chance See he dscusson followng he proposon n he ex QED 0

11 Proof of Proposon 5 Consder he expeced growh formula under ax regme k: E Φ µ k ; e µ I k σ, σ e µ+ σ µ Φ µ k; 0, σ In any H equlbrum, we mus have m H Φ µ H ; 0, σ < 05 and n any L equlbrum, we mus have m L Φ µ L ; 0, σ > 05 I follows ha for any H and L equlbra, µ H > µ L see Fgure 3 Consder any par of equlbrum hresholds µ H > µ L The clam follows from showng ha he funcon F µ Φ µ; σ, σ Φ µ; 0, σ s ncreasng n µ We have f and only f or F µ φ µ; σ, σ Φ µ; 0, σ + Φ µ; σ, σ φ µ; 0, σ Φ µ; 0, σ > 0 φ µ; σ, σ Φ µ; 0, σ + Φ µ; σ, σ φ µ; 0, σ > 0 φ µ; 0, σ Φ µ; 0, σ > φ µ; σ, σ Φ µ; σ, σ φ µ σ ; 0, σ Φ µ σ ; 0, σ, where he las equaly uses he properes of he normal dsrbuon The rao φ µ; 0, σ / Φ µ; 0, σ s he hazard funcon of he normal dsrbuon, whch s ncreasng n µ Thus, hs nequaly s always sasfed, whch confrms he clam QED Proof of Proposon 6: We consder he more general verson n whch G m α e g In hs case, oupu n ax regme k a me s Y + Φ µ k ; σ µ, σ µ α e µ+ σ µ Φ µ k ; 0, µ σ e g e ε + Therefore, E Y + H > α Φ µ H ; σ µ, σ µ e µ+ σ µ Φ µ H ; 0, σ µ e g α Φ µ L ; σ µ, σ µ e µ+ σ µ Φ µ L ; 0, µ σ e g E Y + L holds f and only f Φ µ H; 0, σ µ Φ µ L; 0, σ µ α > Φ µ L; σ µ, σ µ A5 Φ µ H; σ µ, σ µ

12 Ths condon s never sasfed for α 0, as Φ µ ; σ µ, σ µ s ncreasng n µ and µ H > µ L We now show ha s always sasfed for α Indeed, under he assumpon of equlbrum symmery, µ L µ H, ha s, cuoffs are symmerc around 0, whch mples mh < 05 < m L are symmerc around 05 Hence, we can rewre he erm o he lef as Φ µ L; 0, σ µ Φ µ L ; σ µ, σ µ > Φ µ H; 0, σ µ Φ µ H; σ µ, σ µ or F µ H Φ µ H; σ µ, σ µ > Φ µ H; 0, σ µ Φ µ L; σ µ, σ µ F Φ µ L; 0, σ µ µ L The clam follows from he proof of Proposon 5, whch shows ha F µ Φµ;σ µ,σ µ ncreasng funcon of µ and he fac ha µ H > µ L Φµ;0,σ µ s an Fnally, he above argumens show ha condon A5 holds for α and does no hold for α 0 By connuy, here exss a value α < such ha he condon always holds for α > α Ths s he value of α for whch condon A5 holds wh equaly, ha s, QED α log log Φµ L ;σ µ,σ µ Φµ H ;σ µ,σ µ Φµ H ;0,σ µ Φµ L ;0,σ µ Proposon A: The welfare-maxmzng allocaon of human capal s m Φ σ µ, 0; σ µ < 05, whch corresponds o he hreshold µ σ µ deermnng whch agens become enrepreneurs Proof of Proposon A Le µ be he hreshold maxmzng oupu Ths s gven by E Y Φ µ ; σ µ, σ µ e µ+σ µ Φ µ ; 0, σ µ e g The maxmum over µ can be obaned from he frs order condons: E Y φ µ ; σ µ µ, σ µ e µ+σ µ Φ µ ; 0, σ µ e g + Φ µ ; σ µ, σ µ e µ+σ µ φ µ ; 0, σ µ e g 0,

