Temporary financial equilibrium - Semantic Scholar

Temporary financial equilibrium Yves Balasko∗ September 14, 2000

Abstract In a two-period pure exchange economy with financial assets, a temporary financial equilibrium is an equilibrium of the current spot and security markets given forecasts of future prices and returns. The temporary equilibrium model can then be interpreted as a Walrasian model where preferences depend on prices. This idenfication implies, among other consequences, the generic determinateness of the equilibrium solution. It also highlights the mechanism through which forecasts of future prices parameterize current market prices of goods and assets. JEL Classification numbers: D51, D52, D84, G12. Keywords: temporary equilibrium, financial equilibrium, asset pricing.

1. Introduction In this paper, we combine the temporary equilibrium point of view of Lindahl [11] and Hicks [10] (with later contributions by Grandmont, Green, Stigum, and many others—see [8]) with the two-period model with financial assets originally considered by Arrow [1], a model applied to the study of incomplete asset markets by Cass [6], and Werner [14]. See [12] for an account of later contributions to this model. One of the most remarkable feature of the incomplete asset model when returns are denominated in units of accounts is the generic indeterminateness of the equilibrium solution. Roughly speaking, the indeterminacy property stems from the fact that, in this model, the number of independent equations is strictly smaller than the number of unknowns. ∗ Visiting G.S.I.A., Carnegie-Mellon University. A major part of the research reported in this paper was done while visiting the European University Institute, Florence, whose hospitality is gratefully acknowledged. I also benefited from discussions with W. Heller, K. Shell, S. Spear, and J. Werner.

1

With the temporary equilibrium point of view, future prices are made dependent on the current or spot prices while only spot markets have to clear, a consequence being that the number of independent equations is again equal to the number of price unknowns. This observation suggests that the set of temporary equilibria is very likely to be generically discrete. One can then expect the properties of the temporary financial equilibrium model to be significantly different from those of the general equilibrium model with incomplete financial markets. We show in the current paper that the temporary financial equilibrium model can be interpreted as a Walrasian model with price dependent preferences. Therefore, it follows from [4] that the properties of temporary financial equilibria are more similar to those of the Walrasian model than to those of the general equilibrium model with incomplete financial assets. This proves in particular the existence and the generic determinateness of temporary financial equilibria. The paper is organized as follows. In Section 2, we recall the main assumptions and definitions of the two-period asset model. We introduce in Section 3 the temporary equilibrium setup. We show in Section 4 that the temporary financial equilibrium model can be reduced to a Walrasian equilibrium model where preferences are price dependent, which enables us to apply in Section 5 the results of [4]. Concluding comments end this paper with Section 6.

2. Assumptions, definitions, and notation Let us start by recalling the main features of the two-period equilibrium model with incomplete purely nominal financial markets ([5], [6], [7], [12], [14]). Goods Let ` denote the number of physical goods, a number assumed to be the same in period 0 and 1. Let S denote the set of states of nature in period 1. We denote by S their number. The number of contingent goods (delivered in period 1) and the total number of goods are then equal to `S and `(S + 1) respectively. Goods prices We denote by p(0) ∈ R`++ (resp. p(s) ∈ R`++ ) the price vector of the goods delivered in period 0 (resp. in the state of nature s). We pick the first commodity in period 0 as the numeraire, i.e., p1 (0) = 1. We denote by S the set of normalized price vectors of
the goods delivered in period
0. We write the price vector as p = p(0), p[1] with p[1] = p(1), . . . , p(S) , the period 1 component of the price vector p. 2

