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Stabilizing sunspots
Journal of Mathematical Economics 42 (2006) 544–555
Stabilizing sunspots Aditya Goenka a,b,∗ , Christophe Pr´echac c a
Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK b Department of Economics, National University of Singapore, AS2 Level 6, 1 Arts Link, Singapore 117570, Singapore c CERMSEM, Maison des Sciences Economiques, Universit´ e Paris I Panth´eon-Sorbonne, 106-112 Boulevard de l’Hˆopital, 75647 Paris Cedex 13, France Received 18 July 2005; accepted 19 May 2006 Available online 13 July 2006
Abstract This paper analyzes stabilization of sunspot equilibria in an incomplete financial market framework. Indexation of nominal bonds or introduction of real securities can eliminate the sunspot equilibria. However, we show that the utility of one type of consumer may be minimized at a Walrasian allocation relative to other sunspot equilibrium allocations. Thus, if considering stabilization policies from the status quo of incomplete financial markets, there may be no consensus on implementing them. © 2006 Elsevier B.V. All rights reserved. JEL classification: D52; E32; G10; D60 Keywords: Sunspot equilibrium; Incomplete markets; Welfare analysis; Stabilization policy
1. Introduction The sunspot equilibrium literature has shown that if there is any distortion in the economy which renders equilibrium outcomes inefficient, then it is very likely that sunspots will have nontrivial effects (Cass and Shell, 1983; Shell, 1987). Sunspot equilibrium outcomes are a form of pure excess volatility of equilibrium outcomes as allocations vary across intrinsically identical states of nature. A natural question is that if the economy is susceptible to these fluctuations, what is the role of stabilization policy in negating them? The issue is to design policies that will change the set of equilibrium outcomes and rule out the effect of sunspots (early applications of this ∗
Corresponding author. E-mail addresses: [email protected] (A. Goenka), [email protected] (C. Pr´echac)..
0304-4068/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2006.05.003
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approach are Goenka, 1994a,b). This paper raises the issue of whether there can be consensus in the adoption of stabilization policies: It shows that the interests of consumers may be diametrically opposed: While some consumers benefit from stabilization of the endogenous fluctuations, the utility of others is minimized at the stabilized outcomes. We study a situation where consumers trade a nominal bond but markets are incomplete. There can be fluctuations in the price level across states, which in turn causes fluctuations in allocations (Balasko and Cass, 1989; Cass, 1989). These fluctuations can be eliminated by indexation of the nominal bonds. However, even though all consumers are risk averse, in the resulting stabilized Walrasian allocation there can be a decrease in welfare for some of the agents. The result is presented in a stark form so that one consumer will prefer any sunspot allocation to the Walrasian allocation. This result is used to look at the issue of stabilizing the economy against endogenous volatility through monetary policies. Neumeyer (1999) also looks at the issue of stabilizing monetary policy in an incomplete markets framework. In his model there is intrinsic uncertainty. He shows that price stabilization can make nominal financial markets shut down. In our model, there is only extrinsic uncertainty. We show that price stabilization, i.e., elimination of non-trivial sunspot equilibria, can be achieved through indexation of the nominal bond or through introduction of a real bond. If the introduction of new bonds requires participation on both sides of the market—both buyers and sellers—then these markets may not open. The economy has one good in each state, only extrinsic uncertainty, and two types of consumers. Consumers have identical preferences and differ in terms of when their endowments are available. The first type of consumers receive the majority/all of their endowments in the first period, and the second type the majority/all of their endowments in the second period. Thus, the first type can be considered as “old” consumers or “savers” while the second type as “young” consumers or “borrowers”. We show that the utility of the second type of consumers is minimized at the state-symmetric Walrasian allocation. We first give a parametric example and then generalize the result to an open class of preferences (consumers must have a sufficiently high precautionary savings motive (Kimball, 1990)) and general endowments. If consumers have corner endowments then the condition requires consumers to have the Index of Absolute Prudence greater than two. This result is global for preferences that include log-linear and CRRA preferences with the Index of Relative Risk Aversion greater than one, and is local under more general preferences. As we look at the case of one good in each state, effects relating to changes in relative prices within each state are not the key to the welfare effects. To consider stabilization policies we show, in a situation with both nominal and real securities, following the result of Mas-Collel (1992), that if there are as many real securities as the number of goods in each state, the equilibrium allocation has to be state symmetric. Thus, one way in which ‘markets can be completed’ is to index the bond by denominating the return in terms of commodities. The implication of the results is that if the policy maker is constrained to implement policies which are welfare improving for all consumers then stabilization will not be implemented. Alternatively, if the consumers were to vote or have a referendum on whether to adopt the stabilization policy, there will be no consensus on the desirability of the stabilization policy: there are winners and losers to the stabilization policy. Thus, it is important to look at the disaggregated welfare effects of stabilization policies. A minimal requirement of such policies should be that the resulting equilibrium allocation should be in the upper contour set of the status quo. Otherwise tax and transfer schemes are required which are open to incentive problems. The result also indicates that looking at planning problems or looking at stabilization where the policy makers have preferences over inflation rates should be treated with caution in situations
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where heterogeneity of consumers is important. Taking either of these approaches obscures the asymmetric welfare effect on consumers. The result on the welfare properties of the equilibrium outcomes is consistent with what is known about one-good incomplete market economies: the equilibrium outcomes are constrained Pareto efficient (Diamond, 1967). However, as mentioned above we have a stronger statement on the Walrasian allocation vis-a-vis other sunspot equilibrium allocations that may emerge. The plan of the paper is as follows. First, we define the economy with a nominal bond and then present a parametric example where the entire equilibrium set is characterized and the welfare effects identified. This is extended to general preferences, and non-corner endowments as well. Next we consider the economy with both nominal and real assets and show that if there are a sufficient number of these assets, the equilibrium is necessarily state symmetric. We then examine the implications for stabilization policies. 2. Economy with nominal assets Consider a pure-exchange economy with two periods. In period one there is one state s = 0, and S < ∞ states in the second period, s = 1, . . . , S. These are indexed by the superscript s. In each state there is a single consumption good. There are two types consumers indexed by the subscript h = 1, 2 of equal mass. As consumers of a given type are identical and behave as price-takers, without any loss of generality, in what follows we look at a representative consumer of each type. The consumption plan for consumer h is xh = (xh0 , xh1 , xh2 , . . . , xhs , . . . , xhS ). The consumption set for the consumers, Xh , is the S + 1 dimensional positive orthant. Both the consumers have identical preferences represented by the utility function: uh (xh ) = v(xh0 ) +
S 1 s v(xh ). S
(1)
s=1
The sub-utility functions vh (·) are strictly increasing, strictly concave, and thrice-continuously differentiable. The endowments of the two consumers are ω1 = (α, 1 − α, . . . , 1 − α) and ω2 = (1 − α, α, . . . , α) respectively, with α ∈ (0, 1]. The consumers can transfer wealth across the states using a nominal bond. The return matrix of the nominal bond is RN = (−1, 1, . . . , 1)T . The purchase of the nominal bond for consumer h is denoted as θh , and the excess demand for commodities by zh . The prices of the consumption goods are normalized so that p = (1, p1 , . . . , ps , . . . , pS ). The budget constraints are given by: z0h + θh = 0 ps zsh = θh ,
(2) s = 1, . . . , S
(3)
Definition 1. A sunspot (GEI) equilibrium in the economy with nominal bonds is a vector (p, θ1 , θ2 ) such that (i) θh maximizes utility (1) for the consumers subject to the budget constraints (2-3). (ii) The bond market clears, i.e., θ1 + θ2 = 0. Definition 2. Sunspots do not matter, or the sunspot equilibrium is trivial, if the equilibrium allocations are independent of the states in period 2, i.e., if: xh1 = xh2 = · · · = xhS ,
h = 1, 2.