13 or Φ µ ; σ µ, σ µ φ µ ; 0, σ µ φ µ ; σ µ, σ µ Φ µ ; 0, σ µ, or φ µ ; 0, σ µ Φ φ µ ; σ µ, σ µ µ ; 0, σ µ Φ µ ; σ µ, σ µ The densy φ µ ; 0, σ µ s symmerc around zero; herefore, φ µ ; 0, σ µ Φ φ µ ; σ µ, σ µ µ ; 0, σ µ Φ φ µ σ µ; 0, σ µ µ ; σ µ, σ µ Φ µ σ µ; 0, σ µ Noe ha hese are hazard raes, whch are always srcly ncreasng funcons Therefore, hs equaly can hold f and only f µ µ σ µ, or µ σ µ Therefore, he socally opmal allocaon has m Φ σ µ, 0; σ µ < 05 QED Proof of Proposon 7: Because γ L < γ and γ H > γ, here are only wo equlbrum masses of agens, m L Φ µ L, 0, σµ > 05 and m H Φ µ H, 0, σµ < 05 Denong y log Y, we have E y + k g + log Φ µ k ; 0, σµ + log Φ µ k ; σ µ, σ µ + µ + σ µ σ, where we use ε + N σ, σ Recall ha Proposon 6 shows E y + H > E y + L and denoe he dfference y E y + H E y + L Φ µ H ; σ µ, σµ log > 0 Φ µ H ; 0, σ µ 3 log Φ µ L ; σ µ, σµ Φ µ L ; 0, σ µ

14 Le f be he fracon of me spen n he L governmen, on average, n equlbrum and defne he uncondonal average as y E y σ fe y + H + fe y + L σ Suppose pary H s n power The probably of a regme change from H o L s λ HL Pr y + > y Pr E y + H + ε + > y Pr ε + > y E y + H Pr ε + > f E y + H E y + L σ Φ f y σ, σ, σ Φ f y, 0, σ > 05 Now suppose pary L s n power Noe ha E y + L < y Therefore, he probably of a regme change from L o H s λ LH Pr y + < y Pr E y + L + ε + < y Pr ε + < y E y + L Pr ε + < f E y + H + fe y + L σ E y + L Pr ε + < f E y + H E y + L σ Φ f y σ ; σ, σ Φ f y; 0, σ > 0 The ergodc dsrbuon of regme L mples f λ HL λ HL + λ LH Φ f y, 0, σ Φ f y, 0, σ + Φ f y; 0, σ To show ha f 05 s he unque soluon o hs equaon, rewre he equaon as fφ f y; 0, σ Φ f y; 0, σ f 4

15 The symmery of he problem shows ha here s only one soluon, f 05 If f > 05, hen he lef-hand sde s greaer han 05 whle he rgh-hand sde s smaller han 05, and vce versa Subsung f / n λ H,L and λ L,H yelds he clam QED Example : Three values of rsk averson Defne he funcon γ H f y < y γ γ M f y < y < y γ L f y > y wh γ H > γ > γ M > γ > γ L Tha s, when log oupu y logy s very low, agens rsk averson s suffcenly hgh o pu he economy n he unque H-ax equlbrum When oupu s very hgh, rsk averson s suffcenly low o pu he economy n he unque L-ax equlbrum For nermedae oupu, rsk averson s nermedae, leadng o wo possble equlbra In hs case, here s probably p 05 ha H-ax or L-ax equlbra wll be seleced We can selec he oupu hresholds y and y o gve equal uncondonal probables o he H and L equlbra, gven ha n he daa he wo ypes of admnsraons end o have roughly he same presence n offce We now compue he ranson probables, for gven y and y Recall ha aggregae oupu under ax regme τ k and rsk averson γ s Y + e g m k Φ µ k γ,σ µ, σ µ e σ µ+ε + e g Φ µ k γ,0, σ µ Φ µ k γ,σ µ, σ µ e σ µ+ε + where Therefore, f γ γ H > γ, hen here s a unque equlbrum a me and log fnal oupu s y + g + log Φ µ H γ H, 0, σµ + log Φ µ H γ H, σ µ, σ µ + N µ HH y, σ, σ µ + ε + µ HH y g + log Φ µ H γ H, 0, σµ + log Φ µ H γ H, σ µ, σ µ + σ µ σ Smlarly, f γ γ L < γ, hen here s a unque equlbrum a and log fnal oupu s where y + N µ LL y, σ, µ LL y g + log Φ µ L γ L, 0, σµ + log Φ µ L γ L, σ µ, σ µ + σ µ σ 5

