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On Equilibrium Existence in a Finite-Agent, Multi-Asset Noisy Rational Expectations Economy

Ronaldo Carpio and Meixin Guo

Abstract

We introduce a novel method of proving existence of rational expectations equilibria (REE) in multi-dimensional CARA-Gaussian environments. Our approach is to construct a mapping from agents’ initial beliefs (which are characterized by a positive semidefinite matrix), to their updated beliefs, after reaching and observing equilibrium; we then show Brouwer’s fixed point theorem applies. We apply our approach to a finite-market version of Admati (1985), which is a multi-asset noisy REE asset pricing model with dispersed information. We present an algorithm to numerically solve for equilibrium of the finite model, as well as several examples illustrating the difference in equilibrium behavior between the finite and infinite models. Our method can be applied to any multi-dimensional REE model with Gaussian uncertainty and behavior that is linear in agents’ information.

JEL Classification: C62; G12

Appendix

A Proofs

A.1 Preliminaries

Lemma 6.1

(Conditional distribution of Gaussians). Suppose X˜,Y˜ are respectively m, n-dimensional jointly Gaussian random vectors with variance MS+m+n , partitioned as

(52) M=Var(X˜)Cov(X˜,Y˜)Cov(Y˜,X˜)Var(Y˜)=M11M12M12TM22

Then the conditional distribution of X˜ given Y˜ is Gaussian, with mean and variance (Schott (2017), Example 7.4):

(53) E[X˜|Y˜]=E[X˜]+M12M22Y˜E[Y˜]

(54) Var(X˜|Y˜)=M11M12M22M12T

where M22 denotes a generalized inverse of M 22. If M 22 is nonsingular, this is identical to the standard matrix inverse.

Corollary 6.1

(Continuity of conditional Gaussian distribution). The mappings MM12M221 (i. e. the coefficient of Y˜ in E[X˜|Y˜] ) and MM11M12M221M12T=Var(X˜|Y˜) are continuous over {MS+m+n|M22S++n} .

Proof.

The operations involved are: taking a submatrix of M; matrix addition and multiplication; and taking the matrix inverse. The matrix inverse is continuous over the set of nonsingular matrices, which is guaranteed by the constraint M22S++n ; all the other operations are continuous over the set of all real-valued matrices. Therefore, the mappings are continuous.   □

Lemma 6.2

(Positive definiteness and Schur complement). Suppose M=ACCTB is a symmetric, real-valued matrix. Then the following are equivalent: (i) M >  L 0; (ii) B >  L 0 and A>LCB1CT ; (iii) A >  L 0 and B>LCTA1C . (Bernstein (2009), Prop 8.2.4).

Lemma 6.3

(Covariance inequality). Suppose x˜,y˜ are real-valued random variables with finite second moments. Then |Cov(x˜,x˜)|2Var(x˜)Var(y˜) (Mukhopadhyay (2000), Thm 3.9.6).

For AS+n , let λmin(A),λmax(A),dmin(A),dmax(A) denote the minimum and maximum eigenvalues and diagonal entries, respectively, of A.

Lemma 6.4

(Minimum and maximum eigenvalues of positive definite matrices). Suppose A,BS+n . Then:

  1. λmin(A)ILALλmaxI . (Bernstein (2009), Corr. 8.4.2)

  2. 0λmin(A)dmin(A)dmax(A)λmax(A) (Bernstein (2009), Corr. 8.4.7)

  3. AL(<L)Bλmin(A)(<)λmin(B) and λmax(A)(<)λmax(B) (Bernstein (2009), Thm. 8.4.9)

  4. ALBtr(A)tr(B) (Bernstein (2009), Corr. 8.4.10)

  5. The trace of A is equal to the sum of its eigenvalues. (Bernstein (2009), Fact 8.17.8)

Lemma 6.5

(Minimum and maximum eigenvalues of partitioned matrices). Suppose MS++m+n=Var(X˜,Y˜) and is partitioned as in Lemma 6.1. Then:

