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Indeterminacy of rational expectations equilibria in sequential financial markets
Journal of Mathematical Economics 39 (2003) 743–762
Indeterminacy of rational expectations equilibria in sequential financial markets夽 Paola Donati a , Takeshi Momi b,∗ b
a European Central Bank, Frankfurt am Main, Germany Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo–ku, Tokyo, Japan
Received 8 December 2000; received in revised form 4 November 2002; accepted 1 February 2003
Abstract We study a two-period, pure exchange economy with an incomplete nominal asset market where agents have private information and learn from prices. It is well known that the indeterminacy of equilibrium prices, and, generically, allocations, that characterises economies with an incomplete financial market and nominal assets may be exploited to determine the degree of information disclosure by rational expectations equilibrium prices. Here we provide a general characterization of the dimension of the indeterminacy of rational expectations equilibria. © 2003 Elsevier Science B.V. All rights reserved. JEL classification: D52; D80 Keywords: Incomplete markets; Indeterminacy; Information revelation
1. Introduction In intertemporal economies under uncertainty and asymmetric information, individuals with rational expectations formulate forecasts on tomorrow’s equilibrium outcomes after refining their initial beliefs with the information they extract from the prices they observe on the market today. In fact, the rational expectations hypothesis requires that individuals both correctly establish the relationship between the equilibrium price domain and the exogenous economic variables, and they exploit the knowledge of the price map to perfect their private information. If an exhaustive specification of the fundamentals of the economy corresponds 夽 This work merges two papers independently written: “Indeterminacy in sequential financial market economies when prices reveal information” by Paola Donati and “Indeterminacy of rational expectations equilibria” by Takeshi Momi. ∗ Corresponding author. Tel.: +81-35-841-5588. E-mail addresses: [email protected] (P. Donati), [email protected] (T. Momi).
0304-4068/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-4068(03)00023-5
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to each state of aggregate private information, and today’s observed prices convey such joint signal entirely, that is, if prices are fully revealing, individuals with rational expectations may foresee the equilibrium value of tomorrow’s variables exactly. If rational expectations equilibrium prices disclose just some, but not all, of the individual signals, that is, if they are partially revealing, or if they do not provide any private information feedback at all, that is, if they are fully non-revealing, individuals are prevented from completing their knowledge refinement and remain asymmetrically informed at equilibrium. In sequential, general equilibrium economies under uncertainty, the revelation or nonrevelation content of rational expectation equilibrium prices depends on the asset structure. Radner (1979) first showed that in a standard two-period general equilibrium economy where asymmetrically informed individuals receive a finite number of private signals, exchange real assets in an incomplete financial market and the equilibrium price vector has higher dimensionality than the space of individual private information, rational expectation equilibrium prices are fully revealing. Pietra and Siconolfi (1998) and Stahn (2000) extended Radner’s result and established the existence of fully revealing rational expectations equilibria for intertemporal economies in which physical commodities at each spot are strictly more than one. As a consequence, the budget constraints that individuals face when they have to decide on their income allocation across time cannot be trivially reduced to one. In non-trivial sequential economies, in fact, the number of budget constraints in an optimization problem is determined by the level of refinement of individual knowledge. This is defined by the join of the partitions induced on the state space both by private signals and by the information conveyed by equilibrium prices. The finer the joint partition, the finer individual’s knowledge and the lower the number of the budget constraints in his optimization. When individuals transfer revenue across states of the world by means of nominal assets, i.e. assets paying off in terms of unit of account, as opposed to real assets as in Radner’s original setting, the indeterminacy of equilibrium prices, and, generically, allocations,1 can be exploited to fix the degree of information disclosure by rational expectation equilibrium prices. The existence of fully non-revealing rational expectations equilibria was proved by Polemarchakis and Siconolfi (1993). Later, Rahi (1995) by imposing ad hoc restrictions on individual’s information at equilibrium, and Citanna and Villanacci (2000) by means of homotopy methods, showed the existence of partially revealing equilibria. Indeterminacy is an immanent characteristic of the rational expectations equilibrium of any nominal asset economy. Specifically, even when the financial market is potentially complete (S = B), that is, the asset market of each subeconomy is complete after individuals’ private information have been fully revealed, a nominal asset economy still has an indeterminate fully non-revealing and an indeterminate partially revealing equilibrium. As a matter of fact, as shown by the above literature, the existence of a non-revealing equilibrium does not exclude the existence of a fully revealing equilibrium. A natural question which then arises, concerns the dimension of the revealing and non-revealing equilibria (see Polemarchakis and Siconolfi, 1993). This work shows that generically, the dimension expectations equilibrium is: of theh real indeterminacy of non-revealing rational H h SN − H h=1 (N − 1)B − 1 when S > B, and that it is: SN − h=1 (N − 1)B − S if S = B. By contrast, generically, the dimension of the real indeterminacy of the revealing equilibria 1
See Balasko and Cass (1989), Cass (1985), Geanakoplos and Mas-Colell (1989) and Werner (1990).
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is: (S −1)N when S > B, while if, S = B, the real indeterminacy of a revealing equilibrium is 0-dimensional. S, B, H denote the number of states, assets and individuals, respectively, h while N h is the number of the private information of individual h and N = H h=1 N is the total number of the economy-wide information. Finally, the dimension of the real indeterminacy of some special types of partially revealing rational expectations equilibria is estimated. The paper is organized as follows. Sections 2 and 3 present the model, define the notion of equilibrium and describe the parameterization of the space of economies. Section 4 focuses on fully revealing rational expectations equilibria. Section 5 deals with fully non-revealing equilibria and Section 6 with partial revelation. Section 7 concludes.
2. The economy Consider a standard two-period model of a general equilibrium, pure exchange economy under uncertainty, with asymmetric information as in Polemarchakis and Siconolfi (1993). There is a finite set of agents, denoted by the superscript h ∈ H = {1, . . . , H}. At time 0, before trade occurs, each agent h receives a private signal nh ∈ N h = {1, . . . , N h }. No agent can directly observe the private signal received by any another agent. The set of joint signals across agents is N = h∈H N h = {1, . . . , N}, where N = h∈H N h . Each state of aggregate information n ∈ N is represented by the H-dimensional collection (n1 , . . . , nh , . . . , nH ), nh ∈ N h . For every agent, it will be useful to write n = (nh , n−h ) ∈ N h × N −h , where n−h represents signals of agents other than h and N −h =
k∈H \{h} N k . The private information of agent h is alternatively described by the partition Lh = {. . . , Lh , . . . } = {N h × {N −h }} of the set of joint signals. Given the state of information n ∈ N in period 0 and the state of nature s ∈ S = {1, . . . , S} in period 1, there is a spot market where agents trade L finitely many commodities indexed by a subscript l ∈ L = {1, . . . , L}. Commodities are exchanged only on spot markets and there are no markets for contingent commodities. Agent h’s conh (s, n), . . . ) for period 1 and xh (n) = sumption vector at state (s, n) is x1h (s, n) = (. . . , x1l 0 (. . . , x0l (n), . . . ) for period 0. His consumption vector associated with signal n is xh (n) = (x0h (n), x1h (n)) = (x0h (n), (. . . , x1h (s, n), . . . )), while xh = (. . . , xh (n), . . . ). His consumption vectors associated with time period t is denoted by x0h = (. . . , x0h (n), . . . ) ∈ RLN and x1h = (. . . , x1h (n), . . . ) ∈ RLSN . We write x = (x1 , . . . , xH ). We use a similar notation to refer to the commodity price vectors, p, and the initial endowment vectors, eh . We assume that individual initial endowments are strictly positive, and that the initial endowment of every agent in period 0 is signal independent. Thus, eh0 is a vector of period 0 endowment LSN h in RL ++ and e1 is a vector of period 1 endowments in R++ . We parameterize the space of L(1+SN)H the economy in terms of endowments, hence the relevant space is E = R++ . Intertemporal trade occurs only through the exchange of financial assets in period 0. There are B nominal inside assets denoted by a superscript b ∈ B = {1, . . . , B}. The asset price vector given signal n is q(n) = (. . . , qb (n), . . . ), while q = (. . . , q(n), . . . ). We assume that the asset payoffs are signal independent. Asset b pays rsb abstract units of account in period 1 when state s occurs. The S ×B matrix R = (rsb )b∈B s∈S is the payoff matrix. We assume
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that R is of full column rank, and, for simplicity, that R is in general position.2 Agent h’s portfolio, given signal n, is yh (n) = (. . . , ybh (n), . . . ), while yh = (. . . , yh (n), . . . ). We write y = (y1 , . . . , yH ). Let L(p0 , q) = {. . . , θ, . . . } be the partition on N induced by period 0 prices: θ = {n ∈ N |(p0 (n), q(n)) = (p0 (θ), q(θ)) } Let Lh (p0 , q) = Lh ∨L(p0 , q) = {. . . , θ h , . . . } be the joint of agent h’s private information and the price-induced partition agent h rests on to make his investment and consumption plans. Each agent has a prior distribution πh over the state and signal spaces. We assume agent h’s preferences are represented by a state and signal independent von Neumann-Morgenstern utility function3 where cardinal preferences are represented by a smooth and strictly concave utility function uh : RL ++ → R. Therefore, agent h’s ex-ante utility is described as h h h h h h h h h h v (x , L (p0 , q)) = π (n|θ ) u (x0 (n)) + π (s|n)u (x1 (n)) s∈S
θ h ∈Lh (p0 ,q) n∈θ h
given (p, q). 3. Equilibrium A vector ((xh , yh )H h=1 , p, q) is a rational expectations equilibrium if, given p and q, solves the following optimization problem:
(xh , yh )
maxvh (x, Lh (p0 , q)) s.t.
p0 (n)(x0 (n) − eh0 ) = −q(n)y(n),
p1 (n)䊐(x1 (n) − eh1 (n)) = Ry(n), (x0 , y) is Lh (p0 , q)-measurable and markets clear, i.e. (x0h (n) − eh0 ) = 0,
n∈N
n∈N
n∈N
h
(x1h (n) − eh1 (n)) = 0,
n∈N
h
yh (n) = 0,
n∈N
h 2 It is not difficult to consider indeterminacy without this assumption (see Geanakoplos and Mas-Colell, 1989; Werner, 1990). 3 Our result holds also for more general preferences. What is needed are the signal invariance of the utility functions over first period consumption and the signal separability properties. The former is required for our result on the real indeterminacy of FNR equilibrium (see Polemarchakis and Siconolfi (1993)), and the latter for our result on the real indeterminacy on FR equilibrium (see Pietra and Siconolfi (1998)).
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A rational expectations equilibrium is fully revealing (FR) if (p0 (n), q(n)) = (p0 (n ), q(n )) whenever n = n . A rational expectations equilibrium is fully non-revealing (FNR) if (p0 (n), q(n)) is constant across n. A rational expectations equilibrium is partially revealing (PR) otherwise. Thus, at FR equilibria, L(p0 , q) = {{1}, . . . , {n}, . . . , {N}}, while at FNR equilibria, L(p0 , q) = {N}.