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The following example for log-linear preferences and corner endowments enables us to explicitly compute the equilibrium set. Thus, financial and real indeterminacy is shown directly, and the welfare comparisons are made explicitly. 2.1. Leading example In this example, S = 2, α = 1, and the preferences of the two consumers are restricted to be log-linear, i.e., 1 uh (xh ) = log xh0 + (4) log xh1 + log xh2 . 2 To solve for the equilibria first solve for the demand of the two consumers. For consumer 1, substitute the budget equations into the maximand to get Max log(1 − θ1 ) +
θ1 θ1 1 1 log 1 + log 2 ⇔ Max log(1 − θ1 ) + log θ1 2 p 2 p
(5)
Thus, θ1∗ = 21 , which is independent of p1 , p2 . For consumer 2, the maximization problem is: 1 1 θ2 θ2 Max log(−θ2 ) + log 1 + 1 + log 1 + 2 . (6) 2 p 2 p The first order condition for consumer 2 is: 1 1 1 1 1 = + θ2 2p1 (1 + (θ2 /p1 )) 2p2 (1 + (θ2 /p2 )) θ2 θ2 θ2 θ2 or 4p1 p2 1 + 1 1 + 2 = −θ2 2p1 1 + 1 + 2p2 1 + 2 p p p p −
(7)
Market clearing in the bond market implies θ1 = −θ2 , thus θ1∗ = 21 implies that we must have θ2∗ = − 21 . Substitute this into the first order condition above to derive the equilibrium equation: 1 1 (2p1 − 1)(2p2 − 2) − (2p1 − 1) − (2p2 − 1) = 0 2 2 1 1 3 3 1 1 1 2 1 2 or 2p − 1 − 2p − 1 − = ⇒ p − p − = . 2 2 4 4 4 16
(8)
There are, however, two solutions to the equilibrium conditions. The branch through 21 , 21 is not a solution as non-negativity conditions (for consumer 2) are violated. The only solution is the branch through (1, 1). Thus, there is a unique (state) symmetric equilibrium which corresponds to the Walrasian equilibrium. In this economy, the symmetric equilibrium prices are p1 = p2 = 1. Thus, the incompleteness of the markets gives rise to the well known indeterminacy of equilibria. The indeterminacy we have shown is price indeterminacy. One can show that there is real indeterminacy as well. From Balasko and Cass (1989) we know that the equilibrium allocations will lie on the intersection of the offer curves of the consumers. The supporting prices (p and p for the two consumers respectively) however, in general need not be the same. If they are, then the economy is at a
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Walrasian equilibrium. The offer curves of the two consumers in this economy are: 1 p0 p0 0 1 2 : p ,p ,p > 0 , , Ω1 = 2 4p1 4p2 1 p + p2 1 p2 1 p1 0 1 2 , + 1 , + 2 : p , p , p > 0 . Ω2 = 2p0 4 4p 4 4p
(9) (10)
Normalize p0 = p0 = 1. From the market clearing we know that x20 = 21 or from offer curve of consumer 2, p1 + p2 = 1. Substituting this into the offer curve of consumer 2, we have: 1 1 1 − p1 1 p1 0 1 2 , + (11) , + : p ,p ,p > 0 . Ω2 = 2 4 4p1 4 4(1 − p1 ) As we have taken the intersection of the two offer curves into account (the offer curve of consumer 1 being a plane at the x10 coordinate of 21 ), all the information of the equilibrium is contained in this equation. Define the parameter λ = (1 − p1 )/p1 , with λ ∈ (−1, ∞) (for p1 > 0). The offer curve is now given by: 1 1 λ 1 1 , + , + : λ ∈ (−1, ∞) . (12) Ω2 = 2 4 4 4 4λ From this we derive the equation for x21 , x22 by setting the second coordinate of the above equal to x21 and the third coordinate equal to x22 and eliminating λ to obtain: 1 1 1 x21 − x22 − . (13) = 4 16 4 Thus, there are two solutions. The branch through ( 21 , 21 ) satisfies the equilibrium conditions. Thus, there is a one-dimensional real indeterminacy as well. To see examine the welfare effects, one can work with either Eq. (8) or Eq. (13). Working with 1 2 the latter, itis easy to see that the Walrasian equilibrium minimizes x2 x2 subject to the equilibrium 1 restriction x21 − 41 x22 − 41 = 16 , and x21 , x22 > 41 . Alternatively, for consumer 1, the indirect utility function 1 1 1 1 1 + w1 (p1 , p2 ) = log + log log 2 2 2p1 2 2p2
can be reduced to ζ1 (w1 (p1 , p2 )) = −p1 p2 , where ζ1 is a strictly increasing function. The indirect utility of consumer 1 is maximized at the Walrasian prices on the equilibrium set. This can be seen, as p1 = p2 solves: Min p1 p2 s.t. p1 − 43 p2 − 43 =
1 16
p1 , p2 > 43 . For consumer 2, the indirect utility function, up to a strictly increasing transformation is: 1 1 1− 2 . w2 (p1 , p2 ) = 1 − 1 2p 2p
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Using the equilibrium relation between the prices and eliminating p2 , this can be reduced to: m2 (p1 ) =
1 (2p1 − 1)2 . 4 p1 (3p1 − 2)
This function is minimized at the Walrasian prices. To see this compute m2 (p1 ) =
(2p1 − 1)(2p1 − 2) . 4(p1 (3p1 − 2))2
This function is decreasing in the interval
3 4, 1
and increasing in (1, ∞), with a critical point
at 1. In fact, m2 (1) = and limp1 → 3 m2 = limp1 →∞ m2 (p1 ) = 13 . 4 The limiting allocations can also be computed: 1 2 or x1 = 21 , 0, 23 2, 3, 0 1 1 x2 = 21 , 1, 13 or 2, 3, 1 . 1 4,
(p1 )
In this economy the maxima and minima of utility for the two consumers is global. In the one-good economy it is known that the equilibria will be constrained efficient. However, we are saying something more about the nature of the variation of utility of the two consumers on the equilibrium set. The intuition of the example is that in the second period consumer 2 is a net seller of the good. We know that the profit function for a risk-neutral seller is quasi-convex and hence the price fluctuation is desirable from the viewpoint of the seller. What is interesting is that in this economy this effect dominates the loss of welfare due to the risk aversion of consumer 2. 2.2. Main result This example can be generalized and the result does not depend on either log-linear utility or on corner endowments. This is the content of the next result. Once we move away from corner endowments, there is a joint restriction on the prudence (v ) and the size of the endowment which reflects the gain to consumer 2 from being a net seller of the good in the second period from the price fluctuations. In the special case of corner endowments, the condition reduces to one on the size of the Index of Absolute Prudence (Kimball, 1990, p. 61). Proposition 3. Suppose there are S < ∞ states, and endowments are given by: ω1 = (α, 1 − α, . . . , 1 − α) and ω2 = (1 − α, α, . . . , α) with α ∈ (0, 1]. Preferences are identical and given by: uh (xh ) = v(xh0 ) +
S 1 s v(xh ). S s=1
A sufficient condition for the utility of consumer 2 to be minimized at the Walrasian allocation is:
1 v + α − v > 0 2
(14)
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Proof. In the Walrasian equilibrium of the economy given identical preferences and the endowment structure there will be complete risk sharing, i.e., we have: 1 . 2 To study sunspot equilibria consider the budget constraints: xh (0) = xh0 = . . . = xhs = . . . = xhS = xh = xh0 + θh = ωh0 ps xhs
=
ps ωhs
(15) + θh , 1 ≤ s ≤ S.