16 We know from prevous resuls ha µ LL y < µ HH y If nsead γ γ M γ, γ, hen here are wo equlbra, and { N µ LM y + y, σ wh p 05 N µ HM y, σ wh p 05 where µ HM y g + log Φ µ H γ M, 0, σµ + log Φ µ H γ M, σ µ, σ µ + σ µ σ µ LM y g + log Φ µ L γ M, 0, σ µ + log Φ µ L γ M, σ µ, σ µ + σ µ σ Therefore, condonal on each of he four possble evens τ, γ LL, LM, HM, HH, he dsrbuon of oupu s normal The model s hus a four-sae regme shf model whose ranson probables depend on y and y In parcular, we fnd he followng ranson probably marx Λ: 6 4 Transon Marx Λ τ, γ LL LM HM HH 3 h h LL Φ y, µ LL y, σ 05 Φ y, µ LL y, σ Φ y, µ LL y, σ 05 Φ y, µ LL y, σ Φ y, µ LL y, σ Φ y, µ LL y, σ LM Φ y, µ LM y, σ h 05 Φ y, µ LM y, σ Φ y, µ LM y, σ h 05 Φ y, µ LM y, σ Φ y, µ LM y, σ Φ y, µ LM y, σ HM Φ y, µ HM y, σ h 05 Φ y, µ HM y, σ Φ y, µ HM y, σ h 05 Φ y, µ HM y, σ Φ y, µ HM y, σ Φ y, µ HM y, σ 7 HH Φ y, µ HH y, σ h 05 Φ y, µ HH y, σ Φ y, µ HH y, σ h 05 Φ y, µ HH y, σ Φ y, µ HH y, σ 5 Φ y, µ HH y, σ There are four cases of neres: If τ, γ H, H, hen Pr LL Pry + > y Φ y, µ HH y, σ PrHL PrLM 05Pr y < y + < y 05 Φ y, µ HH y, σ Φ y, µ HH y, σ PrHH Pr y + < y Φ y, µ HH y, σ If τ, γ L, L, hen Pr LL Pr y + > y Φ y, µ LL y, σ Pr HL Pr LM 05Pr y < y + < y 05 Φ y, µ LL y, σ Φ y, µ LL y, σ Pr HH Pr y + < y Φ y, µ LL y, σ 3 If τ, γ H, M, hen Pr LL Pry + > y Φ y, µ HM y, σ PrHL PrLM 05Pr y < y + < y 05 Φ y, µ HM y, σ Φ y, µ HM y, σ PrHH Pr y + < y Φ y, µ HM y, σ 6

