n Max (7) where is the Kuhn-Tucker multiplier to reflect the constraint that investor j cannot invest in security k if he does not know about security k. That is =0 if k EJ. and =0 if kE the compliment to J,, k=l,.,n. From(6.a)and (6.b), the first-order conditions for (7) can be written as 0=R k=1 From(8. a)and (8.b),the optimal common-factor exposure and portfolio weights can be written as n+1-RJ/6 w=△、/(6,2),kJ (9b) 0 E (9.c) n+1 1 nE1 (b,-1) From(8. b)and (9.c),we have that (10) =0 for k E j (10), the"shadow cost
-15- -j 3 J J (7) Max [R - 2 Var(R) - kk] (7) {bJ,wi where AX is the Kuhn-Tucker multiplier to reflect the constraint that investor j cannot invest in security k if he does not know about security k. That is, i = 0 if k J. and w = O if k J, the compliment to k k Jj, k = l,...,n. From (6.a) and (6.b), the first-order conditions for (7) can be written as: 0R -R- 6bb (8.a) n+l j(8.a) 0 Ak 6 j- - Ak k 1,...n = (8.b) From (8.a) and (8.b), the optimal common-factor exposure and portfolio weights can be written as: b [Rn+l - R]/j (9.a) w = /(6 a) , k J (9.b) k k jk i wJ -= , k Jc (9.c) k J n wj = b Z wkbk (9.d) n+l kk n - - Z wk (bk - 1) (9.e) n+2 1 k From (8.b) and (9.c), we have that = k Jc (10) k k j and kXj = 0 for k J.. By inspection of (10), the "shadow cost" k 3
16 of not knowing about security k is the same for all investors Having solved for the individual investor optimal demands, we now aggregate to determine equilibrium asset prices and expected returns. To simplify the analysis and focus attention on the effects of incomplete information on equilibrium prices, we make the further assumption that investors have identical preferences and the same initial wealths. (I 6:·6a0dw-w,3·1,…,N.) Under these conditions, it follows from (9. a)that all investors will choose the same exposure to the common-factor =b,j=1,∴,N, and that n+1-R+6b (11) IfD,(Σ denotes the aggregate demand for security k, then it follows from(9. b)and(9.c)that: (12) where nk denotes the number of investors who know about security k(0<N<N),k=l,.,n. From (9.) and (9.e), we have that: (13) nd n+2 NWb- rn+ (14) LetM三 NW denote equilibrium national wealth. If x,(= V,/M is the fraction of the market portfolio invested in security k, then, from the equilibrium condition Vx -Dk and(12), we have that qKs (15)
-16- of not knowing about security k is the same for all investors.16 Having solved for the individual investor optimal demands, we now aggregate to determine equilibrium asset prices and expected returns. To simplify the analysis and focus attention on the effects of incomplete information on equilibrium prices, we make the further assumption that investors have identical preferences and the same initial wealths. (I.e., 6.- 6 and W j W, j = 1,...,N .) Under these conditions, it follows from (9.a) that all investors will choose the same exposure to the common-factor b = b, j = 1,...,N, and that: R = R + 6b (11) N If Dk(- Z wkWi) denotes the aggregate demand for security k, 1 then it follows from (9.b) and (9.c) that: Dk NkWak k k , (12) where Nk denotes the number of investors who know about security k(O < N< N), k = ,...,n. From (9.d) and (9.e), we have that: D = NWb - nD b (13) n+l N k-k and Zn+ (14) n+2 NWb - 1 k Let M W- = NW denote equilibrium national wealth. If Xk(- Vk/M) is the fraction of the market portfolio invested in security k, then, from the equilibrium condition Vk = Dk and (12), we have that: Xk = qkAk/6o 2 (15)
17 where qk=nK/n is the fraction of all investors who know about security k(0 <qk <1),k=1,.n17 Because the market portfolio is a weighted average of investors optimal portfolios and because all investors choose the same common-factor exposure b, it follows that the common-factor exposure of the market port folio is also equal to by assumption, security n+ 1 and security n+ 2 are inside securities and hence, Vn+1-In+1"0 and n+2+2-0. Thus, b b. and M=2 From the definition of 4, in (6.b),(11), and (15), we have that the equilibrium expected return on security k can be written as: 〓R+b.b5+△ B bstitution for b k’and k as defined in (2),into (16)and rearranging terms, we can derive the equilibrium relation between the market value of firm k and the distributional characteristics of its end-of-period cash flow, (Ik, Hk )i the relative size of the investor base who know about the firm, qk and the aggregate-economy variables,(8, b, R and M). Namely, we have that for k= l,,,,n: Armed with (16) and (17), we now explore the effects of incomplete informati on equilibrium expected returns and asset prices To facilitate the analysis, let denote the equilib market value and expected return on firm k if all investors were informed