(16)
Substituting the budget equations into the utility functions and imposing the market clearing equation θ1 = −θ2 = θ, we get the following maximization problems for the two consumers. S 1 θ Consumer 1 : Max v(α − θ) + (17) v 1−α+ s S p s=1
Consumer 2 : Max v(1 − α + θ) +
S 1 θ v α− s S p
(18)
s=1
The set of first order equations define the equilibrium set, E. θ 1 1 φ1 (θ, p) = −v (α − θ) + v 1−α+ s =0 S ps p θ 1 1 φ2 (θ, p) = −v (1 − α + θ) + v α − s = 0. S ps p Let (1 − α) = β. Then, the gradients of these two equations can be written. ⎞ ⎛ 1 θ β + v v (α − θ) + S1 s s 2 p (p ) ⎜ ⎟ ⎟ ⎜ 1 θ θ θ ⎟ ⎜ − 1 2v β + 1 − β + v 1 3 1 S(p ) p S(p ) p ⎟ ⎜ ∇φ1 (θ, p) = ⎜ ⎟ ⎟ ⎜ ... ⎝ ⎠ − S(p1S )2 v β + pθS − S(pθS )3 v β + pθS ⎞ ⎛ 1 θ β − v v (β + θ) − S1 s s )2 p (p ⎜ ⎟ ⎟ ⎜ ⎜ − 11 2 v α − θ1 + θ1 3 v α − θ1 ⎟ S(p ) p S(p ) p ⎟ ∇φ2 (θ, p) = ⎜ ⎟ ⎜ ⎟ ⎜ ... ⎝ ⎠ 1 θ θ θ − S(pS )2 v α − pS + S(pS )3 v α − pS At the Walrasian point, we have ⎞ ⎛ 2v ⎟ ⎜ 1 ⎜ − S v + α − 21 v ⎟ ⎟ ⎜ ∇φ1 (θ, p) = ⎜ ⎟ ⎟ ⎜ ... ⎝ ⎠ − S1 v + (α − 21 )v
(19) (20)
(21)
(22)
(23)
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⎛
⎞ ⎜ 1 ⎟ ⎜ − S v − (α − 21 )v ⎟ ⎜ ⎟ ∇φ2 (θ, p) = ⎜ ⎟ ⎜ ⎟ ... ⎝ ⎠ − S1 v − α − 21 v
551
−2v
(24)
Thus, ∇φ1 is not collinear to ∇φ2 , and E is locally a (S − 1) manifold. The issue is now to see that the utility of consumer 2 is minimized at the Walrasian equilibrium. Min u2 (θ, p) = v(β + θ) + S1 Ss=1 v α − pθs s.t.
φ1 (θ, p) = 0
φ2 (θ, p) = 0. The tangent space at the Walrasian equilibrium, w∗ , is: ˜ p) (θ, ˜ · ∇φ1 (w∗ ) = 0; ⇔
˜ p) (θ, ˜ · (∇φ1 (w∗ ) + ∇φ2 (w∗ )) = 0
˜ p) (θ, ˜ · ∇φ2 (w∗ ) = 0;
This implies: p˜ s = 0
˜ p) (θ, ˜ · (∇φ1 (w∗ ) − ∇φ2 (w∗ )) = 0
θ˜ = 0.
The Lagrangian is: L(θ, p, λ1 , λ2 ) = u2 (θ, p) + λ1 φ1 (θ, p) + λ2 φ2 (θ, p). The first order conditions are: ∂L ∗ (u , λ1 , λ2 ) = 0 ⇔ 0 + λ1 2v + λ2 (−2v ) = 0 ∂θ
(25)
⇔ λ1 = λ2 = λ∗
(26)
θ∗ 1 ∂L ∗ (u , λ1 , λ2 ) = 0 ⇔ v + λ∗ (− 2v ) = 0 s ∂p S S
(27)
⇔ λ∗ =
θ∗ . 2
Now, ∇u2 (θ, p) =
(28)
v (β + θ) −
1 S
θ v (α − pθs ), S(ps )2
1 θ ps v (α − ps )
s≥1
∂ 2 u2 ∗ 2θ θ (u ) = − v + v ∂(ps )2 S S
(29)
2 θ 3θ θ2 ∂ 2 φ1 (W ∗ ) = v + v + v + v s 2 ∂(p ) S S S S
(30)
2 θ 3θ θ 2 ∂ 2 φ2 ∗ (u ) = v − v − v + v . ∂(ps )2 S S S S
(31)
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This implies: 2 θ 2 v + v S S 2 2 θ θ 1 = (v + θv ) = v + (α − )v . S S 2
2θ θ 2 ∂2 L ∗ ∗ (u , λ ) = − v + v +θ ∂(ps )2 S S
For a minimum we want this to be positive, which gives the desired result as
(32) θ2 S
> 0.
2.3. Remarks (1) In the case of corner endowments, i.e., α = 1, we get a sharp bound on the sufficient condition in terms of the Index of Absolute Prudence. Corollary 4. Suppose there are S < ∞ states, and endowments are given by ω1 = (1, 0, . . . , 0) and ω2 = (0, 1, . . . , 1). Then the utility of consumer 2 is minimized at the Walrasian equilibrium if the Index of Absolute Prudence is greater than 2. −
v 2 > 2. v
(33)
Proof. Set α = 1 in Eq. (14) and simplify.