Assets: returns and prices One unit of asset j yields the return ρj (s) in state of nature s ≥ 1; these returns are denominated in units of account. With N denoting the number of assets, the returns of all the N assets in state of nature s are represented by the (row) vector
ρ(s) = ρ1 (s), ρ2 (s), . . . , ρN (s) ∈ RN . For the sake of simplicity, we assume that the returns ρj (s) are all ≥ 0, and for every j = 1, . . . , N different from 0 for at least one s ≥ 1. We denote by qj > 0 the price (in period 0) of asset j. The price vector for the N assets is denoted by q = (qj ) ∈ RN ++ . We define the S × N matrix R as the matrix whose row s is equal to the return vector ρ(s), for 1 ≤ s ≤ S. The consumer’s consumption vector Let m denote the finite number of consumers. Consumer i’s consumption space is `(S+1) X = R++ . Only strictly positive quantities of physical goods can be consumed.
[1] With xi ∈ X, we denote by xi = xi (1), . . . , xi (S) the period 1 component that corresponds to consumer i’s consumption for the S states of nature. We can then write [1]
xi = xi (0), xi . Consumer i’s portfolio is a vector bi ∈ RN . At variance with the consumption of physical (contingent) goods, we do not impose any sign constraints on the portfolios bi . Consumers’ utility functions With S representing the set of states of nature, let Σ denote the unit simplex of RS , i.e. S X Σ = {(π(s)) ∈ RS | π(s) = 1, π(s) ≥ 0, s = 1, . . . , S} s=1

A probability distribution on the set of states of nature S is a point of the simplex Σ. Consumer i’s preferences are represented by a utility function ui : X × Σ → R that is smooth and satisfies the following standard assumptions of smooth equilibrium analysis: the function ui (·, π) is smoothly increasing; smoothly strictly quasi-concave; with indifference surfaces closed in R`(S+1) for any π ∈ Σ. Note that the probability distribution π can be purely subjective. We do not require that the utility function ui belongs to the von Neumann-Morgenstern category, even if functions in that category do satisfy our assumptions. 3

Consumers’ endowments We assume that consumer i is endowed with resources in physical goods and assets `(S+1) represented by the vectors ωi ∈ R++ and βi ∈ RN respectively. Note that at variance with the standard versions of the two-period asset model, we do not assume that consumer i’s endowments in assets represented by the vector βi are necessarily equal to 0. Nevertheless, to avoid issues raised by the existence of “outside money and assets” in the context of a two-period model, we P assume that the sum of asset endowments over all consumers is equal to 0, i.e., i βi = 0. The consumer’s individual maximization problem From now on, as is has become customary in this model, we identify period 0 with a fictitious state of nature denoted by s = 0. With [1] [1]
p[1]  xi − ωi denoting the S × 1 matrix (p(s) · (xi (s) − ωi (s)) for s = 1, . . . , S, we can write consumer i’s S + 1 budget constraints as
p(0) · xi (0) − ωi (0) = −q(bi − βi ) (1)
[1] [1] p[1]  xi − ωi = Rbi (2) Given the price vectors p = (p(0), p[1] ) and q of the physical goods and assets respectively, and the probability distribution π of the S states of nature S, we define: Definition 1. P ROBLEM I consists in the maximization of the utility ui (xi , π) subject to the S + 1 budget constraints (1) and (2). Equilibria of the two-period asset model Definition 2. The pair (p, q) ∈ S × RN ++ is a financial equilibrium if: i) The maximization P ROBLEM I has a solution (xi , bi ) for every i = 1, . . . , m; ii) The solutions (xi , bi ) satisfy the equalities X X X X xi = ωi and bi = βi = 0. i

i

i

i

See for example [12] for a survey of the main properties of financial equilibria. Note that existence of a solution (xi , bi ) to the maximization problem of consumer i depends on the matrix R so that, if a solution exists for some consumer, solutions do exist for all the other consumers: see, e.g., [5], p. 144. 4