17 4 If τ, γ L, M, hen Pr LL Pry + > y Φ y, µ LM y, σ PrHL PrLM 05Pr y < y + < y 05 Φ y, µ LM y, σ Φ y, µ LM y, σ PrHH Pr y + < y Φ y, µ LM y, σ Gven he marx, we can compue he ergodc dsrbuon sasfyng saonary: Gven he ergodc dsrbuon, we oban π Λ E r τ H E r τ H and, smlarly, E Y τ H E Y τ H π HH π HM π HH + π E r τ H, γ H + HM π HH + π E r τ H, γ M HM π HH γ H σ π HH + γ M σ π HH + π HM π HH + π HM π LM π LL π LM + π LLE r τ L, γ M + π LM + π LLE r τ L, γ L π LM γ H σ π LL + γ M σ, π LM + π LL π LM + π LL π HH π HM π HH + π E Y τ H, γ H + HM π HH + π E Y τ H, γ M HM π HH π HH eµhh + πhm π LM y + σ + π HH π HH + π π LL eµhm y + σ HM π LM + π LLE Y τ L, γ M + π LM + π LLE Y τ L, γ L π LM LLeµLH y + π LL π LM σ + LLeµLL y + + π π LM σ + π We choose he wo hresholds y and y so as o oban reasonable values for expeced reurns condonal on he wo regmes, and some amoun of symmery so as o have Pr τ H Pr τ L 05 In parcular, for he parameer values n he ex, we choose y 0979 and y 7539 The ranson marx s Λ τ, γ LL LM HM HH LL LM HM HH

18 The uncondonal probably o rans from H o L and from L o H are, respecvely, Probτ + τ H τ τ L 0505; Probτ + τ L τ τ H 0585 Proof of Proposon 8: The proof for a mxed equlbrum s complcaed by he fac ha ax uncerany a me affecs he sae prce densy and hence he equlbrum prce of he sock when he enrepreneur ssues shares Luckly, he same argumens as n Pásor and Verones 06 go hrough, as we now verfy Le I be he se of agens who choose o become enrepreneurs Le M be he marke value of frm a me The ne-of-ax dvdend pad by frm s D+ τ +G e µ +ε ++ε,+, where we use he subscrp + o emphasze ha hs rae s no known a me when agens make her occupaon choce Proposon A4: In he mxed equlbrum, he sae prce densy a he end of perod begnnng of + s π + h τ + γ e γε + for a consan h known a me Denoe he wo sochasc dscoun facors a and + as π,+ π + E π + ; π +,+ π + E + π +, A6 where + s he announcemen of he pary wnnng he elecon Then he asse prces sasfy In addon, enrepreneur s consumpon a me + s M E π,+ D+ A7 M+ E + π+,+ D+, A8 C yes,+ τ + G e µ +ε + θe ε,+ + θ Proof of Proposon A4: We verfy below ha he sae prce densy depends on only wo shocks, ε + and τ + : π + π ε +, τ +, for some funcon π ε +, τ + 8

19 Gven he conjecured sae prce densy and he defnon of he sochasc dscoun facor π,+ π + /E π +, we can compue he prce of each asse a me as M E π + τ + G e µ +ε + +ε,+ E π + e µ E π ε +, τ + τ + e ε + E e ε,+ E π + e µ E π ε +, τ + τ + e ε + E π + e µ Z, where we defne Z as Z E π ε +, τ + τ + e ε+ E π + e µ E π ε +, τ + τ + e ε++ε + A9 E π +, A0 whch s he me- prce of a secury wh payoff τ + e ε + a me + For laer reference, noe ha he aggregae marke value of he marke porfolo s M P Md Z e µ d and he oal dvdend so ha he marke reurn s I D Mk + τ +e ε+ I I e µ d, R Mk DMk + M P τ +e ε+ Z In he argumens below, we wll also make use of he fac ha each ndvdual sock s nfnesmal, ha s, removng one sock from a connuum does no change he value of he marke porfolo In parcular, we wll use he followng equaly for j : M j dj Md I\ I\j Consder he budge consran of each enrepreneur A me, enrepreneur ssues θ shares of hs own frm From he proceeds, he enrepreneur purchases N j shares of frm j and bonds As we show below, f unresrced θ 0, each enrepreneur would sell all of hs frm N 0 and purchase he marke porfolo, whch would enal an nfnesmal poson n hs own frm The θ consran s always bndng; for any gven θ, each enrepreneur resrcs hs holdngs of hs own frm o exacly θ shares All quanes are expressed n erms of our numerare, whch s he zero-coupon bond wh maury + ha s a clam o one un of capal a + The bond prce 9