  1. λmin(M)λmin(Var(X˜))λmax(Var(X˜))λmax(M) and λmin(M)λmin(Var(Y˜))λmax(Var(Y˜))λmax(M) . (Bernstein (2009), Corr. 8.4.6)

  2. λmax(M)λmax(Var(X˜|Y˜)) , λmax(M)λmax(Var(Y˜|X˜)) , λmin(M)λmin(Var(X˜|Y˜)) , λmin(M)λmin(Var(Y˜|X˜)) . Liu (2005), Corr. 2.3, and note that Var(X˜|Y˜)=M/M22=M11M12M221M12T ).

Lemma 2.1 (Bounded positive semidefinite matrices).

Proof.

(a) (b): By Lemma 6.4b, d and e, the eigenvalues of B are bounded, and therefore the eigenvalues and diagonal elements of A are bounded. By the covariance inequality Lemma 6.3, all off-diagonal entries of A are also bounded. (b) (c): Since tr(A) is bounded, the sum of A’s eigenvalues are bounded and each is non-negative, therefore λ max (A) is bounded. (c) (a): Let B = λ max I; then (a) is satisfied.   □

Lemma 2.2 (Positive definite matrices bounded away from singular).

Proof.

(a) (b): By Lemma 6.4b, d and e, the eigenvalues of B are bounded away from zero, and therefore the eigenvalues of A are bounded away from zero. (b) (c): By Lemma 6.5b. (c) (a): By Lemma 6.6, M11=Var(X˜)LVar(X˜|Y˜) and M22=Var(Y˜)LVar(X˜|Y˜) are both bounded away from singular. Consider the given condition that Var(X˜|Y˜) is bounded away from singular. Then there exists AS++m such that

(55) Var(X˜|Y˜)=M11M12M221M12TLA>L0

We want to show M_S++m+n exists such that MLM_>L0 . Subtract A/2 from both sides of eq. 55 to get (M11A2)M12M221M12TLA2>L0 . Then

(56) M11M12M12TM22LM11A2M12M12TM22>L0

The first inequality holds because M11M12M12TM22M11A2M12M12TM22=A2000 ≥  L 0. The second inequality holds by Lemma 6.2. The second matrix is the desired M_ .   □

Corollary 2.1 (Inverse of bounded matrix is bounded away from singular).

Proof.

The eigenvalues of A  – 1 are the inverse of the eigenvalues of A. λ max (A) is bounded iff λmin(A1)=λmax(A)1 is bounded away from singular.   □

Lemma 6.6

(Loewner ordering of conditional variance) Suppose X˜,Y˜ are jointly Gaussian random vectors. Then Var(X˜|Y˜)LVar(X˜) .

Proof.

By the law of total variance:

(57) Var(X˜)=EY˜[Var(X˜|Y˜)]+VarY˜E[X˜|Y˜]

For a jointly Gaussian distribution, Var(X˜|Y˜) is deterministic, so EY˜[Var(X˜|Y˜)]=Var(X˜|Y˜) . Var(X˜)Var(X˜|Y˜)=VarY˜E[X˜|Y˜] which is positive definite, therefore Var(X˜|Y˜)LVar(X˜) .   □

Lemma 6.7

(Loewner lower bound of a closed set). Suppose X is a closed subset of S++n . Then there exists M_S++n such that for each MX,MLM_>L0 .

Proof.

Let λ_=minMXλmin(M) , the smallest eigenvalue across all MX; then λ_ is bounded away from singular. By Lemma 6.4a, MLλ_I>L0 for all MX.   □

Lemma 6.8

(concavity of conditional variance). Suppose MS++m+n=Var(X˜,Y˜) and is partitioned as in Lemma 6.1. Then:

  1. the map MVar(X˜|Y˜)=M11M12M221M12T is concave.

  2. The upper level set of this map with respect to a fixed AS+m , G(A)={MS+m+n|M22>L0,Var(X˜|Y˜)LA} , is a convex, open set.