3. The temporary equilibrium point of view The temporary equilibrium point of view is based on the assumption that only the markets for the goods delivered in period 0 and the asset markets are open in period 0. The solution of consumer i’s intertemporal maximization problem requires forecasting the prices of the goods for which no markets are open. We model these forecasts by way of probability distributions on future prices. This approach can also accommodate at no additional costs the possibility for the returns of financial assets of being random instead of certain. Probability distributions over the set of prices and returns This approach amounts to taking as sample space the set of future prices and returns. We simplify the probabilistic structure by considering only a finite (but obviously large) set of possible future prices and returns. This set becomes the set of events S. Therefore, an elementary event s ∈ S can be identified with some vector (p(s), ρ(s)). The likelihood of observing the prices p(s) and the returns ρ(s) is expressed by the probability π(s) of state s ∈ S. For example, being certain that the price and return vectors p(s) and ρ(s) will be observed is equivalent to associating the probability 1 to state s, and probability 0 to the others. To sum up, beliefs about future prices and returns become probability distributions over the set of states of nature S, itself identified to the set of future prices and returns. The assumptions that the number S of states of nature is larger than the number of assets N and that the return matrix R has full rank, i.e., a rank equal to N , assumptions that are standard in the literature on incomplete asset markets, are easy to justify in this setup. Forecasts of future prices and returns We consider the forecast functions of every consumers to be exogenously given. They are represented by smooth functions from the set of admissible current asset and commodity prices (to be rigorously defined in the following subsection) into the simplex Σ. Consumer’s maximization problem in the temporary equilibrium setup Consumer i’s maximization problem is similar to the one seen in the two-period model. The S + 1 budget constraints (1) and (2) of consumer i are independent of expectations of future prices and returns. The latter are simply embodied in the dependence of the utility function ui (xi , πi ) on the probability destribution πi on the set of states of nature S. 5

The existence of a solution to the two-period maximization P ROBLEM I of   −q consumer i depends only on the structure of the matrix , not on the precise R specification of the utility function ui (·, πi ). Therefore, the issue of existence of a solution in the temporary equilibrium setup is independent of the actual forecasts of future prices and returns, and existence is guaranteed for all consumers once existence is established for just one consumer. We denote by Q the subset of RN ++ that consists of the asset prices q such that the individual maximization problem has a solution. It follows from [5], p. 144 that the set Q is an open convex polyhedric cone generated by positive linear combinations of the rows of the matrix R. The definition of the set Q enables us to define the forecast function of consumer i as a smooth map ϕi : S × Q → Σ that represents consumer i’s subjective probability distribution given the current spot and asset prices p(0) and q. Proposition 3. The period 0 demand of consumer i is the component (xi (0), bi )
[1] of the solution (xi (0), xi ), bi of consumer i’s maximization PROBLEM I for the probability distribution πi = ϕi (p(0), q). Proof. Obvious. Temporary financial equilibrium Definition 4. The pair (p(0), q) ∈ S × Q is a temporary financial equilibrium (price vector) of the economy defined by the endowments ωi and βi , the utility functions ui (xi , πi ) and the forecast functions ϕi : S × Q → Σ, i = 1, . . . , m, if the equalities X X xi (0) = ωi (0), i

i

X

bi =

i

X

βi = 0,


where xi (0), bi is the period 0 component of the solution to consumer i’s maximization PROBLEM I for every i, are satisfied.

6

4. The temporary equilibrium model: a Walrasian model with price-dependent preferences [1]

In the temporary equilibrium model, the role played by the component xi is purely ancillary. The issue is therefore to express directly the period 0 component (xi (0), bi ) as a function of the current prices and returns (p(0), q). We define P ROBLEM II as: [1]

[1]

Definition 5 (P ROBLEM II). Given (xi (0), bi ), p[1] , ωi , and π, find xi ∈ R`S ++ that maximizes
[1] ui xi (0), xi , π subject to the constraints [1]

[1]

p[1]  (xi − ωi ) = Rbi The solution of P ROBLEM II defines the value function [1]

vi (xi (0), bi , π) = ui (xi (0), xi ), bi , π




for commodity bundles xi (0) and portfolios bi such that P ROBLEM II has a solution. The set of feasible portfolios The existence of a solution to P ROBLEM II depends only on bi , not on xi (0) since the later is not an argument of the constraints. It is also obvious that P ROBLEM II has a unique solution for all bi ’s such that [1]