4. Indeterminacy of FNR equilibrium First we consider the indeterminacy of FNR equilibrium. Theorem 1. If 1 ≤ B < S, H≥ B, and if R is in general position, then, generically in E, FNR equilibrium has SN − h (N h − 1)B − 1 dimensional real indeterminacy: the set of FNR equilibriumcommodity allocations contains the image of a C1 one-to-one function h with domain RSN− h (N −1)B−1 . If B = S, the dimension of the real indeterminacy is h SN − h (N − 1)B − S. Proof. The proof consists of three steps. The first step introduces the notions of modified optimization and constrained equilibrium and presents our first key observation: a FNR equilibrium allocation is identified with a constrained equilibrium allocation satisfying a modified optimization. We also show that the conditions of a constrained equilibrium allocation satisfying a modified optimization are expressed by h (N h − 1)B equations. The second step contains another key point: the constrained economy is identified with a standard incomplete market economy with SN states in period t = 1 and B financial assets. It is well known that the equilibrium of such an economy has a SN − 1 (or SN − S if S = B)-dimensional real indeterminacy (see Geanakoplos and Mas-Colell, 1989). In the third step, combining the above results, we prove that the dimension of the real indeterminacy of FNR equilibrium is SN − h (N h − 1)B − 1 (or SN − h (N h − 1)B − S if S = B). Step 1. We introduce the notions of modified optimization and constrained equilibrium, and show that a FNR equilibrium allocation is identified with a constrained equilibrium allocation satisfying a modified optimization. We distinguish the economy just defined from the economy presented in Section 3 denoting the latter as original. The technique of introducing a modified equilibrium is standard (see Citanna and Villanacci, 2000; Polemarchakis and Siconolfi, 1993). Given non-informative prices (p, q), where (p0 (n), q(n)) = (pˆ 0 , qˆ ) for all n, consider the modified budget constraint: pˆ 0 (x0 (n) − eh0 ) = − qˆ y(n) n∈N
n∈N
p1 (n)䊐(x1 (n) − eh1 (n)) = Ry(n), h
(x0 , y) is L -measurable
n∈N
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The modified budget constraint differs from the budget constraint of the original economy in that the constraints in period 0 associated with a realization of the joint signal have been replaced by their sum. The signal independence property of period 0 endowment and utility function implies that period 0 consumption which solves the modified budget constraint optimization at non-informative prices is also signal independent: x0h (n) = xˆ 0h for all n. Thus, under the modified budget constraint, agent h optimization problem can be rewritten as: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −
1 qˆ y(n) N
p1 (n)䊐(x1 (n) − eh1 (n)) h
(1)
n∈N
= Ry(n),
n∈N
y is L -measurable
where vˆ h (ˆx0 , x1 , Lh ) = vh (. . . , xˆ 0 , . . . , x1 , Lh ). A solution to the modified optimization problem exists and it is unique. It coincides with the solution to the original economy optimization problem at non-informative prices (p, q): (p0 (n), q(n)) = (p, ˆ qˆ ) for all n, whenever yh (n) is n-invariant for all h. The vector h h H h ((ˆx0 , x1 , y )h=1 , pˆ 0 , p1 , qˆ ) is a rational expectations equilibrium for the modified economy if (ˆx0h , x1h , yh )H optimization problem at (pˆ 0 , p1 , qˆ ), and it satisfies the h=1 solves the modified h market clearing conditions: h (ˆx0 − eh0 ) = 0, h (x1h (n) − eh1 (n)) = 0 and h yh (n) = 0 for all n ∈ N . The following lemma shows that yh (n) should in fact be n-invariant at a modified equilibrium, and that the modified equilibria and the FNR equilibria of the original economy actually coincide. Lemma 1. (1) If ((ˆx0h , x1h , yh )H ˆ 0 , p1 , qˆ ) is a modified equilibrium, then for each h, h=1 , p h yh is n-invariant, i.e. yh (n) = yˆ h for all n. Thus, ((x0h , x1h , yh )H h=1 , p, q) where x0 (n) = h xˆ 0 and (p0 (n), q(n)) = (pˆ 0 , qˆ ) for all n is a FNR equilibrium of the original economy. (2) If ((x0h , x1h , yh )H h=1 , p, q) is a FNR equilibrium of the original economy, then, by definition, p0 (n) = pˆ 0 , q(n) = qˆ and, for each agent h, x0h and yh are n-invariant, i.e. x0h (n) = xˆ 0h and yh (n) = yˆ h for all n. Hence, ((ˆx0h , x1h , yh )H ˆ 0 , p1 , qˆ ) is a modified h=1 , p equilibrium. Proof. The proof is essentially the same as in Polemarchakis and Siconolfi (1993). (1) The market clearing conditions of the (modified or original) equilibrium imply that, for h ∈ H , yh (n) = yh (nh , n−h ) = − k∈H \h yk (nh , n−h ) for all (nh , n−h ) ∈ N h × N −h . But for k ∈ H \ h, the Lk (p0 , q)-measurability condition at non-informative prices implies that yk is nh -invariant. Therefore, yh is nh -invariant and, equivalently, it is n-invariant. (2) Since yh is n-invariant we have that yh (n) = yˆ h for all n, and period 0 budget constraint implies that p(x ˆ 0h (n) − e0 ) = −ˆqyˆ h for all n. From our assumptions on the utility function it follows that x0h is n-invariant. 䊐 Lemma 1 motivates the introduction of the notion of constrained optimization. Given (pˆ 0 , p1 , qˆ ), a constrained optimization is an optimization problem which has to satisfy also
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the constraint that y be n-invariant, i.e. that y(n) = yˆ for all n ∈ N : maxˆvh (ˆx0 , x1 , Lh ) pˆ 0 (ˆx0 − eh0 ) = −ˆqyˆ
s.t.
p1 (n)䊐(x1 (n) − eh1 (n))
= Rˆy,
(2) n∈N
The vector ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) is a constrained rational expectations equilibrium h=1 , p if (ˆx0h , x1h , yˆ h )H solves the constrained optimization problem at prices (pˆ 0 , p1 , qˆ ) and if h=1 h h h markets clear: h (ˆx0 − e0 ) = 0, h (x1 (n) − eh1 (n)) = 0 for all n ∈ N , and h yˆ h = 0. Suppose that ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) is a constrained equilibrium. If also h=1 , p h (ˆx0 , x1h , yˆ h )H solves the modified optimization problem at prices (pˆ 0 , p1 , qˆ ), then h=1 h h H h ((ˆx0 , x1 , yˆ )h=1 , pˆ 0 , p1 , qˆ ) is a modified equilibrium, and hence it induces a FNR equilibrium in the original economy. On the other hand, suppose ((xh , yh )H h=1 , p, q) is a FNR rational expectations equilibrium in the original economy. Thus, x0h (n) = xˆ 0h , yh (n) = yˆ h , p0 (n) = pˆ 0 and q(n) = qˆ are satisfied for all n, and ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) is a constrained equilibrium and h=1 , p h h H h ((ˆx0 , x1 , y )h=1 , pˆ 0 , p1 , qˆ ) is a modified equilibrium. Now we specify under what conditions a constrained equilibrium allocation satisfies the modified optimization problem. Compare the first-order conditions of these problems. The first-order conditions of agent h’s constrained optimization problem (2) are: ∂vh ∂xˆ 0h
− µh0 pˆ 0 = 0
∂vh ∂x1h (n)
(3)
− µh (n)䊐p1 (n) = 0,
−µh0 qˆ +
n∈N
(4)
µh (n)R = 0
(5)
n∈N
pˆ 0 (ˆx0h − eh0 ) = −ˆqyˆ h
(6)
p1 (n)䊐(x1h (n) − eh1 (n)) = Rˆyh ,
n∈N
(7)
where µh0 and µh (n) = (. . . , µh (s, n), . . . ), n ∈ N, are the multipliers associated with the constraints of maximization problem (2). Because of the Lh -measurability condition on y in the modified optimization (1), y(n) is constant across n−h , that is, y(nh , n−h ) = y¯ (nh ) for any n−h ∈ N −h , for each nh ∈ N h , and the modified optimization can be rewritten as: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −
1 qˆ y¯ (nh ) Nh h h
n ∈N p1 (nh , n−h )䊐(x1 (nh , n−h ) − eh1 (nh , n−h ))
(8) = R¯y(nh ),
(nh , n−h ) ∈ N h × N −h
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As a consequence, the first-order conditions of the modified maximization problem are: ∂vh − µh0 pˆ 0 = 0 (9) ∂xˆ 0h ∂vh ∂x1h (nh , n−h ) −
µh0 qˆ + Nh
− µh (nh , n−h )䊐p1 (nh , n−h ) = 0,
µh (nh , n−h )R = 0,
(nh , n−h ) ∈ N h × N −h
nh ∈ N h
(10)
(11)
n−h ∈N −h
pˆ 0 (ˆx0h − eh0 ) = −
1 qˆ y¯ h (nh ) Nh h h
(12)
n ∈N
h
−h
p1 (n , n
)䊐(x1h (nh , n−h ) − eh1 (nh , n−h ))
= R¯yh (nh ),
(nh , n−h ) ∈ N h × N −h (13)
µh0
µh (nh , n−h )
(. . . , µh (s, (nh , n−h )), . . . ),
(nh , n−h )
Nh
N −h
and = ∈ × are where the multipliers corresponding to each of the constraints in the modified optimization. Even though we use the same symbol µ to denote the multipliers of the two maximizations, no confusion should arise. Suppose that (ˆx0h , x1h , yˆ h , µh0 , µh (n), n ∈ N ) satisfies (3)–(7). If we set y¯ h (nh ) = yˆ h for all nh , then (ˆx0h , x1h , y¯ h (nh ), nh ∈ N h , µh0 , µh (n), n ∈ N ) satisfies all of the conditions (9) and (13) of the modified optimization problem except for (11). This means that the condition required for the variables in a constrained equilibrium to satisfy the modified optimization problem is that the multipliers µh = (µh0 , µh (n), n ∈ N ) at the constrained equilibrium satisfy (11) for each h ∈ H . Since the µh satisfy (5), then (11) is satisfied if and only if: µh (nh , n−h )R = µh (nh , n−h )R, for any nh , nh ∈ N h (14) n−h ∈N −h
n−h ∈N −h
It is evident that (14) is equivalent to (N h − 1)B equations: µh (nh , n−h )R − µh (nh + 1, n−h )R = 0, n−h ∈N −h
nh = 1, . . . , N h − 1
n−h ∈N −h
Thus, for all h ∈ H , (14) consists of
h∈H (N
h
− 1)B equations.
Step 2. The constrained equilibrium is naturally identified with the equilibrium of a standard incomplete market economy with SN states in period t = 1 and B nominal assets. The equilibrium indeterminacy of a standard incomplete nominal asset market economy has already been treated by Balasko and Cass (1989), Geanakoplos and Mas-Colell (1989) and Werner (1990). By using the same techniques, here we deal with the equilibrium inde terminacy of a standard incomplete market economy which is subject to the h (N h − 1)B additional constraints obtained in Step 1. Now, we introduce a real numeraire asset economy. When we consider not only the states of nature s, but also the states of information n as the index of the states in period t = 1, the n-invariant maximizations (2) can be naturally
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identified with the maximization problem of a standard incomplete asset market economy with SN states in period t = 1, B nominal assets, and the SN × B payoff matrix R . R = .. . R For (s, n) ∈ S × N , we introduce the following lexicographic order: (s, n) < (s , n ) if and only if n < n or n = n and s < s . Once our constrained economy is identified with a standard incomplete asset market economy, we can treat the indeterminacy by applying standard arguments. It is well known that the equilibrium allocations of a nominal asset economy can be identified with those of a real numeraire asset economy. Then, consider an economy with real numeraire assets, which is equivalent to an economy with financial assets characterized by the payoff matrix R. Let, ˜ R(1) ˜ = ... R ˜ R(N) be a SN × B matrix representing the real asset payoffs of the numeraire (commodity one). ˜ S × B matrix R(n) = (˜r b (s, n))b∈B s∈S contains the payoffs corresponding to states (s, n) ∈ b ˜ (S, n), and R(s, n) = (. . . , r˜ (s, n), . . . ) denotes the payoffs of assets at state (s, n). The utility maximization problem of agent h in the real numeraire asset economy is: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −ˆqyˆ
˜ n)ˆy, p1 (s, n)䊐(x1 (s, n) − eh1 (s, n)) = p11 (s, n)R(s,
(s, n) ∈ S × N
where p11 (s, n) is the price of commodity one at state (s, n). To highlight the link with the constrained optimization problem, we rewrite such maximization problem as: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −ˆqyˆ
˜ y, p1 (n)䊐(x1 (n) − eh1 (n)) = [p11 (n)]R(n)ˆ
n∈N
where [p11 (n)] is a diagonal S × S matrix, the elements of which are prices of commodity one, p11 (s, n). We define a real numeraire asset equilibrium ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) h=1 , p as the set of variables satisfying the market clearing conditions and utility optimization problem of all agents. The first-order conditions of agent h’s optimization in the real numeraire asset economy are: ∂ˆvh −µ ˜ h0 pˆ 0 = 0 (15) ∂xˆ 0h ∂vh ∂x1h (n)
−µ ˜ h (n)䊐p1 (n) = 0,
n∈N
(16)
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−µh0 qˆ +
˜ µ ˜ h (n)[p11 (n)]R(n) =0
(17)
n∈N
pˆ 0 (ˆx0h − eh0 ) = −ˆqyˆ h
(18)
h ˜ p1 (n)䊐(x1h (n) − eh1 (n)) = [p11 (n)]R(n)y ,
n∈N
(19)
˜ h (n) = (. . . , µ ˜ h (s, n), . . . ), n ∈ N , are the multipliers corresponding to where µ ˜ h0 and µ each of the constraints in the maximization problem. By comparing the first-order conditions (3)–(7) of the constrained economy optimization and conditions (15)–(19) characterizing the individual maximization problem in the real numeraire asset economy, we can easily see the equivalence of the constrained equilibrium ˜ = ΛR, where Λ and the real numeraire asset economy equilibrium with payoff matrix R is a SN × SN diagonal matrix with positive elements. We write Λ(1) .. Λ= . Λ(N) and
Λ(n) =
Λ(1, n)
..
. Λ(S, n)
Proposition 1. If ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ), with multipliers µh for agent h, is a conh=1 , p h h strained equilibrium, then ((ˆx0 , x1 , yˆ h )H ˜ 0 , p˜ 1 , q˜ ), with multipliers µ ˜ h for agent h, h=1 , p ˜ = ΛR, wherep˜ 0 = pˆ 0 /pˆ 01 , is a real numeraire asset equilibrium with asset return R p˜ 1 (s, n) = p1 (s, n)/p11 (s, n), q˜ = qˆ /pˆ 01 , λ(s, n) = 1/p11 (s, n), µ ˜ h0 = pˆ 01 µh0 and h h µ ˜ (s, n) = p11 (s, n)µ (s, n) for all s, n and h. If ((ˆx0h , x1h , yˆ h )H ˜ 0 , p˜ 1 , q˜ ), with multipliers µ ˜ h for agent h, is a real numeraire asset h=1 , p ˜ = ΛR, and p˜ 01 = 1 and p˜ 11 (s, n) = 1 for all s and n, equilibrium with asset return R h , p ˆ , p , q ˆ then ((ˆx0h , x1h , yˆ h )H 0 1 ), with multipliers µ for agent h, is a constrained equih=1 librium where p1 (s, n) = p˜ 1 (s, n)/λ(s, n), µh (s, n) = λ(s, n)µ(s, ˜ n), µh0 pˆ 0 = µ ˜ h0 p˜ 0 and h h µ0 qˆ = µ ˜ 0 q˜ for all s, n and h. As a consequence, the equilibrium of the constrained economy can be parameterized by Λ of the real numeraire asset economy. Note that, from Proposition 1, ˜ h , n−h ) µ ˜ h (nh , n−h )R(n n−h ∈−h
=
˜ h , n−h ), µ ˜ h (nh , n−h )R(n
for any
nh , nh ∈ N h
n−h ∈−h
at the real numeraire asset equilibrium, means (14) at the constrained equilibrium.