  3. Imposing the additional constraint M22LB for some fixed BS++n results in H(A,B)={MS+m+n|M22LB>L0,Var(X˜|Y˜)LA} , which is a convex, closed set.

Proof.

For (a), see Corollary 1.5.3 in Bhatia (2007). For (b), the upper level set of a concave function is a convex set; it is open since the constraint M22>L0 defines an open set. For (c), we replace the previous constraint with one that defines a closed set.   □

In order to apply Brouwer’s fixed point theorem, we need to show that the set of covariance matrices characterizing agents’ beliefs is convex and compact. Here we present a general result for a set of possible covariance matrices of a Gaussian random vector, subject to two types of constraints: (i) upper and lower bounds on the joint variance of some subset of variables; (ii) a lower bound on the conditional variance of some subset of variables, given another subset. As we will see, in Admati’s model, the set of possible agent beliefs will be defined by constraints of these two types.

Theorem 6.1

(Convexity of set of covariance matrices subject to constraints). In what follows, X˜=(x˜1,...x˜n)Rn is a Gaussian random vector. We impose two types of constraints on the set of possible variance matrices Var(X˜) :

  1. Let X˜I=(x˜i1,...,x˜iI) denote a particular, fixed I-length subset of X˜ , and let A, B be fixed matrices in S+I . Suppose we impose a constraint of the form ALVar(X˜I)LB . The set of Var(X˜I) that satisfies this constraint is a convex, compact subset of S+I . This also implies that each of Var(x˜i1) , ..., Var(x˜iI) is bounded.

  2. Let X˜I=(x˜i1,...,x˜iI) , X˜J=(x˜j1,...,x˜jJ) denote two particular, fixed, disjoint subsets of X˜ of length I and J, respectively, and let A, B be fixed matrices such that AS++J,BS+I . Suppose we impose the following constraints: (i) Var(X˜J)LA>L0 ; (ii) Var(X˜I|X˜J)LBL0 . That is, we impose a fixed lower bound on the conditional variance of X˜I given X˜J ; this requires that Var(X˜J) be nonsingular, which is guaranteed by (i). The set of Var(X˜I,X˜J) that satisfies this constraint is a closed, convex, and unbounded subset of S++I+J .

Suppose we impose any number of constraints of type (a) and type (b); then the set of Var(X˜) that satisfies these constraints is convex and closed. Furthermore, if as a result of conditions of type (a), each of Var(x˜1),...Var(x˜n) is bounded, then it is also bounded.

Proof.

We proceed by induction. Let Vx0 denote the set of possible Var(X˜) before any constraints have been added, and let Vxi denote the set of valid covariance matrices after constraints 1,..., i have been added; we assume Vxi is convex and closed. We form Vxi+1 by defining an additional constraint and taking the intersection of the set it defines with Vxi .

  1. Suppose we add a type (a) constraint: for fixed A,BS+n , we impose ALVar(X˜I)LB . Let C denote the set {Var(X˜I)S+I|ALVar(X˜I)LB} , which is clearly a closed, bounded, and convex subset of S+n . Let Vxi+1=VxiC ; this intersection is closed and convex.

  2. Suppose we add a type (b) constraint: (i) Var(X˜J)LA>L0 ; (ii) Var(X˜I|X˜J)LBL0 . Let C denote the set {Var(X˜,Y˜)|Var(X˜J)LA>L0,Var(X˜I|X˜J)LBL0} . By Lemma 6.8, C is convex and closed, and so is Vxi+1=VxiC .

Vx0=S+n , which is convex and closed. By induction, the property holds for any i.

Furthermore, suppose that after imposing constraints 1,...i, each of Var(x˜1),...Var(x˜n) is bounded. By the covariance inequality (Lemma 6.3), every off-diagonal element must also be bounded, so Vxi is bounded.   □

A.2 Proofs

Lemma 3.3 (Expectation of beliefs and prices).