ωi + R bi > 0. We define the set Bi of feasible portfolios as the set of portfolios bi such that P ROBLEM II has a solution. Proposition 6. The value function vi is defined on R`++ × Bi × Σ. Proof. Obvious. The set Bi plays the same role as a consumption set. (In fact, we will identify in a moment the Cartesian product R`++ × Bi with the consumption space of a consumer with a suitably defined utility function for the bundle (xi (0), bi ).) The properties of the set Bi of feasible portfolios are therefore going to play an important role in the sequel. We have: 7

Proposition 7. The set of feasible portfolios Bi is a (convex) polyhedron whose facets are perpendicular to the positive vectors ρi (s) for s ≥ 1. The origin 0 belongs to Bi . Proof. The facets of Bi are defined by the equations bi · ρ(s) = −p(s) · ωi (s) for s ≥ 1. It follows from the inequalities p(s) · ωi (s) > 0 that the portfolio 0 ∈ RN belongs to the set Bi . Besides, the set Bi is convex as the intersection of (convex) halfspaces. Proposition 8. For any bi ∈ Bi , the inclusion bi + RN + ⊂ Bi is satisfied. Proof. Obvious. Proposition 9. The sets of feasible portfolios Bi have the same recession cone Γ for all i’s. Proof. It suffices to observe that the directions of the facets defining Bi do not depend on consumer i. Remark 1. The set Bi is not necessarily bounded from below. This can easily be checked by considering the following example: N = 2, S = 2, ρi (1) = (1, 1) and ρ1 (2) = (1, 3), the other parameters being arbitrarily chosen. Note also that the set Bi is a convex polyhedron that contains the origin, but that is not necessarily a cone. It suffices to modify the previous example by having S = 3, and to consider ωi (1) = (5, 5), ωi (2) = (5, 5), ωi (3) = (1, 1) and ρi (3) = (1, 2). The indirect utility of current goods and assets Proposition 10. The value function vi is a smooth, smoothly monotonic, smoothly
` strictly quasi-concave function of xi (0), bi ∈ R++ × Bi whose level sets are closed in R` × RN . The proof is postponed to the appendix.

8

5. The temporary model: A Walrasian model with price dependent preferences The (indirect) utility function vi : R`++ × Bi → R defined in the previous section enables us to consider the following maximization problem: Definition 11 (P ROBLEM III). Maximize the (indirect) utility vi (xi (0), bi , πi ), with (xi (0), bi ) ∈ R`++ × Bi , subject to the constraint p(0) · (xi (0) − ωi (0)) + q · (bi − βi ) = 0.

(3)

Consumer i’s P ROBLEM I can then be decomposed into the sequential maximization problem where one starts by solving P ROBLEM II in order to determine the (indirect) utility vi (xi (0), bi , πi ), and one continues by solving P ROBLEM III. This observation is stated in a more mathematical way in the following: [1]
Proposition 12. The element (xi (0), bi ), xi ∈ R`++ × RN × R`S is a solution [1]

of P ROBLEM I if and only if (xi (0), bi ) is a solution of P ROBLEM III and xi a solution of P ROBLEM II given (xi (0), bi ). [1]
Proof. The condition is necessary. Let us assume that (xi (0), bi ), xi ∈ R`++ × RN × R`S is the unique solution of P ROBLEM I. Then, it follows from the constraints (1) and (2) of P ROBLEM I bi necessarily belongs to Bi . It suffices to keep [1] the period 1 component xi of the (unique) solution to P ROBLEM I constant to

[1] observe that the period 0 component xi (0), bi maximizes ui (xi (0), xi ), πi given the period 0 constraint (1), which amounts to (xi (0), bi ) ∈ R`++ × B
i being a solution to P ROBLEM III. By keeping the period 0 component xi (0), bi of the solution to P ROBLEM I constant, we see that the period 1 component must maxi
[1] mize ui xi (0), xi , πi and therefore solves P ROBLEM II subject to the constraint [1] [1]
p[1]  xi − ωi = Rbi .
The condition is sufficient. Let xi (0), bi ∈ R`++ × Bi be a solution of P ROB [1]