20 s hus equal o one a boh mes and + Bonds are n zero ne supply The budge consran s θm I\ N j M j dj + N 0 A Whn each perod, agens only rade once, a me, and hey hold her posons unl me + A me +, agen s consumpon s C,+ θd+ + As we shall see, n equlbrum C,+ > 0 wh probably one I\ N j D j +dj + N 0 A Before we analyze he opmal choce of each ndvdual, we consder he marke-clearng condon Each enrepreneur j ssues exacly θ shares Therefore, we mus have ha n equlbrum all shares ssued are bough by somebody Tha s, he sum of all he j shares bough by agens mus equal θ: θ I\j N j d Compared o he budge equaon, he negral here s over and no over j The bond marke mus clear, oo, and gven ha bonds are n zero ne supply, we mus have N 0 d 0 The uly funcon of enrepreneur I s: I E U C,+ γ E C,+ γ γ E θd+ + I\ γ N j D+dj j + N 0 Consder agan he budge equaon of agen, now rewren as θ M N j M j dj N 0 Subsue for N 0 n he uly funcon o fnd E U C,+ γ E θ D+ M + N j D j + γ Mj dj + M I\ I\ The argumen below also apples o agens wh γ, ha s, log uly nvesors, as he man equaons only depend on margnal uly C γ,t, whch are ndependen of wheher γ or no 0

21 The frs-order condons FOC wh respec o N j are E θ γ D+ M + N j D j + M j dj + M D j + M j 0 I\ We can rewre hs expresson as D E θ + M M + M I\ N j M j M D j + M j dj + M γ D j + M j 0 Facorng M ou of he expecaon and smplfyng, we can rewre he FOC as D E θ + N j M j D j γ + D j + + dj + 0 M I\ M Defne ω j as M j ω j Nj M j M Noe ha for every j, he ne-of-ax arhmec reurn on nvesmen s M j R j + Dj + M j τ + e µ j+ε j,+ +ε + e µ j Z τ + e εj,++ε+ Z Tha s, he reurn R j s he same across frms, excep for he realzaon of he dosyncrac shock ε j,+ All socks have he same expeced reurn equal o E R j + E τ + Z We can rewre he FOC of agen as γ E θr,+ + ω j R j,+dj + R j,+ 0 I\ From he above dscusson, all R j + have he same rsk-reurn characerscs Therefore, he properes of he expecaon are he same, and hence he FOC for each agen or are dencal I follows ha ω j ω j ω j for all : ω j ω j Tha s, each agen nvess he same fracon ω j of her wealh n each sock j Fnally, by mposng marke clearng n he sock and bond marke, we oban he sae prce densy Express frs he number of shares bough N j as a funcon of ω j and hus ω j : ω j ω j Nj M j M for j

22 Solvng for N j, we oban he number of shares bough by each agen : N j ω j M M j for j A3 We now mpose he marke-clearng condon n he sock marke Recall ha he oal number of shares ssued by frm j sasfes Subsue for N j : or θ θ I\j θm j ω j N j d ω j M I\j M j I\j d M d Tha s, for every agen, her exposure o sock j, ω j, mus sasfy ω j θ M j I\j M d, A4 whch mples N j ω j M M j θ M j M I\j Mk dk M j θ M I\j Mk dk A5 Tha s, each agen purchases a number of shares N j proporonal o hs wealh M Consder now he budge equaon of agen : θm I\ N j M j dj + N 0 Subsue for N j or from equaon A5: θ M I\ θ θm θ M M I\j Mk dk Mj dj + N 0, I\ Mj dj I\j Mk dk + N0, or Ths mples ha, for all, θ M θm + N 0 A6 N 0 0