-17- where qk Nk/N is the fraction of all investors who know about security k(O < qk < 1), k =1,...,n.17 Because the market portfolio is a weighted average of investors' optimal portfolios and because all investors choose the same common-factor exposure b, it follows that the common-factor exposure of the market portfolio is also equal to b. Moreover, by assumption, security n + 1 and security n + 2 are inside securities and hence, V xn+ = 0 and V 2 = n+2 = 0. n+l n+l n+2 n Thus, b = ZnXkb and M= ZnV k From the definition of Ak in (6.b), (11), and (15), we have that the equilibrium expected return on security k can be written as: Rk = R + bkSd + k (16) = R + bkb + 6dxkk/qk q k = 1,...,n By substitution for Rk, bk, and ok as defined in (2), into (16) and rearranging terms, we can derive the equilibrium relation between the market value of firm k and the distributional characteristics of its end-of-period cash flow, (Ik,pk,ak,sk); the relative size of the investor base who know about the firm, qk; and the aggregate-economy variables, (6,b,R and M). Namely, we have that for k = ,...,n: Vk = Ik[ k - 6bak - (6sIk)/qkM]/R (17) Armed with (16) and (17), we now explore the effects of incomplete information on equilibrium expected returns and asset prices. To facilitate the analysis, let Vk and Rk denote the equilibrium market value and expected return on firm k if all investors were informed
about firm k (1.e,k- 1). If we hold fixed the aggregate-economy variables (8,b,, R and M), then from (17) we have that: V-6(1-q q (18) By inspection of(18), we have that the market value of firm k will alway s be lower with incomplete information, and the smaller the investor base, the larger is the difference To see the connection between this effect on market price and the shadow cost of incomplete diffusion of information among investors, let AK E/N denote the equilibrium aggregate shadow cost (per investor) for security k. From(10), we have that for k=1,.,n: (N-N,)△,/N (19) q From(15), we have that in equilibrium,k-o because x,>0 Hence, from (19),M>0 with equality holding only if all investors know about security k (i.e, By definition of and It therefore follows from (15),(18), and (19)that Vk/[1+)k/R] (20) Note has dimensions of incremental expected rate of return and r equals one plus the riskless rate of interest. Hence, from (29), the effect of incomplete information on equilibrium price is similar to applying an additional discount rate. Indeed, because R, =I,h/v,, we have from(20) that the incremental equilibrium expected return on security k is
-18- about firm k (i.e., qk = 1). If we hold fixed the aggregate-economy variables (6,b,,R and M), then from (17), we have that: Vk = Vk 6(1- - qk)k I /(qMR) (18) By inspection of (18), we have that the market value of firm k will always be lower with incomplete information, and the smaller the investor base, the larger is the difference. To see the connection between this effect on market price and the shadow cost of incomplete diffusion of information among investors, let k(- ZN/N) denote the equilibrium aggregate shadow cost (per investor) for security k. From (10), we have that for k = l,...,n: Xk =(N - Nk )Ak/N (19) -(19q)A =- (1 - qk)k From (15), we have that in equilibrium, Ak > 0 because xk > 0. Hence, from (19), Xk > 0 with equality holding only if all investors know about security k (i.e., qk = 1). 2and 22= x 2 2 By definition of k nd = V2 and x = V/M. and Xk' Skk kk k k It therefore follows from (15), (18), and (19) that: Vk = Vk/[l + Xk/R] (20) Note: Xk has dimensions of incremental expected rate of returh and R equals one plus the riskless rate of interest. Hence, from (29), the effect of incomplete information on equilibrium price is similar to applying an additional discount rate. Indeed, because Rk IkPk/Vk, we have from (20) that the incremental equilibrium expected return on security k is II