III and xi a solution of P ROBLEM II associated with the right-hand side [1] [1]
constraints p[1]  xi − ωi = Rbi . If P ROBLEM I has a solution, then, it follows from the necessary conditions that the period 0 component of that solution
is indeed the solution to P ROBLEM III and therefore coincides with xi (0), bi . The same line of reasoning based on the uniqueness of the solution to P ROBLEM II [1] implies that the period 1 component of the solution of P ROBLEM I is equal to xi . LEM

9

Theorem 13. The temporary equilibrium model is equivalent to a Walrasian pure exchange model with ` + N goods, the set of prices being the set S × Q, consumer i’s consumption set being R`++ × Bi , his endowments (ωi (0), βi
) ∈ R`++ × Bi , and the price dependent utility function vi (xi (0), bi ), ϕi (p(0), q) . Proof. Follows readily from Proposition 12 combined with the equilibrium conditions of Definition 4. Remark 2. Note that in the equivalent Walrasian pure exchange model, the endowments P in the N goods corresponding to the N assets satisfy the total resource constraint i βi = 0.

6. Properties of the temporary financial equilibria Properties of the temporary financial equilibria are therefore those of the equilibria of the Walrasian model with price dependent preferences. The properties of this model are investigated in [4] for the setup of consumption spaces equal to the whole commodity space. But, the latter assumption is not crucial. Just as when preferences are price independent, the properties of the model remain unchanged if the consumption spaces and preferences are as in the current paper and satisfy the following properties: 1) the recession cone Γ of consumer i’s consumption space Xi is the same for all consumers; 2) for every i, there exists an element xi ∈ R` such that Xi is contained in the set xi + Γ; 3) indifference sets are closed in the commodity space R`+N . Then, one also has to restrict the price set to the interior of the dual of the recession cone Γ. Another (minor) difference with the model studied in [4] is that total resources P are variable in that model. Here, the total resources in physical goods i ωi (0) are P variable but, because of the assumption of no outside money or assets, the sum βi must be equal to 0. This twist prevents us from applying here some of the properties proved in [3] that depend on the global structure of the set of no-trade equilibria. These properties deal mostly with the connectedness of the equilibrium manifold and, more generally, with the diffeomorphism property between the equilibrium manifold and some Euclidean space. (See the examples discussed in the appendix of [3]). The following properties of the equilibrium model with pricedependent preferences are satisfied by the temporary financial equilibria because they only require that the equilibrium set be generically a smooth manifold. It follows from [4] that this is equivalent to having the set of price-income equilibria being a smooth manifold. Since this set can be identified with a set of Pareto optima in the space of indirect utilities ([2] and [4]), we conclude from [13] that this set is a smooth manifold for a generic set of (price-dependent) preferences. 10

Theorem 14. For exogenously given forecast functions of future prices and returns, temporary financial equilibria always exist. Proof. This is Corollary 21 of [4]. Another important result is the following version of Debreu’s theorem on the generic determinateness of the temporary equilibrium solution: Theorem 15. For exogenously given forecast functions of future prices and returns, there exists a generic set of preferences and endowments (ω, β) such that the equilibria are locally isolated. Proof. This is Theorem 17 of [4].

7. Concluding comments Temporary financial equilibria are localy isolated even when asset markets are incomplete and the returns denominated in units of accounts. This property holds true for given individual forecasts of future prices and returns. In other words, any change in the forecast functions induces a change in the temporary equilibrium. This parameterization of temporary equilibria by the forecast functions reconcile the results of the current paper with those on incomplete asset markets and returns denominated in units of accounts. In view of our results, we can interpret the indeterminacy property as the existence of an infinite number of individual forecast functions that are compatible with the equilibrium equation of the two-period model, with the consequence that these (rational) forecast functions parameterize the equilibria of that model.