23 Tha s, all agens have a zero poson n bonds, whch makes sense as all agens have he same rsk averson Thus, he bond marke clears We fnally oban he sae prce densy ha ensures ha he FOC of all agens are sasfed by equaon A4 Consder agan he FOC of agen : γ E θr + + ω j R j +dj + R j + 0 I\ Subsue wha we found earler as he equlbrum wegh of agen no sock j : o fnd ha he FOC s E or or θr + + ω j ω j θ I\ θ M j I\j Mk dk M j I\j Mk dk Rj + dj + γ R j + γ E θr M j + + θ I\ I\j Mk dk Rj +dj + R j + E 0, 0, θr + + θrmk + + γ R j + 0, A7 where R Mk + s he reurn on he marke porfolo: R Mk M j + I I Mk dk Rj +dj M j D j + I I Mk dk I Dj +dj I Mk dk M j Ex ane, all R +, R j + have he same characerscs, as we can wre R + τ + e ε,++ε + Z A8 R Mk + τ + e ε + Z A9 Le us rewre he FOC n erms of dvdends agan: θr E + + θr Mk + + γ D j + 3 M j dj 0

24 For every, we hus have θr E + + θr Mk + + γ D j + E Inegrae across I o oban E θr + + θr Mk + + γ d D j + E I Defne he sae prce densy as π + so ha he above equaon s I θr + + θr Mk + + γ M j I θr + + θr Mk + + γ d M j θr + + θr Mk + + γ d, A0 E π+ D j T E π + M j, whch s he sandard prcng equaon The sochasc dscoun facor s he π,+ π + /E π + We now show ha hs sae prce densy only depends on ε + and τ + as nally conjecured We have π + I I I D θ + θ M γ + θ I Dj +dj I Mk dk + d τ + e ε+ γ + θ + d Z + θ τ +e ε+ γ d Z Z τ + e ε,++ε + Z θ τ +e ε,++ε + τ + e ε+ γ Z γ h τ + γ e γε +, I θe ε,+ + θ γ d where h Z γ θe ε,+ + θ γ d I We can also solve for Z explcly, from s defnon Z E π ε +, τ + τ + e ε+ E π +, bu s no necessary o do so We now show ha prces are well defned a boh mes and + and ha a + agens do no wsh o rebalance her porfolos In parcular, we need o show ha he sae prce densy jus obaned s well defned no only a before he announcemen bu also a + afer he 4

25 announcemen, n he sense ha can sll be derved from agens frs order condons a +, ha sasfes he marngale condon, and ha deflaed prces also sasfy he marngale condons To see all hs, recall he sae prce densy a he end of perod, e, a +, s gven by π + h τ + γ e γε +, where h s a consan From he marngale condon, we have ha he sae prce densy values before me +, a mes + and, are gven by The sock prce mus sasfy E + π + h τ + γ E + e γε + E π + he τ + γ E + e γε + M + E + π + E + π + M + M E π + E π + M + The laer equaon s clearly sasfed by he prcng formula, as hs s how we obaned he sae prce densy o begn wh We now show ha he prcng equaon a he nermedae me + deermnes he sock prce obaned under he full nformaon case: M + E + π + M + E + π + h τ + γ E + e γε + τ + G e µ +ε++ε,+ h τ + γ E + e γε + τ + G e µ E+ e γε +, E + e γε + whch s he same prcng formula we have for he case n whch τ s known from he begnnng e he pure sraegy Nash equlbrum Because also n ha case he sae prce densy s defned from agens frs order condons, follows ha he sae prce densy s well defned on boh mes In parcular, agens do no wan o rebalance her porfolos afer he revelaon of he wnnng pary a + To see hs, we now show ha each agen s wealh a + obaned from he nal nvesmen a also equals he wealh n he pure sraegy equlbrum when he ax s announced a me raher han + Tha s, her uncerany a me does no change he wealh poson a he me of nformaon abou axes, whch n urn mples ha her opmal choce 5