(20)
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Step 3. By following arguments which are standard to deal with the indeterminacy of incomplete financial asset market economies, we estimate the dimension of the indeterminacy of our economy’s equilibrium. Although we have to take into account constraints (20), similar arguments can be used so that we are not going to repeat all the details of the proof. Rather, we will highlight the differences caused by constraints (20). In the following discussion, we focus on the equilibrium allocation of commodities in a real numeraire economy. Hence, we normalize the commodity prices: pˆ 01 = 1 and p11 (s, n) = 1 for all s, n, and let \ \ pˆ 0 and p1 (s, n) denote the prices of commodities other than commodity one at state (s, n) in periods t = 0 and t = 1, respectively. A sufficient condition for different real numeraire asset equilibria corresponding to different parameters is the following. Lemma 2. Let ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) and ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) be real nuh=1 , p h=1 , p meraire asset equilibria with respect to payoff matrices ΛR andΛ R, respectively. Suppose that (i) dim sp[ˆy1 , . . . , yˆ H ] = dim sp[ˆy1 , . . . , yˆ H ] = B. Then, (ˆx0h , x1h ) = (ˆx0h , x1h ) for all h if and only if (pˆ 0 , p1 ) = (pˆ 0 , p1 ). 4 qˆ qˆ . (ii) = sp ΛR ΛR Proof. The “only if ” part is essentially the same as in Geanakoplos and Mas-Colell (1989). The “if” part is an immediate consequence of the utility maximization of all agents. 䊐 Consider the equilibrium of the real numeraire asset economy with constraint (20). Let \ \ (L−1)SN L(SN+1)H f(pˆ 0 , p1 , qˆ , Λ, e), from P ≡ RL−1 ×RB ×RSN to R(L−1)(SN+1) × ++ ×R++ ++ ×R++ B R be the excess demand function of assets and commodities 2, . . . L, of the numeraire asset economy. By Walras’ law, f = 0 implies an equilibrium. The multipliers µ ˜ h0 , µ ˜ h (s, n) of the optimization problem in the real numeraire asset \ \ \ \ economy are functions of (pˆ 0 , p1 , qˆ , Λ, e). Let gh (pˆ 0 , p1 , qˆ , Λ, e) denote the functions h h from P to R(N −1)B defined by gh = (g1h , . . . , gnhh , . . . , gN h −1 ) and gnhh = (gnhh 1 , . . . , gnhh B ) =
˜ h , n−h ) µ ˜ h (nh , n−h )R(n
n−h ∈N −h
−
µ ˜ h (nh + 1, n−h )R(nh + 1, n−h )
n−h ∈N −h
Note that gh = 0 implies (20). We write g = (g1 , . . . , gH ) and F = (f, g) : P → h R(L−1)(SN+1) × RB × R h (N −1)B . Since the utility functions are smooth, F is a smooth function (see Geanakoplos and Polemarchakis, 1986). 4
For any matrixA, we denote by sp[A] the linear subspace spanned by columns of A.
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P. Donati, T. Momi / Journal of Mathematical Economics 39 (2003) 743–762 \
\
Lemma 3. F(pˆ 0 , p1 , qˆ , Λ, e) = 0 implies that rank ∂e F = (L − 1)(SN + 1) + B + h h (N − 1)B. Proof. rank∂e f is (L − 1)(SN + 1) + (B − 1) (see Geanakoplos and Polemarchakis, 1986). We show that by means of endowment perturbations, it is possible to change the value of each of the equations of g independently without changing the other equations. Without loss of generality we consider how to change the value of gnhh 1 by 1. First, define +µ(n ˜ h , n−h ), (nh , n−h ) ∈ N h × N −h so that ˜ +µ ˜ h (1, n−h )R(1, n−h ) n−h ∈N −h
= ··· =
h
h
−h
+µ ˜ (n , n
˜ h , n−h ) = )R(n
n−h ∈N −h
N h − nh , 0, . . . , 0 Nh
(21)
˜ h + 1, n−h ) +µ ˜ h (nh + 1, n−h )R(n
n−h ∈N −h
= ··· =
h
h
−h
+µ ˜ (N , n
˜ h , n−h ) = )R(N
n−h ∈N −h
−nh , 0, . . . , 0 Nh
(22)
Note that (21) and (22) imply that +gnhh = (+gnhh 1 , . . . , +gnhh B ) = (1, 0, . . . , 0), while the other equations in g remain unchanged. Furthermore, (17) continues to be satisfied because ˜ +µ ˜ h (n)R(n) n
=
h
h
−h
+µ ˜ (j, n
˜ n )R(j,
−h
)=
n
h
N
˜ n−h ) +µ ˜ h (j, n−h )R(j,
j=1 n−h ∈N −h
j∈N h n−h ∈N −h
+
˜ n−h ) +µ ˜ h (j, n−h )R(j,
j=nh +1 n−h ∈N −h
= nh
h N h − nh −n , 0, . . . , 0 + (N h − nh ) , 0, . . . , 0 = (0, . . . , 0) Nh Nh
Second, define +x1h (nh , n−h ) so that ∂vh ∂x1h (nh , n−h )
(x1h (nh , n−h ) + +x1h (nh , n−h ))
= (µ ˜ h (nh , n−h ) + +µ ˜ h (nh , n−h ))䊐p1 (nh , n−h ) for all (nh , n−h ) ∈ N h × N −h . Finally, define +eh1 (nn , n−h ) = +x1h (nn , n−h ) for all (nh , n−h ) ∈ N h × N −h . From the above discussion it is easy to see that, for such a perturbation of the initial endowment of
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\
agent h, when F(pˆ 0 , p1 , qˆ , Λ, e) = 0, the value of all of the equations of F other than gnh 1 continue to be 0. By the transversality theorem, for almost all e the set Fe−1 (0) is a Cr -manifold of dimen h sion SN − h (N − 1)B. \ \ Let us consider the set of the variables (pˆ 0 , p1 , qˆ , Λ) in Fe−1 (0) with different corresponding commodity allocations. Consider the conditions in Lemma 2, following Geanakoplos \ \ and Mas-Colell (1989). Let J be the B − 1 sphere and define h(pˆ 0 , p1 , q, Λ, e, z) = B B ( h=1 zh yˆ 1h , . . . , h=1 zh yˆ Bh ) : P × J → RB , where yˆ bh is agent h’s demand for asset \ \ b corresponding to (pˆ 0 , p1 , qˆ , Λ). Define H = (F, h) : P × J → R(L−1)(SN+1) × RB × h R h (N −1)B × RB . Then, as shown in Geanakoplos and Mas-Colell (1989), ∂e H has full rank, hence, for almost all e, He−1 (0) is a SN − h (N h − 1)B − 1 dimensional Cr manifold. Thus, we have a relatively open set V on Fe−1 (0) which is disjoint from He−1 (0). For \ \ (pˆ 0 , p1 , qˆ , Λ) ∈ V , the corresponding {ˆyh , h ∈ H } satisfy (i) in Lemma 2. \ \ For (pˆ 0 , p1 , qˆ , Λ) ∈ V , let \
\
\
\
W(pˆ 0 , p1 , qˆ , Λ) = {(pˆ 0 , p1 , qˆ , Λ ) ∈ V |commodity allocations are as those \
\
corresponding to(pˆ 0 , p1 , qˆ , Λ)} \
\
\
\
We show that dimension of W(pˆ 0 , p1 , qˆ , Λ) is constant for any (pˆ 0 , p1 , qˆ , Λ) in V , the former of which will be written as dimW. Obviously, the dimension we require is dimV − dimW. We obtain the dimension of W by considering condition (ii) of Lemma 2. Remember, that R .. R= . R is not in general position although R is assumed to be so. Now we have to distinguish between two cases: S < B and S = B. The argument is the same as in Geanakoplos and Mas-Colell (1989).5 We first consider the case of S < B. In such case (ii) is satisfied if and only if qˆ = αˆq and Λ = αΛ for some α ∈ R++ . The “if” part is trivial and the“only if” part follows that qˆ qˆ = sp sp Λ(n) R Λ(n)R 5 See Section 3 of Geanakoplos and Mas-Colell (1989), where the case of the payoff matrix not in general position is dealt. Using their symbols and the general theorem, what we show in the following two paragraphs is essentially that dimW = K(R) + Z(R) and that K(R) = 1, Z(R) = 0 when S > B and K(R) = S, Z(R) = 0 when S = B.
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for all n ∈ N is necessary for (ii), that R is in general position and that S < B. Thus, from \ \ \ \ Lemma 2, (pˆ 0 , p1 , qˆ , Λ) and (pˆ 0 , p1 , qˆ , Λ ) in V induce the same commodity allocation \
\
\
\
if and only if (pˆ 0 , p1 ) = (pˆ 0 , p1 ), and qˆ = αˆq and Λ = αΛ for some α ∈ R++ . On the other hand, if (ˆxh , x1h , yˆ h , µh ) is the demand for commodities, assets, and multipliers \ \ \ \ of agent h at (pˆ 0 , p1 , qˆ , Λ), (ˆxh , x1h , 1/αˆyh , µh ) correspond to (pˆ 0 , p1 , αˆq, αΛ) for any α ∈ R++ , as is easily recognized from the first-order conditions (15) and (19). Thus, if \ \ \ \ (pˆ 0 , p1 , qˆ , Λ) is actually in Fe−1 (0), then (pˆ 0 , p1 , αˆq, αΛ) is in Fe−1 (0) for any α ∈ R++ . Therefore, dimW = 1 when S > B. When S = B, (ii) is satisfied if and only if qˆ = qˆ R−1 αR and Λ(n) = Λ(n) α for some diagonal matrix α1 .. α= . αS
of S dimension with positive elements. In the above statement, the “if” part is easily checked for, and “only if” part comes from the following discussion. For (ii) to be satisfied, there must exist y ∈ RJ such that qˆ qˆ y for any y ∈ RJ y= ΛR ΛR Equivalently, qˆ y = qˆ y and Λ(n)Ry = Λ(n) Ry for all n. Then y = (Λ(n) R)−1 Λ(n)Ry = R−1 (Λ(n) )−1 Λ(n)Ry for all n and (Λ(n) )−1 Λ(n) have to be independent of n. Defining this diagonal matrix as α, we obtain Λ(n) = Λ(n) α. Substituting y = R−1 αR into qˆ y = \ \ \ \ qˆ y , we obtain qˆ = qˆ R−1 αR. Thus, from Lemma 2, (pˆ 0 , p1 , qˆ , Λ) and (pˆ 0 , p1 , qˆ , Λ ) \
\
\
\
in V induce the same commodity allocation if and only if (pˆ 0 , p1 ) = (pˆ 0 , p1 ), and qˆ = qˆ R−1 αR and Λ(n) = Λ(n) α for some α ∈ RS++ . As in the above case, it is easy to check \
\
\
\
that such (pˆ 0 , p1 , qˆ , Λ ) is actually on V if (pˆ 0 , p1 , qˆ , Λ) is on V . Therefore, dimW = S when S = B. Thus, dimV − dimW = SN − h (N h − 1)B − 1 if S > B (SN − h (N h − 1)B − S if 䊐 S = B). This ends the proof of Theorem 1.
5. Indeterminacy of FR equilibrium In this section we consider the indeterminacy of the FR equilibrium. Theorem 2. If 1 ≤ B < S, H ≥ B, and R is in general position, then, generically in E, the FR equilibrium contains a submanifold of dimension (S − 1)N. In other words, the set of FR equilibrium commodity allocations contains the image of a C1 one-to-one function with domain R(S−1)N .
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Proof. We consider a fictitious economy where agents have full information. We call this economy a full information economy. If an equilibrium price of the full information economy satisfies (p0 (n), q(n)) = (p0 (n ), q(n )) whenever n = n , this equilibrium is supported as a FR equilibrium of the original economy. Being n the aggregate private signal, the full information economy decomposes into N independent n-subeconomies. Each n-subeconomy is a standard general equilibrium economy with financial assets, which can be again translated into an economy with real numeraire assets through a parameterization implemented by means of the S × S diagonal matrix Λ(n). Following Geanakoplos and Mas-Colell (1989), each n-subeconomy equilibrium has a S − 1 dimensional real indeterminacy. Therefore, it is conjectured that the dimension of the real indeterminacy is in total (S − 1)N. To show that (p0 (n), q(n)) = (p0 (n ), q(n )) and at the same time take into account the n-invariance of first period endowment, we follow Pietra and Siconolfi (1998), who proved that equilibrium prices are generically fully revealing in a real numeraire asset economy. Since our proof is easily obtained by combining already established results and the passages of Theorem 1, we do not go into the details. We will use some of the symbols we have already used in the proof of Theorem 1 to emphasize the similarities of these proofs and facilitate understanding. Agent h’s optimization problem in any n-subeconomy is: maxuh (x0h (n)) +
πh (s|n)uh (x1h (n))
s∈S
s.t.
p0 (n)(x0 (n) − eh0 ) = −q(n)y(n)
p1 (n)䊐(x1 (n) − eh1 (n)) = Ry(n)
h The vector ((x(n)h , y(n)h )H h=1 , p(n), q(n)) is a n-subeconomy equilibrium if (x(n) , H h y(n) )h=1 solves the above maximization problem at prices (p(n), q(n)) and markets clear, i.e. h (x0h (n)−eh0 ) = 0, h (x1h (n)−eh1 (n)) = 0 and h yh (n) = 0. Since a n-subeconomy is obviously a standard nominal asset incomplete market economy, we can identify each n-subeconomy with a real numeraire asset economy by means of a Λ(n) parameterization as in Geanakoplos and Mas-Colell (1989) or in Section 4. h Proposition 2. If ((xh (n), yh (n))H h=1 , p(n), q(n)), with multipliers µ (n) for agent h, is an H h h ˜ q˜ (n)), with multipliers µ ˜ h (n) n-subeconomy equilibrium, then ((x (n), y (n))h=1 , p(n), ˜ for agent h, is a real numeraire n-subeconomy equilibrium with asset return R(n) = Λ(n)R wherep˜ 0 (n) = p0 (n)/p01 (n), p˜ 1 (s, n) = p1 (s, n)/p11 (s, n), q˜ (n) = q(n)/p01 (n), ˜ h0 (n) = p01 (n)µh0 (n) and µ ˜ h (s, n) = p11 (s, n)µh (s, n) for all s λ(s, n) = 1/p11 (s, n), µ and h. If ((xh (n), yh (n))H ˜ 0 (n), q˜ (n)), with multipliers µ ˜ h (n) for agent h, is a real numeraire h=1 , p ˜ = Λ(n)R, and p˜ 01 (n) = 1 and p˜ 11 (s, n) = n-subeconomy equilibrium with asset return R H h h 1 for all s, then ((x (n), y (n))h=1 , p(n), q(n)) with multipliers µh (n) for agent h is an ˜ n), n-subeconomy equilibrium where p1 (s, n) = p˜ 1 (s, n)/λ(s, n), µh (s, n) = λ(s, n)µ(s, µh0 (n)p0 (n) = µ ˜ h0 (n)p˜ 0 (n) and µh0 (n)q(n) = µ ˜ h0 (n)˜q(n) for all s and h.