(2) This condition is satisfied for the following utility functions. (a) The log-linear utility function: log x. (b) The utility function: x1− /(1 − ), with > 0. (3) The role of the Index of Absolute Prudence gives additional intuition for the result: due to the precautionary motive, consumer 2 wants to consume less in period 1 as prices vary due to extrinsic uncertainty. Consumer 1 has fixed demand in period 1. Thus, consumer 1’s endowment is ‘relatively less valuable’ and consumer 2’s endowment is ‘more valuable.’ If the increase in value of endowment is large enough, then this will outweigh the loss in utility due to greater uncertainty. (4) For non-corner endowments, there is a joint restriction on preferences and endowments. If v(x) = x1− /(1 − ), > 0, we have v (x) = x− , v x = − x−1− , v (x) = ( + 1)x− −2 . Then, 1 1 1 1 + α− v = − 2 +1 + α − (2 ( + 1)2 +1 ) v 2 2 2 2 = 2 +1 (−1 + (2α − 1)( + 1))
> 0 iff α >
1 1 + . 2 2( + 1)
(a) If = 2, for u2 to be minimized at the Walrasian point, we need α > 23 . (b) If = 1, then we need, α > 43 . (5) The way in which the result is stated, the result is a local result, i.e., the utility of consumer 2 is a local minima, as we evaluate (14) at the Walrasian equilibrium. However, close reading of the proof will indicate if condition (14) is satisfied for all feasible allocations, i.e., on the interval (0, 1), the result is global.
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Corollary 5. Suppose there are S < ∞ states, and endowments are given by ω1 = (1, 0, . . . , 0) and ω2 = (0, 1, . . . , 1). A sufficient condition for the Walrasian equilibrium allocation to be a global minima for utility of consumer 2 is: −
v 2 (x) > 2, v (x)
∀x ∈ (0, 1).
(34)
(6) This condition (34) is satisfied for the following preferences: (a) The log-linear utility function: log x. (b) The utility function: x(1− ) /(1 − ), with > 1. 3. Stabilizing sunspots Now consider the economy with both nominal and real assets. The set up of the economy is exactly the same as in section 2 except that there is also a security that delivers one unit of the good in each state. This is the indexed bond. We show that if there are at least as many real assets as commodities in each state, which in our case is one, then the allocations will be state symmetric. Thus, either indexing the nominal asset so that its payoffs are denominated in terms of units of the commodity or introducing such a new bond in addition to the existing nominal security will stabilize the economy against sunspots. This result is similar to that in Mas-Collel (1992). To describe the new framework we need to define the new budget sets of the consumers. The consumers can transfer wealth across the states using the indexed bond. The return matrix of the asset is R = (−q, p1 , p2 , p3 )T . Let the purchase of the asset for consumer h be denoted as bh , and the excess demand of commodities in each state by zh . The price of the consumption goods are p = (1, p1 , p2 , p3 ). The nominal bond costs δ to purchase in state 0 and gives a sure return of 1 in all the three other states. Let the demand for the nominal demand by consumer h be denoted by θh . The rest of the economy is as before. We can write the new budget equations which would help us to solve for the consumer demand (or offer curves) and hence, for an equilibrium. For each consumer, h, the budget equations are now: z0h + qbh + δθh = 0
(35)
ps zsh = ps bh + θh s = 1, . . . , S.
(36)
Definition 6. A sunspot (GEI) equilibrium in the economy with nominal and real bonds is a vector (p, q, b1 , b2 , δ, θ1 , θ2 ) such that (i) (θh , bh ) maximize utility (1) for the consumers subject to the budget constraints (35–36). (ii) The nominal bond market clears, i.e., θ1 + θ2 = 0. (iii) The real bond market clears, i.e., b1 + b2 = 0. Proposition 7. If there are as many real securities as commodities in each state, then the equilibrium allocations are state symmetric (in period 1).
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Proof. Suppose this is not the case. Consider an equilibrium (p, q, b, θ, δ), and the corresponding equilibrium allocation x, and construct the alternative allocation: x¯ hS = π(s )xhs ∀s ≥ 1 x¯ h0 = xh0 . s
This allocation is feasible as: h x¯ hs = s π(s ) h xhs = s π(s ) h ωh = h ωh . For each consumer, x¯ h xh with a strict preference for consumers whose allocations were state asymmetric (from strict convexity of preferences). We need to check that we can achieve this allocation using the given set of securities. For this construct the following portfolio. b¯ h = x¯ h − ωh
This is feasible as h b¯ h = 0. Also set θ¯ h = 0 so there is no trading in the nominal asset. It also delivers the desired allocation. Think of a situation where there is only a financial bond so that we are in the environment where there is indeterminacy of equilibrium prices and allocations. A policy maker may want to stabilize the economy thinking that the excessive volatility in allocations will lead to welfare losses as agents are risk averse. This can be done by simply indexing the nominal bond, that is denominating the return in terms of the commodity in each period, or introducing such a bond in addition to the nominal bond being traded. Proposition 2 implies that this simple change will lead to determinacy of allocations and also these will be state symmetric as the Walrasian allocation is achieved. However, Proposition 1 implies that this change will lead to a loss of welfare to consumer 2. If the planner is constrained to implement schemes where there is no welfare loss in equilibrium to any consumer or there is to be unanimity on the part of consumers for the scheme to be implemented, the change will not take place. Thus, if we are in an environment where sunspots already matter, it may be very difficult to neutralize them. Proposition 8. The introduction of an indexed bond will stabilize the economy so that sunspots will not matter. However, there need not be any consensus on the desirability of such a policy. Proof. Immediate from Propositions 1 and 2.
4. Conclusion This paper shows that while stabilization of the economy against sunspot fluctuations may be done easily in principle, getting agreement on their adoption is problematic. In economies with risk averse agents the usual presumption is that any fluctuation is welfare reducing. We show in a simple economy that this intuition is misleading—some consumers may benefit from price fluctuations. Thus, there is a need to think carefully about welfare effects at a decentralized level in discussing the desirability of stabilization policies. Acknowledgements This paper is based on an earlier paper titled “Are sunspots inevitable?”. The research has been funded by Alliance (Franco-British Joint Research Programme) Project No. 97051. We would like to thank Shurojit Chatterji, Piero Gottardi, Chiaki Hara, Atsushi Kajii, Todd Keister, Karl Shell, and an anonymous referee for their helpful comments, as well as seminar audiences at University
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of Essex, Birkbeck College, London Guildhall University, ITAM, and the 14th European General Equilibrium Workshop (2005, Zurich). The usual disclaimer applies. 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., 1989. Sunspots and incomplete financial markets: The leading example. In: Feiwel, G. (Ed.), The economics of imperfect competition and employment: Joan Robinson and beyond. Macmillan London. Cass, D., Shell, K., 1983. Do sunspots matter? Journal of Political Economy 91, 193–227. Diamond, P., 1967. The role of the stock market in a general equilibrium model with technological uncertainty. American Economic Review 57, 759–776. Goenka, A., 1994. Fiscal rules and extrinsic uncertainty. Economic Theory 4, 401–416. Goenka, A., 1994. Rationing and Sunspot Equilibria. Journal of Economic Theory 64, 424–442. Kimball, M.S., 1990. Precautionary savings in the small and in the large. Econometrica 58, 53–73. Mas-Collel, A., 1992. Three observations on sunspots and asset redundancy. In: Daguspta, P., Gale, D., Hart, O., Maskin, E. (Eds.), Economic analysis of Markets and Games. MIT Press, Cambridge. Neumeyer, P.A., 1999. Inflation-stabilization risk in economies with incomplete asset markets. Journal of Economic Dynamics and Control 23, 371–391. Shell, K., 1987. Sunspot equilibrium. In: Eatwell, J., Milgate, M., Newman, P. (Eds.), The New Palgrave: A Dictionary of Economics. Macmillan, London.