Appendix A. Proof of Proposition 10 Another maximization problem
[1] Let us denote by wi = wi (1), . . . , wi (S) ∈ RS a distribution of wealth across the S states of ` S nature. We also define the map A : R`++ × R`S ++ → R++ × R++ by [1]

[1]

A(xi (0), xi ) = (xi (0), wi ), where wi (s) = p(s) · xi (s) for s ≥ 1. The map A is a linear aggregation map in the sense of [3], Definition 2.2. We now consider the following auxiliary maximization problem:

11

[1]

Definition 16 (P ROBLEM IV). For given xi (0) ∈ R`++ and wi [1]
ui xi (0), xi subject to the constraint [1]
[1]
A xi (0), xi = xi (0), wi

∈ RS ++ , maximize the utility (4)

Note that the constraints (4) are in fact equivalent to the S liquidity constraints p(s) · xi (s) = wi (s) for s ≥ 1. We have introduced xi (0) in the constraint (4) in order to make the connection with the results of [3] straightforward. Existence and properties of solutions to P ROBLEM IV are given in the following: [1]
Proposition 17. P ROBLEM IV always has a solution for xi (0), wi ∈ R`++ × RS ++ ; the solution
[1] is unique and is a smooth function of xi (0), wi . Proof. Proposition 17 is a special case of Proposition 4.6 in [3] for the linear aggregation map A.

Another value function [1]

With xi denoting a solution of P ROBLEM IV, define [1]
[1]
Ui xi (0), wi = ui xi (0), xi . The function Ui is therefore the value function associated with P ROBLEM IV and can be interpreted [1]
as the indirect utility of xi (0), wi . [1]
Proposition 18. The (indirect) utility Ui xi (0), wi associated with P ROBLEM IV defines a [1]
smooth, smoothly monotonic, smoothly quasi-concave function of xi (0), wi whose level sets  [1] [1] (xi (0), wi ) ∈ R`+S ++ | Ui (xi (0), wi ) = ui are closed in R`+S for all ui . Proof. The (indirect) utility function Ui corresponds to the composite utility function associated with the linear aggregation map A in the sense of [3]. It then suffices to remark that Proposition 18 amounts to Proposition 4.11 of [3] for the linear aggregation map A. [1]

The map Wi

If consumer
i owns the portfolio of assets bi ∈ RN while being endowed with the physical goods ωi = ωi (s) s≥0 , his expected wealth in the state of nature s ≥ 1 is therefore equal to wi [bi ](s) = pi (s) · ωi (s) + ρi (s) · bi , [1] Wi

N

(5)

S

: R → R the affine map defined by
[1] Wi (bi ) = wi [bi ](s) s≥1 . (6)

given the forecast prices and returns. We denote by

[1]

[1]

[1]

[1]

(Note that the map Wi depends also on ωi , ρi , and pi ; the latter are not included in the notation to keep it as simple as possible.) [1] The following property of the map Wi will be useful:

12

[1]

Lemma 19. The map Wi

: RN → RS is an injection.

[1]

Proof. The map Wi is an affine map; it is an injection if and only if the associated linear map defined by the return matrix R is also an injection. The conclusion follows from elementary linear algebra. It then suffices to remark that we made the assumption rank R = N . Lemma 20. The set of feasible portfolios Bi is the subset of RN that is the preimage of the set RS ++ [1] by the map Wi . Proof. A necessary and sufficient condition for the existence of a solution to the intertemporal maximization PROBLEM I is that consumer i’s wealth is strictly positive in each state of nature.