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Then, consider the set of equilibria in the real numeraire n-subeconomies such that (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) whenever n = n . Note that if (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) for a real numeraire economy, the corresponding prices in period t = 0 for the original n-subeconomy and n -subeconomy also differ, i.e. (p0 (n), q(n)) = (p0 (n ), q(n )).6 As in Lemma 2, the following lemma provides a sufficient condition for different real numeraire asset equilibria to correspond to different parameters. H h h Lemma 4. For each n ∈ N , let ((xh (n), yh (n))H h=1 , p(n), q(n)) and ((x (n), y (n))h=1 , p (n), q (n)) be real numeraire asset equilibria in n-subeconomies with respect to the payoff matrices Λ(n)R and Λ (n)R. Suppose that
(i) dimsp[y1 (n), . . . , yH (n)] = dimsp[y1 (n), . . . , yH (n)] = B. Then, xh (n) = xh (n) for all h if and only ifp(n) = p (n) and q(n) q (n) (ii) sp = sp . Λ(n)R Λ (n)R (L−1)(S+1) L(1+S)H Let fn (p\ (n), q(n), Λ(n), e(n)) : R++ × RB × RS++ × R++ (≡ Qn ) → (L−1)(S+1) B R × R denote the excess demand functions for commodities 2, . . . , L and
assets in the n-subeconomy where we normalize the price of commodity one in each state s to be 1 and p\ (n) denotes the prices of commodities 2, . . . , L. By Walras’ law, fn = 0 implies an equilibrium of the n-subeconomy. Therefore, the aggregate excess demand is defined as F(p\ , q, Λ, e) = (f1 , . . . , fN ) : (L−1)(S+1)N L(1+SN)H R++ × RBN × RSN (≡ Q) → R(L−1)(S+1)N × RBN . Note that by ++ × R++ our n-invariant assumption of the initial endowment in period t = 0, the space of the L(1+SN)H L(1+S)NH endowments is R++ , and not R++ . In spite of the n-invariant constraint on e0 , we can change each equations in F independently by means of endowment e perturbations. See Pietra and Siconolfi (1998) for the proof. Thus, ∂e F has full rank and, for almost all e, Fe−1 (0) is a Cr manifold of dimension SN. \ \ Next, we define g(n,n ) = (p0 (n), q(n)) − (p0 (n ), q(n )) and G(n,n ) = (F, g(n,n ) ) : Q → R(L−1)(S+1)N × RBN × RL−1 × RB . Since ∂p\ (n),q(n) g(n,n ) is evidently of full rank, 0 combined with the fact that ∂e F is of full rank, it follows that G(n,n ) is transverse to 0. r Thus, for almost all e, G−1 (n,n )e (0) is a C manifold of dimension SN − (L − 1) − B. B h h Moreover, let hn (p\ (n), q(n), Λ(n), e(n), z(n)) = ( B h=1 zh y1 (n), . . . , h=1 zh yB (n)) : h B Qn × J → R , where J is the B − 1 sphere and yb (n) is agent h’s demand for asset b 6 As is easily seen from Proposition 2, this is not necessary for (p (n), q(n)) = (p (n ), q(n )) in our orig0 0 inal financial economy because (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) for the real numeraire economy just requires (p0 (n), q(n)) = α(p0 (n ), q(n )), α ∈ R++ , for the corresponding original economy. This is a result of “nominal” indeterminacy intrinsic to nominal asset incomplete market economies (see Geanakoplos and Mas-Colell, 1989). Thus, in our definition of FR prices, we need not consider the condition (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) and our analysis pertaining to this condition is also valid for an alternative definition of FR prices, namely L(p0 , q) = {. . . , θ, . . . } where θ = {n ∈ N |(p0 (n)/p0 (n), q(n)/q(n)) = (p0 (θ)/p0 (θ), q(θ)/q(θ)) }, which is used in Citanna and Villanacci (2000). In any case, this is a technical problem of nominal indeterminacy of our economy and has no importance for our result concerning the dimension of real indeterminacy.
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at (p\ (n), q(n), Λ(n), e(n)) in the n-subeconomy. Define Hn = (F, hn ) : Q × J → R(L−1)(S+1)N × RBN × RB . Following Geanakoplos and Mas-Colell (1989), we can show −1 (0) is a C r manifold of dimension that Hn is transverse to 0. Hence, for almost all e, Hne SN − 1. From these arguments it follows that for almost all e, we can take a relatively open set −1 on Fe−1 (0) disjoint from both G−1 (n,n )e (0) for any pair of (n, n ) and Hne (0) for any n. Note that (p\ , q, Λ) ∈ V , satisfy p(n) = p(n ) whenever n = n and the corresponding {yh (n)} satisfy (i) in Lemma 4 for each n. As in the case of Section 4, we define, for any (p\ , q, Λ) ∈ V ,
W(p\ , q, Λ) = {(p\ , q , Λ ) ∈ V |commodity allocations are as those corresponding to (p, q, Λ) } We prove that the dimension of W(p\ , q, Λ) is constant for any (p\ , q, Λ) in V . We write this dimension as dimW. Consider the conditions of Lemma 4. As shown by Geanakoplos and Mas-Colell (1989), for each n-subeconomy, (ii) is satisfied if and only if q(n) = αq (n) and Λ(n) = αΛ (n) for some α ∈ R++ . Hence (p\ (n), q(n), Λ(n)) and (p\ (n), q , Λ (n)) induce the same commodity allocation if and only if q(n) = αq (n) and Λ(n) = αΛ (n) for some α ∈ R++ . −1 (0), (p\ (n), αq(n), αΛ(n)) Further, it is easily shown that if (p\ (n), q(n), Λ(n)) is in fne(n)
−1 is also in fne(n) (0) for any α ∈ R++ . Therefore, W(p\ , q, Λ) = {(p\ , α1 q(1), . . . , αN q(N), α1 Λ(1), . . . , αN Λ(N)) ∈ V |αi ∈ R++ , i = 1, . . . , N} for any (p\ , q, Λ) ∈ V , and dimW = N. Thus, the dimension of real indeterminacy is dimV − dimW = SN − N. This ends the proof of Theorem 2. 䊐
In the case the asset market in each n-subeconomy is complete, i.e. B = S, the equilibrium in each n-subeconomy is determinate. As a consequence, the FR equilibrium of our economy is determinate with a zero-dimensional real indeterminacy.
6. Indeterminacy of PR equilibrium Even though the indeterminacy of PR equilibrium is an open issue, as a corollary of the results obtained above we discuss some types of PR equilibria. Consider a PR equilibrium such that the number of partitions induced on N by prices in period t = 0 is J: L(p0 , q) = {θ1 , . . . , θJ }. Suppose that each θj is a product of the subsets of agents’ private signals, i.e. (∗)θj = h N hj where N hj ⊂ N h for all h. (See Fig. 1, for example. The partition satisfies the property (∗) in Case 1, but not in Case 2.) Then 7θj = h 7N hj , where 7 denotes the number of the elements. Evidently, j 7θj = N. For PR equilibria satisfying (∗), we can easily obtain the dimension of real indeterminacy, but we will not repeat a rigorous but tedious proof. Given the partition induced by prices, we have θj -subeconomies, and can consider the indeterminacy of equilibrium in each θj -subeconomy independently. Each θj -subeconomy can be considered as an economy where each agent’s private information is given by N hj .
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Fig. 1. Example of the partitions induced by prices.
In each θj -subeconomy, the prices in period t = 0 should be constant across n ∈ θj . Therefore, the desired equilibrium in the subeconomy is regarded as a FNR equilibrium, and we can consider “modified” and “constrained” θj -subeconomies. As in the case of Section 4, the condition such that the constrained θj -subeconomy equilibrium is identified with the original θj -subeconomy equilibrium is expressed by h (7N hj − 1)B equations. On the other hand, the constrained θj -subeconomy is identified with a standard incomplete nominal asset market economy with S(7θj ) states, where the dimension of indeterminacy of equilibrium is S(7θj )−1 (S(7θj )−S if S = B). Thus, the dimension of indeterminacy of the original θj -subeconomy is S(7θj )− h (7N hj −1)B−1 (S(7θj )− h (7N hj −1)B−S if S = B). Hence, the dimension of the real indeterminacy of the PR equilibrium is j {S(7θj ) − h h h h (7N j − 1)B − 1} = SN − j h (7N j − 1)B − J (SN − j h (7N j − 1)B − JS if S = B). Evidently any FNR or FR equilibrium is a special case of a PR equilibrium satisfying (∗). In the case of FNR equilibrium, J = 1 and 7N h1 = 7N h = N h . In the case of FR equilibrium, J = N and 7N hj = 1 for all j and h.
7. Discussion So far, we have relied on the framework given by Polemarchakis and Siconolfi (1993) and have investigated a general case. As shown in Theorems 1 and 2, we cannot provide a general answer to the question of which has larger dimension of real indeterminacy, FR equilibrium or FNR equilibrium. This is because in the general model, possible incompleteness of the market arise from two sources: the states of private information N and the states of nature S.
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We could simplify the model and put emphasis on the role of private information revelation by setting S = B: each n-subeconomy is complete after the information is fully revealed, that is, the market is potentially complete. Further simplification is obtained by setting S = 1, where the state of tomorrow consists of only the state of private information, while that is equal to S × N in the general model. This would be the most useful model for applications. In such a case, it is sufficient to consider just one nominal asset. Thus, S = B = 1 is the minimal setup to produce the non-revelation of the private information. As shown in Theorem 1 and in the comment after the proof of Theorem 2, a non-informative rational expectations equilibrium has real indeterminacy of strictly positive dimension N − h (N h − 1) − 1, whereas a revealing equilibrium is determinate. Even if the economy is potentially complete, the persistence of asymmetric information among agents inevitably leads to the structure of an incomplete market, and becomes the source of real indeterminacy when the assets are nominal. When the agents have private informations and asset are nominal, even if the market is potentially complete, it is possible that the equilibrium prices do not reveal individual private information so that the agents have to trade in a incomplete market. Some technical comment on this simplified case maybe useful. In the general model, as we see in the proof of Theorem 1, we need the rather complicated approach of Geanakoplos and Mas-Colell (1989) to connect the nominal indeterminacy to the real indeterminacy. In the simplified model, however, we can estimate the dimension of the real indeterminacy directly from the dimension of the nominal indeterminacy so that there would be no need of the discussion contained in Step 3. The presence of real indeterminacy in non-informative economies calls therefore for an explicit modeling of how money affects equilibrium outcomes. Future research may then address the question of how a central bank, pursuing possibly a welfare criterion, may select among equilibria and revelation.
Acknowledgements We are grateful to the editor and the co-editor and an anonymous referee for suggesting us to produce this joint paper. Paola Donati wishes to thank Andreu Mas-Colell, Heracles Polemarchakis and Paolo Siconolfi. Takeshi Momi gratefully acknowledges the Grant-in-aid for JSPS Fellows. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Eurosystem or the European Central Bank. References Balasko, Y., Cass, D., 1989. The structure of financial equilibrium with exogenous yields: the case of incomplete markets. Econometrica 57, 135–162. Cass, D., 1985. On the number of equilibrium allocations with incomplete financial markets. CARESS Working Paper, University of Pennsylvania. Citanna, A., Villanacci, A., 2000. Existence and regularity of partially revealing rational expectations equilibrium in finite economies. Journal of Mathematical Economics 34, 1–26.
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Geanakoplos, J., Mas-Colell, A., 1989. Real indeterminacy with financial assets. Journal of Economic Theory 47, 22–38. Pietra, T., Siconolfi, P., 1998. Fully revealing equilibria in sequential economies with asset markets. Journal of Mathematical Economics 29, 211–223. Polemarchakis, H., Siconolfi, P., 1993. Asset markets and information revealed by prices. Economic Theory 3, 645–661. Radner, R., 1979. Rational expectations equilibrium: generic existence, and the information revealed by prices. Econometrica 47, 655–678. Rahi, R., 1995. Partially revealing rational expectations equilibria with nominal assets. Journal of Mathematical Economics 24, 137–146. Stahn, H., 2000. A remark on rational expectation equilibria with incomplete markets and real assets. Journal of Mathematical Economics 33, 441–448. Werner, J., 1990. Structure of financial markets and real indeterminacy of equilibria. Journal of Mathematical Economics 19, 217–232.