Stabilizing sunspots Aditya Goenka a,b,∗ , Christophe Pr´echac c a
Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK b Department of Economics, National University of Singapore, AS2 Level 6, 1 Arts Link, Singapore 117570, Singapore c CERMSEM, Maison des Sciences Economiques, Universit´ e Paris I Panth´eon-Sorbonne, 106-112 Boulevard de l’Hˆopital, 75647 Paris Cedex 13, France Received 18 July 2005; accepted 19 May 2006 Available online 13 July 2006
Abstract This paper analyzes stabilization of sunspot equilibria in an incomplete financial market framework. Indexation of nominal bonds or introduction of real securities can eliminate the sunspot equilibria. However, we show that the utility of one type of consumer may be minimized at a Walrasian allocation relative to other sunspot equilibrium allocations. Thus, if considering stabilization policies from the status quo of incomplete financial markets, there may be no consensus on implementing them. © 2006 Elsevier B.V. All rights reserved. JEL classification: D52; E32; G10; D60 Keywords: Sunspot equilibrium; Incomplete markets; Welfare analysis; Stabilization policy
1. Introduction The sunspot equilibrium literature has shown that if there is any distortion in the economy which renders equilibrium outcomes inefficient, then it is very likely that sunspots will have nontrivial effects (Cass and Shell, 1983; Shell, 1987). Sunspot equilibrium outcomes are a form of pure excess volatility of equilibrium outcomes as allocations vary across intrinsically identical states of nature. A natural question is that if the economy is susceptible to these fluctuations, what is the role of stabilization policy in negating them? The issue is to design policies that will change the set of equilibrium outcomes and rule out the effect of sunspots (early applications of this ∗
Corresponding author. E-mail addresses: [email protected] (A. Goenka), [email protected] (C. Pr´echac)..
0304-4068/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2006.05.003
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approach are Goenka, 1994a,b). This paper raises the issue of whether there can be consensus in the adoption of stabilization policies: It shows that the interests of consumers may be diametrically opposed: While some consumers benefit from stabilization of the endogenous fluctuations, the utility of others is minimized at the stabilized outcomes. We study a situation where consumers trade a nominal bond but markets are incomplete. There can be fluctuations in the price level across states, which in turn causes fluctuations in allocations (Balasko and Cass, 1989; Cass, 1989). These fluctuations can be eliminated by indexation of the nominal bonds. However, even though all consumers are risk averse, in the resulting stabilized Walrasian allocation there can be a decrease in welfare for some of the agents. The result is presented in a stark form so that one consumer will prefer any sunspot allocation to the Walrasian allocation. This result is used to look at the issue of stabilizing the economy against endogenous volatility through monetary policies. Neumeyer (1999) also looks at the issue of stabilizing monetary policy in an incomplete markets framework. In his model there is intrinsic uncertainty. He shows that price stabilization can make nominal financial markets shut down. In our model, there is only extrinsic uncertainty. We show that price stabilization, i.e., elimination of non-trivial sunspot equilibria, can be achieved through indexation of the nominal bond or through introduction of a real bond. If the introduction of new bonds requires participation on both sides of the market—both buyers and sellers—then these markets may not open. The economy has one good in each state, only extrinsic uncertainty, and two types of consumers. Consumers have identical preferences and differ in terms of when their endowments are available. The first type of consumers receive the majority/all of their endowments in the first period, and the second type the majority/all of their endowments in the second period. Thus, the first type can be considered as “old” consumers or “savers” while the second type as “young” consumers or “borrowers”. We show that the utility of the second type of consumers is minimized at the state-symmetric Walrasian allocation. We first give a parametric example and then generalize the result to an open class of preferences (consumers must have a sufficiently high precautionary savings motive (Kimball, 1990)) and general endowments. If consumers have corner endowments then the condition requires consumers to have the Index of Absolute Prudence greater than two. This result is global for preferences that include log-linear and CRRA preferences with the Index of Relative Risk Aversion greater than one, and is local under more general preferences. As we look at the case of one good in each state, effects relating to changes in relative prices within each state are not the key to the welfare effects. To consider stabilization policies we show, in a situation with both nominal and real securities, following the result of Mas-Collel (1992), that if there are as many real securities as the number of goods in each state, the equilibrium allocation has to be state symmetric. Thus, one way in which ‘markets can be completed’ is to index the bond by denominating the return in terms of commodities. The implication of the results is that if the policy maker is constrained to implement policies which are welfare improving for all consumers then stabilization will not be implemented. Alternatively, if the consumers were to vote or have a referendum on whether to adopt the stabilization policy, there will be no consensus on the desirability of the stabilization policy: there are winners and losers to the stabilization policy. Thus, it is important to look at the disaggregated welfare effects of stabilization policies. A minimal requirement of such policies should be that the resulting equilibrium allocation should be in the upper contour set of the status quo. Otherwise tax and transfer schemes are required which are open to incentive problems. The result also indicates that looking at planning problems or looking at stabilization where the policy makers have preferences over inflation rates should be treated with caution in situations
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where heterogeneity of consumers is important. Taking either of these approaches obscures the asymmetric welfare effect on consumers. The result on the welfare properties of the equilibrium outcomes is consistent with what is known about one-good incomplete market economies: the equilibrium outcomes are constrained Pareto efficient (Diamond, 1967). However, as mentioned above we have a stronger statement on the Walrasian allocation vis-a-vis other sunspot equilibrium allocations that may emerge. The plan of the paper is as follows. First, we define the economy with a nominal bond and then present a parametric example where the entire equilibrium set is characterized and the welfare effects identified. This is extended to general preferences, and non-corner endowments as well. Next we consider the economy with both nominal and real assets and show that if there are a sufficient number of these assets, the equilibrium is necessarily state symmetric. We then examine the implications for stabilization policies. 2. Economy with nominal assets Consider a pure-exchange economy with two periods. In period one there is one state s = 0, and S < ∞ states in the second period, s = 1, . . . , S. These are indexed by the superscript s. In each state there is a single consumption good. There are two types consumers indexed by the subscript h = 1, 2 of equal mass. As consumers of a given type are identical and behave as price-takers, without any loss of generality, in what follows we look at a representative consumer of each type. The consumption plan for consumer h is xh = (xh0 , xh1 , xh2 , . . . , xhs , . . . , xhS ). The consumption set for the consumers, Xh , is the S + 1 dimensional positive orthant. Both the consumers have identical preferences represented by the utility function: uh (xh ) = v(xh0 ) +
S 1 s v(xh ). S
(1)
s=1
The sub-utility functions vh (·) are strictly increasing, strictly concave, and thrice-continuously differentiable. The endowments of the two consumers are ω1 = (α, 1 − α, . . . , 1 − α) and ω2 = (1 − α, α, . . . , α) respectively, with α ∈ (0, 1]. The consumers can transfer wealth across the states using a nominal bond. The return matrix of the nominal bond is RN = (−1, 1, . . . , 1)T . The purchase of the nominal bond for consumer h is denoted as θh , and the excess demand for commodities by zh . The prices of the consumption goods are normalized so that p = (1, p1 , . . . , ps , . . . , pS ). The budget constraints are given by: z0h + θh = 0 ps zsh = θh ,
(2) s = 1, . . . , S
(3)
Definition 1. A sunspot (GEI) equilibrium in the economy with nominal bonds is a vector (p, θ1 , θ2 ) such that (i) θh maximizes utility (1) for the consumers subject to the budget constraints (2-3). (ii) The bond market clears, i.e., θ1 + θ2 = 0. Definition 2. Sunspots do not matter, or the sunspot equilibrium is trivial, if the equilibrium allocations are independent of the states in period 2, i.e., if: xh1 = xh2 = · · · = xhS ,
h = 1, 2.