The properties of the indirect utility function vi The properties of the (indirect) utility function vi follow directly from those of Ui . We need, however, to consider still another map, the restriction k of the map ˜ : id ` k R

++

[1]

×Wi

: R`++ × RN → R`++ × RS

to the subset R`++ × Bi , i.e.,
[1] k(xi (0), bi ) = xi (0), Wi (bi ) . We then have vi = Ui ◦ k. The smoothness and the smooth monotonicity properties of vi are then straightforward consequences of Proposition 18 given the fact that the map k is affine with coefficients ≥ 0. (Recall that the returns ρi (s) are ≥ 0.) The preimage of a convex set by an affine map being a convex set, it also follows from Proposition 18 that Vi is quasi-concave. In order to prove that the level sets for Vi are closed in R` × RN , we first observe that, by the definition of Bi , we have

˜−1 xi (0), w[1] = k−1 xi (0), w[1] k i i [1]
for xi (0), wi ∈ R`++ × RS ++ . This therefore implies the equality

˜−1 Ui−1 (ui ) k−1 Ui−1 (ui ) = k for any ui . Since Ui−1 (ui ) is closed in R`+S by Proposition of Ui−1 (ui ) by the
18, the preimage −1 −1 −1 ` ˜ yields the set V (ui ) = k ˜ continuous map k Ui (ui ) that is closed in R × RN . i We now address the issue of the strict quasi-concavity of vi . Since the utility function Ui is strictly [1]
[1]
quasi-concave, the set { xi (0), wi ∈ R`++ × RS ≥ ui } is strictly convex ++ | Ui xi (0), wi [1] for all ui . The intersection of this set with the affine subspace F that is the range of the map Wi is then strictly convex as the intersection of a convex set with a strictly convex set. One concludes [1] by observing that the map defined by Wi viewed as a map from RN into F is an affine bijection whose inverse map is therefore an affine bijection and, as such, preserves strict convexity.

The non-zero Gaussian curvature property In order to be able to proceed with smooth equilibrium analysis, strict quasi-concavity of the (indirect) utility function is not sufficient. What one needs is smooth strict quasi-concavity which, given the smoothness, the smooth monotonicity, and the strict quasi-concavity of the utility function

13

vi , amounts to the level sets (for the function vi ) being smooth hypersurfaces (of R` × RN ) with Gaussian curvature everywhere 6= 0. This is what we now prove. We need to prove two properties: 1) the level sets are smooth hypersurfaces; 2) the Gaussian curvature of the level sets is everywhere 6= 0. Given the affine bijection between the affine subspace F and the domain R` × RN of the map [1] Wi , it suffices that we show that the intersection of the affine subspace F with the (smooth) level hypersurfaces of Ui are in fact smooth hypersurfaces of F in order to prove that the level sets for vi are smooth hypersurfaces. It follows from the transversality theorem (see, e.g., [9]) that a sufficient condition is that the affine subspace F is transverse to the level hypersurfaces associated with Ui . Transversality means that, at every intersection point, the sum of the subspace F and of the tangent space to the level hypersurface is the total space R` × RS . Since the tangent space to the level hypersurface is perpendicular to the gradient vector DUi at that point, it suffices that there is a vector of F that is not perpendicular to DUi to get transversality. Consider, for example, the vector j (0, ρj ) ∈ R` ×RS + for any j; since ρ is 6= 0, the inner product of this vector with DUi is necessarily > 0, and the two vectors cannot be orthogonal. The Gaussian curvature property follows from another line of reasoning. Let us start by recalling the definition of the Gaussian curvature of the hypersurface Z of Rn . One associates with every point of the hypersurface a normal vector. One normalize this normal vector so that its length is equal to 1. This defines the Gauss map t : Z → Sn where Sn denotes the sphere of radius 1 in Rn . The derivative of t is a linear map dt : Tz (Z) → Tt(z) (Sn ) between the two tangent spaces. Using local coordinates, dt is a linear map from Rn−1 into itself. By definition, the Gaussian curvature is the determinant of this map. (Note that the determinant does not depend on the choice of local coordinates.) The Gaussian curvature is not 0 if and only if the tangent map dt is a linear bijection.