Indeterminacy of rational expectations equilibria in sequential financial markets夽 Paola Donati a , Takeshi Momi b,∗ b
a European Central Bank, Frankfurt am Main, Germany Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo–ku, Tokyo, Japan
Received 8 December 2000; received in revised form 4 November 2002; accepted 1 February 2003
Abstract We study a two-period, pure exchange economy with an incomplete nominal asset market where agents have private information and learn from prices. It is well known that the indeterminacy of equilibrium prices, and, generically, allocations, that characterises economies with an incomplete financial market and nominal assets may be exploited to determine the degree of information disclosure by rational expectations equilibrium prices. Here we provide a general characterization of the dimension of the indeterminacy of rational expectations equilibria. © 2003 Elsevier Science B.V. All rights reserved. JEL classification: D52; D80 Keywords: Incomplete markets; Indeterminacy; Information revelation
1. Introduction In intertemporal economies under uncertainty and asymmetric information, individuals with rational expectations formulate forecasts on tomorrow’s equilibrium outcomes after refining their initial beliefs with the information they extract from the prices they observe on the market today. In fact, the rational expectations hypothesis requires that individuals both correctly establish the relationship between the equilibrium price domain and the exogenous economic variables, and they exploit the knowledge of the price map to perfect their private information. If an exhaustive specification of the fundamentals of the economy corresponds 夽 This work merges two papers independently written: “Indeterminacy in sequential financial market economies when prices reveal information” by Paola Donati and “Indeterminacy of rational expectations equilibria” by Takeshi Momi. ∗ Corresponding author. Tel.: +81-35-841-5588. E-mail addresses: [email protected] (P. Donati), [email protected] (T. Momi).
0304-4068/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-4068(03)00023-5
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to each state of aggregate private information, and today’s observed prices convey such joint signal entirely, that is, if prices are fully revealing, individuals with rational expectations may foresee the equilibrium value of tomorrow’s variables exactly. If rational expectations equilibrium prices disclose just some, but not all, of the individual signals, that is, if they are partially revealing, or if they do not provide any private information feedback at all, that is, if they are fully non-revealing, individuals are prevented from completing their knowledge refinement and remain asymmetrically informed at equilibrium. In sequential, general equilibrium economies under uncertainty, the revelation or nonrevelation content of rational expectation equilibrium prices depends on the asset structure. Radner (1979) first showed that in a standard two-period general equilibrium economy where asymmetrically informed individuals receive a finite number of private signals, exchange real assets in an incomplete financial market and the equilibrium price vector has higher dimensionality than the space of individual private information, rational expectation equilibrium prices are fully revealing. Pietra and Siconolfi (1998) and Stahn (2000) extended Radner’s result and established the existence of fully revealing rational expectations equilibria for intertemporal economies in which physical commodities at each spot are strictly more than one. As a consequence, the budget constraints that individuals face when they have to decide on their income allocation across time cannot be trivially reduced to one. In non-trivial sequential economies, in fact, the number of budget constraints in an optimization problem is determined by the level of refinement of individual knowledge. This is defined by the join of the partitions induced on the state space both by private signals and by the information conveyed by equilibrium prices. The finer the joint partition, the finer individual’s knowledge and the lower the number of the budget constraints in his optimization. When individuals transfer revenue across states of the world by means of nominal assets, i.e. assets paying off in terms of unit of account, as opposed to real assets as in Radner’s original setting, the indeterminacy of equilibrium prices, and, generically, allocations,1 can be exploited to fix the degree of information disclosure by rational expectation equilibrium prices. The existence of fully non-revealing rational expectations equilibria was proved by Polemarchakis and Siconolfi (1993). Later, Rahi (1995) by imposing ad hoc restrictions on individual’s information at equilibrium, and Citanna and Villanacci (2000) by means of homotopy methods, showed the existence of partially revealing equilibria. Indeterminacy is an immanent characteristic of the rational expectations equilibrium of any nominal asset economy. Specifically, even when the financial market is potentially complete (S = B), that is, the asset market of each subeconomy is complete after individuals’ private information have been fully revealed, a nominal asset economy still has an indeterminate fully non-revealing and an indeterminate partially revealing equilibrium. As a matter of fact, as shown by the above literature, the existence of a non-revealing equilibrium does not exclude the existence of a fully revealing equilibrium. A natural question which then arises, concerns the dimension of the revealing and non-revealing equilibria (see Polemarchakis and Siconolfi, 1993). This work shows that generically, the dimension expectations equilibrium is: of theh real indeterminacy of non-revealing rational H h SN − H h=1 (N − 1)B − 1 when S > B, and that it is: SN − h=1 (N − 1)B − S if S = B. By contrast, generically, the dimension of the real indeterminacy of the revealing equilibria 1
See Balasko and Cass (1989), Cass (1985), Geanakoplos and Mas-Colell (1989) and Werner (1990).
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is: (S −1)N when S > B, while if, S = B, the real indeterminacy of a revealing equilibrium is 0-dimensional. S, B, H denote the number of states, assets and individuals, respectively, h while N h is the number of the private information of individual h and N = H h=1 N is the total number of the economy-wide information. Finally, the dimension of the real indeterminacy of some special types of partially revealing rational expectations equilibria is estimated. The paper is organized as follows. Sections 2 and 3 present the model, define the notion of equilibrium and describe the parameterization of the space of economies. Section 4 focuses on fully revealing rational expectations equilibria. Section 5 deals with fully non-revealing equilibria and Section 6 with partial revelation. Section 7 concludes.
2. The economy Consider a standard two-period model of a general equilibrium, pure exchange economy under uncertainty, with asymmetric information as in Polemarchakis and Siconolfi (1993). There is a finite set of agents, denoted by the superscript h ∈ H = {1, . . . , H}. At time 0, before trade occurs, each agent h receives a private signal nh ∈ N h = {1, . . . , N h }. No agent can directly observe the private signal received by any another agent. The set of joint signals across agents is N = h∈H N h = {1, . . . , N}, where N = h∈H N h . Each state of aggregate information n ∈ N is represented by the H-dimensional collection (n1 , . . . , nh , . . . , nH ), nh ∈ N h . For every agent, it will be useful to write n = (nh , n−h ) ∈ N h × N −h , where n−h represents signals of agents other than h and N −h =
k∈H \{h} N k . The private information of agent h is alternatively described by the partition Lh = {. . . , Lh , . . . } = {N h × {N −h }} of the set of joint signals. Given the state of information n ∈ N in period 0 and the state of nature s ∈ S = {1, . . . , S} in period 1, there is a spot market where agents trade L finitely many commodities indexed by a subscript l ∈ L = {1, . . . , L}. Commodities are exchanged only on spot markets and there are no markets for contingent commodities. Agent h’s conh (s, n), . . . ) for period 1 and xh (n) = sumption vector at state (s, n) is x1h (s, n) = (. . . , x1l 0 (. . . , x0l (n), . . . ) for period 0. His consumption vector associated with signal n is xh (n) = (x0h (n), x1h (n)) = (x0h (n), (. . . , x1h (s, n), . . . )), while xh = (. . . , xh (n), . . . ). His consumption vectors associated with time period t is denoted by x0h = (. . . , x0h (n), . . . ) ∈ RLN and x1h = (. . . , x1h (n), . . . ) ∈ RLSN . We write x = (x1 , . . . , xH ). We use a similar notation to refer to the commodity price vectors, p, and the initial endowment vectors, eh . We assume that individual initial endowments are strictly positive, and that the initial endowment of every agent in period 0 is signal independent. Thus, eh0 is a vector of period 0 endowment LSN h in RL ++ and e1 is a vector of period 1 endowments in R++ . We parameterize the space of L(1+SN)H the economy in terms of endowments, hence the relevant space is E = R++ . Intertemporal trade occurs only through the exchange of financial assets in period 0. There are B nominal inside assets denoted by a superscript b ∈ B = {1, . . . , B}. The asset price vector given signal n is q(n) = (. . . , qb (n), . . . ), while q = (. . . , q(n), . . . ). We assume that the asset payoffs are signal independent. Asset b pays rsb abstract units of account in period 1 when state s occurs. The S ×B matrix R = (rsb )b∈B s∈S is the payoff matrix. We assume
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that R is of full column rank, and, for simplicity, that R is in general position.2 Agent h’s portfolio, given signal n, is yh (n) = (. . . , ybh (n), . . . ), while yh = (. . . , yh (n), . . . ). We write y = (y1 , . . . , yH ). Let L(p0 , q) = {. . . , θ, . . . } be the partition on N induced by period 0 prices: θ = {n ∈ N |(p0 (n), q(n)) = (p0 (θ), q(θ)) } Let Lh (p0 , q) = Lh ∨L(p0 , q) = {. . . , θ h , . . . } be the joint of agent h’s private information and the price-induced partition agent h rests on to make his investment and consumption plans. Each agent has a prior distribution πh over the state and signal spaces. We assume agent h’s preferences are represented by a state and signal independent von Neumann-Morgenstern utility function3 where cardinal preferences are represented by a smooth and strictly concave utility function uh : RL ++ → R. Therefore, agent h’s ex-ante utility is described as h h h h h h h h h h v (x , L (p0 , q)) = π (n|θ ) u (x0 (n)) + π (s|n)u (x1 (n)) s∈S
θ h ∈Lh (p0 ,q) n∈θ h
given (p, q). 3. Equilibrium A vector ((xh , yh )H h=1 , p, q) is a rational expectations equilibrium if, given p and q, solves the following optimization problem:
(xh , yh )
maxvh (x, Lh (p0 , q)) s.t.
p0 (n)(x0 (n) − eh0 ) = −q(n)y(n),
p1 (n)䊐(x1 (n) − eh1 (n)) = Ry(n), (x0 , y) is Lh (p0 , q)-measurable and markets clear, i.e. (x0h (n) − eh0 ) = 0,
n∈N
n∈N
n∈N
h
(x1h (n) − eh1 (n)) = 0,
n∈N
h
yh (n) = 0,
n∈N
h 2 It is not difficult to consider indeterminacy without this assumption (see Geanakoplos and Mas-Colell, 1989; Werner, 1990). 3 Our result holds also for more general preferences. What is needed are the signal invariance of the utility functions over first period consumption and the signal separability properties. The former is required for our result on the real indeterminacy of FNR equilibrium (see Polemarchakis and Siconolfi (1993)), and the latter for our result on the real indeterminacy on FR equilibrium (see Pietra and Siconolfi (1998)).
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A rational expectations equilibrium is fully revealing (FR) if (p0 (n), q(n)) = (p0 (n ), q(n )) whenever n = n . A rational expectations equilibrium is fully non-revealing (FNR) if (p0 (n), q(n)) is constant across n. A rational expectations equilibrium is partially revealing (PR) otherwise. Thus, at FR equilibria, L(p0 , q) = {{1}, . . . , {n}, . . . , {N}}, while at FNR equilibria, L(p0 , q) = {N}.