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The following example for log-linear preferences and corner endowments enables us to explicitly compute the equilibrium set. Thus, financial and real indeterminacy is shown directly, and the welfare comparisons are made explicitly. 2.1. Leading example In this example, S = 2, α = 1, and the preferences of the two consumers are restricted to be log-linear, i.e., 1 uh (xh ) = log xh0 + (4) log xh1 + log xh2 . 2 To solve for the equilibria first solve for the demand of the two consumers. For consumer 1, substitute the budget equations into the maximand to get Max log(1 − θ1 ) +
θ1 θ1 1 1 log 1 + log 2 ⇔ Max log(1 − θ1 ) + log θ1 2 p 2 p
(5)
Thus, θ1∗ = 21 , which is independent of p1 , p2 . For consumer 2, the maximization problem is: 1 1 θ2 θ2 Max log(−θ2 ) + log 1 + 1 + log 1 + 2 . (6) 2 p 2 p The first order condition for consumer 2 is: 1 1 1 1 1 = + θ2 2p1 (1 + (θ2 /p1 )) 2p2 (1 + (θ2 /p2 )) θ2 θ2 θ2 θ2 or 4p1 p2 1 + 1 1 + 2 = −θ2 2p1 1 + 1 + 2p2 1 + 2 p p p p −
(7)
Market clearing in the bond market implies θ1 = −θ2 , thus θ1∗ = 21 implies that we must have θ2∗ = − 21 . Substitute this into the first order condition above to derive the equilibrium equation: 1 1 (2p1 − 1)(2p2 − 2) − (2p1 − 1) − (2p2 − 1) = 0 2 2 1 1 3 3 1 1 1 2 1 2 or 2p − 1 − 2p − 1 − = ⇒ p − p − = . 2 2 4 4 4 16
(8)
There are, however, two solutions to the equilibrium conditions. The branch through 21 , 21 is not a solution as non-negativity conditions (for consumer 2) are violated. The only solution is the branch through (1, 1). Thus, there is a unique (state) symmetric equilibrium which corresponds to the Walrasian equilibrium. In this economy, the symmetric equilibrium prices are p1 = p2 = 1. Thus, the incompleteness of the markets gives rise to the well known indeterminacy of equilibria. The indeterminacy we have shown is price indeterminacy. One can show that there is real indeterminacy as well. From Balasko and Cass (1989) we know that the equilibrium allocations will lie on the intersection of the offer curves of the consumers. The supporting prices (p and p for the two consumers respectively) however, in general need not be the same. If they are, then the economy is at a
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Walrasian equilibrium. The offer curves of the two consumers in this economy are: 1 p0 p0 0 1 2 : p ,p ,p > 0 , , Ω1 = 2 4p1 4p2 1 p + p2 1 p2 1 p1 0 1 2 , + 1 , + 2 : p , p , p > 0 . Ω2 = 2p0 4 4p 4 4p
(9) (10)
Normalize p0 = p0 = 1. From the market clearing we know that x20 = 21 or from offer curve of consumer 2, p1 + p2 = 1. Substituting this into the offer curve of consumer 2, we have: 1 1 1 − p1 1 p1 0 1 2 , + (11) , + : p ,p ,p > 0 . Ω2 = 2 4 4p1 4 4(1 − p1 ) As we have taken the intersection of the two offer curves into account (the offer curve of consumer 1 being a plane at the x10 coordinate of 21 ), all the information of the equilibrium is contained in this equation. Define the parameter λ = (1 − p1 )/p1 , with λ ∈ (−1, ∞) (for p1 > 0). The offer curve is now given by: 1 1 λ 1 1 , + , + : λ ∈ (−1, ∞) . (12) Ω2 = 2 4 4 4 4λ From this we derive the equation for x21 , x22 by setting the second coordinate of the above equal to x21 and the third coordinate equal to x22 and eliminating λ to obtain: 1 1 1 x21 − x22 − . (13) = 4 16 4 Thus, there are two solutions. The branch through ( 21 , 21 ) satisfies the equilibrium conditions. Thus, there is a one-dimensional real indeterminacy as well. To see examine the welfare effects, one can work with either Eq. (8) or Eq. (13). Working with 1 2 the latter, itis easy to see that the Walrasian equilibrium minimizes x2 x2 subject to the equilibrium 1 restriction x21 − 41 x22 − 41 = 16 , and x21 , x22 > 41 . Alternatively, for consumer 1, the indirect utility function 1 1 1 1 1 + w1 (p1 , p2 ) = log + log log 2 2 2p1 2 2p2
can be reduced to ζ1 (w1 (p1 , p2 )) = −p1 p2 , where ζ1 is a strictly increasing function. The indirect utility of consumer 1 is maximized at the Walrasian prices on the equilibrium set. This can be seen, as p1 = p2 solves: Min p1 p2 s.t. p1 − 43 p2 − 43 =
1 16
p1 , p2 > 43 . For consumer 2, the indirect utility function, up to a strictly increasing transformation is: 1 1 1− 2 . w2 (p1 , p2 ) = 1 − 1 2p 2p
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Using the equilibrium relation between the prices and eliminating p2 , this can be reduced to: m2 (p1 ) =
1 (2p1 − 1)2 . 4 p1 (3p1 − 2)
This function is minimized at the Walrasian prices. To see this compute m2 (p1 ) =
(2p1 − 1)(2p1 − 2) . 4(p1 (3p1 − 2))2
This function is decreasing in the interval
3 4, 1