26 condonal on axes s unchanged compared o he pure sraegy equlbrum: W θm + θm P 05 τ L γ + 05 τ H γ θg e µ 05 τ L γ + 05 τ H e γσ γ 05 τ L γ + 05 τ H γ + θ G m E e µ I 05 τl γ + 05 τ H e γσ γ 05 τ L γ + 05 τ H γ G 05 τl γ + 05 τ H γ e γσ θe µ + θ m E e µ I Agen s wealh a + s W + G τ + γ τ+ γ e γσ θe µ + θm E e µ I θm + + θ M P + Because he marke porfolo and every ndvdual sock prce ncrease or decrease by he same percenage, no rebalancng akes place a me τ + In oher words, he FOC are sll he same for all agens even afer he nformaon release Fnally, we noe ha even wh ax uncerany, consumpon of enrepreneurs a + s he same as n he case wh ax cerany From and C,+ θd + + I\ N j D j +dj + N0 A N 0 0 N j θ M M P θ e µ e µ k dk, we oban C,+ θd + + I\ θ eµ D j + e µ k dk dj θ τ + G e µ +ε,++ε + + e µ I\ θ τ +G e µ j +ε j,++ε + dj e µ k dk τ + G e µ +ε + θe ε,+ + θ A Thus, consumpon C,+ > 0 wh probably one, as clamed earler QED 6

27 Proof of Proposon 8 con d We fnally consruc he mxed equlbrum Frs, le he equlbrum mass be m 05 We keep he general noaon m as wll be useful laer o prove unqueness As n prevous cases, gven m, ruhful vong sll mples enrepreneurs E voe for low axes L and governmen workers G voe for hgh axes H Le p 05 be he probably ha L wns Agens ake no accoun hs uncerany n decdng wheher o be E and voe L or G and voe H In hs case, agens ake no accoun some consumpon uncerany a me + whch wll depend on he vong oucome In parcular, f agen chooses E, hs consumpon s see equaon A: { C yes τ + L G e µ e ε + θe ε,+ + θ wh probably / τ H G e µ e ε + θe ε,+ + θ wh probably / If agen chooses G, hs/her consumpon s Therefore, C no + { τ L G e ε + E e µ j j I m / m wh probably / τ H G e ε + E e µ j j I m / m wh probably / V,yes > V,no f and only f 05 τ L γ M + 05 τ H γ M G γm > e γm µ E γ M 05 τ L γ M + 05 τ H γ M G γ M E e µ γ j j I M E f and only f τ L γ M + τ H γ M e γm µ E θe ε,+ + θ γ e γ M ε + E θe ε,+ + θ γ γ M e γm ε + m m < τ L γ M + τ H γ M E e µ j j I γm m m γ M f and only f log τ L γ M + τ H γ M + γ M µ + log E θe ε,+ + θ γ τ L < log γ M + τ H γ M + γ M log E e µ j j I + γ M m log m f and only f µ K γ M τ L γ M γ M log + τ H γ M + log E e µ j j I τ L γm + τ H γm + γ M log E θe ε,+ + θ γ m + log m 7

28 Therefore, he mass of agens who decde o become enrepreneurs s m γ M d :µ Kγ M Le γ M be such ha m γ M 05 we show below ha such γ M γ, γ exss By consrucon, he medan voer s such ha µ K γ M and hence he s ndfferen beween E and G Such a medan voer flps a con and decdes o become E wh probably 05, supporng he equlbrum We fnally show ha such γ M γ, γ exss Gven he dsrbuon of µ N µ, σ µ, we have m γ M Kγ M In addon, recallng he condonal densy φ µ ; µ, σ µ d Φ K γ M ; µ, σ µ {µ >Kγ M } φµ I φ µ µ K γ M φ µ ; µ, σ µ φ µ Kγ M ; µ, σ µ dµ {µ >Kγ M } {µ >Kγ M } φ µ ; µ, σ µ Φ φ µ ; µ, σ µ, K γ M ;µ, σ µ m we have E e µ j j I Kγ M eµ φ µ ; µ, σ µ d e µ+ Φ K γ M ; µ + σ σ µ, σ µ µ m m In fac, Kγ M e µ φ µ ; µ, σ µ d Kγ M Kγ M e σ µ+µ e µ µ µ σ µ d πσ µ Kγ M e µ +µ µ µ σ µ µ σ µ d πσ µ e µ +µ µ µ+σ µ+µ+σ µ µ+σ µ σ µ d πσ µ Kγ M e µ µ+σ µ σ µ d πσ µ e σ µ+µ Φ K γ M ; µ + σ µ; σ µ 8