4. Indeterminacy of FNR equilibrium First we consider the indeterminacy of FNR equilibrium. Theorem 1. If 1 ≤ B < S, H≥ B, and if R is in general position, then, generically in E, FNR equilibrium has SN − h (N h − 1)B − 1 dimensional real indeterminacy: the set of FNR equilibriumcommodity allocations contains the image of a C1 one-to-one function h with domain RSN− h (N −1)B−1 . If B = S, the dimension of the real indeterminacy is h SN − h (N − 1)B − S. Proof. The proof consists of three steps. The first step introduces the notions of modified optimization and constrained equilibrium and presents our first key observation: a FNR equilibrium allocation is identified with a constrained equilibrium allocation satisfying a modified optimization. We also show that the conditions of a constrained equilibrium allocation satisfying a modified optimization are expressed by h (N h − 1)B equations. The second step contains another key point: the constrained economy is identified with a standard incomplete market economy with SN states in period t = 1 and B financial assets. It is well known that the equilibrium of such an economy has a SN − 1 (or SN − S if S = B)-dimensional real indeterminacy (see Geanakoplos and Mas-Colell, 1989). In the third step, combining the above results, we prove that the dimension of the real indeterminacy of FNR equilibrium is SN − h (N h − 1)B − 1 (or SN − h (N h − 1)B − S if S = B). Step 1. We introduce the notions of modified optimization and constrained equilibrium, and show that a FNR equilibrium allocation is identified with a constrained equilibrium allocation satisfying a modified optimization. We distinguish the economy just defined from the economy presented in Section 3 denoting the latter as original. The technique of introducing a modified equilibrium is standard (see Citanna and Villanacci, 2000; Polemarchakis and Siconolfi, 1993). Given non-informative prices (p, q), where (p0 (n), q(n)) = (pˆ 0 , qˆ ) for all n, consider the modified budget constraint: pˆ 0 (x0 (n) − eh0 ) = − qˆ y(n) n∈N
n∈N
p1 (n)䊐(x1 (n) − eh1 (n)) = Ry(n), h
(x0 , y) is L -measurable
n∈N
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The modified budget constraint differs from the budget constraint of the original economy in that the constraints in period 0 associated with a realization of the joint signal have been replaced by their sum. The signal independence property of period 0 endowment and utility function implies that period 0 consumption which solves the modified budget constraint optimization at non-informative prices is also signal independent: x0h (n) = xˆ 0h for all n. Thus, under the modified budget constraint, agent h optimization problem can be rewritten as: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −
1 qˆ y(n) N
p1 (n)䊐(x1 (n) − eh1 (n)) h
(1)
n∈N
= Ry(n),
n∈N
y is L -measurable
where vˆ h (ˆx0 , x1 , Lh ) = vh (. . . , xˆ 0 , . . . , x1 , Lh ). A solution to the modified optimization problem exists and it is unique. It coincides with the solution to the original economy optimization problem at non-informative prices (p, q): (p0 (n), q(n)) = (p, ˆ qˆ ) for all n, whenever yh (n) is n-invariant for all h. The vector h h H h ((ˆx0 , x1 , y )h=1 , pˆ 0 , p1 , qˆ ) is a rational expectations equilibrium for the modified economy if (ˆx0h , x1h , yh )H optimization problem at (pˆ 0 , p1 , qˆ ), and it satisfies the h=1 solves the modified h market clearing conditions: h (ˆx0 − eh0 ) = 0, h (x1h (n) − eh1 (n)) = 0 and h yh (n) = 0 for all n ∈ N . The following lemma shows that yh (n) should in fact be n-invariant at a modified equilibrium, and that the modified equilibria and the FNR equilibria of the original economy actually coincide. Lemma 1. (1) If ((ˆx0h , x1h , yh )H ˆ 0 , p1 , qˆ ) is a modified equilibrium, then for each h, h=1 , p h yh is n-invariant, i.e. yh (n) = yˆ h for all n. Thus, ((x0h , x1h , yh )H h=1 , p, q) where x0 (n) = h xˆ 0 and (p0 (n), q(n)) = (pˆ 0 , qˆ ) for all n is a FNR equilibrium of the original economy. (2) If ((x0h , x1h , yh )H h=1 , p, q) is a FNR equilibrium of the original economy, then, by definition, p0 (n) = pˆ 0 , q(n) = qˆ and, for each agent h, x0h and yh are n-invariant, i.e. x0h (n) = xˆ 0h and yh (n) = yˆ h for all n. Hence, ((ˆx0h , x1h , yh )H ˆ 0 , p1 , qˆ ) is a modified h=1 , p equilibrium. Proof. The proof is essentially the same as in Polemarchakis and Siconolfi (1993). (1) The market clearing conditions of the (modified or original) equilibrium imply that, for h ∈ H , yh (n) = yh (nh , n−h ) = − k∈H \h yk (nh , n−h ) for all (nh , n−h ) ∈ N h × N −h . But for k ∈ H \ h, the Lk (p0 , q)-measurability condition at non-informative prices implies that yk is nh -invariant. Therefore, yh is nh -invariant and, equivalently, it is n-invariant. (2) Since yh is n-invariant we have that yh (n) = yˆ h for all n, and period 0 budget constraint implies that p(x ˆ 0h (n) − e0 ) = −ˆqyˆ h for all n. From our assumptions on the utility function it follows that x0h is n-invariant. 䊐 Lemma 1 motivates the introduction of the notion of constrained optimization. Given (pˆ 0 , p1 , qˆ ), a constrained optimization is an optimization problem which has to satisfy also
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the constraint that y be n-invariant, i.e. that y(n) = yˆ for all n ∈ N : maxˆvh (ˆx0 , x1 , Lh ) pˆ 0 (ˆx0 − eh0 ) = −ˆqyˆ
s.t.
p1 (n)䊐(x1 (n) − eh1 (n))
= Rˆy,
(2) n∈N
The vector ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) is a constrained rational expectations equilibrium h=1 , p if (ˆx0h , x1h , yˆ h )H solves the constrained optimization problem at prices (pˆ 0 , p1 , qˆ ) and if h=1 h h h markets clear: h (ˆx0 − e0 ) = 0, h (x1 (n) − eh1 (n)) = 0 for all n ∈ N , and h yˆ h = 0. Suppose that ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) is a constrained equilibrium. If also h=1 , p h (ˆx0 , x1h , yˆ h )H solves the modified optimization problem at prices (pˆ 0 , p1 , qˆ ), then h=1 h h H h ((ˆx0 , x1 , yˆ )h=1 , pˆ 0 , p1 , qˆ ) is a modified equilibrium, and hence it induces a FNR equilibrium in the original economy. On the other hand, suppose ((xh , yh )H h=1 , p, q) is a FNR rational expectations equilibrium in the original economy. Thus, x0h (n) = xˆ 0h , yh (n) = yˆ h , p0 (n) = pˆ 0 and q(n) = qˆ are satisfied for all n, and ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) is a constrained equilibrium and h=1 , p h h H h ((ˆx0 , x1 , y )h=1 , pˆ 0 , p1 , qˆ ) is a modified equilibrium. Now we specify under what conditions a constrained equilibrium allocation satisfies the modified optimization problem. Compare the first-order conditions of these problems. The first-order conditions of agent h’s constrained optimization problem (2) are: ∂vh ∂xˆ 0h
− µh0 pˆ 0 = 0
∂vh ∂x1h (n)
(3)
− µh (n)䊐p1 (n) = 0,
−µh0 qˆ +
n∈N
(4)
µh (n)R = 0
(5)
n∈N
pˆ 0 (ˆx0h − eh0 ) = −ˆqyˆ h
(6)
p1 (n)䊐(x1h (n) − eh1 (n)) = Rˆyh ,
n∈N
(7)
where µh0 and µh (n) = (. . . , µh (s, n), . . . ), n ∈ N, are the multipliers associated with the constraints of maximization problem (2). Because of the Lh -measurability condition on y in the modified optimization (1), y(n) is constant across n−h , that is, y(nh , n−h ) = y¯ (nh ) for any n−h ∈ N −h , for each nh ∈ N h , and the modified optimization can be rewritten as: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −
1 qˆ y¯ (nh ) Nh h h
n ∈N p1 (nh , n−h )䊐(x1 (nh , n−h ) − eh1 (nh , n−h ))
(8) = R¯y(nh ),
(nh , n−h ) ∈ N h × N −h
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As a consequence, the first-order conditions of the modified maximization problem are: ∂vh − µh0 pˆ 0 = 0 (9) ∂xˆ 0h ∂vh ∂x1h (nh , n−h ) −
µh0 qˆ + Nh
− µh (nh , n−h )䊐p1 (nh , n−h ) = 0,
µh (nh , n−h )R = 0,
(nh , n−h ) ∈ N h × N −h
nh ∈ N h
(10)
(11)
n−h ∈N −h
pˆ 0 (ˆx0h − eh0 ) = −
1 qˆ y¯ h (nh ) Nh h h
(12)
n ∈N
h
−h
p1 (n , n
)䊐(x1h (nh , n−h ) − eh1 (nh , n−h ))
= R¯yh (nh ),
(nh , n−h ) ∈ N h × N −h (13)
µh0
µh (nh , n−h )
(. . . , µh (s, (nh , n−h )), . . . ),
(nh , n−h )
Nh
N −h
and = ∈ × are where the multipliers corresponding to each of the constraints in the modified optimization. Even though we use the same symbol µ to denote the multipliers of the two maximizations, no confusion should arise. Suppose that (ˆx0h , x1h , yˆ h , µh0 , µh (n), n ∈ N ) satisfies (3)–(7). If we set y¯ h (nh ) = yˆ h for all nh , then (ˆx0h , x1h , y¯ h (nh ), nh ∈ N h , µh0 , µh (n), n ∈ N ) satisfies all of the conditions (9) and (13) of the modified optimization problem except for (11). This means that the condition required for the variables in a constrained equilibrium to satisfy the modified optimization problem is that the multipliers µh = (µh0 , µh (n), n ∈ N ) at the constrained equilibrium satisfy (11) for each h ∈ H . Since the µh satisfy (5), then (11) is satisfied if and only if: µh (nh , n−h )R = µh (nh , n−h )R, for any nh , nh ∈ N h (14) n−h ∈N −h
n−h ∈N −h
It is evident that (14) is equivalent to (N h − 1)B equations: µh (nh , n−h )R − µh (nh + 1, n−h )R = 0, n−h ∈N −h
nh = 1, . . . , N h − 1
n−h ∈N −h
Thus, for all h ∈ H , (14) consists of
h∈H (N
h
− 1)B equations.
Step 2. The constrained equilibrium is naturally identified with the equilibrium of a standard incomplete market economy with SN states in period t = 1 and B nominal assets. The equilibrium indeterminacy of a standard incomplete nominal asset market economy has already been treated by Balasko and Cass (1989), Geanakoplos and Mas-Colell (1989) and Werner (1990). By using the same techniques, here we deal with the equilibrium inde terminacy of a standard incomplete market economy which is subject to the h (N h − 1)B additional constraints obtained in Step 1. Now, we introduce a real numeraire asset economy. When we consider not only the states of nature s, but also the states of information n as the index of the states in period t = 1, the n-invariant maximizations (2) can be naturally
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identified with the maximization problem of a standard incomplete asset market economy with SN states in period t = 1, B nominal assets, and the SN × B payoff matrix R . R = .. . R For (s, n) ∈ S × N , we introduce the following lexicographic order: (s, n) < (s , n ) if and only if n < n or n = n and s < s . Once our constrained economy is identified with a standard incomplete asset market economy, we can treat the indeterminacy by applying standard arguments. It is well known that the equilibrium allocations of a nominal asset economy can be identified with those of a real numeraire asset economy. Then, consider an economy with real numeraire assets, which is equivalent to an economy with financial assets characterized by the payoff matrix R. Let, ˜ R(1) ˜ = ... R ˜ R(N) be a SN × B matrix representing the real asset payoffs of the numeraire (commodity one). ˜ S × B matrix R(n) = (˜r b (s, n))b∈B s∈S contains the payoffs corresponding to states (s, n) ∈ b ˜ (S, n), and R(s, n) = (. . . , r˜ (s, n), . . . ) denotes the payoffs of assets at state (s, n). The utility maximization problem of agent h in the real numeraire asset economy is: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −ˆqyˆ
˜ n)ˆy, p1 (s, n)䊐(x1 (s, n) − eh1 (s, n)) = p11 (s, n)R(s,
(s, n) ∈ S × N
where p11 (s, n) is the price of commodity one at state (s, n). To highlight the link with the constrained optimization problem, we rewrite such maximization problem as: maxˆvh (ˆx0 , x1 , Lh ) s.t.
pˆ 0 (ˆx0 − eh0 ) = −ˆqyˆ
˜ y, p1 (n)䊐(x1 (n) − eh1 (n)) = [p11 (n)]R(n)ˆ
n∈N
where [p11 (n)] is a diagonal S × S matrix, the elements of which are prices of commodity one, p11 (s, n). We define a real numeraire asset equilibrium ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) h=1 , p as the set of variables satisfying the market clearing conditions and utility optimization problem of all agents. The first-order conditions of agent h’s optimization in the real numeraire asset economy are: ∂ˆvh −µ ˜ h0 pˆ 0 = 0 (15) ∂xˆ 0h ∂vh ∂x1h (n)
−µ ˜ h (n)䊐p1 (n) = 0,
n∈N
(16)
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−µh0 qˆ +
˜ µ ˜ h (n)[p11 (n)]R(n) =0
(17)
n∈N
pˆ 0 (ˆx0h − eh0 ) = −ˆqyˆ h
(18)
h ˜ p1 (n)䊐(x1h (n) − eh1 (n)) = [p11 (n)]R(n)y ,
n∈N
(19)
˜ h (n) = (. . . , µ ˜ h (s, n), . . . ), n ∈ N , are the multipliers corresponding to where µ ˜ h0 and µ each of the constraints in the maximization problem. By comparing the first-order conditions (3)–(7) of the constrained economy optimization and conditions (15)–(19) characterizing the individual maximization problem in the real numeraire asset economy, we can easily see the equivalence of the constrained equilibrium ˜ = ΛR, where Λ and the real numeraire asset economy equilibrium with payoff matrix R is a SN × SN diagonal matrix with positive elements. We write Λ(1) .. Λ= . Λ(N) and
Λ(n) =
Λ(1, n)
..
. Λ(S, n)
Proposition 1. If ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ), with multipliers µh for agent h, is a conh=1 , p h h strained equilibrium, then ((ˆx0 , x1 , yˆ h )H ˜ 0 , p˜ 1 , q˜ ), with multipliers µ ˜ h for agent h, h=1 , p ˜ = ΛR, wherep˜ 0 = pˆ 0 /pˆ 01 , is a real numeraire asset equilibrium with asset return R p˜ 1 (s, n) = p1 (s, n)/p11 (s, n), q˜ = qˆ /pˆ 01 , λ(s, n) = 1/p11 (s, n), µ ˜ h0 = pˆ 01 µh0 and h h µ ˜ (s, n) = p11 (s, n)µ (s, n) for all s, n and h. If ((ˆx0h , x1h , yˆ h )H ˜ 0 , p˜ 1 , q˜ ), with multipliers µ ˜ h for agent h, is a real numeraire asset h=1 , p ˜ = ΛR, and p˜ 01 = 1 and p˜ 11 (s, n) = 1 for all s and n, equilibrium with asset return R h , p ˆ , p , q ˆ then ((ˆx0h , x1h , yˆ h )H 0 1 ), with multipliers µ for agent h, is a constrained equih=1 librium where p1 (s, n) = p˜ 1 (s, n)/λ(s, n), µh (s, n) = λ(s, n)µ(s, ˜ n), µh0 pˆ 0 = µ ˜ h0 p˜ 0 and h h µ0 qˆ = µ ˜ 0 q˜ for all s, n and h. As a consequence, the equilibrium of the constrained economy can be parameterized by Λ of the real numeraire asset economy. Note that, from Proposition 1, ˜ h , n−h ) µ ˜ h (nh , n−h )R(n n−h ∈−h
=
˜ h , n−h ), µ ˜ h (nh , n−h )R(n
for any
nh , nh ∈ N h
n−h ∈−h
at the real numeraire asset equilibrium, means (14) at the constrained equilibrium.