and increasing in (1, ∞), with a critical point
at 1. In fact, m2 (1) = and limp1 → 3 m2 = limp1 →∞ m2 (p1 ) = 13 . 4 The limiting allocations can also be computed: 1 2 or x1 = 21 , 0, 23 2, 3, 0 1 1 x2 = 21 , 1, 13 or 2, 3, 1 . 1 4,
(p1 )
In this economy the maxima and minima of utility for the two consumers is global. In the one-good economy it is known that the equilibria will be constrained efficient. However, we are saying something more about the nature of the variation of utility of the two consumers on the equilibrium set. The intuition of the example is that in the second period consumer 2 is a net seller of the good. We know that the profit function for a risk-neutral seller is quasi-convex and hence the price fluctuation is desirable from the viewpoint of the seller. What is interesting is that in this economy this effect dominates the loss of welfare due to the risk aversion of consumer 2. 2.2. Main result This example can be generalized and the result does not depend on either log-linear utility or on corner endowments. This is the content of the next result. Once we move away from corner endowments, there is a joint restriction on the prudence (v ) and the size of the endowment which reflects the gain to consumer 2 from being a net seller of the good in the second period from the price fluctuations. In the special case of corner endowments, the condition reduces to one on the size of the Index of Absolute Prudence (Kimball, 1990, p. 61). Proposition 3. Suppose there are S < ∞ states, and endowments are given by: ω1 = (α, 1 − α, . . . , 1 − α) and ω2 = (1 − α, α, . . . , α) with α ∈ (0, 1]. Preferences are identical and given by: uh (xh ) = v(xh0 ) +
S 1 s v(xh ). S s=1
A sufficient condition for the utility of consumer 2 to be minimized at the Walrasian allocation is:
1 v + α − v > 0 2
(14)
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Proof. In the Walrasian equilibrium of the economy given identical preferences and the endowment structure there will be complete risk sharing, i.e., we have: 1 . 2 To study sunspot equilibria consider the budget constraints: xh (0) = xh0 = . . . = xhs = . . . = xhS = xh = xh0 + θh = ωh0 ps xhs
=
ps ωhs
(15) + θh , 1 ≤ s ≤ S.
(16)
Substituting the budget equations into the utility functions and imposing the market clearing equation θ1 = −θ2 = θ, we get the following maximization problems for the two consumers. S 1 θ Consumer 1 : Max v(α − θ) + (17) v 1−α+ s S p s=1
Consumer 2 : Max v(1 − α + θ) +
S 1 θ v α− s S p
(18)
s=1
The set of first order equations define the equilibrium set, E. θ 1 1 φ1 (θ, p) = −v (α − θ) + v 1−α+ s =0 S ps p θ 1 1 φ2 (θ, p) = −v (1 − α + θ) + v α − s = 0. S ps p Let (1 − α) = β. Then, the gradients of these two equations can be written. ⎞ ⎛ 1 θ β + v v (α − θ) + S1 s s 2 p (p ) ⎜ ⎟ ⎟ ⎜ 1 θ θ θ ⎟ ⎜ − 1 2v β + 1 − β + v 1 3 1 S(p ) p S(p ) p ⎟ ⎜ ∇φ1 (θ, p) = ⎜ ⎟ ⎟ ⎜ ... ⎝ ⎠ − S(p1S )2 v β + pθS − S(pθS )3 v β + pθS ⎞ ⎛ 1 θ β − v v (β + θ) − S1 s s )2 p (p ⎜ ⎟ ⎟ ⎜ ⎜ − 11 2 v α − θ1 + θ1 3 v α − θ1 ⎟ S(p ) p S(p ) p ⎟ ∇φ2 (θ, p) = ⎜ ⎟ ⎜ ⎟ ⎜ ... ⎝ ⎠ 1 θ θ θ − S(pS )2 v α − pS + S(pS )3 v α − pS At the Walrasian point, we have ⎞ ⎛ 2v ⎟ ⎜ 1 ⎜ − S v + α − 21 v ⎟ ⎟ ⎜ ∇φ1 (θ, p) = ⎜ ⎟ ⎟ ⎜ ... ⎝ ⎠ − S1 v + (α − 21 )v
(19) (20)
(21)
(22)
(23)
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⎛
⎞ ⎜ 1 ⎟ ⎜ − S v − (α − 21 )v ⎟ ⎜ ⎟ ∇φ2 (θ, p) = ⎜ ⎟ ⎜ ⎟ ... ⎝ ⎠ − S1 v − α − 21 v
551
−2v
(24)
Thus, ∇φ1 is not collinear to ∇φ2 , and E is locally a (S − 1) manifold. The issue is now to see that the utility of consumer 2 is minimized at the Walrasian equilibrium. Min u2 (θ, p) = v(β + θ) + S1 Ss=1 v α − pθs s.t.
φ1 (θ, p) = 0
φ2 (θ, p) = 0. The tangent space at the Walrasian equilibrium, w∗ , is: ˜ p) (θ, ˜ · ∇φ1 (w∗ ) = 0; ⇔
˜ p) (θ, ˜ · (∇φ1 (w∗ ) + ∇φ2 (w∗ )) = 0
˜ p) (θ, ˜ · ∇φ2 (w∗ ) = 0;
This implies: p˜ s = 0
˜ p) (θ, ˜ · (∇φ1 (w∗ ) − ∇φ2 (w∗ )) = 0
θ˜ = 0.
The Lagrangian is: L(θ, p, λ1 , λ2 ) = u2 (θ, p) + λ1 φ1 (θ, p) + λ2 φ2 (θ, p). The first order conditions are: ∂L ∗ (u , λ1 , λ2 ) = 0 ⇔ 0 + λ1 2v + λ2 (−2v ) = 0 ∂θ
(25)
⇔ λ1 = λ2 = λ∗
(26)
θ∗ 1 ∂L ∗ (u , λ1 , λ2 ) = 0 ⇔ v + λ∗ (− 2v ) = 0 s ∂p S S
(27)
⇔ λ∗ =
θ∗ . 2
Now, ∇u2 (θ, p) =
(28)
v (β + θ) −
1 S
θ v (α − pθs ), S(ps )2
1 θ ps v (α − ps )
s≥1
∂ 2 u2 ∗ 2θ θ (u ) = − v + v ∂(ps )2 S S
(29)
2 θ 3θ θ2 ∂ 2 φ1 (W ∗ ) = v + v + v + v s 2 ∂(p ) S S S S
(30)
2 θ 3θ θ 2 ∂ 2 φ2 ∗ (u ) = v − v − v + v . ∂(ps )2 S S S S
(31)
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This implies: 2 θ 2 v + v S S 2 2 θ θ 1 = (v + θv ) = v + (α − )v . S S 2
2θ θ 2 ∂2 L ∗ ∗ (u , λ ) = − v + v +θ ∂(ps )2 S S
For a minimum we want this to be positive, which gives the desired result as
(32) θ2 S
> 0.