29 Subsung everyhng nsde he hreshold, we fnd K γ M τ L γ M γ M log + τ H γ M τ L γm + τ H γm Φ K γ M ; µ + σ µ, µ σ +µ + σ µ + log m + γ M log E θe ε,+ + θ γm or, defnng µ γ M K γ M µ, we oban µ γ M τ L γ M σ µ + γ M log + τ H γ M τ L γm + τ H γm Φ µ γ M ; σ µ, σµ +log Φ µ γm ; 0, σ µ + γ M log E θe ε,+ + θ γm + log Φ µ γ M ; 0, σµ Φ µ γ M ;0, σ µ, so ha µ γ M σ µ + γ M log + γ M log E τ L γ M + τ H γ M τ L γm + τ H γm θe ε,+ + θ γm + log Φ µ γ M ; σ µ, µ σ Φ µ γ M ;0, σ µ Fnally, γ M s chosen such ha µ γ M 0, so ha m Φ µ γ M ; 0, σ µ Φ 0; 0, σ µ 05, whch mples 0 τ L γ M σ µ + γ M log + τ H γ M Φ 0; σ µ, σµ + log τ L γm + τ H γm 05 + γ M log E θe ε,+ + θ γm 9

30 The exsence and unqueness of a soluon for γ M γ, γ can be obaned as follows Defne µp, γ p τ L γ σ µ + γ log + p τ H γ Φ µp, γ; σ µ, σµ pτ L γ + pτ H γ + log Φ µp, γ; 0, σ µ + γ log E θe ε,+ + θ γ We know ha for any γ γ, γ and p, he equaon s µ, γ τ L σ µ + log + log τ L Φ µ, γ; σ µ, σµ Φ µ, γ; 0, σ µ + γ log E θe ε,+ + θ γ µ L < 0, whereas for p 0, he equaon s µ0, γ τ H σ µ + log + log τ H Φ µ0, γ; σ µ, σµ Φ µ0, γ; 0, σ µ + γ log E θe ε,+ + θ γ µ H > 0 Thus, for any γ γ, γ here exss a p 0, such ha µp, γ 0 A3 We also know ha boh µ, γ and µ0, γ are ncreasng n γ I hen follows ha here s unque value of γ for whch equaon A3 s sasfed for p 05 Ths concludes he proof of he exsence of a mxed equlbrum Announcemen Reurns We fnally oban he resuls abou announcemen reurns Gven he sae prce densy obaned n Proposon A4, we can fnally compue he equlbrum under 30

31 uncerany Le ProbL wns 05 and denoe by γ M he correspondng rsk averson We have M E π + D + E τ + γm e γm ε + τ + G e µ +ε + +ε + E π + E τ + γm e γm ε + E τ + γm e γε+ G e µ E τ + γm e γm ε + 05 τ L γ M + 05 τ H γ M e γm σ + γ M σ G e µ 05 τ L γm + 05 τ H γm e γm σ + γm σ 05 τ L γ M + 05 τ H γ M e γm σ + +γ M γ M σ +γ M σ γm σ G e µ 05 τ L γm + 05 τ H γm 05 τ L γ M + 05 τ H γ M G e µ e γmσ 05 τ L γm + 05 τ H γm G e µ γ M σ ω τ L + ω τ H, where ω τ L γ M τ L γm + τ H γm The resuls follow from he fac ha afer he announcemen, he prce s M + G e µ γ M σ τ +, where τ + denoes he ax rae realzed a me + QED 3

32 REFERENCES Pásor, Ľuboš, and Pero Verones, 06, Income nequaly and asse prces under redsrbuve axaon, Journal of Moneary Economcs 8, 0 Pásor, Ľuboš, and Pero Verones, 07, Polcal cycles and sock reurns, Workng paper 3