(20)
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Step 3. By following arguments which are standard to deal with the indeterminacy of incomplete financial asset market economies, we estimate the dimension of the indeterminacy of our economy’s equilibrium. Although we have to take into account constraints (20), similar arguments can be used so that we are not going to repeat all the details of the proof. Rather, we will highlight the differences caused by constraints (20). In the following discussion, we focus on the equilibrium allocation of commodities in a real numeraire economy. Hence, we normalize the commodity prices: pˆ 01 = 1 and p11 (s, n) = 1 for all s, n, and let \ \ pˆ 0 and p1 (s, n) denote the prices of commodities other than commodity one at state (s, n) in periods t = 0 and t = 1, respectively. A sufficient condition for different real numeraire asset equilibria corresponding to different parameters is the following. Lemma 2. Let ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) and ((ˆx0h , x1h , yˆ h )H ˆ 0 , p1 , qˆ ) be real nuh=1 , p h=1 , p meraire asset equilibria with respect to payoff matrices ΛR andΛ R, respectively. Suppose that (i) dim sp[ˆy1 , . . . , yˆ H ] = dim sp[ˆy1 , . . . , yˆ H ] = B. Then, (ˆx0h , x1h ) = (ˆx0h , x1h ) for all h if and only if (pˆ 0 , p1 ) = (pˆ 0 , p1 ). 4 qˆ qˆ . (ii) = sp ΛR ΛR Proof. The “only if ” part is essentially the same as in Geanakoplos and Mas-Colell (1989). The “if” part is an immediate consequence of the utility maximization of all agents. 䊐 Consider the equilibrium of the real numeraire asset economy with constraint (20). Let \ \ (L−1)SN L(SN+1)H f(pˆ 0 , p1 , qˆ , Λ, e), from P ≡ RL−1 ×RB ×RSN to R(L−1)(SN+1) × ++ ×R++ ++ ×R++ B R be the excess demand function of assets and commodities 2, . . . L, of the numeraire asset economy. By Walras’ law, f = 0 implies an equilibrium. The multipliers µ ˜ h0 , µ ˜ h (s, n) of the optimization problem in the real numeraire asset \ \ \ \ economy are functions of (pˆ 0 , p1 , qˆ , Λ, e). Let gh (pˆ 0 , p1 , qˆ , Λ, e) denote the functions h h from P to R(N −1)B defined by gh = (g1h , . . . , gnhh , . . . , gN h −1 ) and gnhh = (gnhh 1 , . . . , gnhh B ) =
˜ h , n−h ) µ ˜ h (nh , n−h )R(n
n−h ∈N −h
−
µ ˜ h (nh + 1, n−h )R(nh + 1, n−h )
n−h ∈N −h
Note that gh = 0 implies (20). We write g = (g1 , . . . , gH ) and F = (f, g) : P → h R(L−1)(SN+1) × RB × R h (N −1)B . Since the utility functions are smooth, F is a smooth function (see Geanakoplos and Polemarchakis, 1986). 4
For any matrixA, we denote by sp[A] the linear subspace spanned by columns of A.
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\
Lemma 3. F(pˆ 0 , p1 , qˆ , Λ, e) = 0 implies that rank ∂e F = (L − 1)(SN + 1) + B + h h (N − 1)B. Proof. rank∂e f is (L − 1)(SN + 1) + (B − 1) (see Geanakoplos and Polemarchakis, 1986). We show that by means of endowment perturbations, it is possible to change the value of each of the equations of g independently without changing the other equations. Without loss of generality we consider how to change the value of gnhh 1 by 1. First, define +µ(n ˜ h , n−h ), (nh , n−h ) ∈ N h × N −h so that ˜ +µ ˜ h (1, n−h )R(1, n−h ) n−h ∈N −h
= ··· =
h
h
−h
+µ ˜ (n , n
˜ h , n−h ) = )R(n
n−h ∈N −h
N h − nh , 0, . . . , 0 Nh
(21)
˜ h + 1, n−h ) +µ ˜ h (nh + 1, n−h )R(n
n−h ∈N −h
= ··· =
h
h
−h
+µ ˜ (N , n
˜ h , n−h ) = )R(N
n−h ∈N −h
−nh , 0, . . . , 0 Nh
(22)
Note that (21) and (22) imply that +gnhh = (+gnhh 1 , . . . , +gnhh B ) = (1, 0, . . . , 0), while the other equations in g remain unchanged. Furthermore, (17) continues to be satisfied because ˜ +µ ˜ h (n)R(n) n
=
h
h
−h
+µ ˜ (j, n
˜ n )R(j,
−h
)=
n
h
N
˜ n−h ) +µ ˜ h (j, n−h )R(j,
j=1 n−h ∈N −h
j∈N h n−h ∈N −h
+
˜ n−h ) +µ ˜ h (j, n−h )R(j,
j=nh +1 n−h ∈N −h
= nh
h N h − nh −n , 0, . . . , 0 + (N h − nh ) , 0, . . . , 0 = (0, . . . , 0) Nh Nh
Second, define +x1h (nh , n−h ) so that ∂vh ∂x1h (nh , n−h )
(x1h (nh , n−h ) + +x1h (nh , n−h ))
= (µ ˜ h (nh , n−h ) + +µ ˜ h (nh , n−h ))䊐p1 (nh , n−h ) for all (nh , n−h ) ∈ N h × N −h . Finally, define +eh1 (nn , n−h ) = +x1h (nn , n−h ) for all (nh , n−h ) ∈ N h × N −h . From the above discussion it is easy to see that, for such a perturbation of the initial endowment of
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\
agent h, when F(pˆ 0 , p1 , qˆ , Λ, e) = 0, the value of all of the equations of F other than gnh 1 continue to be 0. By the transversality theorem, for almost all e the set Fe−1 (0) is a Cr -manifold of dimen h sion SN − h (N − 1)B. \ \ Let us consider the set of the variables (pˆ 0 , p1 , qˆ , Λ) in Fe−1 (0) with different corresponding commodity allocations. Consider the conditions in Lemma 2, following Geanakoplos \ \ and Mas-Colell (1989). Let J be the B − 1 sphere and define h(pˆ 0 , p1 , q, Λ, e, z) = B B ( h=1 zh yˆ 1h , . . . , h=1 zh yˆ Bh ) : P × J → RB , where yˆ bh is agent h’s demand for asset \ \ b corresponding to (pˆ 0 , p1 , qˆ , Λ). Define H = (F, h) : P × J → R(L−1)(SN+1) × RB × h R h (N −1)B × RB . Then, as shown in Geanakoplos and Mas-Colell (1989), ∂e H has full rank, hence, for almost all e, He−1 (0) is a SN − h (N h − 1)B − 1 dimensional Cr manifold. Thus, we have a relatively open set V on Fe−1 (0) which is disjoint from He−1 (0). For \ \ (pˆ 0 , p1 , qˆ , Λ) ∈ V , the corresponding {ˆyh , h ∈ H } satisfy (i) in Lemma 2. \ \ For (pˆ 0 , p1 , qˆ , Λ) ∈ V , let \
\
\
\
W(pˆ 0 , p1 , qˆ , Λ) = {(pˆ 0 , p1 , qˆ , Λ ) ∈ V |commodity allocations are as those \
\
corresponding to(pˆ 0 , p1 , qˆ , Λ)} \
\
\
\
We show that dimension of W(pˆ 0 , p1 , qˆ , Λ) is constant for any (pˆ 0 , p1 , qˆ , Λ) in V , the former of which will be written as dimW. Obviously, the dimension we require is dimV − dimW. We obtain the dimension of W by considering condition (ii) of Lemma 2. Remember, that R .. R= . R is not in general position although R is assumed to be so. Now we have to distinguish between two cases: S < B and S = B. The argument is the same as in Geanakoplos and Mas-Colell (1989).5 We first consider the case of S < B. In such case (ii) is satisfied if and only if qˆ = αˆq and Λ = αΛ for some α ∈ R++ . The “if” part is trivial and the“only if” part follows that qˆ qˆ = sp sp Λ(n) R Λ(n)R 5 See Section 3 of Geanakoplos and Mas-Colell (1989), where the case of the payoff matrix not in general position is dealt. Using their symbols and the general theorem, what we show in the following two paragraphs is essentially that dimW = K(R) + Z(R) and that K(R) = 1, Z(R) = 0 when S > B and K(R) = S, Z(R) = 0 when S = B.
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for all n ∈ N is necessary for (ii), that R is in general position and that S < B. Thus, from \ \ \ \ Lemma 2, (pˆ 0 , p1 , qˆ , Λ) and (pˆ 0 , p1 , qˆ , Λ ) in V induce the same commodity allocation \
\
\
\
if and only if (pˆ 0 , p1 ) = (pˆ 0 , p1 ), and qˆ = αˆq and Λ = αΛ for some α ∈ R++ . On the other hand, if (ˆxh , x1h , yˆ h , µh ) is the demand for commodities, assets, and multipliers \ \ \ \ of agent h at (pˆ 0 , p1 , qˆ , Λ), (ˆxh , x1h , 1/αˆyh , µh ) correspond to (pˆ 0 , p1 , αˆq, αΛ) for any α ∈ R++ , as is easily recognized from the first-order conditions (15) and (19). Thus, if \ \ \ \ (pˆ 0 , p1 , qˆ , Λ) is actually in Fe−1 (0), then (pˆ 0 , p1 , αˆq, αΛ) is in Fe−1 (0) for any α ∈ R++ . Therefore, dimW = 1 when S > B. When S = B, (ii) is satisfied if and only if qˆ = qˆ R−1 αR and Λ(n) = Λ(n) α for some diagonal matrix α1 .. α= . αS
of S dimension with positive elements. In the above statement, the “if” part is easily checked for, and “only if” part comes from the following discussion. For (ii) to be satisfied, there must exist y ∈ RJ such that qˆ qˆ y for any y ∈ RJ y= ΛR ΛR Equivalently, qˆ y = qˆ y and Λ(n)Ry = Λ(n) Ry for all n. Then y = (Λ(n) R)−1 Λ(n)Ry = R−1 (Λ(n) )−1 Λ(n)Ry for all n and (Λ(n) )−1 Λ(n) have to be independent of n. Defining this diagonal matrix as α, we obtain Λ(n) = Λ(n) α. Substituting y = R−1 αR into qˆ y = \ \ \ \ qˆ y , we obtain qˆ = qˆ R−1 αR. Thus, from Lemma 2, (pˆ 0 , p1 , qˆ , Λ) and (pˆ 0 , p1 , qˆ , Λ ) \
\
\
\
in V induce the same commodity allocation if and only if (pˆ 0 , p1 ) = (pˆ 0 , p1 ), and qˆ = qˆ R−1 αR and Λ(n) = Λ(n) α for some α ∈ RS++ . As in the above case, it is easy to check \
\
\
\
that such (pˆ 0 , p1 , qˆ , Λ ) is actually on V if (pˆ 0 , p1 , qˆ , Λ) is on V . Therefore, dimW = S when S = B. Thus, dimV − dimW = SN − h (N h − 1)B − 1 if S > B (SN − h (N h − 1)B − S if 䊐 S = B). This ends the proof of Theorem 1.
5. Indeterminacy of FR equilibrium In this section we consider the indeterminacy of the FR equilibrium. Theorem 2. If 1 ≤ B < S, H ≥ B, and R is in general position, then, generically in E, the FR equilibrium contains a submanifold of dimension (S − 1)N. In other words, the set of FR equilibrium commodity allocations contains the image of a C1 one-to-one function with domain R(S−1)N .