How to express that the Gaussian curvature is 6= 0 There is a nice way to express the Gaussian curvature of the hypersurface Z in Rn at points where Z can be identified locally with the set of points defined by the explicit equation 1 n−1 xn ). i = ϕ(xi , . . . , xi

(Recall that, because of the monotonicity assumption, it is always possible to write globally and not n−1 1 locally xn for xi = (x11 , . . . , xn i as a smooth convex function of x1 , . . . , xi i ) ∈ Z.) Then, one considers the normal map that associates with the point (x1i , . . . , xn−1 , ϕ(x1i , . . . , xin−1 )) the i normal vector whose coordinates are (

∂ϕ ∂ϕ , . . . , n−1 , −1). ∂x1 ∂x

This vector can be identified with an element of Rn−1 and the normal map as the map from Rn−1 ∂ϕ ∂ϕ into itself that associates with (x1 , . . . , xn−1 ) the element ( 1 , . . . , n−1 ). The tangent map ∂x ∂x is then the Jacobian matrix D2 φ. Strict convexity of the map ϕ implies that the Hessian D2 ϕ is positive semi-definite. In other words, the eigenvalues of the symmetric matrix D2 ϕ that are known to be real are ≥ 0. These eigenvalues are > 0 if and only if the Hessian matrix is invertible (i.e., the quadratic form D2 ϕ positive definite) which means that the Gaussian curvature is 6= 0.

14

Back to the proof of non-zero Gaussian curvature The non-zero Gaussian curvature property is a local property and can be studied in the neighborhood of a given point. There is no loss of generality in taking a coordinate system (x1 , . . . , xn ) such that x1 , . . . , xp generate the linear subspace that intersects transversally the hypersurface. We can assume that the (local) equation of the surface is of the form x1 = ϕ(x2 , . . . , xp , xp+1 , . . . , xn ). Therefore, the intersection with the linear subspace is the hypersurface of Rp defined by the equation x1 = ϕ(x2 , . . . , xp , 0, . . . , 0). This intersection has therefore non-zero Gaussian curvature at the origin if the Hessian matrix defined by the second order derivatives of ϕ with respect to x2 , . . . , xp is positive definite. This matrix is a principal submatrix of the Hessian matrix D2 ϕ that is itself positive definite. It is therefore positive definite. This ends the proof that the indifference surfaces associated with vi have everywhere nonzero Gaussian curvature.

References [1] K. J. Arrow. The role of securities in the optimal allocation of risk-bearing. Review of Economic Studies, 31:91–96, 1964. [2] Y. Balasko. Foundations of the Theory of General Equilibrium. Academic Press, Boston, 1988. [3] Y. Balasko. Markets for composite goods. Journal of Economic Theory, 88:310–339, 1999. [4] Y. Balasko. Smooth equilibrium analysis with price dependent preferences. Working paper, 1999. [5] Y. Balasko and D. Cass. The structure of financial equilibrium with exogenous yields: The case of incomplete markets. Econometrica, 57:135–162, 1989. [6] D. Cass. Competitive equilibria in incomplete financial markets. CARESS working paper #84–09, University of Pennsylvania, 1984. [7] J. Geanokoplos and A. Mas-Collel. Real indeterminacy with financial assets. Journal of Economic Theory, 47:22–38, 1989. [8] J-M. Grandmont. Temporary Equilibrium. Academic Press, San Diego, 1988.

15

[9] V. Guillemin and A. Pollack. Differential Topology. Prentice-Hall, Englewood Cliffs, N.J., 1974. [10] J. R. Hicks. Value and Capital. Clarendon Press, Oxford, 2nd edition, 1946. [11] E. Lindahl. Theory of Money and Capital. Allen and Unwin, London, 1939. [12] M. Magill and W. Shaffer. Incomplete markets. In W. Hildenbrand and H. Sonnenschein, editors, Handbook of Mathematical Economics. Vol. IV, pages 1523–1614. North-Holland, Amsterdam, 1991. [13] S. Smale. Global analysis and economics III. Journal of Mathematical Economics, 1974. [14] J. Werner. Equilibrium in economies with incomplete financial markets. Journal of Economic Theory, 36:110–119, 1985.

16