Proof.

First result: apply the law of iterated expectations to E[E(F˜|Y˜i,P˜)] . Second result: take expectations of eq. 7 and plug in Fˉ for each E[μ˜i] and Zˉ for E[Z˜] :

(58) E[P˜]=(iWi1)1(iWi1E[μ˜i])(iWi1)1E[Z˜]

(59) =(iWi1)1(iWi1Fˉ)(iWi1)1Zˉ

(60) =(iWi1)1(iWi1)Fˉ(iWi1)1Zˉ=Fˉ(iWi1)1Zˉ

            □

Lemma 3.1 (Boundedness of agent’s belief variance).

Proof.

Apply Lemma 6.6 twice. For the first inequality, condition (F˜|Y˜i,P˜) on (Y˜1,...,Y˜I,Z˜) to get Var(F˜|Y˜i,P˜) ≥  L Var(F˜|P˜,Y˜1,...,Y˜I,Z˜) . Since P˜ is completely determined by (Y˜1,...,Y˜I,Z˜) , conditioning on (Y˜1,...,Y˜I,Z˜) is equivalent to conditioning on (P˜,Y˜1,...,Y˜I,Z˜) . Therefore, Vi=Var(F˜|Y˜i,P˜) ≥  L Var(F˜|P˜,Y˜1,...,Y˜I,Z˜) = Var(F˜|Y˜1,...,Y˜I,Z˜) = V_ . For the second inequality, condition F˜ on (Y˜i,P˜) to get Var(F˜|Y˜i,P˜)LVar(F˜)=V .            □

Lemma 3.2 (Variance of price is bounded).

Proof.

By the law of total variance:

(61) Var(F˜)=EY˜i,P˜[Var(F˜|Y˜i,P˜)]+VarY˜i,P˜(E[F˜|Y˜i,P˜])

(62) V=Vi+Var(μ˜i)

Therefore, Var(μ˜i) is bounded. By Lemma 3.1, each V i and Vi1 is bounded each W i and Wi1 is bounded the coefficients of μ˜1,...,μ˜I,Z˜ in eq. 7 are bounded. Since P˜ is a linear combination of random variables with bounded variance and bounded coefficients, Var(P˜) must be bounded.   □

Lemma 3.4 (Variance bounds on private signal, price, and belief).

Proof.

(i) and (iii) follow from the fact that the mapping T involves computing the conditional distribution of (F˜|Y˜i,P˜) in eqs. 12 and 12, and applying Lemma 3.1 and Lemma 3.2. (ii): Consider the matrix Θ given by eq. 18; for any eigenvalue λ i (Θ), we have |λi(Θ)|<1 (see discussion following eq. 23). Then the corresponding eigenvalue of (I – Θ)–1 is 11λi(Θ) , which is bounded away from zero. Therefore, the variance of A2Z˜ , where A2=(IΘ)1(iWi1)1 (eq. 28) is bounded away from singular. Since Var(A2Z˜)=Var(P˜|Y˜1,...Y˜I) , and by Lemma 6.6, Var(P˜|Y˜i)LVar(P˜|Y˜1,...Y˜I) , then Var(P˜|Y˜i) is bounded away from singular. Lemma 2.2 completes the result.   □

Lemma 3.5 (Convexity and compactness of agents’ belief variance space).

Proof.

We apply Theorem 6.1. Constraint 30 is a type (a) constraint equivalent to

(63) VVVTV+SiLVar(F˜,Y˜i)LVVVTV+Si

Constraint 32 is a type (a) constraint. Constraints 31 and 33 are a type (b) constraint. Thus, the result holds.   □

Acknowledgements

We thank the Editor and three anonymous referees for helpful suggestions. We would also like to thank Pengfei Wang, Jinhui Bai, and participants of the brownbag on Macro and International Economics at the PBC School of Finance, Tsinghua University for comments and suggestions.

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Published Online: 2019-10-01

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