2.3. Remarks (1) In the case of corner endowments, i.e., α = 1, we get a sharp bound on the sufficient condition in terms of the Index of Absolute Prudence. Corollary 4. Suppose there are S < ∞ states, and endowments are given by ω1 = (1, 0, . . . , 0) and ω2 = (0, 1, . . . , 1). Then the utility of consumer 2 is minimized at the Walrasian equilibrium if the Index of Absolute Prudence is greater than 2. −
v 2 > 2. v
(33)
Proof. Set α = 1 in Eq. (14) and simplify.
(2) This condition is satisfied for the following utility functions. (a) The log-linear utility function: log x. (b) The utility function: x1− /(1 − ), with > 0. (3) The role of the Index of Absolute Prudence gives additional intuition for the result: due to the precautionary motive, consumer 2 wants to consume less in period 1 as prices vary due to extrinsic uncertainty. Consumer 1 has fixed demand in period 1. Thus, consumer 1’s endowment is ‘relatively less valuable’ and consumer 2’s endowment is ‘more valuable.’ If the increase in value of endowment is large enough, then this will outweigh the loss in utility due to greater uncertainty. (4) For non-corner endowments, there is a joint restriction on preferences and endowments. If v(x) = x1− /(1 − ), > 0, we have v (x) = x− , v x = − x−1− , v (x) = ( + 1)x− −2 . Then, 1 1 1 1 + α− v = − 2 +1 + α − (2 ( + 1)2 +1 ) v 2 2 2 2 = 2 +1 (−1 + (2α − 1)( + 1))
> 0 iff α >
1 1 + . 2 2( + 1)
(a) If = 2, for u2 to be minimized at the Walrasian point, we need α > 23 . (b) If = 1, then we need, α > 43 . (5) The way in which the result is stated, the result is a local result, i.e., the utility of consumer 2 is a local minima, as we evaluate (14) at the Walrasian equilibrium. However, close reading of the proof will indicate if condition (14) is satisfied for all feasible allocations, i.e., on the interval (0, 1), the result is global.
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Corollary 5. Suppose there are S < ∞ states, and endowments are given by ω1 = (1, 0, . . . , 0) and ω2 = (0, 1, . . . , 1). A sufficient condition for the Walrasian equilibrium allocation to be a global minima for utility of consumer 2 is: −
v 2 (x) > 2, v (x)
∀x ∈ (0, 1).
(34)
(6) This condition (34) is satisfied for the following preferences: (a) The log-linear utility function: log x. (b) The utility function: x(1− ) /(1 − ), with > 1. 3. Stabilizing sunspots Now consider the economy with both nominal and real assets. The set up of the economy is exactly the same as in section 2 except that there is also a security that delivers one unit of the good in each state. This is the indexed bond. We show that if there are at least as many real assets as commodities in each state, which in our case is one, then the allocations will be state symmetric. Thus, either indexing the nominal asset so that its payoffs are denominated in terms of units of the commodity or introducing such a new bond in addition to the existing nominal security will stabilize the economy against sunspots. This result is similar to that in Mas-Collel (1992). To describe the new framework we need to define the new budget sets of the consumers. The consumers can transfer wealth across the states using the indexed bond. The return matrix of the asset is R = (−q, p1 , p2 , p3 )T . Let the purchase of the asset for consumer h be denoted as bh , and the excess demand of commodities in each state by zh . The price of the consumption goods are p = (1, p1 , p2 , p3 ). The nominal bond costs δ to purchase in state 0 and gives a sure return of 1 in all the three other states. Let the demand for the nominal demand by consumer h be denoted by θh . The rest of the economy is as before. We can write the new budget equations which would help us to solve for the consumer demand (or offer curves) and hence, for an equilibrium. For each consumer, h, the budget equations are now: z0h + qbh + δθh = 0
(35)
ps zsh = ps bh + θh s = 1, . . . , S.
(36)
Definition 6. A sunspot (GEI) equilibrium in the economy with nominal and real bonds is a vector (p, q, b1 , b2 , δ, θ1 , θ2 ) such that (i) (θh , bh ) maximize utility (1) for the consumers subject to the budget constraints (35–36). (ii) The nominal bond market clears, i.e., θ1 + θ2 = 0. (iii) The real bond market clears, i.e., b1 + b2 = 0. Proposition 7. If there are as many real securities as commodities in each state, then the equilibrium allocations are state symmetric (in period 1).
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Proof. Suppose this is not the case. Consider an equilibrium (p, q, b, θ, δ), and the corresponding equilibrium allocation x, and construct the alternative allocation: x¯ hS = π(s )xhs ∀s ≥ 1 x¯ h0 = xh0 . s
This allocation is feasible as: h x¯ hs = s π(s ) h xhs = s π(s ) h ωh = h ωh . For each consumer, x¯ h xh with a strict preference for consumers whose allocations were state asymmetric (from strict convexity of preferences). We need to check that we can achieve this allocation using the given set of securities. For this construct the following portfolio. b¯ h = x¯ h − ωh
This is feasible as h b¯ h = 0. Also set θ¯ h = 0 so there is no trading in the nominal asset. It also delivers the desired allocation. Think of a situation where there is only a financial bond so that we are in the environment where there is indeterminacy of equilibrium prices and allocations. A policy maker may want to stabilize the economy thinking that the excessive volatility in allocations will lead to welfare losses as agents are risk averse. This can be done by simply indexing the nominal bond, that is denominating the return in terms of the commodity in each period, or introducing such a bond in addition to the nominal bond being traded. Proposition 2 implies that this simple change will lead to determinacy of allocations and also these will be state symmetric as the Walrasian allocation is achieved. However, Proposition 1 implies that this change will lead to a loss of welfare to consumer 2. If the planner is constrained to implement schemes where there is no welfare loss in equilibrium to any consumer or there is to be unanimity on the part of consumers for the scheme to be implemented, the change will not take place. Thus, if we are in an environment where sunspots already matter, it may be very difficult to neutralize them. Proposition 8. The introduction of an indexed bond will stabilize the economy so that sunspots will not matter. However, there need not be any consensus on the desirability of such a policy. Proof. Immediate from Propositions 1 and 2.
4. Conclusion This paper shows that while stabilization of the economy against sunspot fluctuations may be done easily in principle, getting agreement on their adoption is problematic. In economies with risk averse agents the usual presumption is that any fluctuation is welfare reducing. We show in a simple economy that this intuition is misleading—some consumers may benefit from price fluctuations. Thus, there is a need to think carefully about welfare effects at a decentralized level in discussing the desirability of stabilization policies. Acknowledgements This paper is based on an earlier paper titled “Are sunspots inevitable?”. The research has been funded by Alliance (Franco-British Joint Research Programme) Project No. 97051. We would like to thank Shurojit Chatterji, Piero Gottardi, Chiaki Hara, Atsushi Kajii, Todd Keister, Karl Shell, and an anonymous referee for their helpful comments, as well as seminar audiences at University
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