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Proof. We consider a fictitious economy where agents have full information. We call this economy a full information economy. If an equilibrium price of the full information economy satisfies (p0 (n), q(n)) = (p0 (n ), q(n )) whenever n = n , this equilibrium is supported as a FR equilibrium of the original economy. Being n the aggregate private signal, the full information economy decomposes into N independent n-subeconomies. Each n-subeconomy is a standard general equilibrium economy with financial assets, which can be again translated into an economy with real numeraire assets through a parameterization implemented by means of the S × S diagonal matrix Λ(n). Following Geanakoplos and Mas-Colell (1989), each n-subeconomy equilibrium has a S − 1 dimensional real indeterminacy. Therefore, it is conjectured that the dimension of the real indeterminacy is in total (S − 1)N. To show that (p0 (n), q(n)) = (p0 (n ), q(n )) and at the same time take into account the n-invariance of first period endowment, we follow Pietra and Siconolfi (1998), who proved that equilibrium prices are generically fully revealing in a real numeraire asset economy. Since our proof is easily obtained by combining already established results and the passages of Theorem 1, we do not go into the details. We will use some of the symbols we have already used in the proof of Theorem 1 to emphasize the similarities of these proofs and facilitate understanding. Agent h’s optimization problem in any n-subeconomy is: maxuh (x0h (n)) +
πh (s|n)uh (x1h (n))
s∈S
s.t.
p0 (n)(x0 (n) − eh0 ) = −q(n)y(n)
p1 (n)䊐(x1 (n) − eh1 (n)) = Ry(n)
h The vector ((x(n)h , y(n)h )H h=1 , p(n), q(n)) is a n-subeconomy equilibrium if (x(n) , H h y(n) )h=1 solves the above maximization problem at prices (p(n), q(n)) and markets clear, i.e. h (x0h (n)−eh0 ) = 0, h (x1h (n)−eh1 (n)) = 0 and h yh (n) = 0. Since a n-subeconomy is obviously a standard nominal asset incomplete market economy, we can identify each n-subeconomy with a real numeraire asset economy by means of a Λ(n) parameterization as in Geanakoplos and Mas-Colell (1989) or in Section 4. h Proposition 2. If ((xh (n), yh (n))H h=1 , p(n), q(n)), with multipliers µ (n) for agent h, is an H h h ˜ q˜ (n)), with multipliers µ ˜ h (n) n-subeconomy equilibrium, then ((x (n), y (n))h=1 , p(n), ˜ for agent h, is a real numeraire n-subeconomy equilibrium with asset return R(n) = Λ(n)R wherep˜ 0 (n) = p0 (n)/p01 (n), p˜ 1 (s, n) = p1 (s, n)/p11 (s, n), q˜ (n) = q(n)/p01 (n), ˜ h0 (n) = p01 (n)µh0 (n) and µ ˜ h (s, n) = p11 (s, n)µh (s, n) for all s λ(s, n) = 1/p11 (s, n), µ and h. If ((xh (n), yh (n))H ˜ 0 (n), q˜ (n)), with multipliers µ ˜ h (n) for agent h, is a real numeraire h=1 , p ˜ = Λ(n)R, and p˜ 01 (n) = 1 and p˜ 11 (s, n) = n-subeconomy equilibrium with asset return R H h h 1 for all s, then ((x (n), y (n))h=1 , p(n), q(n)) with multipliers µh (n) for agent h is an ˜ n), n-subeconomy equilibrium where p1 (s, n) = p˜ 1 (s, n)/λ(s, n), µh (s, n) = λ(s, n)µ(s, µh0 (n)p0 (n) = µ ˜ h0 (n)p˜ 0 (n) and µh0 (n)q(n) = µ ˜ h0 (n)˜q(n) for all s and h.
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Then, consider the set of equilibria in the real numeraire n-subeconomies such that (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) whenever n = n . Note that if (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) for a real numeraire economy, the corresponding prices in period t = 0 for the original n-subeconomy and n -subeconomy also differ, i.e. (p0 (n), q(n)) = (p0 (n ), q(n )).6 As in Lemma 2, the following lemma provides a sufficient condition for different real numeraire asset equilibria to correspond to different parameters. H h h Lemma 4. For each n ∈ N , let ((xh (n), yh (n))H h=1 , p(n), q(n)) and ((x (n), y (n))h=1 , p (n), q (n)) be real numeraire asset equilibria in n-subeconomies with respect to the payoff matrices Λ(n)R and Λ (n)R. Suppose that
(i) dimsp[y1 (n), . . . , yH (n)] = dimsp[y1 (n), . . . , yH (n)] = B. Then, xh (n) = xh (n) for all h if and only ifp(n) = p (n) and q(n) q (n) (ii) sp = sp . Λ(n)R Λ (n)R (L−1)(S+1) L(1+S)H Let fn (p\ (n), q(n), Λ(n), e(n)) : R++ × RB × RS++ × R++ (≡ Qn ) → (L−1)(S+1) B R × R denote the excess demand functions for commodities 2, . . . , L and
assets in the n-subeconomy where we normalize the price of commodity one in each state s to be 1 and p\ (n) denotes the prices of commodities 2, . . . , L. By Walras’ law, fn = 0 implies an equilibrium of the n-subeconomy. Therefore, the aggregate excess demand is defined as F(p\ , q, Λ, e) = (f1 , . . . , fN ) : (L−1)(S+1)N L(1+SN)H R++ × RBN × RSN (≡ Q) → R(L−1)(S+1)N × RBN . Note that by ++ × R++ our n-invariant assumption of the initial endowment in period t = 0, the space of the L(1+SN)H L(1+S)NH endowments is R++ , and not R++ . In spite of the n-invariant constraint on e0 , we can change each equations in F independently by means of endowment e perturbations. See Pietra and Siconolfi (1998) for the proof. Thus, ∂e F has full rank and, for almost all e, Fe−1 (0) is a Cr manifold of dimension SN. \ \ Next, we define g(n,n ) = (p0 (n), q(n)) − (p0 (n ), q(n )) and G(n,n ) = (F, g(n,n ) ) : Q → R(L−1)(S+1)N × RBN × RL−1 × RB . Since ∂p\ (n),q(n) g(n,n ) is evidently of full rank, 0 combined with the fact that ∂e F is of full rank, it follows that G(n,n ) is transverse to 0. r Thus, for almost all e, G−1 (n,n )e (0) is a C manifold of dimension SN − (L − 1) − B. B h h Moreover, let hn (p\ (n), q(n), Λ(n), e(n), z(n)) = ( B h=1 zh y1 (n), . . . , h=1 zh yB (n)) : h B Qn × J → R , where J is the B − 1 sphere and yb (n) is agent h’s demand for asset b 6 As is easily seen from Proposition 2, this is not necessary for (p (n), q(n)) = (p (n ), q(n )) in our orig0 0 inal financial economy because (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) for the real numeraire economy just requires (p0 (n), q(n)) = α(p0 (n ), q(n )), α ∈ R++ , for the corresponding original economy. This is a result of “nominal” indeterminacy intrinsic to nominal asset incomplete market economies (see Geanakoplos and Mas-Colell, 1989). Thus, in our definition of FR prices, we need not consider the condition (p˜ 0 (n), q˜ (n)) = (p˜ 0 (n ), q˜ (n )) and our analysis pertaining to this condition is also valid for an alternative definition of FR prices, namely L(p0 , q) = {. . . , θ, . . . } where θ = {n ∈ N |(p0 (n)/p0 (n), q(n)/q(n)) = (p0 (θ)/p0 (θ), q(θ)/q(θ)) }, which is used in Citanna and Villanacci (2000). In any case, this is a technical problem of nominal indeterminacy of our economy and has no importance for our result concerning the dimension of real indeterminacy.
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at (p\ (n), q(n), Λ(n), e(n)) in the n-subeconomy. Define Hn = (F, hn ) : Q × J → R(L−1)(S+1)N × RBN × RB . Following Geanakoplos and Mas-Colell (1989), we can show −1 (0) is a C r manifold of dimension that Hn is transverse to 0. Hence, for almost all e, Hne SN − 1. From these arguments it follows that for almost all e, we can take a relatively open set −1 on Fe−1 (0) disjoint from both G−1 (n,n )e (0) for any pair of (n, n ) and Hne (0) for any n. Note that (p\ , q, Λ) ∈ V , satisfy p(n) = p(n ) whenever n = n and the corresponding {yh (n)} satisfy (i) in Lemma 4 for each n. As in the case of Section 4, we define, for any (p\ , q, Λ) ∈ V ,
W(p\ , q, Λ) = {(p\ , q , Λ ) ∈ V |commodity allocations are as those corresponding to (p, q, Λ) } We prove that the dimension of W(p\ , q, Λ) is constant for any (p\ , q, Λ) in V . We write this dimension as dimW. Consider the conditions of Lemma 4. As shown by Geanakoplos and Mas-Colell (1989), for each n-subeconomy, (ii) is satisfied if and only if q(n) = αq (n) and Λ(n) = αΛ (n) for some α ∈ R++ . Hence (p\ (n), q(n), Λ(n)) and (p\ (n), q , Λ (n)) induce the same commodity allocation if and only if q(n) = αq (n) and Λ(n) = αΛ (n) for some α ∈ R++ . −1 (0), (p\ (n), αq(n), αΛ(n)) Further, it is easily shown that if (p\ (n), q(n), Λ(n)) is in fne(n)
−1 is also in fne(n) (0) for any α ∈ R++ . Therefore, W(p\ , q, Λ) = {(p\ , α1 q(1), . . . , αN q(N), α1 Λ(1), . . . , αN Λ(N)) ∈ V |αi ∈ R++ , i = 1, . . . , N} for any (p\ , q, Λ) ∈ V , and dimW = N. Thus, the dimension of real indeterminacy is dimV − dimW = SN − N. This ends the proof of Theorem 2. 䊐
In the case the asset market in each n-subeconomy is complete, i.e. B = S, the equilibrium in each n-subeconomy is determinate. As a consequence, the FR equilibrium of our economy is determinate with a zero-dimensional real indeterminacy.
6. Indeterminacy of PR equilibrium Even though the indeterminacy of PR equilibrium is an open issue, as a corollary of the results obtained above we discuss some types of PR equilibria. Consider a PR equilibrium such that the number of partitions induced on N by prices in period t = 0 is J: L(p0 , q) = {θ1 , . . . , θJ }. Suppose that each θj is a product of the subsets of agents’ private signals, i.e. (∗)θj = h N hj where N hj ⊂ N h for all h. (See Fig. 1, for example. The partition satisfies the property (∗) in Case 1, but not in Case 2.) Then 7θj = h 7N hj , where 7 denotes the number of the elements. Evidently, j 7θj = N. For PR equilibria satisfying (∗), we can easily obtain the dimension of real indeterminacy, but we will not repeat a rigorous but tedious proof. Given the partition induced by prices, we have θj -subeconomies, and can consider the indeterminacy of equilibrium in each θj -subeconomy independently. Each θj -subeconomy can be considered as an economy where each agent’s private information is given by N hj .
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Fig. 1. Example of the partitions induced by prices.
In each θj -subeconomy, the prices in period t = 0 should be constant across n ∈ θj . Therefore, the desired equilibrium in the subeconomy is regarded as a FNR equilibrium, and we can consider “modified” and “constrained” θj -subeconomies. As in the case of Section 4, the condition such that the constrained θj -subeconomy equilibrium is identified with the original θj -subeconomy equilibrium is expressed by h (7N hj − 1)B equations. On the other hand, the constrained θj -subeconomy is identified with a standard incomplete nominal asset market economy with S(7θj ) states, where the dimension of indeterminacy of equilibrium is S(7θj )−1 (S(7θj )−S if S = B). Thus, the dimension of indeterminacy of the original θj -subeconomy is S(7θj )− h (7N hj −1)B−1 (S(7θj )− h (7N hj −1)B−S if S = B). Hence, the dimension of the real indeterminacy of the PR equilibrium is j {S(7θj ) − h h h h (7N j − 1)B − 1} = SN − j h (7N j − 1)B − J (SN − j h (7N j − 1)B − JS if S = B). Evidently any FNR or FR equilibrium is a special case of a PR equilibrium satisfying (∗). In the case of FNR equilibrium, J = 1 and 7N h1 = 7N h = N h . In the case of FR equilibrium, J = N and 7N hj = 1 for all j and h.
7. Discussion So far, we have relied on the framework given by Polemarchakis and Siconolfi (1993) and have investigated a general case. As shown in Theorems 1 and 2, we cannot provide a general answer to the question of which has larger dimension of real indeterminacy, FR equilibrium or FNR equilibrium. This is because in the general model, possible incompleteness of the market arise from two sources: the states of private information N and the states of nature S.
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We could simplify the model and put emphasis on the role of private information revelation by setting S = B: each n-subeconomy is complete after the information is fully revealed, that is, the market is potentially complete. Further simplification is obtained by setting S = 1, where the state of tomorrow consists of only the state of private information, while that is equal to S × N in the general model. This would be the most useful model for applications. In such a case, it is sufficient to consider just one nominal asset. Thus, S = B = 1 is the minimal setup to produce the non-revelation of the private information. As shown in Theorem 1 and in the comment after the proof of Theorem 2, a non-informative rational expectations equilibrium has real indeterminacy of strictly positive dimension N − h (N h − 1) − 1, whereas a revealing equilibrium is determinate. Even if the economy is potentially complete, the persistence of asymmetric information among agents inevitably leads to the structure of an incomplete market, and becomes the source of real indeterminacy when the assets are nominal. When the agents have private informations and asset are nominal, even if the market is potentially complete, it is possible that the equilibrium prices do not reveal individual private information so that the agents have to trade in a incomplete market. Some technical comment on this simplified case maybe useful. In the general model, as we see in the proof of Theorem 1, we need the rather complicated approach of Geanakoplos and Mas-Colell (1989) to connect the nominal indeterminacy to the real indeterminacy. In the simplified model, however, we can estimate the dimension of the real indeterminacy directly from the dimension of the nominal indeterminacy so that there would be no need of the discussion contained in Step 3. The presence of real indeterminacy in non-informative economies calls therefore for an explicit modeling of how money affects equilibrium outcomes. Future research may then address the question of how a central bank, pursuing possibly a welfare criterion, may select among equilibria and revelation.
Acknowledgements We are grateful to the editor and the co-editor and an anonymous referee for suggesting us to produce this joint paper. Paola Donati wishes to thank Andreu Mas-Colell, Heracles Polemarchakis and Paolo Siconolfi. Takeshi Momi gratefully acknowledges the Grant-in-aid for JSPS Fellows. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Eurosystem or the European Central Bank. References Balasko, Y., Cass, D., 1989. The structure of financial equilibrium with exogenous yields: the case of incomplete markets. Econometrica 57, 135–162. Cass, D., 1985. On the number of equilibrium allocations with incomplete financial markets. CARESS Working Paper, University of Pennsylvania. Citanna, A., Villanacci, A., 2000. Existence and regularity of partially revealing rational expectations equilibrium in finite economies. Journal of Mathematical Economics 34, 1–26.
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